X-Planes: Aurora Flight Sciences X-65 CRANE
Filed under: aerospace, history, engineering
This is the sixty-sixth article in the X-Planes series, following the X-1, the X-2, the X-3, the X-4, the X-5, the X-6, the X-7, the X-8, the X-9, the X-10, the X-11, the X-12, the X-13, the X-14, the X-15, the X-16, the X-17, the X-18, the X-19, the X-20, the X-21, the X-22, the X-23, the X-24, the X-25, the X-26, the X-27, the X-28, the X-29, the X-30, the X-31, the X-32, the X-33, the X-34, the X-35, the X-36, the X-37, the X-38, the X-39, the X-40, the X-41, the X-42, the X-43, the X-44, the X-45, the X-46, the X-47, the X-48, the X-49, the X-50, the X-51, the X-52, the X-53, the X-54, the X-55, the X-56, the X-57, the X-58, the X-59, the X-60, the X-61, the X-62, the X-63, and the X-64.
The programme is named for a bird that its own literature cannot find, and the aircraft carries two complete control systems so that it can be compared with itself.
The X-65A was allocated on 24 April 2023 to Aurora Flight Sciences, a Boeing subsidiary, with the engines cell reading 1 Williams FJ44-3A and the sponsor cell reading DARPA, the Defense Advanced Research Projects Agency [Parsch, DOD 4120.15-L Addendum]. The description is one line.
Unmanned demonstrator for CRANE (Control of Revolutionary Aircraft with Novel Effectors) program
The programme aims to fly an aeroplane whose pitch, roll and yaw are commanded by jets of air rather than by moving surfaces [CRANE programme page]. That much is widely reported. What is not reported is that the aircraft carries fourteen active-flow-control effectors and fourteen conventional control surfaces at the same time [X-65 experimental aircraft], and that this makes it the answer to a problem the previous article could not solve.
The Research Question
Active flow control has been demonstrated in wind tunnels for sixty years, and the question this aircraft exists to settle is whether it can be the primary flight control system of a real aeroplane at a speed and a scale that matter. [Abdolahipour 2024]
The physics is not in doubt. A small jet introduced near a surface can reattach a separated boundary layer, and the resulting change in lift is many times larger than the jet’s own reaction. What is in doubt is whether the effect survives the transition from a tunnel model to a flying vehicle, whether the air it needs can be found on board, and whether the authority it delivers is enough at every condition of a flight envelope.
The programme manager named its ancestor himself. He came to DARPA in 1999 to work on Micro Adaptive Flow Control, or MAFC, left in 2003 after managing it, and returned to see that work reach a full-scale aircraft [DARPA moves forward on the X-65]. The interval between the foundational programme and the aircraft’s planned first flight is thus 28 years, and the question of why a demonstrated effect took that long to fly is part of the subject [two-year flight delay].
The questions are therefore these. How much control authority can a jet of air buy, per unit of air, and how does that compare with a hinge? Where does the air come from, and what does the answer imply about the flight envelope? What does the aircraft’s instrumentation have to measure to settle the question? And what does it mean that the vehicle carries two complete and equinumerous control systems rather than one?
Three of the four answers were not what this article expected. The momentum a bleed-fed jet can deliver turns out to be exactly independent of altitude, for two separate cancellations that happen to meet, so the binding condition is Mach number alone. That dependence has an exact minimum at Mach 1.4142, which is beyond this aircraft but not beyond the idea. And the combinatorial figure that looks alarming, being 16,384 distinct lock-out configurations, is a red herring, because the quantity to be identified is a model with 315 terms, and a model of that size is identified in a handful of flights.
The Register Row, and a Turbofan Described as a Turbojet
Andreas Parsch’s addendum to DOD 4120.15-L, a compilation of the designations allocated after August 1998 that he publishes at designation-systems.net from Department of Defense sources, gives the row above and no more. The compiler notes that the official description of an allocation made after October 2018 is not releasable, and he marks this row’s description as his own reading and not official Department of Defense wording [Parsch, DOD 4120.15-L Addendum]. Like most of its recent neighbours the X-65A row carries no note. Of the 31 X-series rows in the copy the compiler last updated on 23 August 2026, 12 carry a note, and of the eleven allocated in 2018 or later, counting the X-56B whose date the register gives as 2019 with a query mark, only the X-76A row does, so the register’s supplementary apparatus is silent here as it is for most recent allocations. That copy is later than this article’s date. The version current on that date, the compiler’s update of 11 August 2025 as the Internet Archive captured it on 25 November 2025, held 526 rows, twenty-nine of them X rows, and lacked fifteen later rows that include the X-68A and the X-76A [Parsch, DOD 4120.15-L Addendum, 11 August 2025]. Under that version 11 of the twenty-nine X rows carried a note and none of the nine allocated in 2018 or later did, so on this article’s date the silence covered every recent allocation.
The encyclopedia entry on the aircraft describes it as having a joined tandem wing in a diamond configuration, twin vertical tails, and a single turbojet engine [Aurora X-65 CRANE]. The engine named in the register’s cell is a turbofan. The test report on that engine by the National Aeronautics and Space Administration, NASA, describes the FJ44-3A as a dual-spool low-bypass-ratio turbofan, certified in 2004 under Federal Aviation Regulations Parts 23 and 25 [FJ44 turbofan engine test], which was read in full for this article. A low-bypass turbofan is the nearest thing to a turbojet that a turbofan can be, so the description errs in a forgiving direction. The error is recorded because the distinction matters to the air budget below.
And the aircraft’s weight is given three ways. DARPA’s release says the aircraft is 7,000+ pound [DARPA moves forward on the X-65], the encyclopedia entry and the Defense News report say 7,000 pound [Aurora X-65 CRANE] [two-year flight delay], and the contractor’s page gives 7,200 pound gross weight [X-65 experimental aircraft]. The spread is 200 pound, which as a share of the lower figure is
or 2.86 percent, and the contractor’s figure is used where a single number is needed, with a note each time.
What is published about the shape of this aircraft is five numbers. A thirty foot span, a gross weight, a limit of Mach 0.7, fourteen effectors and fourteen conventional surfaces, with four interchangeable wing sets of differing sweep [X-65 experimental aircraft] [DARPA moves forward on the X-65]. Wing area, plenum pressure, bleed fraction, effector geometry and mass breakdown are all absent, so each of them is swept in what follows rather than assumed.
Programme Origin, Told Better by the Award Record Than by the Press
The narrative account is short. DARPA began CRANE in 2020, invited three contractors to present concepts in Phase 0, cut to two for a preliminary design in Phase 1, selected Aurora Flight Sciences alone for the detailed design of Phase 2 in January 2023, and awarded Phase 3 to build and flight test the aircraft in January 2024 [Aurora X-65 CRANE] [DARPA moves forward on the X-65].
The federal award record names the three contractors, prices them, and dates the downselect, none of which the narrative does [USAspending].
| Contractor | Instrument | Begins | Ends | Obligated, dollar | Base and all options, dollar | Pricing | Offers received |
|---|---|---|---|---|---|---|---|
| Aurora Flight Sciences | HR001120C0119 | 2020-06-19 | 2023-01-31 | 23,811,305.00 | 28,368,394.00 | Cost plus fixed fee | 25 |
| Lockheed Martin | HR001120C0110 | 2020-06-26 | 2023-02-03 | 18,577,721.00 | 18,577,721.00 | Cost plus fixed fee | 25 |
| Georgia Tech Research Corporation | HR001120C0130 | 2020-07-17 | 2023-10-31 | 6,795,148.00 | 12,376,510.00 | Cost no fee | 25 |
| Aurora Flight Sciences | HR001123C0012 | 2022-12-07 | 2025-10-02 | 94,372,238.00 | 94,372,238.00 | Cost plus fixed fee | 2 |
All three Phase 0 and Phase 1 contracts report twenty-five offers received. So the solicitation drew 25 proposals and funded three, a competition ratio the published accounts do not mention. The Phase 2 and Phase 3 contract reports two offers, the Phase 1 downselect appearing as a procurement fact [USAspending].
And the downselect itself is visible as an exercised option. Aurora’s contract shows an option exercised on 29 June 2021 worth 8,312,272 dollars and Lockheed Martin’s shows one exercised on 12 August 2021 worth 10,135,107 dollars. Georgia Tech Research Corporation has no option exercise at any date. Its last funding action is 13 April 2022 for 192,823 dollars, and 5,581,362 dollars of its ceiling, being 45.1 percent, was never exercised [USAspending]. The record therefore dates the cut from three to two to the middle of 2021, which no source consulted states in words. The unexercised share of a ceiling is one less the ratio of obligation to ceiling, which for Georgia Tech Research Corporation is
\[1 - \frac{6{,}795{,}148}{12{,}376{,}510} = \frac{5{,}581{,}362}{12{,}376{,}510} = 0.451 . \tag{2}\]Georgia Tech was also on different terms. Its contract is priced cost no fee where both companies are cost plus fixed fee, and the record classifies it as an other United States entity where both companies are a United States owned business. A university research corporation took no fee for the same work, and it alone of the three was not carried forward [USAspending].
From the obligations and ceilings in the table above,
\[\frac{23{,}811{,}305}{49{,}184{,}174} = 0.484 , \qquad 1 - \frac{23{,}811{,}305}{28{,}368{,}394} = \frac{4{,}557{,}089}{28{,}368{,}394} = 0.161 , \qquad 49{,}184{,}174 + 94{,}372{,}238 = 143{,}556{,}412 , \tag{3}\]so Aurora took 48.4 percent of the 49,184,174 dollars obligated across the three early contracts, and left 4,557,089 dollars, or 16.1 percent, of its own ceiling unexercised. Total obligations across all four CRANE contracts are 143,556,412 dollars.
The Forty-Two Million Dollar Contract Is Not in the Record
Press accounts report that DARPA awarded Aurora a 42 million dollar contract in January 2023 to begin detailed design [two-year flight delay]. The contract in the award record is 94,372,238 dollars, 2.25 times that figure. It is described as covering Phase 2 and Phase 3 together, and it begins on 7 December 2022, earlier than January 2023 [USAspending].
The two figures are not in conflict, for a reason that deserves stating. Phase 2 and Phase 3 were placed on one instrument, so the announced Phase 2 value and the eventual obligated total on the same contract number are different quantities. The initial obligation on the day the contract began was 6,500,000 dollars, 6.9 percent of what the contract eventually carried. [USAspending] The two ratios are
\[\frac{94{,}372{,}238}{42{,}000{,}000} = 2.25 , \qquad \frac{6{,}500{,}000}{94{,}372{,}238} = 0.0689 . \tag{4}\]| Date | Obligation, dollar | Cumulative, dollar |
|---|---|---|
| 2022-12-07 | 6,500,000.00 | 6,500,000.00 |
| 2023-02-08 | 14,845,397.00 | 21,345,397.00 |
| 2023-02-23 | 6,498,250.38 | 27,843,647.38 |
| 2023-03-13 | 9,942,727.62 | 37,786,375.00 |
| 2023-08-28 | 0.00 | 37,786,375.00 |
| 2023-12-01 | 10,426,578.00 | 48,212,953.00 |
| 2023-12-18 | 15,786,711.00 | 63,999,664.00 |
| 2024-03-20 | 0.00 | 63,999,664.00 |
| 2024-03-25 | 4,267,786.00 | 68,267,450.00 |
| 2024-05-02 | 4,708,392.00 | 72,975,842.00 |
| 2024-05-17 | 4,444,805.00 | 77,420,647.00 |
| 2024-07-30 | 8,000,000.00 | 85,420,647.00 |
| 2024-09-18 | 220,000.00 | 85,640,647.00 |
| 2024-09-24 | 1,000,000.00 | 86,640,647.00 |
| 2024-12-03 | 0.00 | 86,640,647.00 |
| 2025-01-13 | 2,993,728.00 | 89,634,375.00 |
| 2025-01-16 | 4,737,863.00 | 94,372,238.00 |
| 2025-02-11 | 0.00 | 94,372,238.00 |
The shape of that column is the programme’s history. The record holds eighteen actions, of which four move no money at all, and the last dollar is obligated on 16 January 2025. The only action after that date is the last of the eighteen, a zero-dollar change order dated 11 February 2025. [USAspending]
The Pause Is in the Record as an Absence
DARPA told a reporter that it chose to strategically pause the X-65’s development earlier in 2025 because the cost of producing the prototype had exceeded expectations, and that during the pause it restructured the programme so that Aurora became a co-investor, an arrangement finalised in August 2025 [two-year flight delay].
None of that appears in the award record. The period of performance on the Phase 2 and Phase 3 contract ends on 2 October 2025, 69 days before this article’s dateline. No obligation follows 16 January 2025, 328 days before the dateline, and the zero-dollar change order of 11 February 2025 falls 302 days before it. The restructured co-investment appears as no contract, no modification and no obligation [USAspending].
So the federal award record, read on this article’s date, shows CRANE as a completed contract with no aircraft. The record is not at fault. A cost share by a contractor is not federal spending, so a database of federal spending is the wrong instrument to see it with, and what is interesting is that the instrument is silent while remaining correct. A programme can be restructured into a shape that the public record of its funding cannot represent.
Two adjacent awards are relevant. Scientific, engineering and technical assistance support to the DARPA Tactical Technology Office for CRANE was contracted on 20 December 2019, six months before any performer, through a Navy vehicle, and carried 5,679,994.63 dollars with a de-obligation of 220,000 dollars in September 2024. The support was bought before the research was. And the Office of Naval Research funded 1,251,577 dollars of work titled finless unmanned combat air system flight control with active flow control effectors, beginning September 2022. It is the same idea pursued by a different service for 1.33 percent of what the Phase 2 and Phase 3 contract carried [USAspending].
What the Programme’s Name Costs It
CRANE finds nothing aeronautical in the bibliographic index, and novel effectors, the literal phrase in the programme’s title, finds molecular biology, where an effector is a protein that a pathogen injects into a host.
The phrase that does find this aircraft’s prior art is one the programme does not use. hingeless control returns, among others, the experimental development of hingeless aerodynamic control effectors [Kondor et al. 2008], plasma actuators for hingeless aerodynamic control of an unmanned air vehicle [Patel et al. 2007], delta-wing hingeless control by synthetic jet actuation [Traub et al. 2002], and system identification of an unmanned aerial vehicle with hingeless control effectors [Weisman and Valasek 2010]. That is the X-65’s lineage, indexed under a word the programme’s name replaced. The full measurement is in the source base below.
Sizing From First Principles
The Symbols This Article Uses
Every symbol below is declared here and nowhere carries a second meaning.
| Symbol | Meaning | Unit |
|---|---|---|
| $\mathcal{A}$ | amplification, aerodynamic force per unit jet momentum | dimensionless |
| $a$ | speed of sound | metre per second |
| $A^{*}$ | total choked throat area of the effectors | square metre |
| $A_f$ | engine face area | square metre |
| $a_f$ | speed of sound at the engine face | metre per second |
| $\mathcal{A}_{\mathrm{loc}}$ | actuation efficiency, the slope of the lift increment against the momentum coefficient | dimensionless |
| $AR$ | aspect ratio | dimensionless |
| $B$ | engine bypass ratio | dimensionless |
| $\mathbf{B}$ | control effectiveness matrix, three rows by one column per effector | newton metre |
| $b$ | wing span | metre |
| $\beta$ | bleed fraction of engine core mass flow | dimensionless |
| $\mathbf{b}_i$ | one column of the control effectiveness matrix | newton metre |
| $\bar{c}$ | reference chord, taken as area over span | metre |
| $C_1$ | Sutherland’s constant for air, 1.458 times ten to the minus six | kilogram per metre second root kelvin |
| $C_L$ | lift coefficient | dimensionless |
| $C_{\mathcal{M}}$ | moment coefficient referenced to the span | dimensionless |
| $C_\mu$ | momentum coefficient referenced to the wing | dimensionless |
| $C_{\mu,0}$ | threshold momentum coefficient below which nothing happens | dimensionless |
| $C_{\mu,1}$ | threshold momentum coefficient above which the flow is fully attached | dimensionless |
| $C_\mu^{\,1}$ | momentum coefficient a bleed fraction of unity would deliver | dimensionless |
| $C_{\mu,\ell}$ | momentum coefficient referenced to the area one effector influences | dimensionless |
| $c_p$ | specific heat at constant pressure | joule per kilogram kelvin |
| $\delta$ | ratio of compressor-face total pressure to sea-level standard pressure | dimensionless |
| $\Delta C_L$ | lift-coefficient increment produced by an effector | dimensionless |
| $\Delta C_{L,\mathrm{conv}}$ | lift-coefficient increment produced by a conventional surface at full deflection | dimensionless |
| $\Delta F$ | thrust decrement caused by bleed | newton |
| $\Delta L$ | aerodynamic force increment produced by an effector | newton |
| $\Delta L_{\mathrm{AFC}}$ | force increment from an active-flow-control effector, the subscript being the programme’s own abbreviation | newton |
| $\Delta L_{\mathrm{conv}}$ | force increment from a conventional control surface | newton |
| $\Delta T$ | total-temperature rise across a compressor | kelvin |
| $D_f$ | engine face diameter | metre |
| $\eta_c$ | compressor isentropic efficiency | dimensionless |
| $F^{+}$ | reduced excitation frequency | dimensionless |
| $f_e$ | effector excitation frequency | hertz |
| $F_{\mathrm{sls}}$ | sea-level static thrust | newton |
| $G$ | the Mach-number factor of the bleed-fed momentum coefficient | dimensionless |
| $g_0$ | standard gravity, 9.80665 | metre per second squared |
| $\gamma$ | ratio of specific heats, taken as 1.4 for dry air | dimensionless |
| $\mathcal{H}$ | reachable moment set under a shared supply, a convex hull | newton metre squared |
| $h$ | geopotential altitude | metre |
| $h_{\mathrm{trop}}$ | tropopause altitude, 11,000 | metre |
| $J$ | jet momentum flux supplied by the effectors | newton |
| $k$ | usable test points per flight | dimensionless |
| $\kappa$ | concentration factor, wing area over influenced area | dimensionless |
| $\Lambda$ | authority ratio, flow-control force over conventional-surface force | dimensionless |
| $\lambda$ | troposphere lapse rate, minus 0.0065 | kelvin per metre |
| $M$ | flight Mach number | dimensionless |
| $\mathcal{M}$ | control moment | newton metre |
| $\mathbf{m}$ | demanded moment vector | newton metre |
| $\dot{m}_{\mathrm{core}}$ | engine core mass flow | kilogram per second |
| $M_f$ | axial Mach number at the engine face | dimensionless |
| $\dot{m}_j$ | jet mass flow delivered to the effectors | kilogram per second |
| $\dot{m}_{\mathrm{ref}}$ | engine sea-level static mass flow of the bleed source stream | kilogram per second |
| $M_\times$ | crossover Mach number at which the authority ratio reaches unity | dimensionless |
| $\dot{m}_{\mathrm{total}}$ | engine total mass flow | kilogram per second |
| $\mu$ | dynamic viscosity of air | pascal second |
| $n$ | number of active-flow-control effectors, being the fourteen whose derivatives are identified | dimensionless |
| $N_1$ | control derivatives in a linear effector model | dimensionless |
| $N_2$ | terms in a second-order effector model with pairwise interactions | dimensionless |
| $n_a$ | number of controlled axes | dimensionless |
| $N_c$ | distinct locked-or-free configurations across every wing set | dimensionless |
| $n_e$ | number of effectors of both kinds available to the allocator, being twenty-eight on this aircraft | dimensionless |
| $N_f$ | flights needed to over-determine a model | dimensionless |
| $\mathrm{NPR}$ | nozzle pressure ratio, plenum total to ambient static | dimensionless |
| $n_s$ | number of conventional control surfaces | dimensionless |
| $n_w$ | number of interchangeable wing sets | dimensionless |
| $o$ | appropriation outcome for a fiscal year | dollar |
| $P$ | shaft power delivered to the air by a dedicated compressor | watt |
| $p$ | ambient static pressure | pascal |
| $\pi_c$ | compressor total-pressure ratio | dimensionless |
| $p_{\mathrm{sl}}$ | sea-level standard pressure, 101325 | pascal |
| $p_t$ | plenum total pressure | pascal |
| $p_{\mathrm{trop}}$ | pressure at the tropopause | pascal |
| $q$ | free-stream dynamic pressure | pascal |
| $R$ | specific gas constant for dry air, 287.05287 | joule per kilogram kelvin |
| $r$ | appropriation request for a fiscal year | dollar |
| $\mathrm{Re}$ | Reynolds number on the reference chord | dimensionless |
| $\rho$ | ambient air density | kilogram per cubic metre |
| $\rho_f$ | air density at the engine face | kilogram per cubic metre |
| $S$ | wing reference area | square metre |
| $S_\ell$ | area of surface one effector influences | square metre |
| $\sigma$ | inlet total-condition factor, $1 + \left(\gamma-1\right)M^2/2$ | dimensionless |
| $\sigma_f$ | total-condition factor at the engine face | dimensionless |
| $S_\mu$ | Sutherland’s temperature for air, 110.4 | kelvin |
| $T$ | ambient static temperature | kelvin |
| $\tau$ | compressor total-temperature ratio | dimensionless |
| $\theta$ | ratio of compressor-face total temperature to sea-level standard temperature | dimensionless |
| $T_{\mathrm{sl}}$ | sea-level standard temperature, 288.15 | kelvin |
| $T_t$ | plenum total temperature | kelvin |
| $T_{\mathrm{trop}}$ | temperature at the tropopause, 216.65 | kelvin |
| $\mathcal{U}$ | admissible set of effector demands | dimensionless |
| $\mathbf{u}$ | vector of effector demands | dimensionless |
| $\bar{u}$ | the supply limit on effector demand | dimensionless |
| $V$ | true airspeed | metre per second |
| $V_e$ | engine exit velocity at static conditions | metre per second |
| $V_\infty$ | free-stream velocity, used where the contrast with the jet matters | metre per second |
| $V_j$ | effector jet velocity at the orifice | metre per second |
| $\mathrm{VR}$ | jet velocity ratio, jet velocity over free-stream velocity | dimensionless |
| $W$ | aircraft weight | newton |
| $w_c$ | compressor specific work | joule per kilogram |
| $y$ | moment arm from the vehicle centreline | metre |
| $\mathcal{Z}$ | reachable moment set under independent limits, a zonotope | newton metre squared |
The Momentum Coefficient, and the Identity It Hides
The standard figure of merit for a blown flow-control device is the momentum coefficient, being the momentum flux the jet supplies divided by the product of free-stream dynamic pressure and a reference area. Let $\dot{m}_j$ be the jet mass flow in kilogram per second, $V_j$ the jet velocity in metre per second, $q$ the free-stream dynamic pressure in pascal, $S$ the reference area in square metre, and $C_\mu$ the momentum coefficient, dimensionless.
\[J = \dot{m}_j V_j, \qquad C_\mu = \frac{J}{q S} \tag{5}\]where $J$ is the jet momentum flux in newton. Since the numerator and the denominator are both forces, the ratio is dimensionless, and the review cited above attributes the precise formulation to Poisson-Quinton [Abdolahipour 2024].
The force the device produces is a different quantity, an aerodynamic force increment obtained because the jet changed where the flow separated. Let $\Delta C_L$ be the resulting lift-coefficient increment, dimensionless, and $\Delta L$ the force increment in newton.
\[\Delta L = \Delta C_L \, q S \tag{6}\]Dividing one by the other gives an exact identity, and the identity contains the entire question of whether active flow control is a control technology or a thruster in disguise. Let $\mathcal{A}$ be the amplification, being aerodynamic force produced per unit of jet momentum supplied, dimensionless.
\[\mathcal{A} = \frac{\Delta L}{\dot{m}_j V_j} = \frac{\Delta C_L \, q S}{C_\mu \, q S} = \frac{\Delta C_L}{C_\mu} \tag{7}\]The flight condition cancels completely. The amplification, a ratio of two dimensionless numbers, is indifferent to altitude, speed and area. A value of one means the device is a rocket, producing exactly the force its own jet reaction would produce and nothing more. A value far above one means the jet is a trigger rather than a force.
The rows below are illustrative values chosen to show the shape of the regimes, not measurements.
| Regime | Lift-coefficient increment | Momentum coefficient | Amplification |
|---|---|---|---|
| Separation control, weak | 0.050 | 0.005 | 10.00 |
| Separation control, strong | 0.300 | 0.010 | 30.00 |
| Circulation control, modest | 0.800 | 0.100 | 8.00 |
| Circulation control, strong | 2.000 | 0.400 | 5.00 |
| Pure reaction | 0.010 | 0.010 | 1.00 |
The ratio above is a secant and the quantity the literature plots is the tangent, and the distinction matters because the curve is not a line.
\[\mathcal{A} = \frac{\Delta C_L}{C_\mu} \quad \text{against} \quad \mathcal{A}_{\mathrm{loc}} = \frac{\mathrm{d} \, \Delta C_L}{\mathrm{d} C_\mu} \tag{8}\]The review calls the second one the actuation efficiency and reports that it falls through the transition between the two regimes [Abdolahipour 2024]. The two agree only where the curve passes through the origin with constant slope, which it does not. The secant is used throughout, because it converts an available momentum into an available force, and the tangent tells a designer whether more air would help.
Separation control lives at the top of that table and circulation control at the bottom, because of the shape of the underlying curve. At very small momentum coefficient nothing happens, because the jet is too weak to change where the boundary layer leaves the surface. Above a threshold the lift increment rises steeply, and amplification is largest in this regime. At large momentum coefficient the flow is fully attached, further blowing simply adds circulation, and the returns fall toward unity as the device becomes a thruster. So there is an optimal momentum coefficient, neither small nor large, which turns the design problem into one of landing in a band rather than of maximising a quantity. Writing $C_{\mu,0}$ for the threshold below which nothing happens and $C_{\mu,1}$ for the one above which the flow is fully attached,
\[\mathcal{A}_{\mathrm{loc}} \approx 0 \;\; \text{for} \;\; C_\mu < C_{\mu,0}, \qquad \mathcal{A}_{\mathrm{loc}} \gg 1 \;\; \text{for} \;\; C_{\mu,0} < C_\mu < C_{\mu,1}, \qquad \mathcal{A}_{\mathrm{loc}} \to 1 \;\; \text{for} \;\; C_\mu > C_{\mu,1} \tag{9}\]The review puts $C_{\mu,1}$ between three and five percent and reports measured values of $C_{\mu,0}$ well below one percent, both of which are taken up below [Abdolahipour 2024].
The reference case is the physical one reported for the aircraft, being the raw jet reaction of the whole air budget. At the conditions derived below the total jet momentum is 156.1 newton, which is 0.49 percent of the aircraft’s weight. A device that merely pushed with that would be useless. An amplification of thirty makes it a flight control system.
The Unstated Source of the Air
DARPA’s release says the aircraft uses jets of air from a pressurized source [DARPA moves forward on the X-65]. The contractor writes active flow control uses pressurized air for flight control [X-65 experimental aircraft]. Neither names the source of the pressure, the largest quantity the record leaves undetermined, because the two behave oppositely with altitude.
The first architecture bleeds the engine. At a fixed corrected spool speed a gas turbine passes a constant corrected mass flow, so its actual mass flow follows the total pressure at its face divided by the square root of the total temperature there. Let $\dot{m}_{\mathrm{ref}}$ be the engine’s sea-level static mass flow in kilogram per second, $\delta$ the ratio of compressor-face total pressure to sea-level standard pressure and $\theta$ the ratio of compressor-face total temperature to sea-level standard temperature, both dimensionless, and $\beta$ the bleed fraction, dimensionless.
\[\dot{m}_j = \beta \, \dot{m}_{\mathrm{ref}} \, \frac{\delta}{\sqrt{\theta}} \tag{10}\]The second architecture drives a compressor at a fixed shaft power. Let $P$ be the shaft power delivered to the air in watt, $c_p$ the specific heat at constant pressure in joule per kilogram kelvin, and $\Delta T$ the total-temperature rise in kelvin.
\[\dot{m}_j = \frac{P}{c_p \, \Delta T} \tag{11}\]The first depends on ambient pressure and the second does not, and that single difference propagates into opposite conclusions about the flight envelope.
The Atmosphere Every Number Here Is Computed In
Every calculation that follows happens inside the 1976 standard atmosphere. The relations are elementary and they are shown because every figure in every table below depends on them.
In the troposphere the temperature falls linearly. Let $h$ be the geopotential altitude in metre, $\lambda$ the lapse rate in kelvin per metre, $g_0$ the standard gravity in metre per second squared, and $T_{\mathrm{sl}}$ and $p_{\mathrm{sl}}$ the sea-level standard temperature and pressure.
\[T\left( h \right) = T_{\mathrm{sl}} + \lambda h, \qquad p\left( h \right) = p_{\mathrm{sl}} \left( \frac{T\left( h \right)}{T_{\mathrm{sl}}} \right)^{- \frac{g_0}{\lambda R}} \tag{12}\]The exponent is not a fitted constant. It follows from integrating the hydrostatic relation $\mathrm{d}p = - \rho g_0 \, \mathrm{d}h$ against the ideal gas law with a linear temperature profile.
Above the tropopause the temperature is constant and the pressure falls exponentially, with $h_{\mathrm{trop}}$ the tropopause altitude in metre and $T_{\mathrm{trop}}$ and $p_{\mathrm{trop}}$ the conditions there.
\[p\left( h \right) = p_{\mathrm{trop}} \exp\left( - \frac{g_0 \left( h - h_{\mathrm{trop}} \right)}{R \, T_{\mathrm{trop}}} \right), \qquad \rho = \frac{p}{R T}, \qquad a = \sqrt{\gamma R T} \tag{13}\]At thirty thousand feet, taken as a geopotential altitude of 9,144 metre, those give 228.714 kelvin, 30,089.6 pascal, 0.458312 kilogram per cubic metre and a speed of sound of 303.174 metre per second.
The Compressible Identity That Makes the Cancellation Possible
Dynamic pressure is defined as half the density times the square of the speed, and rewriting it in terms of pressure and Mach number is not an approximation. With $\rho$ the density in kilogram per cubic metre, $V$ the true airspeed in metre per second, $M$ the Mach number, dimensionless, $p$ the ambient static pressure in pascal, $T$ the ambient static temperature in kelvin, $R$ the specific gas constant in joule per kilogram kelvin and $\gamma$ the ratio of specific heats, dimensionless,
\[q = \tfrac{1}{2} \rho V^2 = \tfrac{1}{2} \rho \, \gamma R T M^2 = \tfrac{\gamma}{2} \, p \, M^2 \tag{14}\]because $p = \rho R T$ and the speed of sound satisfies $a^2 = \gamma R T$. At a fixed Mach number the dynamic pressure is exactly proportional to the ambient pressure, a fact the bleed architecture exploits. The two routes to $q$ were computed independently and agree to better than one part in $10^{15}$.
Total Conditions and Corrected Flow
The bled mass flow above is written in terms of $\delta$ and $\theta$, which are the compressor-face total conditions normalised on sea level, and the total conditions follow from the flight Mach number alone under full inlet recovery.
\[p_{t} = p \, \sigma^{\frac{\gamma}{\gamma-1}}, \qquad T_{t} = T \sigma, \qquad \delta = \frac{p_t}{p_{\mathrm{sl}}}, \qquad \theta = \frac{T_t}{T_{\mathrm{sl}}} \tag{15}\]The invariant that makes the bled mass flow predictable at all is the corrected flow. At a fixed corrected spool speed a turbomachine passes a fixed value of
\[\dot{m} \frac{\sqrt{\theta}}{\delta} = \text{constant} \tag{16}\]and inverting it gives the bled flow above. At thirty thousand feet and Mach 0.7 the factors are $\sigma$ of 1.0980, $\delta$ of 0.4119 and $\theta$ of 0.8715, so the compressor face sees 41.74 kilopascal and 251.1 kelvin. The corrected flow recomputed from the actual flow at ten altitudes returns 10.0 kilogram per second every time, the invariance the derivation assumes.
The Compressor Ratio, and What the Air Costs in Work
The compressor’s total-temperature ratio follows from its pressure ratio and its efficiency. Let $\tau$ be that ratio, dimensionless, and $w_c$ the specific work in joule per kilogram.
\[\tau = 1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}} - 1}{\eta_c}, \qquad w_c = c_p \, T_{t} \left( \tau - 1 \right) \tag{17}\]At a pressure ratio of 14 and an efficiency of 0.85 the temperature ratio is 2.3241, against 2.1255 for ideal compression, and the work already spent on each kilogram of bled air is 334.1 kilojoule. A bleed throws that work away, so air taken from the discharge of a single-stage centrifugal compressor is the most expensive air in the engine.
The Jet Is Choked, and Its Velocity Depends Only on Its Own Temperature
A nozzle whose plenum exceeds the ambient pressure by more than the critical ratio passes sonic flow at its throat, and the exit velocity is then the local speed of sound there. Let $T_t$ be the plenum total temperature in kelvin.
\[\frac{p_{t}}{p} \ge \left( \frac{\gamma + 1}{2} \right)^{\frac{\gamma}{\gamma - 1}} \approx 1.8929, \qquad V_j = \sqrt{\frac{2 \gamma R \, T_{t}}{\gamma + 1}} \tag{18}\]The jet velocity depends on the plenum total temperature and on nothing else. Not on altitude, not on the plenum pressure, not on the flight speed except through the total temperature. The cancellation needs this second fact.
Air bled from a compressor of pressure ratio $\pi_c$ arrives hot. Let $\eta_c$ be the compressor isentropic efficiency, dimensionless, and let the inlet total-condition factor be $\sigma$.
\[\sigma = 1 + \frac{\gamma - 1}{2} M^2, \qquad T_{t} = T \sigma \left( 1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}} - 1}{\eta_c} \right) \tag{19}\]The nozzle pressure ratio also has a closed form, and the altitude does not appear in it. Let $\mathrm{NPR}$ be the plenum-to-ambient pressure ratio, dimensionless.
\[\mathrm{NPR} = \frac{\pi_c \, p_t}{p} = \pi_c \, \sigma^{\frac{\gamma}{\gamma-1}} \tag{20}\]The ambient pressure divides out before anything is computed.
| Altitude, foot | Ambient pressure, kilopascal | Plenum pressure, kilopascal | Nozzle pressure ratio | Plenum total temperature, kelvin | Jet velocity, metre per second |
|---|---|---|---|---|---|
| 0 | 101.33 | 1,967.7 | 19.419 | 735.3 | 496.2 |
| 15,000 | 57.18 | 1,110.4 | 19.419 | 659.5 | 470.0 |
| 30,000 | 30.09 | 584.3 | 19.419 | 583.7 | 442.1 |
| 45,000 | 14.75 | 286.4 | 19.419 | 552.9 | 430.3 |
The nozzle pressure ratio is 19.419 at every altitude in that table, identically, and the closed form above explains why in one line. The plenum pressure is a fixed multiple of the compressor-face total pressure, and that pressure is a fixed multiple of the ambient pressure at a fixed Mach number, so their ratio cannot depend on altitude. It stands at 10.26 times the critical ratio, so the choked assumption holds across the envelope with an order of magnitude to spare.
Altitude Cancels Exactly, for Two Separate Reasons That Happen to Meet
Assemble the three results. The bled mass flow carries $p \sigma^{\gamma/(\gamma-1)}$ over $\sqrt{T \sigma}$, the choked jet velocity carries $\sqrt{T \sigma \tau}$ where $\tau$ is the compressor total-temperature ratio, and the dynamic pressure carries $p M^2$. Writing $p_{\mathrm{sl}}$ and $T_{\mathrm{sl}}$ for the sea-level standard pressure and temperature,
\[C_\mu = \frac{2 \beta \, \dot{m}_{\mathrm{ref}}}{\gamma \, p_{\mathrm{sl}} \, S} \sqrt{\frac{2 \gamma R \, T_{\mathrm{sl}} \, \tau}{\gamma + 1}} \; \cdot \; \frac{\sigma^{\frac{\gamma}{\gamma-1}}}{M^2} \tag{21}\]Every trace of altitude is gone, and so is the ambient temperature, and so is one of the two appearances of $\sigma$. The ambient pressures cancel because corrected flow makes the bled mass flow proportional to pressure and the compressible identity makes dynamic pressure proportional to pressure. The ambient temperatures cancel for a different reason, because corrected flow divides by the square root of the total temperature while a choked jet fed from that same air multiplies by it. The inlet factor cancels once for the same reason the temperature does, since the same ram rise that warms the compressor face warms the jet.
| Altitude, foot | Dynamic pressure, kilopascal | Bled mass flow, kilogram per second | Momentum coefficient from bleed | Momentum coefficient at fixed shaft power |
|---|---|---|---|---|
| 0 | 34.75 | 1.0590 | 0.000945066 | 0.0015249 |
| 5,000 | 28.92 | 0.8967 | 0.000945066 | 0.0018651 |
| 10,000 | 23.90 | 0.7547 | 0.000945066 | 0.0022978 |
| 15,000 | 19.61 | 0.6311 | 0.000945066 | 0.0028533 |
| 20,000 | 15.97 | 0.5240 | 0.000945066 | 0.0035731 |
| 25,000 | 12.90 | 0.4319 | 0.000945066 | 0.0045157 |
| 30,000 | 10.32 | 0.3530 | 0.000945066 | 0.0057639 |
| 35,000 | 8.18 | 0.2860 | 0.000945066 | 0.0074370 |
| 40,000 | 6.43 | 0.2260 | 0.000945066 | 0.0095018 |
| 45,000 | 5.06 | 0.1778 | 0.000945066 | 0.0120830 |
The momentum coefficient in the fourth column is identical to 12 significant figures from sea level to forty-five thousand feet, and the closed form above reproduces the direct computation to within one part in $10^{15}$. The bled mass flow in the third column falls by a factor of six over that band and the dynamic pressure in the second falls by very nearly the same factor, and the two errors are the same error with opposite signs.
The fifth column, the other architecture, behaves quite differently. A compressor held at constant shaft power delivers a mass flow set by the temperature rise, which barely changes, while the dynamic pressure it is measured against falls with the ambient pressure. Its momentum coefficient rises by a factor of 7.92 over the same band.
So the published phrase a pressurized source conceals a factor of 7.92 in how this aircraft’s control authority varies with altitude, and the two architectures are distinguishable by experiment. Fly the same effector command at two altitudes at the same Mach number and compare the control moments. A ratio near unity is engine bleed. A ratio well above unity is a shaft-driven compressor, reaching 7.92 between sea level and forty-five thousand feet. The analysis thus makes a prediction the flight-test programme can settle in two data points.
The Mach Dependence, and an Exact Minimum at Mach Root Two
What survives is a function of Mach number alone, up to the constant. Let $G$ be that factor, dimensionless.
\[G\left( M \right) = \frac{\sigma^{\frac{\gamma}{\gamma-1}}}{M^2} = \frac{\left( 1 + 0.2 M^2 \right)^{3.5}}{M^2} \tag{22}\]at $\gamma = 1.4$. The numerator is the inlet’s total-pressure recovery, which works in the aircraft’s favour.
| Mach number | Mach factor with inlet recovery | Mach factor without it | Recovery gains | Momentum coefficient at 30,000 feet |
|---|---|---|---|---|
| 0.15 | 45.1484 | 44.4444 | 1.0158 | 0.0150728 |
| 0.20 | 25.7070 | 25.0000 | 1.0283 | 0.0085823 |
| 0.30 | 11.8270 | 11.1111 | 1.0644 | 0.0039484 |
| 0.40 | 6.9784 | 6.2500 | 1.1166 | 0.0023297 |
| 0.50 | 4.7449 | 4.0000 | 1.1862 | 0.0015841 |
| 0.60 | 3.5431 | 2.7778 | 1.2755 | 0.0011828 |
| 0.70 | 2.8308 | 2.0408 | 1.3871 | 0.0009451 |
Going from Mach 0.2 to Mach 0.7 costs a factor of 9.081 in momentum coefficient, where the naive inverse square would have cost 12.250. The inlet recovery therefore mitigates 25.9 percent of the penalty, and the mitigation grows with Mach number because the recovery does.
The factor has a minimum, located exactly at Mach $\sqrt{2}$. Taking logarithms and differentiating,
\[\frac{\mathrm{d} \ln G}{\mathrm{d} M} = \frac{\gamma}{\gamma - 1} \cdot \frac{\sigma'}{\sigma} - \frac{2}{M} = \frac{\gamma M}{\sigma} - \frac{2}{M} \tag{23}\]which vanishes when $\gamma M^2 = 2 \sigma = 2 + \left( \gamma - 1 \right) M^2$, that is when $M^2 = 2$. The specific-heat ratio cancels and the turning point is at Mach 1.4142 whatever the gas is. Since $\sigma\left(\sqrt{2}\right) = \gamma$ exactly, the minimum value has a closed form.
\[G_{\min} = \frac{\gamma^{\frac{\gamma}{\gamma-1}}}{2} = 1.6234 \tag{24}\]The closed form, a direct evaluation and a brute-force search over three hundred thousand Mach numbers agree to one part in $10^{15}$.
Beyond Mach $\sqrt{2}$ a bleed-fed flow-control system gets better with speed again, because the inlet recovery eventually outruns the inverse square. The X-65A’s published limit is Mach 0.7 [DARPA moves forward on the X-65], so this aircraft never reaches the turn. The result is reported because it bears on the idea, although not on this aeroplane.
The Authority Ratio, and Where a Jet Loses to a Hinge
The aircraft’s central question can now be written in one line. Let $\Delta L_{\mathrm{AFC}}$ be the force increment an active-flow-control effector delivers in newton, where the subscript is the programme’s abbreviation for active flow control, $\Delta L_{\mathrm{conv}}$ the force increment a conventional surface delivers in newton, $\Delta C_{L,\mathrm{conv}}$ the lift-coefficient increment that surface delivers at full deflection, dimensionless, and $\Lambda$ the authority ratio, dimensionless.
\[\Lambda = \frac{\Delta L_{\mathrm{AFC}}}{\Delta L_{\mathrm{conv}}} = \frac{\mathcal{A} \, C_\mu \, q S}{\Delta C_{L,\mathrm{conv}} \, q S} = \frac{\mathcal{A} \, C_\mu}{\Delta C_{L,\mathrm{conv}}} \tag{25}\]The dynamic pressure cancels again, so the authority ratio inherits the altitude independence of the momentum coefficient and inherits its Mach dependence entire. A conventional surface produces a moment proportional to dynamic pressure, and so does an active-flow-control effector, for the surprising reason that the engine feeding it swallows more air at lower altitude in almost exactly the same proportion.
The crossover is defined by a condition, the Mach number at which the ratio reaches one, and involves no fitting.
\[\Lambda\left( M_{\times} \right) = 1 \quad \Longleftrightarrow \quad \mathcal{A} \, C_\mu\left( M_{\times} \right) = \Delta C_{L,\mathrm{conv}} \quad \Longleftrightarrow \quad G\left( M_{\times} \right) = \frac{\Delta C_{L,\mathrm{conv}}}{\mathcal{A}} \cdot \frac{G\left( M_{0} \right)}{C_\mu\left( M_{0} \right)} \tag{26}\]for any reference Mach number $M_0$. The root is unique below Mach $\sqrt{2}$ because $G$ is strictly decreasing there, as established above, so a bisection is admissible as well as convenient.
| Amplification | Ratio at Mach 0.7, against 0.02 | Crossover Mach, 0.02 | Ratio, against 0.05 | Crossover Mach, 0.05 | Ratio, against 0.10 | Crossover Mach, 0.10 |
|---|---|---|---|---|---|---|
| 10 | 0.473 | 0.436 | 0.189 | 0.265 | 0.095 | 0.185 |
| 20 | 0.945 | 0.672 | 0.378 | 0.385 | 0.189 | 0.265 |
| 30 | 1.418 | 0.942 | 0.567 | 0.485 | 0.284 | 0.329 |
| 50 | 2.363 | none | 0.945 | 0.672 | 0.473 | 0.436 |
| 100 | 4.725 | none | 1.890 | none | 0.945 | 0.672 |
Take an amplification of 30 against a conventional surface worth 0.05 in lift coefficient. The authority ratio at the top of the envelope is 0.567, so active flow control delivers 56.7 percent of the hinge’s authority at Mach 0.7 and falls short by 43.3 percent. At Mach 0.2 the same hardware delivers 5.15 times the hinge’s authority. The crossover is at Mach 0.485. With the momentum coefficient and the Mach factor from the tables above,
\[\Lambda\left( 0.7 \right) = \frac{30 \times 0.000945066}{0.05} = 0.567 , \qquad \Lambda\left( 0.2 \right) = 0.567 \times \frac{G\left( 0.2 \right)}{G\left( 0.7 \right)} = 0.567 \times \frac{25.7070}{2.8308} = 5.15 , \qquad G\left( M_{\times} \right) = \frac{0.05}{30} \cdot \frac{2.8308}{0.000945066} = 4.992 , \tag{27}\]and the Mach number at which $G$ falls to 4.992 is 0.485.
This inverts the usual control-authority problem, and the inversion is the article’s practical result. For a conventional aeroplane the binding case for control power is low speed, because the surfaces are weakest when the dynamic pressure is least, so approach and engine-out cases size the tail. For a bleed-fed flow-control aeroplane the binding case is high speed, because the effectors deliver a roughly fixed force while the aerodynamic forces they must overcome grow as the square of the Mach number.
One consequence is fortunate. Flight-test programmes expand an envelope from low speed upward, for reasons of safety unrelated to flow control. For this aircraft that happens to mean starting where active flow control is strongest and proceeding toward where it is weakest, so the natural order of an envelope expansion is also the natural order of a risk reduction. A programme that had to begin at the difficult end would be a much worse bet.
To Match a Hinge, and What That Demands
Inverting the ratio gives the amplification required.
\[\mathcal{A}_{\mathrm{req}} = \frac{\Delta C_{L,\mathrm{conv}}}{C_\mu} \tag{28}\]At the reference condition the momentum coefficient is 0.000945, so matching a conventional surface worth 0.02 needs an amplification of 21.2, matching one worth 0.05 needs 52.9, and matching one worth 0.10 needs 105.8. Which of those is reachable is the open question, and the published record brackets it without settling it. The review’s summary of separation-control studies reports a gain in lift of 53 percent on a flapped NACA 0015 with its flap at 35 degrees at a momentum coefficient of 0.3 percent, and gains in maximum lift of 12 to 35 percent at coefficients of 0.082 to 0.14 percent [Abdolahipour 2024]. The review gives those gains relative to each baseline and not as increments, so the amplification they imply is an inference, and on a baseline lift coefficient near one it exceeds a hundred in every one of those cases. By the arithmetic, then, this class of system can replace a weak control surface at cruise at any amplification the literature suggests, and can replace a strong one only if the amplification at cruise reaches the hundred that those low-speed experiments imply, a quantitative version of the claim the programme was built to test.
The Reference Area Is a Choice, the Force Is Not
The denominator of the momentum coefficient is a convention, and the literature acknowledges this in the same breath as it defines the quantity. The review attributes the precise formulation to Poisson-Quinton and writes the denominator as the free-stream dynamic pressure times a reference area [Abdolahipour 2024]. Which reference area is the author’s choice, and the choice decides what question the number answers.
An effector does not act on the whole wing. It acts on the region whose separation behaviour it changes. Let $S_\ell$ be that area in square metre, $C_{\mu,\ell}$ the local momentum coefficient, dimensionless, and $\kappa$ the concentration factor, dimensionless.
\[C_{\mu,\ell} = \frac{\dot{m}_j V_j}{q S_\ell}, \qquad \kappa = \frac{S}{S_\ell} = \frac{C_{\mu,\ell}}{C_\mu} \tag{29}\]The force is unchanged by that choice, a point that is easy to get wrong. The force increment is $\mathcal{A} \dot{m}_j V_j$, the amplification times the momentum supplied, and no reference area appears in it. What the choice changes is which value of the amplification applies, because the amplification is a function of the local momentum coefficient and not of the wing-referenced one. Concentrating the air does not create force. It moves the device along its amplification curve.
Inverting for the area that lands a given local coefficient gives the relation the design has to satisfy.
\[S_\ell = \frac{\dot{m}_j V_j}{C_{\mu,\ell} \, q} \tag{30}\]| Target local momentum coefficient | What it is | Influenced area, square metre | Share of the wing | Concentration factor | Area per effector, square metre |
|---|---|---|---|---|---|
| 0.0008 | the smallest at which a lift response is reported | 18.9013 | 118.13 percent | 0.85 | 1.35009 |
| 0.0034 | a measured synthetic-jet reattachment | 4.4474 | 27.80 percent | 3.60 | 0.31767 |
| 0.0200 | a measured reattachment on a modified coefficient | 0.7561 | 4.73 percent | 21.16 | 0.05400 |
| 0.0300 | the low end of the super-circulation threshold | 0.5040 | 3.15 percent | 31.74 | 0.03600 |
| 0.0500 | the high end of the super-circulation threshold | 0.3024 | 1.89 percent | 52.91 | 0.02160 |
The arithmetic shows plainly why CRANE is a control programme and not a high-lift programme. A high-lift system must change the lift of a whole wing, so it needs a wing-referenced coefficient in the super-circulation regime. Inverting the bleed relation for the fraction of core flow that would take gives the following.
Because the momentum coefficient is exactly proportional to the bleed fraction, the inversion is one line. Let $C_\mu^{\,1}$ be the coefficient a bleed fraction of unity would give.
\[C_\mu = \beta \, C_\mu^{\,1} \quad \Longrightarrow \quad \beta_{\mathrm{req}} = \frac{C_\mu^{\,\mathrm{target}}}{C_\mu^{\,1}}, \qquad \beta_{\mathrm{req}} \le 1 \tag{31}\]The last inequality is no convention. It states that the engine cannot bleed more core flow than it has, and it decides which rows below are physical.
| Target wing-referenced momentum coefficient | Bleed as a share of core flow | Physically available |
|---|---|---|
| 0.0008 | 6.77 percent | yes |
| 0.0034 | 28.78 percent | yes |
| 0.0200 | 169.30 percent | no |
| 0.0300 | 253.95 percent | no |
| 0.0500 | 423.25 percent | no |
Reaching the lowest coefficient at which a lift response has been measured takes 6.77 percent of core flow, and a measured reattachment takes 28.78 percent. Reaching the super-circulation threshold of three percent would take 253.95 percent of core flow, or two and a half times the core flow the engine has, and five percent would take 423.25 percent. The coefficient a bleed fraction of unity would deliver follows from the reference case, and each row is the target over it,
\[C_\mu^{\,1} = \frac{0.000945066}{0.08} = 0.0118133 , \qquad \frac{0.0008}{0.0118133} = 0.0677 , \qquad \frac{0.0034}{0.0118133} = 0.2878 , \qquad \frac{0.03}{0.0118133} = 2.5395 , \qquad \frac{0.05}{0.0118133} = 4.2325 . \tag{32}\]So the assumed bleed fraction of 8 percent is slightly more than the lowest measured threshold needs and far short of super-circulation, which places this aircraft’s air budget in the separation-control regime by arithmetic rather than by assumption. A control system changes a moment rather than a lift, which needs a local effect at a long moment arm, and the concentration factor is what makes that affordable. The same arithmetic that forbids the first permits the second.
Against the Thresholds the Literature Reports, and the One That Matters
A measured lift response appears in the review at a momentum coefficient of 0.08 percent, with the regime boundary at three to five percent [Abdolahipour 2024], and the table below sets this aircraft’s wing-referenced figure against each reported threshold.
The regime boundary is defined phenomenologically as the coefficient that just suffices to suppress separation completely, and the transition across it is gradual, with the fall in actuation efficiency noted where the momentum coefficient was defined. The framing of a curve with a threshold, a favourable band and a saturation is therefore the literature’s and not this article’s.
| Reported threshold | Momentum coefficient | This article’s figure over it | Concentration factor needed | Influenced area, square metre | Per effector, square metre |
|---|---|---|---|---|---|
| NACA 0015 flapped airfoil, periodic excitation, Reynolds 1.5e5 to 6e5 | 0.08 percent | 1.1813 | none needed | 18.901 | 1.3501 |
| NACA 0025, synthetic jet, Reynolds 1e5, reduced frequency 58, drag down 45 per | 0.34 percent | 0.2780 | 3.60 | 4.447 | 0.3177 |
| NACA 0012, modified coefficient, Reynolds 2.3e4, separated below 1.5 percent | 2.00 percent | 0.0473 | 21.16 | 0.756 | 0.0540 |
| the coefficient that just suffices to suppress separation completely | 3.00 percent | 0.0315 | 31.74 | 0.504 | 0.0360 |
| the same, upper end of the reported range | 5.00 percent | 0.0189 | 52.91 | 0.302 | 0.0216 |
Those thresholds come from experiments at Reynolds numbers of twenty-three thousand to six hundred thousand, and this aircraft flies at eleven million [Abdolahipour 2024]. The comparison is valid but not the sharpest available. The sharpest is in the section below on Reynolds number, where a report that tested the method at this aircraft’s Reynolds number quotes the coefficients it used. Against that report the bleed budget falls inside the demonstrated range.
One of the low-Reynolds thresholds carries a warning that applies whatever the scale. A numerical study of momentum injection on a NACA 0012 found the flow separated below a modified coefficient of 1.5 percent, transitional to 2 percent and reattached above, and reported that below the threshold the control injection decreased the lift relative to the uncontrolled case [Abdolahipour 2024]. Blowing below threshold is not merely ineffective. It can be worse than nothing, because low-momentum air added to a boundary layer reduces the momentum that boundary layer carries. The warning concerns a mechanism, whatever the Reynolds number, and for that reason the velocity ratio below is a condition and not a preference.
A Force Is Not a Moment, and the Moment Arm Is Where Control Lives
Every figure so far has been a force and an aeroplane is commanded with moments. The conversion is trivial, and it gives the meaning of the phrase above about a local effect at a long moment arm. Let $y$ be the moment arm in metre, $\mathcal{M}$ the moment in newton metre and $C_{\mathcal{M}}$ its coefficient on the span, dimensionless.
\[\mathcal{M} = \Delta L \, y = \mathcal{A} \, J \, y, \qquad C_{\mathcal{M}} = \frac{\mathcal{M}}{q \, S \, b} \tag{33}\]| Moment arm, metre | Moment at an amplification of thirty, newton metre | Moment coefficient on the span |
|---|---|---|
| 2.000 | 9,364 | 0.00620 |
| 2.286 | 10,703 | 0.00709 |
| 3.000 | 14,045 | 0.00930 |
| 4.000 | 18,727 | 0.01240 |
The whole air budget at an amplification of thirty is 4,682 newton, and placed at a quarter of the span, being 2.286 metre, that is 10,703 newton metre or a rolling-moment coefficient of 0.00709. The jet’s own reaction, without any amplification, is 156.1 newton. With the dynamic pressure of 10,320.7 pascal at the reference condition and the span of 9.144 metre,
\[\mathcal{A} J = 30 \times 156.06 = 4{,}682 , \qquad \mathcal{M} = 4{,}682 \times 2.286 = 10{,}703 , \qquad C_{\mathcal{M}} = \frac{10{,}703}{10{,}320.7 \times 16 \times 9.144} = 0.00709 . \tag{34}\]This is why the moment arm does more work than the amplification in a control application, and why the effectors are on the outboard wings. Doubling the arm doubles the moment for no extra air, where doubling the amplification requires a different flow regime. A high-lift system has no such lever, because lift acts where it acts.
The Velocity Ratio the Momentum Coefficient Conceals
The review cautions explicitly against the use this article has been making of the momentum coefficient. It notes that the coefficient is not an appropriate parameter for comparing the benefits of flow control across different settings and configurations, because many combinations of mass flow and jet velocity give the same value, and that the parameter which separates them is the velocity ratio [Abdolahipour 2024].
Let $\mathrm{VR}$ be the jet velocity ratio, dimensionless, with $V_\infty$ the free-stream velocity in metre per second.
\[\mathrm{VR} = \frac{V_j}{V_\infty} \tag{35}\]The condition the review reports is that the velocity ratio must exceed unity, because otherwise the injected air reduces rather than increases the momentum the boundary layer carries, making separation more likely. At the reference condition the jet leaves at 442 metre per second against a free stream of 212.2 metre per second, so the velocity ratio is 2.083. At Mach 0.2 it is 7.29. The condition is met across the envelope with margin.
At this point the engine’s architecture stops being merely expensive and becomes an advantage. In the review’s summary, when the flow is excited at the same momentum coefficient, a higher velocity ratio yields better outcomes than a higher mass flow at a lower velocity ratio [Abdolahipour 2024]. A single-stage centrifugal compressor forces bleed at full cycle temperature, and a choked jet from hotter air is a faster jet. Air taken at 584 kelvin leaves at 442 metre per second where the same mass flow cooled to ambient would leave at 277 metre per second, a velocity ratio of 2.083 against 1.304, or a factor of 1.597.
The constraint that makes the air expensive in compressor work also makes it effective in the one dimension the literature says the momentum coefficient hides. The point does not favour this engine, but it argues against precooling the bleed, and it follows from two facts about the engine and one sentence in a review.
The Plenum Pressure Sets the Orifice Area
A choked orifice passes a mass flow set by its area and its plenum conditions. Let $A^{*}$ be the total throat area in square metre and $p_t$ the plenum total pressure in pascal.
\[\dot{m}_j = A^{*} \, \frac{p_t}{\sqrt{T_t}} \sqrt{\frac{\gamma}{R}} \left( \frac{\gamma+1}{2} \right)^{-\frac{\gamma+1}{2\left(\gamma-1\right)}} \tag{36}\]The constant evaluates to 0.040415 in square-root kelvin second per metre. The table below computes its inverse.
\[A^{*} = \frac{\dot{m}_j \sqrt{T_t}}{p_t} \left[ \sqrt{\frac{\gamma}{R}} \left( \frac{\gamma+1}{2} \right)^{-\frac{\gamma+1}{2\left(\gamma-1\right)}} \right]^{-1} \; \propto \; \frac{1}{p_t} \tag{37}\]The area is inversely proportional to the plenum pressure at a fixed mass flow and depends on nothing else that varies, so the last column of the table is exactly the reciprocal of the first.
| Nozzle pressure ratio | Plenum pressure, kilopascal | Total throat area, square centimetre | Per effector, square millimetre | Slot height at ten millimetre width, millimetre | Within the published study’s range |
|---|---|---|---|---|---|
| 1.500 | 45.1 | 46.751 | 333.94 | 33.39 | yes |
| 2.000 | 60.2 | 35.063 | 250.45 | 25.05 | yes |
| 3.000 | 90.3 | 23.376 | 166.97 | 16.70 | yes |
| 4.000 | 120.4 | 17.532 | 125.23 | 12.52 | yes |
| 8.000 | 240.7 | 8.766 | 62.61 | 6.26 | no |
| 19.419 | 584.3 | 3.611 | 25.79 | 2.58 | no |
The momentum coefficient is identical in every row of that table, to twelve significant figures, and the throat area varies by a factor of 12.95 across it. Since the area is inversely proportional to the plenum pressure,
\[\frac{A^{*}\left( 1.5 \right)}{A^{*}\left( 19.419 \right)} = \frac{19.419}{1.5} = 12.95 , \qquad \frac{A^{*}\left( 4 \right)}{A^{*}\left( 19.419 \right)} = \frac{19.419}{4} = 4.85 , \tag{38}\]where the argument is the nozzle pressure ratio. A choked jet’s velocity depends on its plenum temperature alone, so regulating the plenum pressure down changes the orifice and changes nothing else. The altitude result therefore survives any plenum schedule that keeps the nozzle choked, a useful robustness, because no plenum pressure for this aircraft is published.
The Paper That Asked This Question and Could Not Be Read
A 2024 paper in the AIAA Journal asks the question this section answers, and its full text is not available. Reddy and Woszidlo set out to determine whether the established models for estimating the momentum coefficient hold as ambient and supply conditions change during flight, simulated a fluidic oscillator and a steady jet nozzle at supply pressure ratios from 1.5 to 4, supply temperatures of 293, 550 and 800 kelvin, and ambient conditions at sea level and ten kilometres, and report that the mass flow collapses as a function of pressure ratio when scaled with one parameter and that the momentum coefficient collapses when scaled with another [Reddy and Woszidlo 2024].
Both registries render those two scaling parameters as [Formula: see text], the paper is closed access, and no repository holds a full text. So only the abstract has been read, and no quantity from inside that paper is quoted here and no claim is made about which scaling parameter it found.
The derivation above is therefore very likely a rediscovery. Its cancellations are exhibited rather than asserted and every step is checkable from the equations given, but a published study that simulates two actuator types across two altitudes and reports that the momentum coefficient collapses under a scaling parameter is, on the face of the abstract, reporting a result of the same family. No novelty is claimed.
What the comparison of parameter ranges does establish is a gap, and that comparison needs only the abstract. The published study’s supply pressure ratios stop at 4 [Reddy and Woszidlo 2024]. A plenum fed from the discharge of a compressor of pressure ratio 14 stands at a nozzle pressure ratio of 19.419 at every altitude, the factor of 4.85 displayed above. Either a real aircraft regulates its plenum far below compressor discharge, in which case the slot areas in the table above are the large ones, or the published scaling study does not cover the regime a bleed-fed aircraft would run in. The plenum total temperature tabulated with the choked jet, 584 kelvin, does fall inside the published 293 to 800 kelvin range.
Dependent Systems
The Engine, and the One Place a Centrifugal Compressor Lets You Bleed
The NASA test report on the FJ44-3A, read in full, describes the engine’s architecture in enough detail to constrain the air budget [FJ44 turbofan engine test]. The low-pressure spool carries an integrally bladed wide-chord fan and blisk compressor stages driven by a two-stage low-pressure turbine. The high-pressure spool carries a single-stage centrifugal compressor driven by a single-stage high-pressure turbine. The combustor is an effusion-cooled annular design, the exhaust is a long duct with an internal daisy-lobe mixer, and the inlet is a pitot design. The base engine weighs 525 pound, is 48 inch long and 23 inch in diameter, in a 3,000 pound-force thrust class.
A single-stage centrifugal high-pressure compressor has exactly one bleed station, being its discharge. An axial compressor of many stages offers a choice of ports at intermediate pressures and temperatures, and a designer wanting moderate-pressure air takes it from a middle stage. This engine offers no such choice. Air taken from the high-pressure spool comes out at the full cycle pressure ratio and the full cycle temperature, the most expensive air in the engine in compressor work and the hottest in the plenum. That is favourable for jet velocity and unfavourable for everything else, the strongest architectural constraint the published record permits deriving.
No mass flow is published for this engine, so it is obtained by two independent routes and the agreement is the check.
The geometric route passes the compressible mass flow through a circular face. Let $D_f$ be the fan diameter in metre and $M_f$ the axial Mach number at the face, dimensionless.
\[\frac{\dot{m}}{A} = \rho_f M_f a_f , \qquad \dot{m} = \frac{\pi D_f^2}{4} \cdot \frac{p_{\mathrm{sl}}}{\sqrt{T_{\mathrm{sl}}}} \sqrt{\frac{\gamma}{R}} \, M_f \, \sigma_f^{- \frac{\gamma+1}{2\left(\gamma-1\right)}} \tag{39}\]where the face quantities are the static conditions reached by expanding sea-level total conditions to $M_f$, and $\sigma_f$ is the total-condition factor at the face. At a face Mach number of 0.55 that is 192.2 kilogram per second per square metre, and the second form agrees with the first to one part in ten to the ninth.
The thermodynamic route inverts the static thrust. At static conditions there is no ram drag to subtract, so the thrust is exactly the exit momentum flux and the inversion is exact. Let $F_{\mathrm{sls}}$ be the sea-level static thrust in newton and $V_e$ the exit velocity in metre per second.
\[F_{\mathrm{sls}} = \dot{m} V_e, \qquad \dot{m} = \frac{F_{\mathrm{sls}}}{V_e} \tag{40}\]| Route | Assumption swept | Mass flow, kilogram per second |
|---|---|---|
| Geometric | fan diameter 20 to 23 inch, face Mach 0.50 to 0.60 | 36.5 to 54.4 |
| Thermodynamic | static thrust over exit velocity 280 to 340 metre per second | 39.2 to 47.7 |
| Taken forward | the value both routes admit | 43 |
The geometric route and the thermodynamic route overlap, and 43 kilogram per second, being 95 pound per second, is the figure both admit.
Bleed comes from the core, so the bypass ratio divides it. Let $B$ be the bypass ratio, dimensionless.
\[\dot{m}_{\mathrm{core}} = \frac{\dot{m}_{\mathrm{total}}}{1 + B}, \qquad \frac{\dot{m}_j}{\dot{m}_{\mathrm{total}}} = \frac{\beta}{1 + B} \tag{41}\]At an assumed bypass ratio of 3.3 the core flow is 10.0 kilogram per second, and a bleed fraction of 8 percent of core flow gives 0.3530 kilogram per second of air at 584 kelvin leaving at 442 metre per second. As a share of total engine flow that bleed is 1.860 percent.
That share is also a lower bound on the thrust the bleed costs, and a lower bound is all that is claimed. The bled air is gone from the exhaust, so at minimum its share of the momentum flux is lost.
\[\frac{\Delta F}{F} \ge \frac{\beta}{1 + B} = 1.860 \text{ percent} \tag{42}\]The true figure is larger and remains uncomputed, because the compressor work of 334.1 kilojoule per kilogram is also wasted and the turbine has less flow with which to drive the fan. The inequality is the defensible statement, and the gap between it and the truth is the size of the cycle rematching, which needs an engine deck.
The engine is 238 kilogram, which against the contractor’s weight is 7.29 percent. DARPA describes the aircraft’s weight, size and speed as similar to those of a military trainer [DARPA moves forward on the X-65], and the engine fraction is offered only as a figure on that comparison.
Two Bandwidths, and Conflating Them Is a Real Error
An active-flow-control effector has two characteristic frequencies and they differ by orders of magnitude and answer to different masters.
The fluid-mechanical frequency is set by the convective time over the surface being controlled. Let $F^{+}$ be the reduced frequency, dimensionless, $f_e$ the excitation frequency in hertz, $\bar{c}$ the reference chord in metre and $V_\infty$ the free-stream velocity in metre per second.
\[F^{+} = \frac{f_e \, \bar{c}}{V_\infty}, \qquad f_e = \frac{F^{+} V_\infty}{\bar{c}} \tag{43}\]The band is not this article’s guess. A comprehensive analysis of airfoil flow-control studies reports that in the majority of cases the optimal reduced frequency for controlling post-stall separation falls between 0.3 and 4, and the same range applies to an excited shear layer and to a deflected flap [Abdolahipour 2024]. Individual optima reported in that review cluster near unity, at 0.65, 0.8, 0.85, 1.284, 1.3 and 1.5 in six separate studies, with one at 6 where the target was a separation bubble. There is also evidence that reduced frequencies of ten and twenty work, which the review attributes to a different group and explains by the spread of surface curvature, upstream conditions and excitation method across experiments.
At a reference chord of 1.750 metre, which follows from the mid-sweep wing area and the published span, the band maps onto the following frequencies.
| Mach number | Free-stream speed, metre per second | Reduced frequency 0.3, hertz | Reduced frequency 4, hertz | Reduced frequency 10, hertz |
|---|---|---|---|---|
| 0.2 | 60.6 | 10.4 | 138.6 | 346.5 |
| 0.3 | 91.0 | 15.6 | 207.9 | 519.8 |
| 0.5 | 151.6 | 26.0 | 346.5 | 866.3 |
| 0.7 | 212.2 | 36.4 | 485.1 | 1,212.8 |
The flight-control frequency is something else entirely. This article assumes that an aeroplane of this size has a short-period mode somewhere between roughly 0.3 and 3.0 hertz, and a control system needs bandwidth of that order to stabilise and command it.
The two bands do not overlap. The excitation band runs from 10.4 to 485.1 hertz across the flight envelope on the published reduced-frequency range, and reaches 1,213 hertz if a reduced frequency of ten is wanted at the top of the envelope. The control band runs from 0.3 to 3.0. At their closest approach they are separated by a factor of 3.47 and at their widest by a factor of 1,617.
The separation makes the architecture tractable and makes a fluidic oscillator the natural device. A sweeping jet generates its own oscillation frequency from its internal geometry and its supply, with no moving part and without being commanded. The high frequency is produced by the device and the low frequency by the valve that feeds it. A flight control computer commands mass flow at a few hertz and the effector supplies hundreds of hertz on its own. An architecture that commanded the excitation frequency directly would need actuators two orders of magnitude faster and would gain nothing.
More deeply, the gap means the two can be designed separately. The effector’s internal geometry is sized against the chord and the flight speed, the valve is sized against the rigid-body dynamics, and neither constrains the other. A single actuator asked to serve both would be sized by the harder of the two requirements and would be wrong for the other.
The Planform and the Four Wing Sets
The aircraft is described as a joined tandem wing in a diamond configuration with twin vertical tails [Aurora X-65 CRANE], as having a distinctive diamond-like wing shape chosen to maximise what can be learned about active flow control at full scale [DARPA moves forward on the X-65], and as having a triangular wing design that enables testing across multiple wing sweeps, with four wings of varying sweep angle and replaceable outboard panels [X-65 experimental aircraft]. Three descriptions of one planform, from three sources, in three vocabularies.
The four wing sets are the part that follows from the research question rather than from aerodynamics. A flow-control effector’s effectiveness depends on the pressure gradient it works against, and the pressure gradient over a swept wing depends on the sweep. So sweep is the independent variable the experiment most wants to vary and the one a fixed airframe cannot vary, and so the aircraft was built to have its outboard wings changed.
No wing area is published. The relations that turn a span and an area into the quantities the tables below carry are elementary. Let $b$ be the span in metre, $AR$ the aspect ratio, dimensionless, $W$ the weight in newton and $W/S$ the wing loading in pascal.
\[\bar{c} = \frac{S}{b}, \qquad AR = \frac{b^2}{S} = \frac{b}{\bar{c}}, \qquad \frac{W}{S}, \qquad C_L = \frac{W}{q S} = \frac{W/S}{q} \tag{44}\]At a reference area of 16 square metres and the published span the mean chord is 1.750 metre, the aspect ratio 5.226, the wing loading 2.00 kilopascal and the lift coefficient in level flight at Mach 0.7 and thirty thousand feet 0.1939. The thrust-to-weight ratio on the contractor’s weight is 0.4167 and the engine is 7.29 percent of it.
| Reference area, square metre | Aspect ratio | Mean chord, metre | Wing loading, kilopascal | Lift coefficient at Mach 0.7 and 30,000 feet |
|---|---|---|---|---|
| 12 | 6.97 | 1.312 | 2.67 | 0.259 |
| 14 | 5.97 | 1.531 | 2.29 | 0.222 |
| 16 | 5.23 | 1.750 | 2.00 | 0.194 |
| 18 | 4.65 | 1.969 | 1.78 | 0.172 |
| 20 | 4.18 | 2.187 | 1.60 | 0.155 |
| 22 | 3.80 | 2.406 | 1.46 | 0.141 |
A joined tandem wing carries lift on two surfaces, so a single reference area is a simplification, used here only as the denominator of a coefficient and making no claim about the aerodynamics.
The Fourteen and the Fourteen
The aircraft carries fourteen active-flow-control effectors and fourteen conventional control surfaces [X-65 experimental aircraft]. The counts are equal, which the published sources state without remarking on it.
The conventional surfaces are not a fallback. They are the experiment’s control group. DARPA’s description is explicit. The aircraft was built with two sets of control actuators so as to minimise risk and maximise insight, the performance with traditional surfaces serves as a baseline, and successive tests selectively lock down the moving surfaces and use the effectors instead [DARPA moves forward on the X-65]. The programme manager called the conventional surfaces training wheels.
This is an experimental design and not a hedge, and the next section explains why it is the right one.
The Measurement Design
Two Vehicles Cannot Be Differenced and One Aircraft Can
The X-64A article reached a negative conclusion about experimental design and this aircraft is the positive answer to it. That article considered whether the value of an altitude-compensating nozzle could be established by flying two vehicles and subtracting, and concluded that it could not, because two vehicles differ in mass, drag, trajectory and instrumentation, and the difference sought was smaller than the differences not sought.
The X-65A performs the same subtraction inside one airframe. The mass is the same, the drag is the same, the trajectory is the same, the transducers are the same, the flight is in some cases the same flight. What changes between the baseline and the test case is which effectors are commanded, and the difference is therefore an estimate of the thing wanted rather than of the things not wanted.
This article takes that to be the consequential design decision, and the published record treats it as risk reduction. It is risk reduction. It is also the reason the programme can produce a number at all. A vehicle with only flow-control effectors would return an absolute measurement of its own handling qualities and no measurement of what flow control is worth. A vehicle with both returns a difference, and the question asks for a difference.
The Combinatorial Number Is a Red Herring
Fourteen surfaces that may each be locked or free give a large number of configurations and the number is misleading.
The configuration count is a power of two and the flight count is a ceiling. Let $n_s$ be the number of conventional surfaces, $n_w$ the number of wing sets, $N_c$ the configuration count and $N_f$ the flights needed at $k$ usable test points each.
\[N_c = 2^{n_s} n_w, \qquad N_f = \left\lceil \frac{N_2}{k} \right\rceil \tag{45}\]| Quantity | Count |
|---|---|
| Active-flow-control effectors | 14 |
| Conventional control surfaces | 14 |
| Wing sets of differing sweep | 4 |
| Distinct locked-or-free configurations of the conventional surfaces | 16,384 |
| The same, once each wing set is counted separately | 65,536 |
| Configurations if the surfaces are locked only in symmetric pairs | 128 |
| Control derivatives in a linear model, three axes | 42 |
| Pairwise interactions among the effectors | 91 |
| Terms in a second-order model, per axis | 105 |
| Terms in a second-order model, three axes | 315 |
| Flights to identify that model at thirty test points each | eleven |
16,384 configurations, or 65,536 once the wing sets are counted, is not the size of the flight-test programme and treating it as one would be an error. An aircraft is not characterised by visiting its configurations. It is characterised by identifying a model, and the model is small.
Let $n$ be the number of effectors, dimensionless, and $n_a$ the number of axes. A linear model in which effectors superpose has one control derivative per effector per axis.
\[N_1 = n \, n_a \tag{46}\]A model that allows every pair of effectors to interact adds one term per pair.
\[N_2 = n_a \left( n + \binom{n}{2} \right) = n_a \left( n + \frac{n \left( n - 1 \right)}{2} \right) \tag{47}\]For fourteen effectors on three axes that is 42 derivatives in the linear case and 315 terms with every pairwise interaction included. At thirty usable test points per flight, identifying the full second-order model needs about eleven flights. A hundred flights would visit 0.61 percent of the lock-out configurations and would over-determine the second-order model many times over.
So the combinatorics do not bind. The aircraft’s availability does, and the quantity to watch in a flight-test programme of this kind is flights flown rather than configurations covered. Locking the surfaces only in symmetric pairs reduces the configuration count to 128, a reminder that most of the apparent combinatorial space is asymmetric cases a flight-test programme has no reason to visit.
The assumption at risk, and the right one to be nervous about, is superposition. The lift a flow-control effector produces saturates as its momentum input rises [Abdolahipour 2024], and effectors sharing a boundary layer may also interact through it, so the pairwise terms are not a formality. The second-order model is the prudent one, and it costs eleven flights rather than eleven thousand.
Twenty-Eight Effectors on Three Axes Is an Allocation Problem
With both sets of effectors available the aircraft has twenty-eight control effectors and three axes to command, so it is over-actuated by a factor approaching ten. That is no nuisance but a recognised problem with a literature of its own [Bodson 2000] [Benson and Orr], in which a control allocator maps a demanded moment onto a redundant effector set subject to rate, position and now mass-flow constraints.
The problem has a standard form. Let $\mathbf{u}$ be the vector of effector demands, dimensionless and of length $n_e$, $\mathbf{B}$ the control effectiveness matrix of three rows and $n_e$ columns, and $\mathbf{m}$ the demanded moment vector in newton metre.
\[\mathbf{m} = \mathbf{B} \mathbf{u}, \qquad \mathbf{u} \in \mathcal{U} \tag{48}\]Three axes can be commanded at all only if the effectiveness matrix has full row rank, so a lock-out that reduces the rank below three removes an axis entirely rather than weakening it.
\[\operatorname{rank} \mathbf{B} = 3 \tag{49}\]The reachable set is the image of the admissible set, being $\mathbf{B} \mathcal{U}$, and in that image this aircraft differs from the classical case.
For this aircraft the admissible set has a feature the classical version does not. Conventional surfaces are constrained individually, each by its own deflection and rate limits, which makes $\mathcal{U}$ a box. The flow-control effectors are constrained collectively, because they share one plenum. Commanding two effectors hard at once does not cost twice the deflection, it costs twice the air, and the air is one budget.
\[\mathcal{U}_{\mathrm{box}} = \left\{ \mathbf{u} : 0 \le u_i \le \bar{u} \right\}, \qquad \mathcal{U}_{\mathrm{plenum}} = \left\{ \mathbf{u} : u_i \ge 0, \; \sum_i u_i \le \bar{u} \right\} \tag{50}\]The first is a box and the second is a simplex, and their images are different shapes. The image of a box under a linear map is a zonotope, being the Minkowski sum of segments along the columns. The image of a simplex is the convex hull of the images of its vertices, which for a simplex with a vertex at the origin is the hull of the origin together with the columns.
Sharing One Plenum Enlarges the Reachable Set
The comparison that matters holds the total air fixed, and when it does the shared plenum wins. Comparing a simplex against a box whose every side is the full supply is comparing against a vehicle with $n_e$ times as much air, which answers no question anybody has. The fair comparison gives the box an equal share of one supply.
For a tractable case take two axes and $n_e$ effectors whose columns are unit vectors spread evenly over a half plane, a half plane because an effector that can only blow has one-sided authority. The zonotope and hull areas then have closed forms.
\[\operatorname{area} \mathcal{Z} = \sum_{i < j} \left\lvert \det \left( \mathbf{b}_i , \mathbf{b}_j \right) \right\rvert, \qquad \operatorname{area} \mathcal{H} = \tfrac{1}{2} \sum_i \left\lvert \det \left( \mathbf{b}_i , \mathbf{b}_{i+1} \right) \right\rvert \tag{51}\]| Effectors | Independent, each at the full supply | Shared supply, demands summing to the supply | Independent, each at an equal share of one supply | Shared over equal shares |
|---|---|---|---|---|
| 3 | 2.0000 | 1.0000 | 0.22222 | 4.5000 |
| 4 | 4.3301 | 1.2990 | 0.27063 | 4.8000 |
| 7 | 14.9282 | 1.5000 | 0.30466 | 4.9236 |
| 14 | 61.7681 | 1.5556 | 0.31514 | 4.9360 |
| 28 | 248.9554 | 1.5673 | 0.31755 | 4.9355 |
| 100 | 3,182.5135 | 1.5705 | 0.31825 | 4.9349 |
At fourteen effectors the shared plenum reaches 4.9360 times the moment area that equal fixed shares of the same total supply would reach, and the ratio converges. It converges to an exact closed form. As the effector count grows the hull tends to a half disc and the equal-share zonotope tends to the reciprocal of pi, so
\[\lim_{n_e \to \infty} \frac{\operatorname{area} \mathcal{H}}{\operatorname{area} \mathcal{Z} / n_e^2} = \frac{\pi / 2}{1 / \pi} = \frac{\pi^2}{2} = 4.9348 \tag{52}\]At a hundred effectors the computed areas are 1.5708 and 0.31831, and the computed ratio is within a part in ten thousand of the limit.
So the plenum is not a penalty. Sharing one supply lets the allocator put all of the air wherever the moment is wanted, where fixed per-effector limits strand air in effectors that are not needed. Sharing costs no reachable area. Its price is that the set is no longer a box, so the allocator cannot be a per-axis gain and has to solve a programme.
The geometry here is illustrative. The effector positions on the X-65A are not published, so the columns are an assumption of even spread rather than a measurement, and the two-axis reduction is for tractability. What does not depend on those choices is that a simplex image is a hull and a box image is a zonotope, a fact about linear maps and not about this aeroplane.
The progressive lock-out programme is therefore a reallocation experiment. Each step removes effectors from the allocator’s disposal and asks whether the remaining set still spans the demanded moments with adequate margin. The question is not whether flow control works. It is whether the reachable set of moments remains large enough as the conventional columns are deleted from the effectiveness matrix.
The Flight Test Record Is Empty
On this article’s dateline the X-65A has not flown, has not been rolled out, and its fuselage is not finished. [two-year flight delay]
The record of what was planned is as follows. Fabrication was under way at facilities in West Virginia and Mississippi by January 2024 [two-year flight delay], rollout was scheduled for early 2025, and first flight for the summer of the same year [DARPA moves forward on the X-65]. Both dates passed. DARPA paused the programme earlier in 2025 because the cost of producing the prototype exceeded expectations, restructured it into a partnership in which the contractor became a co-investor, finalised that arrangement in August 2025, and now expects ground testing in late 2026 or early 2027 and first flight in late 2027 [two-year flight delay]. The fuselage was expected to be complete in January 2026. The aircraft is being built in Bridgeport, West Virginia [two-year flight delay].
The slip is 30 months against the schedule stated when the build contract was announced, which is seven years after the first CRANE contracts and 28 years after the programme its manager named as its ancestor.
There is therefore no flight envelope reached to compare with the envelope designed for. The absence is stated plainly. What can be compared is what was promised against what was recorded, and for that the budget justifications are a better instrument than the press.
What Seven Budget Books Said, Year by Year
The Research, Development, Test and Evaluation justification books are a primary source with an unusual property. Each one states three fiscal years, and the same fiscal year appears in three consecutive books wearing three different hats. The three columns of an exhibit are the prior year’s actual, the current year’s estimate and the budget year’s request. A fiscal year can thus be watched from the moment it is asked for to the moment it is recorded, and every CRANE line in every book from the request for fiscal year 2020 to the request for fiscal year 2026 was read for this article [PB2020] [PB2021] [PB2022] [PB2023] [PB2024] [PB2025] [DARPA RDT&E justification, PB2026].
The quantity tabulated against each year is the outcome against the request, expressed as a fraction so that years of different size are comparable. Let $r$ be the request and $o$ the outcome, both in dollars.
\[\frac{o - r}{r} \tag{53}\]| Fiscal year | Requested | Later estimated | Finally recorded | Outcome against request |
|---|---|---|---|---|
| FY2020 | 13.000 | 20.000 | 23.573 | +81.33 percent |
| FY2021 | 25.000 | 26.000 | 23.614 | -5.54 percent |
| FY2022 | 31.607 | 28.507 | 25.507 | -19.30 percent |
| FY2023 | 52.685 | 40.565 | 40.565 | -23.00 percent |
| FY2024 | 44.500 | 42.500 | 38.264 | -14.01 percent |
| FY2025 | 29.715 | 23.893 | - | -19.59 percent |
| FY2026 | 4.000 | - | - | - |
The first year came in 81.33 percent above its request and every year since has come in below. Fiscal year 2020 was requested at 13.000 million dollars, estimated at 20.000 and recorded at 23.573. From fiscal year 2021 to fiscal year 2025 the programme was funded below its request five times out of five, at a mean shortfall of 16.29 percent. The worst single year is fiscal year 2023, requested at 52.685 million and recorded at 40.565 million, which is -23.00 percent. The first figure and the mean of the five shortfalls are
\[\frac{23.573 - 13.000}{13.000} = 0.8133 , \qquad \frac{0.0554 + 0.1930 + 0.2300 + 0.1401 + 0.1959}{5} = 0.1629 . \tag{54}\]Across fiscal years 2020 to 2025 the programme asked for 196.507 million dollars and is expected to receive 175.416 million, a shortfall of 10.73 percent, and counting the fiscal year 2026 request of 4.000 million in both the totals are 200.507 and 179.416 million. Appropriations through fiscal year 2024 total 151.523 million dollars against 143,556,412 dollars obligated across the four contracts, a ratio of 1.0555, the right sort of agreement, because obligations lag appropriations. Summing the table’s columns,
\[\sum_{2020}^{2025} r = 196.507 , \qquad \sum_{2020}^{2025} o = 175.416 , \qquad 1 - \frac{175.416}{196.507} = 0.1073 , \qquad \sum_{2020}^{2024} o = 151.523 , \qquad \frac{151.523 \times 10^{6}}{143{,}556{,}412} = 1.0555 , \qquad \frac{4.000}{38.264} = 0.1045 . \tag{55}\]The fiscal year 2026 request is 4.000 million, equal to 10.5 percent of what fiscal year 2024 received. It is also the only round number in the series, every other figure carrying three decimal places. A round number among precise ones is the signature of a placeholder.
The Programme Moved Programme Elements
Through fiscal year 2025 CRANE was funded in programme element 0603286E, Advanced Aerospace Systems. From fiscal year 2026 it is funded in programme element 0603468E, Advanced Complex Systems, and the book’s exhibits for both programme elements state the transfer explicitly [DARPA RDT&E justification, PB2026]. An aircraft programme was moved out of the aerospace line and into the complex-systems line.
Promises That Slipped Two Fiscal Years Across Four Books
The plan text is the part of a budget book that ages worst, and reading the same milestone across consecutive books is how the ageing becomes visible.
| Book | Fiscal year it was promised for | The promise |
|---|---|---|
| PB2021 | FY2021 | Complete Preliminary Design Review |
| PB2022 | FY2022 | Conduct system critical design review |
| PB2023 | FY2023 | Conduct system critical design review |
| PB2024 | FY2024 | Complete system critical design review |
| PB2025 | FY2024 | Complete the system Critical Design Review |
| PB2022 | FY2022 | Initiate airworthiness and ground and flight test approvals |
| PB2023 | FY2023 | Initiate airworthiness and ground and flight test approvals |
| PB2024 | FY2024 | Complete airworthiness and ground and flight test approvals |
| PB2025 | FY2024 | Complete airworthiness and ground and flight test approvals |
| PB2026 | FY2025 | Complete airworthiness and flight test approvals |
| PB2024 | FY2024 | Complete fabrication and subsystem integration |
| PB2025 | FY2024 | Complete fabrication and subsystem integration |
| PB2025 | FY2025 | Initiate and complete flight testing |
| PB2026 | FY2025 | Initiate and complete flight testing |
The critical design review was promised for fiscal year 2022, then fiscal year 2023, then fiscal year 2024, then fiscal year 2024 again. Completing fabrication and subsystem integration was a fiscal year 2024 plan in two consecutive books, and the fuselage was not expected to be finished until January 2026 [two-year flight delay]. Initiating and completing flight testing was a fiscal year 2025 plan in two consecutive books. [PB2022] [PB2023] [PB2024] [PB2025] [DARPA RDT&E justification, PB2026]
The last of those merits attention. The book published in June 2025 still lists initiating and completing flight testing of the demonstration aircraft among its fiscal year 2025 plans, four months after the zero-dollar change order that marks the pause and three months before the fiscal year ended [DARPA RDT&E justification, PB2026]. Its fiscal year 2026 entry asks 4.000 million dollars to complete active-flow-control flight testing and data delivery, against a first flight the same agency was describing as late 2027 five months later.
That is not a contradiction and reading it as one would be unfair. This article reads a justification book’s plan text for the current fiscal year as restating the plan on which that year’s appropriation was enacted, which was settled long before the book was printed. The plans column is a record of what was agreed, not a report of what is happening. The right conclusion is narrower and more useful. A budget justification is an excellent source for what a programme was authorised to do and a poor source for what it did, and this article uses it only for the first.
Comparison With Ground Prediction
Nothing has flown, so flight has returned nothing to set beside any ground prediction, and this section records what was predicted so that the comparison can be made when it does. At the time of writing the aircraft has not been rolled out and its fuselage is not finished, and DARPA expects ground testing in late 2026 or early 2027 and first flight in late 2027 [two-year flight delay]. Even the ground testing that would precede flight lies in the future.
The first prediction the record makes is a schedule, the only one that events have already answered. The flight test record above puts the slip at 30 months, and the reading of seven budget books follows the same promises through the justification books.
The aerodynamic predictions are this article’s own, and every one of them is conditional. They are the altitude independence of a bleed-fed momentum coefficient and its fall with Mach number, derived under Sizing From First Principles, and the authority ratios and crossover Mach number displayed with the authority ratio. None of them is a contractor estimate or a published wind tunnel result for this aircraft. The amplification is swept rather than predicted, and the critique of the framing below names it as the weakest link.
One of those predictions is framed so that two data points can test it, being the comparison of control moments at two altitudes and one Mach number set out at the end of the altitude cancellation above. The record leaves the source of the aircraft’s air unstated, so the prediction cannot yet be matched to the design, let alone to a flight.
The ground evidence the programme rests on is tunnel evidence, and the strongest of it was taken at a higher Reynolds number than this aircraft flies at, as the Reynolds-number discussion below sets out. The comparison the aircraft is built to make is internal rather than against a ground prediction, being the difference within one airframe described under the measurement design. None of the flights it needs has been flown, so this section can hold only the predictions.
What the Data Changed, What It Would Change and What It Has Already Changed
An aircraft that has not flown has changed nothing by flying, and its intentions are not counted as results. What CRANE has produced so far is a design, a set of tools and a contract structure, and the budget line states the intended product explicitly. Technologies, design tools and models developed under the programme are to be made available to all the services and to the civilian aerospace sector [DARPA RDT&E justification, PB2026]. The deliverable is named as tools and models, not only as an aeroplane.
Two things have already changed and neither is aerodynamic.
The first is the contracting arrangement. A DARPA technology demonstrator in which the contractor becomes a co-investor to complete the article is not the usual shape, and the agency’s account of why is that the new structure gets costs back to an affordable rate for the government, with the contractor taking over the platform for future test and development once the hard technical challenge is demonstrated [two-year flight delay]. The aircraft is to outlive the programme as a test asset, which the programme manager said at the outset was a design goal and which the modular wing sets exist to serve [DARPA moves forward on the X-65].
The second is that the aircraft became the reason to hold the designation. The X-65A designation was allocated in April 2023, between the Phase 2 award and the Phase 3 award, and the register carries it while the aircraft does not exist. It is allocated to a vehicle that is being built, slowly, with a first flight four and a half years after its designation. [Parsch, DOD 4120.15-L Addendum] [two-year flight delay]
The Contemporary Literature
| Cluster | Records |
|---|---|
| The momentum coefficient itself, which is almost never in a title | 13 |
| Control without a hinge, which is this aircraft’s actual prior art | 81 |
| Allocating a demand across more effectors than axes | 41 |
| The devices, from piezoelectric diaphragms to fluidic oscillators | 814 |
| Blowing, the Coanda surface and the jet flap | 750 |
| Separation and its control, which is what the effectors do | 207 |
| Controlling a vortex, including the forebody jets that flew first | 32 |
| Where the air comes from, and what taking it costs | 43 |
| Reynolds number, and why a tunnel is not an aeroplane | 111 |
| Active flow control as a subject in its own right | 275 |
| Joined, tandem, diamond and tailless, and controlling them | 350 |
| The mechanical control system this aircraft proposes to delete | 85 |
| Identifying an aircraft from its own flight data | 185 |
| Demonstrators, research aircraft and the practice of flying one | 131 |
| Jets in a crossflow | 81 |
| Wall jets and curved wall jets | 69 |
| Shear layers, receptivity and periodic excitation | 30 |
| This programme and the programme it descends from | 2 |
| The medium, admitted by the shared atmosphere family alone | 1 |
| Total | 3,301 |
Every research record in the survey is cited in exactly one cluster below. The clusters run from the specific to the general, so a record that fits a narrow cluster is listed there rather than under the broader one it is also a case of.
The momentum coefficient itself, which is almost never in a title
The parameter this article’s identity is built on. Titles that name the momentum coefficient, and titles on the blowing parameters and the actuation efficiency it summarises.
13 records. [Goodfellow et al. 2011] [Orazi et al. 2011] [Wang et al. 2011] [Goodfellow et al. 2013] [Zhou et al. 2017a] [Saadi and Bahi 2018] [Liu et al. 2020] [Abbasi 2022] [Otto et al. 2022] [Reddy and Woszidlo 2024] [Xu et al. 2025a] [Zhou et al. 2025] [Müller et al. 2026]
Control without a hinge, which is this aircraft’s actual prior art
The prior art the aircraft descends from. Hingeless and innovative control effectors, passive porosity, fluidic thrust vectoring, and the jet and plasma control of unmanned aircraft, missiles and projectiles.
81 records. [Cassel et al. 1969] [Ostroff 1985] [Goodrich and Sliwa 1987] [Goodrich et al. 1989] [Maughmer 1989] [Maughmer et al. 1990] [Bauer and Hemsch 1992] [Wood et al. 1992] [Bauer and Wood 1994] [Wing 1994] [Wood et al. 1994] [Durham and Bordignon 1995] [Barber et al. 1997] [Pedreiro 1997] [Strykowski 1997] [Wing and Giuliano 1997] [Gillard 1998] [Banks et al. 1999a] [Buffington 1999a] [Buffington 1999b] [Gilyard 1999] [Stable Multiple Model Adaptive 1999] [Wood and Bauer 1999] [Addington and Myatt 2000] [Deere 2000] [Erickson 2000] [Moeller and Rediniotis 2000] [Boskovic and Mehra 2001] [Hunter et al. 2001] [Bauer and Wood 2001] [Moeller and Rediniotis 2002] [Tinetti et al. 2002] [Traub et al. 2002] [Wood 2002] [Bauer and Wood 2003] [Chudoba and Cook 2003] [Deere 2003] [Deere et al. 2003] [Erickson 2003] [Waithe and Deere 2003] [Miller et al. 2004] [Raney et al. 2004] [Deere et al. 2005] [Patel et al. 2005] [Flamm et al. 2006] [Liu et al. 2006] [Patel et al. 2006a] [Patel et al. 2006b] [Boskovic et al. 2007] [Deere et al. 2007] [Flamm et al. 2007] [Kutay et al. 2007] [Lechevin et al. 2007] [Oppenheimer and Doman 2007] [Patel et al. 2007] [Staelens et al. 2007] [Zhang et al. 2007] [Kondor et al. 2008] [Neidhoefer et al. 2009] [Glezer et al. 2010] [Wang et al. 2010] [Weisman and Valasek 2010] [Johnson et al. 2011] [Turner 2011] [White et al. 2015] [Erickson 2017] [Kazarin et al. 2017] [Park et al. 2017b] [Matamoros and de Visser 2018] [Warsop and Crowther 2018] [Acheson et al. 2020] [Shayan and Van Kampen 2021] [Liu et al. 2022] [Zelman et al. 2023] [Kara and Kurtuluş 2024] [Li et al. 2024a] [Simon et al. 2025] [Boskovic et al] [Campbell et al] [Sumanik] [Okolo]
Allocating a demand across more effectors than axes
What fourteen effectors and fourteen surfaces on three axes require. Control allocation across redundant effectors, and the reconfigurable flight control that depends on it.
41 records. [Napolitano and Swaim 1989] [Buffington 1997] [Calise et al. 1998] [Wagner et al. 1998] [Page and Steinberg 1999] [Bodson 2000] [Cameron and Princen 2000] [Nam et al. 2000] [Brinker and Wise 2001] [Cotting et al. 2001] [Davidson et al. 2001] [Durham 2001] [Durham and Nelson 2001] [Shore and Bodson 2004] [Shore and Bodson 2005] [Burken et al. 2006] [Doman et al. 2007] [Song 2008] [Frost et al. 2009] [Frost et al. 2010] [Jamil et al. 2010] [Liu and Crespo 2010] [Mahulkar 2010] [Frost et al. 2012] [Waters et al. 2013] [Huijts and Voskuijl 2015] [Control Allocation and Flight 2016] [Xu et al. 2016] [Terrell and Zein-Sabatto 2017] [Yi et al. 2017] [Acheson 2018a] [Acheson 2018b] [Pei et al. 2018] [Vile et al. 2019a] [Vile et al. 2019b] [Litt 2024] [Sarioglu and Dogan 2024] [Benson and Orr] [Miller and Pei] [Acheson and Gregory] [Acheson et al]
The devices, from piezoelectric diaphragms to fluidic oscillators
The effectors themselves. Synthetic jets, sweeping jets and fluidic oscillators, plasma actuators, pulsed and vortex-generator jets, as devices and as applied to wings, bodies, inlets and turbines.
162 of these titles name a sweeping jet or a fluidic oscillator, and 147 of those date from 2010 or later. The device this article argues is the natural one, because it generates its own frequency, is the recent part of the literature.
814 records. [Westerman et al. 1965] [Martinez 1967] [Rehsteiner 1970] [Ito and Suematsu 1974] [Ito and Suematsu 1975] [Oshima and Kinoshita 1975] [Hellbaum and McDermon 1977] [Deadwyler 1980] [Pullin and Simmons 1985] [Roshko 1985] [Johnston and Nishi 1989] [Brenner 1993] [McManus et al. 1994] [McManus et al. 1995] [McManus and Magill 1996] [Sedor 1996] [Johari et al. 1997] [McManus et al. 1997] [Amitay et al. 1998] [Lachowicz et al. 1998] [Seifert et al. 1998] [Amitay et al. 1999] [Glezer 1999] [Lamp and Chokani 1999] [McLean et al. 1999] [Seifert and Pack 1999] [Bons et al. 2000] [Honohan et al. 2000] [Tanaka et al. 2000] [Amitay et al. 2001b] [Amitay et al. 2001a] [Gilarranz and Rediniotis 2001] [Nedungadi et al. 2001] [Tilmann 2001] [Cater and Soria 2002] [Chen and Beeler 2002] [Fasel et al. 2002] [Florent et al. 2002] [Gordon and Soria 2002] [Gregory et al. 2002] [Guo et al. 2002] [Guy et al. 2002] [Heinzen et al. 2002] [Johari and Rixon 2002] [Kim and Williams 2002] [Naughton et al. 2002] [Rediniotis et al. 2002] [Schaeffler et al. 2002] [Smith 2002] [Ashpis et al. 2003] [Gaeta et al. 2003] [Hassan and Roth 2003] [Holman et al. 2003] [Hultgren and Ashpis 2003b] [Hultgren and Ashpis 2003a] [Jiang and Liu 2003] [Johari and Rixon 2003] [Jones and Englar 2003] [Krothapalli et al. 2003] [Liu et al. 2003a] [Mathew et al. 2003] [Milanovic 2003] [Parekh et al. 2003] [Post and Corke 2003] [Rixon and Johari 2003] [Vadillo et al. 2003] [Yamaleev and Carpenter 2003] [Anderson et al. 2004b] [Anderson et al. 2004a] [Cybyk et al. 2004] [Duvigneau and Visonneau 2004] [Enloe et al. 2004] [Gallas et al. 2004] [Goldstein 2004] [Gordon et al. 2004] [Gregory et al. 2004] [Hansen and Bons 2004] [Hultgren and Ashpis 2004] [Lonneman and Cutler 2004] [Milanovic and Zaman 2004] [Mossi and Bryant 2004] [Naitoh and Yamamoto 2004] [Post and Corke 2004a] [Post and Corke 2004c] [Post and Corke 2004b] [Rediniotis 2004] [Rumsey et al. 2004] [Sahu 2004a] [Sahu 2004b] [Schaeffler and Jenkins 2004] [Tan 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al. 2006] [Mehta 2006] [Ogawara et al. 2006] [Orkwis and Daviaux 2006] [Orlov et al. 2006] [Pinheiro 2006] [Porter et al. 2006] [Post and Corke 2006] [Reimann et al. 2006] [Roth and Dai 2006] [Rumsey et al. 2006] [Samimy et al. 2006] [Schaeffler and Jenkins 2006] [Scholz et al. 2006] [Suzuki 2006] [Tamburello and Amitay 2006] [Trávníček et al. 2006] [Vatsa and Turkel 2006] [Webb et al. 2006] [Wilson 2006] [Zhang et al. 2006] [Alvi 2007] [Arienti and Soteriou 2007] [Bourgois et al. 2007] [Corke et al. 2007] [Cybyk et al. 2007] [Deb et al. 2007] [Deng et al. 2007] [Gregory et al. 2007] [Kim et al. 2007a] [Kim et al. 2007b] [Koso and Kinoshita 2007] [Kostas et al. 2007] [Luo et al. 2007] [Ng et al. 2007] [Ortmanns and Kähler 2007] [Post et al. 2007] [Raju et al. 2007] [Rumsey et al. 2007] [Rumsey 2007] [Schaeffler et al. 2007] [Segawa et al. 2007] [Shang 2007] [Sidorenko et al. 2007] [Suzen et al. 2007] [Takeuchi et al. 2007] [Tian et al. 2007] [Vatsa and Turkel 2007] [Xu et al. 2007] [Yamaleev et al. 2007] [Yao et al. 2007] [Akhtar and Nayfeh 2008] [Benard et al. 2008] [Bogdanov et al. 2008] [Braud et al. 2008] [Brown 2008] [Casper et al. 2008] [Coiro et al. 2008] [Deb et al. 2008] [Enloe et al. 2008] [Feikema and Culley 2008] [Garnier et al. 2008] [Gomes and Crowther 2008] [Gunes et al. 2008] [Guo and Zhong 2008] [Higashiura et al. 2008] [Jabbal 2008] [Jolibois et al. 2008] [Kearney-Fischer et al. 2008] [Klimov et al. 2008] [Koso and Kinoshita 2008] [Kotapati et al. 2008] [Liu et al. 2008] [Lucas et al. 2008] [Mittal and Cattafesta 2008] [Nourgostar and Roth 2008] [Okita et al. 2008] [Pinzon and Agarwal 2008] [Pinzon et al. 2008] [Raju et al. 2008b] [Raju et al. 2008a] [Roupassov et al. 2008] [Rusovici et al. 2008] [Santhanakrishnan et al. 2008] [Schatzman and Thomas 2008] [Shah et al. 2008] [Shea and Smith 2008] [Sherman 2008] [Singh and Roy 2008] [Takagaki et al. 2008] [Tereshonok 2008] [Tuck and Soria 2008] [You and Moin 2008] [Zhang et al. 2008] [Agashe et al. 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[Volino and Ibrahim 2012b] [West and Hosder 2012] [Wu et al. 2012] [Xu et al. 2012] [Bernardini et al. 2013] [Bobusch et al. 2013] [Chen et al. 2013a] [Choutapalli 2013] [Dawson and Little 2013] [Gao et al. 2013] [Goh and Lee 2013] [Graff et al. 2013] [Gregory and Tomac 2013] [Hasnain et al. 2013] [Hosseinverdi and Fasel 2013] [Koichi et al. 2013] [Koklu and Melton 2013] [Kordík and Trávníček 2013] [Kotsonis et al. 2013] [Krüger et al. 2013] [Makoto et al. 2013] [Matsuno et al. 2013] [Motoaki et al. 2013] [N. Md et al. 2013] [Norimasa et al. 2013] [Parametric Investigation of Nanosecond 2013] [Sawant et al. 2013] [Schneck and Winfrey 2013] [Seele et al. 2013] [Shimizu et al. 2013] [Sitaraman and Raja 2013] [Takashi et al. 2013] [Tang et al. 2013] [Tesař and Peszynski 2013] [Tesař et al. 2013] [Tetsuya and Hiroaki 2013] [Tomac and Gregory 2013] [Tomohisa et al. 2013] [West and Hosder 2013] [Wilson et al. 2013] [Wu and Ahmed 2013] [Xu and Meng 2013] [Aider et al. 2014] [Albright and Solovitz 2014] [Application of Zero-Net Mass-Flux 2014] [Ashpis and Laun 2014a] [Ashpis and Laun 2014b] [Tathiri et al. 2014] [Friz and Rovey 2014] [Gaertlein et al. 2014] [Gardner et al. 2014a] [González-Espinosa et al. 2014] [Kausar et al. 2014] [Koklu and Owens 2014] [Kozato et al. 2014] [Langston 2014] [Laurendeau et al. 2014] [Likhanskii 2014] [Meier and Heister 2014] [Melton 2014] [Natarajan et al. 2014] [Sato et al. 2014] [Sekimoto et al. 2014] [Shang and Huang 2014] [Stalewski and Sznajder 2014] [Taira 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[Ayers and Johari 2016] [Benard et al. 2016] [Bettrich and Niehuis 2016] [Chae et al. 2016] [Coskun et al. 2016] [Coumar et al. 2016] [Duda et al. 2016] [Fukumoto et al. 2016] [Futrzynski and Efraimsson 2016] [Gil and Strzelczyk 2016] [Greco et al. 2016] [Harinaldi et al. 2016] [Horne and Burnside 2016a] [Horne and Burnside 2016c] [Horne and Burnside 2016b] [Ibrahimoglu et al. 2016] [Itsariyapinyo and Sharma 2016] [Jeyalingam and Jabbal 2016] [Jones et al. 2016] [Kara 2016] [Kato et al. 2016] [Khoshkhoo and Jahangirian 2016] [Kim et al. 2016] [Koklu 2016a] [Koklu 2016c] [Koklu 2016b] [Li et al. 2016a] [Lu et al. 2016a] [Melton and Koklu 2016] [Melton et al. 2016] [Neretti 2016] [Ostermann et al. 2016] [Park and Kim 2016] [Peters et al. 2016] [Phan and Shin 2016] [Sieber et al. 2016] [Slupski and Kara 2016] [Tesař 2016] [Vatsa et al. 2016] [Wang et al. 2016c] [Wang et al. 2016b] [Yan et al. 2016] [Aono et al. 2017] [Bacic 2017b] [Bauer et al. 2017] [Berk et al. 2017] [Boukenkoul et al. 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2018] [Itsariyapinyo and Sharma 2018] [Kanase et al. 2018] [Kanehara et al. 2018] [Kara et al. 2018] [Karyakin et al. 2018] [Kazanskii et al. 2018] [Koklu 2018a] [Koklu 2018c] [Kornyakov et al. 2018] [Messahel et al. 2018] [Miyagi and Kimura 2018] [Miyamoto et al. 2018] [Ogawa et al. 2018] [Palumbo et al. 2018] [Plasma Actuator 2018] [Rice and Amitay 2018] [Salinas et al. 2018] [Sato 2018] [Schlösser and Bauer 2018] [Shen et al. 2018] [Slupski et al. 2018] [Sujar-Garrido et al. 2018] [Synthetic Jet 2018] [Tang and Agarwal 2018] [Trávníček and Broučková 2018] [Vatsa et al. 2018] [Viturro et al. 2018] [Wang and Tang 2018] [Wang et al. 2018a] [Wu et al. 2018b] [Xin et al. 2018] [Zhang 2018] [Zheng et al. 2018] [Zimmerman et al. 2018] [Zong et al. 2018] [Akimoto and Kimura 2019] [Berk and Ganapathisubramani 2019] [Dancova et al. 2019] [Dauengauer and Mullyadzhanov 2019] [Firsov et al. 2019] [Gowda B C et al. 2019] [Handa and Fujimura 2019] [Hasegawa and Obayashi 2019] [Hossain et al. 2019] [Hui et al. 2019] [Isobe et al. 2019] [Kumar and Karn 2019] [Ma et al. 2019] [Mehdipoor et al. 2019] [Messahel et al. 2019] [Ming et al. 2019] [Ostermann et al. 2019] [Ott et al. 2019] [Pack Melton et al. 2019] [Park et al. 2019] [Schatzman et al. 2019] [Seki and Sano 2019] [Shimizu et al. 2019] [Slupski et al. 2019] [Tadjfar and Kamari 2019b] [Tomac and Gregory 2019] [Tomac and Sundström 2019] [Wang et al. 2019a] [Yamashita and Hasegawa 2019] [Ye et al. 2019] [Abdelraouf et al. 2020a] [Abdelraouf et al. 2020b] [Belanger et al. 2020] [Buzica and Breitsamter 2020] [Harinaldi et al. 2020] [Hasan and Atkinson 2020] [Jackson et al. 2020] [Kobayashi et al. 2020] [Kumar and Karn 2020] [Lai et al. 2020] [Le Clainche et al. 2020] [Lei et al. 2020] [Li and Zhang 2020] [Nagasaki and Hasegawa 2020] [Otake et al. 2020] [Oz and Kara 2020] [Palumbo et al. 2020] [Park et al. 2020] [PourRazzaghi and Xu 2020] [Sano and Seki 2020] [Sekimoto et al. 2020] [Shay et al. 2020] [Shi and Sullivan 2020] [Smyk et al. 2020a] [Smyk et al. 2020b] [Sundström and Tomac 2020] [Tomac 2020] [Tomac and Sundström 2020] [Weigel et al. 2020] [Yang and Agarwal 2020] [Yokoyama et al. 2020] [Aiura et al. 2021] [Deb et al. 2021] [Gil et al. 2021] [Jia et al. 2021] [Kim and Kim 2021a] [Kim and Kim 2021b] [Kobayashi and Fuchiwaki 2021] [Li et al. 2021b] [Lim and Lyu 2021] [Löffler et al. 2021] [Koklu 2021b] [Palumbo and de Luca 2021] [Pan et al. 2021] [Peterson et al. 2021] [Pour Razzaghi et al. 2021] [Sarwar et al. 2021] [Sato et al. 2021] [Shimomura et al. 2021] [Sim et al. 2021] [Wen et al. 2021] [Xia et al. 2021] [Zheng 2021] [Zhu et al. 2021] [Alam and Kara 2022] [Arafa et al. 2022] [Ayeni and Cleton 2022] [Azzawi 2022] [Chen and Li 2022] [Chen and Sun 2022] [Farrell et al. 2022] [Sonkaya et al. 2022] [Gungordu 2022] [Zhou et al. 2022] [Itsariyapinyo and Sharma 2022] [Khan et al. 2022] [Kumar Subramanyam et al. 2022] [Kurowski and Szwaba 2022] [Li et al. 2022b] [Lu et al. 2022] [Luo et al. 2022] [Madane et al. 2022] [Maikap et al. 2022] [Mayahara and Sato 2022] [Monat et al. 2022] [Nicholls and Bacic 2022] [Nicholls et al. 2022] [Smyk and Markowicz 2022] [Sundaram et al. 2022] [Tajik and Parezanovic 2022] [Tajik et al. 2022] [Tolkachev 2022] [Vinoth Kumar and Vijayaraghavan 2022] [Wu et al. 2022] [Zhang and Zong 2022] [Zhang et al. 2022b] [Zhang et al. 2022a] [Zhao et al. 2022c] [Aslani et al. 2023] [Awate and Ansell 2023] [Blejchař et al. 2023] [Chen et al. 2023b] [Chernyshev et al. 2023] [Elaissi et al. 2023] [Flow Instability and Momentum 2023] [Gao et al. 2023] [Golubev 2023b] [Golubev 2023a] [Jiang et al. 2023] [Kayvanpour et al. 2023] [Kobayashi and Fuchiwaki 2023] [Kotvitskii et al. 2023] [Liu et al. 2023] [Loe et al. 2023] [Löffler and Weiss 2023] [Maikap et al. 2023] [Mayahara and Sato 2023] [Nicholls and Bacic 2023] [Numerical Investigation of Aerodynamic 2023] [Oka et al. 2023] [Okada et al. 2023] [Tavakkoli et al. 2023] [Truong et al. 2023] [Duy et al. 2023] [Xu et al. 2023] [Yu and Zheng 2023] [Yuura et al. 2023] [zhang et al 2023] [Zhao et al. 2023] [Zhou et al. 2023] [Butler et al. 2024] [Ding et al. 2024] [Wang et al. 2024b] [Hussain and Khan 2024] [Jabbari and Esmaeili 2024] [Karimzadegan and Bergada 2024] [Karimzadegan et al. 2024b] [Karimzadegan et al. 2024a] [Kassem and Kara 2024] [Liu and Liang 2024] [Mahdavi et al. 2024] [Maikap and Rajagopal 2024] [Ogawa et al. 2024] [Serrar et al. 2024] [Sun et al. 2024] [Cheng et al. 2024] [Tocquer et al. 2024] [Viguera et al. 2024] [Yang et al. 2024] [Zhao and Bai 2024] [Zhao et al. 2024] [Zhou et al. 2024] [Zhu et al. 2024] [Aguirre-Villegas et al. 2025] [Ahmed et al. 2025] [Burlett and Mobli 2025] [Duan et al. 2025] [Espinoza Gutierrez et al. 2025] [Gautam et al. 2025] [Gui et al. 2025] [Hammouda et al. 2025] [Hu et al. 2025b] [Hu et al. 2025a] [Investigation of Wing Lift 2025] [Iskandar et al. 2025a] [Kara 2025] [Karthikeyan and Harish 2025] [Kimura et al. 2025] [Kong et al. 2025] [Kopiev 2025] [Lee et al. 2025] [Li et al. 2025] [Liu 2025] [Liu and Chen 2025] [Liu et al. 2025] [Lu et al. 2025] [Maduche and Mongeau 2025] [Martinke et al. 2025] [Mitsumoto et al. 2025] [Lakehal et al. 2025] [Iskandar et al. 2025b] [Price et al. 2025] [Rahni and Jahangirian 2025b] [Rizzetta and Garmann 2025a] [Sato 2025] [Smyk and Stopel 2025] [Viguera et al. 2025] [Wang et al. 2025] [Zengler et al. 2025] [Zhang et al. 2025b] [Zhao and Bai 2025] [Zubair and Wen 2025] [Chen et al. 2026] [D’Onofrio et al. 2026] [Dumrul and Kara 2026] [Dávalos et al. 2026] [Fan et al. 2026] [Gui et al. 2026] [Güngördü 2026] [Jaganath and Steinfurth 2026] [Li et al. 2026b] [Mitsumoto et al. 2026] [Nicholls et al. 2026] [Prabahar and Zha 2026] [Saemian et al. 2026] [Serrar and Khlifi 2026] [Shestakov and Yagodnitsyna 2026] [Shi et al. 2026] [Wylie and Amitay 2026] [Yamada and Norimoto 2026] [Yukuan et al. 2026] [Zhang et al. 2026b] [Zhu et al. 2026a] [Löffler and Weiss 2027] [Ansari Shirvan] [Arcese] [van Dam et al] [de Vries] [Deb et al] [Grosvenor] [Koch et al] [Karimzadegan] [Melton et al] [Melton et al] [Seifert] [Sudak] [Tocquer] [Ukpai and Rediniotis] [Velasquez]
Blowing, the Coanda surface and the jet flap
The oldest literature on controlling a wing with air. Circulation control, the Coanda effect, the jet flap, blown and externally blown flaps, upper-surface and spanwise blowing, and boundary-layer control by blowing and suction, with the acoustics of the blown flap.
317 of these records date from before 1980, which is the era of the NACA boundary-layer-control reports and of the externally blown flaps and upper-surface blowing of the short take-off transport programmes.
750 records. [Seewald 1927] [Reid and Bamber 1928] [Bamber 1932] [Quinn 1944] [Von Doenhoff 1944] [Quinn 1946] [Kohl and Diggs 1947] [Quinn 1947b] [Quinn 1947a] [Schwier 1947] [Huffman et al. 1948] [Pasamanick and Proterra 1948] [Passamanick 1948] [Racisz and Quinn 1948] [Burrows et al. 1949] [Horton and Von Doenhoff 1949] [Braslow and Visconti 1950] [Horton et al. 1950] [Braslow et al. 1951] [Horton et al. 1951] [Blosser 1952] [Cliett 1952] [Rebuffet and Poisson-Quinton 1952] [Blosser 1953] [Cocke and Fink 1953] [Cook et al. 1954] [Kelley and Tolhurst 1955] [Anderson et al. 1956] [Fisher et al. 1956] [Kelly 1956] [Kelly and Tucker 1956] [McLemore and Fink 1956] [Northrop Aircraft INC Hawthorne CA 1956] [Rolls and Innis 1956] [Sferi-Coanda Clichy (France) 1956] [Spreemann and Kuhn 1956] [Stratford 1956b] [Stratford 1956a] [Cerreta 1957] [Davenport 1957] [Dimmock 1957] [Fink and McLemore 1957] [Helmbold 1957] [Hirsch 1957] [James and Maki 1957] [McLemore 1957] [Quigley et al. 1957] [Raspet 1957] [Sferi-Coanda Clichy (France) 1957] [Griffin et al. 1958] [Helmbold 1958] [Hirsch 1958] [Kelly et al. 1958] [McLemore and Fink 1958] [McLemore 1958] [Quigley et al. 1958] [Riebe and Davenport 1958] [Spreemann 1958] [Aoyagi and Hickey 1959] [Maki 1959] [Coanda 1960] [Gault 1960] [Hickey and Aoyagi 1960] [McLemore and Peterson 1960] [Quigley et al. 1960] [Yen 1960] [Grunwald 1961] [Lockwood 1961] [Weiberg and Holzhauser 1961b] [Weiberg and Holzhauser 1961a] [Erickson 1962] [Aoyagi and Hickey 1963] [Korbacher 1963] [White 1963] [Korbacher 1964] [Northrop Corp Hawthorne Ca Norair Div 1964b] [Northrop Corp Hawthorne Ca Norair Div 1964a] [Mehus 1965] [Reba 1966] [Roberts et al. 1966] [Wygnanski 1966] [Yuan et al. 1966] [Chang et al. 1967] [Cordner 1967] [Lissaman 1967] [Pearson 1967] [Vogler 1967] [Corsiglia et al. 1968] [Dunham 1968] [Fink et al. 1968] [Kind 1968] [Miller and Seren 1968] [Aoyagi et al. 1969] [Clever 1969] [Lissaman 1969b] [Lissaman 1969a] [Parlett and Shivers 1969] [Welna et al. 1969] [Bangert 1970] [Beckwith et al. 1970] [Deal et al. 1970] [Englar 1970] [Freeman et al. 1970a] [Freeman et al. 1970b] [Romiti 1970] [Vogler 1970] [Welna et al. 1970] [Williams and Howe 1970] [Aiken and Cook 1971] [Deal et al. 1971] [Dorsch et al. 1971] [Falarski and Aiken 1971] [Falarski and Koenig 1971] [Goodykoontz et al. 1971] [Grafton et al. 1971] [Henderson 1971] [Jenkins 1971] [Kantha et al. 1971] [Parlett et al. 1971] [Smith 1971] [Stabe 1971] [A Structural Design 1972] [Aoyagi et al. 1972] [Bettner 1972] [Design Integration and Noise 1972] [Dorsch et al. 1972a] [Dorsch et al. 1972b] [Falarski 1972] [Falarski and Koenig 1972a] [Falarski and Koenig 1972b] [Goodykoontz et al. 1972a] [Goodykoontz et al. 1972b] [Grantham et al. 1972] [Greene and Perry 1972] [Groeneweg and Minner 1972b] [Groeneweg and Minner 1972a] [Gruschka and Schrecker 1972] [Hayden et al. 1972] [Hughes 1972] [Johnson 1972] [Kier et al. 1972] [Levinsky and Yeh 1972] [Olsen et al. 1972] [Oshima and Kinoshita 1972] [Panitz and Wasan 1972] [Parlett 1972] [Putnam and Lasagna 1972] [Ungar et al. 1972] [Vonglahn et al. 1972a] [Vonglahn et al. 1972b] [Aiken and Cook 1973] [Aiken et al. 1973] [Aoyagi et al. 1973b] [Aoyagi et al. 1973a] [Bettner and Blessing 1973] [Campbell et al. 1973] [Ciffone and Robinson 1973] [Falarski 1973] [Falarski et al. 1973] [Goodykoontz et al. 1973b] [Goodykoontz et al. 1973a] [Hackett and Lyman 1973] [Herold 1973] [Hughes 1973] [Karchmer and Friedman 1973] [Kinoshita and Oshima 1973] [Lansing et al. 1973] [Lissaman 1973] [Mahal and Gilchrist 1973] [McKinzie and Burns 1973] [Morris et al. 1973] [Okumoto and Elsanker 1973] [Parlett et al. 1973] [Phelps and Smith 1973] [Phelps et al. 1973] [Powers and Kier 1973] [Roepcke and Nickson 1973] [Samanich et al. 1973] [Scharton et al. 1973] [Smith 1973] [Theisen et al. 1973] [Capone 1974] [Chin et al. 1974] [Coe and Kulla 1974] [Cole and Rathburn 1974] [Dorsch 1974] [Dorsch et al. 1974] [Gentry 1974] [Gillette et al. 1974] [Goodykoontz et al. 1974] [Grantham et al. 1974] [Harkonen et al. 1974] [Hoad 1974] [Johnson and Phelps 1974] [Johnson and Kardas 1974] [Jones and Heidelberg 1974] [Kinoshita and Oshima 1974] [Middleton 1974] [Middleton and Bergeron 1974] [Parlett et al. 1974] [Quigley et al. 1974] [Scantling and Gloss 1974] [Schoenster 1974] [Schrecker and Maus 1974] [Shivers and Smith 1974] [Smith and White 1974] [Stewart 1974] [Waites and Chin 1974] [Cook et al. 1974] [Abramson 1975] [Addessio and Skifstad 1975] [Aoyagi et al. 1975] [Campbell 1975a] [Campbell 1975b] [Capone 1975a] [Carros et al. 1975] [Chin et al. 1975] [Cochrane and Carros 1975] [Coe et al. 1975] [Cook and Whittley 1975] [Dorsch et al. 1975] [Erickson and Campbell 1975a] [Erickson and Campbell 1975b] [Falarski et al. 1975] [Fink 1975] [Forsyth et al. 1975] [Goodykoontz et al. 1975] [Heidelberg et al. 1975] [Holmberg 1975] [Johnson 1975] [Kinoshita and Oshima 1975] [Koenig and Aoyagi 1975] [Lan and Campbell 1975] [Laub 1975] [Levinsky and Schappelle 1975] [Marton 1975] [Mendenhall et al. 1975] [Middleton et al. 1975] [Miles 1975] [Naeseth and Hoad 1975] [Pennock et al. 1975] [Perry and Greene 1975] [Phelps 1975] [Reddy 1975] [Reddy and Yu 1975] [Renselaer et al. 1975] [Shivers and Smith 1975] [Shivers et al. 1975] [Barger 1976] [Burns et al. 1976] [Clarke 1976] [Coe et al. 1976] [Dods and Watson 1976] [Fournier and Coe 1976] [Gainer et al. 1976] [Gibson 1976] [Goodykoontz et al. 1976] [Hackett et al. 1976] [Hoad 1976] [Johnson 1976] [Kadman 1976] [Kinoshita and Oshima 1976] [Lan 1976] [McKinzie 1976] [McKinzie et al. 1976] [Meleason 1976] [Meleason and Wells 1976] [Mendenhall 1976] [Mendenhall et al. 1976] [Phelps et al. 1976] [Platzer 1976] [Propulsive-Lift Noise 1976] [Shivers et al. 1976] [Sleeman and Phelps 1976] [Tam and Reddy 1976] [Tennant et al. 1976] [Vogler 1976] [Wind-Tunnel Investigation of Aerodynamic 1976] [Abramson 1977] [Addessio and Skifstad 1977] [Baker and Manhardt 1977a] [Baker and Manhardt 1977b] [Becker and Maus 1977a] [Becker and Maus 1977b] [Clark and Yu 1977] [Erickson and Campbell 1977b] [Erickson and Campbell 1977a] [Gilinskii and Lebedev 1977] [Hackett and Boles 1977] [Kaups and Cebeci 1977] [Lan 1977] [Lan et al. 1977b] [Lan et al. 1977a] [Morehouse et al. 1977] [Pennock 1977] [Phelps 1977] [Reubush 1977] [Ryle et al. 1977] [Schoen et al. 1977] [Schoenster and Schroeder 1977] [Seiler 1977] [Fink 1978] [Hackett and Boles 1978] [Lan et al. 1978] [Reddy 1978] [Shovlin 1978] [Vomaske et al. 1978] [Aiken et al. 1979] [Anglin and Satran 1979] [Campbell and Erickson 1979] [Catalano et al. 1979c] [Design of the Circulation 1979] [Erickson 1979] [Hough 1979] [Johns et al. 1979] [Pappa 1979] [Rao 1979] [Bohn and Shovlin 1980] [Furey 1980] [Harris and Falarski 1980] [Hough 1980] [Lorincz 1980] [McKinzie 1980] [Parthasarathy and Vijayaraghavan 1980] [Winter 1980] [Braslow 1981] [Tai et al. 1981] [Wood 1981] [Wukowitz 1981] [Bachalo 1982] [Burley and Hwang 1982] [Cochrane et al. 1982] [Eppel et al. 1982] [McKinzie 1982] [Bevilaqua et al. 1983] [Burley and Hwang 1983] [Hwang 1983] [Seginer and Salomon 1983] [Smeltzer et al. 1983] [Stephenson 1983] [Taback et al. 1983] [Wood and Conlon 1983] [Bevilaqua and Lee 1984] [Biringen 1984] [Burley 1984] [Englar et al. 1984] [Hwang 1984] [Lipera and Sandlin 1984] [Tai and Kidwell 1984] [Wagner et al. 1984] [Wilson and Liperra 1984] [Berman 1985] [Biringen et al. 1985] [Hahn and Wilson 1985] [Harvell and Franke 1985] [Modi 1985] [Panontin 1985] [Salikuddin et al. 1985] [Satran et al. 1985] [Stewart 1985] [Tai and Kidwell 1985] [Tavella 1985] [Carpenter et al. 1986] [Chopra et al. 1986] [Garner 1986] [Krauss et al. 1986] [Lee et al. 1986] [Novak and Cornelius 1986] [Papathanasiou and Nagel 1986] [Pulliam et al. 1986] [Rodman and Wood 1986] [Shrewsbury 1986] [Tavella et al. 1986] [Torres 1986] [Wood 1986] [Wood and Rogers 1986] [Wood and Roberts 1986] [Banks and Paulson 1987] [Bevilaqua and Lee 1987] [Biringen 1987] [Dash et al. 1987] [Dvorak and Dash 1987] [Dvorak et al. 1987] [Englar 1987] [Franke and Harvell 1987] [Gregory-Smith and Gilchrist 1987] [Keener et al. 1987] [Loth 1987] [McLachlan 1987a] [McLachlan 1987b] [Nielsen 1987] [Nielsen and Biggers 1987] [Novak et al. 1987] [Novak and Cornelius 1987] [Pulliam et al. 1987] [Riddle and Eppel 1987] [Salikuddin et al. 1987] [Shovlin et al. 1987] [Shrewsbury 1987] [Spaid and Keener 1987] [Torres 1987] [Viegas et al. 1987] [Wood 1987b] [Wood 1987a] [Gilchrist and Gregory-Smith 1988] [Harvey 1988] [Pfenninger and Vemuru 1988] [Viegas et al. 1988] [Yeh et al. 1988b] [Yeh et al. 1988a] [McLachlan 1989] [Shrewsbury 1989] [Sun and Wang 1989] [Visser et al. 1989] [Yeh et al. 1989] [Lee and Roberts 1990] [Parikh et al. 1990] [Smith et al. 1990] [Williams and Franke 1990] [Danabasoglu et al. 1991] [Ghee and Leishman 1991] [Hunter and Krauss 1991] [Kroo et al. 1991b] [Maddalon 1991] [Paschal et al. 1991] [Spaid and Keener 1991] [Wood and Bauer 1991] [Acharya and Metwally 1992] [Birckelbaw 1992] [Celik and Roberts 1992a] [Celik and Roberts 1992b] [Dirlik et al. 1992] [Ghee and Leishman 1992] [Heinemann and Brown 1992] [Hunter and Cullen 1992] [Miller and Gile 1992] [Rao 1992] [Williams and Franke 1992] [Boalbey et al. 1993] [Ferguson et al. 1993] [Greenwell and Wood 1993] [Linton 1993] [Tso 1993] [Wong et al. 1993] [Celik and Roberts 1994] [Cornelius and Lucius 1994] [Hwang et al. 1994] [Lim 1994] [Meyer and Seginer 1994] [Watmuff 1994] [Wong et al. 1994] [Yang 1994] [Brown 1995] [Emanuel 1995] [Hong and Roberts 1995] [Jingchang et al. 1995] [Johari et al. 1995] [Matsson 1995b] [Mineck 1995] [Pedreiro et al. 1995] [Asami and Sawada 1996] [Englar 1996] [Kraushaar and Chokani 1996] [Mendoza and Ahuja 1996] [Mineck and Hartwich 1996] [Weaver et al. 1996] [Guowei et al. 1997] [Langan et al. 1997] [Sawada and Asami 1997] [Englar 1998] [Guillot et al. 1998] [Matsuo et al. 1998] [Vorobieff and Rockwell 1998] [Weaver et al. 1998] [Banks et al. 1999b] [Braslow 1999b] [Braslow 1999a] [Farcy and Renier 1999] [Ferman and Turner 1999a] [Ferman and Turner 1999b] [Guy et al. 1999a] [Guy et al. 1999b] [Kandil and Yang 1999] [Morton et al. 1999] [Sun and Sheikh 1999] [Grove et al. 2000] [Guy et al. 2000b] [Guy et al. 2000a] [Howe 2000] [Kamishita et al. 2000] [Slotnick et al. 2000] [Sobey 2000] [Speyer and Kim 2000] [Viswanath et al. 2000] [Wang and Sun 2000] [Liu et al. 2001] [Siegel et al. 2001] [Sun and Hamdani 2001] [Englar and Campbell 2002] [Howe 2002] [Jones et al. 2002] [Sellars et al. 2002] [Speyer and Kim 2002] [Sutliff et al. 2002] [Ahuja et al. 2003] [Gross et al. 2003] [Hamdani et al. 2003] [Liu et al. 2003b] [Munro et al. 2003] [Vasilescu and Dancila 2003] [Viswanath and Madhavan 2003] [Abramson 2004] [Frith and Wood 2004] [Gross and Fasel 2004] [Huang et al. 2004] [Hutcheson and Stead 2004] [Hutcheson et al. 2004] [Lopera et al. 2004] [Margaris and Gursul 2004] [Shinozuka et al. 2004] [Viswanath and Madhavan 2004] [Wu et al. 2004] [Zha and Paxton 2004] [Alexander et al. 2005b] [Alexander et al. 2005a] [Bakhle and Srivastava 2005] [Boyd 2005] [Duraisamy and Baeder 2005] [Han et al. 2005] [Hutcheson and Brooks 2005] [Jones 2005] [Jones and Joslin 2005b] [Jones and Joslin 2005a] [McGowan and Gopalarathnam 2005] [Munro et al. 2005] [Muramatsu et al. 2005] [Siller et al. 2005] [Sutliff 2005] [Swanson et al. 2005a] [Swanson et al. 2005b] [Aerodynamic Heat Exchanger 2006] [Coanda Effect and Circulation 2006] [Computational Study 2006] [Cruz and Anders 2006] [Donnelly et al. 2006] [Experimental and Computational Investigation 2006] [Goldstein 2006] [Green and Cruz 2006] [Gross and Fasel 2006b] [Gross and Fasel 2006a] [Jones et al. 2006] [Laible et al. 2006] [Lee-Rausch et al. 2006] [Novel Flow Control Method 2006] [Pneumatic Flap Performance 2006] [Slomski et al. 2006] [Swanson and Rumsey 2006] [Trailing Edge Circulation Control 2006] [Lee 2007b] [Cui et al. 2007] [Dygert and Dang 2007] [Erickson 2007] [Lee 2007a] [Il’inskii and Mardanov 2007] [Margaris and Gursul 2007] [Schlecht and Anders 2007] [Jones et al. 2008] [Layek et al. 2008] [Lin et al. 2008] [Shannon and Morris 2008] [Dygert and Dang 2009] [Englar et al. 2009] [Morris et al. 2009] [Pfingsten and Radespiel 2009] [Singh and Ramamurthi 2009] [Swanson and Rumsey 2009] [Wetzel et al. 2009b] [Wetzel et al. 2009a] [Winkler et al. 2009] [Balzer and Fasel 2010] [Chen et al. 2010] [Edwards 2010] [Guendogdu et al. 2010] [Iguchi and Ilegbusi 2010] [Jensch et al. 2010] [Kweder et al. 2010] [Lin et al. 2010] [Liu et al. 2010] [Madavan and Rogers 2010] [Margaris and Gursul 2010] [Milholen et al. 2010] [Miozzi et al. 2010] [Nichino et al. 2010] [Nishino and Shariff 2010] [Nishino et al. 2010b] [Nishino et al. 2010a] [Pott-Pollenske and Pfingsten 2010] [Sattler et al. 2010] [Wetzel et al. 2010] [Zhang et al. 2010] [Allan et al. 2011] [Florin et al. 2011] [Guo et al. 2011] [Liu et al. 2011] [Lubert and Shafer 2011] [Rumsey and Nishino 2011] [Tan et al. 2011] [Trancossi and Dumas 2011a] [Wetzel and Cattafesta 2011] [Zhang et al. 2011] [Caraway et al. 2012] [Chen et al. 2012] [Djojodihardjo and Hamid 2012] [Drãgan 2012] [Lubert 2012] [Milholen et al. 2012] [Nishino and Shariff 2012] [Paschal et al. 2012] [Watkins et al. 2012] [Winkler et al. 2012] [Arolla and Durbin 2013] [Bobonea and Pricop 2013] [Chen et al. 2013b] [Edstrand and Cattafesta 2013] [Horne and Burnside 2013] [Jones et al. 2013] [Milholen et al. 2013] [Subhash and Dumas 2013] [Trancossi et al. 2013] [Burnazzi and Radespiel 2014] [Chan et al. 2014] [Dumas et al. 2014] [Gerhard et al. 2014] [Kroehnert 2014] [Kumar et al. 2014] [Semaan et al. 2014] [Edstrand and Cattafesta 2015] [El Sayed Mohamed and Semaan 2015] [Jiao and Lu 2015] [Lawrence and Self 2015] [Lee et al. 2016] [Liu and Zha 2016] [Naigle et al. 2016] [Park et al. 2016a] [Zafer and Gürsoy 2016] [Baljit et al. 2017] [Chan et al. 2017] [Feng et al. 2017] [Reger et al. 2017] 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al. 2023] [Nausin et al. 2023] [Jeon et al. 2023] [Lei and Zha 2023b] [Lei and Zha 2023a] [Pan et al. 2023] [Xu and Zha 2023] [Yang et al. 2023] [Zaheer et al. 2023] [Zhang and Zhao 2023] [Zhang et al. 2023b] [Zheng et al. 2023] [Abramova and Soudakov 2024] [Butler and Bons 2024] [Chen et al. 2024] [Jente et al. 2024] [Deere et al. 2024] [Li et al. 2024b] [Rizzetta and Garmann 2024] [Shao et al. 2024] [Soufivand et al. 2024] [Su et al. 2024] [Zhang et al. 2024] [Kumar and Bhumkar 2025] [Pakhomov 2025] [Raj et al. 2025] [Rizzetta and Garmann 2025b] [Soufivand et al. 2025] [Vechalapu et al. 2025] [Yabu et al. 2025] [Zhang et al. 2025a] [Zhao et al. 2025] [Çuhadaroğlu et al. 2025] [Ding et al. 2026] [Liu 2026] [Rackers et al. 2026] [Rizzetta and Garmann 2026] [Shao et al. 2026] [Sheng et al. 2026] [Vechalapu et al. 2026] [Vibhu et al. 2026] [Yang et al. 2026] [Zhao et al. 2026a] [Zhao et al. 2026c] [Zhou et al. 2026] [Churchill] [Couillaud] [McFadden and Bons] [Hard] [Kweder] [McAuliffe] [Starn] [Swan]
Separation and its control, which is what the effectors do
The mechanism the effectors exploit. Control of separation on airfoils, high-lift systems, ramps and humps, including the agency reports this article reads in full and the review it leans on.
207 records. [Foster and Fitzpatrick 1948] [Pratt and Shields 1952] [Salmi 1952] [Bollech and Hadaway 1953] [Childress 1958] [Dike and Dunn 1958] [Peake and Owen 1979] [Sullivan 1979] [Trobaugh et al. 1980] [Sutton 1982] [Zilliac 1983] [Mangalam et al. 1986] [Barnett and Carter 1987] [Durbin and McKinzie 1987] [Huang et al. 1987] [Viken et al. 1987] [Rao and Kariya 1988] [Zaman and McKinzie 1988] [Bar-Sever 1989] [Katz et al. 1989] [Oh and Roberts 1989] [Zaman and McKinzie 1989] [Lin et al. 1990] [Rice and Abbott 1990] [Chow and Chu 1991] [Gad-El-hak and Bushnell 1991] [Lin et al. 1991] [Selby 1991] [Lin et al. 1992] [Yip et al. 1992] [Gally 1994] [Jun-Fei and Shao-Zhi 1994] [Nelson et al. 1994] [Potter and Vandam 1995] [Telionis 1995] [Chandrasekhara et al. 1996] [Rudolph 1996] [Seifert et al. 1996] [Chandrasekhara et al. 1997] [Alam and Sandham 1998] [Hahne and Glaab 1999] [Mochizuki et al. 1999] [Murawski 1999] [Ravindran 1999] [Sengupta et al. 1999] [Sinha 1999] [Song et al. 1999] [Urzynicok and Fernholz 1999] [Abed 2000] [Allan et al. 2000] [Lu 2000] [Pack and Seifert 2000] [Seifert and Pack 2000b] [Separation Control 2000] [Munday and Jacob 2001] [Munday and Jacob 2002] [Munday et al. 2002] [Angele 2003] [Greenblatt and Wygnanski 2003] [Martin et al. 2003] [Pack et al. 2002] [Pack et al. 2003] [Tumin and Ashpis 2003] [Ashpis and Hultgren 2004] [Beck et al. 2004] [Bons 2004] [Greenblatt et al. 2004] [Melton et al. 2004] [Zhang 2004] [Ahmad et al. 2005] [Lee 2005] [Greenblatt 2005b] [Greenblatt 2005c] [Katam et al. 2005] [Logdberg and Alfredsson 2005] [Pern and Jacob 2005] [Seidel et al. 2005] [Sohn et al. 2005] [Yarusevych et al. 2005] [Yoshida et al. 2005] [Zhu and Qiu 2005] [Angele and Grewe 2006] [Greenblatt 2006] [Pern et al. 2006] [Walker 2006] [Abe et al. 2007] [Greenblatt 2007] [Greenblatt et al. 2007] [Khodadoust and Shmilovich 2007] [Khodadoust and Washburn 2007] [Melton et al. 2007] [Abe et al. 2008] [Agarwal et al. 2008] [Favier et al. 2008] [Haucke et al. 2008] [Little et al. 2008] [Mack et al. 2008] [Melton et al. 2006] [You et al. 2008] [Zaman and Culley 2008] [Genç and Kaynak 2009] [Rizzetta and Visbal 2009] [Shmilovich and Yadlin 2009] [Woo et al. 2009] [Eichinger et al. 2010] [Stalnov and Seifert 2009] [Tesař 2010] [Gross and Fasel 2011] [Kühn et al. 2011] [Liang Hua et al. 2011] [Moralev et al. 2011] [Shun and Ahmed 2011] [Arifuzzaman 2012] [Benton et al. 2012] [Desalvo et al. 2012] [Grager et al. 2012] [Gross and Fasel 2012] [Packard and Bons 2012a] [Cadirci and Gunes 2013] [Gross and Fasel 2013a] [Gross and Fasel 2013b] [Gross and Fasel 2013c] [Haucke and Nitsche 2013] [Woo and Glezer 2013a] [Woo and Glezer 2013b] [Yufeng Cheng et al. 2013] [Ahmed et al. 2014] [Clifford et al. 2014] [Desalvo et al. 2014] [Hartwich et al. 2014] [Kato et al. 2014] [Munday and Taira 2014] [van Dommelen and Yapalparvi 2014] [Bayoda et al. 2015] [Fujii 2015] [Guha et al. 2015] [Munday and Taira 2015] [Gupta and Ansell 2016] [Hartwich et al. 2016] [Jairam et al. 2016] [Joy et al. 2016] [Matalanis et al. 2016] [Walker et al. 2016] [Yeh and Taira 2016] [Bodart et al. 2017] [Ciobaca and Dandois 2017] [Guo et al. 2017] [Shahid and Mashud 2017] [Vrchota 2017] [Yeh et al. 2017] [Hosseini et al. 2018] [Koklu 2018b] [Munday and Taira 2018] [Prabhu and Srinivas 2018] [Shimomura et al. 2018] [Stalewski and Stryczniewicz 2018] [Takahashi and Ohnishi 2018] [Esfahani et al. 2019] [Fang et al. 2019] [Gildersleeve and Amitay 2019] [Kobayshi and Sato 2019] [Li et al. 2019a] [Niu et al. 2019] [Oguri et al. 2019] [Ramaswamy and Schreyer 2019] [Siauw and Bonnet 2019] [Tadjfar and Kamari 2019a] [Wang et al. 2019c] [Yeh and Taira 2019] [Obeid et al. 2020] [Shimomura et al. 2020] [Koklu 2021a] [Okoye et al. 2021] [Sato 2021] [Xu et al. 2021] [Chaudhary et al. 2022] [Panigrahi et al. 2021] [Takada et al. 2022] [Control of Flow Separation 2023] [Takada et al. 2023b] [Takada et al. 2023a] [Boutoudj and Tebbiche 2024] [Brandt and Bons 2024] [Dodayav et al. 2024] [Shi et al. 2024] [Gross et al. 2024] [Abdolahipour 2024] [Vander Schaaf et al. 2024] [Ho et al. 2025] [Kamiński et al. 2025] [Rahni and Jahangirian 2025a] [Reddy and Subramanian 2025] [Sheeler et al. 2025] [Song et al. 2025] [Ayok 2026] [Wylie et al. 2026] [Xu et al. 2026] [Yuan 2026] [Zhang et al. 2026a] [Bohmier and Manoharan] [Van Dam et al] [Melton] [Schatz et al] [Hosseini et al] [Hosseini et al] [Hosseini et al] [Hosseini et al]
Controlling a vortex, including the forebody jets that flew first
Control by moving a vortex rather than a surface. Leading-edge, forebody, flap and tip vortices, including the forebody vortex control evaluated in flight.
32 records. [Bradley et al. 1974] [Skow and Penke 1982] [Skow and Peake 1983] [Rao et al. 1986] [Mourtos and Roberts 1987] [Visser et al. 1988] [Ng 1989] [Rao et al. 1989] [El-Hady 1990] [Malcolm and Ng 1990] [Malcolm et al. 1990] [Ng 1990] [Rao 1991] [Bare et al. 1992] [Kandil et al. 1992] [Lemay et al. 1992] [Magness et al. 1992] [Suarez et al. 1992] [Smith et al. 1993] [Suarez et al. 1993b] [Suarez et al. 1993c] [Suarez et al. 1993a] [Alexander and Meyn 1994] [Kramer et al. 1994] [Walchli 1994] [Moreira and Johari 1995] [Bowers 1996] [Johari and Moreira 1996] [Greenblatt 2005a] [Bloxham et al. 2008] [Erickson 2013] [Moralev et al. 2016]
Where the air comes from, and what taking it costs
The air supply. Engine bleed, its extraction and its cost to the engine, and the pneumatic systems that distribute compressed air in an aircraft.
43 records. [Koutz et al. 1950] [Wood and Dallenbach 1950] [Koutz 1951] [Vich 1952] [Gage 1958] [General Dynamics Corp Pomona Ca Pomona Div 1963] [Tagge 1983] [Wiemer 1985] [Zrubek 1985] [Curry and Gilyard 1989] [Yuhas and Ray 1992a] [Yuhas and Ray 1992b] [Collaboration on Bleed Air 2000] [Conan and Savarese 2001] [Guangjun Liu et al. 2004] [Englar and Campbell 2005] [Guangjun Liu et al. 2005] [Bleed Air Cooler 2008] [Shang et al. 2010] [Bergeron et al. 2013] [Guoliang et al. 2016] [Xue et al. 2016] [Pollok and Casella 2017] [Wang et al. 2018b] [Huff et al. 2019] [Takahashi and Stone 2019] [Krishnan et al. 2020] [Zheng et al. 2021] [Singh Negi et al. 2022] [Abdelghany et al. 2023] [Aygun 2023] [Molenaar 2023] [Aygun 2024] [Irmayani et al. 2024] [Ward et al. 2024] [Aircraft Pneumatic Power Systems 2025] [Cavalcante et al. 2025] [Schillaci et al. 2025] [Sharma and Ranganayakulu 2025] [Jiang et al. 2026] [Sharma and Ranganayakulu 2026] [Sharma et al. 2026] [Engine Bleed Air Systems]
Reynolds number, and why a tunnel is not an aeroplane
Whether a tunnel result survives to flight. Reynolds-number and scale effects, the correlation of wind tunnel and flight, and the full-scale and subscale testbeds that bridge the two, with the flight-Reynolds report on separation control.
111 records. [Dearborn and Soule 1937] [Bauer and German 1961] [Barnes 1965] [Soderman and Aiken 1971] [Hicks et al. 1972] [Garcia et al. 1973] [Wilcox and Chamberlin 1974] [Capone 1975b] [Wilcox and Chamberlin 1975] [Richey et al. 1976] [Settles et al. 1976] [Sussman et al. 1976] [Kruse et al. 1977] [Wind Tunnel/Flight Test Correlation 1978] [Daugherty 1979] [Stevens et al. 1979] [Dougherty and Fisher 1980] [Peterson et al. 1980] [Wilson et al. 1980] [Ayers 1981] [Saltzman and Ayers 1981] [Ayers 1982] [Daforno 1982] [Fisher and Dougherty 1982] [Madsen 1982] [McKinney and Baals 1982] [Murman 1982] [Nagel 1982] [Niedling 1982] [Peterson 1982] [Anderson et al. 1983] [Fisher and Dougherty 1983] [Wortman 1984] [Hahne et al. 1987] [Ellzey and Berbee 1988] [Hahne and Jordan 1991] [Meyn et al. 1992] [Bandyopadhyay and Gad-el-Hak 1994] [Williams et al. 1994] [Lamont et al. 1995] [Fisher et al. 1998] [High Reynolds Number Hybrid 1999a] [High Reynolds Number Hybrid 1999c] [High Reynolds Number Hybrid 1999b] [Iwatsuki et al. 1999] [Flow at Large Reynolds 2000] [Ramsey and Chapman 2000] [Reynolds Number Effects 2000] [Seifert and Pack 2000a] [Adrian and Balachandar 2001] [Luckring and Ghee 2001] [van Raalte and van Oudheusden 2001] [Curtin et al. 2002] [Iwamoto et al. 2002] [Song et al. 2002] [Tomek et al. 2002] [Brewer et al. 2004] [Owens et al. 2006b] [Shaw et al. 2006b] [Wake et al. 2006] [Jayaraman et al. 2007] [Jordan and Bailey 2008] [Munson et al. 2008] [Lee et al. 2009] [Marusic et al. 2009] [Taira and Colonius 2009] [Escauriaza and Sotiropoulos 2010] [Marusic et al. 2010] [Thake et al. 2010] [Miller 2011] [Pavlock 2011] [Brown 2013] [Cho and Shyy 2013] [Asada et al. 2014] [Chin et al. 2014] [Miller 2014] [El-Alti et al. 2015] [Lind et al. 2015] [Lu et al. 2015] [Seifert et al. 2015] [Ting et al. 2015] [Lu et al. 2016b] [Mandal and Gu 2016] [Marks et al. 2016] [Whalen et al. 2016] [Zharfa et al. 2016] [Essel and Tachie 2017] [Elahi et al. 2018] [Malekzadeh et al. 2018] [Whalen et al. 2018] [Lin et al. 2019a] [McKenna et al. 2019] [Warsop and Crowther 2019] [Yoon et al. 2019] [Rao and Liu 2020] [Romano 2020] [Wang et al. 2020] [Dougherty 2021] [Sarout et al. 2022] [Yu et al. 2022] [Elahi et al. 2023] [Iyer and Malik 2023] [Ogunka et al. 2023] [Laguarda et al. 2024] [McFadden and Bons 2024] [Zanoun et al. 2024] [Brown et al. 2025] [Soldati et al. 2025] [Zhu et al. 2026b] [Grager] [Langston et al]
Active flow control as a subject in its own right
The field as a field. Reviews, programmes and outlooks for active flow control, laminar flow control by suction, and the integration of flow control into aircraft, inlets and engines.
275 records. [Wheldon and Whites 1966] [Development of Technology 1976] [Sturgeon and Bennett 1976] [Sturgeon et al. 1976] [Swinford 1976] [White et al. 1976] [Bonner et al. 1977] [Meade 1977] [Allison and Dagenhart 1978] [Evaluation of Laminar Flow 1978] [Gratzer and George-Falvy 1978] [Lovell et al. 1978] [Sturgeon 1978] [Turriziani et al. 1978] [Evaluation of Laminar Flow 1979] [Wright 1979] [Sturgeon 1980] [Wilson 1980] [Campbell 1982] [Harvey 1982] [Harvey and Pride 1982] [Pearce 1982] [Fischer et al. 1983] [Pearce 1983] [Jasperson et al. 1984] [Klotzsche et al. 1984] [Lange 1984] [Lineberger 1984a] [Lineberger 1984b] [Pearce et al. 1984] [Braslow and Fischer 1985] [Petley et al. 1985] [Harvey et al. 1986] [Maddalon and Poppen 1986] [Wagner 1986] [Allison and Dagenhart 1987] [Lange 1987] [Harris et al. 1988] [Lange 1988] [Ash and Lakshmanan 1989] [Fischler et al. 1989] [Hardin and Pope 1989] [Harris et al. 1989] [Harvey et al. 1989] [Huff 1989] [Powell et al. 1989] [Maddalon and Braslow 1990] [Shepshelovich and Koss 1990] [Wagner et al. 1990] [Anderson and Farokhi 1991] [Arcara et al. 1991] [Fischer and Vemuru 1991] [Wie et al. 1991] [Woan et al. 1991] [Fischer 1992] [Gee et al. 1992] [Harris et al. 1992] [Krumenacker and Pellicano 1992] [Lanser and Meyn 1992] [Murman et al. 1992] [Wie et al. 1992] [Collier 1993] [Keith and Dewitt 1993] [Lanser and Meyn 1994] [Lo 1994] [Gimbert 1995] [Lo and Wiberg 1995] [Lo and Wiberg 1996] [Wygnanski and Wygnanski 1997] [Roth et al. 1998] [Joslin 1998] [Pack and Joslin 1998] [Sheta et al. 1998] [Anders and Fischer 1999] [Joslin et al. 1999] [Marshall 1999] [Wygnanski et al. 1999] [Frontiers of Flow Control 2000] [Ghee and Hall 2000] [Hamstra et al. 2000] [Marshall 2000] [Abegg et al. 2001] [Buning 2001] [Gorton 2001] [Kahng et al. 2001] [McMichael 2001] [Joslin and Viken 2001] [Telionis 2001] [Vorobieff and Truman 2001] [Young and Fielding 2001] [Carpenter et al. 2002] [Fung and Amitay 2002] [Israel et al. 2002] [Lewington et al. 2002] [Sellers et al. 2002] [Washburn et al. 2002a] [Washburn et al. 2002b] [Bazyma and Rashkovan 2003] [Geiselhart et al. 2003] [Telionis and Vlachos 2003] [Allan et al. 2004] [Anders et al. 2004] [Bower and Kibens 2004] [Fomin et al. 2004] [Glezer 2004] [Gorton et al. 2004] [Lee 2004] [Wong and Maina 2004] [Culley and Bright 2005] [Fleming et al. 2005] [Kimmel et al. 2005] [McMichael et al. 2005] [Suzen et al. 2005] [Brian and Owens 2006] [Owens et al. 2006a] [Palaniswamy et al. 2006] [Pilmoor et al. 2006] [Sahu 2006] [Soria 2006] [Font and Morgan 2007] [Greenblatt and Washburn 2007] [Kiedaisch et al. 2007] [Liu 2007] [Smith et al. 2007] [Bower 2008] [Brzozowski et al. 2008] [Fasel et al. 2008] [Greenblatt and Washburn 2008] [Gross and Fasel 2008] [Kutay et al. 2008] [Lazar et al. 2008] [Liddle and Crowther 2008] [Melton et al. 2008] [Owens et al. 2008] [Potsdam and Le Pape 2008] [Rizzetta et al. 2008] [Rumsey 2008] [Soria 2008] [Vepa 2008] [Vynogradskyy et al. 2008] [Yeshala et al. 2008] [Anderson et al. 2009] [Flow Control Terminology et al. 2009] [Liu et al. 2009a] [Ahmadi et al. 2010] [Bauer et al. 2010] [Grund and Nitsche 2010] [Jabbal et al. 2010] [Kumar and Alvi 2010] [Owens and Allan 2010] [Schaeffler et al. 2010] [Seifert et al. 2010] [Siegel 2010] [Williams et al. 2010] [Haucke and Nitsche 2011] [Li et al. 2011] [Wilkinson 2011] [Hirt et al. 2012] [King et al. 2012] [Lipatov 2012] [Packard and Bons 2012b] [Shea et al. 2012] [Volino 2012b] [Volino 2012a] [Zaman et al. 2012] [Dianics et al. 2013] [King et al. 2013] [Lenahan et al. 2013] [Glezer 2014] [Greenblatt et al. 2014] [Meyer et al. 2014] [Narducci et al. 2014] [Tucker et al. 2014a] [Tucker et al. 2014b] [Junxuan and Xun 2015] [Whalen et al. 2015] [Alexander et al. 2016] [Childs et al. 2016] [Jentzsch et al. 2016] [Lin et al. 2016a] [Lin et al. 2016b] [Tucker et al. 2016] [Whalen 2016] [Bacic 2017a] [Bieler 2017] [Bons 2017b] [Bons 2017a] [Cattafesta 2017] [Ciobaca and Wild 2017] [Eggert and Rumsey 2017] [Li et al. 2017c] [Lin et al. 2017] [Samimy 2017a] [Samimy 2017b] [Sheng and Zhao 2017] [Shmilovich et al. 2017] [Stalnov 2017] [Tadjfar and Asgari 2017] [Yang and Zha 2017] [Agate et al. 2018] [Bishara et al. 2018] [Burrows et al. 2018] [He et al. 2018] [Melton et al. 2018] [Weingaertner et al. 2018] [Andino et al. 2019] [Li 2019] [Lin et al. 2019b] [Vatsa et al. 2019] [Yang et al. 2019] [Lin et al. 2020] [Kamari et al. 2020] [Durhasan 2020] [Tadjfar et al. 2020] [GÜLer et al. 2021] [Tebbiche and Boutoudj 2021] [Yoneda et al. 2021] [Shmilovich et al. 2022b] [Shmilovich et al. 2022a] [Coletti and Faruque 2022] [Lin et al. 2022] [Khalil et al. 2022] [Vijgen et al. 2022] [Woszidlo et al. 2022] [Serrani 2022] [Truong et al. 2022] [Vorozhbit et al. 2022] [Xia et al. 2022] [Zheng et al. 2022] [Shmilovich et al. 2023b] [Shmilovich et al. 2023a] [Feldstein et al. 2023] [Griffith and Faruque 2023] [Hosseinverdi and Fasel 2023] [Mallor et al. 2023] [Anilir et al. 2024] [Chua and Liu 2024] [Karpuk and Mosca 2024] [Ren et al. 2025] [Chen et al. 2025] [Frede and Gatti 2025] [Kalyankar et al. 2025] [Mahns and Amitay 2025] [Wang and Yang 2025] [Yang et al. 2025] [Zhijie et al. 2025] [Ali et al. 2026] [Ch et al. 2026] [Tahir et al. 2026] [Zhao et al. 2026b] [Aircraft laminar flow control] [Chandra et al] [Graff et al] [Nielsen and Jones] [McFadden and Bons] [Jones et al] [Wygnanski] [Paschal et al] [Paschal et al] [Melton et al] [Melton] [Melton et al] [Melton et al] [Melton et al] [Madhan et al] [Andino et al] [Andino et al] [Andino et al] [Vijgen et al] [Perrier] [Sarwar] [Hosseini et al] [Hosseini et al] [Shaw et al] [Vatsa et al]
Joined, tandem, diamond and tailless, and controlling them
The planform family. Joined, tandem and diamond wings, blended wing bodies, flying wings and tailless aircraft, with their aerodynamics, structures, stability and control.
350 records. [Weick and Sanders 1933] [Donlan 1944] [Maggin 1944] [Seacord and Campbell 1945] [Weyl 1945] [Soule 1946] [Sawyer and Trant 1947] [Hieser and Whitcomb 1948] [Cox et al. 1949] [Johnson 1949] [Sawyer and Trant 1950] [Mathews and Kurbjun 1953] [Smith 1953] [Mitcham et al. 1956] [Holdaway et al. 1959] [White and Innis 1959] [Holdaway and Mellenthin 1960b] [Holdaway and Mellenthin 1960c] [Holdaway and Mellenthin 1960a] [Harry and Trobaugh 1966] [Fox and Freeman 1971] [Polhamus 1971] [Donlan and Kuhn 1976] [Jernell 1978a] [Jernell 1978b] [Sears 1980] [Structural Weight Comparison 1981] [Weil 1981] [Samuels 1982] [Clyde et al. 1984] [Nicholas et al. 1984] [Wolkovitch 1984] [Covell 1985] [Moitra 1985] [Perkins 1985] [Wolkovitch 1985] [Hajela and Chen 1986] [Selberg and Cronin 1986] [Wolkovitch 1986] [Smith et al. 1987] [Van der Velden 1987] [Miura et al. 1988] [Ashkenas and Klyde 1989] [Smith and Stonum 1989] [Vandervelden 1989] [Gallman et al. 1990a] [Gallman et al. 1990b] [Guruswamy 1991] [Kroo et al. 1991a] [Burkhalter et al. 1992] [Gallman and Kroo 1992] [Morris 1992] [Fremaux et al. 1993] [Gallman et al. 1993] [NASA Advanced Design Program. 1993] [Simon et al. 1993] [Huber and Schellenger 1994] [Qu and Cutchins 1994] [Huber 1995] [Ross et al. 1995] [Bagwill and Selberg 1996] [Bosworth and Stoliker 1996] [Friehmelt 1996] [Gallman and Kroo 1996] [Mullins et al. 1996] [Bagwill et al. 1997] [Cameron and Cameron 1997] [Gillard et al. 1997] [Paul et al. 1997] [Potsdam et al. 1997] [Sparks and Sparks 1997] [Tournes et al. 1997] [Brinker and Wise 1998] [Buffington et al. 1998] [Liebeck et al. 1998] [Michal 1998] [Myatt et al. 1998] [Protz and Sparks 1998] [Tigner et al. 1998] [Wakayama 1998] [Barker and Balas 1999] [Boskovic and Mehra 1999] [Brinker and Wise 1999] [Eberhardt and Ward 1999] [Hauser 1999] [PI Control 1999] [Shtessel et al. 1999] [Teel 1999a] [Teel 1999b] [Tournes and Johnson 1999] [Wise et al. 1999] [Brinker and Wise 2000] [Calise 2000] [Ghee and Taylor 2000] [Roman et al. 2000] [Wakayama 2000] [Calise et al. 2001] [Ghaffari et al. 2002a] [Ghaffari et al. 2002b] [Gilmore et al. 2002] [Liebeck 2002] [Nangia and Palmer 2002] [Qin 2002] [Shtessel et al. 2002] [Tyler et al. 2002] [Vos et al. 2002] [Wells and Hess 2002] [Carlsson 2003] [Liebeck 2003] [Marisarla et al. 2003] [Nangia et al. 2003b] [Nangia et al. 2003a] [Patil 2003] [Roman et al. 2003] [Sivaji et al. 2003] [Smallwood et al. 2003] [Hur et al. 2004] [Pao et al. 2004] [Rasmussen et al. 2004] [Reich et al. 2004] [Robinson 2004] [Blair et al. 2005] [Campbell et al. 2005] [Carlsson 2005] [Лемко 2005] [Iniguez de Heredia and Friehmelt 2005] [Jung and Lowenberg 2005] [Kaloyanova et al. 2005] [Nangia et al. 2005] [Pao et al. 2005] [Snyder et al. 2005] [Boeing Phantom Works Leading 2006] [Carter et al. 2006] [Jackson and Buttrill 2006] [Kawai et al. 2006] [Kimler and Canfield 2006] [Peigin and Epstein 2006] [Rasmussen et al. 2006] [Saephan and van Dam 2006] [Bond et al. 2007] [Ground Testing of X-48B 2007] [Lee et al. 2007] [Nieto-Wire and Sobel 2007] [Park 2007] [Pereira et al. 2007] [Qin et al. 2007] [Ravindren et al. 2007] [Suleman 2007] [Wong et al. 2007] [Blair et al. 2008] [Boeing Flies Blended Wing 2008] [Bond et al. 2008c] [Bond et al. 2008a] [Bond et al. 2008b] [Carter et al. 2008] [Colgren and Loschke 2008] [LeDoux et al. 2008] [Lee et al. 2008] [Liebeck 2008] [Ma et al. 2008] [Oligney et al. 2008] [Recktenwald and Ahmed 2008] [Reichenbach 2008] [Trittler et al. 2008] [X-48B Blended Wing Body 2008] [Carter et al. 2009] [Cuerno-Rejado et al. 2009] [Di Palma et al. 2009] [Ferrar et al. 2009] [Kim et al. 2009] [Mamla and Galinski 2009] [Nieto-Wire and Sobel 2009] [Roy 2009] [Sotoudeh and Hodges 2009] [Stenfelt and Ringertz 2009] [Vicroy 2009] [Wildschek et al. 2009] [Yue et al. 2009] [Bond et al. 2010] [Buch and George M. 2010] [Chung et al. 2010] [Goldthorpe et al. 2010] [Paletta et al. 2010] [Paranjape and Chung 2010] [Rahman and Whidborne 2010] [Richards et al. 2010] [Roy and Ghosh 2010] [Roysdon and Khalid 2010] [Stenfelt and Ringertz 2010] [Velicki and Thrash 2010] [Westermayer et al. 2010] [Zafirov 2010] [Heeg and Morelli 2011] [Jianghao et al. 2011] [Nieto-Wire and Sobel 2011] [Paranjape et al. 2011] [Peterson and Grant 2011] [Qu et al. 2011] [Ratnayake et al. 2011] [Roysdon and Khalid 2011] [Scott et al. 2011] [Sotoudeh and Hodges 2011] [Vicroy 2011] [Ahn and Lee 2012] [Ali et al. 2012] [Bond et al. 2012] [Cusher and Gopalarathnam 2012] [Grauer et al. 2012] [Meheut et al. 2012] [Brodsky and Luby 2013] [Cardoso et al. 2013] [Demourant and Ferreres 2013] [From Tube and Wing 2013] [Haniš et al. 2013] [Kucera and Hromčík 2013] [Marques 2013] [Nasir and Kuntjoro 2013] [Nieto-Wire and Sobel 2013] [Paulus et al. 2013] [Qu et al. 2013] [Schirrer et al. 2013a] [Schirrer et al. 2013b] [Stroscher et al. 2013] [Suresh et al. 2013] [Vale 2013] [Westermayer et al. 2013] [Wildschek et al. 2013b] [Wildschek et al. 2013a] [Yue et al. 2013] [Zha et al. 2013] [Zhang et al. 2013] [Zheng and Xu 2013] [Ammar and Trépanier 2014] [Gern et al. 2014] [Jung Hoe and Nik Mohd 2014] [Pires et al. 2014] [Reist and Zingg 2014] [Vale et al. 2014] [Wildschek 2014] [Boelens et al. 2015] [Coulson and Scott 2015] [Denieul et al. 2015] [Lambert et al. 2015] [Lemko and Molodchyk 2015] [Meckstroth and Blake 2015] [Kozek and Schirrer 2015] [Molinari et al. 2015] [Nasir et al. 2015] [Reist and Zingg 2015] [Saeed and Graham 2015] [Sotoudeh 2015] [Suleman et al. 2015] [Лемко and Молодчик 2015] [Зінченко and Хомінич 2015] [Yingsong et al. 2015] [Johnston 2016] [Lambert et al. 2016] [Park et al. 2016b] [Reist and Zingg 2016] [Singh et al. 2016] [Teo et al. 2016] [Torenbeek 2016] [Abnous et al. 2017] [Ahn et al. 2017] [Junlei et al. 2017] [Kalinowski 2017b] [Kalinowski 2017a] [Kumar and Khalid 2017] [Park et al. 2017a] [Su et al. 2017] [Brown and Vos 2018] [Buzica et al. 2018] [Molinari et al. 2018] [Qian and Alonso 2018] [Tan et al. 2018] [Vladimir et al. 2018] [Yang et al. 2018] [Zhang et al. 2018b] [Chen et al. 2019] [Dimopoulos et al. 2019] [Nishanth et al. 2019] [Wang et al. 2019b] [Humphreys-Jennings et al. 2020] [Kitsios et al. 2020] [Kumar et al. 2020] [Footohi et al. 2020] [Bainsla et al. 2021] [Gao et al. 2021b] [Li et al. 2021a] [Liu and Whidborne 2021] [Qian and Alonso 2021] [Ruseno 2021] [Sherman et al. 2021] [Contreras-Torres et al. 2022] [DeLaurier 2022] [Fu et al. 2022] [Meng et al. 2022] [Miao et al. 2022] [Okonkwo and Jemitola 2022] [Rossetti and Beblo 2022] [Wang et al. 2022c] [Yang et al. 2022a] [Zhao et al. 2022b] [Adler and Martins 2023] [Arora et al. 2023] [Jackson 2023] [Kryvokhatko 2023b] [Kryvokhatko 2023a] [Razon et al. 2023] [Wang et al. 2023] [Yu et al. 2023] [Fernandes and Maldonado 2024] [He et al. 2024a] [Kryvokhatko 2024b] [Kryvokhatko 2024a] [Li 2024] [Sharma and Hosder 2024] [Ali et al. 2025] [Antoniou et al. 2025] [Anwar 2025] [Bhowmick et al. 2025] [Guo et al. 2025] [Gupta and Khalid 2025] [Hansmann and Stumpf 2025] [Jagtap et al. 2025] [Kerr 2025] [Luu 2025] [Pandi and Kumar 2025] [Saifuddin et al. 2025] [Xu and Yu 2025] [Xu et al. 2025b] [Bhattarai et al. 2026] [Borodulin et al. 2026] [Buddhamatya et al. 2026] [Hansmann and Stumpf 2026] [Li et al. 2026a] [Pérez Segura et al. 2026] [Sinha 2026] [Zhang and He 2026] [Zou et al. 2026] [Aubeelack] [Guiler] [Izquierdo] [Muñoz] [Parello] [Scott] [Sgueglia] [Silva] [Sparks] [Teo]
The mechanical control system this aircraft proposes to delete
What the effectors would replace. Hydraulic, electrohydrostatic and electromechanical flight control actuation, hinge moments, and the effectiveness of the conventional surfaces a jet must match.
85 records. [Strass 1945] [Stone and Burk 1947] [Jones and Sluder 1948] [Kolbe and Tinling 1948] [Thompson 1954] [Wong 1957] [Fournier 1959] [Jenkins et al. 1968] [Rinde 1969] [Edwards 1972] [Bergmann and Sevart 1973] [Tang and Pearson 1973] [Bergmann and Sevart 1975] [Chenoweth and Ryder 1976] [Demarchi and Haning 1978] [Feucht et al. 1978] [Jenney 1979] [Jenney 1980] [Kohnhorst and Magnacca 1980] [Robinson 1980] [Kineyko 1982] [Belmont 1983] [Thompson and Eitenmiller 1983] [Design and Test 1984] [Hornyak 1984] [Jenney and Schreadley 1984] [Defeo and Chen 1987] [Hair 1987] [Vieten 1992] [Alden 1993] [Jategaonkar 1993] [Navarro 1997] [Appa et al. 1998] [Churn et al. 1998] [Oz 1998] [Atallah 1999b] [Atallah 1999a] [Cossar 1999] [Jensen et al. 2000] [Wingett et al. 2004] [Chen et al. 2006] [Wei 2006] [Wingett et al. 2006] [Celmins 2007] [Jayakumar and Das 2007] [Schallert 2007] [Laxminarayana et al. 2011] [Curt and Norman 2011] [Cui et al. 2015] [Garmendia et al. 2015] [Shi et al. 2015] [Fu et al. 2016] [Gaile and Lue 2016] [Joshi et al. 2016] [Li et al. 2016b] [Miller et al. 2016] [Wang et al. 2016a] [Yu et al. 2016] [Wang et al. 2017b] [Wang et al. 2017a] [Bo Yu et al. 2018] [Xiaozhe Sun et al. 2018] [Şener Kaya and Zeki Bilgin 2018] [Ijaz et al. 2019c] [Ijaz et al. 2019b] [Ijaz et al. 2019a] [Zhou et al. 2019] [Sun et al. 2020] [Ercan and Akın 2022] [Wang et al. 2022b] [Eid et al. 2023] [He et al. 2024b] [Wang et al. 2024a] [Bao et al. 2025] [Flight Control Actuation Systems 2025] [Sun et al. 2025] [Aerospace - Flight Control] [Aircraft Flight Control Actuation] [Description of Actuation Systems] [Development Process - Aerospace] [Electrohydrostatic Actuator Applications on] [Flight Control Actuators] [Hisseine] [Historical Design Information of] [Primary Flight Control Hydraulic]
Identifying an aircraft from its own flight data
How the comparison inside one airframe is measured. System and parameter identification from flight data, flight-test technique and instrumentation, and the flight test of control laws.
185 records. [Symington 1940] [Rice 1946] [Pegg 1962] [Godwin et al. 1964] [Nagel and Thomas 1965] [Flight Test Results Pertaining 1970] [Jones 1970] [Kisslinger and Vetsch 1971] [Mehra et al. 1972] [Parrish and Steinmetz 1972] [Simpson 1972] [Steinmetz et al. 1972] [Lyons et al. 1973] [Wilcox 1973] [Wolowicz and Yancey 1973] [Air Force Flight Test Center Edwards AFB CA 1974] [Heffley et al. 1974] [Stepner and Mehra 1974] [Widnall and Grundy 1974] [Burton and Bischoff 1976] [Nicolaides 1976] [Armstrong 1977] [Jenks et al. 1977] [Stein et al. 1977] [Tomaine et al. 1978] [Redd et al. 1979] [Edwards 1981] [Hartmann and Stein 1981] [Miley et al. 1981] [Petersen 1981] [Vincent and Franklin 1981] [Wells et al. 1981] [Compton et al. 1982] [Iliff and Maine 1982] [Edwards 1983] [Iliff and Maine 1983] [Rosenberg and Schuch 1983] [Stewart and Bauer 1983] [Toles and Anderson 1983] [Trankle and Rabin 1983] [Advisory Group for Aerospace Research and Development 1984] [Myers and Burcham 1984] [Anderson and Vincent 1985] [Deets and Brown 1986] [Sefic and Maxwell 1986] [Brown 1987] [Dowden and Bessette 1987] [Gilbert 1987] [Hicks et al. 1987] [Meyer et al. 1987] [Runyan et al. 1987] [Mulder 1988] [Plaetschke and Weiss 1988] [Siddiqi and Abraham 1988] [Davis et al. 1989] [Iliff 1989] [Maine et al. 1990] [Shi Zhong-ke 1990] [Van Graas and Braasch 1991] [Vangraas and Braasch 1991] [Wierzbanowski and Armstrong 1991] [Manning et al. 1992] [Morgenfeld 1992] [Powers et al. 1992] [Van Graas and Braasch 1992] [Napolitano and Spagnuolo 1993] [Brennan and McDaniel 1994] [Canter and Groves 1994] [Kong 1994] [Maine et al. 1994] [Maine et al. 1995] [Morelli 1995] [Schkolnik et al. 1995] [Tischler 1995] [Traven and Whitley 1995] [Batterson and Morelli 1996] [Fisher et al. 1996] [Hamel and Jategaonkar 1996] [Morelli 1996] [Murphy 1996] [Morelli 1998] [Advisory Group for Aerospace Research and Development 1999] [Chen and Valasek 1999] [Mukhopadhyay 1999b] [Mukhopadhyay 1999a] [Murphy 1999] [Ghosh and Raisinghani 2000] [Ghosh et al. 2000] [Moes et al. 2000] [Saraswathi 2000] [Milne et al. 2001] [Napolitano et al. 2001] [Moes and Iliff 2002] [Morelli 2002] [Ray et al. 2002] [Hageman et al. 2003] [Longitudinal Control and Trim 2003] [Newsom et al. 2003] [Smith et al. 2003] [Static Longitudinal Stability Flight 2003] [Klyde et al. 2004] [Mueller et al. 2004] [Stachowiak and Bosworth 2004] [Williams-Hayes 2004] [Burken 2005] [Jategaonkar 2005] [Morelli et al. 2005] [Chowdhary and Jategaonkar 2006] [Sarigul-Klijn and Huber 2006] [Dibley et al. 2007] [Kukreja 2007] [Gangsaas et al. 2008] [Kukreja 2008a] [Kukreja 2008b] [Marwaha et al. 2008] [Murphy and Klein 2008] [Burken and Larson 2009] [DeBusk et al. 2009] [Han et al. 2009] [Shin et al. 2009] [Taylor 2009] [Moua 2010] [Murphy and Klein 2010] [Bachelder et al. 2011] [Cazes et al. 2011] [Gregory et al. 2011a] [Gregory et al. 2011b] [Martos et al. 2011] [McCracken and Jorris 2011] [Silva 2011] [Brown 2012] [Development of a Continuous 2012] [Giordano de Oliveira Silva and Moennich 2012] [Londono and Leonhardt 2012] [Taylor 2012] [Brown and Schaefer 2013] [Siu et al. 2013] [Kuehme et al. 2014] [Herrera et al. 2015] [Sorokowski et al. 2015] [Cumming et al. 2016] [Deiler 2016] [Harris et al. 2016] [Lichota et al. 2016] [Pfifer and Danowsky 2016] [Kumar and Ghosh 2017] [Moszczynski and Grant 2017] [Saderla et al. 2017] [Beh et al. 2018] [Blight et al. 2018] [Kumar and Ghosh 2018] [Morelli and Grauer 2018] [Tischler 2018] [Enciu 2019] [Larsson 2019] [Leung et al. 2019] [Newton 2019] [Waite et al. 2019] [Xu et al. 2019] [Grauer and Boucher 2020] [Gao et al. 2021a] [Goel et al. 2021] [Morelli 2021] [Simmons 2022] [Wang et al. 2022a] [Deepa and Gupta 2023] [Morelli and Grauer 2023] [Nguyen and Xiong 2023] [Patel et al. 2023] [Raol and Singh 2023] [Simmons 2023] [Simmons et al. 2023] [Matt 2024] [Roeser and Mönnich 2024] [Corrigan et al. 2026] [Raab 2026] [Simmons] [Bouadi] [Miller] [Morelli] [Grauer] [Grauer and Boucher] [Grauer] [Waite et al] [Matt]
Demonstrators, research aircraft and the practice of flying one
The research aircraft before this one. Demonstrators and research aircraft from the augmentor-wing and quiet short-haul aircraft to the Active Aeroelastic Wing, the adaptive compliant trailing edge and the X-planes.
131 records. [Baker 1953] [Wright and Loving 1953] [Bellman 1959] [Beeler 1961] [General Electric Co Cincinnati Oh 1964] [Nicholson and lowry 1966] [Pittsburgh Univ Washington DC Research Staff 1966] [Jackson and Patch 1968] [Kemper and Renselaer 1969] [Cleveland 1971] [Quigley et al. 1971] [Rumsey and Spitzer 1971] [Ashleman and Kavdahl 1972] [Innis and Anderson 1972] [Quigley and Vomaske 1972] [Rumsey et al. 1972] [Skavdahl and Patterson 1972] [Spitzer 1972] [Spitzer et al. 1972] [Bartlett et al. 1973] [Whittley 1973] [Amberg et al. 1974] [Marrs et al. 1974] [Quigley and Holzhauser 1974] [Jacobsen and Drinkwater 1975] [Cook and Whittley 1976] [Richardson 1976] [Sim 1976] [Dehoff et al. 1977] [Alexander 1979] [Martin 1979] [McCracken 1979] [Shovlin et al. 1979] [Andrews 1980] [Feinreich and Gevaert 1980] [Green 1980] [Martin 1980] [McArdle et al. 1980] [Cook and Lilley 1981] [Dunn et al. 1981] [Eppel 1981] [Riddle et al. 1981] [Queen and Cochrane 1982] [Cochrane 1983] [Gilyard and Edwards 1983] [Painter 1983] [Curry and Sim 1984] [Sim and Curry 1984] [Sim and Curry 1985] [Larson 1986] [Sefic and Cutler 1986] [Redeker and Voersmann 1988] [Riddle et al. 1988] [Smith and Nelson 1988] [Webb et al. 1988] [Bonnema and Lokos 1989] [Wada 1989] [Stephenson et al. 1990] [Urnes et al. 1990] [Aswani et al. 1991] [Urnes and Yeager 1991] [Vincent 1991] [Dicarlo et al. 1992] [American Institute of Aeronautics and Astronautics 1993] [Stortz and ODonoghue 1995] [Sim 1997] [Walker 1997] [Lind 1999a] [Lind 1999b] [Kota 2000] [Clark 2001] [Lokos et al. 2002b] [Lokos et al. 2002a] [Brenner and Prazenica 2003] [Lee and Song 2003] [Turner 2003] [Hess and Flick 2004] [King and Gopalarathnam 2004] [Kumar and Rao 2004] [Ueno et al. 2004] [Waszak et al. 2004] [Allen et al. 2005] [Clarke et al. 2005] [Cumming and Diebler 2005] [Dibley et al. 2005] [Heeg et al. 2005] [Lizotte and Allen 2005] [Lizotte and Lokos 2005] [Lokos et al. 2005] [Wiesman et al. 2005] [Heeg 2006] [Cosentino 2007] [Lizotte and Allen 2007] [Voracek 2007] [Cosentino 2008] [Skillen and Crossley 2008] [Gandhi et al. 2009] [Di Matteo and Guo 2011] [Yu 2013] [Gardner et al. 2014b] [Miller et al. 2014] [Ryan et al. 2014] [Davis et al. 2015] [Miller et al. 2015] [Cruz and Miller 2016] [Denham and Owens 2016] [Engblom et al. 2016] [Herrera et al. 2016b] [Herrera et al. 2016a] [Jo et al. 2016] [Sato et al. 2016] [Smith et al. 2016] [Öznalbant et al. 2016] [Lechniak and Melton 2017] [Vetrella et al. 2017] [Yamamoto et al. 2017] [Herrera 2018] [Hirshorn and Kemmerly 2018] [Kaygan and Ulusoy 2018] [Thomas et al. 2018] [Xiang et al. 2018] [Garcia et al. 2019] [Dippold 2022] [Bhowmick et al. 2023] [German et al] [Hu] [Quindlen et al] [Heaney] [Heaney and Quindlen] [Arteaga et al] [Standard Practice for Development]
Jets in a crossflow
What an effector issues into the boundary layer. The jet in a crossflow, its vortex structure, trajectory, induced pressures and its computation.
81 records. [Girshovich 1968] [Mosher 1970] [Chang and Werner 1972] [Kamotani and Greber 1972] [Dietz 1975] [Fearn and Weston 1975] [Fearn and Weston 1978] [Yen 1978] [Fearn and Benson 1979] [Murphy et al. 1979] [Hackett et al. 1981] [Perkins and Mendenhall 1981] [Baker and Orzechowski 1982] [Clauss 1982] [Furlong and Fearn 1983] [Snyder and Orloff 1984] [Wilsden and Hackett 1984] [Simon and Ciancone 1985] [Simon and Ciancone 1987] [Fearn 1988] [Roth 1988] [Bray and Knowles 1989] [Kavsaoglu and Schetz 1989] [Roth et al. 1989] [Claus and Vanka 1990] [Krothapalli et al. 1990] [Lytle et al. 1990] [Kim and Benson 1991] [Claus and Vanka 1992] [Kim and Benson 1992] [Roth et al. 1992] [Blake and Stewart 1993] [Chiu et al. 1993] [Dennis et al. 1993] [Dennis 1993] [Krothapalli and Shih 1993] [Margason and Tso 1993] [Manipulation of a Jet 1994] [Liscinsky et al. 1995] [Tso and Margason 1995] [Bain et al. 1996] [Tso and Margason 1996] [Yuan and Street 1996] [Eroglu and Breidenthal 1998] [He 1999] [Wilson et al. 2000] [Xie et al. 2000] [Pandya et al. 2002] [Kiel et al. 2003] [Zaman 2003b] [Zaman 2003a] [Marzouk and Ghoniem 2004] [Bunyajitradulya and Sathapornnanon 2005] [Dano and Liburdy 2005] [Ziefle and Kleiser 2006] [Atkins 2007] [Ivanova et al. 2010] [Zaman et al. 2010b] [Zaman et al. 2010a] [Zaman et al. 2011] [Arunajatesan and McWherter-Payne 2013] [Lefantzi et al. 2013] [Karagozian 2014] [Zhang and Yang 2017a] [Zhang and Yang 2017b] [Borghi et al. 2018] [Dai et al. 2018] [Chang et al. 2019] [Stegmeier et al. 2019] [Zhao et al. 2021] [Lv et al. 2022] [Smith and Yagle 2023] [Gupta and Saha 2024] [Wake Vortex Formation 2024] [Borodulin et al. 2025] [БОРОДУЛИН et al. 2025] [Dhanagopal et al. 2026] [Bouladoux] [Li] [Oertling] [Sakradse]
Wall jets and curved wall jets
The flow a Coanda effector produces. Plane, three-dimensional and curved wall jets, their turbulence, roughness and pressure-gradient effects.
69 records. [Wygnanski and Champagne 1968] [Bajura and Szewczyk 1970] [Bangert 1971] [Kind 1971] [Piva 1972] [Narasimha et al. 1973] [McLean and Herring 1974a] [McLean and Herring 1974b] [Campbell 1975c] [Tsuji et al. 1976] [Alcaraz et al. 1977] [Catalano et al. 1977] [Catalano et al. 1978a] [Catalano et al. 1978b] [Kotansky and Glaze 1978] [Morton et al. 1978] [Catalano et al. 1979b] [Catalano et al. 1979a] [Catalano et al. 1979d] [Horne and Karamcheti 1979] [Saripalli and Simpson 1979] [Kotansky and Glaze 1980] [Maki et al. 1980] [Saripalli and Simpson 1980] [Kotansky and Glaze 1981] [Back and Cuffel 1982] [Gibson and Younis 1982] [Hartmann 1982] [Wall Interference Effects 1983] [Horne 1984] [Roberts 1986] [Roberts 1987] [Rodman et al. 1987b] [Rodman et al. 1987a] [Rodman et al. 1989] [Rask et al. 1990] [Gregory-Smith and Hawkins 1991] [Lim and Roberts 1991] [Park and Rew 1991] [Park and Rew 1992] [Yoon et al. 1993] [Myszko and Knowles 1994] [Matsson 1995a] [Le Cunff and Zebib 1996] [Rew and Park 1996] [Fasel 2000] [Likhachev et al. 2001] [Tachie et al. 2001] [Hossain and Kader 2004] [Imao et al. 2005] [Inoue et al. 2005] [Mochizuki et al. 2005] [Han et al. 2006] [Afzal and Seena 2011] [Kumar and Das 2011] [Rostamy et al. 2011] [Pramanik and Das 2014b] [Pramanik and Das 2014a] [Pramanik and Kumar Das 2014] [Kumar 2015] [Tesař 2015b] [Khayat 2017] [van Hout et al. 2018] [Weidman 2018] [Baleriola et al. 2019] [Hatakawa et al. 2019] [Pease et al. 2021] [McBreen and Zha 2024] [Pajayakrit]
Shear layers, receptivity and periodic excitation
The mechanism oscillatory control exploits. Free and separated shear layers and boundary layers forced by periodic excitation, and their receptivity.
30 records. [Leehey and Shapiro 1980] [Bechert and Stahl 1984] [Bayliss et al. 1985] [Fiedler and Mensing 1985] [Bayliss et al. 1986] [Lam and Ko 1986] [Lam et al. 1986] [Balsa 1988] [Balsa 1989] [Lumley 1989] [Kraus and Cutler 1993] [Strykowski and Krothapalli 1993] [Joslin et al. 1994] [Manuilovich 1995] [Pack and Seifert 1999] [Semionov and Kosinov 2000] [Adelgren et al. 2002] [Billeter 2004] [Kozato and Imao 2005] [Mittal 2005b] [Mittal 2006] [Mittal 2007] [Yehoshua and Seifert 2007] [Samimy et al. 2018] [Borodulin et al. 2020] [Singbeil et al. 2021] [Elsner et al. 2024] [Nakano and Ohtsuki 2024] [Sokolenko et al. 2024] [Kitzinger]
This programme and the programme it descends from
The titles that name CRANE or the work it descends from. No cited title names this aircraft or this contractor, and the homonym measurements under The Source Base show why the programme’s own name finds so little.
2 records. [Tallent and Koss 2004] [Vukasinovic et al. 2023]
The medium, admitted by the shared atmosphere family alone
The atmosphere every number here is computed in. The standard atmosphere tables of a flight-test manual.
1 record. [Air Force Test Pilot School Edwards Afb Ca 1962]
Where the Framing Breaks Down
The Century-Old Paradigm Is Younger Than It Sounds for One Axis
DARPA’s release opens by observing that in December 1903 the Wright brothers flew the world’s first fully controllable aircraft using wing warping, that virtually every aircraft since has used a system of movable external control surfaces, and that the X-65 breaks this century-old design paradigm [DARPA moves forward on the X-65].
The framing is half right, and the wrong half is more interesting than pedantic. The 1903 aircraft was essentially a larger version of the 1902 glider, whose key control refinement was a movable vertical tail, and it carried a forward-mounted horizontal surface for climb and descent [1903 Wright Flyer]. The Flyer therefore used movable external surfaces for pitch and for yaw from the first flight. For roll it did not. It twisted the wing.
Wing warping is not a movable control surface. It is a morphing wing, and the mechanism DARPA credits the Wrights with is the one axis where the original solution was not the paradigm being broken. The paradigm is therefore 122 years old on this article’s dateline for two axes and rather younger for the third, and the X-65 carries traditional flaps and rudders among its conventional surfaces [DARPA moves forward on the X-65], so the aircraft does not break the paradigm on the axis the framing invokes.
The Amplification Curve Is Doing a Great Deal of Work
The article’s central arithmetic depends on a quantity it does not derive. The amplification is treated as a parameter swept from ten to a hundred, and the low-speed separation-control experiments the review summarises imply values at and above the top of that band [Abdolahipour 2024], so no attempt is made to predict it for this aircraft’s geometry.
The treatment is candid, yet the amplification remains the weakest link. The amplification depends on the pressure gradient, the boundary-layer state, the Reynolds number, the effector geometry, its position relative to the separation line and the excitation frequency. None of those is public for the X-65A and several are what the aircraft is meant to measure. So the authority ratios in this article are conditional statements of the form that if the amplification is thirty then the deficit at Mach 0.7 is the 43.3 percent displayed in the section on the authority ratio, and they are not predictions.
Reynolds Number Is Not the Problem
This aircraft’s Reynolds number is one to nearly three orders of magnitude above those of the experiments whose thresholds this article borrows, and that gap does not make the phenomenon doubtful at flight scale.
The Reynolds number on a chord is the ratio of inertial to viscous forces. Let $\mathrm{Re}$ be that number, dimensionless, and $\mu$ the dynamic viscosity in pascal second, which for air follows Sutherland’s law with constants $C_1$ in kilogram per metre second per root kelvin and $S_\mu$ in kelvin.
\[\mathrm{Re} = \frac{\rho V \bar{c}}{\mu}, \qquad \mu\left( T \right) = \frac{C_1 \, T^{3/2}}{T + S_\mu} \tag{56}\]with $C_1$ of $1.458 \times 10^{-6}$ and $S_\mu$ of 110.4 kelvin. That returns 1.7894 times ten to the minus five pascal second at sea level. At Mach 0.7 and thirty thousand feet on a chord of 1.750 metre the X-65A flies at 11.44 million.
Active separation control by oscillatory excitation was demonstrated above that Reynolds number in the year 2000. Seifert and Pack tested the method at chord Reynolds numbers as high as forty million and state in their summary that it was proved successful in delaying boundary layer separation and reattaching separated flows at chord Reynolds numbers of order ten million [Seifert and Pack 2000a]. Their baseline flow was made fully turbulent deliberately, to remove transition as a variable, and they report that the Reynolds number has a very weak effect on the model pressure distributions and spectra regardless of Mach number or sweep.
The Reynolds numbers in the first column are those the review reports for each study [Abdolahipour 2024].
| Study | Reynolds number | The X-65A in flight over it |
|---|---|---|
| Munday and Taira 2018 | 2.30 times ten to the 4 | 497.6 |
| Feero et al. 2015 | 1.00 times ten to the 5 | 114.4 |
| Seifert et al. 1996, low | 1.50 times ten to the 5 | 76.3 |
| Jones and Englar 2003 | 5.00 times ten to the 5 | 22.9 |
| Seifert et al. 1996, high | 6.00 times ten to the 5 | 19.1 |
The X-65A flies at 0.715 times that experiment’s sixteen million and 0.286 times the forty million it quotes. [Seifert and Pack 2000a] It is inside the demonstrated range, not beyond it. So the claim to make is the narrow one. The threshold values borrowed above do come from experiments at a hundred thousand and twenty-three thousand, and transferring a threshold is a different matter from transferring a phenomenon. But the phenomenon is not in doubt at this aircraft’s Reynolds number.
The Better News in the Flight-Reynolds Primary
The same report quotes the oscillatory momentum coefficients it used, and its nomenclature defines the coefficient as the jet momentum over chord times dynamic pressure, the same definition this article uses. [Seifert and Pack 2000a]
| Case the report quotes | Oscillatory momentum coefficient | This article’s figure over it | Concentration still needed |
|---|---|---|---|
| lowest quoted, Mach 0.25 | 0.03 percent | 3.150 | none |
| slot at 64 percent chord, Reynolds 16 million | 0.09 percent | 1.050 | none |
| reduced frequency 1.0 | 0.12 percent | 0.788 | 1.27 |
| wall pressures, Reynolds 16 million, Mach 0.25 | 0.13 percent | 0.727 | 1.38 |
| reduced frequency 0.5 | 0.32 percent | 0.295 | 3.39 |
The wing-referenced coefficient this article computes, 0.0945 percent, sits inside the range that report used, 0.03 to 0.32 percent. It reaches two of the five quoted cases with no concentration at all, and the largest needs a factor of 3.39. The report states its own uncertainty on that coefficient as plus or minus 25 percent of the quoted values [Seifert and Pack 2000a], a primary-sourced error bar and not an assumption of this analysis. The third column is the computed coefficient over the quoted one and the fourth is its reciprocal where that exceeds one, so at the two ends of the range
\[\frac{0.0945}{0.03} = 3.150 , \qquad \frac{0.32}{0.0945} = 3.39 . \tag{57}\]At that report’s Mach number of 0.25 the bleed budget delivers 0.5579 percent, 1.74 times its highest quoted value. So the air budget is not marginal at the condition where the method was demonstrated. It is marginal only where this aircraft wants to fly, and the factor of 5.903 between the two is the Mach dependence derived above and nothing else.
\[C_\mu\left( 0.25 \right) = 0.000945066 \times \frac{G\left( 0.25 \right)}{G\left( 0.7 \right)} = 0.000945066 \times \frac{16.711}{2.8308} = 0.005579 , \qquad \frac{0.5579}{0.32} = 1.74 \tag{58}\]And the Primary Names This Article’s Keystone as the Open Problem
The report’s proposals for future work include overcoming the lack of sufficient control authority, especially at high speeds [Seifert and Pack 2000a].
The report thus names as unsolved, twenty-five years before the X-65A was due to fly, the finding this article derived. The authority ratio falls as the square of the Mach number because the engine delivers a roughly fixed momentum while the aerodynamic forces to be overcome grow with the square of the speed. A primary that tested the method at this aircraft’s Reynolds number, at a third of its Mach number, and concluded that high-speed control authority was the thing still to be found, is the strongest available independent support for the conclusion.
It is also a caution about novelty. The Mach dependence and its exact minimum are derived here, and the observation that high-speed authority is the binding constraint was in the literature first, in the report above.
The Bleed Penalty Is Asserted Rather Than Derived
Taking air from a compressor costs thrust and this article bounds the cost from below without computing it. The mass-flow accounting and the lower bound it gives are exact and are derived in the engine discussion. The thrust consequence is not exact, because it depends on the turbine work no longer available to drive the fan, on the cycle’s rematching, and on the control schedule the engine’s digital controller applies. The single-stage centrifugal architecture makes the answer worse than a multi-stage axial’s would be, because the bled air has had the whole compression done to it, and that much follows from the architecture the NASA report describes [FJ44 turbofan engine test].
Suction Is in the Official Definition and Not in This Analysis
The budget justification defines active flow control as control mechanisms which alter the aerodynamic flow field through ejection or suction of fluid via an orifice on a lifting body [DARPA RDT&E justification, PB2026]. The phrase includes suction, while every step of the momentum accounting assumes blowing.
A suction device removes low-momentum air where a blowing device adds high-momentum air, and its figure of merit is a flow coefficient rather than a momentum coefficient. The identity that makes the amplification exact does not hold for it, because there is no jet momentum in the denominator. The published descriptions speak only of jets of air from a pressurized source [DARPA moves forward on the X-65], which is blowing, and none of them mentions suction, so a system that also used suction would need both accountings. This article’s analysis applies to the blowing case alone.
One Vehicle Still Cannot Do Everything
The differencing argument above is strong, but it has a limit. Locking a conventional surface and commanding an effector instead changes the aircraft’s aerodynamics in ways beyond the intended substitution. A locked surface is still a surface, sitting in a position, with its gaps and its hinge line, present even when unused. An aircraft designed without conventional surfaces would have a different shape, lower weight and no gaps.
The X-65A consequently measures the authority of flow control on an airframe built to accommodate both systems, which is not the same as the authority of flow control on an airframe built for flow control alone. The programme’s framing acknowledges this by calling the conventional surfaces training wheels [DARPA moves forward on the X-65], and the fair reading is that the aircraft measures a lower bound on what a purpose-built vehicle would achieve and an upper bound on the weight saving.
The Source Base
The research survey holds 3,301 research records drawn from the NASA Technical Reports Server, the Defense Technical Information Center and Crossref, the bibliographic index of journal and conference records, of which 1,351 are report primaries, being 40.9 percent, by the identifier rule set out under The Definition of Primary. 3,193 records carry a usable year, running from 1927 to 2027 with a median of 2006. 1,139 of those, 35.7 percent, predate the year 2000, and 956, or 29.9 percent, are 2015 or later.
The heaviest decade is the 2010s, with 854 records, 26.7 percent of the dated set. The 1970s hold 342 records against 63 in the 1960s, and 219 of those 342 are in the cluster on blowing, the Coanda surface and the jet flap, which is the arrival of the circulation-control and powered-lift literature this subject keeps returning to.
| Decade | Records | Share of the dated set |
|---|---|---|
| 1920s | 2 | 0.1 percent |
| 1930s | 3 | 0.1 percent |
| 1940s | 29 | 0.9 percent |
| 1950s | 66 | 2.1 percent |
| 1960s | 63 | 2.0 percent |
| 1970s | 342 | 10.7 percent |
| 1980s | 304 | 9.5 percent |
| 1990s | 330 | 10.3 percent |
| 2000s | 711 | 22.3 percent |
| 2010s | 854 | 26.7 percent |
| 2020s | 489 | 15.3 percent |
What the Reports Server Holds
The reports server holds nothing at all for four phrases, and a null answer from a registry is a measurement. They are hingeless aerodynamic control effector, technology demonstrator aircraft programme, Coanda effect jet attachment and mixing enhancement jet actuator. The first of those is the finding and it is treated separately below.
The Homonyms in This Subject’s Vocabulary
This subject’s vocabulary collides widely with other fields. Up to ten results from the bibliographic index were read for each probe and counted by hand.
| Probe | Results read | On subject | What the rest were |
|---|---|---|---|
CRANE |
6 | 0 | a logging-crane fatality, a mobile-gantry-crane fatality, a civil-engineering journal, CRANE-tools.org, an 1878 book of rhymes and tunes, and Walter Crane’s 1886 A romance of the three Rs |
novel effectors |
10 | 1 | fungal, oomycete and oncological effectors, the one exception being a 2026 paper on hypersonics and directed energy |
effectors |
9 | 1 | TAL effectors, rust effectors, smut fungi and the effectors of plant-defence suppression |
end effector |
10 | 0 | industrial robot wrists and upper-limb kinematics |
actuator |
10 | 0 | road-vehicle air-brake actuator test procedures, eight of the ten |
momentum coefficient |
10 | 1 | tangential momentum accommodation in microchannels, angular momentum coupling, momentum exchange in rarefied flow, and an evolutionary optimiser with a momentum and adaptive decay coefficient, twice |
blowing coefficient |
10 | 0 | elliptic equations with a blowing-up coefficient, plus Blue-blowing and BLOWING ROCK LANDMARKS |
control authority |
10 | 0 | bibliographic authority control, at the National Archives and in Italy |
control effectiveness |
10 | 0 | internal-control auditing and diamondback-moth pest management |
separation control |
10 | 1 | chemical separation processes, dust and pollution control |
fluidic oscillator |
10 | 2 | an antisolvent crystalliser and the American Water Works Association standard for cold-water meters, in two editions |
supercirculation |
8 | 3 | the climate supercirculation of the Earth |
Aurora Flight Sciences |
10 | 2 | Mercury-Atlas 7, a damselfly, a Brazilian philosophy journal and a casket company |
X-65 |
10 | 0 | front matter and acknowledgements |
active flow control |
10 | 10 | nothing off subject |
sweeping jet actuator |
10 | 10 | nothing off subject |
hingeless control |
10 | 7 | a dictionary entry for the word and two papers on hingeless rotors, and 72 records on the reports server, 61 of them with rotorcraft vocabulary in the title |
Three of those rows matter to anyone searching this subject. The programme’s acronym finds nothing aeronautical. The word effector without qualification finds mostly biology. And the article’s own keystone parameter is almost absent from titles, as Thin Subjects below sets out.
A Homonym That Depends on Which Registry You Ask
hingeless control returned ten results from the bibliographic index, of which seven concern control without a hinge, and 72 from the reports server of which 61 carry rotorcraft vocabulary in the title. Six of the seven are flow-control effectors on wings and the seventh is the hingeless flight control of a smart projectile by miniature actuators. The other three are a dictionary entry for the word itself and two papers on hingeless rotors, one on a circulation-control rotor blade and one on higher harmonic control, so the rotor meaning reaches even the journal index.
Eleven do not carry rotorcraft vocabulary, and most of those are rotorcraft work under other words, being higher harmonic control, flap-lag stability in forward flight, ground resonance and a V and short take-off conference. Two are not rotorcraft at all. They are the DARPA, Air Force Research Laboratory, NASA and Northrop Grumman Smart Wing programme, which shares the hingeless vocabulary because it is about a wing with no discrete moving surface and is genuinely adjacent prior art.
A hingeless rotor is a blade retention without flap and lag hinges, which is a structural subject with a large literature from the 1960s and 1970s. A hingeless control effector is an aerodynamic surface replaced by a jet. The same two words name two unrelated subjects and which one you get depends on which registry you ask.
The survey therefore excludes rotorcraft, as the section on what is out of scope records. The phrase that finds this subject differs by registry, being hingeless control in the journal literature and control effectors in the reports, and the second returns the aerodynamic control effectors, the passive porosity effectors and the active flow effectors.
Which Thresholds Are Second-Hand, and Which Originals Can Be Read
Most of the thresholds this article uses reach it through one review. The regime boundary of three to five percent, the reduced-frequency band, the measured response at 0.08 percent, the reattachment at 0.34 percent, the modified-coefficient thresholds, the velocity-ratio condition, the caution against comparing momentum coefficients across configurations and the attribution of the definition to Poisson-Quinton are taken from the review rather than from the works that established them.
The review’s bibliography names the originals. They are Greenblatt and Wygnanski 2000 and Wygnanski 2000 for the regime boundary, the frequency band and the velocity-ratio condition, Seifert and others 1996 for the response at 0.08 percent, Feero and others 2015 for the reattachment at 0.34 percent, Munday and Taira 2018 for the modified-coefficient thresholds, and Joslin and Miller 2009 for the attribution to Poisson-Quinton, while the caution against comparing coefficients across configurations is the review’s own [Abdolahipour 2024]. Every journal item the review names is behind a paywall, so a first-hand reading is available only where the work was also published as a report, and four agency reports on separation control are read in full instead. That asymmetry favours the report literature wherever a first-hand reading matters.
The most consequential of the four is the report on separation control at flight Reynolds numbers [Seifert and Pack 2000a], whose bearing on the Reynolds-number argument, and whose naming of high-speed control authority as an open problem, are set out in the section headed Where the Framing Breaks Down.
The Same Work Twice, Once With a File and Once Without
The reports server holds the flight-Reynolds report of Seifert and Pack under two identifiers and only one of them carries the document. A search returns both. The first has a record and no attachment, so a request for its full text returns nothing at all, and the second carries the portable document and a plain-text rendering of it. A null answer from a registry is usually a measurement about the literature. This one is a measurement about the catalogue, and only the catalogue’s detail record shows which identifier has a file.
What This Article Read in Full
Read in full. The register’s addendum, the encyclopedia entry, the DARPA programme page, the DARPA news release of 3 January 2024, the contractor’s aircraft page, the Defense News report of 20 November 2025, the Smithsonian collection record for the 1903 Flyer, the NASA technical memorandum on the FJ44-3A, the open-access review of flow separation control, the CRANE entries in all seven DARPA justification books from the request for fiscal year 2020 to the request for fiscal year 2026, and four agency reports on separation control, being the flight-Reynolds-number lessons-learned report, the flap-separation report, the slat-shoulder report and the amplitude-scaling workshop paper.
Read to the abstract only, and quoted for nothing inside it. Reddy and Woszidlo 2024, which is closed access, held by no repository, and whose two scaling parameters both registries render as a placeholder [Reddy and Woszidlo 2024].
Held at registry-record strength and quoted for nothing. Five are named by the review, being Seifert and others 1996 in the Journal of Aircraft, Munday and Taira 2018 in the AIAA Journal, Feero and others 2015 in the AIAA Journal [Feero et al. 2015] and 2017 in Experiments in Fluids, and the flow-control terminology chapter of the 2009 AIAA volume. Two more are the MAFC lessons-learned paper of 2004 and Wygnanski’s 1997 conference paper on the periodic addition of momentum. Seven paywalled items whose existence, venue and year are confirmed and whose contents are not. Where this article states a number those works established, it states it on the review’s authority and says so.
The reattachment threshold belongs to the 2015 paper and not to the 2017 one. The review attributes the reattachment at 0.34 percent to Feero and others in 2015, and its bibliography identifies that work as a 2015 AIAA Journal paper on flow reattachment by synthetic jet actuation on a low-Reynolds-number airfoil, distinct from the same group’s 2017 paper on synthetic jet location, which the review cites separately [Abdolahipour 2024].
Read by title only. The other research records, which are cited as a body of literature in the clusters above and not quoted.
The Definition of Primary
Primacy is a property of the document, and this survey derives it from the identifier. A record is a report primary when its address is on the NASA Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. The four agency reports on separation control that this article reads in full are report primaries under that rule. The rule does not reach the journal literature, such as the papers the review names behind paywalls, so 40.9 percent is a lower bound on how much of the survey is first-hand.
Thin Subjects
Six subjects that this article’s derivations depend on have thin literatures. Each count below is the number of research records in this survey whose title carries at least one of the listed terms, with a hyphen read as a space. Where a row joins two groups of terms with the word together, a title must carry a term from each group.
| Subject | Title terms counted | Records |
|---|---|---|
| the momentum coefficient named as a parameter | momentum coefficient, momentum coefficients, blowing coefficient, blowing coefficients | 10 |
| the effect of altitude on what an effector delivers | altitude, altitudes together with flow control, actuator, actuators, actuation, blowing, bleed, momentum | 0 |
| the effect of Mach number on active flow control | Mach, compressibility, compressible, transonic together with active flow control, synthetic jet, synthetic jets, sweeping jet, sweeping jets, fluidic oscillator, fluidic oscillators, separation control, pulsed jets | 11 |
| what engine bleed costs in thrust | bleed, power extraction together with thrust, turbojet, turbofan, engine performance, aircraft performance | 9 |
| the joined diamond planform | diamond together with joined | 3 |
| the programme this aircraft descends from | micro adaptive flow control, microadaptive, MAFC | 2 |
The keystone parameter is almost never in a title, and that is not a thin literature. Ten of the 3,301 cited research titles name the momentum coefficient or the blowing coefficient. The cluster for the parameter holds 13 records, eight of which name it, and the rest are titles on blowing parameters and actuation efficiency. Every paper in active flow control uses the momentum coefficient, and the review this article leans on devotes a numbered section to it [Abdolahipour 2024]. The parameter is a method rather than a subject, so it appears in equations and in axis labels and not in titles. The literature is healthy and the quantity is invisible to a title search.
No cited title asks how altitude changes what a flow-control effector delivers, which is the question this article’s central result answers. The nearest is a 2024 paper whose title asks it of operating conditions in general [Reddy and Woszidlo 2024], and the abstract is all of it that can be read.
The Mach dependence that this article finds binding has a literature of 11 titles, of which two name an airfoil and three name separation control, and none names a control effector’s authority or a control moment. Of the seven that name neither, five characterise actuators in compressible flow or apply them to a transonic compressor, and the others are a test facility and a weapon release.
What a bleed costs an engine is known for few engines. Nine cited titles join bleed or power extraction to thrust, to a turbojet or turbofan, or to engine or aircraft performance. Two of them are NASA records of 1992 on the effect of bleed extraction on the thrust of the F404 [Yuhas and Ray 1992a] [Yuhas and Ray 1992b], and one asks the same question of a commercial engine on an unmanned aircraft [Takahashi and Stone 2019]. None names the FJ44. Asking the reports server for a secondary air system returns two hundred and seven records and they are the proceedings of a seal workshop, because inside a turbine engine secondary air means internal cooling. Asking it for an aircraft air supply system returns electric-propulsion impedance measurements and direct-current power supplies. Asking the defence registry what bleed air costs in engine performance returns two hundred records, of which three concern this article’s subject and none of those three is about bleed air.
The joined diamond planform has three titles of its own, all of them design and aerodynamic studies, and none is about controlling it. The planform’s control is reached instead through the far larger literatures on joined wings and on tailless aircraft.
The CRANE manager named Micro Adaptive Flow Control as the work CRANE descends from, and neither registry holds a report under that name. The reports server returns, for that phrase, two summer faculty fellowship programmes from 1995, a Spacelab life-sciences experiment and a paper on the importance of properties in modelling. The defence registry returns one hundred and twenty-one records, of which two come near this subject and both are about adaptive structures rather than adaptive flow control. The only trace of the programme under its own name is a single conference paper, behind a paywall, whose title is its lessons learned [Tallent and Koss 2004]. Written as one word, the technique’s name finds one defence report, an Army Research Laboratory study of microadaptive flow control applied to a spinning projectile [McMichael et al. 2005], whose title names the technique rather than the programme. A DARPA programme leaves no report literature in a NASA registry because it was never a NASA programme, and the defence registry holds what its contractors delivered rather than what the agency learned. The ancestor of an aircraft is harder to find than the aircraft.
What the Survey Admits
A record is admitted only when a person reading its title finds it on this article’s subject, read against the argument rather than against a word. The subjects are the ones the argument uses, being flow control and its devices, control without a hinge, control allocation, the conventional control system the effectors would replace, the air supply, Reynolds-number and scale effects, the planform family, flight-data identification, research aircraft and demonstrators, the jets, wall jets and shear layers the effectors act through, the atmosphere, and the programme itself.
Many words this subject depends on also name something else, and a record that shares only such a word is excluded. Full scale also names demonstrations in environmental, industrial and civil engineering. The Coanda effect is also named in cardiology, biology, hydraulic engineering, building ventilation and plasma torches. Hingeless also names the hingeless arch of structural engineering and the hingeless rotor. Flow control also names propellant valves, a synthetic jet also matches synthetic jet fuel, a fluidic actuator also names the soft actuators of robotics, and a fluidic oscillator also serves flow meters, drilling hammers, fuel cells and microfluidics. A jet in a crossflow is also a fuel spray and a flame, a wall jet is also a cooling film, and the excitation of a turbulent boundary layer is also the excitation of a panel. Bleed air also carries the literature of cabin air quality and of valve and sensor faults, and a technology demonstrator or a flight test also names work outside aviation. Correction, erratum and withdrawal notices and peer-review records are excluded as well, because a survey counts research works and those are editorial events rather than works, and a document deposited twice under the same title is counted once. A review of a book, a short glossary, dictionary or encyclopedia entry and a notice such as an editorial or a news item are not admitted either, because none of them is a research work.
Three classes are excluded by the scope set out under Out of Scope, being rotorcraft, the control of shock and boundary-layer interaction, and thrust vectoring that deflects the nozzle rather than the jet. The cost of the first is the circulation-control rotor and the X-Wing, and it is stated there.
Some records of doubtful relevance are kept. Environmental control systems fed by bleed air, the bleed-air spoilers of a parafoil, the Coanda effect of a water jet, fluidic amplifiers and fluidic calibration tools, ice protection by synthetic jets, wall-bounded turbulence in pipes, and the tunnel and flight correlation of the Space Shuttle and Apollo each share a device, an air supply or a method with this subject, so each is kept and a reader may weigh it accordingly.
No cluster is a residual. The fluid mechanics the effectors act through is gathered in three named clusters, on jets in a crossflow, on wall jets and on shear layers and their excitation, and a record enters one of them only when its title is one of those flows.
Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.
Epistemic State
Historical fact, from documents read in full. The X-65A designation was allocated on 24 April 2023 to Aurora Flight Sciences, with the engines cell reading 1 Williams FJ44-3A and DARPA as sponsor. CRANE began with three Phase 0 contracts in mid-2020 to Aurora Flight Sciences, Lockheed Martin and Georgia Tech Research Corporation, all three reporting twenty-five offers received. A Phase 2 and Phase 3 contract was placed on Aurora on 7 December 2022 and carried 94,372,238 dollars in eighteen actions, the last dollar obligated on 16 January 2025 and the last action a zero-dollar change order of 11 February 2025. DARPA told Defense News that it paused the programme earlier in 2025, restructured it so the contractor became a co-investor, finalised that in August 2025, and expects first flight in late 2027, and that account rests on the one press report. The aircraft has fourteen flow-control effectors, fourteen conventional control surfaces, four interchangeable wing sets, a thirty foot span and a limit of Mach 0.7. The programme manager named Micro Adaptive Flow Control as its ancestor and said he had come to DARPA in 1999 to work on that programme and left in 2003 after managing it.
Engineering analysis, this article’s own, checkable from the equations given. The amplification identity. The exact cancellation of altitude from the bleed-fed momentum coefficient. The identical nozzle pressure ratio across the envelope. The Mach-number factor, its exact minimum at Mach 1.4142, and the closed form of that minimum. The authority ratio, the crossover Mach number and the amplification required to match a conventional surface. The concentration factor and the effector sizing. The slot areas. The reduced-frequency band mapped onto this aircraft’s chord. The linear and second-order term counts and the flights they imply. All of it is conditional on parameter values that are not published, and every such parameter is swept.
Very likely a rediscovery. The altitude result. A 2024 paper asks whether momentum-coefficient models hold as ambient and supply conditions change in flight, simulates two actuator types at sea level and ten kilometres, and reports that the momentum coefficient collapses under a scaling parameter. Which parameter it is cannot be read from the abstract. No claim of novelty is made.
Engineering inference, narrowed by a primary source. The low-Reynolds experiments whose thresholds this article borrows lie one to nearly three orders of magnitude below this aircraft’s Reynolds number. A report read in full tested the method at chord Reynolds numbers as high as forty million and states that the Reynolds number has a very weak effect on the pressure distributions and spectra of a fully turbulent baseline [Seifert and Pack 2000a]. The X-65A flies inside the range of chord Reynolds numbers that report covers, which its text gives as two to forty million, and inside the range the cited experiments together demonstrate, not beyond it, so the claim made here is the narrow one, that transferring a threshold differs from transferring a phenomenon.
From a secondary source, set against this article’s own arithmetic. The published review reports the separation-control to super-circulation threshold at three to five percent and a measured lift response at 0.08 percent, which the bleed budget reaches wing-referenced. Reaching the super-circulation threshold would take more core flow than the engine has.
Inference, marked as such. That the Phase 0 to Phase 1 downselect occurred in mid-2021 is inferred from two option exercises in June and August 2021 and the absence of any option exercise on the third contract. No source consulted states a date. That Phase 3 is worth about fifty-two million dollars follows from subtracting the reported Phase 2 figure from the obligated total on one contract, and the press figure and the award record are not the same kind of quantity. That the fiscal year 2026 request of 4.000 million dollars is a placeholder is inferred from its being the only round number among seven figures carrying three decimals.
Assumptions this article makes and could be wrong about. That the air comes from engine bleed rather than a dedicated compressor, which is undetermined in the record and which decides whether the altitude result applies at all. That the engine operates at constant corrected flow across the altitude band. That the jet is choked, which the computed nozzle pressure ratio supports with an order of magnitude of margin under the bleed architecture. That the reference area is sixteen square metres, the middle of a sweep. That the effectors blow rather than suck, where the official definition admits both. That the amplification at cruise lies between ten and a hundred, where the low-speed separation-control experiments the review summarises imply a hundred or more.
What the record does not settle. Where the pressurised air comes from. What the wing area, plenum pressure, bleed fraction or effector geometry are. What the aircraft weighs, to within 200 pound. Whether the planform is best described as joined tandem, diamond or triangular, three sources giving three answers. What the bleed costs in thrust. Whether the restructured co-investment is recorded anywhere public. And whether any of this works, the question the aircraft is for.
Written from current knowledge, dated 10 December 2025. The bibliography carries publications later than that dateline, including a substantial 2026 literature on sweeping jets, fluidic oscillators and rudderless flight control. No claim in the prose depends on anything after the dateline, and the survey’s own year range is the one place a later year appears in the body, where it describes the bibliography rather than the aircraft.
Out of Scope
The rotorcraft literature, deliberately. Circulation control on a stopped rotor was flown, and the X-Wing programme is the obvious ancestor of a vehicle controlled by blowing rather than by surfaces. The survey excludes rotorcraft because hingeless on the reports server returns 201 records, 189 of them with rotor or rotary-wing vocabulary in the title, and the exclusion takes the X-Wing with it. The loss is a cost of a necessary guard, and it is stated openly.
Shock and boundary-layer interaction control. The survey excludes it deliberately. The subject is supersonic inlets and hypersonic ramps, and the X-65A is limited to Mach 0.7. Excluding it is a judgement and the judgement is recorded.
Thrust vectoring, except the fluidic kind. Deflecting a propulsion jet is not controlling the flow over a lifting surface, so thrust vectoring is excluded and fluidic thrust vectoring is kept.
The bleed penalty in thrust, quantitatively. Deferred because it needs an engine deck.
Suction-based flow control. Deferred because its figure of merit is a flow coefficient and the identity this analysis uses does not hold for it.
Reynolds-number scaling of the amplification. Deferred because the aircraft is built to answer that question and no analysis can substitute for it.
The structural and aeroelastic consequences of a joined tandem wing. Deferred, and the literature is cited.
Noise. Blowing is loud and the circulation-control literature carries a substantial acoustic component. Out of scope here.
Conclusion
The X-65A exists to find out whether a jet of air can do a hinge’s job, and the arithmetic of the air makes the answer depend almost entirely on Mach number.
The momentum a bleed-fed effector can deliver is exactly independent of altitude, because corrected mass flow and compressible dynamic pressure both scale with ambient pressure, and because a choked jet fed from compressor air gains in velocity exactly what corrected flow loses in the square root of temperature. Two cancellations meet and nothing about the atmosphere survives. What survives is the inlet’s total-pressure recovery divided by the square of the Mach number, a function with an exact minimum at Mach $\sqrt{2}$ that this aircraft never reaches.
So the authority ratio against a conventional surface falls as the aircraft accelerates and is flat as it climbs. At an amplification of 30 against a surface worth 0.05 in lift coefficient, flow control wins below Mach 0.485 and loses above, delivering 5.15 times the hinge’s authority at Mach 0.2 and 56.7 percent of it at Mach 0.7. For a conventional aeroplane the hard case for control power is slow flight. For this one it is fast flight, and a flight-test programme that expands its envelope upward happens to walk from the easy end to the hard one.
The wing-referenced momentum coefficient sits inside the range that oscillatory separation control was demonstrated with at this aircraft’s Reynolds number, 0.03 to 0.32 percent at sixteen million, and at that experiment’s Mach number of 0.25 the same budget delivers 1.74 times its highest quoted value. So the air is sufficient where the method is proved and marginal only where this aircraft wants to fly, and the factor between those two conditions is the Mach dependence and nothing else. The super-circulation regime would still take more core flow than the engine has, the reason this is a control programme and not a high-lift one. The concentration factor is what makes a moment affordable where a lift is not.
And the open problem was named in 2000. The report that demonstrated the method at these Reynolds numbers proposed, among its future directions, overcoming the lack of sufficient control authority especially at high speeds [Seifert and Pack 2000a]. This article derived that constraint and did not discover it.
The aircraft’s most consequential design decision is not aerodynamic. It carries two complete, equinumerous control systems and locks one out progressively. The previous article established that two vehicles cannot be differenced because the difference sought is smaller than the differences not sought. The X-65A performs the subtraction inside one airframe, and can therefore return a number rather than an impression. The 16,384 configurations that follow from fourteen surfaces are a red herring, because the quantity to identify is a model with 315 terms, which about eleven flights can identify.
None of it has flown. The programme has asked for 200.507 million dollars across seven budget years and expects 179.416 million, has been funded below its request in every year since fiscal year 2021 at a mean shortfall of 16.29 percent, promised its critical design review in four consecutive books for three different fiscal years, and was paused and restructured into an arrangement the federal award record cannot represent. Its period of performance expired 69 days before this article’s dateline with the fuselage unfinished.
The last thing to say is about the programme’s name. It is called CRANE, for Control of Revolutionary Aircraft with Novel Effectors, and its acronym returns a logging-crane fatality, a book of rhymes and an illustrated book by Walter Crane. The phrase novel effectors returns fungal pathogens. The words that find this aircraft’s actual prior art are hingeless control effectors, which no document this article read from the programme uses, and even those mean helicopter rotor hubs if you ask the wrong registry. An aircraft meant to break a century-old paradigm was given a name under which the work it descends from cannot be found.
Equations
This section collects the 58 display equations of the article in the order they appear, each under its number and reproduced exactly as it stands in the text. A title in bold names an established result, and a title in italics beginning with the word Bespoke names a construction made for this analysis. Of the 58, 33 are established and 25 are bespoke. The aerodynamic and engine relations are evaluated in the International System, with thirty thousand feet taken as a geopotential altitude of 9,144 metres and the aircraft weight quoted in pounds, the wing area of 16 square metres is a reference value the article takes because none is published, and a few letters need care, so $\delta$ is a compressor-face pressure ratio, $\mu$ is the dynamic viscosity while the subscript in $C_\mu$ marks the momentum coefficient, $B$ is the bypass ratio while the bold $\mathbf{B}$ is the control effectiveness matrix, and $\mathcal{A}$ is the amplification while $A^{*}$ and $A$ are areas.
Equation 1, Spread of the Published Weights. The contractor’s gross weight of 7,200 pounds exceeds the 7,000 pounds given elsewhere by 2.86 percent of the lower figure. X-65 experimental aircraft Aurora X-65 CRANE
\[\frac{7{,}200 - 7{,}000}{7{,}000} = 0.0286 , \tag{1}\]Equation 2, Unexercised Share of the Georgia Tech Ceiling. One less the ratio of obligation to ceiling shows that 45.1 percent of the Georgia Tech Research Corporation ceiling was never exercised. USAspending
\[1 - \frac{6{,}795{,}148}{12{,}376{,}510} = \frac{5{,}581{,}362}{12{,}376{,}510} = 0.451 . \tag{2}\]Equation 3, Aurora’s Share of the Early Contracts and the Total Obligation. Aurora took 48.4 percent of the obligations across the three early contracts and left 16.1 percent of its ceiling unexercised, and the four contracts total 143,556,412 dollars. USAspending
\[\frac{23{,}811{,}305}{49{,}184{,}174} = 0.484 , \qquad 1 - \frac{23{,}811{,}305}{28{,}368{,}394} = \frac{4{,}557{,}089}{28{,}368{,}394} = 0.161 , \qquad 49{,}184{,}174 + 94{,}372{,}238 = 143{,}556{,}412 , \tag{3}\]Equation 4, Phase 3 Obligation Against the Announced Value. The obligated total on the combined Phase 2 and Phase 3 contract is 2.25 times the announced Phase 2 value, and the first obligation was 6.89 percent of it. USAspending
\[\frac{94{,}372{,}238}{42{,}000{,}000} = 2.25 , \qquad \frac{6{,}500{,}000}{94{,}372{,}238} = 0.0689 . \tag{4}\]Equation 5, Jet Momentum Flux and Momentum Coefficient. The momentum coefficient is the jet momentum flux divided by the free-stream dynamic pressure times a reference area. Abdolahipour 2024
\[J = \dot{m}_j V_j, \qquad C_\mu = \frac{J}{q S} \tag{5}\]Equation 6, Aerodynamic Force Increment. The force increment is the lift-coefficient increment times the dynamic pressure and the reference area.
\[\Delta L = \Delta C_L \, q S \tag{6}\]Equation 7, Amplification as Force per Unit Jet Momentum. The amplification, the force produced per unit of jet momentum supplied, is the lift-coefficient increment over the momentum coefficient, with the flight condition cancelling.
\[\mathcal{A} = \frac{\Delta L}{\dot{m}_j V_j} = \frac{\Delta C_L \, q S}{C_\mu \, q S} = \frac{\Delta C_L}{C_\mu} \tag{7}\]Equation 8, Secant Amplification Against the Actuation Efficiency. The amplification is the secant of the lift response curve while the actuation efficiency reported in the literature is its tangent. Abdolahipour 2024
\[\mathcal{A} = \frac{\Delta C_L}{C_\mu} \quad \text{against} \quad \mathcal{A}_{\mathrm{loc}} = \frac{\mathrm{d} \, \Delta C_L}{\mathrm{d} C_\mu} \tag{8}\]Equation 9, Three Regimes of the Lift Response to Blowing. The actuation efficiency is near zero below a threshold momentum coefficient, large in the separation-control band and tends to one once the flow is fully attached. Abdolahipour 2024
\[\mathcal{A}_{\mathrm{loc}} \approx 0 \;\; \text{for} \;\; C_\mu < C_{\mu,0}, \qquad \mathcal{A}_{\mathrm{loc}} \gg 1 \;\; \text{for} \;\; C_{\mu,0} < C_\mu < C_{\mu,1}, \qquad \mathcal{A}_{\mathrm{loc}} \to 1 \;\; \text{for} \;\; C_\mu > C_{\mu,1} \tag{9}\]Equation 10, Bespoke Bleed Mass Flow at Fixed Corrected Spool Speed. Holding the engine at a fixed corrected spool speed, as the article assumes, makes the bled mass flow the bleed fraction times the reference flow times the corrected-flow factor of the compressor face.
\[\dot{m}_j = \beta \, \dot{m}_{\mathrm{ref}} \, \frac{\delta}{\sqrt{\theta}} \tag{10}\]Equation 11, Mass Flow of a Compressor at Fixed Shaft Power. A compressor driven at a fixed shaft power passes a mass flow equal to that power over the specific heat times the total-temperature rise.
\[\dot{m}_j = \frac{P}{c_p \, \Delta T} \tag{11}\]Equation 12, Troposphere Temperature and Pressure. The temperature falls linearly with geopotential altitude and the pressure follows by integrating the hydrostatic relation against the ideal gas law.
\[T\left( h \right) = T_{\mathrm{sl}} + \lambda h, \qquad p\left( h \right) = p_{\mathrm{sl}} \left( \frac{T\left( h \right)}{T_{\mathrm{sl}}} \right)^{- \frac{g_0}{\lambda R}} \tag{12}\]Equation 13, Stratosphere Pressure, Density and Speed of Sound. Above the tropopause the pressure falls exponentially at constant temperature, the density follows from the ideal gas law and the speed of sound from the temperature.
\[p\left( h \right) = p_{\mathrm{trop}} \exp\left( - \frac{g_0 \left( h - h_{\mathrm{trop}} \right)}{R \, T_{\mathrm{trop}}} \right), \qquad \rho = \frac{p}{R T}, \qquad a = \sqrt{\gamma R T} \tag{13}\]Equation 14, Dynamic Pressure in Pressure and Mach Number. The dynamic pressure is exactly half the ratio of specific heats times the static pressure times the square of the Mach number.
\[q = \tfrac{1}{2} \rho V^2 = \tfrac{1}{2} \rho \, \gamma R T M^2 = \tfrac{\gamma}{2} \, p \, M^2 \tag{14}\]Equation 15, Total Conditions at the Compressor Face. With full inlet recovery the compressor-face total pressure and temperature follow from the flight Mach number, and are normalised on sea-level standard values.
\[p_{t} = p \, \sigma^{\frac{\gamma}{\gamma-1}}, \qquad T_{t} = T \sigma, \qquad \delta = \frac{p_t}{p_{\mathrm{sl}}}, \qquad \theta = \frac{T_t}{T_{\mathrm{sl}}} \tag{15}\]Equation 16, Corrected Mass Flow Invariant. At a fixed corrected spool speed a turbomachine passes a fixed corrected mass flow.
\[\dot{m} \frac{\sqrt{\theta}}{\delta} = \text{constant} \tag{16}\]Equation 17, Compressor Temperature Ratio and Specific Work. The total-temperature ratio of a compressor follows from its pressure ratio and isentropic efficiency, and the specific work from that ratio.
\[\tau = 1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}} - 1}{\eta_c}, \qquad w_c = c_p \, T_{t} \left( \tau - 1 \right) \tag{17}\]Equation 18, Choking Condition and Choked Jet Velocity. A nozzle above the critical pressure ratio passes sonic flow at its throat, with a jet velocity set by the plenum total temperature alone.
\[\frac{p_{t}}{p} \ge \left( \frac{\gamma + 1}{2} \right)^{\frac{\gamma}{\gamma - 1}} \approx 1.8929, \qquad V_j = \sqrt{\frac{2 \gamma R \, T_{t}}{\gamma + 1}} \tag{18}\]Equation 19, Bespoke Plenum Total Temperature of Bled Air. Taking the plenum at the compressor discharge temperature with no loss on the way, as the article does, gives the plenum total temperature from the flight Mach number and the compressor pressure ratio and efficiency.
\[\sigma = 1 + \frac{\gamma - 1}{2} M^2, \qquad T_{t} = T \sigma \left( 1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}} - 1}{\eta_c} \right) \tag{19}\]Equation 20, Bespoke Nozzle Pressure Ratio of the Bleed System. Taking the plenum at the compressor discharge pressure with no duct loss gives a nozzle pressure ratio that depends on the compressor pressure ratio and the Mach number alone.
\[\mathrm{NPR} = \frac{\pi_c \, p_t}{p} = \pi_c \, \sigma^{\frac{\gamma}{\gamma-1}} \tag{20}\]Equation 21, Bespoke Momentum Coefficient of an Engine-Bled Effector System. Assembling the article’s bleed flow at fixed corrected speed, the choked jet velocity and the dynamic pressure gives a momentum coefficient from which altitude and ambient temperature cancel.
\[C_\mu = \frac{2 \beta \, \dot{m}_{\mathrm{ref}}}{\gamma \, p_{\mathrm{sl}} \, S} \sqrt{\frac{2 \gamma R \, T_{\mathrm{sl}} \, \tau}{\gamma + 1}} \; \cdot \; \frac{\sigma^{\frac{\gamma}{\gamma-1}}}{M^2} \tag{21}\]Equation 22, Bespoke Mach-Number Factor of the Bleed-Fed Momentum Coefficient. The Mach dependence surviving in the article’s bleed-fed momentum coefficient is the inlet total-pressure factor over the square of the Mach number.
\[G\left( M \right) = \frac{\sigma^{\frac{\gamma}{\gamma-1}}}{M^2} = \frac{\left( 1 + 0.2 M^2 \right)^{3.5}}{M^2} \tag{22}\]Equation 23, Bespoke Turning Point of the Mach Factor. Differentiating the logarithm of the article’s Mach factor shows that it has its minimum at Mach $\sqrt{2}$ whatever the ratio of specific heats.
\[\frac{\mathrm{d} \ln G}{\mathrm{d} M} = \frac{\gamma}{\gamma - 1} \cdot \frac{\sigma'}{\sigma} - \frac{2}{M} = \frac{\gamma M}{\sigma} - \frac{2}{M} \tag{23}\]Equation 24, Bespoke Minimum Value of the Mach Factor. At Mach $\sqrt{2}$ the inlet factor equals the ratio of specific heats, which gives the minimum of the article’s Mach factor in closed form.
\[G_{\min} = \frac{\gamma^{\frac{\gamma}{\gamma-1}}}{2} = 1.6234 \tag{24}\]Equation 25, Bespoke Authority Ratio of Flow Control to a Conventional Surface. The article defines the authority ratio as the force of a flow-control effector over that of a conventional surface at full deflection, from which the dynamic pressure cancels.
\[\Lambda = \frac{\Delta L_{\mathrm{AFC}}}{\Delta L_{\mathrm{conv}}} = \frac{\mathcal{A} \, C_\mu \, q S}{\Delta C_{L,\mathrm{conv}} \, q S} = \frac{\mathcal{A} \, C_\mu}{\Delta C_{L,\mathrm{conv}}} \tag{25}\]Equation 26, Bespoke Crossover Mach Number. The article’s crossover is the Mach number at which its authority ratio reaches one, found from the Mach factor relative to a reference condition.
\[\Lambda\left( M_{\times} \right) = 1 \quad \Longleftrightarrow \quad \mathcal{A} \, C_\mu\left( M_{\times} \right) = \Delta C_{L,\mathrm{conv}} \quad \Longleftrightarrow \quad G\left( M_{\times} \right) = \frac{\Delta C_{L,\mathrm{conv}}}{\mathcal{A}} \cdot \frac{G\left( M_{0} \right)}{C_\mu\left( M_{0} \right)} \tag{26}\]Equation 27, Bespoke Authority Ratio at an Assumed Amplification of Thirty. Taking an amplification of 30 against a conventional surface worth 0.05 in lift coefficient, both values the article assumes, gives 56.7 percent of the hinge’s authority at Mach 0.7 and a crossover at Mach 0.485.
\[\Lambda\left( 0.7 \right) = \frac{30 \times 0.000945066}{0.05} = 0.567 , \qquad \Lambda\left( 0.2 \right) = 0.567 \times \frac{G\left( 0.2 \right)}{G\left( 0.7 \right)} = 0.567 \times \frac{25.7070}{2.8308} = 5.15 , \qquad G\left( M_{\times} \right) = \frac{0.05}{30} \cdot \frac{2.8308}{0.000945066} = 4.992 , \tag{27}\]Equation 28, Bespoke Amplification Required To Match a Conventional Surface. Inverting the authority ratio gives the amplification needed for the article’s momentum coefficient to match a conventional surface.
\[\mathcal{A}_{\mathrm{req}} = \frac{\Delta C_{L,\mathrm{conv}}}{C_\mu} \tag{28}\]Equation 29, Bespoke Local Momentum Coefficient and Concentration Factor. The article references the momentum coefficient to the area one effector influences and defines the concentration factor as the wing area over that area.
\[C_{\mu,\ell} = \frac{\dot{m}_j V_j}{q S_\ell}, \qquad \kappa = \frac{S}{S_\ell} = \frac{C_{\mu,\ell}}{C_\mu} \tag{29}\]Equation 30, Bespoke Influenced Area for a Target Local Coefficient. Inverting the article’s local momentum coefficient gives the area over which a given jet momentum lands a target value.
\[S_\ell = \frac{\dot{m}_j V_j}{C_{\mu,\ell} \, q} \tag{30}\]Equation 31, Bespoke Bleed Fraction Required for a Target Momentum Coefficient. Because the article’s momentum coefficient is proportional to the bleed fraction, the fraction needed for a target coefficient is that target over the coefficient a bleed of unity would give, which cannot exceed one.
\[C_\mu = \beta \, C_\mu^{\,1} \quad \Longrightarrow \quad \beta_{\mathrm{req}} = \frac{C_\mu^{\,\mathrm{target}}}{C_\mu^{\,1}}, \qquad \beta_{\mathrm{req}} \le 1 \tag{31}\]Equation 32, Bespoke Bleed Fractions for the Measured Thresholds. Scaling from the reference case at an assumed bleed fraction of 8 percent, the lowest measured threshold needs 6.77 percent of core flow and the super-circulation threshold of three percent needs 254 percent.
\[C_\mu^{\,1} = \frac{0.000945066}{0.08} = 0.0118133 , \qquad \frac{0.0008}{0.0118133} = 0.0677 , \qquad \frac{0.0034}{0.0118133} = 0.2878 , \qquad \frac{0.03}{0.0118133} = 2.5395 , \qquad \frac{0.05}{0.0118133} = 4.2325 . \tag{32}\]Equation 33, Control Moment and Moment Coefficient. A force increment at a moment arm produces a moment, and dividing by dynamic pressure, wing area and span gives its coefficient.
\[\mathcal{M} = \Delta L \, y = \mathcal{A} \, J \, y, \qquad C_{\mathcal{M}} = \frac{\mathcal{M}}{q \, S \, b} \tag{33}\]Equation 34, Bespoke Rolling Moment at an Amplification of Thirty. The article’s air budget at an assumed amplification of 30, placed at a quarter of the span on the assumed reference area of 16 square metres, gives a rolling-moment coefficient of 0.00709.
\[\mathcal{A} J = 30 \times 156.06 = 4{,}682 , \qquad \mathcal{M} = 4{,}682 \times 2.286 = 10{,}703 , \qquad C_{\mathcal{M}} = \frac{10{,}703}{10{,}320.7 \times 16 \times 9.144} = 0.00709 . \tag{34}\]Equation 35, Jet Velocity Ratio. The jet velocity ratio is the jet velocity over the free-stream velocity, which must exceed one for blowing to energise the boundary layer. Abdolahipour 2024
\[\mathrm{VR} = \frac{V_j}{V_\infty} \tag{35}\]Equation 36, Mass Flow Through a Choked Orifice. A choked orifice passes a mass flow proportional to its throat area and plenum total pressure and inversely proportional to the square root of the plenum total temperature.
\[\dot{m}_j = A^{*} \, \frac{p_t}{\sqrt{T_t}} \sqrt{\frac{\gamma}{R}} \left( \frac{\gamma+1}{2} \right)^{-\frac{\gamma+1}{2\left(\gamma-1\right)}} \tag{36}\]Equation 37, Throat Area for a Given Choked Mass Flow. Inverting the choked flow relation gives a throat area inversely proportional to the plenum pressure at a fixed mass flow and temperature.
\[A^{*} = \frac{\dot{m}_j \sqrt{T_t}}{p_t} \left[ \sqrt{\frac{\gamma}{R}} \left( \frac{\gamma+1}{2} \right)^{-\frac{\gamma+1}{2\left(\gamma-1\right)}} \right]^{-1} \; \propto \; \frac{1}{p_t} \tag{37}\]Equation 38, Bespoke Throat Area Ratios Across Plenum Schedules. Against the nozzle pressure ratio of 19.419 that the article’s assumed compressor gives, regulating the plenum to ratios of 1.5 and 4 enlarges the throat by 12.95 and 4.85 times.
\[\frac{A^{*}\left( 1.5 \right)}{A^{*}\left( 19.419 \right)} = \frac{19.419}{1.5} = 12.95 , \qquad \frac{A^{*}\left( 4 \right)}{A^{*}\left( 19.419 \right)} = \frac{19.419}{4} = 4.85 , \tag{38}\]Equation 39, Compressible Mass Flux Through the Engine Face. The mass flow through a circular face is the face area times the density, Mach number and speed of sound reached by expanding sea-level total conditions to the face Mach number.
\[\frac{\dot{m}}{A} = \rho_f M_f a_f , \qquad \dot{m} = \frac{\pi D_f^2}{4} \cdot \frac{p_{\mathrm{sl}}}{\sqrt{T_{\mathrm{sl}}}} \sqrt{\frac{\gamma}{R}} \, M_f \, \sigma_f^{- \frac{\gamma+1}{2\left(\gamma-1\right)}} \tag{39}\]Equation 40, Static Thrust as Exit Momentum Flux. At static conditions there is no ram drag, so the thrust is the mass flow times the exit velocity and inverts exactly for the mass flow.
\[F_{\mathrm{sls}} = \dot{m} V_e, \qquad \dot{m} = \frac{F_{\mathrm{sls}}}{V_e} \tag{40}\]Equation 41, Core Flow and Bleed Share From the Bypass Ratio. The core flow is the total flow over one plus the bypass ratio, so a bleed fraction of core flow is that fraction over one plus the bypass ratio of total flow.
\[\dot{m}_{\mathrm{core}} = \frac{\dot{m}_{\mathrm{total}}}{1 + B}, \qquad \frac{\dot{m}_j}{\dot{m}_{\mathrm{total}}} = \frac{\beta}{1 + B} \tag{41}\]Equation 42, Bespoke Lower Bound on the Thrust Cost of Bleed. At an assumed bleed fraction of 8 percent and an assumed bypass ratio of 3.3, the momentum lost with the bled air bounds the thrust decrement below at 1.86 percent.
\[\frac{\Delta F}{F} \ge \frac{\beta}{1 + B} = 1.860 \text{ percent} \tag{42}\]Equation 43, Reduced Excitation Frequency. The reduced frequency is the excitation frequency times the chord over the free-stream velocity, which inverts for the frequency. Abdolahipour 2024
\[F^{+} = \frac{f_e \, \bar{c}}{V_\infty}, \qquad f_e = \frac{F^{+} V_\infty}{\bar{c}} \tag{43}\]Equation 44, Mean Chord, Aspect Ratio, Wing Loading and Lift Coefficient. The mean chord and aspect ratio follow from the span and wing area, and the level flight lift coefficient is the wing loading over the dynamic pressure.
\[\bar{c} = \frac{S}{b}, \qquad AR = \frac{b^2}{S} = \frac{b}{\bar{c}}, \qquad \frac{W}{S}, \qquad C_L = \frac{W}{q S} = \frac{W/S}{q} \tag{44}\]Equation 45, Bespoke Configuration Count and Flights Required. The article counts each conventional surface as locked or free on each wing set, giving a power of two, and the flights needed as the second-order term count over the usable test points per flight, rounded up.
\[N_c = 2^{n_s} n_w, \qquad N_f = \left\lceil \frac{N_2}{k} \right\rceil \tag{45}\]Equation 46, Bespoke Derivative Count of a Linear Effector Model. In the linear model the article considers, effectors superpose and contribute one control derivative per effector per axis.
\[N_1 = n \, n_a \tag{46}\]Equation 47, Bespoke Term Count of a Second-Order Effector Model. Adding one interaction term for each pair of effectors gives the article’s second-order model 315 terms for fourteen effectors on three axes.
\[N_2 = n_a \left( n + \binom{n}{2} \right) = n_a \left( n + \frac{n \left( n - 1 \right)}{2} \right) \tag{47}\]Equation 48, Control Allocation Problem. The demanded moment is the control effectiveness matrix times a vector of effector demands drawn from an admissible set.
\[\mathbf{m} = \mathbf{B} \mathbf{u}, \qquad \mathbf{u} \in \mathcal{U} \tag{48}\]Equation 49, Full Row Rank Condition. All three axes can be commanded only if the control effectiveness matrix has full row rank.
\[\operatorname{rank} \mathbf{B} = 3 \tag{49}\]Equation 50, Bespoke Box and Shared-Plenum Admissible Sets. The article models conventional surfaces by individual limits, a box, and the flow-control effectors by one shared air supply, a simplex.
\[\mathcal{U}_{\mathrm{box}} = \left\{ \mathbf{u} : 0 \le u_i \le \bar{u} \right\}, \qquad \mathcal{U}_{\mathrm{plenum}} = \left\{ \mathbf{u} : u_i \ge 0, \; \sum_i u_i \le \bar{u} \right\} \tag{50}\]Equation 51, Areas of a Planar Zonotope and Convex Hull. The area of a planar zonotope is the sum of the absolute determinants of all pairs of generators, and that of a convex polygon fanned from the origin is half the sum over adjacent pairs.
\[\operatorname{area} \mathcal{Z} = \sum_{i < j} \left\lvert \det \left( \mathbf{b}_i , \mathbf{b}_j \right) \right\rvert, \qquad \operatorname{area} \mathcal{H} = \tfrac{1}{2} \sum_i \left\lvert \det \left( \mathbf{b}_i , \mathbf{b}_{i+1} \right) \right\rvert \tag{51}\]Equation 52, Bespoke Limiting Advantage of a Shared Plenum. For the article’s tractable case of unit columns spread evenly over a half plane, the ratio of shared-supply to equal-share moment area tends to $\pi^2 / 2$.
\[\lim_{n_e \to \infty} \frac{\operatorname{area} \mathcal{H}}{\operatorname{area} \mathcal{Z} / n_e^2} = \frac{\pi / 2}{1 / \pi} = \frac{\pi^2}{2} = 4.9348 \tag{52}\]Equation 53, Outcome Against Request. The outcome of each fiscal year is expressed as its difference from the request as a fraction of the request. PB2020 PB2021
\[\frac{o - r}{r} \tag{53}\]Equation 54, First-Year Excess and Mean Later Shortfall. Fiscal year 2020 came in 81.33 percent above its request, and the five later years fell short by a mean of 16.29 percent. PB2020 PB2025
\[\frac{23.573 - 13.000}{13.000} = 0.8133 , \qquad \frac{0.0554 + 0.1930 + 0.2300 + 0.1401 + 0.1959}{5} = 0.1629 . \tag{54}\]Equation 55, Totals of Requests and Outcomes. The column sums give a shortfall of 10.73 percent over fiscal years 2020 to 2025, appropriations 1.0555 times the obligations through 2024, and a 2026 request of 10.45 percent of the 2024 outcome. PB2024 DARPA RDT&E justification, PB2026
\[\sum_{2020}^{2025} r = 196.507 , \qquad \sum_{2020}^{2025} o = 175.416 , \qquad 1 - \frac{175.416}{196.507} = 0.1073 , \qquad \sum_{2020}^{2024} o = 151.523 , \qquad \frac{151.523 \times 10^{6}}{143{,}556{,}412} = 1.0555 , \qquad \frac{4.000}{38.264} = 0.1045 . \tag{55}\]Equation 56, Reynolds Number and Sutherland’s Law. The chord Reynolds number is the ratio of inertial to viscous forces, with the viscosity of air from Sutherland’s law.
\[\mathrm{Re} = \frac{\rho V \bar{c}}{\mu}, \qquad \mu\left( T \right) = \frac{C_1 \, T^{3/2}}{T + S_\mu} \tag{56}\]Equation 57, Bespoke Computed Coefficient Against the Demonstrated Range. The wing-referenced coefficient of 0.0945 percent from the article’s assumed bleed budget is compared with the range of 0.03 to 0.32 percent the report used. Seifert and Pack 2000a
\[\frac{0.0945}{0.03} = 3.150 , \qquad \frac{0.32}{0.0945} = 3.39 . \tag{57}\]Equation 58, Bespoke Momentum Coefficient at Mach 0.25. Scaling the article’s reference coefficient by its Mach factor to the report’s Mach number gives 0.5579 percent, 1.74 times the highest quoted value. Seifert and Pack 2000a
\[C_\mu\left( 0.25 \right) = 0.000945066 \times \frac{G\left( 0.25 \right)}{G\left( 0.7 \right)} = 0.000945066 \times \frac{16.711}{2.8308} = 0.005579 , \qquad \frac{0.5579}{0.32} = 1.74 \tag{58}\]Notation
- $\dot{m}_j$, the jet mass flow delivered to the effectors
- $V_j$, the effector jet velocity at the orifice
- $J$, the jet momentum flux supplied by the effectors
- $q$, the free-stream dynamic pressure
- $S$, the wing reference area
- $C_\mu$, the momentum coefficient referenced to the wing
- $\Delta C_L$, the lift-coefficient increment produced by an effector
- $\Delta L$, the aerodynamic force increment produced by an effector
- $\mathcal{A}$, the amplification, aerodynamic force per unit jet momentum
- $\mathcal{A}_{\mathrm{loc}}$, the actuation efficiency, the slope of the lift increment against the momentum coefficient
- $C_{\mu,0}$, the threshold momentum coefficient below which nothing happens, with $C_{\mu,1}$ the one above which the flow is fully attached
- $\beta$, the bleed fraction of engine core mass flow
- $\dot{m}_{\mathrm{ref}}$, the engine sea-level static mass flow of the bleed source stream
- $\delta$, the ratio of compressor-face total pressure to sea-level standard pressure
- $\theta$, the ratio of compressor-face total temperature to sea-level standard temperature
- $P$, the shaft power delivered to the air by a dedicated compressor
- $c_p$, the specific heat at constant pressure
- $T_{\mathrm{sl}}$, the sea-level standard temperature, with $p_{\mathrm{sl}}$ the sea-level standard pressure
- $\lambda$, the troposphere lapse rate
- $g_0$, standard gravity
- $R$, the specific gas constant for dry air
- $\gamma$, the ratio of specific heats, taken as 1.4 for dry air
- $\sigma$, the inlet total-condition factor
- $p_t$, the plenum total pressure
- $T_t$, the plenum total temperature
- $\tau$, the compressor total-temperature ratio
- $\pi_c$, the compressor total-pressure ratio
- $\eta_c$, the compressor isentropic efficiency
- $w_c$, the compressor specific work
- $\mathrm{NPR}$, the nozzle pressure ratio, plenum total to ambient static
- $G$, the Mach-number factor of the bleed-fed momentum coefficient
- $\Lambda$, the authority ratio, flow-control force over conventional-surface force
- $\Delta C_{L,\mathrm{conv}}$, the lift-coefficient increment produced by a conventional surface at full deflection
- $M_{\times}$, the crossover Mach number at which the authority ratio reaches unity
- $C_{\mu,\ell}$, the momentum coefficient referenced to the area one effector influences
- $S_\ell$, the area of surface one effector influences
- $\kappa$, the concentration factor, wing area over influenced area
- $\mathcal{M}$, the control moment
- $C_{\mathcal{M}}$, the moment coefficient referenced to the span
- $\mathrm{VR}$, the jet velocity ratio
- $A^{*}$, the total choked throat area of the effectors
- $D_f$, the engine face diameter
- $M_f$, the axial Mach number at the engine face
- $F_{\mathrm{sls}}$, the sea-level static thrust
- $V_e$, the engine exit velocity at static conditions
- $B$, the engine bypass ratio
- $\Delta F$, the thrust decrement caused by bleed
- $F^{+}$, the reduced excitation frequency
- $f_e$, the effector excitation frequency
- $\bar{c}$, the reference chord, taken as area over span
- $AR$, the aspect ratio
- $N_c$, the distinct locked-or-free configurations across every wing set
- $N_f$, the flights needed to over-determine a model
- $N_2$, the terms in a second-order effector model with pairwise interactions
- $\mathbf{B}$, the control effectiveness matrix, three rows by one column per effector
- $\mathbf{u}$, the vector of effector demands
- $\bar{u}$, the supply limit on effector demand
- $\mathcal{H}$, the reachable moment set under a shared supply, a convex hull
- $\mathrm{Re}$, the Reynolds number on the reference chord
- $S_\mu$, Sutherland’s temperature for air
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- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-33.” Brendan A R Sechter’s Development Blog, November 8. https://sgeos.github.io/aerospace/history/engineering/2025/11/08/x_planes_lockheed_martin_x33.html
- Sechter, Brendan. 2025. “X-Planes: Orbital Sciences X-34.” Brendan A R Sechter’s Development Blog, November 9. https://sgeos.github.io/aerospace/history/engineering/2025/11/09/x_planes_orbital_sciences_x34.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-35.” Brendan A R Sechter’s Development Blog, November 10. https://sgeos.github.io/aerospace/history/engineering/2025/11/10/x_planes_lockheed_martin_x35.html
- Sechter, Brendan. 2025. “X-Planes: McDonnell Douglas X-36.” Brendan A R Sechter’s Development Blog, November 11. https://sgeos.github.io/aerospace/history/engineering/2025/11/11/x_planes_mcdonnell_douglas_x36.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-37.” Brendan A R Sechter’s Development Blog, November 12. https://sgeos.github.io/aerospace/history/engineering/2025/11/12/x_planes_boeing_x37.html
- Sechter, Brendan. 2025. “X-Planes: Scaled Composites X-38.” Brendan A R Sechter’s Development Blog, November 13. https://sgeos.github.io/aerospace/history/engineering/2025/11/13/x_planes_scaled_composites_x38.html
- Sechter, Brendan. 2025. “X-Planes: X-39, Reserved but Never Assigned.” Brendan A R Sechter’s Development Blog, November 14. https://sgeos.github.io/aerospace/history/engineering/2025/11/14/x_planes_x39_reserved_never_assigned.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-40.” Brendan A R Sechter’s Development Blog, November 15. https://sgeos.github.io/aerospace/history/engineering/2025/11/15/x_planes_boeing_x40.html
- Sechter, Brendan. 2025. “X-Planes: X-41 Common Aero Vehicle.” Brendan A R Sechter’s Development Blog, November 16. https://sgeos.github.io/aerospace/history/engineering/2025/11/16/x_planes_x41_common_aero_vehicle.html
- Sechter, Brendan. 2025. “X-Planes: Orbital Sciences X-42.” Brendan A R Sechter’s Development Blog, November 17. https://sgeos.github.io/aerospace/history/engineering/2025/11/17/x_planes_orbital_sciences_x42.html
- Sechter, Brendan. 2025. “X-Planes: Micro-Craft X-43 Hyper-X.” Brendan A R Sechter’s Development Blog, November 18. https://sgeos.github.io/aerospace/history/engineering/2025/11/18/x_planes_micro_craft_x43_hyper_x.html
- Sechter, Brendan. 2025. “X-Planes: X-44, One Designation and Two Aircraft.” Brendan A R Sechter’s Development Blog, November 19. https://sgeos.github.io/aerospace/history/engineering/2025/11/19/x_planes_x44_one_designation_two_aircraft.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-45.” Brendan A R Sechter’s Development Blog, November 20. https://sgeos.github.io/aerospace/history/engineering/2025/11/20/x_planes_boeing_x45.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-46.” Brendan A R Sechter’s Development Blog, November 21. https://sgeos.github.io/aerospace/history/engineering/2025/11/21/x_planes_boeing_x46.html
- Sechter, Brendan. 2025. “X-Planes: Northrop Grumman X-47.” Brendan A R Sechter’s Development Blog, November 22. https://sgeos.github.io/aerospace/history/engineering/2025/11/22/x_planes_northrop_grumman_x47.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-48.” Brendan A R Sechter’s Development Blog, November 23. https://sgeos.github.io/aerospace/history/engineering/2025/11/23/x_planes_boeing_x48.html
- Sechter, Brendan. 2025. “X-Planes: Piasecki X-49 SpeedHawk.” Brendan A R Sechter’s Development Blog, November 24. https://sgeos.github.io/aerospace/history/engineering/2025/11/24/x_planes_piasecki_x49.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-50 Dragonfly.” Brendan A R Sechter’s Development Blog, November 25. https://sgeos.github.io/aerospace/history/engineering/2025/11/25/x_planes_boeing_x50.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-51 Waverider.” Brendan A R Sechter’s Development Blog, November 26. https://sgeos.github.io/aerospace/history/engineering/2025/11/26/x_planes_boeing_x51.html
- Sechter, Brendan. 2025. “X-Planes: X-52, the Designation Refused.” Brendan A R Sechter’s Development Blog, November 27. https://sgeos.github.io/aerospace/history/engineering/2025/11/27/x_planes_x52_designation_refused.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-53 Active Aeroelastic Wing.” Brendan A R Sechter’s Development Blog, November 28. https://sgeos.github.io/aerospace/history/engineering/2025/11/28/x_planes_boeing_x53_active_aeroelastic_wing.html
- Sechter, Brendan. 2025. “X-Planes: Gulfstream X-54.” Brendan A R Sechter’s Development Blog, November 29. https://sgeos.github.io/aerospace/history/engineering/2025/11/29/x_planes_gulfstream_x54.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-55 ACCA.” Brendan A R Sechter’s Development Blog, November 30. https://sgeos.github.io/aerospace/history/engineering/2025/11/30/x_planes_lockheed_martin_x55_acca.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-56.” Brendan A R Sechter’s Development Blog, December 1. https://sgeos.github.io/aerospace/history/engineering/2025/12/01/x_planes_lockheed_martin_x56.html
- Sechter, Brendan. 2025. “X-Planes: ESAero X-57 Maxwell.” Brendan A R Sechter’s Development Blog, December 2. https://sgeos.github.io/aerospace/history/engineering/2025/12/02/x_planes_esaero_x57_maxwell.html
- Sechter, Brendan. 2025. “X-Planes: X-58, the Slot Taken by XQ-58.” Brendan A R Sechter’s Development Blog, December 3. https://sgeos.github.io/aerospace/history/engineering/2025/12/03/x_planes_x58_slot_taken_by_xq58.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-59 Quesst.” Brendan A R Sechter’s Development Blog, December 4. https://sgeos.github.io/aerospace/history/engineering/2025/12/04/x_planes_lockheed_martin_x59_quesst.html
- Sechter, Brendan. 2025. “X-Planes: Generation Orbit X-60.” Brendan A R Sechter’s Development Blog, December 5. https://sgeos.github.io/aerospace/history/engineering/2025/12/05/x_planes_generation_orbit_x60.html
- Sechter, Brendan. 2025. “X-Planes: Dynetics X-61 Gremlins.” Brendan A R Sechter’s Development Blog, December 6. https://sgeos.github.io/aerospace/history/engineering/2025/12/06/x_planes_dynetics_x61_gremlins.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-62 VISTA.” Brendan A R Sechter’s Development Blog, December 7. https://sgeos.github.io/aerospace/history/engineering/2025/12/07/x_planes_lockheed_martin_x62_vista.html
- Sechter, Brendan. 2025. “X-Planes: ABL Space Systems X-63.” Brendan A R Sechter’s Development Blog, December 8. https://sgeos.github.io/aerospace/history/engineering/2025/12/08/x_planes_abl_space_systems_x63.html
- Sechter, Brendan. 2025. “X-Planes: Invocon X-64.” Brendan A R Sechter’s Development Blog, December 9. https://sgeos.github.io/aerospace/history/engineering/2025/12/09/x_planes_invocon_x64.html
Referenced by
- X-Planes: Synthesis and What the Designation Became 2025-12-16
- X-Planes: Bell Textron X-76 SPRINT 2025-12-15
- X-Planes: X-69 through X-75, the Leapfrogged Block 2025-12-14
- X-Planes: General Atomics X-68 LongShot 2025-12-13
- X-Planes: X-67, the Slot Taken by XQ-67A 2025-12-12
- X-Planes: Boeing X-66 2025-12-11