X-Planes: ABL Space Systems X-63
Filed under: aerospace, history, engineering
This is the sixty-fourth 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, and the X-62.
On 20 April 2022 the register gave out two X numbers with the same description, and the description is identical to the byte.
The X-63A went to ABL Space Systems. The X-64A went to a team of Invocon, KT Engineering and Troy7. Both rows carry the same allocation date, the same sponsor, the same engines cell, and the same 101 characters of description [Parsch, DOD 4120.15-L Addendum].
Demonstrator rocket for AFRL’s ARISE (Aerospike Rocket Integration and Suborbital Experiment) program
Eleven descriptions repeat somewhere in the 539 rows of that register, in the copy the compiler last updated on 23 August 2026. Ten of them are munitions, targets and a ground station, being a laser-guided seventy millimetre rocket that appears four times, a ground control station for an unmanned air vehicle that appears three times, an evolved sea sparrow missile that also appears three times, and then seven that appear twice each, which are an air superiority target replacing a retired fighter, captive missile training, a thrust vector control interface to a vertical launching system, an air defence artillery weapon, a multi-stage supersonic target, a generic sub-scale aerial target and a long range standoff cruise missile prototype. The eleventh is this pair, the only duplicated description among the thirty-one rows of the X- series, which are not the only rows whose designation begins with the letter X [Parsch, DOD 4120.15-L Addendum].
Two X numbers were issued for one programme on one day, and the only thing the register uses to tell them apart is the name of the company building each one.
| Designation | Allocated | Contractor | Engines cell | Sponsor |
|---|---|---|---|---|
| X-60A | 07-Sep-18 | Generation Orbit Launch Services | empty |
USAF |
| X-61A | 06-Jun-19 | Dynetics | 1 F107-WR-105 |
DARPA |
| X-62A | 14-Jun-21 | Lockheed Martin | 1 Pratt & Whitney F100 |
USAF |
| X-63A | 20-Apr-22 | ABL Space Systems | 1 rocket engine |
USAF/USSF |
| X-64A | 20-Apr-22 | Invocon, KT Engineering, Troy7 | 1 rocket engine |
USAF/USSF |
| X-65A | 24-Apr-23 | Aurora Flight Sciences | 1 Williams FJ44-3A |
DARPA |
| X-66A | 03-May-23 | Boeing | 2 Pratt & Whitney |
NASA |
| X-68A | 20-Aug-25 | General Atomics | 1 Williams WJ38-15 |
DARPA |
This article is about the ABL vehicle and about the nozzle both vehicles exist to fly. The X-64A is a separate article and a separate machine, built by a different team to answer the measurement half of the same question. A programme that holds two designations does not divide itself neatly at the seam the register draws, so what follows takes the nozzle and the engine, and leaves the instrumentation and the recovery gear to the other number.
The Research Question
A rocket nozzle has one design altitude and flies through all of them.
The exhaust leaves a bell nozzle at whatever pressure the geometry dictates, and the atmosphere outside is at whatever pressure the altitude dictates, and those two numbers agree at exactly one point on the ascent. Everywhere below that point the jet is over-expanded and the atmosphere pushes back on the nozzle wall. Everywhere above it the jet is under-expanded and energy that could have been thrust leaves as pressure. The mismatch is no defect of manufacture. A fixed exit area entails it.
An aerospike inverts the geometry. Instead of expanding the gas inside a bell, it expands the gas along the outside of a central body, so the outer boundary of the plume is the atmosphere itself. The expansion ends where the ambient pressure ends it, and that pressure differs at every altitude. The idea is old, the hardware has been built several times, small aerospike engines have flown on research and university rockets, and no aerospike engine has flown a trajectory representative of a launch to orbit [ARISE and Fly].
So the questions are these. How much is altitude compensation worth, stated as a quantity rather than as an advantage? What does a fixed nozzle lose, and does that loss make the whole case? What does the fixed nozzle’s optimum look like, and can it be built? And what does it mean to assemble one engine out of many small ones, the other half of what this programme bought?
The answers turn out to be sharper than the subject’s reputation suggests. The loss of a fixed nozzle is an exact geometric object. The optimal fixed nozzle obeys a rule that can be written in one line and that mentions neither the gas nor the engine. And on the trajectory this vehicle’s manufacturer publishes, that rule places the optimum within about a tenth of the largest exit area the nozzle can carry without its flow separating, on one side of that limit or the other depending on a chamber pressure nobody has published.
Programme Origin
The programme’s origin is recorded in three places, and none of them is a contract that can be read. The register allocated both numbers on 20 April 2022 with one description, one sponsor cell and one engines cell [Parsch, DOD 4120.15-L Addendum]. The fact sheet of the Air Force Research Laboratory, AFRL, set out two objectives, an altitude-compensating nozzle and an engine built from many modules [ARISE and Fly]. And the award announcement records two three-year agreements made in December 2019 under an other transaction agreement, which a search of the federal award record does not find [AFRL awards agreements under ARISE]. The register is taken first and the programme second.
The Register and Its Unofficial Wording
The register is compiled by Andreas Parsch from Department of Defense data obtained under the Freedom of Information Act, and it is not itself a Department of Defense publication. The compiler notes that for allocations after October 2018 the official description is no longer releasable to the public, and prints the descriptions of those entries in blue to indicate that they are not official wording [Parsch, DOD 4120.15-L Addendum]. At that point the register stops being a primary source, as the X-61 article also finds. Both rows of this pair carry the marking.
The markup has three states and not two. A row may be wholly official. Its description cell may be marked in full. Or a fragment of the description may be marked while the rest is not. Across the whole register those three states divide 539 rows into 436 unmarked, 86 wholly unofficial and 17 partly unofficial, and across the thirty-one X rows they divide into 21 unmarked, 9 wholly unofficial and 1 partly unofficial [Parsch, DOD 4120.15-L Addendum]. Every count in this article is taken from every row of the copy that the compiler last updated on 23 August 2026, which holds 539 rows, five of them dated 2026 and none of those an X row. That copy is later than this article’s date. The version current on that date, the compiler’s update of 11 August 2025 as captured by the Internet Archive on 6 and 10 December 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 the three states divide the register into 432, 82 and 12 rows and the X rows into 21, 8 and none, eight rows are grey, 435 rows and twenty-five X rows carry an engines cell, twenty-five engines cells use the word rocket, and the Defense Advanced Research Projects Agency sponsors two X rows. Every finding about this pair holds in both versions, being the eleven duplicated descriptions, the two USAF/USSF rows, the seven rows naming the Space Force, the two rows reading 1 rocket engine and the 101-character description. The 436 unmarked rows include eleven that the compiler shows in grey, as designations in use but not confirmed as official allocations, so an unmarked description is not the same thing as an official allocation.
A two-number summary of that split loses a row, and the row it loses is the one whose markup is most easily misread. Counting it unofficial discards the official wording it carries, and counting it official contradicts the compiler’s own markup, which states that part of its wording is a reconstruction. The figures above classify each row by the markup it carries. No other accounting gives every row exactly one place.
| Classification | Whole register | Rows of the X- series |
|---|---|---|
| No part of the description marked as a reconstruction | 436 | 21 |
| Description wholly marked as a reconstruction | 86 | 9 |
| Description partly marked as a reconstruction | 17 | 1 |
| Total | 539 | 31 |
Where the Space Force Appears
The sponsor cell of both rows reads USAF/USSF [Parsch, DOD 4120.15-L Addendum].
Those are the only two rows in the entire register that read exactly that, and among the thirty-one X rows they are the only two that mention the Space Force at all. Seven rows in the whole register name the United States Space Force anywhere in the sponsor cell, and two of the seven are these [Parsch, DOD 4120.15-L Addendum].
The X sponsor column otherwise names the Air Force alone eight times, seven of them as USAF and once spelled out, the National Aeronautics and Space Administration, NASA, six times, the Navy four, the Defense Advanced Research Projects Agency four, the Department of Defense once, and a scattering of joint pairs [Parsch, DOD 4120.15-L Addendum]. A service that did not exist when most of this register was written appears in its experimental aircraft series twice, on the same day, for two halves of one rocket programme [United States Space Force].
An Engines Cell That Names a Class
Both rows say 1 rocket engine [Parsch, DOD 4120.15-L Addendum].
Four hundred and forty-five of the 539 rows carry an engines cell at all, and twenty-seven of the thirty-one X rows do. Those cells almost always name a model. The X-62A of the previous article reads 1 Pratt & Whitney F100. The X-65A reads 1 Williams FJ44-3A. The X-66A reads 2 Pratt & Whitney [Parsch, DOD 4120.15-L Addendum]. Exactly two rows in the register say 1 rocket engine, and they are this pair.
Twenty-six rows of the register use the word rocket in an engines cell somewhere, and only two of them are X rows. The third rocket among the X numbers is the X-60A, whose engines cell is empty [Parsch, DOD 4120.15-L Addendum]. That article found a register calling a rocket an aircraft while the Air Force called it a rocket twelve times. Here the register calls it a rocket in its description and then declines to name the engine, which is defensible, because at the moment of allocation the engine did not have a model name and arguably did not yet exist.
The Programme’s Stated Objectives
The Air Force Research Laboratory runs a portfolio called the Affordable Responsive Modular Rocket, or ARMR. ARISE is the most ambitious thing in it, and it has two objectives [ARISE and Fly].
The first objective is the nozzle. The fact sheet is explicit about what a bell does and why that is a problem.
Since the bell exhaust area is fixed, this optimizes the bell-nozzle for a single altitude. Inversely, an aerospike nozzle expands the exhaust gas on the exterior surface. Thus, the ambient atmosphere controls the exhaust expansion and acceleration.
The second objective is modularity, and the fact sheet insists it is separable from the first.
While modular engines do not require an arrangement via the aerospike nozzle, AFRL’s ARMR portfolio research highlights the inherent advantages of a modular-aerospike configuration for a production engine.
The propulsion system is described as multiple thrusters and turbomachinery units working in concert as a single engine, with multiple propellant pumps and multiple gas generators, manifolded so that the propellant divides evenly among them [ARISE and Fly]. The claim concerns how a large engine might be built out of small ones, and its subject is cost and schedule rather than performance.
The fact sheet also names the three flow regimes the flight is meant to visit, and those regimes are why a flight is needed at all.
Aerospike nozzles operate in three modes (open wake, wake transition, and closed wake) at specific altitudes depending on the engine design, flight trajectory, and ambient atmospheric conditions.
It also states why a test stand cannot substitute.
Realistic flight conditions have not validated in this behavior, since wind tunnel and ground testing cannot provide a truly representative flight environment.
That sentence is missing a word in the original and the meaning survives anyway. What a ground test cannot supply is a falling ambient pressure applied to a full-scale plume for the duration of a real ascent, and altitude compensation is by definition a response to a falling ambient pressure.
One tension in the document needs to be named. The programme is called a suborbital experiment, the fact sheet calls the flight a proof-of-principle test gathering data in the relevant environment of suborbital space, and the same page then says ARISE will fly a launch-to-orbit trajectory to reach all three regimes [ARISE and Fly]. This article reads the phrase as a trajectory representative of a launch to orbit rather than a flight that reaches orbit. That reading agrees with the programme’s name and with every other sentence on the page. It is an inference about wording, marked as one.
The clearance line at the foot of the fact sheet reads 88ABW-2020-0773. The document was cleared for public release in 2020, about two years before either designation was allocated, and it names its two flight-test vehicles as abl and Invocon in a figure caption. Neither X number appears anywhere in it [AFRL ARISE fact sheet].
The Instrument That Bought It
A search of the federal award record finds no trace of this programme under the names it is known by.
A search of the public award database for ABL Space Systems returns no result under procurement contracts, indefinite delivery vehicles, grants or the other financial assistance families. Searches for ARISE aerospike, for Aerospike Rocket Integration and Suborbital Experiment and for Affordable Responsive Modular Rocket return nothing at all [USAspending].
This absence needs no interpretation, because the announcement explains it.
The awards were made using the Space Enterprise Consortium Other Transaction Agreement, executed at Space and Missile Systems Center, Development Corp., at Los Angeles Air Force Base, Calif. The OTA acquisition allows for traditional as well as non-traditional contractors to submit proposals, leveraging unique capabilities from companies that may not be able to participate in a standard FAR based contract acquisition process.
An other transaction agreement is a procurement instrument other than a contract, to which many of the laws and regulations governing federal contracts do not apply, and it exists to reach companies, especially non-traditional ones, that would not otherwise compete for the work. An award made through a consortium is administered by the consortium’s manager under a base agreement [Other Transaction Authority], so a search under the name of the company doing the work need not find it. The previous article found a government that spent twenty-nine million dollars on an aeroplane across seventy-three transactions without once writing the aeroplane’s designation down. This is the same silence, with its instrument named in advance.
The announcement gives the terms it does give. Both awards were made in December 2019, each for three years, and each is for the design, test, manufacture, and launch of a sub-orbital flight of a rocket using a highly instrumented modular aerospike liquid rocket engine [AFRL awards agreements under ARISE]. No dollar figure is stated for either. Three years from December 2019 puts first flight at the end of 2022, and the designations were allocated on 20 April 2022, eight months before that deadline and with no vehicle yet flown.
The announcement also states, in a sentence that decides what this article has to compute, that thrust vector control for the aerospike will be demonstrated.
Three people are quoted. Jade Pomerleau, the programme manager, states that the investigation of a revolutionary, promising, yet unproven architecture is the heart of what AFRL does. Harry O’Hanley, the chief executive of ABL, argues that fast-paced engineering is necessary to maintain the superiority of the U.S industrial base. And Karl Kiefer, the president of Invocon, offers the sentence that sets this article’s historical scale [AFRL awards agreements under ARISE].
The Invocon Team feels that this a great opportunity to accomplish what has eluded the rocket community for nearly 60 years.
Nearly sixty years before December 2019 is about 1960, and the remark, whatever its rhetoric, makes a checkable claim about a literature. It is checked later in this article against the dates of the literature itself.
The Sponsor Spells Its Own Programme Two Ways
One small thing falls out of holding all the primary documents at once. The laboratory’s award announcement writes the programme’s name Sub-orbital, and its fact sheet and programme page write Suborbital.
| Source | Sub-orbital |
Suborbital |
|---|---|---|
| AFRL programme page | 0 | 2 |
| AFRL fact sheet, cleared 2020 | 0 | 2 |
| the award announcement as reprinted | 3 | 1 |
| the X-63A encyclopedia entry | 2 | 1 |
| the X-64A encyclopedia entry | 1 | 1 |
| the register | 0 | 1 |
The two web documents the laboratory publishes are internally consistent and use the unhyphenated form, and so does the register [ARISE and Fly] [AFRL ARISE fact sheet] [Parsch, DOD 4120.15-L Addendum]. The announcement uses the hyphenated form three times, including in its title, and the unhyphenated form once, in a phrase that is not the programme’s name [AFRL awards agreements under ARISE]. The expansion itself never varies. Everyone agrees what the letters mean and the sponsor cannot decide where the hyphen goes.
Sizing From First Principles
No dimension of the X-63A engine is public, so this article sizes the nozzle from the physics and the trajectory instead of from a drawing [ABL Space Systems X-63]. The chamber pressure and area ratio of the aerospike as designed are both unknown, and what follows works in thrust coefficients, which do not depend on how large the engine is. The derivation establishes that thrust is affine in ambient pressure, that the ideal altitude-compensating nozzle is the envelope of all fixed nozzles, that the best fixed nozzle expands to the burn-averaged ambient pressure of its flight, that on the manufacturer’s published ascent [ABL Payload User’s Guide] an ideal spike would deliver between 5.3 and 8.5 percent more first-stage impulse than that nozzle by the gain $\eta$ defined below, and that flow separation rather than performance bounds the fixed nozzle the trajectory wants. The last subsection names the wake regimes that separate the ideal spike from a truncated one.
The Claim, Stated Exactly
What follows depends on one elementary fact about a rocket nozzle, and this subsection writes it down before using it.
Let $F$ be the thrust in newton, $\dot{m}$ the mass flow of propellant in kilogram per second, $v_e$ the exhaust velocity at the exit plane in metre per second, $p_e$ the static pressure of the exhaust at that plane in pascal, $p_a$ the ambient atmospheric pressure in pascal, and $A_e$ the area of the exit plane in square metre. Then
\[F = \dot{m} v_e + \left( p_e - p_a \right) A_e\]which is the momentum flux out of the nozzle plus the net pressure force on its exit plane. Group the terms that do not mention the atmosphere and write $F_{\mathrm{vac}}$ for the thrust the same nozzle would produce against a perfect vacuum, in newton.
\[F_{\mathrm{vac}} = \dot{m} v_e + p_e A_e, \qquad F = F_{\mathrm{vac}} - p_a A_e\]Thrust is an affine function of ambient pressure, and the slope is minus the exit area. Nothing else in the nozzle appears in that slope. A fixed nozzle therefore traces one straight line as it climbs, and two numbers fix its behaviour in the atmosphere, being where the line starts and how steeply it falls.
The Same Statement Without the Engine in It
Coefficients are easier to work with than newtons, because they are independent of the engine’s size.
Let $p_c$ be the chamber pressure in pascal, $A_t$ the area of the throat in square metre, $\varepsilon$ the ratio of exit area to throat area, dimensionless, and $C_F$ the thrust coefficient, also dimensionless. The definitions are these.
\[C_F \equiv \frac{F}{p_c A_t}, \qquad \varepsilon \equiv \frac{A_e}{A_t}\]Dividing the affine law by $p_c A_t$ carries it across unchanged, and the slope becomes the area ratio.
\[C_F\left( \varepsilon, p_a \right) = C_{F,\mathrm{vac}}\left( \varepsilon \right) - \varepsilon \frac{p_a}{p_c}\]Two numbers still describe the nozzle and neither is a length. A fixed nozzle is a point on the vertical axis and a slope, and the rest of the argument concerns which straight line to choose.
Where the Vacuum Coefficient Comes From
The vacuum coefficient is not a free parameter. For a calorically perfect gas expanding isentropically it is fixed by the ratio of specific heats and the area ratio alone.
Let $\gamma$ be the ratio of specific heats, dimensionless, let $M$ be the local Mach number, dimensionless, and let $\Gamma$ be the Vandenkerckhove constant, dimensionless, which collects the throat conditions.
\[\Gamma\left( \gamma \right) = \sqrt{\gamma} \left( \frac{2}{\gamma + 1} \right)^{\frac{\gamma + 1}{2\left( \gamma - 1 \right)}}\]The area ratio and the pressure ratio at any station follow from the Mach number there.
\[\frac{A}{A_t} = \frac{1}{M}\left[ \frac{2}{\gamma + 1}\left( 1 + \frac{\gamma - 1}{2} M^2 \right) \right]^{\frac{\gamma + 1}{2\left( \gamma - 1 \right)}}, \qquad \frac{p}{p_c} = \left( 1 + \frac{\gamma - 1}{2} M^2 \right)^{-\frac{\gamma}{\gamma - 1}}\]The first of those is monotone on the supersonic branch, so an area ratio fixes a Mach number, a Mach number fixes an exit pressure ratio, and the vacuum coefficient follows.
\[C_{F,\mathrm{vac}}\left( \varepsilon \right) = \Gamma \sqrt{ \frac{2 \gamma}{\gamma - 1} \left[ 1 - \left( \frac{p_e}{p_c} \right)^{\frac{\gamma - 1}{\gamma}} \right] } + \varepsilon \frac{p_e}{p_c}\]Every number this article computes about nozzles comes out of those four relations and nothing else [Huzel and Huang 1967] [NASA SP-8120]. The first term is momentum and the second the exit plane pushing on a vacuum. Written the other way, with the constant absorbed, the momentum term is the form most texts print, and the two agree to thirteen decimal places in the cases checked.
The Symbols This Article Uses
Fifty-five relations need a table, and this is it. Two letters carry two meanings each, and the table records both, since $L$ is a thrust loss in the nozzle sections and a moment arm in the sections about steering, and $k$ indexes a module in one place and is a separation threshold in another.
| Symbol | Meaning | Unit |
|---|---|---|
| $ \alpha $ | equivalent gimbal angle | radian |
| $ \alpha_{1T} $ | first stationary point of the Bessel function of order one | dimensionless |
| $ \bar{M} $ | mean molar mass of the combustion products | kilogram per kilomole |
| $ \bar{p}_a $ | burn-averaged ambient pressure | pascal |
| $ \Delta m_p $ | payload mass bought by a gain in first-stage velocity increment | kilogram |
| $ \Delta v_1 $ | ideal velocity increment of the first stage | metre per second |
| $ \Delta v_2 $ | ideal velocity increment of the second stage | metre per second |
| $ \delta $ | available throttle depth | dimensionless |
| $ \dot{m} $ | mass flow of propellant | kilogram per second |
| $ \eta $ | fractional impulse gain of the ideal spike | dimensionless |
| $ \Gamma $ | Vandenkerckhove constant | dimensionless |
| $ \gamma $ | ratio of specific heats | dimensionless |
| $ \lambda $ | temperature lapse rate of an atmospheric layer | kelvin per metre |
| $ \mu $ | margin in area ratio of the trajectory-optimal nozzle below the separation limit | dimensionless |
| $ \phi $ | direction of the commanded moment | radian |
| $ \rho $ | fraction of guaranteed steering authority retained after one module fails | dimensionless |
| $ \sigma $ | contraction ratio, chamber area over throat area | dimensionless |
| $ \tau_k $ | throttle fraction of module k | dimensionless |
| $ \theta_k $ | angular position of module k on the ring | radian |
| $ \varepsilon $ | ratio of exit area to throat area | dimensionless |
| $ \varepsilon^{\star} $ | area ratio the mean-in-time rule selects | dimensionless |
| $ \varepsilon_{\mathrm{sep}} $ | largest area ratio that avoids separation at sea level | dimensionless |
| $ A $ | cross-sectional area at a station in the nozzle | square metre |
| $ A_b $ | area of the exposed base of a truncated plug | square metre |
| $ A_d $ | exit area of a fixed nozzle at its own design point | square metre |
| $ A_e $ | area of the nozzle exit plane | square metre |
| $ A_t $ | area of the nozzle throat | square metre |
| $ c $ | speed of sound in the combustion products | metre per second |
| $ c_{e,2} $ | effective exhaust velocity of the second stage | metre per second |
| $ c^{\star} $ | characteristic velocity | metre per second |
| $ C_F $ | thrust coefficient | dimensionless |
| $ C_F^{\star} $ | thrust coefficient of the ideal altitude-compensating nozzle | dimensionless |
| $ C_{F,\mathrm{vac}} $ | thrust coefficient in vacuum | dimensionless |
| $ D $ | chamber diameter | metre |
| $ F $ | thrust | newton |
| $ F^{\star} $ | upper envelope of the thrust of all fixed nozzles | newton |
| $ F_w $ | axial force on the wetted surface of a plug | newton |
| $ f_{1T} $ | frequency of the first tangential acoustic mode | hertz |
| $ F_{\mathrm{vac}} $ | thrust the same nozzle would produce against a vacuum | newton |
| $ G $ | vacuum thrust regarded as a function of exit area | newton |
| $ g_0 $ | standard gravity | metre per second squared |
| $ H $ | pressure scale height of the atmosphere | metre |
| $ h $ | geometric altitude | metre |
| $ h_0 $ | altitude at the base of an atmospheric layer | metre |
| $ h_b $ | altitude at main engine cut-off | metre |
| $ I $ | delivered impulse in units of chamber pressure times throat area | second |
| $ I_{sp} $ | specific impulse | second |
| $ J $ | time integral of ambient pressure over the burn | pascal second |
| $ j $ | index of the module that failed | dimensionless |
| $ k $ | index over modules, and elsewhere the separation threshold ratio | dimensionless |
| $ L $ | loss of a fixed nozzle against the envelope, and elsewhere the distance from the engine plane to the centre of mass | newton, and metre |
| $ M $ | Mach number | dimensionless |
| $ m $ | number of sides of the achievable moment polygon | dimensionless |
| $ m_0 $ | mass of the second stage with its payload at ignition | kilogram |
| $ m_f $ | mass of the second stage with its payload at burnout | kilogram |
| $ m_p $ | payload mass | kilogram |
| $ M_s $ | Mach number at the point where the flow separates from the wall | dimensionless |
| $ M_{\mathrm{out}} $ | unbalanced moment left by the loss of one module | newton metre |
| $ N $ | number of thruster modules on the ring | dimensionless |
| $ n $ | shape exponent of the ascent profile, and elsewhere a count of trials | dimensionless |
| $ p $ | static pressure at a station | pascal |
| $ p_0 $ | pressure at the base of an atmospheric layer | pascal |
| $ p_a $ | ambient atmospheric pressure | pascal |
| $ p_b $ | pressure acting on the exposed base of a truncated plug | pascal |
| $ p_c $ | chamber pressure | pascal |
| $ p_d $ | ambient pressure at which a fixed nozzle is matched | pascal |
| $ p_e $ | static pressure of the exhaust at the exit plane | pascal |
| $ p_s $ | wall pressure at the point where the flow separates | pascal |
| $ p_{\mathrm{SL}} $ | sea-level pressure of the standard atmosphere | pascal |
| $ q $ | dummy variable of integration over ambient pressure | pascal |
| $ R $ | specific gas constant of air | joule per kilogram kelvin |
| $ r $ | radius of the ring of thruster modules | metre |
| $ R_u $ | universal gas constant | joule per kilomole kelvin |
| $ T $ | temperature | kelvin |
| $ t $ | time from liftoff | second |
| $ T_0 $ | temperature at the base of an atmospheric layer | kelvin |
| $ t_b $ | burn duration of the first stage | second |
| $ T_c $ | chamber temperature | kelvin |
| $ u $ | dummy variable of integration over angle | radian |
| $ v $ | climb rate | metre per second |
| $ v_e $ | exhaust velocity at the exit plane | metre per second |
| $ w $ | gain in the first-stage velocity increment | metre per second |
| $ X $ | number of engines suffering the instability | dimensionless |
Every symbol above appears in the mathematics and every symbol in the mathematics appears above.
From Coefficient to Seconds
One conversion is needed, because specific impulse is the currency this subject is argued in.
Let $\dot{m}$ be the mass flow as before, $c^{\star}$ the characteristic velocity in metre per second, $T_c$ the chamber temperature in kelvin, $\bar{M}$ the mean molar mass of the combustion products in kilogram per kilomole, $R_u$ the universal gas constant in joule per kilomole kelvin, $g_0$ the standard gravity in metre per second squared, and $I_{sp}$ the specific impulse in second.
\[c^{\star} = \frac{p_c A_t}{\dot{m}} = \frac{1}{\Gamma}\sqrt{\frac{R_u T_c}{\bar{M}}}, \qquad I_{sp} = \frac{C_F \, c^{\star}}{g_0}\]The split is clean, and it lets this article work in coefficients. The characteristic velocity carries everything about the propellant and the chamber, the thrust coefficient carries everything about the nozzle and the atmosphere, and they multiply. A claim about altitude compensation concerns the second factor only.
That is already enough to say something the subject’s usual presentation obscures. The entire atmospheric penalty of a fixed nozzle over a burn is its exit area multiplied by the time integral of the ambient pressure. Let $t_b$ be the burn duration in second, and let $J$ be that integral in pascal second.
\[J = \int_{0}^{t_b} p_a(t) \, \mathrm{d}t, \qquad \int_{0}^{t_b} F \, \mathrm{d}t = F_{\mathrm{vac}} t_b - A_e J\]One number describes the nozzle. One number describes the flight. They multiply.
The Identity That Makes Ambient Pressure and Exit Area Conjugate
Now vary the exit area and ask what happens to the vacuum thrust.
For steady one-dimensional flow the momentum equation along the nozzle reads $\dot{m} \, \mathrm{d}v = -A \, \mathrm{d}p$, where $A$ is the local cross-sectional area in square metre and $p$ the local static pressure in pascal. Differentiate $F_{\mathrm{vac}} = \dot{m} v_e + p_e A_e$ with respect to the exit area, holding the chamber conditions fixed, and the first and third terms cancel exactly.
\[\frac{\mathrm{d} F_{\mathrm{vac}}}{\mathrm{d} A_e} = \dot{m} \frac{\mathrm{d} v_e}{\mathrm{d} A_e} + p_e + A_e \frac{\mathrm{d} p_e}{\mathrm{d} A_e} = -A_e \frac{\mathrm{d} p_e}{\mathrm{d} A_e} + p_e + A_e \frac{\mathrm{d} p_e}{\mathrm{d} A_e} = p_e\]The incremental vacuum thrust bought by an increment of exit area is the exit pressure acting on that increment, and nothing else survives. The result is exact, it holds for any gas obeying the one-dimensional equations, and it takes one line.
It also settles the structure of the problem. Write $G$ for the vacuum thrust regarded as a function of exit area, in newton, so that $G(A_e) = F_{\mathrm{vac}}$. Then $G’(A_e) = p_e$, and the thrust against an atmosphere is
\[F(p_a; A_e) = G(A_e) - p_a A_e\]which is $G$ minus a linear term, in the pairing that defines a Legendre transform. Ambient pressure is the variable conjugate to exit area. The correspondence is no analogy. It is the same pairing with the same stationarity condition, which the next paragraph states.
Maximising the thrust over the exit area at a fixed ambient pressure requires $G’(A_e) = p_a$, and $G’(A_e)$ is $p_e$. So the optimum is $p_e = p_a$, the matched-expansion condition that texts commonly state as a separate fact. Far from separate, it is the first-order condition of the transform.
In coefficients the same statement is a derivative with respect to the area ratio, and the later sections use this form.
\[\frac{\partial C_F}{\partial \varepsilon} = \frac{p_e - p_a}{p_c}\]The derivative of the thrust coefficient with respect to area ratio is the pressure difference across the exit plane. Lengthening the nozzle helps while the exhaust is still pushing harder than the atmosphere and hurts once it is not. That one line holds the entire design trade. Central differences confirm the relation in twenty-seven cases to five parts in ten million, which is the step size, and its vacuum case, the conjugacy identity, in twelve cases to 2.1 parts in a hundred thousand million.
The Envelope of All Fixed Nozzles
Every fixed nozzle is one straight line. The family of all fixed nozzles is a family of straight lines, one for each exit area, and the best achievable thrust at any given ambient pressure is the upper envelope of that family. Write $F^{\star}$ for that envelope, in newton.
\[F^{\star}(p_a) = \max_{A_e} \left\{ G(A_e) - p_a A_e \right\}\]The envelope is the Legendre transform of the vacuum thrust and the thrust curve of an ideal altitude-compensating nozzle [Legendre transformation] [convex conjugate]. An aerospike is an attempt to build a machine that sits on the envelope rather than on one of its tangent lines, and the mechanism by which its plume follows the atmosphere down is studied in a separate literature [Ruf and McConaughey 1997].
Two consequences follow immediately and neither needs any hardware.
First, the envelope is convex in ambient pressure, because a maximum of affine functions is always convex. So the ideal spike’s thrust curve bends one way and only one way as the vehicle climbs.
Second, every fixed nozzle touches the envelope at exactly one point and lies strictly below it everywhere else. That one point is its design altitude. This tangency is the entire altitude-compensation argument, and gas dynamics plays no part in it.
That second consequence has a name and an exact statement in the Fenchel-Young inequality.
\[G\left( A_e \right) \le F^{\star}\left( p_a \right) + p_a A_e \quad \text{for every } A_e \text{ and every } p_a\]Equality holds if and only if the nozzle is matched to that ambient pressure. Rearranged, the inequality states that the thrust of any fixed nozzle at any altitude is at most the envelope’s, restating the sentence above with a condition attached to the one case where it is tight. At three area ratios against three ambient pressures each, the gap is positive in every case away from the design point and zero to twelve decimal places at it.
The transform also runs backwards, so the envelope loses nothing.
\[G\left( A_e \right) = \min_{p_a} \left\{ F^{\star}\left( p_a \right) + p_a A_e \right\}\]The family of fixed nozzles and the envelope carry the same information. Knowing the ideal spike’s thrust at every altitude is knowing the vacuum thrust of every bell, and conversely. An aerospike is not a new capability so much as a different point on a structure that was already there.
Two derivatives of the envelope fix its shape between them. Let $A_e(p_a)$ denote the exit area the envelope selects at ambient pressure $p_a$, in square metre.
\[\frac{\mathrm{d} F^{\star}}{\mathrm{d} p_a} = -A_e\left( p_a \right), \qquad \frac{\mathrm{d}^2 F^{\star}}{\mathrm{d} p_a^2} = -\frac{\mathrm{d} A_e}{\mathrm{d} p_a} \ge 0\]The first is the envelope theorem, which says the slope of the envelope is minus the exit area of whichever nozzle is optimal there. Convexity is the other, and it holds for a physical reason rather than a formal one. A higher ambient pressure selects a shorter nozzle, so the exit area falls as the ambient rises, so its negative is non-negative. The envelope bends upward because the optimal nozzle shrinks as the air thickens.
The gap between the envelope and the tangent line has a name. Let $p_d$ be the ambient pressure at which a particular fixed nozzle is matched, in pascal, and let $A_d$ be its exit area in square metre, so that $F^{\star\prime}(p_d) = -A_d$. The loss of that nozzle at some other ambient pressure is
\[L(p_a) = F^{\star}(p_a) - \left[ G(A_d) - p_a A_d \right] = F^{\star}(p_a) - F^{\star}(p_d) - F^{\star\prime}(p_d)\left( p_a - p_d \right)\]That is the Bregman divergence of the convex function $F^{\star}$ generated at the design point [Bregman divergence] [envelope theorem]. It is zero at the design altitude, positive everywhere else, and it grows faster than linearly in the mismatch because the envelope is convex. The performance penalty of a fixed nozzle is not merely analogous to a Bregman divergence. It is one.
Substituting the first derivative above turns that difference into an integral, and the integral is the form that can be read off a picture.
\[L\left( p_a \right) = \int_{p_d}^{p_a} \left[ A_d - A_e\left( q \right) \right] \mathrm{d}q\]The loss is the area between a horizontal line at the fixed nozzle’s exit area and the curve of the exit area the envelope would have chosen. Above the design point the envelope wants a smaller nozzle than the one fitted and the integrand is positive. Below it the envelope wants a larger one and the integrand is negative, but so is the direction of travel, so the loss is positive on both sides and for the same reason.
The order of that subtraction is not free. Written the other way round the same expression is the negative of the loss, so the integrand has to be the fitted nozzle’s exit area minus the one the envelope would have chosen and not the reverse.
A Fixed Nozzle Chosen by a Mean in Time
If a designer must choose one exit area for a whole ascent, what should it be?
One line answers it. The rule appears first as an invariant, the same number coming out of nine separate numerical optimisations that have no reason to agree, and it can then be derived.
Delivered impulse over the burn, in units of chamber pressure times throat area, is the integral of the thrust coefficient. Let $C_F$ be the thrust coefficient, dimensionless, let $C_{F,\mathrm{vac}}$ be its vacuum value, let $p_c$ be the chamber pressure in pascal, and let $\varepsilon$ be the ratio of exit area to throat area, dimensionless. Because $C_F$ is affine in ambient pressure with slope $-\varepsilon$,
\[I(\varepsilon) = \int_{0}^{t_b} C_F \, \mathrm{d}t = t_b \, C_{F,\mathrm{vac}}(\varepsilon) - \frac{\varepsilon}{p_c} \int_{0}^{t_b} p_a(t)\, \mathrm{d}t\]Differentiate with respect to $\varepsilon$ and set the derivative to zero. The derivative of the vacuum thrust coefficient with respect to the area ratio is the exit pressure ratio, by the conjugacy identity above, so
\[t_b \frac{p_e}{p_c} = \frac{J}{p_c} \qquad \Longrightarrow \qquad p_e = \frac{J}{t_b} = \frac{1}{t_b}\int_{0}^{t_b} p_a(t)\, \mathrm{d}t\]The optimal fixed nozzle expands to the time-averaged ambient pressure of its own flight.
Writing $\bar{p}_a$ for that burn-averaged ambient pressure in pascal, and $\varepsilon^{\star}$ for the area ratio the rule selects, dimensionless, the statement constrains the exit pressure alone.
\[\bar{p}_a \equiv \frac{1}{t_b}\int_{0}^{t_b} p_a(t)\, \mathrm{d}t, \qquad p_e\left( \varepsilon^{\star} \right) = \bar{p}_a\]The gas does not appear, and neither do the chamber pressure and the vehicle. The rule is a mean, taken in time and not in altitude, and exact. The question it answers has been asked of other vehicles by other routes [Hiers and Knapke 2019].
The rule also holds numerically. Over four trajectory shapes, five ratios of specific heats and five chamber pressures, one hundred independent searches in all, the exit pressure of each optimum can be set against the burn-averaged ambient pressure computed separately. The optimal area ratios across that grid range from 4.985 to 84.093, a factor of seventeen, and every one of them produces an exit pressure equal to the mean ambient pressure of its own trajectory. The worst relative disagreement across the hundred cases is 4.2 parts in ten million, which is the tolerance of the search rather than an error in the identity.
A mean in time is a strange thing to design a nozzle around, and the strangeness is the point. The atmosphere thins exponentially with altitude but a rocket does not climb at constant speed, so the burn-averaged pressure depends on how quickly the vehicle leaves the dense air rather than on how high it eventually goes. Two vehicles reaching the same altitude on the same burn time, one accelerating hard early and one accelerating hard late, want different nozzles.
The Manufacturer’s Published Ascent
The X-63A was to be a single-stage aerospike variant of the RS1, called the RS1A by its builder, sharing the parent vehicle’s turbopumps, thrusters and injectors [AFRL awards agreements under ARISE]. The parent vehicle’s reference mission is published in the company’s payload user’s guide, which gives the first-stage trajectory as a sequence of timed points [ABL Payload User’s Guide].
Liftoff is at zero. Main engine cut-off and stage separation occur at 160 seconds, at 75 kilometres altitude and 2.6 kilometres per second. Second-stage ignition follows at 163 seconds and 80 kilometres, fairing separation at 200 seconds and 125 kilometres, second-stage cut-off at 508 seconds at 200 kilometres and 7.8 kilometres per second, and payload separation ten seconds after that [ABL Payload User’s Guide].
| Event | Time | Altitude | Speed |
|---|---|---|---|
| Liftoff | 0 seconds | 0 kilometres | 0 kilometres per second |
| Main engine cut-off and stage separation | 160 seconds | 75 kilometres | 2.6 kilometres per second |
| Second-stage start | 163 seconds | 80 kilometres | 2.6 kilometres per second |
| Fairing separation | 200 seconds | 125 kilometres | 2.65 kilometres per second |
| Second-stage cut-off | 508 seconds | 200 kilometres | 7.8 kilometres per second |
| Payload separation | 518 seconds | 200 kilometres | 7.8 kilometres per second |
Those points already say something about the shape of the climb. The mean vertical speed over the first stage is 469 metres per second against a speed at cut-off of 2,600, so the vertical component is about eighteen percent of the total and the trajectory has turned most of the way over by the time the first stage finishes. A first stage is not a lift.
Two published points and a zero initial rate do not determine a trajectory, so the pressure-time integral was computed over a family of monotone profiles. Let $h$ be geometric altitude in metre, $h_b$ the cut-off altitude in metre and $n$ a shape exponent, dimensionless.
\[h(t) = h_b \left( \frac{t}{t_b} \right)^{n}, \qquad 1.5 \le n \le 3.0\]The exponent is a guess and the sweep keeps it honest. A value of 2 is constant acceleration, below 2 is a vehicle that climbs steeply and then pitches over, and above 2 is one that lingers low.
Ambient pressure comes from the 1976 United States Standard Atmosphere, implemented as its seven layers from the defining constants of the standard [U.S. Standard Atmosphere, 1976] [defining constants and abbreviated tables]. Let $\lambda$ be the temperature lapse rate of a layer in kelvin per metre, $T_0$ and $p_0$ the temperature and pressure at its base in kelvin and pascal, $h_0$ its base altitude in metre, $g_0$ the standard gravity and $R$ the specific gas constant of air in joule per kilogram kelvin. A layer with a lapse rate and an isothermal layer take different forms.
\[p(h) = p_0 \left( 1 + \frac{\lambda \left( h - h_0 \right)}{T_0} \right)^{-\frac{g_0}{R \lambda}}, \qquad p(h) = p_0 \exp\left( -\frac{g_0 \left( h - h_0 \right)}{R T_0} \right)\]The isothermal form is the limit of the other as the lapse rate goes to zero. The two forms reproduce the published table values at geopotential altitudes of 5, 11, 15, 20, 32 and 47 kilometres to better than two parts in a hundred thousand. The layers are defined in geopotential altitude and the calculation applies them to the trajectory’s altitude without conversion, which places each pressure slightly low and lowers the burn-averaged pressure below by 0.05 to 0.12 percent across the profiles. The figures below are computed without the conversion, and no conclusion drawn from them depends on a difference of that size. The lineage of the standard is documented as well [the 1976 standard and its predecessors].
The natural length in that expression is the scale height, in metre, and it sets how quickly the question answers itself.
\[H \equiv -\frac{p}{\mathrm{d}p / \mathrm{d}h} = \frac{R T}{g_0}\]At sea level that is 8,434.5 metres, and the lapse rate does not change it there, because the hydrostatic relation makes the local scale height equal to $R T / g_0$ at every altitude, in a layer with a lapse rate as in an isothermal one. What the lapse rate changes is how the scale height varies with altitude, since it varies as the temperature does.
Across that family the burn-averaged ambient pressure runs from 19.4 to 41.6 percent of sea-level pressure, and the value for the constant-acceleration case is 28.06 percent. The unknown shape of the trajectory produces that spread, and the table below gives it in full.
| Profile exponent | Burn-averaged ambient pressure | As a fraction of sea level | Half the integral collected by |
|---|---|---|---|
| 1.5 | 19,620 pascal | 19.4 percent | 17.6 seconds |
| 2.0 | 28,436 pascal | 28.1 percent | 24.0 seconds |
| 2.5 | 35,916 pascal | 35.4 percent | 29.4 seconds |
| 3.0 | 42,178 pascal | 41.6 percent | 34.0 seconds |
A closed form beside the numbers shows what the integral depends on. For an isothermal atmosphere climbed at a constant rate $v$ in metre per second,
\[J = \frac{p_0 H}{v}\left[ 1 - \exp\left( -\frac{v t_b}{H} \right) \right] \;\longrightarrow\; \frac{p_0 H}{v} \quad \text{as } v t_b \gg H\]The atmospheric penalty of a long burn does not depend on how long the burn is. Once the vehicle is several scale heights up, the integral has stopped growing, and what remains is sea-level pressure times the scale height divided by the climb rate. A rocket does not pay for the atmosphere by the second. It pays once, and the bill is set by how fast it left.
The more useful number is where the integral is accumulated. Half of the pressure-time integral of a 160 second burn is collected in the first 17.6 to 34.0 seconds, depending on the profile. The altitude-compensation question is settled in the first 11 to 21 percent of the burn and the remainder barely votes. Ambient pressure is at half of its sea-level value by 5.48 kilometres, at a tenth by 16.1 kilometres, at a hundredth by 31.0 kilometres of geopotential altitude, which is 31.1 kilometres geometric, and at cut-off altitude it is two parts in a hundred thousand of sea level. Ambient pressure is below a tenth of sea level for the last 40 to 64 percent of the burn and below a hundredth for the last 25 to 44 percent, depending on the profile, and the nozzle still has to carry the exit area it chose for the denser air before that.
The Gain of an Ideal Spike on This Trajectory
With the trajectory in hand the gain of ideal altitude compensation can be computed. The comparison is a ratio of two integrals of thrust coefficients, so it requires no dimension of the engine at all, which is fortunate, because none is public.
Writing $\eta$ for the fractional gain, dimensionless,
\[\eta = \frac{\displaystyle\int_{0}^{t_b} C_F^{\star}\left( p_a(t) \right) \mathrm{d}t}{\displaystyle\int_{0}^{t_b} C_F\left( \varepsilon^{\star}, p_a(t) \right) \mathrm{d}t} - 1\]The throat area cancels, the chamber pressure cancels and the mass flow cancels, so a figure can be quoted at all for a vehicle whose engine is not public.
Against the best possible fixed nozzle for this trajectory, chosen by the rule derived above, an ideal altitude-compensating nozzle delivers between 5.3 and 8.5 percent more first-stage impulse, across ratios of specific heats from 1.20 to 1.24 and chamber pressures from 3 to 10 megapascal. Against the naive choice, a fixed nozzle matched at sea level, the gain is between 7.8 and 12.5 percent.
Two remarks keep those numbers honest.
The first is that this is an upper bound and a generous one. An ideal spike has no truncation loss, no base drag, no wake, no cooling penalty and no extra mass. A real one has all five. What the calculation bounds is the size of the prize and not the size of the win.
The second is the shape of the optimum, which is more interesting than its value. The best fixed area ratio for this trajectory is between 2.5 and 2.7 times the sea-level-matched ratio. At a chamber pressure of 6 megapascal and a ratio of specific heats of 1.20, the sea-level-matched nozzle has an area ratio of 7.99 and the trajectory-optimal one has 21.07. The optimal fixed nozzle is heavily over-expanded at liftoff, because most of the burn happens where the air is thin, and the mean that chooses it is dominated by the seconds when the air is not.
| Ratio of specific heats | Chamber pressure | Best fixed area ratio | Sea-level-matched area ratio | Gain over the best fixed nozzle | Gain over the sea-level-matched nozzle |
|---|---|---|---|---|---|
| 1.22 | 10.0 megapascal | 29.63 | 11.26 | 5.74 percent | 8.38 percent |
| 1.22 | 3.0 megapascal | 11.83 | 4.66 | 7.89 percent | 11.69 percent |
| 1.22 | 6.0 megapascal | 20.01 | 7.70 | 6.54 percent | 9.61 percent |
| 1.24 | 10.0 megapascal | 28.02 | 10.79 | 5.31 percent | 7.80 percent |
| 1.24 | 3.0 megapascal | 11.34 | 4.52 | 7.37 percent | 11.00 percent |
| 1.24 | 6.0 megapascal | 19.03 | 7.42 | 6.08 percent | 8.98 percent |
| 1.2 | 10.0 megapascal | 31.38 | 11.75 | 6.22 percent | 9.02 percent |
| 1.2 | 3.0 megapascal | 12.37 | 4.80 | 8.46 percent | 12.45 percent |
| 1.2 | 6.0 megapascal | 21.07 | 7.99 | 7.06 percent | 10.29 percent |
Flow Separation as the Deciding Limit
The mean-in-time rule has a consequence that the usual presentation of altitude compensation never reaches, because the usual presentation stops at the design-point trade.
Recall that the rule sets the exit pressure of the optimal fixed nozzle equal to the burn-averaged ambient pressure. At liftoff the ambient pressure is sea-level pressure, so the exit-to-ambient pressure ratio of the optimal nozzle at liftoff is that burn-averaged pressure divided by the sea-level value, and the chamber pressure has cancelled on the way.
\[\left. \frac{p_e}{p_a} \right|_{t=0} = \frac{\bar{p}_a}{p_{\mathrm{SL}}} = \frac{1}{p_{\mathrm{SL}} t_b} \int_{0}^{t_b} p_a(t) \, \mathrm{d}t\]The condition a designer has to satisfy at liftoff depends on the flight alone. The engine, the propellant and the size of the vehicle do not enter it. For the constant-acceleration model of this vehicle’s first stage that fraction is 0.2806, and the calculation returns 0.28064 for the optimal nozzle at every one of the nine combinations of gas and chamber pressure tested. The invariance is the identity showing itself.
Now set that against the other thing a nozzle designer has to respect. An over-expanded nozzle does not merely lose thrust. Below some exit-to-ambient pressure ratio the boundary layer separates from the divergent wall, the separation line is unsteady, and the resulting side loads are a structural problem rather than a performance one.
Let $k$ be that threshold ratio, dimensionless, and $\varepsilon_{\mathrm{sep}}$ the largest area ratio that respects it at sea level.
\[\frac{p_e}{p_a} \ge k \qquad \Longrightarrow \qquad p_e\left( \varepsilon_{\mathrm{sep}} \right) = k \, p_{\mathrm{SL}}\]Both constraints now bear on the same quantity, the exit pressure at liftoff. One says it should equal the burn-averaged ambient pressure. The other says it must not fall below $k$ times sea level. Whether they can both be met is arithmetic, and the arithmetic depends entirely on what $k$ is.
The Provenance of the Separation Threshold
The four tenths that secondary accounts quote for the onset of separation has a provenance other than the one the number’s ubiquity suggests. The primary that compiles the data changes both the attribution and the answer.
NASA Technical Paper 1207 compiles the available turbulent-flow separation data for over-expanded supersonic nozzles and discusses the correlation techniques [Morrisette and Goldberg 1978]. It gives the value’s history, which a search of the bibliographic index alone does not yield.
a value of $p_s/p_a$ = 0.4 was used as a separation criteria and is still quoted today, although more recent studies have shown it to be inadequate
The paper names no originator for that value. It traces the value to the early studies that found the separation pressure ratio independent of design expansion ratio, divergence angle, ratio of specific heats and gas temperature, among them a 1949 Jet Propulsion Laboratory report by Foster and Cowles, and it names the fourth edition of Sutton and Ross, of 1976, only as an example of a text that still quotes the value [Morrisette and Goldberg 1978]. Summerfield, Foster and Swan, in the Journal of Jet Propulsion for 1954 [Summerfield, Foster and Swan 1954], appear in its reference list among the many experimental studies of separation, which this article has not read and cites at second hand, saying so. A search of that index for the value’s origin returns a 1953 paper by Scheller and Bierlein instead [Scheller and Bierlein 1953], and the compiling paper also explains why that pair is not the origin, because an earlier note by the same authors is the study that conflicted with the others by finding the separation pressure dependent on nozzle divergence angle.
And the four tenths is for the wrong kind of nozzle. The paper separates its data by nozzle type and says that contoured nozzles, which are the case of most interest for modern nozzle design, follow a different correlation from conical ones. For contoured nozzles it fits a second-order curve to the data between separation Mach numbers of 2.4 and 4.5 [Morrisette and Goldberg 1978]. Let $M_s$ be the Mach number at the separation point, dimensionless, and $p_s$ the wall pressure there, in pascal.
That correlation is more permissive than the flat four tenths over most of its range and less permissive at the bottom of it. At separation Mach 2.4, where the fit begins, it gives 0.433, which is stricter than the old rule. It crosses the flat value at Mach 2.59 and falls away from there, giving 0.340 at Mach 3, 0.248 at Mach 4 and 0.230 at Mach 4.5.
So the old rule is 1.74 times as restrictive at the top of the fitted range and slightly less restrictive at the bottom. A summary that the contoured correlation tolerates roughly twice the over-expansion everywhere is true nowhere. A nozzle of the area ratios this article is discussing separates well above Mach 3, so the permissive end is the end that applies here, but the crossing is real.
One further conclusion of that paper matters here, because it licenses something this article does throughout. No obvious effect of the ratio of specific heats was found when the data were correlated as separation pressure ratio against separation Mach number [Morrisette and Goldberg 1978]. Sweeping that ratio, as every calculation in this article does, therefore does not move the separation answer.
Applying the Correlation to This Nozzle
The criterion is local rather than a statement about the exit, so the way to use it is to walk the nozzle from throat to exit and find the first station where the wall pressure falls to the criterion value. If that station lies beyond the exit, the flow is attached.
Doing that for the trajectory-optimal nozzle gives an answer that depends on chamber pressure, and it does not support a flat statement that the optimum cannot be built.
| Ratio of specific heats | Chamber pressure | Trajectory-optimal area ratio | Largest area ratio that stays attached | Optimum as a multiple of it | Does the optimum separate |
|---|---|---|---|---|---|
| 1.2 | 3.0 megapascal | 12.37 | 11.94 | 1.036 | yes |
| 1.2 | 6.0 megapascal | 21.07 | 22.55 | 0.934 | no |
| 1.2 | 10.0 megapascal | 31.38 | 35.46 | 0.885 | no |
| 1.22 | 3.0 megapascal | 11.83 | 11.55 | 1.025 | yes |
| 1.22 | 6.0 megapascal | 20.01 | 21.64 | 0.925 | no |
| 1.22 | 10.0 megapascal | 29.63 | 33.74 | 0.878 | no |
| 1.24 | 3.0 megapascal | 11.34 | 11.18 | 1.014 | yes |
| 1.24 | 6.0 megapascal | 19.03 | 20.78 | 0.916 | no |
| 1.24 | 10.0 megapascal | 28.02 | 32.10 | 0.873 | no |
Under the flat four tenths the optimum is unbuildable at every chamber pressure in the band. Under the contoured-nozzle correlation it is unbuildable only at the low end. Let $\mu$ be the margin in area ratio, dimensionless. The crossover is the chamber pressure at which the two area ratios coincide, and above it the margin is positive.
\[\varepsilon^{\star}\left( p_c \right) = \varepsilon_{\mathrm{sep}}\left( p_c \right), \qquad \mu \equiv 1 - \frac{\varepsilon^{\star}}{\varepsilon_{\mathrm{sep}}}, \qquad 1 - \frac{21.07}{22.55} = 0.066, \qquad 1 - \frac{28.02}{32.10} = 0.127\]The crossover is at 3.3 to 3.7 megapascal, across ratios of specific heats from 1.20 to 1.24, and above it the trajectory-optimal nozzle runs attached with between 6.6 and 12.7 percent of margin in area ratio, the extremes being the cases at 6 megapascal with a ratio of specific heats of 1.20 and at 10 megapascal with 1.24 in the table above.
The chamber pressure recovered from the instability frequency, 2.6 to 8.7 megapascal, straddles that crossover. So for this engine the question is open, and which side of it the real nozzle falls on is decided by a number nobody has published.
The defensible statement is weaker than a flat impossibility and more interesting. It does not hold that the bell cannot have the exit area the trajectory wants. It holds that the trajectory wants an exit area within about a tenth of the largest one that will stay attached, on one side or the other depending on a chamber pressure that is not public. The optimum is not outside the feasible region. It is pressed against its edge, and a designer who wanted it would be trading impulse against a separation margin rather than choosing freely.
Two qualifications hold. The first is that a separation criterion is a design guideline and not a wall, as the design monograph on this subject states, noting that prediction methods have shown agreement with experiment in many cases but that in general predictions are used only as a guide [NASA SP-8120]. The second is that an aerospike does not have the constraint at all, because it has no divergent wall for a boundary layer to separate from. That second point does not depend on which criterion is used, and the programme rests on it.
And the monograph supplies a case this article was otherwise describing in the abstract. The J-2 engine was given a deliberately nonoptimum parabolic contour to raise its exit wall pressure, and was expected to run unseparated at an area ratio of 27 with a chamber pressure of 700 pounds per square inch. A wall-pressure minimum between area ratio 14 and the exit produced unsteady asymmetric separation, worst at startup, and the loads caused thrust-chamber structural failures. A bolt-on diffuser was developed to stop it in mainstage and restraining arms were attached from the test stand to the nozzle skirt to absorb the loads at ignition [NASA SP-8120]. The constraint costs that much when it is met head on, and the margin computed above is therefore something to preserve.
The Three Regimes the Programme Names
The fact sheet names open wake, wake transition and closed wake as the three modes the flight is meant to visit [ARISE and Fly]. Those names describe what happens behind a plug that has been cut off short, which every practical aerospike is.
A full-length plug would be long, heavy and hot. Real ones are truncated, which leaves a blunt base at the downstream end, and the three names classify the flow behind that base.
Truncation splits the thrust into two terms, and the split is exact whatever the flow does. Let $F_w$ be the axial force the expanding gas exerts on the wetted surface of the plug in newton, $p_b$ the pressure acting on the exposed base in pascal, and $A_b$ the area of that base in square metre.
\[F = F_w + p_b A_b\]Cutting the plug short removes part of the first term and creates the second. The base pressure decides how much of the loss the second term returns, and the three regimes describe that pressure. This article can write the decomposition and cannot evaluate it, because $p_b$ is set by a recirculating turbulent flow and no one-dimensional isentropic model produces one.
At low altitude the ambient pressure is high, the plume is squeezed inward against the plug, and the recirculating region behind the truncated base is closed off by the converging flow. That is the closed wake. The base sits at a pressure set by the recirculation rather than by the atmosphere, and the pressure acting on it contributes to thrust.
At high altitude the ambient pressure is low, the plume no longer closes behind the base, and the base region is open to the external flow. That is the open wake, and in it the base pressure is tied to the ambient pressure. Which quantity sets the base pressure distinguishes the two regimes, and the classification turns on that alone.
\[p_b \approx p_a \;\; \text{open wake}, \qquad p_b \neq p_a \;\; \text{closed wake}\]In the open regime the base contributes nothing the atmosphere does not already take back. In the closed regime it contributes a pressure the recirculation sets, which may be above or below ambient, and a truncated spike is designed to keep that term. A normal shock stands in the plume at the transition [Giel and Mueller 1975] [Marsilio 2001].
Between them is a transition, and a transition between two flow topologies is the kind of thing that can be hysteretic, unsteady, or abrupt [Geron and others 2005] [Mizukaki and Watabe 2016] [Taylor and Sato 2008]. A flight is needed for that reason. A ground test can produce either end state in a vacuum chamber. It cannot produce the passage between them at full scale with a real plume and a real trajectory, as the fact sheet states.
The published computational and experimental work on truncated plugs reports that thrust lost by cutting the plug short is partly recovered by pressure acting on the exposed base, that bleeding a small secondary flow into the base region recovers more of it, and that the penalty is larger in over-expanded operation than in under-expanded, where it is reported as small [Sule and Mueller 1973] [Mueller and Sule 1973] [Nasuti and Onofri 2000] [Nair, Suryan and Kim 2017] [Nair, Suryan and Kim 2019]. This article does not compute those quantities. They depend on the base geometry, the bleed rate and the turbulence of the recirculation, none of which is public for this engine, and a base pressure is not something an isentropic one-dimensional model can be made to yield. The three regimes are named here because the programme names them and because they are what the flight would have measured, not because this article can predict them.
Truncation and the base flow account for all of the gap between the ideal spike of the Legendre argument and a real engine, and they make the 5.3 to 8.5 percent that the gain $\eta$ gives an upper bound and no prediction.
Dependent Systems
The record of the hardware around the nozzle comes from the parent vehicle and not from the X-63A. The award announcement describes the RS1A as sharing the RS1’s turbopumps, thrusters and injectors, and it promises that thrust vector control for the aerospike will be demonstrated [AFRL awards agreements under ARISE]. This section takes in turn the steering problem that promise sets for an engine with nothing to gimbal, the parent vehicle’s published specification, the chamber geometry that the company’s incident report gives away, and what the same report says about feeding many engines through one manifold [Flight 2 Static Fire Update].
Steering a Rocket That Has Nothing to Gimbal
The promised thrust vector control sets a problem, because the usual way to steer a rocket is to gimbal the engine, and an annular aerospike is a ring built into the base of the vehicle. There is nothing to swing.
Modularity supplies the alternative. The propulsion system is many thrusters working as one [ARISE and Fly], and thrusters arranged around a ring can be throttled against one another. The idea has precedent. The X-33 was to control its attitude by differentially throttling the two halves of its linear aerospike [Hall and Panossian 1999] [the same work as a report], and the flow of a ring of thrusters expanding onto one plug has been measured [Taniguchi 2005].
This article can answer exactly how much authority the arrangement gives, and the result has a pleasant form.
Let $N$ be the number of modules, dimensionless, equally spaced around a ring of radius $r$ in metre. Let $F$ be the total thrust in newton, so each module produces $F/N$ when fully open. Let $\tau_k$ be the throttle fraction of module $k$, dimensionless, constrained to lie between $1-\delta$ and $1$, where $\delta$ is the available throttle depth, dimensionless. Let $\theta_k = 2\pi k / N$ be the angular position of module $k$ in radian. The moment the ring can apply about a horizontal axis in the direction $\phi$ is
\[M(\phi) = \frac{F r}{N} \sum_{k=0}^{N-1} \tau_k \cos\left( \theta_k - \phi \right)\]The unthrottled ring is balanced, and that is an identity rather than an approximation.
\[\sum_{k=0}^{N-1} \cos\left( \theta_k - \phi \right) = 0 \qquad \text{for every } N \ge 2 \text{ and every } \phi\]A ring of equally spaced thrusters all wide open applies no moment about any axis. That balance makes the arrangement usable at all, and subtracting the identity removes the baseline entirely. What is left is the throttle depth acting on the modules in the forward half only. The optimal throttle pattern is bang-bang, every module in the forward half wide open and every module in the rear half closed to its floor, and the moment it produces is
\[\frac{M(\phi)}{F r} = \frac{\delta}{N} \sum_{k} \max\left( \cos\left( \theta_k - \phi \right), \, 0 \right)\]The available control moment, normalised by thrust and ring radius, is the throttle depth times the support function of a regular polygon.
The Directional Average Is the Throttle Depth Divided by Pi, Exactly
Average that expression over all commanded directions. Each module contributes the mean of $\max(\cos u, 0)$ over a full turn, and that mean is one elementary integral.
\[\frac{1}{2\pi}\int_{0}^{2\pi} \max\left( \cos u, 0 \right) \mathrm{d}u = \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} \cos u \, \mathrm{d}u = \frac{1}{\pi}\]There are $N$ modules and the expression divides by $N$, so the module count cancels exactly.
\[\left\langle \frac{M}{F r} \right\rangle_{\phi} = \frac{\delta}{\pi}\]Not asymptotically. Exactly, for every module count from three upward. Adding modules does not add average control authority at a fixed throttle depth. Averaging numerically over two hundred thousand directions at nine module counts from four to twenty-four returns 0.063661977 at every count, at a throttle depth of two tenths, against $\delta/\pi = 0.063661977$.
So what does the module count provide? It provides evenness. The achievable set of moments is a polygon rather than a disc, and an autopilot can only rely on the worst direction.
Why Nine Modules Steer More Evenly Than Twelve
The polygon’s side count is not the module count, and the difference between them is the result of this subsection.
Let $m$ be the number of sides of the achievable moment polygon, dimensionless. The support function is a pure cosine between the directions at which the set of open modules changes, and it changes whenever a module crosses the boundary of the forward half plane. Those directions are the following.
\[\phi \in \left\{ \theta_k + \frac{\pi}{2} \right\} \cup \left\{ \theta_k - \frac{\pi}{2} \right\}, \qquad k = 0, \dots, N-1\]So the number of sides is the number of distinct members of that union, and enumeration at ten module counts from four to seventeen agrees with the closed form below in every case.
For an even ring the two families of crossings coincide, because a half turn is a whole number of module spacings, so a module enters the forward half at the same instant another leaves it and there are $N$ crossings. For an odd ring a half turn is not a whole number of spacings, the two families interleave, and there are $2N$.
\[m = \begin{cases} N, & N \text{ even} \\ 2N, & N \text{ odd} \end{cases}\]From which the guaranteed authority, the best-case authority and the ratio between them follow in closed form.
\[\frac{M_{\min}}{F r} = \frac{\delta}{m \tan\left( \pi / m \right)}, \qquad \frac{M_{\max}}{F r} = \frac{\delta}{m \sin\left( \pi / m \right)}, \qquad \frac{M_{\max}}{M_{\min}} = \frac{1}{\cos\left( \pi / m \right)}\]A direct numerical search over commanded directions at fourteen module counts and three throttle depths, forty-two cases, agrees with them, and the worst disagreement is below one part in a million million.
Where that polygon sits relative to the modules is a separate question from how many sides it has, and it turns on the module count modulo four rather than on its parity. For an odd ring the greatest authority is attained both at the module directions and midway between them, because the polygon carries two sides per module, and the least a quarter of a spacing from each. For an even ring the answer depends on whether four divides the count, and the reason is that a quarter turn is a whole number of module spacings exactly when it does. At four, eight, twelve, sixteen, twenty and twenty-four modules a module therefore lies on the boundary of the forward half plane whenever the command points straight at another module, and that degenerate direction is the one of least authority. At six, ten, fourteen, eighteen and twenty-two it is the direction of greatest authority instead. The closed forms above are untouched by this, since they depend only on the side count, but a rule carried over from the odd-ring arrangement places the two extremes correctly on one even family and swaps them on the other, which is harder to notice than a wrong number.
The consequence is that an odd ring of $N$ modules behaves exactly like an even ring of $2N$. Nine modules give the steering polygon of eighteen. Eleven give twenty-two. Seventeen give thirty-four.
At a throttle depth of two tenths the guaranteed authority follows from the first closed form with $m$ equal to 8, 18, 12 and 16 sides, and the ratios follow from those.
\[\frac{0.2}{8 \tan\left( \pi / 8 \right)} = 0.0603553, \qquad \frac{0.2}{18 \tan\left( \pi / 18 \right)} = 0.0630142, \qquad \frac{0.2}{12 \tan\left( \pi / 12 \right)} = 0.0622008, \qquad \frac{0.2}{16 \tan\left( \pi / 16 \right)} = 0.0628417\] \[\frac{0.0630142}{0.0622008} = 1.0131, \qquad \frac{0.0630142}{0.0603553} = 1.0441\]Eight modules give 0.0603553. Nine give 0.0630142. Twelve give 0.0622008. Sixteen give 0.0628417. Nine modules guarantee 1.31 percent more control authority than twelve and 4.41 percent more than eight, and nine beat every even module count up to sixteen. The smallest even count that beats nine is twenty. The smallest even count that beats eleven is twenty-four.
This runs the opposite way to the intuition that more thrusters make a smoother ring. Adding a module to an even ring to make it odd nearly doubles the number of sides on its control polygon. Adding one to an odd ring to make it even halves them.
| Modules | Sides of the moment polygon | Guaranteed authority | Best-direction authority | Ratio of best to guaranteed |
|---|---|---|---|---|
| 4 | 4 | 0.050000 | 0.070711 | 1.41421 |
| 5 | 10 | 0.061554 | 0.064721 | 1.05146 |
| 6 | 6 | 0.057735 | 0.066667 | 1.15470 |
| 7 | 14 | 0.062590 | 0.064199 | 1.02572 |
| 8 | 8 | 0.060355 | 0.065328 | 1.08239 |
| 9 | 18 | 0.063014 | 0.063986 | 1.01543 |
| 10 | 10 | 0.061554 | 0.064721 | 1.05146 |
| 11 | 22 | 0.063229 | 0.063879 | 1.01028 |
| 12 | 12 | 0.062201 | 0.064395 | 1.03528 |
| 16 | 16 | 0.062842 | 0.064073 | 1.01959 |
| 17 | 34 | 0.063481 | 0.063753 | 1.00428 |
| 20 | 20 | 0.063138 | 0.063925 | 1.01247 |
| 24 | 24 | 0.063298 | 0.063844 | 1.00863 |
The RS1 first stage from which the X-63A was to be derived carried nine engines in its first block and eleven in its second [ABL Payload User’s Guide] [Flight 2 Static Fire Update]. Both counts are odd, and each is the best available below the next even count that beats it.
Those counts were almost certainly not chosen for this reason. Nine and eleven engines on a first stage are chosen for thrust, for packaging, for engine-out margin and for what the factory can build, and neither the sponsor nor the manufacturer publishes the aerospike variant’s module count, although a launch-vehicle reference lists the RS1A first stage as nine E2 aerospike engines [RS1, Gunter’s Space Page], which no primary document consulted confirms. The result’s claim is narrower, and it still holds. If the ARISE engine’s modules are arranged on a ring and steered by differential throttling, then the parity of the count matters more than the count, and the parity the parent vehicle happens to carry is the favourable one.
Losing One Module
The same geometry answers the engine-out question, and the obvious way to answer it is instructively wrong.
Lose one module of $N$ and two things happen, and the balance identity above gives both at once. Removing module $j$ from a sum that vanishes leaves minus that module’s own term.
\[\frac{\Delta F}{F} = \frac{1}{N}, \qquad \frac{M_{\mathrm{out}}\left( \phi \right)}{F r} = -\frac{1}{N}\cos\left( \theta_j - \phi \right)\]The axial thrust falls by one part in $N$, and the unbalanced moment peaks at one part in $N$ as well. Both penalties are one over the module count and they arrive together.
The surviving ring then has the support function of the remaining modules alone.
\[\frac{M\left( \phi \right)}{F r} = \frac{1}{N}\sum_{k \neq j} \tau_k \cos\left( \theta_k - \phi \right), \qquad \tau_k \in \left[ 0, 1 \right]\]That sum is not the intact one with a smaller coefficient. It is a different sum, missing a term, and the rest of this subsection follows from that difference.
The tempting move is to ask how much authority the ring has left and to answer with the closed form derived above. That formula is for a regular ring and a ring with a hole in it is not regular. In some commanded directions the module that would have pushed hardest is the one that failed, so the degraded polygon is not a scaled copy of the intact one. Solving the degraded problem directly, by taking the support function of the surviving modules over their full throttle range, gives numbers that are smaller by factors of 1.15 to 2.62, and at four modules the discrepancy is total.
A four-module ring loses all of its guaranteed steering authority when one module fails. Three modules remain, none of them positioned to push toward the gap, and the support function in that one direction is exactly zero. The intact formula reports 0.25 where the true figure is zero.
Let $\rho$ be the fraction of guaranteed steering authority that survives the loss of module $j$, dimensionless, taken over the full throttle range so that each module contributes only when it faces the commanded direction. The factor $1/N$ cancels.
\[\rho = \frac{\min_{\phi} \sum_{k \neq j} \max\left( \cos\left( \theta_k - \phi \right), 0 \right)}{\min_{\phi} \sum_{k} \max\left( \cos\left( \theta_k - \phi \right), 0 \right)}, \qquad \rho_8 = \frac{0.176777}{0.301777} = 0.586, \qquad \rho_9 = \frac{0.205648}{0.315071} = 0.653\]The correct figures, as values of $\rho$ from the table below, run as follows. Eight modules keep 58.6 percent. Nine keep 65.3 percent. Eleven keep 71.5 percent. Twelve keep 73.2 percent. Twenty-four keep 86.8 percent.
| Modules | Axial thrust lost | Guaranteed authority intact | Guaranteed authority after one failure | Fraction retained |
|---|---|---|---|---|
| 4 | 25.00 percent | 0.250000 | 0.000000 | 0.0 percent |
| 6 | 16.67 percent | 0.288675 | 0.144338 | 50.0 percent |
| 8 | 12.50 percent | 0.301777 | 0.176777 | 58.6 percent |
| 9 | 11.11 percent | 0.315071 | 0.205648 | 65.3 percent |
| 11 | 9.09 percent | 0.316143 | 0.226192 | 71.5 percent |
| 12 | 8.33 percent | 0.311004 | 0.227671 | 73.2 percent |
| 16 | 6.25 percent | 0.314209 | 0.251709 | 80.1 percent |
| 24 | 4.17 percent | 0.316490 | 0.274823 | 86.8 percent |
Here the parity result reverses. Odd rings steer more evenly when intact, because their control polygon has twice as many sides. But a failure breaks the symmetry that the parity argument depends on, and what matters after a failure is how many modules are left and how finely they surround the gap. Twelve beats nine on this measure even though nine beats twelve on the other.
The two results are both geometric, both exact, and they point in opposite directions. An odd ring is the better steering machine and the worse survivor. The tension is a design tension and no paradox, and it needs no flight to settle.
The Authority as a Gimbal Angle
A moment normalised by thrust and ring radius is hard to compare with anything, so this subsection converts it into the quantity a designer of a conventional stage would quote, which is a gimbal angle.
Let $L$ be the distance from the engine plane to the vehicle’s centre of mass in metre and $\alpha$ the equivalent gimbal angle in radian. A gimballed engine deflected by $\alpha$ applies a moment $F L \sin \alpha$, and setting that equal to the ring’s directional average gives
\[\sin \alpha = \frac{\delta}{\pi} \cdot \frac{r}{L}\]The ring radius divided by the distance to the centre of mass is a small number on a slender vehicle, and the throttle depth divided by pi is smaller still. A throttle range of two tenths on a ring whose radius is a quarter of the arm gives an equivalent deflection of 0.91 degrees. Such a deflection cannot substitute for a gimbal and was never meant to. Differential throttling offers attitude control on a vehicle that has nowhere to put a gimbal, at an authority that is adequate for trim and marginal for anything faster.
A Specification Table That Does Not Close
The encyclopedia entry describes the X-63A, the RS1A of the award announcement, as the first-stage core of the two-stage RS1 [ABL Space Systems X-63]. So the parent vehicle’s published specification is the closest thing to a specification the X-63A has.
That specification contains an error, and the error turns out to be informative.
The payload user’s guide of June 2022 says in prose that the RS1 uses nine sea-level E2 engines on the first stage and one vacuum E2 on the second, that the vehicle is six feet in diameter and eighty-eight feet long when integrated, and that both stages burn liquid oxygen and kerosene through gas-generator cycles [ABL Payload User’s Guide]. Its overview table then gives the first stage an engine thrust of 12,100 pounds force at sea level, an engine quantity of nine, and a total thrust of 133,118 pounds force at sea level.
Nine times 12,100 is 108,900. The table overstates its own product by 24,218 pounds force, which is 22.24 percent, and dividing the stated total by the stated per-engine figure gives a quotient of 11.0015.
\[\frac{133{,}118 - 9 \times 12{,}100}{9 \times 12{,}100} = \frac{24{,}218}{108{,}900} = 0.2224, \qquad \frac{133{,}118}{12{,}100} = 11.0015, \qquad \frac{133{,}118 - 11 \times 12{,}100}{133{,}118} = \frac{18}{133{,}118} = 1.35 \times 10^{-4}\]Eleven engines at 12,100 pounds force give 133,100, which differs from the stated total by 18 pounds force, or 135 parts per million. The total thrust in that table is the eleven-engine number, printed in a row that says nine.
The company’s incident report of 26 August 2024 settles it without ambiguity. Describing a static fire on 19 July 2024 it says the objective was a thirteen-second firing of all 11 first stage engines, that all eleven ignited, and that the vehicle auto-aborted on a low pressure reading from Engine 10 [Flight 2 Static Fire Update]. A vehicle with nine engines does not have an Engine 10. The second block of the RS1 carried eleven [RS1, Gunter’s Space Page], the payload user’s guide describes the first block, and one cell of its table was filled in from the wrong vehicle.
Two other published figures disagree with each other and neither is resolved here. The award announcement gives the RS1 a maximum payload of 1,200 kilograms [AFRL awards agreements under ARISE], and the launch-vehicle reference gives 1,350 kilograms to low Earth orbit [RS1, Gunter’s Space Page], as does the manufacturer’s payload user’s guide [ABL Payload User’s Guide]. The larger exceeds the smaller by twelve and a half percent, since $1{,}350/1{,}200 = 1.125$, and both are described as maxima. No source consulted reconciles them.
What the Fire Gave Away
The E2 engine’s chamber pressure is not published. Neither is its chamber diameter. The payload user’s guide gives thrust, cycle, propellants and feed system, and stops [ABL Payload User’s Guide].
The incident report gives both away without meaning to.
Post-anomaly data review indicates Engines 5 and 8 experienced a 4.5 kHz combustion instability during startup, in alignment with the expected first tangential mode of the E2 thrust chamber.
The first tangential acoustic mode of a cylindrical chamber has a frequency fixed by the chamber diameter and the speed of sound in the burnt gas. Let $f_{1T}$ be that frequency in hertz, $c$ the speed of sound in the combustion products in metre per second, $D$ the chamber diameter in metre, and $\alpha_{1T} = 1.8412$ the first stationary point of the Bessel function of the first kind of order one, dimensionless. Then
\[f_{1T} = \frac{\alpha_{1T} \, c}{\pi D} \qquad \Longrightarrow \qquad D = \frac{\alpha_{1T} \, c}{\pi f_{1T}}\]Naming the frequency names the diameter, to within whatever must be assumed about the gas. The speed of sound follows from the flame temperature, the mean molar mass of the products and the ratio of specific heats, none of which is published either, so all three are swept over a range.
\[c = \sqrt{\frac{\gamma R_u T_c}{\bar{M}}}\]The relation gives the frozen speed of sound, which treats the composition as fixed through the acoustic cycle [NASA SP-8124] [NASA SP-8107]. A real chamber recombines, which raises the effective value, so the diameters below are if anything slightly small. Flame temperatures from 3,200 to 3,800 kelvin, molar masses from 21 to 24 kilogram per kilomole and ratios of specific heats from 1.15 to 1.25 give forty-eight combinations, sound speeds from 1,129 to 1,371 metres per second, and a chamber diameter of
147 to 179 millimetres, with a midpoint of 163.
Chamber pressure needs two more assumptions, being a contraction ratio and a sea-level thrust coefficient. Let $\sigma$ be the contraction ratio, dimensionless, which is the chamber cross-section divided by the throat. The chain from the diameter to the pressure is two lines, and the second rearranges the definition of the thrust coefficient.
\[A_t = \frac{\pi D^2}{4 \sigma}, \qquad p_c = \frac{F}{C_F A_t} = \frac{4 \sigma F}{\pi C_F D^2}\]The chamber pressure goes as the inverse square of the diameter, so the recovered band is wide even though the frequency is published to two significant figures. A twenty percent uncertainty in the diameter is a factor of 1.44 in the pressure before any other assumption is made.
Sweeping contraction ratios from 2 to 4 and thrust coefficients from 1.45 to 1.65 across the diameter band gives thirty-six combinations and a chamber pressure of 2.6 to 8.7 megapascal, with a midpoint of 5.67.
| Quantity | Low | High | Combinations swept |
|---|---|---|---|
| Speed of sound in the burnt gas | 1,129 metres per second | 1,371 metres per second | 48 |
| Chamber diameter | 147 millimetres | 179 millimetres | 48 |
| Chamber pressure | 2.60 megapascal | 8.74 megapascal | 36 |
That band was not assumed. It was recovered. The band assumed on general grounds under The Gain of an Ideal Spike on This Trajectory, without reference to the incident report, is 3 to 10 megapascal for a gas-generator kerosene engine of this thrust class. The two overlap over 3 to 8.74 megapascal, and the overlap as a fraction of each band is
\[\frac{8.74 - 3}{8.74 - 2.60} = \frac{5.74}{6.14} = 0.935, \qquad \frac{8.74 - 3}{10 - 3} = \frac{5.74}{7} = 0.820\]which is 93.5 percent of the recovered band and 82.0 percent of the assumed one. The recovered band does not sit inside the assumed one, because its floor of 2.60 falls below the assumed floor of 3. Two routes sharing no input landed on nearly the same interval, and agreement of that kind appears as an overlap, without containment.
The inference is marked as an inference and its weakest links are named. A thrust chamber is not a plain cylinder, so the mode frequency of a real chamber departs from the textbook value by an amount this article cannot bound. The mode identification is the company’s and not this article’s. And a contraction ratio has to be guessed before a pressure follows. The frequency is not guessed, being published in a sentence written for an entirely different purpose.
The Same Report on Building an Engine Out of Many
The incident report also quantifies the risk that modularity carries, and it does so against the company’s test history.
Two of eleven engines suffered the instability. Inspection found significant erosion of the propellant injectors and liners of Engines 5 and 8, in a pattern the report calls consistent with instability-induced failure. The report also says that in more than three hundred prior tests of the E2 the phenomenon had been observed once, and attributes that single occurrence to an unrelated timing anomaly [Flight 2 Static Fire Update].
Take the historical rate at face value as one in three hundred per engine start. Let $p$ be that per-start probability, dimensionless, and $n$ the number of engines. If the starts were independent, the chance of two or more going unstable would be the binomial tail.
\[\Pr\left[ X \ge 2 \right] = 1 - \left( 1 - p \right)^{n} - n p \left( 1 - p \right)^{n-1}\]At $p = 1/300$ and $n = 11$ that is 6.0 parts in ten thousand, or about one in 1,669. The observed rate of two in eleven is 54.5 times the historical rate, and the expected number of failures at the historical rate is 0.037.
The company’s own leading theory points the same way, that differences in the second block’s propellant feed system produced a higher energy start on the vehicle than on the test stand [Flight 2 Static Fire Update]. The arithmetic adds a measure of how strong that conclusion is. A rate this far above the record does not happen to a batch of eleven by chance.
And it points at the thing modularity has to answer for. A modular engine is many small combustors sharing one manifold, and a shared manifold is a mechanism for correlating their start transients. The independence the calculation above assumes is the assumption the incident falsified. Eleven independent chances of a rare failure is a good trade. Eleven correlated chances is not a trade at all, and the difference between them is a manifold.
The rest of the report is a timeline of a vehicle being lost slowly. Fuel offload began at 4 minutes 20 seconds, liquid oxygen offload at 9 minutes 20 seconds, the mobile water stores ran out at 11 minutes 30 seconds, oxygen and fuel offload capability were lost at 14 minutes 28 and 15 minutes 30, all first-stage telemetry was lost at 20 minutes 43, and at 23 minutes 24 the fire exceeded the thermal capability of the structure and the vehicle buckled onto the pad [Flight 2 Static Fire Update]. The report states that the pad has no direct water supply, and it gives main-stage combustion of about half a second against a planned thirteen. The interval from the water running out to the buckling, and the fraction of the planned firing achieved, are
\[\left( 23 \times 60 + 24 \right) - \left( 11 \times 60 + 30 \right) = 1404 - 690 = 714 \ \mathrm{s}, \qquad \frac{0.5}{13} = 0.038\]so eleven minutes and fifty-four seconds separate the moment the water ran out from the moment the rocket fell over, and the engines achieved 3.8 percent of the test.
The Flight Test Record
The X-63A has no flight test record, and the aerospike flights before it were small, short and far from a launch trajectory. The subsections below set the concept’s long published record against its thin record of flight, and then follow the parent vehicle from its first launch attempt on 10 January 2023 to the company’s renaming on 19 February 2025 [RS1, Gunter’s Space Page] [ABL Space renamed Long Wall].
Sixty Years of Not Flying at Scale
The claim of the president of Invocon, quoted under Programme Origin, that the programme would accomplish what had eluded the rocket community for nearly sixty years reaches back from December 2019 to about 1960, and it is checkable against the literature.
It checks out, and the shape of the literature is the interesting part. The concept has an old and continuous published record and a thin record of flight.
Of this article’s 8,762 surveyed records, 8,568 carry a year, and 488 of those have a title naming a plug, spike or altitude-compensating nozzle, which is 5.7 percent of them, and 17 records without a year name one as well. The earliest is a weather-rocket patent of 1944, and it is followed by a 12-year silence. From 1956 the record is continuous, in the sense that every five-year window from that year to the present contains at least one publication, and the six-year window agrees. The eight-year and ten-year windows agree as well. The founding measurements are a run of reports from one laboratory, Lewis Research Center testing convergent-plug exhaust nozzles against plug design [1954], against outer-shell design [1955], against several design variables at once [1956], and then against an external stream [1959]. The largest gap anywhere after 1956 is 3 years. So the literature begins in earnest 63 years before the sentence that called it nearly sixty, which means the claim was modest by three or four years rather than generous.
| Decade | Records naming a plug, spike or altitude-compensating nozzle |
|---|---|
| 1940s | 1 |
| 1950s | 4 |
| 1960s | 26 |
| 1970s | 47 |
| 1980s | 12 |
| 1990s | 45 |
| 2000s | 106 |
| 2010s | 90 |
| 2020s | 157 |
Read down that column and the flight record’s absence becomes strange rather than unremarkable. The subject was worked on through the nineteen sixties and seventies, went quiet in the nineteen eighties, came back in the nineteen nineties with the single-stage-to-orbit programmes, and has published more in the 2020s so far than in any full decade before it. Seven decades of continuous publication and no flight at the scale of a launch vehicle stage.
The largest aerospike engine this article’s survey documents is the XRS-2200, developed for the X-33 [Rocketdyne XRS-2200]. It was assembled [Kotake, Jones and Cannon 1999], it was hot-fired on a test stand, its control system and its plume were documented [the engine control system] [sea-level plume radiation] [the programme’s engine account] [risk reduction], and the vehicle it was built for was cancelled before it flew, NASA announcing on 1 March 2001 that it would add no further funds and that the X-33 programme would end with its cooperative agreement on 31 March [X-33, X-34 left behind in NASA Space Launch Initiative]. Behind it stands a development record reaching back to the demonstrations of the nineteen seventies [Huang 1974] [Kirby and Martinez 1977]. The nearest thing to a flight test of a launch-vehicle aerospike was the Linear Aerospike SR-71 Experiment, which carried a small linear aerospike pod on the back of an SR-71 and flew seven research flights from October 1997 to November 1998 [LASRE Project]. The pod’s engine was hot-fired twice on the ground and never once in the air [the flight test report] [stability and control results] [hazard mitigation] [the supporting test report]. Liquid oxygen was cycled through it in flight on three occasions, and the hot-fire flight was abandoned because the apparatus leaked [LASRE Project].
Small aerospikes have flown, and the survey holds their reports. A liquid-propellant aerospike engine had been flight-tested on a university and industry sounding rocket by 2004 [Besnard and Garvey 2004], and a multi-chamber liquid oxygen and ethanol aerospike flew later in the same programme [Baker et al 2008]. On 30 and 31 March 2004 two ten-foot solid-fuelled rockets with aerospike nozzles, flown by NASA’s Dryden Flight Research Center, the Air Force Flight Test Center and the Blacksky Corporation, exceeded Mach 1.5 and 26,000 feet [Aerospike Engine Flight Test Successful] [Bui et al 2005]. In the standard atmosphere the pressure at 26,000 feet of geometric altitude is 35.6 percent of sea level, so those plumes did expand against a falling ambient pressure, over nearly two thirds of the fall from sea-level pressure that a launch vehicle meets. On 29 October 2024 a five-metre demonstrator, MIRA II, fired a one-kilonewton liquid oxygen and kerosene linear aerospike for three seconds in flight over the Baltic [POLARIS Spaceplanes Complete First In-Flight Rocket Engine Ignition] [Janardhana 2025]. None of these engines approached the thrust, the burn time or the trajectory of a launch vehicle stage, and none measured altitude compensation across the open, transitional and closed wake.
So the record before ARISE is a concept with sixty-three years of analysis, several pieces of built and successfully fired hardware, one pod carried through the air with cryogenic fluid running through it, and a handful of small aerospikes flown on research and university rockets, none of them on a trajectory representative of a launch. The fact sheet’s claim that ARISE would mark the first representative flight of an aerospike engine after many decades of research and abandoned flight efforts is accurate as stated, and the word representative makes it so [ARISE and Fly].
This article’s arithmetic shows why the gap persisted. The prize, the gain $\eta$ computed under The Gain of an Ideal Spike on This Trajectory, is between 5.3 and 8.5 percent of first-stage impulse against the best fixed nozzle, and it is computed generously. That is a large number for a launch vehicle, and the rocket equation shows how large.
Let $\Delta v_1$ and $\Delta v_2$ be the ideal velocity increments of the two stages in metre per second, $c_{e,2}$ the effective exhaust velocity of the second stage in metre per second, $m_0$ and $m_f$ the mass of the second stage with its payload at ignition and at burnout in kilogram, $m_p$ the payload mass in kilogram, $w$ a gain in $\Delta v_1$ in metre per second, and $\Delta m_p$ the payload that gain buys in kilogram. The rocket equation is linear in exhaust velocity, so a uniform one percent rise in first-stage effective exhaust velocity raises $\Delta v_1$ by one percent. Holding the second stage’s propellant and structure fixed and letting it deliver $w$ less, the rocket equation gives
\[\Delta v_2 = c_{e,2} \ln\frac{m_0}{m_f}, \qquad \frac{\Delta m_p}{m_p} = \frac{w}{c_{e,2}} \cdot \frac{m_0}{m_0 - m_f} \cdot \frac{m_f}{m_p} > \frac{w}{c_{e,2}}\]because both mass ratios exceed one. Taking the staging speed of 2.6 kilometres per second as a floor on $\Delta v_1$, which assumes that gravity and drag losses exceed any help from the Earth’s rotation, and the 3,000 to 3,400 metres per second assumed here for $c_{e,2}$, which is a vacuum specific impulse of about 306 to 347 seconds since the E2’s is not published,
\[w = 0.01 \, \Delta v_1 \ge 0.01 \times 2{,}600 = 26 \ \mathrm{m/s}, \qquad \frac{26}{3{,}400} = 0.0076, \qquad \frac{26}{3{,}000} = 0.0087\]A percent of first-stage exhaust velocity is therefore worth at least three quarters of a percent of payload, and more by the two mass ratios, neither of which is published. The spike’s gain falls mostly in the dense early seconds, when the vehicle is heaviest, so its share of $\Delta v_1$ is smaller than its share of impulse. It is a small number against the cost of qualifying a new nozzle architecture, and a two-stage vehicle can decline to chase it by putting a big nozzle on the upper stage. Altitude compensation is worth most to a single-stage vehicle, and single-stage vehicles keep not being built. The X-33 was one, a single-stage-to-orbit technology demonstrator [stability and control results] [Lockheed Martin X-33]. The RS1A would have been one, for the duration of a suborbital demonstration [AFRL awards agreements under ARISE].
What Happened to It
The RS1 attempted its first orbital launch on 10 January 2023 from Kodiak and failed 10.87 seconds into flight when the first-stage engines shut down [RS1, Gunter’s Space Page]. The second vehicle, an eleven-engine Block 2, was destroyed on the pad on 19 July 2024 in the fire described above [Flight 2 Static Fire Update]. In November 2024 the company announced that it was leaving the commercial launch market [RS1, Gunter’s Space Page], and on 19 February 2025 it announced that it would rename itself Long Wall and turn to missile defence and hypersonic test targets [ABL Space renamed Long Wall].
No source consulted announces the cancellation of the X-63A. What the record shows is a company that no longer builds orbital launch vehicles, a three-year agreement signed in December 2019 whose term expired at the end of 2022, and a demonstrator that has not flown. The inference that the programme is over is drawn by this article and marked as an inference, because an inference from an absence is the weakest kind and the strength of this one comes only from the number of independent things that would have to be true for it to be wrong.
Comparison With Ground Prediction
Nothing flew, so no flight data exist to set beside the ground predictions. The X-63A did not fly within the three-year agreement that ended in 2022, and no source consulted records a flight since. The predictions therefore stand alone, and this section says where each is argued.
The central prediction is an upper bound computed here and not a contractor figure. It is the gain $\eta$ of an ideal altitude-compensating nozzle over the best fixed nozzle on the published ascent, computed under The Gain of an Ideal Spike on This Trajectory. It bounds the prize rather than forecasting the result, because truncation, the base, the wake, cooling and mass all subtract from it. No flight measured the real figure.
The quantity the flight existed to measure has no ground prediction at all. The passage between the open wake, the wake transition and the closed wake on a real ascent is set by a base pressure that a one-dimensional model cannot compute, as The Three Regimes the Programme Names explains, and the flight would have supplied it.
The earlier aerospike record is a ground record as well, at the scale that matters. The small aerospikes flown on research and university rockets gathered nozzle data on short, low flights [Aerospike Engine Flight Test Successful], and neither the XRS-2200 nor the Linear Aerospike SR-71 Experiment, both described under Sixty Years of Not Flying at Scale, produced a flight measurement of altitude compensation against which a ground prediction could be checked.
The record sets test-stand experience beside the vehicle in one place only, the parent stage’s static fire, and that too was a ground test. The excess of its instability rate on the vehicle over the stand’s history, measured by the binomial relation under The Same Report on Building an Engine Out of Many, is attributed by the company’s leading theory to a higher energy start on the vehicle than the test stand produced [Flight 2 Static Fire Update]. The comparison is between a stand and a vehicle on the pad, not between a prediction and a flight.
The parent vehicle flew once and the flight reveals nothing about the aerospike, because the RS1 that failed on 10 January 2023 carried the parent stage and not the aerospike variant [RS1, Gunter’s Space Page].
What the Data Changed
Four results in this article run against what the subject’s usual presentation, or the intuition its vocabulary invites, would lead a reader to expect.
The mean-in-time rule is not the expected answer. A numerical optimisation produces an invariant rather than a number, the same exit-pressure fraction emerging from nine searches that share no parameters, and the derivation shows why. A number that refuses to move is the cheapest possible signal that an identity is hiding underneath.
The separation constraint decides the design, and its answer depends on which criterion is used. On the conical-nozzle threshold that secondary accounts quote, the trajectory-optimal fixed nozzle cannot be built at any chamber pressure considered here. On the correlation that NASA Technical Paper 1207 fits to contoured nozzles it is infeasible only below about 3.5 megapascal. The optimum therefore sits pressed against the limit rather than beyond it.
The parity result reverses the expected answer. The word modular invites the intuition that many small modules steer more finely than few. The directional average is exactly the throttle depth divided by pi at every module count, so more modules do not bring finer steering, and what they do bring depends on whether the count is odd. Nine modules need twenty even ones to beat them.
And the regular-ring formula does not survive the loss of a module. Applying the closed form for a regular ring to a ring with a module missing overstates the surviving authority by up to a factor of 2.62 and, at four modules, by all of it, because a ring with a hole in it is not regular.
The Contemporary Literature
| Cluster | Records |
|---|---|
| Altitude compensation, which is the entire claim | 239 |
| The wake and its closure, which is this nozzle’s own flow | 406 |
| What a fixed nozzle cannot do, being separation and side loads | 365 |
| Bells and cones, being the fixed nozzle itself | 340 |
| Wind tunnel and laboratory nozzles | 54 |
| The spike, the vehicle and the engines this article measures against | 1,050 |
| Many small engines pretending to be one, and what happens when one stops | 186 |
| Steering without a gimbal, and allocating thrust among many | 708 |
| The machinery that feeds the nozzle | 1,791 |
| Liquid rocket engines taken whole | 67 |
| Heat, cooling and erosion in chambers and nozzles | 190 |
| Small thrusters and spacecraft engines | 67 |
| Solid and hybrid motors, on the subjects this article uses | 68 |
| Combined-cycle rocket propulsion | 10 |
| Making it, which is what modular is actually for | 285 |
| The ascent, which is where the integral lives | 1,338 |
| Who is buying this and what they say it is for | 56 |
| Measuring it, because the award says highly instrumented | 422 |
| Plumes and what they do to the vehicle behind them | 698 |
| Exhaust nozzles of aircraft engines, where the plug began | 203 |
| Compressible flow and gas dynamics, the relations the nozzle sections use | 167 |
| Textbooks, histories and surveys of rocket propulsion | 52 |
| Total | 8,762 |
The survey behind this article holds 8,762 research records. 2,546 of them, 29.1 percent, are report primaries, by the identifier rule set out under The Definition of Primary. 8,568 records carry a usable year, running from 1909 to 2027 with a median of 2003. 7,590 of those, 88.6 percent, predate or share the year in which both designations were allocated.
Every record in the survey is cited below, once, in the cluster its title belongs to. The clusters run from the nozzle, its wake and its limits, through the engine, the vehicle and the programme, to the jets, aircraft nozzles, compressible flow and general rocket propulsion on which those subjects rest.
Altitude compensation, which is the entire claim
Plug, aerospike, expansion-deflection, dual-bell and extendible nozzles and the other altitude-compensating concepts, treated as concepts and as flows.
239 records. [Sterbentz and Wilcox 1948] [Ciepluch et al 1954] [Englert et al 1954] [Durham 1955] [Morrell 1956] [The Optimization of Nozzle 1956] [Goldsmith 1958] [Fortini et al 1959] [Musial and Ward 1959] [Aukerman et al 1960] [Corson and Mercer 1960] [Etheridge 1960] [Lovell et al 1960] [Connors et al 1961] [Norton and Swihart 1961] [Rao 1961] [Connors 1962] [Brinich et al 1965] [Holdhusen and Perusse 1965] [Analytical and experimental study 1966] [Aukerman and Trout 1966] [Ward 1966] [Phillips et al 1968] [Quan 1968] [Hosack and Stromsta 1969] [Kors 1970] [Carey et al 1971] [Jeracki and Johns 1971] [Kascak 1971] [Chow and Chang 1972] [Mercer 1972] [Pryor 1972] [Vilenskii et al 1972] [Fuller 1973] [Johns 1973] [Haight and O’Donnell 1974] [Cantrell and Farrow 1975] [Sobin and Martinez 1975] [Maiden 1976] [Maiden 1978] [Jacques 1979] [Petit and Capone 1979] [Meagher 1980] [Obrien and Salmon 1980] [Peters 1980] [Chakravarthy 1981] [Coats 1981] [Obrien 1982] [Heltsley and Crosswy 1983] [Mason and Capone 1983] [Miyajima et al 1983] [Dusa and Wooten 1984] [Krothapalli and Baganoff 1984] [Re and Leavitt 1984] [Wagner and Kassner 1984] [Ewen and Obrien 1986] [Kushida et al 1986] [Pavli et al 1986] [Capone et al 1987] [Carson et al 1987] [Kacynski et al 1987] [Davidian et al 1987] [Kushida et al 1987] [Smith et al 1987] [Eames and Mason 1988] [Idzorek 1988] [Smith 1988] [Yu and Shuen 1988] [Bare and Capone 1989] [Ewen and Obrien 1989] [Martin 1989] [Raiszadeh et al 1989] [Yamamoto et al 1990] [Kim 1991] [McAmis et al 1991] [Reddy and Harloff 1991] [Capone et al 1992] [Martinez 1992] [Parsley and Van Stelle 1992] [Ruffin et al 1992] [Yungster and Trefny 1992] [Carlson and Abdol-Hamid 1993] [Castro and Bustamante 1993] [Immich and Parsley 1993] [Lushchik et al 1993] [Maclean 1993] [Pauley and Dagher 1993] [Smith-Kent et al 1993] [1 Area Ratio Nozzle 1995] [Santiago and Dutton 1995] [Deere and Asbury 1996] [Jankovsky et al 1996] [Ben-Dor et al 1997] [Ellis et al 1997] [Muss et al 1997] [Bykov and Luk’yanov 1998] [Dam et al 1998] [Gilinsky et al 1998] [Graesel and Beylich 1998] [Hamed and Vogiatzis 1998] [Deere and Asbury 1999] [Jankovsky et al 1999] [Dumnov et al 2000] [Soni 2000] [Bulman et al 2001] [Link and Koschel 2001] [Nesman 2001] [Ruf and McDaniels 2002] [Bornhorst 2003] [Hagemann et al 2003] [Jamison 2003] [O’Brien and Farrell 2003] [Yamanaka 2003] [Olles et al 2004] [Yamashita et al 2004] [Marathe and Thiagarajan 2005] [Bulman 2006] [Damira et al 2006] [Davis et al 2006] [Forde et al 2006] [Semenov et al 2006] [Shimizu et al 2006] [Thiagarajan et al 2006] [Tomita et al 2006] [Umbel 2006] [Moríñigo et al 2007] [Tomita et al 2007] [Xiao et al 2007] [Otsubo et al 2008] [Sato et al 2008] [Semenov et al 2008] [Timpano et al 2008] [Davidenko et al 2009] [Fayard et al 2009] [Moshfegh et al 2009] [San et al 2009] [Wang et al 2009] [Solomonov et al 2010] [Tkacik et al 2010] [Verma et al 2010] [Yonezawa et al 2010] [Kim and Lee 2011] [Verma et al 2011] [Jones and Bui 2012] [Sainte-Rose et al 2012] [Ageenko et al 2013] [Analysis of DUAL BELL 2013] [Jones et al 2013] [Jones et al 2013] [Kowsik et al 2013] [Song and Qiang 2013] [Jones et al 2014] [Khandai et al 2014] [Schomberg et al 2014] [Schomberg et al 2014] [Yu et al 2014] [Choi and Huh 2015] [He et al 2015] [Mazurov 2015] [Rajagopal 2015] [Ruf and Jones 2015] [Schneider and Genin 2015] [Zhou et al 2015] [Hwang and Huh 2016] [Schomberg et al 2016] [Semenov et al 2016] [Choi and Huh 2017] [Chutkey et al 2017] [Influence of the Nozzle 2017] [Khan and Mathpal 2017] [Martelli et al 2017] [Moon et al 2017] [Nafis et al 2017] [Shi et al 2017] [Tommila and Hartsfield 2017] [Whitmore and Marshall 2017] [Holder et al 2018] [Moon and Huh 2018] [Ridgway et al 2018] [Barua et al 2019] [Experimental Research on Wall 2019] [Harroun et al 2019] [Laitón et al 2019] [Lee et al 2019] [Mazurov 2019] [Mondal et al 2019] [Reddy et al 2019] [Sun et al 2019] [Zebbiche 2019] [Dhama et al 2020] [He et al 2020] [Samantra et al 2020] [Cimini et al 2021] [Inturi et al 2021] [Kiryushina and Elizarova 2021] [Raju et al 2021] [Application of Double Divergent 2022] [Koyagi and Yokota 2022] [Lee et al 2022] [Levin et al 2022] [Srinivasan et al 2022] [Zachary Muckler 2022] [Comparative Study of OpenFOAM 2023] [Gayathri et al 2023] [Haif et al 2023] [Inturi et al 2023] [Jraisheh et al 2023] [Rathakrishnan 2023] [Swaraj et al 2023] [Tinney 2023] [Yu et al 2023] [Zaheer et al 2023] [Bryan and Langley 2024] [Chang et al 2024] [Duanyang et al 2024] [Ismail et al 2024] [Kirshina et al 2024] [Magalhães and Brójo 2024] [Silva and Brójo 2024] [Song et al 2024] [Trivedi et al 2024] [Xue et al 2024] [Yu et al 2024] [Zeng et al 2024] [Jin 2025] [Kamat 2025] [Kambrath and Thuluvath 2025] [Kaun and Chernyshov 2025] [Levin 2025] [Liu et al 2025] [Nidharia et al 2025] [Tcherak et al 2025] [Xue et al 2025] [Alam et al 2026] [Balusamy and Muruganandam 2026] [James et al 2026] [Murthy et al 2026] [Ornob et al 2026] [Ornob et al 2026]
The wake and its closure, which is this nozzle’s own flow
Base flow, base pressure, base bleed and base heating behind blunt and truncated bodies in supersonic and transonic flow, with the compressible shear layers that bound the recirculation.
406 records. [Chapman 1950] [Bromm and O’Donnell 1954] [Donaldson 1955] [Korst 1956] [Salmi 1956] [Nettle 1958] [Chow 1959] [Cubbage 1959] [Cubbage and Andrews 1960] [Norton and Keith 1960] [Andrews 1961] [Musial and Ward 1961] [Slocumb and Andrews 1961] [Brazzel 1963] [Page and Dixon 1963] [Dahm 1964] [McDonald 1964] [Ying 1964] [Hoglund 1965] [McDonald 1965] [Sergeant 1965] [Wazzan 1965] [Zarin 1965] [Bauer 1966] [Bogdan and Hendershot 1966] [Chang 1966] [Economos 1966] [Emmenthal 1966] [Llinas and Rustay 1966] [Nash 1966] [Panov et al 1966] [Wilson 1966] [Blaha et al 1967] [Hendershot et al 1967] [Mueller 1967] [Roshko and Thomke 1967] [Stahl and Grauer-Carstensen 1967] [Taylor and Tou 1967] [Wilson 1967] [Carpenter and Tabakoff 1968] [Dahm 1968] [Lamb and Hood 1968] [Llinas 1968] [Abbud et al 1969] [Batham 1969] [Huerta 1969] [Johns 1969] [Page 1969] [Addy 1970] [Byington and Hoffman 1970] [Craft and Brazzel 1970] [Demetriades 1970] [Fletcher 1970] [Funk and Johnston 1970] [Hall et al 1970] [Hall et al 1970] [Isaev and Shvets 1970] [Martin and Brazzel 1970] [Page and Reda 1970] [Rom et al 1970] [Gilreath and Schetz 1971] [Greenwald 1971] [Su and Wu 1971] [Kramer 1972] [Pearce 1972] [Spring 1972] [Tanner 1972] [Gerhart 1973] [Gerhart and Korst 1973] [Jakubowski and Lewis 1973] [Messiter et al 1973] [Roache 1973] [Sule and Mueller 1973] [Andrews and Cooper 1974] [Inger 1974] [Chow and Spring 1975] [Gerhart et al 1975] [Giel and Mueller 1975] [Amr and Kassoy 1976] [Approximate Analysis of Base 1976] [Base Combustion Effects on 1976] [Drzewiecki and Foust 1976] [Giel and Mueller 1976] [Theoretical Modeling of the 1976] [Gogish and Pokrovskii 1977] [Starr 1977] [Victor 1977] [White and Agrell 1977] [Cassanto et al 1978] [Diver and Stewartson 1978] [Gogish and Pokrovskii 1978] [Sulyma and Penny 1978] [Tanner 1978] [Williams 1978] [Daniels 1979] [Foust 1979] [Wagner and White 1979] [Daniels 1980] [Schetz et al 1980] [Sulyma 1980] [Wagner and White 1980] [Fox and Bauer 1981] [Gai and Sharma 1981] [Motallebi and Norbury 1981] [Sullins et al 1982] [Sahu et al 1983] [Sahu et al 1983] [Soong and Chu 1983] [Addy et al 1984] [Mueller 1985] [Sinha and Dash 1985] [Addy et al 1986] [Forester and Strom 1986] [Sahu 1986] [Tanaka et al 1987] [Powers 1988] [Ragab 1988] [Sahu and Nietubicz 1988] [Wang 1988] [Bender and Reardon 1989] [Boris et al 1989] [Samimy 1989] [Tang et al 1989] [Demetriades and Brower 1990] [Fourguette et al 1990] [Hunter et al 1990] [Kwok et al 1990] [Marongiu 1990] [Samimy and Abu-Hijleh 1990] [Shen et al 1990] [Zhuang et al 1990] [Clemens et al 1991] [Fourguette et al 1991] [Fourguette et al 1991] [Izadi and Zumwalt 1991] [Leib 1991] [Masalov and Tagirov 1991] [Ota and Goldberg 1991] [Pearce et al 1991] [Wantuck et al 1991] [Erhart 1992] [Kinzie and Martens 1992] [McLaughlin et al 1992] [Moustafa et al 1992] [Nelson 1992] [Poggie et al 1992] [Sahu 1992] [Tanner 1992] [Dougherty and Johnson 1993] [Everson and Nelson 1993] [Everson and Nelson 1993] [Herrin and Dutton 1993] [Hwang et al 1993] [Kumar et al 1993] [Lamb and Oberkampf 1993] [Molezzi and Dutton 1993] [Nietubicz and Gibeling 1993] [Poggie and Smits 1993] [Poggie and Smits 1993] [Reardon and Nelson 1993] [Roshko 1993] [Trifu and Savu 1993] [Tucker and Warsi 1993] [Tucker 1993] [Wang 1993] [Wang 1993] [Williams et al 1993] [Ascoli et al 1994] [Kumar et al 1994] [Martens et al 1994] [Martens et al 1994] [Venkatapathy et al 1994] [Wang 1994] [Xiao and Aggarwal 1994] [Glotov 1995] [Gowda 1995] [Herrin and Dutton 1995] [Lamb and Oberkampf 1995] [Martens and McLaughlin 1995] [Mathur and Dutton 1995] [Najland et al 1995] [Papamoschou 1995] [Park et al 1995] [Sahu and Heavey 1995] [Wang 1995] [Chuang and Chieng 1996] [Clemens et al 1996] [Kaurinkoski 1996] [Loh et al 1996] [Martens et al 1996] [Mathur and Dutton 1996] [Messiter 1996] [Peroomian et al 1996] [Ramaswamy and Loth 1996] [Umemura et al 1996] [Wang 1996] [Freund et al 1997] [Hammer et al 1997] [Kaurinkoski et al 1997] [Lee and Morris 1997] [Peroomian et al 1997] [Sahu and Heavey 1997] [Bannink et al 1998] [Chern et al 1998] [ESDU Data Item estimates 1998] [Kaurinkoski and Hellsten 1998] [Wang 1998] [Wang 1998] [Zhdanov 1998] [Bury and Estivalezes 1999] [Estivalezes and Bury 1999] [Fureby et al 1999] [Saint-Mard and Hendrick 1999] [Tomita et al 1999] [Wang 1999] [Whitmore and Moes 1999] [Forsythe et al 2000] [Wang 2000] [Whitmore and Moes 2000] [Benay and Servel 2001] [Ito and Fujii 2001] [Moraes 2001] [Ottens et al 2001] [Paciorri et al 2001] [Papp and Ghia 2001] [van Raalte and van Oudheusden 2001] [Wang et al 2001] [Baik and Zumwalt 2002] [Bourdon and Dutton 2002] [Fasel 2002] [Forsythe et al 2002] [Ito and Fujii 2002] [Ito and Fujii 2002] [Naughton 2002] [Wei and Chang 2002] [Fureby and Kupiainen 2003] [Ito and Fujii 2003] [Wang et al 2003] [Wang et al 2003] [Yoshizawa et al 2003] [Bourdon and Dutton 2004] [Kastengren and Dutton 2004] [Kawai and Fujii 2004] [Khan and Rathakrishnan 2004] [Kuehner and Dutton 2004] [Subbareddy and Candler 2004] [Wang et al 2004] [August and McKinney 2005] [Choi et al 2005] [DeSpirito 2005] [Kastengren and Dutton 2005] [Kastengren and Dutton 2005] [Kastengren et al 2005] [Kawai and Fujii 2005] [Kuehner and Dutton 2005] [Lee et al 2005] [Park et al 2005] [Tan et al 2005] [Durgesh and Naughton 2006] [Fasel and Sandberg 2006] [Khan and Rathakrishnan 2006] [Lee and Kim 2006] [Ruf 2006] [Shen et al 2006] [Simon et al 2006] [Simon et al 2006] [Sivasubramanian et al 2006] [Sivasubramanian et al 2006] [Turbulent Shear Layers in 2006] [de Feo and Shaw 2007] [Guan et al 2007] [Ko et al 2007] [Kuehner et al 2007] [Lazar et al 2007] [Meliga and Reijasse 2007] [Miki et al 2007] [Gaitonde 2008] [Kim et al 2008] [Nallasamy et al 2008] [Shin et al 2008] [Bonanos et al 2009] [Dharavath et al 2009] [Stark and Wagner 2009] [Venkatakrishnan and Suriyanarayanan 2009] [Verma and Viji 2009] [Maddalena and Dimotakis 2010] [Nallasamy et al 2010] [Bulusu and Garris 2011] [Mehta et al 2011] [Thomas and Unnikrishnan 2011] [Verma and Viji 2011] [Verma and Viji 2011] [Cao et al 2012] [Davis et al 2012] [Dissel et al 2012] [Guo and Wang 2012] [Jeon et al 2012] [Massa 2012] [Hargather et al 2013] [Matsuo et al 2013] [Mehta et al 2013] [Nagao et al 2013] [Belyayev and Kaltayev 2014] [Davis and Kumar 2014] [Mehta et al 2014] [Takahashi et al 2014] [Yu et al 2014] [Zhao-Xin and Bing 2014] [Gentile et al 2015] [Kim and Baek 2015] [Leger and Poggie 2015] [Lin and Yan 2015] [Makasheva et al 2015] [Suryanarayana et al 2015] [Tripathi et al 2015] [Wu et al 2015] [Das and De 2016] [Ecker et al 2016] [Mehta et al 2016] [Patel 2016] [Xue and Yu 2016] [Bulathsinghala et al 2017] [Favale et al 2017] [Guoliang et al 2017] [Kirchner et al 2017] [Koll et al 2017] [Leger et al 2017] [Mehta et al 2017] [Pathan et al 2017] [Shi et al 2017] [Sun 2017] [Ahmed and Khan 2018] [Ilahi Chaudhary et al 2018] [Khan et al 2018] [Khankhasaeva et al 2018] [Saleel et al 2018] [Sethuraman and Khan 2018] [Stack and Gaitonde 2018] [Zha et al 2018] [Bao et al 2019] [Experiment on of Nozzle 2019] [Gu et al 2019] [Kavun et al 2019] [Kirchner et al 2019] [Kirchner et al 2019] [Ko et al 2019] [Nair et al 2019] [Serdarevic-Kadic and Terzic 2019] [Weidner et al 2019] [Zapryagaev et al 2019] [Zhou et al 2019] [Kirchner et al 2020] [Ma et al 2020] [Pu and Jiang 2020] [Scharnowski and Kähler 2020] [Sikroria et al 2020] [Zhang et al 2020] [Zhou et al 2020] [Et. al. 2021] [Karthick 2021] [Soman et al 2021] [Zhou et al 2021] [Ambareen Khan et al 2022] [Bulat et al 2022] [Doshi et al 2022] [Kong et al 2022] [Manu Mohan et al 2022] [Nam et al 2022] [Paul et al 2022] [Smink et al 2022] [Smink et al 2022] [Zhao et al 2022] [Chakraborty 2023] [Husnina Muhamad Zuraidi et al 2023] [Kim et al 2023] [Moffeit et al 2023] [Zhou and Bao 2023] [Heo et al 2024] [Heo et al 2024] [Moffeit et al 2024] [Thakor et al 2024] [Wu et al 2024] [Analysis of Convergent Nozzle 2025] [Ashif et al 2025] [Gianoukakis et al 2025] [Kim and Kim 2025] [Mueller et al 2025] [Murawski and Gaitonde 2025] [Qualters and Zigunov 2025] [Rahma and AlQadi 2025] [Sharma et al 2025] [Sivadas and Deepu 2025] [Sun et al 2025] [Wu and Han 2025] [Kim and Kim 2026] [Kim and Kim 2026] [Krishna et al 2026] [Sun et al 2026] [Wang and Wang 2026] [Zhao et al 2026] [Banton] [Durgesh and Naughton] [Oudheusden and Scarano] [Stein] [Togiti and Lüdeke] [Yan]
What a fixed nozzle cannot do, being separation and side loads
Flow separation in overexpanded nozzles, the side loads it produces, nozzle start and shutdown transients, and supersonic separation stated generally.
365 records. [Cowles and Foster 1949] [McKenney 1951] [Krull and Beale 1956] [Rao 1958] [Green and Nall 1959] [Farley and Campbell 1960] [Holder 1961] [Roschke and Massier 1962] [Glass and Korbacher 1962] [Krayenbuhl 1962] [Boney et al 1963] [Kaufman and Louis G. 1963] [Kaufman and Louis G. 1963] [Application of the method 1964] [Arens 1964] [Craig 1965] [Kalt and Badal 1965] [Callis 1966] [Hoffman 1966] [Hoffman and Thompson 1966] [Johnson 1966] [Back et al 1968] [Land and Tuovila 1968] [Ransom et al 1969] [Chang 1970] [Holden 1970] [Hwang and Murthy 1970] [Snyder and Thompson 1971] [Omori et al 1972] [Ratliff et al 1972] [Smith and Ratliff 1972] [Brilliant and Adamson 1973] [Nave and Coffey 1973] [Rose 1973] [Schmucker 1973] [Schmucker 1973] [Chiou and Hung 1974] [Schmucker 1974] [Token 1974] [Henne 1975] [Wu 1975] [Goetz et al 1976] [Meier 1976] [Abeyounis 1977] [Daines and Boyd 1977] [Seiler 1977] [Wu et al 1977] [Hoffman and Allman 1978] [Holden 1978] [Armstrong 1979] [Chow 1979] [Davin and Petersen 1979] [Miller et al 1979] [Inger 1981] [Carter and Hafez 1982] [Davis 1982] [Rao 1982] [Boraas 1983] [Davis and Abel 1983] [Dawes 1983] [Schmucker 1983] [Ardonceau 1984] [Cline and Wilmoth 1984] [Jurkovich 1984] [Krogmann et al 1984] [Schmucker 1984] [Cline and Wilmoth 1985] [Goodman et al 1985] [Marcum and Hoffman 1985] [Benay and Pot 1986] [Edwards et al 1986] [Mason and Blotter 1986] [McLean and Matoi 1986] [Wang and Hoffman 1986] [Gittler and Kluwick 1987] [Neiland 1987] [Malik and Tagirov 1988] [Shieh 1988] [Danberg and Patel 1989] [Varwig et al 1989] [Edwards and Perkins 1990] [Goeing 1990] [Tagirov 1990] [Chung et al 1992] [Hamed and Kumar 1992] [Inger 1992] [Korte et al 1992] [Rao and Dang 1992] [Guillot and Wilson 1993] [Li and Shi 1993] [Lien and Leschziner 1993] [Schneider 1993] [Chen et al 1994] [Hamed and Kumar 1994] [Ishiguro et al 1994] [Nasuti and Onofri 1994] [Hamed and Vogiatzis 1995] [Szumowski et al 1995] [Bohning 1996] [Dumnov 1996] [Hussaini and Korte 1996] [Hadjadj et al 1997] [Hamed and Vogiatzis 1997] [Forsythe et al 1998] [Hunter 1998] [Igra et al 1998] [Romine 1998] [Wang and Liu 1998] [Asbury and Hunter 1999] [Frey and Hagemann 1999] [Murdock and Welle 1999] [Onofri and Nasuti 1999] [Ostlund and Bigert 1999] [Ostlund and Jaran 1999] [Prozan and Luke 1999] [Reijasse et al 1999] [Zaman and Dahl 1999] [Frey and Hagemann 2000] [Lefrançois et al 2000] [Peters et al 2000] [Morita et al 2001] [Oestlund et al 2001] [Reijasse et al 2001] [Reijasse et al 2001] [Terhardt et al 2001] [Deck and Guillen 2002] [Hagemann et al 2002] [Loh and Zaman 2002] [Nguyen et al 2002] [Tomita et al 2002] [Verma 2002] [Burtschell and Zeitoun 2003] [Chopra et al 2003] [Nguyen et al 2003] [Perigo et al 2003] [Schwane et al 2003] [Tomita et al 2003] [Verma and Ciezki 2003] [Yonezawa et al 2003] [Zerjeski 2003] [Bueno et al 2004] [DalBello et al 2004] [Deck and Nguyen 2004] [Moiseev et al 2004] [Nozzle Design and Optimization 2004] [Papamoschou and Zill 2004] [Pilinski and Nebbache 2004] [Wang 2004] [Wang 2004] [Watanabe et al 2004] [Xia and Schwane 2004] [Östlund et al 2004] [Bourgoing and Reijasse 2005] [Elmiligui et al 2005] [Shi 2005] [Shope and Tatum 2005] [Stark 2005] [Verma and Haidn 2005] [Wang 2005] [Xiao and Tsai 2005] [Yonezawa et al 2005] [Abdol-Hamid et al 2006] [Abdol-Hamid et al 2006] [Dou et al 2006] [Gaffney 2006] [Moríñigo and Salvá 2006] [Nebbache and Pilinski 2006] [Papamoschou and Johnson 2006] [Perrot and Hadjadj 2006] [Shimizu et al 2006] [Verma and Haidn 2006] [Verma et al 2006] [Yonezawa et al 2006] [Yonezawa et al 2006] [Babinsky 2007] [Brown and Ruf 2007] [Smalley et al 2007] [Verma and Haidn 2007] [Weiss et al 2007] [Boccaletto 2008] [Girard and Reijasse 2008] [Inger 2008] [Johnson and Papamoschou 2008] [Moríñigo and Salvá 2008] [Nakakita et al 2008] [Nasuti and Onofri 2008] [Papamoschou et al 2008] [Stark and Wagner 2008] [Verma 2008] [Wang 2008] [Wang and Guidos 2008] [Zhang and Fuchiwaki 2008] [Brown et al 2009] [Debiève and Dupont 2009] [Deck 2009] [Hadjadj and Onofri 2009] [Hadjadj and Perrot 2009] [Ivanov et al 2009] [Mudford et al 2009] [Nagdewe and Kim 2009] [Nagdewe et al 2009] [Nasuti and Onofri 2009] [Pirozzoli et al 2009] [Ponomarev et al 2009] [Ruf et al 2009] [Tomita et al 2009] [Verma and Haidn 2009] [Verma and Haidn 2009] [Baars et al 2010] [Doerffer et al 2010] [Garelli et al 2010] [Johnson and Papamoschou 2010] [Lee et al 2010] [Lijo et al 2010] [Lijo et al 2010] [Martelli et al 2010] [Ruf et al 2010] [Setoguchi et al 2010] [Srivastava et al 2010] [Verma and Haidn 2010] [Verma and Haidn 2010] [Wang et al 2010] [Aghababaie and Taylor 2011] [Baars et al 2011] [Babinsky and Délery 2011] [Baker and Pray 2011] [Dupont et al 2011] [Frate 2011] [Thorsett-Hill et al 2011] [Grilli et al 2011] [Keanini et al 2011] [Lefrançois 2011] [Lefrançois 2011] [Liu and Lu 2011] [Ramsey et al 2011] [Ramsey et al 2011] [Reijasse et al 2011] [Stark and Génin 2011] [Tokura and Maekwa 2011] [Yoon et al 2011] [Baars et al 2012] [Blades et al 2012] [Chaudhuri and Hadjadj 2012] [Fournier et al 2012] [Golub et al 2012] [Lü and Ye 2012] [Olson and Lele 2012] [Shahack et al 2012] [Shams et al 2012] [Stark and Genin 2012] [Wang et al 2012] [Young and Hartfield 2012] [Zhao et al 2012] [Baars and Tinney 2013] [Eyi and Yumusak 2013] [Giglmaier et al 2013] [Sellam et al 2013] [Wang et al 2013] [Benson 2014] [Cuppoletti et al 2014] [Morajkar et al 2014] [Toufique Hasan 2014] [Verma and Haidn 2014] [Wang et al 2014] [Aghababaie and Theunissen 2015] [Bekka et al 2015] [Dippold and Zaman 2015] [Hartfield and Burkhalter 2015] [Jia et al 2015] [Kamali et al 2015] [Lu 2015] [Maicke 2015] [Schomberg et al 2015] [Talsaniya et al 2015] [Yu et al 2015] [Zhang et al 2015] [Chaudhuri and Hadjadj 2016] [Grabe et al 2016] [Jia et al 2016] [Jia et al 2016] [Luchikhina and Tonkov 2016] [Sanal Kumar et al 2016] [Schomberg et al 2016] [Stark and Génin 2016] [Stark and Génin 2016] [Wonnell and Chen 2016] [Bekka et al 2017] [Bekka et al 2017] [Martelli et al 2017] [Martis and Misra 2017] [Meng and Ye 2017] [Verma et al 2017] [Du et al 2018] [Huang et al 2018] [Koike and Babinsky 2018] [Qurooni et al 2018] [Schomberg et al 2018] [Schomberg et al 2018] [Goncalves et al 2019] [Ivanov and Kryukov 2019] [Khobragade et al 2019] [Liu et al 2019] [Martelli et al 2019] [Mishra* et al 2019] [Pulimidi and Lu 2019] [Sansica et al 2019] [Schomberg et al 2019] [Zebiri et al 2019] [Zhu et al 2019] [Daviller et al 2020] [Lee et al 2020] [Lee et al 2020] [Liu and Khine 2020] [Martelli et al 2020] [Panigrahi et al 2020] [Verma and Haidn 2020] [Khaled and Khadidja 2021] [Lee et al 2021] [Peri 2021] [Tavanashad and Shoele 2021] [Abyan et al 2022] [Bhupendra Kumar et al 2022] [Demni et al 2022] [Munro 2022] [Rafiq et al 2022] [Restrepo et al 2022] [Wang et al 2022] [Jones 2023] [Giehler et al 2023] [Hashim and Dharmalingam 2023] [Murugesan et al 2023] [Nedjari et al 2023] [Ramsey et al 2023] [Yang et al 2023] [Zahir et al 2023] [Abada et al 2024] [Allen and Agarwal 2024] [Brójo 2024] [Cerminara et al 2024] [Hu et al 2024] [Jack et al 2024] [Shock-Boundary-Layer Interaction 2024] [Studying the influence of 2024] [Tarnacha 2024] [Tran et al 2024] [Vishwajith 2024] [Cristovão Silva and Brójo 2025] [Dasgupta et al 2025] [Maheswari et al 2025] [Murali 2025] [Nagasai et al 2025] [Ramsey et al 2025] [Stark and Jack 2025] [Yang et al 2025] [Chutkey and Verma 2026] [Tinney and Valdez 2026] [Wei et al 2026] [Zapryagaev et al 2026] [Zhang et al 2026] [Tinney et al] [Mahalingesh] [Mark] [Piquet] [Kolli] [Shock Wave-Boundary Layer Interactions]
Bells and cones, being the fixed nozzle itself
The flow, design and performance of bell, conical and contoured nozzles, which are the fixed nozzles the envelope argument compares against.
340 records. [Bradley and Hale 1909] [Wirt 1924] [Mueller and Peters 1930] [Kantrowitz et al 1943] [Lent 1944] [Puckett 1946] [DeJuhasz 1947] [Puckett 1947] [Isothermal Expansion in Nozzles 1948] [Pinkel 1948] [Altman and Penner 1949] [Oswatitsch and Rothstein 1949] [Penner 1949] [Crown 1950] [Folsom 1950] [McLellan et al 1950] [Krieger 1951] [Seifert and Altman 1952] [Baron 1954] [Dillon and Line 1956] [Meyer 1956] [Rastogi and Pandya 1956] [Dillaway 1957] [Edse 1957] [Branstetter and Setze 1958] [Reynolds 1958] [Rowe 1958] [Feodosiev and Siniarev 1959] [Hardenbergh 1959] [Reynolds and Baldwin 1959] [Analysis Of A New 1960] [Brewer and Levin 1960] [Masugi 1960] [Ahlberg et al 1961] [Lewis and Harrison 1961] [Rao 1961] [Tsuchiya 1961] [Watson 1961] [Bray 1963] [Jack 1963] [Ellison and Johnson 1964] [Moger and Ramsay 1964] [Nozzle performance evaluation program 1964] [SNPO-C nozzle design review 1964] [Stubbs 1964] [Burwell and Hofland 1965] [Krzycki 1965] [Phoebus-2 nozzle design program 1965] [Back and Cuffel 1966] [Crowe and Willoughby 1966] [King 1966] [Manda 1966] [Franciscus and Healy 1967] [Hoffman and Scofield 1967] [Milford and Migdal 1967] [Pirumov 1967] [Crowe et al 1968] [Edenfield 1968] [Frey et al 1968] [Dunn 1969] [Dunn 1969] [Ellis 1969] [Hoffman 1969] [McKevitt 1969] [Mikhailov 1969] [Performance/design and qualification requirements 1969] [Sivells 1969] [Brodsky 1970] [Elliott and Stromsta 1970] [Jaffe and Rao 1970] [Elsbernd and Hoffman 1971] [Glowacki 1971] [Nozzle assembly design status 1971] [Nozzle extension nondestructive test 1971] [Williams 1971] [Back et al 1972] [Dash and Delguidice 1972] [Ellis 1972] [Huff and Groesbeck 1972] [Norton and White 1972] [Sims and Elkin 1972] [Bishop and Sanders 1974] [Cline 1974] [Cline 1974] [Szaniszlo 1974] [Anders 1975] [Brown and Hamilton 1975] [Johnson and Boney 1975] [Sebacher and Lee 1975] [Cline 1976] [Cline 1977] [Thompson and Epstein 1977] [Middlecoff and Thomas 1979] [Chang 1980] [Dutton and Addy 1980] [Chang 1981] [Lin and Shen 1981] [Varner et al 1982] [Fujii and Kutler 1983] [Nachshon 1983] [Shcherbakov 1983] [Tuttle and Blount 1983] [Bevilaqua and Lee 1984] [Wang and Du 1984] [Kwon and Delaney 1985] [Seto et al 1985] [Brown 1986] [Clayton and Back 1986] [Collins 1986] [Desnoyer 1986] [Forde 1986] [Varner 1986] [Bevilaqua and Lee 1987] [Davidian 1987] [Lankford 1987] [Marx 1987] [Schwamborn 1987] [Scott and Visbal 1987] [Thompson and Sutton 1987] [Chang 1988] [Fujiwara et al 1988] [Gregg 1988] [Riedelbauch and Weiland 1988] [Shyne 1988] [Afonin and Butov 1989] [Dang et al 1989] [Griffith 1989] [Long and Hanus 1989] [Ostrander et al 1989] [Vanoverbeke and Shuen 1989] [Baker et al 1990] [Collins et al 1990] [Reddy and Tulasiram 1990] [Berton 1991] [Candler and Perkins 1991] [Davidian and Kacynski 1991] [Dunn et al 1991] [Jacobs and Stalker 1991] [Jacobs and Stalker 1991] [Jentink 1991] [Stubbs et al 1991] [Hackett 1992] [Knoll 1992] [Korte 1992] [Aldo and Argrow 1993] [Davidian and Kacynski 1993] [Keeling 1993] [Kim 1993] [Tsuei and Merkle 1993] [Alcenius and Schneider 1994] [Georgiadis and Yoder 1994] [Kacynski 1994] [Krishnamurthy 1994] [Manski and Hagemann 1994] [Numerical simulation of scarfed 1994] [Rao and Beck 1994] [Yu et al 1994] [Landrum et al 1995] [Pearson et al 1995] [Watanabe et al 1995] [Yagudin 1995] [Follett 1996] [Follett et al 1996] [Landrum and Beard 1996] [Barnhart 1997] [Wu et al 1997] [Chen and Hou 1998] [Cook et al 1998] [Wong et al 1999] [Ahmad and McCool 2000] [Chang 2002] [Lucia et al 2002] [Dyer et al 2003] [Hagemann et al 2003] [Mouronval et al 2003] [Cooper and Shepherd 2004] [Development of an Experiment 2004] [Gamble et al 2004] [Rocket Engine Nozzle Concepts 2004] [TabeJamaat and Pezeshki 2004] [Chang et al 2005] [Senthil Kumar et al 2005] [Chang et al 2006] [LeMoyne 2006] [Zebbiche and Youbi 2006] [Al-Hasan and Schnerr 2007] [Behera and Srinivasan 2007] [Toufik and ZineEddine 2007] [Chang et al 2008] [Gaitonde 2009] [Liang et al 2009] [Metghalchi 2009] [Yoder et al 2009] [Boccaletto and Dussauge 2010] [Ekanayake et al 2010] [Faure et al 2010] [Krushna 2010] [Majdalani and Maicke 2010] [Moríñigo and Salvá 2010] [Yang 2011] [Zebbiche 2011] [Albanese et al 2012] [Chen et al 2012] [Krieg and Mohseni 2012] [Mehta et al 2012] [Naung et al 2012] [Sribonfha et al 2012] [Wu et al 2012] [Eyi and Yumusak 2013] [Kennedy 2013] [Yumusak 2013] [Carter et al 2014] [Charan 2014] [Garelli et al 2014] [Guo et al 2014] [Kumar et al 2014] [Patel 2014] [Srinivas and Potti 2014] [Sun and Wei 2014] [Takahashi and Gibson 2014] [Takovitskii 2014] [Wang et al 2014] [Webster 2014] [Yildizlar et al 2014] [Deshpande and Narappanawar 2016] [Eto et al 2016] [Isono et al 2016] [Jegede and Crowther 2016] [Jeon et al 2016] [Kotteda and Mittal 2016] [Kumar and Arularasu 2016] [Ramji et al 2016] [Athota Rathan Babu et.al. 2017] [Maicke 2017] [Manigandan and Vijayaraja 2017] [A.K. and P.S. 2018] [Francisco 2018] [Jéger and Veress 2018] [Kimura et al 2018] [Lee 2018] [Matveev et al 2018] [Ramesh Kumar and Devarajan 2018] [Sabirzyanov et al 2018] [Zebbiche 2018] [A. Adaikalaraj et al. 2019] [Conte et al 2019] [Control of CD Nozzle 2019] [Control of Nozzle Flow 2019] [Ferrero and Pastrone 2019] [Inspection of Supersonic Flows 2019] [Scott 2019] [Juhasz 2019] [Kimura and Matsuda 2019] [Rocket Nozzle Performance 2019] [Sarweswaran et al 2019] [Sundararaj et al 2019] [Zebbiche 2019] [Zhang et al 2019] [Abada et al 2020] [Harroun et al 2020] [Harroun et al 2020] [Jijo Unnikrishnan 2020] [Krishna 2020] [Kishore 2020] [Pietrykowski et al 2020] [Raja 2020] [Sosa et al 2020] [Stetsyuk et al 2020] [Harikrishnan and Lokavarapu 2021] [Harroun et al 2021] [Jeon and Park 2021] [Kong and Wang 2021] [Koyagi et al 2021] [Kun et al 2021] [Mikhailov 2021] [Mkrtchyan and Kochetkov 2021] [Numerical Simulation of Internal 2021] [Shaidullin et al 2021] [Veeresh Kumar et al 2021] [Gandge and Hampali 2022] [He et al 2022] [Ilie and Sullivan 2022] [Jayaprakash et al 2022] [Kirshina et al 2022] [Liu et al 2022] [Paxson et al 2022] [Voronin 2022] [Wang et al 2022] [Agarwal 2023] [Ahmed and Gross 2023] [Antony and Waldecker 2023] [Computational and Investigational Proportional 2023] [Huang et al 2023] [Huang et al 2023] [Kateryna and Nataliya 2023] [Manoj et al 2023] [Pandit et al 2023] [Parkash and Sharda 2023] [Salhi 2023] [The Pressure Analyzing of 2023] [Zhang et al 2023] [Alili et al 2024] [Chaudhary 2024] [Design and Analysis of 2024] [Dsouza et al 2024] [Gidaspov et al 2024] [Hayashi 2024] [Impact of static pressure 2024] [Nataliya et al 2024] [Paul et al 2024] [Popovich et al 2024] [Sethuraman et al 2024] [Yang et al 2024] [Yunfeng 2024] [Zhang et al 2024] [Cantariño and Pulido 2025] [CFD Based Design Optimization 2025] [Durst et al 2025] [Gummadidala et al 2025] [Nair et al 2025] [Neelakandan et al 2025] [Abada et al 2026] [Alexandrov 2026] [Amir and Abbasi 2026] [Dai et al 2026] [Gadre 2026] [Shek 2026] [Soni 2026] [Tian et al 2026] [Capatina] [Cerantola] [Paxson et al] [Ghosh and Friedrich] [Miki et al] [Lucia et al] [Nak] [Pitot de la Beaujardiere et al] [Quintao] [Rozendaal] [Wide-Throat Nozzle]
Wind tunnel and laboratory nozzles
Nozzles of wind tunnels, shock tunnels and laboratory rigs, where the title studies the nozzle’s own flow.
54 records. [Beckwith et al 1984] [Chen et al 1990] [Chen et al 1991] [Chen et al 1992] [Schmisseur et al 1996] [Palma et al 1998] [Schneider 1998] [Watanabe et al 2001] [Kaneko et al 2002] [Lacey et al 2002] [O’Byrne et al 2002] [OByrne et al 2002] [Chue et al 2003] [Lacey et al 2003] [Palma et al 2003] [Gaffney 2004] [Komuro et al 2005] [O’Byrne et al 2005] [Scott et al 2005] [Gaffey et al 2006] [Gaffney and Norris 2008] [Yen and Martindale 2008] [Candler and McConaughey 2009] [Gaffney 2009] [Bensassi et al 2010] [Wu and Radespiel 2013] [Gong et al 2014] [Axdahl 2015] [Prabhu et al 2015] [Sadraey 2015] [Drozda et al 2017] [Huang et al 2017] [Huang et al 2017] [Yu et al 2017] [Brown 2019] [Gao et al 2019] [Toniato et al 2019] [Gokcen 2021] [Sung and Choi 2021] [Sung and Choi 2021] [Gokcen 2022] [Kovács et al 2022] [Bratman et al 2023] [Novel design of variable 2023] [Sung et al 2023] [Zahir et al 2023] [Gokcen et al 2024] [Jakobs et al 2025] [Shen et al 2025] [Gokcen 2026] [Shen et al 2026] [Davey] [Gokcen et al] [Towsey]
The spike, the vehicle and the engines this article measures against
Aerospike engines and their hardware, the X-33 and its XRS-2200, the Linear Aerospike SR-71 Experiment, and the named engines and launch vehicles this article sets beside the X-63A.
1,050 records. [Walker 1954] [Burdett 1956] [Salmi and Cortright 1956] [Of current interest Liquid 1958] [Krase 1959] [Healy 1960] [Plug nozzle engine 1960] [Stillwell and Larson 1960] [Eggers 1961] [Page and Meyer 1961] [Plug nozzle” rocket engine 1961] [Mercer and Salters 1963] [Peters and Hall 1963] [Rocketdyne Canoga Park Ca 1963] [Rocketdyne Canoga Park Ca 1963] [Rocketdyne Canoga Park Ca 1963] [Scott 1963] [Angelino 1964] [Runkel 1964] [Studhalter 1964] [Wickham 1964] [Bushnell et al 1965] [Kalange and Alcott 1965] [Kreiter 1965] [Nagy 1965] [Postflight Evaluation of Atlas-Centaur 1965] [Rupert 1965] [Williams 1965] [Aerojet-General Corp Sacramento Ca 1966] [Brice et al 1966] [Busse and Leffler 1966] [Cowell and Madison 1966] [Demorest and Wilks 1966] [Kalogeris 1966] [Lee 1966] [Oconnor 1966] [Vehko and Ruhland 1966] [Wolfe 1966] [Alcott and Kalange 1967] [Berrier and Mercer 1967] [Bollermann and Walker 1967] [Bolster 1967] [Borelli and Carroll 1967] [Bushnell et al 1967] [Curry and Uselton 1967] [Frazier 1967] [Haussler 1967] [Kunz 1967] [Lane and Chalfant 1967] [Mueller and Oberkampf 1967] [Preliminary launch vehicle flight 1967] [Barrows 1968] [Bauer 1968] [Bresnahan 1968] [Bresnahan and Johns 1968] [De Pauw et al 1968] [Hall and Mueller 1968] [Hall 1968] [Harrington and Wasko 1968] [Kniss 1968] [Rafferty 1968] [Wasko 1968] [Bender 1969] [Berrier 1969] [Bresnahan 1969] [Cikanek et al 1969] [Flight performance of Atlas-Agena 1969] [Flight performance of the 1969] [Hosack and Stromsta 1969] [Huntley and Samanich 1969] [Mueller and Sule 1969] [Schorr 1969] [Baird and Bergman 1970] [Berrier and Mercer 1970] [Buchanan 1970] [Burley 1970] [Chenoweth and Jeracki 1970] [Clark et al 1970] [Ellsworth and McPherson 1970] [Franklin and Tinsley 1970] [Furst and Hazell 1970] [Harrington 1970] [Harrington 1970] [Lane and Redman 1970] [Rillings and Roy 1970] [Steffen 1970] [Bruckner and Douglas 1971] [Chambellan and Stepka 1971] [Chamberlin and Samanich 1971] [Chenoweth and Lieberman 1971] [Hall and Mueller 1971] [Hoffmann and Humphreys 1971] [Loftus 1971] [Modified RS2101 rocket engine 1971] [Mulready 1971] [Space shuttle main engine 1971] [Berrier 1972] [Brausch 1972] [Bresnahan 1972] [Clark and Lieberman 1972] [Graber and Clark 1972] [Hall and Mueller 1972] [Mueller and Sule 1972] [Muller et al 1972] [Samanich 1972] [Scharton et al 1972] [Berrier 1973] [Carey and Ellis 1973] [Castenholz 1973] [Chamberlin 1973] [Furst 1973] [Kirby 1973] [Lacroix 1973] [Lamont 1973] [Liang and Kleinbub 1973] [Seitz and Searle 1973] [Space shuttle main engine 1973] [Straight and Harrington 1973] [Wheelock 1973] [Burley and Head 1974] [Burley and Johns 1974] [Harrington and Schloemer 1974] [Harrington et al 1974] [Head 1974] [Holland 1974] [Furst 1975] [Harrington et al 1975] [Brusch 1976] [David 1976] [Galanga and Mueller 1976] [Harrington et al 1976] [Lee 1976] [Nosek and Straight 1976] [Riddle and Nebel 1976] [Yamada et al 1976] [David 1977] [Diem and Kirby 1977] [Johnson 1977] 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2011] [Letchworth 2011] [Majamaki and Lee 2011] [Masuda and Kajiwara 2011] [Miller 2011] [Mizukaki et al 2011] [Mobley et al 2011] [Mosley 2011] [Noori and Shahrokhi 2011] [Pao et al 2011] [Reijasse et al 2011] [Simmons and Branam 2011] [Stark and Génin 2011] [Starr et al 2011] [Tomek et al 2011] [Tsuboi et al 2011] [VanHooser and Bradley 2011] [Wagner and Schlechtriem 2011] [Wagner et al 2011] [Whitmore et al 2011] [Yonezawa et al 2011] [Zhang and Liu 2011] [Accettura and Genito 2012] [Butt et al 2012] [Chutkey et al 2012] [Ferlin 2012] [Genin and Stark 2012] [Genin et al 2012] [Holguin 2012] [Lemieux and Murray 2012] [Narimiya et al 2012] [Proschanka et al 2012] [Proshchanka et al 2012] [Rajesh et al 2012] [Takahashi et al 2012] [Yonezawa 2012] [Casiano et al 2013] [Génin et al 2013] [Hulka et al 2013] [Karthikeyan et al 2013] [Kuwamori et al 2013] [Levack and Sack 2013] [Levack and Sack 2013] [Long et al 2013] [Meiss and Besnard 2013] [Miyazawa et al 2013] [Takahashi et al 2013] 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et al 2017] [Viji et al 2017] [Barklage et al 2018] [Baskaran et al 2018] [Chutkey et al 2018] [Dumitrescu et al 2018] [Kim and Huh 2018] [Lai et al 2018] [Masdari et al 2018] [Schnabel and Brophy 2018] [Schneider et al 2018] [Schwer et al 2018] [Takahashi et al 2018] [Tsuboi et al 2018] [Zmijanovic et al 2018] [Ahmad et al 2019] [Akib et al 2019] [Azuma et al 2019] [Boglis and Stoica 2019] [Fotia et al 2019] [Ha et al 2019] [Jourdaine et al 2019] [Kawashima et al 2019] [Kim et al 2019] [Loosen et al 2019] [Sieder et al 2019] [Stark et al 2019] [Weinig 2019] [Whitmore and Armstrong 2019] [Bailey 2020] [Bolgar et al 2020] [Ferlauto et al 2020] [Ferlauto et al 2020] [Ferrero et al 2020] [Génin et al 2020] [Heath and Bell 2020] [Kurita et al 2020] [Lengade 2020] [Loosen et al 2020] [Léger et al 2020] [Shenoy et al 2020] [Stephen A 2020] [Stewart et al 2020] [Tian et al 2020] [Udaiyakumar et al 2020] [Zhu et al 2020] [Sai Shankaran et al 2021] [Balaji et al 2021] [Dakka and Dennison 2021] [Ferlauto et al 2021] [Ferrero et al 2021] [Flight of the aerospike 2021] [Ghosh and Gunasekaran 2021] [Haif et al 2021] [Huang et al 2021] [Jacobs et al 2021] [Khan et al 2021] [Kumar Mishra et al 2021] [Léger et al 2021] [Naseh and Alipoor 2021] [Raja 2021] [Raju et al 2021] [Reynolds et al 2021] [Sankari Ashok Alshiya et al 2021] [Sequeira and Sanjay 2021] [Tobias and Agrawal 2021] [Yokoo et al 2021] [Bonnet et al 2022] [Choi and Huh 2022] [Emelyanov et al 2022] [Emelyanov et al 2022] [Ferrero et al 2022] [Gupta et al 2022] [Ha and Kim 2022] [Haif et al 2022] [Jain and Kumar 2022] [Jency et al 2022] [Langner et al 2022] [Liu et al 2022] [McCormick and Sanders 2022] [Ramanujachari et al 2022] [Sathish et al 2022] [Shaffer and Hairston 2022] [Sriganapathy et al 2022] [Wang et al 2022] [Abdallah Elhirtsi and Zebbiche 2023] [Aithani et al 2023] [Bell and Agrawal 2023] [Di Cicca et al 2023] [Gagliardi 2023] [Gahnolia et al 2023] [Kaun and Chernyshov 2023] [Zaman and Fagan 2023] [Langner et al 2023] [Lee et al 2023] [Legros et al 2023] [Ma et al 2023] [Nagaral et al 2023] [Naik et al 2023] [Numerical Investigation of Supersonic 2023] [Numerical investigation on thrust 2023] [Pyle et al 2023] [Resta 2023] [Soman et al 2023] [Sundaria et al 2023] [Tobias and Agrawal 2023] [Tokoh et al 2023] [Wang et al 2023] [Wu et al 2023] [Zebbiche and Youbi 2023] [Sam Daniel Fenny et al 2024] [Fagan et al 2024] [Balaji et al 2024] [Bayir and Akbıyık 2024] [Bharathwaj et al 2024] [Bindal et al 2024] [Bindal et al 2024] [Carbone et al 2024] [Design and Numerical Analysis 2024] [Di Cicca et al 2024] [Fadigati et al 2024] [Golliard and Mihaescu 2024] [Golliard and Mihaescu 2024] [Kim 2024] [Kret et al 2024] [Langner et al 2024] [Marsilio et al 2024] [Nasuti et al 2024] [Paramo and Arizpe 2024] [Patil et al 2024] [Peshkov and Tret’yakov 2024] [Pharokhipanah and Bouckenooghe 2024] [Resta et al 2024] [Reza et al 2024] [Scarlatella et al 2024] [Scheithauer et al 2024] [Selbmann et al 2024] [Sieder-Katzmann et al 2024] [Strobel and Macneil 2024] [Swathish and Rakesh 2024] [Traudt 2024] [Wang et al 2024] [Wilson et al 2024] [Zaman et al 2024] [Zhang et al 2024] [Zhou et al 2024] [Anafi and Qasem 2025] [Chehat et al 2025] [Di Cicca et al 2025] [Fadigati et al 2025] [Fadigati et al 2025] [Fiore et al 2025] [Golliard and Mihaescu 2025] [Golliard and Mihaescu 2025] [Grishko et al 2025] [Hu 2025] [Inturi et al 2025] [Janardhana 2025] [Jarvis 2025] [Kbab et al 2025] [Zaman et al 2025] [Korth et al 2025] [Langner et al 2025] [Liu et al 2025] [Selvam et al 2025] [Minaz and Meram 2025] [Monokrousos et al 2025] [Motomura 2025] [Nanni et al 2025] [Ozaki et al 2025] [Piparava and Zhang 2025] [Serrato 2025] [Singh and Ravi 2025] [Wang et al 2025] [Wißmann et al 2025] [Wißmann et al 2025] [Zhou et al 2025] [Zhou et al 2025] [Abada et al 2026] [Aslan et al 2026] [Bakker et al 2026] [Benlembarek et al 2026] [Carbone et al 2026] [Chandler et al 2026] [Fagan and Zaman 2026] [Fiore et al 2026] [Golliard and Mihaescu 2026] [Karim et al 2026] [Korth et al 2026] [Legros et al 2026] [Liu et al 2026] [Liu et al 2026] [Lizcano et al 2026] [Magnani et al 2026] [Masse et al 2026] [Nikolayev et al 2026] [Pyle and Jacobs 2026] [Pyle and Jacobs 2026] [Sadiqi and Naimzad 2026] [Sen Ayush et al 2026] [Sharrouf et al 2026] [Steiner and Bauer 2026] [Sun et al 2026] [Tcherak et al 2026] [Wanchoo et al 2026] [Wißmann et al 2026] [Aerospike engine] [Fagan and Zaman] [Fagan et al] [Beebe] [Brennen] [Brock] [Campbell et al] [Case] [Du] [Imbaratto] [Jackson et al] [Korth] [Korth et al] [Zaman et al] [Zaman et al] [Legros] [Mori et al] [Niimi et al] [Shahrokhi and Noori] [Su]
Five titles in this cluster report aerospike engines in flight, from a liquid-propellant engine flight-tested by 2004 to a linear aerospike ignited in flight in 2024 [Besnard and Garvey 2004] [Bui et al 2005] [Baker et al 2008] [Flight of the aerospike 2021] [Janardhana 2025]. Each is a small engine on a short flight, which is why the flight record of the subject is thin rather than empty.
Many small engines pretending to be one, and what happens when one stops
Clustered and multiple engines, modular engines, engine-out, nozzle and thruster arrays, and the reliability of engines and stages.
186 records. [Coplen 1957] [Church et al 1962] [Hale 1963] [Lloyd and Lipow 1963] [Hisler 1964] [Peters 1964] [Cubbage and Weidner 1965] [Multiple exhaust nozzles for 1965] [Chu et al 1966] [Echols 1966] [Hopson and McAnelly 1966] [Kennedy 1966] [Mueller 1966] [Cover and Wasko 1967] [Fergus 1967] [Walker and Zeigler 1968] [Brewer and Craven 1969] [Meadows 1969] [Newman 1969] [Pierman 1969] [Christie and Greenberg 1973] [Vicario et al 1973] [Demetriades 1977] [Thornton 1977] [Fault-tolerant, high reliability electronic 1979] [Newirth and Bosco 1980] [Rabinowitz et al 1980] [Dyson 1981] [Petrie 1982] [Bellows et al 1984] [Russell and Chu 1984] [Cikanek 1986] [Maram and Barkhoudarian 1987] [Nguyen et al 1988] [Powers et al 1988] [Swan and Vizzini 1988] [Wolting 1988] [Kytasty et al 1989] [O’Hara 1989] [Perry 1989] [Royer and Wolting 1989] [Annual Health Monitoring Conference 1990] [Escher et al 1990] [Flatau and Vanmoorhem 1990] [Newlin et al 1990] [Sutharshana et al 1990] [Walker and Baumgartner 1990] [Bickford and Gage 1991] [Cannon et al 1991] [Cruse and Mahadevan 1991] [Gardner et al 1991] [Helmicki 1991] [Millis 1991] [Duyar and Eldem 1992] [Heller et al 1992] [Lee and Steincamp 1992] [Leonard and Kisko 1992] [Mason and Broadhurst 1992] [Rocket engine propulsion ‘system 1992] [Sakala and Raines 1992] [Valenti 1992] [Buelow et al 1993] [Mahadevan et al 1993] [Reichmuth et al 1993] [Sanders and Jerman 1993] [Solanyk 1993] [Wang 1993] [Health monitoring and controls 1994] [Kato et al 1995] [Shepard and Patino 1995] [Anderson et al 1996] [Brown 1996] [Murphy and Voss 1996] [Ungewitter et al 1996] [Wu et al 1996] [Dunn et al 1998] [Huang and Weber 1998] [Sauvageau and Allen 1998] [Zhang et al 1998] [Escher et al 1999] [Huang and Weber 1999] [Liu and Wang 1999] [Pytanowski 1999] [Maggio and Pellacio 2000] [Butas et al 2001] [Gurov 2001] [Kazukiewicz 2001] [Kinzie et al 2001] [Santi 2001] [Daren and Jianbo 2002] [Guikema and Pate-Cornell 2003] [Wu et al 2003] [Research on the High 2004] [The Use of Artificial 2004] [Unal et al 2004] [Buswell 2005] [Hill and Acosta 2005] [Huang et al 2005] [Schwabacher 2005] [Wu 2005] [Hara et al 2006] [Cotae et al 2007] [Krevor and Wilhite 2007] [Maemura and Igarashi 2007] [Martin 2007] [Rusick 2007] [Safie and Weldon 2007] [Safie and Weldon 2007] [Schwabacher et al 2007] [Verma et al 2007] [Anderson et al 2008] [Horisawa et al 2008] [Lurui Xia et al 2008] [Aytulun and Ermis 2009] [Gelder and Bos 2009] [Losik 2010] [Manning et al 2010] [Morse et al 2010] [Gao et al 2012] [Holladay et al 2012] [Li et al 2012] [Mattenberger et al 2012] [Hirata and Yonezawa 2013] [Marcos et al 2013] [Roshanian and Ebrahimi 2013] [Yonezawa et al 2013] [Corl et al 2014] [Huang 2014] [Kim and Hwang 2014] [Safie et al 2014] [Intelligent Fault Diagnosis for 2015] [Daniel et al 2016] [Kanazaki et al 2016] [Zhang 2016] [Du 2017] [Isono et al 2017] [Murugesan et al 2018] [Huang and Aguilar 2019] [Iwaguchi et al 2019] [Li et al 2019] [Podolchak 2019] [Moscow Aviation Institute National Research University et al 2019] [Ireland and Gonzales 2020] [Li et al 2020] [Liang et al 2020] [Richardson et al 2020] [Sensor Failure Management in 2020] [Dolgopolov et al 2021] [Liao et al 2021] [Zeng et al 2021] [Li et al 2022] [Niu et al 2022] [Pedari and Ossareh 2022] [Zhang et al 2022] [Nagappa 2023] [Shi et al 2023] [Xiang 2023] [Biao et al 2024] [Gulczyński et al 2024] [Hornyák 2024] [Huang et al 2024] [Lakshmipuram Raghu et al 2024] [Ma et al 2024] [Murata et al 2024] [Murata et al 2024] [Ren et al 2024] [Cha and Ko 2025] [Kayser et al 2025] [Navarro-Tapia et al 2025] [Ram et al 2025] [Zhang et al 2025] [Chen et al 2026] [Etzenbach et al 2026] [Heldens et al 2026] [Kumar and Shine 2026] [Santhoshkumar et al 2026] [Xiao et al 2026] [Craig et al] [Craig et al] [Benzing et al] [Engine Monitoring System Reliability] [Fernandez] [Liquid Rocket Engine Reliability] [Murata] [Pooley et al] [Pugh]
Steering without a gimbal, and allocating thrust among many
Thrust vector control of every kind, launch vehicle attitude control, thrust allocation among many thrusters, and jets injected into a supersonic crossflow.
708 records. [Burdett 1946] [Blaszak and Fahrenholz 1960] [McAulay and Pavli 1960] [Wu et al 1961] [Drewry and Harmoning 1962] [Fox et al 1962] [Newton and Spaid 1962] [Shandor and Walker 1962] [Broadwell 1963] [Broadwell 1963] [Gagnon et al 1963] [Green and McCULLOUGH 1963] [Karamcheti and Tao-Sze Hsia 1963] [Baxter 1964] [Billet 1964] [Charwat and Allegre 1964] [Conrad 1964] [James and Young 1964] [Kirby and Van Vooren 1964] [Klenk 1964] [Salemka 1964] [Sehgal and Wu 1964] [Shandor 1964] [Baxter 1965] [Guhse 1965] [Hall 1965] [Hollstein 1965] [Jones 1965] [Kilpatrick 1965] [Minimax attitude control of 1965] [Moody and Nakai 1965] [Pneumatic digital actuator for 1965] [Study of selected thrust 1965] [Summer 1965] [Digital compensation of the 1966] [Guhse 1966] [Inouye and Nottage 1966] [Phillips 1966] [Proportional solid propellant secondary 1966] [Robbins and Zebrowski 1966] [Smith 1966] [Summer 1966] [Summer 1966] [Borsody et al 1967] [Budris 1967] [Curtis and Hagstrom 1967] [Curtis and Hagstrom 1967] [Delozier and Kasselmann 1967] [Hansen et al 1967] [Hardy et al 1967] [Lindquist 1967] [McIntosh et al 1967] [Phillips 1967] [Voyager spacecraft system Task 1967] [Blackstone 1968] [Blatter and Keranen 1968] [Borsody and Teren 1968] [Borsody et al 1968] [Chisel and Rose 1968] [Daniele and Teren 1968] [Hung and Thompson 1968] [Investigation of gaseous secondary 1968] [Jenkins and Roy 1968] [Neilson et al 1968] [Neilson et al 1968] [Neilson et al 1968] [Phillips 1968] [Borsody et al 1969] [Cassel et al 1969] [Comparin and Wilson 1969] [Comparin and Wilson 1969] [Harper 1969] [Leondes et al 1969] [Penchuk and Schlundt 1969] [Rivard 1969] [Smith 1969] [Abbott and Walker 1970] [Avduevskii et al 1970] [Boland 1970] [Erickson et al 1970] [Jenkins 1970] [Keltz and Ringwall 1970] [Moody 1970] [Nara et al 1970] [Thompson and Hung 1970] [Thrust Vector Control /TVC/ 1970] [Thrust Vector Control /TVC/ 1970] [Thrust vector control ground 1970] [Widnall 1970] [Wilson and Comparin 1970] [Doyle 1971] [Eatough 1971] [Fleischer and McGlinchey 1971] [Sayles and Levinsky 1971] [Seltzer and Siljak 1971] [Wong 1971] [Yamanaka 1971] [Guhse and Thompson 1972] [Nunn and Brandt 1972] [Burgess 1973] [Cauzza and Mockenhaupt 1973] [Kirkley 1973] [Shrewsberry 1973] [Vandierendonck et al 1973] [Herbert and Reinhardt 1974] [McGlinchey 1974] [Solid rocket thrust vector 1974] [Toney 1974] [Counter and Brinton 1975] [Huskey 1975] [Woodberry 1975] [Zeamer 1975] [Esker 1976] [Masuya et al 1976] [Pickell 1976] [Capone and Maiden 1977] [Experimental studies on thrust 1977] [Levi and Stahlhuth 1977] [Zeamer 1977] [Kosmann et al 1978] [Lloyd and Thorp 1978] [Pagan 1978] [Peters 1978] [Chinn et al 1979] [Effects of Reynolds number 1979] [Finnerty and Price 1979] [Fitzgerald and Kampe 1979] [Fujimoto and Shigemoto 1979] [Pagan 1979] [Space transportation system solid 1979] [Strickland and Shigemoto 1979] [Lominick 1980] [McCool et al 1980] [Price 1980] [Giragosian 1981] [Pagan 1981] [Ellis and Kearney 1982] [Haloulakos 1982] [Jonsson and Malmberg 1982] [Penchuk and Croopnick 1982] [Re and Berrier 1982] [Scarboro 1982] [Turner 1982] [Akiba et al 1983] [Carroll and Cox 1983] [Drzewiecki 1983] [Penchuk and Croopnick 1983] [Redmon 1983] [Bell 1984] [Berrier and Leavitt 1984] [Wooten et al 1984] [Wu and Chow 1984] [Leavitt 1985] [Wie 1985] [Capone and Mason 1986] [Cunningham et al 1986] [Hokenson 1986] [Imlay 1986] [Leitner 1986] [Mason and Burley 1986] [Porzio and Franke 1986] [Simmons 1986] [Wilson and Riccio 1986] [Yezzi and Donguy 1986] [Bare and Reubush 1987] [Branislav et al 1987] [Capone and Bare 1987] [Simmons et al 1987] [Berrier 1988] [Berrier and Mason 1988] [Lambert and Franke 1988] [Mason and Berrier 1988] [Mockenhaupt 1988] [Taylor 1988] [Wilson 1988] [Berdoyes 1989] [Boelitz 1989] [Dhuyvetter and Lam 1989] [Schirmer and Capone 1989] [Talda and Franke 1989] [Wassom et al 1989] [Whitesides et al 1989] [Berrier and Taylor 1990] [Bursey and Dickinson 1990] [Capone et al 1990] [Cavalleri et al 1990] [Friddell and Franke 1990] [Roth 1990] [Soong 1990] [Taylor 1990] [Carpenter et al 1990] [Tiarn and Cavalleri 1990] [Adler et al 1991] [Azevedo 1991] [Carpenter 1991] [Carson and Capone 1991] [Caton and Franke 1991] [Cavalleri et al 1991] [Cowan and Myers 1991] [Dodge and Kana 1991] [Hermanson and Winter 1991] [King et al 1991] [Lutze et al 1991] [Ryan and Franke 1991] [Saugen and Adler 1991] [Wehmueller and Hauser 1991] [White 1991] [Wing and Asbury 1991] [Berdoyes and Ellis 1992] [Capone et al 1992] [Carpenter et al 1992] [Cowan and Myers 1992] [Doonan and Kuchar 1992] [Hamke et al 1992] [Johnson 1992] [Murray 1992] [Prins et al 1992] [Schmidt 1992] [Wing and Asbury 1992] [Aso et al 1993] [Capone and Schirmer 1993] [Chiarelli et al 1993] [Cowan and Weir 1993] [Donat 1993] [Enright 1993] [Lamb 1993] [Weir and Cowan 1993] [Wing and Capone 1993] [Intelligent hierarchical thrust vector 1994] [Whitesides et al 1994] [Wing 1994] [Wing 1994] [Zubkov et al 1994] [Aso et al 1995] [Bechtel and Hall 1995] [Byrd et al 1995] [Federspiel et al 1995] [Gal-Or 1995] [Hawkes and Franke 1995] [Knauber 1995] [Knauber 1995] [Lamb 1995] [Matesanz et al 1995] [Sahinkaya and Cordova 1995] [Shani et al 1995] [Chiang et al 1996] [Huang et al 1996] [Merryman and Hall 1996] [Rahaim et al 1996] [Washington et al 1996] [Giuliano et al 1997] [Reaction Thruster Attitude Control 1997] [Taur et al 1997] [Ulizar 1997] [Wing and Giuliano 1997] [Wing et al 1997] [Deere 1998] [Divo et al 1998] [Dotson et al 1998] [Flamm 1998] [Jetevator for rocket engine 1998] [Orme and Sims 1998] [Schinstock et al 1998] [Alvi and Strykowski 1999] [Divo et al 1999] [Grinstein and DeVore 1999] [Mori 1999] [Yagle et al 2000] [Blanchet and Bartos 2001] [Burnes et al 2001] [Cho et al 2001] [Dovey et al 2001] [Gal-Or 2001] [Palaniswamy et al 2001] [Suzuki 2001] [deChamplain et al 2002] [Fu-Kuang Yeh et al 2002] [Hamel et al 2002] [Hodel et al 2002] [Ko and Yoon 2002] [Rani et al 2002] [Williams and Vittal 2002] [Baksi et al 2003] [Deere et al 2003] [Figueiredo 2003] [Hamel et al 2003] [Harrisson et al 2003] [Hornung et al 2003] [Sung and Hwang 2003] [Waithe and Deere 2003] [Waithe and Deere 2003] [Bukhari et al 2004] [Case Study of Thrust 2004] [Ghasemi Nejhad 2004] [Rainville et al 2004] [Sun and Amano 2004] [TVC Thrust Vector Control 2004] [Yu et al 2004] [Yu et al 2004] [A Study on the 2005] [Attitude Control Design and 2005] [Collins Jr 2005] [Deere et al 2005] [Flamm et al 2005] [Ghenai et al 2005] [Krishnaswamy and Bugajski 2005] [Ma and Ghasemi-Nejhad 2005] [Oh and Bang 2005] [Sackheim et al 2005] [Srinivasan and Bowersox 2005] [Erdem et al 2006] [Flamm et al 2006] [Goldsborough 2006] [Guangshang and Xiaosha 2006] [Mangin et al 2006] [Michael A. Bolender 2006] [Rotunno et al 2006] [Sobester and Keane 2006] [Zahir et al 2006] [Adavbiele and Salami 2007] [Carnevale and Resta 2007] [Das et al 2007] [Deere et al 2007] [Flamm et al 2007] [Lazić and Ristanović 2007] [Neely et al 2007] [Wang et al 2007] [Case 2008] [Erdem et al 2008] [Forliti and Diaz-Guardamino 2008] [Oh et al 2008] [Orbekk 2008] [Adavbiele 2009] [Adavbiele 2009] [Bellandi and Slippey 2009] [Hao Long and Shujie Song 2009] [Khoshnood et al 2009] [Liu et al 2009] [Marcos et al 2009] [Murty et al 2009] [Saito and Fujimoto 2009] [Sung et al 2009] [Wang and Xie 2009] [Wenzhi et al 2009] [Yan et al 2009] [Godard and Kumar 2010] [Hsu 2010] [Jha et al 2010] [Lee et al 2010] [Muse 2010] [Romo and Alvarez 2010] [Tekin et al 2010] [Wenzhi et al 2010] [Yoo 2010] [Chen et al 2011] [Kulyk et al 2011] [Lu 2011] [Mohammadi and Toloei 2011] [Orr 2011] [Orr and Wall 2011] [Rizvi and Weitl 2011] [Xiaofeng et al 2011] [Bates and Young 2012] [Dziubanek 2012] [Liu 2012] [Lu et al 2012] [Pulecchi et al 2012] [Raj et al 2012] [Rubio Hervas and Reyhanoglu 2012] [Sung and Heo 2012] [Wang and Mo 2012] [Yunhua Li et al 2012] [Zhao et al 2012] [Zhou et al 2012] [Bonnici and Starkey 2013] [Fan et al 2013] [Fuxiang Liu and Liumin Wang 2013] [Hu et al 2013] [Hu et al 2013] [Li et al 2013] [Liu 2013] [Liu et al 2013] [Priyant Mark and Julin 2013] [Rizvi and Weitl 2013] [Rubio Hervas et al 2013] [Sankar et al 2013] [Tian and Lu 2013] [Zhang et al 2013] [Zhao-miao and Chuan 2013] [Zmijanovic et al 2013] [Felicetti et al 2014] [Lavrukhin et al 2014] [Liu et al 2014] [Millard and Reed 2014] [Millard 2014] [Orr and Slegers 2014] [Rubio Hervas and Reyhanoglu 2014] [Shi 2014] [Suchitra et al 2014] [Sumathi and Usha 2014] [Suñol and Vucinic 2014] [Suñol et al 2014] [VanZwieten et al 2014] [Wang et al 2014] [Zhang et al 2014] [Živković et al 2014] [Chambon et al 2015] [Chandra Murthy and Chakraborty 2015] [Chitra M V et al 2015] [Hanson et al 2015] [Hasturk 2015] [Khali and Yao 2015] [Kumar V V and Beebi M 2015] [Lin et al 2015] [Mohan et al 2015] [Nauparac 2015] [Numerical Simulation of the 2015] [Ou et al 2015] [Pei and Rothhaar 2015] [Suñol Jiménez et al 2015] [Vv and Laila Beebi 2015] [Yang et al 2015] [Zare-Behtash et al 2015] [Analysis of Transient Flow 2016] [Ansari and Bajodah 2016] [Ansari and Bajodah 2016] [Daljit Majil 2016] [Duraffourg et al 2016] [Graells and Carrabina 2016] [Hall et al 2016] [Heidari and Pouramir 2016] [Jee et al 2016] [Kong et al 2016] [Krishnamani and Mohanraj 2016] [Li et al 2016] [Li et al 2016] [Orr 2016] [Pourhashem and Kalkhoran 2016] [Wekerle et al 2016] [Yan Zhang et al 2016] [Zivkovic et al 2016] [Zmijanovic et al 2016] [Chandra Sekar et al 2017] [Comparative CFD analysis of 2017] [Ding et al 2017] [Ferlauto and Marsilio 2017] [Gudenkauf 2017] [He et al 2017] [Kaijun and Fawei 2017] [Kanyshev et al 2017] [Natarajan et al 2017] [Oh et al 2017] [Pascucci et al 2017] [Sebastian et al 2017] [Shen et al 2017] [Vinod et al 2017] [Wang Pei et al 2017] [Wu 2017] [Yu and Shu 2017] [Ahmed and Dorrah 2018] [Baskaran et al 2018] [Celani 2018] [Cheng et al 2018] [Dey et al 2018] [Fernando 2018] [Francis 2018] [Hu et al 2018] [Ma et al 2018] [Nauparac et al 2018] [Noaman et al 2018] [Resemini and Souza Costa 2018] [Ruetten 2018] [Sakata and Maruyama 2018] [Takase et al 2018] [B. et al 2019] [Beketaeva et al 2019] [Boulahbal et al 2019] [Cong et al 2019] [Duan et al 2019] [Ferlauto and Marsilio 2019] [Lee et al 2019] [Li et al 2019] [Liu et al 2019] [Maruyama et al 2019] [Muthukumar et al 2019] [Nandakumar et al 2019] [Nandakumar et al 2019] [Orr 2019] [Pei et al 2019] [Shakouchi et al 2019] [Söğütcü and Sümer 2019] [Tokareva et al 2019] [Wang et al 2019] [Wang et al 2019] [Xie et al 2019] [Xue et al 2019] [Yang et al 2019] [Zhang et al 2019] [Zhang et al 2019] [Biju Prasad et al 2020] [Boglis and Stoica 2020] [Cossette et al 2020] [Experimental Investigations on the 2020] [McCullough et al 2020] [Kuriakose et al 2020] [Marsilio et al 2020] [Mooij 2020] [Mooij 2020] [Murai et al 2020] [Muthukumar et al 2020] [Akshay et al 2020] [Shefaee Roshan et al 2020] [Shi et al 2020] [Sivadas 2020] [Sun 2020] [Sun 2020] [Thrust Vectoring of a 2020] [Urazbakhtin and Urazbakhtina 2020] [Wu et al 2020] [Wu et al 2020] [Anand 2021] [Biberstein et al 2021] [Brykov et al 2021] [Denisikhin et al 2021] [Emelyanov et al 2021] [Fu et al 2021] [George et al 2021] [Gogoi et al 2021] [Hu et al 2021] [Hu et al 2021] [Hu et al 2021] [Hu et al 2021] [Hu et al 2021] [Huang et al 2021] [Kode and Shtessel 2021] [Li et al 2021] [Mooij and Wang 2021] [Navarro-Tapia et al 2021] [Nguyen et al 2021] [Niu et al 2021] [Numerical Study of Fluidic 2021] [Pang et al 2021] [Pramodh Reddy et al 2021] [Resta et al 2021] [Takahashi and Maruyama 2021] [Tan et al 2021] [Trotta et al 2021] [Vaishnavi and Dhekane 2021] [Varghese et al 2021] [Wu and Kim 2021] [Wu et al 2021] [Zhang et al 2021] [Zhuang and Yulin 2021] [Airoldi et al 2022] [Brykov and Tischenko 2022] [Brykov and Tishchenko 2022] [Chi et al 2022] [Chi et al 2022] [Computational Investigation of Effects 2022] [Launch Vehicle Stability and 2022] [Lin et al 2022] [Linsen et al 2022] [Meng and Lu 2022] [Naik et al 2022] [Dennehy et al 2022] [Resta et al 2022] [Sharma et al 2022] [Song et al 2022] [Sopegno et al 2022] [Srinivas and Sridhar 2022] [VanZwieten et al 2022] [Wang et al 2022] [Xu et al 2022] [Yikilmaz and Üre 2022] [Aamer et al 2023] [Afridi et al 2023] [Afridi et al 2023] [Ahrazoğlu 2023] [Basil et al 2023] [Boryaev 2023] [Das et al 2023] [Diao et al 2023] [He et al 2023] [Henry 2023] [Jia et al 2023] [Jyothy and Wessley 2023] [Jyothy et al 2023] [Lee and Ro 2023] [Li et al 2023] [Mooij and Wang 2023] [Najiya et al 2023] [Neyzan Hosseini and Mahdavy-Moghaddam 2023] [Patrizia Cotugno et al 2023] [Praveen et al 2023] [Mohammed Akshay et al 2023] [Sai and Manideep 2023] [Salimi et al 2023] [Sopegno et al 2023] [Srivastava and Jayachandran 2023] [Study on the Aerodynamic 2023] [Zhang et al 2023] [Zhang et al 2023] [Andréa F. C. de S. Yamada et al 2024] [Bijo and Kumar Ranjith 2024] [Chen et al 2024] [Computational and Experimental Analysis 2024] [Das et al 2024] [Garg et al 2024] [He et al 2024] [Inturi et al 2024] [Jiang et al 2024] [Jung et al 2024] [Kim et al 2024] [Stepp 2024] [Stepp 2024] [Nayebi and Taeibi Rahni 2024] [Orr and Russell 2024] [Park et al 2024] [Qin et al 2024] [Sharjad et al 2024] [Sheth et al 2024] [Shi et al 2024] [Shynkarenko et al 2024] [Thorat et al 2024] [Ugolini et al 2024] [Wang et al 2024] [Wu et al 2024] [Yang et al 2024] [Zhang et al 2024] [Zhang et al 2024] [Zhang et al 2024] [Zhou et al 2024] [Adaptive Fault Tolerant Servo 2025] [Aravindh K 2025] [Balaji et al 2025] [Barry et al 2025] [Benfriha et al 2025] [Bhabar and Mukesh Sharma 2025] [Chen et al 2025] [Chen et al 2025] [Computational Investigation of Fluidic 2025] [Farì et al 2025] [Gupta 2025] [Han 2025] [Hosseini et al 2025] [Kasem and Megahed 2025] [Liu et al 2025] [Liu et al 2025] [Madduri 2025] [Naidu et al 2025] [Negandhi et al 2025] [Nocoń and Szmidt 2025] [Ornob et al 2025] [Panchal et al 2025] [Robust Super-Twisting Sliding-Mode Controller 2025] [Rumpf et al 2025] [Sanchez et al 2025] [Sathish et al 2025] [Kiran et al 2025] [Vohra and Kumar 2025] [Xu et al 2025] [Zhou et al 2025] [Zhu et al 2025] [Zolla et al 2025] [Abdelwahab and Johansen 2026] [Benfriha et al 2026] [Diz et al 2026] [Diz et al 2026] [Dong et al 2026] [Gupta 2026] [Gupta et al 2026] [Hoffmann and Joachimiak 2026] [Huang and Wu 2026] [Huang et al 2026] [Jiang et al 2026] [Jyothy et al 2026] [Liu et al 2026] [Liu et al 2026] [Masyagin 2026] [M․V․V․ and Rajagopal 2026] [Saiki et al 2026] [Santos et al 2026] [Tcherak et al 2026] [Thrust vector control 2026] [Xue et al 2026] [Yadav et al 2026] [Ye et al 2026] [Zhang et al 2026] [Wang et al 2027] [Stuart and McEnulty] [Byrd and Parker] [Chambon] [Russell et al] [Russell et al] [Das et al] [Dong Hyun Kim and Jae Weon Choi] [Franz and Limbert] [Bae and Summers] [Hwang] [Orr and Russell] [Orr et al] [Wall] [Wall et al] [Omoragbon et al] [Sifer et al] [Wu] [Wu] [Zmijanovic]
The machinery that feeds the nozzle
Turbopumps, gas generators, injectors, feed systems, valves, combustion and combustion instability, ignition and start transients, propellants and engine control.
1,791 records. 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Liquid rocket] [Xiaowen Dai et al] [Zapata Usandivaras]
Liquid rocket engines taken whole
Liquid rocket engines described whole, their design, development and performance, where no single named engine is the subject.
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Heat, cooling and erosion in chambers and nozzles
Heat transfer, cooling, ablation, erosion and materials of thrust chambers and nozzles.
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Small thrusters and spacecraft engines
Small chemical thrusters, spacecraft engines and attitude-control engines, as engines.
67 records. [Fio Rito 1964] [Broadwell and Marcisz 1968] [Riebling 1969] [French 1970] [Owens 1971] [Burlage et al 1972] [Evans et al 1972] [Hoehn 1972] [Long and Bjorklund 1972] [Martinez et al 1972] [Smith and Sforzini 1972] [Snoke and Williams 1972] [Holcomb 1973] [Stechman and Campbell 1973] [Nichols 1975] [Bjorklund et al 1979] [Bond 1979] [Stechman and Drenning 1981] [Stechman et al 1982] [Bjorklund 1983] [Berkman 1984] [Stechman 1984] [Liaw 1987] [Whalen 1987] [Finden et al 1988] [Appel et al 1989] [Stechman and Sumpter 1989] [Boyd et al 1990] [Melchior 1990] [Stechman 1990] [Chou et al 1991] [Garrison 1991] [Micci and Long 1991] [Weiss and Merkle 1991] [Chazen et al 1992] [Schneider 1992] [Schneider 1992] [Schwende et al 1992] [Chazen et al 1993] [Chung et al 1993] [Chung et al 1994] [Karpov 1994] [Chazen et al 1995] [Arrington et al 1996] [Jassowski 1997] [Jassowski 1997] [deGroot 1999] [Lee 1999] [Dulligan et al 2000] [Meinhart 2000] [Kujawa et al 2003] [Reed 2003] [Stechman and Lawson 2006] [Greene 2009] [Hagiwara et al 2009] [Dorado et al 2013] [Tong and Fang 2014] [Cai et al 2016] [Fujii and Ikeda 2016] [Parker 2016] [Atwell et al 2017] [Formation and Evaporation of 2018] [Banakar et al 2020] [MIMPS-G Green rocket propulsion 2022] [Shawyer 2023] [Mopuru et al 2025] [Roohi and Mahdavi 2026]
Solid and hybrid motors, on the subjects this article uses
Solid and hybrid motors where the title’s subject is one this article uses, being the nozzle, steering, the plume, chamber acoustics or measurement.
68 records. [The Influence of Erosive 1960] [Kuby 1964] [Nonlinear effects in instability 1964] [Price et al 1964] [Becker and Hamilton 1965] [Coates et al 1965] [Hart and McClure 1965] [Becker and Tang 1966] [Two-hundred sixty-in.-dia motor feasibility 1966] [Gungle 1967] [Summerfield and Krier 1968] [Wong 1968] [marxman and Wooldridge 1969] [Gostintsev et al 1972] [Micci 1975] [Robertson 1975] [Donguy 1977] [Pope and Dirling 1979] [Quilici 1984] [Hwang and Chang 1987] [Edwards 1991] [Clayton 1992] [Whitney and Hennessey 1992] [Smith-Kent et al 1995] [Clinton and Wendel 1998] [Haugen 1998] [Ewing et al 1999] [Lui and Hyer 1999] [Effect of Acoustic Oscillation 2000] [Patel and Chang 2000] [Maw et al 2004] [McLaughlin et al 2004] [Peterson et al 2004] [Erven et al 2006] [Evans et al 2007] [Liggett and Menon 2007] [Thakre and Yang 2007] [Daimon et al 2008] [Dombrovsky 2008] [Dombrovsky 2008] [Cai et al 2009] [Evans et al 2009] [Thakre and Yang 2009] [Design of Nozzle and 2011] [Bianchi and Nasuti 2013] [El-Nady et al 2015] [El-Nady et al 2015] [El-Nady et al 2015] [Joo et al 2015] [Mani et al 2015] [Park et al 2016] [Di Martino et al 2017] [El-Nady et al 2017] [El-Nady et al 2017] [Stephenson 2018] [Alazeezi and Elek 2019] [Nguyen 2020] [de Andrade Janene Gonini et al 2021] [Hyun et al 2021] [AbdelGawad and Guozhu 2022] [Chaiaiad and Thongsri 2023] [Gonini et al 2023] [Grossi et al 2023] [Parhi et al 2023] [Determination of gas flow 2024] [Paul Murugan et al 2025] [Saokhede et al 2025] [Passarani et al 2026]
Combined-cycle rocket propulsion
Rocket propulsion combined with air-breathing modes, as in air-augmented and rocket-based combined-cycle engines.
10 records. [Kors 1990] [Fahahngi 1992] [Farhangi and Trent 1992] [Kalmykov and Yanchilin 1996] [Olds et al 1997] [Hueter 1999] [Smith et al 2002] [Pastrone 2004] [Waung and Etele 2009] [Wang et al 2022]
Making it, which is what modular is actually for
The manufacture and cost of rocket engines and launch vehicles, including additive manufacturing of engine parts and low-cost launch.
285 records. [Baehr 1962] [Edmonds et al 1964] [Friedland 1965] [Agosta and Hammer 1969] [Ashurst et al 1969] [Cacha 1969] [Grim et al 1969] [Hanford 1969] [Ross 1969] [Braun et al 1970] [Goethert 1970] [Mosley 1970] [Ward 1970] [Ciepluch 1971] [Laber et al 1971] [Sohl 1971] [Warga 1971] [Alford et al 1972] [Vanderesch 1972] [Dini and Johnson 1976] [Elegante and Bowman 1978] [Hudson et al 1981] [Fritz and Sackheim 1982] [Lane et al 1982] [Hankins 1984] [Koelle 1984] [Meisl 1986] [French 1987] [Hudson 1987] [Hudson and Hunter 1987] [Meisl 1988] [Koelle 1989] [McKown 1989] [Meisl 1989] [Pearson 1989] [Slayton and Daniels 1989] [Strobl 1989] [Joint DoD/NASA Advanced Launch 1990] [Low-Cost Satellites and Satellite 1990] [Pegasus Key to Low-Cost 1990] [Puening 1990] [Beckman and Volkmann 1991] [Szatkowski and Levin 1991] [Wagner and Cleveland 1991] [Wolfe et al 1991] [Chapman and Nix 1992] [Koelle 1992] [Koelle 1992] [Low-cost expendable launch vehicles 1992] [McKechnie et al 1992] [Sackheim 1992] 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2015] [Karuntzos 2015] [Lee et al 2015] [Mohan Kumar et al 2015] [Vickers 2015] [Vickers 2015] [Bhat et al 2016] [Boone and Miller 2016] [Burgaud et al 2016] [Clinton 2016] [Coopersmith and Davis 2016] [Gradl and Brandsmeier 2016] [Gradl et al 2016] [Jones et al 2016] [O’Neal 2016] [Protz et al 2016] [Sarzi-Amade et al 2016] [Tucker and Schmidt 2016] [West et al 2016] [Bae et al 2017] [Clinton 2017] [Clinton 2017] [Clinton 2017] [Dupont et al 2017] [Gradl et al 2017] [Hayward 2017] [Hutt et al 2017] [Kunka and Jacob 2017] [Bower and Thomas 2018] [Burlingame and Hyatt 2018] [Fikes 2018] [Garcia et al 2018] [Gradl et al 2018] [Gradl et al 2018] [Gradl et al 2018] [Markusic et al 2018] [Weuta 2018] [Zagorski et al 2018] [3D printed rocket combustion 2019] [Coleman 2019] [Gradl et al 2019] [Gradl et al 2019] [Gradl and Protz 2019] [Gradl et al 2019] [Gradl et al 2019] [Gradl et al 2019] [Katsarelis et al 2019] [Stachulla et al 2019] [McClain et al 2019] [Patel et al 2019] [Durkee et al 2020] [Gradl and Protz 2020] [Gradl et al 2020] [Mireles et al 2020] [Bair et al 2021] [Gradl et al 2021] [Grigoryants et al 2021] [Kerstens et al 2021] [Awad Alahbabi et al 2022] [Bao et al 2022] [Launch Vehicle Financial Analysis 2022] [Launch Vehicle Manufacturing 2022] [Malisuwan and Kanchanarat 2022] [Thomas 2022] [Akbulatov et al 2023] [An and Li 2023] [Buettner et al 2023] [Buettner et al 2023] [Cold Flow Characterization of 2023] [Features of the Development 2023] [Kruse et al 2023] [Chen et al 2023] [Ryazantsev et al 2023] [Thomas 2023] [Yi et al 2023] [Çam and Özkol 2023] [Chandler 2024] [Fedotowsky et al 2024] [Kim 2024] [Motta et al 2024] [Gradl et al 2024] [Swain et al 2024] [Vekilov et al 2024] [Wilken 2024] [Barros et al 2025] [Barros et al 2025] [Chamoli 2025] [Chen et al 2025] [Cruz et al 2025] [Cui 2025] [Demeneghi et al 2025] [Chen et al 2025] [Practice of Precise Cost 2025] [Shen et al 2025] [Sliusariev 2025] [Woodger et al 2025] [Dziubek et al 2026] [Lazarets 2026] [Chen et al 2026] [Pustovyi 2026] [Swan 2026] [White 2026] [Yi et al 2026] [Zhang et al 2026] [Goldman et al] [Buck et al] [Katsarelis et al] [Katsarelis et al] [Cost Versus Benefits of] [Thrasher and Lusk] [Michaud et al] [Naden et al] [Gradl] [Gradl] [Gradl] [Gradl] [Gradl] [Gradl and Mireles] [Gradl and Gibson] [Gradl et al] [Gradl et al] [Gradl et al] [Gradl et al] [Gradl et al] [Gradl et al] [Gradl] [Gradl] [Gradl et al] [Gradl et al] [Gradl et al] [Gradl and Mireles] [Gradl et al] [Chen et al] [Schmitt et al] [Fedotowsky et al] [Hudson et al] [Padela et al]
The ascent, which is where the integral lives
Launch vehicle design and staging, ascent trajectories and their optimisation and guidance, ascent aerodynamics, the atmosphere as a flight condition, and launch vehicle programmes and flights.
1,338 records. [Malina and Smith 1938] [Tsien and Evans 1951] [Schurmann 1957] [Newton 1958] [Parkyn 1958] [Struble et al 1958] [Feodosiev and Siniarev 1959] [Liu 1959] [Swanson 1959] [Draim and Stalzer 1960] [Mackay 1960] [Crabill 1961] [Leendertse 1961] [Boynton et al 1962] [Kolenkiewicz and Loughin 1962] [Lovingood and Smith 1962] [Macpherson 1962] [Punga and Campbell 1962] [Tischler 1962] [Schramm 1962] [Bertram and Kuettner 1963] [Hauer et al 1963] [Launch Vehicle Flight Report 1963] [Launch Vehicle Performance 1963] [Luton 1963] [Martin Co Baltimore Md 1963] [Stancil 1963] [Thomae 1963] [Vickery 1963] [Ziegler 1963] [Bos and Morrison 1964] [Dembrow and Jamieson 1964] [Ehricke 1964] [Forrette 1964] [Gunderson et al 1964] [Hammock 1964] [Hillsley and Robbins 1964] [Lester and Morgan 1964] [Levine 1964] [Lunney 1964] [Mueller 1964] [Brereton 1964] [Launch vehicle optimization, phase 1965] [Launch vehicle optimization, phase 1965] [Moser 1965] [Munick 1965] [Pressly 1965] [Roudebush 1965] [Sanders 1965] [Thompson and Woodcock 1965] [Blair et al 1966] [Bush 1966] [Havill and Hornby 1966] [Knaur 1966] [Mayo 1966] [Moch 1966] [Moch 1966] [Pack 1966] [Rosenbaum 1966] [Schroer 1966] [Study of structural bending 1966] [Teren 1966] [Babb and Fuller 1967] [Barrett et al 1967] [Bieber 1967] [Busse and Kreuzer 1967] [Ceresuela 1967] [Cunningham 1967] [Driscoll 1967] [Dunfee and Oss 1967] [Elliott 1967] [Ferris 1967] [Heuston et al 1967] [Hewitt 1967] [Huang and Brown 1967] [Jungmann 1967] [Karel 1967] [Launch and space vehicle 1967] [Mayo 1967] [McGarvey 1967] [Muraca 1967] [Nicolaides et al 1967] [Nolan 1967] [Pines 1967] [Price 1967] [Richards 1967] [Rynaski 1967] [SA-204/LM-1 launch vehicle operational 1967] [Brunk and Gomersall 1968] [Colwell 1968] [Davis et al 1968] [Fuller 1968] [Harris 1968] [Harvey 1968] [Elliott and Rau 1968] [Johnson and L’Ecuyer 1968] [Kork and Schindler 1968] [Launch phase abort analysis 1968] [Marquet and Zeiner 1968] [Morgan 1968] [SA-204/LM-1 launch vehicle dynamics 1968] [Solmon 1968] [Swaim 1968] [Williams 1968] [Bejczy 1969] [Cunningham 1969] [Freeman and Teets 1969] [Hardy 1969] [Hose et al 1969] [Hose et al 1969] [Swaim 1969] [Ammons et al 1970] [Ammons et al 1970] [Cairns 1970] [Coffey and Geye 1970] [Gregory and Wilcox 1970] [Haeussermann 1970] [Ingram et al 1970] [Madden 1970] [McCaskill 1970] [Rice 1970] [Rosenbaum 1970] [Rosenbaum and Willwerth 1970] [Seltzer 1970] [The air-breathing launch vehicle 1970] [Wing 1970] [Bradshaw et al 1971] [Grassl 1971] [McGolrick et al 1971] [Rosenbaum 1971] [Saturn 5 Launch Vehicle 1971] [Saturn 5 launch vehicle 1971] [Sullivan and Uzzell 1971] [Brusch and Reed 1972] [Buchholz and Gamble 1972] [Graves 1972] [Gunn 1972] [Lovingood et al 1972] [Lyons et al 1972] [Maki and Vegte 1972] [Reichert 1972] [Reins and Alvis 1972] [Saturn 5 launch vehicle 1972] [Taylo and Redus 1972] [Trimmer et al 1972] [Ammons 1973] [Bensimon 1973] [Detailed test objectives for 1973] [Gunn 1973] [Jenkins et al 1973] [Larson 1973] [Rehri and Pennanech 1973] [Saturn 1B launch vehicle 1973] [Saturn 1B launch vehicle 1973] [Saturn 5 Launch Vehicle 1973] [Simmons and Hall 1973] [Stofan 1973] [Toelle et al 1973] [Vinh 1973] [Bendot 1974] [Dhuyvetter 1974] [Edge and Powers 1974] [Saturn 1B launch vehicle 1974] [Shen-Orr and Ur 1974] [Bendot et al 1975] [Carter et al 1975] [Johnson 1975] [Tharratt 1975] [Williams et al 1975] [Adimurthy 1976] [Alford and Hoekstra 1976] [Asanuma and Nomura 1976] [Eldred and Gordon 1976] [Flowers 1976] [McGarvey 1976] [Lebeau and Orye 1976] [Lindsay and Fikes 1976] [Notardonato and Masters 1976] [Shrivastava and Reddy 1976] [Ariane Launch Vehicle A 1977] [Bernot 1977] [Conceptual design and preliminary 1977] [Dordain 1977] [Freeman and Fournier 1977] [Haefeli et al 1977] [Haefeli et al 1977] [Haefeli et al 1977] [Jackson et al 1977] [Jaggers 1977] [Johnson 1977] [Martin 1977] [Rehder 1977] [Rehder 1977] [Research Study to Identify 1977] [Solar power satellite. System 1977] [Beichel and Obrien 1978] [Chase 1978] [Freeman and Fournier 1978] [Hepler and Bangsund 1978] [Hepler and Bangsund 1978] [Hepler et al 1978] [Schoyer 1978] [An improved Theoretical Aerodynamic 1979] [Ardema 1979] [Chase 1979] [Churchyard 1979] [Freeman and Wilhite 1979] [Freeman and Powell 1979] [Hanley 1979] [Hepler et al 1979] [Kessler and Carson 1979] [Reilly 1979] [Spearman 1979] [Star-Raker - An airbreather/rocket-powered 1979] [Takenaka et al 1979] [Chase 1980] [Freeman and Powell 1980] [Launch mission summary and 1980] [Small et al 1980] [Hattis 1981] [Koelle 1981] [Lucas 1981] [Barrowman 1982] [Flores and Gurkin 1982] [Grimes and Kraft 1982] [Jayaraman et al 1982] [Jones and Tharayil 1982] [Knauber 1982] [Millard et al 1982] [Patterson 1982] [Schleich 1982] [Schneiders and Ziegltrum 1982] [Wente 1982] [Wilhite 1982] [Bernot 1983] [Hamilton 1983] [Naftel and Powell 1983] [Shuttle derived vehicle analysis 1983] [Surber and Stone 1983] [Assessment of candidate-expendable launch 1984] [Bauer et al 1984] [Gaynor and Cervisi 1984] [Ikeda et al 1984] [Deschamps 1985] [Hertzfeld 1985] [Naftel and Powell 1985] [Parkinson 1985] [Ruppe 1985] [Tewell and Buell 1985] [Tewell and Buell 1985] [Toelle 1985] [Wallace et al 1985] [Eldred and Talay 1986] [Koelle 1986] [Montag et al 1986] [Naftel et al 1986] [Adimurthy 1987] [Dorrington 1987] [Ennix and Coleman 1987] [Nguyen 1987] [Ruppe 1987] [Troutman and Isakowitz 1987] [Davis 1988] [Edmunds 1988] [Holguin and Labbee 1988] [Maitra 1988] [Mihara 1988] [Olds 1988] [Sinha 1988] [Stampfl and Meyer 1988] [Tsiotras and Kelley 1988] [Claggett and Langehough 1989] [Corban et al 1989] [Feaux et al 1989] [Foster 1989] [Foster et al 1989] [Harsh 1989] [Hodges et al 1989] [Liquid Rocket Booster LRB 1989] [Luke 1989] [Pamadi 1989] [Parsley 1989] [Schoess 1989] [Smith et al 1989] [Well 1989] [Wessling and Maybee 1989] [Abbott and Matzenauer 1990] [Calise et al 1990] [Chase 1990] [Dorrington 1990] [Expendable launch vehicles technology 1990] [Greene et al 1990] [ISAS’s New Launch Vehicle 1990] [Johnson 1990] [Joyner and Sabatella 1990] [Koelle 1990] [Krause 1990] [Krieger 1990] [Lepsch et al 1990] [Leung and Calise 1990] [Long March Launch Vehicles 1990] [McKenna 1990] [Palaszewski 1990] [Palaszewski 1990] [Palaszewski 1990] [Powell et al 1990] [Sinha and Shrivastava 1990] [Stanley and Powell 1990] [Sullivan 1990] [U.S. Access to Space 1990] [Whitehead 1990] [Wiswell and Huggins 1990] [Barry et al 1991] [Bless 1991] [Bulirsch and Chudej 1991] [Conceptual design of two-stage-to-orbit 1991] [Deese et al 1991] [Flittie and McFarlane 1991] [Johnson 1991] [Jones and Petley 1991] [Limerick 1991] [Luke 1991] [MacConochie and Stnaley 1991] [Martin et al 1991] [Morgenthaler and Montoya 1991] [Nguyen 1991] [Palaszewski and Powell 1991] [Parkinson 1991] [Plencner 1991] [Powell et al 1991] [Powell et al 1991] [Rice 1991] [Schoettle and Hillesheimer 1991] [Shaughnessy and Gregory 1991] [Sholes 1991] 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[Duffy 1993] [Dutton 1993] [French 1993] [Grimm 1993] [Hardy et al 1993] [Holloway and Limerick 1993] [Jules 1993] [Kugelmann and Pesch 1993] [Leingang et al 1993] [Luke 1993] [Lydon and Simmons 1993] [Lyons 1993] [Naftel and Powell 1993] [Poth et al 1993] [Rutledge 1993] [Ryan and Verderaime 1993] [Ryan and Verderaime 1993] [Ryan and Jewell 1993] [Schmidt et al 1993] [Seywald and Bless 1993] [Snyder 1993] [Stanley and Piland 1993] [Willen and Rude 1993] [Adelfang et al 1994] [Allen et al 1994] [Allen et al 1994] [Ardema et al 1994] [Bekey 1994] [Bond and Yi 1994] [Clapp 1994] [Cobb and Davis 1994] [Daniels and Saavedra 1994] [Dumbacher and Klevatt 1994] [Flittie et al 1994] [Frost and Springer 1994] [Gielda et al 1994] [Immich and Hufenbach 1994] [Kugelmann 1994] [Laursen 1994] [Leung and Calise 1994] [Limerick 1994] [Lovell and Schmidt 1994] [Lovell and Schmidt 1994] [Lu 1994] [Mahanta 1994] [Manski and Fina 1994] [Olds 1994] [On ascent guidance of 1994] [Optimal digital autopilot for 1994] [Park et al 1994] [POGO prevention of H-2 1994] [Siebenhaar et al 1994] [Single stage to orbit 1994] [Spangelo and Klaus 1994] [Springer and Pokora 1994] [Springer and Pokora 1994] [Stanley et al 1994] [Stoyanof and Monk 1994] [Advanced Transportation System Studies 1995] [Advanced transportation system studies 1995] [Bainum and Reiss 1995] [Barret 1995] [Barret 1995] [Braun et al 1995] [Carpenter et al 1995] [Chang 1995] [Chen et al 1995] [Christenson and Holt 1995] [Colasurdo et al 1995] [Cole et al 1995] [Cook 1995] [Freeman et al 1995] [Freeman et al 1995] [Hanson et al 1995] [Heald 1995] [Hinckel 1995] [Hueter 1995] [Hunt 1995] [Ingalls 1995] [Lydon and Va 1995] [Lyon 1995] [Martin 1995] [Mayhew and Norden 1995] [Meyer et al 1995] [Moore et al 1995] [Pamadi 1995] [Park et al 1995] [Phillips and Engelund 1995] [Reusable Launch Vehicle 1995] [Reusable launch vehicle development 1995] [Reusable launch vehicle Technology 1995] [Smith and Adelfang 1995] [Sponable 1995] 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Systems 1997] [Galati and Elkins 1997] [Goracke et al 1997] [Greaves 1997] [Huang et al 1997] [Jankovsky 1997] [Logsdon and Williamson 1997] [Manski et al 1997] [Markopoulos et al 1997] [McCurry and Carter 1997] [McMinn 1997] [Phillips and Engelund 1997] [Reaser 1997] [Shaw et al 1997] [Shtessel and Krupp 1997] [Shtessel et al 1997] [Weilmuenster et al 1997] [Adelfang and Smith 1998] [Andrews 1998] [Barret 1998] [Clarke 1998] [Dasgupta 1998] [Herzog et al 1998] [Huang et al 1998] [Jankovsky et al 1998] [Kohrs and Petersen 1998] [Konno et al 1998] [Martin et al 1998] [McClure 1998] [McClure and Andrews 1998] [Meyerson 1998] [Obermann and Williamson 1998] [Olds and Bellini 1998] [Olds and Bellini 1998] [Palaszewski 1998] [Powell et al 1998] [Rahn et al 1998] [Rasky 1998] [Sachs and Mayrhofer 1998] [Stadler 1998] [Tanck and Steadman 1998] [Tournes and Johnson 1998] [Tuohy 1998] [Wangu and Mouyos 1998] [Azevedo and Buonomo 1999] [Baroth et al 1999] [Casalino and Pastrone 1999] [Dhekane et 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Vaughan 2020] [Nie et al 2020] [Xu et al 2020] [Zardashti and Rahimi 2020] [Zardashti et al 2020] [Zheng et al 2020] [Almeida et al 2021] [Benedikter et al 2021] [Biertümpfel et al 2021] [Chen et al 2021] [Cheng et al 2021] [Cho et al 2021] [Da Cás 2021] [Federici et al 2021] [Fu et al 2021] [Hao and Zhang 2021] [Hu et al 2021] [Huangpeng et al 2021] [Hörschgen-Eggers et al 2021] [Ifenna Okoli et al 2021] [Jo et al 2021] [Justus et al 2021] [Li et al 2021] [Marchetti et al 2021] [Pidvysotskyi 2021] [Rose and Niemeyer 2021] [Ryakam and Crispin 2021] [Shulga and Shcheglov 2021] [Simon and Chudoba 2021] [Sokolov et al 2021] [Sosa and Tuminello 2021] [Tang et al 2021] [Wang et al 2021] [Xiao et al 2021] [Yuwei et al 2021] [Zhang et al 2021] [A Technical History of 2022] [Abdulkerim 2022] [Ascent Trajectory Analysis and 2022] [Benedikter et al 2022] [Canepa et al 2022] [Cherkasov et al 2022] [Dabanović et al 2022] [Deng et al 2022] [Diao et al 2022] [Divakar and Rajalakshmi Samaga 2022] [Fujikawa and Yonemoto 2022] [Han et al 2022] [Haws and Bowman 2022] [Jee et al 2022] [Ju et al 2022] [Lakshminarayana and Chiarelli 2022] [Lan and Li 2022] [Launch Vehicle Performance and 2022] [Launch Vehicle Systems and 2022] [Ma et al 2022] [Ma et al 2022] [Miao et al 2022] [Mo et al 2022] [Nair and Vaidyanathan 2022] [Olejnik et al 2022] [Robust Optimal Trajectory Design 2022] [Sabaghzadeh and Khansari 2022] [Siby and Sreeja 2022] [Space Launch Vehicle Structures 2022] [Villanueva 2022] [Wang and Song 2022] [Bao et al 2023] [Bechasnov and Ilyin 2023] [Cheng et al 2023] [Dalle et al 2023] [Dees and Stockard 2023] [Dinh and Trifoni 2023] [Gettatelli et al 2023] [Gilson et al 2023] [Kuznetsov 2023] [Li et al 2023] [Lu et al 2023] [Mancini et al 2023] [Miao et al 2023] [Perumal 2023] [Radhakrishnan et al 2023] [Redondo Gutierrez et al 2023] [Burth et al 2023] [Shawyer 2023] [Song et al 2023] [Soota and Nadda 2023] [Ukirde and Rathod 2023] [Wang et al 2023] [Wu and Zhang 2023] [Xiang et al 2023] [Xu et al 2023] [Yan and Zhou 2023] [Yan et al 2023] [Yu et al 2023] [Zhang 2023] [Zhang et al 2023] [Boost Trajectory Estimation 2024] [Cau et al 2024] [Cerantola et al 2024] [Chavda 2024] [Dalle et al 2024] [Giannone and Sambasivam 2024] [Rosanova 2024] [Golubek et al 2024] [Gomez Fernandez 2024] [Keba and Kulabukhov 2024] [Krishnan and Thakur 2024] [Krause 2024] [Manwatkar et al 2024] [Martin et al 2024] [Molinari et al 2024] [Nath et al 2024] [Pavanasam et al 2024] [Radhakrishnan et al 2024] [Shaobing and Chi 2024] [Shoyama and Hirakawa 2024] [St-Laurent-Recoura et al 2024] [Sun et al 2024] [Wang and Ma 2024] [Wang et al 2024] [Yu et al 2024] [Zhao et al 2024] [Zhou et al 2024] [Bae 2025] [Benedikter et al 2025] [Mahajan and Condon 2025] [Hu et al 2025] [Hussein and de Weck 2025] [Kim et al 2025] [Lee 2025] [Lee et al 2025] [Lei et al 2025] [Li et al 2025] [Lu et al 2025] [Magistro et al 2025] [Magyar et al 2025] [Wittal et al 2025] [Nair and Kukreja 2025] [Nguyen and Aleti 2025] [Rapp 2025] [Redondo Gutierrez et al 2025] [Reentry Maneuvering Vehicle Ascent 2025] [Sanjay and Kuzhiveli 2025] [Walker 2025] [Wang et al 2025] [Xu et al 2025] [Xu et al 2025] [Xue et al 2025] [Young 2025] [Zhao et al 2025] [DurmuŞ et al 2026] [Kalsekar and Deshpande 2026] [Khan 2026] [Krivenko 2026] [Lei et al 2026] [Raharema et al 2026] [Saju and Vengadeswaran 2026] [Seo and Kim 2026] [Singh et al 2026] [Sun et al 2026] [Velez et al 2026] [Wu 2026] [Yu 2026] [Zhang et al 2026] [Zolla et al 2026] [Zou et al 2026] [Winebarger] [Brevault] [Wu] [Coakley] [Smith] [Dalle et al] [Dorairajan] [Griffin] [Guide for Crewed Suborbital] [Hassall] [Herrell and Zhou] [Horneman] [Launch vehicle dynamics] [Launch Vehicle Systems] [Mackle] [Fritz et al] [Morito et al] [NASA and the Politics] [Birbasov et al] [Oberoi et al] [Packard and Zampino] [Rocket staging] [Satellite Installation and Launch] [Satellite launch vehicles] [Shtessel and Krupp] [Space systems. General test] [Standard Guide for Crewed] [Sun] [Tully] [Uemura] [Wu] [Zhang]
Who is buying this and what they say it is for
The sponsors, programmes and acquisition of rocket propulsion and responsive launch, and the readiness assessment of propulsion technology.
56 records. [Satellites and sounding rockets 1964] [Sterhardt 1965] [NASA sounding rocket survey 1966] [Encyclopedia Satellites and Sounding 1970] [Corliss 1971] [The United States sounding 1971] [Study of solid rocket 1972] [Gregory 1975] [Lucci and Hodson 1975] [Guidotti 1976] [Holtz 1976] [Snyder 1976] [Bumgarner and Gilcrease 1979] [Martin 1979] [Olsen 1979] [Zetterquist 1979] [Lane and Mayo 1980] [Alford et al 1982] [Boscov 1982] [Cooper and Seaton 1982] [Roehrig 1982] [Wilhite 1982] [Kozuma et al 1983] [NASA Orbit Transfer Rocket 1984] [Selden 1989] [Workshop on the Suborbital 1991] [Dawson 1992] [Gurkin 1992] [Gustafson and Harmon 1993] [Integrated propulsion technology demonstrator 1994] [Immich et al 1997] [Clarke 1998] [Flowers and Needleman 1999] [Blair and DeGeorge 2000] [DeGeorge and Fletcher 2002] [Thornburg 2002] [Eberspeaker and Smith 2003] [Harper and Hawk 2004] [Responsive Access to Space 2004] [Hoyt 2006] [Finger et al 2009] [Sauser et al 2009] [Finger et al 2010] [Myrabo et al 2010] [Allen and Roberts 2011] [McNab et al 2011] [Doreswamy 2013] [Jones et al 2014] [Lee and Yi 2020] [Grunert 2022] [Zhang and Li 2022] [Reid and Company 2023] [Assessing the Impact of 2024] [Boldt et al 2025] [Tobiassen 2026] [Davis and Tomlinson]
Measuring it, because the award says highly instrumented
Test stands, altitude test facilities, thrust measurement, instrumentation, telemetry and diagnostics of engines, nozzles and rocket flights.
422 records. 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Plumes and what they do to the vehicle behind them
Rocket and jet plumes, their radiation and impingement, their interaction with the vehicle and the surfaces around it, and free jets and their shock structure.
698 records. 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Exhaust nozzles of aircraft engines, where the plug began
Exhaust, ejector and plug nozzles of aircraft engines and their integration with the airframe, the setting of the plug nozzle’s first measurements.
203 records. [Pinkel et al 1943] [Hearth and Wilcox 1954] [Runckel 1958] [Schmeer et al 1960] [Gates and Cochran 1961] [Scott 1961] [Lockwood 1962] [Addy et al 1964] [Higgins and Wainwright 1964] [Shoemaker 1965] [Higgins et al 1966] [Armstrong and Miller 1967] [Migdal and Greathouse 1968] [Berrier and Wood 1969] [Maiden and Runckel 1970] [Lecuyer 1970] [Groth 1971] [Lawrence et al 1972] [Maiden 1973] [Samanich et al 1973] [Marrs et al 1974] [Schnell 1974] [German 1975] [Robinson and Price 1975] [Capone 1976] [Meleason 1976] [Schlundt 1976] [Von Glahn and Groesbeck 1976] [Berrier et al 1977] [Pendergraft 1977] [Richey et al 1977] [Willard et al 1977] [Calarese and Walterick 1978] [Capone 1978] [Kozlowski and Packman 1978] [Kozlowski and Packman 1978] [Leavitt and Yip 1978] [Lucas et al 1978] [Rushmore and Zelazny 1978] [Benek 1979] [Berrier and Re 1979] [Capone 1979] [Capone et al 1979] [Federspiel 1979] [Groesbeck and Vonglahn 1979] [Mikhail 1979] [Peery and Forester 1979] [Pendergraft 1979] [Putnam 1979] [Thomas 1979] [Wilmoth et al 1979] [Berrier and Re 1980] [Capone and Berrier 1980] [Kozlin 1980] [Plant and Nugent 1980] [Rosenberg and Esker 1980] [Sanders 1980] [Thomas 1980] [Wilmoth 1980] [Capone et al 1981] [Carson and Lee 1981] [Kowalski et al 1981] [Rosenberg and Christiansen 1981] [Schoelen 1981] [Swanson 1981] [Burley and Berrier 1982] [Capone 1982] [Kreskovsky et al 1982] [Lashkov and Shumilkina 1982] [Leavitt and Re 1982] [Pendergraft and Bare 1982] [Roseberg 1982] [Straight and Cullom 1982] [Watson 1982] [Werle and Vasta 1982] [Wilmoth 1982] [Yetter et al 1982] [Bare and Berrier 1983] [Berrier and Mason 1983] [Burstadt and Johns 1983] [Capone and Reubush 1983] [Chiarelli and Compton 1983] [Putnam and Hodges 1983] [Putnam and Strong 1983] [Wilmoth 1983] [Carson et al 1984] [Harris 1984] [Johns and Burstadt 1984] [Pendergraft and Nugent 1984] [Pendergraft and Carson 1984] [Re and Leavitt 1984] [Rowe et al 1984] [Wilmoth and Putnam 1984] [Arbiter 1985] [Berrier 1985] [Goldberg et al 1985] [Leavitt 1985] [Leavitt and Burley 1985] [Mason and Berrier 1985] [Putnam and Bissinger 1985] [Zilz and Bailey 1985] [Burley et al 1986] [Peery and Imlay 1986] [Pendergraft et al 1986] [Re and Leavitt 1986] [Subsonic/Transonic/Supersonic Nozzle Flows and 1986] [Goldberg et al 1987] [Green and Carson 1987] [Henderson and Burley 1987] [Wilmoth and Leavitt 1987] [Naik and Chiarelli 1988] [Emanuel 1989] [Lai and Nelson 1989] [Liang 1989] [Peace 1989] [Tillman et al 1989] [Smith and McArdle 1990] [Baysal and Hoffman 1991] [Carlson and Abdol-Hamid 1991] [Compton and Abdol-Hamid 1991] [Esker and Debonis 1991] [Foley and Pendergraft 1991] [Re and Carson 1991] [Taylor 1991] [Carlson 1992] [Esker and Perusek 1992] [Norton and Kingsley 1992] [Spaid and Keener 1992] [Asbury 1993] [Esker and McArdle 1993] [Meyer and Maclean 1993] [Quinto et al 1993] [Wing et al 1993] [Carlson 1994] [Esch and Glehrl 1994] [Liou and Patton 1994] [Cler 1995] [Debonis et al 1995] [Hunter 1995] [Seiner and Gilinsky 1995] [Wing 1995] [Asbury et al 1996] [Compton 1996] [Lamb et al 1996] [Prediction Capability for Transonic 1996] [Wing 1996] [Asbury 1997] [Orme et al 1998] [Wing 1998] [Gilinsky et al 1999] [Lambert and Mizukami 1999] [Malone and Peavey 1999] [Seiner and Gilinsky 1999] [Gilinsky and Blankson 2000] [Gilinsky et al 2000] [Saiyed et al 2000] [Saiyed 2000] [Saiyed 2000] [Seiner and Gilinsky 2000] [Capone and Deere 2001] [Mikkelsen et al 2001] [Morgan and Gilinsky 2001] [Bridges 2002] [Dalbello et al 2003] [Terrier and Lu 2003] [Doty et al 2004] [Gilinsky et al 2004] [Thayer et al 2004] [Thomas and Kinzie 2004] [Balan and Askew 2005] [Sloan et al 2008] [Halbig 2011] [Kang et al 2011] [Mengle 2011] [Szydlowski and Droit 2011] [Xiaofeng Huang and Yonghong Long 2011] [Mengle and Thomas 2012] [Connolly et al 2014] [Graham and Agarwal 2015] [Wang et al 2015] [Bruce et al 2016] [Ritter 2016] [Ruetten and Woldegiorgis 2016] [Christiansen and Thornock 2017] [Fenrich and Alonso 2017] [Sreenivas et al 2017] [Tinney 2017] [Chen 2018] [O’Dowd and Byerley 2018] [Ding et al 2019] [Gorbovskoi et al 2019] [Kumar et al 2019] [Chen et al 2020] [Ferdaus and Ganapathi 2020] [Mohmed Abdelmotalib et al 2021] [Ge et al 2023] [Menon et al 2023] [Rao et al 2023] [Meng et al 2024] [Meng et al 2024] [Zhu et al 2024] [Li et al 2026] [Waung]
Compressible flow and gas dynamics, the relations the nozzle sections use
Compressible flow and gas dynamics stated generally, including nonequilibrium expansion and the metering of choked flow.
167 records. [Penner 1949] [Ehlers and Cohen 1951] [Kelber et al 1952] [Ferri 1954] [Probstein and Elliott 1954] [Gordon Watson 1955] [Persh 1955] [Meksyn 1958] [Hayman and Stewart 1961] [Mager 1961] [Wegener and Cole 1961] [Pearson 1962] [Clark and Johnson 1963] [Phinney 1963] [Emanuel 1964] [Supersonic annular nozzle as 1964] [Cheng and Lee 1966] [Goldman and Nosek 1966] [Goldman and Nosek 1966] [Hall et al 1966] [Kinslow and Miller 1966] [Sichel and Yin 1966] [Albacete and Glowacki 1967] [Anderson and John D. 1969] [Bushnell et al 1969] [Daum 1970] [Fischer et al 1970] [Ikui and Matsuo 1971] [Kemp and Owen 1971] [Back et al 1972] [Fishburn and Adamson 1972] [Schmidt et al 1972] [Johnson and Rose 1973] [Smith and Mosnier 1973] [Lee and Smith 1974] [Maxwell and Wessling 1974] [Sekundov 1974] [Stainback et al 1974] [Lewis and Williams 1975] [Simoneau 1975] [Weilmuenster 1975] [Kim 1976] [Supersonic Flow and Shock 1976] [Condensation in Expansions Experimental 1977] [Bestion et al 1983] [Dwyer 1983] [Li 1983] [Ota and Chapman 1983] [Rao et al 1983] [Wornom and Hafez 1984] [Dashevskaya and Kholshchevnikova 1986] [Plendl 1986] [Evans 1987] [Sobota and Marble 1987] [Liou 1988] [Schnerr 1988] [Grasso and Speziale 1989] [Kinetic Theory Model for 1989] [Gnoffo 1991] [Jacobs 1991] [Pauley and Dagher 1991] [Riley 1991] [Spekreijse 1991] [Zeitoun et al 1991] [Harloff et al 1992] [Lemonnier and Selmer-Olsen 1992] [Mitsuda and Kurosaka 1992] [Kobayashi and Togashi 1993] [Noguchi and Shiratori 1993] [Demetriades 1994] [Tzuk et al 1994] [Yagudin 1994] [Ducros et al 1995] [Heinbockel and Landry 1995] [Jialing and Weizhong 1995] [Demetriades 1996] [Hanifi et al 1996] [Lin et al 1996] [Tam et al 1996] [Walpot et al 1996] [Spyropoulos et al 1997] [Tam et al 1997] [Brogan and Demetriades 1998] [Bui and Mankbadi 1998] [Inlet, Duct, and Nozzle 1998] [Suzen and Hoffmann 1998] [Akmandor and Nagashima 1999] [Akmandor and Nagashima 2000] [Colonna and Capitelli 2000] [Colonna 2001] [Colonna and Capitelli 2001] [Lucia et al 2001] [Toison and Hureau 2001] [Woodruff et al 2001] [Colonna et al 2002] [Colonna et al 2003] [Foysi and Friedrich 2003] [Mathelin et al 2003] [Zang et al 2003] [Colonna 2005] [Foysi and Friedrich 2005] [Kim et al 2005] [Majdalani and Abu-Irshaid 2005] [Abdol-Hamid et al 2006] [Chen et al 2006] [Kim et al 2006] [Ozalp 2006] [Ghosh et al 2007] [Titov et al 2007] [Titov et al 2008] [Chen et al 2009] [Hong et al 2009] [Josyula and Bailey 2009] [Levin and Skopina 2009] [Li et al 2009] [Marineau and Hornung 2009] [Yu et al 2009] [Gross et al 2010] [Vedula and Josyula 2010] [Yamamoto et al 2010] [Ghosh et al 2011] [Majdalani and Maicke 2011] [Liu and Oh 2012] [Tsuge 2012] [Lin et al 2013] [Wang and Xin 2013] [Critical Flow Venturi Nozzle 2015] [Cuong and Thanh 2015] [Snehil Varghese et al 2015] [Tsuge 2015] [von Lavante et al 2015] [Cyklis and Młynarczyk 2016] [Kiselev et al 2016] [Tsuge 2017] [Chen et al 2018] [Młynarczyk 2018] [Pfeiffer and Nizenkov 2019] [Xiang et al 2019] [Xie et al 2019] [Zhao et al 2019] [Ding et al 2020] [Pujowidodor et al 2020] [Takegawa et al 2020] [Ederouich et al 2021] [Korepanov et al 2021] [Mahmoudian et al 2021] [Tsuge 2021] [Wang et al 2021] [Winn and Touber 2021] [Fang 2022] [Tsuge 2022] [Gao et al 2023] [Ghodbane and Boumeddane 2023] [Huang et al 2023] [Lai 2023] [LeFloch and Thanh 2023] [Liang et al 2023] [Qian 2023] [Man 2024] [zhu 2024] [Chen et al 2025] [Chou et al 2025] [Lin and Zhong 2025] [Roy and Dawar 2025] [Turbulence and Transition in 2026] [Acosta] [Yaping He]
Textbooks, histories and surveys of rocket propulsion
Textbooks, handbooks, histories, surveys and proceedings of rocket propulsion stated in general terms.
52 records. [Some Fundamental Considerations Relating 1958] [Chapter 4. Design of 1959] [Fundamentals of rocket propulsion 1960] [Smith 1962] [Penner 1968] [Dattorre et al 1976] [Dushkin 1977] [Tishkoff et al 1989] [Introduction to Liquid-Propellant Rocket 1992] [Meisl 1992] [Williams 1992] [McConnaughey et al 1993] [Obrien 1993] [Taylor 1996] [Navaz and Dix 1998] [Meyer 1999] [Rothmund and Harlow 1999] [Liou et al 2001] [Hawk 2003] [Sutton 2003] [Anderson 2004] [Pisacane 2005] [Rocket and Spacecraft Propulsion 2005] [Appendix C Data for 2006] [France’s Liquid Propellant Rocket-Engine 2006] [Japan’s Liquid Propellant Rocket-Engine 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket Engines 2006] [Liquid Propellant Rocket-Engine Organizations 2006] [Rocket Propulsion 2006] [Hunley 2008] [Data for Some Liquid-Propellant 2016] [El-Sayed 2016] [El-Sayed 2016] [Emrich 2016] [Classification of Rocket Propulsion 2019] [Rocket Propulsion Classification of 2019] [Chemical Rocket Performance 2021] [Classification of Rocket Propulsion 2021] [Rocket Engines 2021] [Abbas 2024] [Chandler Karp and Jens 2024] [Musielak 2025] [Musielak 2025] [El-Sayed 2026] [Thakur et al 2026] [History and principles of] [The thermal rocket engine]
Where the Framing Breaks Down
The ideal spike of this article does not exist and the gap is not small. A real aerospike is truncated, carries a base, sheds a wake, needs cooling on a surface that faces the plume from the inside, and carries the extra weight of the spike [Aerospike engine]. The 5.3 to 8.5 percent bounds the prize and does not predict the win, and the base flow that decides the difference is beyond a one-dimensional isentropic model.
The trajectory is a model fitted to two points. The manufacturer publishes the altitude at liftoff and at cut-off and no altitude between them [ABL Payload User’s Guide]. The pressure-time integral was computed across a family of profiles and the spread reported, but the family is a guess about the shape of a gravity turn and a real ascent is not a power law.
The module count of the ARISE engine is not published by the programme, and the one reference that states a count, nine, gives no source, so the parity result concerns rings in general and the parent vehicle’s engine count, not the hardware that was to fly. If the modules are not arranged on a ring, the result does not apply at all.
The chamber recovery rests on five assumed properties and a mode identification that is not this article’s. It is offered as a demonstration that the frequency constrains the geometry, and the width of the band it produces measures how much.
And the register is not a primary source here. Both rows are marked as reconstructions by the compiler’s convention, which means the description this article opened with is somebody’s careful reading of open sources rather than a government statement. What is primary is the pair of dates, the pair of contractor names and the fact that two numbers were issued at once.
The Source Base
The Research References
The survey holds 8,762 research records, and every one of them is cited across 22 clusters. They come from the National Aeronautics and Space Administration’s technical reports server, the Defense Technical Information Center’s registry and a journal and conference index. A further 169 definitions are hand-written, being 106 primary documents and encyclopaedic anchors and 63 back-references to earlier articles in this series. The plug nozzle was studied between the late nineteen fifties and the late nineteen seventies by government laboratories and their contractors, who wrote report numbers.
Where the Clusters Publish
The share of report-server records runs from 15.3 to 64.5 percent across the clusters. The thinnest is the fixed-nozzle limits cluster, and the richest is the cluster on the exhaust nozzles of aircraft engines, where the plug nozzle’s first measurements were made in government laboratories. The keystone, altitude compensation, stands at 24.3 percent, against 29.1 percent for the survey as a whole.
The thin report share of the aerospike clusters is matched by a larger share from one publisher. Counting records that carry the digital object identifier prefix of the American Institute of Aeronautics and Astronautics, that society publishes 34 percent of the altitude compensation cluster, 49 percent of the wake cluster and 43 percent of the fixed-nozzle limits cluster, against 36 percent of the survey. Nearly half of the wake literature comes from that one society. The aircraft exhaust nozzle cluster is 24 percent from that publisher and 65 percent reports. The aerospike’s own flows lean on one society’s publications and the older nozzle work around them leans on reports, which suggests that the keystone’s thin report share reflects where this work is published rather than a gap in the archives.
Every Central Noun Is Owned by Another Field
Every central noun of this subject is owned by another field. A search of the bibliographic index for each bare term returns another field’s literature.
spike returns music, wheat, neural spike trains and the coronavirus spike protein, with nothing mechanical in ten results. wake closure returns fatigue crack closure in the crack wake, an entire metallurgical literature, and no rocket result at all. open wake and closed wake return a university and wind-farm wake steering. area ratio is owned by cardiology and thrust coefficient by wind turbines. gas generator is owned by nineteen-seventies coal gasification. altitude compensation returns altitude physiology and a 1976 paper on compensating a carburettor, which is the same idea in a different machine and is not this literature.
So the survey’s patterns pin each noun with a second noun. The compounds that name this subject without collision are truncated plug nozzle, aerospike nozzle, expansion deflection nozzle, plug nozzle altitude compensation and linear aerospike. The only single terms that serve alone are aerospike, turbopump and truncated plug, and aerospike still returns a dictionary headword and a fish.
The proper nouns are worse than the physics. ARISE returns a poem, a hymn and a real European atmospheric-dynamics research infrastructure of the same name. ABL returns Abl tyrosine kinase. ARMR is unusable as a search term, returning six fields and no aerospace, and even the full expansion is not an anchor, since Affordable Responsive Modular Rocket returns affordable housing. RS1 returns the retinoschisin gene. Invocon and Troy7 each return nothing at all.
The Definition of Primary
A research record is a report primary when its identifier is a citation in the National Aeronautics and Space Administration’s technical reports server or a registration under the Defense Technical Information Center’s prefix, 10.21236. By that rule 2,546 of the 8,762 research records, 29.1 percent, are report primaries. The rule classifies by where a record is served and not by who wrote it, so a laboratory’s paper published by a society counts as secondary. The primary documents of the programme itself, being the laboratory’s fact sheet, the award announcement, the manufacturer’s payload user’s guide and incident report, and the register, are hand-written definitions and are not counted among the research records.
Thin Subjects
Seven subjects that this article’s derivations depend on have thin literatures in the survey. Each is counted over the titles of the 8,762 cited research records, a title matching when it carries a term from every group given, with groups separated by semicolons.
| Subject | Title terms | Titles |
|---|---|---|
| differential throttling as a means of steering | differential throttling, differentially throttled, differential throttle | 5 |
| the open, transitional and closed wake of a truncated plug | open wake, closed wake, wake closure, wake transition; plug, spike, aerospike | 0 |
| the area ratio that is optimal over a whole trajectory | optimal expansion ratio, optimum expansion ratio, optimal area ratio, optimum area ratio, trajectory averaged | 1 |
| a criterion for the onset of nozzle flow separation | separation; criterion, criteria; nozzle, nozzles | 1 |
| the first tangential acoustic mode of a chamber | tangential; mode, modes, instability | 9 |
| losing one engine of several | engine out, engine failure, engine failures | 12 |
| the convex duality of thrust and ambient pressure | legendre, convex conjugate, bregman, fenchel; nozzle, nozzles, thrust | 0 |
No title in the survey names the open, transitional or closed wake of a plug. The two titles that name wake transition concern laminar and turbulent transition behind flat plates and bodies, so the three regimes the programme names are carried by the curated papers cited under The Three Regimes the Programme Names rather than by the survey. Differential throttling appears in five titles, all of them on aerospikes, the earliest from 2003 and four of them from 2024 onward. The tangential mode appears in nine titles, two of them on solid motors and the rest on liquid-propellant or unstated rocket chambers. Engine-out appears in twelve titles, eight of them on a launch vehicle or its stage. No title applies Legendre or convex duality to nozzle thrust, and one title optimises a nozzle over a trajectory average, so the mean-in-time rule derived above has almost no company among the survey’s titles.
What the Survey Admits
A record is admitted only when a person reading its title finds it on this article’s subject, meaning a subject the argument uses. Those subjects are altitude-compensating and fixed nozzles and their flows, separation and side loads, base flow and the wake, jets and plumes, thrust vector control and thrust allocation, the liquid rocket engine and its machinery, clustered and modular engines and engine-out, manufacture and cost, launch vehicles and the ascent, measurement and ground test, the atmosphere as a flight condition, the programme and its sponsors, and compressible flow stated generally. The vocabulary of this subject has to be read in compounds, because every central noun of it is owned by another field, as the subsection above shows.
The classes excluded are those that share a word with this subject and nothing else, and those whose domain is distant from it. Nozzles for liquids, sprays, welding, cold spray, isotope separation, turbochargers and the blade rows of turbines share the word nozzle. Mantle plumes and groundwater share the word plume, an astrophysical jet shares the Mach disk, a nose spike on a blunt body shares the word spike without being a nozzle, theatre and literary studies share the words staging and wake, and a database product is named Aerospike. Electric, nuclear and beamed propulsion, solid propellant chemistry and hybrid fuels, re-entry and recovery, spacecraft and satellites, and science carried on rockets are rocket and space subjects the argument does not use, unless the title’s subject is a nozzle, a plume, steering or chamber acoustics. Payload environments, stage structures, jet and rocket noise, aircraft aerodynamics, air-breathing engines and missile ballistics are excluded on the same ground. Correction, erratum, withdrawal and retraction notices, figure and supplementary-material records and the annual subject indexes of journals are excluded as well, because a survey counts research works and those are parts of works or editorial events rather than works. Two records with the same folded title and the same first author are treated as one document, such as a conference paper and its journal version or a report deposited twice, and only the earlier is kept. A review of a book, a reference or definition entry and a notice are not admitted either, since none of them is a research work in its own right. The one exception to keeping the earlier of two records is a preprint, which gives way to its published version.
Some records of doubtful relevance are kept. A title whose subject cannot be told from its words is kept when its words name a rocket propulsion subject or venue. Reusable launch vehicle control titles that do not say whether the flight is ascent or entry are kept as steering, hybrid launch vehicles treated as vehicles are kept as launch vehicles, solid motor manufacture and cost are kept as manufacture, nozzles of wind tunnels are kept where the title studies the nozzle’s own flow, and aerospike nozzles fitted to detonation engines are kept because the nozzle is the subject. No cluster is a residual. The clusters on jets, aircraft nozzles, compressible flow and general rocket propulsion each hold only titles on a subject the argument uses. A reader may weigh each accordingly.
Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement.
What the Award Record Holds
A search of the federal award reporting system finds no record of this programme under the names it is known by, for the reason given under The Instrument That Bought It, which is that an award made through the Space Enterprise Consortium is administered by the consortium’s manager under a base agreement, so a search under the name of the company doing the work need not find it.
What This Article Read in Full
Five primary documents have been read end to end, and every quotation in this article comes from one of them. They are the laboratory’s public fact sheet for the programme, the two-page cleared version of the same document carrying its distribution statement, the award announcement as reprinted verbatim by the base newspaper, the manufacturer’s payload user’s guide of June 2022, and the manufacturer’s own static-fire incident report of 26 August 2024. The register’s officiality classification depends on which element of its markup carries a class attribute rather than on any visible text, so the officiality counts in this article are taken from the markup itself.
Three official mirrors of the award announcement are not retrievable by automated request, which is the documented behaviour of that domain and not evidence about the document. The version cited is a verbatim reprint carrying the same byline and date, and its text agrees with the portions quoted in independent summaries.
Epistemic State
Facts, taken from primary documents and stated as they appear there. The two designations, their shared date of 20 April 2022, their sponsor cell, their engines cell and their identical 101-character description come from the register [Parsch, DOD 4120.15-L Addendum]. The programme’s objectives, the three wake regimes, the modular architecture and the clearance number come from the laboratory’s fact sheet [ARISE and Fly]. The December 2019 award date, the three-year term, the other transaction instrument, the scope wording, the promise of thrust vector control and the three quotations come from the award announcement [AFRL awards agreements under ARISE]. The vehicle’s dimensions, propellants, cycle, engine counts, thrust figures and reference mission come from the manufacturer’s payload user’s guide [ABL Payload User’s Guide]. The static fire date, the eleven engines, Engine 10, the 4.5 kilohertz first tangential mode, the erosion of two injectors, the three hundred prior tests and the full loss timeline come from the manufacturer’s incident report [Flight 2 Static Fire Update].
Derivations, which follow from the facts by mathematics and are confirmed numerically. The affine dependence of thrust on ambient pressure, the conjugacy of ambient pressure and exit area, the Legendre structure of the envelope, the identification of the fixed-nozzle loss as a Bregman divergence, and the rule that the optimal fixed nozzle expands to the burn-averaged ambient pressure are all exact. Central differences reproduce the conjugacy identity in twelve cases to 2.1 parts in a hundred thousand million. One hundred independent optimisations reproduce the mean-in-time rule to 4.2 parts in ten million. A direct directional search reproduces the ring formulae in forty-two cases to below one part in a million million.
Quantities that depend on assumptions, with the assumptions swept and the spread reported. The burn-averaged ambient pressure depends on an ascent profile fitted to two published points, and runs from 19.4 to 41.6 percent of sea level across the family used. The impulse gain of an ideal spike over the best fixed nozzle runs from 5.3 to 8.5 percent across ratios of specific heats from 1.20 to 1.24 and chamber pressures from 3 to 10 megapascal. The recovered chamber diameter runs from 147 to 179 millimetres across forty-eight combinations of gas properties, and the recovered chamber pressure from 2.6 to 8.7 megapascal across thirty-six further combinations. The payload worth of a percent of first-stage exhaust velocity, at least three quarters of a percent, assumes a second-stage effective exhaust velocity of 3,000 to 3,400 metres per second and takes the 2.6 kilometre per second staging speed as a floor on the first stage’s ideal velocity increment.
Inferences, marked as such where they appear. That the fact sheet’s phrase about a launch-to-orbit trajectory means a representative trajectory rather than an orbital flight. That the 133,118 pound force total thrust in a nine-engine row is the eleven-engine figure. That the X-63A programme has ended. That the ARISE engine’s modules, if arranged on a ring, would obey the parity result. None of these is stated as a fact anywhere in this article and each is reversible by one document.
Two conflicts in the sources are recorded and not resolved. The maximum payload of the RS1 is given as 1,200 kilograms in the award announcement and 1,350 kilograms in the launch-vehicle reference [AFRL awards agreements under ARISE] [RS1, Gunter’s Space Page]. And the programme is named a suborbital experiment while its own fact sheet describes a launch-to-orbit trajectory [ARISE and Fly].
One accounting rule governs the register’s officiality counts. The markup has three states, so the counts under The Register and Its Unofficial Wording are given as three numbers, since a two-number summary must misplace the partly marked row.
Two dateline limits. Every register count in this article is as of the compiler’s update of 23 August 2026, which is later than this article’s date, and the counts that differ under the version current on that date are stated beside them under The Register and Its Unofficial Wording [Parsch, DOD 4120.15-L Addendum, 11 August 2025]. The encyclopedia entry cited for the X-63A’s history carries a revision date of 17 December 2025, which is nine days after this article’s date [ABL Space Systems X-63]. Every statement drawn from it concerns events well before that, and the inference about cancellation is made here independently of it. The page’s earlier state was not retrievable from the web archive, and the gap is recorded as a limit and left unresolved.
What is not known and would change things. The module count and arrangement of the ARISE engine, beyond one unsourced reference’s count of nine. The chamber pressure and area ratio of the aerospike as designed. Whether any hardware was built. The dollar value of either agreement. And whether the X-64A, which shares every word of its description with this one, ever reached a different stage.
Out of Scope
The X-64A is a separate article. The Invocon team’s vehicle is described as apparently recoverable, landing on four legs that double as stabilising fins, about 12 metres tall and about 2.4 metres in diameter [Invocon X-64, Directory of U.S. Military Rockets and Missiles, Appendix 4], and the measurement problem it exists to solve is a different one. It is deliberately left alone here beyond the fact that its designation was issued on the same day for the same words.
The internal aerodynamics of the plug contour are not treated. Designing the spike surface is a method-of-characteristics problem with a large literature and it does not bear on the questions asked here.
Base heating, cooling and structural mass are named and not computed, for the reason given above, which is that none of them can be obtained from the model this article uses.
Nothing here is a judgement about whether the money was well spent. The programme bought a flight that did not happen, from a company that left the business, on an instrument that leaves no public trace of its value. Those are three facts and they do not add up to an assessment.
Conclusion
Two X numbers were issued on one day with one description, and the description does not say which vehicle is which.
The thing both were built to fly is older than almost everyone working on it and has never flown at the scale of a launch vehicle. This article’s account of why is that the prize is real, bounded and awkwardly sized. An ideal altitude-compensating nozzle is worth between 5.3 and 8.5 percent of first-stage impulse on this vehicle’s published trajectory, measured by the gain $\eta$ against the best fixed nozzle that trajectory admits. The figure is large for a launch vehicle and small against the cost of a new nozzle architecture.
The mathematics is tidier than the hardware. Thrust is affine in ambient pressure with slope minus the exit area. The incremental vacuum thrust from an increment of exit area is exactly the exit pressure, which makes ambient pressure and exit area conjugate variables. The ideal spike’s thrust curve is the Legendre transform of the vacuum thrust, every fixed nozzle is one of its tangent lines, and the loss of a fixed nozzle is a Bregman divergence. From the same identity comes the design rule, which is that the optimal fixed nozzle expands to the time-averaged ambient pressure of its own flight, a statement containing nothing about the gas, the engine or the vehicle.
And that rule makes the case, though less loudly than a flat separation criterion would suggest. Because half of the pressure-time integral $J$ is collected in the first 11 to 21 percent of the burn, the rule places the optimal nozzle at an exit-to-ambient ratio at liftoff equal to the burn-averaged fraction of sea-level pressure, which is 0.2806 for the central case of Flow Separation as the Deciding Limit.
Whether that separates depends on which criterion is used. Under the correlation that NASA Technical Paper 1207 fits to contoured nozzles, rather than the flat four tenths of the early studies that it calls inadequate, the trajectory optimum separates only below about 3.5 megapascal of chamber pressure, and the pressure recovered for this engine from its own instability frequency straddles that figure. So the optimum is within about a tenth of the largest exit area a bell can carry attached, a real constraint pressed hard, short of an impossibility, and an aerospike does not have it at all because it has no divergent wall for a boundary layer to leave.
The other half of the programme yields a second exact result and it points the other way. Differential throttling of a ring of modules gives a directional average authority of exactly the throttle depth divided by pi, at every module count from three upward, so more modules improve evenness rather than authority. The evenness depends on parity, because an odd ring has no diametrically opposite pair and its control polygon therefore has two sides per module instead of one. Nine modules steer as evenly as eighteen and better than any even count below twenty. After a failure the ranking reverses, because a failure breaks the symmetry the parity argument rests on, and then what matters is only how many are left.
The programme did not fly. The first vehicle of the parent design failed 10.87 seconds after liftoff, the second burned on the pad while the water ran out, and the company renamed itself and went into missile defence. The register still carries both numbers, side by side, with the same hundred and one characters against each. Nothing in it records that the two rows were ever anything but equal.
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