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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Analysis of shuttle oscillation in the Mach Number = 1.7 to Mach Number = 1.0 range

The first five flights of the Space Transportation System (STS) have exhibited unexpected lateral oscillations in the Mach Number = 1.7 to Mach Number = 1.0 region of the descent trajectory. These oscillations can be split into two parts: a predominantly rolling oscillation referred to as the quarter hertz motion and a long-priod, large-amplitude oscillation in sideslips, roll rate, and yaw rate. These motions are analyzed in this paper in order to determine their source and the mechanism driving them. They will first be examined to determine how much of the motion can be explained using the set of equations that contain linear aerodynamic parameters. Then differences between measured moments and those calculated using linear aerodynamics which are considered to be due to unmodeled moments will be exained. Possible error sources such as not inluding nonlinear aerodynamics in the vehicle model, measurement errors, and the effect of pressure gradients will be considered. Based on these analyses, conclusions as to the nature and cause of the Shuttle lateral oscillations will be drawn.

Suit, W. T.↗
NASA NTRS2014

The Origin of Inlet Buzz in a Mach 1.7 Low Boom Inlet Design

Supersonic inlets with external compression, having a good level performance at the critical operating point, exhibit a marked instability of the flow in some subcritical operation below a critical value of the capture mass flow ratio. This takes the form of severe oscillations of the shock system, commonly known as "buzz". The underlying purpose of this study is to indicate how Detached Eddy Simulation (DES) analysis of supersonic inlets will alter how we envision unsteady inlet aerodynamics, particularly inlet buzz. Presented in this paper is a discussion regarding the physical explanation underlying inlet buzz as indicated by DES analysis. It is the normal shock wave boundary layer separation along the spike surface which reduces the capture mass flow that is the controlling mechanism which determines the onset of inlet buzz, and it is the aerodynamic characteristics of a choked nozzle that provide the feedback mechanism that sustains the buzz cycle by imposing a fixed mean corrected inlet weight flow. Comparisons between the DES analysis of the Lockheed Martin Corporation (LMCO) N+2 inlet and schlieren photographs taken during the test of the Gulfstream Large Scale Low Boom (LSLB) inlet in the NASA 8x6 ft. Supersonic Wind Tunnel (SWT) show a strong similarity both in turbulent flow field structure and shock wave formation during the buzz cycle. This demonstrates the value of DES analysis for the design and understanding of supersonic inlets.

Mach 1.7↗
NASA NTRS2014

Grid Resolution Study over Operability Space for a Mach 1.7 Low Boom External Compression Inlet

This paper presents a statistical methodology whereby the probability limits associated with CFD grid resolution of inlet flow analysis can be determined which provide quantitative information on the distribution of that error over the specified operability range. The objectives of this investigation is to quantify the effects of both random (accuracy) and systemic (biasing) errors associated with grid resolution in the analysis of the Lockheed Martin Company (LMCO) N+2 Low Boom external compression supersonic inlet. The study covers the entire operability space as defined previously by the High Speed Civil Transport (HSCT) High Speed Research (HSR) program goals. The probability limits in terms of a 95.0% confidence interval on the analysis data were evaluated for four ARP1420 inlet metrics, namely (1) total pressure recovery (PFAIP), (2) radial hub distortion (DPH/P), (3) ) radial tip distortion (DPT/P), and (4) ) circumferential distortion (DPC/P). In general, the resulting +/-0.95 delta Y interval was unacceptably large in comparison to the stated goals of the HSCT program. Therefore, the conclusion was reached that the "standard grid" size was insufficient for this type of analysis. However, in examining the statistical data, it was determined that the CFD analysis results at the outer fringes of the operability space were the determining factor in the measure of statistical uncertainty. Adequate grids are grids that are free of biasing (systemic) errors and exhibit low random (precision) errors in comparison to their operability goals. In order to be 100% certain that the operability goals have indeed been achieved for each of the inlet metrics, the Y+/-0.95 delta Y limit must fall inside the stated operability goals. For example, if the operability goal for DPC/P circumferential distortion is ≤0.06, then the forecast Y for DPC/P plus the 95% confidence interval on DPC/P, i.e. +/-0.95 delta Y, must all be less than or equal to 0.06.

Grid↗
NASA NTRS1992

Two HSCT Mach 1.7 low sonic boom designs

The objective of this study was to provide low sonic boom concepts, geometry, and analysis to support wind tunnel model designs. Within guidelines provided by NASA, two High Speed Civil Transport (HSCT) configurations were defined with reduced sonic boom that have low drag, high payload, and good performance. To provide information for assessing the feasibility of reduced sonic boom operation, the two designs were analyzed in terms of their sonic boom characteristics, as well as aerodynamics, weight and balance, and performance characteristics. Low drag and high payload were achieved, but both of the blended arrow-wing configurations have deficiencies in high lift capability, fuel volume, wing loading, balance, and takeoff gross weight. Further refinement of the designs is needed to better determine the commercial viability of low boom operation. To help in assessing low boom design technology, the two configurations were defined as wind tunnel models with altered aft-bodies for the wind tunnel sting mounting system.

Haglund, George T.↗
NASA NTRS2023

Propulsion-Airframe Integration for Conceptual Redesign of a Low-Boom Supersonic Transport

A low-boom supersonic transport was generated in a previous low-boom multidisciplinary optimization (MDO) study that used computational fluid dynamics (CFD) off-body pressure to compute the undertrack sonic boom ground signature and calibrated low-fidelity aerodynamics analyses to compute the mission performance metrics. This low-boom aircraft, referred to as the Mach 1.7 40-PAX concept, can carry 40 passengers for a low-boom overland mission with cruise Mach 1.7 for the airport pairs over the continental US and has the potential to achieve an undertrack sonic boom ground noise level below 70 perceived level of decibels at the start of overland cruise (SOC). The engine for the Mach 1.7 40-PAX concept was designed using the Numerical Propulsion System Simulation (NPSS) for the mission analysis and modeled as a flow-through nacelle for CFD-based sonic boom analysis. To understand how the engine plume affects the undertrack ground signature, the Mach 1.7 40-PAX concept is redesigned in this paper after replacing the flow-through nacelles by CFD engines for sonic boom analysis using aeropropulsive CFD simulation. A process is developed for approximation of the NPSS engine at SOC by a CFD engine for aeropropulsive CFD simulation. The generated CFD engine has the identical nozzle boundary conditions and approximately the same mass flow and thrust as those of the NPSS engine at SOC. Then, the outer mold line of the configuration with the CFD engines is redesigned to approximately restore the low-boom characteristics of the Mach 1.7 40-PAX concept at SOC. Finally, the CFD simulation data for the redesigned concept with the CFD engines is used to calibrate the low-fidelity aerodynamic analyses for mission analysis of this concept. The cyclic dependency of the involved disciplinary analyses is resolved using an iteration method for a consistent coupling of the mission analysis, NPSS engine analysis, and low-boom redesign using the aeropropulsive CFD simulation. This low-boom redesign study is used as an example to demonstrate how the propulsion-airframe integration could be implemented for conceptual design of supersonic transports that satisfy both the low-boom and mission performance requirements.

Low-boom supersonic transport↗
NASA NTRS2020

Design Factors for Two-Dimensional, External-Compression Supersonic Inlets

Geometric and aerodynamic design factors were studied for the design of two-dimensional, external-compression inlets operating at a freestream Mach number of Mach 1.7. Computational simulations of the inlet flows were performed to obtain the inlet performance metrics consisting of the inlet flow rates, total pressure recovery, and total pressure distortion at the engine face. The key design factors identified included the external diffuser Mach number, cowl lip interior angle, bleed slot length, throat section aft centerbody slope, and subsonic diffuser length. Using the results of the Mach 1.7 inlet study, inlets were designed for Mach 1.4 and 2.0. The results provide useful insight on the significance of the design factors for the design of such inlets for commercial supersonic aircraft.

Slater, John W.↗
NASA NTRS1991

An investigation of a contoured wall injector for hypervelocity mixing augmentation

An experimental and computational investigation of a contoured wall fuel injector is presented. The injector was aimed at enabling shock-enhanced mixing for the supersonic combustion ramjet engines currently envisioned for applications on hypersonic vehicles. Three-dimensional flow field surveys, and temporally-resolved planar Rayleigh scattering measurements are presented for Mach 1.7 helium injection into Mach 6 air. These experimental data are compared directly with a three-dimensional Navier-Stokes simulation of the flow about the injector array. Two dominant axial vorticity sources are identified and characterized. The axial vorticity produced strong convective mixing of the injectant with the freestream. Shock-impingement was particularly effective as it assured seeding of baroclinic vorticity directly on the helium/air interface. The vorticity coalesced into a counter-rotating vortex pair of a sense which produced migration of the helium away from the wall. The influences of spatial averaging on the representation of the flow field as well as the importance of the fluctuating component of the flow in producing molecularly-mixed fluid are addressed.

Waitz, Ian A.↗
NASA NTRSDate not supplied

CFD Simulation of an Inward-Turning Supersonic Inlet Unstart at Flight Mach Number 1.7

Interest in inward-turning supersonic inlets renewed during NASA’s Commercial Supersonic Technology Project, as a means of reducing disturbances caused by propulsion integration. Inward-turning designs have the potential to greatly reduce the strength of shock waves emanating outward from the nacelle, and therefore their contribution to the overall sonic boom signature. For top-mounted, or over-wing installations, disturbances to aircraft aerodynamics are reduced. The nature of the inward-turning geometry results in a boundary-layer thickening effect, opposite that of the familiar thinning of an outward-turning cone boundary-layer. The application of a small amount of boundary-layer bleed at the throat has been shown to alleviate the negative effects of the boundary-layer on subsonic diffuser performance. Boundary-layer bleed, however, comes at the expense of increased complexity, and external disturbances at bleed exits that would partially defeat the purpose of the inward-turning scheme. Vortex generators were proposed as an alternative to bleed, and an initial CFD study of their effectiveness in this application forms the basis of the present work. A source-term model in the Wind-US CFD code was used to model the effects of an array of vortex generators placed at the entrance to the subsonic diffuser. Baseline results with no vortex generators compared favorably to previously reported results with no bleed at back-pressures near the design point, but stability margin in the present case was limited by a curious non-linear phenomena similar to unstart in mixed-compression inlets. While some improvement was noted with vortex generators, their effectiveness was diminished by the limited stability margin. A time-accurate calculation was subsequently performed to better understand this process. Results of this calculation are presented and instantaneous solutions compared with steady-state, subcritical results previously reported. The stability margin limitation is attributed to the use of a constant-pressure outflow boundary condition, which differed from that used in the prior simulations.

Pre-Compressor Cooling, Water Injection, Hypersoni↗
NASA NTRS1979

Experimental studies of radiative base heating of a Jovian entry model

A scale model of the Jovian entry vehicle of 6.4 cm diameter is tested in an electric-arc-driven shock tube while a sphere model of 5 cm diameter is tested in a combustion-driven shock tube and an electric-arc-driven shock tunnel. The radiative heat-transfer rate and pressure on the front and the base regions are measured in the absence of ablation with sensors imbedded in the models in a stream consisting of 10% hydrogen in a bath of either neon or argon. The measured radiative heat-transfer rates and pressures are up to about 22 kW/sq cm and 12 atm, respectively, at the front stagnation point. The ratio of the base-to-stagnation-point radiative heat-transfer rates is found to be about 1/4 for the sphere at Mach 1.8, about 1/30 for the sphere at Mach 4.8, and about 1/6 for the scale model at Mach 1.7. When the effects of model geometry and Mach number are accounted for, the present experimental results agree well with the theoretical prediction of Park.

Shirai, H.↗
NASA NTRSDate not supplied

Advanced Supersonic Parachute Inflation Research and Experiment-2 (ASPIRE2) Flight Mechanics Modeling and Simulation

Introduction: The Advanced Supersonic Para-chute Inflation Research Experiment-2 (ASPIRE2) program is a sounding rocket flight test to be conducted at Wallops Island, VA in early 2025. This program is a risk mitigation exercise for the Mars Sample Retrieval Lander (MSRL) program and builds upon the success of the 2017 & 2018 ASPIRE program, [1,2]. While the ASPIRE program certified the strengthened 21.5 m diameter disk-gap-band (DGB) deployed at Mach 1.7, ASPIRE2 will certify a 24 m diameter DGB deployed at Mach 2.1. The need to certify this increased parachute performance is driven the increased lander mass for MSRL; over 50% increase in comparison to M2020. Modeling: Like its predecessor, ASPIRE2 is developing a multi-body flight dynamics simulation to predict parachute dynamics and aide in designing the flight test that will target Mars-relevant flight conditions, as shown in Figure 1. This work de-scribes the parachute modeling, flight mechanics simulations (from payload separation to splash-down), and design trades used to prepare for the 2025 ASPIRE2 flight. Discussed herein are comparisons between ASPIRE and ASPIRE2, noting key differences in the parachute modeling and vehicle configuration. A study on the attitude control system performance impacts with respect to payload section design will also be presented. This work will quantify the pre-flight parachute performance in the presence of uncertainties, such as those associated with the separation from the sounding rocket, atmosphere, the parachute system, and vehicle mass. The pre-flight predictions will include Monte Carlo analyses, powered by the flight mechanics simulations to show the ASPIRE2 vehicle performance in meeting program requirements on parachute deployment conditions (Mach & dynamic pressure), parachute loads, vehicle attitude at key milestones, and the vehicle splashdown conditions.

Entry Descent and Landing↗
NASA NTRS1993

Summary of the GASP code application and evaluation effort for scramjet combustor flowfields

Numerical simulations of 3D turbulent mixing and reacting flows have been systematically evaluated by comparison with experimental data and other numerical solutions to provide confidence in the General Aerodynamic Simulation Program (GASP) code version 1.3. Specifically, the GASP flow solver has been used to model high speed flow through scram jet combustors. Unit injector problems evaluated include (1) UVA staged sonic normal injection of air into Mach 2 air stream, (2) the VPI Mach 1.7 low angled (15 deg) flush wall helium injection into an unconfined Mach 6 'cold air' stream, and (3) the HYPULSE angled (30 deg) flush wall hydrogen injection into a Mach 6, high enthalpy (flight Mach 17 simulation) nitrogen and air test gases. Results of the numerical simulation for the first two test cases have been compared with detailed in-stream measurements and with SPARK CFD solutions. The results for the last two cases have been compared with measured wall pressure and heat flux data and with SPARK CFD solutions. The present GASP solutions compare favorably with both the experimental data and the SPARK solutions.

Srinivasan, Shivakumar↗
NASA NTRSDate not supplied

X-59 Air Data Probe Calibration Wind Tunnel Test

NASA's aeronautical innovators are working with Lockheed Martin to design and build a supersonic X-plane, the X-59, under the the Low Boom Flight Demonstrator (LBFD) Project, that produces a gentle thump rather than a loud sonic boom. The plane will be used to collect human response data over select U.S. communities. The data from the X-59 test flights will be provided to U.S. and international regulators to potentially allow supersonic flight over land, drastically reducing travel time within the U.S. and around the world. Two wind tunnel tests were conducted in the NASA Glenn Research Center (GRC) 8- by 6-Foot Supersonic Wind Tunnel to calibrate the nose probe for the X-59. The probe was successfully tested and calibrated at 19 Mach numbers between Mach 0.25 and Mach 1.7. Most data were collected using continuous roll angle sweeps at a set Mach number and pitch angle. About 725 high quality data runs were recorded during testing to be used in the calibration of the probe. A backup nose probe was also run through the same test matrix as the primary probe. The probe will be the primary instrument for the flight test of the X-59 for determining angle of attack, angle of sideslip, airspeed, and pressure altitude of the aircraft in flight. Knowing the aircraft's speed and attitude is critical for flight safety and is also critical data for the Sonic Boom mission. Areas of interest for the test were sea level takeoff and landing at Mach 0.2, subsonic cruise at Mach 0.9, and supersonic cruise at Mach 1.4. During Phase I flight testing, the air data nose probe and air data system as a whole will be further calibrated in flight to account for additional installed effects of the airframe on the air data system.

X-59↗
NASA NTRSDate not supplied

X-59 Air Data Probe Calibration Wind Tunnel Test

NASA's aeronautical innovators are working with Lockheed Martin to design and build a supersonic X-plane, the X-59, under the the Low Boom Flight Demonstrator (LBFD) Project, that produces a gentle thump rather than a loud sonic boom. The plane will be used to collect human response data over select U.S. communities. The data from the X-59 test flights will be provided to U.S. and international regulators to potentially allow supersonic flight over land, drastically reducing travel time within the U.S. and around the world. Two wind tunnel tests were conducted in the NASA Glenn Research Center (GRC) 8- by 6-Foot Supersonic Wind Tunnel to calibrate the nose probe for the X-59. The probe was successfully tested and calibrated at 19 Mach numbers between Mach 0.25 and Mach 1.7. Most data were collected using continuous roll angle sweeps at a set Mach number and pitch angle. About 725 high quality data runs were recorded during testing to be used in the calibration of the probe. A backup nose probe was also run through the same test matrix as the primary probe. The probe will be the primary instrument for the flight test of the X-59 for determining angle of attack, angle of sideslip, airspeed, and pressure altitude of the aircraft in flight. Knowing the aircraft's speed and attitude is critical for flight safety and is also critical data for the Sonic Boom mission. Areas of interest for the test were sea level takeoff and landing at Mach 0.2, subsonic cruise at Mach 0.9, and supersonic cruise at Mach 1.4. During Phase I flight testing, the air data nose probe and air data system as a whole will be further calibrated in flight to account for additional installed effects of the airframe on the air data system.

X-59↗
NASA NTRSDate not supplied

Advanced Supersonic Parachute Inflation Research Experiment 2 (ASPIRE2) Flight Mechanics and Parachute Performance

The Advanced Supersonic Parachute Inflation Re-search Experiment-2 (ASPIRE2) program consists of a sounding rocket flight test to be launched at Wallops Flight Facility (WFF) in 2025. This effort seeks to qualify the supersonic parachute for the Sample Retrieval Lander (SRL) under the Mars Sample Return (MSR) program, building off the successes of its predecessor program, ASPIRE, which launched in both 2017 and 2018. The conops for the ASPIRE2 mission are shown in Figure 1 relative to its predecessor. While the ASPIRE pro-gram qualified a 21.5 m diameter disk-gap band parachute deployed at Mach 1.7, the SRL parachute will be increased to 24 m in diameter and target a Mach 2.1 parachute deployment condition, the largest diameter supersonic parachute deployed at the highest Mach number to date. These requirements are driven by the increased landing mass of the sample retrieval lander, which is roughly a 50% increase over that of the Mars2020 entry vehicle. These changes necessitate ASPIRE2 to certify the parachute performance under similar deployment conditions. This work will cover the 6-DoF para-chute model implementation and flight mechanics performance for the ASPIRE2 campaign and discuss how these results will impact the design of the test article. The ASPIRE2 trajectory is simulated using a multi-body flight dynamics tool, Program to Optimize Simulated Trajectories II (POST2). This work will summarize both the flight mechanics and parachute modeling (from payload separation until splashdown) and performance for this mission. Various design trades will be assessed such as para-chute tuning parameters as well as vehicle center of gravity location.

Evan Roelke↗
NASA NTRSDate not supplied

Propulsion-Airframe Integration for Conceptual Redesign of a Low-Boom Supersonic Transport

A low-boom supersonic transport was designed for a cruise Mach of 1.7 and 40 passengers. This low-boom aircraft, referred to as the Mach 1.7 40-PAX concept, was generated using computational fluid dynamics (CFD) based sonic boom analysis at the start of overland cruise (SOC). The engine for the Mach 1.7 40-PAX concept was designed using the Numerical Propulsion System Simulation (NPSS) and modeled as a flow-through nacelle for CFD analysis. To understand how the engine plume affects the undertrack ground signature, the Mach 1.7 40-PAX concept is redesigned using an aeropropulsive CFD simulation. A process for approximation of the NPSS engine at SOC by a CFD engine is developed. The generated CFD engine has the identical nozzle boundary conditions and approximately the same mass flow and thrust as those of the NPSS engine at SOC. Then, the configuration with the CFD engines is optimized to approximately restore the low-boom characteristics of the Mach 1.7 40-PAX concept. Finally, the CFD simulation data for the optimized concept with the CFD engines is used to calibrate the low-fidelity aerodynamic analyses for mission analysis of this concept. The cyclic dependency of the involved disciplinary analyses is resolved using an iteration method for a consistent coupling of the mission analysis, NPSS engine analysis, and low-boom redesign using the aeropropulsive CFD simulation. This low-boom redesign study is used as an example to demonstrate how the propulsion-airframe integration could be implemented for conceptual design of low-boom supersonic transports.

Low-boom supersonic transport↗
NASA NTRSDate not supplied

Propulsion-Airframe Integration for Conceptual Redesign of a Low-Boom Supersonic Transport

A low-boom supersonic transport was designed for a cruise Mach of 1.7 and 40 passengers. This low-boom aircraft, referred to as the Mach 1.7 40-PAX concept, was generated using computational fluid dynamics (CFD) based sonic boom analysis at the start of overland cruise (SOC). The engine for the Mach 1.7 40-PAX concept was designed using the Numerical Propulsion System Simulation (NPSS) and modeled as a flow-through nacelle for CFD analysis. To understand how the engine plume affects the undertrack ground signature, the Mach 1.7 40-PAX concept is redesigned using an aeropropulsive CFD simulation. A process is developed for approximation of the NPSS engine at SOC by a CFD engine for aeropropulsive CFD simulation. The generated CFD engine has the identical nozzle boundary conditions and approximately the same mass flow and thrust as those of the NPSS engine at SOC. Then, the configuration with the CFD engines is optimized to approximately restore the low-boom characteristics of the Mach 1.7 40-PAX concept. Finally, the CFD simulation data for the optimized concept with the CFD engines is used to calibrate the low-fidelity aerodynamic analyses for mission analysis of this concept. The cyclic dependency of the involved disciplinary analyses is resolved using an iteration method for a consistent coupling of the mission analysis, NPSS engine analysis, and low-boom redesign using the aeropropulsive CFD simulation. This low-boom redesign study is used as an example to demonstrate how the propulsion-airframe integration could be implemented for conceptual design of low-boom supersonic transports.

Low-boom supersonic transport↗
NASA NTRS1992

Ground signature extrapolation of three-dimensional near-field CFD predictions for several HSCT configurations

A CFD analysis of the near-field sonic boom environment of several low boom High Speed Civilian Transport (HSCT) concepts is presented. The CFD method utilizes a multi-block Euler marching code within the context of an innovative mesh topology that allows for the resolution of shock waves several body lengths from the aircraft. Three-dimensional pressure footprints at one body length below three-different low boom aircraft concepts are presented. Models of two concepts designed by NASA to cruise at Mach 2 and Mach 3 were built and tested in the wind tunnel. The third concept was designed by Boeing to cruise at Mach 1.7. Centerline and sideline samples of these footprints are then extrapolated to the ground using a linear waveform parameter method to estimate the ground signatures or sonic boom ground overpressure levels. The Mach 2 concept achieved its centerline design signature but indicated higher sideline booms due to the outboard wing crank of the configuration. Nacelles are also included on two of NASA's low boom concepts. Computations are carried out for both flow-through nacelles and nacelles with engine exhaust simulation. The flow-through nacelles with the assumption of zero spillage and zero inlet lip radius showed very little effect on the sonic boom signatures. On the other hand, it was shown that the engine exhaust plumes can have an effect on the levels of overpressure reaching the ground depending on the engine operating conditions. The results of this study indicate that engine integration into a low boom design should be given some attention.

Siclari, M. J.↗
NASA NTRS1994

Investigation of ramp injectors for supersonic mixing enhancement

A comparative study of wall mounted swept ramp injectors fitted with injector nozzles of different shape has been conducted in a constant area duct to explore mixing enhancement techniques for scramjet combustors. Six different injector nozzle inserts, all having equal exit and throat areas, were tested to explore the interaction between the preconditioned fuel jet and the vortical flowfield produced by the ramp: circular nozzle (baseline), nozzle with three downstream facing steps, nozzle with four vortex generators, elliptical nozzle, tapered-slot nozzle, and trapezoidal nozzle. The main flow was air at Mach 2, and the fuel was simulated by air injected at Mach 1.63 or by helium injected at Mach 1.7. Pressure and temperature surveys, combined with Mie and Rayleigh scattering visualization, were used to investigate the flow field. The experiments were compared with three dimensional Navier-Stokes computations. The results indicate that the mixing process is dominated by the streamwise vorticity generated by the ramp, the injectors' inner geometry having a minor effect. It was also found that the injectant/air mixing in the far-field is nearly independent of the injector geometry, molecular weight of the injectant, and the initial convective Mach number.

Haimovitch, Y.↗