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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Numerical prediction of transition of the F-16 wing at supersonic speeds

A parametric study is being conducted as an effort to numerically predict the extent of natural laminar flow (NLF) on finite swept wings at supersonic speeds. This study is one aspect of a High Speed Research Program (HSRP) to gain an understanding of the technical requirements for high-speed aircraft flight. The parameters that are being addressed in this study are Reynolds number, angle of attack, and leading-edge wing sweep. These parameters were analyzed through the use of an advanced Computational Fluid Dynamics (CFD) flow solver, specifically the ARC 3-D Compressible Navier-Stokes (CNS) flow solver. From the CNS code, pressure coefficients (Cp) are obtained for the various cases. These Cp's are then used to compute the boundary-layer profiles through the use of the 'Kaups and Cebeci' compressible 2-D boundary layer code. Finally, the boundary-layer parameters are processed into a 3-D compressible boundary layer stability code (COSAL) to predict transition. The parametric study then consisted of four geometries which addressed the effects of sweep, and three angles of attack from zero to ten degrees to yield a total of 12 cases. The above process was substantially automated through a procedure that was developed by the work conducted under this study. This automation procedure then yields a 3-D graphical measure of the extent of laminar flow by predicting the transition location of laminar to turbulent flow.

Cummings, Russell M.↗
NASA NTRS1976

Skin friction reduction by slot injection at Mach 0.8

Surface skin friction, boundary layer profiles and turbulent intensity due to axially symmetric tangential slot injection into a transonic boundary layer were measured. Effects of slot height, multiple slot injection, and injection mass flow rate on the surface skin friction downstream of the the slot have been investigated. Surface skin friction was a function of the injection mass flow rate for x/s 40. Large normal pressure gradient and relatively large turbulent intensity were found near the slot with small injection mass flow rate; the region the high turbulent intensity moved downstream with increasing injection mass flow rate. The results with two slot injections indicated that the distance between slots should be less than 30 slot heights in order to achieve some benefits from the first slot. Of significant importance in the present investigation is that the skin friction reduction obtained at transonic speed is of the same order as obtained in the hypersonic regime. Additional work is required in order to formulate a correlation between the turbulent intensity and the injection mass flow rate that may be used in future analysis.

Zakkay, V.↗
NASA NTRS1936

General instability criterion of laminar velocity distributions

The present paper describes the results of a stability investigation on symmetrical velocity profiles in a channel and of boundary-layer profiles. The effect of friction was assumed to be vanishing and did not occur in the stability consideration so far as it had been resorted to for preparatory asymptotic considerations. Proceeding on very general premises as regards the form of the velocity distribution, a proof was deduced of the elementary theorem that velocity profiles with inflection points are unstable.

Tollmien, W↗
NASA NTRSDate not supplied

Wind-US Simulations of the NASA 1507 Inlet Test Case

Steady-state, Reynolds-Averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) simulations were performed of the NASA 1507 Inlet Test Case from the 6th AIAA Propulsion Aerodynamics Workshop (PAW) held in January 2023 using the Wind-US CFD solver. The simulations were performed with bleed regions modeled as surface patches and the Slater bleed boundary condition models were applied within those regions. Both the constant-plenum-pressure and the fixed-plenum-exit bleed models where applied. The vortex generators were modeled using the Bender-Anderson-Yagle (BAY) vortex generator model. The CFD simulations were performed for inlet engine-face flow ratios varying from supercritical to subcritical inlet operation until inlet unstart. The results of the CFD simulations were compared to wind-tunnel data which included boundary layer profiles, axial static pressure distributions, bleed rates, total pressure recovery performance curves, and engine-face total pressure rake profiles. The CFD simulations were able to match most of the boundary layer rake data near the geometric throat, but the boundary layers on the centerbody were thinner than indicated by the wind-tunnel data. For the supercritical CFD simulations, the engine-face flow ratio was at least 2% to 2.5% lower than indicated by the wind-tunnel data, but still consistent with the level of supercritical bleed. The engine-face flow rates of the wind-tunnel tests had an uncertainty of ±2% which could explain the differences. The supercritical engine-face flow rates of the wind-tunnel data indicated a 5% supercritical bleed rate rather than the reported 7.18% bleed rate. The CFD simulations were able to calculate the maximum total pressure recovery of pt2/pt = 0.907 as observed for the wind-tunnel data. The CFD simulations with the constant-pressure bleed model were able to obtain a much lower engine-face total pressure ratio than indicated by both the wind-tunnel data and the CFD simulations using a fixed-plenum-exit bleed model. The CFD simulations indicated a sharp static pressure rise for the terminal shock rather than a more gradual rise indicated by the wind-tunnel data. The CFD simulations were not able to accurately compute the total pressure variation across the engine-face total pressure rake.

Supersonics↗
NASA NTRS1986

A three-dimensional axisymmetric calculation procedure for turbulent flows in a radial vaneless diffuser

An analytical model is proposed to calculate the three-dimensional axisymmetric turbulent flowfield in a radial vneless diffuser. The model assumes that the radial and tangential boundary layer profiles are approximated by power law profiles. Then, using the integrated radial and tangential momentum and continuity equations for the boundary layer and corresponding inviscid equations for the core flow, there results six ordinary differential equation in six unknowns which are easily solved using a Runge-Kutta technique. A model is also proposed for fully developed flow. The results using this technique were compared with the results from a three-dimensional viscous, axisymmetric duct code and with experimental data and good quantitative agreement was obtained.

Schumann, L. F.↗
NASA NTRS1992

Navier-Stokes analysis of an oxidizer turbine blade with tip clearance

The Gas Generator Oxidizer Turbine (GGOT) Blade is being analyzed by various investigators under the NASA MSFC sponsored Turbine Stage Technology Team design effort. The present work concentrates on the tip clearance region flow and associated losses; however, flow details for the passage region are also obtained in the simulations. The present calculations simulate the rotor blade row in a rotating reference frame with the appropriate coriolis and centrifugal acceleration terms included in the momentum equation. The upstream computational boundary is located about one axial chord from the blade leading edge. The boundary conditions at this location were determined by using a Euler analysis without the vanes to obtain approximately the same flow profiles at the rotor as were obtained with the Euler stage analysis including the vanes. Inflow boundary layer profiles are then constructed assuming the skin friction coefficient at both the hub and the casing. The downstream computational boundary is located about one axial chord from the blade trailing edge, and the circumferentially averaged static pressure at this location was also obtained from the Euler analysis. Results were obtained for the 3-D baseline GGOT geometry at the full scale design Reynolds number. Details of the clearance region flow behavior and blade pressure distributions were computed. The spanwise variation in blade loading distributions are shown, and circumferentially averaged spanwise distributions of total pressure, total temperature, Mach number, and flow angle are shown at several axial stations. The spanwise variation of relative total pressure loss shows a region of high loss in the region near the casing. Particle traces in the near tip region show vortical behavior of the fluid which passes through the clearance region and exits at the downstream edge of the gap.

Gibeling, Howard J.↗
NASA NTRS1993

Navier-Stokes analysis of an oxidizer turbine blade with tip clearance with and without a mini-shroud

The Gas Generator Oxidizer Turbine (GGOT) Blade is being analyzed by various investigators under the NASA MSFC-sponsored Turbine Stage Technology Team design effort. The present work concentrates on the tip clearance region flow and associated losses; however, flow details for the passage region are also obtained in the simulations. The present calculations simulate the rotor blade row in a rotating reference frame with the appropriate coriolis and centrifugal acceleration term included in the momentum equations. The upstream computational boundary is located about one axial chord from the blade leading edge. The boundary conditions at this location have been determined by Pratt & Whitney using an Euler analysis without the vanes to obtain approximately the same flow profiles at the rotor as were obtained with the Euler stage analysis including the vanes. Inflow boundary layer profiles are then constructed assuming the skin friction coefficient at both the hub and the casing. The downstream computational boundary is located about one axial chord from the blade trailing edge, and the circumferentially averaged static pressure at this location was also obtained from the P&W Euler analysis. Results obtained for the 3-D baseline GGOT geometry at the full scale design Reynolds number show a region of high loss in the region near the casing. Particle traces in the near tip region show vortical flow behavior of the fluid which passes through the clearance region and exits at the downstream edge of the gap. In an effort to reduce clearance flow losses, the mini-shroud concept was proposed by the Pratt & Whitney design team. Calculations were performed on the GGO geometry with the mini-shroud. Results of these calculations indicate that the mini-shroud does not significantly affect the flow in the passage region, and although the tip clearance flow is different, the mini-shroud does not seem to prevent the above-mentioned vortical flow behavior. Since both flow distortion and total pressure losses are similar for both geometries, the addition of the mini-shroud does not seem to reduce the tip clearance flow effects.

Chan, Tony↗
NASA NTRS1983

Pressures measured in flight on the aft fuselage and external nozzle of a twin-jet fighter

Fuselage, boundary layer, and nozzle pressures were measured in flight for a twin jet fighter over a Mach number range from 0.60 to 2.00 at test altitudes of 6100, 10,700, and 13,700 meters for angles of attack ranging from 0 deg to 7 deg. Test data were analyzed to find the effects of the propulsion system geometry. The flight variables, and flow interference. The aft fuselage flow field was complex and showed the influence of the vertical tail, nacelle contour, and the wing. Changes in the boattail angle of either engine affected upper fuselage and lower fuselage pressure coefficients upstream of the nozzle. Boundary layer profiles at the forward and aft locations on the upper nacelles were relatively insensitive to Mach number and altitude. Boundary layer thickness decreased at both stations as angle of attack increased above 4 deg. Nozzle pressure coefficient was influenced by the vertical tail, horizontal tail boom, and nozzle interfairing; the last two tended to separate flow over the top of the nozzle from flow over the bottom of the nozzle. The left nozzle axial force coefficient was most affected by Mach number and left nozzle boattail angle. At Mach 0.90, the nozzle axial force coefficient was 0.0013.

Nugent, J.↗
NASA NTRS1975

Analytical comparison of hypersonic flight and wind tunnel viscous/inviscid flow fields

Flow fields were computed about blunted, 0.524 and 0.698 radians, cone configurations to assess the effects of nonequilibrium chemistry on the flow field geometry, boundary layer edge conditions, boundary layer profiles, and heat transfer and skin friction. Analyses were conducted at typical space shuttle entry conditions for both laminar and turbulent boundary layer flow. In these calculations, a wall temperature of 1365 K (2000 F) was assumed. The viscous computer program used in this investigation was a modification of the Blottner non-similar viscous code which incorporated a turbulent eddy viscosity model after Cebeci. The results were compared with equivalent calculations for similar (scaled) configurations at typical wind tunnel conditions. Wind tunnel test gases included air, nitrogen, CF4 and helium. The viscous computer program used for wind tunnel conditions was the Cebeci turbulent non-similar computer code.

Fivel, H. J.↗
NASA NTRS1994

Receptivity of flat-plate boundary layer in a non-uniform free stream (vorticity normal to the plate)

Recent progress in both the linear and nonlinear aspects of stability theory has highlighted the importance of the receptivity problem. One of the most unclear aspects of receptivity study is the receptivity of boundary-layer flow normal to vortical disturbances. Some experimental and theoretical results permit the proposition that quasi-steady outer-flow vortical disturbances may trigger by-pass transition. In present work such interaction is investigated for vorticity normal to a leading edge. The interest in these types of vortical disturbances arise from theoretical work, where it was shown that small sinusoidal variations of upstream velocity along the spanwise direction can produce significant variations in the boundary-layer profile. In the experimental part of this work, such non-uniform flow was created and the laminar-turbulent transition in this flow was investigated. The experiment was carried out in a low-turbulence direct-flow wind tunnel T-361 at the Central Aerohydrodynamic Institute (TsAGI). The non-uniform flow was produced by laminar or turbulent wakes behind a wire placed normal to the plate upstream of the leading edge. The theoretical part of the work is devoted to studying the unstable disturbance evolution in a boundary layer with strongly non-uniform velocity profiles similar to that produced by outer-flow vorticity. Specifically, the Tollmien-Schlichting wave development in the boundary layer flow with spanwise variations of velocity is investigated.

Kogan, M. N.↗
NASA NTRS1992

Code validation for the simulation of supersonic viscous flow about the F-16XL

The viewgraphs and discussion on code validation for the simulation of supersonic viscous flow about the F-16XL are provided. Because of the large potential gains related to laminar flow on the swept wings of supersonic aircraft, interest in the applications of laminar flow control (LFC) techniques in the supersonic regime has increased. A supersonic laminar flow control (SLFC) technology program is currently underway within NASA. The objective of this program is to develop the data base and design methods that are critical to the development of laminar flow control technology for application to supersonic transport aircraft design. Towards this end, the program integrates computational investigations underway at NASA Ames-Moffett and NASA Langley with flight-test investigations being conducted on the F-16XL at the NASA Ames-Dryden Research Facility in cooperation with Rockwell International. The computational goal at NASA Ames-Moffett is to integrate a thin-layer Reynolds averaged Navier-Stokes flow solver with a stability analysis code. The flow solver would provide boundary layer profiles to the stability analysis code which in turn would predict transition on the F-16XL wing. To utilize the stability analysis codes, reliable boundary layer data is necessary at off-design cases. Previously, much of the prediction of boundary layer transition has been accomplished through the coupling of boundary layer codes with stability theory. However, boundary layer codes may have difficulties at high Reynolds numbers, of the order of 100 million, and with the current complex geometry in question. Therefore, a reliable code which solves the thin-layer Reynolds averaged Navier-Stokes equations is needed. Two objectives are discussed, the first in greater depth. The first objective is method verification, via comparisons of computations with experiment, of the reliability and robustness of the code. To successfully implement LFC techniques to the F-16XL wing, the flow about the leading edge must be maintained as laminar flow. Therefore, the second objective is to focus on a series of numerical simulations with different values of angle of attack, alpha, and Reynolds numbers. The purpose of the simulations is to study their effects on the two main factors which precipitate transition to turbulence at leading edges of highly swept wings (e.g., 'spanwise contamination' and 'crossflow instability').

Flores, Jolen↗
NASA NTRSDate not supplied

Boundary-Layer Flow Simulations Over Ablating Woven Thermal Protection System Material

Spallation is the mechanical removal of small chunks of material gets removed typically due to high shear conditions of the flow field. This reduces the ability of the thermal protection system (TPS) material to protect the spacecraft as well as cause turbulence in the flow causing higher heating rates. In this work, we focus on the material removal through ablation and high shear flow within the boundary layer region of woven TPS material. Woven TPS (WTPS) material is the latest class of material developed by NASA, to be used within the next generation of space flights. They are complex interlocked weaves designed to create a rigid structure that is highly resistant to heat and can be easily designed and tailored for a wide variety of entry environments. Due to material removal resulting from chemical degradation, the structural integrity of TPS material is affected. Spallation occurs when this structurally compromised material is exposed to the high shear flow conditions within the boundary layer. In order to understand the spallation mechanism within WTPS material, we first perform the material removal simulations which occur primarily through oxidation to obtain the microstructure at various stages of degradation. These simulations are performed using the Porous Microstructure Analysis (PuMA) software developed at NASA Ames. The micro-structure geometry used within these simulations were generated artificially to be similar to the 3D weave architecture of MSR-EEV (Mars Sample Return - Earth Entry Vehicle). The various eroded TPS micro-structures are then subjected to the boundary layer flow conditions to obtain critical surface quantities which contribute to the structural failure mechanism such as heat flux, pressure, and shear stress. The direct simulation Monte Carlo (DSMC) methodology is used to perform these simulations in order to accurately capture the strong gradients within the high-temperature boundary layer flow over the intricate geometry of WTPS material. The boundary layer profile is directly taken from the Computational Fluid Dynamics (CFD) simulation and provided as boundary conditions to the DSMC inlet and outlet. Further, the variation of these properties as the microstructure undergoes changes due to oxidation is also investigated. Finally, these quantities are used as input in PuMA to understand the material expansion/compression and strain within the woven TPS geometry and help in developing a comprehensive spallation and structure failure model.

microstructure↗
NASA NTRSDate not supplied

Boundary-Layer Flow Simulations Over Ablating Woven Thermal Protection System Material

Spallation is the mechanical removal of small chunks of material gets removed typically due to high shear conditions of the flow field. This reduces the ability of the thermal protection system (TPS) material to protect the spacecraft as well as cause turbulence in the flow causing higher heating rates. In this work, we focus on the material removal through ablation and high shear flow within the boundary layer region of woven TPS material. Woven TPS (WTPS) material is the latest class of material developed by NASA, to be used within the next generation of space flights. They are complex interlocked weaves designed to create a rigid structure that is highly resistant to heat and can be easily designed and tailored for a wide variety of entry environments. Due to material removal resulting from chemical degradation, the structural integrity of TPS material is affected. Spallation occurs when this structurally compromised material is exposed to the high shear flow conditions within the boundary layer. In order to understand the spallation mechanism within WTPS material, we first perform the material removal simulations which occur primarily through oxidation to obtain the microstructure at various stages of degradation. These simulations are performed using the Porous Microstructure Analysis (PuMA) software developed at NASA Ames. The micro-structure geometry used within these simulations were generated artificially to be similar to the 3D weave architecture of MSR-EEV (Mars Sample Return - Earth Entry Vehicle). The various eroded TPS micro-structures are then subjected to the boundary layer flow conditions to obtain critical surface quantities which contribute to the structural failure mechanism such as heat flux, pressure, and shear stress. The direct simulation Monte Carlo (DSMC) methodology is used to perform these simulations in order to accurately capture the strong gradients within the high-temperature boundary layer flow over the intricate geometry of WTPS material. The boundary layer profile is directly taken from the Computational Fluid Dynamics (CFD) simulation and provided as boundary conditions to the DSMC inlet and outlet. Further, the variation of these properties as the microstructure undergoes changes due to oxidation is also investigated. Finally, these quantities are used as input in PuMA to understand the material expansion/compression and strain within the woven TPS geometry and help in developing a comprehensive spallation and structure failure model.

microstructure↗
NASA NTRS1990

An experimental investigation of nozzle-exit boundary layers of highly heated free jets

An experimental investigation of the effects of nozzle operating conditions on the development of nozzle-exit boundary layers of highly heated air free jets is reported in this paper. The total pressure measurements in the nozzle-exit boundary layer were obtained at a range of jet Mach numbers from 0.1 to 0.97 and jet total temperatures up to 900 K. The analysis of results shows that the nozzle-exit laminar boundary-layer development depends only on the nozzle-exit Reynolds number. For the nozzle-exit turbulent boundary layer, however, it appears that the effects of the jet total temperature on the boundary-layer integral characteristics are independent from the effects of the nozzle-exit Reynolds number. This surprizing finding has not yet been reported. Further, laminar boundary-layer profiles were compared with the Pohlhausen solution for a flat-wall converging channel and an acceptable agreement was found only for low Reynolds numbers. For turbulent boundary layers, the dependence of the shape factor on relative Mach numbers at a distance of one momentum thickness from the nozzle wall resembles Spence's prediction. Finally, the calculated total pressure loss coefficient was found to depend on the nozzle-exit Reynolds number for the laminar nozzle-exit boundary layer, while for the turbulent exit boundary layer this coefficient appears to be constant.

Lepicovsky, J.↗
DOE OSTI2023

TRACER-Coastal Urban Boundary-Layer Interactions with Convection (TRACER-CUBIC) Field Campaign Report

To better understand the complicated web of processes governing convective cloud life cycle and aerosol-convection interactions, the U.S. Department of Energy (DOE)’s Atmospheric Radiation Measurement (ARM) user facility supported deployment of a variety of advanced atmospheric measurement systems to the greater Houston, Texas, area from 1 October 2021 to 30 September 2022 as part of the Tracking Aerosol Convection Interactions Experiment (TRACER). Houston was selected as a study area because isolated convection and a variety of aerosol conditions are common in this region. This one-year ARM Mobile Facility (AMF) deployment featured a four-month intensive operational period (IOP) during summer 2022 (1 June–30 September). The ARM instrumentation was deployed at three sites along an east-west transect from La Porte, Texas to an ancillary site in a less-polluted rural region southwest of downtown Houston (Figure 1). At the La Porte Site, which is located near the Houston ship channel in an area that experiences significant pollution, the first ARM Mobile Facility (AMF1) was deployed. During the IOP, the ARM tethered balloon system (TBS) operated at the ancillary site. The second-generation C-Band Scanning ARM Precipitation Radar (CSAPR) operated near Pearland, Texas, roughly halfway between the Laporte and ancillary sites. As part of the TRACER- Coastal Urban Boundary-Layer Interactions with Convection (CUBIC) project, three boundary-layer profiling systems) were deployed along a north-south transect spanning from the University of Houston Coastal Center to the Aldine site north of downtown Houston (also blue dot in Figure 1) during the TRACER IOP. These systems included the National Oceanic and Atmospheric Administration (NOAA) National Severe Storms Laboratory CLAMPS2 (C2), which was deployed at the UHCC, the University of Wisconsin SPARC, which was deployed at the ARM CSAPR site near Pearland (orange diamond in middle of map in Figure 1), and the University of Oklahoma CLAMPS1 (C1), which was deployed at Aldine. These three systems have been successfully operated in various field campaigns, providing data sets that collectively offer new insights into atmospheric-boundary-layer (ABL) processes, sea-breeze (SB) circulations, and convection initiation (CI). For the TRACER IOP window, these systems ran continuously between 1 June and 26 September, 2022. Due to commitments to NOAA projects, the Doppler lidar at the UHCC site was not available until 24 June 2022. The CLAMPS and SPARC profiling systems are self-contained platforms that have benefited from several years of development and deployment. Instruments and data processing were maintained remotely, which made the 4-month deployment for the TRACER-CUBIC IOP period possible. The same basic instrument configuration comprises each system: a scanning Doppler wind lidar for flow characterization and passive profiler(s) for characterizing planetary-boundary-layer (PBL) thermodynamic properties. Each platform includes a Halo Streamline Doppler wind lidar, an Atmospheric Emitted Radiance Interferometer (AERI), and a surface meteorology station. CLAMPS1 and CLAMPS2 each also include a microwave radiometer (MWR, Figure 1d). The TRACER-CUBIC hypotheses included (i) Interactions of SB and urban circulations and how they affect the PBL structure in the Houston environment, causing spatially (horizontally and vertically) and temporally highly variable flow patterns, (ii) heat, moisture, and aerosol transport and mixing depend on these flow dynamics, and (iii) an improved understanding of the flow patterns and PBL structure are critical for investigating the processes leading to CI. To test these hypotheses, the project aimed at (i) characterizing SB circulations and their impacts on the diurnal evolution of the structure of the ABL, (ii) studying the evolution of Houston’s complex urban boundary layer, and (iii) identifying effects of urban-induced circulations on pre-convective environments. TRACER-CUBIC observations generally provide good coverage during the summer IOP. Initial screening of the data indicates a good number of cases with bay-breeze (BB) and/or SB signatures, local CI, and interesting boundary-layer features such as strong nocturnal low-level jets (LLJs, Table 1). The numbers listed in rows 3-5 in this table will be further updated as part of ongoing in-depth analyses and systematic identification of local circulations and CI events. More detailed information about the data availability and quality for each instrument is provided in the “readme” files that were submitted to the ARM Data Center along with each archived data sets. These “readme” files also provide instrument descriptions, information about the data collection and processing procedures, data formats, and any additional information relevant for further data analysis.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1992

The hypersonic shock wave-turbulent boundary layer interaction generated by a sharp fin at Mach 8.2

A combined experimental and numerical study has been conducted on the hypersonic shock-wave turbulent-boundary layer interaction at Mach 8.2 generated by a single fin of angles alpha = 10 and 15 deg. Three models are considered: (1) the 3D compressible Reynolds-averaged Navier-Stokes (RANS) equations using the k-epsilon turbulence model, (2) the 3D RANS using the Rodi turbulence model, and (3) the conical RANS using the Baldwin-Lomax algebraic turbulence model. The computations are compared with various experimental data. The computations using models (1) and (2) show quantitatively very similar results and very good agreement with experimental data for surface pressure and skin friction. Comparison with boundary layer profiles of pitot pressure and yaw angle are also generally good, but the peak surface heat transfer is overestimated by up to 48 percent. The effect of the laminar boundary layer on the fin is restricted to the immediate vicinity of the fin surface. Conical calculations using model (3) show substantially poorer agreement with experiment.

Knight, Doyle D.↗
NASA NTRS1994

Effect of density gradients in confined supersonic shear layers, part 1

The effect of density gradients on the supersonic wall modes (acoustic modes) of a 2-D confined compressible shear layer were investigated using linear analysis. Due to the inadequacies of the hyperbolic tangent profile, the boundary layer basic profiles were used. First a test case was taken with the same parameters as in Tam and Hu's analysis with convective Mach number M(sub c) = 1.836 and density ratio of 1.398. Three generalized inflection points were found giving rise to three modes. The first two show similar properties to the Class A and B modes, and the third is an 'inner mode' which will be called a Class C mode. As the density ratio is increased, the smallest of the three neutral phase speeds tends towards the speed of the lower velocity stream, and the other two eventually coalesce and then disappear. These two effects lead to a linear resonance between the Class B modes which increases the cutoff frequency and growth rate of the lowest mode. In fact, growth rates of 2-4 times the test case were found as the density ratio was increased to 7. A similar trend is observed for the Class A modes when the density ratio is decreased from the test case, but the growth rate is not changed by much from the test case.

Peroomian, Oshin↗