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At least 595 records · Page 33

Development of Computational Aeroacoustics Code for Jet Noise and Flow Prediction

Accurate prediction of jet fan and exhaust plume flow and noise generation and propagation is very important in developing advanced aircraft engines that will pass current and future noise regulations. In jet fan flows as well as exhaust plumes, two major sources of noise are present: large-scale, coherent instabilities and small-scale turbulent eddies. In previous work for the NASA Glenn Research Center, three strategies have been explored in an effort to computationally predict the noise radiation from supersonic jet exhaust plumes. In order from the least expensive computationally to the most expensive computationally, these are: 1) Linearized Euler equations (LEE). 2) Very Large Eddy Simulations (VLES). 3) Large Eddy Simulations (LES). The first method solves the linearized Euler equations (LEE). These equations are obtained by linearizing about a given mean flow and the neglecting viscous effects. In this way, the noise from large-scale instabilities can be found for a given mean flow. The linearized Euler equations are computationally inexpensive, and have produced good noise results for supersonic jets where the large-scale instability noise dominates, as well as for the tone noise from a jet engine blade row. However, these linear equations do not predict the absolute magnitude of the noise; instead, only the relative magnitude is predicted. Also, the predicted disturbances do not modify the mean flow, removing a physical mechanism by which the amplitude of the disturbance may be controlled. Recent research for isolated airfoils' indicates that this may not affect the solution greatly at low frequencies. The second method addresses some of the concerns raised by the LEE method. In this approach, called Very Large Eddy Simulation (VLES), the unsteady Reynolds averaged Navier-Stokes equations are solved directly using a high-accuracy computational aeroacoustics numerical scheme. With the addition of a two-equation turbulence model and the use of a relatively coarse grid, the numerical solution is effectively filtered into a directly calculated mean flow with the small-scale turbulence being modeled, and an unsteady large-scale component that is also being directly calculated. In this way, the unsteady disturbances are calculated in a nonlinear way, with a direct effect on the mean flow. This method is not as fast as the LEE approach, but does have many advantages to recommend it; however, like the LEE approach, only the effect of the largest unsteady structures will be captured. An initial calculation was performed on a supersonic jet exhaust plume, with promising results, but the calculation was hampered by the explicit time marching scheme that was employed. This explicit scheme required a very small time step to resolve the nozzle boundary layer, which caused a long run time. Current work is focused on testing a lower-order implicit time marching method to combat this problem.

Keith, Theo G., Jr.↗

Computations of Internal and External Axisymmetric Nozzle Aerodynamics at Transonic Speeds

Computational Fluid Dynamics (CFD) analyses of axisymmetric circular-arc boattail nozzles have been completed in support of NASA's Next Generation Launch Technology Program to investigate the effects of high-speed nozzle geometries on the nozzle internal flow and the surrounding boattail regions. These computations span the very difficult transonic flight regime, with shock-induced separations and strong adverse pressure gradients. External afterbody and internal nozzle pressure distributions computed with the Wind code are compared with experimental data. A range of turbulence models were examined in Wind, including an Explicit Algebraic Stress model (EASM). Computations on two nozzle geometries have been completed at freestream Mach numbers ranging from 0.6 to 0.9, driven by nozzle pressure ratios (NPR) ranging from 2.9 to 5. Results obtained on converging-only geometry indicate reasonable agreement to experimental data, with the EASM and Shear Stress Transport (SST) turbulence models providing the best agreement. Calculations completed on a converging-diverging geometry involving large-scale internal flow separation did not converge to a true steady-state solution when run with variable timestepping (steady-state). Calculations obtained using constant timestepping (time-accurate) indicate less variations in flow properties compared with steady-state solutions. This failure to converge to a steady-state solution was found to be the result of difficulties in using variable time-stepping with large-scale separations present in the flow. Nevertheless, time-averaged boattail surface pressure coefficient and internal nozzle pressures show fairly good agreement with experimental data. The SST turbulence model demonstrates the best over-all agreement with experimental data.

Dalbello, Teryn↗

Time-Accurate Unsteady Pressure Loads Simulated for the Space Launch System at Wind Tunnel Conditions

A transonic flow field about a Space Launch System (SLS) configuration was simulated with the Fully Unstructured Three-Dimensional (FUN3D) computational fluid dynamics (CFD) code at wind tunnel conditions. Unsteady, time-accurate computations were performed using second-order Delayed Detached Eddy Simulation (DDES) for up to 1.5 physical seconds. The surface pressure time history was collected at 619 locations, 169 of which matched locations on a 2.5 percent wind tunnel model that was tested in the 11 ft. x 11 ft. test section of the NASA Ames Research Center's Unitary Plan Wind Tunnel. Comparisons between computation and experiment showed that the peak surface pressure RMS level occurs behind the forward attach hardware, and good agreement for frequency and power was obtained in this region. Computational domain, grid resolution, and time step sensitivity studies were performed. These included an investigation of pseudo-time sub-iteration convergence. Using these sensitivity studies and experimental data comparisons, a set of best practices to date have been established for FUN3D simulations for SLS launch vehicle analysis. To the author's knowledge, this is the first time DDES has been used in a systematic approach and establish simulation time needed, to analyze unsteady pressure loads on a space launch vehicle such as the NASA SLS.

Alter, Stephen J.↗

Monitoring Land Surface Albedo and Vegetation Dynamics Using High Spatial and Temporal Resolution Synthetic Time Series from Landsat and the MODIS BRDF/NBAR/Albedo Product

Seasonal vegetation phenology can significantly alter surface albedo which in turn affects the global energy balance and the albedo warmingcooling feedbacks that impact climate change. To monitor and quantify the surface dynamics of heterogeneous landscapes, high temporal and spatial resolution synthetic time series of albedo and the enhanced vegetation index (EVI) were generated from the 500-meter Moderate Resolution Imaging Spectroradiometer (MODIS) operational Collection V006 daily BRDF (Bidirectional Reflectance Distribution Function) / NBAR (Nadir BRDF-Adjusted Reflectance) / albedo products and 30-meter Landsat 5 albedo and near-nadir reflectance data through the use of the Spatial and Temporal Adaptive Reflectance Fusion Model (STARFM). The traditional Landsat Albedo (Shuai et al., 2011) makes use of the MODIS BRDFAlbedo products (MCD43) by assigning appropriate BRDFs from coincident MODIS products to each Landsat image to generate a 30-meter Landsat albedo product for that acquisition date. The available cloud free Landsat 5 albedos (due to clouds, generated every 16 days at best) were used in conjunction with the daily MODIS albedos to determine the appropriate 30-meter albedos for the intervening daily time steps in this study. These enhanced daily 30-meter spatial resolution synthetic time series were then used to track albedo and vegetation phenology dynamics over three Ameriflux tower sites (Harvard Forest in 2007, Santa Rita in 2011 and Walker Branch in 2005). These Ameriflux sites were chosen as they are all quite nearby new towers coming on line for the National Ecological Observatory Network (NEON), and thus represent locations which will be served by spatially paired albedo measures in the near future. The availability of data from the NEON towers will greatly expand the sources of tower albedometer data available for evaluation of satellite products. At these three Ameriflux tower sites the synthetic time series of broadband shortwave albedos were evaluated using the tower albedo measurements with a Root Mean Square Error (RMSE) less than 0.013 and a bias within the range of 0.006. These synthetic time series provide much greater spatial detail than the 500 meter gridded MODIS data, especially over more heterogeneous surfaces, which improves the efforts to characterize and monitor the spatial variation across species and communities. The mean of the difference between maximum and minimum synthetic time series of albedo within the MODIS pixels over a subset of satellite data of Harvard Forest (16 kilometers by 14 kilometers) was as high as 0.2 during the snow-covered period and reduced to around 0.1 during the snow-free period. Similarly, we have used STARFM to also couple MODIS Nadir BRDF-Adjusted Reflectances (NBAR) values with Landsat 5 reflectances to generate daily synthetic times series of NBAR and thus Enhanced Vegetation Index (NBAR-EVI) at a 30-meter resolution. While normally STARFM is used with directional reflectances, the use of the view angle corrected daily MODIS NBAR values will provide more consistent time series. These synthetic times series of EVI are shown to capture seasonal vegetation dynamics with finer spatial and temporal details, especially over heterogeneous land surfaces.

Vegetation Index↗

Evolving the SpaceFOM: Lessons Learned and Future Development

For over fifty years, simulation has been a cornerstone technology for space missions. In many cases, one single simulator, developed by one team, is not enough to meet the simulation objectives. Astronaut crew training, docking, vehicle testing and complex mission planning all require several simulations to be connected and to interoperate, in order to achieve the goal of the simulation. To facilitate the development of interoperable simulations, an open standard, the SISO SpaceFOM was developed in collaboration between government, industry and academia. The standard was released in 2020. The initial release of SpaceFOM focuses on the core interoperability topics for space simulation: (i) specifying spatial data using several, well defined reference frames, (ii) managing time (including real-time, hard real-time, and as-fast-as-possible execution), (iii) composition of systems with subsystems such as space vehicles, and (iv) execution control, such as initializing, running, pausing, and terminating simulations. The initial release of SpaceFOM focuses on the core interoperability topics for space simulation: (i) specifying spatial data using several, well defined reference frames, (ii) managing time (including real-time, hard real-time, and as-fast-as-possible execution), (iii) composition of systems with subsystems such as space vehicles, and (iv) execution control, such as initializing, running, pausing, and terminating simulations. A number of requirements for the next version of SpaceFOM have been identified and proposed solutions are already under development. One key area is the object classes used for data exchange, where a richer set of classes have been proposed, based on practical use cases. For time management, the use of variable time resolutions and variable time steps is also proposed. For execution control, save and restore of simulations is considered as well as improved robustness, to handle failing components. A number of clarifications are also under consideration. This paper provides both a forward-looking view of the technical developments underway and a retrospective on the standard's evolution. The authors emphasize that prioritizing core challenges and delivering a practical first version were crucial steps. While not every idea was implemented in the initial release, the timely delivery of a functional standard enabled real-world applications, the growth of a user community, and the collection of valuable insights to shape future versions. The authors look forward to working with the next version of SpaceFOM together with the growing user community.

Simulation Interoperability↗

Evolving the SpaceFOM: Lessons Learned and Future Development

For over fifty years, simulation has been a cornerstone technology for space missions. In many cases, one single simulator, developed by one team, is not enough to meet the simulation objectives. Astronaut crew training, docking, vehicle testing and complex mission planning all require several simulations to be connected and to interoperate, in order to achieve the goal of the simulation. To facilitate the development of interoperable simulations, an open standard, the SISO SpaceFOM was developed in collaboration between government, industry and academia. The standard was released in 2020. The initial release of SpaceFOM focuses on the core interoperability topics for space simulation: (i) specifying spatial data using several, well defined reference frames, (ii) managing time (including real-time, hard real-time, and as-fast-as-possible execution), (iii) composition of systems with subsystems such as space vehicles, and (iv) execution control, such as initializing, running, pausing, and terminating simulations. A number of requirements for the next version of SpaceFOM have been identified and proposed solutions are already under development. One key area is the object classes used for data exchange, where a richer set of classes have been proposed, based on practical use cases. For time management, the use of variable time resolutions and variable time steps is also proposed. For execution control, save and restore of simulations is considered as well as improved robustness, to handle failing components. A number of clarifications are also under consideration. This paper provides both a forward-looking view of the technical developments underway and a retrospective on the standard's evolution. The authors emphasize that prioritizing core challenges and delivering a practical first version were crucial steps. While not every idea was implemented in the initial release, the timely delivery of a functional standard enabled real-world applications, the growth of a user community, and the collection of valuable insights to shape future versions. The authors look forward to working with the next version of SpaceFOM together with the growing user community.

FOMs↗

Efficiency and reliability enhancements in propulsion flowfield modeling

The implementation of traditional CFD algorithms in practical propulsion related flowfields often leads to dramatic reductions in efficiency and/or robustness. The present research is directed at understanding the reasons for this deterioration and finding methods to circumvent it. Work to date has focussed on low Mach number regions, viscous dominated regions, and high grid aspect ratios. Time derivative preconditioning, improved definition of the local time stepping, and appropriate application of boundary conditions are employed to decrease the required time to obtain a solution, while maintaining accuracy. A number of cases having features typical of rocket engine flowfields are computed to demonstrate the improvement over conventional methods. These cases include laminar and turbulent high Reynolds number flat plate boundary layers, flow over a backward-facing step, a diffusion flame, and wall heat-flux calculations in a turbulent converging-diverging nozzle. Results from these cases show convergence that is virtually independent of the local Mach number and the grid aspect ratio, which translates to a convergence speed-up of up to several orders of magnitude over conventional algorithms. Current emphasis is in extending these results to three-dimensional flows with highly stretched grids.

Buelow, Philip E. O.↗

Numerical schemes for 3-wave kinetic equations: A complete treatment of the collision operator

In our previous work Walton and Tran (2023), numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. Here we then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.

97 MATHEMATICS AND COMPUTING↗

Ectomycorrhizal effects on decomposition are highly dependent on fungal traits, climate, and litter properties: A model-based assessment. Dataset.

To simulate the effects of mycorrhizal fungi on soil organic matter cycling, we incorporated mycorrhizal processes into the Carbon, Organisms, Rhizosphere, and Protection in the Soil Environment (CORPSE) model to develop a new soil model Myco-CORPSE. The new model was calibrated and evaluated against soil measurements taken at temperate forests in New Hampshire (NH) and Georgia (GA). A series of scenario analysis were also conducted to explore the conditions under which ectomycorrhizal (ECM) N acquisition processes can induce different soil C accumulation in ECM systems compared to arbuscular (AM) systems.In this data package, we included:-The Python codes of the standard Myco-CORPSE model we developed: "Standard Myco_CORPSE python codes.zip". The main program is the "gradient_sim.py" which calculates the bulk soil microbes and CN content along a user defined gradient of clay, soil temperature, soil moisture and mycorrhizal dominance, and relies on two subprograms "CORPSE_deriv.py" and "CORPSE_integrate.py". "CORPSE_deriv.py" calculated the changes in all simulated soil stock within every time step and "CORPSE_integrate.py" integrate the changes in all simulated soil stock within simulated time period. The program "Plot.py" is used to plot the major outputs produced by the main program "gradient_sim.py".-The modified Python codes of Myco-CORPSE models with site-level environmental inputs (in NH and GA) used to conduct simulations in NH and GA sites: "NH_GA model simulations.zip". -The Python codes used to evaluate the Myco-CORPSE simulation outputs in NH and GA sites against site-level measurements: "Plot NH_GA simulation against measurements.zip". It includes both the evaluation Python code, the model outputs on NH and GA sites, and the measured soil properties in both sites.-The modified Python codes of Myco-CORPSE models "Scenario analysis_model simulations.zip" that is used to conduct scenario analysis of how different litter properties, mycorrhizal fungal traits, climate, and seasonal variation in temperature and vegetation phenology impact the mycorrhizal effects on soil CN properties. The sub file folder "Scenario analysis_litter traits" contains the codes for scenario analysis of different litter properties; The sub file folder "Scenario analysis_ECM types" contains the codes for scenario analysis of different ECM fungal traits; The sub file folder "Scenario analysis_climate&seasonality" contains the codes for scenario analysis of different climate and seasonalities;-"Scenario analysis_model results and plotting codes.zip" contains all the output files from the the scenario analysis of Myco-CORPSE model as described above and the plotting codes used to the generate the heatmaps and scatterplots shown in the manuscript "Ectomycorrhizal effects on decomposition are highly dependent on fungal traits, climate, and litter properties: A model-based assessment"The majority of the model outputs did not have specific geographic information or temporal coverage because the analysis we conducted are mainly hypothetical model simulations. We only provided geographic description, coordinates and temporal coverage for those soil measurements which we used for model evaluations (included in the "Plot NH_GA simulation against measurements.zip").

54 ENVIRONMENTAL SCIENCES↗

An efficient and robust algorithm for two dimensional time dependent incompressible Navier-Stokes equations: High Reynolds number flows

An algorithm is presented for unsteady two-dimensional incompressible Navier-Stokes calculations. This algorithm is based on the fourth order partial differential equation for incompressible fluid flow which uses the streamfunction as the only dependent variable. The algorithm is second order accurate in both time and space. It uses a multigrid solver at each time step. It is extremely efficient with respect to the use of both CPU time and physical memory. It is extremely robust with respect to Reynolds number.

Goodrich, John W.↗

An efficient and robust algorithm for two dimensional time dependent incompressible Navier-Stokes equations - High Reynolds number flows

An algorithm is presented for unsteady two-dimensional incompressible Navier-Stokes calculations. This algorithm is based on the fourth order partial differential equation for incompressible fluid flow which uses the streamfunction as the only dependent variable. The algorithm is second order accurate in both time and space. It uses a multigrid solver at each time step. It is extremely efficient with respect to the use of both CPU time and physical memory. It is extremely robust with respect to Reynolds number.

Goodrich, John W.↗

Inlet Spillage Drag Predictions Using the AIRPLANE Code

AIRPLANE (Jameson/Baker) is a steady inviscid unstructured Euler flow solver. It has been validated on many HSR geometries. It is implemented as MESHPLANE, an unstructured mesh generator, and FLOPLANE, an iterative flow solver. The surface description from an Intergraph CAD system goes into MESHPLANE as collections of polygonal curves to generate the 3D mesh. The flow solver uses a multistage time stepping scheme with residual averaging to approach steady state, but R is not time accurate. The flow solver was ported from Cray to IBM SP2 by Wu-Sun Cheng (IBM); it could only be run on 4 CPUs at a time because of memory limitations. Meshes for the four cases had about 655,000 points in the flow field, about 3.9 million tetrahedra, about 77,500 points on the surface. The flow solver took about 23 wall seconds per iteration when using 4 CPUs. It took about eight and a half wall hours to run 1,300 iterations at a time (the queue limit is 10 hours). A revised version of FLOPLANE (Thomas) was used on up to 64 CPUs to finish up some calculations at the end. We had to turn on more communication when using more processors to eliminate noise that was contaminating the flow field; this added about 50% to the elapsed wall time per iteration when using 64 CPUs. This study involved computing lift and drag for a wing/body/nacelle configuration at Mach 0.9 and 4 degrees pitch. Four cases were considered, corresponding to four nacelle mass flow conditions.

Thomas, Scott D.↗

MSFC Stream Model Preliminary Results: Modeling the 1998-2002 Leonid Encounters and the 1993,1994, and 2004 Perseid Encounters

The cometary meteoroid ejection models of Jones (1996) and Crifo (1997) were used to simulate ejection from comets 55P/Tempel-Tuttle during the last 12 revolutions, and the 1862, 1737, and 161 0 apparitions of 1 OSP/Swift-Tuttle. Using cometary ephemerides generated by the JPL HORIZONS Solar System Data and Ephemeris Computation Service, ejection was simulated in 1 hour time steps while the comet was within 2.5 AU of the Sun. Also simulated was ejection occurring at the hour of perihelion passage. An RK4 variable step integrator was then used to integrate meteoroid position and velocity forward in time, accounting for the effects of radiation pressure, Poynting-Robertson drag, and the gravitational forces of the planets, which were computed using JPL's DE406 planetary ephemerides. An impact parameter is computed for each particle approaching the Earth, and the results are compared to observations of the 1998-2002 Leonid showers, and the 1993-1 994 Perseids. A prediction for Earth's encounter with the Perseid stream in 2004 is also presented.

Moser, D. E.↗

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗

A space-marching method for the computation of viscous internal flows

A space-marching method has been developed to compute 3-D viscous flows in internal geometries. The Navier-Stokes equations have been posed as an initial-value problem by neglecting the effects of streamwise diffusion and treating the streamwise pressure gradient as a known source term. The fully coupled system of equations has been solved by a noniterative algorithm at each streamwise step of the computation. A low Mach number formulation of the equations has been used to compute incompressible flow fields. A computer program has been written to implement all aspects of the space-marching algorithm. The program is modular and is easily adapted to the widely varying geometries of internal flows. The space-marching algorithm has been tested by computing simple flows with known analytical solutions. The method has been used to predict complex 3-D turbulent flows. The algorithm is stable and very economical. A single sweep of the flow field by the space-marching method is approximately equivalent to one time-step of the time-marching method.

Govindan, T. R.↗

Unsteady transonic Navier-Stokes computations for an oscillating wing using single and multiple zones

Modern jet transports and maneuvering tactical fighters operating in the transonic regime often give rise to time-dependent fluid physics that interact with flexible structural components, e.g., vortical flow, shocks, and separation. Efficient computational fluid dynamic (CFD) methods are required to study such computationally intensive problems. In this work a numerical method is presented to address this problem. Time-dependent, compressible, Navier-Stokes equations are used to simulate unsteady transonic flow about a three-dimensional rigid wing undergoing a forced periodic motion in angle of attack. An efficient, implicit, diagonal algorithm is utilized because of its low operation count per time step compared to other methods that solve systems of block matrix equations. The formal time accuracy is addressed theoretically and demonstrated numerically by comparison of computational results with experimental data. A zonal grid approach, capable of treating complex geometries, is presented and its time accuracy is demonstrated by comparing a zonal computation with a single grid computation and experimental data.

Chaderjian, Neal M.↗

An unconditionally stable Runge-Kutta method for unsteady rotor-stator interaction

A quasi-three-dimensional analysis has been developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stress-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using an explicit four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing is used to increase the stability limit of the time-accurate computations. The scheme is described, and stability and accuracy analyses are given.

Chima, Rodrick V.↗

Use of finite volume schemes for transition simulation

The use of finite-volume methods in the study of spatially and temporally evolving transitional flows over a flat plate is investigated. Schemes are developed with both central and upwind differencing. The compressible Navier-Stokes equations are solved with a Runge-Kutta time stepping scheme. Disturbances are determined using linear theory and superimposed at the inflow boundary. Time accurate integration is then used to allow temporal and spatial disturbance evolution. Characteristic-based boundary conditions are employed. The requirements of using finite-volume algorithms are studied in detail. Special emphasis is placed on difference schemes, grid resolution, and disturbance amplitudes. Moreover, comparisons are made with linear theory for small amplitude disturbances. Both subsonic and supersonic flows are considered, and it is shown that the locations of branch 1 and branch 2 of the neutral stability curve are well predicted, given sufficient resolution.

Fenno, Charles C., Jr.↗