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Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This talk describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.

Reliability of Complex Nonlinear Numerical Simulations

This work describes some of the procedure to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.

Numerical Investigation of Dual-Mode Scramjet Combustor with Large Upstream Interaction

Dual-mode scramjet combustor configuration with significant upstream interaction is investigated numerically, The possibility of scaling the domain to accelerate the convergence and reduce the computational time is explored. The supersonic combustor configuration was selected to provide an understanding of key features of upstream interaction and to identify physical and numerical issues relating to modeling of dual-mode configurations. The numerical analysis was performed with vitiated air at freestream Math number of 2.5 using hydrogen as the sonic injectant. Results are presented for two-dimensional models and a three-dimensional jet-to-jet symmetric geometry. Comparisons are made with experimental results. Two-dimensional and three-dimensional results show substantial oblique shock train reaching upstream of the fuel injectors. Flow characteristics slow numerical convergence, while the upstream interaction slowly increases with further iterations. As the flow field develops, the symmetric assumption breaks down. A large separation zone develops and extends further upstream of the step. This asymmetric flow structure is not seen in the experimental data. Results obtained using a sub-scale domain (both two-dimensional and three-dimensional) qualitatively recover the flow physics obtained from full-scale simulations. All results show that numerical modeling using a scaled geometry provides good agreement with full-scale numerical results and experimental results for this configuration. This study supports the argument that numerical scaling is useful in simulating dual-mode scramjet combustor flowfields and could provide an excellent convergence acceleration technique for dual-mode simulations.

Mohieldin, T. O.

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations.

Yee, H. C.

Numerical Modeling of Flow Control in a Boundary-Layer-Ingesting Offset Inlet Diffuser at Transonic Mach Numbers

This paper will investigate the validation of the NASA developed, Reynolds-averaged Navier-Stokes (RANS) flow solver, OVERFLOW, for a boundary-layer-ingesting (BLI) offset (S-shaped) inlet in transonic flow with passive and active flow control devices as well as a baseline case. Numerical simulations are compared to wind tunnel results of a BLI inlet experiment conducted at the NASA Langley 0.3-Meter Transonic Cryogenic Tunnel. Comparisons of inlet flow distortion, pressure recovery, and inlet wall pressures are performed. The numerical simulations are compared to the BLI inlet data at a free-stream Mach number of 0.85 and a Reynolds number of approximately 2 million based on the fanface diameter. The numerical simulations with and without tunnel walls are performed, quantifying tunnel wall effects on the BLI inlet flow. A comparison is made between the numerical simulations and the BLI inlet experiment for the baseline and VG vane cases at various inlet mass flow rates. A comparison is also made to a BLI inlet jet configuration for varying actuator mass flow rates at a fixed inlet mass flow rate. Overall, the numerical simulations were able to predict the baseline circumferential flow distortion, DPCP avg, very well within the designed operating range of the BLI inlet. A comparison of the average total pressure recovery showed that the simulations were able to predict the trends but had a negative 0.01 offset when compared to the experimental levels. Numerical simulations of the baseline inlet flow also showed good agreement with the experimental inlet centerline surface pressures. The vane case showed that the CFD predicted the correct trends in the circumferential distortion levels for varying inlet mass flow but had a distortion level that was nearly twice as large as the experiment. Comparison to circumferential distortion measurements for a 15 deg clocked 40 probe rake indicated that the circumferential distortion levels are very sensitive to the symmetry of the flow and that a misalignment of the vanes in the experiment could have resulted in this difference. The numerical simulations of the BLI inlet with jets showed good agreement with the circumferential inlet distortion levels for a range of jet actuator mass flow ratios at a fixed inlet mass flow rate. The CFD simulations for the jet case also predicted an average total pressure recovery offset that was 0.01 lower than the experiment as was seen in the baseline. Comparisons of the flow features for the jet cases revealed that the CFD predicted a much larger vortex at the engine fan-face when compare to the experiment.

Allan, Brian G.

Experimental and Numerical Investigation of the NASA High Efficiency Centrifugal Compressor Vaned Stage Geometry and Aerodynamic Performance

Since its inception in the early 2010s, the NASA High Efficiency Centrifugal Compressor (HECC) has been enigmatic for the propulsion research community: experimental data and numerical simulations of the stage have generally not aligned in their quantifications of performance metrics. Typically, the fault for these disagreements is assigned to the numerical simulations as the simulations are models, and models are inherently incomplete representations of the experiment. This “incompleteness” may manifest in assumptions regarding roughness or heat transfer, simplifications of the flow path (i.e., neglecting bleed flows), or the oft-scapegoated turbulence model. In the case of HECC, recent work showed unexpected discrepancies between the intended impeller geometry defined in the design report (termed Design-Intent) and the manufactured impeller used in the experimental campaigns (termed As-Manufactured). That work used numerical simulations to establish that the geometric differences between the Design-Intent and As-Manufactured impellers were significant enough to result in drastically different performance predictions for the HECC vaneless diffuser configuration. The Design-Intent impeller simulations over predicted the performance relative to the experiment, whereas the As-Manufactured simulations better represented the experimental data, both in terms of one-dimensional performance metrics and spanwise flow profiles. This effort expands on that work by examining in detail the geometry and aerodynamic performance of the HECC vaned diffuser configuration. Further differences between the Design-Intent and As-Manufactured geometries have been discovered in the vaned diffuser and exit guide vanes, and these differences are documented herein. The summations of the geometric differences for all of the components were used to create two numerical models of HECC vaned diffuser configuration: the Design-Intent simulations which are generated from the original geometry definitions given in the design report and the As-Manufactured simulations which are the best available representation of the manufactured compressor hardware used in the experimental test campaigns. In congruence with the earlier vaneless diffuser work, the numerical predictions of the Design-Intent choked mass flow rate, total pressure ratio, and efficiency were notably greater than that of the As-Manufactured simulations. To increase confidence in the experimental dataset, measurements from a recent test campaign conducted in 2024 are used to validate the original experimental data acquired from 2012 to 2014 with good repeatability overall, especially considering the passage of time and differences in the data acquisition systems between the test campaigns. Both numerical simulations were then extensively evaluated against the experimental data. The As-Manufactured simulations provided better estimates of the stage performance than the Design-Intent cases in terms of most performance metrics. Nonetheless, more detailed results still show opportunities for improvement. Despite a more accurate representation of the physical hardware, characterization of the impeller work input remains a challenge even for rigorously developed numerical models.

vaned diffuser

Experimental and Numerical Investigation of the NASA High Efficiency Centrifugal Compressor Vaned Stage Geometry and Aerodynamic Performance

Since its inception in the early 2010s, the NASA High Efficiency Centrifugal Compressor (HECC) has been enigmatic for the propulsion research community: experimental data and numerical simulations of the stage have generally not aligned in their quantifications of performance metrics. Typically, the fault for these disagreements is assigned to the numerical simulations as the simulations are models, and models are inherently incomplete representations of the experiment. This “incompleteness” may manifest in assumptions regarding roughness or heat transfer, simplifications of the flow path (i.e., neglecting bleed flows), or the oft-scapegoated turbulence model. In the case of HECC, recent work showed unexpected discrepancies between the intended impeller geometry defined in the design report (termed Design-Intent) and the manufactured impeller used in the experimental campaigns (termed As-Manufactured). That work used numerical simulations to establish that the geometric differences between the Design-Intent and As-Manufactured impellers were significant enough to result in drastically different performance predictions for the HECC vaneless diffuser configuration. The Design-Intent impeller simulations over predicted the performance relative to the experiment, whereas the As-Manufactured simulations better represented the experimental data, both in terms of one-dimensional performance metrics and spanwise flow profiles. This effort expands on that work by examining in detail the geometry and aerodynamic performance of the HECC vaned diffuser configuration. Further differences between the Design-Intent and As-Manufactured geometries have been discovered in the vaned diffuser and exit guide vanes, and these differences are documented herein. The summations of the geometric differences for all of the components were used to create two numerical models of HECC vaned diffuser configuration: the Design-Intent simulations which are generated from the original geometry definitions given in the design report and the As-Manufactured simulations which are the best available representation of the manufactured compressor hardware used in the experimental test campaigns. In congruence with the earlier vaneless diffuser work, the numerical predictions of the Design-Intent choked mass flow rate, total pressure ratio, and efficiency were notably greater than that of the As-Manufactured simulations. To increase confidence in the experimental dataset, measurements from a recent test campaign conducted in 2024 are used to validate the original experimental data acquired from 2012 to 2014 with good repeatability overall, especially considering the passage of time and differences in the data acquisition systems between the test campaigns. Both numerical simulations were then extensively evaluated against the experimental data. The As-Manufactured simulations provided better estimates of the stage performance than the Design-Intent cases in terms of most performance metrics. Nonetheless, more detailed results still show opportunities for improvement. Despite a more accurate representation of the physical hardware, characterization of the impeller work input remains a challenge even for rigorously developed numerical models.

centrifugal compressor

Pitfalls and guidelines for the numerical evaluation of moderate-order system frequency response

The design and evaluation of a feedback control system via frequency response methods relies heavily upon numerical methods. In application, one can usually develop low order simulation models which for the most part are devoid of numerical problems. However, when complex feedback interactions, for example, between instrument control systems and their flexible mounting structure, must be evaluated, simulation models become moderate to large order and numerical problems become common. A large body of relevant numerical error analysis literature is summarized in a large language understandable to nonspecialists. The intent is to provide engineers using simulation models with an engineering feel for potential numerical problems without getting intertwined in the complexities of the associated mathematical theory. Guidelines are also provided by suggesting alternate state of the art methods which have good numerical evaluation characteristics.

Frisch, H. P.

Finite analytic numerical solution of axisymmetric Navier-Stokes and energy equations

Convective heat transfer for steady state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature and heat transfer coefficients for flow with Reynolds number of 5.0, 100.0, 1000.0, and 2000.0, and Prandtl number of 0.1, 1.0, and 10.0 with uniform grid sizes are reported for axisymmetric pipe with sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic solution is stable, converges rapidly, and simulates the convection of fluid flow accurately since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values thereby minimizing the false numerical diffusion.

Chen, C.-J.

Numerical investigation of separated transonic turbulent flows with a multiple-time-scale turbulence model

A numerical investigation of transonic turbulent flows separated by curvature and shock wave - boundary layer interaction is presented. The free stream Mach numbers considered are 0.4, 0.5, 0.6, 0.7, 0.8, 0.825, 0.85, 0.875, 0.90, and 0.925. In the numerical method, the conservation of mass equation is replaced by a pressure correction equation for compressible flows and thus incremental pressure is solved for instead of density. The turbulence is described by a multiple-time-scale turbulence model supplemented with a near-wall turbulence model. The present numerical results show that there exists a reversed flow region at all free stream Mach numbers considered whereas various k-epsilon turbulence models fail to predict such a reversed flow region at low free stream Mach numbers. The numerical results also show that the size of the reversed flow region grows extensively due to the shock wave - turbulent boundary layer interaction as the free stream Mach number is increased. These numerical results show that the turbulence model can resolve the turbulence field subjected to extra strains caused by the curvature and the shock wave - turbulent boundary layer interaction and that the numerical method yields a significantly accurate solution for the complex compressible turbulent flow.

Kim, S.-W.

A numerical study of mixing in supersonic combustors with hypermixing injectors

A numerical study was conducted to evaluate the performance of wall mounted fuel-injectors designed for potential Supersonic Combustion Ramjet (SCRAM-jet) engine applications. The focus of this investigation was to numerically simulate existing combustor designs for the purpose of validating the numerical technique and the physical models developed. Three different injector designs of varying complexity were studied to fully understand the computational implications involved in accurate predictions. A dual transverse injection system and two streamwise injector designs were studied. The streamwise injectors were designed with swept ramps to enhance fuel-air mixing and combustion characteristics at supersonic speeds without the large flow blockage and drag contribution of the transverse injection system. For this study, the Mass-Averaged Navier-Stokes equations and the chemical species continuity equations were solved. The computations were performed using a finite-volume implicit numerical technique and multiple block structured grid system. The interfaces of the multiple block structured grid systems were numerically resolved using the flux-conservative technique. Detailed comparisons between the computations and existing experimental data are presented. These comparisons show that numerical predictions are in agreement with the experimental data. These comparisons also show that a number of turbulence model improvements are needed for accurate combustor flowfield predictions.

Lee, J.

A numerical study of mixing in supersonic combustors with hypermixing injectors

A numerical study was conducted to evaluate the performance of wall mounted fuel-injectors designed for potential Supersonic Combustion Ramjet (SCRAM-jet) engine applications. The focus of this investigation was to numerically simulate existing combustor designs for the purpose of validating the numerical technique and the physical models developed. Three different injector designs of varying complexity were studied to fully understand the computational implications involved in accurate predictions. A dual transverse injection system and two streamwise injector designs were studied. The streamwise injectors were designed with swept ramps to enhance fuel-air mixing and combustion characteristics at supersonic speeds without the large flow blockage and drag contribution of the transverse injection system. For this study, the Mass-Average Navier-Stokes equations and the chemical species continuity equations were solved. The computations were performed using a finite-volume implicit numerical technique and multiple block structured grid system. The interfaces of the multiple block structured grid systems were numerically resolved using the flux-conservative technique. Detailed comparisons between the computations and existing experimental data are presented. These comparisons show that numerical predictions are in agreement with the experimental data. These comparisons also show that a number of turbulence model improvements are needed for accurate combustor flowfield predictions.

Lee, J.

Dynamical Approach Study of Spurious Numerics in Nonlinear Computations

The last two decades have been an era when computation is ahead of analysis and when very large scale practical computations are increasingly used in poorly understood multiscale complex nonlinear physical problems and non-traditional fields. Ensuring a higher level of confidence in the predictability and reliability (PAR) of these numerical simulations could play a major role in furthering the design, understanding, affordability and safety of our next generation air and space transportation systems, and systems for planetary and atmospheric sciences, and in understanding the evolution and origin of life. The need to guarantee PAR becomes acute when computations offer the ONLY way of solving these types of data limited problems. Employing theory from nonlinear dynamical systems, some building blocks to ensure a higher level of confidence in PAR of numerical simulations have been revealed by the author and world expert collaborators in relevant fields. Five building blocks with supporting numerical examples were discussed. The next step is to utilize knowledge gained by including nonlinear dynamics, bifurcation and chaos theories as an integral part of the numerical process. The third step is to design integrated criteria for reliable and accurate algorithms that cater to the different multiscale nonlinear physics. This includes but is not limited to the construction of appropriate adaptive spatial and temporal discretizations that are suitable for the underlying governing equations. In addition, a multiresolution wavelets approach for adaptive numerical dissipation/filter controls for high speed turbulence, acoustics and combustion simulations will be sought. These steps are corner stones for guarding against spurious numerical solutions that are solutions of the discretized counterparts but are not solutions of the underlying governing equations.

Yee, H. C.

Numerical Modeling of Active Flow Control in a Boundary Layer Ingesting Offset Inlet

This investigation evaluates the numerical prediction of flow distortion and pressure recovery for a boundary layer ingesting offset inlet with active flow control devices. The numerical simulations are computed using a Reynolds averaged Navier-Stokes code developed at NASA. The numerical results are validated by comparison to experimental wind tunnel tests conducted at NASA Langley Research Center at both low and high Mach numbers. Baseline comparisons showed good agreement between numerical and experimental results. Numerical simulations for the inlet with passive and active flow control also showed good agreement at low Mach numbers where experimental data has already been acquired. Numerical simulations of the inlet at high Mach numbers with flow control jets showed an improvement of the flow distortion. Studies on the location of the jet actuators, for the high Mach number case, were conducted to provide guidance for the design of a future experimental wind tunnel test.

Allan, Brian G.

Adaptive Numerical Dissipation Control in High Order Schemes for Multi-D Non-Ideal MHD

The required type and amount of numerical dissipation/filter to accurately resolve all relevant multiscales of complex MHD unsteady high-speed shock/shear/turbulence/combustion problems are not only physical problem dependent, but also vary from one flow region to another. In addition, proper and efficient control of the divergence of the magnetic field (Div(B)) numerical error for high order shock-capturing methods poses extra requirements for the considered type of CPU intensive computations. The goal is to extend our adaptive numerical dissipation control in high order filter schemes and our new divergence-free methods for ideal MHD to non-ideal MHD that include viscosity and resistivity. The key idea consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. These scheme-independent detectors are capable of distinguishing shocks/shears, flame sheets, turbulent fluctuations and spurious high-frequency oscillations. The detection algorithm is based on an artificial compression method (ACM) (for shocks/shears), and redundant multiresolution wavelets (WAV) (for the above types of flow feature). These filters also provide a natural and efficient way for the minimization of Div(B) numerical error.

Yee, H. C.

Controlling Reflections from Mesh Refinement Interfaces in Numerical Relativity

A leading approach to improving the accuracy on numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a generic numerical error which manifests as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh-refinement implementations, potentially limiting the effectiveness of mesh- refinement techniques for some numerical relativity applications. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite-differencing stencil modifications which eliminate this pathology in both our model problem and in numerical relativity examples.

Baker, John G.

Numerical Dissipation and Wrong Propagation Speed of Discontinuities for Stiff Source Terms

In compressible turbulent combustion/nonequilibrium flows, the constructions of numerical schemes for (a) stable and accurate simulation of turbulence with strong shocks, and (b) obtaining correct propagation speed of discontinuities for stiff reacting terms on coarse grids share one important ingredient - minimization of numerical dissipation while maintaining numerical stability. Here coarse grids means standard mesh density requirement for accurate simulation of typical non-reacting flows. This dual requirement to achieve both numerical stability and accuracy with zero or minimal use of numerical dissipation is most often conflicting for existing schemes that were designed for non-reacting flows. The goal of this paper is to relate numerical dissipations that are inherited in a selected set of high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities for two representative stiff detonation wave problems.

Yee, H. C.

Numerical Dissipation and Wrong Propagation Speed of Discontinuities for Stiff Source Terms

In compressible turbulent combustion/nonequilibrium flows, the constructions of numerical schemes for (a) stable and accurate simulation of turbulence with strong shocks, and (b) obtaining correct propagation speed of discontinuities for stiff reacting terms on coarse grids share one important ingredient - minimization of numerical dissipation while maintaining numerical stability. Here coarse grids means standard mesh density requirement for accurate simulation of typical non-reacting flows. This dual requirement to achieve both numerical stability and accuracy with zero or minimal use of numerical dissipation is most often conflicting for existing schemes that were designed for non-reacting flows. The goal of this paper is to relate numerical dissipations that are inherited in a selected set of high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities as a function of stiffness of the source term and the grid spacing.

Yee, H. C.