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DOE OSTI2023

Expanded verification and validation studies of hypersonic aerodynamics with multiple physics-fidelity models

Hypersonic aerothermodynamics is an important domain of modern multiphysics simulation. The Multi-Fidelity Toolkit is a simulation tool being developed at Sandia National Laboratories to predict aerodynamic properties for compressible flows from a range of physics fidelities and computational speeds. These models include the Reynolds-averaged Navier–Stokes (RANS) equations, the Euler equations with momentum-energy integral technique (MEIT), and modified Newtonian aerodynamics with flat-plate boundary layer (MNA+FPBL) equations, and they can be invoked independently or coupled with hierarchical Kriging to interpolate between high-fidelity simulations using lower-fidelity data. However, as with any new simulation capability, verification and validation are necessary to gather credibility evidence. This work describes formal code- and solution-verification activities, as well as model validation with uncertainty considerations. Code verification activities on the MNA+FPBL model build on previous work by focusing on the viscous portion of the model. Viscous quantities of interest are compared against those from an analytical solution for flat-plate, inclined-plate, and cone geometries. The code verification methodology for the MEIT model is also presented. Test setup and results of code verification tests on the laminar and turbulent models within MEIT are shown. Solution-verification activities include grid-refinement studies on simulations that model the HIFiRE-1 wind tunnel experiments. These experiments are used for validation of all model fidelities. A thorough validation comparison with prediction error and uncertainty is also presented. Three additional HIFiRE-1 experimental runs are simulated in this study, and the solution verification and validation work examines the effects of the associated parameter changes on model performance. Finally, a study is presented that compares the computational costs and fidelities from each of the different models.

42 ENGINEERING↗

NASA NTRS1984

Spectral methods for the Euler equations: Chebyshev methods and shock-fitting

The Chebyshev spectral collocation method for the Euler gas-dynamic equations is described. It is used with shock fitting to compute several two-dimensional, gas-dynamic flows. Examples include a shock-acoustic wave interaction, a shock/vortex interaction, and the classical blunt body problem. With shock fitting, the spectral method has a clear advantage over second order finite differences in that equivalent accuracy can be obtained with far fewer grid points.

Hussaini, M. Y.↗

NASA NTRS1987

Shock structure measured in a transonic fan using laser anemometry

Shock structure measurements acquired in a low aspect ratio transonic fan rotor are presented and analyzed. The rotor aspect ratio is 1.56 and the design tip relative Mach number is 1.38. The rotor flowfield was surveyed at near maximum efficiency and near stall operating conditions. Intra-blade velocity measurements acquired with a laser fringe anemometer on blade-to-blade planes in the supersonic region from 10 to 60 percent span are presented. The three-dimensional shock surface determined from the velocity measurments is used to determine the shock surface normal Mach number in order to properly calculate the ideal shock jump conditions. The ideal jump conditions are calculated based upon the Mach numbers measured on a surface of revolution and based upon the normal Mach number to indicate the importance of accounting for shock three dimensionality in turbomachinery design. Comparison of the shock locations with those predicted by a 3-D Euler code showed very good agreement and indicated the usefulness of integrating computational and experimental work to enhance the understanding of the flow physics occurring in transonic turbomachinery passages.

Wood, Jerry R.↗

NASA NTRS1990

Interactive real time flow simulations

An interactive real time flow simulation technique is developed for an unsteady channel flow. A finite-volume algorithm in conjunction with a Runge-Kutta time stepping scheme was developed for two-dimensional Euler equations. A global time step was used to accelerate convergence of steady-state calculations. A raster image generation routine was developed for high speed image transmission which allows the user to have direct interaction with the solution development. In addition to theory and results, the hardware and software requirements are discussed.

Sadrehaghighi, I.↗

NASA NTRS1991

Transonic cascade flow calculations using non-periodic C-type grids

A new kind of C-type grid is proposed for turbomachinery flow calculations. This grid is nonperiodic on the wake and results in minimum skewness for cascades with high turning and large camber. Euler and Reynolds averaged Navier-Stokes equations are discretized on this type of grid using a finite volume approach. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. Jameson's explicit Runge-Kutta scheme is adopted for the integration in time, and computational efficiency is achieved through accelerating strategies such as multigriding and residual smoothing. A detailed numerical study was performed for a turbine rotor and for a vane. A grid dependence analysis is presented and the effect of artificial dissipation is also investigated. Comparison of calculations with experiments clearly demonstrates the advantage of the proposed grid.

Arnone, Andrea↗

NASA NTRS1991

Conical Euler simulation of wing rock for a delta wing planform

Unsteady, vortex-dominated flowfields are presently studied by using the conical Euler equations as an efficient first step toward investigation of the full three-dimensional problem, under the assumption that the supersonic flow about a delta wing is conical and therefore allows the three-dimensional problem to be reduced to a two-dimensional one. Attention is given to the case of a delta wing undergoing wing-rock motion. The code developed has also been modified to allow treatment of the 'free-to-roll' case.

Lee, Elizabeth M.↗

NASA NTRS1991

Higher order accuracy for upwind methods by using the compatibility equations

A new algorithm has been developed for obtaining higher order accuracy in upwind schemes for the Euler and Navier-Stokes equations. In this method, the compatibility relations for the Euler equations are used to construct formulas for the higher order interpolates. Using these formulas, computed results are obtained for steady, inviscid flow through a nozzle and also steady, inviscid and viscous flow over an airfoil. These results are compared with results obtained by using a minmod limiter function. The comparisons show improved accuracy throughout the flow field with the use of the compatibility equations.

Goorjian, Peter M.↗

NASA NTRS2000

Comparison with Analytical Solution: Generation and Radiation of Acoustic Waves from a 2-D Shear Layer

An acoustic source inside of a 2-D jet excites an instability wave in the shear layer resulting in sound radiating away from the shear layer. Solve the linearized Euler equations to predict the sound radiation outside of the jet. The jet static pressure is assumed to be constant. The jet flow is parallel and symmetric about the x-axis. Use a symmetry boundary condition along the x-axis.

Dahl, Milo D.↗

NASA NTRS2004

TetrUSS Capabilities for S and C Applications

TetrUSS is a suite of loosely coupled computational fluid dynamics software that is packaged into a complete flow analysis system. The system components consist of tools for geometry setup, grid generation, flow solution, visualization, and various utilities tools. Development began in 1990 and it has evolved into a proven and stable system for Euler and Navier-Stokes analysis and design of unconventional configurations. It is 1) well developed and validated, 2) has a broad base of support, and 3) is presently is a workhorse code because of the level of confidence that has been established through wide use. The entire system can now run on linux or mac architectures. In the following slides, I will highlight more of the features of the VGRID and USM3D codes.

Frink, Neal T.↗

NASA NTRS1989

Design of optimally smoothing multi-stage schemes for the Euler equations

In this paper, a method is developed for designing multi-stage schemes that give optimal damping of high-frequencies for a given spatial-differencing operator. The objective of the method is to design schemes that combine well with multi-grid acceleration. The schemes are tested on a nonlinear scalar equation, and compared to Runge-Kutta schemes with the maximum stable time-step. The optimally smoothing schemes perform better than the Runge-Kutta schemes, even on a single grid. The analysis is extended to the Euler equations in one space-dimension by use of 'characteristic time-stepping', which preconditions the equations, removing stiffness due to variations among characteristic speeds. Convergence rates independent of the number of cells in the finest grid are achieved for transonic flow with and without a shock. Characteristic time-stepping is shown to be preferable to local time-stepping, although use of the optimally damping schemes appears to enhance the performance of local time-stepping. The extension of the analysis to the two-dimensional Euler equations is hampered by the lack of a model for characteristic time-stepping in two dimensions. Some results for local time-stepping are presented.

Van Leer, Bram↗

DOE OSTI2015

High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

Here, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

65M06↗

NASA NTRS1986

Euler equation analysis of the initial roll-up of aircraft wakes

A new approach has been developed to study the initial roll-up of aircraft wakes. This approach is based on the solution of the three-dimensional Euler equations subject to inflow and initial conditions predicted by a vortex-lattice method. This procedure allows for the solution of this complex problem without any empirical inputs. However, valid solutions are limited to those flowfields generated by wings which are free of flow separation. Results are presented for four configurations: a rectangular wing with part span flap, a rectangular wing, a typical transport-type configuration and a fighter configuration. In addition, an assessment of an 'unsteady 2-D analogy' was undertaken using the 2-D time accurate Euler equation solutions in crossflow planes. In general, predictions from the 2-D unsteady analogy coincided with the 3-D results which were found to be in good agreement with available experimental data.

Mitcheltree, R. A.↗

NASA NTRS1986

Detailed near-wake flowfield surveys with comparison to an Euler method of an aspect ratio 4 rectangular wing

An experimental investigation of the flowfield in the near-wake of an aspect ratio 4 rectangular wing was conducted, providing a complete detailed set of data for use in the validation of computational methods. An angle of attack of 8 degrees and two Reynolds numbers 530,000 and 391,000 were investigated using pitot and six-hole probes. In addition, two types of flow visualization were employed. The data presented includes contours of total pressure, mean velocity, flow angularity, and vorticity distribution data at five chordwise stations of the near-wake ranging from 0.167 to 5.00 chord lengths aft of the trailing edge. The experimental results were compared to the predicted results of a 2-D Euler numerical method. The results predicted by an Euler method failed to accurately define the flowfield. Tangential velocities remained relatively constant over the range of X/C considered though increased in angle of attack and Reynolds number did bring about corresponding increases. Axial velocities also increased with angle of attack and Reynolds number but showed greater sensitivity to increases in X/C. Graphic displays and contours of the total pressure data indicate that roll-up of the wing tip vortex is essentially complete one and one half chords downstream of the trailing edge.

Klinge, M. D.↗

NASA NTRS1995

Methodology for simulation of unsteady flows to determine the time-dependent interference between stationary and moving boundaries

A new methodology is developed to simulate unsteady flows about prescribed and aerodynamically determined moving boundary problems. The method couples the fluid dynamics and rigid-body dynamics equations to capture the time-dependent interference between stationary and moving boundaries. The unsteady, compressible, inviscid (Euler) equations are solved on dynamic, unstructured grids by an explicit, finite-volume, upwind method. For efficiency, the grid adaptation is performed within a window around the moving object. The Eulerian equations of the rigid-body dynamics are solved by a Runge-Kutta method in a non-inertial frame of reference. The two-dimensional flow solver is validated by computing the flow past a sinusoidally-pitching airfoil and comparing these results with the experimental data. The overall methodology is used for two two-dimensional examples: the flow past an airfoil which is performing a three-degrees-of-freedom motion in a transonic freestream, and the free-fall of a store after separation from a wing-section. Then the unstructured mesh methodology is extended to three-dimensions to simulate unsteady flow past bodies in relative motion, where the trajectory is determined from the instantaneous aerodynamics. The flow solver and the adaptation scheme in three dimensions are validated by simulating the transonic, unsteady flow around a wing undergoing a forced, periodic, pitching motion, and comparing the results with the experimental data. To validate the trajectory code, the six-degrees-of-freedom motion of a store separating from a wing was computed using the experimentally determined force and moment fields, then comparing with an independently generated trajectory. Finally, the overall methodology was demonstrated by simulating the unsteady flowfield and the trajectory of a store dropped from a wing. The methodology, its computational cost notwithstanding, has proven to be accurate, automated, easy for dynamic gridding, and relatively efficient for the required man-hours.

Singh, Kamakhya Prasad↗

NASA NTRS1994

Optimized design of a hypersonic nozzle

Conventional procedures for designing nozzles involve the design of an inviscid contour (using the method of characteristics) that is corrected with a displacement thickness calculated from boundary-layer theory. However, nozzles designed using this classical procedure have been shown to exhibit poor flow quality at Mach numbers characteristic of hypersonic applications. The nozzle to be designed will be a part of the NASA HYPULSE facility which is being used for hypervelocity flight research. Thus, the flow quality of the nozzle is a critical question that needs to be addressed. Design of nozzles for hypersonic applications requires a proper assessment of the effects of the thick boundary layer on the inviscid flowfield. Since the flow field is largely supersonic, the parabolized form of the Navier-Stokes (PNS) equations can be used. The requirement of a uniform flow at the exit plane of the nozzle can be used to define an objective function as part of an optimization procedure. The design procedure used in this study involves the coupling of a nonlinear (least-squares) optimization algorithm with an efficient, explicit PNS solver. The thick boundary layers growing on the walls of the nozzle limit the extent of the usable core region (region with uniform flow) for testing models (especially rectangular). In order to maximize the region of uniform flow, it was decided to have the exit plane of this nozzle to be (nearly) rectangular. Thus, an additional constraint on the nozzle shape resulted, namely the nozzle will have a shape transitioning from a circular one at the inlet to that of a rectangle at the exit. In order to provide for a smooth shape transition, the cross sectional contour of the nozzle is defined by a superellipse. The nozzle is taken to be a meter in length. The axial variations of the major and minor radii of the superellipse are governed by cubic splines. The design parameters are the coefficients of the splines associated with the local nozzle wall slopes. Extensive calculations have been made (with a three-dimensional Euler code) to understand the effects of various parameters such as location of the knot points of the spline function, different ways of characterizing the uniformity of the flow in the exit plane, as well as the effect of constraining the area of the nozzle to be invariant. Turbulent flow (measurements indicate that the flow at the nozzle inlet is turbulent) calculations are now being performed (with the inviscidly designed nozzle contours) to assess the flow quality.

Krishnamurthy, Ramesh↗

DOE OSTI2025

A High-Order Discontinuous Galerkin Spectral Element Method for Compressible Reacting Flows

High-order methods have recently been shown to be an effective tool for high-fidelity flow computations like direct numerical simulations and large eddy simulations due to their strong balance between accuracy and computational cost. In this work, a high-order discontinuous Galerkin spectral element method (DGSEM) is developed to solve the chemically reactive Euler equations encountered in high-speed combustion. To handle the disparate length and time scales associated with these equations, we develop a novel method which combines the spectral accuracy of the SEM with the flexibility of DG approach. Thus, the framework is well suited to capture turbulence in smooth regions of the flow, while maintaining numerical stability in the presence of shocks. The numerical method is implemented within the spectral element solver Nek5000. Validation cases are conducted for both non-reactive and reactive discontinuous flows to demonstrate the solver capability. In particular, canonical one-dimensional and two-dimensional detonation simulations are performed and the high-order numerical results are validated against available literature data.

computational fluid dynamics (CFD)↗

NASA NTRS1983

Embedding methods for the steady Euler equations

An approach to the numerical solution of the steady Euler equations is to embed the first-order Euler system in a second-order system and then to recapture the original solution by imposing additional boundary conditions. Initial development of this approach and computational experimentation with it were previously based on heuristic physical reasoning. This has led to the construction of a relaxation procedure for the solution of two-dimensional steady flow problems. The theoretical justification for the embedding approach is addressed. It is proven that, with the appropriate choice of embedding operator and additional boundary conditions, the solution to the embedded system is exactly the one to the original Euler equations. Hence, solving the embedded version of the Euler equations will not produce extraneous solutions.

Chang, S. H.↗

DOE OSTI2021

Parallel exponential time differencing methods for geophysical flow simulations

Two ocean models are considered for geophysical flow simulations: the multilayer shallow water equations and the multilayer primitive equations. For the former, we investigate the parallel performance of exponential time differencing (ETD) methods, including exponential Rosenbrock–Euler, ETD2wave, and B-ETD2wave. For the latter, we take advantage of the splitting of barotropic and baroclinic modes and propose a new two-level method in which an ETD method is applied to solve the fast barotropic mode. Furthermore, these methods could improve the computational efficiency of numerical simulations because ETD methods allow for much larger time step sizes than traditional explicit time-stepping techniques that are commonly used in existing computational ocean models. Several standard benchmark tests for ocean modeling are performed and comparison of the numerical results demonstrates a great potential of applying the parallel ETD methods for simulating real-world geophysical flows.

54 ENVIRONMENTAL SCIENCES↗