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NASA NTRS2001

Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations

The recently developed essentially fourth-order or higher low dissipative shock-capturing scheme of Yee, Sandham and Djomehri (1999) aimed at minimizing nu- merical dissipations for high speed compressible viscous flows containing shocks, shears and turbulence. To detect non smooth behavior and control the amount of numerical dissipation to be added, Yee et al. employed an artificial compression method (ACM) of Harten (1978) but utilize it in an entirely different context than Harten originally intended. The ACM sensor consists of two tuning parameters and is highly physical problem dependent. To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed. The new sensors are derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch to the extra numerical dissipation outside shock layers. The non-dissipative spatial base scheme of arbitrarily high order of accuracy can be maintained without compromising its stability at all parts of the domain where the solution is smooth. Two types of redundant non-orthogonal wavelet basis functions are considered. One is the B-spline wavelet (Mallat & Zhong 1992) used by Gerritsen and Olsson (1996) in an adaptive mesh refinement method, to determine regions where re nement should be done. The other is the modification of the multiresolution method of Harten (1995) by converting it to a new, redundant, non-orthogonal wavelet. The wavelet sensor is then obtained by computing the estimated Lipschitz exponent of a chosen physical quantity (or vector) to be sensed on a chosen wavelet basis function. Both wavelet sensors can be viewed as dual purpose adaptive methods leading to dynamic numerical dissipation control and improved grid adaptation indicators. Consequently, they are useful not only for shock-turbulence computations but also for computational aeroacoustics and numerical combustion. In addition, these sensors are scheme independent and can be stand alone options for numerical algorithm other than the Yee et al. scheme.

Sjoegreen, B.↗

NASA NTRS1973

Numerical solution for the inviscid supersonic flow in the corner formed by two intersecting wedges.

The inviscid, interference corner flow generated by two intersecting wedges immersed in a supersonic stream is obtained by use of a second-order, shock-capturing, finite-difference approach. The governing equations are solved iteratively in conical coordinates to yield the flow structure consisting of multiple shock and slip surfaces. The numerical results for shock wave and slip surface position and structure, pitot pressure traverses, and surface pressure distributions are compared with experimental data obtained over a wide range of Reynolds numbers. The comparisons show the best agreement with the high Reynolds number (greater than 3,000,000) results for which the boundary layer is turbulent.

Kutler, P.↗

DOE OSTI2023

An investigation of shock formation vs shock mitigation of colliding plasma jets

Here, this work studies the interaction between colliding plasma jets to understand regimes in which jet merging results in shock formation vs regimes in which the shock formation is mitigated due to the collisionless interpenetration of the jets. A kinetic model is required for this study because fluid models will always produce a shock upon the collision of plasma jets. The continuum-kinetic, Vlasov–Maxwell–Dougherty model with one velocity dimension is used to accurately capture shock heating, along with a novel coupling with a moment equation to evolve perpendicular temperature for computational efficiency. As a result, this relatively inexpensive simulation can be used for detailed scans of the parameter space toward predictions of shocked vs shock-mitigated regimes, which is of interest for several fusion concepts such as plasma-jet-driven magneto-inertial fusion, high-energy-density plasmas, astrophysical phenomena, and other laboratory plasmas. The initial results obtained using this approach are in agreement with the preliminary outcomes of the plasma liner experiment.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2018

Tetrahedral-Mesh Simulations of Shock-Turbulence Interaction

Despite decades of development of unstructured mesh methods, direct numerical simulations (DNS) of turbulent flows are still predominantly performed on structured or unstructured hexahedral meshes with high-order finite-difference methods, weighted essentially nonoscillatory (WENO) schemes, or hybrid schemes formed by their combinations. Tetrahedral meshes offer easy mesh generation and adaptation around complex geometries and the potential of an orientation-free grid that would benefit the isotropic nature of small-scale dissipation, as well as the solution accuracy of intermediate scales. To advance the state of the art of unstructured-mesh simulation capabilities for shock/turbulence interaction, DNS using pure tetrahedral meshes are carried out with the space-time conservation element, solution element (CESE) method in this research. By its design, the CESE method is constructed based on a non-dissipative scheme and is a genuinely multidimensional numerical framework that is free from the use of an approximate Riemann-solver. The numerical framework also provides the ability to add numerical dissipation (the nondissipative scheme acts as the reference state like that of the reversible state in thermodynamics) when needed (with justification from mathematics/physics). The above-mentioned features along with the CESE method's consistent shock-capturing approach and strong enforcement of flux conservation in spacetime offers a novel method to accurately simulate turbulent flows and their interaction with shocks using tetrahedral meshes. Two canonical problems, namely, isotropic turbulence interaction with a normal shock and a Mach 2.9 turbulent boundary layer flow over a 24deg compression corner are investigated in this study. Computational results show reasonably good agreement with experimental data and results from structured-mesh, high-order simulations available in the literature. Successful validation of these canonical problems demonstrated here paves the way for future high-fidelity supersonic flow simulations involving complex-geometries.

Venkatachari, Balaji Shankar↗

NASA NTRS1990

Heat Pipe Vapor Dynamics

The dynamic behavior of the vapor flow in heat pipes is investigated at startup and during operational transients. The vapor is modeled as two-dimensional, compressible viscous flow in an enclosure with inflow and outflow boundary conditions. For steady-state and operating transients, the SIMPLER method is used. In this method a control volume approach is employed on a staggered grid which makes the scheme very stable. It is shown that for relatively low input heat fluxes the compressibility of the vapor flow is low and the SIMPLER scheme is suitable for the study of transient vapor dynamics. When the input heat flux is high or the process under a startup operation starts at very low pressures and temperatures, the vapor is highly compressible and a shock wave is created in the evaporator. It is shown that for a wide range of input heat fluxes, the standard methods, including the SIMPLER scheme, are not suitable. A nonlinear filtering technique, along with the centered difference scheme, are then used for shock capturing as well as for the solution of the cell Reynolds-number problem. For high heat flux, the startup transient phase involves multiple shock reflections in the evaporator region. Each shock reflection causes a significant increase in the local pressure and a large pressure drop along the heat pipe. Furthermore, shock reflections cause flow reversal in the evaporation region and flow circulations in the adiabatic region. The maximum and maximum-averaged pressure drops in different sections of the heat pipe oscillate periodically with time because of multiple shock reflections. The pressure drop converges to a constant value at steady state. However, it is significantly higher than its steady-state value at the initiation of the startup transient. The time for the vapor core to reach steady-state condition depends on the input heat flux, the heat pipe geometry, the working fluid, and the condenser conditions. However, the vapor transient time, for an Na-filled heat pipe is on the order of seconds. Depending on the time constant for the overall system, the vapor transient time may be very short. Therefore, the vapor core may be assumed to be quasi-steady in the transient analysis of a heat pipe operation.

Issacci, Farrokh↗

NASA NTRS1982

Computational fluid dynamics of airfoils and wings

It is pointed out that transonic flow is one of the fields where computational fluid dynamics turns out to be most effective. Codes for the design and analysis of supercritical airfoils and wings have become standard tools of the aircraft industry. The present investigation is concerned with mathematical models and theorems which account for some of the progress that has been made. The most successful aerodynamics codes are those for the analysis of flow at off-design conditions where weak shock waves appear. A major breakthrough was achieved by Murman and Cole (1971), who conceived of a retarded difference scheme which incorporates artificial viscosity to capture shocks in the supersonic zone. This concept has been used to develop codes for the analysis of transonic flow past a swept wing. Attention is given to the trailing edge and the boundary layer, entropy inequalities and wave drag, shockless airfoils, and the inverse swept wing code.

Garabedian, P.↗

NASA NTRS1985

Shock modelling in transonic and supersonic flow

Numerical simulation studies of problems in transonic flow are conducted in connection with the design of aircraft flying near the speed of sound, while recently there has also been a great deal of interest in supersonic and hypersonic flow calculations. Three important models for inviscid compressible gasdynamics are utilized in the study of these flow problems, taking into account the transonic small disturbance equation (TSD), the transonic full potential equation (FP), and the full Euler equations for compressible gasdynamics (EU). The present paper has the objective to provide a description of shock capturing methods recently developed for each model. The design principles behind the algorithms are also examined, giving attention to consistency, conservative form, monotone and sharp discrete shocks, and a bound on the variation, which prevents spurious oscillations.

Osher, S.↗

NASA NTRS1975

Internal and external axial corner flows

The inviscid, internal, and external axial corner flows generated by two intersecting wedges traveling supersonically are obtained by use of a second-order shock-capturing, finite-difference approach. The governing equations are solved iteratively in conical coordinates to yield the complicated wave structure of the internal corner and the simple peripheral shock of the external corner. The numerical results for the internal flows compare favorably with existing experimental data.

Kutler, P.↗

NASA NTRS1975

Three-dimensional, shock-on-shock interaction problem

The unsteady, three-dimensional flowfield resulting from the interaction of a plane shock with a cone-shaped vehicle traveling supersonically is determined, using a second-order, shock-capturing, finite-difference approach. The time-dependent, inviscid gasdynamic equations are transformed to include the self-similar property of the flow, to align various coordinate surfaces with known shock waves, and to cluster points in the vicinity of the intersection of the transmitted incident shock and the surface of the vehicle. The governing partial differential equations in conservation-law form are then solved iteratively using MacCormack's (1969) algorithm.

Kutler, P.↗

DOE OSTI2022

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Geometry and dynamics of passive scalar structures in compressible turbulent mixing

We present a structure-based numerical analysis of passive scalar mixing in decaying homogeneous isotropic turbulence (DHIT) and shock-turbulence interaction canonical configurations. The analysis focuses on the temporal evolution of ensembles of passive scalar structures, initialized as spheres of different sizes relative to the Taylor microscale. An algorithm is introduced to track the evolution of each individual structure and the interactions with other structures in the ensemble, relating changes in the surface geometry and the underlying physical processes (turbulent transport, scalar dissipation, and shock compression). The tracking algorithm is applied to datasets from shock-capturing direct numerical simulations of DHIT, with Taylor microscale Reynolds number Re λ =40 and turbulence Mach number M t =0.2 ⁠, and STI cases in which the turbulence is processed by a shock wave at Mach numbers M = 1.5 and 3.0. Temporal surface convolution increases for initially larger structures, resulting in a higher probability of locally hyperbolic geometries where breakup into smaller structures occurs. Shock-induced deformation of the structures amplifies breakup processes, enhancing mixing, particularly for larger structures. Mixing enhancement by the shock is manifested as an amplification of the surface-averaged scalar gradient, which increases for initially larger structures. The alignment between the scalar gradient and the most extensional strain-rate eigendirection on the scalar isosurfaces also increases across the shock. Larger magnitudes of the scalar gradient and its alignment with the most compressive strain-rate eigendirection correlate with flatter surface regions. Shock-induced structure compression increases the area coverage of flat regions, where the amplification of scalar gradient is localized.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1993

Computation of supersonic jet noise under imperfectly expanded conditions

The turbulent mixing noise of supersonic jet under imperfectly expanded conditions is calculated for convergent and convergent-divergent (CD) axisymmetric nozzle geometries. The noise prediction incorporates CFD solution of Navier-Stokes equations. The effect of grid resolution on shock structure computation is demonstrated. Mixing noise spectra predicted from fine and coarse grid solutions exhibit little sensitivity to the grid resolution. A proper grid resolution, however, results in a significant improvement in shock capturing capability and helps predictions agree favorably with experimental data. Good agreement between predicted noise spectra and data shows that the CFD-incorporated noise prediction scheme, which was demonstrated for shock-free conditions, works as well for shock-containing flow conditions.

Kim, Chan M.↗

NASA NTRS1993

Computation of supersonic jet noise under imperfectly expanded conditions

Kim, Chan M.↗

NASA NTRS2008

Computational Analyses in Support of Sub-scale Diffuser Testing for the A-3 Facility: Steady Predictions - Part 1

Simulation technology can play an important role in rocket engine test facility design and development by assessing risks, providing analysis of dynamic pressure and thermal loads, identifying failure modes and predicting anomalous behavior of critical systems. Advanced numerical tools assume greater significance in supporting testing and design of high altitude testing facilities and plume induced testing environments of high thrust engines because of the greater inter-dependence and synergy in the functioning of the different sub-systems. This is especially true for facilities such as the proposed A-3 facility at NASA SSC because of a challenging operating envelope linked to variable throttle conditions at relatively low chamber pressures. Facility designs in this case will require a complex network of diffuser ducts, steam ejector trains, fast operating valves, cooling water systems and flow diverters that need to be characterized for steady state performance. In this paper, we will demonstrate with the use of CFD analyses s advanced capability to evaluate supersonic diffuser and steam ejector performance in a sub-scale A-3 facility at NASA Stennis Space Center (SSC) where extensive testing was performed. Furthermore, the focus in this paper relates to modeling of critical sub-systems and components used in facilities such as the A-3 facility. The work here will address deficiencies in empirical models and current CFD analyses that are used for design of supersonic diffusers/turning vanes/ejectors as well as analyses for confined plumes and venting processes. The primary areas that will be addressed are: (1) supersonic diffuser performance including analyses of thermal loads (2) accurate shock capturing in the diffuser duct; (3) effect of turning duct on the performance of the facility (4) prediction of mass flow rates and performance classification for steam ejectors (5) comparisons with test data from sub-scale diffuser testing and assessment of confidence levels in CFD based flowpath modeling of the facility. The analyses tools used here expand on the multi-element unstructured CFD which has been tailored and validated for impingement dynamics of dry plumes, complex valve/feed systems, and high pressure propellant delivery systems used in engine and component test stands at NASA SSC. The analyses performed in the evaluation of the sub-scale diffuser facility explored several important factors that influence modeling and understanding of facility operation such as (a) importance of modeling the facility with Real Gas approximation, (b) approximating the cluster of steam ejector nozzles as a single annular nozzle, (c) existence of mixed subsonic/supersonic flow downstream of the turning duct, and (d) inadequacy of two-equation turbulence models in predicting the correct pressurization in the turning duct and expansion of the second stage steam ejectors. The procedure used for modeling the facility was as follows: (i) The engine, test cell and first stage ejectors were simulated with an axisymmetric approximation (ii) the turning duct, second stage ejectors and the piping downstream of the second stage ejectors were analyzed with a three-dimensional simulation utilizing a half-plane symmetry approximation. The solution i.e. primitive variables such as pressure, velocity components, temperature and turbulence quantities were passed from the first computational domain and specified as a supersonic boundary condition for the second simulation. (iii) The third domain comprised of the exit diffuser and the region in the vicinity of the facility (primary included to get the correct shock structure at the exit of the facility and entrainment characteristics). The first set of simulations comprising the engine, test cell and first stage ejectors was carried out both as a turbulent real gas calculation as well as a turbulent perfect gas calculation. A comparison for the two cases (Real Turbulent and Perfect gas turbulent) of the Ma Number distribution and temperature distributions are shown in Figures 1 and 2 respectively. The Mach Number distribution shows small yet distinct differences between the two cases such as locations of shocks/shock reflections and a slightly different impingement point on the wall of the diffuser from the expansion at the exit of the nozzle. Similarly the temperature distribution indicates different flow recirculation patterns in the test cell. Both cases capture all the essential flow phenomena such as the shock-boundary layer interaction, plume expansion, expansion of the first stage ejectors, mixing between the engine plume and the first stage ejector flow and pressurization due to the first stage ejectors. The final paper will discuss thermal loads on the walls of the diffuser and cooling mechanisms investigated.

Allgood, Daniel C.↗

NASA NTRS2001

A Space-Time Conservation Element and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes

In this paper, we report a version of the Space-Time Conservation Element and Solution Element (CE/SE) Method in which the 2D and 3D unsteady Euler equations are simulated using structured or unstructured quadrilateral and hexahedral meshes, respectively. In the present method, mesh values of flow variables and their spatial derivatives are treated as independent unknowns to be solved for. At each mesh point, the value of a flow variable is obtained by imposing a flux conservation condition. On the other hand, the spatial derivatives are evaluated using a finite-difference/weighted-average procedure. Note that the present extension retains many key advantages of the original CE/SE method which uses triangular and tetrahedral meshes, respectively, for its 2D and 3D applications. These advantages include efficient parallel computing ease of implementing non-reflecting boundary conditions, high-fidelity resolution of shocks and waves, and a genuinely multidimensional formulation without using a dimensional-splitting approach. In particular, because Riemann solvers, the cornerstones of the Godunov-type upwind schemes, are not needed to capture shocks, the computational logic of the present method is considerably simpler. To demonstrate the capability of the present method, numerical results are presented for several benchmark problems including oblique shock reflection, supersonic flow over a wedge, and a 3D detonation flow.

Zhang, Zeng-Chan↗

NASA NTRS1985

Numerically-simulated formation and propagation of interplanetary shocks

The present numerical method for simulating the formation and propagation of interplanetary shocks is based on the shock-capturing finite difference scheme of Lax (1950) and Lax and Wendroff (1960), as well as the recent method of NEAR characteristics of Nakagawa (1980, 1981). Attention is given to examples which strongly suggest that all the shocked solar wind plasma parameters due to given physical perturbations, such as flare-generated shocks, can be predicted through the use of this method; the method is, however, limited to the supersonic and super-Alfvenic flow.

Wu, S. T.↗

NASA NTRS1987

Multiscale turbulence effects in supersonic jets exhausting into still air

A modified version of the multiscale turbulence model of Hanjalic has been applied to the problem of supersonic jets exhausting into still air. In particular, the problem of shock-cell decay through turbulent interaction with the mixing layer has been studied for both mildly interacting and strongly resonant jet conditions. The modified Hanjalic model takes into account the nonequilibrium energy transfer between two different turbulent spectral scales. The turbulence model was incorporated into an existing shock-capturing, parabolized Navier-Stokes computational model in order to perform numerical experiments. The results show that the two-scale turbulence model provides significant improvement over one-scale models in the prediction of plume shock structure for underexpanded supersonic (Mach 2) and sonic (Mach 1) jets. For the supersonic jet, excellent agreement with experiment was obtained for the centerline shock-cell pressure decay up to 40 jet radii. For the sonic jet, the agreement with experiment was not so good, but the two-scale model still showed significant improvement over the one-scale model. It is shown that by relating some of the coefficients in the turbulent-transport equations to the relative time scale for transfer of energy between scales the two-scale model can provide predictions that bound the measured shock-cell decay rate for the sonic jet.

Abdol-Hamid, Khaled S.↗

NASA NTRS2015

Tetrahedral-Mesh Simulation of Turbulent Flows with the Space-Time Conservative Schemes

Direct numerical simulations of turbulent flows are predominantly carried out using structured, hexahedral meshes despite decades of development in unstructured mesh methods. Tetrahedral meshes offer ease of mesh generation around complex geometries and the potential of an orientation free grid that would provide un-biased small-scale dissipation and more accurate intermediate scale solutions. However, due to the lack of consistent multi-dimensional numerical formulations in conventional schemes for triangular and tetrahedral meshes at the cell interfaces, numerical issues exist when flow discontinuities or stagnation regions are present. The space-time conservative conservation element solution element (CESE) method - due to its Riemann-solver-free shock capturing capabilities, non-dissipative baseline schemes, and flux conservation in time as well as space - has the potential to more accurately simulate turbulent flows using unstructured tetrahedral meshes. To pave the way towards accurate simulation of shock/turbulent boundary-layer interaction, a series of wave and shock interaction benchmark problems that increase in complexity, are computed in this paper with triangular/tetrahedral meshes. Preliminary computations for the normal shock/turbulence interactions are carried out with a relatively coarse mesh, by direct numerical simulations standards, in order to assess other effects such as boundary conditions and the necessity of a buffer domain. The results indicate that qualitative agreement with previous studies can be obtained for flows where, strong shocks co-exist along with unsteady waves that display a broad range of scales, with a relatively compact computational domain and less stringent requirements for grid clustering near the shock. With the space-time conservation properties, stable solutions without any spurious wave reflections can be obtained without a need for buffer domains near the outflow/farfield boundaries. Computational results for the isotropic turbulent flow decay, at a relatively high turbulent Mach number, show a nicely behaved spectral decay rate for medium to high wave numbers. The high-order CESE schemes offer very robust solutions even with the presence of strong shocks or widespread shocklets. The explicit formulation in conjunction with a close to unity theoretical upper Courant number bound has the potential to offer an efficient numerical framework for general compressible turbulent flow simulations with unstructured meshes.

Chang, Chau-Lyan↗