Stabilized methods for high-speed compressible flows: toward hypersonic simulations

A stabilized finite element framework for high-speed compressible flows is presented. The Streamline-Upwind/Petrov–Galerkin formulation augmented with discontinuity-capturing (DC) are the main constituents of the framework that enable accurate, efficient, and stable simulations in this flow regime. Full- and reduced-energy formulations are employed for this class of flow problems and their relative accuracy is assessed. In addition, a recently developed DC formulation is presented and is shown to be particularly well suited for hypersonic flows. Several verification and validation cases, ranging from 1D to 3D flows and supersonic to the hypersonic regimes, show the excellent performance of the proposed framework and set the stage for its deployment on more advanced applications.

[1]  C. F. Curtiss,et al.  Molecular Theory Of Gases And Liquids , 1954 .

[2]  Tayfun E. Tezduyar,et al.  SUPG finite element computation of inviscid supersonic flows with YZβ shock-Capturing , 2007 .

[3]  T. Tezduyar,et al.  Element-splitting-invariant local-length-scale calculation in B-Spline meshes for complex geometries , 2020 .

[4]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[5]  Y. Bazilevs,et al.  Residual-based shock capturing in solids , 2020 .

[6]  Dimitri J. Mavriplis,et al.  An implicit block ILU smoother for preconditioning of Newton–Krylov solvers with application in high-order stabilized finite-element methods , 2020 .

[7]  Tayfun E. Tezduyar,et al.  Porosity models and computational methods for compressible-flow aerodynamics of parachutes with geometric porosity , 2017 .

[8]  Alvaro L. G. A. Coutinho,et al.  Compressible Flow SUPG Stabilization Parameters Computed from Degree-of-freedom Submatrices , 2006 .

[9]  A. Mazaheri,et al.  Exploring Hypersonic, Unstructured-Grid Issues through Structured Grids , 2007 .

[10]  B R Hollis Real-Gas Flow Properties for NASA Langley Research Center Aerothermodynamic Facilities Complex Wind Tunnels , 1996 .

[11]  Masahisa Tabata,et al.  On a conservation upwind finite element scheme for convective diffusion equations , 1981 .

[12]  Anders Szepessy,et al.  Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions , 1989 .

[13]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[14]  T. Kubota,et al.  Experimental investigation of supersonic laminar, two-dimensional boundary-layer separation in a compression corner with and without cooling. , 1967 .

[15]  Roy H. Stogner,et al.  Modeling hypersonic entry with the fully-implicit Navier–Stokes (FIN-S) stabilized finite element flow solver , 2014 .

[16]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems , 1986 .

[17]  R. A. Aziz,et al.  Empirical Equations to Calculate 16 of the Transport Collision Integrals Ω(l, s)* for the Lennard‐Jones (12–6) Potential , 1972 .

[18]  Yuri Bazilevs,et al.  Large-Eddy Simulation of Shallow Water Langmuir Turbulence Using Isogeometric Analysis and the Residual-Based Variational Multiscale Method , 2012 .

[19]  Tayfun E. Tezduyar,et al.  Massively parallel finite element simulation Of compressible and incompressible flows , 1994 .

[20]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[21]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[22]  Marek Behr,et al.  Parallel finite-element computation of 3D flows , 1993, Computer.

[23]  C. Johansen,et al.  Investigation of Gas Seeding for Planar Laser-Induced Fluorescence in Hypersonic Boundary Layers , 2015 .

[24]  Tayfun E. Tezduyar,et al.  Parallel finite element computation of missile aerodynamics , 1997 .

[25]  E. Denman,et al.  The matrix sign function and computations in systems , 1976 .

[26]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[27]  Thomas J. R. Hughes,et al.  Symmetrization of conservation laws with entropy for high-temperature hypersonic computations , 1990 .

[28]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[29]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[30]  Tayfun E. Tezduyar,et al.  Flow simulation and high performance computing , 1996 .

[31]  Tayfun E. Tezduyar,et al.  SUPG finite element computation of compressible flows with the entropy and conservation variables formulations , 1993 .

[32]  Tayfun E. Tezduyar,et al.  Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations , 1986 .

[33]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[34]  T. Tezduyar,et al.  Space-time finite element computation of compressible flows involving moving boundaries and interfaces☆ , 1993 .

[35]  T. Tezduyar Computation of moving boundaries and interfaces and stabilization parameters , 2003 .

[36]  Tayfun E. Tezduyar,et al.  Space–time VMS flow analysis of a turbocharger turbine with isogeometric discretization: computations with time-dependent and steady-inflow representations of the intake/exhaust cycle , 2019, Computational Mechanics.

[37]  Tayfun E. Tezduyar,et al.  Multiscale space-time methods for thermo-fluid analysis of a ground vehicle and its tires , 2015 .

[38]  Anindya Ghoshal,et al.  Optimizing Gas Turbine Performance Using the Surrogate Management Framework and High-Fidelity Flow Modeling , 2020 .

[39]  Thomas J. R. Hughes,et al.  A consistent equilibrium chemistry algorithm for hypersonic flows , 1994 .

[40]  Claes Johnson Numerical solution of partial differential equations by the finite element method , 1988 .

[41]  Thomas J. R. Hughes,et al.  Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations , 1984 .

[42]  Masahisa Tabata Uniform convergence of the upwind finite element approximation for semilinear parabolic problems , 1978 .

[43]  Tayfun E. Tezduyar,et al.  Stabilization and shock-capturing parameters in SUPG formulation of compressible flows , 2004 .

[44]  Tayfun E. Tezduyar,et al.  Finite element computation of compressible flows with the SUPG formulation , 1991 .

[45]  Tayfun E. Tezduyar,et al.  Computation of Inviscid Supersonic Flows Around Cylinders and Spheres with the SUPG Formulation and YZβ Shock-Capturing , 2006 .

[46]  Stephen B. Jones,et al.  NO PLIF Study of Hypersonic Transition Over a Discrete Hemispherical Roughness Element , 2009 .

[47]  Claes Johnson,et al.  On the convergence of a finite element method for a nonlinear hyperbolic conservation law , 1987 .

[48]  Tayfun E. Tezduyar,et al.  Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces , 2004 .

[49]  Yuki Ueda,et al.  A node-numbering-invariant directional length scale for simplex elements , 2019, Mathematical Models and Methods in Applied Sciences.

[50]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[51]  Michael Dumbser,et al.  A matrix stability analysis of the carbuncle phenomenon , 2004 .

[52]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations , 1991 .

[53]  Volker Elling,et al.  The carbuncle phenomenon is incurable , 2009 .

[54]  Guillermo Hauke,et al.  Simple stabilizing matrices for the computation of compressible flows in primitive variables , 2001 .

[55]  Karl T. Edquist,et al.  Computations of Viking Lander Capsule Hypersonic Aerodynamics with Comparisons to Ground and Flight Data , 2006 .

[56]  Tayfun E. Tezduyar,et al.  Calculation Of The Stabilization Parameters In Supg And Pspg Formulations. , 2002 .

[57]  Yuri Bazilevs,et al.  Gas turbine computational flow and structure analysis with isogeometric discretization and a complex-geometry mesh generation method , 2020 .

[58]  Tayfun E. Tezduyar,et al.  Compressible-flow geometric-porosity modeling and spacecraft parachute computation with isogeometric discretization , 2018, Computational Mechanics.

[59]  Ming-Chen Hsu,et al.  Immersogeometric analysis of compressible flows with application to aerodynamic simulation of rotorcraft , 2019, Mathematical Models and Methods in Applied Sciences.

[60]  Tayfun E. Tezduyar,et al.  Stabilization and discontinuity-capturing parameters for space–time flow computations with finite element and isogeometric discretizations , 2018 .

[61]  E. Dick,et al.  Introduction to Finite Element Methods in Computational Fluid Dynamics , 2009 .

[62]  Tayfun E. Tezduyar,et al.  Space–time VMS computation of wind-turbine rotor and tower aerodynamics , 2014 .

[63]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[64]  Peter Hansbo,et al.  On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws , 1990 .

[65]  U. Ghia,et al.  Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications , 2007 .

[66]  Anindya Ghoshal,et al.  Compressible flows on moving domains: Stabilized methods, weakly enforced essential boundary conditions, sliding interfaces, and application to gas-turbine modeling , 2017 .

[67]  Thomas J. R. Hughes,et al.  A comparative study of different sets of variables for solving compressible and incompressible flows , 1998 .

[68]  Regina C. Almeida,et al.  An adaptive Petrov-Galerkin formulation for the compressible Euler and Navier-Stokes equations , 1996 .

[69]  Alvaro L. G. A. Coutinho,et al.  Compressible flow SUPG parameters computed from element matrices , 2005 .

[70]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[71]  Tayfun E. Tezduyar,et al.  Element length calculation in B-spline meshes for complex geometries , 2020, Computational Mechanics.

[72]  Tayfun E. Tezduyar,et al.  Tire aerodynamics with actual tire geometry, road contact and tire deformation , 2018, Computational Mechanics.

[73]  G. Sod A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[74]  Tayfun E. Tezduyar,et al.  Turbocharger turbine and exhaust manifold flow computation with the Space–Time Variational Multiscale Method and Isogeometric Analysis , 2019, Computers & Fluids.

[75]  J. E. Carter Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner , 1972 .