Zhaorui Li and Farhad Jaberi Department of Mechanical Engineering Michigan State University East Lansing, Michigan Large-Scale Simulations of High Speed Turbulent Flows

 Develop high-fidelity numerical models for high speed turbulent flows  Fundamental understanding of compressible turbulent flows and shock-turbulence Interactions  Numerical experiments - Analyses of supersonic/hypersonic problems for various flow parameters Objectives: Approach:  High-order numerical methods for LES and DNS of high speed (supersonic) turbulent flows in complex geometries  Existent low-speed SGS models extended and applied to high speed turbulent flows  DNS and experimental data are employed for validation and improvement of LES submodels

LES and DNS of High Speed Turbulent Flows  High-order numerical schemes for compressible turbulent velocity field in complex geometries. Needed for LES & DNS of very high speed supersonic/hypersonic flows.  In LES, large-scale compressibility effects are explicitly calculated. So far, compressible SGS (Dynamic) Gradient, Mixed and MKEV models have been employed. Work is in progress to develop improved deterministic subgrid turbulence and wall models for supersonic and hypersonic flows.  High-order numerical schemes for the filtered scalar field in supersonic turbulent flows. In LES, large-scale (compressible) mixing are explicitly calculated. So far, SGS Gradient models have been employed.  Direct evaluation and improvement of subgrid models via DNS data.  Comparisons with DNS and experimental data for validation of numerical method and SGS models

Application of LES to High Speed Flows Numerical Methods 1) High-order Compact-RK scheme + limiters and/or artificial viscosity (Rizzetta et al. 2001, Cook & Cabot 2005, Kawai & Lele 2007) – 3D code is developed and tested 2)High-order WENO-RK scheme (Shi et al. 2003)- 3D code is developed and tested 3)High-order Monotonicity Preserving (MP)-RK scheme (Huynh 2007) - 3D code is developed and tested) Test Problems 1)1D Problems: Advection (Wave), Burgers, Lax, Shock Tube, (1D calculations with 3D codes) 2)2D Problems: Rayleigh-Taylor Instability, Double Mach Reflection, Isotropic Turbulence (2D calculations with 3D codes) 3)3D Isotropic Turbulence 4)Converging-Diverging Nozzle – 3D LES 5)Supersonic Boundary Layer with Shock wave– 3D DNS and LES 6)Supersonic Mixing Layer – 3D DNS and LES A Single FD code in generalized coordinate system

F ully Compressible Filtered Navier-Stokes Equations in Generalized Coordinates System

F ully compressible Navier-Stokes equations in generalized coordinates system with transformation Solution vector Transformation Jacobian Sixth-order Compact scheme Eighth-order implicit filter WENO5MP5/MP7 Eigensystem in generalized coordinates Here only the format eigenvector for are given (1) Lax-Friedrichs flux splitting (2) Compute left and right eigenvectors of Roe’s mean matrix, transform all quantities needed for evaluating the numerical flux to local characteristic field. (3) Calculate numerical flux in characteristic field (4) Calculate numerical flux in physical space (*) a mirror imagine (with respect to j+1/2) Procedure to that in step(3) is used to calculate (*) only difference between WENO and MP only in step (3) is described in the following (a) Calculate original interface value 7 tt— order scheme 5 tt— order scheme (b) Determine discontinuity (*) limiter needed Where (c) Limiting procedure

1D Advection 1D Advection or Wave Eq. 5 th Order upwind 3 rd order ENO 5 th order WENO, 5 th order MP Schemes Solutions after 10 Periods NX=200CFL=0.4 LES and DNS of High Speed Flows High-Order Numerical Methods for Supersonic Turbulent Flows 5 th order MP 5 th order WENO 3 rd order ENO 5 th order Upwind

1D Burgers Eq. 5 th order WENO and 5 th order WENO and 5 th order MP Schemes Initial condition: 1D Shock- Tube Problem 1D Shock- Tube Problem 5 th order WENO and 5 th order WENO and 5 th order MP Schemes Initial Condition High-Order Numerical Methods for Supersonic Turbulent Flows

High-Order Numerical Methods for Supersonic Flows 2D Inviscid Rayleigh-Taylor Instability Problem 5 th order WENO 5 th order WENOand 5 th order MP Schemes Density Contours

High-Order Numerical Methods for Supersonic Flows 2D Viscous Rayleigh- Taylor Instability Problem 5 th order WENO 5 th order MP Density Contours Re=25,000 5 th order WENO 5 th order MP Re=50,000

High-Order Numerical Methods for Supersonic Flows 2D Viscous Rayleigh-Taylor Instability Problem 5 th order WENO 5 th order WENO 5 th order MP 7 th order MP 7 th order MPSchemes Density X (Y=0.6)

High-Order Numerical Methods for Supersonic Flows Double Mach Problem Initial Condition Ma=10 5 th order WENO 5 th order MP 7 th order MP Density Contours Next Slide

High-Order Numerical Methods for Supersonic Flows Double Mach Problem Density Contours 5 th order WENO 5 th order MP 7 th order MP

High-Order Numerical Methods for Supersonic Flows Supersonic Diverging Nozzle Supersonic Diverging Nozzle Low Back Pressure Mach Number Contours

High-Order Numerical Methods for Supersonic Turbulent Flows Supersonic Diverging Nozzle Supersonic Diverging Nozzle High Back Pressure Mach Number Contours

3D 3D Isotropic Turbulence High-Order Numerical Methods for Supersonic Turbulent Flows Energy Spectrum Enstrophy Enstrophy Dissipation Rate

Incident shock-BL interaction Compression corner-BL interaction Test Case: Supersonic Laminar Flat-Plate Boundary Layer (Anderson, 2000) Computational Details and Problem Setup Numerical Scheme: 5 th order Monotonicity Preserving scheme for inviscid fluxes and 6 th order compact scheme for viscous/scalar fluxes Reference Mach No. : 2.5 Reference Reynolds No. : 582 Shock Wave - Boundary Layer Interactions Pressure Contours temperature Streamwise velocity

Shock-Laminar BL Interactions Expansion Fan Separation region Incident shock β = 30 o Compression Waves Pressure Contours Pressure Distribution in the streamwise direction Streamwise Velocity Contours Computational Details and Problem Setup Numerical Scheme: 5 th order Monotonicity Preserving scheme for inviscid fluxes and 6 th order compact scheme for viscous/scalar fluxes Reference Mach No. : 2.5 Reference Reynolds No. : 582 At y = 2.0, discontinuities which satisfy Rankine- Hugoniot relations are introduced at the inlet.

High-Order Numerical Methods for Supersonic Turbulent Flows 2D Turbulent Mixing Layer – Shock Interactions DNS data for understanding of turbulence-shock interactions and development of improved SGS models Density Contours Pressure Contours wall wall shock shock

High-Order Numerical Methods for Supersonic Turbulent Flows 2D Turbulent Mixing Layer – Shock Interactions Species Mass Fraction Contours

DNS of Spatially Developing 3D Supersonic Mixing Layer Pressure Contours at Z=0.75Lz Re δ/2 =200, Mc=1.2 M1=4.2 M2=1.8

M1=4.2 M2=1.8 Spanwise Vorticity Contours Z=0.75Lz DNS of Spatially Developing 3D Supersonic Mixing Layer

Scalar Mass Fraction Contours Z=0.75Lz

63.5 mm diam center jet Small-scale facility Large-scale facility 10 mm diameter Center jet Supersonic Mixing and Reaction - Co-Annular Jet Experiments (Laboratory and Full-Scale Models) Cutler et al LES of Co-Annular Jet Grid System for LES

Iso-Levels of Mach Number Pressure Temperature LES of Supersonic Co-Annular Jet – Non-Reacting Flow 3D LES Calculations with Compact Scheme

Summary and Conclusions  Robust high-order finite difference methods (i.e. MP, WENO, Compact+limiter) are developed and tested for large-scale and detailed calculations of compressible turbulent flows with/without shock waves in complex geometries  Numerical simulations of various 1D, 2D and 3D flows have been conducted for assessment of numerical schemes and SGS turbulence models  So far, compressible SGS (Dynamic) Gradient and Mixed LES models have been employed  DNS data are being generated/analyzed for shock-turbulent mixing layer and shock-boundary layer interaction problems  Work is in progress to develop improved compressible subgrid turbulence models for supersonic flows.  LES data for supersonic co-annular jet are being compared with experimental data

High-Order Numerical Methods for Supersonic Turbulent Flows Turbulent Mixing Layer – Shock Interactions DNS data for understanding of turbulence-shock- Combustion interactions  and development of improved SGS models  Density  Pressure  wall  shock  Scalar  Scalar Equation

 F ully compressible Navier-Stokes equations in generalized coordinates system with transformation Solution vector Transformation Jacobian 6-order Compact scheme Eighth-order implicit filter    WENO scheme MP Scheme  Eigensystem in generalized coordinates  Here only the format eigenvector for are given (1) Lax-Friedrichs flux splitting (2) Compute left and right eigenvectors of Roe’s mean matrix, transform all quantities needed for evaluating the numerical flux to local characteristic field. (3) Calculate numerical flux in characteristic field  (4) Calculate numerical flux in physical space  (*) a mirror imagine (with respect to j+1/2)  Procedure to that in step(3) is used to calculate  (*) only difference between WENO and MP only in step (3) is described in the following  (a) Calculate original interface value 7 tt— order scheme 5 tt— order scheme  (b) Determine discontinuity  (*) limiter needed  Wh ere  (c) Limiting procedure