Towards robust and accurate computations of shock/turbulence interactions

Johan Larsson

Center for Turbulence Research

Stanford University

Queen’s University, Nov 13, 2007

Outline

• Flows of interest and potential applications• The SciDAC program and my work• Numerical challenges in shock/turbulence interaction• The proposed hybrid method• Evaluation and verification of the method• Summary, the next steps, and some ideas for the future

• Note -- I’m assuming some familiarity with:• Numerical solution of PDEs -- finite difference methods• Fluid mechanics -- shock waves and turbulence• Feel free to interrupt!

Problems of interest

• Flows with interactions between turbulence, shock waves, and material interfaces occur in a wide range of interesting applications• Super/hypersonic flight, shock/boundary layer interaction, inertial

confinement fusion (ICF), supernovae explosions, scramjet combustion, shock wave lithotripsy,…

Turbulent mixing of two fluids with different densities (Rayleigh-Taylor instability)

E.g. early stages of supernova explosion, late stages of ICF

Source: Andy Cook, LLNL

Problems of interest

X-43 (Mach 9.6)

ScramjetSource: www.tipmagazine.com

Source: www.dfrc.nasa.gov/Gallery

Problems of interest

Shock wave passing through a cylinder of heavier gas, generating vorticity and mixing (Richtmeyer-Meshkov instability)

E.g. scramjet combustion (shock interacting with injected fuel)

Source: Andy Cook, LLNL

Problems of interest

Supernovae explosions -- natural convection, shock waves, combustion (fusion)

The turbulent infalling gas starts shaking the core, causing it to pulsate… the oscillations are so intense they send out sound waves. The waves exert a pressure that expels material, reinforcing the shock wave created by the star's collapse. They also amplify the core's vibrations in a runaway reaction, says Burrows, "until the star finally explodes."

National Geographic, March 2007, on supernovae:

Source: flash.uchicago.edu

The SciDAC program

• Dept of Energy ‘Scientific Discovery through Advanced Computing’• Multi-disciplinary program to advance ‘peta-scale’ computational

science• Computer science -- MPI, filesystems, visualization, etc• Applied mathematics -- scalable algorithms, etc• Science -- climate, quantum mechanics, numerical relativity,

astrophysics, biology, etc• Our project

• Stanford, NASA Ames, Lawrence Livermore, UCLA• Initial focus -- numerical methods for shock/turbulence/material

interface interactions• This talk is about my part of this project

shock

isotropic turbulence

Working roadmap

• Hybrid numerical method• Existing methods capture shocks well, but sacrifice accuracy in

treating turbulence -- core problem is numerical dissipation• Proposed method largely eliminates this numerical dissipation

• Verify method on a sequence of problems• Increase complexity step by step

• Canonical shock/turbulence interaction study• Unanswered questions of flow physics• Basic problem for shock/turbulence modeling

• This talk will cover the first 2 items

Introduction -- governing equations

• Navier-Stokes equations for a perfect gas

• Convective terms (LHS) contain amazing range of physics• Shock waves -- discontinuities in the flow field• Vortex stretching etc -- energy transfer towards smaller eddies

• Convective terms also pose greatest numerical challenge• Special ‘shock-capturing’ schemes needed for shocks• Numerical energy transfer (aliasing errors) often cause blow-up

Introduction -- capturing of shock waves

• Shock thickness is roughly the molecular mean-free-path (1 nm in air)• Unfeasible to resolve numerically

• Shock-capturing -- get the correct ‘jump’ on a realistic grid• Need nonlinear (solution-dependent) dissipation• Need conservative form of convective terms -- proven to give

correct weak solution

• Conservative and non-conservative forms:

Introduction -- capturing broadband turbulence

• Stability affected by aliasing error:

• The energy is ‘aliased’ to some unphysical wavenumber• Could lead to catastrophic energy growth and numerical instability

• Linear, ‘dealiasing’ dissipation• Split form of convective terms -- reduces aliasing error

The Taylor-Green vortex (3D)

• Idealized vortex-stretching for nearly incompressible flow• No shock waves, but large aliasing errors• Note instability of conservative form, stability of split form• Note effect on bandwidth: 1/3, 1/2, 2/3 of maximum wavenumber

Energy spectra at t=5Kinetic energy evolution

Introduction -- contradictory requirements

• Need conservative form and shock-capturing dissipation for shocks• Conservative form => need dealiasing dissipation for stability• Dissipation harms accuracy -- only 1/3 or 1/2 of wavenumbers

accurate• ‘Unified’ methods -- same scheme everywhere

• Conservative, both types of dissipation• E.g. hyperviscosity (Cook, PoF 2007), WENO (Martin et al, JCP

2006)• ‘Hybrid’ methods -- different schemes around and away from shocks

• Can use split form for ‘turbulence’ => non-dissipative• Issues of conservation and stability at the interface, where to use

each scheme• With upwinding (Adams and Shariff, JCP 1996), filtering (Rizzetta

et al, AIAA J 2001), central (Pantano et al, JCP 2007)

Hybrid method -- general approach

• Concept: different numerics for different physics• Minimal dissipation through split form• Novelties in present method:

• Conservative coupling for general split schemes• Stability proof at the interface (JCP, under review)

Numerical grid

Hybrid method -- numerical flux framework

• Conservation at interfaces by numerical flux form

with defined

• Hybridize by

• Have reduced problem of interface conservation to finding

Hybrid method -- WENO scheme

• Adaptively chosen weighted combination of candidate fluxes

• Weights chosen based on smoothness

• Candidate stencils and sample weights

• State-of-the-art for shock-capturing, but expensive and dissipative for turbulence

Hybrid method -- central split scheme

• Split convective form by Ducros et al (JCP 2000)

• Derive bilinear interpolation stencils for flux form

such that

yields split form by Ducros et al

Hybrid method -- shock sensor

• Must find regions of shock waves robustly• Many sensors possible, and area in need of improvement• Currently based on comparing dilatation and vorticity

• Dilatation -- small in turbulence, large negative at shocks• Vorticity -- large in turbulence, small at shocks

• Then set

Hybrid method -- final details

• Use 8th order central scheme / 5th order WENO scheme• 4th order Runge-Kutta scheme in time• 8th order scheme for viscous terms

• Compare results to• ‘Pure’ WENO -- 7th order WENO everywhere• Hybrid + 8th order ‘dealiasing’ dissipation of form

• Hybrid with 2nd order central scheme

Test cases -- verification and illustration of method

• 3D Taylor-Green problem• Verify accuracy and stability for broadband ‘turbulence’• Illustrate adverse effect of numerical dissipation

• 1D shock/entropy interaction• Verify shock-capturing and hybrid concept

• 2D shock/vorticity/entropy interaction• Verify method on idealized interaction with shockwave

• 3D isotropic decaying turbulence with shocklets• Verify accuracy for compressible turbulence• Provide inflow condition for full shock/turbulence case

The Shu-Osher shock/entropy interaction in 1D

• Mach 3 shock moves into entropy wave, interaction amplifies entropy waves and creates acoustic waves

• WENO confined to shock waves• No numerical noise at interfaces -- evidence of stability• Hybrid method less dissipative than pure WENO

Entropy profiles

QuickTime™ and a decompressor

are needed to see this picture.

2D shock/vorticity/entropy interaction at Mach 1.5

• Oblique vorticity and entropy waves interacting with a normal shock• Amplification of vorticity and kinetic energy by shock

• Compared to linear theory -- 5th order convergence for amplification ratio (order of the shock-capturing scheme)

Contours of vorticity Vorticity amplification

Isotropic decaying turbulence

• Initial conditions: • Large enough to spontaneously generate shocklets (weak shocks)

• Shock-sensor finds these appropriately• Dilatation flatness much larger than 3 good measure of shocklets

Contours of dilatation

Isotropic decaying turbulence

• Compare on 64^3 grid with filtered DNS on 256^3• Shock-capturing dissipation (pure WENO) overly dissipative• Linear 8th order dissipation same thing, but less severely• Dissipation more important than order of accuracy

Kinetic energy decay Vorticity decay

Isotropic decaying turbulence

• Accurate bandwidth: 1/4, 1/2, 2/3 of maximum wavenumber• Similar to the inviscid Taylor-Green vortex

• Note: addition of explicit subgrid-scale model would only make this worse

Velocity spectra Dilatation spectra

Summary

• Numerical dissipation has negative effect on the accuracy for broadband turbulence -- decreases accurate bandwidth by factor of 2 or more• Would need 8 times more grid points for equivalent accuracy• Dissipation larger than ideal subgrid-scale model -- should not

evaluate models with dissipative numerics• Hybrid approach allows for minimal dissipation by use of split form

• Introduces additional complications, but increased accuracy worth the ‘price’

• Side-benefit of speed -- central scheme 5 times faster than WENO• Overall robust, accurate, and efficient -- good framework for future

DNS and LES studies• Order of accuracy less important -- at least for turbulence statistics,

and for these problems

Future work

• Shock sensor• How to find shocks robustly for more general flows?

• How to parallelize a hybrid method efficiently?• The expensive WENO scheme is used locally, in ‘random’ portions

of the domain -- load balancing non-trivial• Canonical shock/turbulence interaction at high Mach and Reynolds

numbers• How to limit the size of the inflow database?

shock

isotropic turbulence

Some interesting research topics

• Modeling of shock/turbulence interaction in large eddy simulation• Mathematically unresolved shock waves are no different than

unresolved turbulence -- can/should they be modeled jointly/analogously?

• Why are split convective terms more nonlinearly robust?• Two partial explanations exist, neither is complete• 30 years of numerical evidence -- there must be a reason…

• Shock identification• Structure of the velocity gradient tensor?• Jumps in entropy?

Some interesting research topics

• NASA X-43 achieved Mach 9.6 with scramjet engine in 2004• Better understanding and modeling of shock/turbulence

interactions and induced mixing of fuel and oxidizer needed to design better scramjets

• Heat load on high-speed vehicles depend on the transition to turbulence• Discovery shuttle in 2005 -- ground control could not predict the

transition point (Annu. Rev. Fluid Mech. 2006)• Experiments on inertial confinement fusion are ongoing

• Mixing induced by shock/turbulence interactions can severely degrade the fusion process (indeed prevent it completely)

Acknowledgements

• Financial support• NSERC Postdoctoral Fellowship• US Department of Energy SciDAC program• Center for Turbulence Research

• Stimulating discussions• Many people, including Sanjiva Lele, Parviz Moin, Bertil

Gustafsson, Albert Honein, and Andy Cook