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
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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)
Scramjet
Source: 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
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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)
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Unfeasible to resolve numerically
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Shock

capturing

get the correct ‘jump’ on a realistic grid
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Need nonlinear (solution

dependent) dissipation
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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
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No shock waves, but large aliasing errors
•
Note instability of conservative form, stability of split form
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Note effect on bandwidth: 1/3, 1/2, 2/3 of maximum wavenumber
Energy spectra at t=5
Kinetic energy evolution
Introduction

contradictory requirements
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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
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Conservative, both types of dissipation
•
E.g. hyperviscosity (Cook, PoF 2007), WENO (Martin et al, JCP
2006)
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‘Hybrid’ methods

different schemes around and away from shocks
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Can use split form for ‘turbulence’ => non

dissipative
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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
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Concept: different numerics for different physics
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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)
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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
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Dilatation

small in turbulence, large negative at shocks
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Vorticity

large in turbulence, small at shocks
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Then set
Hybrid method

final details
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Use 8th order central scheme / 5th order WENO scheme
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4th order Runge

Kutta scheme in time
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8th order scheme for viscous terms
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Compare results to
•
‘Pure’ WENO

7th order WENO everywhere
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Hybrid + 8th order ‘dealiasing’ dissipation of form
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Hybrid with 2nd order central scheme
Test cases

verification and illustration of method
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3D Taylor

Green problem
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Verify accuracy and stability for broadband ‘turbulence’
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Illustrate adverse effect of numerical dissipation
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1D shock/entropy interaction
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Verify shock

capturing and hybrid concept
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2D shock/vorticity/entropy interaction
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Verify method on idealized interaction with shockwave
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3D isotropic decaying turbulence with shocklets
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Verify accuracy for compressible turbulence
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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
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No numerical noise at interfaces

evidence of stability
•
Hybrid method less dissipative than pure WENO
Entropy profiles
2D shock/vorticity/entropy interaction at Mach 1.5
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Oblique vorticity and entropy waves interacting with a normal shock
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Amplification of vorticity and kinetic energy by shock
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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
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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
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Shock sensor
•
How to find shocks robustly for more general flows?
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How to parallelize a hybrid method efficiently?
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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?
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Two partial explanations exist, neither is complete
•
30 years of numerical evidence

there must be a reason…
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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
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Financial support
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NSERC Postdoctoral Fellowship
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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
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