Towards robust and accurate computations of


Feb 22, 2014 (7 years and 8 months ago)


Towards robust and accurate computations of
shock/turbulence interactions

Johan Larsson

Center for Turbulence Research

Stanford University

Queen’s University, Nov 13, 2007


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


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
Taylor instability)

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

Source: Andy Cook, LLNL

Problems of interest

43 (Mach 9.6)




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

convection, shock waves,
combustion (fusion)


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:


The SciDAC program

Dept of Energy ‘Scientific Discovery through Advanced Computing’

disciplinary program to advance ‘peta
scale’ computational

Computer science

MPI, filesystems, visualization, etc

Applied mathematics

scalable algorithms, etc


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


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


governing equations

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


capturing of shock waves

Shock thickness is roughly the molecular mean
path (1 nm in air)

Unfeasible to resolve numerically


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


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=5

Kinetic energy evolution


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

‘Hybrid’ methods

different schemes around and away from shocks

Can use split form for ‘turbulence’ => non

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

art for shock
capturing, but
expensive and dissipative for

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


small in turbulence, large negative at shocks


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

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)

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

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

Velocity spectra

Dilatation spectra


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’

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

Canonical shock/turbulence interaction at high Mach and Reynolds

How to limit the size of the inflow database?


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

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

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

Heat load on high
speed vehicles depend on the transition to

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)


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