Towards robust and accurate computations of

busyicicleMécanique

22 févr. 2014 (il y a 3 années et 10 mois)

86 vue(s)

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)

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


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=5

Kinetic 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

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