zhaorui li and farhad jaberi department of mechanical engineering michigan state university east...

Zhaorui Li and Farhad JaberiDepartment of Mechanical Engineering

Michigan State UniversityEast Lansing, Michigan

Large-Scale Simulations of High Large-Scale Simulations of High Speed Turbulent FlowsSpeed 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

ObjectivesObjectives::

ApproachApproach::

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 LES and DNS of High Speed Turbulent FlowsFlows 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 Application of LES to High Speed Flows 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

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ˆˆˆ

)()()(ˆ),()()(ˆ

),()()(ˆ

vzvyvxvzvyvx

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),,,,( EwvuU

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Sixth-order Compact scheme

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011 )(

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WENO5 MP5/MP7

Eigensystem in generalized coordinates

0xHere only the format eigenvector for are given

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(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.

UdUFdUUFUFUFUFUFU

/)(ˆmax ),)(ˆ()(ˆ ),(ˆ)(ˆ)(ˆ

2/1ˆ

jF

2/12/12/12/1 )( )/)(ˆ()( jjCjjC RUdUFdL

)(ˆ)()(ˆ2/1 jjCcj UFLUF

(3) Calculate numerical flux in characteristic field k

jk

kcj FF 2/1

3

12/1

ˆ)ˆ(

6/)(ˆ6/)(ˆ53/)(ˆˆ

3/)(ˆ6/)(ˆ56/)(ˆˆ

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212

2121

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(4) Calculate numerical flux in physical space

cjjCj FRF )ˆ()(ˆ2/12/12/1

(*) a mirror imagine (with respect to j+1/2)

Procedure to that in step(3) is used to calculate 2/1

ˆjF

(*) only difference between WENO and MP only in step (3) is described in the following

(a) Calculate original interface value

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420/))(ˆ4)(ˆ38

)(ˆ214)(ˆ319)(ˆ101)(ˆ25)(ˆ3()ˆ(

32

11232/1

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cjcjcjcjcjcj

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UFUFUFUFUFF

77tt—tt—order schemeorder scheme


(b) Determine discontinuity

))ˆ()ˆ)(()ˆ()ˆ(( 2/1,,2/1, c

MPmcjmcjmcjm FFFF

)(ˆ ofcomponent one

is )51( ˆ,

j

jm

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mF

(*) limiter needed ))ˆ()ˆ((,)ˆ()ˆ(minmod)ˆ()ˆ( 1,,,1,, cjmcjmcjmcjmcjmc

MPm FFFFFF Where

(c) Limiting procedure

)]ˆ,ˆ,)ˆmax((),ˆ,)ˆ(,)ˆmin[max((ˆ

)]ˆ,ˆ,)ˆmin((),ˆ,)ˆ(,)ˆmax[min((ˆ

3/4))ˆ()ˆ((5.0)ˆ(ˆ ,5.0ˆˆ

),)ˆ()ˆ((5.0ˆ ),)ˆ()ˆ(()ˆ(ˆ

) , ,4 ,4(minmod ,)ˆ(2)ˆ()ˆ(

,1,,max

,1,,min

42/11,,,

42/1

1,,1,,,

1114

2/1,1,1,

LCm

ULmcjm

MDmcjmcjmm

LCm

ULmcjm

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LCm

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FFFFFFF

dFFFFdFF

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)ˆ,ˆ,)ˆ((median)ˆ( maxmin2/1,2/1, mmcjmcjm FFFF

X-1 -0.5 0 0.5 1

0

0.3

0.6

0.9

1.2 Exact SolutionUnlimited

u(x)

X-1 -0.5 0 0.5 1

0

0.3

0.6

0.9

1.2 Exact SolutionENO3

u(x)

X-1 -0.5 0 0.5 1

0

0.3

0.6

0.9

1.2 Exact SolutionWENO5

u(x)

X-1 -0.5 0 0.5 1

0

0.3

0.6

0.9

1.2 Exact SolutionMP5

u(x)

1D Advection 1D Advection or Wave Eq.or Wave Eq.

55thth Order upwind Order upwind 33rdrd order ENO order ENO

55thth order WENO, order WENO, 55thth order MP order MP

SchemesSchemes

Solutions after 10 Solutions after 10 PeriodsPeriods

NX=200NX=200CFL=0.4CFL=0.4

LES and DNS of High Speed FlowsLES and DNS of High Speed Flows High-Order Numerical Methods for High-Order Numerical Methods for

Supersonic Turbulent FlowsSupersonic Turbulent Flows

55thth o

rder

MP

ord

er M

P

55thth o

rder

WE

NO

ord

er W

EN

O

33rdrd o

rder

EN

O o

rder

EN

O

55thth o

rder

Up

win

d o

rder

Up

win

d

1D Burgers Eq.1D Burgers Eq.

55thth order WENO order WENO and and

55thth order MP order MP SchemesSchemes

Initial condition:Initial condition:

X0 0.5 1 1.5 21.98

1.99

2

2.01

2.02

Exact Solution

WENO5

MP5

NX=100CFL=0.4

u(x,

t)

X0 0.5 1 1.5 21.98

1.99

2

2.01

2.02

Exact Solution

WENO5

MP5

NX=200CFL=0.4

u(x,

t)

)(sin2)0,( 9 xxu

1D Shock- 1D Shock- Tube ProblemTube Problem



Initial ConditionInitial Condition

X-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Exact Solution

WENO5

P

WENO5NX=100,cfl=0.4

X-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Exact Solution

MP5

P

MP5NX=100,cfl=0.4

X-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Exact Solution

MP5

MP5NX=100,cfl=0.4

X-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

Exact Solution

WENO5

WENO5NX=100,cfl=0.4

)1.0 ,0.0 ,125.0(),,( RRR pu

High-Order Numerical Methods for High-Order Numerical Methods for Supersonic Turbulent FlowsSupersonic Turbulent Flows

High-Order High-Order Numerical Numerical

Methods for Methods for Supersonic FlowsSupersonic Flows

2D Inviscid 2D Inviscid Rayleigh-Taylor Rayleigh-Taylor

Instability ProblemInstability Problem



Density ContoursDensity Contours


High-Order Numerical High-Order Numerical Methods for Supersonic Methods for Supersonic

FlowsFlows

2D Viscous Rayleigh-2D Viscous Rayleigh-Taylor Instability Taylor Instability

ProblemProblem

55thth

ord

er W

EN

O o

rder W

EN

O55thth

ord

er M

P o

rder M

P


Re=25,000Re=25,000

55thth

ord

er W

EN

O o

rder W

EN

O55thth

ord

er M

P o

rder M

P

Re=50,000Re=50,000

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=240NY=480NY=960

Re=25000WENO5

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=480NY=960NY=1920

Re=50000WENO5

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=240NY=480NY=960

Re=25000MP5

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=240NY=480NY=960

Re=25000MP7

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=480NY=960NY=1440

Re=50000MP5

0 0.05 0.1 0.15 0.2 0.250.8

1

1.2

1.4

1.6

1.8

2

2.2

NY=480NY=960NY=1440

Re=50000MP7


Methods for Methods for Supersonic FlowsSupersonic Flows

2D Viscous 2D Viscous Rayleigh-Taylor Rayleigh-Taylor

Instability ProblemInstability Problem

55thth order WENO order WENO 55thth order MP order MP 77thth order MP order MP

SchemesSchemes

Den

sit

yD

en

sit

y

X (Y=0.6)X (Y=0.6)


FlowsFlows

Double Mach ProblemDouble Mach Problem

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

WENO5NX=4NY=240

cfl=0.4

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

WENO5NX=4NY=480

cfl=0.4

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

WENO5NX=4NY=960

cfl=0.4

Initial ConditionInitial ConditionMa=10Ma=10

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP5NX=4NY=240

cfl=0.4

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP5NX=4NY=480

cfl=0.4

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP5NX=4NY=960

cfl=0.4

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP7NX=4NY=240

(a)

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP7NX=4NY=480

(b)

X

Y

0 0.5 1 1.5 2 2.5 30

0.5

1

MP7NX=4NY=960

(c)



Next SlideNext Slide


FlowsFlows

Double Mach ProblemDouble Mach Problem

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5WENO5 NX=4NY=480 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5WENO5 NX=4NY=960 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5WENO5 NX=4NY=1920 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP5 NX=4NY=480 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP5 NX=4NY=960 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP5 NX=4NY=1920 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP7 NX=4NY=480 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP7 NX=4NY=960 cfl=0.4

X

Y

2 2.25 2.5 2.750

0.1

0.2

0.3

0.4

0.5MP7 NX=4NY=1920 cfl=0.4

Den

sit

y C

on

tou

rsD

en

sit

y C

on

tou

rs



Methods for Methods for Supersonic Supersonic

FlowsFlows

Supersonic Supersonic Diverging Diverging

NozzleNozzle

Low Back Low Back PressurePressure

Mach Number ContoursMach Number Contours

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=15000

0 2 4 6 8 10-2

-1

0

1

2

Re=10000

WENO5

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=5000

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=2000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=15000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=10000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=5000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=2000


Methods for Methods for Supersonic Supersonic Turbulent Turbulent

FlowsFlows

Supersonic Supersonic Diverging Diverging

NozzleNozzle

High Back High Back PressurePressure

Mach Number ContoursMach Number Contours

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=15000

0 2 4 6 8 10-2

-1

0

1

2

Re=10000

WENO5

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=5000

0 2 4 6 8 10-2

-1

0

1

2

WENO5

Re=2000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=15000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=10000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=5000

0 2 4 6 8 10-2

-1

0

1

2

MP5

Re=2000

3D 3D Isotropic TurbulenceIsotropic Turbulence

0 2 4 6 8 10 120.8

1.6

2.4

3.2

SpectralMP5WENO5MP7

t

(a)

0 2 4 6 8 10 120.03

0.035

0.04

0.045

0.05

SpectralMP5WENO5MP7

t

(b)

10 20 30 40 5010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SpectralMP5WENO5MP7

E(K)

K

(a)

20 25 30 35 40 45 50 5510-10

10-9

10-8

10-7

10-6

10-5

10-4

SpectralMP5WENO5MP7

K

E(K)

(b)


Turbulent FlowsTurbulent Flows

Energy Energy SpectrumSpectrum

Energy Energy SpectrumSpectrum

EnstrophyEnstrophy

Enstrophy Enstrophy Dissipation Dissipation RateRate

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: 5th order Monotonicity Preserving scheme for inviscid fluxes and 6th order compact scheme for viscous/scalar fluxes

• Reference Mach No. : 2.5

• Reference Reynolds No. : 582

Shock Wave - Boundary Layer Interactions

Pressure Contours

temperatureStreamwise velocity

Shock-Laminar BL Interactions

Expansion Fan

Separation region

Incident shock β = 30o

Compression Waves

Pressure Contours

Pressure Distribution in the streamwise direction

Streamwise Velocity Contours

Computational Details and Problem Setup

• Numerical Scheme: 5th order Monotonicity Preserving scheme for inviscid fluxes and 6th 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.



FlowsFlows

2D 2D Turbulent Mixing Turbulent Mixing

Layer – Shock Layer – Shock InteractionsInteractions

DNS data for DNS data for understanding of understanding of turbulence-shock turbulence-shock

interactions interactions and development of and development of

improved SGS modelsimproved SGS models

Den

sity

Con

tou

rsD

en

sity

Con

tou

rsP

ressu

re C

on

tou

rsP

ressu

re C

on

tou

rs

wallwall

shockshock



FlowsFlows

2D 2D Turbulent Mixing Turbulent Mixing

Layer – Shock Layer – Shock InteractionsInteractions

Sp

ecie

s M

ass F

ractio

n C

on

tou

rsS

pecie

s M

ass F

ractio

n C

on

tou

rs

0 50 100 150 200-20

-10

0

10

20

WENO5

0 50 100 150 200-20

-10

0

10

20

WENO5_COMP6(Non-Conser)

0 50 100 150 200-20

-10

0

10

20

WENO5_COMP6(Conser)

0 50 100 150 200-20

-10

0

10

20

MP7_COMP6(Non-Conser)

0 50 100 150 200-20

-10

0

10

20


0 50 100 150 200-20

-10

0

10

20


DNS of Spatially Developing 3D Supersonic Mixing DNS of Spatially Developing 3D Supersonic Mixing LayerLayer

Pre

ssu

re C

on

tou

rs a

t Z=

0.7

5Lz

Pre

ssu

re C

on

tou

rs a

t Z=

0.7

5Lz

ReReδδ/2/2=200, Mc=1.2=200, Mc=1.2

M1=4.2M1=4.2

M2=1.8M2=1.8

ReReδδ/2/2=200, Mc=1.2=200, Mc=1.2

M1=4.2M1=4.2

M2=1.8M2=1.8

Sp

an

wis

e V

ortic

ity C

on

tou

rs Z

=0.7

5Lz

Sp

an

wis

e V

ortic

ity C

on

tou

rs Z

=0.7

5Lz

DNS of Spatially Developing 3D Supersonic Mixing DNS of Spatially Developing 3D Supersonic Mixing LayerLayer

DNS of Spatially Developing 3D Supersonic MixingDNS of Spatially Developing 3D Supersonic Mixing LayerLayer

Scala

r Mass F

ractio

n C

on

tou

rs Z

=0.7

5Lz

Scala

r Mass F

ractio

n C

on

tou

rs Z

=0.7

5Lz

63.5 mm

diam

cen

ter jetCARS/

Rayleighbeams

M=2 vitiated air jet

Burner/ nozzle

CARS/ Rayleighbeams

M=2 vitiated air jet

Burner/ nozzle

Coflownozzle

Facility flange

M=2 setup M=1 setup

SiC liner

Watercooled shell

Small-scale facility Large-scale facility Nozzle (SiC)

Water-cooled combustion chamber

Spark plug

H2 fuel tube

Air+O2

passage

Coflownozzle

Water-cooled injector

10 mm

diam

eter C

enter jet

Supersonic Mixing and Reaction - Co-Annular Jet Experiments (Laboratory and Full-Scale Models) Cutler et al. 2007

LES of Co-Annular Jet

Grid System for Grid System for LESLES

Iso-Levels of Mach Number

Pressure Temperature

LES of Supersonic Co-Annular Jet – Non-Reacting Flow

3D LES Calculations 3D LES Calculations

with Compact Schemewith Compact Scheme

Summary and ConclusionsSummary 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 High-Order Numerical Numerical Methods for Methods for Supersonic Supersonic Turbulent Turbulent FlowsFlows

Turbulent Mixing Turbulent Mixing Layer – Shock Layer – Shock InteractionsInteractions

DNS data for DNS data for understanding of understanding of turbulence-shock-turbulence-shock-Combustion interactions Combustion interactions

and development of and development of improved SGS modelsimproved SGS models

D

en

sity

D

en

sity

P

ressu

reP

ressu

re

wallwall

shockshock

0 50 100 150 200-20

-10

0

10

20

WENO5

0 50 100 150 200-20

-10

0

10

20


0 50 100 150 200-20

-10

0

10

20

WENO5_COMP6(Conser)

0 50 100 150 200-20

-10

0

10

20


0 50 100 150 200-20

-10

0

10

20

MP5_COMP6(Non-Conser)S

cala

rS

cala

r

Scala

r Eq

uatio

nS

cala

r Eq

uatio

n

Fully compressible Navier-Stokes equations in generalized coordinates system with transformation

),,(),,( zyx

JSHGF

t

UJ

ˆˆˆ

)()()(ˆ),()()(ˆ

),()()(ˆ

vzvyvxvzvyvx

vzvyvx

HHJGGJFFJHHHJGGJFFJG

HHJGGJFFJF

),,,,( EwvuU

),,(/),,( zyxJSolution vectorSolution vector Transformation JacobianTransformation Jacobian

6-order Compact scheme

Eighth-order implicit filter

)()( 221111 iiiiiii ffcffbfff )36/(1 ),9/(7 ,3/1 hchb

4

011 )(

2

)(ˆˆˆk

kikiifiif ffka

fff

)147(32

1)2( ),187(

16

1)1( ),7093(

128

1)0( fff aaa

)21(128

1)4( ),21(

16

1)3( ff aa

WENO scheme MP Scheme

Eigensystem in generalized coordinates

0xHere only the format eigenvector for are given

2

b )

2

~

2

b(- )

2

~

2

b(- )

2

~

2

b(-

2

~

2

b

0 ~

~1

~

~~

~

- ~

~~

~

~1~

0 ~

~~

- ~

~1

~

- w~

~~

~

~1~

b b b 1

2

b )

2

~

2

b(- )

2

~

2

b(- )

2

~

2

b(-

2

~

2

b

11112

22

22

11112

11112

c

w

c

v

c

u

c

ξ

ξ-ξv)w

ξ

ξ-u-(ξ

ξ

ξ-ξ)v

ξ

ξ-u-(ξ

bwvubc

w

c

v

c

u

c

L

zyx

x

z

x

zyz

x

zy

x

zz

x

zy

x

yy

x

zy

x

yy

zyx

C

~

~~

~~

~

~

~ 0

~

~ 0

~

~

~

~

~

~1 0 0 1 1

cHuwuvqcH

cwwcw

cvvcv

cuucu

R

zxyx

zxz

yxy

xzyx

C

wvuU

cUcU

zyx

zyxzyx

432

2225

2221 ,,

112

21

222

/1 ,

,/)1( ,2/

bqHqbb

cbwvuq

/~

and

,~~~

~

222zyxxx

zyx wvu

hFFUF jjJ

/)ˆˆ()(ˆ2/12/1

(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.

UdUFdUUFUFUFUFUFU

/)(ˆmax ),)(ˆ()(ˆ ),(ˆ)(ˆ)(ˆ

2/1ˆ

jF

2/12/12/12/1 )( )/)(ˆ()( jjCjjC RUdUFdL

)(ˆ)()(ˆ2/1 jjCcj UFLUF

(3) Calculate numerical flux in characteristic field k

jk

kcj FF 2/1

3

12/1

ˆ)ˆ(

6/)(ˆ6/)(ˆ53/)(ˆˆ

3/)(ˆ6/)(ˆ56/)(ˆˆ

6/)(ˆ116/)(ˆ73/)(ˆˆ

212

2/1

112

2/1

1121

2/1

cjcjcjj

cjcjcjj

cjcjcjj

UFUFUFF

UFUFUFF

UFUFUFF

3.0 ,6.0 ,1.0 where

)(~ ,

~

~

321

3

1

k

kk

i i

kk

4/))(ˆ)(ˆ4)(ˆ3(12/))(ˆ)(ˆ2)(ˆ(13

4/))(ˆ)(ˆ(12/))(ˆ)(ˆ2)(ˆ(13

4/))(ˆ3)(ˆ4)(ˆ(12/))(ˆ)(ˆ2)(ˆ(13

221

2213

211

2112

212

2121

cjcjcjcjcjcj

cjcjcjcjcj

cjcjcjcjcjcj

UFUFUFUFUFUF

UFUFUFUFUF

UFUFUFUFUFUF

(4) Calculate numerical flux in physical space

cjjCj FRF )ˆ()(ˆ2/12/12/1

(*) a mirror imagine (with respect to j+1/2)

Procedure to that in step(3) is used to calculate 2/1

ˆjF

(*) only difference between WENO and MP only in step (3) is described in the following

(a) Calculate original interface value60/))(ˆ3)(ˆ27)(ˆ47)(ˆ13)(ˆ2()ˆ( 21122/1 cjcjcjcjcjcj UFUFUFUFUFF

420/))(ˆ4)(ˆ38

)(ˆ214)(ˆ319)(ˆ101)(ˆ25)(ˆ3()ˆ(

32

11232/1

cjcj

cjcjcjcjcjcj

UFUF

UFUFUFUFUFF



(b) Determine discontinuity

))ˆ()ˆ)(()ˆ()ˆ(( 2/1,,2/1, cMP

mcjmcjmcjm FFFF )(ˆ ofcomponent one

is )51( ˆ,

j

jm

UF

mF

(*) limiter needed ))ˆ()ˆ((,)ˆ()ˆ(minmod)ˆ()ˆ( 1,,,1,, cjmcjmcjmcjmcjmc

MPm FFFFFF Wh

ere(c) Limiting procedure

)]ˆ,ˆ,)ˆmax((),ˆ,)ˆ(,)ˆmin[max((ˆ

)]ˆ,ˆ,)ˆmin((),ˆ,)ˆ(,)ˆmax[min((ˆ

3/4))ˆ()ˆ((5.0)ˆ(ˆ ,5.0ˆˆ

),)ˆ()ˆ((5.0ˆ ),)ˆ()ˆ(()ˆ(ˆ

) , ,4 ,4(minmod ,)ˆ(2)ˆ()ˆ(

,1,,max

,1,,min

42/11,,,

42/1

1,,1,,,

1114

2/1,1,1,

LCm

ULmcjm

MDmcjmcjmm

LCm

ULmcjm

MDmcjmcjmm

Mjcjmcjmcjm

LCm

Mj

AVm

MDm

cjmcjmAV

mcjmcjmcjmUL

m

jjjjjjMjcjmcjmcjmj

FFFFFFF

FFFFFFF

dFFFFdFF

FFFFFFF

dddddddFFFd

)ˆ,ˆ,)ˆ((median)ˆ( maxmin2/1,2/1, mmcjmcjm FFFF

zhaorui li and farhad jaberi department of mechanical engineering michigan state university east...

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