High-order gas evolution model for computational fluid dynamics

Collaborators: Q.B. Li, J. Luo, J. Li, L. Xuan,…

Kun XuHong Kong University of Science and Technology

Fluid flow is commonly studied in one of three ways:– Experimental fluid dynamics.– Theoretical fluid dynamics.– Computational fluid dynamics (CFD).

TheoryExperiment

Scientific Computing

Contents

• The modeling in gas-kinetic scheme (GKS)• The Foundation of Modern CFD • High-order schemes • Remarks on high-order CFD methods• Conclusion

Mean Free Path

Collision

The way of gas molecules passing through the cell interface depends on the cell resolution and particle mean free path

Computation: a description of flow motion in a discretized space and time

5

ContinuumAir at atmospheric condition: 2.5x1019 molecules/cm3, Mean free path : 5x10-8m, Collision frequency : 109 /s

Gradient transport mechanism

Navier-Stokes-Fourier equations (NSF)

RarefactionTypical length scale: LKnudsen number: Kn=/L

High altitude, Vacuum ( ) , MEMS (L ) Kn

Martin H.C. Knudsen (1871-1949)

Danish physicist

Gas properties

6

Fundamental governing equation in discretized space:

12/1

2/12/1

1

2/1),(1)]()([11

n

n

j

jj

n

n j

t

t

x

xxt

t xnj

nj dxdtffQ

xdttuftuf

xff

Take conservative moments to the above equation:

Physical modeling of gas flow in a limited resolution spacef : gas distribution function，W : conservative macroscopic variables

1

)(12/12/1

1n

n

t

t jjnj

nj dtdduffu

xWW

For the update of conservative flow variables, we only needto know the fluxes across a cell interface！PDE-based modeling： use PDE’s local solution to model the physical process of gas molecules passing through the cell interface

7

The physical modeling of particles distribution function at a cell interface

.trajectory particle the is )'(' where

)('),,,','(1),,,,(

2/1

02/10

//)'(2/1

ttuxx

utxfedtevutxgvutxf

j

t

jttt

j

0f

gnt

1nt

)'('2/1 ttuxx j

2/1jx

8

0f

0grg

lg

2/1jx 1jxjx

g

: constructed according toChapman-Enskog expansion

Modeling for continuum flow:

Smooth transition fromparticle free transport to hydrodynamic evolution

),))(H1()(H(

)))(H1)()(()(H))(((

))(H1()(H)()1((

)1/()1(

),,,,(

/

/

0//

0/

0/

2/1

rlt

rrrlllt

rltt

tt

j

gugue

guAtuaguAtuae

uguauatee

gAetge

vutxf

Discontinuous(kinetic scale, free transport) t

t

t

jttt

j utxfedtevutxgvutxf0

2/10//)'(

2/1 )('),,,','(1),,,,(

Hydrodynamicsscale

10

• Numerical fluxes:

• Update of flow variables:

• Prandtl number fix by modifying the heat flux in the above equation

.),,,,(

)(

1

2/1

22221

2/1

dvutxf

vuvu

u

FFFF

j

jE

V

U

.))()(( 2/12/1011 dttFtFww jj

t

xnj

nj

11

Gas-kinetic Scheme ( ) /,/ xt

Upwind Scheme Central-difference

Kinetic scale Hydrodynamic scale

12

M. Ilgaz, I.H. Tuncer, 2009

13

14

15

x/L

Cp

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Present (upside)Present (lower side)Exp (upside)Exp (lower side)

X

Y

Z

Cp10.80.60.40.20

-0.2-0.4-0.6-0.8-1-1.2-1.4

x/L

-Cp

0 0.2 0.4 0.6 0.8 1

-0.8

-0.4

0

0.4

0.8

1.2

S-ASSTMenter TransitionExpExp

Section 3 z/b=0.65

High Mach number flow passing through a double ellipse

M6 airfoil

M=10, Re=10^6, Tin=79K, Tw=294.44K, mesh 15x81x19

Hollow cylinder flare: nitrogen

Mesh61x105x17

temperature

pressure

21

The Foundation of Modern CFD

22

Introduce flow physics into numerical schemes(FDS, FVS, AUSM, ~RPs)

Spatial Limiters (Boris, Book, van Leer,…70-80s)

Modern CFD (Godunov-type methods)Governing equations: Euler, NS, …

（ space limiter）

23

A black cloud hanging over CFD clear sky (1990- now)

Carbuncle Phenomena

Roe AUSM+

24

M=10GKS GRP

25

Godunov’s description of numerical shock wave

Is this physical modeling valid ?

26

Gas kinetic schemeParticle free transport

Physical process from a discontinuity

collision

NS

Euler

Godunov method

Euler

NS

?

(infinite number of collisions)

Riemann solver

High-order schemes (order =>3)

The foundation of most high-order schemes: 1st-order dynamic model: Riemann solver

Reconstruction + Evolution

inviscid

viscous

29

High-order Kinetic Scheme (HBGK-NS)

BGK-NS (2001) HBGK (2009)

High-order gas-kinetic scheme (HGKS)

Gauss-points: Riemann solversfor others

High-order Gas-kinetic scheme:one step integration along the cell interface.

Comparison of gas evolution model: Godunov vs. Gas-Kinetic Scheme

(a): gas-kinetic evolution (b): Riemann solver evolution

Space & time, inviscid & viscous,direction & direction, kinetic &Hydrodynamic, fully coupled !

32

++

+

+

+

+

+

++ + + +

y*

U*

0 2 4 6 8

0

0.2

0.4

0.6

0.8

1

x/L= 0.0247x/L= 0.2625x/L= 0.6239Blasius

+

+ + ++

+

+

+

+

++ + +

y*

V*

0 2 4 6 8

0

0.2

0.4

0.6

0.8

1

x/L= 0.0247x/L= 0.2625x/L= 0.6239Blasius

+

Laminar Boundary Layer 510Re

Viscous shock tube

5th-WENO6th-order viscous

Sjogreen& Yee’s 6th-order WAV66 scheme

500x250 mesh points

Reference solution4000x2000 mesh points

5th-WENO-reconstruction+Gas-Kinetic Evolution

500x250 mesh points

1000x500

Sjogreen& Yee’s 6th-order WAV66 scheme

1000x500Gas Kinetic Scheme

1400x700Gas-kinetic Scheme

Osmp7 (4000x2000）

Remarks on high-order CFD methods

1/2

1/2ˆ( ) 0

i i i

i

Riem iV V V

U f Udx dx f f dxt x t x

Mathematical manipulation physical reality ?

There is no any physical evolution law about the time evolution of derivatives in a discontinuous region !

（ weak solution)

Even in the smooth region, in the update of “slope or high-order derivatives” through weak solution, the Riemann solver (1st-order dynamics) does NOT provide appropriate dynamics.

Example:

Riemann solver only provides u, not at a cell interface

Huynh, AIAA paper 2007-4079 Unified many high-order schemes DG, SD, SV, LCP, …, under flux reconstruction framework

( ) ( )i iF x F x

0)( ,,

dxxdF

dtdu jiiji

Riemann Flux

Interior Flux ( )iF x( )iF x

Z.J. Wang

𝑥 𝑗−1 /2 𝑥 𝑗+1/2

Generalized solutions with piecewise discontinuous initial data

Initial condition at t=0

Solution at t=

Reconstructed new initial conditionfrom nodal values

Update flow variables at nodal points ( , )at next time level,And calculate flux

W(x)=

STRONG Solution from Three Piecewise Initial Data

Control Volume

tx ,

PDE’s local evolution solution (strong solution) is used to

Model the gas flow passing through the cell interface in a discretized space.

PDE-based Modeling

44

Different scale physical modeling

Boltzmann Eqs.

Navier-Stokes

Euler

quantum

Newton

Flow description depends on the scale of the discretized space

Conclusion • GKS is basically a gas evolution modeling in a discretized space.

This modeling covers the physics from the kinetic scale to the hydrodynamic scale.

• In GKS, the effects of inviscid & viscous, space & time, different by directions, and kinetic & hydrodynamic scales, are fully coupled.

• Due to the limited cell size, the kinetic scale physical effect is needed to represent numerical shock structure, especially in the high Mach number case. Inside the numerical shock layer, there is no enough particle collisions to generate the so-called “Riemann solution” with distinctive waves. The Riemann solution as a foundation of modern CFD is questionable.

• In the discontinuous case, there is no such a physical law related to the time evolution of high-order derivatives. The foundation of the DG method is not solid. It may become “a game of limiters” to modify the updated high-order derivatives in high speed flow computation.