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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
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Fluid flow is commonly studied in one of three ways:– Experimental fluid dynamics.– Theoretical fluid dynamics.– Computational fluid dynamics (CFD).
TheoryExperiment
Scientific Computing
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Contents
• The modeling in gas-kinetic scheme (GKS)• The Foundation of Modern CFD • High-order schemes • Remarks on high-order CFD methods• Conclusion
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Mean Free P
ath
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
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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
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Fundamental governing equation in discretized space:
12/1
2/12/1
1
2/1),(
1)]()([
11n
n
j
jj
n
n j
t
t
x
xx
t
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
)(1
2/12/11
n
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
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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
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8
0f
0grg
lg
2/1jx 1jxjx
g
: constructed according toChapman-Enskog expansion
Modeling for continuum flow:
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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
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• 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
vu
v
uu
F
F
F
F
j
jE
V
U
.))()(( 2/12/10
11 dttFtFww jj
t
xnj
nj
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Gas-kinetic Scheme ( ) /,/ xt
Upwind Scheme Central-difference
Kinetic scale Hydrodynamic scale
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12
M. Ilgaz, I.H. Tuncer, 2009
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13
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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
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M=10, Re=10^6, Tin=79K, Tw=294.44K, mesh 15x81x19
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Hollow cylinder flare: nitrogen
Mesh61x105x17
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temperature
pressure
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The Foundation of Modern CFD
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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)
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A black cloud hanging over CFD clear sky (1990- now)
Carbuncle Phenomena
Roe AUSM+
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M=10GKS GRP
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Godunov’s description of numerical shock wave
Is this physical modeling valid ?
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Gas kinetic scheme
Particle free transport
Physical process from a discontinuity
collision
NS
Euler
Godunov method
Euler
NS
?
(infinite number of collisions)
Riemann solver
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High-order schemes (order =>3)
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The foundation of most high-order schemes: 1st-order dynamic model: Riemann solver
Reconstruction + Evolution
inviscid
viscous
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High-order Kinetic Scheme (HBGK-NS)
BGK-NS (2001) HBGK (2009)
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High-order gas-kinetic scheme (HGKS)
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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 !
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+
+
+
+
+
+
+
+
++ + +
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
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Viscous shock tube
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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
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1000x500
Sjogreen& Yee’s 6th-order WAV66 scheme
1000x500Gas Kinetic Scheme
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1400x700Gas-kinetic Scheme
Osmp7 (4000x2000)
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Remarks on high-order CFD methods
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1/2
1/2
ˆ( ) 0i i i
i
Riemi
V V V
U f Udx dx f f dx
t 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)
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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
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Huynh, AIAA paper 2007-4079 Unified many high-order schemes DG, SD, SV, LCP, …, under flux reconstruction framework
( ) ( )i iF x F x
0)( ,,
dx
xdF
dt
du jiiji
Riemann Flux
Interior Flux ( )iF x( )iF x
Z.J. Wang
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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
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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
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Different scale physical modeling
Boltzmann Eqs.
Navier-Stokes
Euler
quantum
Newton
Flow description depends on the scale of the discretized space
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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.
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• 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.