High-order gas evolution model for
computational fluid dynamics
Kun Xu
Hong Kong University of Science and Technology
Collaborators: Q.B. Li, J. Luo, J. Li, L. Xuan,…
Fluid flow is commonly studied in one of three ways:
– Experimental fluid dynamics.
– Theoretical fluid dynamics.
– Computational fluid dynamics (CFD).
Experiment
Theory
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
Computation: a description of flow motion in a
discretized space and time
Collision
The way of gas molecules passing through the cell interface
depends on the cell resolution and particle mean free path
Gas properties
Continuum
Air at atmospheric condition: 2.5x1019 molecules/cm3,
Mean free path : 5x10-8m, Collision frequency  : 109 /s
Gradient transport mechanism
Navier-Stokes-Fourier equations (NSF)
Martin H.C. Knudsen
(1871-1949)
Danish physicist
Rarefaction
Typical length scale: L
Knudsen number: Kn=/L
High altitude, Vacuum ( ) ,
MEMS (L )
Kn 
5
Physical modeling of gas flow in a limited resolution space
f : gas distribution function，
W : conservative macroscopic variables
Fundamental governing equation in discretized space:
n 1
j
f
 f
n
j
1

x
t n 1
t
n
1
[ufx j  1 / 2(t )  uf x j  1 / 2(t )]dt 
x
t n 1
t
n
xj 1 / 2
x
Q(f ,f )dxdt
j 1 / 2
Take conservative moments to the above equation:
W
n 1
j
1 t n1
W 
u  ( f j 1/ 2  f j 1/ 2 )du ddt
n


t
x
n
j
For the update of conservative flow variables, we only need
to 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 6
The physical modeling of particles distribution function at a cell
interface
f(x j 1 / 2 ,t ,u ,v , ) 
where
1

t
(t t ') / 
t / 
g
(
x
'
,
t
'
,
u
,
v
,

)
e
dt
'

e
f0(x j 1 / 2  ut )

0
x ' x j 1 / 2  u(t  t ')
is the particle trajectory.
t n 1
x j 1/ 2  x'u(t  t ' )
g
f0
tn
x j 1/ 2
7
Modeling for continuum flow:
f0
: constructed according to
Chapman-Enskog expansion
gl
g0
gr
g
xj
x j 1 / 2
x j 1
8
Smooth transition from
particle free transport to hydrodynamic evolution
f(x j 1 / 2 ,t ,u ,v , ) 
1

t
(t t ') / 
t / 
g
(
x
'
,
t
'
,
u
,
v
,

)
e
dt
'

e
f0(x j 1 / 2  ut )

0
f ( x j 1/ 2 , t , u, v,  )
 (1  e t / ) g 0   (t /   1  e t / ) Ag 0
 ( (1  e
t / 
)  te
t / 
l
t  
r
)(a H(u )  a (1  H(u ))ug0
Hydrodynamics
scale
 e t / ((a l u (t   )  Al )H(u ) g l  (a r u (t   )  Ar )(1  H(u ))g r )
 e t / (H(u ) g l  (1  H(u ))g r ),
Discontinuous
(kinetic scale, free transport)
t  
:
• Numerical fluxes
 F 


 FU 
F 
 V 
F 
 E 
j 1 / 2



  u

 1 (u 2
2
1


u

f ( x j 1 / 2 , t , u, v,  )d.

v

2
2 
 v   )
• Update of flow variables:
w
n 1
j
w 
n
j
t
1
x 0

( F j 1 / 2 (t )  F j 1 / 2 (t ))dt.
• Prandtl number fix by modifying the heat flux in the above equation
10
Gas-kinetic Scheme ( t /  , x / )
Upwind Scheme
Kinetic scale
Central-difference
Hydrodynamic scale
11
M. Ilgaz, I.H. Tuncer, 2009
12
13
14
15
2
Present (upside)
Present (lower side)
Exp (upside)
Exp (lower side)
Cp
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
x/L
High Mach number flow passing through a double ellipse
Y
Section 3 z/b=0.65
1.2
X
Z
0.8
Cp
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-Cp
0.4
0
S-A
SST
Menter Transition
Exp
Exp
-0.4
-0.8
0
0.2
0.4
0.6
x/L
M6 airfoil
0.8
1
M=10, Re=10^6, Tin=79K, Tw=294.44K, mesh 15x81x19
Hollow cylinder flare: nitrogen
Mesh
61x105x17
temperature
pressure
The Foundation of Modern CFD
21
Modern CFD
(Godunov-type methods)
Governing equations: Euler, NS, …
Introduce flow physics into numerical schemes
(FDS, FVS, AUSM, ~RPs)
Spatial Limiters
(Boris, Book, van Leer,…70-80s)
（space limiter）
22
A black cloud hanging over CFD clear sky (1990- now)
Carbuncle Phenomena
Roe
AUSM+
23
M=10
GKS
GRP
24
Godunov’s description of numerical shock wave
Is this physical modeling valid ?
25
Physical process from a discontinuity
Gas kinetic scheme
Particle free transport
Godunov method
collision
?
NS
NS
Riemann solver
Euler
Euler
(infinite number of collisions)
26
High-order schemes (order =>3)
Reconstruction + Evolution
The foundation of most high-order schemes:
1st-order dynamic model: Riemann solver
inviscid
viscous
High-order Kinetic Scheme (HBGK-NS)
BGK-NS (2001)
HBGK (2009)
29
High-order gas-kinetic scheme (HGKS)
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 !
High-order Gas-kinetic scheme:
one step integration along the
cell interface.
Gauss-points: Riemann solvers
for others
Re  105
Laminar Boundary Layer
1
1
+
+
+
+
0.8
0.8
+
0.6
U*
+
+
+
x/L= 0.0247
x/L= 0.2625
x/L= 0.6239
Blasius
0.4
+
+
0.2
+
V*
+
0.4
0.2
+
+
0
+
x/L= 0.0247
x/L= 0.2625
x/L= 0.6239
Blasius
+
+
0
+
+
+
0.6
+
+
+
0 ++
2
4
y*
6
8
0
+
2
4
y*
6
8
32
Viscous shock tube
500x250 mesh points
5th-WENO
6th-order viscous
Reference solution
4000x2000 mesh points
Sjogreen& Yee’s 6th-order
WAV66 scheme
500x250 mesh points
5th-WENO-reconstruction
+Gas-Kinetic Evolution
1000x500
Sjogreen& Yee’s 6th-order
WAV66 scheme
1000x500
Gas Kinetic Scheme
1400x700
Gas-kinetic Scheme
Osmp7 (4000x2000）
Remarks on high-order CFD methods
Mathematical manipulation
（weak solution)
?
physical reality
i 1/2
U

 U f 
ˆ
V   t  x  dx  V  t dx  ( f Riem ) i1/2  V f x dx  0
i
i
i
There is no any physical evolution law about the
time evolution of derivatives in a discontinuous
region !
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
Fi ( x )  Fi ( x )
Riemann Flux
Fi ( x )
Interior Flux
dui , j
dt

dFi ( xi , j )
dx
0
Fi ( x )
Z.J. Wang
STRONG Solution from Three Piecewise Initial Data
Update flow variables
at nodal points ( , )
at next time level,
And
calculate flux
Solution at t=∆𝑡
Reconstructed new initial condition
from nodal values
Initial condition at t=0
𝑥𝑗−1/2
𝑥𝑗+1/2
Generalized solutions with piecewise discontinuous initial data
W(x)=
𝑊1 𝑥 ,
𝑥 < 𝑥𝑗−1/2,
𝑊2 𝑥 , 𝑥𝑗−1/2 < 𝑥 < 𝑥𝑗+1/2 ,
𝑊3 𝑥 ,
𝑥 > 𝑥𝑗+1/2
𝑡=0
PDE’s local evolution solution (strong solution) is used to
Model the gas flow passing through the cell interface in a
discretized space.
Control Volume
x, t
PDE-based Modeling
Different scale physical modeling
quantum
Boltzmann Eqs.
Newton
Navier-Stokes
Euler
Flow description depends on the
scale of the discretized space
44
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 highorder 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.