Computations of Fluid Dynamics using
the Interface Tracking Method
Zhiliang Xu
Email: zxu2@nd.edu
Department of Mathematics
University of Notre Dame
Outline
Computational Fluid Dynamics




Front Tracking Method


Compressible & incompressible flows
Governing equations
Numerical methodology

Formulation

Improving the accuracy
Conclusions and Future Plans
Compressible & Incompressible Flows
1.
2.
Approximations & Governing equations
•
Continuum assumption
•
The fundamental laws (basis): Conservation
•
Thermo-dynamical equation of state (EOS) e.g. PV=RT
Compressibility
Mach number: M = v/c
M > 0.3: compressible flow
Compressible, inviscid flow: Euler equations
Incompressible viscous flow: Incompressible Navier-Stokes equations
•
No turbulence modeling
Nonlinear Hyperbolic Conservation Laws
Euler equations:
(Gas dynamics)
U F

0
t x
U  (  , u, E )T
F  ( u, p  u 2 , ( E  p)u )T
Equation of state:
Scalar examples:
(Traffic flow)
(Burgers’ equation)
p  p (  , e)
1 2
E  e u
2
t  ( u ) x  0


u (  )  umax (1   /  max ),0     max
ut  (u 2 / 2) x  0
Scalar Conservation Laws
Conservation equation:
ut  f (u ) x  0, f   0
f: Flux function
d x2
u ( x, t )dx  f (u ( x1 , t ))  f (u ( x2 , t )),

dt x1
Linear Advection Equation:
ut  c0u x  0,
C0=const. > 0
u ( x,0)  u0 ( x)
Solution:
u( x, t )  u0 ( x  c0t )
u
u(x,t)
t
x
u(x,0)
Nonlinear Scalar Equation
ut  c(u)u x  0,
Along a characteristic curve
which has slope:
c(u )  f (u )
with
u ( x,0)  u0 ( x)
dx
 c(u )
dt
d
u ( x(t ), t )  x x u   t u  0
dt
The total derivative:
t
where
u
(x,t)
is const. along this curve.
Along this line,
u = u0(x0)
x0
x
Solve
x  x0  c(u0 ( x0 ))t for
u ( x, t )  u0 ( x0 )
x0
Breakup of Continuous Solution
Assume: c (u )  0.
t
dx
du
 c(u ),
0
dt
dt
Characteristics for nonlinear equations
t3
Characteristics
cross, the wave
“breaks”.
tB
t1
x
u
Breaking solution: successive profiles
corresponding to the times 0, t1, tB, t3
u(x,0)
t1
tB
t3
x
Weak Solutions
u
t
tB
x
Weak solutions:
  {u
t
x
 f (u ) x }dxdt  0,
R R
Jump Condition
(Rankine-Hugoniot Condition):
(ul  ur ) (t )  f (ul )  f (ur )
 (t ) : the propagation speed of the discontinui ty
(Lax) Entropy Condition & Shock
t
f (ul )
 (t )
To pick physically relevant solutions.
Ml
f (ul )   (t )  f (ur )
Mr
f (ur )
x
Shock: A discontinuity that satisfies the jump condition
and the entropy condition.
Riemann Problem (Scalar Case)
Init. value problem
with piecewise const. data:
Admit: Similarity solution:
ut  f (u ) x  0, f   0
u , x  0
u0   l
ur , x  0
u ( x, t )  u (x, t ), const   0.
u  u ( ),   x / t
Riemann Solution (Scalar Case)
Case 1: Const. State:
ul  u r
u ( )  const .
Shock speed s
t
Case 2: Shock wave:
ul  u r
ul , x  st
u( x, t )  
ur , x  st
s
f (ul )  f (ur )
ul  u r
x
0
Case 3: Rarefaction wave:
ul  ur
 u  f (u )  0
x  f (ul )t
 ul ,

u ( x, t )  v( x / t ), f (ul )t  x  f (ur )t , f (v( ))  
 u,
x  f (ur )t
 r
Rarefaction wave
t   f (u ( ))
x
0
Numerical Computation
Milestones:
•
Computing discontinuous solutions by Peter Lax (1950s) (Lax-Friedrichs scheme, LaxWendroff scheme) (SIAM Reviews Vol. 11, No. 1. 1969)
•
Godunov’s scheme, upwind schemes
•
High order schemes: TVD, MUSCL, PPM, ENO, WENO, etc
•
Interior or Free Boundary Tracking
1.
1D, 2D interface tracking by Richtmyer and Morton (1960s)
2.
Front tracking by Glimm, McBryan etc. (1980s)
3.
Others (level set, VOF, etc.)
Numerical Solution: Finite Volume Method
uin 1
tn+1
f xi1 / 2
f xi1 / 2
Space-time Boundary of the
Volume
tn
Space-time Volume
Xi : Cell Center
Xi+1/2 : Cell edge
u in
Xi
Xi-1/2
Xi+1/2 Xi+1
1D Finite Volume Scheme
  {ut  f (u ) x }dxdt  0,
uin 1  uin 
1 xi1 / 2
u :
u ( x, t n )dx

x
i

1
/
2
x
n
i
t 1 tn1
1 tn1
[  f (u ( xi 1/ 2 , t )) dt   f (u ( xi 1/ 2 , t )) dt ]
x t tn
t tn
Average of exact flux
u
n 1
i
(Cell average value).
f xi1/ 2
t ˆ
u 
( f i 1/ 2  fˆi 1/ 2 )
x
n
i
Numerical Flux
(1)
Computing Discontinuous Solutions
Conservation:
u
t ˆ
u 
( f i 1/ 2  fˆi 1/ 2 )
x
n 1
i
n
i
Single valued flux on each cell edge (…,Xi+1/2,…).
N
u
n 1
i
i
Consistency:
N
  u in
i
f   f    0
fˆi 1/ 2 (u, u)  f (u)
The Entropy Condition:
The CFL condition:
with
t
f
1
x
( 2)
Computing Discontinuous Solutions
Examples
Godunov’s Method (1959):
Semi-Discrete Method:
fˆi 1/ 2  h(u * (u i1 / 2i , u i1 / 2 ))
du
 L( u )
dt
fˆi 1/ 2 (u i , ui 1 )  f (u* (u i , ui 1 ))

du i
1 
  [ f i 1/ 2  f i 1/ 2 ]
dt
x
Spatial ENO/WENO
reconstruction
Temporal direction: TVD
Runge-Kutta
Dynamic Interface Tracking
Rayleigh-Taylor Mixing
The Level Set Method
Level Set:

 v    0
t

 ( x , t  0)  d
Interface :
(t ) :   0

Discrete Representation of Tracking
y
Volume filling rectangular
x
mesh (Eulerian Coord.)
(N-1) dimensional
Lagrangian mesh
(interface)
ij :
xi 1 / 2 y j 1 / 2
1
 ( x, y )dxdy


x
y
i

1
/
2
j

1
/
2
xy
Y
(i,j)
X
A 2D Representation
Front Tracking: Hybrid method, 2 meshes.
A 3D Interface
Time Marching & Coupling
I n 1
tn+1
tn
Xi-1
Xi I
n
Xi+1
Advancing solution in 1D
To advance the numerical solution in Front Tracking:
(1) Explicit procedure for interface propagation + (2) Updating states (grid cell
center)
Two way coupling:
1.
Interface dynamics to ambient region (interior).
2.
Non-interface solution variation to interface dynamics.
Separation of Interface Propagation
y
x
Normal
Tangent
• Operator Splitting to separate normal and tangential propagation
• Normal propagation to move interface position & coupling
• Tangent propagation to include information flowing tangentially
along the curve.
Normal Propagation of Interface Point
Move the point position and couple the interior wave solution to interface dynamics.
Step 1:
Updated left and
right states of the
point
Step 2:
(Material interface)
t  t
t
Contact
Sl0 Sr0 New position
t  t
Sl
dx
uc
dt
n
S2
S1
S 0 S 0
S
r
0
S1
S2
Left and right states of
the point
t
dx
u
dt
S 1 S f S0 S0
S2
Sr
dx
 u c
dt
n
Sb
S1
S
r
Riemann solution
Method of characteristics
(Coupling)
S2
Advancing Eulerian Grid Solution
Ghost cell method: Coupling interface dynamics to interior
tn+1
: Fluid 1
Extrapolate
: Fluid 2
tn
Xi-1
Xi
: Interface
Xi+1
Cell edge
Ui
n 1

t  L
 Ui 
( Fi 1/ 2  Fi 1/ 2 )
x
n
L
Fi 1/ 2  Fˆ (U in ,U ileft
1 )
U

n 1
i 1
U
n
i 1
R
t 

( Fi 3 / 2  Fi 1/ 2 )
x
R
Fi 1/ 2  Fˆ (U iright ,U in1 )
Conservative Front Tracking - Formulation
Discontinuity
t
Space-time volume
V
Xi

UL UR
V
(
U
   F (U ))dV  0
t
Space-time interface
Xi+1/2
Xi+1
x
A moving discontinuity surface bounds a time-dependent volume V.
Fi L1/ 2  F (U L )   (t ) U L
Fi R1/ 2  F (U R )   (t )  U R
Redefine the flux through the
discontinuity by R-H condition.
2D Space-Time Volumes
Top face
t
y
x
Space-time
hexahedron
Improved Accuracy
Theorem: The conservative tracking method improves
accuracy by at least one order.
1D Numerical Validation
Init. Condition: Shock-Rarefaction
2D Axisymmetric Richtmyer-Meshkov Instability
Light gas
Heavy gas
Material
interface
Shock wave
Init. Condition
(Density Plot)
Conservative
tracking simulation
Non-conservative
tracking simulation
2D Axisymmetric Richtmyer-Meshkov Instability
Conservative Tracking, 100*200 grid
Non-Conservative Tracking, 100*200 grid
Non-Conservative Tracking, 200*400 grid
Amplitude (a): the
height of the interface
perturbation.
a  ( hsp  hbb )
h_sp and h_bb are distances
from origin to the tips of the
spike and the bubble
respectively.
Computations of Incompressible Flows
  

ut  u  u  p  u  f

  (u )  0
What is the role of the pressure?
Hodge Decomposition  Projection Methods
* 
u  u  

where   (u )  0
Projection Method
1.
Advancing the momentum equation in time to determine an
intermediate velocity which is not required to be divergence-free.
2.
Project the intermediate velocity field onto the space of divergencefree field. The gradient part is used to update the pressure.
The Numerical Method

 


u
d
x

L
(
u
,  )   pdS

t ij
 ij

 


L(u ,  )    u  u  dS   D  dS   k s ndx   fdx
 ij
Advancing the front:
Advancing materiel properties:
 ij


The Numerical Method
Compute the intermediate velocity:
Projection:
p
Surface tension:
n 1/ 2
p
n 1/ 2
 t

B
 k s ndx 
A

 knds   (t
A B
B
 tA )
The Blood Flow Modelling
  

 (ut  u  u )  p  u  f

  (u )  0

 c  u  c   D c
t
Conclusions & Future Plans
The
front tracking method to describe the interface.
On
the tracking method:
To achieve uniform high order accuracy.
On
the application:
To develop a blood flow model in the multiscale context