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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 Compressible & incompressible flows Governing equations Numerical methodology
Front Tracking Method Formulation Improving the accuracy
Conclusions and Future Plans
Compressible & Incompressible Flows
1. Approximations & Governing equations
• Continuum assumption
• The fundamental laws (basis): Conservation
• Thermo-dynamical equation of state (EOS) e.g. PV=RT
2. 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
maxmaxmax 0),/1()(
0)(
uu
u xt
T
T
upEupuF
EuU
x
F
t
U
))(,,(
),,(
0
2
Euler equations:
(Gas dynamics)
Equation of state: ),( epp
2
2
1ueE
Scalar examples:
(Traffic flow)
(Burgers’ equation) 0)2/( 2 xt uu
Scalar Conservation Laws
,00 xt ucu C0=const. > 0Linear Advection Equation:
)(),( 00 tcxutxu Solution:
u(x,t)
u(x,0)
0,0)( fufu xt
u
x
t
)),,(()),((),( 21
2
1
txuftxufdxtxudt
d x
x
f: Flux function
)()0,( 0 xuxu
Conservation equation:
Nonlinear Scalar Equation
)()( ufuc ,0)( xt uucu
)(ucdt
dxAlong a characteristic curve
which has slope:
The total derivative:
txucxx ))(( 000
x0
x
t
Along this line,
u = u0(x0)
)()0,( 0 xuxu with
)(),( 00 xutxu
Solve for 0x
is const. along this curve.u(x,t)
where
0)),(( uuxttxudt
dtx
Breakup of Continuous Solution
Characteristics for nonlinear equations
x
t
1t
Bt
3t
x
u
u(x,0)1t
Bt 3t
Characteristics cross, the wave “breaks”.
0),( dt
duuc
dt
dx
Breaking solution: successive profiles corresponding to the times 0, t1, tB, t3
.0)( ucAssume:
Weak Solutions
R R
xt dxdtufu ,0})({Weak solutions:
Jump Condition (Rankine-Hugoniot Condition):
tyiscontinuid of the dation speethe propagt
ufuftuu rlrl
:)(
)()()()(
)(t
t x
rM
lM
ru
lu
Bt x
u
t x
t
x
(Lax) Entropy Condition & Shock
)(t
t x
rM
lM
)(luf
)(ruf
To pick physically relevant solutions.
)()()( rl uftuf
Shock: A discontinuity that satisfies the jump condition and the entropy condition.
Riemann Problem (Scalar Case)
.0 ),,(),( consttxutxu
0,0)( fufu xtInit. value problem
with piecewise const. data:
0 ,
0 ,0 xu
xuu
r
l
txuu /),( Admit: Similarity solution:
)(tt
x
rM
lM
)( luf )( ruf
Riemann Solution (Scalar Case)
))((,
)(
)()(
)(
,
),/(
,
),( vf
tufx
tufxtuf
tufx
u
txv
u
txu
r
rl
l
r
l
stx
stx
u
utxu
r
l
,
,),(Case 2: Shock wave:
t
x
0
Shock speed s
Case 3: Rarefaction wave:t
x0
Case 1: Const. State:
rl
rl
uu
ufufs
)()(
rl uu
rl uu
.)( constu
))(( uf 0)( ufu
rl uu
Rarefaction wave
Numerical Computation
Milestones:
• Computing discontinuous solutions by Peter Lax (1950s) (Lax-Friedrichs scheme, Lax-Wendroff 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
2/1
2/1
),(1
:i
i
x
x nn
i dxtxux
u
])),((1
)),((1
[11
2/12/11
n
n
n
n
t
t i
t
t in
in
i dttxuft
dttxuftx
tuu
1D Finite Volume Scheme
)1(
Average of exact flux
1niu
niu
Space-time Volume
Space-time Boundary of the Volume
(Cell average value).,0})({ dxdtufu xt
2/1ixf
2/1ixf
Xi+1/2 : Cell edge
Xi : Cell Center
tnXi-1/2 Xi+1/2
tn+1
Xi Xi+1
2/1ixf
Numerical Flux
)ˆˆ(uu 2/12/11
iini
ni ff
x
t
Computing Discontinuous Solutions
)ˆˆ(uu 2/12/11
iini
ni ff
x
tConservation:
Single valued flux on each cell edge (…,Xi+1/2,…).
)2(
N
i
ni
N
i
ni uu 1
)(),(ˆ2/1 ufuufi Consistency:
The CFL condition: 1x
tf
The Entropy Condition:
with 0 ff
Computing Discontinuous Solutions
][1u
2/12/1
iii ff
xdt
d
)),((ˆ2/1
2/1*
2/1
iii uuuhfi
Godunov’s Method (1959): ))u,u(()u,u(ˆ1
*12/1 iiiii uff
Semi-Discrete Method:
Spatial ENO/WENO reconstruction
Temporal direction: TVD Runge-Kutta
)u(u
Ldt
d
Examples
Discrete Representation of Tracking
2/1
2/1
2/1
2/1
),(1
:
i
i
j
j
x
x
y
y
ij
dxdyyxyx
Volume filling rectangularmesh (Eulerian Coord.)
(N-1) dimensional Lagrangian mesh (interface)
Front Tracking: Hybrid method, 2 meshes.Hybrid method, 2 meshes.
A 3D InterfaceA 2D Representation
Y
X
(i,j)
x
y
Time Marching & Coupling
1nI
nItn Xi-1 Xi Xi+1
tn+1
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.
Advancing solution in 1D
Separation of Interface Propagation
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.
x
y
Normal Propagation of Interface Point
Move the point position and couple the interior wave solution to interface dynamics.
Riemann solutionMethod of characteristics
(Coupling)
Step 1: Step 2:
SlSr
Left and right states of the point
Updated left and right states of the
point
0S 1S 2S1S2S
n
tt
t
Contact
0S
(Material interface)
New positionSl0 Sr0
udt
dx
0S 1S 2S1S2S bSn
tt
t
cudt
dx
cudt
dx
0SfS
Sl Sr
Advancing Eulerian Grid Solution
Ghost cell method: Coupling interface dynamics to interior
: Fluid 1
: Fluid 2
: Interface
tn+1
tn
Xi Xi+1Xi-1
Extrapolate
Cell edge
)( 2/12/11
iL
in
in
i FFx
tUU
),(ˆ12/1
lefti
ni
Li UUFF
)( 2/12/311
1R
iin
in
i FFx
tUU
),(ˆ12/1
nrighti
Ri i
UUFF
Conservative Front Tracking - Formulation
0))((
dVUFt
UV
x
t
A moving discontinuity surface bounds a time-dependent volume V.
V
Discontinuity
Space-time interface
Redefine the flux through the discontinuity by R-H condition.
LU RU
LLL
i UtUFF )()(2/1 RRR
i UtUFF )()(2/1
Xi Xi+1
Space-time volume
Xi+1/2
Improved Accuracy
Theorem: The conservative tracking method improves accuracy by at least one order.
2D Axisymmetric Richtmyer-Meshkov Instability
Init. Condition (Density Plot)
Conservative tracking simulation
Non-conservative tracking simulation
Heavy gas
Light gas
Shock wave
Material interface
2D Axisymmetric Richtmyer-Meshkov Instability
)( bbsp hha
h_sp and h_bb are distances from origin to the tips of the
spike and the bubble respectively.
Amplitude (a): the height of the interface
perturbation.
Conservative Tracking, 100*200 grid
Non-Conservative Tracking, 100*200 grid
Non-Conservative Tracking, 200*400 grid
Computations of Incompressible Flows
0)(
u
fuuuu pt
What is the role of the pressure?
Hodge Decomposition Projection Methods
0)(where
*
u
uu
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 divergence-free field. The gradient part is used to update the pressure.
The Numerical Method
Advancing the front:
Advancing materiel properties:
xfdxndkdSDdSuuuL
pdSuLxdut
s
ij ij
ij ij
),(
),(
The Numerical Method
Projection:
Compute the intermediate velocity:
tpp nn 2/12/1
Surface tension:
A
B)( AB
BA
s ttkndsxndk
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