Computations of Fluid Dynamics using the Interface Tracking Method

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Computations of Fluid Dynamics using the
Interface Tracking Method

• Zhiliang Xu
• Email zxu2_at_nd.edu
• Department of Mathematics
• University of Notre Dame

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Outline

• Computational Fluid Dynamics
• Compressible incompressible flows
• Governing equations
• Numerical methodology
• Front Tracking Method
• Formulation
• Improving the accuracy
• Conclusions and Future Plans

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Compressible Incompressible Flows

• Approximations Governing equations
• Continuum assumption
• The fundamental laws (basis) Conservation
• Thermo-dynamical equation of state (EOS) e.g.
  PVRT
• Compressibility
• Mach number M v/c
• M gt 0.3 compressible flow
• Compressible, inviscid flow Euler equations
• Incompressible viscous flow Incompressible
  Navier-Stokes equations
• No turbulence modeling

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Nonlinear Hyperbolic Conservation Laws
Euler equations (Gas dynamics)
Equation of state
Scalar examples (Traffic flow) (Burgers
equation)
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Scalar Conservation Laws
f Flux function
Conservation equation
C0const. gt 0
Linear Advection Equation
Solution
u(x,t)
u(x,0)
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Nonlinear Scalar Equation
with
where
Along a characteristic curve which has slope
The total derivative
is const. along this curve.
(x,t)
Along this line, u u0(x0)
Solve
for
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Breakup of Continuous Solution
Assume
Characteristics for nonlinear equations
Characteristics cross, the wave breaks.
Breaking solution successive profiles
corresponding to the times 0, t1, tB, t3
u(x,0)
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Weak Solutions
Weak solutions
Jump Condition
(Rankine-Hugoniot Condition)
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(Lax) Entropy Condition Shock
To pick physically relevant solutions.
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
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Riemann Solution (Scalar Case)
Case 1 Const. State
Shock speed s
Case 2 Shock wave
Rarefaction wave
Case 3 Rarefaction wave
t
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Numerical Computation

• Milestones
• Computing discontinuous solutions by Peter Lax
  (1950s) (Lax-Friedrichs scheme, Lax-Wendroff
  scheme) (SIAM Reviews Vol. 11, No. 1. 1969)
• Godunovs scheme, upwind schemes
• High order schemes TVD, MUSCL, PPM, ENO, WENO,
  etc
• Interior or Free Boundary Tracking
• 1D, 2D interface tracking by Richtmyer and Morton
  (1960s)
• Front tracking by Glimm, McBryan etc. (1980s)
• Others (level set, VOF, etc.)

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Numerical Solution Finite Volume Method
Space-time Volume
Xi Cell Center
Xi1/2 Cell edge
Space-time Boundary of the Volume
1D Finite Volume Scheme
(Cell average value).
Average of exact flux
Numerical Flux
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Computing Discontinuous Solutions
Conservation
Single valued flux on each cell edge
(,Xi1/2,).
with
Consistency
The Entropy Condition
The CFL condition
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Computing Discontinuous Solutions
Examples
Godunovs Method (1959)
Semi-Discrete Method
Spatial ENO/WENO reconstruction
Temporal direction TVD Runge-Kutta
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Dynamic Interface Tracking
Rayleigh-Taylor Mixing
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The Level Set Method
Level Set
Interface
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Discrete Representation of Tracking
Volume filling rectangular mesh (Eulerian Coord.)
(N-1) dimensional Lagrangian mesh (interface)
(i,j)
A 3D Interface
A 2D Representation
Front Tracking Hybrid method, 2 meshes.
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Time Marching Coupling
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
• Interface dynamics to ambient region (interior).
• Non-interface solution variation to interface
  dynamics.

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Separation of Interface Propagation

• 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.

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Normal Propagation of Interface Point
Move the point position and couple the interior
wave solution to interface dynamics.
Step 1
Step 2
Updated left and right states of the point
Left and right states of the point
Riemann solution
Method of characteristics (Coupling)
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Advancing Eulerian Grid Solution

• Ghost cell method Coupling interface dynamics to
  interior

Fluid 1
Fluid 2
Interface
Cell edge
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Conservative Front Tracking - Formulation
Space-time volume
V
Space-time interface
Xi
Xi1
Xi1/2
A moving discontinuity surface bounds a
time-dependent volume V.
Redefine the flux through the discontinuity by
R-H condition.
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2D Space-Time Volumes
Top face
Space-time hexahedron
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Improved Accuracy
Theorem The conservative tracking method
improves accuracy by at least one order.
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1D Numerical Validation
Init. Condition Shock-Rarefaction
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2D Axisymmetric Richtmyer-Meshkov Instability
Light gas
Heavy gas
Material interface
Shock wave
Init. Condition (Density Plot)
Conservative tracking simulation
Non-conservative tracking simulation
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2D Axisymmetric Richtmyer-Meshkov Instability
Conservative Tracking, 100200 grid
Non-Conservative Tracking, 100200
grid Non-Conservative Tracking, 200400 grid
Amplitude (a) the height of the interface
perturbation.
h_sp and h_bb are distances from origin to the
tips of the spike and the bubble respectively.
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Computations of Incompressible Flows
What is the role of the pressure?
Hodge Decomposition ? Projection Methods
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Projection Method

• Advancing the momentum equation in time to
  determine an intermediate velocity which is not
  required to be divergence-free.
• 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
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The Numerical Method
Compute the intermediate velocity
Projection
Surface tension
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The Blood Flow Modelling
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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