1. Computations of Fluid Dynamics using the Interface Tracking Method Zhiliang Xu Email: zxu2@nd.eduzxu2@nd.edu Department of Mathematics University of Notre Dame

2. Outline  Computational Fluid Dynamics  Compressible & incompressible flows  Governing equations  Numerical methodology  Front Tracking Method  Formulation  Improving the accuracy  Conclusions and Future Plans

3. 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 numberMach number: M = v/c M > 0.3: compressible flow Compressible, inviscid flow: Euler equations Incompressible viscous flow: Incompressible Navier-Stokes equations No turbulence modeling

4. Nonlinear Hyperbolic Conservation Laws Euler equations: (Gas dynamics) Equation of state: Scalar examples: (Traffic flow) (Burgers’ equation)

5. Scalar Conservation Laws C 0 =const. > 0Linear Advection Equation: Solution: u(x,t) u(x,0) u x t f : Flux function Conservation equation:

6. Nonlinear Scalar Equation Along a characteristic curve which has slope: The total derivative: x0x0 x t Along this line, u = u 0 (x 0 ) with Solvefor is const. along this curve.(x,t) where

7. Breakup of Continuous Solution Characteristics for nonlinear equations x t x u u(x,0) Characteristics cross, the wave “breaks”. Breaking solution: successive profiles corresponding to the times 0, t 1, t B, t 3 Assume:

8. Weak Solutions Weak solutions: Jump Condition (Rankine-Hugoniot Condition): x u

9. (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) I nit. value problem with piecewise const. data: Admit: Similarity solution:

10. Riemann Solution (Scalar Case) Case 2: Shock wave: t x 0 Shock speed s Case 3: Rarefaction wave: t x 0 Case 1: Const. State: Rarefaction wave

11. 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.)

12. Numerical Solution: Finite Volume Method 1D Finite Volume Scheme Average of exact flux Space-time Volume Space-time Boundary of the Volume (Cell average value). X i+1/2 : Cell edge X i : Cell Center tntn X i-1/2 X i+1/2 t n+1 XiXi X i+1 Numerical Flux

13. Computing Discontinuous Solutions Conservation: Single valued flux on each cell edge (…,X i+1/2,…). Consistency: The CFL condition: The Entropy Condition: with

14. Computing Discontinuous Solutions Godunov’s Method (1959): Semi-Discrete Method: Spatial ENO/WENO reconstruction Temporal direction: TVD Runge-Kutta Examples

15. Dynamic Interface Tracking Rayleigh-Taylor Mixing

16. The Level Set Method Level Set: Interface :

17. Discrete Representation of Tracking Volume filling rectangular mesh (Eulerian Coord.) (N-1) dimensional Lagrangian mesh (interface) Hybrid method, 2 meshes. Front Tracking: Hybrid method, 2 meshes. A 3D Interface A 2D Representation Y X (i,j) x y

18. Time Marching & Coupling tntn X i-1 XiXi X i+1 t n+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

19. Separation of Interface Propagation NormalTangent 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

20. Normal Propagation of Interface Point Move the point position and couple the interior wave solution to interface dynamics. Riemann solution Method of characteristics (Coupling) Step 1: Step 2: (Material interface)New positionS l0 Sr0Sr0 SlSl SrSr Left and right states of the point Updated left and right states of the point (Material interface) New positionS l0 S r0 SlSl SrSr

21. Advancing Eulerian Grid Solution Ghost cell method: Coupling interface dynamics to interior : Fluid 1 : Fluid 2 : Interface t n+1 tntn XiXi X i+1 X i-1 Extrapolate Cell edge

22. Conservative Front Tracking - Formulation 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. XiXi X i+1 Space-time volume X i+1/2

23. 2D Space-Time Volumes x y t Space-time hexahedron Top face

24. Improved Accuracy Theorem: The conservative tracking method improves accuracy by at least one order.

25. 1D Numerical Validation Init. Condition: Shock-Rarefaction

26. 2D Axisymmetric Richtmyer-Meshkov Instability Init. Condition (Density Plot) Conservative tracking simulation Non-conservative tracking simulation Heavy gas Light gas Shock wave Material interface

27. 2D Axisymmetric Richtmyer-Meshkov Instability 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

28. Computations of Incompressible Flows What is the role of the pressure? Hodge Decomposition  Projection Methods

29. 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:

30. The Numerical Method Projection: Compute the intermediate velocity: Surface tension: A B

31. The Blood Flow Modelling

32. 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