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R.R.Nourgaliev , T.N.Dinh, and T.G.Theofanous Center for Risk Studies and Safety, UCSB

The “Numerical Acoustic Relaxation (NAR)” Method for Time-Dependent Incompressible Single- and Multiphase Flows. R.R.Nourgaliev , T.N.Dinh, and T.G.Theofanous Center for Risk Studies and Safety, UCSB. OUTLINE:. WHY ? - Motivation… HOW ? - Basic Idea … Interpretation …

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R.R.Nourgaliev , T.N.Dinh, and T.G.Theofanous Center for Risk Studies and Safety, UCSB

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  1. The “Numerical Acoustic Relaxation (NAR)” Method for Time-Dependent Incompressible Single- and Multiphase Flows R.R.Nourgaliev, T.N.Dinh, and T.G.Theofanous Center for Risk Studies and Safety, UCSB

  2. OUTLINE: • WHY?- Motivation… • HOW?- Basic Idea … Interpretation … • HOW WELL?- Validation & Demonstration… • CONCLUSION- … Pros/Cons … Area of application … Future development…

  3. MATHEMATICAL MODEL: • Solenoidal velocity field; • Acoustic effects are neglected; • “Implicit” methods; • “Explicit” methods; • (“Particle” methods); NUMERICAL METHODS: WHY? - Motivation Incompressible flow

  4. Poisson Equation: Acoustic modes Recover solenoidal velocity I."IMPLICIT" METHODS EXAMPLE: Projection Method • “Predictor” (Advection) • “Corrector” (Projection): Variable Density (Multiphase Flow): • Hodge-Helmholtz Decomposition: • Poisson Equation: Velocity Correction Fluid-2 Fluid-1 WHY? - Motivation

  5. Solvability of Linear Algebra Slow convergence with increasing grid size N POISSON EQUATION (...PROBLEMS...) WHY? - Motivation Multigrid Methods NOT EFFICIENT: Variable-Density Flows… Particulate Flows… Porous Media… Complex-Geometry Flows… Parallelization…

  6. Poisson Equation WHY? - Motivation II."EXPLICIT" METHODS Chorin’ Method of Artificial Compressibility (AC) Lattice Boltzmann Equation Method (LBE) Numerical Acoustic Relaxation (NAR)

  7. HOW? – Basic Idea (LBE) STEP I: "A-FLUID": Governing Equations for “A-Fluid”: “Macroscopic-Level” equations of the LBE model by He&Luo (…Except “Artifacts”…+Flexibility to vary viscosity)

  8. HOW? – Basic Idea Pressure waves (“A-waves”) travel in STEP II: "Stretched pseudo-time": (Steady-State) • Chorin’ AC • Roger&Kwak’ (Each time step dt: to steady-state dtP) Scaling Analysis: “Numerical Mach Number”:N(M)

  9. HOW? – Basic Idea Extention to Multifluid Flows STEP III: I. Variable-Density Extension: II. Level-Set Method:“Capturing” interface. III. “Ghost-Fluid Method” (GFM):Coupling at the interface.

  10. HOW? – Basic Idea Numerics STEP IV: • “Hyperbolic part”:Characteristic-Based Approach (WENO5, 5th-order). • “Viscous part”:4th-order central difference. • “Time-Stretching”:Implicit Trapezoidal (IT, 2nd-order). • “Level Set”: • PARALLELIZATION:MPI, Domain Decomposition. • WENO5/RK3 . • “Re-initialization” (PDE-based WENO5/RK3 ). • ”Extension-Velocity Technique” (PDE-based).

  11. HOW? – Basic Idea STEP V: "Interpretation":

  12. HOW WELL? Validation & Demonstration 1. Lid-Driven Cavity: 2. Doubly-Periodic Shear Layer: 3. Rayleigh-Taylor Instability: 4. Collapse of Water Column:

  13. 1. Lid-Driven Cavity: Z Re=400

  14. 1. Lid-Driven Cavity: X Re=1,000

  15. “Sin” perturbation 2. Doubly-Periodic Shear Layer: Formulation: “Thin Layer”: 128x128

  16. 2. Doubly-Periodic Shear Layer: 128x128 256x256

  17. 2. Doubly-Periodic Shear Layer:

  18. 2. Doubly-Periodic Shear Layer:

  19. 2. Doubly-Periodic Shear Layer:

  20. Artificial vortex t=1 64x64 128x128 256x256 2. Doubly-Periodic Shear Layer:

  21. 3. Rayleigh-Taylor Instability: Initial Growth Rate Heavy fluid Development of “bubble” At=0.5 Re=256 64x128 Development of “spike” Light fluid

  22. 3. Rayleigh-Taylor Instability: NAR: 128x256; 1:2 He-Chen-Zhang LBE: 256x1056; 1:4

  23. 3. Rayleigh-Taylor Instability:

  24. Initial Perturbation For Multi-Mode 3. Rayleigh-Taylor Instability: 200x400

  25. Initial Perturbation For MultiMode 3. Rayleigh-Taylor Instability:

  26. 4. Collapse of Water Column:

  27. t=0.05sec t=0.20sec t=0.10sec t=0.30sec 4. Collapse of Water Column:

  28. 4. Collapse of Water Column:

  29. 4. Collapse of Water Column:

  30. CONCLUSION FUTURE DEVELOPMENT: • MASSIVE PARALLEL COMPUTATIONof LARGE-DENSITY-DIFFERENCE and COMPLEX-FLOW-GEOMETRY flows. • Multiphase flows consist of COMPRESSIBLE-and-INCOMPRESSIBLE fluids. • Incompressible TWO-FLUID MODEL of multiphase flows.

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