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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

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
OUTLINE: Time-Dependent Incompressible Single- and Multiphase Flows

  • WHY?- Motivation…

  • HOW?- Basic Idea … Interpretation …

  • HOW WELL?- Validation & Demonstration…

  • CONCLUSION-

    … Pros/Cons … Area of application … Future development…


Why motivation

MATHEMATICAL MODEL: Time-Dependent Incompressible Single- and Multiphase Flows

  • Solenoidal velocity field;

  • Acoustic effects are neglected;

  • “Implicit” methods;

  • “Explicit” methods;

  • (“Particle” methods);

NUMERICAL

METHODS:

WHY? - Motivation

Incompressible flow


Why motivation1

Poisson Equation: Time-Dependent Incompressible Single- and Multiphase Flows

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


Why motivation2

Solvability of Linear Algebra Time-Dependent Incompressible Single- and Multiphase Flows

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…


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

Poisson Equation Time-Dependent Incompressible Single- and Multiphase Flows

WHY? - Motivation

II."EXPLICIT" METHODS

Chorin’ Method of Artificial Compressibility (AC)

Lattice Boltzmann Equation Method (LBE)

Numerical

Acoustic

Relaxation

(NAR)


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW? – Basic Idea Time-Dependent Incompressible Single- and Multiphase Flows

(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)


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW? – Basic Idea Time-Dependent Incompressible Single- and Multiphase Flows

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)


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW? – Basic Idea Time-Dependent Incompressible Single- and Multiphase Flows

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.


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW? – Basic Idea Time-Dependent Incompressible Single- and Multiphase Flows

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).


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW? – Basic Idea Time-Dependent Incompressible Single- and Multiphase Flows

STEP V:

"Interpretation":


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

HOW WELL? Time-Dependent Incompressible Single- and Multiphase Flows

Validation & Demonstration

1. Lid-Driven Cavity:

2. Doubly-Periodic Shear Layer:

3. Rayleigh-Taylor Instability:

4. Collapse of Water Column:


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

1. Lid-Driven Cavity: Time-Dependent Incompressible Single- and Multiphase Flows

Z

Re=400


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

1. Lid-Driven Cavity: Time-Dependent Incompressible Single- and Multiphase Flows

X

Re=1,000


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

“Sin” perturbation Time-Dependent Incompressible Single- and Multiphase Flows

2. Doubly-Periodic Shear Layer:

Formulation:

“Thin Layer”: 128x128


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

2. Doubly-Periodic Shear Layer: Time-Dependent Incompressible Single- and Multiphase Flows

128x128

256x256


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

2. Doubly-Periodic Shear Layer: 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. Doubly-Periodic Shear Layer: 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. Doubly-Periodic Shear Layer: Time-Dependent Incompressible Single- and Multiphase Flows


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

Artificial vortex Time-Dependent Incompressible Single- and Multiphase Flows

t=1

64x64

128x128

256x256

2. Doubly-Periodic Shear Layer:


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

3. Rayleigh-Taylor Instability: Time-Dependent Incompressible Single- and Multiphase Flows

Initial Growth Rate

Heavy fluid

Development of “bubble”

At=0.5

Re=256

64x128

Development of “spike”

Light fluid


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

3. Rayleigh-Taylor Instability: Time-Dependent Incompressible Single- and Multiphase Flows

NAR:

128x256; 1:2

He-Chen-Zhang

LBE:

256x1056; 1:4


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

3. Rayleigh-Taylor Instability: Time-Dependent Incompressible Single- and Multiphase Flows


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

Initial Perturbation For Multi-Mode Time-Dependent Incompressible Single- and Multiphase Flows

3. Rayleigh-Taylor Instability:

200x400


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

Initial Perturbation For MultiMode Time-Dependent Incompressible Single- and Multiphase Flows

3. Rayleigh-Taylor Instability:


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

4. Collapse of Water Column: Time-Dependent Incompressible Single- and Multiphase Flows


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

t=0.05sec Time-Dependent Incompressible Single- and Multiphase Flows

t=0.20sec

t=0.10sec

t=0.30sec

4. Collapse of Water Column:


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

4. Collapse of Water Column: Time-Dependent Incompressible Single- and Multiphase Flows


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

4. Collapse of Water Column: Time-Dependent Incompressible Single- and Multiphase Flows


R r nourgaliev t n dinh and t g theofanous center for risk studies and safety ucsb

CONCLUSION Time-Dependent Incompressible Single- and Multiphase Flows

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.