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Delft University of Technology

Numerical simulation of two-phase flows using acombined VOF / Levelset method

Computational Science 2005

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Research team

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Sander van der Pijl, Kees Vuik & Guus Segal

Department of Applied Mathematical Analysis

Faculty of Electrical Engineering, Mathematics and Computer Science

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Benjamin Vrolijk & Frits Post

Department of Mediamatics

Faculty of Electrical Engineering, Mathematics and Computer Science

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Emil Coyajee & Bendiks Jan Boersma

Laboratory of Aero- & Hydrodynamics

Faculty of Mechanical Engineering

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Outline

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Introduction

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Theory

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Numerical Method

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Parallel Implementation

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Visualization

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Closing remarks

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Introduction

Incompressible two-phase flows

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gas-liquid

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liquid-liquid

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Applications

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Chemical industry (separation, boiling, ..)

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Combustion (fuel injectors)

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Printing industry (inkjets)

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Coating (spray paint, ..)

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Maritime application (green water loading, water waves)

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Applications: Spray combustion

(Ham et al. 2003)

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Theory

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Theory (cont’d.)

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Conservation of mass in phase "0" and "1"

��� ���� � �

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Conservation of momentum in phase "0" and "1"

���� �� � ���� � � � �� � � �

���� �

��� �� � � ��� �� � � � ���� � � � ���� � � �

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Coupling between phase "0" and "1" through interface conditions

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Interface conditions

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Continuity of velocity

� � � � �

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Continuity of stresses

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� � � � �� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� �� ��

� � �� ��� � �

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Interface conditions (cont’d).

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In principle � is discontinuous and thus also �

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Regularization of � gives

� � � � � � � � � �

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Reduction of interface conditions to

� � � � � � � � �� � �

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Surface tension force regularized into a volume force (Brackbill et al., 1992):

� � � � � � � � �� ��

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Interface normal � and curvature � have to be known away from the interface

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Computational method

Spatial discretization, with variable � and �:

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Cartesian and uniform mesh

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Marker and Cell layout

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Discontinuous density, water/air � � � �� � � � �

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Regularization of viscosity

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Continuous surface force approach � no interfaceconditions

x∆

z∆

y∆

z

y

x

u

w

v

p ρ µ, , , ,φ ψ

Explicit time-integration for fluid flow and interface advection equations

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Navier-Stokes: pressure correction method

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Surface representation

Required for calculation of ���� and �

Front tracking:

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Marker particles (Tryggvason and coworkers)

Front capturing:

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Volume of Fluid (VOF) (Rider & Kothe 1998, Scardovelli & Zaleski 1999,Renardy & Renardy 2002, Pilliod & Puckett 2004)

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Levelset (LS) (Sussman et al. 1994, Chang et al. 1996, Sethian 1999)

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LS/VOF (Sussman & Puckett 2000)

Front tracking/capturing:

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LS/Marker particles (Enright et al. 2003)

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Volume of Fluid

Advantage

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Mass conserving interface advection (numerically by construction)

Disadvantage

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Elaborate reconstruction of interface position and curvature, i.e. density, viscosityand surface tension.

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Levelset

Advantage

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Straightforward extraction of interface position, computation of curvature, i.e.density, viscosity and surface tension

Disadvantage

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Numerical implementation of interface advection is not mass conserving

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LS advection

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Numerical implementation � dissipation

�� � �

� � � �� ��� � ����� �

��

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Numerical dissipation � Mass loss/gain

Advection

Exact

Numeric

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Apply small corrections to LS function

� � � � �� � � �

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How to calculate

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VOF reconstructed from LS

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VOF advection is mass conserving by construction

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Invert (Newton-Raphson) with

� �

as initial guess

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Mass conservation, up to a specified �

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Comp. Meth. Overview

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Velocity update � ��

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LS advection

� ��

� � � ��

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VOF advection

� ��

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LS correction

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Poisson equation

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Solution with PCG

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Pressure correction � � � � � �

A Mass-Conserving Level-Set (MCLS) Method for Modeling of Multi-Phase Flows, S.P. van der Pijl, A. Segal, C.Vuik, &

P. Wesseling (accepted: Int. Jour. for Num. Meth. in Fluids)

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Results: Falling water drop

Serial code, numerical resolutions possible up to

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gridpoints

Relative mass error <

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Stationary bubble/Laplace problem

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Exact solution: Pressure constant, velocity zero

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Numerical solution: Pressure perturbed, velocity non-zero

-0.5 -0.25 0 0.25 0.5

x

-0.5

-0.25

0

0.25

0.5

z

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Surface tension

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Similar implementation/problems for all structured-grid methods

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Surface tension in N.S. equations: � � ��

(Brackbill et al. 1992)

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Sources of error:

1. Delta function approximation of the discontinuity

2. Computation of curvature: � � ���� �

� � � � effect of� �

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Resulting symptoms: Parasitic currents for a stationary bubble (Laplace problem)

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Complete VOF/Level-Set reconstruction (1)

Simultaneously solve:

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� � �

and

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� ��

� � � � �� � � �� � ��� �

� � � � � � �

solved by 1st order Fast Marching method (Sethian 1999)

� � � � � ��

� � �

solved up to machine-precision

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Complete VOF/Level-Set reconstruction (2)

x

0

0.2

0.4

0.6

0.8

1

y

0

0.2

0.4

0.6

0.8

1

vof0

0

0.2

0.4

0.6

0.8

1

X Y

Z

vof0

10.90.80.70.60.50.40.30.20.10

x

0

0.2

0.4

0.6

0.8

1

y

0

0.2

0.4

0.6

0.8

1

ph

i

-0.4

-0.2

0

0.2

X Y

Z

phi

0.20.150.10.050

-0.05-0.1-0.15-0.2-0.25-0.3-0.35-0.4

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Remarks

� � � � � � �

ensures a unique solution for Level-Set function

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’Classic’ re-initialization (Sussman 1994) no longer required

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Surface tension representation improved, but not sufficient yet

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Immediate future: obtain higher order solution to

� � � � � �

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Parallelization of the code

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Parallel code required for meshes larger than

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up to

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Approach: Message Passing Interface (MPI) library with domain decomposition

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Parallel Poisson solver

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CG without pre-conditioner

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Quality of initial guess important

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Code runs on SGI Origin 3800 or SGI Altix 3700 (Teras & Aster) at Sara

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Parallelization: Domain decomposition

x

y z

x

y z

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Parallel performance

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"Home made Beowulf cluster" with gigabit over copper

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Supercomputers TERAS/ASTER

#NCPU Beowulf Aster

1 44

2 40

4 30

8 17

16 7

32 8

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Visualization

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Visualization of very large time-dependent data sets is a huge problem.

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To visualize the boundaries between fluids (phase fronts) we need

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interactive isosurface extraction and rendering of large time-varying data sets.

Time

Iso surf. value

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Data structure

A data structure for

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Interactive isosurface extraction

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Time-dependent data sets

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“Incremental” surfaces

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Use of temporal coherence

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Fast rendering

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No need to keep original data in memory

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Temporal Hierarchical Index Tree (Shen, 1998)

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Each node represents a certain time range.

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Each node contains “constant” cells for that time range.

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Cells in one node need not be stored below that node.

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The difference between (consecutive) time steps can be found by backtracking upthe tree.

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In each node, a (possibly large) number of cells must be stored.

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Out-of-core tree building

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During creation of an index tree, we need the entire temporal evolution of every cell,because we want to make use of temporal coherence as much as possible.

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Instead of using

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� � -files, with each file representing a different time step, weuse

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� �

-files

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All time-dependent data for a cell is in one single file.

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Split the data set in �-direction and create multiple trees.

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For example, for a

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data set, we could create

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trees of

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.

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Out-of-core visualization

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During visualization, all sub-trees have to be read to reconstruct the entire spatialdomain, but not complete.

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A time window in is kept main memory, centered around the current time step.

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This approach, alleviates the huge memory requirements for the visualization

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Data sets

Data set Bubbles Clouds

Resolution

� � � � � �� � � � � � � � � � � � � � # Time steps

�� � Raw data size

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MB

� MB

# THI Trees

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xy-resolution

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z-resolution

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# Time steps

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Total size

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MB

� � � MB

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MB

� �

MB

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Visualization tool

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Rendering benchmarks

0 1 2 3 4 5

x 105

0

50

100

150

200

250

Number of cells

Fra

mer

ate

(FP

S)

Rendering framerate as a function of the number of cells

Fast Point−based RendererShell Renderer

� � �

FPS for over

� � �

cells

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FPS for over

�� �

cells

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Even

� �

FPS for

� ��

cells

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Playing benchmarks

0 0.5 1 1.5 2 2.5

x 105

0

20

40

60

80

100

120

Number of cells

Fra

mer

ate

(FP

S)

Average playing framerate as a function of the number of cells

Fast Point−based RendererShell Renderer

Extraction and rendering from all time steps

� � �

FPS (= time steps per second) for over

� � �

cells

� �

FPS (= time steps per second) for over

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cells

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As high as

� �

FPS (= time steps per second) for

� ��

cells

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Some high resolution results

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(1)

X Y

Z

g

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Some high resolution results

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(2)

X Y

Z

g

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Closing remarks

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A mass conserving VOF/LS method has been developed

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Large scale flow simulations of complicated two-phase problems can be performed

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Collaboration between three disciplines has been very productive

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Scientific "freedom" of the NWO-CS program is very stimulating, new research linescan be developed.