New insights from Navier-Stokes modeling of free surface …folk.uio.no/jks/talk1.pdf · New insights from Navier-Stokes modeling of free surface ... The free surface movement is

New insights from Navier-Stokesmodeling of free surface flows

J. Kristian Sveen

[email protected]

Dept. of Mathematics, University of Oslo

New insights from N-S – p.1/38

Overview

Navier-Stokes equation

Define a free surface

How to model a free surface numericallyDifferent models reviewed (MAC, VOF, Level-set,SPH ++)

Examples � new insight?Commercial and “academic” codes


Navier-Stokes equation

Incompressible fluid: ��

� ��

� ��

��

No mass flux across surface:

�� on surface

Fluid


Free surfaces

Computational domain discretized

Air above neglected - calculations only in fluid domain

Fluid

Floryan and Rasmussen [1989], Scardovelli and Zaleski[1999], Tsai and Yue [1996]


Free surfaces




Free surfaces




Free surfaces

interface crosses grid


Free surfaces - moving grid

Lagrangian grid aligned w surface

new problem: reconnecting of surface (ex: breakingwave) - large deformations to grid


What has been done?

Huge amount of work done fortwo phase flowsrigid boundaries

less for free surfacesOpen channel flows, turbulenceBreaking waves, tsunamis etcDrops and bubbles

New insights from N-S, Free surface flows – p.7/38

Solving N-S

Yacht design for Americas Cup (using Fluent + academiccode (Princeton)):

http://alinghi.epfl.ch


Tracking the free surface

The free surface movement is not explicit in the N-Sequations

Surface tracked via other approach“Marker And Cell” (google: 575 hits)“Volume-Of-Fluid” (google: 12.400 hits)“Direct Numerical Simulation” (google: 11.700)“level-set”, front-tracking (google: 77.400)combinations of the above


A note on boundary conditions

The implementation of free surface boundary conditionsplay a crucial role in the numerical implementation

Chen et al. [1995] - discussion over symmetry:

SMAC SM New version

Gibou et al. [2002]New insights from N-S, Free surface flows – p.10/38

How to model the free surface


Methods

Marker methods

Volume-of-Fluid

Level-set

Smoothed Particle Hydrodynamics (SPH)

Direct Numerical Simulations (DNS)

Lattice (-gas, -Bolzman) models + quick note onCellular Automata


Marker methods


Surface Marker techniques

Overview

Harlow and Welch [1965]

Massless particles introduced in the fluid

Particles transported according to the velocity field andtracked as part of the solution procedure

Cells with markers are fluid cells, fluid cells borderingempty cells � surface cells.

Extension by Chen et al. [1997] - track only particlesnear the surface

Recent applications: Bidoae et al. [2003],Popinet and Zaleski [2002], Christensen [2001]

New insights from N-S, surface marker – p.14/38

Marker techniques - basics

MAC SM


Marker Techniques - example

Bidoae et al. [2003] - results used to estimate forces onbuildings subject to Tsunami-impact.

+ investigate different damping mechanisms via breakwaters


Volume-of-Fluid methods


Volume-Of-Fluid methods

Idea: introduce scalar defining the filling degree of eachcell

A value of 1 indicates a full cell and 0 that the cell isvoid

Integrated/moved in time by solving a transportequation

Hirt and Nichols [1981], Floryan and Rasmussen [1989] and

Gueyffier et al. [1998], Scardovelli and Zaleski [1999]

New insights from N-S, VOF – p.18/38

VOF

Combined w markers: Aulisa et al. [2003] and

“baby-cells” (cells within cells) by Kim and Lee [2003]

Widely used in engineering - Commercial codes suchas CFX and Fluent


VOF - example

Model of oilrig in a wave-tank

wavemaker zoom

courtesy of fluent

Problem: code not verified for this setup


Level-set methods


Level-set

Osher and Sethian [1988], Osher and Fedkiw [2001],Sethian [2001], Osher and Fedkiw [2003],Sethian and Smereka [2003]

Applications and extensions, see forexample Iafrati and Campana [2003], Bourlioux [1995],Sussman and Smereka [1997], Sussman and Puckett[2000], Sharif et al. [2001], Tsai [2002]

Also in combinations w. markers or VOF

New insights from N-S, level-set – p.22/38

Level-set

Idea: introduce scalar

�

which is a distance functionshortest distance in the domain to the fluid surfaceimplicit representation of the interface

integrated/moved in time by solving a transportequation

Problem: Level-set methods do not conserve mass


Level-set, explicit surface

Following Osher and Fedkiw [2003]

In one spatial dimension, divide the real line into threedistinct pieces:

1−1 8−8

We refer to

� � � � � �

as the inside portion and the rest asbeing outside

The two points� � � � � �

can now be called an interfaceexplicit representation


Level-set, implicit surface

Implicit representation - we define the interface as anisocontour of some function

The zero isocontour of

� �

��

is exactly

� � � � � �


Hybrid particle level-set method

Foster and Fedkiw [2001], Enright et al. [2002, 2003]

combines “surface marker”-technique with level-set.

Solves the problem of mass conservation

(...+ view in real time)


Level-set - new insights

Frequent use in special effects in movies

In use in fields such as: Imaging, vision, graphics,computational mechanics

Science (fluid mech): codes verified againstexperiments, theory and other numerics

Long time evolution often neglectedTypically only one test-case and few parametersconsidered

Promising results (as all new methods tend to give)


Smoothed particle hydrodynamics

New insights from N-S, meshless – p.28/38

Smoothed Particle Hydrodynamics

Overview

Introduced by Gingold and Monaghan [1977]

Originally used to model astrophysical processes

Free surface flows by Monaghan [1994]

New insights from N-S, Meshless, SPH – p.29/38

SPH - Idea

Continuum approximated by a finite number of particles

Properties of a fluid at any point estimated by taking aweighted average over a surrounding volume

� ��

2h

i


SPH - basics

Mesh-free method

Particles carry all computational information

Field variables found by averaging (smoothing) fieldvariables over region of interest

Spatial derivatives of field variable � interpolatingformula analytically differentiated


SPH - new insights?

Large amount of publications - few on free surfaces.

Courtesy of Fontaine [2000] (see also Monaghan [1994])


Direct Numerical Simulation


Direct Numerical Simulation

Huge amount of work done in the absence of freesurface (primarily on turbulence)

Free surface turbulence:Shen et al. [1999, 2002, 2003], Pan and Banerjee[1995], Fulgosi et al. [2003]

Typically related to modeling drag on ships andoffshore structures

Multiphase flow: see for example Tryggvason et al.[2001] - (not free surface)

New insights from N-S, DNS modelling – p.34/38

DNS - example

Shen et al. [1999]:

surface layer of a shear flow

linearized surface (no-slip condition)- no waves


What have we learned?


Summary

Large number of publications on numerical methods,comparatively few on “new insights”

Codes typically tested on a few “special cases”The “broken-dam” syndrome

Long time evolution often neglected

Only a few parameters checked

Good models for surface waves?


Summary contd

Increased use of “Physical Modeling” in movies andcomputer games to create “real” behaviour

Physical modeling in animated movies:

Foster and Fedkiw [2001] New insights from N-S, DNS modelling – p.38/38

References

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New insights from Navier-Stokes modeling of free surface …folk.uio.no/jks/talk1.pdf · New insights from Navier-Stokes modeling of free surface ... The free surface movement is