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New insights from Navier-Stokesmodeling of free surface flows
J. Kristian Sveen
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
New insights from N-S – p.2/38
Navier-Stokes equation
Incompressible fluid: ��� � � ���
� �� �� � � � �
� ��
�� � � � ��
No mass flux across surface:
���� �� � � � ��� on surface
Fluid
New insights from N-S – p.3/38
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]
New insights from N-S – p.4/38
Free surfaces
Computational domain discretized
Air above neglected - calculations only in fluid domain
New insights from N-S – p.4/38
Free surfaces
Computational domain discretized
Air above neglected - calculations only in fluid domain
New insights from N-S – p.4/38
Free surfaces - moving grid
Lagrangian grid aligned w surface
new problem: reconnecting of surface (ex: breakingwave) - large deformations to grid
New insights from N-S – p.6/38
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
New insights from N-S, Free surface flows – p.8/38
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
New insights from N-S, Free surface flows – p.9/38
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
Methods
Marker methods
Volume-of-Fluid
Level-set
Smoothed Particle Hydrodynamics (SPH)
Direct Numerical Simulations (DNS)
Lattice (-gas, -Bolzman) models + quick note onCellular Automata
New insights from N-S, Free surface flows – p.12/38
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 - example
Bidoae et al. [2003] - results used to estimate forces onbuildings subject to Tsunami-impact.
+ investigate different damping mechanisms via breakwaters
New insights from N-S, surface marker – p.16/38
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
New insights from N-S, VOF – p.19/38
VOF - example
Model of oilrig in a wave-tank
wavemaker zoom
courtesy of fluent
Problem: code not verified for this setup
New insights from N-S, VOF – p.20/38
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
New insights from N-S, level-set – p.23/38
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
New insights from N-S, level-set – p.24/38
Level-set, implicit surface
Implicit representation - we define the interface as anisocontour of some function
The zero isocontour of
� �
�� � �� � �
is exactly
� � � � � �
New insights from N-S, level-set – p.25/38
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)
New insights from N-S, level-set – p.26/38
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)
New insights from N-S, level-set – p.27/38
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
New insights from N-S, Meshless, SPH – p.30/38
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
New insights from N-S, Meshless, SPH – p.31/38
SPH - new insights?
Large amount of publications - few on free surfaces.
Courtesy of Fontaine [2000] (see also Monaghan [1994])
New insights from N-S, Meshless, SPH – p.32/38
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
New insights from N-S, DNS modelling – p.35/38
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?
New insights from N-S, DNS modelling – p.37/38
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
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