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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Intro to Computational Fluid Dynamics
Brandon LloydCOMP 259April 16, 2003
Image courtesy of Prof. A. Davidhazy at RIT. Used without permission.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Overview
• Understanding the Navier-Stokes equations Derivation (following [Griebel 1998]) Intuition
• Solving the Navier-Stokes equations Basic approaches Boundary conditions
• Tracking the free surface
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Operators - gradient div - divergence2- Laplacian… - Hand waving / Lengthy
math compression
Foundations
yu
xuuu
div
2
2
2
22
yu
xuu
y
ux
uu ∂∂,∂
∂
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Transport Theorem
xdtxufft
xdtxfdt
dt t
),()div(),(
),(),( tct
txu
),( tcx
c
0
),( tc
t
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Conservation of Mass
densityis;),()0,(mass0
t
xdtxxdx
0),()div(),(
xdtxu
txdtx
dt
dt t
0)div(
ut
Transport theorem
0div u
Integrand vanishes
is constantfor incompressiblefluids
Continuity equation
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Conservation of Momentum
t
xdtxutx
),(),(momentum
forcesactingmomentuminchange
t
xdtxftx
),(),(:forcesbody
t
dsntx),(:forcessurface σ
ttt
dsntxxdtxftxxdtxutxdt
d ),(),(),(),(),( σ
normal:tensorstress: n
σ
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Conservation of Momentum
0divdiv)())(()( σguuuuudt
d
Transport theorem Divergencetheorem
fupuudt
ud
21
)(
Momentum equation
…
ttt
dsntxxdtxftxxdtxutxdt
d ),(),(),(),(),( σ
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Navier-Stokes Equations
fupuudt
ud
u
21
)(
0
convection viscosityexternalforcespressure
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Solving the equations
Basic Approach1. Create a tentative velocity
field.a. Finite differencesb. Semi-Lagrangian method (Stable
Fluids [Stam 1999])
2. Ensure that the velocity field is divergence free:
a. Adjust pressure and update velocitiesb. Projection method
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tentative Velocity Field
Finite differences – mechanical translation of equations.
n
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tentative Velocity Field
Limits on time step• CFL conditions – don’t move
more than a single cell in one time step
• Diffusion term
ytvxtu maxmax ,
1
22
112
yx
t
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tentative Velocity Field
Stable Fluids Method1. Add forces: 2. Advection3. Diffusion
)()()(~01 xftxuxu
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tentative Velocity Field
AdvectionFinite differences is unstable for large Δt.Solution: trace velocities back in time.
Guarantees that the velocities will never blow up.
)),((~)(~12 txpuxu
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tentative Velocity Field
DiffusionDiscretizing the viscosity term spreads velocity
among immediate neighbors. Unstable when time step too small, grid spacing too large, or viscosity is high.
Solution: Instead of using an explicit time step use an implicit one.
This leads to a large but sparse linear system.
)(~)(~)( 232 xuxut
I
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Satisfying the Continuity Eq.The tentative velocity field is not necessarily
divergence free and thus does not satisfy the continuity equation.
Three methods for satisfying the continuity equation:
1. Explicitly satisfy the continuity equation by iteratively adjusting the pressures and velocities in each cell.
2. Find a pressure correction term that will make the velocity field divergence free.
3. Project the velocities onto their divergence free part.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Explicitly Enforcing u=0
Since we have not yet added the pressure term, we can use pressure to ensure that the velocities are divergence free.
u>0 increased pressure and subsequent outfluxu<0 decreased pressure and subsequent influx
Relaxation algorithm1. Correct the pressure in a cell2. Update velocities 3. Repeat for all cells until each has u<ε
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Solving for pressure
Another approach involves solving for a pressure correction term over the whole field such that the velocities will be divergence free and then update the velocities at the end.
)1()()1( 1~
nnn
pt
uu
dt
ud
)1()()1( ~
nnn pt
uu
0~ )1(2)()1(
nnn pt
uu
Discretize in time
Rearrange terms
Satisfy continuity eq.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Solving for pressure
We end up with the Poisson equation for pressure.
This is another sparse linear system. These types of equations can be solved using iterative methods.
Use pressures to update final velocities.
)()1(2 ~ nn ut
p
)1()()1( ~
nnn pt
uu
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Projection Method
The Helmholtz-Hodge Decomposition Theorem states that any vector field can be decomposed as:
where u is divergence free and q is a scalar field defined implicitly as:
We can define an operator P that projects a vector field onto its divergence free part:
quw
2qw
qwwu
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Projection Method
Applying P to both sides of the momentum equation yields a single equation only in terms of u:
Thus for the last step :
Look familiar? The scalar field q is actually related to pressure!
quuqu ~~ 2
))(P( 2 fuuudt
ud
)1()()1( ~
nnn pt
uu
)()1(2 ~ nn ut
p
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
The Bottom Line
All three methods are equivalent!
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Boundary Conditions
• No slip: Set velocity to 0 on the boundary. Good for obstacles.
• Free slip: Set only the velocity in the direction normal to the boundary to zero. Good for setting up a plane of symmetry.
• Inflow: Specified positive normal velocity. Good for sources.
• Outflow: Specified negative normal velocity. Good for sinks.
• Periodic: Copy the last row and column of cells to first row and column. Good for simulating an infinite domain.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Staggered GridThe staggered grid provides
velocities immediately at cell boundaries, is convenient for finite differences, and avoids oscillations.
Consider problem of a 2D fluid at rest with no external forces. The continuous solution is:
On a discretized non-staggered grid you can have:
jip , jip ,1jip ,1
1, jip
1, jip
21, jiv
21, jiv
jiu ,21 jiu ,2
1
constant00 pvu
oddforeven,for
00
21,
,,
jiPjiPp
vu
ji
jiji
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Tracking the Free SurfaceThe movement of the free surface
is not explicit in the Navier-Stokes equations.
Three methods for tracking the free surface:
1. Marker and cell (MAC) method2. Front tracking3. Particle level set method
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
MAC
Due to [Harlow and Welch 1965].Track massless marker particles to
determine where the free surface is located.
Markers are transported according to the velocity field.
Cells with markers are fluid cells. Fluid cells bordering empty cells are surface cells.
There are boundary conditions that must be satisfied at the surface.
Extended by [Chen et al. 1997] to track particles only near the surface.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
MAC
Problems:• Can lead to mass dissipation, especially with
stable fluid style advection.• No straight forward way to extract a smooth
surface.
Image from [Griebel 1998].
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Front Tracking
Proposed by [Foster and Fedkiw 2001]Front tracking uses a combination of a level set
and particles to track the surface. The particles are used to define an implicit
function. An isocontour of this function represents the liquid surface.
The isocontour yields a smoother surface than particles alone.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Front Tracking
Using the level set method, the isocontour can be evolved directly over time by using the fluid velocities.
Particles and level set evolution have complementary strengths and weaknesses Level set evolution suffers volume loss Particles can cause visual artifacts Level sets are always smooth. Particles retain details.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Front Tracking
Combine the two techniques by giving particles more weight in areas of high curvature. Particles escaping the level set are rendered directly as splashing droplets.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Particle Level Set MethodPresented by [Enright et
al 2002].Implicit surface loses
detail on coarse grids.Particles keep the
surface from crossing them but can’t keep it from drifting away.
Add particles to both side of the implicit surface.
Escaped particles indicate the location of errors in the implicit surface so it can be rebuilt.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Particle Level Set MethodExtrapolated velocities at the surface
give more realistic motion.
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
References
CHEN, J., AND LOBO, N. 1994. Toward interactive-rate simulation of fluids with moving obstacles using the navier-stokes equations. Computer Graphics and Image Processing, 107–116.
CHEN, S., JOHNSON, D., RAAD, P. AND FADDA, D. 1997. The surface marker and micro cell method. International Journal of Numerical Methods in Fluids, 25, 749-778.
FOSTER, N., AND METAXAS, D. 1996. Realistic animation of liquids. Graphical Models and Image Processing, 471–483.
FOSTER, N., AND FEDKIW, R. 2001. Practical animation of liquids. In Proceedings of SIGGRAPH 2001, 23–30.
GRIEBEL, M., DORNSEIFER, T., AND NEUNHOEFFER, T. 1998. Numerical Simulation in Fluid Dynamics: A Practical Introduction. SIAM Monographs on Mathematical Modeling and Computation. SIAM
KASS, M., AND MILLER, G. 1990. Rapid, stable fluid dynamics for computer graphics. In Computer Graphics (Proceedings of SIGGRAPH 90), vol. 24, 49–57.
O’BRIEN, J., AND HODGINS, J. 1995. Dynamic simulation of splashing fluids. In Proceedings of Computer Animation 95, 198–205.
STAM, J. 1999. Stable fluids. In Proceedings of SIGGRAPH 99, 121-128.
