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Introduction to Modeling Fluid Dynamics in

MATLAB

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Different Kind of Problem• Can be particles, but lots of

them• Solve instead on a uniform

grid

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No Particles => New StateParticle• Mass• Velocity• Position

Fluid• Density• Velocity Field• Pressure• Viscosity

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No Particles => New EquationsNavier-Stokes equations for

viscous, incompressible liquids.

fuuuu

u

pt 1

0

2

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What goes in must come outGradient of the velocity field= 0

Conservation of Mass

fuuuu

u

pt 1

0

2

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Time derivativeTime derivative of velocity field

Think acceleration

fuuuu

u

pt 1

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au

t

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Advection termField is advected through itself

Velocity goes with the flow

fuuuu

u

pt 1

0

2

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Diffusion termKinematic Viscosity times Laplacian of u

Differences in Velocity damp out

fuuuu

u

pt 1

0

2

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Pressure termFluid moves from high pressure to low pressureInversely proportional to fluid density, ρ

fuuuu

u

pt 1

0

2

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External Force TermCan be or represent anythying

Used for gravity or to let animator “stir”

fuuuu

u

pt 1

0

2

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Navier-StokesHow do we solve these equations?

fuuuu

u

pt 1

0

2

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Discretizing in space and time• We have differential

equations• We need to put them in a

form we can compute

• Discetization – Finite Difference Method

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Discretize in Space

X VelocityY VelocityPressure

Staggered Grid vs Regular

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Discretize the operators• Just look them up or derive

them with multidimensional Taylor Expansion

• Be careful if you used a staggered grid

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Example 2D Discetizations

-1 0 1

1

-1

1 -4 1

1

1

Divergence Operator Laplacian

Operator

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Make a linear systemIt all boils down

to Ax=b.

dddd nnxnnx

bb

x

xx

2

1

2

1

??

?????

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Simple Linear System• Exact solution takes O(n3)

time where n is number of cells

• In 3D k3 cells where k is discretization on each axis

• Way too slow O(n9)

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Need faster solver• Our matrix is symmetric and

positive definite….This means we can use♦ Conjugate Gradient

• Multigrid also an option – better asymptotic, but slower in practice.

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Time Integration• Solver gives us time

derivative• Use it to update the system

stateU(t+Δt)

U t

U(t)

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Discetize in Time• Use some system such as

forward Euler.• RK methods are bad because

derivatives are expensive • Be careful of timestep

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Time/Space relation?• Courant-

Friedrichs-Lewy (CFL) condition

• Comes from the advection term

uxt

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Now we have a CFD simulator• We can simulate fluid using only

the aforementioned parts so far• This would be like Foster &

Metaxas first full 3D simulator

• What if we want it real-time?

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Time for Graphics Hacks• Unconditionally stable

advection♦ Kills the CFL condition

• Split the operators♦ Lets us run simpler solvers

• Impose divergence free field♦ Do as post process

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Semi-lagrangian Advection

CFL Condition limits speed of information travel forward in time

Like backward Euler, what if instead we trace back in time?

p(x,t) back-trace

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Divergence Free Field• Helmholtz-Hodge Decomposition

♦ Every field can be written as

• w is any vector field• u is a divergence free field• q is a scalar field

quw

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Helmholtz-Hodge

STAM 2003

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Divergence Free Field• We have w and we want u

• Projection step solves this equation

q

q

q

2

2

wuw

uw

qwu

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Ensures Mass Conservation• Applied to field before

advection• Applied at the end of a step

• Takes the place of first equation in Navier-Stokes

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Operator Splitting• We can’t use semi-lagrangian

advection with a Poisson solver• We have to solve the problem

in phases

• Introduces another source of error, first order approximation

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Operator Splitting

0 u

uu u2 p1 ftu

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Operator Splitting1. Add External

Forces

2. Semi-lagrangian advection

3. Diffusion solve

4. Project field

f

uu

u20 u

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Operator Splitting

u2f

uu

W0 W1 W2 W3 W4

u(x,t)

u(x,t+Δt)

0 u

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Various Extensions• Free surface tracking• Inviscid Navier-Stokes• Solid Fluid interaction

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Free Surfaces• Level sets

♦ Loses volume♦ Poor surface detail

• Particle-level sets♦ Still loses volume♦ Osher, Stanley, & Fedkiw, 2002

• MAC grid♦ Harlow, F.H. and Welch, J.E., "Numerical

Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965).

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Free Surfaces

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MAC Grid Level Set

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Inviscid Navier-Stokes• Can be run faster• Only 1 Poisson Solve needed• Useful to model smoke and

fire♦ Fedkiw, Stam, Jensen 2001

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Solid Fluid Interaction• Long history in CFD• Graphics has many papers on

1 way coupling♦ Way back to Foster & Metaxas, 1996

• Two way coupling is a new area in past 3-4 years♦ Carlson 2004

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Where to get more info• Simplest way to working fluid

simulator (Even has code)♦ STAM 2003

• Best way to learn enough to be dangerous♦ CARLSON 2004

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ReferencesCARLSON, M., “Rigid, Melting, and Flowing Fluid,” PhD Thesis, Georgia Institute of Technology, Jul. 2004.

FEDKIW, R., STAM, J., and JENSEN, H. W., “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pp. 15–22, Aug. 2001.

FOSTER, N. and METAXAS, D., “Realistic animation of liquids,” Graphical Models and Image Processing, vol. 58, no. 5, pp. 471–483, 1996.

HARLOW, F.H. and WELCH, J.E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965).

LOSASSO, F., GIBOU, F., and FEDKIW, R., “Simulating water and smoke with an octree data structure,” ACM Transactions on Graphics, vol. 23, pp. 457–462, Aug. 2004.

OSHER, STANLEY J. & FEDKIW, R. (2002). Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag.

STAM, J., “Real-time fluid dynamics for games,” in Proceedings of the Game Developer Conference, Mar. 2003.