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Stable Fluids
A paper by Jos Stam
Contributions
Real-Time unconditionally stable solver for
Navier-Stokes fluid dynamics equations
Implicit methods allow for large timesteps
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Excessive damping – damps out swirling vortices
Easy to implement
Controllable (?)
Some Math(s)
Nabla Operator:
Laplacian Operator:
Gradient:
More Math(s)
Vector Gradient:
Divergence:
Directional Derivative:
Navier-Stokes Fluid Dynamics
Velocity field u, Pressure field p
Viscosity v, density d (constants)
 External force f
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Navier-Stokes Equation:
Mass Conservation Condition:
Navier-Stokes Equation
Derived from momentum conservation condition
4 Components:
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Advection/Convection
Diffusion (damping)
Pressure
External force (gravity, etc)
Mass Conservation Condition
Velocity field u has zero divergence
Net mass change of any sub-region is 0
 Flow in == flow out
 Incompressible fluid
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Comes from continuum assumption
Enforcing Zero Divergence
Pressure and Velocity fields related
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Say we have velocity field w with non-zero divergence
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Can decompose into
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Helmholtz-Hodge Decomposition
u has zero divergence
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Define operator P that takes w to u:
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Apply P to Navier-Stokes Equation:
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(Used facts that
and
)
Operator P
Need to find
Implicit definition:
Poisson equation for scalar field p
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Neumann boundary condition
Sparse linear system when discretized
Solving the System
Need to calculate:
Start with initial state
Calculate new velocity fields
New state:
Step 1 – Add Force
Assume change in force is small during timestep
Just do a basic forward-Euler step
Note: f is actually an acceleration?
Step 2 - Advection
Method of Characteristics
p is called the characteristic
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Partial streamline of velocity field u
Can show u does not vary along streamline
Determine p by tracing backwards
Unconditionally stable
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Maximum value of w2 is never greater
than maximum value of w1
Step 3 – Diffusion
Standard diffusion equation
Use implicit method:
Sparse linear system
Step 4 - Projection
Enforces mass-conservation condition
Poisson Problem:
Discretize q using central differences
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Sparse linear system
Maybe banded diagonal…
Relaxation methods too inaccurate
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Method of characteristics more precise for divergence-free field
Complexity Analysis
Have to solve 2 sparse linear systems
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Theoretically O(N) with multigrid methods
Advection solver is also O(N)
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However, have to take lots of steps in particle tracer, or
vortices are damped out very quickly
So solver is theoretically O(N)
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I think the constant is going to be pretty high…
Periodic Boundaries
Allows transformation into Fourier domain
In Fourier domain, nabla operator is equivalent to ik
New Algorithm:
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2)
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Compute force and advection
Transform to Fourier domain
Compute diffusion and projection steps
•
4)
Trivial because nabla is just a multiply
Transform back to time domain
Diffusing Substances
Diffuse scalar quantity a (smoke, dust, texture coordinate)
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Advected by velocity field while diffusing
ka is diffusion constant, da is dissipation rate, Sa is source term
Similar to Navier-Stokes
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Can use same methods to solve equations, Except dissipation term
The End