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Real-Time Fluid Dynamics for Games

Before writing Real-Time Fluid Dynamics for Games, Jos Stam published Stable Fluids in 1999, introducing the ideas that made his approach to fluid simulation so influential. From what I understand so far, this later paper is a much smaller and more approachable introduction to those same ideas, aimed at getting something running quickly rather than diving into all the theory.

The goal isn’t scientific accuracy. Instead, it’s to produce visually convincing smoke and fluid motion while remaining stable and fast enough for real-time applications. Better yet, the paper includes the complete C implementation, making it possible to follow both the mathematics and the code side by side.

That’s why I’m starting here. Rather than treating the paper as something to blindly implement, I’m using it as an excuse to finally understand the mathematics behind fluid simulation. I’ll almost certainly branch into other papers, books, and techniques as I learn more, but this feels like a good place to start.

The big bad wolf. The million-dollar equation (literally — there’s a Millennium Prize for anyone who can solve the Navier–Stokes existence and smoothness problem).

Stam writes, in the most compact vector form, the equation for the velocity :

$$ \frac{\partial \mathbf{u}}{\partial t} = -(\mathbf{u} \cdot \nabla)\mathbf{u} + \nu \nabla^2 \mathbf{u} + \mathbf{f} $$

Here $\mathbf{u}$ is the velocity field — a vector field telling you the velocity of the fluid at each point in space. The way I see it, this equation is kind of saying: the change in velocity over time (the left side) comes from three main things: how the fluid moves itself around, how it diffuses or spreads out due to viscosity ($\nu \nabla^2 \mathbf{u}$ term), and whatever external forces are acting on it ($\mathbf{f}$).

The second equation describes how a density field (like a continuous smoke density cause we are not modeling every particle) moves through the velocity field.

$$ \frac{\partial \rho}{\partial t} = -(\mathbf{u} \cdot \nabla)\rho + \kappa \nabla^2 \rho + S $$

The main idea is that the density $\rho$ gets carried around by the velocity field (first term), it diffuses with some diffusion constant $\kappa$, and there might be sources $S$ that add more density (like smoke being emitted).

In Fluid Simulation for Computer Graphics, Bridson phrases the Navier-Stokes equations like so:

“Most fluid flow of interest in animation is governed by the famous in-compressible Navier-Stokes equations, a set of partial differential equations that are supposed to hold throughout the fluid”

Momentum Equation:

$$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla p + \mathbf{g} + \nu \nabla^2 \mathbf{u} $$

  • $\mathbf{u}$ — velocity of the fluid
  • $p$ — pressure (force per unit area that the fluid exerts on anything)
  • $\rho$ — density of the fluid (mass per unit volume $kg/m^3$)
  • $\mathbf{g}$ — gravity and any additional external forces
  • $\nu$ — kinematic viscosity (how “thick” or resistant to stirring the fluid is)

Incompressibility Condition:

$$ \nabla \cdot \mathbf{u} = 0 $$

This expresses that the fluid is incompressible — its volume doesn’t change, and there’s no net gain or loss of fluid at any point.

So, depending on what assumptions you make — incompressible vs. compressible flow, constant viscosity, ignoring temperature, etc. — you can write these equations in a bunch of different ways. The ones Stam uses are simplified for real-time visual simulation: he’s not trying to simulate airplane aerodynamics, just something that looks like smoke or water moving naturally.

For now, I’m not worrying about all the physics behind them — like where exactly they come from or what all the simplifications mean. I just want to understand how the equations turns into the little C code blocks in Stam’s paper.

Solver Basic Structure

The discrete representation is a 2D grid where each cell stores quantities like velocity and density. Values are assumed to be constant within each cell and represented at the center (later Bridson explains methods where not storing at the center is a much better idea).

Bridson calls this Eulerian Viewpoint TODO continue

We surround our grid simulation region with an aditional layer of cells that act as a buffer region to handle boundary conditions. In practice we can store everything in a one dimensional grid $size = (N+2)*(N+2)$, the paper describes how.

Our solver begins with an initial state for both velocity and density fields, then updates them based on interactions over time. In practice, this means we repeatedly take the current state of the fluid, apply forces, inject new densities, and step the simulation forward by a small time interval $dt$ which in our case will be our frame rate.