Fluid dynamics and CFD

What is it?

Formulas

$$\frac{\partial u}{\partial t} + (u\cdot\nabla)u = -\frac{1}{\rho}\nabla p + \nu\,\Delta u + f, \qquad \nabla\cdot u = 0$$

The mathematics behind it

• Vector fields★★★★★fundamental

  The state of a fluid simulation is a velocity field $u(x, t)$.

• Divergence★★★★★fundamental

  Incompressibility $\nabla\cdot u = 0$ is enforced each step by a pressure projection.

• Curl★★★★★fundamental

  Vorticity $\omega = \nabla\times u$ describes eddies; "vorticity confinement" adds back swirls lost to numerical damping in film VFX.

• Divergence theorem (Gauss)★★★★★fundamental

  Finite-volume CFD is the divergence theorem applied cell by cell — the proof idea above, as an algorithm.

• Partial differential equations★★★★★fundamental

  The Navier–Stokes equations are a nonlinear system of PDEs.

• Numerical differentiation★★★★★frequent

  Grid-based fluid solvers discretize derivatives with finite differences on staggered grids.

• Laplacian★★★★★fundamental

  The pressure projection solves a Poisson equation $\Delta p = \nabla\cdot u^\ast$ every time step.

• Surface integrals and flux★★★★★frequent

  Finite-volume solvers update each cell by the fluxes across its faces.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Weather and climate modelling