Navier-Stokes simulation
/ nav-YAY STOHKS /
If you want to predict how a fluid actually moves — not in cartoon, but in detail — there is one set of equations almost everyone starts from: the Navier-Stokes equations. They are just Newton's law, force equals mass times acceleration, written for every tiny parcel of fluid, plus the bookkeeping that mass is conserved. A Navier-Stokes simulation is the act of solving those equations numerically for a particular flow, in a particular shape, with particular boundaries.
The equations say that a fluid parcel accelerates because of pressure differences pushing it, viscous friction dragging it, and any external force like gravity, all while the velocity field stays divergence-free (incompressible fluid neither piles up nor leaves gaps). The hard part, and the reason no general formula exists, is a single nonlinear term — the fluid carries its own momentum along with it (convection) — which couples every scale of motion to every other and breeds turbulence. To simulate, you discretize the domain into a grid, replace the derivatives with finite-difference, finite-volume, or spectral approximations, and at each time step solve a pressure equation (a Poisson problem) to keep the flow incompressible, then update the velocities. The pressure solve is typically the most expensive piece and leans on fast Poisson solvers, multigrid, or Krylov methods with good preconditioners.
These simulations forecast weather, design pipelines and pumps, model ocean currents, and animate believable water and smoke in films. But honesty compels two warnings. First, whether smooth 3D solutions always exist for all time — Navier-Stokes global regularity — is one of the great open problems of mathematics, a Clay Millennium Prize question. Second, fully resolving turbulence (direct numerical simulation) is so costly that for nearly all engineering flows we instead model the small scales rather than compute them, which is itself the subject of turbulence modeling.
Lid-driven cavity flow is the classic test case: fluid in a square box is dragged by the top wall sliding sideways. A Navier-Stokes solver on, say, a 256-by-256 grid reproduces the central vortex and the small counter-rotating eddies in the corners — results that match published benchmarks digit for digit, which is exactly how new solvers are verified.
The lid-driven cavity: a standard benchmark whose vortices new Navier-Stokes codes must reproduce.
Whether smooth 3D Navier-Stokes solutions exist for all time is OPEN — a Clay Millennium Prize problem. We routinely simulate the equations without knowing, in full generality, that the thing we are simulating cannot blow up.