Fluid Dynamics

The Navier–Stokes Equations

Fluid motion is governed by the Navier–Stokes equations, derived by Claude-Louis Navier and George Gabriel Stokes independently in the early 19th century. For an incompressible viscous fluid, they express conservation of momentum:

ρ(Du/Dt) = −∇p + μ∇²u + ρf

where ρ is density, u is the velocity field, p is pressure, μ is dynamic viscosity, and f is external body force. A second equation ensures incompressibility: ∇·u = 0.

Whether these equations always have smooth solutions is one of the seven Millennium Prize Problems — unsolved, worth $1 million.

The Reynolds Number

The Reynolds number Re = ρvL/μ is the most important dimensionless quantity in fluid mechanics. It compares inertial forces to viscous forces:

• Re < 2,300 — laminar flow: smooth, parallel streamlines
• 2,300 < Re < 4,000 — transitional flow: unstable and intermittent
• Re > 4,000 — turbulent flow: chaotic eddies and vortices

In this simulation, reducing viscosity increases the effective Reynolds number, driving the transition from smooth to chaotic flow. Observe how vortices form and cascade to smaller scales — the energy cascade of turbulence.

Numerical Simulation Methods

This simulation uses a grid-based Eulerian approach, solving the velocity and pressure fields on a fixed grid:

• Advection — moving velocity along itself (semi-Lagrangian method)
• Diffusion — spreading momentum due to viscosity (Gauss–Seidel relaxation)
• Pressure projection — enforcing incompressibility (∇·u = 0) with a Poisson solve

This approach, pioneered by Jos Stam in his 1999 paper “Stable Fluids”, produces unconditionally stable simulations regardless of timestep size.

Fluid Dynamics in Science

Navier–Stokes equations underpin much of modern engineering and science:

• Aerodynamics — aircraft wing design, Formula 1 cars, wind turbines
• Oceanography — ocean circulation, tsunami propagation, mixing
• Meteorology — weather prediction, storm modelling, climate simulation
• Biomedical — blood flow in arteries, drug delivery optimisation
• Industrial — chemical reactor design, heat exchangers, HVAC

Modern computational fluid dynamics (CFD) runs on supercomputers solving these equations billions of times per timestep, enabling the design of jets, ships, and buildings.