Navier-Stokes 2D
Stable 2D fluid simulation using velocity advection, viscous diffusion, and iterative pressure projection. Click to add or remove obstacles. Watch vortex shedding, turbulence, and boundary layer separation emerge from a handful of equations.
ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u — incompressible Navier-Stokes
About this experiment
The Navier-Stokes equations describe the motion of viscous fluids and are among the most important equations in physics and engineering. For an incompressible fluid, they express Newton's second law for fluid parcels: the rate of change of momentum equals the pressure gradient force plus the viscous diffusion force. The incompressibility condition ∇·u = 0 (divergence-free velocity field) is enforced separately via a pressure projection step. This simulation uses the stable fluids method introduced by Jos Stam in 1999, which unconditionally stable for any time step by using semi-Lagrangian advection (tracing fluid parcels backward in time).
The Reynolds number Re = ρUL/μ is the dimensionless ratio of inertial to viscous forces, where U is the characteristic flow speed, L is the characteristic length scale (obstacle size), and μ is the dynamic viscosity. At low Re (Re < 1), viscous forces dominate and flow is laminar and symmetric — this is Stokes flow, where the Navier-Stokes equations reduce to the linear Stokes equations. At moderate Re (Re ~ 40–200), a recirculation zone forms behind obstacles and eventually sheds into the famous Kármán vortex street. At high Re, flow becomes turbulent — chaotic, three-dimensional, multi-scale. All of this from the same two equations.
Vorticity ω = ∇×u (the curl of velocity) measures local rotation in the fluid. Positive vorticity (shown in red/orange) indicates counterclockwise rotation; negative vorticity (blue/cyan) indicates clockwise rotation. Vortex shedding — the alternating detachment of vortices from either side of a bluff body — is visible in the vorticity visualization. The frequency of shedding is characterized by the Strouhal number St = fL/U ≈ 0.21, remarkably constant across a wide range of Reynolds numbers for circular cylinders.
The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute, with a $1 million prize for a solution. The question is whether smooth, globally defined solutions always exist in three dimensions, or whether singularities (points of infinite velocity) can develop in finite time. In 2D, global smooth solutions are known to exist. In 3D, the answer remains unknown after more than a century of effort by the world's best mathematicians — a testament to the depth of these deceptively simple-looking equations.