26.1. The Navier–Stokes equations

The equations of motion for a viscous fluid, and why their turbulent solutions are so costly to compute.

For an incompressible fluid with constant density \(\rho\) and kinematic viscosity \(\nu\), the velocity field \(\mathbf{u}(\mathbf{x}, t)\) and pressure \(p(\mathbf{x}, t)\) satisfy

\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}, \qquad \nabla \cdot \mathbf{u} = 0. \]

The first equation is Newton’s second law for a small parcel of fluid: its acceleration, including the nonlinear transport term \((\mathbf{u} \cdot \nabla)\mathbf{u}\), balances the pressure gradient, viscous diffusion and any body force \(\mathbf{f}\) such as gravity. The second states that the flow conserves mass.

26.1.1. Turbulence

How turbulent a flow is depends on the Reynolds number

\[ \mathrm{Re} = \frac{U L}{\nu}, \]

the ratio of inertial to viscous forces for a characteristic speed \(U\) and length \(L\). At high Reynolds numbers the flow breaks into eddies over a wide range of sizes. A direct numerical simulation must resolve them all, down to the smallest (Kolmogorov) scale; the number of grid points grows roughly as \(\mathrm{Re}^{9/4}\) and the total work as about \(\mathrm{Re}^{3}\), which is why DNS of real engineering flows remains out of reach of classical computers.