Direct numerical simulation

Direct numerical simulation of a turbulent fluid numerically solves the specified governing equations while resolving the dynamically relevant turbulent and dissipative scales. It does not replace unresolved turbulent motions with a subgrid-scale model. The calculation still assumes a continuum model, boundary conditions and a finite numerical representation; direct does not mean exact or atomistic.

A published example solves the coupled velocity, thermal and compositional equations for fingering convection with a pseudospectral method and triply periodic perturbations. The authors explicitly require resolving the dissipation scale because they use no subgrid-scale model, and inspect composition and vorticity fields to assess resolution. The simulations inform a reduced transport prescription for stellar calculations, rather than resolving an entire star.^[ref]

An exact laminar check helps explain numerical verification. For unforced incompressible Navier-Stokes equations with constant kinematic viscosity $\nu$, take constant pressure andin a periodic box containing whole wavelengths. The divergence vanishes and $u\cdot \nabla u=0$, so the equation reduces to ${\partial }_{t}u=\nu {\partial }_y^{2}u$. The mean kinetic energy per unit mass is thereforeA solver should recover this decay and energy budget under numerical refinement. Passing the check is useful but does not establish that a turbulent calculation resolves its smallest relevant scales. Direct numerical simulation also needs adequate box size, time resolution and statistical sampling.