= 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>.<Chemical Transport and Spontaneous Layer Formation in Fingering Convection in Astrophysics by Justin M. Brown and others>[Section 2, equations 6–9 and numerical-method paragraphs, manuscript pages 9–10]{ref}
An exact laminar check helps explain numerical verification. For unforced incompressible <Navier-Stokes equations> with constant <kinematic viscosity> $\nu$, take constant <pressure> and
$$
u(x,y,z,t)=\bigl(Ue^{-\nu k^2t}\sin(ky),0,0\bigr)
$$
in a periodic box containing whole wavelengths. The <divergence> vanishes and $u\cdot\nabla u=0$, so the equation reduces to $\partial_tu=\nu\partial_y^2u$. The mean <kinetic energy> per unit mass is therefore
$$
E(t)=\frac{U^2}{4}e^{-2\nu k^2t},\qquad
-\frac{dE}{dt}=\nu\langle|\nabla u|^2\rangle.
$$
A 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.
Back to article page
