A direct numerical simulation (DNS) is a simulation in computational fluid dynamics in which the Navier-Stokes equations are numerically solved without any turbulence model. This means that the whole range of spatial and temporal scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov scales), given by

$\backslash eta=(\backslash nu^\{3\}/\backslash varepsilon)^\{1/4\}$

where ν is the kinematic viscosity and ε the kinetic energy dissipation, up to the integral scale L. To satisfy these conditions, the number N of points along a given mesh direction with increments h must satisfy

$Nh\; >\; L$

and

$h\; \backslash leq\; \backslash eta$.

Since

$\backslash varepsilon\; \backslash approx\; u\text{'}^\{3\}/L$,

where u' is the root mean square (RMS) of the velocity, the previous relations imply that a three-dimensional DNS requires a number of mesh points $N^\{3\}$ satisfying

$N^\{3\}\backslash ge\; \backslash mathrm\{Re\}^\{9/4\}$

where Re is the turbulent Reynolds number

$\backslash mathrm\{Re\}=\backslash frac\{u\text{'}L\}\{\backslash nu\}$.

Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the most powerful computer currently available. However, direct numerical simulation is a useful tool in fundamental research in turbulence. Using DNS it is possible to perform "numerical experiments", and extract from them information difficult or impossible to obtain in the laboratory, allowing a better understanding of the physics of turbulence. In addition, direct numerical simulations are useful in the development of turbulence models for practical applications, such as sub-grid scale models for Large eddy simulation (LES) and models for methods that solve the Reynolds-averaged Navier-Stokes equations (RANS). This is done by means of "a priori" tests, in which the input data for the model is taken from a DNS simulation, or by "a posteriori" tests, in which the results produced by the model are compared with those obtained by DNS. The biggest DNS in the world, up to this date, used $4096^\{3\}$ mesh points. It was carried out in the Earth Simulator supercomputer,in Japan (the computer has 40 TFlops of computational speed and a memory of 10 Tbytes).

See also

• Large eddy simulation
• Reynolds-averaged Navier-Stokes equations

External link

• DNS page at CFD-Wiki

Fluid dynamics