Direct numerical simulation

A direct numerical simulation (DNS)[1][2] is a simulation in computational fluid dynamics (CFD) 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.

is the rate of kinetic energy dissipation.

To satisfy these resolution requirements, the number of points

is the root mean square (RMS) of the velocity, the previous relations imply that a three-dimensional DNS requires a number of mesh points

is the turbulent Reynolds number: Hence, the memory storage requirement in a DNS grows very fast with the Reynolds number.

In addition, given the very large memory necessary, the integration of the solution in time must be done by an explicit method.

This means that in order to be accurate, the integration, for most discretization methods, must be done with a time step,

, small enough such that a fluid particle moves only a fraction of the mesh spacing

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 computers 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.

Also, 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.