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Concept
Direct numerical simulation
Summary
A direct numerical simulation (DNS) 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. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov microscales), up to the integral scale , associated with the motions containing most of the kinetic energy. The Kolmogorov scale, , is given by
where is the kinematic viscosity and is the rate of kinetic energy dissipation. On the other hand, the integral scale depends usually on the spatial scale of the boundary conditions.
To satisfy these resolution requirements, the number of points along a given mesh direction with increments , must be
so that the integral scale is contained within the computational domain, and also
so that the Kolmogorov scale can be resolved.
Since
where is the root mean square (RMS) of the velocity, the previous relations imply that a three-dimensional DNS requires a number of mesh points satisfying
where 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 in each step. That is,
( is here the Courant number). The total time interval simulated is generally proportional to the turbulence time scale given by
Combining these relations, and the fact that must be of the order of , the number of time-integration steps must be proportional to . By other hand, from the definitions for , and given above, it follows that
and consequently, the number of time steps grows also as a power law of the Reynolds number.
Official source
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