Direct Numerical Simulation Part I – Spectral Methods

A plea for a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The Reynolds Number.

It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers  in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.

Nevertheless DNS proposes a indispensable tool for academic studies of understanding turbulent flow structure and even stepped ground in some various engineering fields for small-scale calculations.

When confronted CFD in general and especially DNS the range of scales to be accurately represented in a computation is represented by the physics of the problem at hand. the grid arrangement gives light to the scales to be represented and accuracy, which is not less of importance for a valid DNS is determined by the numerical method.

Depending on the large-scale features of the problem are determined by features such as inhomogeneous directions,mixing layer thickness, boundary layer thickness, etc…
The Kolmogorov length scale :


Is essentially the smallest scale to be modelled, although future DNS showed it may certainly be larger for some cases.

For achieving the “highest order” of numerical modeling the best route to follow is by spectral methods.

in the following a short insight about spectral methods and their features is going to be presented along with this CFD blogger favourites DNS codes.

Turbulent Boundary Layer (P. Schlatter and D. Henningson of KTH)


As explained in the above paragraph the resolution requirements are a direct result of the numerical scheme employed. To get the best for simple large-scale geometry settings that support a lighter grid resolution most DNS imply spectral methods.
One should remember as far as discretisation error is concerned the two sources are the nonlinearity of the governing equations and the differentiation error.

To quantify the differentiation error, Fourier analysis and the concept modified wave number (which shall be explained thoroughly in future following post) is the useful procedure.

If one shall consider a single fourier mode in one dimension:


and discretisation it on a domain of 2*Pi, whilst using a uniform mesh of N points,
The mesh spacing which shall be h=2*Pi/N. It’s immediately seen by differentiation that the exact first derivative in the n node is:


The numerically computed derivatives take the form of:


k’ is dependent on k and h. It’s called the modified wave number for the first derivative operation. The difference between k and k’ demonstrates and provides the differentiation error. if we were to choose. This could be calculated and actually is a part of basic CFD courses to find the differentiation error of lower numerical schemes (see Lorena Barba CFD course lesson 9)

A must read on spectral methods for incompressible flows:

Spectral Methods for Incompressible Viscous Flow (Applied Mathematical Sciences)

Recommended book with in-depth insight to various approaches to conduct DNS and LES:

Numerical Techniques for Direct and Large-Eddy Simulations (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)

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