Spatial Resolution

The range of scales that need to be accurately represented in a computation is dictated by the physics. The grid determines the scales that are represented, while the accuracy with which these scales are represented is determined by the numerical method. The Kolmogorov length scale, $\eta = (\nu^3/\epsilon)^{1/4}$, is commonly quoted as the smallest scale that needs to be resolved. However, this requirement seems to be too stringent, and it is observed that the smallest resolved length scale is required to be of $O(\eta)$ and not equal to $\eta$. Spectral DNS shows very good agreement with experiments even though the Kolmogorov scale is not resolved. The smallest length scale that must be accurately resolved depends on the energy spectrum, and is typically greater than the Kolmogorov length scale; e.g. Moser & Moin [6] noted that most of the dissipation in the curved channel occurs at scales greater than $15\eta$ (based on average dissipation).

The resolution requirements are also fairly influenced by the numerical method used. Differencing schemes with larger numerical error would require higher resolution to achieve the same degree of accuracy compared to the spectral methods and the other more accurate finite differencing counterparts. Other factors which influence the spatial resolution are the differentiation error and the errors associated with the nonlinearity of the governing equations (triadic interaction between the scales, and aliasing), which should be sufficiently small.

And then of course, the Reynolds number plays the most important role. An acknowledged limitation of DNS is its restriction (by cost considerations) to low Reynolds number. For channel flow, the approximate number of grid points needed can be estimated from the expression by Wilcox [17]

$$N_{DNS} = (0.088 {\rm Re}_h)^{9/4}$$

where ${\rm Re}_h$ is the Reynolds number based on the mean channel velocity and channel height. According to the above expression, to compute a flow with Reynolds number of 10⁶ which we encounter in real-life, we would require approximately 133 billion grid points which is astronomical. To reduce this cost somewhat, Reynolds number scaling is used whenever possible depending on the observed dependence of the flow on the Reynolds number without changing the essential physics. Thus the choice of optimum Reynolds number for DNS is dependent on the application, as DNS need not obtain real-life Reynolds numbers to be useful in the study of real-life applications.