Assuming that a direct numerical simulation is performed, what is a good method for estimate the impact of small scale on large scale in fluid dynamics ? For example is it pertinent to compare two run with different grid size or two run with different viscosity ? Is there some relevant statistical tools for this ?
The large scale field can be defined as a coarse-grained field \begin{equation} \overline{q}_l(t,\mathbf{x})=\int G_l(\mathbf{y}) q(t,\mathbf{y}+\mathbf{x})d\mathbf{y} \end{equation} where $G_l$ is a normalized convolution kernel of scale $l$. For example the forme of $G_l$ can be $G_l(y)=l^{-3}/\sqrt{2 \pi} \exp(-((y/l)^2/2)$.
The small scales field is defined as \begin{equation} q'_l=q-\overline{q}_l \end{equation}
If at some scale $l$ we can remove the small scale of the dynamic, the impact of the small scale on the large scale, will be the difference between the field of the full dynamical system with the field of the truncated dynamical system.