Reynolds number with hyper-viscosity

Question

Is it possible to evaluate a Reynolds number when viscosity operator is substituted by hyper-viscosity operator at the power H (Laplacien to the power H) in the incompressible Navier-Stokes equations ?

Answer 1

For the equation: \begin{equation} \partial_t u_i + u_j \partial_j u_i=-\partial_i p+ \nu_{hyper} \Delta^H u_i \end{equation} with $u$ the velocity in $m.s^{-1}$ and is characteristic order $U$, $p$ the pressure in $m^{2}.s^{-2}$, $\nu$ the hyper-viscosity in $m^{2H}.s^{-1}$. The characteristic length scale is note $L$ in $m$.

Following the non-dimensionalizing, the number Reynolds like is: \begin{equation} Re_{hyper}=\frac{U L^{2H-1}}{\nu_{hyper}} \end{equation}