Scaling

Nassu uses a grid refinement by a factor of two \((\Delta x_f = \Delta x_c / 2)\), which gives a general spatial resolution of

(1)\[\Delta x_n = \frac{\Delta x_0}{2^n}\]

For the temporal scaling, there are two popular options. The diffusive scaling is \(\Delta t \sim \Delta x^2\), it gives a constant relaxation frequency between levels. The acoustic scaling is \(\Delta t \sim \Delta x\), it gives a constant mesoscopic velocity \((\Delta x / \Delta t)\) between levels. Nassu uses the latter:

(2)\[\Delta t_n = \frac{\Delta t_0}{2^n}\]

This means that each iteration over a coarser level requires two iterations over a finer level. The acoustic scaling keeps the Mach number constant between levels, because the mesoscopic velocity is the same.

In order to keep the Reynolds number constant, there is a requirement on the mesoscopic viscosity:

(3)\[\nu_{0,f} = 2\nu_{0,c}\]

Similarly, the mesoscopic force density is scaled such that its non-dimensional (physical) value is constant throughout all levels:

(4)\[F_{\alpha, n} = \frac{\Delta t_n}{\Delta t_0}F_{\alpha, 0} = \frac{F_{\alpha, 0}}{2^n}\]

Notice that this scaling ensures \(F_n/\Delta t_n = F_0/\Delta t_0\). The scaling performed at the interface between levels must assure that the macroscopic moments (\(\rho\), \(\rho u_{\alpha}\) and \(\sigma_{\alpha\beta}\)) remain continuous thorough the simulation domain.

The LBM populations at interface are built through summation of their equilibrium and non-equilibrium populations \(\left(f_{i}=f_{i}^{\mathrm{eq}}+f_{i}^{\mathrm{neq}}\right)\), which can be calculated from the macroscopics \(\left(\rho,u_{\alpha},S_{\alpha\beta}\right)\).