-------
Scaling
-------

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

.. math::
    \Delta x_n = \frac{\Delta x_0}{2^n}
    :label: mb_dx

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

.. math::
    \Delta t_n = \frac{\Delta t_0}{2^n}
    :label: mb_dt

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: 

.. math::
    \nu_{0,f} = 2\nu_{0,c}
    :label: visc_lvls

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

.. math::
    F_{\alpha, n} = \frac{\Delta t_n}{\Delta t_0}F_{\alpha, 0} = \frac{F_{\alpha, 0}}{2^n}
    :label: mb_scaled_force

Notice that this scaling ensures :math:`F_n/\Delta t_n = F_0/\Delta t_0`. 
The scaling performed at the interface between levels must assure that the macroscopic moments (:math:`\rho`, :math:`\rho u_{\alpha}` and :math:`\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 :math:`\left(f_{i}=f_{i}^{\mathrm{eq}}+f_{i}^{\mathrm{neq}}\right)`, which can be calculated from the macroscopics :math:`\left(\rho,u_{\alpha},S_{\alpha\beta}\right)`. 

.. toctree::
   :maxdepth: 1
   :hidden:

   LES SGS scaling<scaling.les.rst>