Macroscopics

This page collects the macroscopic quantities the solver works with and gives each one its discrete moment definition. It is the practical companion to the Chapman-Enskog derivation: that page proved which equations the lattice recovers and why; this page lists what you read off the populations to evaluate them. The kinetic primer set the pattern that every macroscopic is a velocity moment of the distribution; here that pattern becomes finite sums over the discrete populations.

Through a Chapman-Enskog expansion [1] it is possible to recover the following macroscopic balance equations from LBE:

(1)\[\begin{split} \begin{aligned} &\partial_t\rho + \partial_\alpha\left( \rho u_\alpha \right) = 0 \\ &\partial_t\left(\rho u_\alpha\right) + \partial_\beta\left(\rho u_\alpha u_\beta\right) = - \partial_\alpha p + \partial_\beta\left[\eta\left(\partial_\alpha u_\beta + \partial_\beta u_\alpha\right)\right] + F_\alpha \end{aligned} \end{split}\]

Note

These equations use index (Einstein) notation: a Greek subscript such as \(\alpha\) runs over the spatial directions \(x, y, z\), and any index repeated within a term is implicitly summed over, so \(\partial_\alpha u_\alpha\) means \(\partial_x u_x + \partial_y u_y + \partial_z u_z\). It is a compact way to write vector and tensor equations; see Einstein notation.

such that the fluid’s dynamic viscosity can be related to LBM mesoscopic variables as:

(2)\[ \eta = \rho c_s^2\Delta t\left(\frac{1}{\omega} - \frac{1}{2}\right) \]

Valid only at low Mach

This recovery holds only at low Mach number, \(\mathrm{Ma} = u_{\mathrm{max}}/c_s\). The compressibility error grows as \(\mathrm{Ma}^2\), so the maximum velocity must be controlled; the practical working threshold used throughout the solver is \(\mathrm{Ma} < 0.1\), derived on the Chapman-Enskog page.

Each of the four macroscopics below plays a distinct role, and each is a moment of the populations, in the exact sense introduced in the kinetic primer:

  • Density \(\rho\) is the zeroth moment. It also fixes the pressure through the ideal-gas relation \(p = \rho c_s^2\) recovered in the Chapman-Enskog analysis, so there is no separate pressure variable to solve for.

  • Velocity \(u_\alpha\) is the first moment, corrected by half a force step (the Guo correction) so that the second-order accuracy of the scheme is preserved when a body force is present.

  • Non-equilibrium stress \(\Pi_{\alpha\beta}^{\mathrm{neq}}\) is the second moment of the non-equilibrium part of the populations. The Chapman-Enskog analysis identified this object as the carrier of the viscous stress (it is the \(\Pi_{\alpha\beta}^{(1)}\) of (9)).

  • Rate-of-strain \(S_{\alpha\beta}\) is obtained algebraically from \(\Pi_{\alpha\beta}^{\mathrm{neq}}\), with no finite differences. It is the input the Smagorinsky subgrid model needs, which is why an LBM-LES can compute its turbulent viscosity locally and cheaply.

Macroscopics such as the density \(\rho\) and velocity are recovered from the population moments:

(3)\[ \sum_{i=0}^{q-1}f_{i}=\rho, \]
(4)\[ \sum_{i=0}^{q-1}f_{i}c_{i,\alpha}=\rho u_{\alpha}-\frac{\Delta t}{2}F_{\alpha}, \]

We also define the second-order moment of non-equilibrium populations:

(5)\[ \Pi_{\alpha\beta}^{\mathrm{neq}} = \sum_{i} f_{i}^{\mathrm{neq}}c_{i\alpha}c_{i\beta} \]

and the rate-of-strain is computed from the second-order moment of non-equilibrium populations:

(6)\[ S_{\alpha\beta} = - \frac{\omega}{2 \rho c_s^2 \Delta t} \left[ \Pi_{\alpha\beta}^{\mathrm{neq}} + \frac{\Delta t}{2}\left( F_\alpha u_\beta + F_\beta u_\alpha \right) \right] \]

Note

The rate-of-strain tensor \(S_{\alpha\beta}\) measures how fast a fluid element is deforming: it is the symmetric part of the velocity-gradient tensor, \(S_{\alpha\beta} = \tfrac{1}{2}(\partial_\alpha u_\beta + \partial_\beta u_\alpha)\). Its magnitude is what drives the Smagorinsky subgrid viscosity in the LES model. See strain-rate tensor.

In return, the second-order moment of non-equilibrium populations \(\Pi_{\alpha\beta}^{\mathrm{neq}}\) can be calculated from the rate-of-strain tensor with:

(7)\[ \Pi_{\alpha\beta}^{\mathrm{neq}} = - \frac{2 \rho c_s^2 \Delta t}{\omega}S_{\alpha\beta} - \frac{\Delta t}{2}\left( F_\alpha u_\beta + F_\beta u_\alpha \right) \]

This last relation is the bridge back to the LES model: the Smagorinsky subgrid viscosity is computed from \(S_{\alpha\beta}\), and the modified relaxation it implies is fed back through \(\Pi_{\alpha\beta}^{\mathrm{neq}}\).

Next steps

Everything so far has been written with \(\Delta x = \Delta t = 1\). The next page makes that convention explicit and shows how to convert between these lattice units and physical metres and seconds for a real simulation.