Modified BGK form of Collision Term

With the basic equations and methodology for obtaining moments of the collision operator laid out, a modified BGK form of the collision term is suggested:

$$\Omega= -\frac{1}{\lambda}(P-P^{eq'}),$$ (18)

Here we take the "equilibrium" distribution, P^eq' to have the following form:

$$P^{eq'}= \frac {n'}{(2 \pi mKT')^{D/2}}\exp(\frac{(\vec{k}-\vec{k}')^2}{2mKT'})$$ (19)

and $n', \vec {k}',$ and T' are determined by the moment equations.

Using the modified BGK collision operator in 17a gives n'=n (the number of particles per unit volume). The momentum moment of Eq. 18 obtains:

$$\int \!\! d^3k \, \vec{k} \Omega = -\int \!\! d^3k \vec{k} ... ...<\vec{k}>-\vec{k}') = -\frac{\rho}{\lambda}(\vec{v}-\vec{v}').$$ (20)

where $\vec{v}=<\vec{k}>/m$ is the average velocity and $\rho=mn$ is the density. Then Eq. 17b implies

$$\vec{v}'= \vec{v}- \frac{\lambda}{\rho} \nabla \cdot \stackrel {\leftrightarrow}{W}.$$ (21)

Finally noting that the kinetic energy moment of P^eq' is

$$\int \!\! d^3k \, \ \frac{k^2}{2m} P^{eq'}= \frac{3\rho KT'}{2m}+ \frac{1}{2}\rho {\vec{v}}^{'2},$$ (22)

Eq. 17c gives

$$KT'= \frac{ 2 m\lambda}{3 \rho}(- \partial_t \phi - \vec {\na... ... {\nabla} \cdot \vec{Z})+ \frac {m}{3}(<v^2>- {\vec{v}}^{'2})$$ (23)

Hence to preserve global energy conservation the local temperature must be rescaled.

In addition, this modified BGK form, in the isothermal limit, can recover the Shan Chen [4] model where, as in Eq. 21, the equilibrium velocity is shifted to account for non-local interactions. In their model, $\stackrel {\leftrightarrow}{W}$ was approximated by an effective potential which is a function of density:

$$\vec {\nabla} \cdot \stackrel {\leftrightarrow}{W} \simeq \vec {\nabla} V(\rho(\vec{x}_1),\rho(\vec{x}_2)).$$ (24)

The modified BGK collision operator (Eq. 19) can now be incorporated into a discrete Boltzmann model for numerical simulation by, for example, expanding P^eq' in terms of Hermite polynomials and discretizing the velocity space by quadrature methods [12]. Similarly, other realizations of a thermal equilibrium distribution can be easily adopted to this approach by appropriately shifting the velocity and temperature in a manner described above.

While the above modified BGK formalism can accommodate the three conserved quantities: mass, momentum, and energy, information from other moments of the collision operator may not be present. For example, correlations in the velocity field are not properly accounted for in the Gaussian form of the equilibrium distribution. Extra constraints may be accounted for in a more general locally anisotropic "equilibrium" distribution (the ES or ellipsoidal statistical model [18]:

$$\begin{eqnarray*}{P_{es}}^{eq'} =n\frac{\sqrt{det A}}{\pi^{\frac{3}{2}}} exp[-A_{ij}(k_i-k'_i)(k_j-k'_j)]. \end{eqnarray*}$$

The A_ij are determined from the 2nd order moment equations. Note, the ES distribution was originally used to model systems with variable Prandtl number [18].

For the case when the molecular chaos approximation is valid, the moment

$$\int d^3kk_ik_j\Omega = -m\int d^3r_1 P(\vec{r}) P(\vec{r}_1)... ...delta_{2j}v_i)\hat{j}+ (\delta_{3i}v_j+\delta_{3j}v_i)\hat{k}]$$ (25)

may be properly accounted for in the BGK form.