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Stochastic Fields Equations

The final equations to be solved are

Continuity

$$\frac{d \varrho^n}{d t} + \frac{\partial \varrho^n \mathscr{U}^n_j }{\partial x_j} = 0 \label{finalcontinuitySPDE}$$

Momentum

$$\frac{d \varrho^n \mathscr{U}^n_i}{d t} + \frac{\partial \varrho^n \mathscr{U}^n_j \mathscr{U}^n_i}{\partial x_j} = -\frac{\partial \mathscr{P}^n}{\partial x_i} + \frac{\varrho^n}{\overline{\rho}} \frac{\partial \widetilde{\tau}_{ij}}{\partial x_i} + \varrho^n G_{ij} \left( \mathscr{U}^n_j - \widetilde{u}_j \right) + \varrho^n \sqrt{C_0 \frac{\epsilon_{sgs}}{\overline{\rho}}} \frac{d W^n_i}{d t}$$

Mass Fraction of specie k

$$\frac{\partial \varrho^n \mathscr{Y}^n_k}{\partial t} +\frac{\partial \varrho^n \mathscr{U}^n_i \mathscr{Y}^n_k}{\partial x_i} = \frac{\varrho^n}{\overline{\rho}}\frac{\partial \widetilde{J}_{k,i} }{\partial x_i} + \varrho^n \dot{\omega}_k - \frac{1}{2} C_{Y_k} \frac{\epsilon_{sgs}}{k_{sgs}} \varrho^n \left( \mathscr{Y}^n_k - \widetilde{Y}_k \right)$$

Energy

$$\frac{\partial \varrho^n \mathscr{E}^n_t}{\partial t} + \frac{\partial \varrho^n \mathscr{U}^n_i \mathscr{E}^n_t}{\partial x_i} = \frac{\varrho^n}{\overline{\rho}}\frac{\partial \widetilde{q}_i}{\partial x_i} - \frac{\varrho^n}{\overline{\rho}}\frac{\partial \overline{p} \widetilde{u}_i}{\partial x_i} + \frac{\varrho^n}{\overline{\rho}}\frac{\partial \widetilde{\tau}_{ij}\widetilde{u}_j}{\partial x_i} - \frac{1}{2} C_{e_t} \frac{\epsilon_{sgs}}{k_{sgs}} \varrho^n \left( \mathscr{E}^n_t - \widetilde{e_t} \right)$$

The employed closure relation for the dissipation of the sub-grid kinetic energy is

$$\epsilon_{sgs} = C_\epsilon k_{sgs}^{3/2}/\Delta$$

where the constant $$C_\epsilon = 1.05$$ . The micro-mixing constants $$C_{Y_k} = 2$$ and the Langevin constant is set to 2.1,

$$k_{sgs}= \frac{1}{2} \left( \frac{1}{N_f} \sum_{n=1}^{N_f} (\mathscr{U}^n_i-\widetilde{u}_i)^2 \right)$$

The filtered variables can be obtained from the average of the Eulerian stochastic fields.

$$\overline{Q} = \frac{1}{N_f}\sum_{n=1}^{N_f} Q^n; \ \ \ \ \widetilde{Q} = \frac{\sum_{n=1}^{N_f} \varrho^n Q^n}{\sum_{n=1}^{N_f} \varrho^n}$$

Implementation Notes