LES Documentation

Governing Equations

Continuity

$$\frac{\partial \rho }{\partial t} + \frac{\partial \rho u_j }{\partial x_j} = 0$$

Momentum

$$\frac{\partial \rho u_i }{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j}$$

Mass Fraction of specie k

$$\frac{\partial \rho Y_k }{\partial t} + \frac{\partial \rho u_j Y_k}{\partial x_j} = \frac{\partial J_j^k}{\partial x_j} + \rho \dot{\omega}_k$$

Total Energy

$$\frac{\partial E_t }{\partial t} + \frac{\partial (E_t + p) u_j}{\partial x_j} = \frac{\partial u_i \tau_{ij}}{\partial x_j}+ \frac{\partial q_j }{\partial x_j} + \sum_k \frac{ \partial h_k J_j^k}{\partial x_j}$$

Closures

Transport fluxes

Newtonian flows, Fourier and Fickian diffusion and also include (optionally) Soret diffusion

$$q_j = -\lambda \frac{\partial T }{\partial x_j}$$

$$\tau_{ij} = 2 \mu S_{ij}$$

$$J_j^k = - D_k \frac{\partial X_k }{\partial x_j}$$

Mixing Rules

Equations of state

$$p = f(\rho,T,Y_k)$$

Perfect gas, ideal gas, Van der Waals and Soave-Redlich-Kwong

The calorific equation of state is

$$e_i = f(\rho,T,Y_k)$$

Filtered Equations

Assiming the filter function conmutes with spatial derivatives

Continuity

$$\frac{\partial \overline{\rho} }{\partial t} + \frac{\partial \overline{\rho u} }{\partial x} + \frac{\partial \overline{ \rho v} }{\partial y} + \frac{\partial \rho w }{\partial z} = 0$$

or using Favre (or density weighted) filtering $$\overline{\rho \phi} = \bar{\rho} \widetilde{\phi}$$

we obtain

$$\frac{\partial \bar{\rho} }{\partial t} + \frac{\partial \bar{\rho} \widetilde{u} }{\partial x} + \frac{\partial \bar{ \rho} \widetilde{v} }{\partial y} + \frac{\partial \bar{\rho} \widetilde{ w} }{\partial z} = 0$$

Momentum

$$\frac{\partial \overline{\rho u_i} }{\partial t} + \frac{\partial \overline{\rho u_i u_j}}{\partial x_j} = -\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij} }{\partial x_j}$$

or

$$\frac{\partial \bar{\rho} \widetilde{u_i} }{\partial t} + \frac{\partial \bar{\rho} \widetilde{ u_i u_j}}{\partial x_j} = -\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial \overline{\tau}_{ij} }{\partial x_j}$$

Conventional Sub-grid Closures