Equations

Cerisse solves equations in the general form

$$\frac{\partial \mathbf{U} }{\partial t} + \frac{\partial \mathbf{F}_j }{\partial x_j} + \frac{\partial \mathbf{F}_{visc} }{\partial x_j} = \mathbf{S}$$

Where $\mathbf{U}$ is the array in conservative variables

$$\mathbf{U} = \begin{bmatrix} \rho \\ \rho u_i \\ E_t \\ \rho Y_1 \\ \rho Y_2 \\ ... \end{bmatrix}$$

and $\mathbf{F}$ represent the fluxes of the conserved variables across a surface. For example in x-direction.

$$\mathbf{F}_x = \begin{bmatrix} \rho u \\ \rho u_2 + p \\ \rho u v \\ \rho u w \\ u(E_t + p) \\ \rho u Y_1 \\ \rho u Y_2 \\ ... \end{bmatrix}$$

These are called the Euler fluxes. If Euler fluxes are the only present, the resultant equations are the Euler Equations.

$\mathbf{F}_{visc}$ represents the fluxes due to molecular transport (viscous streess, heat fluxes, mass disffusion, etc). While $\mathbf{S}$ is a generic source term. While combining these terms appropiately we can build differnt type of equations (check definition in PROB), such as Euler, Navier-Stokes, Reactive Navier-Stokes, etc. The general equations can be seen in DNS .

Finite Volume Method

The expression can be written in divergence form

$$\frac{\partial \mathbf{U} }{\partial t} = -\nabla \cdot \left( \mathbf{F} + \mathbf{F}_{visc} \right) + \mathbf{S} = -\nabla \cdot \mathbf{F}^\ast + \mathbf{S}$$

Integrating over cell and dividing over control volume $V_{ijk}$

$$\frac{1}{V_{ijk}} \int \frac{\partial \mathbf{U} }{\partial t} dV = \frac{1}{V_{ijk}} \int -\nabla \cdot \mathbf{F}^\ast \, dV + \frac{1}{V_{ijk}} \int \mathbf{S} dV$$

Defining the cell-averaged value as

$$\phi_{ijk} = \frac{1}{V_{ijk}} \int \phi dV$$

and using divergence theorem

$$\frac{\partial \mathbf{U}_{ijk} }{\partial t} = \frac{1}{V_{ijk}} \oint_{\delta V} \mathbf{F}^\ast \cdot \mathbf{n} \, dA + \mathbf{S}_{ijk}$$

Using polyhedral cells, the following expression follows by summing over the cells faces ()

$$\frac{\partial \mathbf{U}_{ijk} }{\partial t} = \frac{1}{V_{ijk}} \sum_{f=1}^{Nfaces} \mathbf{F}_f^\ast \cdot \mathbf{n}_f \, A_f + \mathbf{S}_{ijk}$$

The above formulation ensures global conservation as the fluxes of shared faces between two control volumes cancel out.