DNS

Continuity

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

Momentum

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

Chemical species transport

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

The code solves for the partial densities $\rho_k = \rho Y_k$

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

Total Energy

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

where $E_t = \rho e + \rho k$ is the total energy plus kinetic energy.

Closures

Transport fluxes

The main closures assume Newtonian flows, Fourier heat transfer and Fickian diffusion and also (optionally) Soret diffusion. The shear stress is given by

$$\tau_{ij} = 2 \mu S_{ij} + (\mu_b - \frac{2}{3} \mu)\nabla \cdot \vec{v} \; \delta_{ij}$$

where $\mu$ and $\mu_b$ are the dynamic viscosity (or coefficient of shear viscoisty) and coefficient of bulk viscosity respectively. The heat flux follows Fourier's law

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

where $\lambda$ is the conductivity.

In the diffusion flux is common to use the Hirschfelder-Curtiss approximation that simplifies the Maxwell-Stefan equation by introducing an effective diffusion coefficient of specie-k in a mixture of gases. The molar flux is (neglecting pressure and temperature gradients)

$$\overline{J}_j^k = - D_k \frac{\partial X_k }{\partial x_j}$$

and in mass fraction

$$J_j^k = \rho \frac{M_k}{M} \overline{J}_j^k = - \rho \frac{M_k}{M} D_k \frac{\partial X_k }{\partial x_j}$$

If the mean molecular weight does not vary greatly in space, this simplifies to Ficks law :

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

The diffusion coefficient of specie-k in the mixture is

$$D_k = \frac{1 - Y_k}{\sum_j X_j/\mathcal{D}_{jk}}$$

based on binary diffusion coefficients $\mathcal{D}_{ij}$

Equations of state

The equations of state are of the form

Ideal gas

For a mixture of ideal gases

and the mass specific enthalpy

$$h = h_k^0 Y_k + \int_{T^0}^T C_p dT$$

and the specific internal energy can be directly obtained by $e = h - p/\rho$. The current implementations include perfect gas, ideal gas and Soave-Redlich-Kwong (through PelePhysics), including mixtures.