Governing Equation

CompFlowLab solves the coupled conservation equations for mass, momentum, total energy, and species mass fractions:

$$\frac{\partial q}{\partial t} + \frac{\partial f}{\partial x} - \frac{\partial f_v}{\partial x} = H$$

where $q$ is the vector of conserved variables defined as

$$q = \begin{pmatrix} \rho \\ \rho u \\ \rho h^0 - p \\ \rho Y_l \end{pmatrix}$$

Here, $\rho$ is the density, $u$ is the velocity in the x-direction, and $Y_l$ is the mass fraction of the $l^{th}$ species. The total specific enthalpy $h^0$ is defined as

$$h^0 = h + \frac{1}{2} u^2 = \sum_{l}^N h_l Y_l + \frac{1}{2} u^2$$

Fluxes

The fluxes are decomposed into inviscid $f$ and viscous $f_v$ components:

$$f = \begin{pmatrix} \rho u \\ \rho u^2 + p \\ \rho u h^0 \\ \rho u Y_l \end{pmatrix}, \quad f_v = \begin{pmatrix} 0 \\ \tau_{xx} \\ u \tau_{xx} - q_x \\ \rho D_l \frac{\partial Y_l} { \partial x} \end{pmatrix}$$

Here, $D_l$ denotes the effective diffusion coefficient of the $l^th$ species into the mixture. In practice, this approximates multicomponent diffusion as the binary diffusion of each species into a mixture.

Heat Flux

The heat flux in the x-direction, $q_x$, is defined as

$$q_x = -K \frac{\partial T}{\partial x} + \sum_{l=1}^N \rho D_l \frac{\partial Y_l}{\partial x} h_l + q_{source}$$

where:

• $K$ is the thermal conductivity,
• $T$ is the temperature,
• $Y_l$ are the species mass fractions,
• $h_l$ are the species specific enthalpies,
• $q_{source}$ represents volumetric heat sources (e.g., radiation or external heating).
  
  These three terms correspond to conduction, species diffusion, and volumetric heat generation, respectively.\

Shear Stress

The shear stress $\tau_{xx}$ is defined in terms of the molecular viscosity and velocity gradient:\

$$\tau_{xx} = \frac{4}{3} \mu \frac{\partial u}{\partial x}$$

where $\mu$ is the molecular viscosity.

Source term

The source term $H$ accounts for chemical reactions and includes one entry per species equation representing the production or destruction rate of the $l^th$ species, $\omega_l$:

$$H = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \omega_l \end{pmatrix}$$