2 Governing Carrier-Flow Equations

2.1 Control-volume form

Let \(\Omega_P\) be a finite control volume with boundary \(\partial\Omega_P\) and outward area vector \(d\mathbf{S}=\mathbf{n}\,dS\). The compressible carrier equations solved by CMPS are written in conservative balance form. For compactness, define \[\mathbf{U}=\begin{bmatrix} \rho \\ \rho\mathbf{u} \\ \rho E \end{bmatrix},\] where \(\rho\) is density, \(\mathbf{u}\) is absolute velocity, and \(E=e+\tfrac12\mathbf{u}\cdot\mathbf{u}\) is specific total energy. Turbulence and species equations extend this carrier vector when active.

The integral balance is \[\frac{d}{dt}\int_{\Omega_P}\mathbf{U}\,d\Omega +\oint_{\partial\Omega_P} \left(\mathbf{F}^{c}-\mathbf{F}^{v}\right)\cdot d\mathbf{S} =\int_{\Omega_P}\mathbf{S}_{U}\,d\Omega. \](2.1)

2.2 Continuity

The mass equation is \[\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0,\] or, for one control volume, \[\frac{d}{dt}\int_{\Omega_P}\rho\,d\Omega +\oint_{\partial\Omega_P}\rho\mathbf{u}\cdot d\mathbf{S}=0. \](2.2)

The constant-density artificial-compressibility regime does not use Eq. (2.2) as a physical density evolution equation. Its pressure/continuity equation is given in Chapter 4.

2.3 Momentum

For a Newtonian fluid, \[\frac{\partial(\rho\mathbf{u})}{\partial t} +\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u}+p\mathbf{I}) -\nabla\cdot\boldsymbol{\tau} =\mathbf{S}_{m}, \](2.3) with viscous stress \[\boldsymbol{\tau} =\mu_{\mathrm{eff}} \left[ \nabla\mathbf{u}+(\nabla\mathbf{u})^T -\frac{2}{3}(\nabla\cdot\mathbf{u})\mathbf{I} \right]. \](2.4) Here \(\mu_{\mathrm{eff}}\) is the molecular viscosity for laminar flow and contains the modeled turbulent contribution when the active turbulence closure requires it. Equation (2.4) is kept in tensor form in this guide; replacing it by a Laplacian is only valid under additional assumptions on viscosity and divergence and is not used as a general identity.

The control-volume momentum flux through a face is therefore \[\left[ \rho\mathbf{u}(\mathbf{u}\cdot\mathbf{n}) +p\mathbf{n} -\boldsymbol{\tau}\cdot\mathbf{n} \right]A_f.\] In CMPS the inviscid part is replaced by a numerical face flux, while viscous terms are evaluated from reconstructed gradients and effective transport properties.

2.4 Total energy

The compressible total-energy equation is \[\frac{\partial(\rho E)}{\partial t} +\nabla\cdot\left[(\rho E+p)\mathbf{u}\right] -\nabla\cdot(\boldsymbol{\tau}\cdot\mathbf{u}) +\nabla\cdot\mathbf{q} =S_E, \](2.5) where Fourier heat conduction is \[\mathbf{q}=-k_{\mathrm{eff}}\nabla T.\] For a reacting or multi-species formulation, the energy flux may additionally contain enthalpy carried by species diffusion. CMPS constructs these contributions in the same coupled residual so that their state dependence can enter the AD Jacobian.

Using total enthalpy \[H=E+\frac{p}{\rho},\] the inviscid energy flux is \(\rho H(\mathbf{u}\cdot\mathbf{n})\). This is the form used naturally by AUSM-family fluxes.

2.5 Species transport

For \(N\) chemical species, CMPS may solve \(N-1\) independent mass-fraction equations and close the final fraction by \[Y_N=1-\sum_{s=1}^{N-1}Y_s. \](2.6) For an independent species \(s\), \[\frac{\partial(\rho Y_s)}{\partial t} +\nabla\cdot(\rho\mathbf{u}Y_s) +\nabla\cdot\mathbf{J}_s =\dot\omega_s, \](2.7) with diffusive mass flux \(\mathbf{J}_s\) and source \(\dot\omega_s\). Passive-species mode sets the chemical source to zero. The \(N-1\) representation preserves the algebraic sum of mass fractions but requires diffusion and thermochemistry terms to remain consistent with the dependent species.

2.6 Turbulence extension

When GE \(k\)–\(\omega\) RANS is enabled, transport equations for turbulent kinetic energy \(k\) and specific dissipation rate \(\omega\) are added to the coupled block. Turbulent stress and effective transport properties then modify Eqs. (2.3) and (2.5). CMPS also includes the turbulent kinetic-energy contribution in the carrier total enthalpy used by the GE-RANS flux path. The detailed closure is given in the GE-RANS chapter.

2.7 Finite-volume residual

For a cell \(P\), the spatial residual used throughout the remainder of this guide is defined as \[\mathbf{R}_P(\mathbf{q}) =\sum_{f\in\partial P} \left(\mathbf{F}^{c}_f-\mathbf{F}^{v}_f\right)A_f -\Omega_P\mathbf{S}_P. \](2.8) The sign convention used by the current CMPS matrix assembly is \[\mathbf{B}_P=-\mathbf{R}_P, \qquad \mathbf{A}=-\frac{\partial\mathbf{B}}{\partial\mathbf{q}} =\frac{\partial\mathbf{R}}{\partial\mathbf{q}}, \](2.9) so the Newton-like correction is \(\mathbf{A}\,\delta\mathbf{q}=\mathbf{B}\). This convention is developed in detail in Chapter 3.