CMPS Theory Manual › Species Transport and Reacting Flow

Species Transport and Reacting Flow

Species conservation

For a mixture with \(N_s\) chemical species, CMPS transports \(N_s-1\) independent mass fractions and reconstructs the dependent species from \(\sum_iY_i=1\). The conservative species equation is

\[\frac{\partial(\rho Y_i)}{\partial t}+\nabla\cdot(\rho\mathbf uY_i)= -\nabla\cdot\mathbf j_i+\dot\omega_i.\]

The last species is dependent so that the composition sum remains exact after admissibility control.

Diffusive species flux

The diffusive mass flux is represented by an effective Fickian coefficient,

\[\mathbf j_i=-\rho D_{eff}\nabla Y_i.\]

For GE turbulence,

\[D_{eff}=D_l+D_t,\qquad D_t=\frac{\mu_t}{Sc_t\rho},\qquad Sc_t=0.9.\]

The finite-volume face operator uses harmonic interpolation plus a non-orthogonal correction. Species sensible-enthalpy diffusion returns to the mixture energy equation through

\[\mathbf q_{h,Y}=\sum_i h_i\mathbf j_i.\]

Finite-rate chemistry

For reaction \(r\), the forward Arrhenius constant is

\[\boxed{k_{f,r}=A_rT^{b_r}\exp\left(-\frac{E_{a,r}}{R_uT}\right)}.\]

With molar concentration \(C_i=\rho Y_i/W_i\), the forward rate of progress is

\[\mathcal R_{f,r}=k_{f,r}\prod_{i\in\mathcal R_r}C_i^{\nu'_{ir}}.\]

For the current third-body branch, the forward rate is multiplied by

\[M_r=\rho\sum_i\eta_{ir}Y_i,\qquad \mathcal R_{f,r}\leftarrow M_r\mathcal R_{f,r},\]

where \(\eta_{ir}\) is the stored third-body efficiency.

For a reversible reaction, the source computes

\[\Delta G_r^\circ=\Delta H_r^\circ-T\Delta S_r^\circ,\qquad K_{eq,r}=\exp\left(-\frac{\Delta G_r^\circ}{R_uT}\right),\qquad k_{r,r}=\frac{k_{f,r}}{K_{eq,r}},\]

and subtracts the product-side rate

\[\mathcal R_r=\mathcal R_{f,r}-k_{r,r}\prod_{i\in\mathcal P_r}C_i^{\nu''_{ir}}.\]

Species and energy sources

The stoichiometric molar production rate of species \(i\) is

\[\dot\omega_i^{mol}=\sum_r(\nu''_{ir}-\nu'_{ir})\mathcal R_r,\]

and the corresponding mass source is

\[\dot\omega_i=W_i\dot\omega_i^{mol}.\]

The current split stiff-source path advances the local composition over an interval \(\tau\) and constructs the CFD source from the integrated change,

\[\boxed{\dot\omega_i=\rho f\frac{Y_i^{*}-Y_i^0}{\tau}}.\]

The energy source is reconstructed from species enthalpies,

\[\dot q_{chem}=\sum_i\dot\omega_i h_i(T).\]

Cell-integrated contributions are obtained by multiplying these volumetric source rates by the control-volume measure.

Stiff source integration

The chemistry subproblem is a local ODE system for composition and temperature, schematically

\[\frac{d}{dt}\begin{bmatrix}Y_1&\cdots&Y_{N_s}&T\end{bmatrix}^T=\mathbf S_{chem}(p,\mathbf Y,T).\]

A stiff backward-difference integration path advances this system from \(\mathbf y^0\) to \(\mathbf y^*\). The integrated state change is then converted to the finite-volume source rather than assuming that the beginning-of-step instantaneous rate remains constant over the chemistry interval.

Turbulent finite-rate closure

When turbulent finite-rate chemistry is active with a turbulent flow model, the local chemistry interval is tied to a Kolmogorov-scale time,

\[\tau_{chem}=C_t\tau_\eta.\]

The source is additionally multiplied by a fine-scale volume fraction. With kinematic viscosity \(\nu=\mu/\rho\), turbulence time scale \(\tau_\eta\), length scale \(\ell_\eta\), and model constant \(C_l\), the current expression is

\[f=\left(\left[C_l\left(\frac{\nu\tau_\eta}{\ell_\eta^2}\right)^{1/4}\right]^{-3}-1\right)^{-1}.\]

The effective chemistry source used by the CFD cell is therefore proportional to \(f(Y_i^*-Y_i^0)/\tau_{chem}\).

Mechanism and thermochemical data

Reaction mechanisms, species molecular weights and polynomial thermochemical data are stored in the CMPS mixture/chemical-mechanism model. Standard CHEMKIN-style mechanism and thermodynamic inputs are supported by the project parsers, while the solver consumes the normalized internal species/reaction representation.

Current coupling limits

The source archive contains laminar finite-rate and turbulent finite-rate volumetric reaction modes. Split volumetric reaction integration is currently disabled when homogeneous VOF is active; nonreacting species transport remains a separate coupled capability.