diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index e55507d8..e8d406d5 100644 --- a/_docs_v7/Theory.md +++ b/_docs_v7/Theory.md @@ -9,6 +9,8 @@ This page contains a very brief summary of the different governing equation sets - [Compressible Navier-Stokes](#compressible-navier-stokes) - [Compressible Euler](#compressible-euler) +- [Thermochemical Nonequilibrium Navier-Stokes](#thermochemical-nonequilibrium-navier-stokes) +- [Thermochemical Nonequilibrium Euler](#thermochemical-nonequilibrium-euler) - [Incompressible Navier-Stokes](#incompressible-navier-stokes) - [Incompressible Euler](#incompressible-euler) - [Turbulence Modeling](#turbulence-modeling) @@ -87,6 +89,68 @@ Within the `EULER` solvers, we discretize the equations in space using a finite --- +# Thermochemical Nonequilibrium Navier-Stokes # + +| Solver | Version | +| --- | --- | +| `NEMO_NAVIER_STOKES` | 7.0.0 | + + +To simulate hypersonic flows in thermochemical nonequilibrium, SU2-NEMO solves the Navier-Stokes equations for reacting flows, expressed in differential form as + +$$ \mathcal{R}(U) = \frac{\partial U}{\partial t} + \nabla \cdot \bar{F}^{c}(U) - \nabla \cdot \bar{F}^{v}(U,\nabla U) - S = 0 $$ + +where the conservative variables are the working variables and given by + +$$U = \left \{ \rho_{1}, \dots, \rho_{n_s}, \rho \bar{v}, \rho E, \rho E_{ve} \right \}^\mathsf{T}$$ + +$$S$$ is a source term composed of + +$$S = \left \{ \dot{w}_{1}, \dots, \dot{w}_{n_s}, \mathbf{0}, 0, \dot{\theta}_{tr:ve} + \sum_s \dot{w}_s E_{ve,s} \right \}^\mathsf{T}$$ + +and the convective and viscous fluxes are + +$$\bar{F}^{c} = \left \{ \begin{array}{c} \rho_{1} \bar{v} \\ \vdots \\ \rho_{n_s} \bar{v} \\ \rho \bar{v} \otimes \bar{v} + \bar{\bar{I}} p \\ \rho E \bar{v} + p \bar{v} \\ \rho E_{ve} \bar{v} \end{array} \right \}$$ + +and + +$$\bar{F}^{v} = \left \{ \begin{array}{c} \\- \bar{J}_1 \\ \vdots \\ - \bar{J}_{n_s} \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \sum_k \kappa_k \nabla T_k - \sum_s \bar{J}_s h_s \\ \kappa_{ve} \nabla T_{ve} - \sum_s \bar{J}_s E_{ve} \end{array} \right \}$$ + +In the equations above, the notation is is largely the same as for the compressible Navier-Stokes equations. An individual mass conservation equation is introduced for each chemical species, indexed by $$s \in \{1,\dots,n_s\}$$. Each conservation equation has an associated source term, $$\dot{w}_{s}$$ associated with the volumetric production rate of species $$s$$ due to chemical reactions occuring within the flow. + +Chemical production rates are given by $$ \dot{w}_s = M_s \sum_r (\beta_{s,r} - \alpha_{s,r})(R_{r}^{f} - R_{r}^{b}) $$ + +where the forward and backward reaction rates are computed using an Arrhenius formulation. + +A two-temperature thermodynamic model is employed to model nonequilibrium between the translational-rotational and vibrational-electronic energy modes. As such, a separate energy equation is used to model vibrational-electronic energy transport. A source term associated with the relaxation of vibrational-electronic energy modes is modeled using a Landau-Teller formulation $$ \dot{\theta}_{tr:ve} = \sum _s \rho_s \frac{dE_{ve,s}}{dt} = \sum _s \rho_s \frac{E_{ve*,s} - E_{ve,s}}{\tau_s}. $$ + +Transport properties for the multi-component mixture are evaluated using a Wilkes-Blottner-Eucken formulation. + +--- + +# Thermochemical Nonequilibrium Euler # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER` | 7.0.0 | + + +To simulate inviscid hypersonic flows in thermochemical nonequilibrium, SU2-NEMO solves the Euler equations for reacting flows which can be obtained as a simplification of the thermochemical nonequilibrium Navier-Stokes equations in the absence of viscous effects. They can be expressed in differential form as + +$$ \mathcal{R}(U) = \frac{\partial U}{\partial t} + \nabla \cdot \bar{F}^{c}(U) - S = 0 $$ + +where the conservative variables are the working variables and given by + +$$U = \left \{ \rho_{1}, \dots, \rho_{n_s}, \rho \bar{v}, \rho E, \rho E_{ve} \right \}^\mathsf{T}$$ + +$$S$$ is a source term composed of + +$$S = \left \{ \dot{w}_{1}, \dots, \dot{w}_{n_s}, \mathbf{0}, 0, \dot{\theta}_{tr:ve} + \sum_s \dot{w}_s E_{ve,s} \right \}^\mathsf{T}$$ + +and the convective and viscous fluxes are + +$$\bar{F}^{c} = \left \{ \begin{array}{c} \rho_{1} \bar{v} \\ \vdots \\ \rho_{n_s} \bar{v} \\ \rho \bar{v} \otimes \bar{v} + \bar{\bar{I}} p \\ \rho E \bar{v} + p \bar{v} \\ \rho E_{ve} \bar{v} \end{array} \right \}$$ + # Incompressible Navier-Stokes # | Solver | Version |