add nemo ns and nemo euler to theory.md

Signed-off-by: jtneedels <[email protected]>

Author: jtneedels

_docs_v7/Theory.md

@@ -10,6 +10,7 @@ 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)
@@ -113,29 +114,42 @@ $$\bar{F}^{c}   = \left \{ \begin{array}{c} \rho_{1} \bar{v} \\ \vdots \\ \rho_{
 
 and 
 
-$$\bar{F}^{v} = \left \{ \begin{array}{c} \\- \bar{J}_1 \\ \vdots \\ - \bar{J}_{n_s}  \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \kappa \nabla T  \end{array} \right  \}$$
+$$\bar{F}^{v} = \left \{ \begin{array}{c} \\- \bar{J}_1 \\ \vdots \\ - \bar{J}_{n_s}  \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \sum_k \kappa_k \nabla T_k - \sum_s \bar{J}_s h_s \\ \kappa_{ve} \nabla T_{ve} - \sum_s \bar{J}_s E_{ve}  \end{array} \right  \}$$
 
-where $$\rho$$ is the fluid density, $$\bar{v}=\left\lbrace u, v, w \right\rbrace^\mathsf{T}$$ $$\in$$ $$\mathbb{R}^3$$ is the flow speed in Cartesian system of reference, $$E$$ is the total energy per unit mass, $$p$$ is the static pressure, $$\bar{\bar{\tau}}$$ is the viscous stress tensor, $$T$$ is the temperature, $$\kappa$$ is the thermal conductivity, and $$\mu$$ is the viscosity. The viscous stress tensor can be expressed in vector notation as
+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.
 
-$$\bar{\bar{\tau}}= \mu \left ( \nabla \bar{v} + \nabla \bar{v}^{T} \right ) - \mu \frac{2}{3} \bar{\bar I} \left ( \nabla \cdot \bar{v} \right )$$
+Chemical production rates are given by $$ \dot{w}_s = M_s \sum_r (\beta_{s,r} - \alpha_{s,r})(R_{r}^{f} - R_{r}^{b})  $$
 
-Assuming a perfect gas with a ratio of specific heats $$\gamma$$ and specific gas constant $$R$$, one can close the system by determining pressure from $$p = (\gamma-1) \rho \left [ E - 0.5(\bar{v} \cdot \bar{v} ) \right ]$$ and temperature from the ideal gas equation of state $$T = p/(\rho R)$$. Conductivity can be a constant, or we assume a constant Prandtl number $$Pr$$ such that the conductivity varies with viscosity as $$\kappa = \mu c_p / Pr$$. 
+where the forward and backward reaction rates are computed using an Arrhenius formulation.
 
-It is also possible to model non-ideal fluids within SU2 using more advanced fluid models that are available, but this is not discussed here. Please see the tutorial on the topic.
+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 $$ \dot{\theta}_{tr:ve} = \sum _s \rho_s \frac{de^{ve}_{s}}{dt} = \sum _s \rho_s \frac{e^{ve*}_{s} - e^{ve}_{s}}{\tau_s}. $$
 
-For laminar flows, $$\mu$$ is simply the dynamic viscosity $$\mu_{d}$$, which can be constant or assumed to satisfy Sutherland's law as a function of temperature alone, and $$Pr$$ is the dynamic Prandtl number $$Pr_d$$. For turbulent flows, we solve the Reynolds-averaged Navier-Stokes (RANS) equations. In accord with the standard approach to turbulence modeling based upon the Boussinesq hypothesis, which states that the effect of turbulence can be represented as an increased viscosity, the viscosity is divided into dynamic and turbulent components, or  $$\mu_{d}$$ and $$\mu_{t}$$, respectively. Therefore, the effective viscosity in becomes
+Transport properties for the multi-component mixture are evaluated using a Wilkes-Blottner-Eucken formulation.
 
-$$\mu =\mu_{d}+\mu_{t}$$
+---
 
-Similarly, the thermal conductivity in the energy equation becomes an effective thermal conductivity written as
+# Thermochemical Nonequilibrium Euler #
 
-$$\kappa =\frac{\mu_{d} \, c_p}{Pr_{d}}+\frac{\mu_{t} \, c_p}{Pr_{t}}$$
+| Solver | Version | 
+| --- | --- |
+| `NEMO_EULER` | 7.0.0 |
 
-where we have introduced a turbulent Prandtl number $$Pr_t$$. The turbulent viscosity $$\mu_{t}$$ is obtained from a suitable turbulence model involving the mean flow state $$U$$ and a set of new variables for the turbulence. 
 
-Within the `NAVIER_STOKES` and `RANS` solvers, we discretize the equations in space using a finite volume method (FVM) with a standard edge-based data structure on a dual grid with vertex-based schemes. The convective and viscous fluxes are evaluated at the midpoint of an edge. In the `FEM_NAVIER_STOKES` solver, we discretize the equations in space with a nodal Discontinuous Galerkin (DG) finite element method (FEM) with high-order (> 2nd-order) capability.
+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 #