From 09be3ef0158b6046ed8b53dfcb43a7a56680aeb7 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Tue, 3 Jan 2023 15:01:00 -0800 Subject: [PATCH 1/6] initial commit Signed-off-by: jtneedels --- _docs_v7/Theory.md | 46 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 46 insertions(+) diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index e55507d8..bf5e36cc 100644 --- a/_docs_v7/Theory.md +++ b/_docs_v7/Theory.md @@ -9,6 +9,7 @@ This page contains a very brief summary of the different governing equation sets - [Compressible Navier-Stokes](#compressible-navier-stokes) - [Compressible Euler](#compressible-euler) +- [Reacting, Compressible Navier-Stokes](#reacting-compressible-navier-stokes) - [Incompressible Navier-Stokes](#incompressible-navier-stokes) - [Incompressible Euler](#incompressible-euler) - [Turbulence Modeling](#turbulence-modeling) @@ -87,6 +88,51 @@ Within the `EULER` solvers, we discretize the equations in space using a finite --- +# Reacting, Compressible Navier-Stokes # + +| Solver | Version | +| --- | --- | +| `NEMO_NAVIER_STOKES`, `NEMO_RANS` | 7.0.0 | + + +In order to simulate continuum hypersonic flows in thermochemical nonequilibrium, SU2-NEMO solves the Navier-Stokes equation 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 + +$$U = \left \{ \rho_{1}, ..., \rho_{n_s}, \rho \bar{v}, \rho E \right \}^\mathsf{T}$$ + +$$S$$ is a generic source term, and the convective and viscous fluxes are + +$$\bar{F}^{c} = \left \{ \begin{array}{c} \rho \bar{v} \\ \rho \bar{v} \otimes \bar{v} + \bar{\bar{I}} p \\ \rho E \bar{v} + p \bar{v} \end{array} \right \}$$ + +and + +$$\bar{F}^{v} = \left \{ \begin{array}{c} \cdot \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \kappa \nabla T \end{array} \right \}$$ + +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 + +$$\bar{\bar{\tau}}= \mu \left ( \nabla \bar{v} + \nabla \bar{v}^{T} \right ) - \mu \frac{2}{3} \bar{\bar I} \left ( \nabla \cdot \bar{v} \right )$$ + +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 $$\kappa = \mu c_p / Pr$$. + +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. + +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 + +$$\mu =\mu_{d}+\mu_{t}$$ + +Similarly, the thermal conductivity in the energy equation becomes an effective thermal conductivity written as + +$$\kappa =\frac{\mu_{d} \, c_p}{Pr_{d}}+\frac{\mu_{t} \, c_p}{Pr_{t}}$$ + +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. + +--- + # Incompressible Navier-Stokes # | Solver | Version | From a4eadd37ba5e054f6c547fd60f74e1bad9cd8ea4 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Tue, 3 Jan 2023 15:03:25 -0800 Subject: [PATCH 2/6] reverting --- _docs_v7/Theory.md | 46 ---------------------------------------------- 1 file changed, 46 deletions(-) diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index bf5e36cc..e55507d8 100644 --- a/_docs_v7/Theory.md +++ b/_docs_v7/Theory.md @@ -9,7 +9,6 @@ This page contains a very brief summary of the different governing equation sets - [Compressible Navier-Stokes](#compressible-navier-stokes) - [Compressible Euler](#compressible-euler) -- [Reacting, Compressible Navier-Stokes](#reacting-compressible-navier-stokes) - [Incompressible Navier-Stokes](#incompressible-navier-stokes) - [Incompressible Euler](#incompressible-euler) - [Turbulence Modeling](#turbulence-modeling) @@ -88,51 +87,6 @@ Within the `EULER` solvers, we discretize the equations in space using a finite --- -# Reacting, Compressible Navier-Stokes # - -| Solver | Version | -| --- | --- | -| `NEMO_NAVIER_STOKES`, `NEMO_RANS` | 7.0.0 | - - -In order to simulate continuum hypersonic flows in thermochemical nonequilibrium, SU2-NEMO solves the Navier-Stokes equation 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 - -$$U = \left \{ \rho_{1}, ..., \rho_{n_s}, \rho \bar{v}, \rho E \right \}^\mathsf{T}$$ - -$$S$$ is a generic source term, and the convective and viscous fluxes are - -$$\bar{F}^{c} = \left \{ \begin{array}{c} \rho \bar{v} \\ \rho \bar{v} \otimes \bar{v} + \bar{\bar{I}} p \\ \rho E \bar{v} + p \bar{v} \end{array} \right \}$$ - -and - -$$\bar{F}^{v} = \left \{ \begin{array}{c} \cdot \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \kappa \nabla T \end{array} \right \}$$ - -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 - -$$\bar{\bar{\tau}}= \mu \left ( \nabla \bar{v} + \nabla \bar{v}^{T} \right ) - \mu \frac{2}{3} \bar{\bar I} \left ( \nabla \cdot \bar{v} \right )$$ - -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 $$\kappa = \mu c_p / Pr$$. - -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. - -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 - -$$\mu =\mu_{d}+\mu_{t}$$ - -Similarly, the thermal conductivity in the energy equation becomes an effective thermal conductivity written as - -$$\kappa =\frac{\mu_{d} \, c_p}{Pr_{d}}+\frac{\mu_{t} \, c_p}{Pr_{t}}$$ - -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. - ---- - # Incompressible Navier-Stokes # | Solver | Version | From 5c38024beed1f689f8e12ec9325a8a26abb7a5e9 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Wed, 4 Jan 2023 10:54:55 -0800 Subject: [PATCH 3/6] incorporate Wally edits Signed-off-by: jtneedels --- .DS_Store | Bin 0 -> 6148 bytes _docs_v7/Theory.md | 50 +++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 50 insertions(+) create mode 100644 .DS_Store diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000000000000000000000000000000000000..5656c4e348f0fd8c2a43f6875e70dac6dbbe547a GIT binary patch literal 6148 zcmeHKyH3ME5S)b+7Bnd-?-!_Ottfm!J^&;F2q{V=qNH}mZ!`Muh zJcZW_z!rz~Eiebr(H-&OVQRkbKCz3+I9i;s#{-VIJFechlj`#c=XPYRvBMKa{LOCr zn&;wbwEKcLZ19Y|-!ZB=f4Z_#Knh3!DIf);z)uC(v)#JuL`5ke1*E{Y0{(qybjQAM zNQ_ShLyQ2#8Pj2Wj#+}(JVERWheT#*mQ-R=twsz>I`ggS`obYG>9Dw&=hV$s6N<&{ z%(p0q^@)m7Knk2HaGA@w_y2qPFZ2H?Nh>KJ1^$%+He0Ti3%*kI*2&9xuWj^4`j@db n${C^+6QdP#;jQ>)RM-5O`+eb%7mri^f1$u9y`vlM literal 0 HcmV?d00001 diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index e55507d8..4d349967 100644 --- a/_docs_v7/Theory.md +++ b/_docs_v7/Theory.md @@ -9,6 +9,7 @@ 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) - [Incompressible Navier-Stokes](#incompressible-navier-stokes) - [Incompressible Euler](#incompressible-euler) - [Turbulence Modeling](#turbulence-modeling) @@ -87,6 +88,55 @@ Within the `EULER` solvers, we discretize the equations in space using a finite --- +# Thermochemical Nonequilibrium Navier-Stokes # + +| Solver | Version | +| --- | --- | +| `NEMO_NAVIER_STOKES`, `NEMO_RANS` | 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 + +$$\bar{F}^{v} = \left \{ \begin{array}{c} \\- \bar{J}_1 \\ \vdots \\ - \bar{J}_{n_s} \\ \bar{\bar{\tau}} \\ \bar{\bar{\tau}} \cdot \bar{v} + \kappa \nabla T \end{array} \right \}$$ + +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 + +$$\bar{\bar{\tau}}= \mu \left ( \nabla \bar{v} + \nabla \bar{v}^{T} \right ) - \mu \frac{2}{3} \bar{\bar I} \left ( \nabla \cdot \bar{v} \right )$$ + +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 $$\kappa = \mu c_p / Pr$$. + +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. + +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 + +$$\mu =\mu_{d}+\mu_{t}$$ + +Similarly, the thermal conductivity in the energy equation becomes an effective thermal conductivity written as + +$$\kappa =\frac{\mu_{d} \, c_p}{Pr_{d}}+\frac{\mu_{t} \, c_p}{Pr_{t}}$$ + +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. + +--- + # Incompressible Navier-Stokes # | Solver | Version | From 713e46ad729a694abd35f45da6833f731d29cca9 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Wed, 4 Jan 2023 11:37:24 -0800 Subject: [PATCH 4/6] add nemo ns and nemo euler to theory.md Signed-off-by: jtneedels --- _docs_v7/Theory.md | 38 ++++++++++++++++++++++++++------------ 1 file changed, 26 insertions(+), 12 deletions(-) diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index 4d349967..9ff7a69b 100644 --- a/_docs_v7/Theory.md +++ b/_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 \}$$ # Incompressible Navier-Stokes # From 3a33c9daf290a4a79a4e07720192b9eebcd290e6 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Wed, 4 Jan 2023 12:13:33 -0800 Subject: [PATCH 5/6] remove ds file Signed-off-by: jtneedels --- .DS_Store | Bin 6148 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 .DS_Store diff --git a/.DS_Store b/.DS_Store deleted file mode 100644 index 5656c4e348f0fd8c2a43f6875e70dac6dbbe547a..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeHKyH3ME5S)b+7Bnd-?-!_Ottfm!J^&;F2q{V=qNH}mZ!`Muh zJcZW_z!rz~Eiebr(H-&OVQRkbKCz3+I9i;s#{-VIJFechlj`#c=XPYRvBMKa{LOCr zn&;wbwEKcLZ19Y|-!ZB=f4Z_#Knh3!DIf);z)uC(v)#JuL`5ke1*E{Y0{(qybjQAM zNQ_ShLyQ2#8Pj2Wj#+}(JVERWheT#*mQ-R=twsz>I`ggS`obYG>9Dw&=hV$s6N<&{ z%(p0q^@)m7Knk2HaGA@w_y2qPFZ2H?Nh>KJ1^$%+He0Ti3%*kI*2&9xuWj^4`j@db n${C^+6QdP#;jQ>)RM-5O`+eb%7mri^f1$u9y`vlM From ccf2544e9b157e0c5c49a8b3d70e40bf8058cedd Mon Sep 17 00:00:00 2001 From: jtneedels Date: Wed, 4 Jan 2023 12:18:35 -0800 Subject: [PATCH 6/6] remove nemo_rans Signed-off-by: jtneedels --- _docs_v7/Theory.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_docs_v7/Theory.md b/_docs_v7/Theory.md index 9ff7a69b..e8d406d5 100644 --- a/_docs_v7/Theory.md +++ b/_docs_v7/Theory.md @@ -93,7 +93,7 @@ Within the `EULER` solvers, we discretize the equations in space using a finite | Solver | Version | | --- | --- | -| `NEMO_NAVIER_STOKES`, `NEMO_RANS` | 7.0.0 | +| `NEMO_NAVIER_STOKES` | 7.0.0 | To simulate hypersonic flows in thermochemical nonequilibrium, SU2-NEMO solves the Navier-Stokes equations for reacting flows, expressed in differential form as @@ -122,7 +122,7 @@ Chemical production rates are given by $$ \dot{w}_s = M_s \sum_r (\beta_{s,r} - 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 $$ \dot{\theta}_{tr:ve} = \sum _s \rho_s \frac{de^{ve}_{s}}{dt} = \sum _s \rho_s \frac{e^{ve*}_{s} - e^{ve}_{s}}{\tau_s}. $$ +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.