From 4c4fb2b9f1cd3f4f3014b6c29fb9bf42d8b7f699 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 18:47:38 -0800 Subject: [PATCH 01/16] init commit Signed-off-by: jtneedels --- _data/docs_v7.yml | 1 + _docs_v7/Thermochemical-Nonequilibrium.md | 59 +++++++++++++++++++++++ 2 files changed, 60 insertions(+) create mode 100644 _docs_v7/Thermochemical-Nonequilibrium.md diff --git a/_data/docs_v7.yml b/_data/docs_v7.yml index 6f4de818..672311d1 100644 --- a/_data/docs_v7.yml +++ b/_data/docs_v7.yml @@ -30,6 +30,7 @@ docs_v7: - Theory - Streamwise-Periodicity + - Thermochemical-Nonequilibrium - title: Users Guide docs_v7: diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md new file mode 100644 index 00000000..3a68a11e --- /dev/null +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -0,0 +1,59 @@ +--- +title: Thermochemical Nonequilibrium +permalink: /docs_v7/Thermochemical-Nonequilibrium/ +--- + +This page contains a summary of the physical models implemented in the NEMO solvers in SU2 designed ot simulate hypersonic flows in thermochemical nonequilibrium. This includes detials on thermodynamic and chemistry models, as well as transport properties and boundary conditions. + +--- + +- [Thermodynamic Model](#thermodynamic-model) + +--- + +# Thermodynamic Model # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + +A rigid-rotor harmonic oscillator (RRHO) two-temperature model is used to model the thermodynamic state of continuum hypersonic flows. Through the independence of the energy levels, the~total energy and vibrational--electronic energy per unit volume can be expressed as +$$ \rho e = \sum_s \rho_s \left(e_s^{tr} + e_s^{rot} + e_s^{vib} + e_s^{el} + e^{\circ}_s + \frac{1}{2} \bar{v}^{\top} \bar{v}\right), +$$ + +and +$$ + \rho e^{ve} = \sum_s \rho_{s} \left(e_s^{vib} + e_s^{el}\right). +$$ + +Considering a general gas mixture consisting of polyatomic, monatomic, and free electron species, expressions for the energy stored in the translational, rotational, vibrational, and electronic modes are given as +$$ + e^{tr}_s =\begin{cases} + \frac{3}{2} \frac{R}{M_s} T & \text{for monatomic and polyatomic species,}\\ + 0 & \text{for electrons,} + \end{cases} +$$ +\begin{equation} + e^{rot}_s =\begin{cases} + \frac{\xi }{2} \frac{R}{M_s} T & \text{for polyatomic species,}\\ + 0 & \text{for monatomic species and electrons,} + \end{cases} +\end{equation} +where $\xi$ is an integer specifying the number of axes of rotation, +\begin{equation} + e^{vib}_s =\begin{cases} + \frac{R}{M_s} \frac{\theta^{vib}_s}{exp\left( \theta^{vib}_s / T^{ve}\right) - 1} & \text{for polyatomic species,}\\ + 0 & \text{for monatomic species and electrons,} + \end{cases} +\end{equation} +where $\theta^{vib}_s$ is the characteristic vibrational temperature of the species, and~\begin{equation} + e^{el}_s =\begin{cases} + \frac{R}{M_s}\frac{\sum_{i=1}^{\infty} g_{i,s}{\theta^{el}_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})}}{\sum_{i=0}^{\infty} g_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})} & \text{for polyatomic and monatomic species,}\\ + \frac{3}{2} \frac{R}{M_s} T^{ve} & \text{for electrons,} + \end{cases} +\end{equation} + +\noindent where $\theta^{el}_s$ is the characteristic electronic temperature of the species and $g_i$ is the degeneracy of the $i^{th}$ state. + +--- + From 72c55d30206ffaabc0e6aa453aca27fda9884fcb Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 19:05:01 -0800 Subject: [PATCH 02/16] finite rate chemistry Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 58 +++++++++++++++++++---- 1 file changed, 48 insertions(+), 10 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 3a68a11e..62ef5e35 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -8,6 +8,7 @@ This page contains a summary of the physical models implemented in the NEMO solv --- - [Thermodynamic Model](#thermodynamic-model) +- [Finite Rate Chemistry](#finite-rate-chemistry) --- @@ -17,10 +18,9 @@ This page contains a summary of the physical models implemented in the NEMO solv | --- | --- | | `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | -A rigid-rotor harmonic oscillator (RRHO) two-temperature model is used to model the thermodynamic state of continuum hypersonic flows. Through the independence of the energy levels, the~total energy and vibrational--electronic energy per unit volume can be expressed as +A rigid-rotor harmonic oscillator (RRHO) two-temperature model is used to model the thermodynamic state of continuum hypersonic flows. Through the independence of the energy levels, the total energy and vibrational--electronic energy per unit volume can be expressed as $$ \rho e = \sum_s \rho_s \left(e_s^{tr} + e_s^{rot} + e_s^{vib} + e_s^{el} + e^{\circ}_s + \frac{1}{2} \bar{v}^{\top} \bar{v}\right), $$ - and $$ \rho e^{ve} = \sum_s \rho_{s} \left(e_s^{vib} + e_s^{el}\right). @@ -33,27 +33,65 @@ $$ 0 & \text{for electrons,} \end{cases} $$ -\begin{equation} +$$ e^{rot}_s =\begin{cases} \frac{\xi }{2} \frac{R}{M_s} T & \text{for polyatomic species,}\\ 0 & \text{for monatomic species and electrons,} \end{cases} -\end{equation} +$$ where $\xi$ is an integer specifying the number of axes of rotation, -\begin{equation} +$$ e^{vib}_s =\begin{cases} \frac{R}{M_s} \frac{\theta^{vib}_s}{exp\left( \theta^{vib}_s / T^{ve}\right) - 1} & \text{for polyatomic species,}\\ 0 & \text{for monatomic species and electrons,} \end{cases} -\end{equation} -where $\theta^{vib}_s$ is the characteristic vibrational temperature of the species, and~\begin{equation} +$$ +where $\theta^{vib}_s$ is the characteristic vibrational temperature of the species, and + + +$$ e^{el}_s =\begin{cases} - \frac{R}{M_s}\frac{\sum_{i=1}^{\infty} g_{i,s}{\theta^{el}_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})}}{\sum_{i=0}^{\infty} g_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})} & \text{for polyatomic and monatomic species,}\\ + \frac{R}{M_s}\frac{\sum_{i=1}^{\infty} g_{i,s}{\theta^{el}_{i,s} \exp(-\theta^{el}_{i,s}/T_{ve})}}{\sum_{i=0}^{\infty} g_{i,s} exp(-\theta^{el}_{i,s}/T_{ve})} & \text{for polyatomic and monatomic species,}\\ \frac{3}{2} \frac{R}{M_s} T^{ve} & \text{for electrons,} \end{cases} -\end{equation} +$$ + +where $\theta^{el}_s$ is the characteristic electronic temperature of the species and $g_i$ is the degeneracy of the $i^{th}$ state. + +--- + +# Finite Rate Chemistry # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + +The source terms in the species conservation equations are the volumetric mass production rates which are governed by the forward and backward reaction rates, $R^f$ and $R^b$, for a given reaction $r$, and can be expressed as +$$ + \dot{w}_s = M_s \sum_r (\beta_{s,r} - \alpha_{s,r})(R_{r}^{f} - R_{r}^{b}). +$$ + +From kinetic theory, the forward and backward reaction rates are dependent on the molar concentrations of the reactants and products, as well as the forward and backward reaction rate coefficients, $k^f$ and $k^b$, respectively, and can be expressed as +$$ + R_{r}^f = k_{r}^f \prod_s (\frac{\rho_s}{M_s})^{\alpha_{s,r}}, +$$ +and +$$ + R_{r}^b = k_{r}^b \prod_s (\frac{\rho_s}{M_s})^{\beta_{s,r}}. +$$ + +For an Arrhenius reaction, the forward reaction rate coefficient can be computed as +$$ + k_{r}^f = C_r(T_r)^{\eta_r} exp\left(- \frac{\epsilon_r}{k_B T_r}\right), +$$ +where $C_r$ is the pre-factor, $T_r$ is the rate-controlling temperature for the reaction, $\eta_r$ is an empirical exponent, and $\epsilon_r$ is the activation energy per molecule. + +The rate-controlling temperature of the reaction is calculated as a geometric average of the translation--rotational and vibrational--electronic temperatures, +$$ + T_r = (T)^{a_r}(T^{ve})^{b_r}, +$$ + -\noindent where $\theta^{el}_s$ is the characteristic electronic temperature of the species and $g_i$ is the degeneracy of the $i^{th}$ state. --- From 89dd8b2853350abc464ee8981f11157785009e29 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 22:35:52 -0800 Subject: [PATCH 03/16] vib relaxation and start transport coeffs Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 78 ++++++++++++++++++++++- 1 file changed, 76 insertions(+), 2 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 62ef5e35..8bebd2c1 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -9,6 +9,10 @@ This page contains a summary of the physical models implemented in the NEMO solv - [Thermodynamic Model](#thermodynamic-model) - [Finite Rate Chemistry](#finite-rate-chemistry) +- [Vibrational Relaxation](#vibrational-relaxation) +- [Transport Coefficients](#transport-coefficients) + -[Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) + -[Gupta-Yos](#gupta-yos) --- @@ -86,12 +90,82 @@ $$ $$ where $C_r$ is the pre-factor, $T_r$ is the rate-controlling temperature for the reaction, $\eta_r$ is an empirical exponent, and $\epsilon_r$ is the activation energy per molecule. -The rate-controlling temperature of the reaction is calculated as a geometric average of the translation--rotational and vibrational--electronic temperatures, +The rate-controlling temperature of the reaction is calculated as a geometric average of the translation-rotational and vibrational-electronic temperatures, $$ - T_r = (T)^{a_r}(T^{ve})^{b_r}, + T_r = (T)^{a_r}(T^{ve})^{b_r}. $$ +The value of he equilibrium constant $K_{eq}$ is expressed as +$$ + K_{eq} = \exp( A_0 T_m + A_1 + A_2 \log(1/T_m) + A_3 (1/T_m) + A_4 (1/T_m)^2 ), +$$ + +where $T_m$ is a modified temperature and $A_1 - A_4$ are constants dependent on the reaction. These reaction constants, the rate constrolling temperature and Arrhenius parameters are stored within the fluid model class in SU2 NEMO. + +--- + +# Vibrational Relaxation # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + +Vibrational relaxation is computed using a standard Landau-Teller relaxation time with a Park high-temperature correction +$$ + \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}, +$$ +where $\tau_s$ is computed using a combination of the Landau-Teller relaxation time, $\langle \tau_s \rangle_{L-T}$, and a limiting relaxation time from Park, $\tau_{ps}$ using +$$ + \tau_s = \langle \tau_s \rangle_{L-T} + \tau_{ps}, +$$ +and +$$ + \langle \tau_s \rangle_{L-T} = \frac{\sum_r X_r}{\sum_r X_r/\tau_{sr}}. +$$ +The interspecies relaxation times are taken from experimental data from Millikan and White, expressed as +$$ + \tau_{sr} = \frac{1}{P}exp\left[A_sr\left(T^{-1/3} - 0.015\mu_{sr}^{1/4}\right) - 18.42\right]. +$$ +A limiting relaxation time, $\tau_{ps}$, is used to correct for under-prediction of the Millikan--White model at high temperatures. $\tau_{ps}$ is defined as +$$ + \tau_{ps} = \frac{1}{\sigma_s c_s n}, +$$ + +where $\sigma_s$ is the effective collision~cross-section. --- +# Transport Coefficients # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + + +Mass, momentum, and energy transport in fluids are all governed by molecular collisions, and expressions for these transport properties can be derived from the kinetic theory. The mass diffusion fluxes, $\mathbf{J}_s$, are computed using Fick's Law of Diffusion: +$$ + \mathbf{J}_s = \rho D_s \nabla(c_s), +$$ + +where $c_s$ is the species mass fraction and $D_s$ is the species multi-component diffusion coefficient. The values of $D_s$ are computed as a weighted sum of binary diffusion coefficients between all species in the mixture. These are obtained by solving the Stefan--Maxwell equations under the Ramshaw approximations. The viscous stress tensor is written as +$$ + \boldsymbol{\sigma} = \mu \left( \nabla \mathbf{u} + \nabla {\mathbf{u}}^\mathsf{T} - \frac{2}{3} \mathbf{I} (\nabla \cdot \mathbf{u}) \right), +$$ +where $\mu$ is the mixture viscosity coefficient. The conduction heat flux for each thermal energy mode, $\mathbf{q}^{k}$, is assumed to be given by Fourier’s Law of heat conduction: +$$ +\mathbf{q}^{k} = \kappa^{k} \nabla(T^k), +$$ + +where $\kappa^{k}$ is the thermal conductivity associated with energy mode $k$. + +$D_s$, $\mu$, and $\kappa$ can be evaluated using either a Wilkes-Blottner-Eucken or Gupta-Yos transport models, with the implemntation details and reocmmendations on use given in the sections below. + + +## Wilkes-Blottner-Eucken ## + + + +## Gupta-Yos ## + + From c63133da5f7281858adfe9f4f47551da82a7dbd9 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 22:56:16 -0800 Subject: [PATCH 04/16] partial WBE Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 40 ++++++++++++++++++++--- 1 file changed, 36 insertions(+), 4 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 8bebd2c1..e894f58c 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -10,9 +10,9 @@ This page contains a summary of the physical models implemented in the NEMO solv - [Thermodynamic Model](#thermodynamic-model) - [Finite Rate Chemistry](#finite-rate-chemistry) - [Vibrational Relaxation](#vibrational-relaxation) -- [Transport Coefficients](#transport-coefficients) - -[Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) - -[Gupta-Yos](#gupta-yos) +- [Viscous Phenomena and Transport Coefficients](#viscous-phenomena-and-transport-coefficients) + - [Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) + - [Gupta-Yos](#gupta-yos) --- @@ -136,7 +136,7 @@ where $\sigma_s$ is the effective collision~cross-section. --- -# Transport Coefficients # +# Viscous Phenomena and Transport Coefficients # | Solver | Version | | --- | --- | @@ -164,6 +164,38 @@ $D_s$, $\mu$, and $\kappa$ can be evaluated using either a Wilkes-Blottner-Eucke ## Wilkes-Blottner-Eucken ## +The mixture dynamic viscosity and thermal conductivity are computed using Wilke's semi-empirical mixing rule as + +$$ +\mu = \sum_s \frac{X_s \mu_s}{\phi_s}, +$$ + +and + +$$ +\kappa = \sum_s \frac{X_s \kappa_s}{\phi_s}, +$$ + +where $X_s$ is the mole fraction of species $s$. The species dynamic viscosity is computed using Blottner's three paramter curve fit for high temperature air, + +$$ +\mu_s = 0.1 \exp [(A_s\log(T) + B_s)\log(T) + C_s]. +$$ + +The species thermal conductivities are computed according to Eucken's formula as + +$$ +\kappa^{tr}_s = \mu_s \left( \frac{5}{2} C_{v_s}^{trans} + C_{v_s}^{rot} \right), +$$ + +$$ +\kappa^{ve}_s = \mu_s C^{ve}_{v_s}. +$$ + +And the term $\phi_s$ is given by +$$ +\phi_s = \sum_r X_r \left[ 1 + \sqrt{\frac{\mu_r}{\mu_s}}\left( \frac{M_r}{M_s} \right)^{1/4} \right]^{2} \left[ \sqrt{8 \left(1 + \frac{M_s}{M_r} \right)} \right]^{-1}. +$$ ## Gupta-Yos ## From dde84c57771f66728042cdc725bdc7e3ee717db9 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 23:39:54 -0800 Subject: [PATCH 05/16] finish wbe Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 31 ++++++++++++++++++++--- 1 file changed, 27 insertions(+), 4 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index e894f58c..62524b2d 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -10,7 +10,7 @@ This page contains a summary of the physical models implemented in the NEMO solv - [Thermodynamic Model](#thermodynamic-model) - [Finite Rate Chemistry](#finite-rate-chemistry) - [Vibrational Relaxation](#vibrational-relaxation) -- [Viscous Phenomena and Transport Coefficients](#viscous-phenomena-and-transport-coefficients) +- [Viscous Phenomena and Transport Coefficients](#viscous-phenomena-and-transport-coefficients) - [Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) - [Gupta-Yos](#gupta-yos) @@ -98,10 +98,10 @@ $$ The value of he equilibrium constant $K_{eq}$ is expressed as $$ - K_{eq} = \exp( A_0 T_m + A_1 + A_2 \log(1/T_m) + A_3 (1/T_m) + A_4 (1/T_m)^2 ), + K_{eq} = \exp( A_0 \left(\frac{T^c}{10,000}\right) + A_1 + A_2 \log \left( \frac{10,000}{T^c} \right) + A_3 \left( \frac{10,000}{T^c} \right) + A_4 \left( \frac{10,000}{T^c} \right)^2 ), $$ -where $T_m$ is a modified temperature and $A_1 - A_4$ are constants dependent on the reaction. These reaction constants, the rate constrolling temperature and Arrhenius parameters are stored within the fluid model class in SU2 NEMO. +where $T^c$ is a controlling temperature and $A_0 - A_4$ are constants dependent on the reaction. These reaction constants, the rate constrolling temperature and Arrhenius parameters are stored within the fluid model class in SU2 NEMO. --- @@ -145,7 +145,7 @@ where $\sigma_s$ is the effective collision~cross-section. Mass, momentum, and energy transport in fluids are all governed by molecular collisions, and expressions for these transport properties can be derived from the kinetic theory. The mass diffusion fluxes, $\mathbf{J}_s$, are computed using Fick's Law of Diffusion: $$ - \mathbf{J}_s = \rho D_s \nabla(c_s), + \mathbf{J}_s = - \rho D_s \nabla(Y_s) + Y_s \sum_k \rho D_k \nabla(Y_k) $$ where $c_s$ is the species mass fraction and $D_s$ is the species multi-component diffusion coefficient. The values of $D_s$ are computed as a weighted sum of binary diffusion coefficients between all species in the mixture. These are obtained by solving the Stefan--Maxwell equations under the Ramshaw approximations. The viscous stress tensor is written as @@ -197,7 +197,30 @@ $$ \phi_s = \sum_r X_r \left[ 1 + \sqrt{\frac{\mu_r}{\mu_s}}\left( \frac{M_r}{M_s} \right)^{1/4} \right]^{2} \left[ \sqrt{8 \left(1 + \frac{M_s}{M_r} \right)} \right]^{-1}. $$ +The effective species diffusion coefficeint is copmuted as a weighted sum of the species binary diffusion coefficients + +$$ +\frac{(1 - X_i)}{D_i} = \sum_{i\neqj} \frac{X_j}{D_{ij}}, +$$ + +where the binary diffusion coefficients are computed as + +$$ +\rho D_{ij} = 1.1613 \times 10^{-25} \frac{M \sqrt{T \left( \frac{1}{M_i} + \frac{1}{M_j} \right) }}{\Omega_{ij}^{(1,1)}}, +$$ + +and the collision integral for the mass diffusion coefficient is computed as + +$$ +\pi \Omega_{ij}^{(1,1)} = D T^{A(\log(T))^2 + B \log(T) + C}, +$$ + +where A-D are constants. + +The Wilkes-Blottner-Eucken model is generally efective up to temperatures of 10,000 K. Above these temperatures it is recommended to use the Gupta-Yos model. ## Gupta-Yos ## + + From 009f0f9c711da2a194533f8cdb8ee9fffbd97005 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Thu, 19 Jan 2023 23:52:26 -0800 Subject: [PATCH 06/16] start gy Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 62524b2d..7caf7ca3 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -221,6 +221,19 @@ The Wilkes-Blottner-Eucken model is generally efective up to temperatures of 10, ## Gupta-Yos ## +Aother model develped by Gupta focuses on the transport properties of weakly ionized flows, and is generally more accurate than the Wilkes-Blottner-Eucken model at temperatures above 10,000 K. +The mixutre viscoisty is computed as + +$$ +\mu = \sum_{s \neq e} \frac{m_s \gamma_s}{\sum_{r \neq e} \gamma_r \Delta_{s,r}^{(2)}(T_{tr}) + \gamma_r \Delta_{e,r}^{(2)}(T_{ve})} + \frac{m_e \gamma_e}{\sum_r \gamma_r \Delta_{e,r}^{(2)}(T_{ve}) } +$$ + +where + +$$ +\gamma_s = \frac{\rho_s}{\rho M_s} +$$ + From ddd6552c4be0806e6b4e0d5c8a642c32a56939ea Mon Sep 17 00:00:00 2001 From: jtneedels Date: Fri, 20 Jan 2023 10:28:12 -0800 Subject: [PATCH 07/16] finish gy Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 70 ++++++++++++++++++++++- 1 file changed, 68 insertions(+), 2 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 7caf7ca3..65e5c9d4 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -200,7 +200,7 @@ $$ The effective species diffusion coefficeint is copmuted as a weighted sum of the species binary diffusion coefficients $$ -\frac{(1 - X_i)}{D_i} = \sum_{i\neqj} \frac{X_j}{D_{ij}}, +\frac{(1 - X_i)}{D_i} = \sum_{i \neq j} \frac{X_j}{D_{ij}}, $$ where the binary diffusion coefficients are computed as @@ -223,8 +223,16 @@ The Wilkes-Blottner-Eucken model is generally efective up to temperatures of 10, Aother model develped by Gupta focuses on the transport properties of weakly ionized flows, and is generally more accurate than the Wilkes-Blottner-Eucken model at temperatures above 10,000 K. +The forumalae for the transport coefficients are dependent on the collision terms +$$ +\Delta_{s,r}^{(1)}(T) = \frac{8}{3} \left[ \frac{2M_s M_r}{\pi R T (M_s + M_r)} \right]^{1/2} \pi {\Omega_{s,r}^{(1,1)}} +$$ +and +$$ +\Delta_{s,r}^{(2)}(T) = \frac{16}{5} \left[ \frac{2M_s M_r}{\pi R T (M_s + M_r)} \right]^{1/2} \pi {\Omega_{s,r}^{(2,2)}}. +$$ The mixutre viscoisty is computed as $$ @@ -234,6 +242,64 @@ $$ where $$ -\gamma_s = \frac{\rho_s}{\rho M_s} +\gamma_s = \frac{\rho_s}{\rho M_s}. +$$ + +Thermal conductivity is computed in terms of different energy modes. The contribution due to translation modes is expressed as + +$$ +\kappa_t = \frac{15}{4} k_{B} \sum_{s \neq e} +\frac{\gamma_s}{\sum_{r \neq e} a_{s,r} \gamma_r \Delta_{s,r}^{(2)}(T_{tr}) + 3.54 \gamma_e \Delta_{s,e}^{(2)}(T_{ve})}, +$$ + +where + +$$ +a_{s,r} = 1 + \frac{\left[1 - (m_s/m_r) \right] \left[ 0.45 - 2.54(m_s/m_r) \right] }{\left[1 + (m_s/m_r) \right]^2} +$$ + +and where + +$$ +m_s = \frac{M_s}{N_{av}} +$$ + +with $N_{av}$ being Avogadro's Number. The thermal conductivity for the rotational modes is expressed as + +$$ +\kappa_r = k_{B} \sum_{s \neq e} +\frac{\gamma_s}{\sum_{r \neq e} \gamma_r \Delta_{s,r}^{(1)}(T_{tr}) + \gamma_e \Delta_{s,e}^{(1)}(T_{ve})}. $$ +The mixture translational/rotational thermal conductivity can then be expressed as + +$$ +\kappa_{tr} = \kappa_t + \kappa_r. +$$ + +The vibrational/electronic mode thermal conductivity is + +$$ +\kappa_{ve} = k_{B} \frac{C_{ve}}{R} \sum_{s \in molecules} \frac{\gamma_s} +{\sum_{r \neq e} \gamma_r \Delta_{s,r}^{(1)}(T_{tr}) + \gamma_e \Delta_{s,r}^{(1)}(T_{ve}) } +$$ + +and the thermal conductivity for electrons is given by + +$$ +\kappa_e = \frac{15}{4} k_{B} \frac{\gamma_e}{\sum_r 1.45 \gamma_r \Delta_{e,r}^{(2)}(T_{ve})}. +$$ + +Finally, the binary diffusion coefficient for heavy particles is given by + +$$ +D_{s,r} = \frac{k_{B} T_{tr}}{p \Delta_{s,r}^{(1)}(T_{tr})}, +$$ + +and for electrons, + +$$ +D_{e,r} = \frac{k_{B} T_{ve}}{p \Delta_{e,r}^{(1)}(T_{ve})}. +$$ + +--- From 62fe15d5ef435f9c1645a8fc91676bb16b8b61d6 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Fri, 20 Jan 2023 10:37:41 -0800 Subject: [PATCH 08/16] adding section headers Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 65e5c9d4..bb19ee21 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -13,6 +13,9 @@ This page contains a summary of the physical models implemented in the NEMO solv - [Viscous Phenomena and Transport Coefficients](#viscous-phenomena-and-transport-coefficients) - [Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) - [Gupta-Yos](#gupta-yos) + - [Sutherland Viscosity Model](#sutherland-viscosity-model) +- [Gas Surface Interaction](#gas-surface-interaction) +- [Slip Flow](#slip-flow) --- @@ -302,4 +305,8 @@ $$ D_{e,r} = \frac{k_{B} T_{ve}}{p \Delta_{e,r}^{(1)}(T_{ve})}. $$ +## Sutherland Viscosity Model ## + +In addition to the + --- From 2ecd392a0cf32c5fba01b5a82310546255d9dad9 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Fri, 20 Jan 2023 11:05:05 -0800 Subject: [PATCH 09/16] add sutherland, add slip flow Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 43 +++++++++++++++++++++-- 1 file changed, 40 insertions(+), 3 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index bb19ee21..aa9243eb 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -14,8 +14,8 @@ This page contains a summary of the physical models implemented in the NEMO solv - [Wilkes-Blottner-Eucken](#wilkes-blottner-eucken) - [Gupta-Yos](#gupta-yos) - [Sutherland Viscosity Model](#sutherland-viscosity-model) -- [Gas Surface Interaction](#gas-surface-interaction) - [Slip Flow](#slip-flow) +- [Gas-surface Interaction](#gas-surface-interaction) --- @@ -162,7 +162,7 @@ $$ where $\kappa^{k}$ is the thermal conductivity associated with energy mode $k$. -$D_s$, $\mu$, and $\kappa$ can be evaluated using either a Wilkes-Blottner-Eucken or Gupta-Yos transport models, with the implemntation details and reocmmendations on use given in the sections below. +$D_s$, $\mu$, and $\kappa$ can be evaluated using the models discussed below by selecting the appropriate options in the configuration file. ## Wilkes-Blottner-Eucken ## @@ -307,6 +307,43 @@ $$ ## Sutherland Viscosity Model ## -In addition to the +In addition to the two models discussed above, there is the option to use a Sutherland model to calculate the flow viscosity. The Sutherland model is not applicable at high temperatures. + +In this case the viscosity is computed as + +$$ +\mu = \mu_{0} \left( \frac{T}{T_{0}} \right)^{3/2} \frac{T_0 + S_{\mu}}{T + S_{\mu}}, +$$ + +where $T_0$ is a reference temperature (273.15 K), $\mu_0$ is a reference viscosity, and $S_{\mu}$ is the Sutherland constant. + +If the Sutherland model is selected with a NEMO solver, species diffusion coefficients and thermal conductivity are computed using the models described in the Wilkes-Blottner-Eucken section. + +--- + +# Slip Flow # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + +SU2-NEMO uses the Maxwell velocity and Smoluchowski temperature jump equations to compute the velocity and temperature of the gas in contact with the surface. The equations are given as +$$ +v_s = \frac{2 - \sigma}{\sigma} \lambda \frac{\partial v}{\partial n } + \\ +\frac{3}{4} \frac{\mu}{\rho T} \frac{\partial T}{\partial x}, +$$ + +and +$$ +T - T_w = \frac{2 - \alpha}{\alpha} \lambda \frac{2\gamma}{(\gamma + 1 )Pr} \frac{\partial T}{\partial n}, +$$ + +respectively, where $\mu$ is the flow viscosity, $\rho$ is the mixture density, $Pr$ is the Prandtl number, $\gamma$ is the specific heat ratio, $T$ is the temperature of the gas, $T_w$ is the temperature of the surface, and~= $\lambda$ is the mean free path, calculated as +$$ + \lambda = \frac{\mu}{\rho} \frac{\pi}{\sqrt{2RT}}. +$$ + +The coefficients $\sigma$ and $\alpha$ are referred to as the Tangential Momentum Accommodation Coefficient (TMAC) and the Thermal Accommodation Coefficient (TAC), respectively. The values of the accommodation coefficients depend on the physical characteristics of the surface, and are usually determined empirically. + --- From 253663292f2dae9168d7644eac9368c1694d5609 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Fri, 20 Jan 2023 11:14:40 -0800 Subject: [PATCH 10/16] add gsi Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 30 +++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index aa9243eb..da8f1d5c 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -346,4 +346,34 @@ $$ The coefficients $\sigma$ and $\alpha$ are referred to as the Tangential Momentum Accommodation Coefficient (TMAC) and the Thermal Accommodation Coefficient (TAC), respectively. The values of the accommodation coefficients depend on the physical characteristics of the surface, and are usually determined empirically. +--- + +# Gas-surface Interaction # + +| Solver | Version | +| --- | --- | +| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | + +Mechanisms of gas-surface interaction are implemented as specific boundary conditions within the SU2-NEMO computational suite. The net result of recombination reactions occurring on the surface is a production of chemical species due to catalytic reactions, $\dot{\omega}_s^{cat}$, that must be balanced by the normal diffusive and convective flux at the wall. For steady flow and a no-slip boundary, this can be expressed as + +$$ + \mathbf{J}_s \cdot \mathbf{n} = \dot{\omega}_s^{cat}. +$$ + +In SU2-NEMO, the chemical production of species due to catalytic processes is included in the computation of the viscous component of the residual, as an additional diffusive flux equivalent to the chemical source term computed due to catalytic reactions. Gradients of species density are then computed directly as part of the SU2-NEMO computational routine, which are used to compute gradients of species mass fraction at wall vertices. + +Options in SU2-NEMO include a super-catalytic wall in which species concentrations are set to specify full recombination to a specified equilibrium concentration (typically the free-stream conditions) + +$$ +Y_{w,s} = Y_{eq,s}, +$$ + +as well as a partiall catalytic wall using a specified reaction efficiency model + +$$ + \dot{\omega}_s^{cat} = \gamma_s Y_s \rho_w \sqrt{\frac{R_s T_w}{2\pi}}, +$$ + +where $\gamma_{s}$ is the species catalytic efficiency, and represents the proportion of incident mass flux of monatomic species $s$ which recombines into its heteronuclear diatomic molecule at the wall. + --- From ffc4e120aec438ca37dc97e05746a25023428748 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Fri, 20 Jan 2023 12:47:11 -0800 Subject: [PATCH 11/16] add collision cross section description Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 29 +++++++++++++++++++---- 1 file changed, 24 insertions(+), 5 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index da8f1d5c..291a849b 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -209,16 +209,32 @@ $$ where the binary diffusion coefficients are computed as $$ -\rho D_{ij} = 1.1613 \times 10^{-25} \frac{M \sqrt{T \left( \frac{1}{M_i} + \frac{1}{M_j} \right) }}{\Omega_{ij}^{(1,1)}}, +\rho D_{ij} = 1.1613 \times 10^{-25} \frac{M \sqrt{T \left( \frac{1}{M_i} + \frac{1}{M_j} \right) }}{\Omega_{ij}^{(1,1)}}. $$ -and the collision integral for the mass diffusion coefficient is computed as +Collision integrals are computed using a four parameter curve fit for neutral-neutral, neutral-ion, and electron-ion collisions $$ -\pi \Omega_{ij}^{(1,1)} = D T^{A(\log(T))^2 + B \log(T) + C}, +\pi \Omega_{ij}^{(n,n)} = D T^{A(\log(T))^2 + B \log(T) + C}, $$ -where A-D are constants. +where A-D are constants. Ion-ion, electron-ion, and electron-electron collisions modeled using a shielded Coulomb potential as + +$$ +\pi \Omega_{ij}^{(n,n)} = 5.0 \times 10^{15} \pi (\lambda_D / T)^2 \log \{D_n T^{*} \left[ 1 - C_n \exp\left( -c_n T^{*} \right) \right] + 1 \} +$$ + +where + +$$ +T^{*} = \frac{\lambda_D}{e^2_{CGS} / (k_{B,CGS} T) } +$$ + +and the Debye length $\lambda_D$ is defined as + +$$ +\lambda_D = \sqrt{\frac{k_{B,CGS} T}{4 \pi n_{e,CGS} e^2_{CGS}}}. +$$ The Wilkes-Blottner-Eucken model is generally efective up to temperatures of 10,000 K. Above these temperatures it is recommended to use the Gupta-Yos model. @@ -234,8 +250,11 @@ $$ and $$ -\Delta_{s,r}^{(2)}(T) = \frac{16}{5} \left[ \frac{2M_s M_r}{\pi R T (M_s + M_r)} \right]^{1/2} \pi {\Omega_{s,r}^{(2,2)}}. +\Delta_{s,r}^{(2)}(T) = \frac{16}{5} \left[ \frac{2M_s M_r}{\pi R T (M_s + M_r)} \right]^{1/2} \pi {\Omega_{s,r}^{(2,2)}}, $$ + +where the collision cross-sections are computed as described in the Wilkes-Blottner-Eucken section. + The mixutre viscoisty is computed as $$ From 314bb8e69bf97af679fe668bd1ed5950262fdf95 Mon Sep 17 00:00:00 2001 From: Jacob Needels <44848904+jtneedels@users.noreply.github.com> Date: Fri, 20 Jan 2023 20:33:57 -0800 Subject: [PATCH 12/16] Update _docs_v7/Thermochemical-Nonequilibrium.md Co-authored-by: Wally Maier --- _docs_v7/Thermochemical-Nonequilibrium.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 291a849b..bd8702fd 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -346,7 +346,7 @@ If the Sutherland model is selected with a NEMO solver, species diffusion coeffi | --- | --- | | `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | -SU2-NEMO uses the Maxwell velocity and Smoluchowski temperature jump equations to compute the velocity and temperature of the gas in contact with the surface. The equations are given as +SU2-NEMO uses the Maxwell velocity and Smoluchowski temperature jump equations to compute the velocity and temperature of the gas in contact with the surface. The equations are given as $$ v_s = \frac{2 - \sigma}{\sigma} \lambda \frac{\partial v}{\partial n } + \\ \frac{3}{4} \frac{\mu}{\rho T} \frac{\partial T}{\partial x}, From 3ac2259bab4108b3c67e371a9be697310ca569f3 Mon Sep 17 00:00:00 2001 From: Jacob Needels <44848904+jtneedels@users.noreply.github.com> Date: Fri, 20 Jan 2023 20:34:08 -0800 Subject: [PATCH 13/16] Update _docs_v7/Thermochemical-Nonequilibrium.md Co-authored-by: Wally Maier --- _docs_v7/Thermochemical-Nonequilibrium.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index bd8702fd..07a1283a 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -328,7 +328,7 @@ $$ In addition to the two models discussed above, there is the option to use a Sutherland model to calculate the flow viscosity. The Sutherland model is not applicable at high temperatures. -In this case the viscosity is computed as +In this case, the viscosity is computed as $$ \mu = \mu_{0} \left( \frac{T}{T_{0}} \right)^{3/2} \frac{T_0 + S_{\mu}}{T + S_{\mu}}, From 3cec6f5af4eedf01c86272119251a4c152b08ba6 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Sun, 22 Jan 2023 00:00:21 -0800 Subject: [PATCH 14/16] correcting models only in ns solver Signed-off-by: jtneedels --- _docs_v7/Thermochemical-Nonequilibrium.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_docs_v7/Thermochemical-Nonequilibrium.md b/_docs_v7/Thermochemical-Nonequilibrium.md index 291a849b..a3d8794c 100644 --- a/_docs_v7/Thermochemical-Nonequilibrium.md +++ b/_docs_v7/Thermochemical-Nonequilibrium.md @@ -143,7 +143,7 @@ where $\sigma_s$ is the effective collision~cross-section. | Solver | Version | | --- | --- | -| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | +| `NEMO_NAVIER_STOKES` | 7.0.0 | Mass, momentum, and energy transport in fluids are all governed by molecular collisions, and expressions for these transport properties can be derived from the kinetic theory. The mass diffusion fluxes, $\mathbf{J}_s$, are computed using Fick's Law of Diffusion: @@ -344,9 +344,9 @@ If the Sutherland model is selected with a NEMO solver, species diffusion coeffi | Solver | Version | | --- | --- | -| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | +| `NEMO_NAVIER_STOKES` | 7.0.0 | -SU2-NEMO uses the Maxwell velocity and Smoluchowski temperature jump equations to compute the velocity and temperature of the gas in contact with the surface. The equations are given as +SU2-NEMO uses the Maxwell velocity and Smoluchowski temperature jump equations to compute the velocity and temperature of the gas in contact with the surface. The equations are given as $$ v_s = \frac{2 - \sigma}{\sigma} \lambda \frac{\partial v}{\partial n } + \\ \frac{3}{4} \frac{\mu}{\rho T} \frac{\partial T}{\partial x}, @@ -371,7 +371,7 @@ The coefficients $\sigma$ and $\alpha$ are referred to as the Tangential Momentu | Solver | Version | | --- | --- | -| `NEMO_EULER`, `NEMO_NAVIER_STOKES` | 7.0.0 | +| `NEMO_NAVIER_STOKES` | 7.0.0 | Mechanisms of gas-surface interaction are implemented as specific boundary conditions within the SU2-NEMO computational suite. The net result of recombination reactions occurring on the surface is a production of chemical species due to catalytic reactions, $\dot{\omega}_s^{cat}$, that must be balanced by the normal diffusive and convective flux at the wall. For steady flow and a no-slip boundary, this can be expressed as From 24d23bb0381897f944665b0830465eff288a1c53 Mon Sep 17 00:00:00 2001 From: jtneedels Date: Sun, 22 Jan 2023 16:53:07 -0800 Subject: [PATCH 15/16] add detail on transport coeff model to phys def Signed-off-by: jtneedels --- _docs_v7/Physical-Definition.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/_docs_v7/Physical-Definition.md b/_docs_v7/Physical-Definition.md index 39440704..bf8d67bd 100644 --- a/_docs_v7/Physical-Definition.md +++ b/_docs_v7/Physical-Definition.md @@ -18,6 +18,7 @@ SU2 offers different ways of setting and computing this definition. This documen - [Free-Stream Definition (Thermochemical Nonequilibrium)](#free-stream-definition-thermochemical-nonequilibrium) - [Free-Stream Temperatures](#free-stream-temperatures) - [Chemical Composition and Mass Fractions](#chemical-composition-and-mass-fractions) +- [Transport Coefficient Model (Thermochemical Nonequilibrium)](#transport-coefficient-model-thermochemical-nonequilibrium) - [Flow Condition (Incompressible)](#flow-condition-incompressible) - [Thermodynamic and Gauge Pressure](#thermodynamic-and-gauge-pressure) - [Initial State and Non-Dimensionalization](#initial-state-and-non-dimensionalization) @@ -104,6 +105,18 @@ A chemistry model, consisting of a set of flow species, thermochemical propertie Free-stream mass fractions must also be specified in list using the option `GAS_COMPOSITION`. The mass fractions are specified as decimal values in the order of the species in the gas model. For example, an AIR-5 mixture of 77% oxygen and 23% nitrogen would be expressed as (0.77, 0.23, 0.00, 0.00, 0.00). +## Transport Coefficient Model (Thermochemical Nonequilibrium) ## + +| Solver | Version | +| --- | --- | +| `NEMO_NAVIER_STOKES` | 7.0.0 | + +A transport coefficient model must be specified for viscous simulations with the NEMO solver, using the `TRANSPORT_COEFF_MODEL` config option. Available options are the Wilkes-Blottner-Eucken, Gupta-Yos, and Sutherland viscosity models, specified by `WILKE`, `GUPTA-YOS`, and `SUTHERLAND`, respectively. The default option for transport coefficient model is Wilkes-Blottner-Eucken. + +It should be noted the Sutherland model is only used to evaluate viscosity, and the Wilkes-Blottner-Eucken model is used to evaluate diffusion coefficient and thermal conductivity. + +--- + ## Flow Condition (Incompressible) ## | Solver | Version | From baf42433e6a9dfd071f3a268aa4fa6474c1a0e3a Mon Sep 17 00:00:00 2001 From: Jacob Needels <44848904+jtneedels@users.noreply.github.com> Date: Mon, 20 Feb 2023 14:40:31 -0800 Subject: [PATCH 16/16] specify su2_noneq lib --- _docs_v7/Physical-Definition.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_docs_v7/Physical-Definition.md b/_docs_v7/Physical-Definition.md index bf8d67bd..1bb6589a 100644 --- a/_docs_v7/Physical-Definition.md +++ b/_docs_v7/Physical-Definition.md @@ -111,7 +111,7 @@ Free-stream mass fractions must also be specified in list using the option `GAS_ | --- | --- | | `NEMO_NAVIER_STOKES` | 7.0.0 | -A transport coefficient model must be specified for viscous simulations with the NEMO solver, using the `TRANSPORT_COEFF_MODEL` config option. Available options are the Wilkes-Blottner-Eucken, Gupta-Yos, and Sutherland viscosity models, specified by `WILKE`, `GUPTA-YOS`, and `SUTHERLAND`, respectively. The default option for transport coefficient model is Wilkes-Blottner-Eucken. +A transport coefficient model must be specified for viscous simulations with the NEMO solver, using the `TRANSPORT_COEFF_MODEL` config option. Available options when using the SU2TCLib thermochemical library are the Wilkes-Blottner-Eucken, Gupta-Yos, and Sutherland viscosity models, specified by `WILKE`, `GUPTA-YOS`, and `SUTHERLAND`, respectively. The default option for transport coefficient model is Wilkes-Blottner-Eucken. It should be noted the Sutherland model is only used to evaluate viscosity, and the Wilkes-Blottner-Eucken model is used to evaluate diffusion coefficient and thermal conductivity.