Derivations for LowMachNavierStokes Jacobians

Makefile

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+texsources := $(shell find -name '*.tex')
+
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 figures: $(figures)
 
-$(texfinal): *.tex $(bibfiles) $(figures) $(clsfiles) $(styfiles)
+$(texfinal): $(texsources) $(bibfiles) $(figures) $(clsfiles) $(styfiles)
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derivations/low_mach_navier_stokes_jacobians/low_mach_navier_stokes.tex

@@ -0,0 +1,140 @@
+\subsection{Background Information}
+The derivation detailed herein is for computing analytical Jacobians for the Low Mach Navier-Stokes equations. The final results are then implemented in GRINS src/physics/src/low\_mach\_navier\_stokes.C, with the goal being that there
+will be a significant speedup in computation over using the standard finite difference approximation. The derivation is broken up into 3 subsections: mass (P residual), momentum (U residual), and energy (T residual).
+Note that a sign convention has been specified for purposes of optimization.\\
+Also note that, in this case, the following parameters are considered functions of temperature, T:
+\begin{itemize}
+    \item density, $\rho$
+    \item specific heat, $c_p$
+    \item viscosity, $\mu$
+    \item thermal conductivity, $k$
+\end{itemize}
+For the formulations, \textbf{u}=\{$P,U,T$\} and the test functions \textbf{$\Phi$}=\{$\xi,\phi,\psi$\}.\\
+The variable names used in the code follow this template: $Kxy$, where $x$ is the residual and $y$ is the derivation variable.  For example, $KPT$ is the derivative of the P residual with respect to temperature,
+while $KuP$ is the derivative of the $u$-velocity component of the U residual with respect to pressure. \newline
+Lastly, a note on notation. Index notation is used throughout this section in order to make the derivations clearer. Indices $a-h$ are used for vector components, while $i-r$ are used for degrees of freedom.
+Also note that Cartesian coordiantes are assumed for these derivations. As such, in the momentum equation in particular, terms of the following form may appear: $\phi_i^{a,b}$. In the index notation used herein for the case
+where both sub- and superscripts appear on a single variable, 
+subscripts refer exclusively to degrees of freedom, while superscripts are used exclusively for vector operations (i.e. divergence or gradient).
+
+\newpage
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Mass Equation}
+The given P residual:
+\begin{equation}
+    F_{p,i} = (-(U \cdot \nabla T) T^{-1} + \nabla \cdot U)\xi_i
+\end{equation}
+
+Now converting to tensor notation
+
+\begin{equation}
+    F_{p,i} = -u_a T_{,a} T^{-1} \xi_i + u_{b,b}\xi_i
+\end{equation}
+
+Galerkin approximation:
+\begin{align*}
+    u_a &= \sum_i u_i \phi_i^a\\
+    T &= \sum_i T_i \psi_i
+\end{align*}
+
+\begin{equation}
+    F_{p,i} = -u_k \phi_k^a T_l \psi_l^{,a} (T_m \psi_m)^{-1} \xi_i + u_n \phi_n^{b,b} \xi_i
+\end{equation}
+
+The P residual is a function of velocity, $U$, and temperature, $T$, so derivatives must be taken with respect to both of them using the following form:
+
+\begin{align*}
+    \frac{\partial}{\partial u_j} (u_i \phi_i^a) &= \phi_j^a \\
+    \frac{\partial}{\partial T_j} (T_i \psi_i) &= \psi_j
+\end{align*}
+
+\begin{align}
+    \frac{\partial F_{p,i}}{\partial u_j} = &-\phi_j^{a} T_l \psi_l^{,a} (T_m \psi_m)^{-1} \xi_i + \phi_j^{b,b} \xi_i \nonumber \\
+    \frac{\partial F_{p,i}}{\partial u_j} = &-\phi_j^{a} T_{,a} T^{-1} \xi_i + \phi_j^{b,b} \xi_i \label {fp_du}\\ 
+    \nonumber \\
+    \frac{\partial F_{p,i}}{\partial T_j} = &-u_k \phi_k^a \xi_i \left [ T_l
+\psi_l^{,c} (-1) (T_m \psi_m)^{-2} (\psi_j) + (T_n \psi_n)^{-1} \psi_j^{,d} \right ] + 0 \nonumber \\
+    \frac{\partial F_{p,i}}{\partial T_j} = &u_c T_{,c} T^{-2} \psi_j \xi_i - u_d T^{-1} \psi_j^{,d} \xi_i \label{fp_dT}
+\end{align}
+
+Equations \ref{fp_du} and \ref{fp_dT} above are the generalized Jacobian terms for the P residual. When implemented in LMNS.C, the following terms will result:\\
+$KPu, KPv, KPw, KPT$
+
+\newpage
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Momentum Equation}
+The U residual is given in vector and tensor notation as follows:
+\begin{align}
+    F_{U,i} &= -\rho u \cdot \nabla u \cdot \phi_i + P(\nabla \cdot \phi_i) - \mu \left [\nabla u : \nabla \phi_i + (\nabla u)^T : \nabla \phi_i - \frac{2}{3} (\nabla \cdot u) I : \nabla \phi_i \right ] + \rho \mathbf{g} \cdot \phi_i \\
+    F_{U,i}^a &= -\rho u_b u_{a,b} \phi_i^a + P\phi_i^{a,a} - \mu \left [u_{a,c} \phi_i^{a,c} + u_{d,a} \phi_i^{a,d} - \frac{2}{3} u_{e,e} \delta_{a,f} \phi_i^{a,f} \right ] + \rho g_a \phi_i^a \ (no \ sum \ on \ a) \label{Fui_tensor}
+\end{align}
+
+Note that in \ref{Fui_tensor}, $\delta_{a,f}$ is the Kronecker delta function. \\
+
+Now, the Galerkin approximation
+
+\begin{align*}
+    u_a &= \sum_i u_i \phi_i^a\\
+    P &= \sum_i P_i \xi_i\\
+    T &= \sum_i T_i \psi_i
+\end{align*}
+
+\begin{align}
+    F_{U,i}^a =  -\rho u_k \phi_k^b u_l \phi_l^{a,b} \phi_i^a + P_m \xi_m \phi_i^{a,a} - \mu \left [u_n \phi_n^{a,c} \phi_i^{a,c} + u_o \phi_o^{d,a} \phi_i^{a,d} - \frac{2}{3} u_p \phi_p^{e,e} \delta_{a,f} \phi_i^{a,f} \right ] + \rho g_a \phi_i^a \label{u_resid}
+\end{align}
+
+Equation \ref{u_resid} above is a function of velocity, pressure, and temperature (due to $\rho = \rho(T)$ and $\mu = \mu(T)$ \ ), so derivatives must be taken with respect to all of them.\\
+\textbf{Also note that, as stated in \ref{Fui_tensor}, there is no sum over index $a$}.
+
+\begin{align}
+   \frac{\partial F_{U,i}^a}{\partial u_j} = &-\rho \left [ u_k \phi_k^b \phi_j^{a,b} + u_l \phi_l^{a,g} \phi_j^g \right ] \phi_i^a 
+        + 0 - \mu \left [\phi_j^{a,c} \phi_i^{a,c} + \phi_j^{d,a} \phi_i^{a,d} - \frac{2}{3} \phi_j^{e,e} \delta_{a,f} \phi_i^{a,f} \right ] \nonumber \\     
+   \frac{\partial F_{U,i}^a}{\partial u_j} = &-\rho \left [ u_b \phi_j^{a,b} + u_{a,g} \phi_j^g \right ] \phi_i^a 
+        - \mu \left [\phi_j^{a,c} \phi_i^{a,c} + \phi_j^{d,a} \phi_i^{a,d} - \frac{2}{3} \phi_j^{e,e} \delta_{a,f} \phi_i^{a,f} \right ] \label{du_du} \\
+   \nonumber \\ 
+   \frac{\partial F_{U,i}^a}{\partial P_j} = &\xi_j \phi^{a,a} \label{du_dp} \\
+   \nonumber \\
+   \frac{\partial F_{U,i}^a}{\partial T_j} = &-\frac{d\rho}{dT} \frac{d}{dT_j}(T_k \psi_k) u_l \phi_l^b u_m \phi_m^{a,b} \phi_i^a
+        + 0 \nonumber \\ &- \left[ 0 + \frac{d\mu}{dT} \frac{d}{dT_j}(T_n \psi_n) [u_o \phi_o^{a,c} \phi_i^{a,c} + u_p \phi_p^{d,a} \phi_i^{a,d} - \frac{2}{3} u_q \phi_q^{e,e} \delta_{a,f} \phi_i^{a,f} ] \right ] \nonumber
+        \\ &+ \frac{d\rho}{dT} \frac{d}{dT_m}(T_r \psi_r) g_a \phi_a \nonumber \\       
+   \frac{\partial F_{U,i}^a}{\partial T_j} = &-\frac{d\rho}{dT} \psi_j u_b u_{a,b} \phi_i^a
+        - \frac{d\mu}{dT} \psi_j \left [u_{a,c} \phi_i^{a,c} + u_{d,a} \phi_i^{a,d} - \frac{2}{3} u_{e,e} \delta_{a,f} \phi_i^{a,f} \right ]
+        + \frac{d\rho}{dT} \psi_j g_a \phi_i^a \label{du_dT} 
+\end{align}
+
+Equations \ref{du_du}, \ref{du_dp}, and \ref{du_dT} above can then be implement in LMNS.C to create the following terms:\\
+$Kuu, Kuv, Kuw, Kvu, Kvv, Kvw, Kwu, Kwv, Kww, KuT, KvT, KwT, KuP, KvP, KwP$
+
+\newpage
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Energy Equation}
+The T residual is given as follows:
+\begin{align}
+    F_{T,i} &= -\rho c_p (u \cdot \nabla T) \psi_i - k ( \nabla T \cdot \nabla \psi_i) \\
+    F_{T,i} &= -\rho c_p (u_a T_{,a})\psi_i - k T_{,b} \psi_i^{,b}
+\end{align}
+
+Again, note that $\rho = \rho(T)$, $c_p = c_p(T)$, and $k = k(T)$
+
+\begin{align}
+    F_{T,i} = -\rho c_p u_k \phi_k^a T_l \psi_l^{,a} \psi_i - k T_m \psi_m^{,b} \psi_i^{,b}
+\end{align}
+
+Noting $F_{T,i}$ is a function of velocity and temperature.
+
+\begin{align}
+    \frac{\partial F_{T,i}}{\partial u_j} = &-\rho c_p \phi_j^a T_l \psi_l^{,a} \psi_i - 0 \nonumber \\
+    \frac{\partial F_{T,i}}{\partial u_j} = &-\rho c_p \phi_j^a T_{,a} \psi_i \label{dT_du} \\
+    \nonumber \\
+    \frac{\partial F_{T,i}}{\partial T_j} = &-\psi_i \left [ \rho [ c_p u_k \phi_k^a \psi_j^{,a} + u_l \phi_l^b T_m \psi_m^{,b} \frac{d c_p}{dT} \psi_j ] + 
+    [ c_p u_n \phi_n^{c} T_o \psi_o^{,c} ] \frac{d \rho}{dT} \psi_j  \right ] \nonumber \\
+                                      &- k \psi_j^{,d} \psi_i^{,d} - T_p \psi_p^{,e} \psi_i^{,e} \frac{dk}{dT} \psi_j \nonumber \\
+    \frac{\partial F_{T,i}}{\partial T_j} = &-\psi_i \left [ \rho [ c_p u_a \psi_j^{,a} + u_b T_{,b} \frac{d c_p}{dT} \psi_j ] + 
+    [ c_p u_c T_{,c} ] \frac{d \rho}{dT} \psi_j  \right ] \nonumber \\
+                                      &- k \psi_j^{,d} \psi_i^{,d} - T_{,e} \psi_i^{,e} \frac{dk}{dT} \psi_j \label{dT_dT}
+\end{align}
+
+Equations \ref{dT_du} and \ref{dT_dT} above will result in the following terms for LMNS.C:\\
+$KTu, KTv, KTw, KTT$
+
+\end{document}

main.tex

@@ -75,4 +75,10 @@ \chapter{Stokes}
 \bibliographystyle{abbrv}
 \bibliography{refs}
 
+\chapter{Low Mach Navier Stokes}
+
+\section{Analytic Jacobians}
+
+\input{./derivations/low_mach_navier_stokes_jacobians/low_mach_navier_stokes.tex}
+
 \end{document}