From 2012 Petrova, "Finite Volume Methods- Powerful Means of Engineering Design"

$\bar\phi = \frac{1}{\Delta t} \int_{t_0}^{t_0 + \Delta t} \phi dt = $ Reynolds averaging over time.

$\tilde \phi = \frac{1}{\bar{\rho}} \frac{1}{\Delta t} \int_{t_0}^{t_0 + \Delta t} (\rho \phi) dt = $ Favre averaging.

Notice this means $\tilde{\phi} = \frac{ \overline{\rho \phi} }{ \bar{\phi} }$ and $\bar{\rho} \tilde{\phi} = \overline{\rho \phi}$.

Does this mean that $\frac{\partial \tilde \phi}{\partial \bar \rho} = -\frac{1}{\bar{\rho}} \tilde \phi$, or should we pretend these are different variables altogether? It does if I use $\overline{\rho \phi}$ instead of $\tilde{\phi}$ as primitive variables. I'm going to just say they are different variables.
Also, how does that $\rho$ inside the integral affect $\frac{\partial \tilde \phi}{\partial \bar \rho}$?

conserved quantities:
$U^I = \left[\begin{matrix} \bar{\rho} \\ \bar{\rho} \tilde{v}^i \\ \bar{\rho} \tilde e_{total} \\ \bar{\rho} k \\ \bar{\rho} \omega \end{matrix}\right] = \left[\begin{matrix} \bar{\rho} \\ \overline{\rho v^i} \\ \overline{\rho e_{total}} \\ \bar{\rho} k \\ \bar{\rho} \omega \end{matrix}\right]$

primitive variables:
$W = \left[\begin{matrix} \bar{\rho} \\ \tilde{v}^i \\ P^* \\ k \\ \omega \end{matrix}\right]$

If the derivative of the Favre-average wrt the Reynolds-averaged density is non-zero, then perhaps the Favre-averaged variable shouldn't be the primitive variable?
Maybe it is safest to simply not consider primitive variables? But then won't the eigensystem be complicated to compute, without using the acoustic matrix?

$\bar{\rho} = $ Reynolds averaged density.
$\tilde{v}^i = $ Favre averaged velocity.
$\tilde e_{total} = $ specific total energy.
$P^* = $ static pressure (?).
$k =$ turbulent kinetic energy.
$\omega =$ specific turbulent dissipation rate.

converting between variables:
$\tilde e_{total} = \tilde e_{int} + \frac{1}{2} \tilde{v}^2 + k$
$\tilde e_{int} = C_v \tilde T$
$\tilde e_{total} = C_v \tilde T + \frac{1}{2} \tilde{v}^2 + k$
$\tilde T = \frac{1}{C_v} \tilde e_{int}$
$\bar P = \bar \rho R \tilde T$
$\tilde e_{total} = \frac{C_v}{\bar \rho R} \bar P + \frac{1}{2} \tilde{v}^2 + k$

$P^* = \bar P + \frac{2}{3} \bar \rho k$
$P^* = \bar \rho R \tilde T + \frac{2}{3} \bar \rho k$
$P^* = \frac{R}{C_v} \bar \rho \tilde e_{int} + \frac{2}{3} \bar \rho k$
$P^* = \frac{R}{C_v} (\bar \rho \tilde e_{total} - \frac{1}{2} \bar{\rho} \tilde{v}^2 - \bar{\rho} k) + \frac{2}{3} \bar \rho k$
$\bar P = P^* - \frac{2}{3} \bar \rho k$
$\tilde e_{total} = \frac{C_v}{\bar \rho R} (P^* - \frac{2}{3} \bar \rho k) + \frac{1}{2} \tilde{v}^2 + k$
$\tilde e_{total} = \frac{1}{2} \tilde{v}^2 + \frac{C_v}{\bar \rho R} P^* + (1 - \frac{2 C_v}{3 R}) k$

Conserved Quantities wrt Primitive Variables:

$U^I = \left[\begin{matrix} \bar{\rho} \\ \bar{\rho} \tilde{v}^i \\ \bar{\rho} \tilde e_{total} \\ \bar{\rho} k \\ \bar{\rho} \omega \end{matrix}\right] = \left[\begin{matrix} \bar{\rho} \\ \bar{\rho} \tilde{v}^i \\ \frac{C_v}{R} P^* + (1 - \frac{2}{3} \frac{C_v}{R}) \bar \rho k + \frac{1}{2} \bar{\rho} \tilde{v}^2 \\ \bar{\rho} k \\ \bar{\rho} \omega \end{matrix}\right]$

$\frac{\partial U^I}{\partial W^J} = \left[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ \tilde{v}^i & \bar \rho \delta^i_j & 0 & 0 & 0 \\ \frac{1}{2} \tilde{v}^2 + (1 - \frac{2 C_v}{3 R}) k & \bar \rho \tilde{v}_j & \frac{C_v}{R} & (1 - \frac{2 C_v}{3 R}) \bar \rho & 0 \\ k & 0 & 0 & \bar \rho & 0 \\ \omega & 0 & 0 & 0 & \bar \rho \end{matrix}\right]$

$\frac{\partial W^I}{\partial U^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ -\frac{1}{\bar{\rho}} \tilde{v}^i & \frac{1}{\bar{\rho}} \delta^i_j & 0 & 0 & 0 \\ \frac{R}{2 C_v} \tilde{v}^2 & - \frac{R}{C_v} \tilde{v}_j & \frac{R}{C_v} & \frac{2}{3} - \frac{R}{C_v} & 0 \\ -\frac{1}{\bar{\rho}} k & 0 & 0 & \frac{1}{\bar{\rho}} & 0 \\ -\frac{1}{\bar{\rho}} \omega & 0 & 0 & 0 & \frac{1}{\bar{\rho}} \end{matrix}\right] }$

Flux:

$F^I = \downarrow I \left[\begin{matrix} \bar{\rho} \tilde{v}^k n_k \\ \bar{\rho} \tilde{v}^i \tilde{v}^k n_k + P^* n^i \\ \tilde{v}^k n_k (\bar{\rho} \tilde e_{total} + P^*) \\ \bar{\rho} \tilde{v}^k n_k k \\ \bar{\rho} \tilde{v}^k n_k \omega \end{matrix}\right] $

$F^I = \downarrow I \left[\begin{matrix} \bar{\rho} \tilde{v}^k n_k \\ \bar{\rho} \tilde{v}^i \tilde{v}^k n_k + P^* n^i \\ \tilde{v}^k n_k (\bar{\rho} ( \frac{1}{2} \tilde{v}^2 + \frac{C_v}{\bar \rho R} P^* + (1 - \frac{2 C_v}{3 R}) k ) + P^*) \\ \bar{\rho} \tilde{v}^k n_k k \\ \bar{\rho} \tilde{v}^k n_k \omega \end{matrix}\right] $

$F^I = \downarrow I \left[\begin{matrix} \bar{\rho} \tilde{v}^k n_k \\ \bar{\rho} \tilde{v}^i \tilde{v}^k n_k + P^* n^i \\ \frac{1}{2} \bar{\rho} \tilde{v}^2 \tilde{v}^k n_k + \frac{C_v}{R} \tilde{v}^k n_k P^* + (1 - \frac{2 C_v}{3 R}) \bar{\rho} \tilde{v}^k n_k k + P^* \\ \bar{\rho} \tilde{v}^k n_k k \\ \bar{\rho} \tilde{v}^k n_k \omega \end{matrix}\right] $

Flux derivative with respect to primitive variables:

$\frac{\partial F^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix} \tilde{v}^k n_k & \bar{\rho} n_j & 0 & 0 & 0 \\ \tilde{v}^i \tilde{v}^k n_k & \bar{\rho} ( \delta^i_j \tilde{v}^k n_k + \tilde{v}^i n_j ) & n^i & 0 & 0 \\ \frac{1}{2} \tilde{v}^2 \tilde{v}^k n_k + (1 - \frac{2 C_v}{R}) \tilde{v}^k n_k k & \bar{\rho} ( \tilde{v}_j \tilde{v}^k n_k + \frac{1}{2} \tilde{v}^2 n_j ) + \frac{C_v}{R} P^* n_j + (1 - \frac{2 C_v}{3 R}) \bar{\rho} n_j k & \frac{C_v}{R} \tilde{v}^k n_k & -\frac{2 C_v}{3 R} \bar{\rho} \tilde{v}^k n_k & 0 \\ \tilde{v}^k n_k k & \bar{\rho} n_j k & 0 & \bar{\rho} \tilde{v}^k n_k & 0 \\ \tilde{v}^k n_k \omega & \bar{\rho} n_j \omega & 0 & 0 & \bar{\rho} \tilde{v}^k n_k \end{matrix}\right] }$

Acoustic matrix:

${A^I}_J + \delta^I_J = \frac{\partial W^I}{\partial U^K} \cdot \frac{\partial F^K}{\partial W^J}$

${A^I}_J + \delta^I_J = \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ -\frac{1}{\bar{\rho}} \tilde{v}^i & \frac{1}{\bar{\rho}} \delta^i_k & 0 & 0 & 0 \\ \frac{R}{2 C_v} \tilde{v}^2 & - \frac{R}{C_v} \tilde{v}_k & \frac{R}{C_v} & \frac{2}{3} - \frac{R}{C_v} & 0 \\ -\frac{1}{\bar{\rho}} k & 0 & 0 & \frac{1}{\bar{\rho}} & 0 \\ -\frac{1}{\bar{\rho}} \omega & 0 & 0 & 0 & \frac{1}{\bar{\rho}} \end{matrix}\right] } \cdot \downarrow K \overset{\rightarrow J}{ \left[\begin{matrix} \tilde{v}^l n_l & \bar{\rho} n_j & 0 & 0 & 0 \\ \tilde{v}^k \tilde{v}^l n_l & \bar{\rho} ( \delta^k_j \tilde{v}^l n_l + \tilde{v}^k n_j ) & n^k & 0 & 0 \\ \frac{1}{2} \tilde{v}^2 \tilde{v}^l n_l + (1 - \frac{2 C_v}{R}) \tilde{v}^l n_l k & \bar{\rho} ( \tilde{v}_j \tilde{v}^l n_l + \frac{1}{2} \tilde{v}^2 n_j ) + \frac{C_v}{R} P^* n_j + (1 - \frac{2 C_v}{3 R}) \bar{\rho} n_j k & \frac{C_v}{R} \tilde{v}^l n_l + 1 & (1 - \frac{2 C_v}{3 R}) \bar{\rho} \tilde{v}^l n_l & 0 \\ \tilde{v}^l n_l k & \bar{\rho} n_j k & 0 & \bar{\rho} \tilde{v}^l n_l & 0 \\ \tilde{v}^l n_l \omega & \bar{\rho} n_j \omega & 0 & 0 & \bar{\rho} \tilde{v}^l n_l \end{matrix}\right] } $

${A^I}_J + \delta^I_J = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix} \tilde{v}^l n_l & \bar{\rho} n_j & 0 & 0 & 0 \\ 0 & \delta^i_j \tilde{v}^l n_l & \frac{1}{\bar{\rho}} n^i & 0 & 0 \\ \frac{1}{6} \tilde{v}^l n_l k & P^* n_j & \frac{R}{C_v} (1 - \tilde{v}_k n^k) + \tilde{v}^k n_k & 0 & 0 \\ \end{matrix}\right] }$

Flux derivative with respect to direction:

${F^{Ij}}_{,j} = \left[\matrix{ \bar\rho_{,j} \tilde v^j + \bar\rho \tilde {v^j}_{,j} \\ \bar\rho_{,j} \tilde v^i \tilde v^j + \bar\rho \tilde {v^i}_{,j} \tilde v^j + \bar\rho \tilde v^i \tilde {v^j}_{,j} + \delta^{ij} {P^*}_{,j} \\ \tilde {v^j}_{,j} (\bar\rho \tilde e_{total} + P^*) + \tilde v^j ( (1 + \frac{C_v}{R}) {P^*}_{,j} + (1 - \frac{2 C_v}{3 R}) (\bar\rho_{,j} k + \bar\rho k_{,j}) + \frac{1}{2} \bar\rho_{,j} \tilde v^2 + \bar\rho \tilde v^j \tilde v_{,j} ) \\ \bar\rho_{,j} \tilde v^j k + \bar\rho \tilde {v^j}_{,j} k + \bar\rho \tilde v^j k_{,j} \\ \bar\rho_{,j} \tilde v^j \omega + \bar\rho \tilde {v^j}_{,j} \omega + \bar\rho \tilde v^j \omega_{,j} }\right]$

$\frac{\partial F^{Ij}}{\partial x^j} = \frac{\partial F^{Ij}}{\partial W^K} \frac{\partial W^K}{\partial x^j}$
$= \left[\matrix{ \tilde v^j & \bar\rho \delta^j_k & 0 & 0 & 0 \\ \tilde v^i \tilde v^j & \bar\rho (\tilde v^j \delta^i_k + \tilde v^i \delta^j_k) & \delta^{ij} & 0 & 0 \\ ((1 - \frac{2 C_v}{3 R}) k + \frac{1}{2} \tilde v^2) \tilde v^j & (\bar\rho \tilde e_{total} + P^*) \delta^j_k + \bar\rho \tilde v^j \tilde v_k & (1 + \frac{C_v}{R}) \tilde v^j & (1 - \frac{2 C_v}{3 R}) \tilde v^j & 0 \\ \tilde v^j k & \bar\rho \delta^j_k k & 0 & \bar\rho \tilde v^j & 0 \\ \tilde v^j \omega & \bar\rho \delta^j_k \omega & 0 & 0 & \bar\rho \tilde v^j }\right] \left[\matrix{ \bar\rho \\ \tilde v^k \\ P^* \\ k \\ \omega }\right]_{,j}$
$= \frac{\partial F^{Ij}}{\partial W^K} \frac{\partial W^K}{\partial U^L} \frac{\partial U^L}{\partial x^j}$
$= \left[\matrix{ \tilde v^j & \bar\rho \delta^j_k & 0 & 0 & 0 \\ \tilde v^i \tilde v^j & \bar\rho (\tilde v^j \delta^i_k + \tilde v^i \delta^j_k) & \delta^{ij} & 0 & 0 \\ ((1 - \frac{2 C_v}{3 R}) k + \frac{1}{2} \tilde v^2) \tilde v^j & (\bar\rho \tilde e_{total} + P^*) \delta^j_k + \bar\rho \tilde v^j \tilde v_k & (1 + \frac{C_v}{R}) \tilde v^j & (1 - \frac{2 C_v}{3 R}) \tilde v^j & 0 \\ \tilde v^j k & \bar\rho \delta^j_k k & 0 & \bar\rho \tilde v^j & 0 \\ \tilde v^j \omega & \bar\rho \delta^j_k \omega & 0 & 0 & \bar\rho \tilde v^j }\right] \left[\matrix{ 1 & 0 & 0 & 0 & 0 \\ -\frac{1}{\bar\rho} \tilde v^k & \frac{1}{\bar\rho} \delta^k_l & 0 & 0 & 0 \\ \frac{R}{2 C_v} \tilde v^2 & - \frac{R}{C_v} \tilde v_l & \frac{R}{C_v} & \frac{2}{3} - \frac{R}{C_v} & 0 \\ -\frac{1}{\bar\rho} k & 0 & 0 & \frac{1}{\bar\rho} & 0 \\ -\frac{1}{\bar\rho} \omega & 0 & 0 & 0 & \frac{1}{\bar\rho} }\right] \left[\matrix{ \bar\rho \\ \bar\rho \tilde v^l \\ \bar\rho \tilde e_{total} \\ \bar\rho k \\ \bar\rho \omega }\right]_{,j}$
$= \left[\matrix{ 0 & \delta^i_l & 0 & 0 & 0 \\ - \tilde v^i \tilde v^j + \frac{R}{2 C_v} \tilde v^2 \delta^{ij} & \tilde v^j \delta^i_l + \tilde v^i \delta^j_l - \frac{R}{2 C_v} \tilde v_l \delta^{ij} & \frac{R}{C_v} \delta^{ij} & (\frac{2}{3} - \frac{R}{C_v}) \delta^{ij} & 0 \\ \tilde v^j ( (1 - \frac{1}{\bar\rho}) (1 - \frac{2 C_v}{3 R}) k + \frac{R}{2 C_v} \tilde v^2 - \frac{1}{\bar\rho} (\bar\rho \tilde e_{total} + P^*) ) & \frac{1}{\bar\rho} (\bar\rho \tilde e_{total} + P^*) \delta^j_l - \frac{R}{C_v} \tilde v^j \tilde v_l & (1 + \frac{R}{C_v}) \tilde v^j & (-\frac{1}{3} - \frac{R}{C_v} + \frac{2}{3} \frac{C_v}{R}) \tilde v^j + \frac{1}{\bar\rho} (1 - \frac{2 C_v}{3 R}) \tilde v^j & 0 \\ - \tilde v^j k & \delta^j_l k & 0 & \tilde v^j & 0 \\ - \tilde v^j \omega & \delta^j_l \omega & 0 & 0 & \tilde v^j }\right] \left[\matrix{ \bar\rho \\ \bar\rho \tilde v^l \\ \bar\rho \tilde e_{total} \\ \bar\rho k \\ \bar\rho \omega }\right]_{,j}$