25. Density-Pressure Linearization In Pressure Correction

Compressible pressure-correction equations need a relation between a pressure correction and the density change it induces. During the pressure-correction solve, the non-pressure thermodynamic / scalar state is treated as fixed or lagged, so the pressure-sensitive part of the density response can be linearized.

For cell \(P\), define the EOS-updated density at the perturbed pressure but lagged thermodynamic / composition state by

(25.1)\[\widetilde{\rho}_P \equiv \rho\!\left(p_P^*+p_P',s_P^*,Y_P^*\right), \qquad \rho_P^* \equiv \rho\!\left(p_P^*,s_P^*,Y_P^*\right).\]

The first-order Taylor expansion of the equation of state about the current pressure-correction reference state is

(25.2)\[\widetilde{\rho}_P = \rho_P^* + \left. \left( \frac{\partial \rho}{\partial p} \right)_{s,Y} \right\rvert_{(p_P^*,\,s_P^*,\,Y_P^*)} p_P' + O\!\left((p_P')^2\right).\]

The density correction is the change relative to the reference state,

(25.3)\[\rho_P' \equiv \widetilde{\rho}_P-\rho_P^*,\]

so the generic first-order relation is

(25.4)\[\rho_P' \approx \left. \left( \frac{\partial \rho}{\partial p} \right)_{s,Y} \right\rvert_{(p_P^*,\,s_P^*,\,Y_P^*)} p_P'.\]

The symbol \(s\) specifies the thermodynamic tangent used in the local EOS response. It does not introduce entropy as a primary solved variable in the pressure-correction equation, nor does it assert that the global flow evolution is isentropic. The acoustic sound speed is defined by

(25.5)\[a_{s,P}^2 = \left. \left( \frac{\partial p}{\partial \rho} \right)_{s,Y} \right\rvert_{(p_P^*,\,s_P^*,\,Y_P^*)}.\]

Taking the reciprocal gives

(25.6)\[\left. \left( \frac{\partial \rho}{\partial p} \right)_{s,Y} \right\rvert_{(p_P^*,\,s_P^*,\,Y_P^*)} = \frac{1}{a_{s,P}^2}.\]

Substitution into Eq. (25.4) gives the acoustic pressure-correction relation used in the main derivation:

(25.7)\[\rho_P' \approx \frac{p_P'}{a_{s,P}^2}.\]

If one instead writes continuity in the normalized volumetric form

(25.8)\[\frac{1}{\rho} \left[ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) \right] =0,\]

then the pressure-correction linearization is a relative density change rather than an absolute one. To keep that normalization explicit and linear, the density in the denominator is taken from the current reference state, \(\rho_P^*\), not from the updated quantity \(\widetilde{\rho}_P\). Using \(\widetilde{\rho}_P\) would put the unknown corrected density in the denominator and would go beyond the present first-order linearization. With that convention,

(25.9)\[\frac{\rho_P'}{\rho_P^*} \approx \frac{p_P'}{\rho_P^*a_{s,P}^2}.\]

This is an EOS-general acoustic linearization. The equation of state enters through the thermodynamic sound speed, so the relation is not limited to ideal gases. It is also not the same as using an arbitrary thermodynamic derivative such as \((\partial \rho/\partial p)_{T,Y}\) or \((\partial \rho/\partial p)_{h,Y}\). For ideal gases, \(1/a_{s,P}^2 = 1/(\gamma R_m T_P)=\rho_P/(\gamma p_P)\), so the acoustic linearization reduces to the familiar ideal-gas special case.

Two qualifications are important.

First, Eq. (25.7) is a local first-order linearization used inside the pressure-correction step. It is not an assertion that the full nonlinear density update is always exactly \(p'/a_s^2\). In the segregated algorithm, the pressure-correction solve treats the current thermodynamic / scalar state as lagged while estimating the density change induced by a pressure correction. Density changes driven by updates to temperature, species, volume fraction, chemistry, or other closure variables are accounted for elsewhere in the nonlinear iteration sequence.

Second, the numerical importance of the compressible correction terms is not controlled by “density gradient” alone. The controlling scale in the pressure-correction equation is the local acoustic compressibility \(\chi_p \sim 1/a_s^2\) together with the flux and time factors that multiply it. In low-Mach or very stiff states with large \(a_s\), these terms may be small. When the local sound speed is small, when the relative face flux is large, when the timestep makes the unsteady compressible diagonal significant, or when the EOS causes strong variation in \(a_s\) from one region to another, these terms can materially affect the assembled coefficients and the robustness of the pressure-velocity coupling.