A general formulation for pseudo-compressibility methods for the simulation of fluid flow

In this study, we revisit the general pressure evolution equation for compressible fluids and its algebraic variants, clarify their derivation and interpretation, and propose a generalized form of the pseudo-compressibility method for incompressible fluids. Starting from the pressure evolution equation for compressible fluids, we demonstrate that an entropy-production term naturally arises in the GPE from the dissipative nature of the flow. Earlier studies modeled this term using a pressure-dissipation closure, where the diffusivity was expressed in terms of intrinsic thermodynamic properties, such as the specific heat ratio, molecular Prandtl number, and Reynolds number. Our analysis shows that such a closure is strictly valid only at the physical Mach number: when the Mach number is artificially increased by reducing the speed of sound, as in pseudo-compressibility models, the entropy production may be misrepresented, because the reduced sound speed requires consistent adjustments to other thermodynamic properties—adjustments that are typically neglected in practice. This mismatch introduces modeling errors that are difficult to quantify. Motivated by these findings, we argue that the entropy-production term is better treated as a conceptually motivated, rather than physically exact, quantity—one that should include a free parameter to be calibrated for specific applications. A generalized GPE formulation is finally proposed in which an isentropic weakly compressible continuity equation is augmented with a pressure-damping model.