26. How Does Continuity Know The Provisional Velocity Is Incorrect?

In a compressible flow, a nonzero face-flux imbalance is not automatically wrong. The continuity equation balances density storage against net mass flux. For a control volume \(P\) with a possible mass source \(S_{m,P}\), a first-order representative balance is

(26.1)\[\frac{\Omega_P}{\Delta t} \left( \rho_P^{n+1}-\rho_P^n \right) + \sum_f \dot{m}_f^{\,n+1} = S_{m,P}.\]

Therefore the provisional face fluxes are not judged by asking whether they are divergence-free. They are judged by asking whether the density storage term and net face-flux term are compatible with the same thermodynamic state.

In a pressure-based compressible formulation, density is typically not an independent solution variable. It is obtained from the thermodynamic state, schematically

(26.2)\[\rho_P^{n+1} = \rho\!\left( p_P^{n+1}, T_P^{n+1}, Y_P^{n+1} \right).\]

The density change required by continuity is therefore not arbitrary. It must be the density change implied by the updated pressure, temperature, composition, and any other thermodynamic closure variables.

After the momentum predictor, the solver has a provisional velocity field and a corresponding provisional face mass flux,

(26.3)\[\dot{m}_f^* = \rho_f^* \vec{u}_f^{\,*}\cdot\vec{A}_f.\]

The provisional state is consistent only if the discrete continuity balance is satisfied with the density supplied by the current thermodynamic state:

(26.4)\[\frac{\Omega_P}{\Delta t} \left( \rho_P^*-\rho_P^n \right) + \sum_f \dot{m}_f^* - S_{m,P} = 0.\]

If this residual is nonzero, the provisional velocity field, pressure field, and thermodynamic density field are not mutually consistent. The density cannot simply be changed to absorb the imbalance unless the thermodynamic state also changes in a way that makes that density change valid.

The pressure-correction equation supplies that coupled change. The correction modifies the face fluxes through the momentum correction and modifies density through the equation of state:

(26.5)\[p^{n+1} = p^* + p', \qquad \vec{u}^{\,n+1} \approx \vec{u}^{\,*} - d\nabla p', \qquad \rho^{n+1} \approx \rho^* + \left(\frac{\partial \rho}{\partial p}\right)_{\mathrm{lagged}} p'.\]

The derivative in Eq. (26.5) represents the pressure response used during the pressure-correction solve while the non-pressure thermodynamic state is held fixed or otherwise lagged. In a simple idealized discussion this may be written as a constant-temperature / constant-composition derivative; in acoustic pressure-correction forms it is represented through the local sound-speed relation.

The corrected state is accepted when the density storage term and the corrected face fluxes satisfy Eq. (26.1). Thus, in a compressible pressure-based method, continuity does not validate a velocity field by requiring zero divergence. It validates the compatibility of the velocity field, pressure field, and EOS density field. A provisional velocity field is acceptable only if its face fluxes produce the density change implied by the other solved equations and the equation of state.