4 Artificial Compressibility for Constant-Density Flow

The incompressible carrier solver is a dedicated constant-density formulation that reuses the coupled finite-volume infrastructure without treating pressure as an equation-of-state variable. This chapter documents the state, pseudo-time equation, inviscid flux, and the main formulation consequences.

4.1 Primitive and stored states

For dimension and species, the active primitive block has the form The physical carrier density is constant. The stored conservative-like carrier components are (4.1) The contribution in is present when GE-RANS is active.

Equation (4.1) is an important user interpretation: the pressure slot stores . It is not . The artificial-compressibility scale enters the pseudo-time pressure equation described next.

4.2 Artificial pressure equation

The steady incompressible constraint is NeuralFlow creates a pseudo-time pressure-velocity coupling through (4.2) where is pseudo time. Consequently the pressure-row pseudo-time Jacobian contains .

The local artificial speed is bounded away from zero, (4.3) with a positive reference-speed floor. The formulation also incorporates boundary-pressure and physical dual-time information when preparing local preconditioning/time-scale data. The floor in Eq. (4.3) prevents the pseudo-acoustic scale from vanishing at a stagnation point.

4.3 Interior inviscid flux

Let be the numerical carrier mass flux per unit area and the numerical mechanical pressure. For an interior face with outward unit normal , the current incompressible carrier flux has the structure (4.4) GE-RANS uses the same numerical mass flux for and advection. Its mechanical pressure includes the modeled contribution and the carrier enthalpy contains .

Using one numerical mass flux in continuity, momentum advection, total enthalpy, turbulence, and species transport is important for discrete consistency. The upwind side for transported quantities is selected from the sign of .

4.4 Artificial-compressibility AUSM⁺-up splitting

The active incompressible carrier scheme follows the AUSM⁺-up construction (Liou 2006), but its characteristic speed is the artificial speed rather than the thermodynamic sound speed. A shared face value defines the normal pseudo-Mach numbers The face Mach number is assembled from fourth-order split functions plus the AUSM⁺-up pressure-difference correction, and the face pressure is assembled from fifth-order pressure splits plus the velocity-difference correction, The formulation uses the standard AUSM⁺-up constants associated with its active path (, , , and ). The pressure splitting is gauge-centred: adding the same constant to both pressure states leaves the numerical mass flux unchanged and produces .

For constant density, with the sign of controlling the upwind state in Eq. (4.4).

4.5 Gauge pressure

The primitive pressure is not clipped to a positive absolute-pressure floor in the incompressible update. Negative gauge pressure is therefore representable. For a uniform gauge shift the face mass flux remains invariant and the mechanical face pressure shifts by the same constant. This gives the expected gauge covariance in the momentum/continuity subsystem.

The energy variable deserves a qualification. Because contains , an individual face enthalpy flux changes under a gauge shift. At a converged incompressible solution that change cancels with discrete continuity when the complete residual is considered. During finite pseudo-iterations, however, energy residual histories can depend on the selected pressure datum. Pressure reference and surface-force reference values should therefore be treated deliberately in gauge-pressure calculations.

4.6 Closed-domain pressure datum

A fully closed incompressible domain has the familiar arbitrary pressure constant. The current pseudo-pressure diagonal makes the linear update nonsingular during pseudo-time iteration and preserves the initialized gauge, but the formulation does not impose a separate pressure pin or a global mean-zero constraint. Consequently two independently initialized closed-domain runs need not converge to exactly the same additive pressure datum even when the velocity field is equivalent.

4.7 Physical-time treatment

For transient incompressible calculations, momentum, thermal, turbulence, particle, and species rows receive the selected BDF physical-time residual. The artificial pressure row does not receive a physical pressure-storage term. It remains the pseudo-time constraint used to converge each physical time step. The resulting algorithm is a dual-time method: physical accuracy is set by the BDF discretization, while artificial compressibility controls convergence of the inner pressure-velocity iterations.

4.8 4.8 User implications and checks

For a constant-density material, NeuralFlow automatically selects the artificial-compressibility formulation. The pressure variable is a gauge pressure and can therefore be negative relative to the selected reference level. The artificial acoustic controls affect pseudo-time pressure propagation and convergence speed; they do not represent a physical sound speed of the liquid or constant-density gas.

For user qualification, verify that a uniform state remains uniform, that the solution is insensitive to an arbitrary gauge-pressure offset in problems where only pressure differences matter, and that mass conservation and pressure-driven engineering quantities are converged. For transient calculations, also check physical time-step sensitivity because the artificial-compressibility parameter should primarily control inner convergence rather than the resolved physical time scale.