6 Carrier Flux Discretization on Interior Faces

6.1 Face geometry and orientation

Consider an interior face shared by cells and . The unit normal is oriented from to and the area is . A conservative numerical flux must satisfy the orientation identity (6.1) so that the same face contribution enters neighbouring cells with opposite sign.

6.2 Compressible AUSM-family flux

For a compressible aerodynamic state, an AUSM-family inviscid face flux can be separated into a numerical mass flux and mechanical pressure, (6.2) with appended turbulence/species components transported by the appropriate numerical mass flux. AUSM⁺-up constructs from split Mach functions and a pressure-difference correction, and constructs from split pressure functions and a velocity-difference correction (Liou 2006). This separation is valuable for a primitive-variable coupled solver because pressure-to-continuity and velocity-to-momentum couplings appear directly in the face Jacobian.

For a moving/rotating coordinate representation the convective normal speed is formed relative to the grid/frame velocity. The energy flux must then be derived consistently for the selected absolute- or relative-energy formulation. The current NeuralFlow formulation status of this term is discussed in Chapter 20.

6.3 Constant-density artificial-compressibility flux

For the incompressible regime, the pressure-index row is the numerical mass flux rather than a density flux. The current row structure is (6.3) where is mass flux per unit area. Turbulence and species rows use the same . Because the physical density is constant, pressure influences continuity through the AUSM⁺-up pressure-difference correction and through the pseudo-time pressure coefficient, not through an equation-of-state derivative .

6.4 Viscous momentum flux

The viscous momentum flux is with from Eq. (2.4). On an unstructured cell-centred grid, is reconstructed at the face. For non-orthogonal grids a simple two-point normal difference is insufficient by itself; the reconstruction must retain the tangential/non-orthogonal gradient contribution used by the active NeuralFlow gradient path.

6.5 Energy and species diffusion

The viscous/thermal energy flux contains heat conduction and, where enabled, viscous work and species enthalpy diffusion, The exact active terms depend on solver options and material/species configuration. A dependent-species formulation requires the enthalpy-diffusion correction to be constructed consistently with Eq. (2.6).

6.6 AD face Jacobian

For each face, the complete expression is differentiated with respect to both cell states, For cell , these derivatives are accumulated into its diagonal and -neighbour matrix blocks. For cell , the same conservative face flux is assembled with opposite orientation. This produces the block-sparse nearest-neighbour coupling used by the global implicit solver.

6.7 HLLC carrier flux

The current compressible carrier solver also contains an HLLC flux path. NeuralFlow forms density-weighted Einfeldt-type left and right signal estimates. With face-normal velocities and , the Roe-like normal velocity and acoustic estimate are

The bounding wave speeds are and . The contact speed is determined from pressure and momentum continuity across the star region, and the final HLLC flux selects the left, left-star, right-star or right state according to the signs of , and . Species are convected with the selected mass flux. The homogeneous VOF production path currently uses the AUSM carrier flux rather than HLLC.