1 Introduction

NeuralFlow is a cell-centred, unstructured finite-volume solver for coupled fluid-flow and multiphysics calculations. The governing equations are assembled as finite-volume residuals and solved with a fully implicit coupled correction. The linear system uses a block-sparse Jacobian whose columns are the active primitive solution variables, so pressure, velocity, temperature and any additional active model variables are corrected together rather than one equation at a time.

The solver architecture is intended for problems in which the characteristic time scales and physical couplings differ strongly. Typical examples combine high-speed compressible flow, low-speed regions, heat transfer, turbulence, passive species, and a dilute dispersed phase. A segregated equation-by-equation procedure can become slow or fragile for such systems because a correction to one field immediately changes the residuals of several other fields. NeuralFlow instead constructs one coupled update, (1.1) where the block rows correspond to the active conservation equations and the block columns correspond to primitive-variable corrections such as velocity, pressure, temperature, turbulence quantities, particle variables, and independent species mass fractions.

1.1 Flow regimes

The current solver contains two distinct carrier-flow formulations.

• Compressible formulation. Density follows the material equation of state. Carrier convection is evaluated with the selected compressible numerical flux, including AUSM-family methods and HLLC where configured. Low-Mach stiffness can be reduced using time-derivative preconditioning while retaining a density-based conservative residual.

• Constant-density incompressible formulation. Physical carrier density is fixed and pressure-velocity coupling is created with an artificial-compressibility equation. The active carrier flux is a dedicated AUSM⁺-up-type artificial-compressibility flux. Gauge pressure is retained as a signed primitive variable, including negative gauge values. The artificial-compressibility coefficient belongs to the pseudo-time mass matrix; the stored pressure state itself is the pressure , not .

Both carrier-flow regimes use the same coupled implicit solution framework, viscous and volumetric-source treatment, multilevel acceleration and parallel domain decomposition. Their continuity/pressure equations are nevertheless mathematically different, so the constant-density formulation is described separately in Chapter 4 rather than as a limiting equation-of-state case.

1.2 Why primitive-variable coupling is used

A conservative finite-volume residual does not require the Newton correction itself to be expressed in conservative variables. NeuralFlow uses primitive-variable columns because pressure and temperature are directly useful thermodynamic and coupling variables. The residual rows nevertheless retain their physical conservative meaning. For the aerodynamic block in spatial dimensions, but the corresponding rows represent momentum equations, continuity in the pressure-index row, and total energy in the temperature-index row. Turbulence, particle, and species equations extend this block when enabled.

This distinction is important when interpreting the equations and convergence histories: the pressure-index column contains the primitive pressure correction, while the row occupying the same block position represents mass/continuity. NeuralFlow is not solving a separate pressure-conservation law.

1.3 Numerical fluxes and pressure-velocity coupling

For compressible flow, NeuralFlow uses upwind flux formulations appropriate to wave propagation. AUSM⁺-up is especially useful because mass and pressure fluxes are constructed with separate Mach- and pressure-splitting functions and include low-speed pressure/velocity dissipation terms. The implicit Jacobian is consistent with the complete face-flux expression, so pressure-to-mass-flux and velocity-to-momentum couplings are retained in the coupled correction.

For the constant-density incompressible regime, an artificial acoustic speed replaces the physical acoustic scale in the pressure-velocity coupling. This preserves a finite pseudo-time pressure equation at stagnation and allows a density-based coupled infrastructure to solve a constant-density system without introducing an equation-of-state density response.

1.4 Multiphysics scope

The NeuralFlow equation set can be extended by GE - turbulence transport, independent species equations, dilute dispersed-phase transport, interfacial-area transport, body-force terms and wall heat-transfer models. A boundary-attached one-dimensional charring-material response is also available for supported compressible turbulent-flow cases. Each option must be interpreted with its physical assumptions: the dilute dispersed model remains a dilute model even when packing protection is activated, and the charring model predicts internal conversion/thermal response without geometric surface recession.

1.5 Parallel and multilevel execution

For parallel calculations, NeuralFlow decomposes the finite-volume mesh into numerical partitions that exchange interface data and participate in a distributed coupled solve. Multigrid levels follow the same selected physical model on their coarse representations. CPU execution through Intel MKL and NVIDIA CUDA GPU execution are available for the supported linear-algebra paths; changing the execution device changes performance and memory behavior, not the governing finite-volume model.

1.6 Reference scope and qualification terminology

This manual is a formulation and user reference rather than a validation report. The following terms are used consistently:

• Model choice: whether the selected equations and closures represent the intended physical regime.
• Numerical convergence: residual reduction together with stable engineering quantities and conservation balances.
• Resolution: mesh, time-step, and discretization sensitivity of quantities used in engineering decisions.
• Model sensitivity: dependence of the result on turbulence, wall, multiphase, chemistry, or material closures.

Known restrictions that affect selectable model combinations are stated in the relevant theory sections and summarized in the model-availability reference. They define the supported formulation for the current release.