Let be a finite control
volume with boundary and outward area vector
. The
compressible carrier equations solved by NeuralFlow are written in
conservative balance form. For compactness, define where is density, is absolute velocity, and
is
specific total energy. Turbulence and species equations extend this
carrier vector when active.
The integral balance is (2.1)
2.2 Continuity
The mass equation is or, for one control volume,
(2.2)
The constant-density artificial-compressibility regime does
not use Eq. (2.2)
as a physical density evolution equation. Its pressure/continuity
equation is given in Chapter 4.
2.3 Momentum
For a Newtonian fluid, (2.3) with viscous stress (2.4) Here is the molecular
viscosity for laminar flow and contains the modeled turbulent
contribution when the active turbulence closure requires it. Equation (2.4) is kept
in tensor form in this guide; replacing it by a Laplacian is only valid
under additional assumptions on viscosity and divergence and is not used
as a general identity.
The control-volume momentum flux through a face is therefore In NeuralFlow the inviscid part is replaced by a
numerical face flux, while viscous terms are evaluated from
reconstructed gradients and effective transport properties.
2.4 Total energy
The compressible total-energy equation is (2.5) where Fourier heat conduction is
For a reacting or multi-species formulation, the energy flux
may additionally contain enthalpy carried by species diffusion. NeuralFlow
constructs these contributions in the same coupled residual so that
their state dependence can enter the AD Jacobian.
Using total enthalpy the inviscid energy
flux is . This is the form used naturally
by AUSM-family fluxes.
2.5 Species transport
For chemical species, NeuralFlow may
solve independent mass-fraction
equations and close the final fraction by (2.6) For an independent species , (2.7) with diffusive mass flux and source . Passive-species mode sets
the chemical source to zero. The representation preserves the
algebraic sum of mass fractions but requires diffusion and
thermochemistry terms to remain consistent with the dependent
species.
2.6 Turbulence extension
When GE – RANS is enabled, transport
equations for turbulent kinetic energy and specific dissipation rate are added to the coupled block.
Turbulent stress and effective transport properties then modify Eqs. (2.3)
and (2.5). NeuralFlow also
includes the turbulent kinetic-energy contribution in the carrier total
enthalpy used by the GE-RANS flux path. The detailed closure is given in
the GE-RANS chapter.
2.7 Finite-volume residual
For a cell , the spatial
residual used throughout the remainder of this guide is defined as (2.8) The sign convention used by
the current NeuralFlow matrix assembly is (2.9) so the Newton-like correction is
.
This convention is developed in detail in Chapter 3.