6. Detailed Governing Equation Forms#
This appendix collects fuller forms of the core governing equations so that individual terms can be identified without jumping immediately into the finite-volume assembly. The first goal is to state the continuum carrier-phase equations with explicit source terms. The second is to show how some of those terms change appearance when the equations are rewritten for the pressure-based solver.
The source terms are kept generic on purpose. They can represent Lagrangian-particle coupling, porous-media forcing, radiation, phase change, user-specified volumetric forcing, or other model contributions. Setting them to zero gives the uncoupled carrier-flow equations. Keeping them visible is especially helpful for particle calculations, where the particle phase can deposit mass and species, exert drag on the Eulerian field, and exchange heat or phase-change energy with the carrier gas.
The intent is that, if one encounters a term such as species enthalpy diffusion, buoyancy work, a particle energy source, or a \(Dp/Dt\) coupling term while reading the theory guide or the solver implementation, there is a single place to look up what that term means and why it appears.
6.1. Common Definitions#
Unless stated otherwise, the equations in the first part of this appendix are written for an inertial frame on a fixed grid. Solver-specific moving-grid and pressure-based forms are discussed later.
The principal thermodynamic quantities are
The material derivative following the Eulerian carrier velocity is
For a Newtonian fluid, the viscous stress tensor is
When turbulent transport is modeled through an eddy-viscosity assumption, the same structure is often retained with an effective viscosity \(\mu_{\mathrm{eff}}=\mu+\mu_t\).
For multispecies flows, the species diffusion flux is written as
where \(\vec{V}_k\) is the diffusion velocity of species \(k\). The total heat flux is then
The first term is Fourier heat conduction. The second term is the species enthalpy diffusion term: it accounts for the fact that diffusing species carry their own enthalpy with them, so species diffusion also transports energy. Throughout this appendix, \(\vec{q}\) means the total heat flux in Eq. (6.1.5), not Fourier conduction alone.
It is also useful to define the viscous-dissipation function
6.2. Generic Source-Term Notation#
The equations below include generic conservative source terms. These sources are written as volumetric continuum rates and are positive when they add mass, momentum, or energy to the Eulerian carrier phase. In the finite-volume implementation they may be stored temporarily as cell-integrated rates; dividing by the control-volume size recovers the continuum rate used here.
The notation used here is:
\(S_{\rho}\): carrier-phase mass source. In a particle calculation, positive \(S_{\rho}\) means mass has moved from the Lagrangian phase into the Eulerian carrier.
\(S_{\rho Y_k}\): species mass source for species \(k\), with \(S_{\rho}=\sum_k S_{\rho Y_k}\) when the transported species set accounts for all mass added to the carrier.
\(\vec{S}_{\rho\vec{v}}\): conservative momentum source, with units of force per volume. Particle drag, momentum carried by transferred mass, porous-media resistance, and imposed forcing all enter this bucket.
\(S_{\rho E}\) and \(S_{\rho H}\): conservative total-energy and total-enthalpy sources, with units of power per volume. These may include convective particle/fluid heat exchange, phase-change energy, radiation, chemistry heat release, mechanical work, or energy carried by transferred mass. For the conservative total-enthalpy equation written below, the conserved quantity is \(\rho H-p=\rho E\), so \(S_{\rho H}\) and \(S_{\rho E}\) represent the same physical source bookkeeping. Separate symbols are retained only to make contact with solver forms that use different primary variables.
\(S_{\rho e}\) and \(S_{\rho h}\): static internal-energy and static-enthalpy sources. These are useful after a total-energy or total-enthalpy equation has been reduced to a thermal equation; they are not necessarily independent source models.
These symbols are deliberately generic. A Lagrangian particle module is one contributor, but other models can contribute to the same source buckets. A nonzero \(S_{\rho}\) is therefore not a statement that mass is created globally; in a coupled calculation it is usually an exchange between the Eulerian carrier and another modeled subsystem.
One bookkeeping point is important throughout this appendix. A conservative source \(S_{\rho\phi}\) is the source for the conserved variable \(\rho\phi\). If the same equation is rewritten for the specific variable \(\phi\), continuity contributes a correction and the source appears as \(S_{\rho\phi}-\phi S_{\rho}\). This correction is not a new physical model. It is the algebraic consequence of solving for a specific variable while the carrier continuity equation contains a mass source.
6.3. Continuity Equation#
The fully compressible continuity equation is
This equation enforces conservation of carrier-phase mass. For a closed single-phase carrier calculation, \(S_{\rho}=0\). For a coupled particle calculation, \(S_{\rho}>0\) in a cell means mass has been transferred from the Lagrangian phase into the Eulerian carrier during that instant, while \(S_{\rho}<0\) would represent removal from the carrier.
For compressible flows, density is not independent; it is obtained from an equation of state,
For ideal-gas mixtures this reduces to \(\rho = p/(R_m T)\), while for real-fluid calculations the same continuity equation remains valid but the thermodynamic closure is more general.
6.4. Momentum Equation#
A convenient full form of the momentum equation is
The terms on the right-hand side are:
\(-\nabla p\): pressure-gradient force.
\(\nabla\cdot\boldsymbol{\tau}\): viscous diffusion of momentum.
\(\vec{f}_b\): body-force contribution associated with gravity or buoyancy.
\(\vec{f}_{ni}\): non-inertial reference-frame forces such as Coriolis, centripetal, angular-acceleration, or translational-acceleration terms.
\(\vec{S}_{\rho\vec{v}}\): any additional model-dependent conservative momentum source, including particle drag, momentum carried by mass transfer, porous-media resistance, or user-specified forcing.
A dedicated code-facing summary of the non-inertial kinematics and source-term form is given in the non-inertial reference-frame appendix.
For buoyant flows, the same physics is sometimes written as \(\rho\vec{g}\) and sometimes as \((\rho-\rho_{\mathrm{ref}})\vec{g}\), depending on how the hydrostatic part of the pressure field is handled. The governing physics is the same; only the pressure-force decomposition changes.
For particle coupling, \(\vec{S}_{\rho\vec{v}}\) is the momentum received by the carrier per unit volume and time. With the sign convention above, a drag force exerted by the particles on the gas enters with the sign of the force on the gas.
6.5. Total Enthalpy Equation#
The total enthalpy form is often the most direct way to see where pressure work has gone. A useful continuum form is
Here \(\vec{q}\) is the total heat flux defined in Eq. (6.1.5). In multispecies calculations, this means that the compact term \(-\nabla\cdot\vec{q}\) already includes the species enthalpy diffusion contribution associated with the species fluxes \(\vec{J}_k\) from Eq. (6.1.4).
Substituting the total heat-flux definition gives
The important physical terms are:
\(\nabla\cdot(\lambda\nabla T)\): thermal conduction.
\(-\nabla\cdot\left(\sum h_k \vec{J}_k\right)\): species enthalpy diffusion.
\(\nabla\cdot(\boldsymbol{\tau}\cdot\vec{v})\): viscous work.
\(\vec{f}_b\cdot\vec{v}\) and \(\vec{f}_{ni}\cdot\vec{v}\): work done by body forces.
\(S_{\rho H}\): any additional conservative total-enthalpy source such as chemistry, radiation, phase change, particle/fluid heat exchange, or user-specified source models.
If a model transfers mass into the carrier, the enthalpy source should be interpreted as the conservative enthalpy rate delivered to the carrier before any later specific-variable correction is applied. For example, an evaporating particle may contribute both species mass and the enthalpy associated with the vapor it deposits.
Because the conserved variable in this form is \(\rho H-p=\rho E\), this conservative source is the same physical source that appears in the total-energy equation, i.e. \(S_{\rho H}\equiv S_{\rho E}\) for the continuum equation as written here. Differences between total-enthalpy and total-energy solver rows arise later from rewriting the equation for a specific primary variable and from moving pressure terms between the left- and right-hand sides.
6.6. Total Energy Equation#
The total energy equation can be written as
As in the total enthalpy form, \(\vec{q}\) is the total heat flux in Eq. (6.1.5). The expanded form therefore contains both Fourier heat conduction and species enthalpy diffusion:
An equivalent split form that makes the pressure-work term explicit is
This equation and the total enthalpy equation are algebraically equivalent; the difference is only how the pressure-related transport is grouped. The heat-flux meaning is unchanged: \(-\nabla\cdot\vec{q}\) still includes the species enthalpy diffusion term from Eq. (6.1.5).
For particle coupling, \(S_{\rho E}\) is the conservative total-energy rate delivered to the carrier. It may contain thermal exchange, phase-change energy, energy carried by transferred mass, and mechanical work associated with interphase momentum exchange. Which part of that total source later appears as heat in the temperature equation depends on the kinetic-energy bookkeeping described next.
6.7. Temperature Equation#
The temperature equation is not a separate conservation law. It is obtained by starting from the total-energy equation, removing the kinetic-energy equation, and then using thermodynamic identities to replace static internal energy by temperature. This is the same EOS-general route used in standard derivations of the temperature form of the energy equation, such as Panton’s discussion of using temperature as the dependent variable[Pan96].
This derivation is the cleanest place to see how source terms carry through the equation forms. The total-energy source first separates into kinetic and static-energy parts. The static-energy source is then converted to a temperature source through the equation of state. If continuity has a mass source, that conversion also produces the \(-eS_{\rho}\) and \(-(p/\rho)S_{\rho}\) corrections shown below.
6.7.1. From Total Energy To Static Internal Energy#
Let
Using the split total-energy form and dotting the momentum equation with \(\vec{v}\) gives a kinetic-energy equation,
The source \(S_K\) is the part of the conservative source bookkeeping that belongs to carrier kinetic energy. The two terms in the usual expression come from dotting the momentum equation with the carrier velocity and then using the source-bearing continuity equation:
Therefore, for the kinetic energy implied by the carrier momentum equation,
The first term is the power associated with the conservative momentum source when it acts on the carrier velocity. The second term is the specific-variable correction caused by the mass source in continuity. This expression does not assume that the mass enters with the carrier velocity; it uses whatever conservative momentum source \(\vec{S}_{\rho\vec{v}}\) the particle or other coupling model supplies.
The required consistency condition is simply that the total-energy source be split into kinetic and static parts:
Using this carrier kinetic-energy source gives
Now consider only the mass-transfer part of an LPT source. Suppose particles deposit mass into the carrier at rate \(\dot{m}=S_{\rho}^{\,m}\) with source velocity \(\vec{v}_m\). For an evaporating droplet, \(\vec{v}_m\) might be modeled as the particle velocity \(\vec{v}_p\); a different model might instead assume that vapor is born already equilibrated with the carrier, \(\vec{v}_m=\vec{v}\). If \(K_m=\frac{1}{2}\vec{v}_m\cdot\vec{v}_m\) and \(S_{\mathrm{stat},m}\) denotes the non-kinetic static-energy contribution supplied by the mass-transfer model, then the conservative source pieces are
Here \(S_{\mathrm{stat},m}\) may include deposited species enthalpy, latent heat, convective heat exchange, chemistry, or any other thermochemical bookkeeping. The last line assumes that the conservative total-energy source includes the kinetic energy carried by the deposited mass. If a particle model assigns that energy differently, the same algebra should be applied to that model’s actual \(S_{\rho E}^{\,m}\).
Substituting the mass-transfer momentum and mass sources into \(S_K\) gives
The static-energy source that remains after subtracting carrier kinetic energy is then
Thus, if evaporated mass is deposited with the particle velocity and the particle velocity differs from the Eulerian carrier velocity, the relative kinetic energy remains in the static-energy bookkeeping unless the particle model deliberately assigns it somewhere else. If the deposited mass is assumed to enter at the carrier velocity, this relative kinetic-energy term is zero.
This same result is what appears inside the temperature source term used later. For the mass-transfer contribution alone,
This is the place where a momentum source, such as particle drag, enters the energy bookkeeping. If the total-energy source contains exactly the carrier-velocity mechanical work \(\vec{v}\cdot\vec{S}_{\rho\vec{v}}\) and there is no mass source, then that work changes carrier kinetic energy and cancels out of the static-energy equation. It does not directly heat the carrier. A temperature source appears only from the part of the model energy transfer that remains after this kinetic-energy bookkeeping, such as convective heat transfer, latent/phase-change energy, internal energy carried by deposited mass, or any slip-work dissipation that the particle model assigns to the Eulerian phase.
Subtracting the kinetic-energy equation from the total-energy equation leaves the static internal-energy balance
Here \(S_{\rho e}=S_{\rho E}-S_K\) is the net conservative source that remains in the static-energy equation after kinetic-energy bookkeeping has been done. For pure heat addition, it is simply the volumetric heat source. For a coupled particle model, it is the part of the interphase exchange that belongs in the carrier static-energy balance, such as convective heat transfer, phase-change energy, thermalized slip work, or internal energy carried by transferred mass.
If the continuity equation has a mass source, the conservative left-hand side is expanded with the product rule:
Therefore the material static-energy equation is obtained by substituting this identity into the conservative balance and moving the extra \(eS_{\rho}\) term to the right-hand side:
The term \(-eS_{\rho}\) is the specific-variable correction described earlier. It appears because the conservative left-hand side has been rewritten as a material derivative of the specific static internal energy.
6.7.2. Static Energy To A Temperature Equation#
The thermodynamic step is independent of any ideal-gas assumption. Because the pressure-based solver works naturally with pressure and temperature, we write the static internal energy as \(e=e(T,p,Y)\) from the start. The differential below is written at fixed composition \(Y\) to isolate the pressure-temperature structure of the EOS. In multispecies calculations, composition changes are supplied by the species equations, and their energy transport appears through species enthalpy diffusion and through the model source terms:
where
The appearance of \(c_p\) in this identity does not mean that the derivation has switched to an enthalpy balance. It is only a thermodynamic derivative that appears when static internal energy is expressed using pressure and temperature as the independent variables.
Taking a material derivative gives
Using the continuity equation with source,
so the pressure-volume part of the thermodynamic identity is
Substituting this into the static internal-energy equation cancels the \(p\nabla\cdot\vec{v}\) term and gives the EOS-general pressure-temperature equation
This is the form most directly connected to the temperature equation assembled in the pressure-based solver. The pressure derivative is a thermodynamic term; it is not an artificial source introduced by the discretization. The source correction also came from static energy: it is the conservative static-energy source minus the static energy and pressure-volume energy carried by the mass source. Since \(h=e+p/\rho\), the correction can also be read as \(S_{\rho e}-hS_{\rho}\).
Thus, if the raw model sources are supplied as conservative total energy, momentum, and mass, and if the carrier-velocity work split above is appropriate, the source part of the temperature equation can be read as
This expanded form is only a bookkeeping aid. A particle model may choose a more detailed partition of interphase work, slip kinetic energy, and thermalized energy; in that case the temperature equation should use the resulting \(S_{\rho e}=S_{\rho E}-S_K\).
For the LPT mass-deposition example above, substituting \(S_{\rho}^{\,m}=\dot{m}\), \(\vec{S}_{\rho\vec{v}}^{\,m}=\dot{m}\vec{v}_m\), and \(S_{\rho E}^{\,m}=S_{\mathrm{stat},m}+\dot{m}K_m\) into this numerator gives \(S_{\mathrm{stat},m}-h\dot{m}+\frac{1}{2}\dot{m}|\vec{v}_m-\vec{v}|^2\). This is why the relative kinetic energy shows up in the temperature-source bookkeeping only when the deposited mass carries kinetic energy at a velocity different from the carrier velocity.
Using the heat-flux definition
the \(c_p\) form becomes
6.7.3. Solver-Oriented Specific-Temperature Forms#
Dividing by \(c_p\) gives the specific-temperature form
The last term is the variable-\(c_p\) chain-rule correction that appears when \((1/c_p)\nabla\cdot(\lambda\nabla T)\) is rewritten as \(\nabla\cdot[(\lambda/c_p)\nabla T]\). If \(c_p\) is constant, it vanishes.
If one instead writes the left-hand side in conservative \(\rho T\) form, the continuity source contributes one more bookkeeping term:
The \(+T S_{\rho}\) term is tied only to this conservative \(\rho T\) left-hand side. If \(S_{\rho}=0\), this term is simply zero. If a mass source is present and the same equation is then assembled in a continuity-added form, the solver subtracts \(T\) times the continuity equation from the transport operator:
When continuity contains a source, the bracketed continuity term is \(S_{\rho}\), so the continuity-added manipulation contributes \(-T S_{\rho}\) and cancels the conservative-form \(+T S_{\rho}\) above. In the same way, a \(\rho c_p T\) form would generate a \(+c_p T S_{\rho}\) term, and a continuity-added manipulation with \(\phi=c_p T\) would cancel that particular term. This cancellation does not remove the thermodynamic mass-source correction already present in
That correction belongs to the static-energy to temperature conversion. The disappearing term is only the extra \(+\phi S_{\rho}\) introduced by choosing a conservative \(\rho\phi\) left-hand side after the equation had already been written for the specific variable.
6.7.4. Conservative \(\rho c_p T\) Temperature Form#
The form=temperature_rcpt option uses a conservative row built around \(\rho c_p T\) rather than \(\rho T\). This form keeps the linear-system coefficients in energy units. Starting from the \(c_p\) temperature equation above, define
where \(h=e+p/\rho\). Then the material equation is
The conservative \(\rho c_p T\) left-hand side expands as
Therefore the conservative \(\rho c_p T\) form is
The two terms that are unique to this rewritten form are product-rule terms:
The first term appears because \(c_p\) is inside the transported product. In the implementation it is represented by the ttSourceTerm rules that use the current and previous values of cp and the gradient of cp. The second term is the conservative-left-hand-side mass bookkeeping term. It is analogous to the \(+T S_{\rho}\) term in the \(\rho T\) form, but it is multiplied by \(c_p\) because the transported variable is \(c_pT\).
For the Lagrangian-particle coupling, the model source contribution to the \(\rho c_p T\) source row is the energy-unit version of the specific-temperature source. Using the total-energy, momentum, and mass sources supplied by the coupling model,
Here PDEFactor=0 corresponds to the conservative assembly, so the \(+c_pT S_{\rho}\) term remains. PDEFactor=1 corresponds to the continuity-added assembly, where the row has already subtracted the continuity equation multiplied by \(c_pT\), so that conservative bookkeeping term is canceled. In both cases the thermodynamic source correction \(S_{\rho e}-hS_{\rho}\) remains; it is not canceled by the continuity-added rewrite.
For the evaporating-mass example where \(S_{\rho}^{\,m}=\dot{m}\), \(\vec{S}_{\rho\vec{v}}^{\,m}=\dot{m}\vec{v}_m\), and \(S_{\rho E}^{\,m}=S_{\mathrm{stat},m}+\dot{m}K_m\), this source becomes
Thus the ttSourceTerm particle source uses the same static-energy numerator as the tSourceTerm particle source, but it is not divided by \(c_p\). The only additional LPT mass-source term in the \(\rho c_p T\) row is the conservative-form bookkeeping term \((1-\mathrm{PDEFactor})c_pT S_{\rho}\).
A compact way to keep the bookkeeping straight is to write the continuity residual as
The continuity-added equation is then obtained by adding or subtracting a multiple of \(R_{\rho}\), depending on the residual sign convention used by the assembled row. Because \(R_{\rho}\) includes the mass source, this manipulation is algebraically valid whether the Lagrangian-particle module is active or not. With no particle mass source, \(S_{\rho}=0\) and the source-related correction vanishes. With particle mass deposition, the conservative-form \(+\phi S_{\rho}\) term and the continuity-added \(-\phi S_{\rho}\) term cancel each other, while the actual model sources \(S_{\rho\phi}\) and the thermodynamic conversion terms remain.
The nonconservative specific-temperature form is the cleanest way to read the Lagrangian-particle source correction in the implementation:
Those last two terms are not ideal-gas assumptions and they are not inherited from an enthalpy-equation derivation. They come from the static-energy thermodynamic conversion and from the fact that the continuity equation has a source. This is why a particle mass source that enters continuity also produces internal-energy and \(p/\rho\) corrections in the temperature-form source assembly.
This is also why the total-energy/static-energy route is the clean reference derivation, while the temperature equation is the place where the thermodynamic derivative structure of the equation of state becomes most visible. In the solver, the temperature equation is written in rewritten forms such as the specific-temperature and \(\rho c_p T\) forms. The specific-temperature form exposes the divided-by-\(c_p\) diffusion and source terms; the \(\rho c_p T\) form introduces additional product-rule terms involving \(Dc_p/Dt\). In both cases, terms associated with \(Dp/Dt\), \(\nabla c_p\), \(Dc_p/Dt\), or \(S_{\rho}\) are rewriting terms required to preserve the original static-energy balance.
6.8. Species Equation#
For each transported species,
The term \(-\nabla\cdot\vec{J}_k\) is species diffusion, \(\dot{\omega}_k\) is the net production or destruction rate due to chemistry or other inter-species source mechanisms, and \(S_{\rho Y_k}\) is a generic species mass source. In a particle evaporation calculation, for example, the particle model deposits mass into the gas species selected by the evaporation or boiling model. The species equations are coupled back into the energy equations through the diffusion fluxes \(\vec{J}_k\), the species enthalpies \(h_k\), and any enthalpy associated with \(S_{\rho Y_k}\).
The usual consistency conditions are
6.9. Solver-Form Notes#
The equations above are the continuum reference forms. In the solver, additional terms can appear when the equations are written in forms that are more convenient for pressure-based finite-volume discretization. Those terms are not new physics; they arise from rewriting, linearization, or discrete bookkeeping.
6.9.1. Moving-Grid Form#
On moving or deforming grids, the convective transport is written in terms of the velocity relative to the grid,
so a term like \(\nabla\cdot(\rho\vec{v}\phi)\) becomes \(\nabla\cdot(\rho\vec{u}_r\phi)\). This is the ALE form of the transport equations.
For the generic transport identities below, define the corresponding transport derivative
On a fixed grid, \(\vec{u}_r=\vec{v}\) and \(D_r/Dt\) reduces to the material derivative \(D/Dt\) used earlier.
6.9.2. Continuity-Added Transport Form#
For a generic transported variable \(\phi\),
If \(S_{\rho}=0\), the bracketed term vanishes at the continuum level by continuity. In a discrete solver, however, one may choose either the conservative form or the continuity-added form because the resulting linear systems have different properties. The extra terms that show up in the discrete equations are therefore associated with the chosen transport form, not with additional physical modeling.
When continuity has a source, the bracketed term is \(S_{\rho}\) rather than zero. Therefore a conservative transport equation of the form
can be written in a specific-variable form as
This is the source-bookkeeping pattern used by the Lagrangian-particle coupling. The particle module contributes conservative mass, species, momentum, and energy transfer rates; the solver then adds the corresponding \(-\phi S_{\rho}\) correction when the equation is assembled for a specific variable such as velocity, total enthalpy, total energy, temperature, or species mass fraction.
6.9.3. Why Pressure-Time Terms May Appear Explicitly#
In the total enthalpy equation, the conserved quantity is \(\rho H-p\). When this equation is discretized, it is common to keep the transported unknown as \(H\) and move the pressure part to the right-hand side. That is why a temporal pressure-derivative contribution can appear explicitly in the assembled equation even though it originated from the basic continuum form.
Similarly, in the temperature equation the \(Dp/Dt\) term is not an artificial source term. It is part of the thermodynamic temperature equation itself. What changes in the solver is only how that term is evaluated and linearized.
6.9.4. Why Variable-\(c_p\) Terms Appear In Temperature Forms#
If the temperature equation is divided by \(c_p\), or if a conservative \(\rho c_p T\) form is used, additional chain-rule terms can appear. Typical examples are:
terms involving \(\nabla c_p\) when a diffusion term is written in divided-by-\(c_p\) form,
terms involving \(Dc_p/Dt\) when a \(\rho c_p T\) equation is expanded.
These are rewriting terms. They are not separate physical source mechanisms, but they should be retained if one wants the rewritten equation to remain equivalent to the original temperature equation when \(c_p\) varies in space or time.
6.9.5. Species Enthalpy Diffusion In The Energy Equations#
The species enthalpy diffusion term in the energy equations should be interpreted as the energy counterpart of species diffusion. Once the species equations include diffusion fluxes \(\vec{J}_k\), the energy equations inherit the companion transport term \(\sum h_k\vec{J}_k\). In a consistent finite-volume implementation, the same species diffusion fluxes used in the species equations should feed the energy-equation term as well.
At a finite-volume face \(f\), this means the discrete species-enthalpy diffusion contribution should be built as
where \(J_{k,f}\) is the signed species diffusion flux already used by the species equation. The face species enthalpy \(h_{k,f}\) is a thermodynamic multiplier on a diffusive flux, not an advected variable. Therefore it should not be selected by the advective upwind direction of the mass or volume flux. A face-flux reversal should reverse the convective transport, but it should not abruptly switch the thermodynamic state used to weight a diffusive species flux.
The face value \(h_{k,f}\) should instead be compatible with the thermodynamic face-state policy used for the energy or enthalpy equation. If the solver constructs a full face EOS state, then the mixture enthalpy \(h_f\), species enthalpies \(h_{k,f}\), density, temperature, and composition should all be derived from that same face EOS query, following the single-face-state rule in the additive-volume EOS appendix and Eq. (8.8.2). If no full face EOS state is constructed for this diffusive term, a centered face value such as
is more consistent with the diffusive character of \(J_{k,f}\) than an advective upwind choice. Here \(h_{k,f}^{(L)}\) and \(h_{k,f}^{(R)}\) denote the left- and right-side reconstructed species enthalpies. The key requirement is not that the mixture enthalpy and each species enthalpy be numerically equal; rather, they should be chosen from a compatible face thermodynamic state or reconstruction policy so that the energy equation and the species diffusion term do not mix incompatible face states.
6.9.6. Body-Force Work Terms#
If the momentum equation includes gravity, buoyancy, or non-inertial reference-frame forces, the corresponding work terms appear in the total enthalpy and total energy equations through \(\vec{f}\cdot\vec{v}\). These are physical terms, not discretization artifacts, and they are the correct energy counterparts of the momentum sources.
6.10. Summary#
The main purpose of this appendix is to separate three different ideas that are easy to mix together:
the continuum governing equation itself,
the physical source terms that belong in that equation,
the additional terms that appear when the equation is rewritten for a particular pressure-based finite-volume formulation.
Reading the equations in that order usually makes it much easier to identify what a given term is doing.
6.11. References#
Ronald L. Panton. Incompressible Flow. Wiley-Interscience, 2nd edition, 1996.