6. Pressure-Correction Equation on Unstructured Grids#

The momentum predictor and momentum-weighted interpolation provide a provisional cell velocity, a provisional face flux, and the face response to a pressure change. This chapter inserts those quantities into discrete continuity, linearizes the density and flux corrections, and collects the result into the sparse algebraic equation solved for pressure correction.

In the algorithmic roadmap of Chapter 3, this is the stage after the momentum predictor has supplied provisional face fluxes and before the pressure, velocity, and face-flux corrections are applied. Put differently, this section is where the continuity balance is rewritten into the linear pressure-correction row for \(p^{'}\).

Three ingredients have to be made explicit:

the unsteady density contribution,
the pressure-induced face-flux correction,
the bookkeeping that sends each face contribution into the diagonal, neighbor, and right-hand-side entries.

The goal is the standard diagonal–neighbor–source form used by a scalar finite-volume linear system. Its implementation mapping is deferred to the coding appendix.

6.1. State Labels and Continuity Balance#

Before breaking down the terms, fix the state labels used throughout this chapter. The superscript \(n\) denotes the previous converged time level. The superscript \(k\) denotes the current outer-iteration estimate at the new time level, and \(k+1\) denotes the next outer-iteration estimate at that same time level. A superscript \(*\) denotes a provisional quantity produced by the momentum predictor using coefficients and closure data lagged at iteration \(k\). A prime denotes the pressure-induced correction solved for during the pressure-correction stage. Thus, one may read \(p^*\equiv p^k\), whereas \(\vec{u}^*\) and \(\hat{m}_f^*\) are newly predicted quantities formed from that pressure reference. These stages occur inside one physical time step; they are not additional time-discretization levels. Once the outer iteration converges, its final state becomes the new time level \(n+1\).

The density notation needs one further distinction. Let \(\widehat{\rho}^{\,k+1}\) denote the density estimated inside the pressure-correction equation by linearizing only its pressure response. The hat is important: \(\widehat{\rho}^{\,k+1}\) is not, in general, the complete thermodynamic density \(\rho^{k+1}\) that will be rebuilt from the EOS after the corrected pressure, energy or temperature, composition, and any other required state variables are available.

Begin from the semi-discrete continuity balance for one cell after the momentum predictor but before the pressure correction is applied. At this stage, the provisional face fluxes do not, in general, satisfy discrete continuity. Let \(\hat{m}_f^*\) denote the predicted face flux appearing in the selected continuity form. In the ordinary mass-based form this is the predicted face mass flux \(\dot{m}_f^*\); normalized and moving-grid forms use the corresponding normalized or grid-relative flux. The derivation below uses the ordinary mass balance on a fixed control volume, so \(\hat m_f^*\) has mass-flow units. Normalized continuity forms are developed in Pressure-Correction Equation Forms. For a moving control volume, the time derivative acts on \(\rho\Omega\) and the face flux is measured relative to grid motion. The pressure-correction equation seeks a face correction \(\dot{m}_f'\) such that the pressure-linearized cell balance is satisfied:

Face fluxes retain the stored left-to-right orientation. In a balance for cell \(P\), multiply them by the incidence sign \(\sigma_{Pf}\), as defined in Loci Face Orientation and Cell Incidence.

(6.1.1)#\[\left[ \frac{\widehat{\rho}_P^{\,k+1} - \rho_P^{n}}{\Delta t} \right]\Omega_P + \sum_{f\in\mathcal F_P}\sigma_{Pf}\left( \hat{m}_f^{*} + \dot{m}_f^{'} \right) = 0\]

This is the starting continuity balance for the pressure-correction derivation, not yet the final linear equation. The next subsections express the unknown pressure response of density and face flux in terms of \(p'\), then collect those terms into the algebraic row.

6.2. Unsteady Density Contribution#

For compressible flow, split the pressure-linearized density estimate into the known density at the current outer iteration and an unknown pressure-induced increment:

(6.2.1)#\[\widehat{\rho}_P^{\,k+1} - \rho_P^{n} = \left( \widehat{\rho}_P^{\,k+1} - \rho_P^k \right) + \left( \rho_P^k - \rho_P^n \right) = \rho_P^{'} + \left( \rho_P^k - \rho_P^n \right)\]

Here

(6.2.2)#\[\rho_P' \equiv \widehat{\rho}_P^{\,k+1}-\rho_P^k\]

is an auxiliary density correction used to form the pressure equation. It models only the part of the density response assigned to \(p'\) in this linearization. It is neither the complete outer-iteration change \(\rho_P^{k+1}-\rho_P^k\) nor the complete time-step change \(\rho_P^{n+1}-\rho_P^n\).

Thus the unsteady contribution in the cell balance becomes

(6.2.3)#\[\left[ \frac{\widehat{\rho}_P^{\,k+1} - \rho_P^{n}}{\Delta t} \right]\Omega_P = \left[ \frac{\rho_P^{'} + \left( \rho_P^k - \rho_P^n \right)}{\Delta t} \right]\Omega_P.\]

The unknown part now requires a local relation between pressure and density. Write the EOS in a form suited to the acoustic tangent,

(6.2.4)#\[\rho_P = \rho(p_P,s_P,Y_{1,P},\ldots,Y_{N_s,P}).\]

A complete first-order thermodynamic change about the current nonlinear state would contain pressure, entropy, and composition contributions:

(6.2.5)#\[\delta\rho_P \approx \left(\frac{\partial \rho}{\partial p}\right)_{s,Y} \delta p_P + \left(\frac{\partial \rho}{\partial s}\right)_{p,Y} \delta s_P + \sum_{i=1}^{N_s} \left(\frac{\partial \rho}{\partial Y_i}\right)_{p,s,Y_{j\ne i}} \delta Y_{i,P}.\]

The segregated pressure-correction solve does not solve for all of these increments simultaneously. It retains one pressure-sensitive density response implicitly and lags the remaining thermodynamic and composition changes until their corresponding updates and the subsequent EOS reconstruction.

Density-Linearization Assumption

With entropy and composition frozen for this local pressure response, the auxiliary density correction is

(6.2.6)#\[\rho_P' \approx \psi_P p_P', \qquad \psi_P = \left(\frac{\partial \rho}{\partial p}\right)_{s,Y} = \frac{1}{a_{s,P}^2}.\]

This acoustic tangent is EOS-generic. It is not an isothermal derivative, and using it does not assert that the complete flow or complete outer iteration is isentropic.

The corresponding first-order pressure-only estimate about iteration \(k\) is

(6.2.7)#\[\widehat{\rho}_P^{\,k+1} = \rho_P^k + \left.\left( \frac{\partial \rho}{\partial p} \right)_{s,Y}\right|_k p_P' + O\!\left((p_P')^2\right),\]

where the sound speed supplied by the current EOS state gives

(6.2.8)#\[a_{s,P}^2 = \left.\left( \frac{\partial p}{\partial \rho} \right)_{s,Y}\right|_k, \qquad \left.\left( \frac{\partial \rho}{\partial p} \right)_{s,Y}\right|_k = \frac{1}{a_{s,P}^2}.\]

Substitution into the unsteady term gives, for the first-order time-discrete form used to expose the algebra,

(6.2.9)#\[\left[ \frac{\widehat{\rho}_P^{\,k+1} - \rho_P^{n}}{\Delta t} \right]\Omega_P = \frac{\Omega_P}{\Delta t\,a_{s,P}^2} p_P^{'} + \frac{\rho_P^k-\rho_P^n}{\Delta t}\Omega_P\]

Final Unsteady Density Contribution

For the compressible pressure-correction row, the unsteady density contribution is

(6.2.10)#\[{}^{(p')}\!a_{P,\mathrm{time}} p_P^{'} = \sum_N {}^{(p')}\!a_{P,N,\mathrm{time}} p_N^{'} + {}^{(p')}\!b_{P,\mathrm{time}} ,\]

with

(6.2.11)#\[{}^{(p')}\!a_{P,\mathrm{time}} = \frac{\Omega_P}{\Delta t\,a_{s,P}^2}, \qquad {}^{(p')}\!a_{P,N,\mathrm{time}} = 0, \qquad {}^{(p')}\!b_{P,\mathrm{time}} = -\frac{\rho_P^k-\rho_P^n}{\Delta t}\Omega_P.\]

That is, the acoustic density response contributes an implicit diagonal term, while the lagged time-level defect remains on the right-hand side.

Higher-order time schemes change the temporal weights but not this separation: the pressure-dependent acoustic response is implicit, while the continuity defect formed from the current iterate and previous time levels is known. The time-discretization coefficients are developed in Chapter 10.

6.3. EOS Reconstruction After Pressure Correction#

The pressure equation has now used \(\widehat{\rho}^{\,k+1}=\rho^k+p'/a_s^2\) only as a local linear model. After the pressure, velocity, energy or temperature, species, and other transported quantities have been updated, the next nonlinear density is reconstructed from the complete EOS relation:

(6.3.1)#\[\rho_P^{k+1} = \mathcal{R}\!\left( p_P^{k+1}, \Theta_P^{k+1}, \boldsymbol{Y}_P^{k+1}, \ldots \right),\]

where \(\Theta\) denotes whichever thermal variable closes the selected equation set. If total energy or total enthalpy is transported, the corrected velocity is needed to remove the kinetic contribution before the EOS is evaluated. If temperature is transported directly, that conversion is unnecessary.

Pressure-Equation Density Versus Iterated EOS Density

The quantity \(\rho'=p'/a_s^2\) closes the linear pressure-correction equation; it does not prescribe the full density update. The actual next-iteration density is rebuilt from the EOS using the corrected thermodynamic state. This distinction allows temperature, composition, phase, and other closure changes to affect \(\rho^{k+1}\) without pretending that all of them were solved implicitly in the pressure equation.

In Stream’s iteration labels, the pressure equation is assembled from the current rho{n,it} and soundSpeed{n,it}. The corrected thermodynamic state then produces rhoStar{n,it}, which advances to rho{n,it+1}. The code name rhoStar therefore denotes the EOS-rebuilt next-iteration density; it should not be identified with either the provisional superscript \(*\) used for the momentum predictor or the hatted density estimate introduced above.

6.4. Splitting the Face Flux#

Write the corrected flux as the sum of the provisional flux and its pressure-induced change:

(6.4.1)#\[\dot m_f=\hat m_f^*+\dot m_f'\]

For one interior face, label its cells in stored order: \(P=\operatorname{cl}(f)\) and \(N=\operatorname{cr}(f)\). The compact pressure correction derived in the momentum-interpolation chapter is

(6.4.2)#\[\dot m_f'=-d_f(p_N'-p_P') \qquad d_f>0\]

Here \(d_f\) is the effective coefficient used in the pressure equation. It equals the direct coefficient \(d_f^{\mathrm{dir}}\) when the induced cell terms cancel. If those terms are approximated by a two-cell pressure difference, \(d_f\) includes that approximation. For example, Eq. (5.9.5.7) gives the SIMPLEC face approximation. The direct coefficients themselves are given in Eqs. (5.5.3.1) and (5.8.5.1).

6.5. Assembling One Cell Row#

For the row of an arbitrary cell \(P\), let \(N\) be the neighbor across face \(f\). Convert the stored correction to the outward direction:

(6.5.1)#\[\sigma_{Pf}\dot m_f'=d_f(p_P'-p_N')\]

This expression holds whether \(P\) is the stored left or right cell. The coefficient of \(p_P'\) enters the diagonal. Moving the term containing \(p_N'\) to the right gives the positive neighbor coefficient \(d_f\). The known provisional flux contributes \(-\sigma_{Pf}\hat m_f^*\) to the right-hand side.

A specified-pressure boundary contributes \(d_fp_P'\) to the left and \(d_fp_{\mathrm{bnd}}'\) to the right. If the predictor already satisfies the prescribed pressure, \(p_{\mathrm{bnd}}'=0\). A prescribed-flux boundary has no flux correction; take \(d_f=0\) there in the sums below. Its known flux still enters continuity.

Substitute the density linearization and the outward face corrections into Eq. (6.1.1). For the case \(p_{\mathrm{bnd}}'=0\) at specified-pressure boundaries,

(6.5.2)#\[\frac{\Omega_P}{\Delta t\,a_{s,P}^2}p_P' +\frac{\rho_P^k-\rho_P^n}{\Delta t}\Omega_P +\sum_{f\in\mathcal F_P}\sigma_{Pf}\hat m_f^* +\sum_{f\in\mathcal F_P}d_fp_P' -\sum_{f\in\mathcal F_P^{\mathrm{int}}}d_fp_N' =0\]

Collect the terms multiplying \(p_P'\) on the left and move the known terms and neighbor corrections to the right:

(6.5.3)#\[\left(\frac{\Omega_P}{\Delta t\,a_{s,P}^2} +\sum_{f\in\mathcal F_P}d_f\right)p_P' =\sum_{f\in\mathcal F_P^{\mathrm{int}}}d_fp_N' -\sum_{f\in\mathcal F_P}\sigma_{Pf}\hat m_f^* -\frac{\rho_P^k-\rho_P^n}{\Delta t}\Omega_P\]

In each interior-face sum, \(N\) is the cell across that face. If several faces connect the same pair of cells, add their coefficients into one neighbor entry, as in the transport equation.

Final Standard Pressure-Correction Row

(6.5.4)#\[{}^{(p')}\!a_Pp_P' =\sum_{N\in\mathcal N(P)}{}^{(p')}\!a_{P,N}p_N' +{}^{(p')}\!b_P\]

The coefficients are

(6.5.5)#\[\begin{split}\begin{aligned} {}^{(p')}\!a_P &={}^{(p')}\!a_{P,\mathrm{time}}+\sum_{f\in\mathcal F_P}d_f \\ {}^{(p')}\!a_{P,N} &=\sum_{f\in\mathcal F(P,N)}d_f \\ {}^{(p')}\!b_P &=-\sum_{f\in\mathcal F_P}\sigma_{Pf}\hat m_f^* +{}^{(p')}\!b_{P,\mathrm{time}}+{}^{(p')}\!b_{P,\mathrm{bc}} \end{aligned}\end{split}\]

Here \(\mathcal F(P,N)\) contains the faces shared by \(P\) and \(N\). The temporal coefficient and source were derived above. The boundary source is the sum of known \(d_fp_{\mathrm{bnd}}'\) terms; it is zero when all specified-pressure boundaries have \(p_{\mathrm{bnd}}'=0\).

For fixed density, the temporal density correction vanishes. The compact face correction then gives the pressure equation directly. For compressible flow, the face density also changes with pressure and adds the contribution derived next.

6.6. Additional Compressible Density-Correction Term at Faces#

To first order, the face mass-flux change contains a velocity change at fixed density and a density change at fixed provisional velocity:

(6.6.1)#\[\dot m_f' =\rho_f^k u_{n,f}'A_f+\rho_f'u_{n,f}^*A_f =-d_f(p_N'-p_P')+\rho_f'\frac{\hat m_f^*}{\rho_f^k}\]

In this face relation, \(P\) and \(N\) again denote the stored left and right cells. The second term uses the same frozen face density \(\rho_f^k\) that relates provisional velocity to mass flux. With an upwind acoustic density correction,

(6.6.2)#\[\begin{split}\rho_f'\approx\frac{p_{\mathrm{up}}'}{a_{s,\mathrm{up}}^2} \qquad \mathrm{up}= \begin{cases} P & \hat m_f^*>0 \\ N & \hat m_f^*<0 \end{cases}\end{split}\]

Define the nonnegative magnitude of the density-flux coefficient by

(6.6.3)#\[ c_f=\frac{|\hat m_f^*|}{\rho_f^k a_{s,\mathrm{up}}^2}\]

The stored density-flux correction is \(\operatorname{sgn}(\hat m_f^*)c_fp_{\mathrm{up}}'\). For a cell row, multiply this term by \(\sigma_{Pf}\). Outflow carries \(p_P'\) and adds a positive diagonal coefficient; inflow carries the neighbor correction and adds a positive coefficient on the right. At zero provisional flux the contribution is zero.

Interior Compressible Coefficient Update

(6.6.4)#\[\begin{split}\begin{aligned} {}^{(p')}\!a_{P,\mathrm{comp}} &=\sum_{f\in\mathcal F_P^{\mathrm{int}}} \begin{cases} c_f & \sigma_{Pf}\hat m_f^*>0 \\ 0 & \sigma_{Pf}\hat m_f^*\leq0 \end{cases} \\ {}^{(p')}\!a_{P,N,\mathrm{comp}} &=\sum_{f\in\mathcal F(P,N)} \begin{cases} c_f & \sigma_{Pf}\hat m_f^*<0 \\ 0 & \sigma_{Pf}\hat m_f^*\geq0 \end{cases} \\ {}^{(p')}\!b_{P,\mathrm{comp}}&=0 \end{aligned}\end{split}\]

These additions use the same stored face coefficient in both cells. Boundary density corrections follow the prescribed boundary state and are derived in Boundary Conditions.

Adding these terms to the base row gives

(6.6.5)#\[\begin{split}\begin{aligned} {}^{(p')}\!a_P &={}^{(p')}\!a_{P,\mathrm{time}} +\sum_{f\in\mathcal F_P}d_f+{}^{(p')}\!a_{P,\mathrm{comp}} \\ {}^{(p')}\!a_{P,N} &=\sum_{f\in\mathcal F(P,N)}d_f+{}^{(p')}\!a_{P,N,\mathrm{comp}} \\ {}^{(p')}\!b_P &=-\sum_{f\in\mathcal F_P}\sigma_{Pf}\hat m_f^* +{}^{(p')}\!b_{P,\mathrm{time}}+{}^{(p')}\!b_{P,\mathrm{bc}} \end{aligned}\end{split}\]

The pressure-correction matrix-storage appendix describes the corresponding sparse matrix storage convention.

6.7. Face Coefficient Supplied by Momentum Interpolation#

For the compact face correction, momentum interpolation supplies the effective coefficient \(d_f\) used above. Its value includes the specified momentum-response, temporal and SIMPLEC approximations. The direct coefficient \(d_f^{\mathrm{dir}}\) is only one contribution. The derivation in Connection to the Pressure-Correction Equation identifies when cell-pressure or geometric terms also contribute.

If those couplings are retained, insert the complete face correction from Eq. (5.9.8.2) into continuity. This adds pressure connections beyond the two-cell terms assembled here. Omitting or approximating these changes leaves their remaining effect to the outer iteration. The applied flux correction must use the same relation as the assembled pressure equation.

6.8. Implementation Mapping#

The mathematical pressure-correction row is complete at this point. Its mapping to the Stream sparse-matrix stores, including the stored off-diagonal sign convention, temporary accumulation rules, SIMPLEC scaling, temporal right-hand-side terms, and code landmarks, is preserved in the pressure-correction matrix-storage appendix. Keeping that mapping in the coding reference allows the derivation here to remain focused on the numerical method.

6.9. Applying the Pressure Correction#

Once the pressure correction field is computed by solving Eq. (6.5.4), the mass flux, pressure, and velocity fields are corrected. For the SIMPLE and SIMPLEC families, these updates have the schematic form below. SIMPLER obtains its primary pressure from a preliminary pressure equation, while PISO and PIMPLE accumulate staged corrections; those distinctions are derived in Chapter 7.

(6.9.1)#\[p_P = p_P^{*} + \alpha_p p_P^{'} \]

(6.9.2)#\[\vec{u}_P = \vec{u}_P^{*} - \frac{D_P}{\Omega_P} \sum_f \sigma_{Pf}p_f^{'} \vec{A}_f \]

(6.9.3)#\[\dot{m}_f = \hat{m}_f^{*} + \dot{m}_f^{'} \]

where \(\dot{m}_f^{'}\) is given by Eq. (6.4.2) or Eq. (6.6.1). In these update equations, \(p^*\) is the pressure reference used to form \(p'\), while \(\vec{u}^*\) and \(\hat{m}_f^*\) are predicted quantities built from that reference state. \(\alpha_p\) is the under-relaxation factor for pressure, and \(D_P\) is the selected cell momentum-response coefficient. Linearizing the momentum pressure reconstruction gives Eq. (5.9.3.2). A simpler pressure-correction gradient is an additional approximation; omitted reconstruction changes remain to be resolved by the outer iteration. For SIMPLE, \(D_P=\Omega_P/a_P^{\mathrm{pred}}\); SIMPLEC uses the consistent response that also approximates the neighboring velocity corrections. Both responses, including the effect of momentum under-relaxation, are derived in Chapter 7. Variables with subscript \(P\) are cell-centered and those with subscript \(f\) are face-centered. For compressible flow, these primary-field corrections are followed by the complete EOS reconstruction described in EOS Reconstruction After Pressure Correction.

6.10. Boundary Conditions#

The different types of boundaries commonly encountered in a computational domain can be classified as: inflow, outflow, wall and symmetry boundaries. The boundary conditions are imposed in terms of the primitive variables, either by assigning them fixed values or extrapolating from the interior of the domain, depending on the local flow regime. Once the appropriate boundary mass fluxes are computed from boundary density and velocity components, the contribution to the discretized momentum and energy equations by the boundary faces is no different than any interior face. The contributions to the coefficients of the pressure correction equation, however, depend on the type of boundary, as discussed below.

At wall boundaries, the fluid velocity is the same as that of the wall which can be stationary or moving; the mass flux is zero. At symmetry boundaries, the flow is parallel to the boundary; thus, all the variables are extrapolated from the interior and the mass flux is set to zero. The inflow and outflow boundaries need special treatment depending on the flow regime and are discussed below.

6.10.1. One-Sided Momentum-Interpolation Coefficients#

A boundary coefficient follows from the prescribed pressure or flux and the one-sided momentum relation. The following are the base SIMPLE coefficients; another cell momentum approximation requires its corresponding one-sided response. An interior two-cell coefficient does not determine this boundary relation. For a classical/Pascau fixed-pressure boundary, the complete adjacent-cell diagonal gives

(6.10.1.1)#\[d_{\mathrm{bnd}} =\rho_{\mathrm{bnd}}\alpha_v \frac{\Omega_P}{{}^{(u)}\!a_P} \frac{A_{\mathrm{bnd}}}{\ell_{\mathrm{bnd}}}\]

Here \(\ell_{\mathrm{bnd}}\) is the positive normal distance to the boundary. Denner’s one-sided construction instead uses the non-temporal diagonal and its face time factor, as follows. The definitions of \(\lambda_0\) and the geometric factor \(\beta\) are those in the momentum-interpolation chapter.

At a boundary face there is one adjacent cell. With

(6.10.1.2)#\[\mathcal{D}_{\mathrm{bnd}} = \frac{\Omega_P^n}{\widetilde a_P} \qquad F_{\mathrm{bnd}} = \frac{\mathcal{D}_{\mathrm{bnd}}} {1+\rho_{\mathrm{bnd}}\lambda_0 \mathcal{D}_{\mathrm{bnd}}}\]

the corresponding coefficient is

(6.10.1.3)#\[d_{\mathrm{bnd}} = \rho_{\mathrm{bnd}} \alpha_v \beta_{\mathrm{bnd}} F_{\mathrm{bnd}} \frac{A_{\mathrm{bnd}}} {\lVert\vec{x}_{\mathrm{bnd}}-\vec{x}_P\rVert}\]

For a specified-pressure boundary, this coefficient contributes to the pressure-correction diagonal. For a boundary whose mass flux is prescribed, the pressure correction does not alter that flux, so no corresponding unknown face-pressure correction is assembled.

For a prescribed boundary correction \(p_{\mathrm{bnd}}'\), the compact face relation adds \(d_{\mathrm{bnd}}p_P'\) to the pressure-row left side and \(d_{\mathrm{bnd}}p_{\mathrm{bnd}}'\) to its right side. If the predictor already satisfies a fixed boundary pressure, \(p_{\mathrm{bnd}}'=0\). A prescribed-flux boundary supplies a known flux and no unknown pressure-induced flux correction.

6.10.2. Inlet: Fixed Mass Flux#

At an inlet, a mass flux and the flow angle can be specified. Static or total temperature is specified. Pressure can be specified or extrapolated from the interior depending on the flow regime. Three flow regimes can exist:

Constant density flow: density and velocity components are fixed. Pressure is extrapolated from the interior.
Subsonic flow: density is not fixed at the inlet. Pressure is extrapolated from the interior cells and then density is computed from the equation of state using the boundary temperature. Velocity components are then computed from the specified mass flux and flow angle.
Supersonic flow: density, velocity components and pressure are all fixed.

In all of the above situations, the mass flux at a boundary face is given by

(6.10.2.1)#\[\dot{m}_{\mathrm{bnd}} = \rho_{\mathrm{bnd}} \vec{u}_{\mathrm{bnd}} \cdot \vec{A}_{\mathrm{bnd}} = \rho_{\mathrm{bnd}} V_{n,\mathrm{bnd}} A_{\mathrm{bnd}}\]

which goes into the pressure correction equation as a source term. The mass flux correction is zero; therefore, the contribution from the boundary face to the diagonal and neighbor coefficients of the pressure correction equation given by Eq. (6.5.4), is also zero. It should be noted that the contribution from \(\rho^{'}\) for compressible flows, given by Eq. (6.6.4) is also zero for a fixed mass flux boundary.

6.10.3. Outlet: Fixed or Extrapolated Pressure#

At outflow boundaries, the velocity vector \(\vec{u}_{\mathrm{bnd}}\) is extrapolated from the interior and temperature is extrapolated. Pressure can be specified or extrapolated depending on the flow regime. Density is extrapolated from the interior. Mass flux is given by Eq. (6.10.2.1) and the mass flux correction can be written as:

(6.10.3.1)#\[\dot{m}_{\mathrm{bnd}}^{'} = \rho_{\mathrm{bnd}} V_{n,\mathrm{bnd}}^{'} A_{\mathrm{bnd}} + \rho_{\mathrm{bnd}}^{'} V_{n,\mathrm{bnd}}^{*} A_{\mathrm{bnd}}\]

The normal velocity correction at the boundary in terms of pressure correction at the boundary and the adjacent cell, analogous to correction at an interior face given by Eq. (6.4.2), can be written as:

(6.10.3.2)#\[V_{n,\mathrm{bnd}}' =-\frac{d_{\mathrm{bnd}}}{\rho_{\mathrm{bnd}}A_{\mathrm{bnd}}} \left(p_{\mathrm{bnd}}'-p_P'\right) \]

Two situations can arise at an outlet:

Fixed Pressure Outlet: If the flow at the outlet is incompressible or subsonic, the static pressure is fixed which implies \(p_{\mathrm{bnd}}^{'} = 0\) .
Extrapolated Pressure Outlet: If the flow is supersonic at the outlet, then the pressure is extrapolated from the interior which implies \(p_{\mathrm{bnd}}^{'} = p_P^{'}\) .

For a fixed-pressure outlet, \(p_{\mathrm{bnd}}'=0\), so the pressure-induced boundary density correction is also zero. For an extrapolated-pressure outlet, \(p_{\mathrm{bnd}}'=p_P'\) and the acoustic density correction can be nonzero. To stay consistent with the EOS-generic linearization used throughout this chapter, write the extrapolated boundary density correction as

(6.10.3.3)#\[\rho^{'}_{\mathrm{bnd}} = \left.\left( \frac{\partial \rho}{\partial p} \right)_{s,Y}\right|_{\mathrm{bnd}} p_{\mathrm{bnd}}^{'} \approx \frac{1}{a_{s,\mathrm{bnd}}^2} p_{\mathrm{bnd}}^{'}.\]

and the mass flux correction is given by:

(6.10.3.4)#\[\dot{m}_{\mathrm{bnd}}^{'} = -d_{\mathrm{bnd}}\left(p_{\mathrm{bnd}}'-p_P'\right) + \rho_{\mathrm{bnd}}^{'} V_{n,\mathrm{bnd}}^{*} A_{\mathrm{bnd}}\]

The contributions to the pressure-correction equation from outflow boundary faces are obtained directly from Eq. (6.10.3.4). If one later specializes to an ideal-gas EOS, the acoustic derivative above reduces to the familiar ideal-gas expression, but that specialization is no longer assumed implicitly here.