22. Nonreflecting Open Boundary Conditions#

This appendix documents the nonreflecting open-boundary formulation used for compressible open boundaries in the current solver. It supplements the generic inlet and outlet discussion in Chapter 6 by focusing on how the boundary primitive state itself is constructed when the objective is to reduce spurious acoustic reflection.

The discussion here is intentionally mathematical. User-side specification of the target data is covered in the stream guide; in this appendix, the required target state is assumed to already be known.

The equations below are not meant as ad hoc boundary formulas. They belong to the broader family of characteristic open-boundary conditions introduced for hyperbolic systems by Thompson (1987) and adapted to compressible Navier-Stokes flow by Poinsot and Lele (1992). For viscous, reacting, and multidimensional open boundaries, the present formulation follows the same line of development as Sutherland and Kennedy (2003) and Yoo and Im (2007).

Three ideas organize the formulation:

  1. the boundary update is split into supersonic, subsonic, and backflow branches,

  2. the subsonic branch uses one-dimensional characteristic compatibility in the boundary-normal direction,

  3. the subsonic branch is relaxed toward a target state over a user-controlled length scale and, at outlets, corrected by multidimensional transverse terms.

22.1. Characteristic Background#

At an open compressible boundary, the key question is not “what value should be imposed for each primitive variable?” but rather “which characteristic information is entering the computational domain, and which characteristic information is leaving it?”

The usual starting point is a locally one-dimensional inviscid analysis in the outward normal direction \(\mathbf n\). In that approximation, the acoustic part of the boundary data can be written in terms of the characteristic combinations

(22.1.1)#\[W^{\pm} = p \pm Z u_n, \qquad Z=\rho a,\]

where \(u_n=\mathbf u\cdot\mathbf n\) and \(Z\) is the acoustic impedance. The signs here are tied to the outward-pointing normal: at a subsonic boundary, one acoustic characteristic carries information from the interior toward the boundary, while the other carries information from the exterior or target state into the domain.

This observation is the reason the NRBC is formulated differently at inlet and outlet:

  1. at a subsonic inlet, one acoustic relation must remain compatible with the interior state, while the remaining incoming information is supplied by the target inflow state,

  2. at a subsonic outlet, the incoming acoustic content is associated primarily with pressure, while the outgoing information is taken from the interior flow,

  3. at supersonic boundaries, the characteristic picture becomes one-way: all information enters at a supersonic inflow, and all information leaves at a supersonic outflow.

The one-dimensional characteristic picture alone is not sufficient for a practical multidimensional CFD boundary condition. The formulation below therefore augments the normal-direction compatibility relation with explicit relaxation toward a target state and, at subsonic outlets, with transverse multidimensional corrections. Those additions are the parts of the model most closely associated with the later NSCBC developments of Sutherland and Kennedy (2003) and Yoo and Im (2007).

22.2. Notation#

At one boundary face, let \(\mathbf n\) be the outward unit normal. The adjacent interior-cell state is written

(22.2.1)#\[\left(\rho_i,\mathbf u_i,p_i,T_i,a_i\right),\]

where \(a_i\) is the acoustic speed associated with the interior state.

The target state prescribed for the boundary model is written

(22.2.2)#\[\left(\rho_{\mathrm{ref}},\mathbf u_{\mathrm{ref}},p_{\mathrm{ref}},T_{\mathrm{ref}},a_{\mathrm{ref}}\right).\]

For outlet boundaries, the target pressure is denoted by

(22.2.3)#\[p_{\mathrm{tar}}.\]

Because the solver is nonlinear and pressure-based, the NRBC update also uses a lagged reference during a time step’s nonlinear iteration sequence. The velocity and pressure components are face values, while density and acoustic speed are taken from the adjacent cell:

(22.2.4)#\[\left(\rho^{\ell},\mathbf u^{\ell},p^{\ell},a^{\ell}\right).\]

This lagged reference plays a stabilizing role. At the first inlet iteration, the target state supplies these values; at the first outlet iteration, the adjacent-cell state does. Later outer iterations use values captured on the first outer iteration of that time step. For the outlet formulation, lagged face gradients are also used so that the boundary state is not reconstructed from the same just-updated face state.

The normal velocity components are

(22.2.5)#\[u_{i,n} = \mathbf u_i \cdot \mathbf n, \qquad u_{\mathrm{ref},n} = \mathbf u_{\mathrm{ref}} \cdot \mathbf n, \qquad u_n^{\ell} = \mathbf u^{\ell} \cdot \mathbf n.\]

The tangential part of a vector is written with the projector

(22.2.6)#\[\mathbf P_t = \mathbf I - \mathbf n \otimes \mathbf n, \qquad \mathbf u_t^{\ell} = \mathbf P_t \mathbf u^{\ell}.\]

The acoustic impedances used in the characteristic relations are

(22.2.7)#\[Z_i=\rho_i a_i, \qquad Z^{\ell}=\rho^{\ell}a^{\ell}.\]

Two user-controlled parameters enter the relaxation terms:

(22.2.8)#\[\sigma \ge 0, \qquad L > 0.\]

Here \(\sigma\) is a dimensionless relaxation strength and \(L\) is a boundary length scale. The present formulation also uses representative patchwise normal-Mach measures:

  1. \(M_{\max}\), which modulates the strength of the relaxation term,

  2. \(M_{\mathrm{avg}}\), which modulates the strength of the outlet transverse correction.

With that notation, the subsonic relaxation coefficient used repeatedly below is

(22.2.9)#\[\kappa = \max\left[ 0, \frac{\sigma a^{\ell}}{L} \left(1-M_{\max}^2\right) \right].\]

For subsonic inflow, a second relaxation rate is used:

(22.2.10)#\[\alpha_{\rho} = \frac{\sigma a^{\ell}}{L}.\]

22.3. Boundary-State Philosophy#

The NRBC model does not attempt to prescribe all primitive variables blindly at every open boundary. Instead, it distinguishes between outgoing and incoming characteristic content.

At a subsonic inlet, only one acoustic characteristic leaves the domain, so the boundary pressure is determined from a compatibility relation, while density and velocity are relaxed toward a target state.

At a subsonic outlet, one acoustic characteristic enters the domain, so the pressure is relaxed toward the target outlet pressure while the velocity and density are reconstructed from the resulting pressure change and from the interior state.

At supersonic boundaries, the branching is simpler: all characteristics enter at a supersonic inflow, while all characteristics leave at a supersonic outflow.

22.4. Inlet Formulation#

22.4.1. Supersonic Inflow#

If the prescribed inflow speed is supersonic relative to the target acoustic speed,

(22.4.1.1)#\[-u_{\mathrm{ref},n} > a_{\mathrm{ref}},\]

the full target state is imposed:

(22.4.1.2)#\[\mathbf u_b = \mathbf u_{\mathrm{ref}}, \qquad p_b = p_{\mathrm{ref}}, \qquad T_b = T_{\mathrm{ref}}.\]

This is the expected characteristic result: for a supersonic inflow, all primitive information is incoming and can be prescribed.

22.4.2. Subsonic Inflow#

For a subsonic inlet, the acoustic picture is mixed: one wave leaves the domain and must remain compatible with the adjacent interior state, while the incoming acoustic content is associated with the target inflow state. In the local one-dimensional approximation, this is expressed by taking the outgoing characteristic from the interior and the incoming characteristic from the target state,

(22.4.2.1)#\[W_b^{+}=W_i^{+}, \qquad W_b^{-}=W_{\mathrm{ref}}^{-}.\]

The implementation limits the interior pressure used by this compatibility relation to a band around the target pressure,

(22.4.2.2)#\[\widetilde p_i = \min\left[ \max\left(p_i,0.9p_{\mathrm{ref}}\right), 1.1p_{\mathrm{ref}} \right].\]

Using Eq. (22.1.1) with the interior impedance \(Z_i=\rho_i a_i\), the resulting compatible boundary pressure is

(22.4.2.3)#\[ p_b = \frac{1}{2} \left[ \left(\widetilde p_i + Z_i u_{i,n}\right) + \left(p_{\mathrm{ref}} - Z_i u_{\mathrm{ref},n}\right) \right],\]

or equivalently,

(22.4.2.4)#\[p_b = \frac{1}{2} \left[ p_{\mathrm{ref}} + \widetilde p_i - \rho_i a_i \left(u_{\mathrm{ref},n}-u_{i,n}\right) \right].\]

This is the essential reason pressure is not prescribed freely at a subsonic inlet: the boundary pressure is the value compatible with one outgoing interior acoustic signal and one incoming target acoustic signal. The pressure band is an implementation safeguard for strongly mismatched startup states; it is not part of the underlying linear characteristic relation.

If the adjacent-cell velocity reverses so that \(u_{i,n}\ge 0\), the face is no longer behaving as an inlet. The implementation then applies the full target inlet state instead of amplifying the reversal through the subsonic characteristic update.

Once \(p_b\) is known, the remaining variables are not reconstructed from a purely one-dimensional inviscid model alone. Following the practical NSCBC open-boundary philosophy of relaxing the boundary state toward the target data over a finite length scale emphasized by Sutherland and Kennedy (2003), density is updated by combining acoustic compressibility with explicit relaxation toward the target density:

(22.4.2.5)#\[\rho_b = \frac{ \rho^{\ell} + \Delta t\,\alpha_{\rho}\rho_{\mathrm{ref}} + \dfrac{p_b-p^{\ell}}{\left(a^{\ell}\right)^2} }{ 1+\Delta t\,\alpha_{\rho} }.\]

The boundary velocity is then relaxed toward the target velocity, while still responding to the acoustic pressure correction in the normal direction:

(22.4.2.6)#\[\mathbf u_b = \frac{ \mathbf u^{\ell} + \Delta t\,\kappa \mathbf u_{\mathrm{ref}} + \mathbf n \dfrac{p_b-p^{\ell}}{\rho^{\ell}a^{\ell}} }{ 1+\Delta t\,\kappa }.\]

The boundary temperature follows from the equation of state after \((\rho_b,p_b)\) has been determined:

(22.4.2.7)#\[T_b = T(\rho_b,p_b,Y_b),\]

where \(Y_b\) denotes the boundary composition used by the mixture model.

The role of the inlet formulation can be summarized compactly:

  1. use characteristic compatibility to obtain the admissible boundary pressure,

  2. relax density toward the target thermodynamic state,

  3. relax velocity toward the target inflow direction and magnitude,

  4. recover temperature from the EOS.

22.5. Outlet Formulation#

22.5.1. Backflow and Supersonic Branches#

The outlet treatment first checks whether the face is actually behaving like an outlet.

If

(22.5.1.1)#\[u_{i,n} \le 0,\]

the present formulation does not apply the nonreflecting update. Instead, it switches to a robust backflow fallback. Conceptually, this branch treats the face as a temporarily ill-posed outlet and suppresses the outgoing update.

If

(22.5.1.2)#\[u_{i,n} \ge a_i,\]

the outlet is supersonic and all characteristics leave the domain. The boundary state is therefore extrapolated from the interior:

(22.5.1.3)#\[\mathbf u_b = \mathbf u_i, \qquad p_b = p_i, \qquad T_b = T_i.\]

22.5.2. Subsonic Outlet#

For a subsonic outlet, the characteristic roles are reversed: the outgoing information is carried mainly by the interior flow, while the incoming acoustic information is associated with pressure. The practical consequence is that the pressure update becomes the primary boundary relation, and the outlet density and velocity are reconstructed afterward from that pressure change.

The subsonic outlet branch is therefore the main nonreflecting relation. It uses the lagged state

(22.5.2.1)#\[\left(\rho^{\ell},\mathbf u^{\ell},p^{\ell},a^{\ell}\right)\]

and the normal-velocity change

(22.5.2.2)#\[\Delta u_n = u_{i,n} - u_n^{\ell}.\]

If only the one-dimensional characteristic picture were retained, the outlet pressure would be updated from the outgoing acoustic compatibility and a relaxation toward the target outlet pressure. In multidimensional viscous/reacting flow, however, purely normal-direction compatibility is not sufficient. The outlet model therefore also includes a transverse correction of the kind introduced by Yoo and Im (2007). Let

(22.5.2.3)#\[\nabla_t p^{\ell} = \mathbf P_t \nabla p^{\ell}\]

denote the tangential pressure gradient. Let

(22.5.2.4)#\[\mathbf d_n^{\ell} = \nabla_t u_n^{\ell}, \qquad \theta_t^{\ell} = \nabla_t\cdot \mathbf u_t^{\ell}\]

denote the tangential gradient of the normal velocity and the tangential divergence of the tangential velocity. The transverse contribution is then written as

(22.5.2.5)#\[\mathcal T_t^{\ell} = -\frac{1}{2} \left[ \mathbf u_t^{\ell}\cdot \left( \nabla_t p^{\ell} - \rho^{\ell}a^{\ell}\mathbf d_n^{\ell} \right) + \gamma_i p^{\ell}\,\theta_t^{\ell} \right],\]

where \(\gamma_i\) is the effective ratio of specific heats inferred from the adjacent interior thermodynamic state. This term is the part of the outlet model that accounts for multidimensional wave content that a purely one-dimensional LODI relation would miss.

The strength of the transverse term is modulated by

(22.5.2.6)#\[\beta_{\mathrm{NR}} = M_{\mathrm{avg}}.\]

With these definitions, the subsonic outlet pressure update is

(22.5.2.7)#\[p_b = \frac{ p^{\ell} + \rho^{\ell}a^{\ell}\Delta u_n + \Delta t\,\kappa p_{\mathrm{tar}} - \Delta t\,\beta_{\mathrm{NR}} \mathcal T_t^{\ell} }{ 1+\Delta t\,\kappa }.\]

Each term in Eq. (22.5.2.7) has a clear origin:

  1. \(p^{\ell}\) is the lagged boundary pressure about which the new update is formed,

  2. \(\rho^{\ell}a^{\ell}\Delta u_n\) is the one-dimensional acoustic compatibility contribution,

  3. \(\Delta t\,\kappa p_{\mathrm{tar}}\) is the explicit relaxation toward the desired outlet pressure,

  4. \(-\Delta t\,\beta_{\mathrm{NR}}\mathcal T_t^{\ell}\) is the multidimensional transverse correction.

Once \(p_b\) is known, the boundary velocity is reconstructed from the acoustic pressure jump:

(22.5.2.8)#\[\mathbf u_b = \mathbf u_i + \mathbf n \frac{p_i-p_b}{\rho_i a_i}.\]

The boundary density is then updated through the acoustic linearization

(22.5.2.9)#\[\rho_b = \rho_i + \frac{p_b-p_i}{a_i^2},\]

and the temperature again follows from the EOS:

(22.5.2.10)#\[T_b = T(\rho_b,p_b,Y_b).\]

This construction is consistent with the intended role of a subsonic outlet: pressure carries the incoming characteristic information, while velocity and density adjust to that pressure change and to the interior state already inside the domain.

22.6. Interpretation of the Relaxation Parameters#

The two user-controlled quantities \(\sigma\) and \(L\) appear only through ratios of the form \(a/L\), so their interpretation is straightforward:

  1. larger \(\sigma\) strengthens relaxation toward the target state,

  2. smaller \(L\) strengthens the same relaxation by shortening the nominal damping distance,

  3. \(\sigma=0\) removes the explicit relaxation-to-target contribution and leaves only the compatibility update.

The parameter \(\kappa\) in Eq. (22.2.9) also carries the factor \((1-M_{\max}^2)\), so the explicit relaxation weakens as the characteristic normal Mach number of the boundary patch approaches unity from below. The nonnegative limit prevents a locally supersonic patch value from turning relaxation into anti-damping while a face is still evaluated by the subsonic branch.

The factor \(\beta_{\mathrm{NR}}\) in Eq. (22.5.2.6) increases the influence of the transverse term as the mean outlet Mach number grows. In that sense, it acts as a multidimensional correction weight rather than as a separate target-state relaxation parameter.

22.7. Closing Remarks#

The present NRBC formulation is designed to reduce reflections at open compressible boundaries, not to guarantee a perfectly reflection-free boundary for arbitrary nonlinear waves. In particular:

  1. the subsonic inlet relies on a normal-direction characteristic compatibility relation plus relaxation toward a target state,

  2. the subsonic outlet relaxes pressure toward a target outlet pressure while adding transverse-gradient corrections,

  3. the supersonic branches reduce to full prescription at inflow and pure extrapolation at outflow,

  4. the outlet branch includes a separate backflow fallback because a genuine backflow event is not a one-way-wave outflow problem.

Once the boundary primitive state has been constructed by these relations, the corresponding boundary mass fluxes and pressure-correction contributions fit into the finite-volume framework developed in Finite-Volume Transport Assembly on Unstructured Grids and Pressure-Correction Equation on Unstructured Grids.

22.8. Literature Context#

The equations in this appendix should therefore be read as one practical member of the characteristic open-boundary family:

  1. Thompson, K.W. (1987), “Time-Dependent Boundary Conditions for Hyperbolic Systems,” Journal of Computational Physics, 68(1), 1-24.

  2. Poinsot, T.J. and Lele, S.K. (1992), “Boundary Conditions for Direct Simulations of Compressible Viscous Flows,” Journal of Computational Physics, 101(1), 104-129.

  3. Sutherland, J.C. and Kennedy, C.A. (2003), “Improved Boundary Conditions for Viscous, Reacting, Compressible Flows,” Journal of Computational Physics, 191(2), 502-524.

  4. Yoo, C.S. and Im, H.G. (2007), “Characteristic Boundary Conditions for Simulations of Compressible Reacting Flows with Multi-Dimensional, Viscous and Reaction Effects,” Combustion Theory and Modelling, 11(2), 259-286. Older code comments sometimes cite this work as 2006 because that corresponds to the accepted-manuscript date rather than the final journal issue.