# Coupling the Numerical Solution of the Partial Differential Equations of the Viscous Flow with the Integral Equations of the Outer Inviscid Flows

9.1.3.1 Coupling the Numerical Solution of Boundary Layer Equations with Integral Representation of the Inviscid Flows:

Incompressible Flows

For incompressible flows, the outer flow region is approximated by integrals over the body and the wake in terms of the shape of the body and the displacement thickness, as discussed before. The boundary layer equations are discretized using finite difference methods. The velocity u at the outer edge of the boundary layer is related to the distribution of the displacement thickness. Iterative methods must be used to solve these coupled nonlinear equations. Many sweeps of the boundary layer are required and upstream effects are recovered through the coupling with the inviscid flow integral representation. Separated flows can be simulated using upwinding scheme for the convective terms of the streamwise momentum equation. This procedure is used by many authors, see for example Lock and Williams [30]. A similar procedure is used also in triple deck theory, see Stewartson’s review [31].

9.1.3.2 Coupling the Numerical Solution of Boundary Layer Equations with Integral Representation of the Inviscid Flows: Supersonic Flows

For supersonic flows, the inviscid outer region can be represented by the Ackeret formula. The finite difference approximations of the boundary layer equations can be solved by marching in the direction of the flow in case of no separation. local iterations at each step are still needed. This procedure was used in the sixties by Reynher and Flugge-Lotz [32] to study shock wave/ boundary layer interaction. For separated flows, they switch off the uux term in the momentum equation since u is negative but very small in the separated region. Such strategy (FLARE approximation) has been adopted by others to avoid numerical instability of the calculations of separated flows, although the results are at best first order accurate this way. Also, the simple relation between the pressure and the deflection angle for the outer flow region limits the validity of the calculations to Small disturbance linearized supersonic aerodynamic problems.

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