# Coupling Numerical Solutions of Partial Differential

Equations of both the Inviscid and Viscous Flow Regions

9.1.4.1 Coupling the Numerical Solution of Boundary Layer Equations with the Numerical Solution of the Inviscid Flow Equations

In this case, some coupling procedures discussed before as inverse and semi-inverse or simultaneous iterations are used. For incompressible flows, Moses et al.  intro­duced a simple effective method where the inviscid flow and viscous flow equations are solved coupled along vertical lines extending from the boundary layer to the outer inviscid flow region, governed by elliptic equation for the streamfunction (or potential function). hence, the line relaxation method of the inviscid flow equation is augmented at each streamwise location with the discrete boundary layer equa­tions, coupled through the displacement thickness and many sweeps are required for convergence. The parabolic nature of the boundary layer equations in the case of attached flows, is not utilized in this method. However, the method can be used for the simulation of separated flows.

Also, the method can be extended to transonic flows where the inviscid flow equations are nonlinear and of mixed-type for both potential and Euler formulations. In this regard, one should mention the successful method introduced by Drela and Giles .

Higher order boundary layer equations, including curvature effects and pressure variation normal to the main flow direction, have been used and have led to the triple deck theory (see Stewartson , Neiland  and Sychev ). The latter has been proved to be successful for the trailing edge and separated laminar flows. Attempts to extend the theory to turbulent flows are faced with the problem of turbulence modeling. Extensions to unsteady flows are in progress (see Ryzhov ).