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Viscous/inviscid coupling
The solution failure at separation is traceable to the neglect of the viscous displacement mechanism. In brief, there is one unique value of due/ds which is admissible by the boundary layer at the separation point, so this value cannot be imposed via the input ue(s). The problem is eliminated if the displacement effect is incorporated into the potential flow problem, which is termed viscous/inviscid coupling. The boundary layer flow can now influence the potential flow’s ue(s) distribution, and can thus enforce the requirement of the unique due/ds value at the separation point.
A practical consequence of incorporating a viscous displacement model into a potential flow calculation is that now the potential and the boundary layer flow problems are two-way coupled, as diagrammed in
Figure 4.27. Specifically, the potential and boundary layer problems now depend on each other and cannot be solved in the simple sequential manner of the classical case diagrammed in Figure 4.26.
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Viscous / Inviscid Flow Problem for ф(Г), e(s), 8*(s)… wall BC p дф/bn = dm/ds і V—————— 1 Boundary layer equations m ODEs in e(s), 8*(s), Ue = дф/ds |
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A possible solution approach is to iterate between the potential and boundary layer equations, as suggested by the dotted arrows in Figure 4.27, which is known as direct viscous/inviscid iteration. This is not satisfactory since it tends to be unstable, as analyzed by Wigton and Holt [29]. The boundary layer problem will also fail outright if separation is encountered. Other iteration schemes have been proposed, such as the one by Veldman [30] and LeBalleur [23], with various degrees of success. The most reliable approach has been to solve the inviscid and viscous equations simultaneously by the Newton method. The XFOIL [5] and MSES [6] codes are two 2D implementations of this approach. Example results have been shown in Figures 3.6, 3.7, 4.9, 4.10. An example of simultaneously-coupled 2D viscous and 3D inviscid methods is the TRANAIR code, as reported by Bieterman et al [31].
Figure 4.27: Two-way coupling between potential-flow equations and boundary layer equations occurs if a displacement model is incorporated into the potential flow problem. The direct vis – cous/inviscid iteration suggested by the dotted arrows will fail if separation is present. A simultaneous solution of all the equations is most effective at avoiding this solution failure.
