Weak Viscous/Inviscid Interactions

The standard boundary layer theory does not take into account normal gradients outside the boundary layer. A higher order boundary layer theory was developed by Van Dyke in [135]. The first order outer region equations are the standard Euler equations for inviscid flows, while the first order inner region equations are the standard boundary layer equations. The second order outer region equations are the linearized Euler equations, while the second order inner region equations are linear perturbation of the boundary layer equations with several source terms.

Because of linearity, the boundary conditions and the source terms can be split to different second order effects including curvature terms, displacement, entropy and total enthalpy gradients. Notice, the second order external flow is affected by the displacement of the first order boundary layer. The inner and outer solutions must match and a single composite formula valid in the whole domain can be obtained.

Recently, Cousteix developed a hypersonic interaction theory based on the defect approach [136]. (The defect approach was developed by Le Balleur for transonic flows). Cousteix used in the inner region the difference between the physical variables and the external solutions. In the outer region, the defect variables vanish and the equations for the outer flow region are identical to Van Dyke’s. The equations in the inner region are different. For example, the pressure in the first order boundary layer varies and is equal to the local inviscid flow pressure.

In the defect approach, better matching is obtained between the inner and outer solutions than in Van Dyke’s method, at the expense of calculating the inviscid flow on the boundary layer grid.

Successful applications of Euler/boundary layer coupling via defect approach were reported by Monnoyer [137] who demonstrated that proper accounting for entropy and vorticity variations in the inviscid flow outer region is an important aspect of hypersonic flow simulations.