Copyright © A. Filippone (1999-2003). All Rights Reserved.
Computational Aerodynamics

Viscous-Inviscid Interaction


The viscous-inviscid coupling techniques have been developed rather late, if compared to the process of development of both inviscid and viscous solutions. The progress of this approach relies primarily on the coupling algorithm.


Development of a coupling algorithm is a very difficult task, that is somewhat dissociated from the domain of validity of the equations that describe the viscous flow.

Among the leat successful methods there is the Inverse Method, which is not practical because of the problem of defining an inverse formulation of the inviscid flow.

More successful was the Semi-Inverse method developed by Carter, 1979, with some empirical background. The method is based on the under-relaxation on the boundary conditions, and does converge for unseparated or mildly separated flow at low angle of attack. It generally requires hundreds of iterations and the empirical choice of the relaxation parameter.

Presently there are three alternatives. Le Balleur’s Semi-Inverse scheme with stability analysis (Le Balleur, 1978) remains a classic robust formulation for airfoil flows. It has been extended to the computation of flows with massive separation, multi-element airfoils, three-dimensional flows.

Another method, called Quasi-Simultaneous, developed originally by Veldman, 1981, consists in coupling simultaneously a linear combination of pressure and displacement thickness with the triple-deck boundary layer theory. This method advances in the streamwise direction, in a fashion similar to parabolic problems. The Quasi- Simultaneous method has been applied to airfoil flows and fixed-wing flows by a number of authors.

The final alternative is the Fully-Simultaneous method. It consists in solving simultaneously the inviscid and viscous flow equations, and coupling them through the residuals with a Newton-type of iteration (Drela-Giles, 1987). This scheme, that is probably the most robust viscous-inviscid formulation for airfoil flows, presents difficulties in three dimensions, due to the enormous size of the jacobian matrix that has to be inverted.

The methods just mentioned have been developed in the attempt of calculating steady flows. Another method due to Le Balleur, 1985, called Semi-Implicit, was developed for the computation of unsteady flows at transonic speeds. This was the first attempt to solve time-consistent flows, with both viscous and inviscid solvers implicit in time, and the coupling algorithm performed by a relaxation at each time-step. After some initial success, this method has not been applied, and Le Balleur himself has worked on improving his Semi-Inverse scheme.

Governing Equations

The governing equations for the inviscid outer flow go from the panel methods all the way up to the compressible Euler equations, including transonic small perturbation equations and full potential equation.

For the viscous flow the boundary layer equations, either in integral or differential form, are widely used, although they should be restricted to flows with limited separation and mild normal pressure gradients.

It is not uncommon to find calculations that extend to conditions involving large trailing edge separation and deep stall. The Reynolds-averaged Navier-Stokes, or parabolized equations, are more appropriate to these types of flows, and have become increasingly attractive.


The Reynolds numbers involved always require appropriate turbulence modelling and turbulent transition criteria. The differential equations have been solved with eddy viscosity terms as modelled by Cebeci-Smith, 1974, Baldwin-Lomax, 1978, and Johnson-King, 1984. The integral equations, instead, need closure relations, often referred to as lag-entrainment equations (Head, 1958).

In recent years it has appeared that higher order terms are more easily incorporated in the integral equations than in the corresponding differential equations, although the latter ones are nominally more correct.


At the beginning of the development of the VII it was considered a satisfactory goal to compute separated boundary layers. A landmark in this direction is the integration of the boundary layer equations past the separation point, Goldstein’s singularity (Catherall-Mangler, 1966).

The direct solution of the boundary layer equations in separated regions proved to be an ill-conditioned problem, therefore some methods have been developed for their inverse integration. The switch between direct/inverse mode in attached/separated region has been applied by several authors.

The research converged to the problem of predicting the maximum lift coefficient (Le Balleur, Drela,, to computing the shock-wake boundary layer interaction (Delery-Marvin, 1986), multi-component airfoils, spoiler devices and blunt bodies (Lock-Williams, 1987), axisymmetric and three-dimensional flows over wings (Wigton-Yoshihara, 1983; Le Balleur, 1983).

Most of these goals have been reached by different means, but the problem of computing unsteady flows remains open.

Range of Applications

Figs. 1 and 2 show the range of application of successful VII techniques for both airfoil flows and fixed wings. Such polars do not exist for rotating blades, because the problem is largely unexplored.

Figure 1: Range of application of VII techniques for airfoils

Figure 2: Range of application of VII techniques for wings

For details see:

Selected References

  • Cebeci T, Smith AMO. Analysis of Turbulent Boundary Layers. McGraw-Hill, New York, 1974.

  • Lock RC, Williams BR. Viscous-Inviscid Interaction in External Aerodynamics. Progress in Aerospace Sciences, Vol. 24, No 2, pages 51-160, 1987.

  • Proceedings of 2th Symp. on Numerical and Physical Aspects of Aerodynamic Flows. Springer-Verlag, Long Beach, Ca, Jan. 1983.

  • Proceedings of 3th Symp. on Numerical and Physical Aspects of Aerodynamic Flows. Springer-Verlag, Long Beach, Ca, Jan. 1986.

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Copyright © A. Filippone (1999-2003). All Rights Reserved.