Viscous-Inviscid Interaction
Summary
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.
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, et.al), 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.
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.
Overview
Governing Equations
Turbulence
State-of-the-Art
Range of Applications
For details see:
Selected References