Computational approaches based on previous nonlinear theoretical findings are developed for a Composite system, the Navier-Stokes system, and the Euler system of equations, in turn, with applications to incompressible boundary-layer transition and dynamic stall. The emphasis is on schemes appropriate for medium-to-large Reynolds numbers, and so as a start the computations are kept to two spatial dimensions. The Composite scheme is developed first, including significant normal-pressure-gradient effects as suggested by the theory. The same scheme is then modified to accommodate, iteratively, the Navier-Stokes form and then the Euler form, for comparison. The agreement between the three sets of results tends to be very close in the parameter ranges studied. The necessary extensions of the work are also discussed.
1. I n t r o d u c t i o n
Computational studies of unsteady disturbed flow produced within a boundary layer are described in this article, the computations being based on Composite, Navier-Stokes-and Euler-equation representations. The potential applications to understanding and prediction of boundary-layer transition and to unsteady airfoil computations are uppermost in mind here, along with a number of other motivations and issues which are addressed subsequently. These include the questions of whether a time-marching computational approach can possibly be used as an engineering tool at large Reynolds numbers, e.g. for transition prediction, whether in particular a two-dimensional time-marching treatment is able to capture significant aspects of transition to turbulence (see also below), and whether the approach can provide any fresh insight into transition prediction and transition criteria. Continuing new insight, especially of a nonlinear kind as here, is certainly needed in this area of transition and especially for by-pass transition processes, which partly form the background for the present investigation; there is also much interest in, and connection with, the closely related areas of unsteady separation and dynamic stall. Again, it is well known that the global effects of unsteady boundary-layer displacement, receptivity, and most of all transition, are often considerable in unsteady airfoil oscillations.
Most aerodynamic interest is in the medium-to-high Reynolds number range, within the present contexts. For that reason a Composite scheme, suggested by asymptotic scaling theories to a great extent, is applied and tested first, rather than the more conventional Navier-Stokes DNS approaches which tend to be restricted to relatively low Reynolds numbers, from accuracy and/or cost considerations, or Euler approaches which miss the viscous production. This Composite scheme (which tackles a reduced set of equations, as in Smith et al. [1]) is then used subsequently as the springboard for an apparently novel NS approach also, as well as for an Euler-equation approach, given the advantages of such reduced-equation schemes demonstrated in steady aerodynamic computations. Again, there are areas of unsteady