Numerical studies on laminar internals seperated flows

Abstract

In order to understand the nature of steady laminar incompressible separated flows, numerical solutions have been constructed for plane two-dimensional internal flows over a wide range of Reynolds numbers. Three aspects of the problem are considered in the study: (i) the methods of accelerating the convergence of iterative solutions of the finite difference analogues of the full Navier-Stokes and energy equations; (ii) the asymptotic nature of separated flows at high Reynolds numbers; (iii) the response of the separated region for changes in physical parameters for laminar flows which are physically realizable.

Among the many acceleration procedures tested in the study, a new implicit method – the Alternating Direction Line Iterative (ADLI) method – developed in the present work proves to be the most economical method of solving the Navier-Stokes equations for problems with well-defined boundaries. The method results in substantial savings in computational time when compared to the well-known Successive Over Relaxation (SOR) method. The accuracy of the method is comparable to that of SOR. The second-order procedure – Aitken's extrapolation technique – turns out to be the most efficient method of solving the energy equation.

The solutions of the flow in a square cavity show that only at very high Reynolds numbers (Re 30000) does the flow completely conform to that assumed by Batchelor's model. For the first time, it is demonstrated that the downstream secondary eddy grows and decays in a manner similar to the upstream one. The solutions of the flow over a backward-facing step exhibit two significant features which have not been observed before: (a) the separated region behind the step decreases in size for Re > 1000; (b) at such high Reynolds numbers, a second separated region appears on the upper wall of the duct.

The solution of the flow over a thin obstruction does not show a decrease in the length of the separated region up to Re = 1500. However, the rate of increase with Re shows a significant decrease at high Reynolds numbers. These results indicate that the characteristic behaviour of the internal separated flow is represented by the secondary eddies of the cavity flow. It should, however, be noted that the range of Reynolds numbers in which a separated region grows and decays is strongly dependent on the main flow conditions and geometry. None of these separated regions exhibit an inviscid core of constant vorticity, as envisaged by Batchelor's model. Moreover, the growth and decay of the separated regions is at variance with the assumptions of the other existing asymptotic models. Thus, none of the separated flow models seems to correctly represent the asymptotic behaviour of at least the class of separated flows considered here.

In the range of moderate Reynolds numbers, the influences of the type of separation inducer, the inlet flow conditions, suction, and injection on the separated region are studied. Also, the effects of thermal boundary conditions, Prandtl number, and conduction in the obstruction on heat transfer to the separated regions are considered. The solutions indicate that not all separated regions grow (i.e., increase in length) linearly with increase in Re even at moderate Reynolds numbers. They support the idea that the separated region as a whole, being completely viscous, behaves like a boundary layer. Lastly, the possibility of obtaining an approximate solution for a class of separated flows is examined.