Steady two-dimensional flow through a row of normal flat plates

Abstract

A numerical and experimental study is described for the two-dimensional steady flow through a uniform cascade of normal flat plates. The Navier–Stokes equations are written in terms of the stream function and vorticity and are solved using a second-order-accurate finite-difference scheme which is based on a modified procedure to preserve accuracy and iterative convergence at higher Reynolds numbers. The upstream and downstream boundary conditions are discussed and an asymptotic solution is employed both upstream and downstream. A frequently used method for dealing with corner singularities is shown to be inaccurate and a method for overcoming this problem is described. Numerical solutions have been obtained for blockage ratio of 50 % and Reynolds numbers in the range 0 [les ]R[les ] 500 and results for both the lengths of attached eddies and the drag coefficients are presented. The calculations indicate that the eddy length increases linearly withR, at least up toR= 500, and that the multiplicative constant is in very good agreement with the theoretical prediction of Smith (1985a), who considered a related problem. In the case ofR= 0 the Navier–Stokes equations are solved using the finite-difference scheme and a modification of the boundary-element method which treats the corner singularities. The solutions obtained by the two methods are compared and the results are shown to be in good agreement. An experimental investigation has been performed at small and moderate values of the Reynolds numbers and there is excellent agreement with the numerical results both for flow streamlines and eddy lengths.

The stability of laminar symmetric separated wakes

“…Numerical solutions for circular cylinder cascades have been presented by Fornberg (1991) (with Re 6 400 in the present terms and 5 6 H 6 100) and, very recently and to much higher Reynolds numbers, by Gajjar & Azzam (2004). Similar computations for flat-plate cascades have been by presented by Natarajan, Fornberg & Acrivos (1993) (with Re 6 400 and 5 6 H 6 25) and Ingham, Tang & Morton (1990) (with Re 6 500 and H = 2). Chernyshenko's (1988) asymptotic theory was extended to the cascade case by Chernyshenko & Castro (1993) and numerical computations gave results which were shown to tend qualitatively towards those expected, but only at the very highest Reynolds numbers.…”

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

“…As an alternative to the least-square matching, the exact matching strategy, which solves the same number of matching equations as the coe cients, namely n = m+1 in Equation (22), were applied in References [17,19]. Nevertheless, we found that this method is not robust since the convergence speed and the results can be very sensitive to the matching locations.…”

“…This method was applied to treat the stress singularity in die swell problems in Stokes ows [6,7,18]. Several studies [17,19,20] have employed the local asymptotic solution for the Stokes ow to overcome the vorticity singularity in the numerical solution of the NavierStokes equations adopting the stream function-vorticity formulation. They assigned the local asymptotic solution for the Stokes ow in the neighbourhood of an angular point as the boundary condition for the ow in the remaining region, which was solved using a ÿnite-di erence scheme.…”

“…However, the convergence of the numerical solution using this strategy strongly depends on the implementation. For example, Ingham et al [19] assigned the local solution to three grid points closest to the tip of a vertical at plate and determined the unknown coe cients of the local solution from the numerical solution at the adjacent points. They reported that a very small under-relaxation factor has to be applied in order to achieve convergence.…”

“…In et al [20] considered the same problem. They introduced a sub-zone matching procedure rather than the point matching as in Reference [19] and achieved robust convergence. The accuracy of the numerical solution was not reported in both studies.…”