Flow past a two– or three–dimensional steep–edged roughness

Abstract

Flow past a single small planar or three-dimensional roughness mounted on a smooth surface is investigated theoretically for various edge steepnesses, the oncoming planar motion being within a boundary layer or other near-wall shear. Nonlinear edge properties at large Reynolds numbers largely control the flow responses at the threedimensional roughness wing-tips and the impacts of separation(s), among other features. From analysis and computation, criteria are found for the generation of nonlinear upstream influence, downstream influence and separations, for two-and threedimensional roughnesses, as well as wing-tip separations. In particular, it is predicted that with a severe edge (e.g. a 90• forward-facing step) the ratio of the upstream separation distance over the roughness edge height is a constant times Re 1/4 W in two dimensions, the constant being approximately 0.142 and the Reynolds number Re W being based on the roughness edge height and the incident velocity slope at the surface. In three dimensions Re W is multiplied by sin ψ, as expected physically, where ψ is the tangent angle of the roughness planform. The ratio prediction above is very general, applying not only for any incident shear flow, but also for any front-edge geometry. Other separation and reattachment properties, extensions and a comparison with an experiment, are also discussed.

Bifurcation of nonlinear Tollmien–Schlichting waves in a high-speed channel flow

“…This phenomenon is similar to that observed in calculations of the interactive boundary-layer flow over a surface roughness, where regular separation is encountered as the roughness height parameter approaches a critical value (e.g. see Smith & Walton 1998). We can see from both the Navier-Stokes and asymptotic results that the flow is becoming localised: this seems to suggest that the asymptotic theory will eventually break down as it is based on an assumed slow streamwise development of the flow.…”

Computational analysis of the two-dimensional–three-dimensional transition in forward-facing step flow

“…In addition, Stüer's detailed velocity measurements show the three-dimensional flow ahead of the step to be an open-type separation rather than a closed separation as encountered in two dimensions. The streamline pattern computed from his experimental data reveal contracting spiralling motions in the separation zone, which resemble the flow structures ahead of three-dimensional blunt obstacles described by Smith & Walton (1998). A simplified sketch of this flow pattern at the step is provided in figure 3, which illustrates how fluid entrained into the separation region is transported in the spanwise direction, before being released over the step corner.…”

“…This is reflected, for example, in the stability properties of separated boundary layers at high Reynolds numbers (Re), where three-dimensional modes may become preferentially excited as a consequence of nonparallelism of the base flow (Stewart & Smith 1987; see also the overview article Smith 2000 and the references cited therein). For forward-facing steps and similar configurations, experiments as well as numerical simulations clearly revealed that the separation zone ahead of the step develops a three-dimensional structure (see Chiba et al 1995;Pollard et al 1996;Smith & Walton 1998;Stüer et al 1999;Chou & Chao 2000). Note, however, that this three-dimensionality not only occurs at high Reynolds numbers; it is also observed with flows at low Reynolds numbers down into the viscous regime.…”

On ‘spot’ evolution under an adverse pressure gradient

“…At increased amplitudes, or later times, the direct inertial effect of (6.6) may be connected with an empirical suggestion by Jones mentioned in § 1. Simultaneously, the mean-pressure property (6.8a) may tend to cause separation after the trailing edge passes by, the mechanism then being not unlike that in the upstream separations in channel flows (Smith 1978), in roughness flows (Smith & Walton 1998) Other features may be considered also, in a cautious manner. First, § § 3 and 4 indicate a 'spot' trailing-edge velocity equal to half that of the leading edge, a value which is not inconsistent with the above experiments.…”