Based on the hypothesis that patterns of skin-friction lines and external streamlines reflect the properties of continuous vector fields, topology rules define a small number of singular points (nodes, saddle points, and foci) that characterize the patterns on the surface and on particular projections of the flow (e.g., the crossflow plane). The restricted number of singular points and the rules that they obey are considered as an organizing principle whose finite number of elements can be combined in various ways to connect together the properties common to all steady three dimensional viscous flows. Introduction of a distinction between local and global properties of the flow resolves an ambiguity in the proper definition of a three dimensional separated flow. Adoption of the notions of topological structure, structural stability, and bifurcation provides a framework to describe how three dimensional separated flows originate and succeed each other as the relevant parameters of the problem are varied.

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Topology of Three-Dimensional Separated Flows

The flow field around surface-mounted, prismatic obstacles with different spanwise dimensions was investigated using the crystal violet, oil-film and laser-sheet visualization techniques as well as by static pressure measurements. The aim of this study is to highlight the fundamental differences between nominally two-dimensional and fully three-dimensional obstacle flows. All experiments were performed in a fully developed channel flow. The Reynolds number, based on the height of the channel, lay between 8 X 10[sup 4] and 1.2 X 10[sup 5]. Results show that the middle region of the wake is nominally two-dimensional for width-to-height ratios (W/H) greater than 6. The separated region in front of wider obstacles is characterized by the appearance of a quasi-regular distribution of saddle and nodal points on the forward face of the obstacles. These three-dimensional effects are considered to be inherent to such separating flows with stagnation.

The Flow Around Surface-Mounted, Prismatic Obstacles Placed in a Fully Developed Channel Flow (Data Bank Contribution)

This paper describes the flow structure observed over a bell-shaped hill with height h (the profile of which is the reciprocal of a fourth-order polynomial) when it was placed first in a large towing tank containing stratified saline solutions with uniform stable density gradients and second in an unstratified wind tunnel. (A similarly shaped model hill was also studied in a small towing tank.) Observations were made at values of the Froude number F (≃ U/Nh) in the range 0·1 to 1·7 and at F = ∞, where U is the towing speed and N is the Brunt-Vaisala frequency, and at values of the Reynolds number from 400 to 275000. For F [lsim ] 0·4, the observations verify Drazin's (1961) theory for low-Froude-number flow over three-dimensional obstacles and establish limits of applicability. For Froude numbers of the order of unity, it is found that a classification of the lee-wave patterns and separated-flow regions observed in two-dimensional flows also appears to apply to three-dimensional hills.Flow-visualization techniques were used extensively in obtaining both qualitative and quantitative information on the flow structure around the hill. Representative photographs of dye tracers, potassium permanganate dye streaks, shadowgraphs, surface dye smears, and hydrogen-bubble patterns are included here. While emphasis is centred on obtaining a basic understanding of the flow around three-dimensional hills, the results are applicable to the estimation of air pollutant dispersion around hills.

Experiments on stably and neutrally stratified flow over a model three-dimensional hill

This is a review of the mechanisms that control neutrally stable turbulent boundary-layer flow over hills and waves, their relative magnitudes, and how they exert their greatest effects in different regions of the flow. We compare calculations based on various analytical and computational models with each other and with relevant experimental data. We discuss practical applications of these studies.

Turbulent flow over hills and waves

By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at post-stall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by periodic blowing–suction near the leading edge with low-level power input. This unsteady forcing can modulate the evolution of the separated shear layer to promote the formation of concentrated lifting vortices, which in turn interact with trailing-edge vortices in a favourable manner and thereby alter the global deep-stall flow field. In a certain range of post-stall angles of attack and forcing frequencies, the unforced random separated flow can become periodic or quasi-periodic, associated with a significant lift enhancement. This opens a promising possibility for flight beyond the static stall to a much higher angle of attack. The same local control also leads, in some situations, to a reduction of drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for high-α buffet control. The computations are in qualitative agreement with several recent post-stall flow control experiments. The physical mechanisms responsible for post-stall flow control, as observed from the numerical data, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The leading-edge shear layer and vortex shedding from the trailing edge are two basic constituents of unsteady post-stall flow and its control. Since the former has a rich spectrum of response to various disturbances, in a quite wide range the natural frequency of both constituents can shift and lock-in to the forcing frequency or its harmonics. Thus, most of the separated flow becomes resonant, associated with much more organized flow patterns. During this nonlinear process the coalescence of small vortices from the disturbed leading-edge shear layer is enhanced, causing a stronger entrainment and hence a stronger lifting vortex. Meanwhile, the unfavourable trailing-edge vortex is pushed downstream. The wake pattern also has a corresponding change: the shed vortices of alternate signs tend to be aligned, forming a train of close vortex couples with stronger downwash, rather than a Karman street.

Post-stall flow control on an airfoil by local unsteady forcing

In flows around three-dimensional surface obstacles in laminar or turbulent streamsthere are a number of points where the shear stress or where two or more component,s of the mean velocity are zero. In the first part of this paper we summarize and extend the kinematical theory for the flow near these points, particularly by emphasizing the topological classification of these points as nodes or saddles. We show that the zero-shear-stress points on the surface and on the obstacle must be such that the sum of the nodes ΣN and the sum of the saddles Σs satisfy
 \[
 \Sigma_N -\Sigma_S = 0.
 \]
 If the obstacle has a hole through it, such as a passageway under a building,
 \[
 \Sigma_N -\Sigma_S =-2.
 \]
 If the surface is a junction between two pipes,
 \[
 \Sigma_N -\Sigma_S =-1.
 \]
 We also consider, in two-dimensional plane sections of the flow, the points where the components of the mean velocity parallel to the planes are zero, both in the flow and near surfaces cutting the sections. The latter points are half-nodes N′ or half-saddles S′. We find that
 \[
 (\Sigma_N +{\textstyle\frac{1}{2}}\Sigma_{N^{\prime}}-(\Sigma_{S^{\prime}}+{\textstyle\frac{1}{2}}\Sigma_{S^{\prime}}) = 1-n,
 \]
 where n is the connectivity of the section of the flow considered.In the second part new flow-visualization studies of laminar and turbulent flows around cuboids and axisymmetric humps (i.e. model hills) are reported. A new method of obtaining a high resolution of the surface shear-stress lines was used. These studies show how enumerating the nodes and saddle points acts as a check on the inferred flow pattern.Two specific conclusions drawn from these studies are that:
 for all the flows we observed, there are no closed surfaces of mean streamlines around the separated flows behind three-dimensional surface obstacles, which con-tradicts most of the previous suggestions for such flows (e.g. Halitsky 1968);the separation streamline on the centre-line of a three-dimensional bluff obstacle does not, in general, reattach to the surface.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112078001068

Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization

A pair of curved gauzes with non-uniform porosity was used to generate a strong constant-shear flow with low turbulence intensity. The complex features of this flow around a circular cylinder with its axis normal to the vorticity and mean velocity vectors were studied. Methods such as the use of end plates, inclusion of gaps at the junctions of the cylinder with the end plates and fluid withdrawal were applied to minimise end effects. Exploratory studies were made to investigate shear effects on the vortex wake formation region. Effects of the steepness factor on vortex formation were explained.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112089001862

J. A. Peterka

Papers

Kinematical studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization

Secondary flows and vortex formation around a circular cylinder in constant-shear flow