Flow visualization studies of the zero-pressure-gradient turbulent boundary layer over the Reynolds-number range 500 < Reθ < 17500 have shown large Reynolds-number effects on boundary-layer structure.At high Reynolds numbers (Reθ > 2000, say) the layer appears to consist very largely of elongated hairpin vortices or vortex pairs, originating in the wall region and extending through a large part of the boundary-layer thickness or beyond it; they are for the most part inclined to the wall at a characteristic angle in the region of 40–50°. Large-scale features, which exhibit a slow overturning motion, appear to consist mainly of random arrays of such hairpin vortices, although there is some evidence of more systematic structures.At low Reynolds numbers (Reθ < 800, say) the hairpin vortices are very much less elongated and are better described as horseshoe vortices or vortex loops; large-scale features now consist simply of isolated vortex loops (at the very lowest Reynolds numbers), or of several such loops interacting strongly, and show a relatively brisk rate of rotation.

We use flow visualization to examine the end-wall boundary layer as it passes through a rotating blade row of an axial-flow compressor; we find that concentrations of vorticity, such as the trailing vortex shed from the blade tip, or the horseshoe vortex from the blade leading edge, which we may expect to be present in the end-wall boundary layer, are swamped and rapidly dispersed by the large-scale motions in the turbulent end-wall layer. We investigate the effect of varying tip clearance on the boundary layer and find that increasing tip clearance and blade loading both cause the layer to increase in thickness. We also investigate the effect on the boundary layer of an end-wall casing treatment of the radial type; this leads to an increase in large-scale turbulent motions in the end-wall layer and, for a given clearance and blade loading, to an increase in the thickness of the boundary layer. One vortex is evident with either a treated or solid end-wall: this is the scraping vortex due to the ploughing effect of the blade as it passes through the end-wall boundary layer. This vortex remains evident well downstream.

SummaryAlthough the Cebeci-Smith method of calculating turbulent boundary layers is widely used and generally gives acceptably accurate results, highly inaccurate skin-friction values are obtained for relaxing flows and equilibrium layers in strong adverse pressure gradient. In the present paper, these anomalies are removed by suitable modifications to the basic eddy-viscosity model.

SummaryA free-vortex flow over a stationary disc produces a boundary layer on the surface which proceeds inwards towards the centre under the action of the imposed radial pressure gradient. Close to the centre the boundary layer leaves the surface to form a rising core. The present paper uses a control-volume approach and earlier calculations of laminar boundary-layer development on the disc to determine the characteristics of the core-formation process.

SummaryEvidence is presented to show that the universal law of the wall has a wider range of validity than the assumptionl= ky, with k a universal constant. If an effective value of k is defined for the wall region its value is shown to vary between wide limits, and keffcan be correlated with other parameters describing the flow in the wall region.

A well-tested integral method has been used to calculate turbulent boundary-layer development for the distribution of external velocity given byU∝x−0.255. The results suggest that different values of the initial momentum thickness, so long as this is below some critical value, produce a range of equilibrium layers having widely different values of the form parameterG. For values of the initial momentum thickness greater than the critical value, layers are produced which proceed more or less rapidly to separation. These results provide a plausible explanation for conflicting experimental observations made in the past.Additional calculations for the flowsU∝x−0.15andU∝x−0.35suggest that, in the first case, a unique equilibrium condition is approached whatever the initial momentum thickness unless this exceeds some critical value; in the second case no equilibrium condition appears possible.

SummaryObserved variations of eddy viscosity in the outer regions of turbulent boundary layers are here explained in terms of the reorientation and stretching of vortex elements. The explanation, which is purely qualitative, gives a clear and plausible physical picture.

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Two implicit finite-difference schemes, one iterative and one non-iterative, have been used to obtain improved solutions for the laminar boundary layer on a finite disc beneath a potential-flow vortex.

SummaryTo obtain profiles of shear stress and eddy viscosity, the boundary-layer equations in finite-difference form have been applied to published turbulent-boundary-layer developments measured in nominally two-dimensional conditions. In applying this procedure, measured velocity profiles have been represented by members of Thompson’s profile family. Except close to separation, these representations are very satisfactory, and derived shear-stress profiles are generally in good agreement with direct measurements. Various eddy-viscosity and mixing-length models are compared with the results of the analysis and are found in general to differ widely among themselves and from the present results. The widely used assumption,l= ky in the wall region, appears to be invalid.

SummaryThe properties of equilibrium turbulent boundary layers have been examined using Thompson’s family of velocity profiles along with alternative π – G relationships. The relationship which is in best agreement with measurements of equilibrium layers confirms an earlier suggestion that (νT/Uδ*)max is not a universal constant for such layers but decreases for small and negative values of π. A close relationship is established between eddy viscosity and entrainment, and it is shown that veδ/νT is effectively constant for π > 2.

SummaryFor the turbulent boundary layer it is shown that, if an initial velocity profile is given, along with the local pressure gradient and shear-stress distribution through the layer, then the shape of the velocity profile a short distance downstream is unaffected by flow convergence or divergence, provided this is constant through the layer. For flow approaching an obstacle, increased divergence close to the surface is shown to account for the marked changes in profile shape that have been observed.

In certain accelerated flows the entrainment in the boundary layer, as normally defined, may be either zero or negative; on the other hand, there is no reason to suppose, on physical grounds, that the spread of mean or fluctuating vorticity should cease or become negative in such flows. This paradox is resolved in the present paper. It is also shown that in the equilibrium turbulent sink-flow boundary layer, where the entrainment as normally defined is zero, the reduced advection along streamlines in the outer part of the layer comes about mainly through increased dissipation: there is no reason to assume any radical change in the turbulence structure.

SummaryTwo alternative presentations are given of Patel’s Preston tube calibration. In the first, Δp/τw is tabulated as a function of Δpd2/ρν2; in the second, cf is plotted as a function of Δp/½ρU2 and Ud/ν, so that values of cf, can be read off directly.

SummaryEarlier papers described a method of calculating three-dimensional turbulent boundary layers based on the use of momentum-integral equations in the streamwise and cross-flow directions. Here the method is applied to a problem which is initially formulated in a coordinate system appropriate to the somewhat complex body geometry. Transformation to a streamline coordinate system is then made before the application of a rapidly converging iterative method of solution. The calculations, which are confined to single Reynolds number and a particular value of the rotation parameter, show the very large increases in drag and torque that accompany early transition.

SummaryBradshaw’s method of calculating the development of two-dimensional turbulent boundary layers involves the simultaneous solution of partial differential equations of mean motion and turbulent kinetic energy. The present approach avoids the computational complexities of this procedure.The use of Thompson’s two-parameter family of velocity profiles and associated skin-friction law enables the momentum integral equation to be satisfied, along with Bradshaw’s version of the turbulent kinetic-energy equation at a specified fraction of the boundary layer thickness. This fraction (y/δ = 0·5) is chosen as representing the position in the boundary layer where Bradshaw’s equation, which contains several empirical functions, is shown by comparisons with experiment to hold with greatest accuracy. Thus the present simplified approach leads not only to a reduction in computing time but also to an appreciable increase in the general accuracy of prediction.