Mechanism of reversion in highly accerlerated turbulent boundary layers

Abstract

This thesis consists of two related parts which are complementary to each other but can also be read independently. The first part is a study of the development of both the mean and fluctuating flow quantities in an initially turbulent boundary layer undergoing reversion to laminar flow (or relaminarization) as a consequence of a large favourable pressure gradient beginning at a certain point x1x_1x1?. The mean flow development is examined through two analyses expected a priori to be valid respectively towards either end (in the streamwise direction) of the relaminarizing boundary layer. One of these analyses formulates a quasi-laminar flow in the later stages of reversion, where the Reynolds stresses have by definition no significant effect on the mean flow; this flow is described by an asymptotic theory constructed for large values of a pressure-gradient parameter, scaled on a characteristic Reynolds stress gradient. Based on the concept of stress-freezing (justified in the second part of the thesis) the Reynolds shear stress used in it is estimated by its maximum value at x1x_1x1? which, by the assumption of equilibrium at x1x_1x1?, is proportional to the wall stress there. The limiting flow so constructed consists of an inner laminar boundary layer and a matching inviscid (but rotational) outer layer. There is consequently no entrainment to the lowest order in A?1A^{-1}A?1, and the boundary layer thins down to conserve outer vorticity. Further, the predictions of the theory for the common measures of boundary layer thickness, and for the velocity profiles, are in very good agreement with experimental results, almost all the way from x1x_1x1?. In fact, they often overlap with the second analysis, consisting of calculations assuming fully developed turbulent flow and expected to be valid in some neighbourhood of x1x_1x1?. On the other hand, the development of wall parameters like the skin-friction suggests the concept of a short bubble-shaped reverse-transitional region on the wall, where neither turbulent nor quasi-laminar calculations are valid; however, a simple interpolation between the two limiting analyses is found to be adequate. If the prediction of these wall parameters by the quasi-laminar theory is taken as the index, reversion is complete where ??50\Lambda \approx 50??50. Regarding the turbulent fluctuations, it is argued that following the quick reduction in the rate of turbulent bursting (observed by Schraub & Kline, 1965), the Reynolds stresses that are found to persist in the relaminarizing boundary layer are merely inherited from the original turbulence. This argument is shown here to lead to a decay with distance of the streamwise fluctuations in the inner layer, according to inverse-power laws characteristic of quasi-steady perturbations on a laminar flow. In the outer part, however, it is shown that the dominant physical mechanism operating is one of rapid-distortion of the inherited turbulence structure, with viscous and inertia forces playing an essentially secondary role. All the evidence available suggests that retransition to turbulence quickly follows the onset of instability in the inner layer. It is concluded that reversion in highly accelerated flows is essentially the domination of pressure forces over the slowly responding Reynolds stresses inherited from an originally turbulent flow, accompanied by the generation of a new laminar boundary layer stabilized by the favourable pressure gradient. In the second part of the thesis, the results on Reynolds stress variations are complemented by a detailed assessment of the relevance of the rapid-distortion limit to shear flows subjected to large pressure gradients. It is first shown that the conditions required for the validity of the limit are less difficult to fulfil in such shear-flow situations than in the classical wind tunnel studies of grid turbulence. A generalization of the rapid-distortion theory of Batchelor & Proudman (1954) is developed to estimate the effect of sudden but arbitrary three-dimensional distortion on homogeneous and initially axisymmetric turbulence. Each component energy after distortion can in general be expressed as a sum of the (known) isotropic result and a suitable correction term. The correction terms, evaluated here completely and in closed form for two non-trivial special forms of the axisymmetric spectral tensor, are not significant if the initial deviation from isotropy as well as the total imposed strain are not large. The analysis in these two special cases is carried out to the same degree of completion as in isotropic turbulence. It is concluded that the simpler of the two cases can in fact be used almost always to estimate the departure from the Batchelor-Proudman results. Next, rapid distortion of turbulence in the presence of a small inhomogeneity is studied. A convenient measure of this inhomogeneity is the ratio (say A?1A^{-1}A?1) of the appropriate rate of strain in the normal direction to that in the streamwise direction. To the lowest order in A?1A^{-1}A?1, an approximate solution for the component energies is the same as given by an extension to two-dimensional flows of Prandtl’s classical arguments: the shear stress initially present is frozen along streamlines, thus explaining an important experimental observation, and justifying the assumption made in scaling AAA in the first part of the thesis. It is shown that the difference between the present limit (or equivalently the Prandtl limit) and the Batchelor-Proudman limit is essentially in their treatment of the pressure-velocity terms, which are ignored in the former limit but retained in the latter. Lack of experimental information makes it difficult to assess the consistency of either approach; neither are the experiments in suddenly accelerated boundary layers conclusive in this regard. To the next order in A?1A^{-1}A?1, the normal and spanwise components of energy remain unaffected, unlike the streamwise component and the shear stress. In particular, the increment in shear stress depends linearly on the initial total turbulent energy, the degree of departure from isotropy at x1x_1x1?, and the magnitude and sign of the local mean vorticity; this prediction is in qualitative agreement with experiment.