Lateral straining of turbulent boundary layers. Part 1. Streamline divergence

Abstract

Extensive experimental studies are presented of the effects of prolonged streamline divergence on developing turbulent boundary layers. The experiment was arranged as source flow over a flat plate with a maximum divergence parameter of about 0.075. Mild, but alternating in sign, upstream-pressure-gradient effects on diverging boundary layers are also discussed.It appears that two overlapping stages of development are involved. The initial stage covers a distance of about 20 initial boundary-layer thicknesses (δ0) from the start of divergence, where the coupled effects of pressure gradient and divergence are present. In this region there is a fairly large reduction in divergence parameter, Rθ (Reynolds number based on momentum thickness) remains constant (≈ 1400) and the boundary-layer properties change rapidly. In the second region, which lasts nearly to the end of the diverging section, the pressure-gradient effects are negligible, the rate of decrease in divergence parameter is very small and Rθ increases gradually. Up to the last measurement station (≈ 100δ0) the flow is still considered to be at a low Reynolds number (Rθ ≈ 2000). For almost the entire length of this region, the profiles of non-dimensional eddy viscosity appear to be self-similar, but have larger values than for the unperturbed flow. Also in this region, beyond 35δ0, the wake parameter, which has reduced significantly, becomes nearly constant and independent of Rθ. On the other hand the entrainment rate attains a constant value at around 50δ0. It appears that the boundary layer reaches a state of equilibrium. It is suggested that this is the result of an enhanced turbulent diffusion to the outer layer. Spectral measurements show that divergence affects mainly the low-wavenumber, large-scale motions. However, there is no change in large-eddy configurations, since the dimensionless structure parameters show only negligible deviations from the unperturbed values.

A numerical study of laterally strained wall-bounded turbulence

“…(The profiles of k and −u v at At = 0.19 are included in Appendix B.) The Case SD results are consistent with the observations of Saddoughi & Joubert (1991), Pauley et al (1993) and Pompeo et al (1993), all of whom also found (for similar strength positive ∂W/∂z) that k and −u v increase slightly in the outer layer. At the time, there was significant uncertainty surrounding this result, due to the magnitude of the changes involved, and especially the potential influence in the experiments of streamwise and spanwise pressure gradients (see discussion in Pauley et al 1993).…”

“…Physically, this quantity can be viewed as a measure of how 'efficiently' the turbulence is able to utilize its kinetic energy to mix momentum across the mean velocity gradient, and thus how effectively the velocity fluctuations extract energy from the mean flow. As found in the experiments, both divergence (Saddoughi & Joubert 1991) and convergence (Panchapakesan et al 1997) cause outer-layer reductions in this quantity. The amount of the reduction found here varies with the sign and strength of A 33 (but is not proportional to |dβ * D /dt|).…”

“…Recall that u c remains constant and δ * changes in proportion to h(t) = h(0) exp(A 22 t) during the ZPG lateral straining. The behaviour of the slower-strain cases agrees with the mildly perturbed experiments, in that divergence leads to increased ν T /u c δ * (Saddoughi & Joubert 1991), while convergence does the opposite (Pompeo et al 1993). For the Cases FD and FC strains, however, ν T is roughly the same fraction of u c δ * at At = 0.38 as it is in the unstrained initial flow.…”

“…It appears that either a constant ZPG lateral divergence does not always drive the flow to a constant-β D condition, or that a total strain of exp(0.38) = 1.46 is not sufficient for one to develop. Given that, as mentioned earlier, At = 0.38 corresponds to travelling about 15 % of the total distance through the Saddoughi-Joubert duct (Saddoughi & Joubert 1991), the latter option is perhaps the most likely (cf. figure 11 of Panchapakesan et al 1997).…”

Transformation of a laterally diverging boundary layer flow to a two-dimensional boundary layer flow

“…Lateral divergence of boundary layer flows had been the subject of many investigations in the past [1][2][3][4][5]. One of the important conclusions of these studies [3][4][5] is the observed similarity with two-dimensional (2-D) flows even when the flow is turbulent (see for example, Pauley et al [4]), but with a thinner boundary layer compared to 2-D flows.…”