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Transient growth induced by surface roughness in a Blasius boundary layer

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The Nature of High Reynolds Number Turbulence

Introduction There has been much recent interest in transient growth both theoretically [1-4] and experimentally [2,5]. The present work stems from an interest in model reduction for flow control: starting with a linear physical model representative of near-wall turbulence, the aim of this experiment is to devise a wall-based estimator. The inviscid transient growth instability has an optimal form of spanwise periodic streamwise vortices resulting in high/low speed streaks. Slow growth and decay in x and spanwise periodicity suggest a reduced requirement for mode observability. Equations (1) and (2) describe the generation of optimal modes through the coupling term, , viz.

Experimental Arrangement The Blasius base flow is developed on a vertical cast aluminium plate with a sharp leading edge in a very low-turbulence intensity (around 0.05%) wind tunnel. A periodic array of roughness elements at 200 mm from the leading edge is used to perturb the base state and introduces steady steamwise vortex pairs approximating the optimum mode. The height, k, spanwise separation, Δz and diameter, d, of the roughness elements are all adjustable. For the present work, 0.5 ≤ k ≤ 1.5, 10 ≤ Δz ≤ 30 and 2.5 ≤ d ≤ 5.0 mm and the measurements are conducted over the central six elements. Two hot wires, one normal and one slanted by 45 degrees to the mean flow, are used to obtain u and w in the range 250 ≤ x ≤ 700 mm.

Results and Discussion Figure 1 present typical results for these experiments. Figure 1(a) shows the u-disturbance spectra as a function of spanwise-wavenumber. The spectrum is taken at the wall-normal location where the maximum rms of the disturbance occurs below that predicted by theory [3]. For this case, the first harmonic of the disturbance introduced is discernible for the first four streamwise locations. The energy of the disturbance is observed to grow at the first streamwise location and peaks when eta is equal to 0.45 before it decays monotonically. Figures 1(b) and© show the corresponding disturbance contours at x = 300. The double positive peaks in the u-disturbance merge into a single peak further downstream of the roughness elements. The shape of the w-disturbance is more complicated and suggests that the streamwise vortices produced by the roughness elements are very weak by this location.

The u-component of the disturbance shows the effects of a spanwise shear at larger x, the shear increasing with streak amplitudes. In common with other experiments [2,5], the evolution of the disturbance shows significant deviation to that predicted by theory, suggesting a suboptimal initial growth. This raises questions regarding the receptivity mechanism of laminar boundary layers subjected to different types of perturbations, such as free stream turbulence or wall roughness [6]. In future work, we intend to investigate this further as well as estimating the v component from continuity. Simultaneous measurements of the two components of surface skin friction will also be made to enable the design of an estimator for control by surface deformation.

This talk is part of the Isaac Newton Institute Seminar Series series.

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