Wall shear stress caused by small amplitude perturbations of turbulent boundary-layer flow: an experimental investigation

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Abstract
The oscillation of the wall shear stress caused by imposing sound on a turbulent boundary-layer flow constitutes a boundary condition for the solution of the acoustic wave equation. The no-slip condition at the wall requires the excitation of a shear wave which is superimposed on the sound wave. The shear wave propagates into the turbulent medium. The wall impedance (shear stress/velocity) of streamwise polarized shear waves has been measured in two different ways, namely (a) by evaluating the phase velocity and the attenuation of a plane sound, wave which propagates in turbulent pipe flow, and (b) by evaluating the resonance frequency and the quality factor of a longitudinally vibrating glass pipe which carries turbulent flow. The results, which were obtained over a wide range of Strouhal numbers, exhibit very good agreement between the two measuring methods. The wall shear stress impedance is strongly affected by the turbulence. This indicates that the turbulent shear stress is modulated by the shear wave. At all measuring conditions, the propagation of the shear wave was confined essentially to the inner portion of the turbulent boundary layer. In principle, two different Strouhal numbers, based on inner and outer variables respectively, describe the dynamics of the Reynolds stress, even in the inner layer (Laufer & Badri Narayanan 1971). However, it turns out that the outer Strouhal number (based on the diameter and the centre-line velocity) has no noticeable effect on the wall shear stress impedance. The dependence of the impedance on the inner Strouhal number (based on the friction velocity and the viscosity) reveals that the shear wave is strongly reflected at the edge of the viscous sublayer. It is concluded that the stress-to-strain ratio at the edge of the viscous sublayer corresponds either to a viscoelastic medium or even to a medium with negative viscosity.