On the formation of longitudinal vortices in a turbulent boundary layer over wavy terrain

Abstract

Parallel inviscid O(1) shear interacting with O(ε) spanwise-independent neutral rotational Rayleigh waves are used to model turbulent boundary layer flow over small-amplitude rigid wavy terrain. Of specific interest is the instability of the flow to spanwise-periodic initially exponentially growing longitudinal vortex modes via the Craik–Leibovich CL2-O(1) instability mechanism and whether it is this instability mechanism that gives rise to longitudinal vortices evident in the recent experiments of Gong et al. (1996). In modelling the flow, wave and turbulence length scales are assumed sufficiently disparate to cause minimal interaction. This allows the primary mean velocity profile to be specified. Two profiles were chosen: a power law and the logarithmic law of the wall. Important in wave–mean interactions of this class are the effect of wave-induced fluctuations upon the mean state and the influence of the developing mean flow on the fluctuating part of the motion. The former is described by a generalized Lagrangian-mean formulation; the latter by a modified Rayleigh equation. Together they comprise an eigenvalue problem for the growth rate appropriate to the initial stages of the instability. Both primary mean flows are unstable to longitudinal vortex form in the presence of Rayleigh waves whose amplitudes diminish with altitude. Moreover the interaction is most unstable for streamwise wavenumbers α = O(1), the growth rate increasing with increased spanwise wavenumber. In comparing the results with experiment, it is first shown that spanwise-independent waves excited in Gong et al.'s experiment depict velocity fluctuations whose amplitudes diminish with altitude in accord with those for appropriate Rayleigh waves. Concordantly, the longitudinal vortices depict transverse velocity components that are weaker by a factor of ε than the axial perturbation and are observed to grow at a rate consistent with exponential growth. All are key features of CL2-O(1), although the observed growth rate is not in accord with the maximal suggested by inviscid instability theory. Rather it appears that the spanwise wavenumber takes a value at which energy is extracted from the mean motion in an optimal volume-averaged sense while minimizing energy loss to both viscous dissipation and small-scale turbulence. It is concluded that the CL2-O(1) instability mechanism is physically realizable and that the data of Gong et al. represent the first documented observations thereof.