Weak mean flows induced by anisotropic turbulence impinging onto planar and undulating surfaces

Prandtl's secondary mean motions of the second kind are driven by the variation of Reynolds stresses near resistive boundaries. In the flows considered here the turbulence is generated away from the boundary in the absence of a mean flow and then impacts onto a rigid surface placed into the flow at $t\,{=}\,0$. The initial development of the distorted flow is obtained using the linear approximation and the statistical analysis of rapid distortion theory, following Hunt & Graham (1978) assuming homogeneous stationary high-Reynolds-number turbulence with an integral length scale $L_\infty$ and r.m.s. velocity $v_\infty\prime$. First, the effects of axisymmetric anisotropy and of different forms of the spectra are analysed for turbulence impinging onto a plane surface lying at an angle $\alpha$ to the unit vector $\bf e$ of the axis of symmetry of the energy spectrum tensor $\Phi_{ij}(\mbox{{\boldmath k}})$. $R$ is defined as the ratio of the largest to smallest variances of the velocity components. The surface blocking leads to gradients of Reynolds shear stresses normal to the surface in the source layer $B^{(s)}$ with thickness of order $L_\infty$ and thence to a mean velocity $U(t) \,{\sim}\,{-}tv_\infty\prime^2 \sin 2\alpha (1\,{-}\,1/R) /L_\infty$ along the slope in the opposite direction of the projection of ${\boldmath e}$ onto the plane (i.e. in the direction ($\mbox{{\boldmath e}}\wedge\mbox{{\boldmath n}}$)$\wedge\mbox{{\boldmath n}}$ where ${\boldmath n}$ is the normal into the flow). $U$ is greatest near the surface where $y \,{\ll}\, L_\infty$. As a result of shear stresses being induced by the mean velocity gradient, a steady flow results over a time scale $T_L\,{=}\,L_\infty/v_\infty\prime$ – an order of magnitude estimate for the steady-state mean velocity is thence $U(t/T_L \,{\to}\, \infty) \,{\sim}\, v_\infty\prime(\sin 2\alpha (1\,{-}\,1/R))^{1/2}$. Secondly, the effect of a curved surface is studied by analysing isotropic turbulence near an undulating surface of wavelength $\Lambda$ and amplitude $H$, with a low slope so that $H \,{\ll}\, \Lambda$. The boundary condition of zero normal velocity at the curved surface generates larger irrotational fluctuations in the troughs, smaller fluctuations over the crest, and shear stresses over the slopes. The curl of the gradients of Reynolds normal and shear stresses within $B^{(s)}$ cause the growth of a mean vorticity which induces a mean velocity of order $-tv_\infty\prime^2/L_\infty$ within $B^{(s)}$ and a weaker recirculating velocity of order $-tv_\infty\prime^2/\Lambda$ in a deeper wave layer, $B^{(w)}$, with thickness of order $\Lambda$ outside $B^{(s)}$. The wavelength of the mean motion is $\Lambda$, with downward motions over the troughs and upward motion over the crest. As in the first case, a steady flow is predicted when $t/T_L \,{\gg}\, 1$. Anisotropic free-stream turbulence also induces mean motions on undulating surfaces with the same wavelength $\Lambda$ as that of the undulating surface, but the directions of these mean motions can be towards or away from the troughs/crests depending on the orientation of the anisotropy of the free stream. Flow visualization experiments conducted in a mixing box with oscillating anisotropic and isotropic grids demonstrated the existence of these mean flows and that they reach a steady state with an intensity and length scale comparable to those predicted. These results are also consistent with numerical simulation of Krettenauer & Schumann (1992) of convective turbulence over an undulating surface.