High-Reynolds-number turbulence in a shear-free boundary layer: revisiting the Hunt–Graham theory

The capability of rapid distortion theory to predict the long-time evolution of shearless turbulence close to an impermeable surface has been seriously questioned in recent years. However, experiments and large-eddy simulations performed at high Reynolds number show that second-order turbulence statistics follow closely the predictions of the theory elaborated by Hunt & Graham (1978). To clarify this issue, a theoretical analysis is carried out in order to determine the relative magnitude of the vortical corrections which were not taken into account in the original theory. By evaluating the various terms of the enstrophy balance in the near-surface region, it is shown that this relative magnitude is a decreasing function of the turbulent Reynolds number, an argument reconciling most existing results. Hence the Hunt & Graham theory appears to be a leading-order approximation capable of describing short- and long-time evolutions of shear-free boundary layers in the limit of large Reynolds number. The expression for the pressure fluctuation corresponding to this approximation is then derived and approximate Reynolds stress budgets are obtained. These budgets are used to predict and discuss the characteristics of the intercomponent energy transfer near a flat surface in both time-decaying and spatially decaying turbulence. In agreement with available results, predictions reveal that tangential velocity components transfer energy towards the normal component in the former case, while they generally receive energy from this component in the latter case.