Form and wave drag due to stably stratified turbulent flow over low ridges

Abstract

We investigate how stable stratification affects the aerodynamic force, or form drag, induced by turbulent boundary-layer flow over two-dimensional hills. Both analytical and numerical models are used to calculate the flow and thence the form drag on the hill. In the analytical model we use a two-layer, truncated mixing-length, turbulence model, which is consistent with scaling arguments and which produces reliable estimates of the form drag for the neutral flow. The form drag is also calculated analytically using an eddy-viscosity model, and the results compare well with values computed with the nonlinear numerical model that uses a similar turbulence model. The leading-order contribution to the form drag is from a non-separated sheltering mechanism, which is similar to the mechanism in neutral flow. Stable stratification changes the magnitude of this mechanism through several effects. For weak stratification the predominant effect is an increase in shear in the upstream wind profile across a middle layer, which increases the form drag by a factor of two or more. There is indirect experimental evidence to support this finding. If the stratification is more stable, then the shear across the middle layer becomes limited because the boundary layer has a finite depth. Then the dynamical effect of buoyancy on the pressure perturbation becomes important and reduces the form drag, eventually to zero. For still more stable stratification, gravity waves and associated wave drag are produced. The analysis shows that the appropriate scaling velocity for wave drag is the approach-flow wind speed evaluated at the middle-layer height. The relationship between the form and wave drag components is investigated by evaluating the analytical formula for the drag on isolated hills of two idealized shapes. For weak stratification the form drag dominates, but as the stratification becomes more stable the wave drag increases and first equals the form drag, at a value that depends on the hill shape, and then dominates.