Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid

Abstract

Late wakes (Nt> 20) of towed spheres in a stably stratified fluid were analysed in a plane using a reliable, customized DPIV technique that provides sufficient spatial and temporal resolution to cover all important scales of motion in this freely decaying geophysical flow. Systematic experiments were conducted with independent variation ofRe∈ [103, 104] andF∈ [1, 10] (F ≡ 2U/NDis an internal Froude number based on the buoyancy frequency,N, and the sphere radius,D/2), and for selected {Re, F} pairs above this range.The normalized wake width grows at approximately the same rate as in a three-dimensional unstratified wake, but it becomes narrower, not wider, with decreasingF(i.e. as stratification effects become more important). The centreline defect velocity, on the other hand, reaches values an order of magnitude above those measured for three-dimensional unstratified wakes at equivalent downstream locations. Both observations are argued to be consequences of the very high degree of order and coherence that emerge in the late-wake vortex structures.Streamwise-averaged turbulence quantities, such as the velocity fluctuation magnitude, and mean-square enstrophy, show similar power law behaviour for allRe≤ 5 × 103, with exponents equal to those expected in three-dimensional axisym-metric turbulent wakes. There is no obvious physical reason why three-dimensional arguments are so successful in such a flow, and at such long evolution times. The scaling collapses none of the cases forRebelow 4 – 5 × 103, appearing to establish a minimumRefor a class of self-similar stratified wake flows that evolve from fully turbulent initial conditions.Individual vortex cross-sections appear to be well approximated by Gaussian distributions at allRe, F andNtstudied here. The scaling behaviour of individual vortices mimics that of the statistical, wake-averaged quantities, and differs measurably from a simple two-dimensional viscous diffusion model. The importance of formulating a realistic three-dimensional model is discussed, and some limited steps in this direction point to future useful experiments and modelling efforts.