Evolution of the Turbulent Far Wake of a Sphere

Abstract

The classic turbulent axisymmetric wake derivation for the spreading of wake half-width, δ, and maximum mean velocity, ūmax decay comes from arguments of a high local Reynolds number, Re, and thus negligible viscosity. If instead one assumes the local Reynolds number is small, then at some distance sufficiently far downstream the turbulent production term in the Reynolds shear stress equation will decay and a new similarity solution will arise: as shown by [2, 4]. This solution features the scaling of δ ∼ (x/d)1/2 and ūmax ∼ (x/d)−1. In other words, the turbulent wake is scaling itself at rates that match the theoretical laminar wake, yet with a local Reynolds number high enough for the turbulent fluctuations to be non-negligible. Whilst the derivation of a low Reynolds number solution is a mathematical exercise, obtaining data to confirm or deny its existence has proved difficult. No experiment has been conducted at a combination of high enough initial Reynolds number and far enough downstream to capture this transition behaviour. Furthermore, only the DNS study of Gourlay [3] has been able to achieve this behaviour; leading some researchers to question whether this decay state would occur or if the wake instead would relaminarise [7]. This paper presents results for a towed a sphere through water at a Reynolds number, based on sphere diameter, of 13000. Our experiments have been able to capture the wake transitioning from the high local Reynolds number solution to the low local Reynolds number solution via high-speed time-resolved PIV. The value of local Reynolds number that exhibits itself in the extreme far wake during the low local Reynolds number solution suggests the wake is still turbulent, supporting the claim of [2, 4].