A Look at the Turbulent Wake Using Scale-By-Scale Energy Budgets

It is now well established that coherent structures exist in the majority of turbulent flows and can affect various aspects of the dynamics of these flows, such as the way energy is transferred over a range of scales as well as the departure from isotropy at the small scales. Reynolds and Hussain (J Fluid Mech 54:263–288, 1972) were first to derive one-point energy budgets for the coherent and random motions respectively. However, at least two points must be considered to define a scale and allow a description of the mechanisms involved in the energy budget at that scale. A transport equation for the second-order velocity structure function, equivalent to the Karman-Howarth (1938) equation for the two-point velocity correlation function, was written by Danaila et al. (1999) and tested in grid turbulence, which represents a reasonable approximation to (structureless) homogenous isotropic turbulence. The equation has since been extended to more complicated flows, for example the centreline of a fully developed channel flow and the axis of a self-preserving circular jet. More recently, we have turned our attention to the intermediate wake of a circular cylinder in order to assess the effect of the coherent motion on the scale-by-scale energy distribution. In particular, energy budget equations, based on phase-conditioned structure functions, have revealed additional forcing terms, the most important of which highlights an additional cascade mechanism associated with the coherent motion. In the intermediate wake, the magnitude of the maximum energy transfer clearly depends on the nature of the coherent motion.