Turbulent plane wakes subjected to successive strains

Abstract

Six direct numerical simulations of turbulent time-evolving strained plane wakes have been examined to investigate the response of a wake to successive irrotational plane strains of opposite sign. The orientation of the applied strain field has been selected so that the flow is the time-developing analogue of a spatially developing wake evolving in the presence of either a favourable or an adverse streamwise pressure gradient. The magnitude of the applied strain rate $a$ is constant in time $t$ until the total strain e$^{at}$ reaches about 4. At this point, a new simulation is begun with the sign of the applied strain being reversed (the original simulation is continued as well). When the total strain is reduced back to its original value of 1, yet another simulation is begun with the strain again being reversed back to its original sign. This is done for both initially ‘favourable’ and initially ‘adverse’ strains, providing simulations for each of these strain types from three different initial conditions. The evolution of the wake mean velocity deficit and width is found to be similar for all the ‘adversely’ strained cases, with both measures rapidly achieving exponential growth at the rate associated with the cross-stream expansive strain e$^{at}$. In the ‘favourably’ strained cases, the wake widths approach a constant and the velocity deficits ultimately decay rapidly as e$^{-2at}$. Although all three of these cases do exhibit the same asymptotic exponential behaviour, the time required to achieve this is longer for the cases that have been previously adversely strained (by $at \,{\approx}\, 1$). The evolution described above is not consistent with the predictions of classical self-similar analysis; a more general ‘equilibrium similarity solution’ is required to describe the results. Examination of these simulations confirms that the wake width and mean velocity deficit evolutions observed in Rogers (2002) are not a result of the particular initial condition used in that work. At least for the cases considered here, the wake Reynolds number and the ratio of the turbulent kinetic energy to the square of the wake mean velocity deficit are determined nearly entirely by the total strain. For these measures, the order in which the strains are applied does not matter and the changes brought about by the strain are nearly reversible. The wake mean velocity deficit and width, on the other hand, differ by about a factor of 3 when the total strain returns to 1, depending on whether the wake was first ‘favourably’ or ‘adversely’ strained. The strain history is important for predicting the evolution of these quantities.