Scale-by-scale energy transfer in a dual-plane jet flow

Abstract

We numerically investigate the scale-by-scale energy transfer along all three directions (i.e., streamwise, vertical, and spanwise directions) at three specifically selected centerline locations (i.e., X/L0 = 7, 12, and 20) in a dual-plane jet flow by using the Karman–Howarth–Monin–Hill (KHMH) equation, where X and L0 are the streamwise distance from the inlet and the initial jet width, respectively. Unlike the well-known Karman–Howarth equation, the fully generalized KHMH equation allows us to study the scale-by-scale energy balance without any assumptions (i.e., homogeneity and isotropy). We calculate each term in the KHMH equation by using the data from a direct numerical simulation [Zhou et al., “Dual-plane turbulent jets and their non-Gaussian velocity fluctuations,” Phys. Rev. Fluids 3, 124604 (2018)]. At X/L0 = 7, where the flow is inhomogeneous and anisotropic, the scale-by-scale energy transfer is quite different in the streamwise, vertical, and spanwise directions. One interesting finding is that a negative production term in the vertical direction can be found in the reversal flow region. Unlike most flows previously investigated, the linear energy cascade plays an important role in the energy cascade. The linear energy transfer is heavily dependent on the direction: in the streamwise direction, it is forward (from the large scale to the small scale), but in the vertical direction, it is backward (from the small scale to the large scale). A physical model is proposed and also verified, which suggests that the forward linear energy cascade corresponds to fluid compression, and the backward linear energy cascade corresponds to the fluid stretching. At X/L0 = 12, where the energy spectrum exhibits a well-defined −5/3 scaling, there is no equilibrium energy cascade. The linear energy cascade is still dominant and shows a backward cascade in the streamwise direction and a forward cascade in the vertical direction, while the non-linear energy cascade remains forward. At X/L0 = 20, where the flow becomes much more homogeneous and isotropic, within a short scale-range around the Taylor microscale, the dissipation term can be balanced by the combination of the non-linear energy transfer term and advection term. This observation to some extent echoes Kolmogorov’s hypothesis, but limited only to length scales around the Taylor microscale, and the persistence of the advection term can find its root in the low local Reynolds number at that length scale.