Penta-decomposition of instantaneous field in spanwise-rotating turbulent plane Couette flow

Spanwise-rotating turbulent plane Couette flow (RPCF) is one of the fundamental prototypes for wall-bounded turbulent flows in rotating reference frames. In this turbulent problem, there are large-scale roll cells which are widely studied. In this paper, a penta-decomposition method is proposed to separate the instantaneous velocity and the total kinetic energy into five parts, i.e., a mean part, a streamwise part and a cross-flow part of the secondary flow, and a streamwise part and a cross-flow part of the residual field. The transport equations for the last four shares, which contribute the total turbulent kinetic energy, are derived. According to these transport equations, the mechanisms of energy transfer among different fractions of turbulent kinetic energy can be revealed clearly. Our objective is to explore the energy balance and transfer among different fractions of the turbulent kinetic energy in RPCF based on a series of direct numerical simulation databases at a Reynolds number Re-w - U(w)h/v - 1300 (here, U-w is half of the wall velocity difference, and h is the channel half-width) and rotation number Ro = 2 Omega h/U-w (Omega(Z) is the constant angular velocity in the spanwise direction) in a range of 0 <= Ro <= 0.9. The results show that the energy is transferred between the streamwise part/cross-flow part of secondary flows and the residual field through the correlation between the vorticity of secondary flows and the shear stress of residual field. The rotation term acts as a bridge to transfer the energy between the streamwise part and the cross-flow part of either the secondary flows or the residual field. Moreover, pressure-strain redistribution term also plays an important role in the energy transfer between streamwise part and cross-flow part in residual field. For the streamwise part of residual field, in certain rotate rates, the energy obtained from the streamwise part of secondary flows by the correlation between the vorticity of secondary flows and the shear stress of residual field is larger than that obtained from mean flow through mean shear, implying that the streamwise motions of secondary flows have a significant influence on the streamwise motions of residual field.