Fourier analysis of the roll-up and merging of coherent structures in shallow mixing layers

The current study investigates the role of nonlinearity in the development of two-dimensional coherent structures (2DCS) in shallow mixing layers. A nonlinear numerical model based on the depth-averaged shallow water equations is used to investigate temporal shallow mixing layers, where the mapping from temporal to spatial results is made using the velocity at the center of the mixing layer. The flow is periodic in the stream-wise direction and the transmissive boundary conditions are used in the cross-stream boundaries to prevent reflections. The numerical results are examined with the aid of Fourier decomposition. Results show that the previous success in applying local linear theory to shallow mixing layers does not imply that the flow is truly linear. Linear stability theory is confirmed to be only valid within a short distance from the inflow boundary. Downstream of this linear region, nonlinearity becomes important for the roll-up and merging of 2DCS. While the energy required for the merging of 2DCS is still largely provided by the velocity shear, the merging mechanism is one where nonlinear mode interaction changes the velocity field of the subharmonic mode and the gradient of the along-stream velocity profile which, in turn, changes the magnitude of the energy production of the subharmonic mode by the velocity shear implicitly. The nonlinear mode interaction is associated with energy up-scaling and is consistent with the inverse energy cascade which is expected to occur in shallow shear flows. Current results also show that such implicit nonlinear interaction is sensitive to the phase angle difference between the most unstable mode and its subharmonic. The bed friction effect on the 2DCS is relatively small initially and grows in tandem with the size of the 2DCS. The bed friction also causes a decrease in the velocity gradient as the flow develops downstream. The transition from unstable to stable flow occurs when the bed friction balances the energy production. Beyond this point, the bed friction is more dominant and the 2DCS are progressively damped and eventually get annihilated. The energy production by the velocity shear plays an important role from the upstream end all the way to the point of transition to stable flow. The fact that linear stability theory is valid only for a short distance from the inflow boundary suggests that some elements of nonlinearity is incorporated in the mean velocity profile in experiments by the averaging process. The implicit nature of nonlinear interaction in shallow mixing layers and the sensitivity of the nonlinear interaction to phase angle difference between the most unstable mode and its subharmonic allows local linear theory to be successful in reproducing features of the instability such as the dominant mode of the 2DCS and its amplitude.