Turbulent mixing and passive scalar transport in shallow flows

A depth-integrated model including subgrid scale mixing effects for turbulent transport by long waves and currents is presented. A fully nonlinear, depth-integrated set of equations for weakly dispersive and rotational flow is derived by the long wave perturbation approach. The same approach is applied to derive a depth-integrated scalar transport model which can accommodate small vertical variation of a weakly unsteady scalar. The proposed equations are solved by a fourth-order accurate finite volume method. The depth-integrated flow and transport models are applied to typical problems which have different mixing mechanisms. From the simulations, several important conclusions are obtained. (i) From simulation of a mixing layer generated by internal transverse shear, it is revealed that the dispersive stress implemented with a stochastic backscatter model (BSM) can play an important role for energy transfer in a shallow mixing layer. (ii) From a comparison of the characteristic width of a scalar plume in shallow and uniform flow, the proposed depth-integrated transport model coupled with the depth-integrated flow model can predict the passive scalar transport physically-that is, based on the turbulent intensity-without relying on a coarse empirical constant. (iii) From the same simulation, the inherent limitation of the two-dimensional horizontal model to capture vertical structure is recognized in near field. (iv) If the main mechanism of flow instability originates from relatively large-scale bottom topography features, then the effects of the dispersive stresses (i.e., BSM) are less important.