The model of coastal waves based on the depth-averaging of the large eddy simulation equations (Kazakova & Richard 2019) is extended to the case of regular and irregular wave trains. To take into account a stronger turbulence, the third moment of the horizontal velocity is modelled with a gradient-diffusion hypothesis. The effect of this new diffusive term is to smooth and regularize the solutions. An asymptotically equivalent model including the improvement of the dispersive properties is solved with a Discontinuous Galerkin numerical scheme. The model has a low sensitivity to the space discretization parameters. Several classical test cases of wave trains are used to validate the model. In the shoaling zone, the model is similar to the Serre-Green-Naghdi equations but the inclusion of a variable called enstrophy to take into account the large-scale turbulence and the nonuniformity of the mean velocity improves the predictive ability in the inner surf zone. In particular, the turbulent energy of the model is dissipated within one wave cycle and is transported shoreward in the case of waves with a long period whereas, in the case of short periods, it is mostly transported seaward because its dissipation is far from being complete within one period. The case of an irregular wave train propagating over a submerged bar is simulated without any breaking criterion. This benchmark test case validates further the model's ability in predicting the nonlinear effects due to shoaling, breaking, propagation in a shallow horizontal part and in a deeper region.