The scope of this review is to draw some recent numerical models and related approximation methods for a wide range of free surface water flow simulations. Basically, the models are derived from unsteady free surface full 3D Navier‐Stokes equations considered as a satisfied description of the physical features concerning several hydro phenomena as for example lake eutrophication remedial by air injection, transport of pollutant, flood in rivers, watershed, etc. The computational cost generated by such 3D global model being important, we focus the effort by combining the use of 2D sophisticated models keeping the most significant physical aspects, appropriate coupling models or multi‐models of different dimensions, numerical efficient schemes using a priori or a posteriori error estimates and parallel algorithmic through HPCN facilities. The common numerical methods and techniques used are based on the characteristics method for the time discretization of the convection terms occurring in the mass, free surface flow or momentum equations presented under different formulation; in the momentum equations, this leads to steady Quasi‐Stokes problems discretized by finite element mixed formulation using a conjugate gradient Uzawa algorithm. In this review, we present the methods of solution and some related numerical simulations using the above mentioned common techniques of various water free surface problems under different formulation: St‐Venant velocity‐height, two phase water‐air bubble Navier‐Stokes velocity‐pressure or stream vector‐vorticity.

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