Numerical simulations of Francis and Kaplan turbines are accomplished using a higher-order accurate, conservative, and robust technique based on the discontinuous Galerkin method. These numerical simulations are typically used in conjunction with experimental measurements on turbine models to design hydraulic machinery having higher efficiency operating under off-design conditions. Due to the complicated geometries of Francis and Kaplan runners, mesh generation is a very difficult task, and the resulting meshes are very distorted with highly skewed elements. The discontinuous Galerkin method uses a compact or elementwise representation of field variables, therefore it can deliver high accuracy on skewed meshes because it does not rely on reconstruction techniques (as Finite Volume methods) or on large stencils (as Finite Difference methods), techniques that are strongly dependent on the quality of the underlying mesh/grid. A new high-order accurate discontinuous Galerkin technique for convection-diffusion systems has been recently developed as an extension of the classical discontinuous Galerkin technique for convection dominated problems. This extension addresses problems involving diffusion (second-order derivatives) and is based on a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across interelement and domain boundaries. The approximation of field variables is local to each element in the mesh, without continuity requirements across elements, therefore, this method can incorporate local adaptive refinements of the mesh very easily. Adaptive refinement of the mesh is extensively used when the solution requires better resolution, or simply when the initial mesh is too coarse- to deliver accurate solutions. Moreover, because of the elementwise representation of field variables, the order of polynomial approximation can be adapted element by element, without continuity constraints which are common to classical continuous approximations. There is a cost associated with handling interelement discontinuities in the basis functions, but the benefits by far compensate this additional cost. The robustness of this technique hinges precisely in using discontinuous functions to represent steep gradients, which typically create oscillations in classical high-order approximations when the underlying mesh is not fine enough. Numerical simulations of Francis and Kaplan turbines suggest that the method is very robust, and capable of delivering highly accurate solutions.
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