This thesis presents an implicit monolithic formulation for two-equation turbulence models and their sensitivities. The Sensitivity Equations Method is a tool for analysis and optimal design of complex flows. The classic solution algorithm for two-equation turbulence models is efficient in terms of memory but expensive with regards to the calculation time. The sensitivity analysis suffers from the time inefficiency of this decoupled approach, particularly in the unsteady regime. The sustained increase in the computer speed and memory opens the door to the development of monolithic, fully coupled formulations which are slightly more expensive in memory requirements but considerably faster. An adaptative Finite Elements code served as the basis for the development of the coupled approach. The large size of matrix systems generated by the coupled approach motivated the use of the more efficient UMFPACK linear solver over the existing Skyline solver. The k -- epsilon two-equation model is used for closure of the Reynolds Averaged Navier Stokes equations. The correctness of the implementation is verified by the Method of the Manufactured Solution. The performances of the algorithms in terms of required memory and calculation time are then assessed through different applications. Depending on the case, the coupled technique is from 2.5 to 20.0 times faster but necessitates 2.4 to 3 times the memory required by its decoupled counterpart. Also, the comparison of results from linear solvers showed that the dependence of the resources (time and memory) on the size of the problem (nodes number) is quadratic for Skyline but linear for UMFPACK. We conclude that the aim of the thesis is thus achieved: an implicit monolithic formulation is developed serving as a performant tool for the efficient study of the turbulent flows and of their sensitivities.
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