New algorithms to speed up RANSE computations in hydrodynamics

Free-surface capturing methods (volume of fluid, level-set formulation) have become more and more popular among the CFD developers involved in viscous naval hydrodynamics. This increasing interest is due to the fact that this approach is more robust than those based on a free-surface fitting methodology since no regridding is necessary. Moreover, the merging or breakup of the interface is also handled in a natural way. To achieve such computations, specific compressive discretisation schemes are used to solve the concentration transport equation and keep the sharpness of the interface [1, 2]. Even if the capacities and the flexibility of such an approach is unquestionable, two features (easily assimilated as drawbacks) can be highlighted :

  • the formulation is intrinsically unsteady since the concentration is convected by the flow. Up to now, any steady formulation has ever been successful, to the knowledge of the authors,
  • the compressive property has numerically severe Courant number limitation.

So, this is quite wastefulness to use such an approach when dealing with physical steady cases. It is all the more a pity since for an implicit solver, the concentration equation is the sore equation to have such a Courant number limitation ! This issue was first underlined by Ubbink in the conclusions of his PhD thesis [1] : « the Courant limitation is not insurmountable because it should be possible to apply a technique of sub-cycling where the time step of the main loop is divided into smaller steps in order to advect the volume fractions… ».

Therefore, to reduce this Courant number limitation, such an original time-splitting (which can also be noted time sub-cycling) procedure for the concentration equation has been successfully developed and validated for steady state cases. For instance, it enables to increase the global time step while keeping the Courant number constant. It raises no problem for fixed bodies. However, when using the Newton’s law for solved motions, the large time steps lead to a divergent flow/motion coupling, due to the added mass effects. To avoid this problem, a quasi-static approach has been developed to reach an equilibrium position. The latter has been successfully combined with the time-splitting procedure for the concentration without problems of stability. Compared to a classical unsteady approach using the Newtow’s law, this new numerical procedure to deal with steady cases for hydrodynamics applications enables to reduce significantly the CPU time.

After a brief description of the RANSE solver in which this work has been implemented, this article outlines the time-splitting procedure as well as the quasi-static approach. Then, a test-case is shown, demonstating the capability and the efficiency of these techniques.

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