A Conservative CIP Method for Violent Free Surface Flows

In the past several years, continuous effort has been made in RIAM, Kyushu University to develop an efficient Cartesian grid method for violent free surface problems in ship and ocean engineering. Our researches have been focused on (1) Accurate and robust flow solver for multi-phase flows with solid bodies. (2) Conservative and smearingless interface capturing scheme. In the numerical method, violent wave-body interaction is treated as multi-phase problem and solved numerically in a Cartesian grid. The CIP (Constrained Interpolation Profile) algorithm [1] was adopted as the base scheme for the flow solver. It has been proved by a number of numerical examples that the CIP method is accurate and robust enough for such multi-phase problems [2][3][4]. For capturing the immersed boundaries (free surfaces), we have improved and applied the CIP-CSL3 scheme [3] and THINC scheme [4] to our CFD code and their efficiency was investigated. Both schemes are conservative and smearingless. In this research we consider to improve the conservation property of flow solver since the CIP scheme we are using for advection calculation is not a conservative one. In some marine applications, e.g., simulation of violent tank sloshing in which long physical time phenomena and free surface impact phenomena are need to consider, conservation property of the CFD code, both for the flow solver and the interface capturing method, is an important factor that affects the computation quality. In a violent tank sloshing experiment [3], we have measured the pressure on the sidewall. For some cases two pressure peaks in one oscillation period were observed while in the computation the second peak could not be properly predicted. We considered that the use of non-conservative CIP scheme may cause it because some of the momentum of fluid might be lost during the impact of the fluid onto the wall in the computation. As the framework of our CIP based CFD code works well so far, in the current development we retain all the features of the code except for the flow solver in which the CIP scheme is replaced by the CIP-CSL2 (Constrained Interpolation Profile – Conservative Semi-Lagrangian scheme with 2 order polynomial function) scheme [5]. The mathematical formulation and numerical details are described in this paper and some preliminary validation cases are presented.