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In this paper, we derive a limiting condition for three-dimensional compressible flows and present the multi-dimensional limiting process for three-dimensions. The basic idea of the multi-dimensional limiting condition is that the vertex values interpolated at a grid point should be within the maximum and minimum cell-average values of neighboring cells for the monotonic distribution. By applying the MLP (Multi-dimensional Limiting Process), we can achieve monotonic characteristic, which results in the substantial enhancement of solution accuracy and convergence behavior. IntroductionSince the late 1970s, numerous ways to control oscillations have been studied and several limiting concepts have been proposed. Most representatives would be TVD [1, 2], TVB [4] and ENO [3]. The concept of TVD (Total Variation Diminishing) was proven to be extremely successful in solving hyperbolic conservation laws. Most oscillation-free schemes have been based on the mathematical analysis of one-dimensional convection equation and applied to systems of equations with the help of some linearization step. They are also applied to multi-dimensional applications with dimensional splitting. Though they may work successfully in many cases, it is insufficient or almost impossible to control oscillations near shock discontinuity in multiple space dimensions. For that reason, the need of oscillation control method for multi-dimensional applications is obvious.In order to find out the criterion of oscillation control for multiple dimensions, Kim and Kim [5] extended the one-dimensional monotonic condition to two dimensions and presented the two-dimensional limiting condition successfully. With the limiting condition, a multi-dimensional limiting process (MLP) is proposed which gives more accurate results for two-dimensional Euler and Navier-Stokes equations. It is this approach which prompts the work of the present paper. Basically, it extends the idea of MLP to three dimensions. Thus, in this paper, we derive a three-dimensional limiting condition and present the multi-dimensional limiting process for two-and three-dimensional situations.

a b s t r a c tThe capabilities of CFDShip-Iowa V4.5 and V6.2 simulations for wave run-up for single/multiple surface-piercing cylinders in regular waves are assessed in conjunction with the ITTC OEC Workshop on VIV and Wave Run-up held in Nantes, France October 17-18, 2013. Statistical convergence and grid/domain/turbulence model sensitivity studies show uncertainty U I values are less than 1% for both the experiment/simulation data and grid/domain/turbulence model sensitivities are less than the facility bias U FB . Simulations are qualitatively validated using U FB and error E between experiment and simulation. Errors for single/four cylinder(s) are comparable with U FB and average errorĒ for the other CFD data of the workshop participants. Results from both V4.5 and V6.2 agree well with experiment and also show capabilities to apply for complex/large-scale problem and to describe detailed wave breaking/spraying for V4.5 and V6.2, respectively. Study on run-up profile shows mean wave elevations are similar to free surface elevations for a cylinder in a steady current. For wave steepness/wave length effect, 2nd to 1st harmonic ratio on the cylinder weather face decreases with steepness for short wave, but increases for long wave. Wave diffraction analysis shows that 2nd/3rd to 1st harmonic ratio in diffracted wave contour reach up to 18-41%/9-16% respectively. Due to interaction between cylinders, crest/trough heights increase 5-17%/4-10% for 4C, respectively. Study on vortex shedding shows short wave has insufficient time to develop periodic vortex shedding compared to long wave.

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