## Abstract

In the paper entitled ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’ (Wang & Wang 2009 *Proc. R. Soc. A* **465**, 1843–1868 (doi:10.1098/rspa.2008.0410)), the present authors' earlier work (Qian *et al*. 2006, *Proc. R. Soc. A* **462**, 21–42 (doi:10.1098/rspa.2005.1528)) has been specifically applied to the study of water-entry and -exit of solid objects. An extended boundary condition, retaining the term owing to acceleration of moving boundaries in the momentum equation, has been implemented for calculating the pressure gradient at solid surfaces and, based on their numerical experiments, it was concluded by Wang and Wang that without this term the calculation will substantially under-predict the impact forces and may even break down. Therefore, a more complex procedure based on the exact solution of a Riemann problem for moving boundaries was implemented. In this short comment, by applying the authors' free surface capturing code to the same flow problem of water-entry of a wedge, it can, however, be demonstrated that the results from implementing the new pressure boundary condition are nearly identical to that of employing the original boundary condition without the acceleration term, indicating that its effects on the simulation results are minimal. A further examination of the implementation details on the pressure boundary condition also supports this conclusion.

Despite rapid developments in the field of computational fluid dynamics, an accurate and efficient simulation of free surface flows with embedded moving boundaries and/or solid bodies remains a difficult task. It requires a numerical method that can adequately handle both a moving and possibly breaking free surface and solid bodies of arbitrary shape whose motions can be either prescribed or induced by the surrounding fluid. Various mesh-based and meshless approaches (e.g. Kleefsman *et al*. 2005; Oger *et al*. 2006) have recently been developed with a varying degree of success for such flow problems. In Qian *et al.* 2006, the present authors have described an effective alternative to these methods in which a high-resolution Riemann solver-based approach has been used for capturing the interface between water and air and a Cartesian cut cell mesh generation technique has been adopted for the description of the moving solid boundaries. Many examples of practical importance such as wave generation in numerical wave tanks using a piston-type paddle and water entry of solid objects have been presented, demonstrating the applicability of the method to a variety of free surface flow problems.

Subsequently, the method has been specifically extended and applied to water-entry problems by Wang & Wang 2009, in which several improvements to the original method were claimed by the authors, including, most notably, the implementation of a new pressure boundary condition at moving solid surfaces. The unsteady term owing to the acceleration of solid bodies in the momentum equations was retained when calculating the pressure gradient at the surface of a solid boundary. Further to this and as an attempt to improve the numerical stability of their code, the exact solution of a Riemann problem for moving boundaries has been used for calculating the fluxes at the boundary (Causon *et al*. 2001). According to the authors (Wang & Wang 2009), the main reason for the inclusion of the acceleration term in the pressure boundary condition was that without this term their method would substantially under-predict the impact forces and even diverge at some point in the numerical simulation. Although it is well known that the acceleration term can be important for some flow problems where the solid boundary undergoes rapid change in velocity, it can be demonstrated that for the wedge water-entry problem as described in the experimental and theoretical work of Zhao *et al*. 1997, this term will not have much effect on the simulation results.

Generally, the pressure boundary condition at a moving solid boundary can be written as
1.1where *V* _{n}=*u*_{b}*n*_{x}+*v*_{b}*n*_{y} is the normal component of the velocity at the solid boundary with an outward normal unit ** n**=

*n*

_{x}

**+**

*i**n*

_{y}

**. This condition can be used for calculating the pressure values at both the ghost cells and the surface points, which in turn are needed to work out the pressure gradient at boundary cells and the flux at the solid boundary, respectively. For the specific case of water entry of a wedge where only the vertical motion is permitted, this condition can be simplified as 1.2By using this condition, the pressure value at surface point**

*j**b*of a cut cell can be extrapolated from the pressure value at its centre

*c*(figure 1), 1.3where Δ

_{cb}is the length of the line segment from point

*c*to

*b*. To calculate the instantaneous velocity of the wedge, at every time step, the pressure at the solid surface

*S*

_{b}is integrated to get the total vertical force exerted by the surrounding fluid: 1.4Then, through Newton's second law, its moving velocity can be calculated by integrating the following ordinary differential equation: 1.5where

*m*

_{b}is the mass of the wedge.

To show the effect of the acceleration term (the second term on the right-hand side in equation (1.3)) on the calculation results, simulations have been undertaken for the same flow problem using the original code developed by the current authors. A wedge with a deadrise angle of 30^{°} is allowed to fall into the initially calm water under gravity and, according to the experiment (Zhao *et al*. 1997), the downward velocity at which the apex of the wedge comes into contact with the water surface is 6.15 m s^{−1}. The wedge is 0.5 m wide and has a mass of 241 kg. The computational domain of 2 m×2 m is divided into 200×200 uniformly square mesh cells of the size 0.01 m×0.01 m, which is the same as used by Wang & Wang (2009). In the simulation, the densities for water and air are taken as 1000 and 1 kg m^{−3}, respectively, and the value for the artificial compressibility parameter *β* is taken as 1000. The time step used for the calculation is 0.05 ms and a total of 0.25 s has been simulated for the initial water entry of the wedge. Two pressure boundary conditions with and without the acceleration term (denoted as boundary condition A and boundary condition B, respectively, herein) have been applied separately for the flow problem. In figure 2, the results for the vertical impact forces and the resultant moving velocities of the wedge are compared with the experimental ones for the duration of the simulation. From these results, it can be seen that the results from using the two pressure boundary conditions are nearly identical, although both calculations have slightly under-predicted the impact force before it reaches the peak value, and then over-predicted it. A comparison of the pressure contours from using the different pressure boundary conditions as shown in figure 3 for *t*=0.01358 s at which time the impact force and hence the acceleration of the wedge are nearly reaching their maximum values also confirms this.

These results can be further explained as follows. In equation (1.3), the second term on the right-hand side arises from the acceleration of the moving boundary and for the initial water-entry problem, the value of the acceleration ∂*v*_{n}/∂*t* can reach as high as −100 m s^{−2} and therefore the term *ρ*∂*v*_{n}/∂*t* can be large. However, as the term is multiplied by Δ_{cb}*n*_{y} (figure 1) whose value is restricted by the side length of the mesh cell and therefore is very small, its value will be small compared with the pressure value at the centre of a cut cell. For example, in the current simulation, a uniform mesh of cell size d*x*=d*y*=0.01 m has been used and this means that a value of Δ_{cb}*n*_{y} will be less than 0.005 m. By assuming the density of water to be 1000 kg m^{−3}, the maximum value of the second term on the right-hand side of equation (1.3) can be approximated to be 1000×100×0.005=500 Pa. Comparing this to the average pressure value around the wetted surface of the wedge, which can be estimated to be over 5×10^{4} Pa from the pressure contour plots in figure 3, the pressure increment owing to body acceleration will be less than 1 per cent of the pressure value at the centre of the cut cell and is therefore small. This means that the effect of this term and in fact also the term owing to gravitational acceleration (the last term on the right-hand side of equation (1.3)) on the evaluation of pressure near the boundary can be generally ignored if the mesh cells surrounding the solid boundary are fine enough.

Based on the numerical experiments and a discussion of the implementation details on the pressure boundary conditions, it can be concluded that for the simulation of water entry of a free falling wedge and indeed for most applications involving moving boundaries, the inclusion of the acceleration term owing to body motion in the pressure boundary condition is generally not needed and its effects on the accuracy of the simulation are likely to be minimal. The contradictory results reported in Wang & Wang (2009) may indicate errors from other sources unrelated to the acceleration term in the pressure boundary condition.

- Received June 13, 2011.
- Accepted September 9, 2011.

- This journal is © 2011 The Royal Society