# Free Surface Flow Simulation by a Viscous Numerical Cylindrical Tank  
[PERSON]  
1 College of Civil Engineering and Architecture, Hainan University, Haikou 570228, China;  
[EMAIL_ADDRESS] (X.R. ); [EMAIL_ADDRESS] (FX.) [PERSON]  
1 College of Civil Engineering and Architecture, Hainan University, Haikou 570228, China;  
[EMAIL_ADDRESS] (X.R. ); [EMAIL_ADDRESS] (FX.) [PERSON]  
2 School of Hydraulic Engineering, Changsha University of Science & Technology, Changsha 410114, China  
2 Department of Civil Engineering, Nagoya University, Nagoya 464-8602, Japan;  
[EMAIL_ADDRESS]  
[PERSON]  
3 Correspondence: [EMAIL_ADDRESS]; Tel. : +86-0898-6627-92323  
To simulate the flow movement in a cylindrical geometry, the use of the cylindrical coordinate system, if possible, is the most relevant and desirable in the formulation and discretization of the Navier-Stokes equations. For instance, in [6], [PERSON] and [PERSON] developed an incompressible flow simulation model in cylindrical coordinates by an explicit projection method, and [PERSON] et al. [7] studied the natural convection heat transfer and fluid flow in a vertical cylindrical envelope by a numerical model built in a cylindrical coordinate. However, the resolution of the Navier-Stokes equations in cylindrical coordinates involves some specific difficulties, especially when the computational domain contains the axis \(r=0\). When the cylindrical center is within the computational domain, terms like \(1/r\) or \(\partial\phi/\partial r\) become undefined, which causes a singularity at the cylindrical center. To overcome this problem, many attempts have been made. [PERSON] et al., with a Cartesian mesh in the axis of the cylindrical coordinates, built a numerical model to simulate pipe flow in [8]. In [9], [PERSON] and [PERSON] avoided this problem by rewriting the Navier-Stokes equations with special variables, which are always zero on the axial, and [PERSON] suggested a discretization without nodes on the axis in [10]. Moreover, another problem arises since the mesh size varies with the radial coordinate when the regular orthogonal grid system is applied. Therefore, investigating the flow phenomena within a large-diameter container, the control cells that are far from the origin are much coarser than those that are near the origin, so an extremely small time step should be applied to satisfy the numerical stability. In order to obtain a grid independent solution in cylindrical coordinates, the zonal embedded grid technique, as shown in Figure 1, was implemented in the present paper by locally refining the grid size by blocks, which was originally proposed by [PERSON] et al. in [11] for the accurate computation of a boundary layer above a flat plate, and then developed to a grid system in the two-dimensional polar coordinates in [12] by [PERSON] and [PERSON]. Flow simulation performed in the cylindrical coordinates is not new. However, the two-phase incompressible flow model implemented in the cylindrical coordinate system has attracted less attention, and implementing free surface tracking method in the cylindrical coordinates, like the volume of fluid (VOF) method presented in [13], is novel since the surface construction in the VOF method was extremely cumbersome in cylindrical coordinates due to its geometric properties. The VOF method in cylindrical coordinates has been adopted in [14] to analyze the bubble rising problems inside a cylindrical container. In their work, to avoid the complex surface reconstruction process of the VOF method, the interpolated volume fraction at the faces of control volumes was calculated by the first order upwind or downwind scheme according to the free surface orientation. Following this idea, a more accuracy VOF method was proposed in [15] by [PERSON] and [PERSON] using the high-resolution based CICSAM scheme (Compressive Interface Capturing Scheme for Arbitrary Mesh) to compute the volume fraction, which could be applied in the cylindrical coordinates.