# Effect of Lateral End Plates on Flow Crossing a Yawed Circular Cylinder
## 2 Computational Methodology  
The dimensionless transient equations of continuity and momentum for 3D incompressible flow of a Newtonian fluid are presented as follows [13]:  
\[\
abla\cdot u=0 \tag{1}\]  
\[\frac{\mathrm{D}u}{\mathrm{D}t}=-\
abla p+\frac{1}{Re}\
abla^{2}u \tag{2}\]  
where the dimensionless variables are expressed as  
\[p=\frac{\overline{p}}{\rho\mathrm{L}_{0}},\quad x=\frac{\overline{x}}{D},\quad t =\frac{\overline{t}\mathrm{L}_{0}}{D},\quad u=\frac{\overline{u}}{\mathrm{L}_ {0}} \tag{3}\]  
The calculation configuration for flow crossing a yawed cylinder is built up with \(35\)\(D\times n\)\(D\times 20\)\(D\) in three dimensions, as shown in Figure 1a. The width of the calculation domain is taken as \(n\)\(D=9\times D\) for the nonslip wall boundary condition and \(n\)\(D=6\times D\) for the periodic wall boundary condition in order to save computational resource, which is enough to capture the feature of the wake flow for the periodic wall boundary condition. Two coordinate systems are defined, as shown in Figure 1b. One is the global coordinate system of Oxyz, in which the origin is located at the center of the midspan cross section of the cylinder, and the x-, y- and z-directions are defined respectively as the streamwise direction, the vertical direction, and the spanwise direction. The other is a cylinder-oriented coordinate system Ox'y'z', in which the origin is overlapped with that of the global coordinate system, and the x'-, y'- and z'-directions are defined as the direction of the normal to the frontal stagnant edge, the same direction as y-, and the axial direction aligned with the cylinder axis. The yaw angle \(\alpha\) is defined as an angle between the normal to the cylinder x' and the incoming flow direction x, where \(\alpha=0^{\circ}\) represents orthogonal cross-flow of a cylinder. A cylinder is clamped between two lateral end-plates. Its upstream end is attached with a base end-plate, and its downstream end is attached with a tip end-plate. The two vertical constraints of the domain are respectively named as ceiling and floor. The boundary conditions for the governing equations are set as follows. In the following simulation, our calculations are developed on the platform of the open source CFD software OpenFOAM(r). A uniform velocity is used to define the inlet condition, where the streamwise velocity component is set to a free uniform stream velocity \(U_{0}\) while the spanwise and vertical velocity components are set to zero. A Neumann boundary condition of pressure is imposed on the outlet with enough downstream length (18 times the diameter of the cylinder) to assure no backward flow on the outlet. The free-slip boundary condition is set on the ceiling and floor. The no-slip boundary condition is applied on the  
Figure 1: Definition of calculation configuration and coordinate system. cylinder wall. On the two lateral end-plate walls, both nonslip and periodic boundary conditions are used to investigate the effect on the flow characteristics in the wake. The numerical methodology is briefly presented as follows. The convective fluxes are discretized using conservative skew-symmetric split-centered formulation. The viscous fluxes are discretized using the fourth order compact central differences scheme [14]. Time advancement is assessed by a standard explicit fourth-order Runge-Kutta algorithm. Turbulence is calculated using the large eddy simulation (LES) approach with a dynamic Smagorinsky-Lilly subgrid-scale model, which had been proven to be in very good agreement with experiments when solving flow orthogonally crossing a circular cylinder [15; 16]. The subgrid-scale turbulent stress tensor is computed from  
\[\tau_{ij}=-2\mu_{i}\overline{S}_{ij}+\frac{1}{3}\tau_{kk}\delta_{ij} \tag{4}\]  
where \(\tau_{kk}\) is the isotropic part of the subgrid-scale stress.