## 2 Materials and Methods  
### Computational Method and Feasibility Analysis  
VLES was proposed in 1998 as a hybrid turbulence method that combined the advantages of diverse turbulence modelings of RANS (Reynolds-averaged Navier-Stokes), LES, and DNS. [18; 19; 20] It is also known as the flow simulation methodology (FSM) which exhibits significant potential in some application. This method essentially depends on the resolution control function \(Fr\) to damp the Reynolds stresses, aimed at modeling the subscale stress tensor, which is expressed as:  
\[\tau_{ij}^{sub}=Fr\tau_{ij}^{RANS} \tag{1}\]  
In this paper, a specific VLES method established on the standard \(k\)-\(\varepsilon\) RANS turbulence model is adopted, whose essence is to modify the formulation of the turbulent viscosity by using a control function \(Fr\)[18]:  
\[\mu_{t}=Fr\rho C_{\mu}k^{2}/\varepsilon. \tag{2}\]  
In this VLES method, the transport equations are expressed as:  
\[\frac{D\rho k}{Dt}=P_{k}-\rho\varepsilon+\frac{\partial}{\partial x_{j}} \Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{ \partial x_{j}}\Bigg{]}, \tag{3}\]  
\[\frac{D\rho\varepsilon}{Dt}=\frac{\varepsilon}{k}(C_{\varepsilon 1}P_{k}-C_{ \varepsilon 2}\rho\varepsilon)+\frac{\partial}{\partial x_{j}}\Bigg{[} \bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial \varepsilon}{\partial x_{j}}\Bigg{]}. \tag{4}\]  
where these transport equations are precisely the same as in the standard \(k\)-\(\varepsilon\) model, \(k\) is the modeled turbulent kinetic energy, and \(\varepsilon\) is the turbulent dissipation rate. In addition, \(P_{k}\) is the kinetic energy production item and is defined as:  
\[P_{k}=2\mu_{t}S_{ij}S_{ij}, \tag{5}\]  
\[S_{ij}=\frac{1}{2}\bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\bigg{)}, \tag{6}\]  
In a computational domain, the distribution of the \(Fr\) value depends on the variation of two turbulence length scales and constant terms in the \(Fr\) expression which can realize the transformation of the calculation model between LES and RANS. The \(Fr\) is defined as:  
\[Fr=\ \min\Bigg{(}1.0,\bigg{[}\frac{1.0-exp(-\beta L_{c}/L_{k})}{1.0-exp(-\beta L _{i}/L_{k})}\bigg{]}^{n}\Bigg{)} \tag{7}\]  
where \(\beta\) is the model constant and equals \(2.0\times 10^{-3}\). In Equation (7), \(Fr\) is approximately understood as the ratio of the unsolved turbulent kinetic energy to the total turbulent kinetic energy, and therefore it is between 0 and 1. Here, \(L_{i}\) is the integral length scale, \(Lc\) is the turbulent cutoff length scale, and \(L_{k}\) is the Kolmogorov length scale, which are defined as, respectively:  
\[L_{c}=C_{x}(\Delta x\Delta y\Delta z)^{\frac{1}{3}}, \tag{8}\]  
\[L_{k}=v^{\frac{3}{2}}/\varepsilon^{\frac{1}{4}}, \tag{9}\]  
\[L_{i}=k^{\frac{3}{2}}/\varepsilon. \tag{10}\]where \((\Delta x\Delta y\Delta z)^{1/3}\) is the equivalent length of a mesh resolution, \(n=2\), \(Cx=0.61\), and \(\
u\) is the kinematic viscosity of the fluid. In the regions near the wall, \(Fr\) is close to 1, due to \(L_{\rm c}>L_{i}\), meaning the RANS model is approximately adopted. However, a VLES model is applied in the region away from the wall, due to \(Fr<1\), or \(L_{\rm c}<L_{i}\). The VLES method is through a resolution control function \(Fr\) to determine the computational model, therefore, the user defined function (UDF) in ANSYS Fluent is required to realize an intended function. The VLES method has both the accuracy of the LES model for calculating the large-scale vortex structures and the high efficiency of the RANS model for calculating the flows near the wall. More importantly, it also has good predictability for a coarse mesh. [PERSON] et al. introduced the VLES method into the dynamic grid technology to calculate the wake structure of an oscillating hydrofoil, and verified the feasibility and high efficiency of the VLES method applied in the dynamic grid technology by comparisons with LES and RANS models [20].