Direct numerical simulation of three-dimensional flow past a yawed circular cylinder of infinite length
Introduction
When a cylinder is exposed to a steady approaching flow, the wake structure downstream of the cylinder is three-dimensional (3-D) as long as the Reynolds number (Re) is larger than about 170, even before it becomes turbulent (Roshko, 1954; Bloor, 1964; Williamson, 1988; Norberg, 2001). The Reynolds number is defined by Re=UD/ν, where U is the incoming velocity in the streamwise direction, D is the cylinder diameter, ν is the fluid kinematic viscosity. The wake flow is in the transitional regime when Reynolds number is between 170 and 300 and becomes fully turbulent as Reynolds number is larger than about 400. In most of civil and mechanical engineering applications, the Reynolds number is usually much larger than 400. Therefore, the wake flow is normally turbulent.
Extensive studies on turbulent wake flows past a circular cylinder have been conducted using both experimental and numerical methods. Williamson, 1988, Williamson, 1991, Williamson, 1992 investigated the three-dimensional transition of the flow behind a circular cylinder. It was found that the three-dimensionality and turbulence in a wake are triggered by instabilities within the vortex formation region. The instabilities include the generation of the small-scale streamwise vortices, large-scale vortex dislocations and small-scale shear-layer instability vortices. When Reynolds number is in the turbulent regime, the hydrodynamic forces on the circular cylinder fluctuate with time due to the vortex shedding. Schewe (1983) reported the experimental results of the drag coefficient CD and lift coefficient CL over a wide range of Reynolds number. The CD and CL are defined by CD=FD/(ρDU2/2) and CL=FL/(ρDU2/2), where ρ is the fluid density, FD and FL are drag and lift forces on a unit length of a cylinder in the flow direction and that in the cross-flow direction, respectively. According to Schewe's results, the force coefficients and the vortex shedding frequency are not sensitive to Reynolds number as long as the latter is in the subcritical regime (300<Re<3×105).
Since the wake flow behind a circular cylinder is three-dimensional, it is desirable that the flow is simulated by a 3-D model in order to obtain a full understanding of the wake flow. Two-dimensional numerical models tend to overestimate the lift coefficient and the vortex shedding frequency because the variation of flow in the cylinder's spanwise direction (Kang, 2006; Zhao et al., 2007; Tsutsui et al., 1997) is ignored. Karniadakis and Triantafyllou (1992) and Zhang and Dalton (1998) simulated the 3-D flow past a circular cylinder using three-dimensional models for Reynolds number in the range of 100–500. Their findings about the transition of the flow to turbulence agree well with those found in experiments (Williamson, 1992). Lei et al. (2001) studied the effect of the computational domain size on the accuracy in the simulation of 3-D flow past a circular cylinder. They found that the size of the computational domain in the spanwise direction must be at least four times the cylinder diameter in order to simulate the 3-D wake flow accurately.
In many engineering applications, such as the flow past cables, subsea pipelines, risers, etc., the direction of the flow may not be perpendicular to the structure. This kind of flows can be represented by a wake flow downstream of a yawed cylinder in laboratory studies and numerical simulations. In this case, the fluid velocity in the spanwise as well as in the cross-flow directions may be of similar magnitudes. It is expected that the three-dimensional effect and the wake flow patterns of a yawed cylinder will be stronger than that of a wake when the cylinder is perpendicular to the flow. Flows past a yawed cylinder have been studied by a number of investigators both experimentally [e.g. King (1977); Ramberg (1983); Kozakiewicz et al. (1995); Thakur et al. (2004)] and numerically [e.g. Chiba and Horikawa (1987); Marshall (2003); Lucor and Karniadakis (2003)]. Experimental results showed that the force coefficients and the Strouhal number, which are normalized by the velocity component perpendicular to the cylinder, are approximately independent on the yaw angle. This is often called the independence principle or the cosine law in the literature. Kozakiewicz et al. (1995) found that the independence principle can be applied to stationary cylinders in the vicinity of a plane wall for a yawed angle between 0° and 45°. In case of flow past a yawed cylinder of finite length, it was shown that the wake vortices far from the upstream end of the cylinder are approximately parallel to the cylinder. The vortices near the upstream end of the cylinder are aligned at an angle larger than the cylinder yaw angle (Ramberg, 1983; Thakur et al., 2004). Lucor and Karniadakis (2003) simulated flow past a yawed cylinder of infinite length at two large yaw angles, namely 60° and 70°. They reported that the vortex shedding angles of the vortices in the wake of a yawed cylinder are somewhat less than the cylinder's yaw angle.
In the present study, flow past an infinitely long stationary circular cylinder at yawed angles in the range of 0–60° was investigated numerically. The definition of the coordinate system and the yaw angle α are given in Fig. 1, where α=0° represents the right attack angle case (i.e. the flow direction is perpendicular to the cylinder). Direct simulation of the Navier–Stokes equations was performed without employing any turbulent models. The Reynolds number Re was 1000. This Reynolds number was selected based on following considerations. Firstly, the wake flow at this Reynolds number is fully turbulent according to previous studies. It has been shown that the hydrodynamic forces are not sensitive to the Reynolds number as long as the later is in the subcritical regime (300<Re<3×105). Secondly a relative small value of Reynolds number in the subcritical regime allows a direct simulation of the Navier–Stokes equations being carried out with affordable computational costs. Therefore, the present choice of the Reynolds number was a compromise of the flow regimes and the computational cost. The effects of the yaw angles on the wake flow, the hydrodynamic force and the vortex shedding frequency were examined.
Section snippets
Governing equations and boundary conditions
The nondimensional Navier–Stokes equations and the continuity equation in Cartesian coordinate system x′y′z′ (Fig. 1) arewhere ui is the velocity component in the xi-direction, (x1,x2,x3)=(x′,y′,z′), the subscripts in f,t and f,i represent the derivatives of f with respect to time t and x′i, respectively, and p is the pressure. The computational domain is shown in Fig. 1(b). In the present study, flow past a cylinder of infinite length was simulated by a
Numerical results
Numerical simulations were carried out at Reynolds number Re=1000 and yaw angles ranging from 0° to 60°. The cases for α>60° were not covered in this study because of the difficulties in generating quality computational meshes. It is expected that the dimensional force in the cross cylinder direction for α>60° is much smaller than in other cases. So, flows with α>60° may not be of as much engineering interests as those with smaller yaw angles. Parallel computational program code is developed
Conclusions
Flow past a circular cylinder at yaw angles in the range of 0–60° was investigated numerically by solving the three-dimensional Navier–Stokes equations directly using a finite element method. The results were compared with the flow visualization in the experiments. The effects of the yaw angle on vortex structures and the force coefficients were examined. The results are summarized as follows:
- (1)
In the numerical simulation the flow in the early stage of the transition period from 2-D to 3-D is
Acknowledgement
The authors would like to acknowledge the supports from Australia Research Council through ARC Discovery Projects Program Grant no. DP0557060 and CSIRO Flagship Collaboration Cluster on Subsea Pipelines.
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