Hydrodynamic and heat transfer characteristics of vortex- induced vibration of square cylinder with various flow approach angle

Abstract

In this work, the effect of flow approach angle (α) on forced convective heat transfer from an elastically mounted vibrating rigid square cylinder is studied numerically. The Navier-Stokes equations along with the two-dimensional equation of motion of elastically mounted rigid square cylinder are solved using finite difference method. An Arbitrary Lagrangian Euler (ALE) method is used to capture the relation between fluid flow and vibrating cylinder. Numerical simulations are conducted for different Reynolds number $\in$ [60–100], reduced velocity (U_red $\in [3-8])$ and flow approach angle ($\mathrm{\alpha }\in [0^{\text{o}}-{45}^{\text{o}}]$). Prandtl number (Pr) = 7.1 and Reduced mass (M_red) = 2 are kept constant and damping coefficient, ${\mathrm{\xi }}^{\text{'}}$ = 0. The detailed kinematics of the flow and temperature fields is visualized in terms of instantaneous vorticity and isotherm contours. Examining the distribution of local Nusselt number (Nu) along the faces of the heated square develops further detailed insights. Overall gross characteristics are reported in terms of the time average drag coefficient and Nusselt number. Average Nusselt number ($\overline{Nu}$) scales linearly with Reynolds number (Re) while both U_red and α has quadratic relationship. Different vortex shedding modes viz; 2S, 2S* and P + S have also been observed for different flow approach angle. At α $={30}^o$, the average Nusselt number ($\overline{Nu}$) is maximum for U_red = 6, 8 and at all Reynolds numbers as the flow shows 2S* mode of vortex shedding.

Introduction

Vortex-induced vibration (VIV) has been the topic of intense research due to its practical importance in various engineering areas, for example, chimney stacks, long bridges and wind engineering. VIV is also essential for designing heat exchanger tubes, offshore structure such as riser tubes, sea elevators. Flow induced vibration is utilized for augmentation of convective heat transfer [1]. Izadpanah et al. [2] reported the heat transfer enhancement with the help of finned vibrating cylinder. Lee et al. [3] introduced the concept of VIVACE that is used to harness energy of flow stream and convert it into electrical energy. Khan et al. [4] investigated the augmentation of VIVACE by means of thermal buoyancy. Whenever fluid flows over a bluff body, process of vortex shedding appears beyond critical Reynolds number. Due to the shedding phenomenon, the body experiences a periodic force on it. When natural frequency of the body matches the vortex shedding frequency, a well-known lock-in phenomenon arises and the body starts vibrating at the higher amplitude leading catastrophic effects on it. Study of vortex induced vibration attracts the attention of researchers due to the flow physics involved at the downstream side of the bluff bodies. Bishop and Hasan [5], Feng [6], Griffin and Ramberg [7] were the pioneer researchers in the field of VIV. An exhaustive review of Sarpkaya [8], Bearman [9] and Govardhan and Williamson [10] give a thorough understanding of vortex-induced vibration. Williamson and Roshko [11] have classified various vortex shedding modes like 2P and 2S modes for vibrating circular cylinder on the basis of shedding vortices from the cylinder in one period of vibration. Earlier studies show that displacement regime of circular cylinder was categorised mainly into three different branches, initial, lower and upper branch [[12], [13], [14], [15]]. For high Reynolds number flow, generally, upper branch exists, however, for low Reynolds number upper branch is observed to be absent. Reduced velocity in Flow-induced vibration can be used as an independent parameter as it is described in the literature of Singh and Mittal [16] and Zhao et al. [17].

In the numerical study of Zhao et al. [17] the VIV phenomenon of a cylinder with square shape at Re = 100, reduced mass (M_red) = 3 and at three different flow approach angle varies from 0° to 45° was investigated. In their study, response of the square cylinder with flow approach angle has been examined and it is concluded that flow approach angle affects the amplitude response as well as the lock-in regime of the cylinder. Effect of mass ratios on vibration of the square cylinder is reported by Sen and Mittal [18]. In their study the cylinder was moving in-line as well as in transverse direction and damping coefficient was set to be zero. They conducted numerical simulation for various mass ratios (1, 5, 10 and 20) and observed that galloping occurs at higher mass ratios.

Nemes et al. [19] experimentally examined the effect of flow approach angle on flow-induced vibration of the square cylinder. In their study, it was shown that the square cylinder with low reduced mass can experience VIV and galloping phenomena. They reported that both phenomena galloping and VIV were the function of the orientation of the square cylinder. Newman and Karniadakis [20] reported that the cylinder displacement increases as the damping decreases. Cui et al. [21] have conducted the two-dimensional numerical simulations on a square and rectangular cylinder. In their study, the effect of angle of incidence on these cylinders has been reported. They reported that the galloping was present at an angle of incidence = 0° whereas vortex-induced vibration phenomenon was existing at an angle of incidence = 22.5°, 45° for the square shape cylinder. Joly et al. [22] observed that the galloping phenomenon of a two-dimensional square cylinder. This numerical study includes the effect of Re on the onset of galloping and displacement of a square cylinder with a high mass ratio. In recent research, Massai et al. [23] examined the Flow-Induced Vibration of sharp edge rectangular cylinder with low mass ratio and the effect of angle of attack on it.

Wan and Patnaik [24] have studied vortex shedding suppression phenomenon of vibrating circular cylinder for Reynolds number 150 in mixed convection regime. Fluid flow is aligned opposite to the direction of gravity and cylinder was moving in the x direction. They have conducted their numerical experiments for different mass ratios with different reduced velocities and concluded that higher Richardson number is required to suppress the vortex shedding of the cylinder with higher displacement. Izadpanah et al. [25] comprehensively investigated the effect of vortex-induced vibration on heat transfer from a circular cylinder for fixed Re = 150 and Pr = 7.1. Variation of mean Nusselt number for various reduced velocities, U_red (3–8) with different damping ratios was investigated and they concluded that Nusselt number peaks when the cylinder displacement is maximum. Garg et al. [26] in their numerical study reported the critical Reynolds number for onset of VIV in the presence of thermal buoyancy. Thermal buoyancy is induced by the two plates that is placed above and below the circular cylinder symmetrically. Garg et al. [27] also studied the VIV of circular cylinder in the presence of negative thermal buoyancy. They reported that cylinder displacement increases when it is subjected to negative thermal buoyancy. There are few studies which have been carried out on the effect of heat transfer from a stationary square cylinder in the literature [[28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]]. It is evident from the literature survey that the convective heat transfer from vibrating square cylinder seems to be missing.

To the author's best knowledge heat transfer from vibrating square cylinder have not been studied in detail for different flow approach angles and this serves as the motivation for the present study. In this study, numerical experiments are performed at low Reynolds numbers (Re = 60, 80 and 100) for a two-dimensional vibrating square cylinder having two degree of freedom placed in water (Pr = 7.1). Moreover, the variation of coefficient of drag, Nusselt number and frequency ratios have been studied at Re = 60, 80 and 100. The effect of flow approach angle on these parameters has been carried out in detail. The flow approach angle is varied in the range of 0°–45°. The above discussed flow parameters has been comprehensively studied for transverse vibration (α = 90°). The paper is divided in the following section: Mathematical model is described in section 2, governing equations, numerical aspects and validation study is discussed in subsequent sections and in section 6 results and discussion have been presented.