Influence of thermal buoyancy on vortex shedding behind a circular cylinder in parallel flow

Abstract

In this study, we perform a numerical investigation to better comprehend the domination and the effect of strong thermal buoyancy on hydrodynamic and thermal characteristics, vortex shedding in particular, of a horizontal circular heated cylinder subject to a vertically upward laminar flow stream. The Reynolds number is varied in the range 20 $\le Re\le$ 150 while maintaining the Prandtl number constant at $Pr$ = 7.1. The effect of thermal buoyancy on the system is determined by varying the Richardson number up to $Ri$ = 5. The behaviour is determined by solving the unsteady laminar two-dimensional Navier–Stokes and standard energy equations numerically, utilising the spectral-element method with buoyancy considered using the Boussinesq approximation. Representative vorticity, streamline, and thermal patterns are presented, and the average Nusselt numbers are plotted as a function of the Richardson number for a set of Reynolds numbers to explain, in detail, the role of superimposed thermal buoyancy on the wake flow and rates of heat transfer. The predictions demonstrate that the wake flow exhibits unsteady periodic characteristics for the selected moderate Reynolds numbers, and as the buoyancy parameter increases, the fluid flow and temperature fields become less stable. Subsequently, it was observed that heat transfer augmented considerably, the vortex shedding stopped abruptly, the wake converted to steady twin vortices beyond certain critical Richardson numbers, and the Nusselt numbers suddenly reduced to minimum values. These critical values are determined to increase with the Reynolds number. The results also confirm that for increased heating, the recirculating flow in the cylinder near the wake disappears and flow separation occurs only at the backward stagnation point.

Introduction

One of the most interesting phenomena observed in viscous flows past bluff bodies is that of vortex shedding, with the wake structure predominantly called a von-Kármán vortex street. The characteristics of vortex shedding behind circular cylinders or similar two-dimensional obstacles have been extensively discussed; relevant reviews were provided by Bergere and Wille [1], and later by Williamson [2]. This phenomenon has significant implications for numerous thermal applications such as chimney stacks, cooling towers, fuel rods of nuclear reactors, and heat exchanger tubes, where a circular cylinder exists as one of the fundamental structural components. For applications, it must be recognised that buoyancy forces can influence the thermal and flow fields considerably under moderate Reynolds numbers and with a relatively high temperature discrepancy between the cylinder surface and flowing fluid. Indeed, the situation becomes physically complex when the heat transfer strongly influences the nature of the wake behind the hot cylinder owing to the temperature-induced buoyancy forces.

In mixed convective flows past cylinders, the heat transfer characteristics depend mainly on the Reynolds, Grashof or Richardson, and Prandtl numbers, and the angle between the forced flow and buoyancy force direction. These flows can be classified on the basis of the direction of the main flow to the buoyancy-induced flow into two primary forms, horizontal cross flows and vertical flows. In particular, the wake flow behaviour is completely different for the situation of a vertical flow under the influence of vertical buoyancy compared to that for a horizontal cross-flow configuration. The behaviour behind a circular cylinder within mixed convective horizontal cross flows has been reasonably studied, including in Biswas and Sandip [3], Ankit and Amit [4], Jiansheng and Yunjian [5], Jiansheng and Chen [6], Kumar and Sameen [7], Sumit and Dominic [8], Vivien et al. [9], Aleksyuk and Osiptsov [10], Amit et al. [11], Sonal et al. [12], and Zheng et al. [13].

Moreover, in a vertical flow configuration, the forced flow and buoyancy force can be in the same (parallel-flow) or opposing (counter-flow) directions. In fact, several numerical and experimental studies have been conducted previously for the situation wherein, the forced flow is pointed vertically upward (aiding-flow), which is the case considered in this study. Oosthuizen and Madan [14] conducted pioneering experimental work on the problem of mixed convective parallel flow for Reynolds numbers over the range 100 $\le Re\le$ 300, and proposed an empirical relationship for the Nusselt number. Jackson and Yen [15] also established a correlation that was determined to be suitable for high Grashof numbers. Other authors, for example, Acrivos [16], Joshi and Sukhatme [17], Nakai and Okazaki [18], Sparrow and Lee [19], Merkin [20], Badr [21,22], and Syakila et al. [23] numerically studied the steady-state condition of combined convection from a circular cylinder under boundary-layer flows using approximate methods. However, the aforementioned investigations were directed at studying only steady flows, disregarding the periodic flow behaviour and determining the critical Grashof number where the flow abruptly transforms from steady to unsteady.

The transient behaviour of mixed convective flow about a circular cylinder was investigated numerically by Jain and Lohar [24] and Noto and Matsumoto [25]. They revealed that the increase in the cylinder temperature increases the vortex shedding frequency. These studies, however, were performed while the flow was developing; it is not clear if they apply to a fully developed flow. Following this, Farouk and Guceri [26] analysed numerically mixed convective flow about a heated cylinder positioned between vertical parallel insulated plates for a low Reynolds number of 6.20. However, their study focused only on the effect of varying the separation distance between the plates. Ho et al. [27] reported a numerical investigation on buoyancy-aided convective flow around a cylinder placed inside an open vertical insulated duct for Reynolds numbers of 20, 40, and 60. Vortex shedding was not observed at a Reynolds number of 60, presumably owing to the considerable blockage caused by the narrow channel wall spacing. Therefore, they investigated the combined influence of the buoyancy force and channel wall spacing on the steady wake and heat transfer enhancement. It was determined that a considerable augmentation in heat transfer can be obtained through blockage variation.

Singh et al. [28] identified unsteady (full-periodic) flow and temperature fields about a hot/cold cylinder within an adiabatic vertical duct for Richardson numbers over the range −1 $\le Ri\le$ 1, and at a fixed blockage ratio of 0.25 and Reynolds number of 100. The effect on the cylinder heating/cooling was related to the positive/negative effect of the buoyancy forces. It was demonstrated that the shedding incident persists to characterise the flow for Richardson numbers approaching 0.15; however, the aiding-buoyancy alters the dynamics entirely by stopping the shedding thereafter. They predicted that over the range −1 $\le Ri\le$ 0.15, the increase in the magnitude of Richardson number tends to marginally decrease the average Nusselt number. However, beyond the critical value of 0.15, the increase in Richardson number serves to enhance the average Nusselt number. A similar problem was also examined by Gandikota et al. [29]. A comprehensive investigation was performed for several Reynolds numbers over the range 50 $\le Re\le$ 150, for Richardson numbers over the range −0.5 $\le Ri\le$ 0.5, and for two blockage ratios of 0.02 and 0.25. For $Re$ = 100, their results indicated that wake unsteadiness stops entirely at critical Richardson numbers of 0.15 and 0.18 at the above mentioned blockage parameters, with a higher critical value for the lower blockage ratio. They indicated that the proximity of the side wall restricts the fluid flow neighbouring the cylinder, which stabilises against the tendency to shed vortices, and consequently lower cylinder heating is necessitated to stabilise the flow. Remarkably, the average Nusselt number was found to persistently increase with Richardson number over its entire negative and positive ranges in both channels with a lesser gradient below the critical value.

The current literature reveals that only two studies, Chang and Sa [30] and Hatanaka and Kawahara [31], have investigated full-periodic parallel flows (aiding-flows) over a circular cylinder in the absence of side walls (unconfined flows). They analysed the behaviour of near-wake vortices behind a hot and/or cold circular cylinder for Richardson numbers over the range −1 $\le Ri\le$ 1 and for a constant Reynolds number of 100. These studies used different numerical methods; however, both predicted that the cooling of the cylinder surface strengthens the shear layer and the periodic flow becomes more activated. However, heating was found to suppress the frequent flow into a stable twin-eddy structure at a critical Richardson number of 0.15, identifying a collapse of the Kármán vortex street. Hatanaka and Kawahara [31] did not document the heat transfer enhancement/diminution, as their focus was only on the changing dynamic and thermal patterns. Surprisingly, Chang and Sa [30] reported that the trend of the average Nusselt number demonstrated interesting and sensitive features in the unsteady flow region before the suppression of the vortex shedding. Thus, in the negative range of the Richardson numbers −1 $\le Ri<$ 0, the average Nusselt number increases notably; however, in the positive range of the Richardson numbers 0 $<Ri<$ 0.15, it decreases marginally to compose a localised minimum prior to the discontinuity at $Ri$ = 0.15. Moreover, in the region of steady flow with twin stationary vortices connected to the cylinder, for $Ri\ge$ 0.15, the average Nusselt number was demonstrated to increase again.

In the aforementioned investigations of Singh et al. [28], Gandikota et al. [29], and Chang and Sa [30], there are apparently conflicting conclusions concerning the reliance of the wall convective heat transfer on the Richardson number in the unsteady flow regime, where the flow structure is dynamically altered and characterised by the vortex shedding. Therefore, the first objective of the current study is to investigate thoroughly the effect of this parameter, and provide a deeper understanding of the augmentation or diminution of the convective heat transfer with the Richardson number, with special attention near its critical value. Secondly, the role of the Reynolds number was ignored in these investigations. Importantly, a higher critical Richardson number could occur at a higher Reynolds number, or vice versa. This could be an indication that the critical Richardson number is not a universal property for the suppression of vortex shedding; rather, it could be Reynolds number dependent. Furthermore, after the stage of shedding suppression, stable twin vortices are formed and attached to the aft of the cylinder. A complete picture and detailed information regarding the dependency of the Nusselt number on the decaying and then vanishing of these dual vortices by the heating effect have not been provided satisfactorily. For these reasons, the present study further investigates and analyses the transient buoyant upward flows over a heated unconfined circular cylinder to better realise the connection between the temperature-induced buoyancy forces and vortex shedding patterns changing in the combined forced and free convection regime, destruction of stable wakes by the heating influence, and the convection heat transfer characteristics, for different Reynolds numbers.