Elsevier

Ocean Engineering

Volume 73, 15 November 2013, Pages 41-54
Ocean Engineering

Poiseuille flow across an eccentrically confined stationary/rotating cylinder

https://doi.org/10.1016/j.oceaneng.2013.07.025Get rights and content

Highlights

  • Flow regime map is presented at various rotational-velocity and eccentricity-ratio for Re = 100.

  • For stationary cylinder, the direction of lift force changes with increasing eccentricity.

  • For rotating cylinder, stabilization effect is more at positive as compared to negative eccentricity.

  • Flow stabilization mechanism is different for positive and negative eccentricity.

  • Stabilization effect of rotation and eccentricity can be used for flow control.

Abstract

Effect of eccentricity-ratio and rotation on a Poiseuille flow across a stationary/rotating cylinder is studied numerically for various Reynolds number (Re=50–150) at a blockage ratio of β=20%. Several interesting flow-patterns and flow-transition, for various eccentricity-ratio (75%γ+75%) and rotational-velocity (0α2) at Re=100 are noted. The flow patterns are understood with the help of streamlines and vorticity contours. The ensuing flow is analyzed with the help of separation point on the surface of the cylinder, strength and trajectory of the shed-vortices and flow distribution above and below the cylinder. Flow-transition from unsteady to steady flow is reported as a flow-regime map, at various values of α and γ. Stabilization effect is more at positive as compared to negative eccentricity, for counter clockwise (CCW) rotating cylinder. It is established that the mechanism of flow-stabilization is different for positive and negative eccentricity, for rotating cylinder. For stationary cylinder, increase in eccentricity causes a reduction in the drag-coefficient and Strouhal-number; and a change in the direction of lift force at higher eccentricity ratio. Under the influence of rotation, a reduction (enhancement) in the drag-force is obtained at a constant positive (negative) eccentricity. Stabilization effect of rotation and eccentricity can be used for flow control.

Introduction

Viscous flow across a cylinder is a classical fluid-dynamic problem as the vorticity generated due to no-slip condition causes various flow phenomena (such as development of boundary-layer, free-shear layer and wake) as well as flow-transitions with increasing Reynolds number (Re=uD/ν). For free stream cross-flow across a rotating cylinder, rotation induced asymmetry in the flow generates a lift-force which is proportional to circulation; according to the Kutta–Joukowski theorem. Asymmetry in the viscous flow across a stationary cylinder also develops when cylinder is placed off-center in a channel (called here as eccentrically channel-confined flow), with a fully developed flow at the inlet. Due to the eccentricity, the strength of the shear-layer and vorticity-generation from the top and bottom surfaces of the cylinder/channel are different. The interest in the present study lies in studying the combined effect of rotation and eccentricity. The interaction of the shear-layer and wall vorticity developed due to eccentricity, viscosity, rotation and channel-walls is therefore studied in detail in the current investigation. Such a study has several ocean systems related applications such as flow across bundle of risers which links the seabed to the offshore platforms used for oil exploration, propeller shaft of a ship, submarine cables, bridge-piers and off-shore pipelines near the seabed. However, the applications are more for stationary as compared to rotating cylinder.

Fluid flow is generally classified as external or internal. The channel-confined flow across a cylinder has characteristics of both the types of flows and is widely encountered in engineering applications such as heat exchangers. In the past, the flow has been studied more for a stationary as compared to a rotating cylinder in spite of its various applications such as spinning mills. In case of flow across eccentrically confined stationary/rotating cylinder in a channel, Reynolds number, blockage-ratio (β=D/H), non-dimensional rotational velocity (α=Dω/2u) and eccentricity-ratio [γ=(2βYc1)/(1β)] are the governing parameters. The blockage ratio represents the extent of confinement and the rotational velocity represents the cylinder surface tangential velocity in terms of u (mean-velocity in the channel). The eccentricity ratio represents the extent of proximity of cylinder to the channel wall; +100% and −100% represents the cylinder touching the top and bottom wall, respectively.

There are numerous studies on a centrally/symmetrically channel-confined flow across a stationary (Chen et al., 1995, Sahin and Owens, 2004, Chakraborty et al., 2004, Khan et al., 2004, Rehimi et al., 2008, Singha and Sinhamahapatra, 2010) and for unconfined flow across a rotating (Tang and Ingham, 1991, Kang and Choi, 1999, Mittal and Kumar, 2003, Sengupta et al., 2003, Stojkovic et al., 2003, Lam, 2009, Paramane and Sharma, 2009) cylinder. A detailed literature survey is reported on the channel-confined (Prasad et al., 2011) and the unconfined (Paramane and Sharma, 2009) flows in the recent work from our research group. The critical Reynolds number (based on the mean inlet velocity facing the cylinder ucyl), for the onset of VS, is found to increase with increasing blockage ratio for channel-confined flow across a stationary cylinder. For unconfined flow across a rotating cylinder, a detailed study was done by Paramane and Sharma (2009) for wide range of Reynolds number (20Re160) and rotation parameters (0α6) in the two-dimensional laminar flow regime. They reported suppression of VS at certain critical rotational velocity which increases with increasing Re (60). With further increase in α, the flow remains steady except at a narrow range of intermediate α. Thus, the channel-confinement and the rotation leads to stabilization of a free-stream flow across a stationary cylinder.

There are a few work on the eccentrically channel-confined flow across stationary (Zovatto and Pedrizzeti, 2001, Mettu et al., 2006) and centrally channel-confined flow across a rotating (Prasad et al., 2011) cylinder. Eccentrically channel-confined flow across a stationary cylinder was studied by Zovatto and Pedrizzeti (2001) for 0γ87.5% and 40Re200 at a constant β=20%. With increasing γ, they found that the onset of vortex-shedding (VS) at a larger Reynolds number and the pattern of VS changes from Von Karman street to a single row of same sign vortices. They reported that the effect of eccentricity is to stabilize the flow. This was studied in more detail by Mettu et al. (2006), at various eccentricity-ratio (0γ87.5%), Reynolds-number (10Re500) and blockage-ratio (10%β40%). With increasing γ, they found that the drag-coefficient (CD=FD/12ρu2D) and Strouhal number decreases.

Centrally channel-confined flow across a rotating cylinder was studied by Cliffe and Tavener (2004) for α1.2 and β=0.7. They found that the critical Reynolds number (Rec) for the onset of VS increases with increasing rotational velocity at higher Reynolds number (ReRec). They further showed that the VS suppressed at higher Reynolds number (ReRec). A more detailed study was reported, recently by Prasad et al. (2011), at various rotational-velocity (0α2), blockage-ratio (0%β50%) and Reynolds number (35Re170). They found that the critical Reynolds number for the onset of VS increases, with increasing blockage ratio and increasing rotational velocity for stationary/rotating cylinder in a channel. Finally, they showed that channel-confinement leads to enhancement and rotation results in a reduction in the drag force as well as heat transfer.

The individual effect of eccentricity, channel-confinement and rotation on flow-stabilization is same but different on engineering parameters such as lift and drag coefficient. Thus, it would be interesting to study their combined effect. No such results are found in the published literature. Thus, the objective of the present work is to study the combined effect of eccentricity, channel-confinement and rotation on the flow-transition and engineering parameters for the flow across a cylinder; at various Reynolds number in the two-dimensional laminar flow regime. Furthermore, the objective is to perform a detailed fluid-dynamic analysis and provide reasons for flow transitions.

Section snippets

Physical description of the problem

The schematic of Poiseuille flow across an eccentrically confined stationary/rotating cylinder (of diameter D) in a channel (of height 5D) is shown in Fig. 1(a). The cylinder is placed at a non-dimensional distance of Yc(yc/D) from the bottom-wall. The non-dimensional distance between the center of cylinder and horizontal centerline of the channel is defined as eccentricity-ratio, given asγyc0.5H0.5(HD)=2βYc1(1β)The distance of the center of the cylinder from the horizontal centerline of

Mathematical and numerical details

The governing equations, boundary conditions, numerical details and range of governing parameter, employed in the present work, are discussed in this section.

Effect of eccentricity and rotation on flow-transition

Fig. 3(a–f) shows the temporal variation of lift-coefficient at Re=100, for various eccentricity ratio and rotational velocity of the cylinder. It is observed that the fluid flow stabilizes, with increasing eccentricity and rotational velocity of cylinder; summarized and presented as flow regime map in Fig. 3(g).

Although the rotation stabilizes the flow, it is interesting to see in Fig. 3(d) that the amplitude of variation of lift-coefficient increases with increasing rotational-velocity, for

Flow patterns

In this section, the effect of eccentricity on the flow pattern near the stationary/rotating cylinder is discussed with the help of streamlines, vorticity-contours and flow-distribution above and below the cylinder; for Re=100. For stationary cylinder, variation in the separation-point and separated region on the cylinder surface is obtained to provide information about wake size and its variation with eccentricity-ratio. Vorticity information is supplemented by vortex trajectory and strength

Flow parameters

Variation of engineering parameters such as Strouhal number, mean and rms value of lift/drag coefficients, at various Reynolds-number, rotational-velocity and eccentricity-ratio, are presented in this section. Forces acting on cylinder surface are calculated by integrating cylinder surface pressure and shear stress.

Summary

The tools (streamline-patterns, vorticity-contours, trajectory as well as strength of shed vortices and flow distribution) employed in the previous sections have revealed several new and interesting results about the flow. These results are summarized in Table 1.

Reasons for flow-transition

The possible reasons for the flow transition (refer Fig. 3(g)) is provided separately for eccentrically placed stationary and rotating cylinder. It is observed that increase in eccentricity as well as rotational velocity of the cylinder

Conclusions

Flow regime map is presented demarcating steady and unsteady flow at various rotational-velocity and eccentricity-ratio for Re=100. Suppression of vortex-shedding (VS) is found with increasing eccentricity-ratio and rotational-velocity. For stationary cylinder, the suppression of VS takes place at the same magnitude of positive and negative eccentricity. Whereas, for cylinder rotating at constant velocity, larger magnitude of negative as compared to positive eccentricity is required for the

Acknowledgments

First author is sincerely grateful to Finolex Academy of Management and Technology, Ratnagiri for providing study leave to carry out this research work. We are extremely thankful to the anonymous reviewers for their comments, which have helped us to bring out a substantially improved manuscript.

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