Triggering asymmetry for flow past circular cylinder at low Reynolds numbers

doi:10.1016/j.compfluid.2012.02.006

Computers & Fluids

Volume 59, 30 April 2012, Pages 44-60

https://doi.org/10.1016/j.compfluid.2012.02.006 Get rights and content

Abstract

The flow past a circular cylinder above some critical Reynolds number (Re_c) is known to be asymmetric and, as a consequence, vortex shedding occurs naturally in physical situations. But in numerical simulations, especially in finite difference set up, it is well known that the shedding process needs to be initiated artificially for flows in the threshold regime in the vicinity of Re_c. In the present study, we discuss four different techniques, including an untested one that trigger asymmetry into the flow in this regime with the help of a recently developed finite difference scheme for the biharmonic pure stream function form of the Navier–Stokes equations. In particular, we choose Reynolds number Re = 44 and 50; in case of the former, the flow eventually retains its symmetry and for the latter, it settles into a periodic state. We further narrow down the range of Re to establish that the critical Reynolds number resides in the regime 46.5 < Re_c ⩽ 47 and finally provide an estimation of Re_c as well. In order to ascertain credibility to our numerical simulation, we also compute flows for Re slightly beyond this regime, namely Re = 40 and 80 for which the flow pattern is well established. For Re = 44 and 80, we replicate the streakline patterns of some earlier experiments for which no such numerical simulation are seen before. In all the cases considered here, our numerical results are in excellent match with available experimental and numerical results. Finally, we provide a brief comparison of the four perturbation techniques.

Highlights

► A new pure stream function based finite difference scheme is used. ► Robustness of 4 perturbation techniques including an untested one is established. ► The techniques trigger asymmetry in the threshold regime in the vicinity of Rec. ► Experimental results are replicated for Re = 44 and 80 by streakline simulation. ► By direct numerical time integration, we have found a very narrow range for Rec.

Introduction

The flow around a circular cylinder has been the subject of intense research in the last century and numerous theoretical, numerical and experimental investigations have been reported in the literature [1], [4], [8]. It is a complicated phenomenon and is characterized mainly by the Reynolds number (Re). The time development of an incompressible viscous flow induced by an impulsively started circular cylinder is now a classical problem in fluid mechanics. It displays almost all the fluid mechanical phenomena for incompressible viscous flows in the simplest of geometric settings. It is universally accepted that at small Re the flow is laminar, steady and remains attached to the cylinder. At Re ≈ 6 although the flow separates from the cylinder but is still steady and laminar. With the increase in Reynolds number, expectedly the flow becomes more complicated due to the appearance of secondary and tertiary vortices. Flows above some critical Reynolds number (Re_c) eventually become periodic and are known to develop vortex shedding represented by the von K $\overset{´}{a}$ rm $\overset{´}{a}$ n vortex street. Traditionally both numerical and experimental studies have been made to pinpoint critical Reynolds number and experimental efforts to locate Re_c is scattered over a wide interval (30, 65) [8], [9], [10], [11], [12]. Numerical simulation of flow past impulsively started circular cylinder in the region which is far beyond Re_c can be found in plenty [13], [14], [15], [16], [17], [18], [19], [20]. But simulation around Re_c is tricky as normal computational procedure may lead to the capture of some un-physical behavior of the flow. Also such computation puts a high demand in CPU time, the growth rate at the onset of instability being extremely slow. Most often, numerical analysts have studied the linear instability of this flow as a Hopf bifurcation problem. A look at the literature suggests that the data reported by various researchers, about the threshold value of Re_c at which Hopf bifurcations takes place, are scattered in the range 43 ⩽ Re_c ⩽ 50 [21], [22], [23], [24], [25], [26], [27]. However, recent studies indicate some agreement among the researchers to Re_c ≲ 47 [28], [29], [30]. In this regard it is worth mentioning that in a recent investigation, Sengupta et al. [31], [32] have concluded that the reporting of different Re_c by different experimental and numerical facilities may be related to the receptivity of the flow field to background disturbances during the linear temporal growth of the disturbance field for post-critical Reynolds numbers. In their study they have made a detailed analysis of various results available in literature and have concluded that there may not be any fixed critical Reynolds number in the sense that such a value may be scheme and/or facility dependent. Nevertheless, numerical results [15], [33], [34] describing the flow properties accurately in the range of 45 ⩽ Re ⩽ 50 are scanty.

In this study we report evolution and properties of flow field around the narrow threshold value of Re_c obtained by numerically solving the Navier–Stokes (N–S) equation. The value of Re_c being highly sensitive, we use a newly developed pure stream function form of the N–S equation for non-rectangular domain where stream function and velocity boundary conditions are sufficient to carry out the simulation. The scheme [35] being compact it further circumvents the need for any unphysical or artificial boundary conditions. As one is required to deal with a single equation having only one dependent variable, the computation is seen to be much more efficient.

A nominally two-dimensional flow past a circular cylinder maintains its symmetry for a long time because of the absence of a preferred direction in the flow field. Note that a circular cylinder has perfect symmetry with identical body curvature everywhere and in this sense the circular cylinder is a special geometry. Further if the computational domain is also considered to be circular then the flow does not have a preferred direction. Thus it is necessary to introduce some artificial perturbation in order to trigger asymmetry into the flow behind the cylinder, especially for Re close to Re_c. Although some form of the immersed interface methods [33], [34] are seen to capture asymmetry in the flow around the threshold regime without introducing perturbation, this may be due to the fact that they unknowingly introduce asymmetry through inherent numerical error. The objective of the present study is threefold:

(i)
To apply a newly developed pure streamfunction based finite difference (FD) scheme to study the evolution of the flow in the threshold region. As mentioned above, simulation of the flow in the threshold region through FD is scanty; long time computation of the flow invariably results in high demand in CPU time. The approach used here eliminates the need to compute pressure and vorticity as a part of the computational process and therefore computationally much economical than the primitive variable and streamfunction–vorticity (ψ–ω) formulations.
(ii)
To explore the effects of the perturbation induced through four different techniques on the eventual behavior of the flow; we are specifically interested in the perturbation effect on the flow induced through the transverse oscillation of the cylinder and, simultaneous blowing and suction for which no other numerical results are seen in the literature.
(iii)
To delve into the flow features in a narrow subrange of the threshold region to confirm the critical Reynolds number.

Initially we compute the flow for two Reynolds numbers which are outside the threshold regime and the flow pattern is well established. We start our simulation with Re = 80 where flow is known to be periodic and several experimental and numerical results are available. For such an Re, which is well beyond the threshold regime, there is no need to artificially trigger asymmetry and hence no perturbation is required. This simulation also shows the efficiency of our scheme in capturing flow phenomena reported by experimentalists. We then proceed to simulate the flow for Re = 40 where the fully developed flow field is perfectly symmetric and no vortex shedding takes place. We compare our results with the benchmark results available in the literature. Eventually, we turn our attention for flow for two Re in the threshold regime, namely, Re = 44 and Re = 50 where the four different perturbation techniques are employed to trigger asymmetry into the flow. We see that the flow for Re = 44 eventually retains its symmetry in all the cases while for Re = 50 the introduced perturbations lead to the periodic vortex shedding. Finally, we narrow down our study to the long time computation of the flow for Re = 46.5 and Re = 47 and ascertain that the critical Re_c ∈ (46.5, 47]. The different characteristics of the flow field are discussed in details for all the Reynolds numbers considered here. Finally we make few observations on the effects of the four perturbation techniques that are being used to trigger asymmetry into the flow.

The paper is divided into the seven sections. In Section 2 we deal with mathematical formulations and discretization procedures, Section 3 concerns with the problem and the numerical issues, in Section 4, we talk about the perturbation technique, the numerical results have been presented in Section 5, the effect of the perturbation techniques in Section 6 and finally the work is summarized in Section 7.

Section snippets

Mathematical formulation

The biharmonic pure stream formulation of the Navier–Stokes equations for unsteady 2D incompressible viscous flows in non-dimensional (x, y) co-ordinate system can be written as: $\frac{\partial}{\partial t} (\nabla^{2} ψ) + [(\nabla^{⊥} ψ) \cdot \nabla] \nabla^{2} ψ = \frac{1}{Re} \nabla^{4} ψ$ where the velocity field is $(u, v) = \nabla^{⊥} ψ = (\frac{\partial ψ}{\partial y}, - \frac{\partial ψ}{\partial x}),$ ψbeing the stream function. We non-dimensionalise the flow variables by considering x = x^∗/a, y = y^∗/a, u = u^∗/U_∞, v = v^∗/U_∞, t = t^∗U_∞/D, Re = DU_∞/ν; u^∗, v^∗ being the dimensional velocities along x^∗−, y^∗− directions respectively and ν is the kinematic

The problem and the numerical issues

We assume the cylinder to be of unit radius placed in an infinite domain. The schematic diagram is presented in Fig. 1. At the far-field R_∞, a potential flow is assumed with uniform free-stream velocity U_∞ = 1. A typical computational grid of size 61 × 101 is shown in Fig. 2. In their work Lee and Tsuei [36] observed that large solution error is likely to propagate near the rotating boundary. We employ a uniform grid spacing along the cross radial direction and nonuniform grid spacing in the radial

Perturbation technique

It is well established that immediately after the fluid motion starts, the flow is irrotational everywhere. But gradually, due to the fluid motion vorticity is generated on the surface of the cylinder and slowly it is transported to the rear stagnation point and flow reversal takes place. After a short period of time for flows with Re ≈ 6, a recirculating zone appears and it continues to grow eventually leading to the flow separation. Here the flow separates from the cylinder and two symmetric

Flows for Re = 80

We begin our computation for Re = 80 where available experimental and the numerical results document convincingly the process of vortex shedding for this Re value represented by the von K $\overset{´}{a}$ rm $\overset{´}{a}$ n vortex street. The reason for starting with a Reynolds number other than a one in the threshold region 43 ⩽ Re ⩽ 50 is that the confidence resulting from accurate validation of our numerical results with the established ones could be carried over to the threshold region where perturbation needs to be

Effect of the perturbation techniques

A close look at the evolution of the lift coefficients for Re = 44, 46.5, 47 and 50 (Fig. 13, Fig. 19, Fig. 22, Fig. 23) reveals that in all the cases asymmetry sets in instantly after the perturbation is introduced. However, its intensity gradually increases as one moves from the first technique to the second one and then to the third as the magnitude of the lift coefficient suggests. The intensity of the fourth perturbation technique is found to be less severe compared to the third one but is

Conclusion

For an impulsively started circular cylinder, above some critical Reynolds number Re_c, asymmetry sets in and as time progresses, vortex shedding occurs naturally in physical situations. However, in the range of 43 ⩽ Re ⩽ 50 in the vicinity of Re_c, called by many as the threshold regime, only a few numerical simulations are available that describe the physics of the flow accurately. Numerical computation is tricky in this range; normal computational procedure sometimes leads to the capture of some

Acknowledgments

Both the authors are thankful to the anonymous reviewers for their suggestions and comments which have helped in furthering the quality of the manuscript.

The first author is thankful to the Department of Science and Technology, Government of India for supporting a part of the work by providing financial support in the form of a project (Project No. SR/S4/MS: 468/07).

The second author is thankful to the University Grants Commission (UGC), Government of India for supporting a part of the work by

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Triggering asymmetry for flow past circular cylinder at low Reynolds numbers

Abstract

Highlights

Introduction

Section snippets

Mathematical formulation

The problem and the numerical issues

Perturbation technique

Flows for Re = 80

Effect of the perturbation techniques

Conclusion

Acknowledgments

J Wind Eng Ind Aerodynam

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

Comput Methods Appl Mech Eng

J Fluid Struct

Comput Methods Appl Mech Eng

Comput Fluids

J Comput Phys

J Comput Phys

J Comput Phys

Vortex dynamics in the cylinder wake

Ann Rev Fluid Mech

Modelling free-stream turbulence based on wind tunnel and flight data for instability studies

Int J Emerg Multidiscip Fluid Sci

Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder

J Fluid Mech

A study of two-dimensional flow past an oscillating cylinder

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Non-parallel stability of a flat-plate boundary layer using the complete Navier–Stokes equations

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Int J Numer Methods Fluids