A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder

Abstract

The flow around an infinitely long circular cylinder at Reynolds numbers between $5⩽\mathrm{Re}⩽250$ is investigated numerically by means of a spectral element method. Careful studies of the effect of resolution and extension of the computational domain on drag and lift forces, base-pressure coefficient and Strouhal number are performed in the laminar, two-dimensional regime. Asymptotic results are obtained by increasing the size of the computational domain to several thousands of cylinder diameters. It is shown that, in contrast to the Strouhal number, the force coefficients and the base-pressure coefficient are strongly dependent on the resolution and even more on the size of the computational domain. For both the asymptotic and finite domains, the Reynolds number relationships are compared to numerical and experimental data from the literature. The results accurately reproduce experimental findings and explain deviations of former numerical investigations. Our database is useful both for validation of numerical codes and measurement verifications where the separation of physical features and effects of experimental arrangements are frequently an open question.

Introduction

The flow around an unconfined circular cylinder of infinite length is a classical problem of fluid dynamics. Despite the simple geometry, the cylinder flow shows many phenomena in a relatively small range of Reynolds numbers that also occur in more complex cases. Examples are flow separation, the primary instability with development of the unsteady von Kármán vortex street, transition to three-dimensionality with interaction of ‘mode A’ and ‘mode B’ instabilities, the occurrence of turbulence in the wake, and the Kelvin–Helmholtz instability of the separated shear layer. The cylinder flow has been investigated for more than one century, starting with fundamental work of Strouhal (1878) on the dependency between frequency of vortex shedding, free-stream velocity, and cylinder diameter. Other experiments, including those by Wieselsberger (1921), Roshko (1954), Tritton (1959), and Berger (1964), revealed qualitative and quantitative features of the flow at low Reynolds numbers. Numerical investigations, starting from steady and two-dimensional, aimed at validating the measured results. Extensive reviews can be found in Braza et al. (1986) and Lange (1997).

A strong revival of interest on flow around a cylinder could be noted at the end of last century in the form of extensive experimental and numerical investigations. In particular, better experimental techniques allowed a deeper insight into the structure of the cylinder wake. Influences from the cylinder ends, length-to-diameter ratio, free-stream turbulence, blockage, vibrations, and three-dimensional effects were identified to cause the large scatter in earlier experimental results (Gerich and Eckelmann, 1982, Williamson, 1989, Williamson, 1992, Norberg, 1994).

Numerical works were first targeted at steady cylinder flows to calculate the drag, pressure and velocity (resp. vorticity) distributions, as well as the separation lengths. Later, with the expansion to unsteady simulations, time-dependent quantities including the Strouhal number were determined. Mostly, only a few selected Reynolds numbers were investigated. Extensive studies on resolution as well as blockage effects, i.e. the ratio of domain extension and cylinder diameter, were not carried out, since computer capacity was strongly limited in the past. The main interest was frequently a validation and demonstration of new computer codes and techniques and not the accurate computation of flow quantities. From that, the large scatter in the available database can be explained.

Examples for analyzes of the influence of computational domain extensions on the drag at low Reynolds numbers can be found in Fornberg (1980), Behr et al. (1995), Anagnostopoulos et al. (1996), Lange (1997), and Kravchenko et al. (1999). Fornberg ($\mathrm{Re}=20$), Behr et al. ($\mathrm{Re}=100$), Anagnostopoulos et al. ($\mathrm{Re}=106$) as well as Kravchenko et al. ($\mathrm{Re}=100$) concentrated on one Reynolds number, whereas Lange investigated the flow around a circular cylinder at very low Reynolds numbers ($\mathrm{Re}<10$). On the other hand, Rosenfeld (1994) and Yang et al. (1994) varied the number of grid points but retained a fixed size domain. Persillon and Braza (1998) varied, both the number of grid points and the domain extension; however, this was accomplished simultaneously.

Numerous experimental as well as numerical works dealt with the Reynolds number dependency of flow quantities like the base-pressure coefficient and the Strouhal number. Newer examples on the experimental side can be found in Williamson, 1989, Williamson, 1992 or Norberg (1994). On the numerical side, Reynolds number dependencies of the Strouhal number and the base-pressure coefficient, that can easily be used to compare with experimental results, are computed together with the drag and lift forces that cannot be measured accurately. Examples can be found in Franke et al. (1990) and Yang et al. (1994), where only steady results were given. Henderson (1995) published two-dimensional results for the variation of drag components and the base-pressure coefficient with Reynolds number in the form of curve-fits. However, because of the small domain extension, results are not grid independent. A similar work was published by Park et al. (1998), where the flow quantities at Reynolds numbers between 2 and 160 are given. Blockage effects are diminished compared to the work of Henderson because of a larger domain extension. Since the domain-independence has been investigated by looking at the Strouhal number in most numerical works, considerable errors can result in flow quantities like lift, drag, and especially the base-pressure coefficient, because the Strouhal number is least effected by the domain size (see Section 4).

In this paper, two-dimensional numerical simulations of the flow around a circular cylinder at Reynolds numbers between 5 and 250 using a spectral element method are presented. Details of the code are shown in Section 2. One of the qualities of this global discretization scheme is the simplicity of grid refinement, e.g. improvement of resolution, by increasing the order of polynomial basis within the spectral elements. Resolution as well as extension of the computational domain are varied systematically and separately to receive domain-independent flow quantities. Because the independence from the domain extension seems to be an arbitrary assumption of the different authors, asymptotic solutions were searched by using extremely large domain sizes. This was accomplished for the steady regime as shown in Section 3, as well as for the unsteady cylinder flow in Section 4. The effect of domain extension is shown for the drag, base-pressure coefficient, lift, and Strouhal number (the two last-mentioned only in the unsteady case). The domain dependency of flow quantities is given as polynomial approximations and compared to data from literature.

Finally, two sizes of the computational domain are used to determine the Reynolds number relationship of the flow quantities in the steady and unsteady regime, as shown in Section 5. The results of the present work are compared to experimental as well numerical data. In Section 6 the Strouhal–Reynolds number relationship is investigated. For it, a new formulation of curve fitting is proposed and compared to traditional polynomial approximations.