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LAMINAR VORTEX SHEDDING BEHIND A CIRCULAR CYLINDER

Urban Svensson

Computer-aided Fluid Engineering AB

April 1998

ABSTRACT

The laminar vortex street behind a circular cylinder is simulated with PHOENICS version 3.1. A series of embedded grids and the ASAP (Arbitrary Source Allocation Procedure) method are employed in the problem setup.

All results from the simulations are in general agreement with available experimental evidence. In particular, it is demonstrated that the transition Reynolds number, from a steady flow field to the vortex shedding regime, is correctly predicted and that the Strouhal-Reynolds number relation is in fair agreement with experimental data.

CONTENTS LIST

1. INTRODUCTION
2. MATHEMATICAL FORMULATION
3. PHOENICS SETTINGS
4. RESULTS
5. DISCUSSION AND CONCLUSIONS
6. REFERENCES

1 INTRODUCTION

The classical fluid mechanics problem of flow around a circular cylinder is a popular test case for computational fluid dynamics. In the present paper we will study the laminar vortex shedding behind a cylinder, found in the Reynolds number regime .

The Reynolds number is defined as , where u is the free stream velocity, d the diameter of the cylinder and v the ki

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mage200.gif" WIDTH=56 HEIGHT=18> the flow is steady, i.e. no vortex street is formed and above the vortex street becomes unstable and irregular. The frequency, n, of the vortex shedding is traditionally expressed in non-dimensional form through the Strouhal number . In the Reynolds number range studied in this paper, it is well know that the Strouhal number increases with the Reynolds number. An outline of the situation studied is given in Figure 1.

The objective of this study is to verify that PHOENICS version 3.1, and the new computational techniques embodied in this version, predicts the laminar vortex street behind a circular cylinder correctly. In particular we want to demonstrate that the embedded grid technique and the ASAP-method (Arbitrary Source Allocation Procedure, see the PHOENICS documentation) offer a powerful and accurate approach to the problem studied.

2 MATHEMATICAL FORMULATION

It will be assumed that the flow is transient, two-dimensional and laminar. We will further assume that the fluid is incompressible, with constant density, , and kinematic viscosity, v. The following set of equations then results:

Conservation of mass:

(1)

Conservation of momentum:

(2)

were p is pressure, denotes velocity and a coordinate.

Boundary conditions are given as a specified inflow a , see Figure 1, and a prescribed pressure at the outflow boundary. Zero friction is specified for all other boundaries as well as for the cylinder surface.

1. PHOENICS SETTINGS
2. The PHOENICS settings (q1 and ground) for the problem are given in Appendix A. The q1-file is generated by the VR-interface (Virtual Reality) which is used for the problem specification.

The computational domain (see Figure 1) is m2 (x and y direction); this domain has a grid spacing of 0.005 metres. The first embedded grid has the dimensions m2 and a grid spacing of 0.0025 metres. This grid covers the cylinder and the expected wake-region. The second embedded grid covers the cylinder and the immediate region after the cylinder. This grid has t " WIDTH=73 HEIGHT=18> m2 and a grid spacing of metres. The total number of grid cells is 4740.

The fluid is set to be water a 20° C, which gives a density of 998.2 kg/m3 and a kinematic viscosity of m2/s.

In order to reduce numerical diffusion the harmonic QUICK scheme was used for the momentum equations.

3. RESULTS
4. Results will be presented for the Reynolds number interval . In all simulations the cylinder diameter and fluid viscosity were kept constant and the various Reynolds numbers were thus obtained by changing the free stream velocity.

In Figure 2 the vortex street at is illustrated. The colour filled contours were obtained from the solution of a scalar equation which had a value of 1.0 for the lower half of the inlet and 0.0 for the rest of the inlet.

A detailed view of the flow field behind the cylinder is given in Figure 3. For it is known that two small circulation regions form behind the cylinder (Tritton (1977)) and at these eddies have grown to a length of about two cylinder diameters. At the eddies become unstable and a vortex street is formed. The simulations g HEIGHT=18> and a periodic one at and hence predicts the instability at , without any initialisation or perturbation. At the vortex street is well developed as can be seen in Figure 3.

The transverse velocity component was recorded, at the centreline, at the distance of about eight diameters downstream the cylinder. In Figure 4 we see the result for two Reynolds numbers. As can be seen, a weak periodic flow is established also for .

The transverse velocity recordings are used to determine the frequency, n, of the vortex shedding. With known cylinder diameter and free stream velocity we can then calculate the Strouhal number. In Thompson et al (1994) the experimentally determined relation between the Reynolds and Strouhal numbers is given as:

(3)

where , and . This relation is believed to be accurate to in the Reynolds number interval considered here. In Figure 5 this relation is shown together with present simulations. A fair agreement is found.

5. DISCUSSION AND CONCLUSIONS

All results presented look plausible and are also in general agreement with experimental findings. In particular one can note the following features:

• The wake shape behind the cylinder for Reynolds numbers 10 and 30 (see Figure 3) is according to experimental evidence.
• The transition from a steady flow to the vortex shedding regime is predicted to take place between and ;

also this is in agreement with experimental data.

• The variation of the Strouhal number with the Reynolds number is correctly predicted (see Figure 5).

The last point may need some further comments. As can be seen in Figure 5, the predictions are always above the experimental curve. This may be due to the transverse dimension of the domain, as it is known (Thompson et al (1994)) that the Strouhal number increases if the width decreases. We may thus have a too small width for the situation to be independent of the width. However, in this study no evaluation of the domain-size has been carried out. Further, solutions have been tested for time-step independence but not for grid independence. Some further work is thus required to establish that the solutions are domain-size and grid independent.

The general conclusion from the study is anyway that the laminar vortex shedding behind a circular cylinder is well predicted by PHOENICS 3.1.

1. REFERENCES

POLIS, 1998. The online PHOENICS-documentation. PHOENICS version 3.1.

Thompson M, Hourigan K, Sheridan J, 1994. Three-dimensional instabilities in the wake of a circular cylinder. Paper presented at the International Colloquium on Jets, Wakes and Shear Layer, DBCE, CSIRO, Highett, Victoria Australia.

Tritton D.J, 1977.Physical Fluid Dynamics. Van Nostrand Reinhold.

Figure 1. Schematic outline of situation studied.

Figure 3a. Flow field behind the cylinder .

Figure 3b. Flow field behind the cylinder .

Figure 3c. Flow field behind the cylinder .

Figure 4. Transverse velocity eight cylinder diameters behind the cylinder, at Reynolds number = 50 and 100

¾ experimental data, as given by Thompson et al (1994).

• present simulations.