Transactions on Modelling and Simulation vol 16, 1997 WIT Press, ISSN X

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1 Numerical and experimental investigation of oscillating flow around a circular cylinder P. Anagnostopoulos*, G. Iliadis* & S. Kuhtz^ * University of Thessaloniki, Department of Civil Engineering, Thessaloniki 54006, Greece anagnostopou@olymp. ccf. auth. gr * University of Basilicata, Department of Environmental Physics and Engineering, Potenza, Italy silvana@hp712.unibas.it Abstract The objective of this paper is the numerical and experimental investigation of oscillating flow around a circular cylinder. The finite element method was used for the solution, whereas the experiments were conducted in a U-shaped tube. The computation and experiment were performed at Keulegan-Carpenter numbers extending up to 10 and frequency parameters equal to 34 and 53. From the traces of the in-line force exerted on the cylinder the force coefficients were evaluated. Good agreement was found between computational results and experimental measurements. 1 Introduction Oscillatory flow around a circular cylinder is a flow phenomenon which has proved to be a challenging area for research, since it provides a simplified tool for the investigation of flow around a cylinder immersed in a wave environment. The phenomenon is controlled by two dimensionless numbers, the Keulegan-Carpenter number (KC) and the Reynolds number (Re). KC is defined as D

2 342 Computer Methods and Experimental Measurements where U^ is the maximum flow velocity, T the period of flow oscillation and D the cylinder diameter. The Reynolds number is given from *.W (2, where v is the kinematic viscosity of the fluid. The ratio of these two numbers is known as the frequency parameter, /J, and is defined as P=_^=^ (3) KC v7 Several experimental investigations of the phenomenon have been conducted throughout a wide range of Reynolds and Keulegan-Carpenter numbers. Experiments at low KC have revealed that the flow can be classified into a number of different flow regimes governed mainly by KC and dependent also on Re (Bearman et al. [1], Williamson [2], Sarpkaya [3]). At KC < 1 the flow remains attached, symmetrical and two-dimensional. As KC increases, the flow separates from the cylinder and remains symmetrical until KC reaches a critical threshold, whose value depends on the frequency parameter. If this critical KC is exceeded the flow becomes asymmetric and various vortex shedding flow regimes are observed, at which the number of vortices shed in each oscillation cycle increases with the Keulegan-Carpenter number. Although several investigators have proposed ranges of KC for particular types of vortex-shedding, more than one mode of shedding is possible even when KC and 0 are fixed as reported by Bearman et al. [1], and the flow may switch between different modes. Williamson [2] observed the transverse street (7<KC<13), the single pair (13 <KC<15), the double pair (15 <KC< 24) and the three pair (24<KC<32) regimes for KC increasing from 7 to 32. Tatsuno & Bearman [4] conducted a flow visualization study for a large number of KC and 0 pairs, where KC was extending up to 15 and 0 was lower than 160. Their investigation revealed eight flow regimes within this KC and 0 range, and the great significance of 0 in the form of the flow pattern for constant KC. It is interesting to note that some of the regimes detected by Tatsuno & Bearman do not seem to occur at higher values of 0 parameter. The attached oscillatory flow at very small KC was studied analytically by Stokes [5]. Wang [6] obtained a solution of the Navier-Stokes equations valid for KC <# 1 and j8 > 1, and proposed formulae for the drag and inertia coefficients. With the increase of efficiency of digital computers, the numerical solution of the phenomenon in two-dimensions became feasible. Baba & Miyata [7], Murashige et al. [8], Wang & Dalton [9] and Justesen [10] presented finite difference solutions. Skomedal et al. [11], Graham & Djahansouzi [12] and Smith & Stansby [13] used discrete vortex methods for the calculation of two-dimensional flow. On the other hand Anagnostopoulos et al. [14] employed a finite element scheme. In the present study the finite element method was employed for the

3 Computer Methods and Experimental Measurements 343 solution of viscous oscillatory flow around a circular cylinder, and the experiments were conducted in a U-shaped tube. The computation and experiment were performed at Keulegan-Carpenter numbers extending up to 10 and frequency parameters equal to 34 and 53. From the traces of the in-line force exerted on the cylinder the force coefficients were evaluated and are presented in relevant diagrams. 2 The Numerical Solution The mathematical model of the problem consists of the Navier-Stokes equations, in the formulation where the stream function 9 and the vorticity f are the field variables. Considering the values of # and f at two successive time steps n and n + 1, the governing equations become 6% 8% (5) The pressure distribution throughout the flow field can be calculated from the Poisson's equation - } (6) dy dx dx dy) where p is the fluid density and u and v the two components of the fluid velocity, calculated from the values of the stream function. Applying the Galerkin's method to equations (4), (5) and (6) for each element and assembling for the whole continuum the following equations are obtained (9) where [KJ, [IC,], [KJ, [KJ and [KJ are square matrices, {RJ, %}, {R,}, {RJ and {R$} are column matrices, whereas f represents the derivative of f with respect to time. The time-dependent free stream velocity U(t) of the oscillatory flow is defined as

4 344 Computer Methods and Experimental Measurements (10) where U^ is the maximum velocity and T is the oscillation period. On the inflow boundaries the stream function varies linearly according to the relationship (11) where y is the ordinate of the point considered, while is maintained constant along the upper and lower boundaries. The vorticity was assumed equal to zero throughout the outer boundaries, while along a no-slip boundary can be calculated from the formula which is easily derived from the relationship between the vorticity and circulation over an element, ft, ft and ft are the vorticity values at the three nodes of the element (e) considered, while the two components of the fluid velocity along the element boundary are denoted as u and v, and A is the area of the element. The integration in equation (12) along the perimeter of the element was conducted interpolating linearly u and v from the nodal velocities. The solution algorithm consists of the following steps: a) Evaluation of the stream function at time t+at from eq. (7). The values of vorticities are those calculated in the previous time step, or, in particular, the initial conditions. b) The vorticity values at the no-slip boundary are corrected from eq. (12). c) The vorticities at time t+at are calculated from eq. (8). 3 The Experimental Arrangement The oscillatory flow around a circular cylinder was studied experimentally using a special U-shaped tube depicted in figure 1, with a working section of 0.6m x 0.6m and 1.5m long. After a series of preliminary experiments with pure water as the working fluid it was realized, that the best way to achieve low values of the 0 parameter compatible with acceptably large hydrodynamic forces to permit accurate measurement, was to modify the viscosity of the fluid to a higher value than that of water. In order to reduce the Reynolds number and the 0 parameter, glycol was added to the water in the tube at specified proportion, resulting to the increase of the kinematic viscosity v of the mixture by up to approximately 6 times. For the measurement of the unsteady hydrodynamic forces exerted on the test cylinder of 3cm diameter spanning the tube walls strain gauges were employed, as described by Kuhtz et al. [15].

5 Computer Methods and Experimental Measurements -BLOWER DEPTH PROBE 1.22 WINDOW OF WORKING SECTION Figure 1. The U-tube. 4 Results Computations and experiments of oscillating flow around a circular cylinder were conducted at 0 = 34 and 13=53, for various KG extending up to 10. At low KC the flow remains symmetrical. As the KC increases, the flow becomes asymmetric. The asymmetry has as effect the generation of a transverse force acting on the cylinder. If the KC is increased still further, the shedding of vortices becomes irregular. Each pair of KC and (3 yields a different flow pattern, which is not periodic at consecutive flow cycles. The non-periodicity of the flow pattern has as effect the generation of intermittent traces of the hydrodynamic forces exerted on the cylinder. For 0 = 34 the flow becomes asymmetric at KC=5.5 and aperiodic at KC = 8, while for 0=53 the asymmetric flow is first observed at KC=4.5 and the aperiodic at KC=7. The time history of the in-line force normalized by 0.5pU^D as computed and measured experimentally for KC=4.76 and 0 = 34 (Re =162) is shown in figure 2. The trace of the oscillating stream velocity is also depicted. Figure 2 reveals that the amplitude of the in-line force measured experimentally is slightly higher than the computed. The free stream velocity lags behind the inline force by approximately 90. Morison et al. [16] proposed that the total in-line force per unit length of a cylinder of diameter D in an oscillating flow can be expressed as the sum of two components as

6 Computer Methods and Experimental Measurements U (13) where p is the fluid density and CD and CM the drag and inertia coefficients. The Fourier averaged coefficients CD and CM for a cylinder immersed in an oscillating flow, as defined by equation (13), are given by [17] 2%. ~ sm6 c =1 (14) **D 271 COS0 (15) The rms value of the total in-line force coefficient is defined as C- (rms) = dt (16) i The drag and inertia coefficients together with the rms coefficient of the total in-line force as functions of KC for 0 = 34 and 0 = 53 are depicted in figures 3 and 4. Figures 3 and 4 reveal very good agreement of the computed CD with Wang's [6] analytical results for KC< 1. As KC is increased over 1, the numerical CD values become slightly higher than those predicted by Wang's theory, most likely due to flow separation effects. It is interesting to note the agreement between the computed CD values with those measured experimentally at KC<8, and the discrepancy for KO8 when 0 equals 53. KC=4.76, (3=34 (comp.) (exper.) Figure 2. Traces of the computed and measured in-line force.

7 Computer Methods and Experimental Measurements Experiment Wang's analysis [6] Experiment Wang's analysis [6] r\l/ Experiment Inviscid theory KU Figure 3. Force coefficients and rms value of the in-line force coefficient as functions of KC, for 0 = 34.

8 Computer Methods and Experimental Measurements " - Experiment Wang's analysis [6] Experiment Wang's analysis [6] KG Experiment Inviscid theory Figure 4. Force coefficients and rms value of the in-line force coefficient as functions of KC, for /3 = 53.

9 Computer Methods and Experimental Measurements 349 The computed CM values at the lower KC regime (KC< 1) are slightly higher than those predicted by Wang's analysis in both diagrams, while for KC ranging between 1 and 2 the computed CM values become closer to the theoretical. For KC higher than 2 the computed CM values depart from those predicted by Wang's analysis. In the KC range between 2 and 8 small discrepancies are observed between the computed CM values and the experimental, which increase drastically as KC becomes greater than 8 at 0=53. Figure 4 dictates that the discrepancy between computed and measured force coefficients occurs predominantly in the asymmetric flow regime. As resulted from the present computation and confirmed also by Tatsuno & Bearman's [4] visual study, the flow for 0=53 and KO8 is aperiodic, which causes an intermittent time history of the in-line force. Apparently, the non-periodicity of the force trace reflects on the Fourier averaged values of the drag and inertia coefficients, in spite of the similarity of the rms values of the total inline force. Figures 3 and 4 show very good agreement between computed and measured F%(rms) values. The F^(rms) values computed herein are slightly higher than the prediction of inviscid theory, according to which Cpx(rms)=2*V/KC for small KC values. An interesting result is that the rms value of the total in-line force coefficient decreases with increasing KC and is almost independent of /3. Conclusions The oscillating flow around a circular cylinder at Keulegan-Carpenter numbers extending up to 10 and frequency parameters equal to 34 and 53 was studied numerically and experimentally. The computational results are in good agreement with experimental evidence at similar conditions, which confirms the accuracy of the solution. Small discrepancies between computed and measured force coefficients are observed when the flow is aperiodic. The nonperiodic character of flow in the aperiodic regime reflects on the traces of the hydrodynamic forces providing explanation of the discrepancy. Acknowledgement The present project was supported financially by the Science Programme of the European Community, Contract No SCT-CT References 1. Bearman, P.W., Graham, J.M.R., Naylor, P. & Obasaju, E.D. The role of vortices in oscillatory flow about bluff bodies, in Proc. Intl Symp. on Hydrodynamics in Ocean Engng, The Norwegian Inst. of Technology, pp , Trondheim, Norway, 1981.

10 350 Computer Methods and Experimental Measurements 2. Williamson, C.H.K. Sinusoidal flow relative to circular cylinders, /. Fluid Mech., 1985, 155, Sarpkaya, T. Forces on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers, /. Fluid Mech., 1986, 165, Tatsuno, M. & Bearman, P.W. A visual study of the flow around an oscillating cylinder at low Keulegan-Carpenter number and low Stokes numbers, /. Fluid Mech., 1990, 211, Stokes, G.G. On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Phil Soc., 1851, 9, Wang, C.-Y. On the high frequency oscillating viscous flows, /. Fluid Mech., 1968, 32, Baba, N., & Miyata, H. Higher-order accurate difference solutions of vortex generation from a circular cylinder in an oscillatory flow, J. Comput. Phys., 1987, 69, Murashige, S., Hinatsu, M. & Kinoshita, T. Direct calculations of the Navier-Stokes equations for forces acting on a cylinder in oscillatory flow, in Proc. Eighth Intl Conf. Offshore Mech. and Arctic Engng, Vol. 2, pp , The Hague, Wang, X. & Dalton, C. Oscillating flow past a rigid circular cylinder: A finite difference calculation, ASMEJ. Fluids Engng, 1991, 113, Justesen, P. A numerical study of oscillating flow around a circular cylinder, /. Fluid Mech., 1991, 222, Skomedal, N.G., Vada, T. & Sortland, B. Viscous forces on one and two circular cylinders in planar oscillatory flow, Appl. Ocean Res., 1989, 11, Graham, J.M.R. & Djahansouzi, B. Hydrodynamic damping of structural elements, in Proc. Eighth Intl Conf. Offshore Mech. and Arctic Engng, Vol. 2, pp , The Hague, Smith, P.A. & Stansby, P.K. Viscous oscillatory flows around cylindrical bodies at low Keulegan-Carpenter numbers using the vortex method, J. Fluids and Struct., 1991, 5, Anagnostopoulos, P., Iliadis, G. & Rasoul, J. Numerical solution of oscillatory flow around a circular cylinder at low Reynolds and Keulegan- Carpenter numbers, in Proc. Eight Intl Conf. Finite Elements in Fluids, pp , Barcelona, Kuhtz, S., Bearman, P.W. & Graham, J.M.R. Problems encountered in measuring forces on immersed bodies, Experimental Techniques, in press. 16. Morison, J.R., O'Brien, M.P., Jonshon, J.W. & Schaff, S.A. The force exerted by surface waves on piles, Petrol. Trans., 1950, 189, Sarpkaya, T. Vortex shedding and resistance in harmonic flow about smooth and rough circular cylinders at high Reynolds numbers, Tech. Rep. No. NPS-59SL76021, 1976, Naval Postgrad. School, Monterey, CA.