Conference on

Modelling Fluid Flow

CMFF'12

September 4-7, 2012

The 15th Event of international Conference Series

on Fluid Flow Technologies Held in Budapest

CONFERENCE PROCEEDINGS

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J. Vad

Department of Fluid Mechanics

Budapest University of Technology and Economics

2012

Proceedings of the Conference on Modelling Fluid Flow

Budapest University of Technology and Economics, Hungary 2012

Edited by J. Vad

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Local Organising CommitteeChairman: Dr. J. Vad, Budapest (H)Secretary: Ms. A. Rákai, Budapest (H)Members: Dr. Gy. Paál, Budapest (H)

Prof. Sz. Szabó, Miskolc (H)Prof. T. Lajos, Budapest (H)

Organising InstitutionDepartment of Fluid Mechanics, Budapest University of Technology and Economics

Co-operating OrganisationsDepartment of Hydrodynamic Systems, Budapest University of Technology and EconomicsDepartment of Fluid and Heat Engineering, University of MiskolcScienti�c Society of Mechanical Engineers (Flow Technology Section)Committee of Fluid Mechanics and Thermodynamics of the Hungarian Academy of SciencesThe Japan Society of Mechanical EngineersVisualization Society of Japan

EditorAssociate Prof. János Vad, PhD.Department of Fluid Mechanics, Budapest University of Technology and Economics

Conference OrganisationInternational Scienti�c and Programme Committee (ISPC)Chairman: Prof. D. Thevenin, Magdeburg (D)Honorary Chairman: Univ. Prof. Dr.-Ing. habil. R. Schilling, Munich (D)Review Chairman: Prof. L. Baranyi, Miskolc (H)Members:

Prof. R. S. Abhari, Zürich (CH)Prof. B. J. Boersma, Delft (NL)Prof. A. R. J. Borges, Lisbon (P)Dr. B. P. M. van Esch, Eindhoven (NL)Dr. Á. Fáy, Miskolc (H)Prof. L. Fuchs, Lund (S)Dr. T. Gausz, Budapest (H)Prof. B. J. Geurts, Twente (NL)Prof. V. Goriatchev, Tver (RUS)Prof. R. Grundmann, Dresden (D)Prof. G. Halász, Budapest (H)Prof. H. Jaberg, Graz (A)Dr. L. Kalmár, Miskolc (H)Prof. G. Kosyna, Braunschweig (D)Prof. K. Kozel, Prague (CZ)Dr. G. Kristóf, Budapest (H)Prof. H. Kuhlmann, Vienna (A)Dr. L. Kullmann, Budapest (H)

Prof. J. Kumicak, Kosice (SK)Prof. I. R. Lewis, Newcastle-upon-Tyne (UK)Prof. M. Leschziner, London (UK)Prof. N. C. Markatos, Athens (GR)Prof. M. Nedeljkovic, Belgrade (SRB)Prof. H. Nørstrud, Trondheim (N)Prof. A. Okajima, Kanazawa (J)Prof. F. Rispoli, Rome (I)Prof. R. Rohatynski, Zielona Góra (PL)Prof. W. Schneider, Vienna (A)Dr. E. Shapiro, Cran�eld (UK)Prof. M. Shirakashi, Nagaoka (J)Prof. S. J. Song, Seoul (KR)Prof. R. Susan-Resiga, Timisoara (RO)Prof. T. Takahashi, Nagaoka (J)Dr. T. Verstraete, Sint-Genesius-Rode (B)Prof. T. Weidinger, Budapest (H)

Conference on Modelling Fluid Flow (CMFF’12)

The 15th International Conference on Fluid Flow Technologies

Budapest, Hungary, September 4-7, 2012

NUMERICAL INVESTIGATION OF MECHANICAL ENERGY TRANSFER

BETWEEN THE FLUID AND A CYLINDER OSCILLATING TRANSVERSE TO

THE MAIN STREAM

László BARANYI1, László DARÓCZY2

1 Corresponding Author. Department of Fluid and Heat Engineering, University of Miskolc, Miskolc-Egyetemváros, H-3515, Hungary Tel.: +36 46 565 154, Fax: +36 46 565 471, E-mail: arambl@uni-miskolc.hu 2 University of Miskolc. E-mail: daroczylaszlo@gmail.com

ABSTRACT

This paper deals with the two-dimensional numerical simulation of low-Reynolds number flow past a circular cylinder forced to oscillate transverse to the main stream. The study concentrates on the investigation of mechanical energy transfer E

between the fluid and a transversely oscillating cylinder. When E is negative the fluid works to dampen the cylinder oscillation. When E is positive, work is done on the cylinder and this can be a source of vortex-induced vibration (VIV) for free vibration cases. The object of this paper is to identify subdomains in the parameter domain of Reynolds number, oscillation amplitude and frequency ratio (Re, Ay, f/St0) where the mechanical energy transfer is positive.

Computations are carried out using in-house code based on the finite difference method. Since carrying out the computation even for one point of the three-dimensional domain of (Re, Ay, f) is computationally expensive, the analysed space is limited to Re = 100-180, Ay = 0.1-1.0 and f = (0.6-1.2) St0, where St0 is the dimensionless vortex shedding frequency from a stationary cylinder at the same Reynolds number.

Keywords: CFD, circular cylinder, lift, lock-in,

low-Reynolds number flow, mechanical energy

transfer

NOMENCLATURE

A [-] oscillation amplitude, non- dimensionalised by d

CD [-] drag coefficient, 2D /(�U2d)

CL [-] lift coefficient, 2L /(�U2d)

Cp [-] static pressure coefficient D [-,N/m] dilation or divergence, drag force

per unit length E [-] mechanical energy transfer L [N/m] lift force per unit length

Re [-] Reynolds number, Ud/ vSt [-] Strouhal number, fvd/U

T [-] cycle period, non-dimensionalised by d/U

U [m/s] free stream velocity a0x, a0y [-] cylinder acceleration in x and y

directions, non-dimensionalised by U2

/d

d [m] cylinder diameter, length scale fv [s-1] vortex shedding frequency FR [-] frequency ratio, f/St0

p [-] pressure, non-dimensionalised by �U

2

t [-] time, non-dimensionalised by d/U u,v [-] velocities in x, y directions, non- dimensionalised by U

x,y [-] Cartesian co-ordinates, non- dimensionalised by d�t [-] time step, non-dimensionalised by by d/U� [kg/m3] fluid density � [-] vorticity, �=�v/�x-�u/�y, non- dimensionalised by U/dSubscripts and Superscripts

D drag fb fixed body L lift pb base pressure rms root-mean-square value v vortex x, y components in x and y directions 0 for stationary cylinder

1. INTRODUCTION

Flow past oscillating cylinders has been studied widely because of its practical implications. Structures subjected to wind or underwater flows can oscillate, which can have unwanted consequences. Vibration of stacks, silos or tube bundles of heat exchangers are some examples of

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fluid-structure interaction. If a cylinder is in forced motion, then the vortex shedding and the interaction between the fluid and the cylinder is affected by factors such as cylinder forcing frequency and the amplitude and direction of oscillation, in addition to the Reynolds number (Re) of the flow. Transverse oscillation is most often studied due to its relevance to real life.

When vortices are shed from a bluff body, the induced periodic lift force may lead to large amplitude oscillations, especially when the damping is small and the vortex shedding frequency is near to the natural frequency of the body. As mentioned in [1], since many phenomena in flow-induced vibrations are only weakly dependent on Re, they can be fairly accurately simulated even at relatively low Reynolds numbers, such as those in this study.

There are a huge number of papers available for flow past a circular cylinder oscillating transverse to the free stream. In their very famous paper based on a low-Reynolds number experimental investigation of flow past a cylinder in forced motion, Williamson and Roshko [2] determined a map of vortex shedding modes. Blackburn and Henderson for Re=500 [3] and for Re=200 [4] and Lu and Dalton [5] for Re=185 found some vortex switches in transverse oscillation for frequency ratios over 1. Low-Reynolds number numerical studies, based on finite difference method, found no vortex switches for transverse oscillation either against oscillation amplitude [6] or against frequency ratio below subharmonic forcing [7]. The same computational procedure was applied and its results were compared with those of a finite volume based commercial software package for cylinder oscillation in [8], investigating the effect of oscillation amplitude on the force coefficients and on energy transfer. Neither method found vortex switches. For transverse oscillation, both positive and negative values of mechanical energy transfer were obtained. Very good agreement was found between the results of the two methods.

The aim of this study is a systematic investigation in a relatively wide parameter domain of the effect of three important parameters on the force coefficients and on the mechanical energy transfer between the cylinder and the fluid for low Reynolds numbers. These three parameters are the Reynolds number Re, oscillation amplitude Ay, and frequency ratio f/St0. To the best knowledge of the authors such a systematic investigation in the phase space of (Re, Ay, f/St0) is not available in the literature.

2. COMPUTATIONS

Computations were carried out using an in-house code based on the finite difference method.

2.1. Computational method

A non-inertial system fixed to the cylinder is used to compute 2D low-Reynolds number unsteady flow around a circular cylinder placed in a uniform stream and forced to oscillate transversely to the uniform free stream flow. The governing equations are the Navier-Stokes equations for incompressible constant-property Newtonian fluid written in the non-inertial system fixed to the accelerating cylinder, the equation of continuity and a Poisson equation for pressure. All quantities in these equations are non-dimensional. The physical domain consists of two concentric circles with dimensionless radii of R1 and R2, where R1

represents the cylinder surface and R2 the far field. No-slip boundary condition is used for the

velocity; a Neumann-type boundary condition is applied for pressure on the cylinder surface and potential flow is assumed in the far field. Boundary-fitted co-ordinates are used to accurately impose the boundary conditions. The physical domain is mapped into a rectangular shape computational domain (Fig. 1). With this the mesh of logarithmically spaced elements on the physical plane are transformed to an equidistant mesh on the computational plane. The transformed governing equations with boundary conditions are solved by a finite difference method [9]. Space derivatives are approximated by fourth-order central difference except for the convective terms for which a third-order modified upwind scheme is used. The Poisson equation for pressure is solved by the successive over-relaxation (SOR) method. The equation of motion is integrated explicitly and the continuity equation was satisfied at every time step. For further details see [9].

Figure 1. Physical and computational domains

The 2D code developed by the first author has been tested extensively against experimental and computational results (see [9] for details) for both stationary and oscillating cylinders.

The non-dimensional displacement of the centre of the transversely oscillated cylinder is described by

)2sin()(0 tfAty yy π−= , (1)

270

where t is the dimensionless time, Ay and fy = f are the dimensionless transverse amplitude and frequency of cylinder oscillation, respectively. In this study only locked-in cases are considered, when the frequency of vortex shedding synchronizes with that of the cylinder oscillation.

2.2. Computational setup

In this study we investigated the behaviour of flow past a circular cylinder placed in a uniform stream with its axis perpendicular to the velocity of the main flow. The cylinder is oscillated mechanically transverse to the uniform free stream.Here we concentrate on domains where lock-in occurs and where the mechanical energy transfer is positive. We also investigate the force coefficients within a wide parameter domain (or phase space) of (Re, Ay, f/St0). Here St0 is the dimensionless vortex shedding frequency or Strouhal number for a stationary cylinder at that Re. Force coefficients investigated under lock-in conditions are the time mean (TM) and root-mean-square values (rms) of lift (CL), of drag (CD), and of base pressure (Cpb).

Throughout this paper the lift and drag coefficients used do not contain the inertial forces originated from the non-inertial system fixed to the accelerating cylinder. Coefficients obtained by removing the inertial forces are often termed ‘fixed body’ coefficients [5]. The relationship between the two sets of coefficients can be written as [10]

xDfbDyLfbL aCCaCC 00 2,

2

ππ+=+= , (2)

where subscript ‘fb’ refers to the fixed body (understood in an inertial system fixed to the stationary cylinder). In Eq. (2) a0x and a0y denote the acceleration of the cylinder in x and y directions, respectively. Since the inertial terms are T-periodic functions, their time-mean values vanish, resulting in identical TM values for lift and drag in the inertial and non-inertial systems. Naturally the rms value of CL will be different in the two systems, while CDrms remains the same, as a0x=0 for transverse cylinder oscillation (see Eq. (2)).

2.3. Phase space (Re, Ay, FR) and simulation settings

A large number of computations were carried out in order to obtain an appropriate resolution of the three-dimensional phase space (Re, Ay, f/St0). Computations were carried out for all possible combinations of Re=100, 120, 140, 160 and 180; Ay=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0; and frequency ratio f/St0=0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2. The frequency ratio f/St0 will be denoted by FR for the sake of simplicity. The St0 values from [11] are St0= 0.1644, 0.1735, 0.1806, 0.1864, 0.1913 for Re=100, 120, 140, 160, 180, respectively. Besides

these points, computations were also carried out for a further 36 points in order to improve the precision of the analysed three dimensional surfaces, making a total of 386 computational points.

All other settings were the same for the simulations. A mesh of 361x292 (peripheral x radial) was used with the ratio of radii of R2/R1=160 and the dimensionless time step �t=0.0005 was used to make the first order temporal discretization scheme accurate. Comparisons with second-order Runge-Kutta discretization results showed that at this small time step the results agree well with each other [12]. Simulations were run for the non-dimensional time of t=600-1000.

Results from the in-house Fortran code were post-processed. The computed results were the mechanical energy transfer E, the time-mean of CL, CD, -Cpb and the root-mean-square of CL,fb, CD,fb

only for cases where periodic solution were achieved, i.e. for the lock-in region. Values were collected in an Excel file, automatically exported to an in-house database handler program, and selected cases were exported to a spreadsheet for post-processing with Matlab, where the original set of cases was interpolated to a much finer mesh using the command GRIDDATAN and the command ISOSURFACE was used to visualize surfaces at constant values of the analyzed coefficients [13].

3. RESULTS

Computational results are presented with a focus on mechanical energy transfer between the fluid and the cylinder within the locked-in domain in the investigated phase space.

3.1. Lock-in domain

Figure 2. Lock-in region

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Out of the 386 analyzed cases lock-in occurred in 252 cases. The lock-in domain displayed in three-dimensional (3D) form in Fig. 2 was plotted only approximately, with the isosurface being drawn for the value of 0.5 (where 1 represented cases in the lock-in region, and 0 otherwise). Naturally, due to computing limitations, the interpolated surfaces are not precise representations of the lock-in–not lock-in boundary, but still these rough boundaries are basically sufficient for our needs. The refinement of these boundaries could be a further step in our research. A more careful determination of the lock-in boundaries for a transversely oscillating cylinder at Re=180 is found e.g., in [14].

3.2. Mechanical energy transfer

The mechanical energy transfer E between fluid and a transversely oscillated cylinder is determined when the fluid is already periodic. E was defined in [3] as

(3)

where v0y is the cylinder velocity, t and T are the dimensionless time and time period, respectively. Eis positive when work is done on the cylinder and negative when work is done on the fluid by the cylinder.

Figure 3 shows the mechanical energy transfer E against frequency ratio FR for different amplitude values Ay within the lock-in domain at Re=180. As can be seen in the figure, E is negative for the largest part of the phase space. For large amplitude values E is negative and its absolute value increases with FR. Small positive values can be seen only for FR between 0.8 and 1.0 and amplitudes below 0.6. Curves for smaller Re numbers are similar.

Figure 3. Mechanical energy transfer; Re=180

Figure 4 shows part of the same diagram for positive E values. E increases with increasing FR

and rather surprisingly the effect of oscillation amplitude reveals no clear tendency: the lowest and second lowest amplitudes belong to 0.5 and 0.1, respectively. Figures 5 and 6 show the

corresponding curves for Re=140 and 100, respectively. A careful look at the three figures reveals that the tendencies are basically the same;this was true for all Reynolds numbers investigated.

Figure 4. Positive mechanical energy transfer;

Re=180

Figure 5. Positive mechanical energy transfer;

Re=140

Figure 6. Positive mechanical energy transfer;

Re=100

Figure 7 shows E against oscillation amplitude Ay for different frequency ratios FR within the lock-in domain at Re=100. As can be seen, E is negative again for the largest part of the phase space. For large amplitude values E is negative and its absolute value increases with FR. Small positive values can be seen only for amplitudes below 0.6. Curves for larger Re numbers are similar. Interestingly, there is

( ) ( ) , d 0y

0

L ttvtCE

T

�=

272

a special point in the figure which practically all curves go through (Ay�0.65 and E� -0.6).

Figure 7. Mechanical energy transfer; Re=100

Figure 8 shows the part of Fig. 7 where E

values are positive. It can be seen that E has a maximum for each curve at around Ay =0.3 and that E increases with increasing FR (as can also be seen in Figs. 4-6). The largest computed E value in the investigated domain can be seen in Fig. 8; its value is 0.4794 and belongs to Re=100, FR=1.1 and Ay=0.3. The same point can also be seen in Fig. 5. Figures 9 and 10 show the corresponding curves for Re=140 and 180, respectively. It can be seen that Figs. 8-10 are very similar to each other, and that the Reynolds number effect is not strong at all, supporting the findings in [1]. Curves belonging to the same frequency ratio shift to somewhat higher Evalues for larger Re values, forming a clear trend, unlike that seen for Ay in Figs. 4-6. Figures belonging to all Re values investigated are similar to each other.

Figure 8. Positive mechanical energy transfer;

Re=100

The clear tendency can be deducted from the figures that larger amplitudes usually result in larger energy transfer from the cylinder to the fluid. However, at lower amplitudes this behaviour

becomes less obvious, and a domain of positive energy transfer can be identified.

Figure 9. Positive mechanical energy transfer;

Re=140

Figure 10. Positive mechanical energy transfer;

Re=180

The mechanical energy transfer results introduced in Figs. 3-10 are also represented by isosurfaces in the 3D phase space shown in Fig. 11. Isosurfaces of E = -7, -4, -1.2, -0.5, 0, 0.1, 0.25 and 0.4 are shown in the figure, as well as the border of the lock-in region. Figure 12 shows the same E

isosurfaces from a different view. Interestingly, the region of phase space where

the mechanical energy transfer is positive and relatively large is close to the upper border of the lock-in region. This behaviour may contribute to vortex-induced vibrations.

273

Figure 11. Isosurfaces of mechanical energy

transfer

Figure 12. Isosurfaces of mechanical energy

transfer

3.3. Time-mean of drag coefficient

The TM of drag in the lock-in domain increases quite regularly when plotted against oscillation amplitude, as can be seen in the example for an intermediate Re value of 140, shown in Fig. 13. The curves belonging to different frequency ratios show increasing drag with increasing FR. Figure 14 displays the TM of drag against the frequency ratio for different oscillation amplitude values. With the exception of the curve for Ay=0.8 at higher FR

values, once again the drag tends to increase with increasing Ay and FR. The effect of Reynolds number is rather weak, as can be seen in the isosurfaces of drag shown in Fig. 15.

Figure 13. Drag coefficient against oscillation

amplitude, Re=140

Figure 14. Drag coefficient against frequency

ratio, Re=140

Figure 15. Isosurfaces of time-mean of drag

3.4. Fixed-body lift

The TM of lift was 0 for all cases computed, in accordance with earlier findings [6]. Figure 16 shows the rms of fixed-body lift CLfb,rms against

274

frequency ratio for different oscillation amplitudevalues. It can be seen from the figure that CLfb,rms increases gradually with increasing FR and Ay.

Figure 16. Rms of fixed-body lift against

frequency ratio; Re=140

Figure 17. Rms of fixed-body lift against

oscillation amplitude; Re=140

Figure 18. Isosurfaces of rms of fixed-body lift

CL,fb,rms

Figure 17 shows CLfb,rms against Ay for different FR values. This representation of the function CLfb,rms(Ay, FR) basically conveys the same message: an increase in CLfb,rms with both Ay and FR. One exception is found for FR=1.1 in the range of Ay=0.6-0.7, where the curve is almost constant.

The isosurfaces of rms values of the fixed-body lift coefficient are plotted in Fig. 18, showing also the effect of Re, which appears to be minimal over much of the phase space.

3.4. Base pressure coefficient

The isosurfaces of base pressure coefficient Cpb

are plotted in Fig. 19. Similarly to previous results, -Cpb (minus sign used for positive values) increases with the increase in the frequency and amplitude, while Reynolds number has only a weak effect on the coefficient over much of the phase space.

Figure 19. Isosurfaces of the base pressure

coefficient

3.5. Vortex structure

Throughout the phase space vortex structure was identified as 2S, i.e., a single vortex is shed from each side in one cycle. A typical example can be seen in Fig. 20, where the grey colour indicates negative (clockwise rotation) vorticity values, while the black is positive (anti-clockwise).

Figure 20. Vortex contours at Re=100, FR=0.8,

Ay=0.5

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4. CONCLUSIONS

Flow properties were investigated in the three-dimensional phase space (Re, Ay, f/St0) under lock-in condition. The main emphasis was on the mechanical energy transfer E between the fluid and a cylinder mechanically oscillated transverse to the free stream.

In the largest part of the phase space, E was found to be negative, meaning that energy is extracted from the cylinder. Large-magnitude negative values were found at large oscillation amplitude Ay values. However, at Ay < 0.6, and at frequency ratios f/St0 = 0.8-1.1, positive E values were found at all Reynolds numbers investigated (Re=100-180); this is near the upper boundary of the lock-in region.

The time-mean of drag and base pressure coefficients and the rms of the lift coefficient increased with both oscillation amplitude and frequency ratio, supporting previous findings.

For all parameters, including E, the effect of Reynolds number was found to be weak across most of the phase space.

Vortex structure was found to be 2S (one vortex is shed from the cylinder in each half cycle).

Further investigation might include the more accurate determination of lock-in boundaries and a more detailed investigation of positive mechanical energy transfer near the boundary of lock-in.

ACKNOWLEDGEMENTS

The support provided by the Hungarian Scientific Research Fund under contract No. OTKA K 76085 is gratefully acknowledged. The work was carried out as part of the TÁMOP-4.2.1.B-10/2/KONV-2010-0001 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union, co-financed by the European Social Fund.

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