A numerical study of flow past a cylinder with cross flow and inline oscillation

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Computers & Fluids 35 (2006) 393–415

A numerical study of flow past a cylinder with crossflow and inline oscillation

M.R.H. Nobari *, H. Naderan

Department of Mechanical Engineering, AmirKabir University of Technology, 424 Hafaz Ave.,

P.O. Box 15875-4413, Tehran, Iran

Received 7 March 2004; received in revised form 31 August 2004; accepted 11 February 2005Available online 19 July 2005

Abstract

Two-dimensional fluid flow around an oscillating circular cylinder is studied numerically at different val-ues of oscillation frequency and amplitude. A novel finite element method which uses discretization alongthe characteristic line is used for simulation. The solver is coupled to a mesh movement scheme using theArbitrary Lagrangian–Eulerian (ALE) formulation to account for body motion in the flow field. Two casesof cylinder motion have been studied, cross flow and inline oscillation. In both cases, occurrence of lock onis investigated and the bounds of the lock on region are determined. A comparison of the numerical resultswith the experimental data indicates that 2D simulation is valid up to Re = 300. Beyond that, 3D effectsappear. By using flow visualization, effect of a cylinder oscillation on the flow field and wake patternhas been studied. Also, variation of the mean drag coefficient against the oscillation parameters is discussed.The numerical results are in good agreement with the experimental data available in the literature.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

The problem of flow around oscillating bodies has many attractions, regarding the physics ofthe flow field as well as practical engineering applications. The nonlinear interaction of the body

0045-7930/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2005.02.004

* Corresponding author.E-mail address: [email protected] (M.R.H. Nobari).

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394 M.R.H. Nobari, H. Naderan / Computers & Fluids 35 (2006) 393–415

motion and the incident flow, gives rise to a number of interesting phenomena, such as vortex lockon, hysteresis and bifurcation, alteration of the vortex shedding patterns, chaotic behaviour of thewake and many more. From the engineering point of view, this subject is of main interest to thefields of wind engineering, offshore exploration, marine structures and nuclear and conventionalpower plants.

To analyse this problem, CFD is an invaluable tool. In this paper, a novel finite element ap-proach is used to investigate the problem. Historically, the finite element methods, in contrastto its great success in the field of mechanics of solids, have received less interest in the CFD com-munity. Some of this is due to the problems of the convection dominated flows, which requiresome sort of stabilization. Two major stabilization methods have been used extensively in the lit-erature; first the Petrov–Galerkin formulation with upwind test functions, and second, the leastsquare formulation. Although these methods have been used successfully, they have a highcomputational cost. The Petrov–Galerkin formulation produces system of equations with non-symmetrical mass matrix, which is much more costly to solve, compared to system of equationswith a symmetric coefficient matrix. In the least-square formulation, some additional nonlin-ear terms are involved. These facts limit the use of the mentioned approaches for transientproblems.

In the present work, a semi-implicit projection method with characteristic-Galerkin formula-tion, proposed by Zienkiewicz is used to simulate flow around a cylinder with inline and cross flowoscillations. This new scheme which is fitted to a moving unstructured grid for the first time inthis study, offers the advantage of no limit in the geometry, number of bodies and the body mo-tion. First, flow over a fixed cylinder is studied for different Reynolds numbers in the range of 40–1000, and hence the upper limit for the Reynolds number to consider 2D simulation with goodaccuracy is determined. Subsequently, the numerical results are provided at Reynolds numbersof 100 and 300 for the oscillating cylinder cases. The effects of amplitude and frequency of oscil-lations on the wake flow patterns and hydrodynamics forces are studied in detail to evaluate thelock on region and the mean drag coefficients. Also, the accuracy of the code implemented isinvestigated by comparing the computations with the experimental data and other numericalresults available.

2. Numerical method

Using indicial notation, the convection–diffusion equation for a general scalar variable / with asource term is written as follows

o/ot

þ ujo/oxj

¼ o

oxjko/oxj

� �� s ð1Þ

The above equation is not self-adjoint and for this reason, the use of standard Galerkin formu-lation does not give an optimal result. By defining dx0i ¼ dxi � ui dt, it transforms into Eq. (2)which is self-adjoint

o

ot/ðx0i; tÞ ¼

o

ox0jk

o

ox0j/ðx0i; tÞ

!� sðx0i; tÞ ð2Þ

u tδ = ∆

nφ

1nφ +

n

x δ−φ

x

t

Fig. 1. Nodes and values used to discretize Eq. (2) in the characteristic direction for 1D case.

M.R.H. Nobari, H. Naderan / Computers & Fluids 35 (2006) 393–415 395

Now, applying the standard Galerkin formulation leads to optimal result. If k and s are zero, thesolution to Eq. (2) will simply be /ðx0i; tÞ ¼ const. This means that in the presence of k and s, thesolution of a diffusion equation gives the desired result, but it must be done along the x0i ¼ constlines, i.e. the nodes must be moved along these lines.

Temporal discretization of Eq. (2), along the characteristic line, using a central difference (seeFig. 1 for the one dimensional case), gives

1

Dtð/nþ1 � /jnx�dÞ �

1

2

o

ox0jko/ox0j

!� s

" #nþ1

þ 1

2

o

ox0jko/ox0j

!� s

" #nx�d

ð3Þ

where x, is the location of a nodal point at n + 1st step and d is the displacement vector giving thefoot of the characteristics at level n. To alleviate the need for moving nodes, following Zienkiewiczand Taylor [13] a local Taylor series expansion around /n

x is written. Since a transient analysis isdesired, we do not hesitate to use explicit time stepping.

/jnx�d ¼ /n � djo/n

oxjþOðDt2Þ ð4Þ

o

oxiko/oxi

� ��nx�d

¼ o

oxiko/oxi

� �n

� djo

oxj

o

oxiko/oxi

� �n� þOðDt2Þ

sjnx�d ¼ sn � djosn

oxjþOðDt2Þ

In Eq. (4), di ¼ �uiDt and �ui are the ith component of displacement vector and average (defined inEq. (6)) velocity, respectively. By substituting the terms in Eq. (3) with their corresponding valuesfrom Eq. (4), Eq. (5) is reached

/nþ1 � /n ¼ �Dt �ujo/n

oxj� o

oxiko/oxi

� �nþ1=2

þ snþ1=2

" #� Dt2

2�uj

o

oxj

o

oxiko/oxi

� �� s

�n

ð5Þ

At this stage, by using different approximations for �ui, various stabilization terms will appear.The choice in the current work is Eq. (6), proposed by Zienkiewicz and Codina [14]. Substitutingit into Eq. (5) and neglecting higher order terms, Eq. (7) is obtained


�uj ¼ 12ðunþ1

j þ ujjnx�dÞ ð6Þ

/nþ1 � /n ¼ �Dt unþ1=2j

o/n

oxj� o

oxiko/oxi

� �nþ1=2

þ snþ1=2

" #

þ Dt2

2ujoukoxj

o/oxk

� �n

þ ðujukÞnþ1=2 o2/n

oxjoxk� unþ1=2

jo

oxj

o

oxiko/oxi

� �n� þ unþ1=2

josn

oxj

�ð7Þ

A further approximation of ( )n+1/2 = ( )n + O(Dt) gives the final equation for convection–dif-fusion transport of property / in a fully explicit form as Eq. (8).

/nþ1 � /n ¼ �Dt ujo/oxj

� o

oxiko/oxi

� �þ s

� nþ Dt2

2unk

o

oxkujo/oxj

� o

oxiko/oxi

� �þ s

� nð8Þ

Eq. (8) is valid for any scalar property, including the components of the momentum vector. So itwill be used for the spatial discretization of the momentum equation.

Navier–Stokes equations in the incompressible form with zero body force are written asfollows:

oUi

otþ uj

oUj

oxj¼ � op

oxiþ m

o2Ui

oxjoxjð9Þ

oUi

oxi¼ 0 ð10Þ

In Eqs. (9) and (10), Ui is the ith component of the momentum vector. Discretizing Eq. (9) alongthe characteristic line and treating the pressure gradient term as a part of the momentum sourceterm, we have

Unþ1i � Un

i ¼ Dt �ujoUi

oxjþ m

o2Ui

oxjoxj� opoxi

� nþ Dt2

2uk

o

oxkujoUi

oxj� opoxi

� nð11Þ

Applying the projection method by removing the pressure terms and splitting the above equa-tion into two complementary equations with the aid of auxiliary velocity U*, we get

U i � Un

i ¼ Dt �ujoUi

oxjþ m

o2Ui

oxjoxj

� nþ Dt2

2uk

o

oxkujoUi

oxj

� nð12Þ

Unþ1i � U

i ¼ �Dtopoxi

n

� Dt2

2uk

o2pnþ1

oxkoxið13Þ

Enforcing the incompressibility condition on Un+1 from Eq. (10), a Poisson equation is foundfor calculation of the pressure field.

Dto2pnþ1

oxioxi¼ oU

i

oxið14Þ


By solving Eqs. (12), (14) and (13) sequentially, the flow field at time step n + 1 is calculated fromknown values of u and p at step n.

Approximating velocity and pressure variables with standard shape functions Nu and Np,applying the Galerkin weighted residual formulation to Eq. (12)–(14) and integrating by partsthe viscous, stabilizing and Laplacian terms we have

MðeU� eUn

Þ ¼ �Dt CueU þ Ks~u� f � DtðKu

eU � fsÞh in

ð15Þ

DtH~pnþ1 ¼ GeU� fp ð16Þ

MðeUnþ1� eUn

Þ ¼ �Dt GT~pnþ1 þ Dt2P~pn

� �ð17Þ

In which denotes nodal values and the matrices and vectors are defined in the appendix. Theabove formulation and splitting of momentum equation, circumvents the Babuska–Brezzi condi-tion. Zienkiewicz and Taylor [13] have shown that for every degree of velocity and pressure shapefunctions, the BB condition is satisfied. In the present work, first order shape functions for bothvelocity and pressure are employed.

Since it is desired to solve the above equations in a time varying domain, they are modified andreformulated using the ALE scheme. By using ALE formulation, it appears that the general con-vection–diffusion equation for / which is

o/ot

þ uio/oxi

¼ s ð18Þ

will turn into

o/ot

þ cio/oxi

¼ s ð19Þ

where ci = ui � wi is the convection velocity relative to the grid and wi is the ith component of thegrid velocity vector. It should be noticed that in the above equation, x is the coordinate systemattached to the grid. The above in mind, the solution sequence involves applying boundary dis-placements, solving mesh movement equations (calculation of mesh velocities), computing relativeconvective velocities, applying flow boundary conditions, and solving modified ALE flow equa-tions (calculation of fluid velocities). Sarrate et al. [15] and Koobus and Farhat [16] discuss theALE formulation in details.

For the purpose of mesh movement, the torsion spring analogy, proposed by Farhat et al. [17]has been used. The reason for choosing this method was its robustness in handling large boundarymovements, compared to other mesh movement schemes like Laplacian operator [18,19], bihar-monic operator [20] and lineal spring analogy [21]. The addition of torsion spring prevents theelement from collapsing. As a result, large boundary movements can be performed without specialprecautions. Farhat el al. [17] gives an excellent description of the method and compares its per-formance with other grid movement techniques.


3. Previous works

Research on the subject of flow and moving body interaction began when Strouhal (1878) pub-lished the results of his experiments about Aeolian tones. In these experiments, Strouhal mounteda piece of thin wire on a frame and the assembly was rotated around its axis parallel to thewire. He noted that the frequency of sound generated was dependent on wire diameter andvelocity and was independent of its length and tension. Furthermore, when the frequency ofgenerated sound coincided on the natural frequency of wire, the intensity of sound was amplified[1].

Later Gerrard [2] studied the mechanism of vortex shedding from circular cylinders and pre-sented a model to describe it. According to this model, the separated shear layer feeds the vortexbehind the cylinder with circulation until it draws the opposite shear layer into itself. The oppositesigned vorticity cuts the circulation feed and the vortex detaches. Gerrard has emphasized twolength scales in his model. The first one is the length of the vortex formation region which corre-sponds to the distance of a point in the wake where vorticity is maximized. The second, the lengthof shear layer diffusion in the vortex formation region. According to the experimental results, areduction in the length of vortex formation region or the shear layer diffusion length will increasevortex shedding frequency. Using this, Gerrard explains why the Strouhal number is nearly con-stant across a wide range of Reynolds number.

Koopman [1] investigated the effect of cross flow oscillation of cylinder on vortex shedding fre-quency at low Reynolds number and determined conditions which shedding frequency coincidedwith oscillation frequency. By varying the oscillation frequency, Koopman obtained the region inwhich lock on occurred. He observed that lock on changed the angle of vortices with respect to thecylinder axis. For a fixed cylinder, the axis of shed vortices is slant relative to the cylinder axis.When cylinder oscillates in the lock on region, this angle disappears and shed vortices are parallelto the cylinder. Koopman mentioned that this configuration is stable and does not change in thepresence of external disturbance. Also there is a 10% diameter threshold for oscillation amplitudeunder which oscillation has no effect on the flow field.

Griffin and Ramberg [3] studied the effect of longitudinal oscillations on the vortex sheddingfrom circular cylinder. They observed that lock on happened in this case similar to the cross flowcase but the lock on region is around twice the Strouhal frequency. Also the amplitude thresholdwas lower and frequency span of the region was wider than the cross flow case.

By using visualization techniques, Griffin and Ramberg obtained interesting information aboutvortex structure in the wake. When the cylinder oscillates in the frequency range of F = 1.7–2.2, aregular vortex street is formed like the fixed cylinder, where F is the ratio of the oscillation to theStrouhal frequency. Longitudinal spacing of vortices had an inverse relation with oscillation fre-quency. Besides this arrangement, there was another stable arrangement in which two vorticeswith opposite signs are shed simultaneously, in each cycle. Downstream interaction of vortices re-sulted in a wake with two rows of vortices, one on each side of the cylinder, containing pairs ofopposite signed vortices.

Williamson and Roshko [4] have studied the effect of amplitude of oscillation on the wake. Atlow values of amplitude, the vortex street is similar to the fixed cylinder case with a single vortexon each side. By increasing the amplitude, vortex shedding pattern changes to a form with apair of vortices on one side and a single vortex on the other side. The former pattern was termed


as 2S and the later as P + S. These wake patterns have been also observed by Griffin and Ramberg[3].

Zdravkovich [5] reproduced the results of experiments performed by Den Hartog (1934) andobserved that in the vicinity of Strouhal frequency, there is a jump in the phase angle betweenvortex shedding and cylinder displacement. This phase change leads to reversing of energy ex-change direction among body and fluid. With reference to Bearman and Davies [6] the after bodyshape has a great influence on the magnitude of phase change. For circular and triangular crosssections, phase change is large while for square and rectangle is small.

Barbi et al. [7] examined oscillatory flow around a fixed cylinder. The question was whether thiscase leads to vortex lock on or not. According to the previous works (Hartfield and Morkovin,1973) the answer was negative but the results of Barbi et al. showed the opposite. One possiblereason for the fault in Hartfield and Morkovin results is a too small amplitude and frequencyof oscillation. Barbi et al. performed their experiments in two different laboratories with two dif-ferent fluids, yet they observed occurrence of lock on. They also noticed that outside the lock onregion, the frequency of vortex shedding gradually moved towards the oscillation frequency. Theobserved vortex pattern was consistent with results of Griffin and Ramberg [3].

One of the most comprehensive studies conducted on the subject of flow over a cylinder withcross flow oscillation is the work of Ongoren and Rockwell [8,9] which covered a range ofReynolds number from 580 to 1300. Apart from validation of the previous results they noticedthat there exist two stable states of vortex lock on. In a frequency of about half the Strouhal fre-quency, vortex shedding synchronizes itself to the cylinder motion always detaching from oneside. By increasing the frequency to near the Strouhal frequency, the classical state of lock onis retained.

Karniadakis and Triantafyllou [10] used the spectral element method to simulate the flowacross cylinder. Their calculations were in two parts. In the first part, they simulated flow arounda fixed cylinder and by the use of linear stability theory, showed that the vortex street formation isdue to the global instability in the flow field. Also they did not observed any sign of chaos. While,Tritton (1959) observed in his experiments the discontinuity in the diagram of Strouhal numbervariations against Reynolds number and attributed these cases to the modes of chaos in the flow.Results of Karniadakis and Triantafyllou [10] denied the existence of chaotic modes and certifiedthe view point of Van ata and Gharib (1987) which related the discontinuities to unwanted oscil-lation of cylinder during the experiment.

In the second part, they simulated the flow around a fixed cylinder with an oscillatory pointdisturbance upstream. Results of this simulation showed that outside but near the upper boundof the lock on region, the effect of external disturbance and the natural shedding mechanism werein competition with each other and this resulted in a chaotic state of vortex shedding. By movingaway from this region, the effect of disturbance diminishes and vortex shedding frequency returnsto Strouhal frequency.

Hall and Griffin [11] simulated the oscillatory flow around a fixed cylinder using the spectralelement method. Unlike the work of Karniadakis and Triantafyllou [10] which modeled the effectof local disturbance, Hall and Griffin applied a sinusoidal horizontal velocity component on theincoming flow and by varying the frequency they observed the occurrence of lock on. The boundsof the lock on region they obtained agreed with the results of Barbi et al. [7] and Karniadakis andTriantafyllou [10].


Blackburn and Henderson [12] studied the phase shift in 2D flow around a cross flow oscillat-ing cylinder using the spectral element method. They limited their study to one value of Red = 500and A = 0.25 but with different values of F, where A is the ratio of oscillation amplitude to thecylinder diameter. Their results showed occurrence of hysteresis and bifurcation in the vicinityof Strouhal frequency, which was observed in the experiments of Ongoren and Rockwell [9].Also they observed transition from 2S to P + S pattern as mentioned by Williamson and Roshko[4].

They related the phase shift and change of vortex shedding pattern to the competition of twodifferent vorticity generation mechanisms, where one of them is the pressure gradient and theother is the oscillatory motion of the cylinder.

4. Grid independence test

Here, the grid independence test is carried out with respect to the time-averaged drag coeffi-cients and the Strouhal number. For the grid independence study, three different meshes, as shownin Fig. 2, are deployed for simulation at a Reynolds number of 300 for the fixed cylinder and at aReynolds number of 100 for the cross flow oscillating cylinder in lock on region. The number ofnodes and elements for these meshes are given in Table 1.

The comparisons of the numerical results obtained on these three meshes are listed in Table 2.As it is evident, no significant variations are shown in the time-averaged parameters for the dif-ferent meshes in two cases.

For all the numerical results will be reported in the next section, mesh 1 with 3244 nodes and6359 triangular elements is employed. Accordingly, this mesh is used to study the effect of timestep size on the above mentioned time-averaged parameters. To do so, the computations are car-ried out for three different Courant numbers at the Reynolds number of 300 for the fixed cylinder.The outcome is shown in Table 3, indicating a good accuracy of the code and its well conservativeproperty concerning both different grid sizes and time step values.

5. Results and discussion

Using the solution method described in the previous section, a computer program was devel-oped with C++ programming language. In the preprocessor section, this program accepts an ini-tial unstructured triangular mesh and a boundary condition file.

The code was verified by simulating flow around a fixed circular cylinder. The geometry and themesh used are shown in Fig. 2. It consists of 3244 nodes and 6359 triangular elements. To capturethe steep gradients near the surface, layers of stretched elements are wrapped around the cylinder.

Fig. 3 shows the calculated Strouhal number as a function of Reynolds number. Vortex shed-ding frequency is calculated by spectral analysis of the lift coefficient time variation.

The lift and drag coefficient history curves for Red = 500 is plotted in Fig. 4. Comparison ofexperimental and numerical values of drag coefficient is given in Fig. 5. The difference with exper-imental results for Red > 300 is due to the strengthening of three dimensional effects in the realflow.

Fig. 2. The three meshes used in simulating flow around a fixed cylinder. (a) Mesh 1, (b) Mesh 2, (c) Mesh 3, (d)stretched elements near the surface for Mesh 1.


In order to study the flow around an oscillating body, the two cases of cross flow and inlineoscillation of a circular cylinder is simulated. The computational grid used for these two cases,is the same as that for the stationary case, Fig. 2.

Table 1Number of nodes and elements of three meshes

Mesh No. nodes No. elements

1 3244 63592 5037 99343 13,526 26,890

Table 2Grid independence test

Mesh type Fixed cylinder at Re = 300 Cross flow oscillating cylinder at Re = 100

Mean drag coefficient Strouhal number Mean drag coefficient Strouhal number

1 1.370 0.202 1.95 0.192 1.366 0.207 1.93 0.1873 1.365 0.207 1.87 0.185

Table 3Time step size independence test for mesh 1 at Re = 300

Courant number Mean drag coefficient Strouhal number

0.7 1.370 0.2020.5 1.368 0.2030.3 1.365 0.202

Fig. 3. Variation of calculated Strouhal number (S) vs. Reynolds number (Re) for a fixed cylinder. (h) Computational,(n) experimental results from Roshko [22].


Fig. 4. Variation of lift (Cl) and drag (Cd) coefficients vs. t ¼ tU1d .

Fig. 5. Drag coefficient vs. Reynolds number. (h) Computational, (n) experimental results from Weisselberger,adapted from Schlichting [23].


5.1. Cross flow oscillations

To simulate oscillatory motion of the cylinder, the coordinates of center of the cylinder is de-fined as:

Table 4Values of amplitudes and frequencies used for simulation, cross flow oscillation, Red = 100

Parameter Value

A 0.2, 0.4F 0.6, 0.8, 0.9, 1.05, 1.3

Fig. 6of Ko


x ¼ const

y ¼ Y sinð2pftÞð20Þ

where Y and f are the amplitude and frequency of oscillation, respectively. Simulation is per-formed at Red = 100 for values of A = Y/d and F = f/fs presented in Table 4, where fs is the Strou-hal frequency.

These points are shown in Fig. 6 accompanied with the results of Koopman [1], showingbounds of the lock on region. Filled points indicate occurrence of lock on and hollow ones showthat lock on has not happened.

Occurrence of lock on is discovered by spectral analysis of temporal variations of the lift coef-ficient. Fig. 7 shows the lift coefficients and power spectra for A = 0.2, F = 0.6 and F = 1.05.

When lock on does not occur, two peaks can be seen in the diagram (Fig. 7c), which correspondto the oscillation frequency (peak 1) and the Strouhal frequency (peak 2). In the case lock on oc-curs, the peak at Strouhal frequency disappears (Fig. 7d). Near the upper bound of the lock onregion, the peak has been widened, indicating presence of chaos (Fig. 8). This situation has beenstudied by Karniadakis and Triantafyllou [10].

. Simulation points and bounds of the lock on region. (h) Lock on does not occur, (j) lock on occurs, (+) resultsopman [1] for Red = 100. The continuous lines are drawn according to Koopman results.

Fig. 7. Temporal variation (a, b) and power spectra (c, d) of the lift coefficient. Cross flow oscillation at Red = 100,A = 0.2. (a) and (c) F = 0.6, (b) and (d) F = 1.05.

Fig. 8. Power spectrum of the lift coefficient for cross flow oscillating cylinder. Red = 100, A = 0.2, F = 1.3.


Fig. 9. Contours of vorticity for cross flow oscillating cylinder at Red = 100 and A = 0.2 for two consecutive cycles ofcylinder motion. Left column (a–c) F = 0.8. Right column (d–f) F = 0.9.


The effect of lock-on on the flow field and vortex shedding can be clearly seen by plotting vor-ticity contours. Fig. 9 shows this for two consecutive cycles of cylinder motion with A = 0.2. Theleft column is the case when lock on does not happen (F = 0.8). The intermittency in shed vorticesis a result of presence of two frequencies. The right column is for the case where lock on occurs(F = 0.9). As it is evident, in contrast to the left column, vortices are shed in a regular way which isa result of synchronization of vortex shedding with cylinder motion.

In the lock on region, changing the amplitude of oscillation has a dramatic effect on the vortexwake pattern. Fig. 10 shows a snapshot of the flow field at Red = 300 for a fixed value of F = 0.9and three different values of A = 0.2,0.4,0.65.

At low values of A (Fig. 10a), the wake is similar to the Von-Karman vortex street with alter-native shedding of single vortices from the cylinder. As the amplitude of oscillation grows, thelateral spacing of the vortices shrinks (Fig. 10b), until the wake appears as a row of opposite signvortices. Further increase in the amplitude, leads to the formation of secondary vortices andchanges the wake pattern (Fig 10c). In this arrangement, a pair of opposite sign vortices travelsat one side and a single one at the other side of the wake. Griffin and Ramberg [3] have observedthis transition in their experiments, and more recently Williamson and Roshko [4] have named theformer pattern as 2S and the later as P + S.

The behavior of the mean drag coefficient is interesting in the lock on region. As shown inFig. 11, far below the Strouhal frequency, the drag coefficient is nearly the same as the fixedcylinder and the slight increase is the result of higher incident velocity due to cylinder motion.As the frequency of oscillation approaches the Strouhal frequency, the drag coefficient rises withits peak at natural shedding frequency. Crossing the upper bound of the lock on region, the drag

Fig. 10. Contours of vorticity for cross flow oscillating cylinder at Red = 300.

Fig. 11. Variation of the mean drag coefficient with oscillation parameters, Red = 100.


Table 5Mean drag coefficient for fixed and oscillating cylinder at Red = 100

Pressure drag coeff. Viscous drag coeff. Total drag coeff.

Fixed cylinder 1.105 0.605 1.72Oscillating cylinder (A = 0.4, F = 1.05) 1.55 0.76 2.31


coefficient starts to decrease and sets back to its normal value. Table 5 compares the values of vis-cous, pressure and total drag coefficients at Red = 100 for a fixed and an oscillating cylinder. Theoscillating cylinder has an amplitude and frequency of oscillation equal to 0.4d and 1.05fsrespectively.

While the 10% increase in viscous drag is the result of increase in the relative velocity of fluidand body, the major part of increase is due to the pressure drag and stems from the vortex syn-chronization phenomenon. In Fig. 12, contours of pressure and streamlines for one cycle of vortexdetachment have been shown for both cases mentioned in Table 5. Streamlines for the right col-umn which is for the oscillatory body case, are drawn in the reference frame attached to thecylinder.

As can be seen, when the cylinder changes direction at its upper extreme point of oscillationpath (Fig. 12h), a vortex is shed from the upper half surface of the cylinder and the boundary layeradheres to the lower half surface of the cylinder, but at the lower extreme point of oscillation path(Fig. 12j), the reverse phenomenon happens. This will result in spreading of the low pressure re-gion (lightly shaded region) to the back of cylinder, reducing the back pressure (Fig. 12h and k).This process happens in reverse for the next half cycle.

The time averaged result, indicates a more pronounced pressure reduction, compared to thefixed cylinder case in the left column, and an increase in pressure drag.

5.2. Inline oscillation

Inline oscillation of the cylinder is defined as:

x ¼ X sinð2pftÞy ¼ const

ð21Þ

The flow is simulated for values of A = X/d and F = f/fs shown in Table 6 at Red = 300.In Fig. 13, the points of simulation and bounds of the lock on region, adapted from the results

of Griffin and Ramberg [3] are plotted. Identification of lock on is performed by spectral analysisof temporal lift coefficient similar to the cross flow case. Although this diagram is similar to theone in Fig. 6, two major differences are noticeable. Firstly, the frequency of occurrence of lock onwhich is twice the Strouhal frequency. Secondly, the lower level of external disturbance (A) whichcauses lock on.

Fig. 14 shows the power spectrum of the lift coefficient for cases A = 0.05, F = 1.4 and F = 2.3.When lock on does not take place (F = 1.4), the dominant frequency of the flow is located atStrouhal frequency (Fig. 14a and c). To provide the lock on region for inline oscillation case,the oscillation frequency of the cylinder is set about twice as much as the Strouhal frequency

Fig. 12. Streamlines and pressure contours for fixed (a–f) and oscillating (g–l) cylinder. Oscillation parameters are thesame as that mentioned in Table 5.


Table 6Values of amplitudes and frequencies used for simulation, inline oscillation, Red = 300

Parameter Value

A 0.05, 0.1F 1.4, 1.6, 1.8, 2.0, 2.1, 2.3, 2.5

Fig. 13. Lock on (shaded) and nonlock on states for inline oscillation of cylinder at Red = 300. (+) Results of Griffinand Ramberg [3], Red = 190 and · results of Hall and Griffin [11], Red = 200.


(F = 2.3), as shown in Fig. 14b and d. The power spectrum analysis indicates that the dominantfrequency for inline oscillation case occurs at half of the oscillation frequency of the cylinderwhich is equal to 1.15 times the Strouhal frequency (Fig. 14d).

Vortex synchronization can be seen by visualizing the flow field. Fig. 15 illustrates contours ofvorticity for two consecutive cycles of oscillation in two cases, F = 1.4, A = 0.05 and F = 1.8,A = 0.1. Outside lock on region, similar to the cross flow case which is shown in the left columnof Fig. 9, vortices are shed intermittently. However, regular shedding takes place when lock onhappens (right column).

By increasing the amplitude of oscillation, a change of wake pattern takes place. Fig. 16 showsthe wake for F = 2.0 and A = 0.05,0.1,0.15 at Red = 300. As the amplitude increases, lateral spac-ing of vortices decrease (Fig 16a) until they become collinear. Further increase will result in breakdown of vortices and appearance of secondary vortices (Fig 16b and c).

Variation of the mean drag coefficient is plotted against oscillation parameters in Fig. 17. Thegeneral trend of the curve is similar to the cross flow case, suggesting that the same phenomenahappen here. What should be noted is the higher value of relative velocity which causes the vis-cous drag to increase.

Fig. 14. Temporal variation (a, b) and power spectra (c, d) of the lift coefficient. Inline oscillation at Red = 300,A = 0.05. (a) and (c) F = 1.4, (b) and (d) F = 2.0.

Fig. 15. Contours of vorticity for inline oscillating cylinder at Red = 300 for two consecutive cycles of cylinder motion.Left column (a–c) A = 0.05, F = 1.4. Right column (d–f) A = 0.1, F = 1.8.


Fig. 16. Contours of vorticity for inline oscillating cylinder at Red = 300.

Fig. 17. Variation of the mean drag coefficient with oscillation parameters, Red = 300.


6. Summary and conclusions

In this paper, we have numerically studied the flow across an oscillating cylinder, using a newcharacteristic-Galerkin finite element method. The advantages of this method are needless to the


staggered meshes, high quality capturing of sharp gradients, generation of symmetric matrices,and one code for both compressible and incompressible flows [14]. To the best of our knowl-edge, there is no literature that applies this method to the moving boundary problems. The solverwas coupled, using the ALE formulation, with a grid movement technique which takes advan-tage of torsion spring analogy. With this, large boundary movements are handled easily andefficiently.

Implementing the solver and grid movement method, using the C++ programming language,simulation of flow around any number of arbitrary shaped 2D objects with independent arbitrarymotion became possible. Flow around circular cylinder with forced cross flow and inline oscilla-tory motion was simulated with this program at various values of amplitude and frequency ofoscillation.

Even though the code we developed is able to simulate the flow field at high Reynolds numbers,both laminar and turbulent flows, we analysed our results up to Re = 300. The reason for thisstems from the comparison of 2D results with the experimental data. It indicates that beyondRe = 300 three dimensional effects appear, and 2D simulation starts deviating from the real phys-ics of the problem.

For the cross flow case, vortex lock on was observed with frequency of oscillation near theStrouhal frequency. On the other hand, lock on took place near twice the Strouhal frequencyfor the inline oscillation case. For both cases several points were simulated and bounds of the lockon region were roughly obtained and compared with available experimental results. The effect oflock-on on the wake was investigated and secondary vortex shedding due to the increased ampli-tude was observed. In the case where lock on did not take place the intermittency of the vortexstreet was illustrated. Also, variation of the mean drag coefficient vs. the oscillation parameterswas studied and an explanation was given for its behaviour.

Appendix A

The various matrices resulting from the Galerkin weak formulation are defined as follows

M1ij ¼

ZXNi

uNju dX; C1

ij ¼Z

XNi

uukoNj

u

oxkdX;

Hij ¼Z

X

oNiu

oxk

oNju

oxkdX; K1

ij ¼ � 1

2

ZXuluk

oNiu

oxl

oNju

oxkdX;

f mi ¼

ZCNi

uloumoxj

njdC �Z

XNi

uqgmdX

� n; f sm

i ¼ � 1

2

ZXuloNi

u

oxlqgmdX

� n;

Gmij ¼

ZX

oNip

oxmNj

udX; Pmij ¼

ZXuk

oNiu

oxk

oNjp

oxmdX;

f pi ¼

ZCNi

p U j � Dt

opnþ1

oxj

� �njdC;

i; j ¼ 1; 2; . . . ; number of nodes per element

l;m ¼ 1; 2


The above formulae, define the scalar components of the submatrices M1, C1, K1, H, Gm, Pm andthe subvectors fm and fs

m. Using them, the actual stiffness, mass and stabilising matrices and force

vectors can be written as below. Each zero is zero matrix of appropriate size.

M ¼ M1 0

0 M1

" #Cu ¼

C1 0

0 C1

" #

Ks ¼ lH 0

0 H

� Ku ¼

K1 0

0 K1

" #

G ¼ ½G1 G2 P ¼ P1

P2

" #

f ¼ f1

f2

( )fs ¼ fs

1

fs2

( )

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A numerical study of flow past a cylinder with cross flow and inline oscillation

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