Numerical simulation of viscous flow past a circular cylinder subject to a circular motion

European Journal of Mechanics B/Fluids 49 (2015) 121–136

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Numerical simulation of viscous flow past a circular cylinder subject to a circular motion Qasem M. Al-Mdallal ∗ Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, Abu Dhabi, United Arab Emirates

highlights • • • •

The problem of a cylinder undergoing controlled circular motions in a uniform stream has been numerically investigated. The phenomenon of sudden jump in mean drag coefficient and RMS drag coefficient within these lock-in regions is observed. Different locked-on vortex shedding modes are observed when either the radius of the circular path or the forcing frequency increases. The occurrence of multiple lock-in regions is demonstrated.

article

info

Article history: Received 10 February 2014 Received in revised form 26 August 2014 Accepted 27 August 2014 Available online 8 September 2014 Keywords: Vortex formation Locked-in Quasi-locked-in Circular path Hydrodynamic forces

abstract This paper presents a computational study of the two-dimensional flow of a viscous incompressible fluid past a circular cylinder subject to a circular motion. We used the Fourier spectral analysis with finite difference approximations to integrate the two-dimensional unsteady Navier–Stokes equations which govern the motion. The numerical simulations reveal the occurrence of multiple lock-in regions. In addition, an increase and inflation in the values of the mean and RMS drag coefficients within these lock-in regions, is observed. Comparisons with previous numerical and experimental results verify the accuracy and the validity of the present study. © 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction The investigation of the flow around oscillating bluff bodies has been a primary task in the design of numerous cylindrical structures. The key feature of interest for flow around oscillating bluff bodies is the periodic vortex shedding phenomenon and how it can lead to a better understanding of structure failures. Hence, due to the practical significance of this type of flow, numerous fundamental experimental, analytical and numerical studies have been performed over the years; see, for example, the recent works of Sarpkaya [1], Bearman [2], Williamson and Govardhan [3], and Sumer and Fredsøe [4]. A large number of research papers have been published over the past decades on the problem of a cylinder vibrating transversely to a uniform flow; see, by way of example not exhaustive enumeration, Bishop and Hassan [5], Williamson and Roshko [6], Lu and Dalton [7], Guilmineau and Queutey [8], Sarpkaya [1],

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Tel.: +971 3 7136397; fax: +971 3 7671291. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.euromechflu.2014.08.008 0997-7546/© 2014 Elsevier Masson SAS. All rights reserved.

Williamson and Govardhan [9] and Barrero-Gil and FernandezArroyo [10]. In contrast, the literature reveals, relatively, less papers focusing on the case of a cylinder vibrating streamwise with an incident uniform flow; see for example the works of Griffin and Ramberg [11], Ongoren and Rockwell [12], Cetiner and Rockwell [13], Guilmineau and Queutey [8], Al-Mdallal et al. [14], Góis and Souza [15], Marzouk and Nayfeh [16], Suthon and Dalton [17] and Leontini et al. [18,19]. A major aim of these studies was to explore the phenomenon of vortex lock-in or ‘‘synchronization’’ (where the shedding frequency is dictated by the vibration frequency) via, partly, the analysis of hydrodynamics forces acting on the cylinder. The pioneering work of Ongoren and Rockwell [12] demonstrated, in the case of a cylinder performing forced streamwise oscillation, the existence of two lock-in regions when f ≈ 2f0 and f ≈ 3f0 , which was also confirmed by Al-Mdallal [20]. Later on, Cetiner and Rockwell [13] proved that, in the case of a cylinder performing forced streamwise oscillation, lock-in region occurs when f ≈ f0 at higher amplitudes of oscillation. For comprehensive summary of findings on a forced streamwise or transverse oscillations of a circular cylinder in a uniform flow, the reader is referred to the works of Williamson and Govardhan [9] and Leontini et al. [18,19].

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It is clear that, combining streamwise and transverse motions could generate different types of paths such as circular, elliptical and figure-eight-type. To the best knowledge of the author, only very limited number of investigations have been carried out on the problem of a circular cylinder undergoing a circular, elliptical or figure-eight-type motion in a uniform stream [21–29]. Karanth et al. [21] presented preliminary computational results on the case of an orbital motion of a circular cylinder in a uniform stream at a Reynolds number of 100; the streamwise cylinder displacement was set to 20% of the diameter of the cylinder (d) whereas transverse cylinder displacement was chosen in the range from 20% to 80% of d. Here Reynolds number is defined as Re = Ud/ν , where ν is the kinematic viscosity of the fluid and U is the uniform flow velocity. Their study demonstrated that the lock-in range of frequency increases as transverse cylinder displacement increases. Later in 2007, Didier and Borges [23] studied the problem of a uniform stream past a circular cylinder undergoing a circular motion. Their investigation is performed at a fixed Reynolds number, Re = 300, amplitudes of streamwise and transverse velocities of the cylinder were set to 10% of the free-stream velocity. They reported that the lock-in occurs in two frequency bands (around f0 and 2f0 ). In addition, they display the phenomenon of sudden jump in mean drag coefficient within these lock-in regions. The problem of a circular cylinder undergoing an orbital motion in a uniform stream was also studied numerically by Baranyi [25,26] at low Reynolds numbers, Re = 120–180. A major objective of his works was to investigate the effect of ellipticity values on the cylinder force coefficients and energy transfer. Recently, Al-Mdallal [29] discussed numerically the initial flow past a circular cylinder with orbital motion using a perturbation theory with the collocation method. The time of first separation and the fluid forces at early stages were successfully predicted. The problem of a circular cylinder undergoing a figure-eight-type motion in a uniform stream was also investigated by Jeon and Gharib [30], Baranyi [27] and Reid [28]. Jeon and Gharib [30] found, experimentally, that even small amounts of streamwise motion (i.e. streamwise amplitude is equal to 0.1 of transverse amplitude) can inhibit the formation of the 2P mode (in which two vortices are alternately shed from each side of the cylinder per cycle). However, Reid [28] confirms, computationally, the existence of different locked-in and quasi-locked-in vortex shedding modes at certain flow parameters at a Reynolds number of Re = 200. Notably, the literature includes relevant studies on the problem of flow behind a cylinder forced by a combination of oscillatory translational and rotational motion; see the recent works of Al-Mdallal [20] and Nazarinia et al. [31,32]. Although, the main objective of this study is to investigate the problem of flow past a circular cylinder undergoing circular motion, it is necessary to mention several investigations which relate to the study undertaken here. For instance, an orbiting cylinder, following a circular path only, immersed in quiescent fluid had been investigated numerically by Teschauer et al. [33]. Their study focused on the effect of both Keulegan–Carpenter number, KC, and Reynolds number, Re, on the flow pattern. Chen et al. [34] presented computational investigations on flow structure for the case of a stationary circular cylinder placed in an orbital flow. The case of a cylinder vibrating streamwise with an incident non-uniform flow is investigated by Konstantinidis and Liang [35]. Finally, but not least, the problem of an elastically mounted circular cylinder with low-mass damping which follows an orbital path in a free stream has also attracted much attention in recent years; the reader is suggested to refer to the review paper of Williamson and Govardhan [9] and references therein. Within this study, we refer to the modes of vortex shedding according to the number of shedding vortices from each side of the cylinder over a single vortex shedding cycle, Tv = nT , where n is

either a fraction or an integer number, and T is the cylinder oscillation period. For example, the 2S mode per Tv , refers to the alternate shedding counter-rotating single vortex from each side of the cylinder over Tv . It should be noted that if Tv = T in the 2S mode, then we have the classical Kármán vortex street. However, the P + S per Tv mode refers to the shedding of two vortices from one side of the cylinder followed by a single vortex from the other side per Tv . The coalescence phenomenon which is defined as the merging of vortices immediately behind the cylinder in the vortex shedding layers is also taken into account in this study. Following the notation of Reid [28], coalescence occurrence is identified by an uppercase ‘‘C’’ that is written before the vortex shedding mode. For instance, the C(2S) mode per Tv refers to the alternate shedding counter-rotating single vortex from each side of the cylinder over Tv , where at least one of the shed vortices was formed by the coalescence of vortices. In addition, under certain circumstances, a switching in the vortex shedding mode in the near wake region may occur. Thus, the notation A → B refers to the switching from A-mode to B-mode. It is worth mentioning that, in classifying these modes, only the near-wake region is considered as in the work of Ongoren and Rockwell [12], while the notation of Williamson and Roshko [6] is used. Notice that vortex-shedding classification terminology employed has been recently used by several researchers such as Al-Mdallal et al. [14] and Reid [28]. The main objective of this paper is to numerically investigate the development of the near wake structure behind a circular cylinder undergoing a circular motion in the presence of a uniform stream of a viscous incompressible fluid. The numerical simulations are conducted at a fixed Reynolds number, Re = 200, and at displacement amplitude-to-cylinder diameter of A = 0.05 and 0.25. While, the ratio of oscillation frequency, f , to Kármán vortexshedding frequency, f0 is chosen from f /f0 = 1 to 3, to classify regions in which fundamental, sub-harmonic and super-harmonic synchronization exists. The frequency-lock-in regions are determined using the near-wake vorticity contours and Lissajous patterns of unsteady lift coefficient, following the methodology of Cetiner and Rockwell [13] and Al-Mdallal et al. [14]. The modes of vortex shedding in the lock-in regions are presented and classified. The variation of the forces acting on the cylinder as a function of forcing frequency and radius of the circular path are studied in detail. 2. Problem statement and governing equations Initially, at dimensional time t ∗ = 0, an infinitely long circular cylinder of diameter d starts to move in a viscous incompressible fluid of a uniform velocity U with combined streamwise and transverse oscillation to follow a circular trajectory. Hence, the motion of the cylinder center is described by the following harmonic oscillations: X ∗ (t ∗ ) = A∗x cos(2π fx∗ t ∗ ), ∗

Y ∗ (t ∗ ) = A∗y sin(2π fy∗ t ∗ )

∗

(1)

where X and Y represent the instantaneous dimensional streamwise and transverse cylinder displacements, respectively. Here fx∗ and fy∗ , A∗x and A∗y , respectively, represent the dimensional frequencies and amplitudes of the harmonic motions. In this study, we assume that fx∗ = fy∗ = f ∗ and A∗x = A∗y = A∗ . Hence, Eq. (1) produces an anticlockwise orbital circular motion. Using the cylinder diameter d and free stream velocity U, the motion can be described by the following dimensionless parameters: Reynolds number, Re = Ud/ν , where ν is the kinematic viscosity of the fluid; the radius of the path, A = A∗ /d; and the forcing frequency ratio, f /f0 , with f = df ∗ /U. Thus, the dimensionless streamwise and transverse cylinder displacements are, respectively, given by X (t ) = A cos(2π ft ),

Y (t ) = A sin(2π ft ),

Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

where t = t ∗ U /d is the dimensionless time. Obviously, the dimensionless streamwise and transverse cylinder velocities are, respectively, given by X˙ (t ) = −Uv sin(2π ft ),

Y˙ (t ) = Uv cos(2π ft ),

where Uv = 2π fA. To model the fluid motion, we use the unsteady Navier–Stokes equations together with the continuity equation in the noninertial reference frame, attached to the cylinder. Since the flow is assumed two-dimensional in the xy-plane, these equations in the dimensionless form are given by

∂ v⃗ ⃗ v = −1∇ ⃗ p + 2 ∇ 2 v⃗ + a⃗, + (⃗v · ∇)⃗ ∂t 2 Re ⃗ · v⃗ = 0 ∇

p∗ = 1/2 ρ U 2 p.

(5)

In this work we adopted the modified polar coordinates (ξ , θ ) where x + iy = exp(ξ + iθ ) with ξ = ln(r ). Consequently, Eqs. (2) and (3) in stream function-vorticity formulation are 2 ∂ζ = e2ξ ∂t Re



∂ 2ζ ∂ 2ζ + ∂ξ 2 ∂θ 2



∂ψ ∂ζ ∂ψ ∂ζ + − , ∂ξ ∂θ ∂θ ∂ξ

∂ ψ ∂ ψ + = e2ξ ζ , 2 ∂ξ ∂θ 2 2

(6)

2

(7)

where the domain of the problem is D(ξ ,θ,t ) = {(ξ , θ , t ) : 0 ≤ ξ ≤ ∞, 0 ≤ θ < 2π , t ≥ 0}. Here ζ and ψ represent vorticity and stream function, respectively. Note that, in principle, the dimensionless radial and transverse components of the velocity (uξ , vθ ) are related to the stream function by

∂ψ uξ (ξ , θ , t ) = e−ξ ,

∂ψ vθ (ξ , θ , t ) = −e−ξ , ∂θ ∂ξ and the function ζ is defined by   ∂ ξ ∂ ξ ζ (ξ , θ , t ) = e−2 ξ (e uξ ) − (e vθ ) . ∂θ ∂ξ

2π



0

2π

∞



e(2−p)ξ ζ (ξ , θ , t ) cos(pθ )dθ dξ = 2π Y˙ (t )δ1,p ,

surface, i.e. (8)

0

e(2−p)ξ ζ (ξ , θ , t ) sin(pθ )dθ dξ

0

= 2π (1 − X˙ (t ))δ1,p ,

the cylinder, i.e.

∂ψ → 1 − X˙ (t ) sin θ + Y˙ (t ) cos θ as ξ → ∞, ∂ξ   ∂ψ e−ξ → 1 − X˙ (t ) cos θ − Y˙ (t ) sin θ as ξ → ∞, ∂θ

(12b)

for p = 0, 1, . . . , where δ1,p is the Kronecker delta symbol. Obviously, initial conditions at t = 0 are not explicitly given which cause a major difficulty in the direct use of numerical methods. Therefore, we implement the boundary-layer theory to provide not only initial conditions but also to determine the entire initial flow at small times. Determination of the initial flow We implement the boundary-layer theory by introducing a boundary-layer coordinate z through the transformation

ξ = kz ,

(13) 1 2

where k = (8t /Re) . We also scale ψ and ζ using k as follows:

ψ = kΨ ,

ζ = ω/k.

(14)

Consequently, Eqs. (6)–(7) and the required conditions (8), (10)– (12) in terms of these boundary-layer coordinates are, respectively, given by

  ∂ω ∂ 2ω 2kz + 2e z +ω ∂ z2 ∂z   ∂ω ∂ Ψ ∂ω ∂ Ψ ∂ω ∂ 2ω = 4t e2kz + − − k2 2 , ∂t ∂θ ∂ z ∂ z ∂θ ∂θ

∂Ψ = 0 when z = 0, ∂z ω → 0 as z → ∞, ω(z , θ , t ) = ω(z , θ + 2 π , t ), Ψ (z , θ , t ) = Ψ (z , θ + 2 π , t ), ∞



(15)

(16)

0

(9b)

(18) (19)

e2kz ω(z , θ , t )dθ dz = 0,

(20a)

e(2−p)kz ω(z , θ , t ) cos(pθ )dθ dz = 2π Y˙ (t )δp,1 ,

(20b)

0

∞



2π



0

(9a)

(17)

0

∞



2π



0

• The free stream conditions and the decay of vorticity away from 

(12a)

0

Ψ = 0,

• The no-slip and impermeability conditions on the cylinder

e−ξ

∞

subject to

The associated boundary conditions are as follows:



for t > 0 and ξ ∈ [0, ∞).

2 ∂ 2Ψ 2∂ Ψ + k = e2kz ω, ∂ z2 ∂θ 2

Boundary conditions

∂ψ ψ= = 0, when ξ = 0, ∂ξ for t > 0 and θ ∈ [0, 2π ].

(11)

It may be shown, following the works of Dennis and Quartapelle [36] and Dennis and Kocabiyik [37], that (8) and (9) can be used to derive from (7) the following set of integral conditions:

(3)

⃗ is the translational acceleration arising from Further, the term a the noninertial reference frame of the oscillating cylinder and it is given by ⃗ = (−X¨ (t ), −Y¨ (t ), 0). a

ζ (ξ , θ , t ) = ζ (ξ , θ + 2 π , t ), ψ(ξ , θ , t ) = ψ(ξ , θ + 2 π , t ),



(4)

(10)

for t > 0 and θ ∈ [0, 2π ]. • The dependent variables ψ and ζ should be periodic functions of θ with period 2π , i.e.

(2)

where v ⃗ = (u, v, 0) and ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 . Notice that all quantities in Eqs. (2) and (3) are dimensional, their dimensionless counterparts being given by

v⃗ ∗ = U v⃗ ,

ζ → 0 as ξ → ∞,

123

2π



e(2−p)kz ω(z , θ , t ) sin(pθ )dθ dz

0

= 2π (1 − X˙ (t ))δp,1 ,

(20c)

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Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

for all integers p ≥ 1. The exact initial expressions for ω and Ψ at t = 0 can be obtained from (15) and (16) and are given by 4

2

ω(z , θ, 0) = √ Y˙ (0)e−z cos(θ ) π

4 2 + √ (1 − X˙ (0))e−z sin(θ ), π   1 −z 2 ˙ − 1) cos(θ ) Ψ (z , θ, 0) = 2Y (0) z erf(z ) + √ (e π   1 2 + 2(1 − X˙ (0)) z erf(z ) + √ (e−z − 1) sin(θ ), π

(21)

∞  ∞ 

Ψ (z , θ, t ) =



n  ((2 − l)z )n−q

(n − q)!   = 2π δm,0 − βm δl,1 δn,0 ,

0

0

(22)

 ωq,m sin(lθ )dθ dz

q=0

(30c)

where βm and σm are the components of the series expansions for oscillatory velocities ∞ 

βm t m ,

Y˙ (t ) =

m=0

∞ 

σm t m .

(31)

m =0

The unknown functions ωn,m and Ψn,m can be decomposed into a finite set of Fourier cosine and sine components in the coordinate θ as follows: m+1

ωn,m (z , θ ) =



Gpn,m (z ) cos(pθ ) + gnp,m (z ) sin(pθ ) ,



(32)

Fnp,m (z ) cos(pθ ) + fnp,m (z ) sin(pθ ) .

(33)

p=0 m+1

Ψn,m (z , θ ) =

ωn,m (z , θ )t m kn ,

n=0 m=0

∞  ∞ 

2π



X˙ (t ) =

for n = 0, 1, . . . . On the other hand, the initial flow solution is determined following the methodology of Collins and Dennis [38] by building up a perturbation solution in powers of t and k as follows:

ω(z , θ, t ) =

∞



(23)

Ψn,m (z , θ )t k . m n

n=0 m=0

If we substitute expressions (23) into Eqs. (15) and (16) and then equate the coefficients of the successive powers of k and t to zero we obtain equations for both ωn,m and Ψn,m

∂ωn,m ∂ 2 ωn,m + 2z − [4m + 2(n − 1)]ωn,m = Rn,m (z , θ ), 2 ∂z ∂z n ∂ 2 Ψn,m (2z )n−q ∂ 2 Ψn−2,m  = − + ωq,m , ∂ z2 ∂θ 2 (n − q)! q =0

(24) (25)



p=0 p

p

p

The equations and conditions for Gn,m (z ), Fn,m (z ), gn,m (z ) and p fn,m (z ) can be obtained by substituting (32)–(33) in (24)–(30) and using standard orthogonality conditions. In fact, finding the exact expressions for all ωn,m and Ψn,m is too lengthy and far from trivial. Herein, we only obtain the exact solutions for ω0,1 , Ψ0,1 , ω0,2 , Ψ0,2 , ω1,0 , Ψ1,0 , ω1,1 , Ψ1,1 , ω1,2 , Ψ1,2 , ω2,0 , Ψ2,0 , ω2,1 , Ψ2,1 . The compact expressions for most of these terms are given in the Appendix. We thus finally obtain an expression for the vorticity

ζ (ξ , θ , t ) ≈

1 k

ω0,0 + ω0,1 t + ω0,2 t 2

 + k(ω1,0 + ω1,1 t + ω1,2 t 2 ) + k2 (ω2,0 + ω2,1 t ) (34)

where n−1   ∂ 2 ωn−2,m (2z )n−q Rn,m (z , θ ) = − − 2 2 ∂θ (n − q)! q=0   ∂ωq,m − (2m + q − 1)ωq,m × z ∂z  n m −1    ∂ Ψq,j ∂ωn−q,m−j−1 + 4 ∂θ ∂z q =0 j =0   ∂ Ψq,j ∂ωn−q,m−j−1 − . ∂z ∂θ



which is valid for small t and large R. It should be noted that the analytical solution (34) will be used to check the accuracy of the numerical results at small times. 3. A numerical approach

(26)

In these equations the functions with a negative subscript are taken to be identically zero. The associated boundary and integral conditions in Eqs. (24)–(25) are deduced from (17)–(20); they are given by

The numerical algorithm utilized for solving (6)–(7) subject to (8) and (10)–(12) is originally referred to Collins and Dennis [38]. Over the years, this algorithm has been effectively used to discuss problems of a cylinder undergoing different types of oscillations (see, for example, [39–44,20,45,14]). Essentially, the stream function ψ and vorticity ζ are expressed as a truncated Fourier series of the form

ζ (ξ , θ , t ) ≃

1 2

G0 (ξ , t ) N 

+

[Gn (ξ , t ) cos(nθ ) + gn (ξ , t ) sin(nθ )] ,

(35a)

n =1

∂ Ψn,m = 0 when z = 0, ∂z ωn,m → 0 when z → ∞, ωn,m (z , θ ) = ωn,m (z , θ + 2 π ),

Ψn,m = 0,

Ψn,m (z , θ ) = Ψn,m (z , θ + 2 π ),   ∞  2π  n (2z )n−q ωq,m dθ dz = 0, (n − q)! 0 0 q =0   ∞  2π  n ((2 − l)z )n−q ωq,m cos(lθ )dθ dz (n − q)! 0 0 q =0

= 2πσm δl,1 δn,0 ,

(27)

ψ(ξ , θ , t ) ≃

1 2

F0 (ξ , t )

(28)

+

N 

[Fn (ξ , t ) cos(nθ ) + fn (ξ , t ) sin(nθ )] ,

(35b)

n =1

(29) (30a)

(30b)

where N is the order of truncation. Substituting (35a,b) into (6) and (7) gives 4N + 2 interdependent partial differential equations. The boundary conditions associated with the resultant equations follow directly from (8), (10) and (12). The solution through time domain is then divided into two distinct zones (0, ts ] and (ts , tmax ] where ts represents the switching time from the boundary-layer coordinate to the physical coordinate and tmax represents the terminal time. Since the computational domain in the spatial directions is unbounded, we choose a

Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

125

large enough artificial outer boundary, z∞ , to represent infinity for the numerical treatment. The solution in the first zone is obtained by solving the equations in D(z ,t ) domain whereas the solution in the second zone is obtained by solving the equations in D(ξ ,t ) . We firstly assume that Gn , gn , Fn and fn for n = 0, . . . , N are known at particular time tj . Then the solutions for Gn and gn for n = 0, . . . , N are determined by utilizing the implicit Crank–Nicolson scheme to advance the solution to the next time step t = tj+1 . Further, we use the central finite difference formula to approximate spatial derivatives. The numerical solution for Fn and fn are obtained using stable integration procedures given by Dennis and Chang [46]. 4. Calculation of forces acting on the cylinder The main forces exerted on the cylinder surface are represented by the drag and lift coefficients, defined as D 1 2

,

ρ U2 d

CL =

L 1 2

ρ U2 d

,

(36)

where D and L represent the drag and lift forces exerted on a unit length of the cylinder, respectively. The lift and drag coefficients are derived by integrating the normal and shear stresses around a closed contour containing the cylinder, and their final forms are given by CL = CLF + CLP + π Y¨ (t ),

CD = CDP + CDF + π X¨ (t ),

(37)

where the terms π Y¨ (t ) and π X¨ (t ) are due to the acceleration imposed on the flow. Notice that CLF and CLP are the dimensionless viscous (friction) and pressure lift coefficients, respectively, given by CLF = − CLP =



2 Re

2 Re

2π

0 2π

 0

ζ |ξ =0 cos(θ )dθ ,  ∂ζ  cos(θ )dθ . ∂ξ ξ =0

(38)

However, CDF and CDP are the dimensionless viscous and pressure drag coefficients, respectively, given by

CDP

2



2π

ζ |ξ =0 sin(θ )dθ , Re 0   −2 2π ∂ζ  sin θ dθ . = Re 0 ∂ξ ξ =0

CDF =

CD, CL

CD =

Fig. 1. The time variation of the drag coefficient, CD , for Re = 200, A = 0.1 and f /f0 = 2.0; by using ξ∞ = 6 (∗ ), ξ∞ = 7 ( ) and ξ∞ = 8 ( ).

(39)

5. Validation In this section we shall demonstrate the performance and efficiency of the present numerical scheme. Firstly, in order to obtain accurate numerical results, we have to pay attention to the selection of the numerical algorithm parameters N , z∞ , ξ∞ and the time steps △tj+1 . This goal is achieved by conducting a series of numerical simulations to determine the ‘‘optimal choice’’ of these parameters. Herein, for example, we present three simulation cases on ξ∞ = 6, 7 and 8 to calculate the drag coefficient, CD , for the case of uniform flow past a circular cylinder subject to a circular motion when Re = 200, A = 0.1 and f /f0 = 2.0 as seen in Fig. 1. It is observable that there is a minimal effect on the accuracy of these curves when we change ξ∞ from 7 to 8. Thus, in this case we would choose ξ∞ = 7 as our optimal value. Following the same steps, the numerical experimentations suggest that the ‘‘optimal’’ values for the numerical parameters are: ks ≈ 0.65, hz = 0.02 and z∞ = 7. The order of truncation of the Fourier expansion is arranged automatically by initial setting N = 2, then, one more

Fig. 2. The time variation of the lift and drag coefficients for the case of uniform flow past a stationary cylinder at Re = 200. Table 1 Comparisons of f0 , CL,max and CD for the case of uniform flow past a fixed cylinder at Re = 200.

Present Poncet [47] Henderson [48] Wen and Lin [49] Guilmineau and Queutey [8] Meneghini and Bearman [50]

f0

CL,max

CD

0.1954 0.1990 0.1972 – 0.1950 0.1960

0.69 0.70 – – – 0.59

1.31 1.34 1.34 1.30 – 1.23

term is added when the last term exceeds 10−3 with the maximum number of terms N = 90. The dimensionless time step was chosen as 1tj+1 = 10−4 for the first 100 steps, then was increased to 1tj+1 = 10−3 for the next 100 steps and finally 1tj+1 = 10−2 for the rest of the calculations. Fig. 2 shows the lift and drag coefficients for the case of uniform flow past a stationary cylinder (no forced oscillations) at Re = 200. The predicted values of the natural shedding frequency, f0 , the maximum lift coefficient, CL,max , and the mean drag coefficient, CD , for the case of uniform flow past a fixed cylinder at Re = 200 are compared in Table 1 with the available previous results. Excellent agreements are obtained. A further comparison is done with Guilmineau and Queutey [8] and Mironova [51] for the case of uniform flow past a cylinder subject to transverse oscillation. In this test, the cylinder motion is described by Y (t ) = −Ay cos(2π ft ) with Ay = 0.2 and f /f0 = 1.1. The calculated values of the RMS lift coefficient, CL,rms , the RMS drag coefficient, CD,rms , and the mean drag coefficient, CD , are shown in Table 2. As clearly seen in this table, quite reasonable agreement has been obtained. Note that Guilmineau and Queutey [8] and Mironova [51] have, respectively, used the values of f0 as 0.1950 and 0.2140. It is reasonable, from computational point of view, to explain the slight discrepancy between the present results and those obtained by Guilmineau and Queutey [8] and Mironova [51] as being due to using different numerical schemes. It may be noted that, even if the time-mean of the drag

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a

b

Fig. 3. Comparison of the equivorticity lines for the case of uniform flow past a cylinder subject to streamwise oscillations when Re = 100, A = 0.14 and f /f0 = 2.0: Su et al. [52] (left); present results (right).

Fig. 4. Variation of the drag and lift coefficients with time at early times for R = 500, Ax = Ay = 0.1, f = 0.2; —, numerical solutions; ◦, analytical solutions.

Table 2

Table 3

Comparisons of CL,rms , CD,rms and CD for the case of a cylinder vibrating transversely to a steady flow at Re = 185, Ay = 0.2 and f /f0 = 1.1.

Comparisons of CD for uniform flow past a cylinder undergoing a circular motion at Re = 300, and Ax = Ay = 0.1/8π f0 .

Present Guilmineau and Queutey [8] Mironova [51]

CL,rms

CD,rms

CD

f /f0

0.5

1.0

1.5

2.0

2.5

0.903 0.895 0.897

1.414 1.431 1.440

1.407 1.422 1.420

Present Didier and Borges [23] Baranyi [25]

1.34 1.34 1.36

1.41 1.46 1.43

1.33 1.37 1.36

1.45 1.45 1.48

1.34 1.36 1.38

and lift coefficients (originating from the non-inertial system used) is zero, their contribution to the rms values is not negligible. Fig. 3 shows a comparison between the near-wake structures obtained in the present study for uniform flow past a cylinder subject to streamwise oscillations with that obtained numerically by Su et al. [52]. In this comparison, the cylinder motion is described by X (t ) = −Ax cos(2π ft ) with Ax = 0.14 and f /f0 = 2.0 (f0 = 0.1680). It is apparent that these comparisons are in good agreement. The mean drag coefficient, CD , for uniform flow past a cylinder undergoing a circular motion at Re = 300 is compared with the computational results of Didier and Borges [23], and Baranyi [25] in Table 3. In each of these studies, a different numerical technique is used to integrate the two-dimensional unsteady Navier–Stokes equations. (Didier and Borges [23] used a fully coupled finite volume method, while, Baranyi [25] employed the finite difference method.) In their studies, numerical simulations are performed at Re = 300, Ax = Ay = 0.1/8π f0 and f /f0 = 0.5, 1.0, 1.5, 2.0, 2.5 with f0 = 0.214. Satisfactory agreement is observed. As further checks, the results of the drag and lift coefficients obtained from the numerical solutions at small values of t and obtained from the analytical solution (34), for the case of uniform flow past a circular cylinder subject to a circular motion, are given in Fig. 4. It is clearly seen that the comparison is extremely good at small times.

6. Results and discussions The present calculations are performed for Re = 200, 0.05 ≤ A ≤ 0.25 and 1.0 ≤ f /f0 ≤ 3.0. It is worth mentioning that some researchers have reported three-dimensional effects in the far wake at Re > 178; see Williamson [53] and Henderson [48]. However, forcing the cylinder to oscillate in a uniform stream is a type of wake control which suppresses the three-dimensionality and produces a two-dimensional flow at least in the near-wake region of the cylinder. Therefore, since the main goal of this study is to investigate the development of the flow in the near wake region of oscillating cylinder, we may assume that the flow remains laminar and two-dimensional. 6.1. Vortex formation modes We begin with detailed discussion of the effect of the frequency ratio, f /f0 , on the wake of an oscillating circular cylinder at the oscillation amplitude A = 0.05. Then, the results at different oscillation amplitudes (A > 0.05) are presented in the same manner but with less details. The time history of the lift coefficients up to t = 90 and Lissajous curves for A = 0.05 and selected values of cylinder excitation frequency, 1.75 ≤ f /f0 ≤ 2.1, are shown in Fig. 5. It is noted that the lift coefficients have a repetitive behavior every two cycles

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127

Fig. 5. Left: the time variation of CL , Middle: the Lissajous patterns (CL vs. X ), and Right: the Lissajous patterns (CL vs. Y ) for Re = 200, A = 0.05 and 1.75 ≤ f /f0 ≤ 2.1.

of oscillations which inspires us to conclude that the vortex shedding in the near-wake region is locked-in to the cylinder oscillation frequency over two periods, 2T , when the oscillation frequency is near twice the natural shedding frequency, f ≈ 2f0 . As already mentioned in the Introduction, this conclusion has also been reported recently by many researchers for the case of uniform flow past a cylinder undergoing forced streamwise oscillations. A further comment on the behavior of the lift coefficient is that it exhibits a regular sign of influence from a higher harmonic, in which, its beating frequency increases as f /f0 increases. Hence, we may predict the development of different sizes of vortices from both sides of the cylinder, which certainly indicates the appearance of vortex shedding modes other than the classic Kármán vortex street, 2S per T . Fig. 6 shows typical equivorticity patterns for

A = 0.05 and 1.75 ≤ f /f0 ≤ 2.1 over two periods of cylinder oscillations. The snapshots are taken at the instant (X (t ), Y (t )) = (A, 0) and every one oscillation cycle thereafter. The snapshots at t = 0 and t = 2T for all cases are identical, which support our previous claim that the vortex shedding modes in the near-wake region at the considered excitation frequencies are locked-in to the cylinder oscillation frequency over two periods, 2T . Further investigation shows that the vortex shedding mode is the lockedin P + S → 2S mode, per 2T for 1.75 ≤ f /f0 ≤ 2 in which two positive vortices (large and secondary) are developed in the lower shear layer of the cylinder followed by a single vortex from the top side over 2T . Note that the secondary vortex vanishes with the elongation of the shedding layer of the first vortex which produces the locked-in 2S mode, per 2T . However, when the forcing

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Fig. 6. Equivorticity lines over two periods of oscillation, 2T , for Re = 200, A = 0.05 and 1.75 ≤ f /f0 ≤ 2.10.

frequency increases to f /f0 = 2.1, the wake structure appears to be the locked-in 2S mode, per 2T . A closer look at the snapshots in Fig. 6 leads us to conclude that as f /f0 increases, the velocity of the cylinder increases, which causes a notable decrease in the formation length and the size of the large vortices. As a result, the second weaker vortex cannot be seen any more when 2 < f /f0 ≤ 2.1. An important remark about the lift coefficient is that the size of the vortex shed is linked to the local maximum and minimum of the lift coefficient. Consequently, as shown in Figs. 5 and 6, the decrease in the value of one the two peaks in the lift coefficients, as the frequency ratio increases, is due to the vanishing of the secondary vortex. The mode transition phenomenon has been captured and analyzed by several researchers in the case of uniform flow past an oscillating cylinder, see for example the recent work of Baranyi [26] and the references therein. Remarkably, the flow of frequency ratio f /f0 = 2.6, outside the lock-in region, exhibits periodic behavior every eight periods of cylinder oscillation, 8T , as shown in Fig. 7 of the lift coefficient. Furthermore, the structure of the equivorticity patterns in the near wake region for f /f0 = 2.6 (see Fig. 8) confirms the existence of the locked-in C(2P + 4S) mode per 8T . In this mode, shedding of a vortex pair as well as two single vortices from each side of

the cylinder, within 8T , is observed. It has also been shown in this figure that some vortices are larger and stronger than others because they are developed by mixing more small vortices. This discrepancy in the size of the shedding vortices within the lockin region causes the appearance of different peaks with different maxima and minima. It is worth mentioning that the CL traces for frequency ratios, f /f0 = 1.3, 1.6, 2.4, 2.5, 2.7 and 2.9 (which are not shown here for economy of space) display quasi-periodic signatures every 4T , 5T , 12T , 5T , 14T and 3T , respectively. The wake structure at the oscillation amplitude A = 0.1 looks quite similar to the case A = 0.05. In this case, the main lockin region is found to be f /f0 ∈ [1.5, 2.1], as shown in Fig. 9, which displays the typical equivorticity patterns over two periods of cylinder oscillations with the corresponding lift coefficient for f /f0 = 1.5, 1.75, 1.9, 2, 2.1. The snapshots are taken at the instant (X (t ), Y (t )) = (A, 0) and every half oscillation cycle thereafter. A substantial decrease in vortex formation length as f /f0 increases is observed which, essentially, affects the vortex formation modes as described in the preceding case at A = 0.05. For instance, the locked-in P + S → 2S mode per 2T is observed for 1.5 ≤ f /f0 < 1.75. However, when the frequency ratio increases in the range 1.75 ≤ f /f0 < 2, the secondary bottom

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129

Fig. 7. Left: the time variation of CL , Middle: the Lissajous patterns (CL vs. X ), and Right: the Lissajous patterns (CL vs. Y ) for Re = 200, A = 0.05 and f /f0 = 2.6.

Fig. 8. Equivorticity lines over eight periods of oscillation, 8T , for Re = 200, A = 0.05 and f /f0 = 2.6.

vortex becomes stronger and more intensive to shed away downstream. Thus, the locked-in P + S mode per 2T is clearly captured for f /f0 ∈ [1.75, 2). When 2 ≤ f /f0 ≤ 2.1, the vortex formation length becomes shorter which causes the suppression of the secondary vortices, and hence, the locked-in 2S mode per 2T is produced. Notably, the locked-in C(2P + 4S) mode per 8T is observed at f /f0 = 2.5 which has quite similar description to that obtained at (A, f /f0 ) = (0.05, 2.6). In extra investigation (not shown here), the CL coefficients for f /f0 = 1.4, 2.2, 2.8, 2.9 display quasi-periodic signatures every 3T , 2T , 3T , 3T , respectively. As the oscillating amplitude, A, increases to 0.15, the main lockin regions occur over the ranges f /f0 ∈ [1.75, 2.0] and f /f0 ∈ [2.6, 2.8]. As shown in Fig. 10, the vortex shedding mode, for 1.75 ≤ f /f0 ≤ 2.0, is the locked-in P + S mode per 2T in which two strong positive vortices are shed from the bottom of the cylinder

followed by a single vortex from the top side within two cycles. We observe that, at f /f0 = 1.9 and f /f0 = 2.0, the two shed vortices from the bottom side of the cylinder coalesce downstream at or before a distance 10a to form a single vortex (with the core of the first vortex) during the second period of oscillation. The vortex shedding, for 2.6 ≤ f /f0 ≤ 2.8, produces the locked-in C(2P) → P + S mode per 3T . This mode is described in Fig. 11 at f /f0 = 2.6. Three vortices are developed on each side of the cylinder where the first vortex in each shed pair is just a result of coalescence of the first two vortices. Furthermore, the shed bottom vortex pair coalesces in the region very close to the cylinder surface to form a single large vortex in the second period. Obviously, tracking the lift coefficients within three periods of oscillations shows three positive and three negative peaks which confirms our description of the vortex shedding mode. However,

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a

b

c

d

e

Fig. 9. Equivorticity lines over two periods of oscillation, 2T , for Re = 200, A = 0.1: (a) f /f0 = 1.5; (b) f /f0 = 1.75; (c) f /f0 = 1.9; (d) f /f0 = 2.0; (e) f /f0 = 2.1.

a

b

c

Fig. 10. Equivorticity lines over two periods of oscillation, 2T , for Re = 200, A = 0.15: (a) f /f0 = 1.75; (b) f /f0 = 1.9; (c) f /f0 = 2.

searching outside the lock-in regions aided in finding out the locked-in C(5S) mode per 3T at f /f0 = 1.3, see Fig. 12. In this mode, three vortices are developed on each side of the cylinder over three cycles, but due to the coalescence effect in the upper vortex shedding layer only two negative vortices convect downstream. Furthermore, the shed bottom vortex pair coalesces in the region

very close to the cylinder surface to form a single large vortex in the second period. Further investigations show that the vortex shedding exhibits quasi-locked-in modes when f /f0 = 1.2, 1.4, 1.5, 1.6, 2.1, 2.3 per 4T , 2T , 2T , 4T , 2T , 5T , respectively. When the oscillation amplitude further increases to the value of A = 0.2, the vortex locked-in phenomenon is, mainly, identified

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131

Fig. 11. Equivorticity lines over three periods of oscillation, 3T , for Re = 200, A = 0.15 and f /f0 = 2.6.

Fig. 12. Equivorticity lines over three periods of oscillation, 3T , for Re = 200, A = 0.15 and f /f0 = 1.3.

in the ranges 1.75 ≤ f /f0 ≤ 2.0, 2.2 ≤ f /f0 ≤ 2.3 and 2.5 ≤ f /f0 ≤ 2.8. The dominant vortex shedding modes in these regions are found to be the locked-in P + S mode per 2T for 1.75 ≤ f /f0 ≤ 2.0, the locked-in C(2P + 2S) mode per 5T for 2.2 ≤ f /f0 ≤ 2.3 and the locked-in C(2P) mode per 3T for 2.5 ≤ f /f0 ≤ 2.8. Since the description of the locked-in P + S mode is very similar to that discussed above, we only discuss the

vortex shedding modes C(2P + 2S) and C(2P). Hence, for better description of the C(2P + 2S) mode, equivorticity patterns for f /f0 = 2.2 over five periods of cylinder oscillations are displayed in Fig. 13. The snapshots are taken at the instant the cylinder reaches the position (X (t ), Y (t )) = (A, 0) and every one oscillation cycle thereafter. In this mode, five vortices are developed on each side of the cylinder over five cycles. However, due to the merging of

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Fig. 13. Equivorticity lines over five periods of oscillation, 5T , for Re = 200, A = 0.2, f /f0 = 2.2.

Fig. 14. Equivorticity lines over three periods of oscillation, 3T , for Re = 200, A = 0.2, f /f0 = 2.6.

Fig. 15. Equivorticity lines at an instant corresponding to the cylinder position (X (t ), Y (t )) = (A, 0) for Re = 200, A = 0.25: (a) f /f0 = 2.4, (b) f /f0 = 2.5, (c) f /f0 = 2.6, (d) f /f0 = 2.7.

vortices in the vicinity of the cylinder, we observe the shedding of a vortex pair as well as a single vortex from each side of the cylinder. On the other hand, a brief description of the locked-in C(2P) mode is given in Fig. 14 at f /f0 = 2.6 by means of equivorticity contours. In this mode, three vortices are developed on each side of the cylinder. As a result of coalescence effect, the first vortex in each shed pair is a result of the mergence of the first two vortices. It is also noticed that the vortex shedding exhibits the locked-in C(5S)

mode per 3T at f /f0 = 1.2. As a closing remark on the case A = 0.2, the flow exhibits quasi-locked-in mode for f /f0 = 1.6 per 2T . As the oscillation amplitude increases to A = 0.25, the flow characteristics in the near wake region have been changed markedly so that the lock-in region is not concentrated around 2f0 but it is shrunk to f /f0 ∈ [2.4, 2.7]. The dominant vortex shedding mode in this region is found to be the locked-in C(2P) mode per 3T . To support this claim, Fig. 15 displays typical equivorticity

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133

Fig. 16 displays an overview of locked-in states in the frequency–amplitude plane for Re = 200, 0.05 ≤ A ≤ 0.25 and 1 ≤ f /f0 ≤ 3.0. This diagram categorizes the vortex formation into three states: lock-in, quasi-lock-in and unlock-in. It is clearly seen that vortex lock-in occurs, in general, in the region close to f /f0 ≈ 2. This leads to the conclusion that the streamwise motion is dominant in the considered range of flow parameters. Moreover, this diagram reveals the existence of multiple lock-in regions. A final comment on Fig. 16 is: we believe that any transition from unlock-in state to lock-in state (or vice versa) should go first through quasi-lock-in state. 6.2. Fluid forces acting on the surface of the cylinder

Fig. 16. Overview of vortex lock-in states for Re = 200, 0.05 ≤ A ≤ 0.25, and 1.0 ≤ f /f0 ≤ 3.0: lock-in ( ), quasi-lock-in ( ), and unlock-in ( ).

contours for f /f0 = 2.4, 2.5, 2.6, 2.7 at the instant corresponding to the cylinder position (X (t ), Y (t )) = (A, 0). A reduction in the upper and lower vortex shedding layers is captured as f /f0 increases. In addition, it is noted that the two vortices shed from the upper side of the cylinder immediately coalesce in the near-wake region to form a single large vortex. Furthermore, the flow associated with the frequency ratio f /f0 = 1.2, exhibits the locked-in C(2P + 2S) mode per 5T . A final remark on this case, quasi-lockedin modes are captured for f /f0 = 1.2, 1.75, 1.8, 1.9, 2.3, 2.8 per 3T , 2T , 2T , 2T , 3T , 3T , respectively.

Based on the present simulations, it is noted that the values of the maximum lift coefficient, CL,max , vary within the range 0.7877 ≤ CL,max ≤ 3.3487 with the minimum value at (A, f /f0 ) = (0.05, 1.8) and the maximum value at (A, f /f0 ) = (0.2, 3.0). Additionally, the values of both minimum lift coefficient, CL,min , and minimum drag coefficient, CL,min , are, generally, decreasing as the oscillating amplitude, A, decreases regardless of the value of f /f0 . It is worth mentioning that negative values of drag coefficient, CD , (thrust) are observed for A > 0.05 at relatively high values of forcing frequency, f /f0 . Negative values of the drag coefficient, CD , are also reported in previous studies of flow past a rotating cylinder, see for example the recent work of Du [54] and Mittal and Kumar [55]. A remarkable result, comes out of intensive investigations, is that the pressure components in the main forces, CDP and CLP , have much more contributions to the main forces, CD and CL , than the frictional components, CDF and CLF . This can be seen in the variations of CDP , CDF , CLP and CLF over six periods of oscillations

Fig. 17. Variations of CDP , CLP , CDF and CLF for Re = 200, f /f0 = 2 and A = 0.05 (), 0.15 () and 0.25 (◦) (T ≈ 5.12, 76.77 ≤ t ≤ 107.47).

134

Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

a

b

c

d

L , CD,rms and CD for Re = 200, 1.0 ≤ f /f0 ≤ 3.0 and A = 0.05 (◦), 0.1 ( Fig. 18. Predicted values of CL,rms , C

for A = 0.05, 0.15, 0.25 at f /f0 = 2 as displayed in Fig. 17. Another result can be seen in Fig. 17 is that the influence of pressure and friction components increases as the oscillation amplitude increases. Fig. 18 displays the variation of RMS lift coefficient, CL,rms , mean L , RMS drag coefficient, CD,rms , and mean drag lift coefficient, C coefficient, CD , for Re = 200, A = 0.05–0.25, and f /f0 = 1.0–3.0. As shown in this figure, more significant effects on the fluid forces are found at high values of oscillation amplitude, A. Moreover, Figs. 18(a) and (b) demonstrate that increasing the oscillation L regardless of the amplitude, A, increases CL,rms and decreases C value of f /f0 . In addition, Fig. 18(a) shows a steady increase of CL,rms in the lock-in region. Inflation of the quantities CD,rms and CD (Figs. 18(c) and (d), respectively) is observed. This variation is more pronounced at high values of oscillating amplitude, A. Deeper investigation shows that these quantities steady increase in the lock-in region until the end of the region at which a fall down occurs. Surprisingly, when A = 0.25, a drop down in the values of CD,rms and CD is also observed at the forcing frequency f /f0 = 1.9 which is the end of the quasi-periodic region. Conclusion The problem of a cylinder undergoing controlled circular motions in a uniform stream has been numerically investigated. The numerical simulations are conducted at Re = 200, 0.05 ≤ A ≤ 0.25 and 1.0 ≤ f /f0 ≤ 3.0, where f0 is the natural shedding frequency from the stationary cylinder. Multiple lock-in regions are verified in this study. It is found that the main lock-in regions occur (for A ≤ 0.2) when the forcing frequency, f , is close to twice the natural shedding frequency, f0 . This ensures the dominance

), 0.15 (

), 0.2 ( ) and 0.25 (

).

of streamwise motion in the considered range of flow parameters. Furthermore, different locked-in vortex shedding modes are observed when either the radius of the circular path or the forcing frequency increases. At relatively low radius of circular path (A ≤ 0.2), lock-in transition from the S + P to the 2S mode per 2T is observed in the range around twice 2f0 . This is partially consistent with the numerical findings of Al-Mdallal et al. [14] in the case of a cylinder vibrating streamwise with an incident uniform flow. Due to coalescence effect, five lock-in vortex shedding modes are captured; these modes are C(5S) per 3T , C(2P + 4S) per 8T , C(2P + 2S) per 5T , C(2P) → P + S and C(2P) per 3T . Additionally, the effect of increasing either the radius of the circular path or the forcing frequency on the lift and drag coefficients is also investigated. Acknowledgment The author would like to express his appreciation to the Faculty of Science at United Arab Emirates University for providing the computing facilities that this research heavily depended upon. Appendix The exact solutions for ω0,1 , Ψ0,1 , ω1,0 , Ψ1,0 , ω1,1 , Ψ1,1 , ω2,0 , Ψ2,0 , and ω2,1 are given by 1 −1 2 ω0, 1 (z , θ ) = 12 erf(z ) z erfc(z ) + 6 erf(z ) (2 z 2 − 1) e−z π 2 3   2 2 + −16 z e−z + 12 z e−2 z + 16 z erfc(z ) π −1    −3 −1 2 2 − 16 e−z π 2 + 8 −z erfc(z ) + e−z π 2

Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

× (β1 sin(θ ) − σ1 cos(θ )) 1 + −12 z (erf(z ) + 2) erfc(z )

+

45

+ 2

− 6 [(2 z 2 − 1) erf(z ) − 2] e−z π 2

−1 2

+

19 6



+ 2 σ0 (1 + β0 ) cos(2 θ )]. 1 −3 z erfc(z ) ((−3 + 2 z 2 ) erf(z ) + 4 z 2 ) Ψ0, 1 (z , θ ) = 9  √ √ + 24 2 erf( 2 z ) − 12 erf(z )  −1 2 − 6 + ((−33 + 12 z 2 ) erf(z ) + 6 + 6 z 2 ) e−z π 2   −3 2 + 8 −1 + ( 1 + z 2 ) e − z π 2    2 + −4 z (3 + 2 z 2 ) erfc(z ) + 6 z e−2 z π −1 × [(−(1 + β0 )2 + σ02 ) sin(2 θ ) + 2 σ0 (1 + β0 ) cos(2 θ )]  1  3 −1 2 4 z erfc(z ) − 6erf(z ) z + 4 (1 + (−1 − z 2 ) e−z )π 2 × 3

× (σ1 cos(θ ) − β1 sin(θ ))   1 2 ω1, 0 (z , θ ) = −erfc(z ) + (4 z 3 + 2 z ) e−z π − 2 × (−(1 + β0 ) sin(θ ) + σ0 cos(θ )).

3√ 1 3√ 45 − z4 + 2 erfc(z ) − 2 2 64 2 2

√ √ √  2  z −3 erf(z ) − 12 2 + 9 + 4 2 erf( 2 z ) 3



2

+ (16 e−z − 12 e−2 z − 16 erfc(z )) z π −1  −3 2 + 16 e−z π 2 [(σ02 − (1 + β0 )2 ) sin(2 θ )

+

−

−erf(z ) + 2 z 2 erfc(z )  1 2 + (−6 z e−z + 8 z )π − 2 4

× ((1 + β0 ) sin(θ ) − σ0 cos(θ )).  −1 2 ω1, 1 (z , θ ) = (2 z 2 + 1) erfc(z ) − 6 z e−z π 2  1 × (β1 sin(θ ) − σ1 cos(θ )) + − erfc(z ) 

147 32

12

z (249 − 288

√

2 + 8 z2)

2

e −z π

−1

9

 + +

+

4 3

−2 z 2

(3 z 2 + 2) (z 2 + 1) e

π −1 −

3

−z 2

ze

π

−3



2

× [(σ0 2 − (1 + β0 )2 ) sin(2 θ ) + 2 σ0 (1 + β0 ) cos(2 θ)].  1 1 Ψ1,1 = (−3z 2 + 7z 4 )erfc(z ) + erf(z ) 6

+

1 12

1

π−2

3 2

1 18

z2 +

4

16 3



e −2 z

60

2



3

15

π −1 3

2

[32z + (−23z + 10 z 3 )e−z ]π − 2



× (−(1 + β0 ) sin(θ ) + σ0 cos(θ )) 1  Ψ2, 0 (z , θ ) = −6 z erf(z ) + 4 z 3 erfc(z ) − 8 z 3 + 24 z 2 − 1 24  1 2 + (−12 z 4 − 8 z 2 + 1) e−z π − 2

√ 2] +

1 12

[(24 z 8 + 40 z 6

2

(947 z 3 + 420 z 7 + 518 z 5 + 522 z ) e(−z  1 3 5 7 (−2 z 2 ) + (32 z + 14 z + z + 12 z ) e π −1

2)

315 6

15

20

2

× (2 σ0 (1 + β0 ) cos(2 θ ) + ( σ0 2 − (1 + β0 )2 ) sin(2 θ )). 1  −12 z erfc(z ) ω2, 0 (z , θ ) = − 12  1 2 + (24 z 6 − 4 z 4 − 6 z 2 − 3) e−z π − 2

−

2

3

e −z

z

63

  32 16 4 32 2 + 2 (2 z 2 + 1) erfc(z ) + − z 2 − z − e −z 5

2

48



+ 66 z 4 − 54 z 2 − 33) erf(z ) − 32 z 4 − 8 z 6 − 338 z 2  √ 8 1 2 − 401 + 144 2]e(−z ) π − 2 + (360 z + 247 z 3 ) erfc(z )





+5

 

    5 4 1 79 4 2 64 2 2 + − z +z + erfc(z ) − + − z − e −z

3

1

√

  √ 37 − + 2 2 z3

erf(z ) +

+ 380 z 2 + 711 − 288

√ √ × [(−154 + 192 2 − 16 erf(z )) z 2 + 96 2 − 77]  2 + (3z + 6 z 5 + 13 z 3 ) erf(z ) −



12

8



z + 2 z3

× (−(1 + β0 ) sin(θ ) + σ0 cos(θ )). 1  ω2, 1 (z , θ ) = (4 z 3 + 6 z ) erfc(z ) 6  −1 2 + (−12 z 4 − 8 z 2 + 1) e(−z ) π 2 (σ1 cos(θ )  1 z erfc(z ) [2 erf(z ) (15 + 28 z 2 ) − β1 sin(θ )) +

1 

Ψ1, 0 (z , θ ) =

−

64



3

135



−

1 21

3

(28 z 4 + 668 z 2 + 591) e(−z ) π − 2 2



× [((1 + β0 )2 − σ0 2 ) sin(2 θ ) − 2 σ0 (1 + β0 ) cos(2 θ )]. References

8

[−16z + (−14z + 13z )e 3

× (σ1 cos(θ ) − β1 sin(θ )) +

−z 2

]π

− 21



  13 1 − + z 4 erf(z )2 8

2

    √ √ 37 77 4 z + − 6 2 z2 + −2 2 + 48

16

[1] T. Sarpkaya, A critical review of the intrinsic nature of vortex induced vibrations, J. Fluids Struct. 19 (4) (2004) 389–447. [2] P. Bearman, Circular cylinder wakes and vortex-induced vibrations, J. Fluids Struct. 27 (5–6) (2011) 648–658. [3] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [4] B.M. Sumer, J. Fredsøe, Hydrodynamics around Cylindrical Structures, World Scientific, Singapore, 1997. [5] R. Bishop, A. Hassan, The lift and drag forces on a circular cylinder oscillating in a flowing fluid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 277 (1964) 51–75.

136

Q.M. Al-Mdallal / European Journal of Mechanics B/Fluids 49 (2015) 121–136

[6] C. Williamson, A. Roshko, Vortex formation in the wake of an oscillating cylinder, J. Fluids Struct. 2 (1988) 355–381. [7] X. Lu, C. Dalton, Calculation of the timing of vortex formation from an oscillating circular cylinder, J. Fluids Struct. 10 (1996) 527–541. [8] E. Guilmineau, P. Queutey, A numerical simulation of vortex shedding from an oscillating circular cylinder, J. Fluids Struct. 16 (6) (2002) 773–794. [9] C.H.K. Williamson, R. Govardhan, A brief review of recent results in vortexinduced vibrations, J. Wind Eng. Ind. Aerodyn. 96 (6) (2008) 713–735. [10] A. Barrero-Gil, P. Fernandez-Arroyo, Fluid excitation of an oscillating circular cylinder in cross-flow, Eur. J. Mech. B Fluids 29 (5) (2010) 364–368. [11] O.M. Griffin, S.E. Ramberg, Vortex shedding from a cylinder vibrating in line with incident uniform flow, J. Fluid Mech. 75 (1976) 257–271. [12] A. Ongoren, D. Rockwell, Flow structure from an oscillating cylinder part 2: Mode competition in the near wake, J. Fluid Mech. 191 (1988) 225–245. [13] O. Cetiner, D. Rockwell, Streamwise oscillations of a cylinder in a steady current. Part 1: Locked-on stages of vortex formation and loading, J. Fluid Mech. 427 (2001) 1–28. [14] Q. Al-Mdallal, K.P. Lawrence, S. Kocabiyik, Forced streamwise oscillations of a circular cylinder: lock-on modes and resulting forces, J. Fluids Struct. 23 (5) (2007) 681–701. [15] E. Góis, L. Souza, An Eulerian Immersed Boundary Method for flow simulations over stationary and moving rigid bodies, J. Braz. Soc. Mech. Sci. Eng. 32 (2010) 477–484. [16] O.A. Marzouk, A.H. Nayfeh, Characterization of the flow over a cylinder moving harmonically in the cross-flow direction, Int. J. Non-Linear Mech. 45 (8) (2010) 821–833. [17] P. Suthon, C. Dalton, Streakline visualization of the structures in the near wake of a circular cylinder in sinusoidally oscillating flow, J. Fluids Struct. 27 (7) (2011) 885–902. [18] J.S. Leontini, D.L. Jacono, M.C. Thompson, A numerical study of an inline oscillating cylinder in a free stream, J. Fluid Mech. 688 (2011) 551–568. [19] J.S. Leontini, D.L. Jacono, M.C. Thompson, Wake states and frequency selection of a streamwise oscillating cylinder, J. Fluid Mech. 730 (2013). [20] Q. Al-Mdallal, Analysis and computation of the cross-flow past an oscillating cylinder with two degrees of freedom (Ph.D. thesis), Memorial University of Newfoundland, St. John’s, Canada, 2004. [21] D. Karanth, G.W. Rankin, K. Sridhar, Computational study of flow past a cylinder with combined in-line and transverse oscillation, Comput. Mech. 16 (1995) 1–10. [22] N. Jauvtis, C.H.K. Williamson, The effect of two degrees of freedom on vortexinduced vibration at low mass and damping, J. Fluid Mech. 509 (2004) 23–62. [23] E. Didier, A.R.J. Borges, Numerical predictions of low Reynolds number flow over an oscillating circular cylinder, J. Comput. Appl. Mech. 8 (1) (2007) 39–55. [24] A. Sanchis, G. Sælevik, J. Grue, Two-degree-of-freedom vortex-induced vibrations of a spring-mounted rigid cylinder with low mass ratio, J. Fluids Struct. 24 (6) (2008) 907–919. [25] L. Baranyi, Numerical simulation of flow around an orbiting cylinder at different ellipticity values, J. Fluids Struct. 24 (6) (2008) 883–906. [26] L. Baranyi, Orbiting cylinder at low Reynolds numbers, in: IUTAM Symposium on Unsteady Separated Flows and their Control, in: IUTAM Bookseries, vol. 14, 2009, pp. 161–165. [27] L. Baranyi, Simulation of a low-Reynolds number flow around a cylinder following a figure-8-path, Int. Rev. Appl. Sci. Eng. 3 (2) (2012) 133–146. [28] G. Reid, On the application of active open-loop and closed-loop controls on a circular cylinder in the presence and absence of a free surface (Ph.D. thesis), Memorial University of Newfoundland, St. John’s, Canada, 2010. [29] Q. Al-Mdallal, A numerical study of initial flow past a circular cylinder with combined streamwise and transverse oscillations, Comput. & Fluids 63 (2012) 174–183. [30] D. Jeon, M. Gharib, On circular cylinders undergoing two-degree-of-freedom forced motions, J. Fluids Struct. 15 (2001) 533–541. [31] M. Nazarinia, D.L. Jacono, M. Thompson, J. Sheridan, Flow behind a cylinder forced by a combination of oscillatory translational and rotational motions, Phys. Fluids 21 (5) (2009) 51701.

[32] M. Nazarinia, D.L. Jacono, M. Thompson, J. Sheridan, The three-dimensional wake of a cylinder undergoing a combination of translational and rotational oscillation in a quiescent fluid, Phys. Fluids 21 (6) (2009) 064101. [33] I. Teschauer, M. Schäfer, A. Kempf, Numerical simulation of flow induced by a cylinder orbiting in a large vessel, J. Fluids Struct. 16 (2002) 435–451. [34] M.M. Chen, C. Dalton, L. Zhuang, Force on a circular cylinder in an elliptical orbital flow at low Keulegan–Carpenter numbers, J. Fluids Struct. 9 (1995) 617–638. [35] E. Konstantinidis, C. Liang, Dynamic response of a turbulent cylinder wake to sinusoidal inflow perturbations across the vortex lock-on range, Phys. Fluids 23 (2011) 075102. [36] S.C.R. Dennis, L. Quartapelle, Some used of Green’s theorem in solving the Navier–Stokes equations, Internat. J. Numer. Methods Fluids 9 (1989) 871–890. [37] S.C.R. Dennis, S. Kocabiyik, An asymptotic matching condition for unsteady boundary-layer flows governed by the Navier–Stokes equations, IMA J. Appl. Math. 47 (1991) 81–98. [38] W.M. Collins, S.C.R. Dennis, Flow past an impulsively started circular cylinder, J. Fluid Mech. 60 (1973) 105–127. [39] H.M. Badr, S.C.R. Dennis, Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder, J. Fluid Mech. 158 (1985) 447–488. [40] H.M. Badr, M. Coutanceau, S.C.R. Dennis, C. Menard, Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104 , J. Fluid Mech. 220 (1990) 459–484. [41] S.C.R. Dennis, P. Nguyen, S. Kocabiyik, The flow induced by a rationally oscillating and translating circular cylinder, J. Fluid Mech. 407 (2000) 123–144. [42] F. Mahfouz, H. Badr, Flow structure in the wake of a rotationally oscillating cylinder, Trans. ASME, J. Fluids Eng. 122 (2000) 290–301. [43] S. Kocabiyik, F.M. Mahfouz, Q. Al-Mdallal, Numerical simulation of the flow behind a circular cylinder subject to small-amplitude recti-linear oscillations, Adv. Eng. Softw. 35 (2004) 619–631. [44] K.P. Lawrence, Computation of unsteady viscous incompressible flow around an obliquely oscillating circular cylinder using a parallelized finite difference algorithm (Master’s thesis), Memorial University, Newfoundland, Canada, 2004. [45] Q. Al-Mdallal, S. Kocabiyik, Rotational oscillations of a cylinder in cross-flow, Int. J. Comput. Fluid Dyn. 20 (5) (2006) 293–299. [46] S.C.R. Dennis, G.Z. Chang, Numerical Integration of the Navier–Stokes Equations, Technical Summary Report No. 859, Mathematical Research Center, University of Wisconsin, 1969. [47] P. Poncet, Topological aspects of three-dimensional wakes behind rotary oscillating cylinders, J. Fluid Mech. 517 (2004) 27–53. [48] R.D. Henderson, Nonlinear dynamics and patterns in turbulent wake transition, J. Fluid Mech. 352 (1997) 65–112. [49] C.-Y. Wen, C.-Y. Lin, Two-dimensional vortex shedding of a circular cylinder, Phys. Fluids 13 (2001) 557–560. [50] J.R. Meneghini, P.W. Bearman, Numerical simulation of high amplitude oscillatory flow about a circular cylinder, J. Fluids Struct. 9 (1995) 435–455. [51] L. Mironova, Accurate computation of free surface flow with an oscillating circular cylinder based on a viscous incompressible two-fluid model (Ph.D. thesis), Memorial University of Newfoundland, St. John’s, Canada, 2008. [52] S. Su, M. Lai, C. Lin, An immersed boundary technique for simulating complex flows with rigid boundary, Comput. & Fluids 36 (2) (2007) 313–324. [53] C.H.K. Williamson, The existence of two stages in the transition to threedimensionality of a cylinder wake, Phys. Fluids 31 (1988) 3165–3168. [54] L. Du, Numerical study of flow past rotating and rotationally oscillating cylinders (Ph.D. thesis), University of Houston, Houston, Texas, United States of America, 2011. [55] S. Mittal, B. Kumar, Flow past a rotating cylinder, J. Fluid Mech. 476 (2003) 303–334.