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CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 15, Number 3, Fall 2007 NUMERICAL SIMULATIONS OF FLOW PAST AN OBLIQUELY OSCILLATING ELLIPTIC CYLINDER S. J. D. D’ALESSIO AND SERPIL KOCABIYIK ABSTRACT. The present work deals with the numerical investigation of the unsteady flow created by oblique transla- tional oscillations of an inclined elliptic cylinder placed in a steady uniform flow of a viscous incompressible fluid. The mo- tion is assumed to start impulsively from rest at t = 0. The flow is two-dimensional and the harmonic oscillations act in a direc- tion 45 to the uniform oncoming flow. The unsteady Navier- Stokes equations, expressed in terms of stream function and vorticity, are solved using an implicit spectral finite-difference procedure. Examined in this study is the wake evolution for inclinations η =0, π/4 and times 0 <t 12 for a Reynolds number of 10 3 and a fixed minor-major axis ratio of 0.5. The effect of the oblique translational oscillations of the cylinder on the hydrodynamic forces has been determined and contrasted with the corresponding transverse and inline oscillation cases. 1 Introduction Flows past bluff bodies are important from the standpoint of fundamental research and in the design and maintenance of engineering structures. In the case of uniform flow past a station- ary cylinder, the forces that are experienced by the cylinder tend to be steady at Reynolds numbers below 40. At higher Reynolds num- bers the flow in the wake of the cylinder becomes unsteady and a von arm´ an vortex street develops. As a result, the forces that are imposed by the fluid upon the body become oscillatory in nature. This leads to, in most cases, the generation of marked vibrations on the cylindri- cal body. In actual engineering situations, the oscillation is caused by periodic fluctuations in the the external flow or by forced oscillations of the body itself. In order to design engineering structures to withstand the vibration, it is necessary to investigate the effects of translational cylinder oscillations. The circular cylinder has been the generic bluff Keywords: viscous, incompressible, unsteady, elliptic cylinder, oblique transla- tional oscillation, spectral finite-difference scheme. Copyright c Applied Mathematics Institute, University of Alberta. 247

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Page 1: NUMERICAL SIMULATIONS OF FLOW PAST AN OBLIQUELY ... · NUMERICAL SIMULATIONS OF FLOW PAST AN OBLIQUELY OSCILLATING ELLIPTIC CYLINDER S. J. D. D’ALESSIO AND SERPIL KOCABIYIK ABSTRACT

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 15, Number 3, Fall 2007

NUMERICAL SIMULATIONS OF FLOW

PAST AN OBLIQUELY OSCILLATING

ELLIPTIC CYLINDER

S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

ABSTRACT. The present work deals with the numericalinvestigation of the unsteady flow created by oblique transla-tional oscillations of an inclined elliptic cylinder placed in asteady uniform flow of a viscous incompressible fluid. The mo-tion is assumed to start impulsively from rest at t = 0. The flowis two-dimensional and the harmonic oscillations act in a direc-tion 45 to the uniform oncoming flow. The unsteady Navier-Stokes equations, expressed in terms of stream function andvorticity, are solved using an implicit spectral finite-differenceprocedure. Examined in this study is the wake evolution forinclinations η = 0, π/4 and times 0 < t ≤ 12 for a Reynoldsnumber of 103 and a fixed minor-major axis ratio of 0.5. Theeffect of the oblique translational oscillations of the cylinder onthe hydrodynamic forces has been determined and contrastedwith the corresponding transverse and inline oscillation cases.

1 Introduction Flows past bluff bodies are important from thestandpoint of fundamental research and in the design and maintenanceof engineering structures. In the case of uniform flow past a station-ary cylinder, the forces that are experienced by the cylinder tend tobe steady at Reynolds numbers below 40. At higher Reynolds num-bers the flow in the wake of the cylinder becomes unsteady and a vonKarman vortex street develops. As a result, the forces that are imposedby the fluid upon the body become oscillatory in nature. This leadsto, in most cases, the generation of marked vibrations on the cylindri-cal body. In actual engineering situations, the oscillation is caused byperiodic fluctuations in the the external flow or by forced oscillations ofthe body itself. In order to design engineering structures to withstandthe vibration, it is necessary to investigate the effects of translationalcylinder oscillations. The circular cylinder has been the generic bluff

Keywords: viscous, incompressible, unsteady, elliptic cylinder, oblique transla-tional oscillation, spectral finite-difference scheme.

Copyright c©Applied Mathematics Institute, University of Alberta.

247

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248 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

body used for simulations of flows around bluff bodies. However, the as-sociated wake structures within a hundred diameters downstream onlyencompass a subset of wake structures associated with bluff bodies (seeJohnson et al. [10]). Consequently, other simple bluff bodies should alsobe considered in order to fully understand bluff body wake dynamics.An elliptic geometry represents an obvious and welcome extension, al-lowing a wide range of cross-sections ranging from a circular cylinderto a flat plate depending on the minor-major axis ratio. In addition,the angle of attack of the ellipse acts as another parameter which canalter the wake structure. For example, at low angles of attack and forthin elliptic cylinders the flow generally remains attached to the bodysurface and behaves in a similar manner to that of a conventional air-foil. Whereas, at high angles of attack and for thicker ellipses the flowseparates and a bluff-body flow regime results. The present study dealswith a numerical investigation of a class of flows produced by obliquetranslational oscillations of an elliptic cylinder placed in a cross-flow.Numerical solutions of uniform flow past elliptic cylinders at various an-gles of attack were obtained by Staniforth [18], Lugt and Haussling [12],Patel [17], Mittal and Balachandar [14], and Nair and Sengupta [15].Badr et al. [2] have summarized these studies. For flows induced by anelliptic cylinder undergoing translational oscillations in the presence ofan oncoming uniform stream references may only be made to the worksof Okajima et al. [16], D’Alessio and Kocabiyik [5], and Kocabiyik andD’Alessio [11]. In these studies inline or transverse oscillations of anelliptic cylinder were considered.

In the present work we consider the two-dimensional flow caused by aninfinitely long elliptic cylinder impulsively set in motion and translatingwith uniform velocity U∞. In addition, the cylinder is also undergoingharmonic oscillations in a direction of 45 with the horizontal free-streamdirection. The cylinder is inclined at an angle η with the horizontal. Theellipse has major and minor axis of lengths 2a and 2b, respectively, andthe cylinder oscillates with the velocity U cosωt∗ where ω = 2πf withf denoting the forced frequency of oscillation. The Reynolds number isdefined by R = 2cU∞/ν where c =

√a2 − b2 is the focal length and ν

is the kinematic viscosity. The velocity ratio, α = U/U∞, the forcingStrouhal number, Ω = c ω/U∞, the angle of inclination, η, and theminor-to-major axis ratio of the ellipse, r = b/a, serve as dimensionlesscontrol parameters.

The method of solution is an extension of the method developed byStaniforth [18] that takes into account cylinder oscillations in a direc-tion of 45 with the horizontal free stream. In the works of D’Alessio et

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NUMERICAL SIMULATIONS OF FLOW 249

al. [4], D’Alessio and Kocabiyik [5] and Kocabiyik and D’Alessio [11]the numerical technique of Staniforth was successfully extended to com-pute the development of the flows imposed by oscillatory motion, eitherrectilinear or rotational, of an inclined elliptic cylinder. In D’Alessio andKocabiyik [5] the problem of an elliptic cylinder subject to transverseoscillations was solved while in Kocabiyik and D’Alessio [11] the caseof inline oscillations was addressed. D’Alessio et al. [4], on the otherhand, considered a uniform flow past a thin inclined elliptic cylinderunder rotary oscillations. In a subsequent study, the early stages of flowdevelopment over elliptic airfoils oscillating in pitch at large angles ofattack was simulated by Akbari and Price [1].

The goal of the present study is to investigate the effects of the ellipseinclination angle, η, on the flow structure in the near-wake region aswell as on the hydrodynamic forces acting on the cylinder for a fixedReynolds number of R = 103, forcing Strouhal number of Ω = π andvelocity ratio of α = 0.25. Numerical calculations are performed formoderate times 0 < t ≤ 12 and for inclinations η = π/4 and η = 0 foran ellipse having r = 0.5. Noticeable changes in the near-wake and inthe forces take place as η varies and are reported. It is noted that Ωand α are maintained at Ω = π and α = 0.25 for the present study sinceflow structure in such cases is characterized by the formation of vortexpairs which convect away from the body, forming wakes. In general, theeffect of the decrease of the oscillation amplitude is to reduce the sizeof the separated region. However, for the sufficiently small oscillationamplitude range, α/Ω 1, when no flow separation takes place, we havethe unexpected result that jets issue from the cylinder surface followinga boundary-layer collision. The emergence of a thin round jet along theaxis of oscillation was first predicted and visualized by Davidson andRiley [6] for the case of purely translational oscillations of an ellipticcylinder placed in a quiescent viscous fluid.

The underlying assumptions made in this study, as in previous ones,are that the flow remains two-dimensional and laminar. One can arguethat for the Reynolds number regime considered three-dimensional ef-fects and turbulence may significantly alter the flow. In fact, experimen-tal work conducted by Williamson [20] for the case of a circular cylindersuggests that a three dimensional transition occurs for Reynolds num-bers R > 178. This was also confirmed by Zhang et al. [21]. Szepessyand Bearman [19] measured a fluctuating lift on a thin section of a largeaspect-ratio fixed-circular cylinder and found that two-dimensional sim-ulation schemes generally overestimate the root-mean-square value ofthe fluctuating lift. This observation has been substantiated by Gra-

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250 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

ham [9], who gathered numerical predictions for circular cylinder flowand compared them with experimental results. He found that above aReynolds number of about 150 the mean and fluctuating forces were gen-erally overpredicted, with largest differences occurring in the fluctuatinglift. It should also be noted that measured time histories of the fluctu-ating lift show a pronounced amplitude modulation whereas simulatedtime histories mostly display a constant amplitude, once the flow has set-tled. However, forcing a bluff body to oscillate introduces a mechanismfor synchronizing the moment of shedding along its length. With thisconsideration, two dimensional numerical simulations should be reliablein terms analyzing flow details, at least in the near wake region. Forexample, the work of Blackburn and Henderson [3] supports the notionthat cylinder vibrations tend to suppress the three-dimensionality andproduce flows that are more two-dimensional than their fixed cylindercounterparts.

2 Formulation and governing equations In the present paperwe consider the two-dimensional flow generated by an infinitely longelliptic cylinder whose axis coincides with the z-axis placed in a viscousincompressible fluid. The cylinder is inclined at an angle η with thehorizontal and the major and minor axes are taken to lie along the x andy axes, respectively. Initially, the cylinder is at rest and at time t = 0 itsuddenly starts to translate horizontally with uniform velocity U∞ andalso oscillates harmonically in a direction of 45 with the horizontal.Equivalently, as shown in Figure 1, we take the cylinder to oscillate andthe fluid to flow past it with uniform velocity U∞.

A mathematically convenient non-inertial frame of reference whichtranslates and oscillates with the cylinder is employed. In this frame theunsteady dimensionless equations for a viscous incompressible fluid inprimitive variables can be written in vector form as

∂~v

∂t= −~∇

(

p+1

2| ~v |2

)

− 2

R~∇× ~ω + ~a,(1)

~∇ · ~v = 0.(2)

Here, t is the non-dimensional time defined by t = U∞t∗/c with t∗

denoting the dimensional time. For two-dimensional flow taking placein the xy-plane, the velocity is ~v = (u, v, 0) and the vorticity is given by

~ω = ~∇×~v = (0, 0, ζ). The term ~a is the translational acceleration arisingfrom the non-inertial reference frame of the vibrating cylinder. This

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NUMERICAL SIMULATIONS OF FLOW 251

FIGURE 1: The coordinate system and the flow configuration.

translational acceleration is easily derived as follows. If ~u and ~v (not tobe confused with the scalar velocity components u and v) denote the non-dimensional velocities in the fixed and non-inertial frames, respectively,then we have that

(3) ~u = ~v + αh(cos η,− sin η) cos(Ωt) + αv(sin η, cos η) cos(Ωt)

where Ω is the non-dimensional angular frequency of oscillation andαh, αv denote non-dimensional peak velocities of oscillation in the hor-izontal and vertical directions respectively. The term ~a is then relatedto the time derivative of the last two terms in (3) and is hence given by

~a = Ωαh(cos η,− sin η) sin(Ωt) + Ωαv(sin η, cos η) sin(Ωt).

We note that the special cases of transverse and inline oscillations arerecovered by setting αh = 0 and αv = 0, respectively. These cases arereported in the studies of D’Alessio and Kocabiyik [5] and Kocabiyikand D’Alessio [11]. To orchestrate oscillations in a direction of 45 withthe oncoming flow we simply set αh = αv ≡ α.

Since the appropriate coordinates for the present problem are theelliptic coordinates (ξ, θ), we use the following conformal transformationwhich relates the elliptic coordinates (ξ, θ) to the Cartesian coordinates(x, y):

x = cosh(ξ + ξ0) cos θ, y = sinh(ξ + ξ0) sin θ.

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252 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

Here, the constant ξ0 = tanh−1(b/a), and ξ = 0 defines the surface ofthe cylinder. Using the elliptic coordinate system, with the origin at thecenter of the cylinder, the equations of motion can be written in termsof the vorticity, ζ, and the stream function, ψ, in dimensionless form as

∂2ψ

∂ξ2+∂2ψ

∂θ2= M2ζ,(4)

∂ζ

∂t=

1

M2

[

2

R

(

∂2ζ

∂ξ2+∂2ζ

∂θ2

)

+

(

∂ψ

∂θ

∂ζ

∂ξ− ∂ψ

∂ξ

∂ζ

∂θ

)]

.(5)

The dependent variables ψ, ζ in these equations are defined in terms ofthe usual dimensional quantities as ψ∗ = U∞cψ, ζ∗ = U∞ζ/c and theJacobian of the above transformation, M 2, is given by

(6) M2 =1

2[cosh 2(ξ + ξ0) − cos 2θ] .

Expressions for the dimensionless velocity components (vξ , vθ) in thedirections of increase of (ξ, θ) in terms of the stream function ψ aregiven by

vξ = − 1

M

∂ψ

∂θ, vθ =

1

M

∂ψ

∂ξ,

and the vorticity ζ is defined in terms of the velocity components as

ζ =1

M2

(

− ∂

∂θ(Mvξ) +

∂ξ(Mvθ)

)

.

The boundary conditions for t > 0 and 0 ≤ θ ≤ 2π are the impermeabil-ity and no-slip conditions on the cylinder surface given by

(7) ψ =∂ψ

∂ξ= 0 when ξ = 0.

The far-field conditions can be derived by first noting that in the fixedframe ~u→ (− cos η, sin η) as x2 + y2 → ∞. Then from (3) we obtain

~v =

(

−∂ψ∂y

,∂ψ

∂x

)

→ (− cos η, sin η) − αh(cos η,− sin η) cos(Ωt)

− αv(sin η, cos η) cos(Ωt)

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NUMERICAL SIMULATIONS OF FLOW 253

as x2 + y2 → ∞. In terms of elliptic coordinates the above conditionsare expressed as

∂ψ

∂ξ→ 1

2eξ+ξ0 sin(θ + η) +

1

2eξ+ξ0 [αh sin(θ + η)(8)

− αv cos(θ + η)] cos(Ωt) as ξ → ∞,

∂ψ

∂θ→ 1

2eξ+ξ0 cos(θ + η) +

1

2eξ+ξ0 [αh cos(θ + η)(9)

+ αv sin(θ + η)] cos(Ωt) as ξ → ∞,

or equivalently as

ψ → 1

2eξ+ξ0 sin(θ + η) +

1

2eξ+ξ0 [αh sin(θ + η)(10)

− αv cos(θ + η)] cos(Ωt) as ξ → ∞.

The far-field vorticity, on the other hand, satisfies

(11) ζ → 0 as ξ → ∞.

The surface boundary conditions given by (7) for the stream functionare overspecified. Boundary condition (11) gives a requirement for thevorticity in the far field, but there is no explicit condition for the vor-ticity on the cylinder surface. In principle, the surface vorticity canbe computed from the known stream function by applying equation (4),however the large velocity gradient at the surface reduces the accuracy ofsuch computations. In this study integral conditions are used to predictthe surface vorticity. Following the works of Dennis and Quartepelle [7],and Dennis and Kocabiyik [8], the conditions (7)–(10) for the streamfunction are transformed into a set of global integral conditions for thevorticity using equation (4). These conditions are derived by applyingGreen’s second identity for the Laplacian operator, namely

∫∫

V

(φ∇2ψ − ψ∇2φ) dV =

S

(

φ∂ψ

∂n− ψ

∂ψ

∂n

)

ds ,

to the flow domain V exterior to the cylinder. Here, the boundary S ofthe flow domain is the contour of the cylinder itself together with a con-tour at a large distance, ~n refers to the outward pointing normal to theboundary S of the flow domain, and s is measured along contour. Taking

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254 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

φ to be the set of harmonic functions φ = 1, e−nξ cosnθ, e−nξ sinnθ :n = 1, 2, ... and using ∇2ψ = M2ζ from (4), it follows after someintegration by parts and making use of the far-field conditions that:

0

∫ 2π

0

M2ζ(ξ, θ, t) dθ dξ = 0,(12)

0

∫ 2π

0

e−nξM2ζ(ξ, θ, t) cos(nθ)dθ dξ(13)

= πeξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)] δ1,n,

0

∫ 2π

0

e−nξM2ζ(ξ, θ, t) sin(nθ)dθ dξ(14)

= πeξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)] δ1,n,

for all integers n ≥ 1. These are employed in the solution procedure toensure that all necessary conditions of the problem are satisfied. Here,δm,n is the Kronecker delta symbol defined by

δm,n = 1 if m = n, and δm,n = 0 if m 6= n.

The use of integral conditions can be found in the works of Stani-forth [18], D’Alessio et al. [4], Badr et al. [2], Mahfouz and Kocabiyik [13],and Kocabiyik and D’Alessio [11], to mention a few of the various ap-plications.

Lastly, an initial condition is necessary to start the flow. Boundary-layer theory for impulsively started flows is used to provide this by uti-lizing the boundary-layer transformation

(15) ξ = kz, ψ = kΨ, ζ =ω

k, k = 2

(

2t

R

)1/2

,

which maps the initial flow onto the scale of the boundary-layer thick-ness. The governing equations and the boundary and integral conditionsare first transformed using (15). The equations and boundary conditionson the cylinder surface satisfied by Ψ are given by D’Alessio and Ko-cabiyik [5] and the integral conditions in the present case take the form

0

∫ 2π

0

M2 ω(z, θ, t) dθ dz = 0,(16)

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NUMERICAL SIMULATIONS OF FLOW 255

0

∫ 2π

0

e−nkzM2 ω(z, θ, t) cos(nθ)dθ dz(17)

= πeξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)] δ1,n,

0

∫ 2π

0

e−nkzM2 ω(ξ, θ, t) sin(nθ)dθ dz(18)

= πeξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)] δ1,n,

for all integers n ≥ 1. It is noted that the integral conditions givenby (17) and (18) differ from those given in D’Alessio and Kocabiyik [5]and Kocabiyik and D’Alessio [11] owing to the difference in cylindermotions. The initial solution at t = 0 is obtained following the work byStaniforth [18]. This initial solution is given by

ω0(z, θ, 0) =2√π

eξ0

M0

[(1 + αh) sin(θ + η) − αv cos(θ + η)]e−M2

0z2

,(19)

Ψ0(z, θ, 0) =eξ0

M0

[(1 + αh) sin(θ + η) − αv cos(θ + η)](20)

×[

M0z erf(M0z) −1√π

(1 − e−M2

0z2

)

]

,

where erf(M0z) denotes the error function andM 20 = [cosh 2ξ0−cos 2θ]/2.

This initial solution forms the starting point of the numerical integrationprocedure which is outlined in the following section.

3 Numerical solution summary The transformed vorticity trans-port equation for ω in terms of the coordinates (z, θ) is solved by fi-nite differences using a Gauss-Seidel iterative procedure with under-relaxation applied only to the surface vorticity. Since the procedureis similar to that used in the studies of D’Alessio et al. [4], D’Alessioand Kocabiyik [5], and Kocabiyik and D’Alessio [11], we will brieflydescribe the numerical technique. The computational domain, boundedby 0 ≤ z ≤ z∞ and 0 < θ < 2π, is first discretized into a network ofL× P equally spaced grid points located at

zi = ih, i = 0, 1, . . . , L where h = z∞/L,

θj = jλ, j = 0, 1, . . . , P where λ = 2π/P.

Here z∞ refers to the outer boundary approximating infinity.

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256 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

We express the stream function in the form of a truncated Fourierseries

(21) Ψ(z, θ, t) =1

2F0(z, t) +

N∑

n=1

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

The equations governing the Fourier coefficients are

(22)

∂2Fn

∂z2− n2k2Fn = sn(z, t); n = 0, 1, . . . ,

∂2fn

∂z2− n2k2fn = rn(z, t); n = 1, 2, . . . ,

where

(23)

sn(z, t) =1

π

∫ 2π

0

M2ω(z, θ, t) cosnθdθ,

rn(z, t) =1

π

∫ 2π

0

M2ω(z, θ, t) sinnθdθ.

Boundary conditions for the Fourier components of Ψ are

F0(0, t) = Fn(0, t) = fn(0, t) = 0,

∂F0

∂z=∂Fn

∂z=∂fn

∂z= 0 when z = 0,

and as z → ∞,

e−kzF0 → 0, e−kz ∂F0

∂z→ 0,

e−kzFn → 1

2keξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)]δn,1,

e−kz ∂Fn

∂z→ 1

2eξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)]δn,1,

e−kzfn → 1

2keξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)]δn,1,

e−kz ∂fn

∂z→ 1

2eξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)]δn,1,

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NUMERICAL SIMULATIONS OF FLOW 257

for all integers n ≥ 1. The integral conditions can be formulated interms of the functions rn(z, t) sn(z, t) as follows:

0

s0(z, t)dθ dz = 0,(24)

0

e−nkzsn(z, t) dz(25)

= eξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)]δ1,n,∫

0

e−nkzrn(z, t) dz(26)

= eξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)]δ1,n,

for all integers n ≥ 1. These conditions play an important role in thedetermination of the surface vorticity as we shall shortly see. They are,in fact, equivalent to the one-dimensional form of the Green’s theoremconstraint given by Dennis and Quartapelle [7].

The above equations (22) for a given n at a fixed time are of the form

(27) h′′(z) − β2 h(z) = g(z)

where β = nk and the prime refers to differentiation with respect toz. A special scheme is used for integrating these ordinary differentialequations using step-by-step formulae. As previously mentioned, thevorticity transport equation is solved by finite differences. The schemeused to approximate this equation is very similar to the Crank-Nicolsonimplicit procedure. The specific details will not be presented, but can befound in [18, 5, 11] The surface vorticity, which is needed to completethe integration procedure, is determined by inverting (23) and is givenby the following expression

ω(0, θ, t)

=1

M20

1

2s0(0, t) +

N∑

n=1

[rn(0, t) sin(nθ) + sn(0, t) cos(nθ)]

.

(28)

It may also be necessary to subject the surface vorticity to under-relaxation in order to obtain convergence.

The integration procedure is initiated using the initial solution (19)and (20) at t = 0. The use of this initial solution is essential for obtaining

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258 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

accurate results at small times. The potential flow solution, in this casegiven by

ψ(ξ, θ, t) = eξ0 sinh ξ sin(θ + η) + eξ0 sinh ξ

× [αh sin(θ + η) − αv cos(θ + η)] cos(Ωt),

(29)

has also been used as an initial condition at t = 0 in previous relatedstudies. This, however, will definitely lead to inaccurate results fol-lowing the start of the fluid motion. It is noted that the potential flowsolution (29) can easily be obtained by solving the stream function equa-tion (4), after setting ζ = 0, subject to the impermeability and far-fieldconditions.

Because of the impulsive start, small time steps were needed to getpast the singularity at t = 0. Initially, ∆t = 10−4 was used; as timeincreased the time step was gradually increased until reaching t = 0.01.For t > 0.01 the time step ∆t = 0.01 was used. The grid size ∆zin the coordinate z is more or less independent of R. The number ofpoints in the z-direction is taken to be 201 with a uniform spacing of∆z = 0.06. This sets the outer boundary of our computational domain(z∞ = 12) at a physical distance of about 40 major axis lengths away fora Reynolds number R = 103 and time t = 12. Placing z∞ well outsidethe growing boundary-layer enables us to enforce the far-field conditions(10)–(11) along the outer edge of our computational domain z = z∞so that the application of the far-field conditions does not affect thesolution in the viscous region near the cylinder surface. We point outthat the physical coordinate ξ = kz is a moving coordinate and hencethe outer boundary ξ∞ = kz∞ is constantly being pushed further awayfrom the cylinder surface with time. For this reason we are justified insaying that the vorticity, by the mechanism of convection, will not reachthe outer boundary ξ∞. The maximum number of terms retained in theseries (21) was 51 which corresponds to an upper limit of N = 25 in thesums. Checks were made for R = 103 at several times to ensure that Nis large enough. This was done by increasing N and observing that thesolution did not change appreciably. The computational parameters areto some extent chosen to be comparable with those used by D’Alessioand Kocabiyik [5], since these were found to be satisfactory and werechecked carefully. Moreover, this scheme is tested against the resultsof Staniforth [18] for the non-oscillating (i.e., purely translating) caseusing similar Reynolds numbers; tests indicate that the solutions arequite accurate.

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NUMERICAL SIMULATIONS OF FLOW 259

4 Results and discussion The numerical results are grouped intwo cases, (i) η = π/4 and (ii) η = 0, to illustrate the effect of inclinationon the ensuing flow. In the case of η = π/4 the cylinder oscillates ina direction of the major axis of the ellipse while for the case η = 0 theoscillations are at 45 to the direction of the major axis. For each ofthese cases the other parameters characterizing the flow were fixed andthe values were R = 103, Ω = π, r = 0.5 and αh = αv ≡ α = 0.25.The results are presented in the form of streamline patterns as wellas time variations of the drag and lift coefficients and surface vorticitydistributions. The special cases of transverse and inline oscillations withη = π/4 were also computed for comparison purposes. A brief derivationof the formulae for the force coefficients is outlined in Section 4.1.

We note that the two cases considered in the present work deal withthe same oscillation frequency of f = 0.5 and therefore have a periodT = 1/f = 2. Thus, a complete cycle consists of the following fourstages: at t = 0 the ellipse starts to oscillate with its maximum velocityin a direction of 45 to the horizontal free stream. At t = 0.5 the ellipsereaches its maximum oblique displacement in this direction and is in aninstantaneous state of rest. At t = 1 the ellipse is in its equilibriumposition and again attains maximum velocity in the opposite direction.Then, at t = 1.5 the ellipse occupies its other extremum displacementposition and is again in an instantaneous state of rest. Finally, at t = 2the ellipse is in its starting position and this pattern repeats itself.

The plots to be presented are from the vantage point of the non-inertial reference frame of the cylinder; consequently, the oncoming flowdirection (from right to left) will appear to periodically rotate. At eventimes it will appear to approach the cylinder from above while at oddtimes from below. At half times (i.e., t = 6.5, 7.5, 10.5, 11.5) the oncom-ing flow approaches the cylinder horizontally since at these times thecylinder is momentarily at rest. What will also be noticed in the flowpatterns is a cyclic variation in the spacing between consecutive stream-lines. This is a result of the periodic changes in the relative velocitybetween the cylinder and the oncoming flow caused by the oscillations.At even times the spacing is smallest due to the higher relative velocitywhile at odd times the spacing is largest in accordance with a lower rela-tive velocity. With the above points in mind, we now proceed to presentand discuss the flow patterns for the two cases: η = π/4 and η = 0.Lastly, the flow patterns for each case were plotted at the same times soas to make comparisons and differences easier to report.

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260 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

4.1 Derivation of the formulae for force coefficients, CD and

CL If L and D are the dimensional lift and drag on the cylinder, thedimensionless drag, CD, and lift, CL, coefficients are then defined byCD = D/ρU2

∞c and CL = L/ρU2

∞c, respectively. The drag and lift

coefficients were computed using

CD =

(

2 sinh ξ0R

∫ 2π

0

(

∂ζ

∂ξ

)

0

sin θdθ(30)

− 2 cosh ξ0R

∫ 2π

0

ζ0 sin θdθ

)

cos η

+

(

2 cosh ξ0R

∫ 2π

0

(

∂ζ

∂ξ

)

0

cos θdθ

− 2 sinh ξ0R

∫ 2π

0

ζ0 cos θdθ

)

sin η

− παhΩ cosh ξ0 sinh ξ0 sin(Ωt),

CL =

(

− 2 cosh ξ0R

∫ 2π

0

(

∂ζ

∂ξ

)

0

cos θdθ(31)

+2 sinh ξ0

R

∫ 2π

0

ζ0 cos θdθ

)

cos η

+

(

2 sinh ξ0R

∫ 2π

0

(

∂ζ

∂ξ

)

0

sin θdθ

− 2 cosh ξ0R

∫ 2π

0

ζ0 sin θdθ

)

sin η

− παvΩ sinh ξ0 cosh ξ0 sin(Ωt).

The first integral in each expression of CD and CL represents the coeffi-cient due to pressure and the second terms are due to friction. The lastterm in the expressions for CD and CL represents the inviscid contribu-tion which results from the acceleration due to time dependent cylinderoscillations. We point out that at t = 0 both CD and CL are infinitein magnitude due to the impulsive start delivered to the cylinder att = 0; afterwards CD and CL decrease rapidly. There are two dom-inating flow fields affecting the boundary-layer region. The first is apotential flow field while the second is that resulting from vortical mo-tion. In the present problem, the second field has a negligible effect at

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NUMERICAL SIMULATIONS OF FLOW 261

the start of the motion but as time increases so does its influence. Sucha field continues to evolve with time until eventually reaching a nearperiodic behaviour after many oscillations. The frictional forces makesmall contributions to the total force in comparison with the inviscidcontributions due to the moderately large Reynolds number considered.

A brief derivation of the formulae (30) and (31) can be outlined asfollows. In terms of elliptic coordinates, the θ-component of the dimen-sionless momentum equation (1) in a reference frame that translates andoscillates with the cylinder becomes

∂vθ

∂t= − 1

M

∂θ

(

p+1

2

[

v2ξ + v2

θ

]

)

− Ω

Mcosh(ξ + ξ0) sin θ(αh cos η + αv sin η) sin(Ωt)

Msinh(ξ + ξ0) cos θ(−αh sin η + αv cos η) sin(Ωt)

+2

MR

∂ζ

∂ξ.

(32)

On the cylinder surface (32) simplifies greatly owing to the imperme-ability and no-slip conditions, and becomes

(

∂P

∂θ

)

0

=2

R

(

∂ζ

∂ξ

)

0

− Ω cosh ξ0 sin θ(αh cos η + αv sin η) sin(Ωt)

+ Ω sinh ξ0 cos θ(−αh sin η + αv cos η) sin(Ωt).

(33)

The forces in the horizontal and vertical directions, X and Y , respec-tively, can be obtained by integrating the pressure and frictional stresseson the surface. This leads to

X = sinh ξ0

∫ 2π

0

(

∂P

∂θ

)

0

sin θdθ − 2 cosh ξ0R

∫ 2π

0

ζ0 sin θdθ,(34)

Y = − cosh ξ0

∫ 2π

0

(

∂P

∂θ

)

0

cos θdθ +2 sinh ξ0

R

∫ 2π

0

ζ0 cos θdθ.(35)

Making use of (33), the relations

CD = X cos η − Y sin η and CL = X sin η + Y cos η,

and simplifying then yields equations (30)–(31).

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262 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

4.2 Streamline patterns and force coefficients for R = 103, Ω =π, α = 0.25, r = 0.5 and η = π/4 Figures 2a–2l show instantaneoussnap shots of the flow field for the time interval 4 ≤ t ≤ 12 and capturesthe details of the flow patterns during three of the six cycles of oscilla-tion. The vortices in the near wake are simply the result of one vortexin each half oscillation cycle. Close to the cylinder, Figure 2a at t = 4,a counterclockwise vortex pair exists in the upper half of the cylinderand is convected downstream with the aid of the cylinder motion alongits major axis. This continues until t = 5 and Figure 2b shows theformation of a triple-vortex arrangement behind the cylinder with thebottom vortices rotating clockwise and the top vortex rotating counter-clockwise. At t = 6, Figure 2c, a single large vortex forms indicatingthat the other vortices weakened as they were advected downstream. Att = 6.5, Figure 2d, the large vortex has just been shed and a new vortexhas formed near the leading edge. Figures 2e and 2f, at times t = 7 andt = 7.5, reveal that another vortex has formed near the leading edge toreplace the previous one which has also been shed. The triple-vortexarrangement seems to reappear again at t = 7. Figures 2h–2l show fivesnapshots of the flow covering the sixth complete cycle for the time in-terval 10 ≤ t ≤ 12. Figures 2a, 2c, 2g, 2h and 2l at t = 4, 6, 8, 10 and12 exhibit streamlines which are closely packed together as previouslyexplained. Similarly, a large spacing between consecutive streamlines inFigures 2b, 2e and 2j, at t = 5, 7 and 11, is observed. Figures 2g and2h, which show the flow field at the beginning and the end of the fifthcomplete cycle (t = 8 and t = 10), are similar to the situation at the endof the sixth cycle (t = 12) shown in Figure 2l. The minor differencesbetween these figures reflect the continuous development of the flow fieldaway from the cylinder because of vortex shedding and the interactionwith the free stream. This flow field has not yet become periodic andrequires many more oscillations before a quasi-steady state is reached.

Time variations of the surface vorticity distribution during the sixthoscillation cycle at times 10.5, 11, 11.5 and 12 are shown in Figure 3.These plots reveal rapid variations, especially in the vicinity of the tips ofthe cylinder, and suggest that a periodic pattern in the surface vorticitydistribution may be emerging. Figure 4 illustrates the periodic variationin both the drag and lift coefficients, CD and CL. The fluctuations inthese coefficients appear to be periodic with a period equal to that ofthe forced oscillations.

For comparison purposes the flow was also computed for the specialcases of inline and transverse oscillations. Streamline patterns at se-lected times during the sixth cycle of oscillation for these cases are shown

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NUMERICAL SIMULATIONS OF FLOW 263

FIGURE 2a: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 4.

FIGURE 2b: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 5.

in Figures 5 and 6. To orchestrate these oscillations in a direction of 0

and 90 (i.e., inline and transverse, respectively) with the oncoming flowwe simply set αv = 0 and αh = 0, respectively. In the near wake thevortex patterns are synchronized with the cylinder oscillations and aresimilar to those for oblique oscillations in a direction of 450 with the on-coming flow. Comparison of Figures 2g–2l with the corresponding onesfor the transverse and inline cases indicates that the vortices in the nearwake are simply the result of single vortex shedding in each half oscil-lation cycle. For the case of oblique oscillations, during each cycle thecounter-rotating vortices in the near wake grow to produce an almostsymmetric pattern at times when cylinder is momentarily at rest (i.e.,t = 10.5 and t = 11.5). This is not so for the inline and transverse cases.As the oscillation angle increases from 0 to 90, the size of the separated

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264 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 2c: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 6.

FIGURE 2d: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 6.5.

FIGURE 2e: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 7.

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NUMERICAL SIMULATIONS OF FLOW 265

FIGURE 2f: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 7.5.

FIGURE 2g: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 8.

FIGURE 2h: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 10.

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266 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 2i: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 10.5.

FIGURE 2j: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 11.

FIGURE 2k: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 11.5.

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NUMERICAL SIMULATIONS OF FLOW 267

FIGURE 2l: Streamline plot for the oblique oscillation case with

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at t = 12.

FIGURE 3: Surface vorticity distributions for the case R = 103,

r = 0.5, Ω = π, αh = αv = 0.25, η = π

4at times t = 10.5, 11, 11.5, 12.

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268 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 4: Time variation of the drag and lift coefficients for the case

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = π

4.

flow region appears to decrease while the vortex shedding process seemsto speed up slightly. This increases the rate at which the flow reachesa quasi-periodic pattern. The only other point worth emphasizing isthat the lateral spacing of the vortex street seems to be increasing asthe oscillation angle increases from 0 to 90. Figures 7 and 8 contrastthe time variations in the drag and lift coefficients for the oblique, inlineand transverse cases, respectively. The periodic variations in CD andCL display three different amplitudes. The amplitudes of CD tend todecrease with the oscillation angle whereas the opposite occurs with CL.It is interesting to point out that beyond the initial transition period(i.e., t > 6) the fluctuations in CD for the oblique case are essentiallyin phase with those for the inline case and out of phase with those forthe transverse case. The situation is different with the fluctuations inCL; here, the fluctuations for the oblique case are in phase with thosefor the transverse case and out of phase with those of the inline case.Lastly, we note that the fluctuations in CD and CL are in phase witheach other for the oblique case (see Figure 4), whereas for the transverseand inline cases they are out of phase with each other (see [5, Figures 8and 9] and [11, Figure 7]).

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NUMERICAL SIMULATIONS OF FLOW 269

FIGURE 5a: Streamline plot for the inline oscillation case with

αh = 0.25, αv = 0, R = 103, r = 0.5, Ω = π, η = π

4at t = 10.

FIGURE 5b: Streamline plot for the inline oscillation case with

αh = 0.25, αv = 0, R = 103, r = 0.5, Ω = π, η = π

4at t = 10.5.

FIGURE 5c: Figure 5c: Streamline plot for the inline oscillation case

with αh = 0.25, αv = 0, R = 103, r = 0.5, Ω = π, η = π

4at t = 11.

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270 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 5d: Streamline plot for the inline oscillation case with

αh = 0.25, αv = 0, R = 103, r = 0.5, Ω = π, η = π

4at t = 11.5.

FIGURE 5e: Streamline plot for the inline oscillation case with

αh = 0.25, αv = 0, R = 103, r = 0.5, Ω = π, η = π

4at t = 12.

FIGURE 6a: Streamline plot for the transverse oscillation case with

αh = 0, αv = 0.25, R = 103, r = 0.5, Ω = π, η = π

4at t = 10.

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NUMERICAL SIMULATIONS OF FLOW 271

FIGURE 6b: Streamline plot for the transverse oscillation case with

αh = 0, αv = 0.25, R = 103, r = 0.5, Ω = π, η = π

4at t = 10.5.

FIGURE 6c: Streamline plot for the transverse oscillation case with

αh = 0, αv = 0.25, R = 103, r = 0.5, Ω = π, η = π

4at t = 11.

FIGURE 6d: Streamline plot for the transverse oscillation case with

αh = 0, αv = 0.25, R = 103, r = 0.5, Ω = π, η = π

4at t = 11.5.

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272 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 6e: Streamline plot for the transverse oscillation case with

αh = 0, αv = 0.25, R = 103, r = 0.5, Ω = π, η = π

4at t = 12.

FIGURE 7: Comparison of the time variation in the drag coefficient

for the oblique, inline and transverse oscillation cases for R = 103,

r = 0.5, ω = π, η = π

4.

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NUMERICAL SIMULATIONS OF FLOW 273

FIGURE 8: Comparison of the time variation in the lift coefficient

for the oblique, inline and transverse oscillation cases for R = 103,

r = 0.5, ω = π, η = π

4.

5 Streamline patterns and force coefficients for R = 103, Ω =π, α = 0.25, r = 0.5 and η = 0 In this case the computations arecarried out for six complete cycles. The streamline patterns emergingfor the case η = 0, portrayed in Figures 9a and 9e, illustrate the flow atfive instances in time during the sixth cycle spanning the time interval10 ≤ t ≤ 12. Comparing these figures with those corresponding tothe case having η = π/4 indicates that as η decreases from π/4 to 0 thesize of the separated vortex region also decreases. The flow field featuresshown in Figure 9 reveal some similarities with the previous case but alsosome fundamental differences. The most noticeable difference is that thetriple-vortex arrangement is clearly missing. Since the cylinder is morestreamlined here, the vortices are not as protected as in the case havingη = π/4. This makes the vortices more vulnerable to advection andprevents the opportunity for the three vortices to coexist. The sheddingfrequency, however, appears to be the same. This is supported by theperiodic variations in CD , CL shown in Figure 11. Here, the fluctuationsin CL are much larger than those in CD and are again nearly in phase.Finally, the surface vorticity distributions displayed in Figure 10 showthat apart from the tips of the cylinder the distribution is much flatterthan that with η = π/4. The only other point worth emphasizing is thatthe lateral spacing of the vortex street seems to decrease as η decreases.

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274 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 9a: Streamline plot for the case R = 103, r = 0.5, Ω = π,

αh = αv = 0.25, η = 0 at t = 10.

FIGURE 9b: Streamline plot for the case R = 103, r = 0.5, Ω = π,

αh = αv = 0.25, η = 0 at t = 10.5.

FIGURE 9c: Streamline plot for the case R = 103, r = 0.5, Ω = π,

αh = αv = 0.25, η = 0 at t = 11.

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NUMERICAL SIMULATIONS OF FLOW 275

FIGURE 9d: Figure 9d: Streamline plot for the case R = 103, r = 0.5,

Ω = π, αh = αv = 0.25, η = 0 at t = 11.5.

FIGURE 9e: Streamline plot for the case R = 103, r = 0.5, Ω = π,

αh = αv = 0.25, η = 0 at t = 12.

FIGURE 10: Surface vorticity distributions for the case R = 103,

r = 0.5, Ω = π, αh = αv = 0.25, η = 0 at times t = 10.5, 11, 11.5, 12.

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276 S. J. D. D’ALESSIO AND SERPIL KOCABIYIK

FIGURE 11: Time variation in the drag and lift coefficients for the case

R = 103, r = 0.5, Ω = π, αh = αv = 0.25, η = 0.

6 Conclusions Numerically analyzed in this study were the near-wake structure behind an inclined obliquely oscillating elliptic cylinderand the corresponding hydrodynamic forces acting on the cylinder. Thecylinder underwent translational oscillations in a direction of 45 to theuniform oncoming flow. The effect of the inclination of the ellipse andin particular its orientation with respect to the cylinder oscillations wasdiscussed. Significant differences were observed in the flow patterns forthe inclinations η = 0 and η = π/4. An obvious effect of the obliqueoscillations is to induce vortex shedding from the tips of the cylinderat a frequency equal to that of the oscillation frequency. This effectsuperposes itself on the usual vortex street formed from a purely trans-lating elliptic cylinder. In addition, the effects of oblique oscillationshave been contrasted with those corresponding to transverse and inlineoscillations. It was found that oblique oscillations cause fluctuating dragand lift forces which are in phase with each other, whereas transverseand inline oscillations produce fluctuating drag and lift forces which areout of phase. Furthermore, in the oblique oscillation case the wake struc-ture did not involve double co-rotating vortices which were observed inboth the transverse and inline oscillation cases ([5, 11]).

Acknowledgements The support of the Natural Sciences and En-gineering Research Council of Canada is gratefully acknowledged.

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NUMERICAL SIMULATIONS OF FLOW 277

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Department of Applied Mathematics,University of Waterloo,Waterloo, Ontario, N2L 3G1, Canada

Department of Mathematics and Statistics,Memorial University of Newfoundland,St. John’s, Newfoundland, A1C 5S7, CanadaE-mail address: [email protected]