NUMERICAL SIMULATIONS OF FLOW PAST AN OBLIQUELY OSCILLATING ELLIPTIC CYLINDER

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 15, Number 3, Fall 27 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 translational oscillations of an inclined elliptic cylinder placed in a steady uniform flow of a viscous incompressible fluid. The motion is assumed to start impulsively from rest at t =. The flow is two-dimensional and the harmonic oscillations act in a direction 5 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 =, π/ and times < t 12 for a Reynolds number of 1 3 and a fixed minor-major axis ratio of.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 stationary cylinder, the forces that are experienced by the cylinder tend to be steady at Reynolds numbers below. At higher Reynolds numbers the flow in the wake of the cylinder becomes unsteady and a von Kármán 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 cylindrical 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 translational oscillation, spectral finite-difference scheme. Copyright c Applied Mathematics Institute, University of Alberta. 27

2 28 S. J. D. D ALESSIO AND SERPIL KOCABIYIK body used for simulations of flows around bluff bodies. However, the associated wake structures within a hundred diameters downstream only encompass a subset of wake structures associated with bluff bodies (see Johnson et al. [1]). Consequently, other simple bluff bodies should also be considered in order to fully understand bluff body wake dynamics. An elliptic geometry represents an obvious and welcome extension, allowing a wide range of cross-sections ranging from a circular cylinder to a flat plate depending on the minor-major axis ratio. In addition, the angle of attack of the ellipse acts as another parameter which can alter the wake structure. For example, at low angles of attack and for thin elliptic cylinders the flow generally remains attached to the body surface and behaves in a similar manner to that of a conventional airfoil. Whereas, at high angles of attack and for thicker ellipses the flow separates and a bluff-body flow regime results. The present study deals with a numerical investigation of a class of flows produced by oblique translational oscillations of an elliptic cylinder placed in a cross-flow. Numerical solutions of uniform flow past elliptic cylinders at various angles of attack were obtained by Staniforth [18], Lugt and Haussling [12], Patel [17], Mittal and Balachandar [1], and Nair and Sengupta [15]. Badr et al. [2] have summarized these studies. For flows induced by an elliptic cylinder undergoing translational oscillations in the presence of an oncoming uniform stream references may only be made to the works of Okajima et al. [16], D Alessio and Kocabiyik [5], and Kocabiyik and D Alessio [11]. In these studies inline or transverse oscillations of an elliptic cylinder were considered. In the present work we consider the two-dimensional flow caused by an infinitely long elliptic cylinder impulsively set in motion and translating with uniform velocity U. In addition, the cylinder is also undergoing harmonic oscillations in a direction of 5 with the horizontal free-stream direction. The cylinder is inclined at an angle η with the horizontal. The ellipse has major and minor axis of lengths 2a and 2b, respectively, and the cylinder oscillates with the velocity U cos ωt where ω = 2πf with f denoting the forced frequency of oscillation. The Reynolds number is defined by R = 2cU /ν where c = a 2 b 2 is the focal length and ν is the kinematic viscosity. The velocity ratio, α = U/U, the forcing Strouhal number, Ω = c ω/u, the angle of inclination, η, and the minor-to-major axis ratio of the ellipse, r = b/a, serve as dimensionless control parameters. The method of solution is an extension of the method developed by Staniforth [18] that takes into account cylinder oscillations in a direction of 5 with the horizontal free stream. In the works of D Alessio et

3 NUMERICAL SIMULATIONS OF FLOW 29 al. [], D Alessio and Kocabiyik [5] and Kocabiyik and D Alessio [11] the numerical technique of Staniforth was successfully extended to compute the development of the flows imposed by oscillatory motion, either rectilinear or rotational, of an inclined elliptic cylinder. In D Alessio and Kocabiyik [5] the problem of an elliptic cylinder subject to transverse oscillations was solved while in Kocabiyik and D Alessio [11] the case of inline oscillations was addressed. D Alessio et al. [], on the other hand, considered a uniform flow past a thin inclined elliptic cylinder under rotary oscillations. In a subsequent study, the early stages of flow development over elliptic airfoils oscillating in pitch at large angles of attack was simulated by Akbari and Price [1]. The goal of the present study is to investigate the effects of the ellipse inclination angle, η, on the flow structure in the near-wake region as well as on the hydrodynamic forces acting on the cylinder for a fixed Reynolds number of R = 1 3, forcing Strouhal number of Ω = π and velocity ratio of α =.25. Numerical calculations are performed for moderate times < t 12 and for inclinations η = π/ and η = for an ellipse having r =.5. Noticeable changes in the near-wake and in the forces take place as η varies and are reported. It is noted that Ω and α are maintained at Ω = π and α =.25 for the present study since flow structure in such cases is characterized by the formation of vortex pairs which convect away from the body, forming wakes. In general, the effect of the decrease of the oscillation amplitude is to reduce the size of the separated region. However, for the sufficiently small oscillation amplitude range, α/ω 1, when no flow separation takes place, we have the unexpected result that jets issue from the cylinder surface following a boundary-layer collision. The emergence of a thin round jet along the axis of oscillation was first predicted and visualized by Davidson and Riley [6] for the case of purely translational oscillations of an elliptic cylinder 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 argue that for the Reynolds number regime considered three-dimensional effects and turbulence may significantly alter the flow. In fact, experimental work conducted by Williamson [2] for the case of a circular cylinder suggests that a three dimensional transition occurs for Reynolds numbers R > 178. This was also confirmed by Zhang et al. [21]. Szepessy and Bearman [19] measured a fluctuating lift on a thin section of a large aspect-ratio fixed-circular cylinder and found that two-dimensional simulation schemes generally overestimate the root-mean-square value of the fluctuating lift. This observation has been substantiated by Gra-

4 25 S. J. D. D ALESSIO AND SERPIL KOCABIYIK ham [9], who gathered numerical predictions for circular cylinder flow and compared them with experimental results. He found that above a Reynolds number of about 15 the mean and fluctuating forces were generally overpredicted, with largest differences occurring in the fluctuating lift. It should also be noted that measured time histories of the fluctuating lift show a pronounced amplitude modulation whereas simulated time histories mostly display a constant amplitude, once the flow has settled. However, forcing a bluff body to oscillate introduces a mechanism for synchronizing the moment of shedding along its length. With this consideration, two dimensional numerical simulations should be reliable in terms analyzing flow details, at least in the near wake region. For example, the work of Blackburn and Henderson [3] supports the notion that cylinder vibrations tend to suppress the three-dimensionality and produce flows that are more two-dimensional than their fixed cylinder counterparts. 2 Formulation and governing equations In the present paper we consider the two-dimensional flow generated by an infinitely long elliptic cylinder whose axis coincides with the z-axis placed in a viscous incompressible fluid. The cylinder is inclined at an angle η with the horizontal and the major and minor axes are taken to lie along the x and y axes, respectively. Initially, the cylinder is at rest and at time t = it suddenly starts to translate horizontally with uniform velocity U and also oscillates harmonically in a direction of 5 with the horizontal. Equivalently, as shown in Figure 1, we take the cylinder to oscillate and the fluid to flow past it with uniform velocity U. A mathematically convenient non-inertial frame of reference which translates and oscillates with the cylinder is employed. In this frame the unsteady dimensionless equations for a viscous incompressible fluid in primitive variables can be written in vector form as (1) (2) v t = (p + 12 ) v 2 2 R ω + a, v =. Here, t is the non-dimensional time defined by t = U t /c with t denoting the dimensional time. For two-dimensional flow taking place in the xy-plane, the velocity is v = (u, v, ) and the vorticity is given by ω = v = (,, ζ). The term a is the translational acceleration arising from the non-inertial reference frame of the vibrating cylinder. This

5 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 to be confused with the scalar velocity components u and v) denote the nondimensional velocities in the fixed and non-inertial frames, respectively, then we have that (3) u = v + α h (cos η, sin η) cos(ωt) + α v (sin η, cos η) cos(ωt) where Ω is the non-dimensional angular frequency of oscillation and α h, α v denote non-dimensional peak velocities of oscillation in the horizontal and vertical directions respectively. The term a is then related to the time derivative of the last two terms in (3) and is hence given by a = Ωα h (cos η, sin η) sin(ωt) + Ωα v (sin η, cos η) sin(ωt). We note that the special cases of transverse and inline oscillations are recovered by setting α h = and α v =, respectively. These cases are reported in the studies of D Alessio and Kocabiyik [5] and Kocabiyik and D Alessio [11]. To orchestrate oscillations in a direction of 5 with the oncoming flow we simply set α h = α v α. Since the appropriate coordinates for the present problem are the elliptic coordinates (ξ, θ), we use the following conformal transformation which relates the elliptic coordinates (ξ, θ) to the Cartesian coordinates (x, y): x = cosh(ξ + ξ ) cos θ, y = sinh(ξ + ξ ) sin θ.

6 252 S. J. D. D ALESSIO AND SERPIL KOCABIYIK Here, the constant ξ = tanh 1 (b/a), and ξ = defines the surface of the cylinder. Using the elliptic coordinate system, with the origin at the center of the cylinder, the equations of motion can be written in terms of the vorticity, ζ, and the stream function, ψ, in dimensionless form as () (5) 2 ψ ξ ψ θ 2 = M 2 ζ, ) + ζ t = 1 [ ( 2 2 ζ M 2 R ξ ζ θ 2 ( ψ ζ θ ξ ψ ξ )] ζ. θ The dependent variables ψ, ζ in these equations are defined in terms of the usual dimensional quantities as ψ = U cψ, ζ = U ζ/c and the Jacobian of the above transformation, M 2, is given by (6) M 2 = 1 2 [cosh 2(ξ + ξ ) cos 2θ]. Expressions for the dimensionless velocity components (v ξ, v θ ) in the directions of increase of (ξ, θ) in terms of the stream function ψ are given by v ξ = 1 M ψ θ, v θ = 1 M ψ ξ, and the vorticity ζ is defined in terms of the velocity components as ζ = 1 ( M 2 θ (Mv ξ) + ) ξ (Mv θ). The boundary conditions for t > and θ 2π are the impermeability and no-slip conditions on the cylinder surface given by (7) ψ = ψ ξ = when ξ =. The far-field conditions can be derived by first noting that in the fixed frame u ( cos η, sin η) as x 2 + y 2. Then from (3) we obtain v = ( ψ y, ψ ) x ( cos η, sin η) α h (cos η, sin η) cos(ωt) α v (sin η, cos η) cos(ωt)

7 NUMERICAL SIMULATIONS OF FLOW 253 as x 2 + y 2. In terms of elliptic coordinates the above conditions are expressed as (8) (9) ψ ξ 1 2 eξ+ξ sin(θ + η) eξ+ξ [α h sin(θ + η) α v cos(θ + η)] cos(ωt) as ξ, ψ θ 1 2 eξ+ξ cos(θ + η) eξ+ξ [α h cos(θ + η) + α v sin(θ + η)] cos(ωt) as ξ, or equivalently as (1) ψ 1 2 eξ+ξ sin(θ + η) eξ+ξ [α h sin(θ + η) α v cos(θ + η)] cos(ωt) as ξ. The far-field vorticity, on the other hand, satisfies (11) ζ as ξ. The surface boundary conditions given by (7) for the stream function are overspecified. Boundary condition (11) gives a requirement for the vorticity in the far field, but there is no explicit condition for the vorticity on the cylinder surface. In principle, the surface vorticity can be computed from the known stream function by applying equation (), however the large velocity gradient at the surface reduces the accuracy of such computations. In this study integral conditions are used to predict the surface vorticity. Following the works of Dennis and Quartepelle [7], and Dennis and Kocabiyik [8], the conditions (7) (1) for the stream function are transformed into a set of global integral conditions for the vorticity using equation (). These conditions are derived by applying Green s second identity for the Laplacian operator, namely ( (φ 2 ψ ψ 2 φ) dv = φ ψ n ψ ψ ) ds, n V to the flow domain V exterior to the cylinder. Here, the boundary S of the flow domain is the contour of the cylinder itself together with a contour at a large distance, n refers to the outward pointing normal to the boundary S of the flow domain, and s is measured along contour. Taking S

8 25 S. J. D. D ALESSIO AND SERPIL KOCABIYIK φ to be the set of harmonic functions φ = {1, e nξ cos nθ, e nξ sin nθ : n = 1, 2,...} and using 2 ψ = M 2 ζ from (), it follows after some integration by parts and making use of the far-field conditions that: (12) (13) (1) 2π 2π 2π M 2 ζ(ξ, θ, t) dθ dξ =, e nξ M 2 ζ(ξ, θ, t) cos(nθ)dθ dξ = πe ξ [sin η + (α h sin η α v cos η) cos(ωt)] δ 1,n, e nξ M 2 ζ(ξ, θ, t) sin(nθ)dθ dξ = πe ξ [cos η + (α h cos η + α v sin η) cos(ωt)] δ 1,n, for all integers n 1. These are employed in the solution procedure to ensure that all necessary conditions of the problem are satisfied. Here, δ m,n is the Kronecker delta symbol defined by δ m,n = 1 if m = n, and δ m,n = if m n. The use of integral conditions can be found in the works of Staniforth [18], D Alessio et al. [], Badr et al. [2], Mahfouz and Kocabiyik [13], and Kocabiyik and D Alessio [11], to mention a few of the various applications. Lastly, an initial condition is necessary to start the flow. Boundarylayer theory for impulsively started flows is used to provide this by utilizing the boundary-layer transformation (15) ξ = kz, ψ = kψ, ζ = ω ( ) 1/2 2t k, k = 2, R which maps the initial flow onto the scale of the boundary-layer thickness. The governing equations and the boundary and integral conditions are first transformed using (15). The equations and boundary conditions on the cylinder surface satisfied by Ψ are given by D Alessio and Kocabiyik [5] and the integral conditions in the present case take the form (16) 2π M 2 ω(z, θ, t) dθ dz =,

9 NUMERICAL SIMULATIONS OF FLOW 255 (17) (18) 2π 2π e nkz M 2 ω(z, θ, t) cos(nθ)dθ dz = πe ξ [sin η + (α h sin η α v cos η) cos(ωt)] δ 1,n, e nkz M 2 ω(ξ, θ, t) sin(nθ)dθ dz = πe ξ [cos η + (α h cos η + α v sin η) cos(ωt)] δ 1,n, for all integers n 1. It is noted that the integral conditions given by (17) and (18) differ from those given in D Alessio and Kocabiyik [5] and Kocabiyik and D Alessio [11] owing to the difference in cylinder motions. The initial solution at t = is obtained following the work by Staniforth [18]. This initial solution is given by (19) (2) ω (z, θ, ) = 2 e ξ [(1 + α h ) sin(θ + η) α v cos(θ + η)]e M 2 π M Ψ (z, θ, ) = eξ [(1 + α h ) sin(θ + η) α v cos(θ + η)] M [ M z erf(m z) 1 ] (1 e M 2 z2 ), π z2, where erf(m z) denotes the error function and M 2 = [cosh 2ξ cos 2θ]/2. This initial solution forms the starting point of the numerical integration procedure which is outlined in the following section. 3 Numerical solution summary The transformed vorticity transport equation for ω in terms of the coordinates (z, θ) is solved by finite differences using a Gauss-Seidel iterative procedure with underrelaxation applied only to the surface vorticity. Since the procedure is similar to that used in the studies of D Alessio et al. [], D Alessio and Kocabiyik [5], and Kocabiyik and D Alessio [11], we will briefly describe the numerical technique. The computational domain, bounded by z z and < θ < 2π, is first discretized into a network of L P equally spaced grid points located at z i = ih, i =, 1,..., L where h = z /L, θ j = jλ, j =, 1,..., P where λ = 2π/P. Here z refers to the outer boundary approximating infinity.

10 256 S. J. D. D ALESSIO AND SERPIL KOCABIYIK We express the stream function in the form of a truncated Fourier series (21) Ψ(z, θ, t) = 1 2 F (z, t) + N [F n (z, t) cos(nθ) + f n (z, t) sin(nθ)]. n=1 The equations governing the Fourier coefficients are (22) 2 F n z 2 n2 k 2 F n = s n (z, t); n =, 1,..., 2 f n z 2 n2 k 2 f n = r n (z, t); n = 1, 2,..., where (23) s n (z, t) = 1 π r n (z, t) = 1 π 2π 2π M 2 ω(z, θ, t) cos nθdθ, M 2 ω(z, θ, t) sin nθdθ. Boundary conditions for the Fourier components of Ψ are and as z, F (, t) = F n (, t) = f n (, t) =, F z = F n z = f n = when z =, z e kz F, e kz F z, e kz F n 1 2k eξ [sin η + (α h sin η α v cos η) cos(ωt)]δ n,1, e kz F n z 1 2 eξ [sin η + (α h sin η α v cos η) cos(ωt)]δ n,1, e kz f n 1 2k eξ [cos η + (α h cos η + α v sin η) cos(ωt)]δ n,1, e kz f n z 1 2 eξ [cos η + (α h cos η + α v sin η) cos(ωt)]δ n,1,

11 NUMERICAL SIMULATIONS OF FLOW 257 for all integers n 1. The integral conditions can be formulated in terms of the functions r n (z, t) s n (z, t) as follows: (2) (25) (26) s (z, t)dθ dz =, e nkz s n (z, t) dz = e ξ [sin η + (α h sin η α v cos η) cos(ωt)]δ 1,n, e nkz r n (z, t) dz = e ξ [cos η + (α h cos η + α v sin η) cos(ωt)]δ 1,n, for all integers n 1. These conditions play an important role in the determination of the surface vorticity as we shall shortly see. They are, in fact, equivalent to the one-dimensional form of the Green s theorem constraint 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) β 2 h(z) = g(z) where β = nk and the prime refers to differentiation with respect to z. A special scheme is used for integrating these ordinary differential equations using step-by-step formulae. As previously mentioned, the vorticity transport equation is solved by finite differences. The scheme used to approximate this equation is very similar to the Crank-Nicolson implicit procedure. The specific details will not be presented, but can be found in [18, 5, 11] The surface vorticity, which is needed to complete the integration procedure, is determined by inverting (23) and is given by the following expression (28) ω(, θ, t) = 1 { 1 M 2 2 s (, t) + N n=1 } [r n (, t) sin(nθ) + s n (, t) cos(nθ)]. It may also be necessary to subject the surface vorticity to underrelaxation in order to obtain convergence. The integration procedure is initiated using the initial solution (19) and (2) at t =. The use of this initial solution is essential for obtaining

12 258 S. J. D. D ALESSIO AND SERPIL KOCABIYIK accurate results at small times. The potential flow solution, in this case given by (29) ψ(ξ, θ, t) = e ξ sinh ξ sin(θ + η) + e ξ sinh ξ [α h sin(θ + η) α v cos(θ + η)] cos(ωt), has also been used as an initial condition at t = in previous related studies. This, however, will definitely lead to inaccurate results following the start of the fluid motion. It is noted that the potential flow solution (29) can easily be obtained by solving the stream function equation (), after setting ζ =, subject to the impermeability and far-field conditions. Because of the impulsive start, small time steps were needed to get past the singularity at t =. Initially, t = 1 was used; as time increased the time step was gradually increased until reaching t =.1. For t >.1 the time step t =.1 was used. The grid size z in the coordinate z is more or less independent of R. The number of points in the z-direction is taken to be 21 with a uniform spacing of z =.6. This sets the outer boundary of our computational domain (z = 12) at a physical distance of about major axis lengths away for a Reynolds number R = 1 3 and time t = 12. Placing z well outside the growing boundary-layer enables us to enforce the far-field conditions (1) (11) along the outer edge of our computational domain z = z so that the application of the far-field conditions does not affect the solution in the viscous region near the cylinder surface. We point out that the physical coordinate ξ = kz is a moving coordinate and hence the outer boundary ξ = kz is constantly being pushed further away from the cylinder surface with time. For this reason we are justified in saying that the vorticity, by the mechanism of convection, will not reach the outer boundary ξ. The maximum number of terms retained in the series (21) was 51 which corresponds to an upper limit of N = 25 in the sums. Checks were made for R = 1 3 at several times to ensure that N is large enough. This was done by increasing N and observing that the solution did not change appreciably. The computational parameters are to some extent chosen to be comparable with those used by D Alessio and Kocabiyik [5], since these were found to be satisfactory and were checked carefully. Moreover, this scheme is tested against the results of Staniforth [18] for the non-oscillating (i.e., purely translating) case using similar Reynolds numbers; tests indicate that the solutions are quite accurate.

13 NUMERICAL SIMULATIONS OF FLOW 259 Results and discussion The numerical results are grouped in two cases, (i) η = π/ and (ii) η =, to illustrate the effect of inclination on the ensuing flow. In the case of η = π/ the cylinder oscillates in a direction of the major axis of the ellipse while for the case η = the oscillations are at 5 to the direction of the major axis. For each of these cases the other parameters characterizing the flow were fixed and the values were R = 1 3, Ω = π, r =.5 and α h = α v α =.25. The results are presented in the form of streamline patterns as well as time variations of the drag and lift coefficients and surface vorticity distributions. The special cases of transverse and inline oscillations with η = π/ were also computed for comparison purposes. A brief derivation of the formulae for the force coefficients is outlined in Section.1. We note that the two cases considered in the present work deal with the same oscillation frequency of f =.5 and therefore have a period T = 1/f = 2. Thus, a complete cycle consists of the following four stages: at t = the ellipse starts to oscillate with its maximum velocity in a direction of 5 to the horizontal free stream. At t =.5 the ellipse reaches its maximum oblique displacement in this direction and is in an instantaneous state of rest. At t = 1 the ellipse is in its equilibrium position and again attains maximum velocity in the opposite direction. Then, at t = 1.5 the ellipse occupies its other extremum displacement position and is again in an instantaneous state of rest. Finally, at t = 2 the ellipse is in its starting position and this pattern repeats itself. The plots to be presented are from the vantage point of the noninertial reference frame of the cylinder; consequently, the oncoming flow direction (from right to left) will appear to periodically rotate. At even times it will appear to approach the cylinder from above while at odd times from below. At half times (i.e., t = 6.5, 7.5, 1.5, 11.5) the oncoming flow approaches the cylinder horizontally since at these times the cylinder is momentarily at rest. What will also be noticed in the flow patterns is a cyclic variation in the spacing between consecutive streamlines. This is a result of the periodic changes in the relative velocity between the cylinder and the oncoming flow caused by the oscillations. At even times the spacing is smallest due to the higher relative velocity while at odd times the spacing is largest in accordance with a lower relative velocity. With the above points in mind, we now proceed to present and discuss the flow patterns for the two cases: η = π/ and η =. Lastly, the flow patterns for each case were plotted at the same times so as to make comparisons and differences easier to report.

14 26 S. J. D. D ALESSIO AND SERPIL KOCABIYIK.1 Derivation of the formulae for force coefficients, C D and C L If L and D are the dimensional lift and drag on the cylinder, the dimensionless drag, C D, and lift, C L, coefficients are then defined by C D = D/ρU 2 c and C L = L/ρU 2 c, respectively. The drag and lift coefficients were computed using (3) (31) ( 2 sinh ξ C D = R C L = 2π ( 2 cosh ξ R ( ) ζ ξ 2 cosh ξ R ( 2 cosh ξ + R 2 sinh ξ R sin θdθ 2π 2π 2π ) ζ sin θdθ cos η ( ) ζ ξ ζ cos θdθ cos θdθ ) sin η πα h Ω cosh ξ sinh ξ sin(ωt), 2π ( ) ζ cos θdθ ξ + 2 sinh ξ R ( 2 sinh ξ + R 2 cosh ξ R 2π 2π 2π ) ζ cos θdθ cos η ( ) ζ ξ ζ sin θdθ sin θdθ ) sin η πα v Ω sinh ξ cosh ξ sin(ωt). The first integral in each expression of C D and C L represents the coefficient due to pressure and the second terms are due to friction. The last term in the expressions for C D and C L represents the inviscid contribution which results from the acceleration due to time dependent cylinder oscillations. We point out that at t = both C D and C L are infinite in magnitude due to the impulsive start delivered to the cylinder at t = ; afterwards C D and C L decrease rapidly. There are two dominating flow fields affecting the boundary-layer region. The first is a potential flow field while the second is that resulting from vortical motion. In the present problem, the second field has a negligible effect at

15 NUMERICAL SIMULATIONS OF FLOW 261 the start of the motion but as time increases so does its influence. Such a field continues to evolve with time until eventually reaching a near periodic behaviour after many oscillations. The frictional forces make small contributions to the total force in comparison with the inviscid contributions due to the moderately large Reynolds number considered. A brief derivation of the formulae (3) and (31) can be outlined as follows. In terms of elliptic coordinates, the θ-component of the dimensionless momentum equation (1) in a reference frame that translates and oscillates with the cylinder becomes (32) v θ t = 1 M θ ( p + 1 [ v 2 2 ξ + vθ 2 ] ) Ω M cosh(ξ + ξ ) sin θ(α h cos η + α v sin η) sin(ωt) + Ω M sinh(ξ + ξ ) cos θ( α h sin η + α v cos η) sin(ωt) + 2 ζ MR ξ. On the cylinder surface (32) simplifies greatly owing to the impermeability and no-slip conditions, and becomes ( ) P = 2 ( ) ζ (33) Ω cosh ξ sin θ(α h cos η + α v sin η) sin(ωt) θ R ξ + Ω sinh ξ cos θ( α h sin η + α v cos η) sin(ωt). The forces in the horizontal and vertical directions, X and Y, respectively, can be obtained by integrating the pressure and frictional stresses on the surface. This leads to (3) (35) 2π ( ) P X = sinh ξ θ 2π ( ) P Y = cosh ξ θ Making use of (33), the relations sin θdθ 2 cosh ξ R cos θdθ + 2 sinh ξ R 2π 2π ζ sin θdθ, ζ cos θdθ. C D = X cos η Y sin η and C L = X sin η + Y cos η, and simplifying then yields equations (3) (31).

16 262 S. J. D. D ALESSIO AND SERPIL KOCABIYIK.2 Streamline patterns and force coefficients for R = 1 3, Ω = π, α =.25, r =.5 and η = π/ Figures 2a 2l show instantaneous snap shots of the flow field for the time interval t 12 and captures the details of the flow patterns during three of the six cycles of oscillation. The vortices in the near wake are simply the result of one vortex in each half oscillation cycle. Close to the cylinder, Figure 2a at t =, a counterclockwise vortex pair exists in the upper half of the cylinder and is convected downstream with the aid of the cylinder motion along its major axis. This continues until t = 5 and Figure 2b shows the formation of a triple-vortex arrangement behind the cylinder with the bottom vortices rotating clockwise and the top vortex rotating counterclockwise. At t = 6, Figure 2c, a single large vortex forms indicating that the other vortices weakened as they were advected downstream. At t = 6.5, Figure 2d, the large vortex has just been shed and a new vortex has formed near the leading edge. Figures 2e and 2f, at times t = 7 and t = 7.5, reveal that another vortex has formed near the leading edge to replace the previous one which has also been shed. The triple-vortex arrangement seems to reappear again at t = 7. Figures 2h 2l show five snapshots of the flow covering the sixth complete cycle for the time interval 1 t 12. Figures 2a, 2c, 2g, 2h and 2l at t =, 6, 8, 1 and 12 exhibit streamlines which are closely packed together as previously explained. Similarly, a large spacing between consecutive streamlines in Figures 2b, 2e and 2j, at t = 5, 7 and 11, is observed. Figures 2g and 2h, which show the flow field at the beginning and the end of the fifth complete cycle (t = 8 and t = 1), are similar to the situation at the end of the sixth cycle (t = 12) shown in Figure 2l. The minor differences between these figures reflect the continuous development of the flow field away from the cylinder because of vortex shedding and the interaction with the free stream. This flow field has not yet become periodic and requires many more oscillations before a quasi-steady state is reached. Time variations of the surface vorticity distribution during the sixth oscillation cycle at times 1.5, 11, 11.5 and 12 are shown in Figure 3. These plots reveal rapid variations, especially in the vicinity of the tips of the cylinder, and suggest that a periodic pattern in the surface vorticity distribution may be emerging. Figure illustrates the periodic variation in both the drag and lift coefficients, C D and C L. The fluctuations in these coefficients appear to be periodic with a period equal to that of the forced oscillations. For comparison purposes the flow was also computed for the special cases of inline and transverse oscillations. Streamline patterns at selected times during the sixth cycle of oscillation for these cases are shown

17 NUMERICAL SIMULATIONS OF FLOW 263 FIGURE 2a: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t =. FIGURE 2b: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 5. in Figures 5 and 6. To orchestrate these oscillations in a direction of and 9 (i.e., inline and transverse, respectively) with the oncoming flow we simply set α v = and α h =, respectively. In the near wake the vortex patterns are synchronized with the cylinder oscillations and are similar to those for oblique oscillations in a direction of 5 with the oncoming flow. Comparison of Figures 2g 2l with the corresponding ones for the transverse and inline cases indicates that the vortices in the near wake are simply the result of single vortex shedding in each half oscillation cycle. For the case of oblique oscillations, during each cycle the counter-rotating vortices in the near wake grow to produce an almost symmetric pattern at times when cylinder is momentarily at rest (i.e., t = 1.5 and t = 11.5). This is not so for the inline and transverse cases. As the oscillation angle increases from to 9, the size of the separated

18 26 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 2c: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 6. FIGURE 2d: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 6.5. FIGURE 2e: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 7.

19 NUMERICAL SIMULATIONS OF FLOW 265 FIGURE 2f: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 7.5. FIGURE 2g: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 8. FIGURE 2h: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 1.

20 266 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 2i: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 1.5. FIGURE 2j: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 11. FIGURE 2k: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 11.5.

21 NUMERICAL SIMULATIONS OF FLOW 267 FIGURE 2l: Streamline plot for the oblique oscillation case with R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at t = 12. FIGURE 3: Surface vorticity distributions for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π at times t = 1.5, 11, 11.5, 12.

22 268 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE : Time variation of the drag and lift coefficients for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = π. flow region appears to decrease while the vortex shedding process seems to speed up slightly. This increases the rate at which the flow reaches a quasi-periodic pattern. The only other point worth emphasizing is that the lateral spacing of the vortex street seems to be increasing as the oscillation angle increases from to 9. Figures 7 and 8 contrast the time variations in the drag and lift coefficients for the oblique, inline and transverse cases, respectively. The periodic variations in C D and C L display three different amplitudes. The amplitudes of C D tend to decrease with the oscillation angle whereas the opposite occurs with C L. It is interesting to point out that beyond the initial transition period (i.e., t > 6) the fluctuations in C D for the oblique case are essentially in phase with those for the inline case and out of phase with those for the transverse case. The situation is different with the fluctuations in C L ; here, the fluctuations for the oblique case are in phase with those for the transverse case and out of phase with those of the inline case. Lastly, we note that the fluctuations in C D and C L are in phase with each other for the oblique case (see Figure ), whereas for the transverse and inline cases they are out of phase with each other (see [5, Figures 8 and 9] and [11, Figure 7]).

23 NUMERICAL SIMULATIONS OF FLOW 269 FIGURE 5a: Streamline plot for the inline oscillation case with α h =.25, α v =, R = 1 3, r =.5, Ω = π, η = π at t = 1. FIGURE 5b: Streamline plot for the inline oscillation case with α h =.25, α v =, R = 1 3, r =.5, Ω = π, η = π at t = 1.5. FIGURE 5c: Figure 5c: Streamline plot for the inline oscillation case with α h =.25, α v =, R = 1 3, r =.5, Ω = π, η = π at t = 11.

24 27 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 5d: Streamline plot for the inline oscillation case with α h =.25, α v =, R = 1 3, r =.5, Ω = π, η = π at t = FIGURE 5e: Streamline plot for the inline oscillation case with α h =.25, α v =, R = 1 3, r =.5, Ω = π, η = π at t = 12. FIGURE 6a: Streamline plot for the transverse oscillation case with α h =, α v =.25, R = 1 3, r =.5, Ω = π, η = π at t = 1.

25 NUMERICAL SIMULATIONS OF FLOW 271 FIGURE 6b: Streamline plot for the transverse oscillation case with α h =, α v =.25, R = 1 3, r =.5, Ω = π, η = π at t = 1.5. FIGURE 6c: Streamline plot for the transverse oscillation case with α h =, α v =.25, R = 1 3, r =.5, Ω = π, η = π at t = 11. FIGURE 6d: Streamline plot for the transverse oscillation case with α h =, α v =.25, R = 1 3, r =.5, Ω = π, η = π at t = 11.5.

26 272 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 6e: Streamline plot for the transverse oscillation case with α h =, α v =.25, R = 1 3, r =.5, Ω = π, η = π at t = 12. FIGURE 7: Comparison of the time variation in the drag coefficient for the oblique, inline and transverse oscillation cases for R = 1 3, r =.5, ω = π, η = π.

27 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 = 1 3, r =.5, ω = π, η = π. 5 Streamline patterns and force coefficients for R = 1 3, Ω = π, α =.25, r =.5 and η = In this case the computations are carried out for six complete cycles. The streamline patterns emerging for the case η =, portrayed in Figures 9a and 9e, illustrate the flow at five instances in time during the sixth cycle spanning the time interval 1 t 12. Comparing these figures with those corresponding to the case having η = π/ indicates that as η decreases from π/ to the size of the separated vortex region also decreases. The flow field features shown in Figure 9 reveal some similarities with the previous case but also some fundamental differences. The most noticeable difference is that the triple-vortex arrangement is clearly missing. Since the cylinder is more streamlined here, the vortices are not as protected as in the case having η = π/. This makes the vortices more vulnerable to advection and prevents the opportunity for the three vortices to coexist. The shedding frequency, however, appears to be the same. This is supported by the periodic variations in C D, C L shown in Figure 11. Here, the fluctuations in C L are much larger than those in C D and are again nearly in phase. Finally, the surface vorticity distributions displayed in Figure 1 show that apart from the tips of the cylinder the distribution is much flatter than that with η = π/. The only other point worth emphasizing is that the lateral spacing of the vortex street seems to decrease as η decreases.

28 27 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 9a: Streamline plot for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at t = 1. FIGURE 9b: Streamline plot for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at t = 1.5. FIGURE 9c: Streamline plot for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at t = 11.

29 NUMERICAL SIMULATIONS OF FLOW 275 FIGURE 9d: Figure 9d: Streamline plot for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at t = FIGURE 9e: Streamline plot for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at t = 12. FIGURE 1: Surface vorticity distributions for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η = at times t = 1.5, 11, 11.5, 12.

30 276 S. J. D. D ALESSIO AND SERPIL KOCABIYIK FIGURE 11: Time variation in the drag and lift coefficients for the case R = 1 3, r =.5, Ω = π, α h = α v =.25, η =. 6 Conclusions Numerically analyzed in this study were the nearwake structure behind an inclined obliquely oscillating elliptic cylinder and the corresponding hydrodynamic forces acting on the cylinder. The cylinder underwent translational oscillations in a direction of 5 to the uniform oncoming flow. The effect of the inclination of the ellipse and in particular its orientation with respect to the cylinder oscillations was discussed. Significant differences were observed in the flow patterns for the inclinations η = and η = π/. An obvious effect of the oblique oscillations is to induce vortex shedding from the tips of the cylinder at a frequency equal to that of the oscillation frequency. This effect superposes itself on the usual vortex street formed from a purely translating elliptic cylinder. In addition, the effects of oblique oscillations have been contrasted with those corresponding to transverse and inline oscillations. It was found that oblique oscillations cause fluctuating drag and lift forces which are in phase with each other, whereas transverse and inline oscillations produce fluctuating drag and lift forces which are out of phase. Furthermore, in the oblique oscillation case the wake structure did not involve double co-rotating vortices which were observed in both the transverse and inline oscillation cases ([5, 11]). Acknowledgements The support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

31 NUMERICAL SIMULATIONS OF FLOW 277 REFERENCES 1. M. H. Akbari and S. J. Price, Simulation of the flow over elliptic airfoils oscillating at large angles of attack, J. Fluids Structures 1 (2), H. M. Badr, S. C. R. Dennis and S. Kocabiyik, Numerical simulation of the unsteady flow over an elliptic cylinder at different orientations Internat. J. Numer. Methods Fluids 37 (21), H. M. Blackburn and R. D. Henderson, A study of two-dimensional flow past an oscillating cylinder, J. Fluid Mech. 385 (1999), S. J. D. D Alessio, S. C. R. Dennis and P. Nguyen, Unsteady viscous flow past an impulsively started oscillating and translating elliptic cylinder, J. Engrg. Math. 35 (1999), S. J. D. D Alessio and S. Kocabiyik, Numerical simulation of the flow induced by a transversely oscillating inclined elliptic cylinder, J. Fluids Structures 15 (21), B. J. Davidson and N. Riley, Jets induced by oscillatory motion, J. Fluid Mech. 53 (1972), S. C. R. Dennis and L. Quartapelle, Some uses of Green s Theorem in solving the Navier-Stokes equations, Internat. J. Numer. Methods Fluids 9 (1989), S. C. R. Dennis and S. Kocabiyik, An asymptotic matching condition for unsteady boundary-layer flows governed by the Navier-Stokes equations, IMA J. Appl. Mech. 7 (1991), J. M. R. Graham, Comparing computation of flow past circular cylinders with experimental data, in Bluff Body Wakes, Dynamics and Instabilities, pp , New York, Springer, S. A. Johnson, M. C. Thompson and K. Hourigan, Predicted low frequency structures in the wake of elliptical cylinders, European J. Mech. B/Fluids 23 (2), S. Kocabiyik and S. J. D. D Alessio, Numerical study of flow around an inclined elliptic cylinder oscillating in line with an incident uniform flow, European J. Mech. B/Fluids 23 (2), H. J. Lugt and H. J. Haussling, Laminar flow past an abruptly accelerated elliptic cylinder at 5 incidence, J. Fluid Mech. 65 (197), F. M. Mahfouz and S. Kocabiyik, Transient numerical simulation of buoyancy driven flow adjacent to an elliptic tube, Internat. J. Heat Fluid Flow 2 (23), R. Mittal and S. Balachandar, Direct numerical simulation of flow past elliptic cylinders, J. Comput. Phys. 12 (1996), M. T. Nair and T. K. Sengupta, Unsteady flow past elliptic cylinders, J. Fluids Structures 11 (1997), A. Okajima, H. Takata and T. Asanuma, Viscous flow around a transversally oscillating elliptic cylinder, Institute of Space and Aeronautical Science, University of Tokyo, Report No. 533 (1975). 17. V. A. Patel, Flow around the impulsively started elliptic cylinder at various angles of attack, Comput. & Fluids 9 (1981), A. N. Staniforth, Ph.D. thesis, University of Western Ontario, London, Ontario, Canada (1972). 19. S. Szepessy and P. W. Bearman, Aspect ratio and end plate effects on vortex shedding from a circular cylinder, J. Fluid Mech. 23 (1992), C. H. K. Williamson, The existence of two stages in the transition to threedimensionality of a circular wake, Phys. Fluids 31 (1988),

32 278 S. J. D. D ALESSIO AND SERPIL KOCABIYIK 21. H.-Q. Zhang, U. Fey, B. R. Noack, M. Konig and H. Eckelmann, On the transition of the cylinder wake, Phys. Fluids 7 (1995), 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, Canada address: serpil@math.mun.ca