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 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 = 0. The flow
is two-dimensional and the harmonic oscillations act in a direction 45◦ to the uniform oncoming flow. The unsteady NavierStokes 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 103 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 stationary cylinder, the forces that are experienced by the cylinder tend to
be steady at Reynolds numbers below 40. 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.
c
Copyright Applied
Mathematics Institute, University of Alberta.
247
248
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. [10]). 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 [14], 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 45◦ 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 = a2 − b2 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 45◦ with the horizontal free stream. In the works of D’Alessio et
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 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. [4], 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 = 103 , forcing Strouhal number of Ω = π and
velocity ratio of α = 0.25. Numerical calculations are performed for
moderate times 0 < t ≤ 12 and for inclinations η = π/4 and η = 0 for
an ellipse having r = 0.5. 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 = 0.25 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 [20] 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-
250
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 150 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 = 0 it
suddenly starts to translate horizontally with uniform velocity U∞ and
also oscillates harmonically in a direction of 45◦ 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
1
2~
2
~
= −∇ p + | ~v | − ∇
×ω
~ + ~a,
∂t
2
R
~ · ~v = 0.
∇
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, 0) and the vorticity is given by
~ v = (0, 0, ζ). The term ~a is the translational acceleration arising
ω
~ = ∇×~
from the non-inertial reference frame of the vibrating cylinder. This
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 + αh (cos η, − sin η) cos(Ωt) + αv (sin η, cos η) cos(Ωt)
where Ω is the non-dimensional angular frequency of oscillation and
αh , αv 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 = 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 are
recovered by setting αh = 0 and αv = 0, 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 45◦ with
the oncoming flow we simply set αh = αv ≡ 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 + ξ0 ) cos θ, y = sinh(ξ + ξ0 ) sin θ.
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S. J. D. D’ALESSIO AND SERPIL KOCABIYIK
Here, the constant ξ0 = tanh−1 (b/a), and ξ = 0 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
(4)
(5)
∂2ψ ∂2ψ
+
= M 2 ζ,
∂ξ 2
∂θ2
2
∂ζ
1
2 ∂ ζ
∂2ζ
∂ψ ∂ζ
∂ψ ∂ζ
.
= 2
+
−
+
∂t
M R ∂ξ 2
∂θ2
∂θ ∂ξ
∂ξ ∂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
M2 =
(6)
1
[cosh 2(ξ + ξ0 ) − cos 2θ] .
2
Expressions for the dimensionless velocity components (vξ , vθ ) in the
directions of increase of (ξ, θ) in terms of the stream function ψ are
given by
1 ∂ψ
1 ∂ψ
,
vθ =
,
vξ = −
M ∂θ
M ∂ξ
and the vorticity ζ is defined in terms of the velocity components as
1
∂
∂
ζ = 2 − (M vξ ) +
(M vθ ) .
M
∂θ
∂ξ
The boundary conditions for t > 0 and 0 ≤ θ ≤ 2π are the impermeability and no-slip conditions on the cylinder surface given by
(7)
ψ=
∂ψ
= 0 when ξ = 0.
∂ξ
The far-field conditions can be derived by first noting that in the fixed
frame ~u → (− cos η, sin η) as x2 + y 2 → ∞. Then from (3) we obtain
~v =
∂ψ ∂ψ
−
,
∂y ∂x
→ (− cos η, sin η) − αh (cos η, − sin η) cos(Ωt)
− αv (sin η, cos η) cos(Ωt)
NUMERICAL SIMULATIONS OF FLOW
253
as x2 + y 2 → ∞. In terms of elliptic coordinates the above conditions
are expressed as
(8)
1
1
∂ψ
→ eξ+ξ0 sin(θ + η) + eξ+ξ0 [αh sin(θ + η)
∂ξ
2
2
− αv cos(θ + η)] cos(Ωt) as ξ → ∞,
(9)
1
1
∂ψ
→ eξ+ξ0 cos(θ + η) + eξ+ξ0 [αh cos(θ + η)
∂θ
2
2
+ αv sin(θ + η)] cos(Ωt) as ξ → ∞,
or equivalently as
ψ→
(10)
1 ξ+ξ0
1
e
sin(θ + η) + eξ+ξ0 [αh sin(θ + η)
2
2
− 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 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 (4),
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)–(10) for the stream
function are transformed into a set of global integral conditions for the
vorticity using equation (4). These conditions are derived by applying
Green’s second identity for the Laplacian operator, namely
ZZ
I ∂ψ
∂ψ
2
2
ds ,
−ψ
(φ ∇ ψ − ψ ∇ φ) dV =
φ
∂n
∂n
V
S
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
254
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 (4), it follows after some
integration by parts and making use of the far-field conditions that:
(12)
Z
(13)
Z
∞
0
∞
0
Z
Z
2π
M 2 ζ(ξ, θ, t) dθ dξ = 0,
0
2π
e−nξ M 2 ζ(ξ, θ, t) cos(nθ)dθ dξ
0
= πeξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)] δ1,n ,
(14)
Z
∞
0
Z
2π
e−nξ M 2 ζ(ξ, θ, t) sin(nθ)dθ dξ
0
= 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 to
ensure 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 Staniforth [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 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,
ω
ζ= ,
k
2t
k=2
R
1/2
,
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)
Z
∞
0
Z
2π
M 2 ω(z, θ, t) dθ dz = 0,
0
NUMERICAL SIMULATIONS OF FLOW
(17)
Z
∞
0
Z
255
2π
e−nkz M 2 ω(z, θ, t) cos(nθ)dθ dz
0
= πeξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)] δ1,n ,
(18)
Z
∞
0
Z
2π
e−nkz M 2 ω(ξ, θ, t) sin(nθ)dθ dz
0
= 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 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 = 0 is obtained following the work by
Staniforth [18]. This initial solution is given by
2 2
2 e ξ0
(19) ω0 (z, θ, 0) = √
[(1 + αh ) sin(θ + η) − αv cos(θ + η)]e−M0 z ,
π M0
(20) Ψ0 (z, θ, 0) =
e ξ0
[(1 + αh ) sin(θ + η) − αv cos(θ + η)]
M0
1
−M02 z 2
× M0 z erf(M0 z) − √ (1 − e
) ,
π
where erf(M0 z) denotes the error function and M02 = [cosh 2ξ0 −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. [4], D’Alessio
and Kocabiyik [5], and Kocabiyik and D’Alessio [11], we will briefly
describe the numerical technique. The computational domain, bounded
by 0 ≤ z ≤ z∞ and 0 < θ < 2π, is first discretized into a network of
L × 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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S. J. D. D’ALESSIO AND SERPIL KOCABIYIK
We express the stream function in the form of a truncated Fourier
series
(21)
N
X
1
Ψ(z, θ, t) = F0 (z, t) +
[Fn (z, t) cos(nθ) + fn (z, t) sin(nθ)] .
2
n=1
The equations governing the Fourier coefficients are
(22)
∂ 2 Fn
− n2 k 2 Fn = sn (z, t);
∂z 2
n = 0, 1, . . . ,
∂ 2 fn
− n2 k 2 fn = rn (z, t);
∂z 2
n = 1, 2, . . . ,
where
sn (z, t) =
1
π
Z
rn (z, t) =
1
π
Z
(23)
2π
M 2 ω(z, θ, t) cos nθdθ,
0
2π
M 2 ω(z, θ, t) sin nθdθ.
0
Boundary conditions for the Fourier components of Ψ are
F0 (0, t) = Fn (0, t) = fn (0, t) = 0,
∂F0
∂Fn
∂fn
=
=
=0
∂z
∂z
∂z
when z = 0,
and as z → ∞,
e−kz F0 → 0,
e−kz
∂F0
→ 0,
∂z
1 ξ0
e [sin η + (αh sin η − αv cos η) cos(Ωt)]δn,1 ,
2k
∂Fn
1
e−kz
→ eξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)]δn,1 ,
∂z
2
1 ξ0
e [cos η + (αh cos η + αv sin η) cos(Ωt)]δn,1 ,
e−kz fn →
2k
∂fn
1
e−kz
→ eξ0 [cos η + (αh cos η + αv sin η) cos(Ωt)]δn,1 ,
∂z
2
e−kz Fn →
NUMERICAL SIMULATIONS OF FLOW
257
for all integers n ≥ 1. The integral conditions can be formulated in
terms of the functions rn (z, t) sn (z, t) as follows:
(24)
(25)
Z
Z
∞
s0 (z, t)dθ dz = 0,
0
∞
e−nkz sn (z, t) dz
0
= eξ0 [sin η + (αh sin η − αv cos η) cos(Ωt)]δ1,n ,
(26)
Z
∞
e−nkz rn (z, t) dz
0
= 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 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
h00 (z) − β 2 h(z) = g(z)
(27)
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(0, θ, t)
=
1
M02
N
X
1
s0 (0, t) +
[rn (0, t) sin(nθ) + sn (0, t) cos(nθ)] .
2
n=1
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 (20) at t = 0. The use of this initial solution is essential for obtaining
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) = eξ0 sinh ξ sin(θ + η) + eξ0 sinh ξ
× [αh sin(θ + η) − αv cos(θ + η)] cos(Ωt),
has also been used as an initial condition at t = 0 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 (4), after setting ζ = 0, 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 = 0. Initially, ∆t = 10−4 was used; as time
increased 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 ∆z
in the coordinate z is more or less independent of R. The number of
points 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 for
a Reynolds number R = 103 and time t = 12. Placing z∞ well outside
the 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 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 = 103 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.
NUMERICAL SIMULATIONS OF FLOW
259
4 Results and discussion The numerical results are grouped in
two cases, (i) η = π/4 and (ii) η = 0, to illustrate the effect of inclination
on the ensuing flow. In the case of η = π/4 the cylinder oscillates in
a direction of the major axis of the ellipse while for the case η = 0 the
oscillations are at 45◦ 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 = 103 , Ω = π, r = 0.5 and αh = αv ≡ α = 0.25.
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/4 were also computed for comparison purposes. A brief derivation
of the formulae for the force coefficients is outlined in Section 4.1.
We note that the two cases considered in the present work deal with
the same oscillation frequency of f = 0.5 and therefore have a period
T = 1/f = 2. Thus, a complete cycle consists of the following four
stages: at t = 0 the ellipse starts to oscillate with its maximum velocity
in a direction of 45◦ to the horizontal free stream. At t = 0.5 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, 10.5, 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/4 and η = 0.
Lastly, the flow patterns for each case were plotted at the same times so
as to make comparisons and differences easier to report.
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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, the
dimensionless drag, CD , and lift, CL , coefficients are then defined by
2
2
CD = D/ρU∞
c and CL = L/ρU∞
c, respectively. The drag and lift
coefficients were computed using
(30)
CD =
2 sinh ξ0
R
Z
2π
0
∂ζ
∂ξ
sin θdθ
0
Z
2 cosh ξ0 2π
ζ0 sin θdθ cos η
R
0
Z
2 cosh ξ0 2π ∂ζ
cos θdθ
+
R
∂ξ 0
0
Z
2 sinh ξ0 2π
−
ζ0 cos θdθ sin η
R
0
−
(31)
CL =
− παh Ω cosh ξ0 sinh ξ0 sin(Ωt),
Z
2 cosh ξ0 2π ∂ζ
−
cos θdθ
R
∂ξ 0
0
Z
2 sinh ξ0 2π
+
ζ0 cos θdθ cos η
R
0
Z
2 sinh ξ0 2π ∂ζ
+
sin θdθ
R
∂ξ 0
0
Z
2 cosh ξ0 2π
ζ0 sin θdθ sin η
−
R
0
− παv Ω sinh ξ0 cosh ξ0 sin(Ωt).
The first integral in each expression of CD and CL represents the coefficient due to pressure and the second terms are due to friction. The last
term in the expressions for CD and CL represents the inviscid contribution which results from the acceleration due to time dependent cylinder
oscillations. We point out that at t = 0 both CD and CL are infinite
in magnitude due to the impulsive start delivered to the cylinder at
t = 0; afterwards CD and CL 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
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 (30) 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
1 ∂
1 2
∂vθ
=−
vξ + vθ2
p+
(32)
∂t
M ∂θ
2
Ω
cosh(ξ + ξ0 ) sin θ(αh cos η + αv sin η) sin(Ωt)
M
Ω
+
sinh(ξ + ξ0 ) cos θ(−αh sin η + αv cos η) sin(Ωt)
M
2 ∂ζ
+
.
M R ∂ξ
−
On the cylinder surface (32) simplifies greatly owing to the impermeability and no-slip conditions, and becomes
2 ∂ζ
∂P
(33)
=
− Ω cosh ξ0 sin θ(αh cos η + αv sin η) sin(Ωt)
∂θ 0
R ∂ξ 0
+ Ω sinh ξ0 cos θ(−αh sin η + αv 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
Z 2π Z
∂P
2 cosh ξ0 2π
(34) X = sinh ξ0
ζ0 sin θdθ,
sin θdθ −
∂θ 0
R
0
0
Z 2π Z
∂P
2 sinh ξ0 2π
(35) Y = − cosh ξ0
ζ0 cos θdθ.
cos θdθ +
∂θ 0
R
0
0
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).
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 instantaneous
snap shots of the flow field for the time interval 4 ≤ 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 = 4,
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 10 ≤ t ≤ 12. Figures 2a, 2c, 2g, 2h and 2l at t = 4, 6, 8, 10 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 = 10), 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 10.5, 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 4 illustrates the periodic variation
in both the drag and lift coefficients, CD and CL . 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
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 = π4 at t = 4.
FIGURE 2b: Streamline plot for the oblique oscillation case with
R = 103 , r = 0.5, Ω = π, αh = αv = 0.25, η = π4 at 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 flow
we simply set αv = 0 and αh = 0, 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 450 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 = 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
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 = π4 at t = 6.
FIGURE 2d: Streamline plot for the oblique oscillation case with
R = 103 , r = 0.5, Ω = π, αh = αv = 0.25, η = π4 at 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 = π4 at t = 7.
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 = π4 at 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 = π4 at t = 8.
FIGURE 2h: Streamline plot for the oblique oscillation case with
R = 103 , r = 0.5, Ω = π, αh = αv = 0.25, η = π4 at t = 10.
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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 = π4 at 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 = π4 at t = 11.
FIGURE 2k: Streamline plot for the oblique oscillation case with
R = 103 , r = 0.5, Ω = π, αh = αv = 0.25, η = π4 at t = 11.5.
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 = π4 at t = 12.
FIGURE 3: Surface vorticity distributions for the case R = 103 ,
r = 0.5, Ω = π, αh = αv = 0.25, η = π4 at times t = 10.5, 11, 11.5, 12.
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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 = π4 .
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 0◦ to 90◦ . 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 CD and
CL display three different amplitudes. The amplitudes of CD tend to
decrease 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 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
CL ; 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 CD and CL are in phase with
each other for the oblique case (see Figure 4), 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]).
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 = π4 at 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 = π4 at 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 = π4 at t = 11.
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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 = π4 at 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 = π4 at 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 = π4 at t = 10.
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 = π4 at 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 = π4 at 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 = π4 at t = 11.5.
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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 = π4 at 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 = π4 .
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 = π4 .
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 are
carried out for six complete cycles. The streamline patterns emerging
for the case η = 0, portrayed in Figures 9a and 9e, illustrate the flow at
five instances in time during the sixth cycle spanning the time interval
10 ≤ t ≤ 12. Comparing these figures with those corresponding to
the case having η = π/4 indicates that as η decreases from π/4 to 0 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/4. 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 CD , CL shown in Figure 11. Here, the fluctuations
in CL are much larger than those in CD and are again nearly in phase.
Finally, the surface vorticity distributions displayed in Figure 10 show
that apart from the tips of the cylinder the distribution is much flatter
than that with η = π/4. The only other point worth emphasizing is that
the lateral spacing of the vortex street seems to decrease as η decreases.
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.
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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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 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 45◦ 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 = 0 and η = π/4. 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.
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, Canada
E-mail address: [email protected]