Flow around a helically twisted elliptic cylinder
Woojin Kim, Jungil Lee, and Haecheon Choi
Citation: Physics of Fluids 28, 053602 (2016); doi: 10.1063/1.4948247
View online: http://dx.doi.org/10.1063/1.4948247
View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/28/5?ver=pdfcov
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PHYSICS OF FLUIDS 28, 053602 (2016)
Flow around a helically twisted elliptic cylinder
Woojin Kim,1 Jungil Lee,2 and Haecheon Choi1,3,a)
1
Department of Mechanical and Aerospace Engineering, Seoul National University,
Seoul 08826, South Korea
2
Department of Mechanical Engineering, Ajou University, Suwon 16499, South Korea
3
Institute of Advanced Machines and Design, Seoul National University,
Seoul 08826, South Korea
(Received 21 January 2016; accepted 13 April 2016; published online 10 May 2016)
In the present study, we conduct unsteady three-dimensional simulations of flows
around a helically twisted elliptic (HTE) cylinder at the Reynolds numbers of 100
and 3900, based on the free-stream velocity and square root of the product of the
lengths of its major and minor axes. A parametric study is conducted for Re = 100
by varying the aspect ratio (AR) of the elliptic cross section and the helical spanwise
wavelength (λ). Depending on the values of AR and λ, the flow in the wake contains
the characteristic wavelengths of λ, 2λ, 6λ, or even longer than 60λ, showing a wide
diversity of flows in the wake due to the shape change. The drag on the optimal
(i.e., having lowest drag) HTE cylinder (AR = 1.3 and λ = 3.5d) is lower by 18%
than that of the circular cylinder, and its lift fluctuations are zero owing to complete
suppression of vortex shedding in the wake. This optimal HTE configuration reduces
the drag by 23% for Re = 3900 where the wake is turbulent, showing that the HTE
cylinder reduces the mean drag and lift fluctuations for both laminar and turbulent
flows. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4948247]
I. INTRODUCTION
The flow behind a two-dimensional (2D) bluff body contains vortex shedding when the Reynolds
number is larger than a critical value. This vortex shedding induces high mean drag and lift fluctuations on the body, which increases the fuel consumption of vehicles and causes severe structural
vibrations. Therefore, many active and passive control methods have been developed to suppress the
vortex shedding.1
For two-dimensional bluff-body flows, the controls for suppressing vortex shedding may be
classified into two-dimensional (2D) and three-dimensional (3D) forcings, respectively.1 Here, the
2D forcing imposes the disturbances that are uniform along the spanwise direction. Examples of
this forcing include the end plate,2,3 splitter plate,4–6 base bleed,7,8 secondary small cylinder,9,10
active blowing/suction based on the flow sensing,11,12 to name a few. These controls suppress or
weaken vortex shedding and achieve reductions of the mean drag and lift fluctuations. On the other
hand, in the 3D forcing, disturbances imposed vary along the spanwise direction. This forcing may
be more efficient than the 2D forcing, in that the control power required for the first is less than
that for the latter. For example, Kim and Choi13 applied a 3D forcing, sinusoidal blowing/suction
along the spanwise direction, to flow over a circular cylinder at Re = u∞d/ν = 100, and showed
that vortex shedding was completely suppressed at the forcing amplitude of 0.08u∞, whereas a
base bleeding (a 2D forcing) required the forcing magnitude of 0.2u∞.14 Here u∞ is the free-stream
velocity, d is the cylinder diameter, and ν is the kinematic viscosity. Hwang et al.15 suggested from
a stability analysis that the high efficiency of 3D forcing is attributed to highly distorted mean flow
by the 3D forcing.
Unlike active controls, passive controls such as geometric modifications do not require any
power input. Many 3D passive controls suggested such as the segmented trailing edge,16–18 wavy
a) Author to whom correspondence should be addressed. Electronic mail: [email protected]
1070-6631/2016/28(5)/053602/16/$30.00
28, 053602-1
Published by AIP Publishing.
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Phys. Fluids 28, 053602 (2016)
trailing edge,19 wavy stagnation face,20 wavy front and flat rear faces,21 and small tabs22 successfully attenuated vortex shedding behind a bluff body having fixed separation line. For a bluff body
having moving separation line, a circular cylinder with a sinusoidal axis along the spanwise direction,23 and a circular cylinder using small tabs24 attenuated vortex shedding. To avoid the effects
of flow incidence angle, different geometric modifications have been also introduced, e.g., helical
strakes on a circular cylinder,25 a wavy cylinder with sinusoidally varying diameter along the spanwise direction,26,27 hemispherical bumps attached to a circular cylinder,28 and a helically twisted
elliptic (HTE) cylinder with an elliptic cross section rotating about the spanwise axis.29 Extensive
parametric studies on the 3D forcings have been conducted at Re = 100, and maximum drag reductions of 16%–20%, together with complete suppression of vortex shedding, have been reported at
the forcing spanwise wavelengths of 4–6d, where d is the diameter of a circular cylinder or the
height of a square cylinder.13,20,27 The reductions of the mean drag and root-mean-square (rms) lift
fluctuations by these controls are reported to be caused by direct wake modifications rather than by
separation delay.
Similar geometric modifications in the spanwise direction are also found in nature. For example,
the stem of a daffodil has an elliptic cross section with pointed ends on the major axis rotating along
the stem axis.30 Hanke et al.31 observed the vibrissae of a harbor seal largely perpendicular to the
swimming direction and made a modeled harbor seal vibrissae having an elliptic cross section with
varying the major and minor diameters and the angle of attack along the spanwise direction. Although
vortex shedding was not completely suppressed, the modeled harbor seal vibrissae produced reductions of the mean drag by 40% and the rms lift fluctuations by 90% at Re = 500, as compared to those
on a circular cylinder. The 2D vortex rollers in the wake were replaced by 3D vortical structures,
and these structures had longer spanwise wavelengths than the geometric spanwise wavelength of
the vibrissae and were similar to the two-cell antisymmetric mode observed in the wake of a bluff
body with a wavy trailing edge.19
The geometric characteristics of the daffodil stem, i.e., a HTE cylinder, were studied by Kim
et al.32 and Jung and Yoon.29 Kim et al.32 investigated the effects of the amplitude and wavelength
of helically twisted ellipse on the flow around a HTE cylinder at Re = 100 and showed a preliminary result that the strength of vortex shedding and mean drag was significantly reduced. Jung
and Yoon29 conducted large eddy simulation (LES) of flow over a HTE cylinder at Re = 3000. In
their study, the parameter chosen for the optimization was the aspect ratio of the ellipse by fixing
the wavelength of HTE cylinder as done in the work of Lam and Lin33 for the wavy cylinder,
and they obtained the mean drag reductions of 13% and 5% as compared to those for the circular
and wavy cylinders, respectively. Hence, their main objective was to investigate the effect of the
cross-sectional shape of the cylinder on the flow in the wake. We notice that the maximum drag
reduction from a HTE cylinder by Jung and Yoon29 at Re = 3000 is approximately half of that by
the distributed forcing (blowing/suction) obtained by Kim and Choi13 at Re = 3900. This is possibly
because the wavelength of the HTE cylinder was fixed to be 2.275d.29 Therefore, in the present
study, we conduct an extensive parametric study by varying the spanwise wavelength as well as the
aspect ratio of the HTE cylinder at Re = 100 and examine their effects on the mean drag, lift fluctuations, and flow structures. Then, the optimal configuration obtained for minimum drag at Re = 100
is applied to the HTE cylinder at Re = 3900, expecting a significant amount of drag reduction even
if the flow in wake is turbulent.
In Sec. II, the geometry of the HTE cylinder is described, and numerical aspects are given
in Sec. III. In Sec. IV, the variations of mean drag and rms lift coefficients are presented and
modified flow structures in the wake are classified into three different modes at Re = 100. In Sec. V,
given optimal aspect ratio obtained at Re = 100, the flows at Re = 3900 for various wavelengths are
investigated, followed by conclusions in Sec. VI.
II. GEOMETRY OF THE HELICALLY TWISTED ELLIPTIC (HTE) CYLINDER
The present HTE cylinder has an elliptic cross section which rotates clockwise along the
spanwise direction (Fig. 1). The shape of the HTE cylinder is given as
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Phys. Fluids 28, 053602 (2016)
FIG. 1. Schematic diagram of the helically twisted elliptic (HTE) cylinder.
(x cos θ + y sin θ)2 (−x sin θ + y cos θ)2 1
+
= ,
(1)
4
a2
b2
πz
θ=
,
(2)
λ
where a and b are the major and minor axes, respectively, and λ is the wavelength (half twisted
length).√The aspect ratio of elliptic cylinder is AR (=a/b) and the characteristic length is taken to
d 2 = (a + b)/2 as the characteristic length, but we prefer
be d = ab. Jung and Yoon29 defined
√
the characteristic length of d = ab to that of d 2 = (a + b)/2. Although these two definitions, d
and d 2, provide the same characteristic length for the circular cylinder (i.e., a = b = 2r), the cross
section area of HTE cylinder (=πab/4) with the present definition becomes equal to that of a
circular cylinder regardless of AR. So, two parameters, AR and λ, determine the shape of the HTE
cylinder. A parametric study on these two parameters is conducted for Re = 100 to find the optimal
geometry of the HTE cylinder providing minimum mean drag and lift fluctuations. The parameter
ranges considered are AR = 1.1–1.5 and λ = 2–7d. This wavelength range includes the optimal
wavelengths found in the previous studies for different geometries, λ opt = 5.6d (square cylinder
with a wavy stagnation face20), 4–5d (distributed forcing13), 6d (wavy cylinder27), and 4d (circular
cylinder using tabs24). The result of the parametric study for Re = 100 is then applied to the case of
Re = 3900.
III. NUMERICAL METHOD
The governing equations for unsteady incompressible viscous flow are the continuity and
Navier-Stokes equations normalized by the characteristic length (d) and velocity (u∞),
∂ui ∂ui u j
∂p
1 ∂ 2ui
+
=−
+
+ f i,
∂t
∂xj
∂ x i Re ∂ x j ∂ x j
(3)
∂ui
− q = 0,
∂ xi
(4)
where t is the time, x i = (x, y, z) are the Cartesian coordinates (streamwise, transverse, and spanwise directions, respectively), ui = (u, v, w) are the corresponding velocity components, and p is the
pressure. Here, f i is the momentum forcing applied at the grid cell faces on or inside the immersed
boundary (i.e., the HTE cylinder surface) to satisfy the no-slip boundary condition there, and q is
the mass source/sink applied at the grid cell centers to satisfy local mass conservation for the cells
containing the immersed boundary (see the work of Kim et al.34 for details). Flows at two different
Reynolds numbers, Re = 100 and 3900, are simulated. For Re = 3900, large eddy simulation (LES)
with a dynamic global subgrid-scale model35,36 is performed.
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Phys. Fluids 28, 053602 (2016)
TABLE I. Flow statistics for a circular cylinder (E: experiment; L: LES).
In the present computation, the domain size is 30d (x) × 50d (y) × πd (z)
and the number of grids is 465 (x) × 329 (y) × 64 (z).
Son and Hanratty38 (E)
Norberg39 (E)
Norberg40 (L)
Kravchenko and Moin41 (L)
Kim and Choi13 (L)
You and Moin42 (L)
Lam et al.43 (E)
Lam et al.43 (L)
Present (L)
Re
CD
C Lrms
−C pb
θ s (deg)
5000
3000
3900
3900
3900
3900
4000
3900
3900
...
...
...
1.04
1.06
1.01
1.03
1.03
1.05
...
...
0.3
...
0.27
0.12
0.16
0.12
0.25
...
0.84–0.89
...
0.94
0.95
0.92
...
...
0.98
86
...
...
88
...
...
...
...
85
For time advancement, a semi-implicit fractional step method37 is used: a second-order CrankNicolson method and a third-order Runge-Kutta method are used for the diffusion and convection terms, respectively. For spatial discretization, a second-order central difference is used for
all spatial derivative terms. At the inlet, a Dirichlet boundary condition, u = u∞ and v = w = 0,
is given. On the top and bottom boundaries, ∂u/∂ y = v = ∂w/∂ y = 0 is applied, and the periodic boundary condition is imposed on the spanwise direction. At the exit, a convective boundary
condition is given as ∂ui /∂t + c∂ui /∂ x = 0, where c is the plane-averaged streamwise velocity
at the exit. For Re = 100, the computational domain sizes are 90d (x) × 100d ( y) × nλ(z) (n =
1, 2, 3, 6, 12, 30, 60; λ = 2–7d), and the numbers of grid points are 385 (x) × 217 ( y) × 32n (z). For
Re = 3900, the computational domain sizes are 30d (x) × 50d ( y) × nλ (z) (n = 1, 2; λ = 2–7d),
and the numbers of grid points are 465 (x) × 329 ( y) × 16 nλ/d (z). These grid resolutions resulted
from an extensive grid independence study: grid resolution in each direction was increased by
1.5–2 times and the changes in the mean drag coefficient were less than 1.5%. The size of computational time step is obtained from maximum Courant-Friedrichs-Lewy (CFL) number condition,
CFL = ∆t (|u| /∆x + |v | /∆ y + |w| /∆z) ≤ 1. The flow statistics such as the mean drag coefficient
(C D ), rms lift fluctuation coefficient (CLrms), mean base pressure coefficient (C pb), and mean separation angle (θ s ) for the circular cylinder at Re = 3900 are given in Table I, together with those from
2
previous experiments and numerical simulations at and near Re = 3900, where CD = 2FD/ρu∞
L z d,
2
CL = 2FL /ρu∞ L z d, FD and FL are the drag and lift forces, respectively, L z is the spanwise domain
size, d is the diameter of the circular cylinder, and the overbar indicates the time averaging. As
shown, the present results agree well with those of previous experimental and numerical studies.
FIG. 2. Effect of the spanwise domain size (L z ) on the mean drag and rms lift fluctuation coefficients for the HTE cylinder
(Re = 100): (a) mean drag coefficient, CD; (b) rms lift fluctuation coefficient, C Lrms. In (b), the results for the cases of AR = 1.2
nearly overlap with those for the cases of AR = 1.3. Here, L z /λ considered are 1, 2, 3, 6, 12, 30, and 60.
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Kim, Lee, and Choi
Phys. Fluids 28, 053602 (2016)
FIG. 3. Time histories of the drag and lift coefficients for the case of AR = 1.1 and λ = 5d (Re = 100): (a) C D ; (b) C L .
L z /λ = 2 (red line); 3 (yellow line); 6 (green line); 12 (light blue line); 30 (dark blue line); 60 (black line).
Because of the periodic boundary condition imposed in the spanwise direction and also
the wavelength of the HTE cylinder λ, flow fields obtained significantly depend on the choice
of the spanwise domain size, nλ. We systematically increase the domain size until the change in
the predicted mean drag coefficients is less than 1.5%. Then, we double the domain size further
to see if there are vortical structures whose length scale is bigger than the previous domain size.
Figure 2 shows the effect of the spanwise domain size on the mean drag and rms lift fluctuation
coefficients for the HTE cylinder at Re = 100. As shown, the drag and lift forces strongly depend on
the spanwise domain size; their values and the characteristic length scale of vortical structures (see
Sec. IV B) converge when L z = 2λ (AR = 1.1 and λ = 7d; AR = 1.2 and 1.3) and 6λ (AR = 1.1 and
λ = 6d), but do not fully converge even at L z = 60λ for the case of AR = 1.1 and λ = 5d. Figure 3
shows the time histories of the drag and lift coefficients for AR = 1.1 and λ = 5d. The results
strongly depend on the choice of the spanwise domain size; with increasing L z , the lift fluctuations
keep decreasing, although the magnitude of drag converges at L z = 12λ (see also Fig. 2).
Figure 4 shows the time histories of the drag coefficient for different HTE cylinder configurations at Re = 100. Here, we consider two different initial conditions at t = 0 to see if they change
the simulation results: in Fig. 4(a), the initial condition is the steady flow over a circular cylinder
at Re = 10 with random perturbations on the velocity and pressure with an amplitude of 0–0.2,
and in Fig. 4(b) it is an instantaneous flow over a circular cylinder at Re = 100. It is surprising
to note that the latter initial condition requires an enormous amount of time integration to reach a
statistically steady state, whereas the flow reaches the statistically steady state relatively faster with
the first initial condition. With the latter initial condition (Fig. 4(b)), the drag coefficient converges
to one state but redevelops to another state after long time, suggesting that some other instability
FIG. 4. Effect of initial conditions on the drag coefficient (Re = 100): (a) steady flow over a circular cylinder at Re = 10 with
random perturbations; (b) an instantaneous flow over a circular cylinder at Re = 100. Here, L z = 2λ (AR = 1.1 and λ = 7d;
AR = 1.2 and 1.3 and λ = 4d and 5d) and 6λ (AR = 1.1 and λ = 6d).
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FIG. 5. Instantaneous vortical structures in the wake behind the HTE cylinder (AR = 1.1 and λ/d = 6): (a) L z /λ = 1; (b) 2;
(c) 3; (d) 6.
mechanism may exist for the HTE cylinder. However, the final states from two initial conditions are
identical, as shown in Figs. 4(a) and 4(b).
Figure 5 shows the effect of the spanwise domain size on instantaneous vortical structures
for the case of AR = 1.1 and λ = 6d. Instantaneous vortical structures are identified using the
method by Jeong and Hussain.45 With L z = λ, the flow reaches a steady state, whereas distorted
vortex rollers with the wavelength of 2λ are formed in the wake with L z = 2λ. On the other hand,
with L z = 3λ, vortical structures in the wake have the wavelength of 3λ. Thus, a simulation with
L z = 6λ is performed, and in this case, the flow contains dominant vortical structures with the
wavelength of 2λ. These flow structures appear to be similar to those with L z = 2λ, but their corresponding mean drag and rms lift fluctuation coefficients are very different (see Fig. 2). The vortical
structures with L z = 12λ are nearly the same as those with L z = 6λ (Fig. 15(b)). Figure 6 shows
the effect of the spanwise domain size for the case of AR = 1.2 and λ = 4d. Similar to the previous
case shown in Fig. 5, flow becomes steady with L z = λ. On the other hand, the flow structures
computed with L z = 2λ and 6λ are very similar to each other, and their mean drag and rms lift
fluctuation coefficients are also nearly the same (Fig. 2). These two examples clearly indicate that
an appropriate choice of the spanwise domain size (L z ) is very important in understanding the
flow over a HTE cylinder. In Sec. IV, we present the results that are fully converged from careful
selections of numerical parameters.
IV. RESULTS AT Re = 100
A. Drag and lift coefficients
Figure 7 shows the variations of the mean drag and rms lift fluctuation coefficients with the
wavelength and aspect ratio at Re = 100. The drag reaches minimum at λ = 3.5–4d except for the
case of AR = 1.1 where minimum drag occurs at λ ∼ 5d, and another local minimum is observed at
λ ∼ 6d for the cases of AR = 1.2 and 1.3. The maximum reduction of drag by the HTE cylinder is
FIG. 6. Instantaneous vortical structures in the wake behind the HTE cylinder (AR = 1.2 and λ/d = 4): (a) L z /λ = 1; (b) 2;
(c) 6.
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Kim, Lee, and Choi
Phys. Fluids 28, 053602 (2016)
FIG. 7. Variations of the mean drag and rms lift fluctuation coefficients with the wavelength (λ) and aspect ratio (AR) at
Re = 100: (a) CD; (b) C Lrms.
, Circular cylinder;
, AR = 1.1;
, AR = 1.2;
, AR = 1.3;
, AR = 1.4;
, AR = 1.5. Here, L z = 2λ (AR = 1.1 and λ = 7d; AR = 1.2 and 1.3 and λ = 4d, 4.5d, and 5d), 6λ (AR = 1.1 and
λ = 6d), 60λ (AR = 1.1 and λ = 5d), and λ (otherwise).
about 18% for the cases of λ = 3.5–4d and AR = 1.3–1.4, as compared to that of a circular cylinder.
This amount of drag reduction is similar to those by the distributed forcing,13 the wavy cylinder,27
and a circular cylinder using tabs.24 The present optimal spanwise wavelength (λ opt = 3.5–4d) is
also similar to those by the distributed forcing (4–5d)13 and a circular cylinder using tabs (4d),24 but
shorter than that by the wavy cylinder (6d).27 The rms lift fluctuation coefficients are almost zero
at λ = 3.5–5d except for the case of AR = 1.1. This different behavior for AR = 1.1 is because the
geometric modification for this case is not so significant, and thus, the flow modification is more
gradual than those for other cases.
Figure 8 shows the mean skin friction lines and the contours of the mean pressure coefficient on
the surfaces of circular and HTE cylinders. The mean skin friction lines indicate that the separation
angle for the circular cylinder is 118◦, which agrees well with the result of Park et al.,44 and that for
the HTE cylinder is delayed near z = 0 (where the major axis of the elliptic cross section is aligned
with the streamwise direction) and is advanced near z = 0.5λ (where the minor axis is aligned with
the streamwise direction). Note also that the largest and smallest separation angles are not observed
at z = 0 and 0.5λ, respectively, but at slightly shifted spanwise locations owing to clockwise rotation of the ellipse along the spanwise axis. The delay and advance of the separation angle along
the spanwise direction indicates that the drag reduction is achieved through wake modifications
rather than by the separation delay. On the other hand, the surface pressure on the HTE cylinder
FIG. 8. Mean skin friction lines and contours of the mean surface pressure coefficient (top view; not to scale): (a) circular
cylinder; (b) HTE cylinder with AR = 1.3 and λ = 2d; (c) HTE cylinder with AR = 1.3 and λ = 3.5d. Here, L z = λ (AR = 1.3
and λ = 2d and 3.5d).
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FIG. 9. Spanwise variations of the mean base pressure and recirculation line for the HTE cylinder (AR = 1.3): (a) Cpb; (b) Lr.
Here, Cpb and Lr of the circular cylinder are −0.713 and 1.917, respectively.
3.5;
, 4;
, 5;
, 6;
, λ/d = 2;
, 2.5;
, 3;
,
, 7. Here, L z = λ (λ/d = 2, 2.5, 3, 3.5, 6, and 7) and 2λ (λ/d = 4 and 5).
with AR = 1.3 and λ = 3.5d is recovered more than that with AR = 1.3 and λ = 2d, consistent with
the drag-reduction results shown in Fig. 7. The increase in the base pressure is attributed to the
suppression of vortex shedding in the wake (see below).
Figure 9 shows the spanwise variations of the mean base pressure coefficient and recirculation
line for AR = 1.3. Here the mean recirculation line is the line on which the mean streamwise
velocity is zero at the center plane ( y = 0). The mean base pressure on the HTE cylinder is recovered from that on the circular cylinder (C pb = −0.713) and is the highest at λ = 3.5d where the
maximum drag reduction is obtained. The mean recirculation line is also positioned farthest from
the cylinder at λ = 3.5d in 0.2 < z/λ < 0.8, but the recirculation does not exist at z/λ < 0.2 and
z/λ > 0.8. A similar spanwise distribution of the mean recirculation line has been observed for a
wavy cylinder27 suggesting the existence of two counter-rotating transverse vortices observed by
Darekar and Sherwin20 which induce backward flow at z/λ = 0.5 and forward flow at z/λ = 0
and 1.
B. Flow modes
In this section, we classify the flow structures in the wake of the HTE cylinder into three modes
(modes α, β, and γ) according to their spanwise characteristic length (l m ), i.e., l m = λ for mode α,
l m = 2λ for mode β, and l m > 2λ for mode γ, respectively. Figure 10 shows the map of modes α, β,
and γ on the (λ–AR) plane. For the parameter ranges considered, mode α is widely observed. Most
of the previous studies13,20,24,27 observed mode α only, but Tombazis and Bearman19 found mode
β for a bluff body with a wavy trailing edge. In the present study, mode β is found at λ/d = 4–5
and AR = 1.2–1.3, and at λ/d = 7 and AR = 1.1, whereas mode γ is observed at λ/d = 5–6 and
AR = 1.1.
1. Mode α
In mode α, the spanwise characteristic length of vortical structures in the wake is λ. Mode α
is subdivided into three modes α-1, α-2, and α-3, and some of the corresponding flow fields are
shown in Fig. 11. In mode α-1 (Figs. 11(a)-11(c)), vortex rollers in the near wake are distorted along
the spanwise direction and their strength becomes weaker in downstream. For λ = 2d, the distorted
vortex rollers return to be two-dimensional in an early wake region. In this mode, the mean drag
and rms lift fluctuations decrease more with larger λ and AR according to distorted two-dimensional
structures in the wake (see also Fig. 7).
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Kim, Lee, and Choi
FIG. 10. Flow mode map with the wavelength and aspect ratio:
mode β-2; , mode γ.
Phys. Fluids 28, 053602 (2016)
, mode α-1;
, mode α-2;
, mode α-3;
, mode β-1;
,
In mode α-2 (Fig. 11(d)), the flow is steady and vortex shedding is completely suppressed, which
results in a significant mean-drag reduction and zero lift fluctuation. In mode α-3 (Figs. 11(e)-11(f)),
three-dimensional vortical structures, which are very different from the distorted vortex rollers
observed in mode α-1, are generated in the wake. These three-dimensional vortical structures are
stronger and more tilted in the streamwise direction with increasing λ. Therefore, the mean drag and
rms lift fluctuations increase from their minimum values of mode α-2 with increasing λ (see Fig. 7).
2. Mode β
In mode β, the spanwise characteristic length of vortical structures in the wake is 2λ. Mode β is
divided into two modes β-1 and β-2. In mode β-1, vortex shedding occurs alternatingly in the spanwise direction (with the period of 2λ) in addition to the common vortex shedding in the transverse
direction (see Fig. 12(a)). Therefore, the transverse velocity in the first wavelength (i.e., z/λ = 0–1)
is 180◦ out of phase with that in the second wavelength (i.e., z/λ = 1–2) (Fig. 12(b)). Owing to
FIG. 11. Instantaneous vortical structures of mode α: (a) (λ/d, AR) = (2, 1.3), (b) (3, 1.3), (c) (3, 1.5), (d) (3.5, 1.5), (e) (5,
1.5), (f) (7, 1.5). (a)-(c) Mode α-1; (d) mode α-2; (e) and (f) mode α-3. In this figure, flow structures are drawn only for
0 ≤ z ≤ λ (L z = 2λ).
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FIG. 12. Mode β-1 at (λ/d, AR) = (4, 1.2): (a) instantaneous vortical structures at tu∞/d = 4203; (b) time traces of the transverse velocity, v, at (x, y, z) = (2d, 0.6d, 0.5λ) (solid line) and (2d, 0.6d, 1.5λ) (dashed line). Note that this computation is
carried out at L z = 12λ.
this phase difference along the spanwise direction, the instantaneous lift force is zero despite the
increase in the mean drag from that of mode α-1 (λ/d = 3.5, AR = 1.2). This mode β-1 corresponds
to the two-cell antisymmetric mode observed by Tombazis and Bearman.19
Figure 13(a) shows the time traces of the transverse velocity at (x, y, z) = (2d, 0.6d, 0.5λ)
and (2d, 0.6d, 1.5λ). The amplitude modulations are clearly observed, and their corresponding
frequency is f A d/u∞ = 0.0145 (Fig. 13(b)). There are two other peak frequencies in the energy
spectrum, f B d/u∞ = 0.1454 and f C d/u∞ = 0.1600. These frequencies are lower than that of the
circular cylinder, f d/u∞ = 0.165, and their difference corresponds to the modulation frequency, f A.
The phase difference between the transverse velocities in the first- and second-wavelength domains
(i.e., z/λ = 0–1 and 1–2, respectively) varies between 110◦ and 248◦. The reason why the two
velocities are not exactly 180◦ out of phase is that vortex dislocation occurs alternatingly in the first
and second-wavelength domains (see arrows in Fig. 13(c)).
3. Mode γ
In mode γ, the spanwise characteristic length of vortical structures in the wake is larger than
2λ. Unlike those in modes α and β, the vortical structures in mode γ show various flow patterns
even instantaneously. Figure 14 shows vortical structures in the wake at two instants of time. In
this figure, each instantaneous flow field is split into four pieces because of very long spanwise
domain size used (L z = 60λ). Vortex rollers distorted along the spanwise direction (blue arrows
in this figure) and vortical structures with alternating vortex shedding in the spanwise direction
(red arrows) are simultaneously observed at each instant. Also, temporal transition between these
flow structures occurs at multiple regions in the wake (see red and blue arrows at z/L z = 0.47,
respectively, in Figs. 14(a) and 14(b)), which decreases drag and lift fluctuations.
FIG. 13. Mode β-2 at (λ/d, AR) = (7, 1.1): (a) time traces of the transverse velocity, v, at (x, y, z) = (2d, 0.6d, 0.5λ) (red
solid line) and (2d, 0.6d, 1.5λ) (blue dashed line); (b) energy spectrum; (c) instantaneous vortical structures at ① t u ∞/d =
1610 and ② 1640. In (b), f A d/u ∞ = 0.0145, f B d/u ∞ = 0.1454, and f C d/u ∞ = 0.1600. Note that this computation is carried
out at L z = 6λ.
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FIG. 14. Mode γ (λ/d = 5, AR = 1.1, L z = 60λ): instantaneous vortical structures at (a) t u ∞/d = 622; (b) t u ∞/d = 762.
Figure 15(a) shows the time histories of the drag and lift coefficients at AR = 1.1 and λ = 6d.
The drag and lift coefficients contain very low frequencies, together with vortex shedding frequencies. Figure 15(b) shows the instantaneous vortical structures in the wake. At tu∞/d = 1566, vortex
rollers in the near wake are distorted along the spanwise direction (see an arrow in Fig. 15(b)①). At
this instant, these rollers produce high drag and high lift fluctuations. At tu∞/d = 1650, the vortex
FIG. 15. Mode γ (λ/d = 6, AR = 1.1, L z = 12λ): time histories of the (a) drag (dashed line) and lift (solid line) coefficients
and (b) vortical structures in the wake at different instants, ① t u ∞/d = 1566 and ② 1650. The vortical structures in upper
(y > 0) and lower (y < 0) parts are drawn in blue and gray colors, respectively.
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FIG. 16. Mode analysis (mode γ; λ/d = 6, AR = 1.1, L z = 12λ): (a) streamwise velocity fluctuations, u ′; (b) transverse
velocity fluctuations, v ′. (Red line), mode 0 (m = 0); (green line), mode 1 (m = 1); (light blue line), mode 2 (m = 2); (dark
blue line), mode 6 (m = 6); (purple line), mode 12 (m = 12). Black lines in this figure correspond to modes having relatively
lower energy.
rollers are split into multiple 2λ structures even in the near wake (see an arrow in Fig. 15(b)②),
decreasing the drag and the lift fluctuations. To identify dominant wavelengths of vortical structures, we conduct a mode analysis.46 Figure 16 shows the distributions of energy of streamwise and
transverse velocity fluctuations among different modes. The energy of each mode m is defined as
 L y /2
Ẽ (x, m) =
E (x, y, m) d y,
(5)
−L y /2
where m is the spanwise wavenumber (m = L z /λ m ), λ m is the wavelength, E (x, y, m) =
û ′ (x, y, m,t) û ′∗ (x, y, m,t) or v̂ ′ (x, y, m,t) v̂ ′∗ (x, y, m,t), û ′ and v̂ ′ are the discrete Fourier coefficients of the streamwise and transverse velocity fluctuations, u ′ and v ′, respectively, and the overbar
denotes the time averaging. As shown, the modes 0 (m = 0) and 6 (m = 6) are dominant near the
cylinder and far wake, respectively, for the streamwise velocity, whereas the mode 0 is dominant
everywhere for the transverse velocity. The vortical structures with λ m = 2λ corresponding to
m = 6 are observed farther away from the cylinder in Fig. 15(b).
V. RESULTS AT Re = 3900 (TURBULENT FLOW IN THE WAKE)
At Re = 100 (laminar flow in the wake), vortex shedding was completely suppressed and rms
lift fluctuations became zero for the case of AR = 1.3 and λ = 3.5d. So, in this section, we fix the
aspect ratio of the HTE cylinder to be AR = 1.3 and vary the wavelength from λ = 2d to 7d at
Re = 3900 (turbulent wake). At Re = 3900, the mean drag and rms lift fluctuation coefficients of the
HTE cylinder with L z = λ are nearly same as those with L z = 2λ, showing less sensitivity on the
spanwise domain size for turbulent flow.
FIG. 17. Variations of the mean drag and rms lift fluctuation coefficients with the wavelength (AR = 1.3; Re = 3900): (a) CD;
(b) C Lrms.
, Circular cylinder;
, AR = 1.3. Here, L z = λ.
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FIG. 18. Spanwise variations of the mean base pressure coefficient and recirculation line for the HTE cylinder (AR = 1.3;
Re = 3900): (a) Cpb; (b) Lr. Here, Cpb and Lr of the circular cylinder are −0.98 and 1.76, respectively.
, λ/d = 2;
,
3;
, 4;
, 5;
, 6;
, 7. Here, L z = λ.
Figure 17 shows the variations of the mean drag and rms lift fluctuation coefficients with the
wavelength for AR = 1.3 at Re = 3900, together with those of the circular cylinder. The amount
of drag reduction in percentage at Re = 3900 is larger than that at Re = 100. For example, the
drag reduction at Re = 3900 is 23% for λ = 4d, but it is 18% for λ = 3.5d at Re = 100. On the
other hand, the rms lift fluctuation coefficients at Re = 3900 are quite small but not zero for all
λ’s considered here, whereas they are zero for λ = 3.5d at Re = 100. This indicates that the HTE
cylinder at Re = 3900 does not annihilate vortex shedding in the wake, but it produces more drag
reduction because vortex shedding increases the form drag and the ratio of form drag to total drag is
higher at Re = 3900 than that at Re = 100.
Figure 18 shows the spanwise variations of the mean base pressure coefficient and recirculation
line for AR = 1.3 at Re = 3900. The mean base pressure on the HTE cylinder is higher than that
on the circular cylinder. The highest mean base pressure occurs at λ = 4d at which maximum drag
reduction is obtained. The increase in the mean base pressure is accompanied by the increase in the
mean recirculation line, i.e., C pb = −C/L r (Lam and Lin;33 Lam et al.47). Here, the value of C varies
depending on the shape of a bluff body, and they are 1.72 and 2.09–2.51 for the circular and HTE
cylinders, respectively. These values agree well with the results of Jung and Yoon.29
Figure 19 shows instantaneous vortical structures for the circular and HTE cylinders. The flow
over the circular cylinder shows the alternating vortex shedding and three-dimensional vortical
FIG. 19. Instantaneous vortical structures at Re = 3900 (side and top views): (a) circular cylinder; (b) HTE cylinder with
AR = 1.3 and λ = 2d; (c) HTE cylinder with AR = 1.3 and λ = 4d. Note that the computations in (b) and (c) are carried out
with L z = 2λ.
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FIG. 20. Time traces of the transverse velocity in the shear layer at Re = 3900: (a) circular cylinder; (b) HTE cylinder with
λ = 4d and AR = 1.3. Here, L z = λ.
structures in the spanwise direction. On the other hand, the vortices, existing right behind the circular cylinder, disappear in the near wake behind the HTE cylinder and the alternating vortex shedding
is significantly weakened, which results in drag reduction and reduced lift fluctuations by the HTE
cylinder. Furthermore, the shear-layer vortices at z = 0 start to evolve farther downstream in the
wake than at other spanwise locations. This slow evolution of the shear layer at z = 0 is similar to
that at the location of suction in the distributed forcing.13
Figure 20 shows the temporal evolutions of the transverse velocity in the shear layer for the
circular and HTE cylinders. Owing to different shear layer evolutions along the spanwise direction,
the transverse velocities show different behaviors. In the case of circular cylinder, the transverse
velocities show two distinct frequencies (Fig. 20(a)). The low-frequency behavior is associated with
the vortex shedding, whereas the high-frequency one is due to the evolution of shear layer vortices.
On the other hand, in the case of the HTE cylinder, vortex shedding in the wake is significantly
weakened and the shear layer vortices evolve from x ≈ 0.7d at z ≈ 0.5λ but from x ≈ 1.5d at
z ≈ 0 (see Fig. 19(c)). Therefore, as shown in Fig. 20(b), the transverse velocity at z ≈ 0 has very
small fluctuations, whereas that at z ≈ 0.5λ contains fluctuations associated with the shear-layer
evolution.
VI. CONCLUSIONS
In the present study, we investigated the flows around HTE cylinders at Re = 100 (laminar
wake) and 3900 (turbulent wake). A parametric study on the aspect ratio and wavelength of the HTE
cylinder was conducted at Re = 100, and flow structures in the wake were classified into modes α
(l m = λ), β (l m = 2λ), and γ (l m > 2λ), where l m is the spanwise characteristic length of vortical
structures. In mode α, flow modes were subdivided into mode α-1 (distorted vortex rollers), α-2
(steady flow), and α-3 (fully three-dimensional vortical structures). In mode β, the phase differences between two consecutive vortex rollers in the spanwise direction were 180◦ in mode β-1 and
110◦–248◦ in mode β-2. In mode γ, the flow structures in the wake were very long in the spanwise
direction, flow characteristics similar to modes α and β were simultaneously observed, and temporal
transition between these two modes occurred (AR = 1.1 and λ = 5d). Maximum drag reduction of
18% and zero lift fluctuation were obtained at AR = 1.3 and λ = 3.5d (mode α-2). This optimal
HTE geometry (AR = 1.3) obtained for Re = 100 was applied to turbulent flow at Re = 3900 varying λ/d = 2–7. As a result, the mean drag reduction of 23% and very small lift fluctuations were
obtained for the HTE cylinder with AR = 1.3 and λ = 4d.
In general, drag reduction is accompanied by suppression of the classical Karman vortex shedding. The present HTE cylinder not only attenuates the Karman vortex shedding but also produces
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alternating vortex shedding in the spanwise direction (modes β and γ), thus weakening the strength
of vortical structures in the wake and reducing the mean drag and lift fluctuations.
ACKNOWLEDGMENTS
This work was supported by the NRF programs (Grant Nos. 2011-0028032 and
2014M3C1B1033848) of MSIP, Korea.
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