Oblique Cylinder Wake Flow: 3D Characteristics & Axial Pattern

Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Contents lists available at ScienceDirect
Journal of Wind Engineering & Industrial Aerodynamics
journal homepage: www.elsevier.com/locate/jweia
Three-dimensional characteristics and axial flow pattern in the wake flow of
an oblique circular cylinder
Rui Wang b, c, Dabo Xin a, *, Jinping Ou b, c
a
b
c
School of Civil Engineering, Northeast Forestry University, Harbin, 150040, China
School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China
Key Lab of Structures Dynamic Behavior and Control (Harbin Institute of Technology), Ministry of Education, Harbin, 150090, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Oblique circular cylinder
Computational fluid dynamics
Three-dimensional flow characteristics
Axial flow pattern
Yaw angle
Wake flow
Many circular cylindrical engineering structures submerged in fluid are not always orthogonal to flow direction.
Oblique circular cylinder is a more general model to study the flow around circular cylindrical engineering
structures. Comparing with a normal circular cylinder, there are more three-dimensional effects reflected in the
wake flow and aerodynamic forces of an oblique circular cylinder. Moreover, axial flow is a unique and important
flow phenomenon in the oblique circular cylinder wake flow. This study introduces a numerical study based on an
oblique circular cylinder model and unsteady Reynolds-Averaged Navier-Stokes equations (URANS) to reveal the
three-dimensional external flow and axial flow. Effects of the yaw angles take the most attentions. The analyzed
flow features include aerodynamic force coefficients, wake flow velocities, correlations between aerodynamic
forces and wake flow structures, three-dimensional vortex structures and three-dimensional axial flow. Spectrum
signatures of wake flow fluctuation and time-average axial flow pattern are verified by a visualized wind tunnel
experiment. Numerical results prove that there are obvious three-dimensional effects in wake flow velocities and
aerodynamic forces. Yaw angle, as a crucial parameter, determines the configuration of vortices. Furthermore,
yaw angle affects the spectrum signatures of wake flow fluctuation and aerodynamic force fluctuation, and change
the spanwise correlation of wake flow. A novel time-average axial flow velocity-distribution pattern is observed in
the numerical results and experiment results. Its mechanism is analyzed based on the wake vortex structures.
1. Introduction
As a classical fluid problem, external flow around a circular cylinder
was usually studied based on two-dimensional flow before, which means
that the flow velocity and fluid dynamic force along the cylinder axis
were neglected in flow field analysis. However, there are indeed threedimensional flow structures in cylinder wake flow within specific Reynolds number ranges (Hama, 1957; Grant, 1958). Williamson (1996) and
Wu et al. (1996) proposed three-dimensional instability in circular cylinder’s wake flow transition regime. They found that there are pairs of
streamwise vortices with certain spanwise distances in circular cylinder
wake flow, which is typically a three-dimensional instability phenomenon. Barkley and Henderson (1996) used the Floquet stability analysis to
calculate the spanwise distances of the streamwise vortex pairs. Especially, the streamwise vortices with spanwise periodicity, which cannot
be reflected in two-dimensional flow field, have suppressing effect on
spanwise vortex shedding (Zhang and Lee, 2005; Kim and Choi, 2005;
Hwang et al., 2013). Thus, three-dimensional flow structures should be
considered seriously in the analysis of circular cylinder’s external flow.
Many circular cylindrical engineering structures, such as cables,
pipelines, and electric transmission lines, are not always orthogonal to
flow streamwise. For an oblique circular cylinder, three-dimensional
flow characteristics are more obvious and complex. Shlrakashi et al.
(1986) observed three-dimensional flow characteristics in circular cylinder wake flow by a series of wind tunnel tests. They noticed that vortex
shedding frequency declined with the increasing of cylinder’s yaw angle,
and ascribed this phenomenon to the three-dimensional secondary flow
in cylinder wake. Wang et al. (2011) also observed the decline of vortex
shedding frequency and three-dimensional flow by wind tunnel experiments. Except for experimental researches, many numerical simulation
investigations were performed to reveal the details of three-dimensional
flow structures, and the spanwise components of velocity were always
observed in the oblique circular cylinder wake flow (Yeo and Jones,
2008; Zhao et al., 2009; Liang et al., 2015; Zhao, 2015; Wang et al., 2018,
* Corresponding author.
E-mail address: [email protected] (D. Xin).
https://doi.org/10.1016/j.jweia.2020.104381
Received 20 February 2020; Received in revised form 8 September 2020; Accepted 8 September 2020
Available online 23 September 2020
0167-6105/© 2020 Elsevier Ltd. All rights reserved.
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
wind-induced vibration of an elastic oblique circular cylinder could be
disturbed by the three-dimensional flow structures. Vortex-induced vibration (VIV) amplitudes of an oblique circular cylinder could be smaller
than a normal cylinder (Franzini et al., 2009). Dry-galloping could be
observed for a dry oblique circular cylinder submerged in critical flow,
which is a diverging wind-induced vibration with large amplitudes and
low frequency (Matsumoto et al., 2010; McTavish et al., 2018). Wind
tunnel experiments indicated that dry-galloping of an oblique circular
cylinder can be predicted by Den Hartog criterion (Macdonald and Larose, 2006; Cheng et al., 2008a, b).
Axial flow is an obvious three-dimensional flow phenomenon in the
wake flow of an oblique circular cylinder. Shlrakashi et al. (1986) found
it by a wind tunnel experiment and described it as a downstream flow
along cylinder axis. Matsumoto et al. (1988, 2001, 2010) observed axial
flow by light strings in a stay-cable’s wake flow and connected this with
the dry-galloping of an inclined circular cylinder. They used artificial jet
flow to simulate the axial flow and reproduced the circular cylinder
dry-galloping in a wind tunnel experiment. They found that the axial flow
possibly plays a role like a splitter plate in wake flow, which interrupt the
communication between upper and lower separated shear layers. Wang
et al. (2018) also conducted a numerical study at the Reynolds Number of
3900 to prove that. Hoftyzer and Dragomirescu (2010) attempted to
reveal the relationship between three-dimensional flow patterns and
axial flow by numerical simulation, and described the possible distribution characteristics of axial flow by analyzing surface pressure. Hu et al.
(2015, 2016) also observed axial flow in the wake flow of an inclined
square cylinder by numerical simulations and wind tests. They found that
the streamwise vortices generated by cylinder ends caused axial flow,
which possibly induced square cylinder galloping with the similar
mechanism proposed by Matsumoto et al. (2010). However, for an inclined circular cylinder, the mechanism of axial flow formation is still
unclear. And the relationship between axial flow and dry-galloping is
ambiguous. Li et al. (2017) found that the axial flow was weakened with
the wind attack angle between 30 and 35 , in which dry-galloping still
possibly happened. Their unusual results indicated the complexity of the
axial flow phenomenon.
This work focuses on the three-dimensional characteristics and axial
Fig. 1. Oblique cylinder model and coordinate systems.
2019). Intense three-dimensional characteristics of the flow around an
oblique circular cylinder cause the differences of surface pressure distribution and aerodynamic forces between an inclined cylinder and a
normal cylinder. Some wind tunnel test results and numerical simulations demonstrated that independence principle (IP) was inaccurate for
predicting the aerodynamic forces of an oblique circular cylinder with
larger yaw angle (Poulin and Larsen, 2007; Vakil and Green, 2009;
Hoang et al., 2015; Seyed-Aghazadeh and Modarres-Sadeghi, 2018). Yeo
and Jones (2011, 2012) simulated a three-dimensional yawed and inclined circular cylinder at Re ¼ 1.4 105, and revealed the
three-dimensional spatial and temporal distribution of the aerodynamic
force coefficients of every circular cylinder sections. Furthermore, the
Fig. 2. Computational domain and boundary conditions, (a) front view and (b) side view.
2
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 3. Meshing approach for an oblique cylinder, (a) front view, (b) local sectional view in x-y plane (z ¼ 0 m), (c) local sectional view in a vertical plane through
cylinder axis (x ¼ -tanβ⋅z).
coordinate system determines the definitions of vectors in flow field. A
global coordinate system Oxyz is used for building the numerical model
of an oblique circular cylinder in a three-dimensional calculation
domain. According to Fig. 1, the orientation of the circular cylinder axis
can be determined by α and β. Meanwhile, the aerodynamic forces and
wind velocities along x, y and z directions are defined based on Oxyz.
These data are used to compare the current numerical results and the
experimental results provided by Hoang et al. (2015). However, for the
complexity of the relationship between the global coordinate system
Oxyz and the oblique circular cylinder model, it is difficult to contact the
flow structures in the flow around the cylinder with the physical vectors
along x, y, and z directions.
Therefore, a local coordinate system containing axes along iaxis, idrag
and ilift is constructed based on the cylinder axis and incoming flow direction, which is suitable for analyzing the relationship among the threedimensional flow structures, the aerodynamic forces and the wake flow
velocities, so that the flow regime can be understood. In the 3-D space
containing the oblique cylinder model, any vector can be decomposed
into a vector parallel with the cylinder axis and a vector orthogonal with
the cylinder axis. If the velocity vector of incoming flow is decomposed
according to above approach, the vector component parallel with the
cylinder axis can be defined as the axial direction iaxis, and the vector
component orthogonal with the cylinder axis can be defined as the drag
direction idrag. The lift direction ilift is defined as the direction along lift
vector, which is orthogonal with iaxis and idrag. Thus, ilift can be determined with iaxis and idrag by the right-hand-rule. The unit vectors consisting the local coordinate system can be obtained with the unit vectors
consisting the global coordinate system and the angles of α and β. The
equations are listed as following:
Table 1
Time-average CD of the oblique circular cylinder model with different cell
numbers and turbulence models.
Cell
number
Time
step
(Δt⋅f)
Turbulence
model
Time-average
Cx (Current
simulation)
Timeaverage Cx
(Hoang
et al.,
2015)
Deviation
5,635,548
8.2 104
8.2 104
8.2 104
8.2 104
8.2 104
8.2 104
DES k-ω SST
0.83
1.02
18.6%
RSM stress-
1.04
2.0%
DES k-ω SST
0.83
18.6%
RSM stress-
1.04
2.0%
DES k-ω SST
0.86
15.7%
RSM stress-
1.05
2.9%
5,635,548
7,930,260
7,930,260
10,578,256
10,578,256
ω
ω
ω
flow in an oblique circular cylinder wake flow. A numerical simulation
was performed based on computational fluid dynamics (CFD). Effects of
yaw angles are analyzed on several flow characteristics, including aerodynamic force coefficients, wake flow velocities, correlations between
aerodynamic forces and wake flow velocities, 3-D vortex structures and
3-D axial flow. A novel regular spanwise distribution pattern of the axial
flow velocity cores is observed, and the flow mechanism of this phenomenon is discussed. A theory of formation of axial flow is proposed in
this investigation. Finally, a flow visualized wind tunnel experiment is
executed to verify some of the simulated results.
iaxis ¼ sin β cos αix þ sin αiy þ cos β cos αiz
2. Numerical approaches
idrag ¼
2.1. Definitions of coordinate systems
For the simulation of three-dimensional numerical models, the
3
(1)
ðcos2 β þ sin2 βsin2 αÞix þ ðsin β sin α cos αÞiy þ ðsin β cos βcos2 αÞiz
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2 β þ sin2 βsin2 α
(2)
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 4. Wall y þ distribution on the cylinder model surface, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
Fig. 6. Comparison of St between the current simulation and reference experiment.
2.2. The numerical model of the three-dimensional oblique circular
cylinder
Fig. 5. Comparison of aerodynamic force coefficients between the current
simulation and reference experiment.
cos β cos αiy sin αiz
ilift ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2 βcos2 α þ sin2 α
The test cases’ setup is mainly based on two engineering status, i.e.
cable’s rain-wind induced vibration (RWIV) and dry-galloping. These
two vibration phenomena are highly relevant with the flow around an
oblique circular cylinder, and their three-dimensional flow features are
obvious. RWIV often is found in the range of 25 < α < 35 and 30 < β <
35 (Gu and Du, 2005; Li et al., 2010), and the cable dry-galloping often
is found in the range of 25 < α < 40 and 15 < β < 45 (Matsumoto
et al., 2010; Tanaka et al., 2016; Vo et al., 2017). Considering the limitation of the computing power, four oblique cases are discussed in this
work, which contain β of 0 , 15 , 30 , and 45 , while α is maintained with
30 . Considering the isotropy of the incoming flow in plane y-z, a yaw
angle ϕ is defined as the angle between cylinder axis and x-axis (Fig. 1),
(3)
where ix, iy, and iz are the unit vectors along the x, y, z coordinates in
Fig. 1, iaxis is the unit vector along the circular cylinder axis, idrag is the
unit vector along the drag direction, and ilift is the unit vector along the
lift direction. iaxis, idrag and ilift constitute a three-dimensional Cartesian
coordinates system which depends on the cylinder orientation.
4
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
as zero-shear slip walls. The front and back boundaries are set as a pair of
periodic boundaries, which aims to avoid the effect of the spanwise
cylinder ends by assuming periodicity in the spanwise direction (Gallardo et al., 2014; Wang et al., 2019). Mesh refining region is separated
from outside domain by a pair of interfaces.
2.3. Computational configuration
The current CFD simulation is performed by using FLUENT, which is
based on the finite-volume method. Transient uncompressible flow is
solved by the unsteady Reynolds-Averaged Navier-Stokes equations
(URANS), and the turbulence model used is the Stress-Omega Reynolds
Stress Model (RSM). Delayed detached eddy-simulation (DDES) is also
considered as the turbulence model and it is excluded for its insufficient
precision in current working cases (Details in section 2.4). SIMPLEC algorithm is chosen for pressure-velocity coupling. Second order upwind
scheme is used for the spatial discretization of momentum, turbulence
dissipation rate, and Reynolds stresses. Least square cell based is chosen
for the discretization of gradient, and second order is chosen for the
discretization of pressure. Time integration is solved by first order implicit scheme. Time step size is set as Δt ¼ 5 105 s.
Fig. 7. Comparison of the pressure coefficients distribution between the current
simulation and reference experiment (Cantwell and Coles, 1983; Nishimura and
Taniike, 2001; Norberg, 1993; Weidman, 1968).
2.4. Computational validation
which can mark the four cases as ϕ ¼ 90 , ϕ ¼ 77.05 , ϕ ¼ 64.34 and ϕ
¼ 52.24 . Lower yaw angle means more obvious obliquity of the circular
cylinder model, and ϕ ¼ 90 means the cylinder axis is orthogonal with
incoming flow. The relationship of α, β and ϕ can be demonstrated by
below equation:
cos φ ¼ sin β cos α
The computational validation is verified by mesh independence, wall
yþ, and calculation accuracy. Mesh independence and calculation accuracy are validated by comparing the time-average aerodynamic forces
coefficients of the cylinder model with the wind tunnel test results provided by Hoang et al. (2015). It is noticeable that the aerodynamic force
coefficients provided by Hoang et al. (2015) are based on the global
coordinate system. In the global coordinate system, the aerodynamic
force coefficients are defined as following:
(4)
According to above four cases, four circular cylinder models with the
diameter D of 0.1 m are constructed. Their aspect ratios increase with the
decline of ϕ, and range from 11.55 to 16.33. As shown in Fig. 2, calculation domain is a parallelepiped and four shortest edges are parallel with
the circular cylinder axis. The calculation domain has a length of 100D, a
height of 30D and a width of 10D. The oblique circular cylinder model is
located at the position illustrated in Fig. 2. A square cylinder domain with
a width of 10D is constructed around the oblique circular cylinder as a
mesh-refining region, which is parallel with the circular cylinder axis.
The meshing approach is illustrated in Fig. 3. Totally four calculation
domains containing 5,203,256, 5,635,548, 5,721,990, and 5,572,680
hexahedral grids are constructed for the cases of ϕ ¼ 90 , ϕ ¼ 77.05 , ϕ
¼ 64.34 , and ϕ ¼ 52.24 respectively.
The calculation domain contains six face boundaries. The inlet
boundary is set as a velocity-inlet, which provides a uniform airflow with
a speed of 7.72 m/s along x-direction and a turbulence intensity of 0.1%.
Thus, the Reynolds number is about 5.3 104 in this simulation. The
outlet boundary is a pressure-outlet which keeps 0 Pa on its plane. The
top and bottom boundaries are set as symmetry, which can be regarded
Cx ¼
Fx
1=2ρU 2 A
(5)
Cy ¼
Fy
1=2ρU 2 A
(6)
Cz ¼
Fz
1=2ρU 2 A
(7)
where Cx, Cy, and Cz are the aerodynamic force coefficients along the
directions of ix, iy, and iz respectively. Fx, Fy, and Fz are the aerodynamic
forces along the directions of ix, iy, and iz respectively. ρ is the density of
air, which is set as 1.225 kg/m3. U is the incoming wind velocity. A is
reference area, which can be calculated by equation (8):
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A ¼ DL 1 þ tan2 α=cos2 β
(8)
where D is the circular cylinder diameter and L is the circular cylinder
Fig. 8. Statistic values of aerodynamic force coefficients, (a) Time-average aerodynamic force coefficients and (b) RSM of aerodynamic force coefficients.
5
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 9. Non-dimensional PSD of aerodynamic force coefficients with Re ¼ 5.3 104. When the frequency is normalized based on St, there are (a) PSD of Clift, (b) PSD
of Cdrag, and (c) PSD of Caxis. And when the frequency is normalized based on Stn, there are (d) PSD of Clift, (e) PSD of Cdrag, and (f) PSD of Caxis.
Fig. 10. Positions of wake velocity monitoring point, (a) view in x-z plane, and (b) view in y-z plane.
6
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 11. Non-dimensional statistic values of wake velocities, (a) mean vlift, (b) mean vdrag, (c) mean vaxis, (d) RMS of vlift, (e) RMS of vdrag, and (f) RMS of vaxis.
yaw angle. Fig. 6 provides the comparison of Strouhal number (St). In this
part, Strouhal number (St) is defined according to the reference
experiment:
length.
Mesh independence test is based on the working case with wind velocity Re ¼ 5.3 104 and ϕ ¼ 77.05 , which has the vortex shedding
frequency of f ¼ 16.3 Hz. Table 1 lists the calculated results of timeaverage Cx of the oblique circular cylinder model with different cell
numbers and turbulence models. Obviously, the calculation cases with all
meshes and stress-ω RSM have results with the deviations under 3%, and
stress-ω RSM has better behavior than k-ω SST DES. Thus, the mesh with
cell number 5635548 is chosen to be used in the calculation, and stress-ω
RSM is chosen as the turbulence model.
Wall y-plus is a parameter indicating the quality of the initial level of
cells on the circular cylinder surface, which are shown in Fig. 4. Almost
the y þ values of all the cells on the circular cylinder surface are less than
3, which means the cells in near-wall region are refined enough to
simulate the boundary layer accurately.
Calculation accuracy test is based on all oblique cases. The items
compared include the three-dimensional aerodynamic force coefficients,
the Strouhal number (St) and the pressure distribution on cylinder surface. Fig. 5 demonstrates that the time-averaged Cx, Cy, and Cz calculated
by current simulation agree well with the experimental results with any
St ¼
fD
U
(9)
where f is the prominent frequency of lift forces, which represent axial
vortex shedding frequency. Fig. 6 indicates that simulated St is slightly
larger than the referenced experimental results, and the deviation is no
more than 6%, which is an acceptable deviation. Fig. 7 shows the mean
pressure coefficients (Cp) on cylinder surface. According to Zhao et al.
(2009), the mean pressure coefficients on an oblique circular cylinder
surface agree well with the Independence Principle (IP) when yaw angle
ϕ belongs to the range of 30 –90 . Thus, the definition of Cp is according
to the following equation in this part to treat the oblique circular cylinders as the cylinder normal to the streamwise:
Cp ¼
7
p p0
1=2ρUdrag 2
(10)
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 12. PSD of wake flow velocities in monitoring point 3 when Re ¼ 5.3 104. When the frequency is normalized based on St, there are (a) PSD of vlift, (b) PSD of
vdrag, and (c) PSD of vaxis. And when the frequency is normalized based on Stn, there are (d) PSD of vlift, (e) PSD of vdrag, and (f) PSD of vaxis.
where p is the time-averaged pressure on model surface, p0 is the freestream static pressure. Udrag is the component of U orthogonal with cylinder axis, as described by equation (11):
Udrag ¼ U
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2 β þ sin2 αsin2 β
Caxis ¼
Faxis
2
1=2ρUdrag
DL
(12)
Fdrag
2
1=2ρUdrag
DL
(13)
Cdrag ¼
(11)
Fig. 7 indicates that the curves of the time-averaged Cp almost agree
with the curves collected by referenced experiments. The separation
points and the lowest pressure appear at the same angle with the
experimental data.
Clift ¼
Flift
2
1=2ρUdrag
DL
(14)
where Caxis, Cdrag, and Clift are the aerodynamic force coefficients along
the directions of iaxis, idrag, and ilift respectively. Faxis, Fdrag, and Flift are
the aerodynamic forces along the directions of iaxis, idrag, and ilift
respectively.
The yaw angle’s effects on the statistic results of aerodynamic force
coefficients are illustrated in Fig. 8. Mean Caxis and mean Clift of the circular cylinder remain about 0 with ϕ ranging between 52.24 and 90 .
This means that there are no obvious static axial force and static lift force
acting on the circular cylinder. The RMS of Caxis also remains 0 with every
test yaw angles, which indicates that there is no obvious axial force
fluctuation. The RMS of Clift jumps from 0.2 to 0.4 and remains at about
0.4 with the rising of ϕ. This indicates that the yawed orientation of a
3. Numerical results
3.1. Aerodynamic force coefficients
Aerodynamic force coefficients of the oblique cylinders with different
yaw angles were monitored with the sampling interval of 5 105 s.
More than 2 105 samples were analyzed statistically for each coefficient. The aerodynamic force coefficients are defined as following
equations based on the local coordinate system:
8
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 13. Spanwise correlation coefficients of wake flow velocities, (a) spanwise correlation coefficients of vlift, (b) spanwise correlation coefficients of vdrag, (c)
spanwise correlation coefficients of vaxis.
Fig. 14. Correlation between wake velocities and aerodynamic forces, (a) correlation coefficients of Clift and vlift, (b) correlation coefficients of Cdrag and vdrag, (c)
correlation coefficients of Caxis and vaxis.
vortex shedding features and fluctuation frequencies. Fig. 9. provides the
non-dimensional power spectral density (PSD) of three aerodynamic
force coefficients when Re ¼ 5.3 104. In Fig.9 ~ac, the frequency is
normalized based on St defined by equation (9), which is helpful to
distinguish the effects of yaw angles. In Fig.9 d ~ f, the frequency is
normalized based on Stn defined by following equation (15), which is
circular cylinder has few effects on the lift fluctuation when ϕ is limited
in a certain range from 64.34 to 90 . Mean Cdrag decreases slightly with
the increase of ϕ, which means an oblique circular cylinder could suffer
larger drag force than a cylinder orthogonal with flow. The RMS of Cdrag
indicates that the drag fluctuation is not sensitive to ϕ.
The power spectrums of Clift, Cdrag, and Caxis can reflect the axial
9
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 15. Three-dimensional vortex structures identified by Q criterion, ISO surfaces of Q ¼ 400 with Re ¼ 5.3 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and
(d) ϕ ¼ 52.24 .
Fig. 16. Axial vortices contours in S1, S2, and S3 at t ¼ 13 s, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
10
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 17. Axial vortices evolution in S1, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
prominent PSD peaks in the curves of the case of ϕ ¼ 90 (i.e. the circular
axis is orthogonal with incoming flow). This means that smaller yaw
angle induces more obvious periodic fluctuation of the axial aerodynamic
force. When ϕ ¼ 90 , the main prominent frequency of Caxis are same
with the basic frequency of Clift. When ϕ ¼ 52.24 and 64.34 , the main
prominent frequency of Caxis are same with the basic frequency of Cdrag.
And when ϕ ¼ 77.05 , the basic frequencies of both Clift and Cdrag are
observed. This indicates that with yaw angle declining, there is a transition of the main frequency of axial force from the basic frequency of Clift
to the basic frequency of Cdrag. The periodic fluctuation of axial aerodynamic force cannot be ascribed to the axial vortex shedding in the
oblique cylinder wake flow. There should be other flow regime causing
this phenomenon.
helpful to verify the IP of axial vortex shedding frequencies.
Stn ¼
fD
Udrag
(15)
There are one or more peak values shown by the PSD curves of
aerodynamic force coefficients, which mark the prominent frequencies.
Among the three aerodynamic force coefficients, the prominent frequencies of Clift reflect the axial vortex shedding frequencies directly. The
curves of Clift show two prominent frequencies (Figs. 9a & 6d). The lower
frequency is the main and basic frequency, which is same with the axial
vortex shedding frequency. And the other one is three times of the axial
vortex shedding frequency, which is the high-order harmonic frequency
probably caused by nonlinear components in flow field (Baek and Karniadakis, 2009; Ling and Zhao, 2009; Dahl, 2008). Fig. 9a indicates that
the axial vortex shedding frequency increase with ϕ increasing. Meanwhile, Fig. 9d demonstrates that the axial vortex shedding frequency can
be predicted well by IP, which also has been reported by Wang et al.
(2019). The prominent frequency of five times of basic frequency is
observed when ϕ ¼ 52.24 , which means there are more nonlinear factors in the axial vortex shedding of an oblique circular cylinder with
smaller yaw angle. PSD of Cdrag shows similar features with the PSD of Clift
(Fig. 9b & e). The basic frequency of Cdrag is twice axial vortex shedding
frequency. The basic frequency of drag is also caused by axial vortex
shedding. A complete axial vortex shedding process contains two times of
vortex shedding and each of them generates a period of drag force fluctuation. PSD of Caxis shows some different features (Fig. 9c & f). The peak
value of the PSD of Caxis decreases with the increase of ϕ, and there are no
3.2. Wake flow velocity characteristics
Wake flow velocities are collected by seven monitoring points
(MP1~MP7). Fig. 10 shows the positions of these seven monitoring
points, which are located on a line at the leeward side of the model and
parallel with the circular cylinder axis. All monitoring points are marked
by their z-coordinates.
Statistic values of wake flow velocities are displayed by Fig. 11. The
mean values of the statistic values of seven monitoring points are connected by solid lines. According to Fig. 11a and b, mean vlift keeps about
0 with different ϕ, and mean vdrag has a rising trend with the increasing of
ϕ. Fig. 11c indicates that there is prominent axial flow when yaw angle is
not 90 . Mean axial flow velocities decrease approximately linearly with
ϕ increasing. It should be noticed that the curves of mean vaxis agree well
11
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 18. Three-dimensional transient streamlines around oblique cylinders, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
Fig. 19. Δvaxis contours in S1, S2, and S3 at t ¼ 13 s with Re ¼ 5.3 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
wake flow velocities. The RMS of vlift shows little variation when ϕ is
between 64.34 and 90 , while it drops suddenly when ϕ declines to
52.24 . This means the status of axial vortex shedding changes a lot when
the yaw angle is reduced to a certain case. The RMS of vdrag shows no
obvious trend with the increasing of ϕ. Meanwhile, the RMS of vaxis has
an increasing trend with the decline of ϕ, which indicates smaller yaw
angle induces more intensity fluctuation on axial flow velocity.
with the curves of the Uaxis, which is defined as the component of U
parallel with the cylinder axis in equation (16)
Uaxis ¼ Uðsin β cos αÞ
(16)
Therefore, it is reasonable to ascribe the axial flow to the component
of incoming flow parallel with the cylinder axis.
Fig. 11d, e, and 11f reflect the RMS of the fluctuation of the three
12
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 20. Δvaxis contours in S1, S2, and S3 at t ¼ 13 s with Re ¼ 10.0 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
Fig. 21. Evolution process of the axial flow in cylinder wake flow with ϕ decreasing.
The power spectrums of vlift, vdrag, and vaxis reveal the fluctuation
frequencies of axial-vortex-relevant structure and axial flow directly. The
results of monitoring point MP3 are captured as the typical examples and
illustrated in Fig. 12. Similar with the PSD of aerodynamic forces, the
PSD of velocities also normalize the frequencies based on St and Stn
respectively. vlift reflects the evolution of the axial-vortex-relevant flow
structures directly. Figs. 12a and d indicates that there is a single
prominent frequency of vlift when ϕ ¼ 90 , 77.05 and 64.34 that
corresponds to the axial vortex shedding frequency. It is observed that
the peak values of the curves decrease with the decline of ϕ, which means
that the fluctuation energy captured by the prominent frequency tend to
be reduced during the leaning process of the circular cylinder. Moreover,
signal bandwidth of the PSD of vlift becomes larger with the decline of ϕ,
i.e. yaw angle affects the fluctuation energy distribution obviously in
frequency-domain. When ϕ ¼ 52.24 , there are several peaks on the
curves of the PSD of vlift while no one of them corresponds to the axial
13
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 22. Δvaxis in cylinder wake flow, Re ¼ 5.3 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
with the flow structures around a bluff body. A general example is that
the axial vortex shedding causes the vortex-induced forces on a cylinder.
Analyzing the correlations of aerodynamic forces and wake flow velocities is helpful for understanding the relationship between flow structures
and aerodynamic forces. Along the directions of drag and lift, the correlation between aerodynamic forces and wake velocities is enhanced for
increasing ϕ (Fig. 14a & b). This means that with a smaller yaw angle, the
axial vortices in wake flow contribute less to the aerodynamic force
fluctuations of the oblique circular cylinder. This conclusion also explains
the reason why there are differences between the prominent frequencies
of the aerodynamic forces and the wake velocities when ϕ ¼ 52.24 .
Moreover, along the direction of cylinder axis, the aerodynamic force has
less correlation with the axial velocity in the wake flow (Fig. 14c).
vortex shedding frequency. It demonstrates an intense variation in axial
vortex structures. Similar features also can be observed in the PSD curves
of vdrag (Fig. 12b & e). Both the results of vlift and vdrag indicate that the
axial-vortex-relevant structure loses its determinate place in the fluctuation of wake flow velocity gradually with the decline of ϕ. For vaxis
(Fig. 12c & f), only when ϕ ¼ 52.24 , there are prominent frequencies
observed, while they do not correspond to the axial vortex shedding
frequencies. Except for this case, PSD curves of the vaxis of the last cases
show no prominent frequencies. Thus, it is reasonable to consider that
there are certain flow structures appearing in cylinder wake region when
ϕ ¼ 52.24 which causes the inharmonic fluctuation of vaxis.
The spanwise correlation of wake flow velocities is analyzed by the
correlation coefficients between the middle monitoring point MP4 and
the other monitoring points (Fig. 13). Figs. 13a and b displays the
spanwise correlation coefficients of vdrag and vlift. The curves indicate that
the spanwise correlation of vdrag and vlift have a rising trend with ϕ
increasing. Therefore, the formation of spanwise vortices is probably
disturbed and the spanwise correlation of wake flow is weakened when
the flow passes an oblique circular cylinder. Fig. 13c illustrates that the
spanwise correlation coefficients of vaxis keep approximately 0 in all cases
of ϕ. That means the axial velocities at different spanwise positions in the
wake flow have little spanwise correlation with each other.
3.4. Three-dimensional vortex structures
Three-dimensional vortex structure is the main flow structure in the
wake flow of a cylinder, which is highly relevant with the fluctuation of
aerodynamics forces. The fluctuation features and spanwise correlation
of velocities are reflected directly by the three-dimensional vortex
structure. The transient three-dimensional vortex structures identified by
the Q criterion (Chakraborty et al., 2005) are shown in Fig. 15. With the
decline of ϕ, the axial vortices are curved and distorted prominently and
leave the location parallel with the cylinder axis. It is obvious that the
curved and distorted axial vortex structures cause the decline of the
spanwise correlations of wake flow and the redistribution of the
3.3. Correlations between wake flow velocities and aerodynamic forces
The aerodynamic forces’ fluctuation of a bluff body is highly relevant
14
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 23. Δvaxis in cylinder wake flow, Re ¼ 10.0 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
fluctuation energy in frequency-domain. Meanwhile, the axes of secondary vortices still keep roughly orthogonal with distorted spanwise
vortices, so that the distributions of secondary vortices tend to be irregular. The secondary vortices in wake flow are parallel with streamwise
direction when the cylinder is perpendicular to incoming flow, and
deflect from streamwise direction with ϕ decreasing. The secondary
vortices with irregular arrangement also cause the fluctuation of vlift and
vdrag in wake flow, which disturbs original basic velocity fluctuation
frequency and affects the distribution of the fluctuation energy in
frequency-domain.
For more details of axial vortex shedding characteristics, three slices
(S1, S2, &S3) are located along cylinder axis with same distances, and all
of them are orthogonal with z-coordinate axis (Fig. 10b). Fig. 16 lists the
transient axial vortex shedding in all slices with different oblique cases
respectively. Axial vortex shedding in different slices show similar status
when ϕ ¼ 90 , while there are obvious differences among the three slices
when ϕ < 90 . With the decline of ϕ, inconsonant vortex shedding in
different slices is increasingly obvious. Inconsonant axial vortex shedding
in different slices causes the curved and distorted vortex structures, and
generates the vortex-induced forces with poorer spanwise correlations.
Furthermore, the poorer spanwise correlations of vortex-induced forces
causes the smaller fluctuation of aerodynamic forces and the lower peak
value of the PSD curves when ϕ ¼ 52.24 .
The yaw angle’s effect on axial vortex shedding process is also
analyzed. For the convenience of comparing all oblique cases, dimensionless time t* is introduced into the description of axial vortex shedding
process. The dimensionless time t* is defined as the following equation.
t* ¼
t t0
¼ ðt t0 Þf
T
(17)
where t is the simulation time, t0 is the initial time of the snapshots, T is
the period of axial vortex shedding, and f is the frequency of axial vortex
shedding. t* ¼ 1 means a complete axial vortex shedding period.
Fig. 17 displays the snapshots of S1 in a complete axial vortex shedding period. With the decline of ϕ, the axial vortices tend to be extended
and split into series of small eddies. This behavior can obviously cause
the redistribution of the fluctuation energy of the wake flow in
frequency-domain, which induces wider signal bandwidth and lower
peak value in PSD further. It is similar with the phenomenon observed by
Matsumoto et al. (2001) in a flow visualization test. Matsumoto considered the series of small eddies as separated axial vortexes, which shed in
different vortex shedding periods. However, this results indicate that the
small eddies are split from one axial vortex during one vortex shedding
period.
3.5. Three-dimension axial flow characteristics
Three-dimensional streamline is a direct way to reveal the flow
structures surrounding the oblique circular cylinders, which is shown in
Fig. 18. It is observed that there are several streamlines extending along
the axis direction in the cylinder wake flow when ϕ < 90 . The axial parts
of streamlines indicate that there is axial flow. Comparing the four
15
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 24. Time-average streamwise vorticities in cylinder wake flow, Re ¼ 5.3 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
oblique cases, the axial flow is more obvious with the decline of ϕ. This
means yaw angle ϕ determines the features of axial flow. Another
noticeable phenomenon is the deflection of the streamlines when they
pass the oblique cylinder. The flow tends to turn into the direction of
cylinder axis in the windward side of the oblique cylinder, and then tend
to flow according to the direction orthogonal with the cylinder axis. This
means the three-dimensional flow surrounding an oblique cylinder
cannot be simplified as two-dimensional flow surrounding an elliptic
cylinder.
Transient axial flow velocity contains too many fluctuations, which
make its flow structure difficult to be clarified. Therefore, the timeaveraged axial flow structure became the suitable analysis target. According to Fig. 11c, axial flow velocities are close to the axial component
of the incoming flow velocity. The axial component of the incoming flow
velocity can be regarded as the background axial velocity. It should be
noticeable that the background axial velocity is relevant with yaw angles.
Thus, to remove the background axial velocity is a necessary approach to
analyze and compare the structure of the axial flow with different yaw
angles. Aiming at this, a parameter Δvaxis is defined by the following
equation as the index to clarify the axial flow structures.
Δvaxis ¼ vaxis Uaxis
where positive value means an increment of the axial velocities along
iaxis, and vice versa. There are two positive regions of Δvaxis appearing
gradually at both upper and lower borders of the oblique cylinder sections with the decline of ϕ, and a negative core of Δvaxis appearing in the
center area of the cylinder wake flow when ϕ ¼ 52.24 . Considering the
suddenness of the appearing of the negative cores, a simulation based on
the original numerical model and Re ¼ 10.0 104 is performed to check
this phenomenon, and the results are displayed in Fig. 20. Like Fig. 19,
there is obvious negative core of Δvaxis appearing in the center area of the
cylinder wake flow when ϕ ¼ 52.24 . Moreover, the negative core also
appears in the wake flow of the case of ϕ ¼ 64.34 . The negative cores
can be observed with both two Re, which indicates that it is not an isolated phenomenon.
In Fig. 21, the details of the three-dimensional streamlines shown in
Fig. 18 are combined to reveal the flow mechanism of the unique distribution of Δvaxis. When ϕ ¼ 90 , streamlines indicate that the flow
around the cylinder can be almost considered as 2-D flow, and there are
no obvious organized distribution of Δvaxis in the slices. With ϕ < 90 , the
streamlines tend to turn into the direction orthogonal with the cylinder
axis when they pass the oblique cylinder. The flow marked by the upswept streamlines causes the two positive regions of Δvaxis at both upper
and lower borders of the oblique cylinder sections. It can be observed
that the length of the upswept streamlines is extended with the decline of
ϕ, which explains the reason of the extending trend of the positive regions of Δvaxis. The area with smaller value of Δvaxis between the two
positive regions is the location of axial flow, which tends to be more
(18)
where vð Þaxis is the time-average axial flow velocity, and Uaxis is defined
by equation (16).
The contours of Δvaxis in the slices S1 to S3 are listed in Fig. 19, in
16
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 25. Time-average streamwise vorticities in cylinder wake flow, Re ¼ 10.0 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
flow is a noticeable phenomenon that deserves to be analyzed. Fig. 24
illustrates the contours of time-average streamwise vortices on the slice
S4. With the decline of yaw angle, the time-average streamwise vortices
are increasingly obvious. When ϕ ¼ 52.24 , there are two rows of
streamwise vortex cores distributed along the cylinder axis with the
approximate distance of 1.8D, which is same with the negative cores of
Δvaxis. It should be noticed that the 1.8D is a rough distance and not
uniform absolutely. It is difficult to ensure the accurate locations of the
vortex cores, which makes the distance difficult to be confirmed. Moreover, the limit number of the flow field samples also induces the fuzzy
time-averaged results of vorticity field. This phenomenon indicates that
the distributions of the axial flow structures and streamwise vortices are
highly relevant with each other. Similar distribution of streamwise vortex cores is also observed in the cases with Re ¼ 10.0 104 (Fig. 25). It is
noticeable that the positive time-averaged streamwise vortices are
generated at the upper border of the oblique circular cylinder, and the
negative time-averaged streamwise vortices are generated at the lower
border of the oblique circular cylinder. The flow between two vortices
with opposite directions of rotation will be accelerated by the two
vortices, which is a probably flow mechanism of the negative cores of
Δvaxis.
The arrangement of the time-averaged streamwise vortex cores is
highly relevant with the arrangement of time-average axial velocity
cores, while the formation regime of the arrangement of time-averaged
streamwise vortex cores is still unclear. To clarify this issue, Fig. 26
shows the snapshots of the transient streamwise vortices S4 in a complete
obvious with the decline of ϕ. Nevertheless, the flow regime of the
negative core of Δvaxis appearing when ϕ ¼ 52.24 is not be revealed by
the three-dimensional streamlines.
Aiming at clarifying the flow regime of the negative cores of Δvaxis
appearing when ϕ ¼ 52.24 , the flow field is sliced at the downstream
side of the oblique cylinder. The slice S4 (Fig. 10a) is parallel with the
cylinder axis by a distance of 1.5D and orthogonal with x-z plane. Contours of Δvaxis in the slice are shown in Fig. 22. A pair of dashed lines
marks the projections of the oblique cylinder outlines on the slices. It is
found that there are a series of negative cores of Δvaxis locating along the
cylinder axis with certain distance when ϕ ¼ 52.24 . The distances of the
adjacent negative cores are about 1.4D in the y-z plane, which represents
about a spanwise distance of 1.8D in the 3-D space. Except for the case of
ϕ ¼ 52.24 , there are no obvious organized distribution of Δvaxis. The
results of cases with Re ¼ 10.0 104 are also illustrated in Fig. 23.
Similar organized distribution of the negative cores of Δvaxis is observed
when ϕ ¼ 52.24 , and the distance is also 1.8D. There are also a series of
negative cores of Δvaxis appearing in the case of ϕ ¼ 64.34 , while their
distance is not a certain value. It should be noticed that Δvaxis is a timeaverage variable, which indicates that there are quasi-static axial flow
structures distributed with a certain spanwise distance in the oblique
cylinder wake flow.
Based on the arrangement features of the negative cores of Δvaxis, it is
reasonable to connect the flow structure that has similar spanwise distribution with this phenomenon. Accordingly, the streamwise vortex
motivated by the three-dimensional instability of circular cylinder wake
17
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 26. Axial vortices evolution in slice S4 with Re ¼ 5.3 104, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
period of axial vortex shedding. When ϕ ¼ 90 , the streamwise vortices
oscillate between the upper border and the lower border of the oblique
cylinder, and the period of the oscillation corresponds to the axial vortex
shedding. There are both positive and negative streamwise vortices arranged in a row of vortex cores. With the decline of ϕ, the positive
vortices tend to be gathered near the upper outline and negative vortices
tend to be gathered near the lower outline. The oscillation of the
streamwise vortices is weakened gradually until it is disappeared.
Comparing with other cases, the streamwise vortices of the case of ϕ ¼
52.24 tend to be more stable in a period of axial vortex shedding, which
probably induces the arrangement of the time-average streamwise vorticities after considering large number of flow field samples. Therefore,
the mode of time-average streamwise vorticities displayed in Figs. 24 and
25 is formed.
Summarizing the phenomena about the arrangement of axial velocity
cores, the possible mechanism of axial flow is deduced here and illustrated by Fig. 27. In the core area of the oblique cylinder wake flow, the
flow velocities along idrag and ilift are blocked by the cylinder and the
axial vortices. The main component of the axial flow in an oblique cylinder wake flow is the axial component of incoming flow. Meanwhile,
18
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
different modes in the wake flow of an oblique circular cylinder.
4. Experiment results
4.1. Experiment setup
A particle image velocimetry (PIV) experiment is performed to verify
the axial flow features observed in above numerical simulation. This
experiment is executed in the wind tunnel laboratory of Northeast
Forestry University at Harbin, China, which has a test section of 0.8 m
(wide) 1.0 m (height) and can provides the incoming flow with turbulence intensity less than 0.5%. Four steel tubes with a diameter of 32
mm are used as the rigid cylinder models, which are installed in wind
tunnel with β between 0 and 45 by interval of 15 and α of 30
respectively. The four steel tubes correspond to the previous four oblique
cases, i.e. ϕ ¼ 90 , ϕ ¼ 77.05 , ϕ ¼ 64.34 and ϕ ¼ 52.24 . Their aspect
ratios range from 33.9 to 48.0 for their different yaw angles. Test wind
speed is 24.1 m/s, and the Reynolds Number is about 5.3 104, which is
same with the numerical simulation. Both ends of the oblique cylinder
model are fixed on the wall of the wind tunnel, as shown in Fig. 28.
A two-dimensional PIV system consists of a digital CCD camera
(Powerview 4 MP-LS), a double-pulsed Nd:YAG laser (380 mJ, 532 nm
wavelength), a LaserPulse computer controlled synchronizer, and a
server as data acquisition and processing terminal. The system captures
2-D flow field with a frequency of 5 Hz in this test. The laser sheet is a
plane through the cylinder axis and parallel with the incoming flow direction (Fig. 28b). Oil droplets with diameters in the range of 0.5 μm–10
μm are produced as the tracer particles.
A one-dimensional hot-wire probe is used for capturing axial vortex
shedding frequency. The probe is installed orthogonally with incoming
flow and near the separate point of the cylinder, so that separate flow
could pass the probe. The probe’s sampling frequency is 10,000 Hz, and
the sampling time is 40 s.
Fig. 27. Flow mechanism of time-average axial flow velocity cores.
there are series of secondary vortices generated for the three-dimensional
instability regime, which provide the quasi-static streamwise vortices
with the certain spanwise distance of 1.8D when ϕ ¼ 52.24 . There is an
interaction between the axial flow and streamwise vortices. When the
streamwise vortices and axial flow keep in the same flow direction, the
axial flow and streamwise vortices are accelerated with each other, and
axial velocity cores are generated. When the streamwise vortices and
axial flow keep in opposite flow directions, the axial flow is weakened
and the streamwise vortices are suppressed. Thus, there are only positive
streamwise vortices near cylinder upper surface, and only negative
streamwise vortices near cylinder lower surface. And the accelerated
axial flow regions are also arranged along the cylinder axis with a certain
spanwise distance of 1.8D. It should be noticed that the certain distance
of 1.8D means that the streamwise vortices correspond to neither of
mode A and mode B mentioned by Williamson (1996). There may be
Fig. 28. Wind tunnel test models and experiment set up, (a) front view and (b) side view.
Fig. 29. PSD of wind velocity, (a) measured by hot-wire probe, (b) simulation results.
19
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 30. Vectors of time-average velocities in cylinder wake flow, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
about 1.8D. The experimental results verify that there are organized axial
flow structures with the spanwise distance of about 1.8D in the oblique
cylinder wake flow, like the structures shown in Fig. 22d.
4.2. Vortex shedding frequency characteristics
The non-dimensional PSDs of the wind velocities measured by a hotwire probe (vtest) are shown in Fig. 29a. With the decline of yaw angle, the
signal bandwidth tends to be wider. The prominent frequencies of all
oblique cases can be predicted well by IP. Similar trends can also be
observed in the simulation results (Fig. 29b). There are some differences
between the simulated and experimental results, which are probably
caused by the lack of flow details in the simulation based on URANS and
the difference of the monitoring points’ positions. Both the numerical
and experimental results indicate that ϕ is a key parameter for the fluctuation energy distribution of the axial vortex shedding in frequency
domain. The vortex shedding in oblique cylinder wake flow takes up a
wider frequency domain than a normal cylinder.
5. Conclusions
In the current study, a CFD simulation on 3-D oblique circular cylinders with different oblique orientations is conducted to reveal the
relationship between yaw angles and several flow characteristics.
Analyzed events include aerodynamic force coefficients, wake flow velocities, correlations between aerodynamic forces and wake flow velocities, 3-D vortex structures and 3-D axial flow. A novel time-average axial
flow distribution pattern is observed in the CFD simulation and verified
by a PIV experiment. The possible formation mechanism of axial flow and
its time-average characteristics is proposed.
Firstly, independence principle is still effective for predicting the
prominent frequencies of the aerodynamic forces and axial flow velocity
of an oblique circular cylinder. The prominent frequencies of the aerodynamic forces of an oblique cylinder are mainly depended on Udrag, so
that the prominent frequencies decrease with the decline of ϕ. Axial flow,
as a 3-D flow characteristic, is probably caused by the component of
incoming flow parallel with the cylinder axis. Thus, axial flow speed
increases with the decline of ϕ.
For an oblique circular cylinder, yaw angle is a crucial parameter,
which determines vortex shedding patterns and affects aerodynamic
force coefficients further. In the wake flow of an oblique circular cylinder, 3-D spanwise vortices are curved and distorted prominently. Based
on the distorted vortex configuration, inconsonant vortex shedding in
different sections is increasingly obvious. Inconsonant axial vortex
4.3. Axial flow characteristics
For each test case, 2000 transient flow field sections are collected, and
the time-average flow field is calculated based on the 2000 samples.
Fig. 30 illustrates the vectors of time-average velocities in the cylinder
wake flow measured by the PIV system. Axial flow appears and its velocity is intensified with the decline of ϕ, which corresponds to the trend
illustrated by Figs. 11c and 18. Fig. 31 provides the contours of -Δvaxis in
cylinder wake flow measured by PIV system. -Δvaxis is displayed as the
result because the opposite direction of axial velocity is defined in the PIV
post-process. Obviously, there are no clear distribution features of -Δvaxis
in measured regions with ϕ of 90 , 77.05 and 64.34 . Meanwhile, when
ϕ ¼ 52.24 , there are several positive cores of -Δvaxis, i.e. negative cores of
Δvaxis, which arrange along the cylinder axis with a spanwise distance of
20
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Fig. 31. -Δvaxis in cylinder wake flow, (a) ϕ ¼ 90 , (b) ϕ ¼ 77.05 , (c) ϕ ¼ 64.34 , and (d) ϕ ¼ 52.24 .
Declaration of competing interest
shedding causes the decline of the spanwise correlations of the cylinder
wake flow and the weakening of the vortex-induced forces’ fluctuations.
The distorted vortex configuration and inconsonant vortex shedding also
induce changes on the distribution of wake flow fluctuation energy in
frequency spectrum. Specifically, smaller yaw angle causes wider signal
bandwidth and lower PSD peak value of wake flow fluctuation and
aerodynamic force fluctuation. And smaller ϕ causes more obvious
inharmonic fluctuations of axial velocities. Furthermore, with a smaller
ϕ, vortex shedding contributes less on aerodynamic force fluctuations,
which is the reason that the correlation between wake flow velocities and
aerodynamic forces declines with ϕ decreasing.
When ϕ ¼ 52.24 , there is a novel time-average axial flow distribution pattern. There are several of high velocity cores of time-average axial
flow arranging along the cylinder axis with a certain spanwise distance of
about 1.8D in the cylinder wake flow. Similarly, time-averaged streamwise vortex cores are also arranged with the certain spanwise distance of
about 1.8D. This distribution pattern is probably due to the interaction
between the axial flow and the streamwise vortex pairs caused by threedimensional instability. Time-average axial velocity cores appear at the
locations where the streamwise vortices and the axial flow keep in the
same flow direction.
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (No. 51878131).
References
Baek, H., Karniadakis, G.E., 2009. Suppressing vortex-induced vibrations via passive
means. J. Fluid Struct. 25 (5), 848–866. https://doi.org/10.1016/
j.jfluidstructs.2009.02.006.
Barkley, D., Henderson, R.D., 1996. Three-dimensional Floquet stability analysis of the
wake of a circular cylinder. J. Fluid Mech. 322, 215–241. https://doi.org/10.1017/
S0022112096002777.
Cantwell, B.J., Coles, D., 1983. An experimental study of entrainment and transport in the
turbulent near wake of a circular cylinder. J. Fluid Mech. 136 (-1), 321–374. https://
doi.org/10.1017/S0022112083002189.
Chakraborty, P., Balachandar, S., Adrian, R.J., 2005. On the relationships between local
vortex identification schemes. J. Fluid Mech. 535, 189–214. https://doi.org/
10.1017/S0022112005004726.
Cheng, S., Larose, G.L., Savage, M.G., Tanaka, H., Irwin, P.A., 2008a. Experimental study
on the wind-induced vibration of a dry inclined cable—Part I: Phenomena. J. Wind
Eng. Ind. Aerod. 96 (12), 2231–2253. https://doi.org/10.1016/j.jweia.2008.01.008.
Cheng, S., Irwin, P.A., Tanaka, H., 2008b. Experimental study on the wind-induced
vibration of a dry inclined cable—Part II: proposed mechanisms. J. Wind Eng. Ind.
Aerod. 96 (12), 2254–2272. https://doi.org/10.1016/j.jweia.2008.01.007.
Dahl, J.J.M., 2008. Vortex-induced Vibration of a Circular Cylinder with Combined InLine and Cross-Flow Motion. Doctoral dissertation. Massachusetts Institute of
Technology. http://hdl.handle.net/1721.1/44747.
CRediT authorship contribution statement
Rui Wang: Methodology, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization.
Dabo Xin: Conceptualization, Validation, Project administration, Funding acquisition. Jinping Ou: Resources, Supervision.
21
R. Wang et al.
Journal of Wind Engineering & Industrial Aerodynamics 206 (2020) 104381
Norberg, C., 1993. Pressure forces on a circular cylinder in cross flow. In: Eckelmann, H.,
Graham, J.M.R., Huerre, P., Monkewitz, P.A. (Eds.), Bluff-body Wakes, Dynamics and
Instabilities. International Union of Theoretical and Applied Mechanics. Springer,
Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-00414-2_60.
Poulin, S., Larsen, A., 2007. Drag loading of circular cylinders inclined in the along-wind
direction. J. Wind Eng. Ind. Aerod. 95 (9–11), 1350–1363. https://doi.org/10.1016/
j.jweia.2007.02.011.
Seyed-Aghazadeh, B., Modarres-Sadeghi, Y., 2018. An experimental study to investigate
the validity of the independence principle for vortex-induced vibration of a flexible
cylinder over a range of angles of inclination. J. Fluid Struct. 78, 343–355. https://
doi.org/10.1016/j.jfluidstructs.2018.01.004.
Shlrakashi, M., Hasegawa, A., Wakiya, S., 1986. Effect of the secondary flow on karman
vortex shedding from a yawed cylinder. Bulletin of JSME 29 (250), 1124–1128.
https://doi.org/10.1299/jsme1958.29.1124.
Tanaka, T., Matsumoto, M., Ishizaki, H., Kibe, H., 2016. Dry galloping characteristic and
vibration control of inclined stay cable. In: Title JAXA Special Publication:
Proceedings of the First International Symposium on Flutter and its Application,
pp. 639–648. https://pdfs.semanticscholar.org/f94f/fc9b0499138889075fd81
295eacada0426bd.pdf.
Vakil, A., Green, S.I., 2009. Drag and lift coefficients of inclined finite circular cylinders at
moderate Reynolds numbers. Comput. Fluids 38 (9), 1771–1781. https://doi.org/
10.1016/j.compfluid.2009.03.006.
Vo, D., Katsuchi, H., Yamada, H., 2017. March). Dry galloping of surface modification
cable in low Scruton number range. In: Proc. 1st Int. Symp. Flutter its Appl,
pp. 629–638. https://jaxa.repo.nii.ac.jp/?
action=pages_view_main&active_action=repository_
view_main_item_detail&item_id=3460&item_no=1&page_id=13&block_id=21.
Wang, H.F., Razali, S.M., Zhou, T.M., Zhou, Y., Cheng, L., 2011. Streamwise evolution of
an inclined cylinder wake. Exp. Fluid 51 (2), 553–570.
Wang, R., Cheng, S., Ting, D.S., 2018. Simulating the role of axial flow in stay cable
vibrations via a perforated wake splitter plate. In: Wind Engineering for Natural
Hazards: Modeling, Simulation, and Mitigation of Windstorm Impact on Critical
Infrastructure, pp. 111–132. https://doi.org/10.1061/9780784415153.ch06.
Wang, R., Cheng, S., Ting, D.S., 2019. Effect of yaw angle on flow structure and cross-flow
force around a circular cylinder. Phys. Fluids 31 (1). https://doi.org/10.1063/
1.5079750, 014107.
Weidman, Patrick D., 1968. Wake Transition and Blockage Effects on Cylinder Base
Pressures. Engineer’s thesis. California Institute of Technology. https://doi.org/
10.7907/95CE-1189. https://thesis.library.caltech.edu/4577/1/Weidman_pd
_1968.pdf.
Williamson, C.H.K., 1996. Three-dimensional wake transition. J. Fluid Mech. 328,
345–407. https://doi.org/10.1017/S0022112096008750.
Wu, J., Sheridan, J., Welsh, M.C., Hourigan, K., 1996. Three-dimensional vortex
structures in a cylinder wake. J. Fluid Mech. 312, 201–222. https://doi.org/10.1017/
S0022112096001978.
Yeo, D., Jones, N.P., 2008. Investigation on 3-D characteristics of flow around a yawed
and inclined circular cylinder. J. Wind Eng. Ind. Aerod. 96 (10–11), 1947–1960.
https://doi.org/10.1016/j.jweia.2008.02.040.
Yeo, D., Jones, N.P., 2011. Characterization of flow oblique to a circular cylinder with low
aspect ratio using 3-D detached eddy simulation. J. Wind Eng. Ind. Aerod. 99 (11),
1117–1125. https://doi.org/10.1016/j.jweia.2011.08.002.
Yeo, D., Jones, N.P., 2012. Aerodynamic forces induced by vertically oscillating incoming
flow on a yawed horizontal circular cylinder. J. Wind Eng. Ind. Aerod. 104, 188–195.
https://doi.org/10.1016/j.jweia.2012.02.032.
Zhang, W., Lee, S.J., 2005. PIV measurements of the near-wake behind a sinusoidal
cylinder. Exp. Fluid 38 (6), 824–832.
Zhao, M., 2015. The validity of the independence principle applied to the vortex-induced
vibration of an inclined cylinder in steady flow. Appl. Ocean Res. 53, 155–160.
https://doi.org/10.1016/j.apor.2015.08.005.
Zhao, M., Cheng, L., Zhou, T., 2009. Direct numerical simulation of three-dimensional
flow past a yawed circular cylinder of infinite length. J. Fluid Struct. 25 (5), 831–847.
https://doi.org/10.1016/j.jfluidstructs.2009.02.004.
Franzini, G.R., Fujarra, A.L.C., Meneghini, J.R., Korkischko, I., Franciss, R., et al., 2009.
Experimental investigation of Vortex-Induced Vibration on rigid, smooth and inclined
cylinders. J. Fluids Struct. 25 (4), 742–750. https://doi.org/10.1016/
j.jfluidstructs.2009.01.003.
Gallardo, J.P., Andersson, H.I., Pettersen, B., 2014. Turbulent wake behind a curved
circular cylinder. J. Fluid Mech. 742, 192–229. https://doi.org/10.1017/
jfm.2013.622.
Grant, H.L., 1958. The large eddies of turbulent motion. J. Fluid Mech. 4 (2), 149–190.
https://doi.org/10.1017/S0022112058000379.
Gu, M., Du, X., 2005. Experimental investigation of rain–wind-induced vibration of cables
in cable-stayed bridges and its mitigation. J. Wind Eng. Ind. Aerod. 93 (1), 79–95.
https://doi.org/10.1016/j.jweia.2004.09.003.
Hama, R., 1957. Three-dimensional vortex pattern behind a circular cylinder.
J. Aeronaut. Sci. 24, 156–158.
Hoang, M.C., Laneville, A., L
egeron, F., 2015. Experimental study on aerodynamic
coefficients of yawed cylinders. J. Fluid Struct. 54, 597–611. https://doi.org/
10.1016/j.jfluidstructs.2015.01.002.
Hoftyzer, M., Dragomirescu, E., 2010. Numerical investigation of flow behaviour around
inclined circular cylinders. In: Proceedings of the Fifth International Symposium on
Computational Wind Engineering (CWE2010).
Hu, G., Tse, K.T., Kwok, K.C., Zhang, Y., 2015. Large eddy simulation of flow around an
inclined finite square cylinder. J. Wind Eng. Ind. Aerod. 146, 172–184. https://
doi.org/10.1016/j.jweia.2015.08.008.
Hu, G., Tse, K.T., Kwok, K.C., 2016. Aerodynamic mechanisms of galloping of an inclined
square cylinder. J. Wind Eng. Ind. Aerod. 148, 6–17. https://doi.org/10.1016/
j.jweia.2015.10.011.
Hwang, Y., Kim, J., Choi, H., 2013. Stabilization of absolute instability in spanwise wavy
two-dimensional wakes. J. Fluid Mech. 727, 346–378. https://doi.org/10.1017/
jfm.2013.270.
Kim, J., Choi, H., 2005. Distributed forcing of flow over a circular cylinder. Phys. Fluids
17 (3). https://doi.org/10.1063/1.1850151, 033103.
Liang, H., Jiang, S.Y., Duan, R.Q., 2015. Spanwise characteristics of flow crossing a yawed
circular cylinder of finite length. Procedia Engineering 126, 83–87. https://doi.org/
10.1016/j.proeng.2015.11.183.
Li, H., Chen, W.L., Xu, F., Li, F.C., Ou, J.P., 2010. A numerical and experimental hybrid
approach for the investigation of aerodynamic forces on stay cables suffering from
rain-wind induced vibration. J. Fluid Struct. 26 (7–8), 1195–1215. https://doi.org/
10.1016/j.jfluidstructs.2010.06.006.
Li, S., Zeng, Q., Wen, X., Chen, Z., 2017. Numerical simulations and tests for dry galloping
mechanism of stay cables. J. Vib. Shock 11 (36), 100–105. https://doi.org/
10.13465/j.cnki.jvs.2017.11.015.
Ling, G.C., Zhao, H.L., 2009. Vortex dislocations in wake-type flow induced by spanwise
disturbances. Phys. Fluids 21 (7). https://doi.org/10.1063/1.3192652, 073604.
Macdonald, J.H., Larose, G.L., 2006. A unified approach to aerodynamic damping and
drag/lift instabilities, and its application to dry inclined cable galloping. J. Fluid
Struct. 22 (2), 229–252. https://doi.org/10.1016/j.jfluidstructs.2005.10.002.
Matsumoto, M., Shiraishi, N., Kitazawa, M., Knisely, C., Shirato, H., Kim, Y., Tsujii, M.,
1988. Aerodynamic behavior of inclined circular cylinders-cable aerodynamics. Wind
Engineers, JAWE 1988 (37), 103–112. https://doi.org/10.5359/jawe.1988.37_103.
Matsumoto, M., Yagi, T., Shigemura, Y., Tsushima, D., 2001. Vortex-induced cable
vibration of cable-stayed bridges at high reduced wind velocity. J. Wind Eng. Ind.
Aerod. 89 (7–8), 633–647. https://doi.org/10.1016/S0167-6105(01)00063-0.
Matsumoto, M., Yagi, T., Hatsuda, H., Shima, T., Tanaka, M., Naito, H., 2010. Dry
galloping characteristics and its mechanism of inclined/yawed cables. J. Wind Eng.
Ind. Aerod. 98 (6–7), 317–327. https://doi.org/10.1016/j.jweia.2009.12.001.
McTavish, S., Raeesi, A., D’Auteuil, A., Yamauchi, K., Sato, H., 2018. An investigation of
the mechanisms causing large-amplitude wind-induced vibrations in stay cables using
unsteady surface pressure measurements. J. Wind Eng. Ind. Aerod. 183, 19–34.
https://doi.org/10.1016/j.jweia.2018.10.004.
Nishimura, H., Taniike, Y., 2001. Aerodynamic characteristics of fluctuating forces on a
circular cylinder. J. Wind Eng. Ind. Aerod. 89 (7–8), 713–723. https://doi.org/
10.1016/S0167-6105(01)00067-8.
22