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: xindabo@nefu.edu.cn (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