Large eddy simulations of flow past an inclined circular cylinder: Insights into the three-dimensional effect

Flow past a stationary circular cylinder can be encountered in many engineering applications, such as towing cables, subsea pipelines and suspension bridges. This problem is typically characterized by the Reynolds number Re = U_∞D/ν, where U_∞ is the free stream velocity, D is the cylinder's diameter, and ν is the kinematic viscosity of the fluid. For most engineering applications, Re is larger than 350. Above this threshold the three-dimensionality of the wake flow can be clearly observed, as explained in the work of Williamson.¹ The inclination angle α is the angle between the free-stream flow direction and the perpendicular plane of the cylinder axis. In the case of a vertical cylinder, the cylinder is normal to the incoming flow α = 0°. However, in most realistic flow scenarios, the flow is not perfectly normal to the cylinder's main axis. Compared to the purely vertical case of α = 0°, the cylinder inclination results in the development of an axial flow traveling in the spanwise direction. The existence of this spanwise flow is the main factor that influences the three-dimensional wake flow and has a significant effect on important hydrodynamic phenomena such as vortex-induced vibration, heat transfer, and vortex-induced noise.

The primary issue for flow past an inclined cylinder is the dynamics of the vortex shedding from the cylinder, which is a major factor affecting the flow-induced forces acting on the cylinder. Najafi et al.² conducted an experiment for the flow past an inclined cylinder at Re = 5000. They employed both flow visualization and velocity field sampling by particle image velocimetry (PIV) technique. They revealed two distinct wake flow patterns, depending on the range of inclination angle, one corresponding to α = 0°–20° and the second corresponding to α = 35°–45°. The dependence of vortex structures on the angle of inclination was also investigated by Lam et al.³ who used large eddy simulations (LES). The spanwise vortices were found to be shed obliquely from the cylinder in the cases of α > 45° as revealed by the instantaneous wake patterns at different locations along the cylinder span. However, quantifying the shedding angle of vortices separating from an inclined cylinder is challenging because the vortex shedding and its orientation are not steady and difficult to quantificationally identify, especially at high Re and large α.⁴ A feasible way to identify the vortex shedding pattern is to analyze the force variation on the cylinder. Yeo and Jones,^5,6 Hogan and Hall,⁷ Lucor and Karniadakis,⁸ Zhao et al.,⁹ Wang et al.,¹⁰ and Zhao et al.¹¹ presented the spatial-temporal contours of the pressure and lift coefficients on the cylinder. The results showed inclined stripes in the spatiotemporal domain at α ≥ 30°, representing a spanwise traveling mode in the vortex shedding behind an inclined cylinder.

For flows past nominally two-dimensional bodies, such as circular cylinders with an infinite length, the correlation length in the spanwise direction is an important measurement for describing the three-dimensionalities in the near-wake. A higher spanwise correlation indicates that the vortex shedding tends to occur uniformly along the spanwise direction. A good understanding of the spanwise correlation is not only essential for predicting the spatial distribution of the vortical structures but also can be used to determine a proper spanwise length of the computational domain to achieve a reasonable balance between the computational cost and the size of the domain sufficient to capture the essential flow physics for numerical simulations. For a vertical cylinder within the subcritical flow regime approximately from Re = 350 to Re = 3  $×$ 10⁵, the correlation length of 2D–3D has been well documented and the spanwise length of H/D = 4–6 of the computational domain has been extensively adopted by numerous works, such as Kravchenko and Moin,¹² Lei et al.,¹³ Prsic et al.,¹⁴ Tian and Xiao,¹⁵ Jiang and Cheng,¹⁶ and Janocha et al.¹⁷ On the other hand, the spanwise correlation length of an inclined cylinder has rarely been studied. The numerical results in the study of Yeo and Jones⁶ at Re = 1.4  $×$ 10⁵ revealed that force fluctuations are still correlated within a finite length of 10D along the inclined cylinder. Based on the results of direct numerical simulations (DNS) by Zhao et al.,^18,20 the correlation length of the axial vortices at the angle of inclination α = 45° was measured to be 3.2 and 4.0D for Re = 300 and 400, while the correlation length for Re ≥ 500 has not been confirmed as the vortex structures were not clearly identified. The spanwise flow characteristics of an infinite cylinder with both inclined and yawed angles at Re = 5.3  $×$ 10⁴ were studied by Wang et al.¹⁹ The time-averaged streamwise vortices indicated the length between two vortex cores is 1.8D. The experiment at Re = 5.61  $×$ 10⁴ in Hogan and Hall⁷ quantitively proved that the correlation length decreases rapidly from 3.3D to 1.1D with α increasing from 0° to 30°. For larger angles of inclination, the spanwise characteristic length of the vortical structures has not been thoroughly investigated to the authors' knowledge.

In addition to the three-dimensional characteristics of the wake flow structures, it is also important to quantitatively investigate the variation of the vortex-induced forces acting on the cylinder with the angle of inclination. The independence principle (IP), also known as the cosine law, was proposed as an estimation method for hydrodynamic characteristics. This theory assumes that the force coefficients and the vortex shedding frequency are equivalent to those in the vertical cases when they are normalized by the velocity component perpendicular to the cylinder axis u_n, regardless of the inclination angle α. The application of IP can greatly simplify the analysis of flow past cylinders with an arbitrary angle of inclination. However, the validity of IP is still arguable. Zhao et al.^18,20 performed a DNS study on the inclined cylinder with 0° ≤ α ≤ 60° at Reynolds numbers covering 100 ≤ Re ≤ 1000. They found that the Strouhal number St = fD/U_∞ (where f is the vortex shedding frequency) and the mean drag coefficient can be well represented by the IP when α ≤ 30°, while the root mean square lift coefficient is highly dependent on the varying α. Lam et al.³ used LES to study the case of an inclined cylinder at Re = 3900. Their results suggested that the IP is valid up to α = 45°, and a similar conclusion was also drawn by Liang and Duan²¹ at the same Re. Najafi et al.² and Zhou et al.²² conducted an experimental investigation on the wake characteristics of an inclined cylinder at Re = 5000 and Re = 7200, respectively. The Strouhal number St, as well as the directions of separated shear layers and the spanwise vortices, was found to obey the IP when α ≤ 40°–45°, while other features, such as velocity components, were dependent on α. On the other hand, results reported by Hogan and Hall⁷ support that the vortex shedding frequency of an inclined cylinder can be predicted using IP with reasonable accuracy only when α ≤ 20°. The study at Re = 3900 using LES by Zhou et al.²³ also held the viewpoint that hydrodynamic force coefficients and Strouhal number predicted by LES do not agree with the values predicted using IP, despite that these results predicted by IP remain constant at 15° ≤ α ≤ 60°. Wang et al.¹⁰ concluded that the IP can reasonably predict the Strouhal number and the pressure distribution at some parts of the cylinder surface, while the drag force was underpredicted with a similar magnitude for all inclination angles.

Marshell²⁴ pointed out that IP is basically a two-dimensional method and only considers the contribution of the velocity component normal to the cylinder axis u_n. This may explain the limited applicability of IP at higher angles of inclination and higher Reynolds numbers, where the three-dimensional effect induced by the axial flow in the spanwise direction of the cylinder cannot be ignored. Although the three-dimensional nature of the vortices has been widely studied, as mentioned above, including the characteristics of vortex shedding and spanwise instabilities in shear layers, its effect on the hydrodynamic forces remains unexplored. In this research, we aim to comprehensively study the spatial and temporal characteristics of the three-dimensional wake flow past an inclined cylinder and quantitively investigate the correlations between the force on the cylinder and the flow characteristics. These detailed quantitative conclusions have been rarely considered in the existing works to authors' best acknowledge. Four inclination angles ranging from 0° to 60° are selected in this study. In such cases, the forces normal to the cylinder axis are anticipated to be more pronounced compared to situations where the inclination angles exceed 60°. Therefore, those angles are of much engineering interest.¹⁸ Three specific cases with the inclination angles of 30°, 45°, and 60° are representative to show the variance in the wake flow behind the cylinder and have been widely considered in the existing works.^9–11 In the first step, the coefficients normalized by u_n are discussed to verify the IP. The influence of the inclination angle on the three-dimensional effects in the wake is addressed by analyzing the velocity and the Reynolds stress distributions, anisotropy, and spanwise length scales of the wake vortices. Finally, to quantify the origin of the vortex-induced force at different angles of inclination, the vorticity field is decomposed into the spanwise and the cross-spanwise parts, and their respective contributions to the hydrodynamic forces are calculated. The rest of this paper is organized as follows: The numerical methods applied in this study are described in Sec. II. The convergence and validation studies are presented in Sec. III. The results and discussions of flow analyses at four angles of inclination are covered in Sec. IV. Section V summarizes the most important findings and conclusions of the present study.

The flow past an infinite cylinder with an angle of inclination α is simulated using a rectangular computational domain, as shown in Fig. 1. The domain size is 6D in height, 20D in width, and 35D in length. The side planes and the inlet boundary are 10D away from the axis of the cylinder. The distance between the outlet boundary and the axis of the cylinder is 25D. In this study, the coordinate system is defined by locating the origin at the geometric center of the cylinder and fixing x, y, z axes to the streamwise, transverse, and spanwise directions at α = 0°, respectively. Therefore, in the following discussion, u, v, w in the velocity vector U represent the velocity components in x, y and z axis directions, respectively.

The boundary conditions used in the simulations are as follows. The lateral boundaries parallel to the horizontal xy-plane are imposed with the periodic boundary conditions, and the symmetry boundary conditions are employed on the vertical planes parallel to the xz-plane. The non-slip boundary condition is set on the cylinder surface. The inlet boundary is specified with uniform flow U = (u, v, w) = (U_∞ cos α, 0, U_∞sinα) and pressure gradient ∂p/∂n = 0, while reference pressure p = 0 and ∂U/∂n = 0 are set on the outlet boundary.

Lumley's triangle introduces a map constructed by the above invariants, which can be used to identify all realizable turbulence states. The physical definition of Lumley's triangle is introduced as follows. The left and right boundaries defined by η =  $−$|ξ| and $−$ 1/6  $≤$ ξ $≤$ 1/3 represent the axisymmetric turbulence structures, where the oblate-shaped turbulence with two major eigenvalues of the anisotropy tensor is located at the left boundary, and the prolate-shaped turbulence with only one major eigenvalue lies on the right boundary. The upper boundary refers to the essentially two-dimensional turbulence. In this state, the turbulent fluctuations are significant only in two directions. The three vertices of Lumley's triangle at right, left, and bottom represent the turbulence with the line shape of the one-dimensional feature, the disk shape with two-dimensional axisymmetric property, and the sphere shape with isotropy, respectively.

As the angle of inclination increases, it is expected that the dominating component of vorticity in the wake region may also change, which consequently leads to differences in the development of vortex-induced forces. Therefore, one of the primary objectives of this paper is to characterize the orientation of the three-dimensional vortex structures behind a cylinder at different angles of inclination and further quantify the relationship between the flow structures and the forces on the cylinder. For this reason, the force partitioning method^39–43 is employed. This method simplifies a complex fluid flow problem by decomposing the total force into parts related to viscosity, vorticity, and added mass.

The mesh topology adopted in this study is shown in Fig. 4, presenting the horizontal cross-section of the domain. The O-grid area surrounding the cylinder is formed by an overlap of four circles of 15D in diameter whose centers are 10D away from the cylinder axis. A number of 320 nodes are equally distributed on the circumference of the cylinder and radially extruded inside the O-grid zone. The first node next to the cylinder surface is placed according to the average dimensionless distance y⁺ = u_f h/ν < 0.3, where u_f is the friction velocity and h is the distance in the normal direction to the cylinder's surface. The mesh size increases in both x and y directions from the cylinder surface to the boundary of the O-grid zone. The meshes adjacent to the O-grid area smoothly transition to an H-grid topology and then gradually extrude to reduce the cell number away from the cylinder. The length of the longest cells in the far field is kept under 0.4D. In most studies on the flow past a vertical cylinder at Re = 3900 using LES, the spanwise resolution Δz is approximately from 0.047D to 0.065D.^{12,16,17,44–46} The reported spanwise resolutions for the cases of an inclined cylinder at Re = 3900 range from 0.09D to 0.11D.^21,23 In this study, 96 layers of nodes are equally distributed along the spanwise direction of the computation domain to capture the axial flow, corresponding to Δz = 0.063D. The results are sampled over non-dimensional time tu_n/D > 250, covering at least 50 vortex shedding periods.

In the following discussion, all spanwise-averaged results are denoted by angle brackets $·$ and all time-averaged results are denoted with an overline $· ¯$⁠. Table I shows three flow cases around a cylinder at α = 0° for the mesh independence test. As the normal velocity u_n = U_∞ at α = 0°, the subscript n of all coefficients in the convergence studies is omitted. Three mesh schemes are used, where the cell number increases at a rate of approximately 30%. This increment is carried out by expanding the node number in the radial direction within the O-grid zone, progressing from 150 to 200 and eventually to 250. The mesh size in the O-grid zone increases at a ratio less than 1.02, to guarantee the dimensionless distance y⁺ < 0.3 to the wall.

All three mesh variants give a similar Strouhal number St = fD/U_∞ around 0.21, where f is the vortex shedding frequency. The average drag coefficient $C d ¯$⁠, the root mean square of the lift coefficient $C l rms$⁠, the base pressure coefficient $− C p b ¯$⁠, the separation angle $θ sep ¯$⁠, and the recirculation length L_rec predicted by the simulation using the medium mesh M2 agree very well with those predicted by simulation using the finest mesh M3. A detailed mesh convergence study, including comparisons of velocity profiles and pressure distributions on the cylinder, can be found in  Appendix. Based on the obtained results, it can be concluded that a reasonable mesh convergence is obtained by M2.

The convergence test for the time step Δt is then performed using mesh M2. Table II shows that St is independent of the time step size in the investigated range of Δt, and the rest of the results in the three cases are also close to each other. Considering the balance of accuracy and efficiency, the time step scheme T2 (ΔtU_∞/D = 3.9  $×$ 10⁻³) is chosen for the remaining simulations.

The flow past a cylinder at various angles of inclination is modeled by modifying the inlet boundary condition to represent the oblique flow instead of rotating the cylinder and remeshing the whole domain.^10,11 A general flow feature can be identified by analyzing the instantaneous streamlines of the flow around the cylinder. When the flow first reaches the cylinder with a nonzero angle of inclination, it moves a small distance along the cylinder span due to the velocity component w > 0. After that, the flow passes the cylinder obliquely upwards on both sides of the cylinder and then separates from the cylinder surface at the separation point (denoted by the dashed lines in Fig. 10). The axial flow is clearly visible in the near wake after the shear layer separates. Figure 10 shows the instantaneous streamlines with seed points distributed along the line z/D =  $−$ 3H in the plane y/D = 0 in three cases of α = 30°, 45°, and 60°. The lower part of the cylinder is not traced due to the location of the seed points of the upstream streamlines. Despite this, the above-mentioned flow pattern behind an inclined cylinder is precisely reproduced by the present simulation, and similar patterns were also reported by Najafi et al.² and Lam et al.³ 

As shown in Fig. 12, the time-averaged streamwise velocity component for two investigated α is sampled along the $x ′$ axis direction at $y ′$ = 0 and z/D = 0, $−$1, and $−$2, respectively. $u x ′ ¯$ is zero at the cylinder surface and then gradually increases to about 0.8D with the increasing $x ′$⁠. A local minimum caused by strong recirculation in the near wake is visible in each plot. As the flow characteristics are statistically homogeneous along the z axis rather than $z ′$ axis, a phase difference occurs inevitably at different z-locations. Despite this, the present simulation is able to capture the profile of $u x ′ ¯$⁠, as well as the velocity magnitude at the far field, in both investigated cases (α = 30° and 45°). The results of the present validation study support the ability of the present model to predict accurately $u x ′ ¯$ profiles for inclined cylinder flow cases.

The independence principle (IP) is a convenient approach for estimating the hydrodynamic features of flow past an inclined cylinder. It assumes that the features normalized by the velocity component perpendicular to the cylinder axis, u_n = U_∞cosα, are independent of the angle of inclination α. To validate the applicability of IP, the time- and spanwise-averaged coefficients, shown in Table III, are used to analyze the flow past a cylinder with different angles of inclination. The main objective of this study is to illustrate the detailed insights of the flow past an inclined cylinder, which can be widely seen in various engineering applications, such as towing cables, subsea pipelines, and suspension bridges. The selected four angles ranging from 0° to 60° can cover most of those scenarios. The inclination angles 30°, 45°, and 60° are also representative to show the differences in the wake flow of the cylinder and have been widely concerned in the existing works. Therefore, those angles are selected in the present study.

In comparison with the vertical case of α = 0°, the Strouhal number St_n is similar in all investigated cases. However, the normalized drag and lift coefficients $C d n ¯$ and $C l n rms$ are lower in the inclined cases. The coefficient $C z n ¯$ of the spanwise force increases with the increasing angle of inclination. Figure 13 shows the pressure coefficient distribution on the surface of the cylinder, where θ is the angular coordinate with θ = 0° being the stagnation point. Similar to the drag and lift forces, the normalized pressure distribution is similar in all three cases of α ≥ 30°, while their magnitudes are smaller than that in the vertical case.

The present results show that St_n remains relative stable in all four cases, which indicates that IP can be used to reasonably predict the vortex shedding frequency. This conclusion is similar with those in the experimental study of Najafi et al.² and Zhou et al.²² On the other hand, the drag coefficients $C d n ¯$ and lift coefficients $C l n rms$ have obvious differences between the vertical and inclined cases. This observation suggests the force predictions using IP are inaccurate, which agrees well with Zhou et al.²³ and Wang et al.¹⁰ The discrepancy in force coefficient predictions by IP is because it only considers the contribution of the velocity in the 2D xy-plane. However, the flow three-dimensionality induced by the axial flow along the cylinder span should not be neglected when the angle of inclination is larger than 30°. The increasing importance of flow three-dimensionality is indicated by the increasing value of $C z n ¯$ with the increasing α. A detailed discussion of the three-dimensional effects is given in Subsections IV B–IV F.

Figure 14 shows the spanwise-averaged drag and lift coefficients as a function of time, where low and high drag regimes are specified by $C d n ¯ + C d n min/2$ and $C d n ¯ + C d n max/2$⁠, respectively. It should be clarified that $C z n$ does not show significant time variability throughout the present simulations, and it is not shown in the plots. In the case of α = 0°, apparent low and high drag regimes are spotted, and these regimes are correlated with small and large amplitudes of the lift coefficient. Low and high drag regimes still exist in the inclination cases, but the differences between their magnitudes are much less noticeable. Those differences in the two regimes are consistent with the variance of $C l n rms$ with α shown in Table III, where the smallest difference occurs at α = 45° and it is similar at α = 30° and α = 60°. On the other hand, the frequency of the drag coefficient is approximately twice that of the lift coefficient in each case of α. This indicates that the periodic vortex shedding at both sides of the cylinder is still the main reason for the periodic variation in forces, regardless of whether the cylinder is vertical or inclined.

In order to investigate the variation of vortex shedding frequency in both high and low regimes in all four cases, wavelet transform analysis is conducted to illustrate the amplitude St_n in the time series. Figure 15 shows the time-frequency representation of the lift coefficient after wavelet transform. The peak frequencies are distributed near St_n in all four cases, denoting that the vortex shedding frequency is temporally stable and is independent of the occurrence of low/high drag phenomenon. However, the enhancement and suppression in the amplitude can be observed corresponding to the high and low drag regimes and this amplitude modulation is attenuated with the increasing incline angle.

The wall stress distribution indicates the fluid motion on the cylinder surface. Additionally, the separation of the boundary layer occurs at the point where the wall shear stress becomes zero. This separation phenomenon further impacts the subsequent vortex shedding behind the cylinder. Therefore, it is imperative to conduct a thorough investigation of the wall stress distribution on the cylinder surface. Figures 16 and 17 present the time- and spanwise-averaged wall shear stress components normalized by U_∞ and u_n. Both components of wall shear stress distribution along the cylinder circumference vary significantly between different angles of inclination. Figure 16 shows that the magnitude of the tangential component represented by the magnitude of $| τ w x ¯, τ w y ¯|$ normalized by u_n at α = 30° is close to that in the vertical case and increases significantly in α = 45° and α = 60° cases. This indicates that the flow characteristics remain relatively similar in the xy-plane between the cases of α = 0° and α = 30°.

Considering the time-averaged spanwise component of shear stress $τ w z ¯$⁠, Fig. 17 shows that $τ w z ¯$ is zero only in the cases of α = 0°, and the $τ w z ¯$ values become nonzero for inclined cylinder cases due to spanwise flow. The spanwise flow is the main origin of enhanced three-dimensionality of inclined cylinders' near wake and decreased IP accuracy at large inclination angles. The largest magnitude of $τ w z ¯$ is located at θ = 0° where the incoming flow reaches the cylinder, and it reduces with the increasing θ. After the normalization by u_n, the largest magnitude at this position is observed in the case of α = 60°, followed by α = 45° and α = 30°. This observation is consistent with the observation of streamlines in Fig. 10. Close to the separation point where $| τ w x ¯, τ w y ¯|$ is zero, the value of $τ w z ¯$ reaches its minimum but remains greater than zero. After the separation point, the value of $τ w z ¯$ continues to increase, and this increment is the most apparent in the case of α = 60° after the normalization by u_n.

The three-dimensional effect in the surrounding flow at different angles of inclination is first illustrated by the averaged results. Figure 18 shows the time- and spanwise-averaged velocity component $u ¯$ in the x axis direction. The recirculation lengths L_rec/D (defined by the distance between two points where $u ¯$ = 0) are 1.69, 1.84, and 1.69 in the cases of α = 30°, 45°, and 60°, respectively. They are all longer than 1.31, observed in the vertical case. The detailed contours of the velocity component and the averaged streamlines in the x axis direction in the case of a vertical cylinder are shown in Fig. 19, as well as the results in the case of α = 60° representing the inclination cases. In the case of α = 0°, the backward flow behind the cylinder is along the negative x axis direction, while in the inclination case, the spanwise flows occur inside the recirculation zone, which results in the nonzero $τ w z ¯$ behind the separation point.

The present results can initially explain the difference in drag coefficient $C d n ¯$ among the four cases. In the inclination cases of α ≥ 30°, the recirculation length is larger than for the vertical case, which leads to an increase in the distance between the cylinder and the location of the lowest pressure indicated by the recirculation core. This results in a higher pressure at the cylinder back (θ > 90°), as shown in Fig. 13, and further leads to a lower pressure difference between the front and back sides of the cylinder. It is the root cause of lower drag coefficients in the cases of α ≥ 30° compared with the vertical case.

The time- and spanwise-averaged Reynolds shear stress distributions in Fig. 20 provide a quantitative description of velocity fluctuations in the wakes of the analyzed configurations. All the results are normalized using u_n. The locations of the peak values of the Reynolds stress components are similar in three inclination cases. Those locations can also be indicated from the similar lengths of the recirculation zones in three inclination cases in Fig. 18, all of which are greater than that in the vertical case. It can be seen from Fig. 20 that $u ′ v ′ ¯$ is the largest of the three components, and its magnitude decreases with the increasing α. The overall spatial distributions of $u ′ v ′ ¯$ are similar for different α. An interesting phenomenon can be observed in the correlations with the spanwise fluctuation $w ′$ of $u ′ w ′ ¯$ and $v ′ w ′ ¯$⁠. Their magnitudes are relatively smaller than $u ′ v ′ ¯$⁠, and they increase with the angle of inclination α, while $u ′ v ′ ¯$ decreases with increasing α inversely. This observation suggests that with the increase in α, the correlation between $u ′$ and $v ′$ decreases, while their correlations to the $w ′$ are, respectively, enlarged. With the increasing α within the recirculation region (except for the shear layer region), the amplitudes of $u ′ w ′ ¯$ and $v ′ w ′ ¯$ increase with α, indicating a strong secondary flow in the spanwise direction induced by the spanwise fluctuations.

The Reynolds stress shows general variations in turbulent motions in three directions, as explained above. In this subsection, the Lumley's triangle anisotropy map^26–33 is further employed for quantifying the anisotropy of the Reynold stress. In the following analysis, all the data are sampled along the y axis, from y/D = 0 to y/D = 6, as the fluctuations of the wake flow are intense within this range.³³ 

In the vertical case of α = 0° in Fig. 21(a), the vortical structure distribution at x/D = 1 first shows an oblate shape near y/D = 0. As the distance from the centerline increases in y axis, the shape generally changes into a prolate-like shape, accompanied by an increase in anisotropy. At x/D = 3, the velocity fluctuation distributions from y/D = 0 to y/D = 6 are starting from a prolate-like shape, followed by an oblate-like shape, and finally showing an anisotropic prolate shape. For the further downstream x/D = 7, the vortices at y/D = 0 generally display a prolate shape with high anisotropy in the vertical case, and the vortices become two-component axisymmetric disk-shaped in the area near y/D = 6. In the case of α = 30° in Fig. 21(b), the variation in the vortices shape is generally similar to that in the vertical case, except that the overall anisotropy is much stronger, especially comparing the flow states at x/D = 1. The increasing anisotropy is more evident in the area away from the centerline at y/D > 2. Moreover, the occurrence of oblate-shaped vortices at x/D = 3 is also closer to the centerline compared with α = 0°. The vortex structures at x/D = 7 at y/D = 0 show a relatively high anisotropy. However, the characteristic shape far away from the cylinder is a one-dimensional line-shaped with high anisotropy, which makes the main difference in vortical anisotropy between the inclined and the vertical cases.

When the angle of inclination further increases to α = 45°, the general anisotropy is further enhanced, as shown in Fig. 21(c). The occurrence of oblate-shaped vortices at x/D = 3 is less noticeable. For the case of α = 60° in Fig. 21(d), the vortices at x/D = 1 generally show the strongest anisotropy, and oblate-shape vortices are less apparent. When the flow state at y/D = 0 moves from x/D = 1 to x/D = 3, the vortices represent an oblate shape, and this shape quickly becomes prolate at x/D = 3. This indicates the occurrence of oblate-shaped flow structures is the least obvious in four cases. At downstream area, the vortices become oblate-shaped within a small area near centerline y/D = 0 at α = 45°, while all the vortices show a prolate- and line-shaped with a high anisotropy in the case of α = 60°.

In general, the provided Lumley's triangle anisotropy maps reveal that most of the vortices in the near wake of an inclined cylinder are prolate- and line-shaped, especially in the cases of α = 45° and 60°. It is different from the oblate-shaped vortices commonly observed in the vertical case. Moreover, the oblate-shaped vortex structures also become less apparent in the far wake behind an inclined cylinder. A possible reason is that the vortices are gradually stretched into a long shape with the increasing α and the distance to the cylinder due to the existence of the axial flow and the velocity component in the spanwise direction. Thus, the strength of the vorticity fluctuation is more likely to become pronounced in only one single direction. As a result, all the vortex structures show extreme anisotropy in the area far away from the inclined cylinder with a larger angle of inclination.

In this section, the Hilbert transform is used to quantify the spanwise length scales in the wake of the cylinder. The temporal and spatial variations of the flow structures along the cylinder span at specific locations in the flow field can be studied using this method. Two sampling locations at y/D = 0 are selected, as shown in Fig. 22.

The temporal variations of the spanwise length scale λ_z and the amplitude of ω_y at the first sampling location within the recirculation zone are shown in Fig. 23. At this location close to the cylinder, the distributions of $P λ z$ (the PDF of λ_z) are approximately continuous along the temporal axis. High values of $P λ z$ are correlated with large amplitudes of $C l n$⁠, which becomes more evident with the increasing α. The highest probabilities are generally located around λ_z/D ≈ 0.63 in all four cases of α. In the low-drag/low-lift regimes, the peak probabilities are much smaller. High values of $P λ z$ indicate that the vortex structures are well organized and tend to be aligned in similar spatial directions. The results shown in Fig. 23 suggest that well-organized vortices indicated by high values of $P λ z$ generally occur with a low pressure in the wake region behind the cylinder. This leads to a high-pressure difference between the front and back sides of the cylinder, which finally results in the high-drag/high-lift regimes as shown in the time histories of $C l n$ in Fig. 23 and in Fig. 14. Considering $A ω y$ (the amplitudes of ω_y), it appears that the occurrence of high peaks of $A ω y$ is further correlated with the occurrence of high $P λ z$ for the inclined cylinder cases. In the vertical case, the peaks of $A ω y$ randomly occur both in space and in time. When the angle of inclination further increases, $A ω y$ gradually decreases, indicating the strength of vortices becomes smaller. Moreover, the tilted stripes in the three cases of inclination are due to the spanwise traveling of the vortices near the cylinder. With the increasing α, the tilted stripes become thinner and more sparsely scattered in space and time.

The vortices outside of the recirculation zone are then analyzed by sampling the data at x/D = 3 and y/D = 0, as shown in Fig. 24. At this location, the peak values of $P λ z$ and $A ω y$ periodically occur with the similar frequencies of $C l n$⁠. This indicates that the vortex shedding behind the cylinder in all four cases of different angles is periodic and may have a major effect on the temporal variations of the forces on the cylinder. The correlation lengths indicated by the peak locations of the $P λ z$ contours at this location in the four cases are also around λ_z/D ≈ 0.63. On the other hand, $A ω y$ shows less continuity in the slightly inclined stripes. This suggests that the spanwise traveling of the vortices away from the inclined cylinder is less evident.

The above results have shown the differences in the time-averaged and instantaneous flow features observed in the near wakes of cylinders with different angles of inclination. However, it is still of great significance to understand the quantitative correlations between the three-dimensional vortices and the forces on the cylinder at different angles of inclination. Therefore, the force partitioning method^39–43 is adopted to decompose the drag and lift forces into the contributions of the volume integration of the vortices associated with Q and the surface integration of the vorticity on the cylinder due to the viscosity, respectively. Then, the contribution of the vortices in the flow field is further decomposed into the parts of the spanwise vorticity and the cross-spanwise vorticity according to the orientation of the vorticity vector. By analyzing the contours of the vortex-induced forces together with the vortical structures, the quantitative contribution of the vortices can be identified.

Figure 25 shows an overall view of the time-histories of each force component after spanwise averaging. In the four cases of different inclination angles, the contribution of vortex-induced drag $C d n ω$ to the total drag $C d n$ is higher than 90%. In contrast, the contribution of the viscosity-induced drag $C d n v$ associated with the vorticity on the surface of the cylinder is approximately less than 5%. Therefore, the force on the cylinder is quantitatively proved to be dominated by the vortices in the wake region, regardless of the angle of inclination. However, when the contribution of the vortices is further decomposed into the parts of spanwise vorticity $C d n ω z$ and the cross-spanwise vorticity $C d n ω x y$⁠, their contributions to the drag force exhibit noticeable differences for different α. For the vertical case, the dominant factor is $C d n ω z$ associated with the spanwise vorticity, and the magnitude of $C d n ω x y$ related to the cross-spanwise vorticity is close to zero. The contribution of $C d n ω x y$ becomes nonzero for the inclined cylinder cases, and the growth of $C d n ω x y$ fluctuations generally synchronizes with the decay of $C d n ω z$ fluctuations, as observed in their temporal evolutions. The value of $C d n ω x y$ increases and becomes larger than that of $C d n ω z$ when the angle of inclination increases to α = 30°. As the angle further increases to α = 45°, $C d n ω z$ gradually becomes the dominant contribution to the vortex-induced drag force again. For the largest investigated α = 60°, the cross-spanwise vortex-induced $C d n ω x y$ even makes a significant negative contribution to $C d n$⁠. Considering the lift forces on the cylinder, the vortex-induced lift is also dominant in all four cases, with its contribution accounting for almost the entire lift force. Moreover, the spanwise vortex-induced $C d n ω z$ is found to be the main contribution to $C l n$ in all cases, while the cross-spanwise vortex-induced $C l n ω x y$ is out of phase with $C l n$⁠. Furthermore, with the increasing α, the amplitudes of $C l n ω x y$ become comparable to $C l n$⁠.

In the following figures, a detailed explanation of the variations in forces is given by plotting the temporal vortex-induced force evolution along the cylinder span. Figure 26 shows the decomposed forces in the vertical case of α = 0°, together with the spanwise-averaged coefficients. For the dominant component $C d n ω z$⁠, its magnitude is generally larger than zero, while $C d n ω x y$ has irregular scattering fluctuations between negative and positive peak values along both the spanwise and temporal directions. This indicates the spanwise vortex formed behind the cylinder can only increase the drag, while the cross-spanwise vorticity can have both positive and negative influence on the drag. The contours of $C l n ω z$ show clear periodicity with positive and negative magnitudes, and $C l n ω x y$ also displays localized fluctuations with weak temporal periodicity. The peaks of $C l n ω z$ occur at the same time as troughs of $C l n ω x y$⁠, and vice versa. This indicates that the cross-spanwise vortices suppress the amplitude of lift force.

Since the forces on the cylinder have been proven to be highly related to the vortex shedding, the vortex-induced forces on the cylinder are then analyzed together with the three-dimensional vortex structures. Figure 27 shows the instantaneous iso-surfaces of spanwise and cross-spanwise vortices at time step t = t₀ denoted in Fig. 26, colored by their contributions to the vortex-induced force coefficients. For the drag force, the dominant positive contribution of spanwise vortices –Q_zφ_x is from the spanwise shear layer, as shown in Fig. 27(a). In the far wake region, the spanwise vortices decay and their influence is less significant. Figure 27(b) shows that for the cross-spanwise vortices, the large contribution to the drag force comes mainly from the highly three-dimensional vortices within the recirculation region. In the far wake region, there are streamwise oriented vortices with a high spatial density. However, their contributions to the drag force are low due to the distance to the cylinder. For the lift force, the contributions of vortices are similar to those observed for the drag force as shown in Figs. 27(c) and 27(d). The only difference is due to the distribution of the potential φ. Therefore, for the other angles of inclination, we only focus on the analysis of the contribution of the three-dimensional vortex-induced drag forces.

The force decomposition in the inclined case of α = 30° is presented in Fig. 28. Different from the vertical case, the overall magnitude of $C d n ω z$ is much smaller within both low and high drag regimes, while $C d n ω x y$ time series shows more positive peaks than the corresponding time series for the vertical case. In this study, the spanwise vortices are identified by the angle between the local vorticity vector and the cylinder span being smaller than 45°. Therefore, the increase in the magnitude of $C d n ω x y$ in Fig. 28(b) can be explained by the tilting of spanwise vortices into the cross-spanwise direction. In addition, the stripes in the contours of forces are inclined with respect to the time axis. This indicates the vortices are obliquely traveling along the cylinder span. On the other hand, the vortex-induced lift force at α = 30° shows more temporal periodicity than the vertical case. There are much fewer localized positive peaks in $C l n ω x y$ compared with those for α = 0° while more negative events of $C l n ω x y$ than the positive ones; therefore, it is indicated that the overall cross-spanwise vortices in this case have a suppressive effect on the lift force.

To provide a three-dimensional view of the vortex structures behind an inclined cylinder, the instantaneous iso-surfaces of the normalized Q at the time step t = t₃₀ denoted in Fig. 28 are shown in Fig. 29. The dominant spanwise contribution to drag force −Q^zφ_x still comes from the shear layer as shown in Fig. 29(a), while the strength of the shear layer is reduced due to the decreased normal velocity u_n compared with the case of α = 0°. On the other hand, the spatial scales of the spanwise vortices in the near wake become larger compared with those for α = 0°, while their contrition to the drag force is still small. As shown in Fig. 29(b), the contribution −Q^xyφ_x of the cross-spanwise vortices to the drag force is negative around the stagnation point of the cylinder, which is related to the spanwise motion found in Fig. 10. The positive contribution is mainly within the recirculation zone, and it is larger compared with α = 0° case. The spatial scale of the cross-spanwise vortices is larger in the far wake, while their contribution is small due to the distance. A close inspection of the 2D contours on two xy-planes at A30 and B30 is further given in Fig. 29, where the contribution of the entire vorticity field is shown. The general contribution of spanwise vortices −Q^zφ_x is similar in both contours in Fig. 29(a). It can be also observed that the location where −Q^zφ_x reach its maximum occurs approximately at shear layer rollup. These two factors result in a spanwise uniformity of $C d n ω z$ contours in Fig. 28. Due to the strong cross-spanwise vortices at x/D > 2 at A30 in Fig. 29(b), their contribution to the drag force is high. Compared with B30, there are also stronger highly three-dimensional small-scale cross-spanwise vortices within the recirculation region at A30. These two factors result in increase in $C d n ω x y$ at A30.

The vortex-induced forces on the cylinder at α = 45° are shown in Fig. 30. In comparison against the case of α = 30°, the differences in drag forces are that the overall magnitude of $C d n ω z$ is larger at α = 45°, and both strong positive and negative peaks with similar amplitudes occur in $C d n ω x y$⁠. Considering the lift forces, the general patterns are similar to those observed at α = 30°. The only difference is that the variation between the positive and negative peaks is smaller at α = 45°. This is also related to the largest distance of the recirculation zone at α = 45° that reduces the influence of vortices on the forces.

Figure 31 shows the iso-surfaces of the vortex structures at t = t₄₅. For the spanwise vortices in Fig. 31(a), their spatial density becomes even smaller compared with α = 30°. According to the present instantaneous iso-surface of the normalized Q, the strength of the spanwise shear layer is smaller in comparison with α = 0° and 30°, and thus the main positive contribution to the drag force −Q^zφ_x originates from the location of the shear layer rollup. The effect of large inclination angle on cross-spanwise vortices can be seen in Fig. 31(b); specifically, vortex structures become aligned in the spanwise direction and well organized. The negative cross-spanwise contribution upstream the cylinder increases with the increasing angle of inclination. Comparing two 2D contours at A45 and B45, contributions of cross-spanwise vortices at B45 are stronger, which results in the intensified positive drag force at the corresponding location. To explain the stripes observed in the force contours, a time-series of 3D cross-spanwise vortices around t = t₄₅ is shown in Fig. 32. A spanwise traveling of local strong cross-spanwise vortices near B45 can be observed as marked by the blue circles. This can be identified as oblique stripes denoted by the blue arrow in Fig. 30.

The results of force partitioning in the case of α = 60° are presented in Fig. 33. Considering the drag forces, the overall magnitude of $C d n ω z$ is positive and is much larger compared with other inclined cylinder cases, while the space-time representation of $C d n ω x y$ contains predominantly negative values with only a few localized positive streaks. The contributions to the drag force of $C d n ω z$ and $C d n ω x y$ are attributed to the change in the spatial organizations of the vortices. Due to the angle of the incident flow, some contributions from the cross-spanwise vortices are extracted and turned into the spanwise direction at α = 60°, which is consistent with the experimental observation in Najafi et al.² In the time-series of vortex-induced lift force, a clear temporal periodicity for the positive and negative $C l n ω z$ is visible, and the positive and negative peak values of $C d n ω x y$ seen as stripes in Fig. 33(d) tend to be longer in the spatial-temporal contours compared with other investigated cases.

Figure 34 presents the instantaneous iso-surfaces at t = t₆₀ in the case of α = 60°. As shown in Fig. 34(a), the spanwise uniform vortices in the wake region almost disappear. The vortex structures almost fragment and break down into smaller structures, which is consist with the anisotropy observed in the Lumley's triangle map. Moreover, the strength of the spanwise shear layer indicated by the iso-surfaces of normalized Q significantly decreases with the increasing angle of inclination, and thus its contribution to the drag force almost disappears. The amount of the cross-spanwise vortices also reduces, and their orientation tends to be tilted in the spanwise direction as illustrated on the xz-view in Fig. 34(b). Although the cross-spanwise vortices still make some positive contribution to the drag force, due to their low spatial density and a stronger negative contribution around the stagnation point, the overall contribution of $C d n ω x y$ becomes negative, as observed in Fig. 33.

The present study utilizes large eddy simulations to investigate the three-dimensional effects of wake flow behind inclined circular cylinder. Initially, a convergence study is conducted at a Re = 3900, aiming to determine the most appropriate grid and temporal resolution. Following the convergence studies, the resultant data from simulations of flow past a vertical cylinder are validated with existing published data. This comparison reveals a good agreement between the present simulations and the published results, in terms of both cylinder pressure and velocity distribution in the wake region. Subsequent investigations involve simulating the flow past inclined cylinders with varying angles of inclination. To simulate the flow past inclined cylinders, the inlet condition is modified to achieve an oblique incoming flow profile. The reliability of this method is further validated by comparing the obtained streamlines and velocity distributions with those in previously published studies. A parametric examination is then conducted, exploring four specific inclination angles of α = 0°, 30°, 45°, and 60° at Re = 3900. The validity of the independence principle (IP) is evaluated at different α. In order to explain the origins of discrepancies between the predictions made using IP and the present simulation results, a thorough analysis of the three-dimensional wake features is performed. The main conclusions of the present study are summarized as follows:

1. The vortex shedding frequencies normalized by u_n observed in four cases of inclination angle are similar, and their temporal variations are stable. It appears that IP can be used to predict the Strouhal number of the inclined cylinder flows at Re = 3900 reasonably well. However, both the drag and lift coefficients at α ≥ 30° normalized by u_n are smaller than those in the vertical case, indicating deficiencies of IP for predicting force coefficients. It is a consequence of simplifications of the IP method, which is essentially based on a two-dimensional flow and only considers the contribution of the velocities in the cross-spanwise 2D plane. However, the three-dimensional effect induced by the axial flow in the inclination cases cannot be neglected for larger angles of inclination (α ≥ 30°). The importance of three-dimensional effects is clearly illustrated by the evolution of the spanwise force coefficient and the magnitude of the spanwise wall shear stress with the angle of inclination.

2. The mean drag coefficient and the rms lift coefficient normalized by u_n are not decreasing linearly with the increasing α, which is related to the recirculation length L_rec of the wake flow. It increases from L_rec = 1.31 to L_rec = 1.69 with the angle increasing from α = 0° to 30°, reaches its maximum of L_rec = 1.84 at α = 45°, and is L_rec = 1.69 at α = 30°. With the center of the low-pressure zone moving away from the cylinder, the pressure difference between the front and back sides of the cylinder reduces. Moreover, the variation in vortices also has less influence on the force when L_rec increases. These lead to a decreasing $C d n ¯$ and $C l n rms$ from α = 0° to 45°. With the angle further increases to 60°, L_rec decreases, and thus $C d n ¯$ and $C l n rms$ are larger than those in the case of α = 45°.

3. The force partitioning analysis reveals that both the drag and lift forces on the cylinder are mainly induced by the vortex shedding behind the cylinder, regardless of the inclination angles. Furthermore, the lift force is found to be mainly affected by the vortices in the spanwise direction, while the origins and location of the dominant drag force contributor are highly affected by the angle of inclination. In the vertical case, the drag is mainly caused by the shear layer and the vortices in the spanwise direction. When the angle of inclination is α = 30°, the vortices are mainly in the cross-spanwise direction, and the strength of the spanwise shear layer also decreases. Thus, the cross-spanwise contribution to the drag is larger than that of the spanwise vortices. The strength of the spanwise shear layer further decreases, and the proportions of spanwise and cross-spanwise vortices are similar at α = 45°. Thus, the contributions of spanwise and cross-spanwise vortices to the drag force are of similar magnitudes in the case of α = 45°. When α = 60°, the contribution of the spanwise vortices to the drag increases and the contribution of the cross-spanwise vortices becomes negative. Since the drag force on an inclined cylinder is highly related to the orientations of the wake vortices, the techniques for modifying the drag on an inclined cylindrical body can be proposed and optimized by modifying the spatial characteristics of the wake vortices in specific directions based on the present conclusion.

4. The spanwise length of the vortices at the centerline is approximately 0.63D in all cases of α = 0°, 30°, 45°, and 60°, whether it is in the recirculation zone (x/D = 1) or out of the zone (x/D = 3). However, the local strong vortices are found to travel along the cylinder span in all investigated cases, manifested as stripes in the contours of both the amplitude of the vortices and their induced force coefficients. On the other hand, based on the instantaneous iso-surfaces of normalized Q, it is also found that the spatial density of both spanwise and cross-spanwise vortices decreases with the increasing inclination angle. The orientation of cross-spanwise vortices tends to be tilted, and the vortex structures in both spanwise and cross-spanwise directions fragment and break down into small structures at α = 60°. Moreover, the strength of the vortices is also found to decrease with the increasing angles of inclination according to the amplitude of ω_y. For the characteristic shape of vortices, the vortices are gradually stretched to the prolate shape with the increasing inclination angle, and the vorticity fluctuation is more pronounced in only one single direction, especially for large inclination angles of α = 45° and 60°. These results prove that the vortices display significant anisotropy in the far wake region in the investigated inclination cases.

This study was supported with computational resources provided by UNINETT Sigma 2—the National Infrastructure for High Performance Computing and Data Storage in Norway under Project No. NN9372. The first author also acknowledges the support of the China Scholarship Council (CSC) and the Cultivation Program for the Excellent Doctoral Dissertation of Dalian Maritime University (Grant No. 2022YBPY002).

The authors have no conflicts to disclose.

Gen Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Wenhua Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Marek Jan Janocha: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Guang Yin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Muk Chen Ong: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The detailed results of the convergence study are presented in this section. The convergence test for the mesh is first conducted. Figure 35 shows the pressure distribution on the cylinder using coarse, medium, and fine meshes of M1, M2, and M3, where the medium and fine meshes show a good agreement. Figure 36 shows that the predicted wall shear stress along the cylinder, and the obtained separation points are almost the same using all three meshes. The results of the streamwise velocity component in Fig. 37 indicate that the medium mesh M2 and the fine mesh M3 capture the same variation in the wake flow field and further obtain similar recirculation lengths. Figure 38 shows the Reynolds stress components at different locations of x/D obtained by using the three mesh schemes, where the turbulence characteristics represented by velocity fluctuations are in good agreement by the medium and fine meshes. Based on those results, it can be concluded that M2 has obtained convergence and thus it is used for the following convergence study for the time step.

The results of the convergence test for the time step are presented in the following figures: Figures 39 and 40 indicate that the pressure and the wall shear stress along the cylinder are not affected by the variation in the time step. Figure 41 shows the streamwise velocity distribution along the centerline y = 0, and Fig. 42 is the velocity fluctuations. The results of T1, T2, and T3 are also in good agreement. Therefore, the mesh scheme M2 and the time step T2 are applied for the rest of the parametric studies considering the balance between the computational cost and the accuracy.