2.1 Flow configuration

We consider a quarter-ring curved cylinder with straight extensions appended to both of its ends. The diameter of the cylinder is denoted  $D$ . The radius of the ring, i.e. radius of the centreline of the curved part of the cylinder, is $R=12.5D$ . A perspective view of the geometry, together with a not-to-scale sketch of the computational domain, is shown in figure 1. The main features of this wake flow, as learned from earlier low- $Re$ investigations (see Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ; Shang et al. Reference Shang, Stone and Smits2018) are also sketched. The Reynolds number $Re$ is defined based on the free-stream velocity $U_{0}$ and cylinder diameter  $D$ , i.e. $Re=U_{0}D/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}$ is the kinematic fluid viscosity.

The coordinate system is defined with $x$ in the streamwise direction, $y$ in the cross-flow direction and $z$ in the vertical direction. The origin $O$ of the coordinate system is at the centre of curvature (see figure 1). The computational domain has lengths $L_{x}$ , $L_{y}$ and $L_{z}$ in the corresponding directions. In order to eliminate the free-end effects in the simulations, we append straight extensions at both ends of the curved part. They are referred to as vertical extension and horizontal extension, and their lengths are denoted $L_{v}$ and $L_{h}$ , respectively. Both extensions are important for the present configuration, as discussed by Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ). Moreover, both the straight extensions have a clear real-life significance, because curved cylinder structures, as a part of either a catenary riser or a deep-sea pipeline, never appear isolated.

2.2 Governing equations and boundary conditions

In a DNS study, the governing equations are the mass conservation equation (2.1) and the time-dependent 3-D full Navier–Stokes (N–S) equation (2.2) for incompressible flows:

(2.1) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}=\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D70C}$ denotes the constant fluid density, and $\unicode[STIX]{x1D735}$ is the gradient operator. The gravity force is omitted in (2.2). We apply the following boundary conditions in the simulations.

1. (i) At the inlet: uniform incoming velocity $\boldsymbol{u}=(u,v,w)=(U_{0},0,0)$ .

2. (ii) At the outlet: Neumann boundary condition $(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x=0)$ and zero pressure ( $p=0$ ).

3. (iii) At the other four boundaries of the domain: free-slip boundary conditions.

4. (iv) At the surface of the cylinder: no-slip and impermeability wall boundary condition.

2.3 Numerical methods

The governing equations (2.1) and (2.2) are directly solved without introducing any turbulence model. A well-verified second-order finite-volume direct numerical simulation/large-eddy simulation (DNS/LES) code MGLET (Manhart, Tremblay & Friedrich Reference Manhart, Tremblay, Friedrich, Jenssen, Kvamsdal, Andersson, Pettersen, Ecer, Periaux, Satofuka and Fox2001) is utilized to conduct all the simulations. In MGLET, equations (2.1) and (2.2) are discretized on a 3-D staggered Cartesian mesh. The discretized equations are integrated in time with Williamson’s third-order low-storage Runge–Kutta scheme (Williamson Reference Williamson1980), while pressure corrections are achieved by Stone’s strongly implicit procedure (SIP) (Stone Reference Stone1968). The representation of the cylindrical geometry inside the Cartesian mesh is accomplished by a direct-forcing immersed boundary method (IBM). The details of this IBM implementation and its validation can be found in Peller et al. (Reference Peller, Le Duc, Tremblay and Manhart2006). The code has recently been used for DNS/LES of the same curved cylinder configuration (Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ) as well as various other flow configurations (e.g. Gallardo et al. Reference Gallardo, Pettersen and Andersson2014a ; Tian et al. Reference Tian, Jiang, Pettersen and Andersson2017; Strandenes et al. Reference Strandenes, Jiang, Pettersen and Andersson2019).

MGLET uses a multi-level 3-D Cartesian mesh, and mesh generation is based on a zonal grid algorithm (Manhart Reference Manhart2004). The overall mesh consists of various numbers of cubic grid boxes (grids) at different levels. Each grid consists of $N\times N\times N$ uniform Cartesian cells, regardless of the size of the grid. In other words, every single cell has an ideal aspect ratio of 1 : 1 : 1. The grids are otherwise distributed in an unstructured manner, such that local refinement can be achieved with the zonal grid algorithm. For each refinement, the parent-level grid is split into eight equal child-level grids. Normally, the most-refined cells are applied to regions of interest, because full DNS resolution over the whole computational domain is unaffordable in wake simulations. The regions of interest are selected based on certain criteria, both manually and automatically. For the present simulations, we ensure that all regions with time-averaged vorticity magnitude above $0.1U_{0}/D$ are refined (automatically). In addition, the near wake (including the boundary layers) is resolved with DNS resolution down to $x/D=16.5$ , and with the second finest resolution down to $x/D\approx 23.5$ (manually fixed). This mesh generation method offers the possibility to refine only the flow regions of interest without sacrificing either grid quality or wasting any cells. A slice of the mesh structure of case VE-16D taken in the geometrical symmetry plane (the $(x,z)$ -plane at $y/D=0$ ) is shown in figure 2 to illustrate the zonal grid algorithm.

Figure 2.

A slice of the VE-16D mesh (see table 1) in the symmetry plane. Each square represents one grid box that consists of $N\times N\times N$ uniform Cartesian cells, regardless of its size. The coarsest four levels (out of in total six levels) of grids are marked with numbers.

Table 1.

Mesh parameters of all simulations in the present study. The mesh for case Fine and case VE-6D is the same. It is used for both the grid study and the vertical extension study, and therefore has two labels to ease the discussions. The case Coarse has five levels of grids, and the other cases all have six levels of grids. For all cases in this table, $Re=3900$ and $L_{h}/D=10$ . From the data in this table, one sees that the largest blockage in the $y$ direction is less than 3 %.

In table 1, we summarize the computational parameters of all simulations in the present study. Five DNS are required to address two computational issues: grid study with cases Coarse, Medium and Fine, and vertical extension study with cases VE-6D, VE-12D and VE-16D. The mesh for case Fine is also used in the vertical extension study (VE-6D) and therefore has two different labels for the convenience of discussions. In-depth investigations of the wake dynamics will be based on the results from case VE-16D. In all simulations, the time step $\unicode[STIX]{x0394}t$ is chosen to ensure that the maximum Courant–Friedrichs–Lewy (CFL) number is smaller than 0.5. It is worth mentioning that the (multi-level) mesh topology of Coarse, Medium and Fine meshes are kept the same, but the minimum cell size $\unicode[STIX]{x1D6E5}_{c}/D$ is varied to consider different resolutions in the grid convergence study.

2.4 Grid convergence and statistical convergence

Although it is difficult to find directly comparable experimental results for the present configuration in the literature, we did careful validations on straight circular cylinder flows with the same code. These comparisons were presented in Gallardo et al. (Reference Gallardo, Pettersen and Andersson2014a ) and Strandenes et al. (Reference Strandenes, Pettersen, Andersson and Manhart2017), for example, and serve as a good basis for the present work. We still need a systematic mesh study.

Three meshes with different grid resolutions ( $\unicode[STIX]{x1D6E5}_{c}/D$ ) were considered to investigate grid convergence, all with the shortest vertical extension $L_{v}/D=6$ . One notices that the domain size is not fixed for the three grid study simulations, because $L_{x}$ , $L_{y}$ and $L_{z}$ are not necessarily integers in our mesh generation strategy. On the other hand, they must fit to an integer multiple of the edge length of the coarsest grid box, i.e. following

(2.3) $$\begin{eqnarray}L_{i}=\text{integer}\times N\times \unicode[STIX]{x1D6E5}_{c}\times 2^{N_{L}-1},\quad i=x,y,z,\end{eqnarray}$$

where $N$ is the number of cells on one edge of the grid box, $N_{L}$ is the level of grids in a mesh and $\unicode[STIX]{x1D6E5}_{c}$ is the minimum grid cell size. The case Coarse has $N_{L}=5$ , while the other four cases have $N_{L}=6$ . This is also demonstrated in figure 2.

Table 2 includes the body force coefficients obtained from all simulations. The drag coefficients $C_{Fx}$ are time-averaged values, while the side force coefficient $C_{Fy}$ is the root-mean-square (r.m.s.) value because we have zero mean side force from the simulations. Three drag coefficients are listed in table 2: $\langle C_{Fxc}\rangle$ is the mean drag coefficient of the curved part of the cylinder, $\langle C_{Fxv}\rangle$ is of the vertical extension part and $\langle C_{Fxt}\rangle$ is the drag coefficient of both parts. The drag coefficients are normalized as

(2.4) $$\begin{eqnarray}\langle C_{Fxj}\rangle =\frac{\langle F_{xj}\rangle }{0.5\unicode[STIX]{x1D70C}DL_{pj}U_{0}^{2}},\quad j=c,v,t,\end{eqnarray}$$

where the subscript $c$ stands for the curved part, $v$ for the vertical part and $t$ for the curved plus vertical parts. The r.m.s. lift coefficient is normalized in the same way. Notice that the projection length $L_{pj}$ (vertical length of the cylinder’s projection into the $(y,z)$ -plane) varies for different parts, as listed in table 2.

Table 2.

Calculated force coefficients for the curved part, vertical part and both parts, respectively. The drag coefficients are defined in (2.4). Subscript $c$ stands for curved part, $v$ for vertical part and $t$ for curved plus vertical parts;  $L_{pj}$ stands for the projection area of the corresponding parts into the $(y,z)$ -plane; and  $C_{Fyt}$ is the r.m.s. value.

The data in table 2 show converging trends of all force coefficients when the mesh is gradually refined. Besides, in figure 3, we plot the time-averaged streamwise velocity $\langle u\rangle /U_{0}$ and the time-averaged pressure coefficient $\langle C_{p}\rangle$ (2.5) along a sampling line that follows the cylinder span in the symmetry plane, with $h/D=0.4$ apart from the cylinder:

(2.5) $$\begin{eqnarray}\langle C_{p}\rangle =\frac{\langle p\rangle -p_{0}}{0.5\unicode[STIX]{x1D70C}U_{0}^{2}},\end{eqnarray}$$

where $\langle p\rangle$ is the time-averaged pressure and $p_{0}$ is the pressure at the inlet boundary. A sketch depicting the location of the sampling line is included in the lower left corner of figure 3.

Figure 3.

Time-averaged pressure coefficient $\langle C_{p}\rangle$ and streamwise velocity $\langle u\rangle /U_{0}$ plotted along a sampling line. This sampling line is located in the symmetry plane, and $h/D=0.4$ apart from the cylinder, indicated by the dashed line in the lower left. Results from all three grid study simulations are collected. The abscissa is divided into two parts: the curved part is measured by angle $\unicode[STIX]{x1D703}$ (defined in figure 1), while the straight vertical part is measured by height $z/D$ . The length scales in these two parts are not necessarily the same; $L_{v}/D=6$ .

Flow around the horizontal extension part has been thoroughly discussed in Jiang et al. (Reference Jiang, Pettersen and Andersson2018a ), so in this paper we focus on flow around the curved and vertical extension parts. Figure 3 shows that the Medium mesh gives close results to those from the Fine mesh, for both the velocity and pressure plots. In most of this figure, we barely see any differences between the Medium and Fine mesh results. The results from the Coarse mesh, however, deviate more in some parts, e.g. the negative peak values at $\unicode[STIX]{x1D703}\approx 65^{\circ }$ in both the velocity and pressure plots. Yet results from the coarsest mesh show the same trends as those from the other two meshes. Moreover, the location of the peaks are correctly captured even by the coarsest mesh. The force coefficients in table 2, together with the flow field plots in figure 3, show a clear tendency of grid convergence, and that the Medium mesh gives promisingly close solutions to the Fine mesh.

In order to examine whether the finest mesh gives DNS-quality resolution, we have carefully checked the local grid cell size versus the Kolmogorov length scale  $\unicode[STIX]{x1D702}$ . The results show that, in the finest grid level, the local grid cell size is always smaller than  $2\unicode[STIX]{x1D702}$ . This assures that the grid is sufficiently fine to resolve all essential velocity gradients.

A statistical convergence examination is also necessary because the results in §§ 3–5 will be presented based on both instantaneous and time-averaged flow results. The VE-16D case was simulated first for 525 time units ( $D/U_{0}$ ) to ensure that the flow is properly developed, and then continued for another $600D/U_{0}$ to collect the statistical data. A careful check has confirmed that the statistical results no longer vary after this long sampling time.

Based on the results presented in this section, we conclude that the Fine mesh (see table 1) is sufficiently good for reliable DNS simulations in the present study. This mesh has six levels of grids, with minimum cell size $\unicode[STIX]{x1D6E5}_{c}/D=0.00625$ . It has in total 11 714 grids, each of which has $56\times 56\times 56$ uniform Cartesian cells, resulting in a total of $2.06\times 10^{9}$ grid cells. For the vertical extension study (see table 1), we will apply the same grid resolution and mesh structure for cases VE-12D and VE-16D. One sees from table 1 that all three vertical extension study cases use more than two billion grid cells; the largest simulation VE-16D uses in total $2.91\times 10^{9}$ cells.

All simulations were performed on an SGI ALTIX ICE X SLES – 11sp3 cluster at NTNU. To run case VE-16D, we used 3200 processors (2 GB memory per processor), with the recent improved parallel efficiency of MGLET (Strandenes et al. Reference Strandenes, Pettersen, Andersson and Manhart2017), it took less than 2 s to march the simulation one time step forwards. We have run at least 750 000 time steps in the VE-16D simulation in order to properly collect the statistical data; therefore, this single case cost in total approximately 1.35 million CPU hours.

Figure 4.

Perspective view of the instantaneous wake obtained from the VE-16D simulation, taken at an arbitrary moment after the flow is fully developed. The wake structures are shown by the isosurface of $\unicode[STIX]{x1D706}_{2}=-2$ , and coloured by the instantaneous vorticity magnitude $|\unicode[STIX]{x1D714}|D/U_{0}$ . Here we use the transitional regime to indicate the region between the shedding and non-shedding regimes.

4.1 Flow regimes and vortex dislocations

An eye-catching feature of this wake flow is the distinct frequencies in different flow regimes, accompanied by massive vortex dislocations. This is easily observed in figure 5(a), where distinct flow structures are observed in different coexisting flow regimes. Slightly inclined alternating red (positive  $v$ ) and blue (negative  $v$ ) strips manifest themselves in the upper part ( $z/D>2$ ) of the snapshot, revealing regular von Kármán vortex shedding, with a modest oblique shedding angle. In the lower part of the snapshot in figure 5(a), however, the flow is more complex. In a certain area around $z/D\approx 0$ , the alternating velocity strips are distinctly wider and more tilted than in the upper part, indicating more oblique shedding with a longer period.

Figure 9.

(a) The power spectral density of all three velocity components at sampling point A, indicated in figure 5(a), whose coordinates are $(x/D,y/D,z/D)=(15,0,10)$ . (b) The same as in panel (a) but for sampling point B with coordinates $(15,0,-2)$ .

Figure 10.

The power spectral density of $v/U_{0}$ plotted over a vertical sampling line in the symmetry plane, located at $x/D=15.0$ . The $z/D$ value varies from $-6.0$ to 16.0.

Figures 9(a) and 9(b) show the power spectral density of the three velocity components at two selected sampling points in the symmetry plane, i.e. points A and B, respectively. The locations of these two sampling points are marked as large dots in figure 5(a), where we see that they are positioned in different flow regimes. Velocity signals are collected every second time step, giving a sampling period of $0.003D/U_{0}$ . In total 131 072 sampling values were collected to ensure long enough sampling time ( ${\approx}393.2D/U_{0}$ ). Welch’s method (Welch Reference Welch1967) was used to calculate the spectral density in figure 9(a,b). A Hanning window was employed with window width of 65 536 to reduce background artifacts in amplitudes. Figure 9(a) clearly shows a dominating frequency of $fD/U_{0}=0.21$ at point A, which is close to the parallel shedding frequency for a straight circular cylinder at the same $Re=3900$ (see e.g. Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). The dominating frequency $fD/U_{0}=0.09$ read from figure 9(b) is, on the other hand, distinctly lower than the parallel shedding frequency, which is in qualitative agreement with the observations in figure 5(a) that the vortex shedding in the lower wake has longer period. We note that in figure 9(b), the dominating frequency in $P_{uu}$ and $P_{ww}$ ( $fD/U_{0}=0.18$ ) is twice that in  $P_{vv}$ . This is because the sampling point is located in the symmetry plane.

Figure 10 gives a more comprehensive picture of the frequency differences in the wake, by plotting the power spectral density of $v/U_{0}$ over a vertical sampling line in the symmetry plane (the $(x,z)$ -plane at $y/D=0$ ) located at $x/D=15.0$ . Notice that we do not use a log scale in figure 10, and only show the spectra in the range $fD/U_{0}\leqslant 0.35$ , in order to focus on the interesting frequencies. In figure 10, we observe that the dominating frequency $fD/U_{0}=0.09$ in the lower wake ( $z/D\lessapprox 2.5$ ) is constant, while the dominating frequency in the upper wake experiences a mild gradual increase from $fD/U_{0}\approx 0.19$ to 0.21. A similar scenario was reported at $Re=400$ and 500 for the same geometrical configuration in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ).

An abrupt frequency change shows up at $z/D\approx 2.5$ (see figure 10) along the sampling line (at $x/D=15$ ). This is consistent with the snapshot in figure 5(a), where vortex dislocations are observed in the area indicated by a dashed ellipse. The mechanism behind vortex dislocations is the phase difference between neighbouring vortices (Williamson Reference Williamson1996). In the wake behind various cylinder configurations, the phase differences mostly originate either from spanwise non-uniformities (see e.g. Tian et al. Reference Tian, Jiang, Pettersen and Andersson2017) or from end effects (see e.g. Williamson Reference Williamson1989). In our case, we clearly observe from figure 5(a) that dislocations are closely related to the strong oblique shedding in the lower wake. This has been accepted to be a result of the strong axial flow in the concave curved cylinder wake (see e.g. Miliou et al. Reference Miliou, de Vecchi, Sherwin and Graham2007; Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ; Shang et al. Reference Shang, Stone and Smits2018). The strong axial flow, as a representative characteristic of this wake, essentially originates from the curved span of the cylinder (Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ). Therefore, although the dislocations in the present case only show up behind the vertical extension ( $z/D>0$ ) whose span is uniform, their origin is still the spanwise non-uniformity of the curved cylinder. Nevertheless, due to the relatively large frequency difference in the present case, the dislocations become severe in this area. In figure 5(a), we also observe that the dislocations happen at different $z/D$ positions as $x/D$ varies. The animation in supplementary movie 1 (available at https://doi.org/10.1017/jfm.2019.648) presents a sequence of similar snapshots as figure 5(a), from which the dynamics can be more clearly understood.

4.2 New characteristics and perspectives of the axial flow

Strong axial flow in the concave curved cylinder wake at low $Re$ has been reported and explained in previous studies (see Miliou et al. Reference Miliou, de Vecchi, Sherwin and Graham2007; Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ; Shang et al. Reference Shang, Stone and Smits2018), and considered as the main reason why oblique shedding and various flow regimes coexist in the flow. However, by plotting the time-averaged axial velocity $\langle u_{ax}\rangle /U_{0}$ along several lines in the near wake (figure 11), we observe some distinct and noteworthy features compared to our previous knowledge at lower $Re$ . Earlier in figure 8, we already observed a high vertical velocity $\langle w\rangle$ in the near wake. Yet, since the cylinder span is curved, the streamwise velocity $\langle u\rangle$ also contributes to the actual axial velocity. In figure 11 $\langle u_{ax}\rangle$ is therefore calculated as $u_{ax}=u\cos \unicode[STIX]{x1D703}+w\sin \unicode[STIX]{x1D703}$ .

Figure 11.

Time-averaged axial velocity $\langle u_{ax}\rangle /U_{0}$ (solid lines) and streamwise velocity $\langle u\rangle /U_{0}$ (dashed lines) plotted over four sampling lines in the symmetry plane along the cylinder span. Results from simulation VE-16D. The sampling lines are arranged in a similar way as sketched in figure 8, where $h/D$ denotes the distance from the cylinder surface.

We note two distinct new phenomena in figure 11. First, two peaks ( $P_{1}$ and $P_{2}$ ) can be spotted in the $\langle u_{ax}\rangle$ distributions, while only one peak (corresponding to  $P_{1}$ ) was reported in the lower- $Re$ flows (see figures 5 and 7 in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b )). The physics behind peak $P_{1}$ has been well understood from earlier studies (e.g. Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ) and attributed to the generation and development of a vortex sheet pair. However, the physics behind the newly observed peak $P_{2}$ is unclear and will be investigated in this subsection. The previously known peak $P_{1}$ has $\langle u_{ax}\rangle$ values comparable in magnitude to the incoming velocity  $U_{0}$ , both at $Re=3900$ and at lower $Re$ . These are already high axial velocity values. However, the $\langle u_{ax}\rangle$ values at  $P_{2}$ , as read from figure 11, are even higher, and reach up to  $1.4U_{0}$ .

Secondly, we observe high streamwise velocity $\langle u\rangle$ values at approximately the same spanwise location as $P_{2}$ for $\langle u_{ax}\rangle$ , i.e. at $\unicode[STIX]{x1D703}\approx 65^{\circ }$ . If we recall the $\langle w\rangle$ results in figure 8, we also notice a local peak in the $\langle w\rangle$ distributions at roughly the same location. Neither of them were observed in the lower- $Re$ flows of the same configuration (e.g. Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ). The second peak in figure 8 is more easily overlooked due to the continuously high $\langle w\rangle$ values (larger than  $U_{0}$ ) over a wider span compared to $\langle u\rangle$ and $\langle u_{ax}\rangle$ shown in figure 11.

The first peak $P_{1}$ is in the non-shedding regime, where the physical meaning behind the variations in $\langle w\rangle$ and $\langle u\rangle$ can be explained by the vortex sheet production and cylinder shelter effect, respectively, as addressed in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ) for lower- $Re$ flows. It is therefore reasonable to further explain $P_{1}$ in $\langle u_{ax}\rangle$ as a joint effect of $\langle w\rangle$ and $\langle u\rangle$ . However, this does not necessarily mean that $\langle u_{ax}\rangle$ is a result of $\langle u\rangle$ and $\langle w\rangle$ ; they are just different measures of the near-wake velocity in different coordinate systems. In fact, it becomes confusing if we try to explain $P_{2}$ in $\langle u_{ax}\rangle$ in the same way as we did for  $P_{1}$ . Although we observe separated peaks for $\langle u\rangle$ (in figure 11) and $\langle w\rangle$ (in figure 8) at $\unicode[STIX]{x1D703}\approx 65^{\circ }$ , they are somewhat opposed to the general understanding of the wake flow, as explained in the following.

The production process of the vortex sheets described in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ) ends as the vortex sheets start to shed. This is indicated by the small drop in $\langle w\rangle$ between the two peaks (see figure 8). An intuitive scenario is like what we observed in the low- $Re$ wakes, i.e. this drop continues for all three velocity components $\langle u\rangle$ , $\langle w\rangle$ and $\langle u_{ax}\rangle$ (see figure 12 in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b )). But at $Re=3900$ , all three velocities increase again. The peak of $\langle u\rangle$ at $\unicode[STIX]{x1D703}\approx 65^{\circ }$ is especially confusing. We note that $\unicode[STIX]{x1D703}\approx 65^{\circ }$ (which corresponds to $z/D\approx -6$ in the Cartesian coordinate system) is already in the shedding regime, because we have detected a prominent dominating frequency $fD/U_{0}=0.09$ at $z/D=-6$ (see figure 10). Consequently, we would expect a recirculation bubble in the time-averaged flow field. Considering that our sampling locations are all very close to the cylinder ( $h/D\leqslant 0.4$ ), and a conventional recirculation area behind a circular cylinder at $Re=3900$ can reach approximately $1.5D$ behind the cylinder (Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008), we should have expected negative $\langle u\rangle$ here instead of a local peak value even higher than $0.5U_{0}$ . This peak value in $\langle u\rangle$ is therefore rather abnormal. In other words, we need to seek the physics behind $P_{2}$ in the $\langle u_{ax}\rangle$ distribution (figure 11) from another point of view.

Figure 12.

Time-averaged base pressure $C_{pb}$ and stagnation pressure $C_{ps}$ distributions along the span of the cylinder. Pressure values are interpolated $h=0.015D$ apart from the base (stagnation) line. The spanwise location is indicated in different scales for the curved and vertical extension parts, as in previous figures. But $z/D$ values are additionally marked for the curved part in this figure. They are calculated based on the base pressure line, i.e. with radius $13.015D$ . This is to give a quick reference to the positions in the Cartesian coordinate system.

Figure 12 shows the time-averaged base pressure $C_{pb}$ and stagnation pressure $C_{ps}$ distributions along the cylinder span (except for the horizontal part). It is worth mentioning that, with IBM, smooth surface pressure distributions are difficult. This is a known shortcoming of IBM due to the way a curved surface is interpreted. The ‘surface pressure coefficients’ plotted in figure 12 are interpolated values at two lines $0.015D$ away from the base and stagnation lines, respectively. The distance $h=0.015D$ is chosen because it is slightly larger than the smallest cell’s diagonal ( $\sqrt{3}\unicode[STIX]{x1D6E5}_{c}<h=0.015D<1.5\sqrt{3}\unicode[STIX]{x1D6E5}_{c}$ , where $\unicode[STIX]{x1D6E5}_{c}=0.00625D$ ), such that we safely avoid wiggles possibly caused by cells directly cut by the solid surface, and still stay as close as possible to the surface.

In figure 12, we observe that the base pressure experiences a sharp drop in the range from $\unicode[STIX]{x1D703}\approx 55^{\circ }$ to $\unicode[STIX]{x1D703}\approx 65^{\circ }$ . At $\unicode[STIX]{x1D703}<55^{\circ }$ we are in the non-shedding regime where the wake is dominated by a counter-rotating vortex sheet pair. In a circular cylinder wake, it is well known that the base pressure reaches a minimum at the Reynolds number when the wake transits from steady to unsteady (see e.g. figure 1 in Roshko (Reference Roshko1993) and figure 4.12 in Zdravkovich (Reference Zdravkovich1997)). After the wake becomes unsteady, the base pressure increases with $Re$ in a certain range. A similar tendency is observed in figure 12 around point $Tr$ indicated therein. We can therefore reasonably assume that the vortex sheet pair transits from a steady non-shedding regime to an unsteady shedding regime at $\unicode[STIX]{x1D703}\approx 55^{\circ }$ .

It is worth mentioning that we can also observe a mild favourable pressure gradient at $\unicode[STIX]{x1D703}<40^{\circ }$ in figure 12. However, this favourable pressure gradient does not lead to acceleration of the flow for two reasons. First, in the near wake when viscous effects are strong, the Bernoulli equation is not necessarily valid. Second, even if the Bernoulli equation is valid, this favourable pressure gradient is small compared to that around $\unicode[STIX]{x1D703}\approx 60^{\circ }$ . Therefore, it could not lead to a sufficient velocity increase to compensate for the velocity reduction induced by the shelter effect of the cylinder, as explained in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ).

Figure 13.

Instantaneous $\unicode[STIX]{x1D714}_{z}$ snapshots plotted in three horizontal planes at $z/D=-9.0$  (a),  $-7.0$  (b) and $-6.0$  (c). The locations of the three horizontal planes are marked as $S_{1}$ , $S_{2}$ and $S_{3}$ on the base pressure distribution in figure 12. Panels (a–c) use the same colour scale, as shown to the upper right. Time is $t=1125D/U_{0}$ .

More detailed flow visualizations are presented in figure 13(a–c), where the instantaneous vertical vorticity component $\unicode[STIX]{x1D714}_{z}D/U_{0}$ is plotted in three selected horizontal planes (referred to as $z$ -cuts in this paper) at $z/D=-9.0$ , $-7.0$ and $-6.0$ , respectively. The corresponding locations of these three horizontal planes are marked with black dots in figure 12, and respectively denoted as $S_{1}$ , $S_{2}$ and  $S_{3}$ . The plane in figure 13(a) is in the non-shedding regime. Here, we clearly see that the wake is laminar and symmetric behind the elliptical cross-section. Figure 13(b) is the plane at $z/D=-7.0$ , corresponding to $S_{2}$ ( $\unicode[STIX]{x1D703}=57.5^{\circ }$ ) in figure 12, just after the transition point $Tr$ . From figure 13(b), we observe that the wake just starts to become unsteady, the two vortex sheets tend to oscillate in the $y$ direction, but shedding barely happens. Further checks reveal that in the plane $z/D=-7.0$ , only a small part of the vortex tail is shed into the wake, while the two main vortices with concentrated  $\unicode[STIX]{x1D714}_{z}$ , representing the vortex sheet pair in the plane, remain attached to the cylinder. An animation of the wake dynamics in the $z/D=-7.0$ plane is provided as supplementary movie 2, which gives a better impression of the dynamics. By comparing figure 13(a) and 13(b), our assumption that point $Tr$ (in figure 12) is where the vortex sheet pair transits from steady to unsteady is confirmed.

Figure 13(c) shows $\unicode[STIX]{x1D714}_{z}$ contours in the $z/D=-6.0$ plane, corresponding to $\unicode[STIX]{x1D703}=62.5^{\circ }$ (and point  $S_{3}$ ) in figure 12. This is close to where the $C_{pb}$ curve reaches its minimum and suction is maximum. We note that figure 13(c) shows a similar pattern as the primary instability in a straight cylinder laminar wake. It is therefore surprising to see very clear turbulence in the downstream region. An explanation of this unexpected scenario will be presented in § 4.3. One can also get an intuitive impression of the dynamics in the $z/D=-6.0$ plane via the animation in supplementary movie 3. Moreover, we have noticed earlier from figure 5(a) that the flow is less recognizable in the area around $z/D=-6$ , namely the transitional regime. All these results strongly suggest that the transitional regime is special; the flow dynamics therein is very complicated and perhaps detailed attention has not been paid to this in earlier studies. This will be discussed in more detail in § 4.3.

In figure 14 we outline a 3-D picture of the high-axial-flow region by plotting the isosurface of $\langle w\rangle /U_{0}=0.6$ coloured by $\langle u\rangle /U_{0}$ , from which we see that this region is confined in the very near wake.

Figure 14.

Isosurface of $\langle w\rangle /U_{0}=0.6$ coloured by $\langle u\rangle /U_{0}$ , to give a 3-D perspective picture of the high axial flow we discuss.

From the results and discussions in this subsection, we learned that the strong favourable base pressure gradient along the span of the curved cylinder, in the range from $\unicode[STIX]{x1D703}\approx 55^{\circ }$ to $\unicode[STIX]{x1D703}\approx 65^{\circ }$ , provides a reasonable explanation for why the second peak $P_{2}$ shows up in the axial velocity distribution in figure 11. This second peak in the axial velocity distribution is a new phenomenon that has not been reported and investigated before.

4.3 One frequency – two mechanisms

As discussed in § 4.1, vortex dislocations are prominent in the present wake and can easily be identified as the ‘border’ between the parallel shedding and the oblique shedding regimes. In figure 5(a), we notice that this ‘border’ is not parallel to the $x$ direction, but is slightly tilted, due to the strong axial flow in the near wake. At $x/D=15.0$ , vortex dislocations occur at around $z/D=2.5$ . Below this $z$ position, the wake in the shedding regime is dominated by a single uniform low frequency $fD/U_{0}=0.09$ . This is clearly illustrated in figures 9(b) and 10. In the low- $Re$ concave curved cylinder wake studied earlier, this frequency is related only to the oblique shedding (see figure 10 in Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b )). However, at $Re=3900$ in the present study, we observe that the wake between the parallel shedding and non-shedding regimes is more complicated. In figure 13(c), we also observed an atypical shedding pattern. It is therefore important to explore the wake dynamics in this area in more detail.

Figure 15.

Four $\unicode[STIX]{x1D714}_{z}D/U_{0}$ snapshots plotted in a horizontal plane at $z/D=-2.0$ over one time period $T$ (determined by the corresponding frequency $fD/U_{0}=0.09$ ). This plane is in the oblique shedding regime. In this figure, $t_{0}\approx 1128D/U_{0}$ . All snapshots use the same colour scale.

Figure 15 shows four snapshots of the $\unicode[STIX]{x1D714}_{z}$ distribution in the $z/D=-2.0$ plane over one period of time  $T$ , which is determined by the dominating frequency $fD/U_{0}=0.09$ . With reference to figure 5(a), we know that the horizontal plane at $z/D=-2.0$ is in the oblique shedding regime. Despite the relatively long period, the snapshots in figure 15 outline a turbulent von Kármán type shedding, but not similar to that in the circular cylinder wake at the same $Re=3900$ (see e.g. Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). This is understandable because we are not in a parallel shedding regime. However, the relatively short shear layers and the turbulence close to the cylinder in figure 15 are apparently unexpected, because they are typical characteristics of a circular cylinder wake in the upper subcritical state, or more precisely in TrSL3 ( $Re>20\,000$ ) as referred to in Zdravkovich (Reference Zdravkovich1997). Reynolds number $Re=3900$ is otherwise considered to be in the TrSL2 state. One may claim that the borders between these two states are not strict, but considering that we are in an oblique shedding regime and the cross-section of the cylinder at $z/D=-2.0$ is still slightly elliptic (i.e. more slender than a circular cylinder), we should clearly expect the wake state to be closer to the lower boundary of the subcritical state TrSL1 rather than to the upper boundary towards TrSL3. This assumption is also supported by the fact that oblique shedding postpones the onset of the shear layer instability in a cylinder wake (Prasad & Williamson Reference Prasad and Williamson1997). Anyway, we could not find the shedding pattern in figure 15 to resemble typical TrSL1 flow visualizations (see e.g. figure 40 in Gerrard (Reference Gerrard1978)), meaning that the oblique shedding here is affected by other physics and thus becomes atypical. At lower $z$ positions, the shedding in a $z$ -cut (an $(x,y)$ -plane) becomes even more different from a typical von Kármán shedding pattern (as already observed in figure 13(c) at $z/D=-6$ ). This again suggests that the flow in this area is probably dominated by different dynamics, for which we need to seek explanations from the non-shedding regime.

The main wake structures in the laminar non-shedding regime are the rolled-up vortex sheets on the two sides of the curved cylinder. Similar vortex sheet structures are reported in various wakes, e.g. sharp-nosed cylinders at high incidence (Thomson & Morrison Reference Thomson and Morrison1971), and prolate spheroids at high incidence (Jiang et al. Reference Jiang, Gallardo, Andersson and Zhang2015, Reference Jiang, Gallardo, Andersson and Okulov2016), etc. A common feature of these wakes is the continuously inclined or curved separation line. When the vortex sheets separate from the object and shed into the wake, they naturally evolve into a counter-rotating helical vortex pair. More interestingly, this kind of helical vortex pair is mostly asymmetric and non-periodic. Zdravkovich (Reference Zdravkovich1997, p. 983) summarized, based on earlier research, that there may appear several such asymmetric helical vortex pairs along the span of an inclined cylinder. Each pair stays almost at a fixed position to form an alternating vortex street that can be observed in a cross-sectional plane. The helical vortex pair in an inclined prolate spheroid wake is also asymmetric: when the symmetry of the pair is lost, the wake randomly deflects to one side or the other and never turns back again, resulting in a substantial time-averaged side force acting on the spheroid, as comprehensively reviewed in Andersson, Jiang & Okulov (Reference Andersson, Jiang, Okulov and Gelfgat2019). In both situations, the asymmetric helical vortex pairs are mostly in the streamwise direction rather than in the spanwise direction. This leads us to explore the flow in vertical planes normal to the inflow direction, namely ( $y,z$ )-planes ( $x$ -cuts).

The time-averaged streamwise vorticity distribution in a ( $y,z$ )-plane located at $x/D=14.0$ was presented earlier in figure 7 but not discussed in detail. In figure 16, we show six snapshots of the instantaneous streamwise vorticity $\unicode[STIX]{x1D714}_{x}$ plotted in the same $(y,z)$ -plane to show the wake dynamics over one time period  $T$ . Notice that $T$ is determined by the same frequency $fD/U_{0}=0.09$ as in figure 15.

Figure 16.

Six $\unicode[STIX]{x1D714}_{x}D/U_{0}$ snapshots plotted in a vertical ( $y,z$ )-plane positioned at $x/D=14.0$ over one time period  $T$ (determined by the corresponding frequency $fD/U_{0}=0.09$ ). All snapshots are observed from a downstream position, and use the same colour scale. In this figure $t_{0}\approx 1125D/U_{0}$ .

The snapshot in figure 16(a) is arbitrarily chosen at $t=t_{0}\approx 1125D/U_{0}$ , from which we immediately identify the asymmetric counter-rotating helical vortex pair as marked by $L_{1}$ and  $R_{2}$ , respectively. Vortex $L_{1}$ has positive $\unicode[STIX]{x1D714}_{x}$ and rotates anticlockwise (figure 16 is observed from a downstream position), while vortex $R_{2}$ has negative $\unicode[STIX]{x1D714}_{x}$ and rotates clockwise. Vortex $L_{1}$ is larger in size and is located at a higher $z$ position; $R_{2}$ is located at a lower $z$ position, and is smaller in size, but $\unicode[STIX]{x1D714}_{x}$ is very concentrated. In figure 16(b), i.e. after $0.25T$ , $R_{2}$ disappears and $L_{1}$ is heavily disintegrated. Meanwhile, a concentrated negative $\unicode[STIX]{x1D714}_{x}$ region ( $R_{1}$ ) emerges. It is interesting to notice the striking similarities between figure 16(b) in this paper and figure 4 in Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015); the latter is in the wake behind an inclined prolate spheroid. This might indicate that we observe the same type of vortical structures. However, we must notice that figure 16(b) is an instantaneous snapshot, while figure 4 in Jiang et al. (Reference Jiang, Gallardo, Andersson and Zhang2015) is a time-averaged plot. This major difference tells us that the vortices in these two wakes evolve differently with time.

In figure 16(c), at $t=t_{0}+0.375T$ , we clearly see two new concentrated asymmetric counter-rotating vortices $R_{1}$ and  $L_{2}$ . When half a period has passed, in figure 16(d), the positions of the two vortices are nearly perfectly mirrored from those in figure 16(a), and the signs of $\unicode[STIX]{x1D714}_{x}$ are switched. This process reverses itself in the second half-period, and eventually at $t=t_{0}+T$ , as shown in figure 16(f), the snapshot almost duplicates that in figure 16(a). An animation showing the vortex motions in this plane is available as supplementary movie 4. Now we can explain the $\langle \unicode[STIX]{x1D714}_{x}\rangle$ plots in figure 7 better. The two pairs of counter-rotating vortices in the time-averaged flow field reveal a fascinating periodic motion in the transitional regime, and the two pairs are $L_{1}$ versus $R_{2}$ and $R_{1}$ versus  $L_{2}$ . We can now address the abnormal scenario shown in figure 13(c), which is due to the breakdown of the helical vortex pair. This turns out to be similar to that described in Jiang et al. (Reference Jiang, Gallardo, Andersson and Okulov2016). This is also evident if one observes the vortical structures in the corresponding regions in figures 4 and 6.

The discussions of figures 15 and 16 show two distinct mechanisms in the oblique shedding and transitional regimes, respectively. It is noteworthy that, in our previous low- $Re$ concave curved cylinder wake study (Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ), a transitional regime does not exist. From the experimental investigation, Shang et al. (Reference Shang, Stone and Smits2018) showed that streamwise vortex regimes may exist below the oblique shedding regime, in certain configurations. However, that regime is non-shedding, and therefore also different from the dynamics illustrated in figure 16. The transitional regime and the detailed dynamics discussed herein have, to the authors’ knowledge, never been reported before.

One remaining question is why these two distinct mechanisms occur with identical frequency. We speculate that the oblique shedding triggers the periodic motion of the asymmetric helical vortex pair in the transitional regime; meanwhile the helical vortex pair influences the oblique shedding. The interactions and balance between these two mechanisms eventually compromise to a single frequency. We give the following justifications for this speculation.

First, based on earlier research referred to in the discussion, we can reasonably assume that the asymmetric helical vortex pair which originated from the vortex sheets is intrinsically non-periodic. This assumption should at least be valid in the Reynolds-number range we consider here. In Thomson & Morrison (Reference Thomson and Morrison1971), the alternating asymmetric flow pattern remained steady up to $Re\sim 10^{4}$ , whereas in the inclined spheroid wake, we observed a mean side force up to $Re=4000$ (Strandenes et al. Reference Strandenes, Jiang, Pettersen and Andersson2019). However, in the present wake, the helical vortex pair directly meets the periodic oblique shedding, and the latter triggers the former to have a periodic motion.

Second, we notice that the single dominating frequency in this region is quite low ( $fD/U_{0}=0.09$ ) compared to what we learned from previous studies. If we compare it with the parallel shedding frequency in this wake, i.e. $fD/U_{0}=0.21$ read from figure 9(a), it gives $\arccos \,(0.09/0.21)\approx 65^{\circ }$ , which ideally can be considered as the oblique shedding angle. However, both Jiang et al. (Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ) and Shang et al. (Reference Shang, Stone and Smits2018) reported oblique shedding angles ${\leqslant}45^{\circ }$ in lower- $Re$ wakes, and both showed that the frequency in the oblique shedding region is directly related to the oblique shedding angle. A $65^{\circ }$ -inclined line is drawn in figure 5(a). By comparing the slope of this line with the inclination of the velocity strips, which essentially represent the inclination of the vortices, we find that they do not match perfectly. The oblique angle of vortices further downstream roughly follow this line, but the oblique angle of vortices close to the cylinder is observably smaller. In fact, at point B in figure 5(a), the inclination of the velocity strip is only approximately  $45^{\circ }$ , but we could not detect a higher shedding frequency corresponding to this oblique angle. These abnormal discrepancies reveal that the oblique shedding is also heavily influenced by the periodic asymmetric helical vortex pair.

5.1 Flow features in different wake regimes

From the discussions in § 4, one already gets an impression of this interesting wake, especially the different wake dynamics in different regimes. We now summarize these different features and outline a more comprehensive picture of the overall wake. A straightforward comparison of the distinct flow features at different spanwise locations is made in figure 17. Here, we choose three horizontal planes at $z/D=10.0$ , 0.0 and $-5.0$ , respectively, and plot instantaneous $\unicode[STIX]{x1D714}_{z}$ and time-averaged $\langle u\rangle$ distributions. The three planes are in qualitatively different flow regimes.

Figure 17.

Instantaneous vertical vorticity $\unicode[STIX]{x1D714}_{z}D/U_{0}$ and time-averaged streamwise velocity $\langle u\rangle /U_{0}$ plotted in three horizontal planes positioned in different flow regimes. The left column (a,c,e) are $\unicode[STIX]{x1D714}_{z}$ plots, and the right column (b,d,f) are $\langle u\rangle$ plots. The three horizontal planes are located at $z/D=10.0$ (a,b), 0.0 (c,d) and $-5.0$ (e,f), respectively. For instantaneous plots, $t=1125D/U_{0}$ , the same as in figure 13.

Figure 17(a,b) is taken at $z/D=10.0$ and in the parallel shedding regime. Both the snapshots of $\unicode[STIX]{x1D714}_{z}$ in figure 17(a) and the $\langle u\rangle$ distribution in figure 17(b) show characteristic turbulent von Kármán shedding behind circular cylinders. However, compared to the circular cylinder wake at the same $Re$ (see figure 4 in Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008)), the free shear layers in figure 17(a) are longer. Moreover, the recirculation region in figure 17(b) is also longer (see figure 16 in Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008)), meaning that, even in the parallel shedding regime, the flow is still not identical to that behind a straight circular cylinder at the same $Re$ (this will be further illustrated later in figure 18).

Figure 18.

Plots of the $\langle u\rangle /U_{0}$ and $\langle u^{\prime }u^{\prime }\rangle /U_{0}^{2}$ distributions along the wake centreline (  $y/D=0$ ) at different $z$ positions. The abscissa is scaled by $(x-x_{b})/D$ , where $x_{b}/D=\sqrt{13.0^{2}-(z/D)^{2}}$ denotes the $x$ coordinate of the rear of the cylinder at the corresponding $z$ position. Panels (a,b) show $\langle u\rangle /U_{0}$ results, while (c,d) show $\langle u^{\prime }u^{\prime }\rangle /U_{0}^{2}$ results. Ten $z$  positions are chosen from different flow regimes; see legends in the plots.

Figure 17(c,d) is taken at $z/D=0.0$ , namely at the intersection between the curved part and the vertical straight extension. This horizontal plane is located in the oblique shedding regime. The snapshot in figure 17(c) shows similar features as that in figure 15, also located in the oblique shedding regime ( $z/D=-2.0$ ). This means that the flow in this regime is more uniform, unlike in the transitional regime. In figure 17(d) we observe a recirculation area with negative $\langle u\rangle$ values in the near wake, but this area does not extend to the cylinder. In short, flow in the oblique shedding regime is more similar to the ordinary von Kármán shedding, although somewhat atypical, due to the influence from the transitional regime, as we argued in § 4.3.

Figure 17(e,f) is plots at $z/D=-5.0$ in the transitional regime. We see from figure 17(e) that the cross-flow motion in the near wake is strong; shear layers are barely visible. At around $x/D=15.0$ , we observe a large area with concentrated vorticity, which is quite distinguishable from other small-scale turbulent vortical structures around it. From our discussions in § 4.3, we know that this is one of the two helical vortices. Notice that, although the main vorticity component in the helical vortices is  $\unicode[STIX]{x1D714}_{x}$ , $\unicode[STIX]{x1D714}_{z}$ is also substantial because the vortex sheets are generated along the curved span. In figure 17(f) we do not observe any recirculation region because the time-averaged streamwise velocity $\langle u\rangle$ is positive all over the plane. This again tells us that the flow in the transitional regime is fundamentally different from the ordinary von Kármán shedding behind most cylindrical configurations. Furthermore, we recall that $z/D=-6.0$ in figure 13(c) is also in the transitional regime, yet the flow therein is clearly different from figure 17(e). In figure 13(c), the vortex sheets are still laminar in the near wake, but in figure 17(e), the flow becomes mostly turbulent. This suggests that the flow transition in the transitional regime is very abrupt and dramatic.

In figure 18, we plot $\langle u\rangle /U_{0}$ and $\langle u^{\prime }u^{\prime }\rangle /U_{0}^{2}$ distributions along the wake centreline ( $y/D=0$ ) at different $z$ positions. Plane $z/D=-8.0$ is in the non-shedding regime. The corresponding $\langle u\rangle$ curve in figure 18(a) indicates positive $\langle u\rangle$ in the very near wake, followed by a substantial recirculation region until $(x-x_{b})/D\approx 1.45$ . The minimum negative velocity at $z/D=-8.0$ is the largest in magnitude among all the results in figure 18(a,b), and reaches $\langle u\rangle =-0.46U_{0}$ . Downstream of the recirculation area, the $\langle u\rangle$ curve at $z/D=-8.0$ has the fastest recovery back to  $U_{0}$ , meaning that the wake in the $z/D=-8.0$ plane is more confined to areas close to the cylinder, and meets the characteristics of vortex sheets (see e.g. figure 13 a, which is also in the non-shedding regime). The $\langle u^{\prime }u^{\prime }\rangle$ curve at $z/D=-8.0$ in figure 18(c) is zero everywhere, as expected, because the flow is steady in the non-shedding regime.

The $z/D=-6.0$ and $-5.0$ planes are in the transitional regime. The $\langle u\rangle$ curve at $z/D=-6.0$ , in figure 18(a), shows a limited region of negative values, outside of which the values are all positive. Since this negative $\langle u\rangle$ region is small and relatively far behind the cylinder, and we already understand that flow in this regime is not dominated by von Kármán shedding, we hesitate to classify this region as a recirculation region. Anyway, such a distribution is totally different from that behind a straight circular cylinder. The $\langle u\rangle$ curve at $z/D=-5.0$ is even more atypical. It shows no negative values at all. Moreover, $\langle u\rangle$ values are relatively high ( ${>}0.6U_{0}$ ), which is in good agreement with the observations in figure 17(b). In figure 18(c), we observe that the $\langle u^{\prime }u^{\prime }\rangle$ distribution at $z/D=-5.0$ is higher than that at $z/D=-6.0$ over most of the range. These results again reveal the complexity and dramatic changes of the flow in the transitional regime.

We note that $z/D=-4.0$ is probably around the border between the transitional and oblique shedding regimes. Therefore, the flow regime to which $z/D=-4.0$ belongs is not obvious. We see, anyway, that the distributions of both $\langle u\rangle$ and $\langle u^{\prime }u^{\prime }\rangle$ at $z/D=-4.0$ in figure 18 are more similar to those at $z/D=-5.0$ . Nevertheless, since the flows in both the transitional and oblique shedding regimes are complex and heavily influenced by the periodic asymmetric counter-rotating helical vortex pair, it may not be of importance to identify a distinct border between them.

The four horizontal planes at $z/D\geqslant 3.0$ are all in the parallel shedding regime. We first see, from figure 18(b,d), that both $\langle u\rangle$ and $\langle u^{\prime }u^{\prime }\rangle$ distributions at these four planes are similar to those behind a straight circular cylinder at the same $Re$ (Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). Secondly, the curves at these four $z$ positions are close to each other, implying a higher degree of spanwise uniformity of the flow in this regime.

Figure 19.

(a) Recirculation length $L_{r}$ at various $z$ positions. The dashed horizontal line indicates a reference PIV result for a straight cylinder at $Re=3900$ , i.e. $L_{r}/D=1.51$ , from Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008). (b) Vortex formation length $L_{vf}$ at various $z$ positions. The dashed horizontal line indicates a reference PIV result for a straight cylinder at $Re=3900$ , i.e. $L_{vf}/D=0.87$ , from Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008).

The last two positions shown in figure 18, i.e. $z/D=-1.0$ and 1.0, are in the oblique shedding regime. We see from figure 18 that the $\langle u\rangle$ curves at these two $z$ positions resemble von Kármán shedding. However, the recirculation region at $z/D=-1.0$ is much shorter compared to that at $z/D=1.0$ . Moreover, the location of the highest peak of the $\langle u^{\prime }u^{\prime }\rangle$ curve at $z/D=-1.0$ is much closer to the cylinder compared to that at $z/D=1.0$ . This is reasonable after we have explained the distinct flow dynamics in the two neighbouring regimes. The oblique shedding regime in the present wake is relatively narrow compared to that in the lower- $Re$ wakes (Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ), as already observed in figures 4 and 5(a).

From the results in figure 18, we also obtain two commonly considered parameters in bluff-body wakes, namely the recirculation length $L_{r}$ and the vortex formation length  $L_{vf}$ . The former denotes the length of the recirculation region at the wake centreline, while the latter denotes the distance between the cylinder rear and the location of the highest $\langle u^{\prime }u^{\prime }\rangle$ value at the wake centreline. The results are plotted in figure 19 by collecting these two wake parameters at every integer $z$ position from $z/D=-8.0$ to 14.0 in the present wake. We note that $L_{r}$ in figure 19(a) is measured from $(x-x_{b})=0$ to where $\langle u\rangle$ turns from negative to positive values. This means that the small positive $\langle u\rangle$ region in the near wake in the non-shedding regime (see e.g. the $z/D=-8.0$ distribution in figure 18 a), caused by the strong axial flow, is not excluded from  $L_{r}$ . Moreover, as just discussed, we do not define $L_{r}$ in the transitional regime.

Figure 20.

Plots of the $\langle u\rangle /U_{0}$ distributions at three $x$ locations in the wake. (a) At $(x-x_{b})/D=0.5$ , namely $0.5D$ behind the cylinder rear. (b) At $1.0D$ behind the cylinder rear. (c) At $1.5D$ behind the cylinder rear. In each panel, distributions at four different $z$ positions are shown together for comparison: $z/D=-8.0,~-5.0,~-1.0$ and 4.0 are representative of the non-shedding, transitional, oblique shedding and parallel shedding regimes, respectively. The circles in panels (b) and (c) are reference PIV results in a circular cylinder wake at $Re=3900$ , taken at $(x-x_{b})/D=1.06$ and 1.54, respectively. The PIV results are from Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008).

From figure 19(a), we also observe the exceptionally large $L_{r}$ increment in the oblique shedding regime. From $z/D=-2.0$ to $z/D=2.0$ , $L_{r}/D$ increases from 0.73 to 2.40, i.e. by more than a factor of three. In the parallel shedding regime of the present wake, $L_{r}/D$ is stabilized at around 2.5. The very short $L_{r}$ at the beginning of the oblique shedding regime has been attributed to the influence from the transitional flow, as already discussed. However, we notice that $L_{r}$ in the parallel shedding regime in the present wake is exceptionally long, knowing that $L_{r}/D\approx 1.5$ is a typical value in a circular cylinder wake at $Re=3900$ (see table II in Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008)). In addition, we notice from figure 19(b) that $L_{vf}/D\approx 2.5$ in the present wake is remarkably longer compared to $L_{vf}/D\approx 0.9$ in the circular cylinder wake at $Re=3900$ (see Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). The reference values of $L_{r}$ and $L_{vf}$ taken from Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008) are depicted as horizontal dashed lines in figure 19(a) and 19(b), respectively.

The parallel shedding regime is directly influenced not by the periodic asymmetric helical vortex pair, but by the strong axial flow discussed earlier in § 4.2. We read from figure 11 that, at $z/D=0$ , $\langle u_{ax}\rangle$ in the near wake is still as high as $0.4U_{0}$ for all $h/D$ distances. If we assume that the wake behind the vertical extension in the present configuration is isolated, then there exists an upward vertical inflow of approximately $0.4U_{0}$ at its lower boundary. This is indeed a strong effect. We interestingly noticed similar values for both $L_{r}$ and $L_{vf}$ in a wall-bounded cylinder wake (see figure 8 in Strandenes et al. (Reference Strandenes, Pettersen, Andersson and Manhart2017)). These authors concluded that the no-slip boundary condition applied at both cylinder ends heavily influenced $L_{r}$ and $L_{vf}$ even when the cylinder is $24D$ long.

Figure 20 shows the flow features in different coexisting flow regimes from another aspect, by showing the $\langle u\rangle$ distribution at three $x$ locations, i.e. $0.5D$ , $1.0D$ and $1.5D$ behind the cylinder rear, and at four different $z$ locations (one in each flow regime). Reference values from the particle image velocimetry (PIV) test in Parnaudeau et al. (Reference Parnaudeau, Carlier, Heitz and Lamballais2008) are also included for comparison. We observe that the flow in the parallel shedding regime (i.e. $z/D=4.0$ ) shows the clearest change from a U-shaped to a V-shaped profile from figure 20(a–c), similar as in a circular cylinder wake. The other three regimes show different features. In the oblique shedding regime, i.e. $z/D=-1.0$ , the $\langle u\rangle$ profiles at all three $(x-x_{b})$ locations are V-shaped, reflecting the much shorter recirculation region. This agrees well with earlier discussions in this section. The wake in the non-shedding regime, represented by $z/D=-8.0$ profiles, is narrow but has the largest negative $\langle u\rangle$ . The wake in the transitional regime, represented by $z/D=-5.0$ , is wide. Interestingly, in figure 20(a), we observe that $\langle u\rangle$ is maximum at $y/D=0$ in the $z/D=-5.0$ plane. In other words, the $\langle u\rangle$ profile in the near wake of the transitional regime manifests a jet-like profile. This does not happen often in a wake flow, but has been described in the helical vortex behind an inclined spheroid (Jiang et al. Reference Jiang, Gallardo, Andersson and Okulov2016). This interesting jet-like profile thus becomes easier to understand since we have already concluded that the transitional regime is dominated by periodic asymmetric helical vortices.

Table 3.

Parameters and flow features in different flow regimes.

Based on the discussions in this section, we can summarize the features of the different flow regimes in the present wake as in table 3.

5.2 Analogies with other cylindrical structure wakes

In this subsection, we address the analogies proposed in § 1.3.

We first note that the constant elliptic cross-section cylinder analogy is not valid. In the non-shedding regime, we do observe a narrower wake, but as it comes to the transitional regime, the wake becomes very wide and the shedding frequency is dramatically lower compared to the parallel shedding frequency. This is contrary to the features of vortex shedding behind an elliptic cylinder with its major axis aligned with the incoming flow, where a narrower wake and higher shedding frequencies are expected.

Secondly, we note that the splitter plate analogy may not be valid either. The wake behind a circular cylinder with a splitter plate is characterized by longer recirculation region and lower shedding frequency compared to the circular cylinder wake, as mentioned in § 1.3. However, in the concave curved cylinder wake, although the axial flow is very strong, we could not observe similarities to the splitter plate wake. In the transitional regime where the axial flow is strongest, we do not even have recirculation in the wake, as revealed by figure 17(b) and figure 20. Moreover, we observe strong cross-flow interactions in the transitional regime, as shown in figure 17(a). This is totally opposed to what we would expect with a splitter plate. In the parallel shedding regime, the axial velocity is relatively low in magnitude, and we observe a longer recirculation region as discussed in association with figure 19. The shedding frequency is, however, equal to parallel shedding behind a circular cylinder at the same $Re$ , so this analogy still does not hold. The only region where the flow features agree with the splitter plate analogy is perhaps in the upper part of the oblique shedding regime, namely $1\leqslant z/D\leqslant 3$ , where the recirculation region is longer ( $L_{r}>1.5D$ ) and the shedding frequency is low ( $fD/U_{0}=0.09$ ). However, this is only a very limited area, and, as we have discussed, the oblique shedding regime is somehow a buffer area between the transitional and parallel shedding regimes. Therefore, we do not consider the splitter plate analogy to be valid in the present wake.

Last, but not least, the inclined cylinder analogy seems to be valid in the concave curved cylinder wake. As we move along the span from $\unicode[STIX]{x1D703}=0^{\circ }$ to $90^{\circ }$ , we first observe attached flow, and then steady symmetric vortex sheets ( $\unicode[STIX]{x1D703}<55^{\circ }$ ). The vortex sheets become unsteady as $\unicode[STIX]{x1D703}$ increases further. After entering the transitional regime, we have shown that the dominating wake dynamics is the periodic asymmetric helical vortex pair. Although this is not exactly the same as the steady asymmetric helical vortices behind an inclined cylinder at high angles of attack, we have shown that this helical vortex pair is intrinsically non-periodic but passively becomes periodic because of the oblique shedding. In the inclined cylinder case, the attack angle is fixed, so each of the four flow types can appear independently as the attack angle is changing. However, in the present curved cylinder configuration, the local attack angle continuously changes along the span; therefore, each type of flow is influenced by its neighbourhood. As we move along the cylinder and leave the transitional regime, we observe unsteady shedding at the end of the curved part. Therefore, along the curved cylinder span in the present wake, we have seen the coexistence of all four types of flow that have been documented to appear separately in an inclined cylinder wake as the angle of attack gradually increases. It is noteworthy that, although earlier studies on concave curved cylinder wakes (Jiang et al. Reference Jiang, Pettersen, Andersson, Kim and Kim2018b ; Shang et al. Reference Shang, Stone and Smits2018) all adopted the flow regime definitions from the inclined cylinder study (Ramberg Reference Ramberg1983), the asymmetric helical vortex pair, i.e. the transitional regime here, has never been documented in the concave curved cylinder case before. Therefore, we believe that this inclined cylinder analogy has not been perfect until now, when we clearly identify all four types of flow regimes in the present study.