3.1 The effect of $Re$ on $C_{D}$ and $St$

The dependence of $C_{D}$ and $St$ on $Re$ has been well documented for the polygonal cylinders of $N\leqslant 4$ and $N=\infty$ . This work will be focused on the polygonal cylinders of $N>4$ , for which the information on $C_{D}$ and $St$ is scarce in the literature. Table 1 lists the presently measured $C_{D}$ and $St$ at $Re=1.0\times 10^{4}$ . The data for $N\leqslant 4$ and $N=\infty$ are also included for the purpose of comparison. There is in general a reasonable agreement between the present data and those in Tropea et al. (Reference Tropea, Yarin and Foss2007). There is a significant discrepancy in $C_{D}$ and $St$ for the case of 3C, marked by #. This is because the cross-sectional shape of the cylinder in Tropea et al. (Reference Tropea, Yarin and Foss2007) is a $90^{\circ }$ wedge, different from the present equilateral triangle. The present $St$ for 2F is lower than that in Tropea et al. (Reference Tropea, Yarin and Foss2007), marked by an asterisk, but is closer to that (0.13) reported in Blevins (Reference Blevins1990).

Table 1.

The present $C_{D}$ and $St$ values obtained at $Re=1.0\times 10^{4}$ . Also included are the reported data in Tropea et al. (Reference Tropea, Yarin and Foss2007).

Figure 4 shows how $C_{D}$ and $St$ change as $Re$ varies from $1.0\times 10^{4}$ to $5.8\times 10^{4}$ , for all polygonal cylinders at both principal orientations, including the flat plate (2F) and the circular cylinder. The $C_{D}$ of the N12 and N16 cylinders declines rather rapidly from $Re\approx 3.0\times 10^{4}$ to $4.0\times 10^{4}$ and meanwhile the corresponding $St$ rises sharply. The observation points to a boundary transition from laminar to turbulent state. The effect of the transition on the changes of $C_{D}$ and $St$ can differ significantly when the cylinder changes its orientation. For example, $C_{D}$ decreases by 25 % and $St$ increases by 15 % from $Re\approx 2.4\times 10^{4}$ to $3.0\times 10^{4}$ for the 12C orientation, but the changes contract to 15 % and 5 %, respectively, for the 12F orientation. Furthermore, the $Re$ at which the transition starts to take place also varies from one orientation to the other. It can be inferred that the flow characteristics can be very different between the two orientations. On the other hand, $C_{D}$ and $St$ for $N\leqslant 8$ cylinders vary little, within $\pm 8\,\%$ and $\pm 4\,\%$ , respectively, over the entire $Re$ range, suggesting no change of the boundary layer state.

Figure 4.

Dependence of $C_{D}$ and $St$ on $Re$ for both principal orientations. (a)  $C_{D}$ for NC; (b)  $C_{D}$ for NF; (c)  $St$ for NC; (d)  $St$ for NF; the cases for the flat plate and the circular cylinder are also included, denoted as 2F and Cir., respectively, in (b) and (d).

Figure 5(a) summarises the variation in the critical Reynolds number $Re_{c}$ , at which the boundary layer transition from laminar to turbulent state occurs, with increasing $N$ for the NC cases. The N24 case from Tian & Li (Reference Tian and Li2007) and the circular cylinder case ( $N=\infty$ ) from White (Reference White2001) are also included. The $Re_{c}$ is defined at the middle of the rapidly falling $C_{D}$ region in figure 4(a). The $C_{D}$ values before and after the transition, i.e.  $C_{D\text{-}sub}$ and $C_{D\text{-}sup}$ , are also presented for each case. The NF cases are not shown since the transition effect on 12F is not very strong. Clearly, both $Re_{c}$ and $\unicode[STIX]{x0394}C_{D}=(C_{D\text{-}sub}-C_{D\text{-}sup})$ rise with increasing $N$ . The dependence of $Re_{c}$ on $N$ may be used to estimate $Re_{c}$ for the polygons that are not measured presently. For instance, the $Re_{c}$ of the 20C case is predicted to be about $7.0\times 10^{4}$ .

Figure 5.

(a) Dependence of the critical Reynolds number $Re_{c}$ (●), $C_{D\text{-}sub}$ (○) and $C_{D\text{-}sup}$ on the polygon side number $N$ , for $N\geqslant 12$ for the corner orientation. (b) Dependence of $Re_{c}$ on the roughness height $\unicode[STIX]{x1D700}/D$ ; ○: polygons, ▪: circular cylinders with roughness elements.

Given an adequately large $N$ , the corners of a polygonal cylinder may be considered to be wedge-shaped roughness elements evenly distributed on the surface of a circular cylinder with a radius $R_{i}$ . Denoting the distance between one corner and the polygon centroid as $R_{o}$ , the roughness height $\unicode[STIX]{x1D700}$ can be written as:

In an attempt to understand the positive correlation between $Re_{c}$ and $N$ shown in figure 5(a), $Re_{c}$ is plotted against $\unicode[STIX]{x1D700}/D$ in (b). One may recognise that the variation in $Re_{c}$ against $\unicode[STIX]{x1D700}/D$ for large $N$ polygonal cylinders agrees reasonably well both qualitatively and quantitatively with its counterpart for the rough-surfaced circular cylinders. It may be inferred that, given an adequately large  $N$ , $Re_{c}$ of the polygonal cylinder may be estimated from that of the rough-surfaced circular cylinder, subjected to a small uncertainty. Güven, Farell & Patel (Reference Güven, Farell and Patel1980) observed that larger roughness elements on the surface of a circular cylinder gave rise to a thicker and more retarded boundary layer. As a result, the transition to turbulence occurred at lower $Re$ . As the roughness elements diminished in size, the drop in $C_{D}$ from subcritical to supercritical regime became more pronounced. The result shown in figure 5, i.e. increasing $Re_{c}$ and $\unicode[STIX]{x0394}C_{D}$ with larger $N$ or smaller $\unicode[STIX]{x1D700}/D$ , is fully consistent with their observation. The result further points to an analogy between polygonal cylinders of large $N$ and roughened circular cylinders.

Figure 6.

Dependence of $C_{D}$ (a,b) and $St$ (c,d) on the polygon side number $N$ for $Re<Re_{c}$ (a,c) and $Re>Re_{c}$ (b,d). The transition occurs at $N=12$ , 16. Data points 9C and 10C in (b) are two examples of the predicted results. The dash-dot lines denote the indicative trend for the polygons of large $N$ . The data points of 2F in (a) and (b) are from Tropea et al. (Reference Tropea, Yarin and Foss2007) and from Blevins (Reference Blevins1990) in (c) and (d). The two $+$ symbols in (b) and (d) indicate the turning points of the indicative trend lines from $N=16$ to the circular cylinder.

Note that figure 5(b) includes only the cases of $N\geqslant 12$ . For $N\leqslant 8$ , the corner number of the polygon is so low that $(R_{o}-R_{i})$ becomes relatively large compared to the characteristic dimension $R_{i}$ of the polygon and cannot be approximated as roughness elements any more. Therefore, the analogy between the polygon corners and roughness elements is no longer valid.

3.2 Dependence on $N$ of $C_{D}$ , $St$ and $C_{L}^{\prime }$

Figure 6 presents the dependence on $N$ of the measured $C_{D}$ and $St$ for both orientations of the polygons at $Re=2.0\times 10^{4}$ and $5.0\times 10^{4}$ , which correspond to the flow before and after the transition, respectively, of the N12 and N16 polygons. One may note that at small $N$ the $C_{D}$ or $St$ values differ greatly between the two orientations. However, this difference diminishes as $N$ becomes large, regardless of the orientation. While $C_{D}$ in general tends to decline with increasing $N$ , $St$ rises. There is another interesting observation, that is, the NF and NC data points ( $C_{D}$ and $St$ ) appear to form a distorted Cartesian mesh pattern, as highlighted by the solid and dashed lines. It is worthwhile conducting an analysis of the mesh pattern, which can be of practical importance, i.e. predicting $C_{D}$ and $St$ for the polygonal cylinders not measured presently. Four solid lines can be identified for each $Re$ , viz.

1. (i) $\text{I}_{1}$ , $N=2n+1$ ; C oriented for odd $n$ ; F oriented for even $n$ ;

2. (ii) $\text{II}_{1}$ , $N=2n$ ; C oriented for even $n$ ; F oriented for odd $n$ ;

3. (iii) $\text{I}_{2}$ , $N=2n+1$ ; F oriented for odd $n$ ; C oriented for even $n$ ;

4. (iv) $\text{II}_{2}$ , $N=2n$ ; F oriented for even $n$ ; C oriented for odd $n$ .

On each of the four solid lines, the F- and C-orientated data points are separated from each other, provided that the untested cases (e.g.  $N=9,10$ ), which can be estimated from the available data, are also included. The estimation of the untested cases will be explained below. There are actually more dashed lines than those marked in figure 6. For example, 2F–3F–4F can also form a dashed line. Unlike the solid lines, the data points along each of the dashed lines are of either uniformly $F$ or uniformly $C$ orientation. This particular pattern is essentially connected to the geometry of the polygons near the lateral maximum width points (the top and the bottom point) and the position of the flow separation point if the transition takes place ( $N=12,16$ ), which will be discussed in § 3.4.

The pattern shown in figure 6 can be used to predict approximately $C_{D}$ and $St$ for the cases that are not tested in the current experiment. For instance, based on the characteristics of the solid lines listed from item (i) to item (iv) above, it can be predicted from figure 6(b) that case 10C should be on the solid line $\text{I}_{2}$ and can be determined at the intersection of $\text{II}_{2}$ and $N=10$ . Connecting points 7C, 8C and 10C smoothly forms a new dashed line as shown in figure 6(b), which displays a similar trend to the other two dashed lines. The intersection of this newly constructed dashed line and $N=9$ gives the position of 9C. Obviously, curve $\text{II}_{2}$ , if extrapolated, passes almost precisely point 9C. As a result, $C_{D}$ of 9C at $Re=5.0\times 10^{4}$ is estimated to be 1.25 and $C_{D}$ of 10C to be 1.40, with a caveat of a small uncertainty. Following the same procedure, we may also estimate from figure 6(d) that $St$ of $9\text{C}\approx 0.170$ and $St$ of $10\text{C}\approx 0.165$ at $Re=5.0\times 10^{4}$ .

It is more difficult to predict $C_{D}$ and $St$ for polygons of $N\geqslant 16$ , which requires an extrapolation of the known data. However, it is possible to predict a general trend of $C_{D}$ and $St$ for the tested $Re$ range. Recall that circular cylinders can be treated as $N=\infty$ . Figure 5 can be used to estimate the $Re_{c}$ of large $N$ polygons. Given $N=18$ , $Re_{c}$ is found to be approximately $5.0\times 10^{4}$ and a larger $N$ indicates an even higher $Re_{c}$ . We may thus deduce that the flow at $Re=2.0\times 10^{4}$ (figure 6 a,c) is in the subcritical regime, implying that the two points between $N=16$ and $\infty$ can be connected smoothly, as shown in figure 6(a,c).

At $Re=5.0\times 10^{4}$ (figure 6 b,d), however, the variation in $C_{D}$ and $St$ is more complicated once the transition has taken place for the polygon of large $N$ , say $N=18$ . There are two turning points in each $C_{D}$ and $St$ line, as marked by $+$ symbols in figure 6(b,d). As $Re_{c}$ is defined at the middle of a transitional $Re$ range, typically over $Re_{c}-1.0\times 10^{4}\sim Re_{c}+1.0\times 10^{4}$ . As such, the transition is likely to start at $Re\approx 4.0\times 10^{4}$ for an $N=18$ polygon. Thus $C_{D}$ and $St$ for $N=18$ polygon should follow the trend of the N12 and N16 polygons where the transition occurs, as indicated by the symbol $+$ at $N=18$ in figure 6(b,d). Note that $Re_{c}$ is even higher for $N>18$ . For example, as estimated in figure 5, $Re_{c}\approx 6.0\times 10^{4}$ for $N=19$ . Therefore, at $Re=5.0\times 10^{4}$ , the N19 cylinder is less likely to be in the transition, which implies that $St$ would drop and $C_{D}$ would increase markedly, compared to their counterparts at $N=18$ . Moreover, as $N$ increases, $C_{D}$ and $St$ will asymptotically approach their counterparts of the circular cylinder and their values will vary little because the subcritical flow condition remains unchanged when $N\rightarrow \infty$ . Therefore, another turning point $+$ at $N=19$ is predicted, which is connected smoothly to the value of $N=\infty$ in figure 6(b,d).

Figure 7 presents the dependence of the fluctuating lift coefficient $C_{L}^{\prime }$ on $N$ . In order to avoid the influence of the synchronisation between vortex shedding and the natural frequency of the cylinder, the $C_{L}^{\prime }$ measurement is limited to $U_{\infty }$ at which $f_{s}<80$  Hz. The data at $Re=1.6\times 10^{4}$ are illustrated. A number of observations can be made. Firstly, the present $C_{L}^{\prime }$ on the circular cylinder agrees very well with Norberg (Reference Norberg2003), which provides a validation for our measurements. Secondly, for either orientation, $C_{L}^{\prime }$ rises rapidly first with increasing $N$ , reaching the maximum at 4F and 6C, respectively, and then drops quickly before approaching the value of $N=\infty$ . Finally, $C_{L}^{\prime }$ varies greatly from one orientation to the other for $N<7$ but has the same value for both orientations for $N\geqslant 12$ , implying that $C_{L}^{\prime }$ becomes independent of the cylinder orientation for large $N$ . The result is consistent with the proposed analogy between the polygonal cylinder of $N\geqslant 12$ and the roughened circular cylinder.

Figure 7.

Dependence of the lift coefficient $C_{L}^{\prime }$ on $N$ . Note that the data of the circular cylinder from Norberg (Reference Norberg2003) are also included, which collapse almost completely with the present measurement.

3.3 Wake characteristics

In this section, a number of wake characteristic length scales are extracted from the time-averaged velocity fields obtained from the PIV data. The length scales are further used to define the wake Strouhal number and the wake drag coefficient.

Figure 8 uses the 5C case to illustrate the definitions of wake length scales, where asterisks denote normalisations by $D$ and/or $U_{\infty }$ . One important quantity is the flow separation angle $\unicode[STIX]{x1D703}_{s}$ , which is defined as the angle between $U_{\infty }$ and $\overline{U}_{sp}^{\ast }$ , where $\overline{U}_{sp}^{\ast }$ is the tangential velocity at the separation point along the separation streamline. For the 5C case, the two angles $\unicode[STIX]{x1D703}_{o}$ (the separation point location) and $\unicode[STIX]{x1D703}_{s}$ are related by:

where $\unicode[STIX]{x1D711}=(N-2)\unicode[STIX]{x03C0}/N$ . The general relation of $\unicode[STIX]{x1D703}_{s}$ to the geometry and the orientation is not so straightforward. Note that the separation angle is defined differently from Tropea et al. (Reference Tropea, Yarin and Foss2007), where $\unicode[STIX]{x1D703}_{o}$ is defined as the separation angle. The reason for this different definition and a more detailed discussion of $\unicode[STIX]{x1D703}_{s}$ will be given in § 3.4. The marked streamlines pass through the separation points, which are also the free streamlines closest to the centreline. The reversed flow zone, defined by $\overline{U}_{x}^{\ast }=0$ , is marked by the streamline passing through the wake stagnation point located at $x_{R}^{\ast }$ downstream of the polygon centroid $O$ . The maximum wake width $D_{w}^{\ast }$ is measured at $x_{w}^{\ast }$ downstream of $O$ . $\overline{U}_{w}^{\ast }$ is the tangential velocity at the maximum wake width point ( $x_{w}^{\ast },y_{w}^{\ast }$ ) along the streamline through the separation point.

Figure 8.

Definitions of various characteristic length and velocity scales in the dimensionless form for polygonal cylinder wake. Asterisks denote normalisations by $D$ and/or $U_{\infty }$ . Note that the free streamlines and reversed flow zone are drawn based on the PIV data obtained at $Re=1.0\times 10^{4}$ . The bottom streamline uses the mirror image of the top one due to the shadow of the cylinder (laser illumination comes from above).

Figure 9 presents the dependences of $x_{R}^{\ast }$ , $D_{w}^{\ast }$ , $x_{w}^{\ast }$ on $N$ , which are quite similar to each other. Not surprisingly, $x_{R}^{\ast }\approx 2x_{w}^{\ast }$ . Different orientations result in a difference in $x_{R}^{\ast }$ , $D_{w}^{\ast }$ or $x_{w}^{\ast }$ , especially for $4\leqslant N\leqslant 8$ . Even though the blockage ratios are the same for all the cylinders, a different degree in bluffness may also have an impact on the back pressure, the separation shear layers and hence the structure of the wake (Roshko Reference Roshko1955). For $N\geqslant 12$ , $D_{w}^{\ast }$ remains roughly constant and is almost the same as that in the circular cylinder wake, but the streamwise length scales $x_{R}^{\ast }$ and $x_{w}^{\ast }$ are both significantly smaller. Note that $Re$ in figure 9 is well below  $Re_{c}$ .

Figure 9.

The dimensionless characteristic length scales ( $x_{R}^{\ast },D_{w}^{\ast },x_{w}^{\ast }$ , defined in figure 8) as functions of $N$ , at $Re=1.0\times 10^{4}$ .

Based on the momentum equation, the drag on a bluff body can be calculated from its wake parameters (length and velocity scales) and independently on the geometry of the bluff body itself (Antonia & Rajagopalan Reference Antonia and Rajagopalan1990). Roshko (Reference Roshko1955) proposed that there exists a universal similarity for the vortex-shedding patterns and the wake structures among differently shaped bluff bodies. Once $(C_{p})_{b}$ is determined, $St^{R}$ in (1.1) is found to be

where $l$ is the longitudinal vortex spacing, and $u$ is the convection velocity of the vortices. Williamson & Brown (Reference Williamson and Brown1998) proposed a corrected universal Strouhal number $St^{W}$ , viz.

where $\unicode[STIX]{x1D6FF}$ is the characteristic shear layer thickness. Ahlborn et al. (Reference Ahlborn, Seto and Noack2002) suggested another correction:

where $D_{v}$ is the equivalent diameter of the shed vortices. The resultant universal Strouhal number from different corrections vary quite significantly, which is partly due to the ways of determining $D_{w}$ and  $U_{s}$ .

The use of the above mentioned universal Strouhal number reduces and sometimes eliminates the direct influence of the bluff body geometries, though inevitably involving more quantities to be measured with more uncertainties and measurement error. For simplicity, we replace only the length scale by $D_{w}^{\ast }$ and define alternative $C_{D}^{+}$ and $St^{+}$ , i.e.

The dependences of $C_{D}^{+}$ and $St^{+}$ on $N$ are presented in figure 10, along with those of $C_{D}$ and $St$ . The distorted Cartesian pattern observed in figure 6 disappears for both $C_{D}^{+}$ and $St^{+}$ ; rather, the data points are much less scattered. The $C_{D}^{+}$ data occur about 1.0 and $St^{+}$ near $0.21\sim 0.23$ , except for $N\leqslant 4$ . It seems plausible that using the wake scales does have potential in getting the data collapsed and in establishing a possible correlation between $C_{D}^{+}$ and $St^{+}$ (a correlation, if any, seems to be rather weak from figure 10). Nevertheless, some data points still display a relatively large scatter, especially at small $N$ . Note that involving the wake characteristics implies additional measurements to be conducted. It would be desirable to establish a direct correlation between $C_{D}$ and $St$ . We will propose a new scaling factor in § 3.5, which proves to work well for polygonal cylinders.

Figure 10.

Dependence of the universal $C_{D}^{+}$ and $St^{+}$ , as compared to $C_{D}$ and $St$ , on $N$ . $Re=1.0\times 10^{4}$ .

3.4 Flow separation angle

In the context of a polygonal cylinder, the flow separation angle $\unicode[STIX]{x1D703}_{s}$ is in fact the angle between the particular polygon side upstream of the separation point and the incoming flow direction, that is, this angle may be directly determined from the separation point $\unicode[STIX]{x1D703}_{o}$ , if known. The location of $\unicode[STIX]{x1D703}_{o}$ and hence $\unicode[STIX]{x1D703}_{s}$ can be fairly accurately found out by carefully examining the flow visualisation images, as illustrated in figure 11. The separation points of the polygonal cylinders, including the 2F case, all occur at one of the corners (above the wake centreline), except perhaps at very large $N$ , such as the circular cylinder. The determined $\unicode[STIX]{x1D703}_{o}$ and $\unicode[STIX]{x1D703}_{s}$ at $Re=1.0\times 10^{4}$ are indicated schematically in figure 12. The $\unicode[STIX]{x1D703}_{s}$ for the circular cylinder is the angle between the incoming flow direction and the tangent line at the separation point. Note that, in cases of 5F and also 8C, flow separates first at corner A before reattaching to the polygon surface edge AB, and then separates again at B. Therefore, corner B is taken as the separation point.

Figure 11.

Smoke streak flow visualisation images for all the testing cases at $Re=1.0\times 10^{4}$ .

Figure 12.

Schematic diagrams indicating the location of the separation point above the centreline, marked by a circle, and the separation angle $\unicode[STIX]{x1D703}_{s}$ , marked by $\measuredangle$ , as determined at $Re=1.0\times 10^{4}$ .

It has been well established that $C_{D}$ of the circular cylinder depends on the position of the separation point. For $Re<Re_{c}$ , the laminar boundary layer separates fairly early, at $\unicode[STIX]{x1D703}_{o}\approx 79^{\circ }$ and $\unicode[STIX]{x1D703}_{s}\approx 11^{\circ }$ . For $Re>Re_{c}$ , the transition occurs in the boundary layers and the flow separation is postponed, resulting in a lower  $C_{D}$ .

For the polygons of $N\leqslant 4$ , $\unicode[STIX]{x1D703}_{o}$ is fixed and independent of  $Re$ . $C_{D}$ changes little with  $Re$ .

For the polygons of $5\leqslant N\leqslant 8$ , $\unicode[STIX]{x1D703}_{o}$ is also fixed. To facilitate discussions, denote the point at the maximum lateral width as $T_{o}$ and take the front corner if two corners correspond to the maximum width position, as is the case for 6C, 8F, 12F and 16F (figure 12). Use $T_{1}$ and WSP to represent the corner upstream of $T_{o}$ and the windward stagnation point on the centreline, respectively. Four scenarios have been observed:

1. (i) flow separates at $T_{o}$ and there is no corner between $T_{o}$ and WSP, as for the cases for 5C, 6C and $N\leqslant 4$ , which are associated with the dashed line 3C–5C in figure 6;

2. (ii) flow separates at $T_{o}$ and there is one corner between $T_{o}$ and WSP (6F, 7F, 8F);

3. (iii) flow separates at $T_{1}$ and the boundary layer reattaches and separates again just before $T_{o}$ (5F, 8C);

4. (iv) flow separates at $T_{1}$ but does not reattach (7C).

For $N\geqslant 12$ , the wake pattern is similar to that behind the circular cylinder, where the transition may have a pronounced effect on the flow. Nevertheless at $Re<Re_{c}$ , the separation point is fixed at one corner. A few examples of flow around the polygonal cylinders of $N\geqslant 12$ at $Re=5.0\times 10^{4}$ when the transition occurs (refer to figure 4) are shown in figure 13.

Figure 13.

Smoke streak flow visualisation for (a) 12C, (b) 12F, (c) 16C and (d) 16F when the transition occurs. $Re=5.0\times 10^{4}$ . The arrows indicate the approximate mean separation point, since instantaneously the separation point oscillates from one instant to the next.

The dependence of $\unicode[STIX]{x1D703}_{s}$ on $N$ shows again a distorted Cartesian mesh pattern (not shown) similar to figure 6, which suggests a relation between $C_{D}$ and $\unicode[STIX]{x1D703}_{s}$ . As a matter of fact, $\unicode[STIX]{x1D703}_{s}$ and $C_{D}$ fall approximately on a linear correlation for most of the cases, as shown in figure 14. There are only a few cases that deviate from this linear correlation. It is found that the separation point occurs at $T_{o}$ (see figures 11 and 12) for all the cases reasonably following the linear correlation. For 7F and 8C cases, $T_{o}$ is actually the separation point of the reattached flow which initially separated at $T_{1}$ . For these two cases, the flow is more sensitive to $Re$ and the data $(\unicode[STIX]{x1D703}_{s},C_{D})$ deviate from the linear relation in figure 14. The other three data points which deviate further correspond to the cases 7C, 12C and 16C, where the flow separates at $T_{1}$ without reattachment. In the cases showing large deviation to the linear relation, the lateral distance between the two separation points (above and below the centreline) is not the same as the characteristic length scale  $D$ .

Figure 14.

Correlation between $C_{D}$ and $\unicode[STIX]{x1D703}_{s}$ at $Re=1.0\times 10^{4}$ .

In order to account for such deviations, we propose to define a new angle $\unicode[STIX]{x1D709}$ associated with the point $T_{o}$ , between two directions:

1. (i) the direction from $T_{1}$ to $T_{o}$ and the incoming flow direction in the absence of the transition;

2. (ii) the direction from $T_{o}$ to TP (the point where the transition occurs) and the incoming flow direction when the transition occurs (if TP is downstream of $T_{o}$ , $\unicode[STIX]{x1D709}$ is negative);

3. (iii) the tangent line at the separation point and the incoming flow direction for the circular cylinder case.

Thus defined, $\unicode[STIX]{x1D709}$ is underpinned by two observations: (i) in the absence of the transition, $C_{D}$ depends only on $N$ and the orientation, not on $Re$ ; (ii)  $T_{o}$ essentially defines the entire windward surface, which effectively describes the degree of bluffness for a regular polygon of any $N$ . In the absence of the transition, $\unicode[STIX]{x1D709}$ can be mathematically written, in units of degrees, as:

where $\lceil \,\rceil$ denotes the nearest larger integer. Note that $\unicode[STIX]{x1D709}$ is calculated based on $T_{o}$ above the centreline, which is not always the separation point. Therefore, $\unicode[STIX]{x1D709}$ is sometimes different from $\unicode[STIX]{x1D703}_{s}$ . Figure 15 presents the results calculated from (3.8), which displays a distorted Cartesian pattern similar to that in figure 6(a); see the dashed lines. When $Re\geqslant Re_{c}$ , for $N=12$ and $N=16$ cases, $\unicode[STIX]{x1D709}$ depends also on $Re$ and no analytical solution can be obtained simply on the geometry and orientation. The true value of $\unicode[STIX]{x1D709}$ when the transition occurs can be measured from experiments. It is worth mentioning that beyond $Re_{c}$ , the separation point is no longer fixed at a particular point; rather it oscillates around a mean location, which is determined from the average of the 100 smoke streak images.

Figure 15.

Dependence of the angle $\unicode[STIX]{x1D709}$ on $N$ , based on (3.8), in the absence of the transition.

Figure 16(a) presents the dependence of $C_{D}$ on $\unicode[STIX]{x1D709}$ , which shows clearly that $C_{D}$ and $\unicode[STIX]{x1D709}$ are almost linearly related for the circular cylinder in the absence of the transition and the polygons of $N\leqslant 8$ . The data may be least-square-fitted to the following straight lines:

Figure 16(b) combines three $Re$ cases and both NC and NF orientations. The data for $Re=1.0\times 10^{4}$ were obtained in the wake of the polygons of $D=50$  mm. Evidently, the linear relation remains valid for all the data. The empirical correlation (3.9) does not take the effect of $Re$ into account, that is, $\unicode[STIX]{x1D709}$ remains the same for a given polygon in the subcritical regime and so does $C_{D}$ . In fact, $C_{D}$ for most polygons subtly decreases as $Re$ increases, which could be considered in (3.9) by including a $Re$ -related correction factor. However, this correlation term is found to be very small compared to the $\unicode[STIX]{x1D709}$ term, essentially within the uncertainty of (3.9). For example, $C_{D}$ reduces by ${\approx}10\,\%$ as $Re$ changes from $2.0\times 10^{4}$ to $1.0\times 10^{5}$ . Therefore, the $Re$ correction is dropped.

Figure 16.

Dependence on $\unicode[STIX]{x1D709}$ of (a) $C_{D}$ at $Re=5.0\times 10^{4}$ ; (b)  $C_{D}$ at $Re=2.0\times ,5.0\times ,10.0\times 10^{4}$ ; (c)  $St$ at $Re=5.0\times 10^{4}$ ; (d)  $St$ at $Re=2.0\times ,5.0\times ,10.0\times 10^{4}$ . The dashed lines are the least-square-fits to the data of the circular cylinder and $N\leqslant 8$ in (b,d), where no transition takes place. The data obtained in the wake of a circular cylinder at $Re=40.0\times 10^{4}$ from Wieselsberger (Reference Wieselsberger1921) and Bearman (Reference Bearman1969) are also included for comparison. The left-pointing arrows in (a,c) illustrate the effects of the transition on the estimate of $\unicode[STIX]{x1D709}$ , that is, $\unicode[STIX]{x1D709}$ changes from the black colour symbol to the red once the transition occurs.

Figure 16(a) also reflects the importance of determining the true $\unicode[STIX]{x1D709}$ for the case when the transition takes place, as indicated by the left-pointing arrows. Apparently, if $\unicode[STIX]{x1D709}$ in the subcritical regime is used for that in the critical regime, the data points will deviate further from the fitting curve. Taking the $N=16$ polygon for example, the transition incurs a change in $\unicode[STIX]{x1D709}$ from $11.25^{\circ }$ to $-11.25^{\circ }$ for 16C and from $22.5^{\circ }$ to $-22.5^{\circ }$ for 16F. Once the correct $\unicode[STIX]{x1D709}$ is used, the corresponding $C_{D}$ collapses onto the fitting curves reasonably well, as illustrated in figure 16(b), that is, equation (3.9) may also apply to cases in the critical regime, as long as the correct $\unicode[STIX]{x1D709}$ is used. This implies that (3.9) can be used to predict the true separation point location and the separation angle for polygons given the flow regime.

The data from Wieselsberger (Reference Wieselsberger1921) for a circular cylinder in the critical flow regime ( $Re=4.0\times 10^{5}$ ) are also included for comparison, which deviate significantly from the fitting curve. This is not unexpected since the $Re$ of the data is beyond the $Re$ range over which the correlation is obtained.

The $St$ data corresponding to $C_{D}$ shown in figure 16(a,b) are presented in figure 16(c,d). It is obvious that the $St-\unicode[STIX]{x1D709}$ correlation is not quite linear and the data are least-square-fitted to a second-order polynomial, viz.

As shown in figure 16(c), the data collapse quite well onto this fitting curve when the correct $\unicode[STIX]{x1D709}$ is used. Compared to $C_{D}$ , the variation in $St$ with $Re$ is even smaller, within $\pm 4\,\%$ . Thus the $Re$ correction factor is also not necessary.

Figure 17.

Empirical relation between $St$ and $C_{D}$ for all the cases tested at $Re=(1.0\sim 6.0)\times 10^{4}$ . $\unicode[STIX]{x1D701}=C_{D}^{0.6}St=0.2$ is the fitting curve based on the least-squares technique, with $\unicode[STIX]{x1D701}=0.22$ and 0.18 corresponding to the upper and the lower envelopes, respectively. Correlations by Hoerner (Reference Hoerner1965), Ahlborn et al. (Reference Ahlborn, Seto and Noack2002) and Alam & Zhou (Reference Alam and Zhou2008) are also included for the purpose of comparison.

Equations (3.9) and (3.10) may be used to estimate $C_{D}$ and $St$ of the untested cases. Take the 10C case for example. In the subcritical regime, $\unicode[STIX]{x1D709}(10\text{C})$ can be calculated from (3.8) to be $36^{\circ }$ . By substituting $\unicode[STIX]{x1D709}(10\text{C})$ into (3.9) and (3.10), we obtain $C_{D}(10\text{C})\approx 1.36$ and $St(10\text{C})\approx 0.168$ , which agree very well with the predictions in § 3.2 (figure 6).

3.5 Relation between $C_{D}$ and $St$

The empirical relations of equations (3.9) and (3.10) could be combined to connect directly $C_{D}$ and $St$ , although additional uncertainties may be induced as some data deviate from the fitting curves more than the others. As such, we fit, based on the least-squares technique, all the data from the measurements directly to the correlation form proposed by Hoerner (Reference Hoerner1965), viz.

where $\unicode[STIX]{x1D701}$ is a constant. Figure 17 indicates that $\unicode[STIX]{x1D701}=0.2$ appears to be the best fit to all the tested polygons and the $Re$ range ( $1.0\times 10^{4}\sim 6.0\times 10^{4}$ ). The upper and lower envelopes, given by $\unicode[STIX]{x1D701}=0.22$ and 0.18, enclose most of the data points except 2F, 3C and some 4C data which occur above the upper envelope. A common feature for 2F and 3C polygons is the lack of leeward body volume or afterbody leading to a higher back pressure coefficient and hence a larger $C_{D}\cdot St$ value (Bearman Reference Bearman1967). Using (3.11) to calculate $C_{D}$ from $St$ or vice versa yields a maximum error of approximately $\pm 10\,\%$ . This error may be further reduced by fine tuning the $\unicode[STIX]{x1D701}$ value. For example, letting $\unicode[STIX]{x1D701}$ to be 0.23 will enclose most of the 2F and 3C data inside the envelopes.

A number of empirical $C_{D}\sim St$ relations proposed by previous researchers are also plotted in figure 17 for the purpose of comparison, i.e.

where $k$ is an energy parameter which varies in general with different bluff body shapes. Almost all the present data, including the two envelopes ( $\unicode[STIX]{x1D701}=0.22,0.18$ ), fall between the two curves corresponding to $k=12$ and 16, respectively, of (3.13). Equation (3.12) does not seem to represent the present data very well for $C_{D}<1$ . Equation (3.14) exhibits a deviation from the data for $C_{D}<1$ and $C_{D}>2$ because this equation is derived based on the assumption of energy conservation in spite of inevitable energy exchange between the cylinder and the supporting frames in actual experiments.

It is worth making a comparison between the presently proposed (3.11) and (3.12)–(3.14) based on a collection of the data in the literature. Take $N=4$ and $N=\infty$ for example to estimate $St$ from $C_{D}$ and vice versa. The results are given in figure 18. In figure 18(a) the $C_{D}$ data of a circular cylinder summarised in Tropea et al. (Reference Tropea, Yarin and Foss2007) are used to calculate the corresponding $St$ from (3.12)–(3.14). In view of the data distribution for a circular cylinder in figure 17, the $\unicode[STIX]{x1D701}$ value in (3.11) is chosen to be 0.21 to represent the best $C_{D}$ on the circular cylinder in figure 18(a). The $k$ value in (3.13) is chosen to be 14.5, which gives the best fit. Note that Ahlborn et al. (Reference Ahlborn, Seto and Noack2002) suggested $k=12.7$ , which produces a larger deviation from the experimental data. Equation (3.11), albeit obtained from the data over a limited $Re$ range, gives the best fit to the experimental data from Tropea et al. (Reference Tropea, Yarin and Foss2007) for a wider $Re$ range and is even in agreement with the data of $Re>Re_{c}$ . Comparing to (3.12), equation (3.11) is equally good for $Re<Re_{c}$ but better for $Re>Re_{c}$ . Equation (3.14) displays a large deviation for $10^{3}<Re<10^{4}$ and (3.13) agrees well with the experimental data for the entire $Re$ range presented, similarly to (3.11).

Figure 18.

Evaluation of various $St\sim C_{D}$ relations (3.11)–(3.14): (a)  $St$ calculated from $C_{D}$ on a circular cylinder summarised in Tropea et al. (Reference Tropea, Yarin and Foss2007); (b)  $C_{D}$ calculated from $St$ on a square cylinder summarised in Blevins (Reference Blevins1990). Experimental data from Okajima (Reference Okajima1982), White (Reference White2001) and Alam & Zhou (Reference Alam and Zhou2008) are included for the purpose of comparison.

Figure 18(b) presents the $C_{D}$ data for the $N=4$ polygon, calculated based on (3.11) to (3.14), from $St$ summarised in Blevins (Reference Blevins1990). Comparison is further made between the calculated $C_{D}$ values and those reported in the literature. The $\unicode[STIX]{x1D701}$ value in (3.11) is chosen to be 0.19. In (3.13), $k$ is set to be 12, which leads to a better fit to the data than $k=11.5$ proposed in Ahlborn et al. (Reference Ahlborn, Seto and Noack2002). The $C_{D}$ data calculated from (3.11) agree qualitatively with the available experimental data in Okajima (Reference Okajima1982) for $Re<1.0\times 10^{3}$ , but increase from 1.9 to 2.2 for $1.0\times 10^{4}<Re<1.0\times 10^{5}$ , exceeding the experimental data reported in White (Reference White2001), with the largest deviation up to 10 %. Assessment cannot be made for other $Re$ ranges due to a lack of experimental data. The prediction from (3.12) is quite similar to that from (3.11).

Table 2 summarises $C_{D}$ and $St$ , reported in the literature, for the two-dimensional bodies of various cross-sectional geometry or irregular bluff bodies. The $\unicode[STIX]{x1D701}$ value is calculated based on $C_{D}$ and $St$ from (3.11). While most bluff bodies fall in between the envelopes ( $0.18\leqslant \unicode[STIX]{x1D701}\leqslant 0.22$ ), $\unicode[STIX]{x1D701}$ tends to be small for more streamline-shaped bodies. For example, the rectangular cylinder with an aspect ratio 2 : 1 corresponds to $\unicode[STIX]{x1D701}=0.11$ , smaller than 0.24, with an aspect ratio of 1 : 2. Table 2 and also figure 18 indicate that $\unicode[STIX]{x1D701}$ is independent of $Re$ for cylinders with mild bluffness, at least for the $Re$ range examined presently. Furthermore, the experimental data (not shown) confirm that $\unicode[STIX]{x1D701}$ changes in the range of $0.20\pm 0.02$ as the angle of attack of the polygonal cylinder varys between the corner and the face orientations at $Re=5.0\times 10^{4}$ .

Table 2.

The $C_{D}$ and $St$ values of various irregular polygonal cylinders reported in the literature and their correlation coefficient $\unicode[STIX]{x1D701}$ calculated from (3.11). In the table, CW, FW and CL denote the windward-facing curved surface, the windward-facing flat surface and the leeward-facing curved surface, respectively; RC and CC represent the rounded corner and the chamfered corner, respectively; $r$ , $d$ and $c$ are the abbreviations for the radius of the rounded corner, the side length of the square cylinder and the size of the chamfered corner, respectively.

Figure 17 indicates that $St$ and $C_{D}$ are inversely related. This can be demonstrated analytically.

Assuming that the vortices shed from the cylinders are two-dimensional and can be approximately described by the Lamb–Oseen vortex model (Saffman Reference Saffman1995), the azimuthal velocity $u_{\unicode[STIX]{x1D703}}$ of an isolated vortex can be written as:

where $t$ is time ( $\sqrt{4\unicode[STIX]{x1D708}t}$ is the characteristic vortex size), $\unicode[STIX]{x1D6E4}_{o}$ is the vortex strength and $r_{s}$ is the radius to the vortex centre. The kinetic energy of the flow field associated with an isolated Lamb–Oseen vortex may be written as:

where $\unicode[STIX]{x1D70C}$ is the fluid density. From the energy budget point of view, one pair of vortices form over one period of vortex shedding, which produces a drag force $F_{D}$ , i.e. $KE=F_{D}U_{\infty }\unicode[STIX]{x1D70F}$ , where $\unicode[STIX]{x1D70F}=D/U_{\infty }$ is the characteristic time scale of vortex shedding. Defining a dimensionless circulation $K=\unicode[STIX]{x1D6E4}_{o}/\unicode[STIX]{x03C0}U_{\infty }D$ and writing the integral in (3.16) as $\unicode[STIX]{x1D702}(t)$ , $C_{D}$ is then related to $St$ as:

The formation of a vortex is the roll up of a shear layer, and vortex shedding occurs when fluid engulfed in carrying vorticity with opposite signs cuts off further the entrainment of circulation (Gerrard Reference Gerrard1966). Taking the average velocity in the boundary layer to be $U_{\infty }/2$ and $\unicode[STIX]{x1D6FF}$ to be the characteristic boundary layer thickness, consider a shear layer with vorticity $\unicode[STIX]{x1D714}\sim U_{\infty }/\unicode[STIX]{x1D6FF}$ rolling up over time $\unicode[STIX]{x1D70F}$ . Then

Note that, depending on the degree of bluffness and hence the wake width, only a fraction of the circulation contributes to the vortex formation (Bearman Reference Bearman1984). Equation (3.18) then needs to be revised such that

The $Re$ effect may be reflected in $D_{w}^{\ast }$ . Substituting (3.19) into (3.17) yields

The integrand of $\unicode[STIX]{x1D702}(t)$ is bounded over $r_{s}\in [0,\infty )$ and the integral $\unicode[STIX]{x1D702}(t)$ is not very sensitive to $t$ . Therefore, $\unicode[STIX]{x1D702}(t)$ is a weak function of the characteristic vortex size $\sqrt{4\unicode[STIX]{x1D708}t}$ . Taking a realistic vortex size to be of the order of $D/2$ , $\unicode[STIX]{x1D702}$ is typically around 7.0. Substituting the known numerical values into  $f$ , we may finally obtain

As the empirical correlation (3.11) agrees well with the present experimental data, we may infer $f(D_{w}^{\ast })\sim \unicode[STIX]{x1D701}C_{D}^{-0.4}$ . Note that $C_{D}$ is an increasing function of $D_{w}^{\ast }$ , that is, $C_{D}$ increases with larger $D_{w}^{\ast }$ . Using the $D_{w}^{\ast }$ data at $Re=1.0\times 10^{4}$ shown in figure 9 and the corresponding $C_{D}$ , we may find $D_{w}^{\ast }\sim C_{D}^{0.8}$ . Therefore, $f(D_{w}^{\ast })\sim (D_{w}^{\ast })^{-1/2}$ .