4.2.1. Steady flows

First, let us discuss the steady flows that one can obtain at different values of $Re$. For the cylinder with ${\small \text{AR}}=1$ and $Re$ being larger than approximately $10$, the flow begins to separate on the ends of the cylinder, a closed recirculation zone begins to form, and a pair of counter-rotating vortices appear in the wake. At this time, there are two mutually perpendicular symmetric planes ($x$–$y$ and $x$–$z$ passing through the origin of the coordinate system) in the wake, and the centroid of the cylinder is on these two planes. We call this flow pattern P1. The corresponding Reynolds number at the onset of pattern P1 is $Re_{c0}$. In general, the ${\small \text{AR}}$ is inversely correlated with $Re_{c0}$. The smaller ${\small \text{AR}}$ is, the closer $Re_{c0}$ is to 20 in the case of the sphere wake (Johnson & Patel Reference Johnson and Patel1999), and the larger ${\small \text{AR}}$ is, the closer $Re_{c0}$ is to 4 in the case of infinite-length 2-D cylinder flow. Specifically, the critical Reynolds numbers of the finite cylinder with ${\small \text{AR}}=0.5, 0.75, 1.0, 1.5$ and $2$ are $Re_{c0}\approx 18, 15, 10, 8$ and $6$, respectively.

A streamline diagram is used to characterise the typical wake structure of the pattern P1 for ${\small \text{AR}}=1$ and $Re=100$ in figure 5. The specific cross-section is the $x$–$y$ and $x$–$z$ planes of the flow. Unless stated otherwise, the streamwise direction of the flow field diagram in this article is from left to right. The topological traits of the pattern P1, such as the separation position on the cylinder's end ($\theta _s$), the vortex centre point ($x_b,y_b$) or ($x_b,z_b$) and the length of the separation bubble ($x_s$), have also been shown in figure 5. The fluid separates from the cylinder's end at the angle $\theta _s$ (the vertex of the angle is at the origin, one edge ends at the forward stagnation point, and the other at the separation point), and converges at a point $x_s$ on the $ox$-axis, forming a closed separation bubble in the $x$–$z$ plane. The coordinates of the centre of the separation vortex pair in the $x$–$y$ and $x$–$z$ planes are $(x_b,y_b)$ and $(x_b,z_b)$, respectively, and the velocity at the vortex centre point ($x_b,y_b,z_b$) is zero. All the lengths are non-dimensionalised by the cylinder diameter. With these parameters, we can quantify the effect of $Re$ on the pattern P1, as follows.

Figure 5.

A typical steady wake pattern P1 with two symmetrical planes perpendicular to each other, at $Re=100$ and ${\small \text{AR}}=1$. ($a$) Streamlines on plane $x$–$y$, ($b$) streamlines on $x$–$z$ plane.

Previous works on flow past a sphere (Fornberg Reference Fornberg1988; Johnson & Patel Reference Johnson and Patel1999; Tomboulides & Orszag Reference Tomboulides and Orszag2000) found and confirmed that when $Re>75$, the separation angle and the separation bubble length are logarithmically related to the Reynolds number. Following this idea, we found that a logarithmic relation also holds in the flow past short cylinders, as illustrated in figure 6. For the steady flow, the separation bubble length can be approximately related to $Re$ by $x_s=a\ln (Re+b)+c$ (see the dotted curves in panel ($a$) and for the unsteady flow, the separation angle and separation bubble length are measured in the mean flow). The goodness of fit $R^2$ for ${\small \text{AR}}=1$ is 0.9988, and exceeds $0.999$ for the other four cylinders. The turning points of curve $x_s$–$Re$ in figure 6($a$) correspond to the transition of the wake mode (from steady flows to time-periodic flows), which also can be observed in $\bar {C}_{ly}$–$Re$ (figure 4$b$). One can see that the $x_s$–$Re$ relation curve for the flow past a sphere is close to the ${\small \text{AR}}=0.75$ data. As pointed out by one of the referees, this aspect ratio is notable because, interestingly, it has a projected frontal area that is very close to that of a sphere (i.e. ${\rm \pi} /4\approx 0.785$), which can also explain that the curve $\bar {C}_{ly}$–$Re$ (${\small \text{AR}}=0.75$) in figure 4($b$) almost coincides with that of the sphere.

Figure 6.

($a$) Separation bubble length and ($b$) separation angle as a function of the Reynolds number. The dotted lines represent fitting functions $x_s=a\ln (Re+b)+c$ and $\theta _s =aRe^b+c$ (see figure 5($a$) for the schematic definitions). The fitting coefficients $(a, b, c)$ are shown in the legends. In panel ($b$) the black, red and green dashed lines are the averaged value of the separation angles in the $x$–$y$ and $x$–$z$ planes for the short cylinders with ${\small \text{AR}}=0.5,0.75$ and $1$, respectively.

In figure 6$(b)$ the relationship between the separation angle and the Reynolds number is approximately a power function $\theta _s=aRe^b+c$ within the range of $Re<600$. It turns out that the separation angle is less sensitive to the transition, and almost no visible kinks are found in these results. At a first glance, the results of the sphere from Johnson & Patel (Reference Johnson and Patel1999) deviate significantly from the result of ${\small \text{AR}}=0.75$ (note that these results in figure 6($b$) are only for $\theta _s$ in the $x$–$y$ plane). A more fair comparison entails an additional consideration of the separation angle in the $x$–$z$ plane. With reference to figure 5($b$) (plane $x$–$z$), the separation angles at the sharp trailing corners of the short cylinder are fixed at 135 $^\circ$ (${\small \text{AR}}=1$), $143.13^\circ$ (${\small \text{AR}}=0.75$) and $153.44^\circ$ (${\small \text{AR}}=0.5$). Hence, we can simply average the two angles in the $x$–$z$ and $x$–$y$ planes and plot the new results as dashed lines in figure 6($b$). One can see that in this case, the results of short cylinders (for ${\small \text{AR}}=0.5,0.75,1$) indeed straddle that of the sphere, consistent with the previous simple argument as suggested by the referee.

When $Re$ further increases, larger than a certain critical $Re_{c1}$, the current DNS results show that the pattern P1 will be unstable and transition to another steady wake pattern P2 (figure 7) with only one symmetrical plane ($x$–$y$), simultaneously inducing non-zero values of $\bar {C}_{ly}$. The bifurcation of the short cylinder at $Re_{c1}$ is steady, time-independent without vortex shedding ($St=0$), so the flow experiences a pitchfork bifurcation. It is noted that the pattern P2 may not appear for certain values of ${\small \text{AR}}$. As we will show in the end of the next section, ${\small \text{AR}}=1.75$ is such an example. We will present more systematic results of the effect of ${\small \text{AR}}$ in § 4.4. This is similar to the flow past a sphere, where there is a regular bifurcation before the Hopf bifurcation. The difference is that the wake of the sphere retains a symmetric plane that is randomly oriented, which can be related to the numerical uncertainty or the perturbations in the experimental environment (Tomboulides & Orszag Reference Tomboulides and Orszag2000). In the literature there is no discussion about the P2-type flow pattern (Inoue & Sakuragi Reference Inoue and Sakuragi2008 did not report this) or its associated regular bifurcation for the flow past a short cylinder with its axis being perpendicular to the streamwise direction (in the following, for the ease of discussion, we will call this flow a radial flow; likewise, we call the flow past a short cylinder with its axis parallel to the streamwise direction an axial flow). Recently, Pierson et al. (Reference Pierson, Auguste, Hammouti and Wachs2019) conducted a numerical simulation of an axial flow past a short cylinder at ${\small \text{AR}}=1, 20\leq Re\leq 420$ and ${\small \text{AR}}=3, 25\leq Re\leq 250$. They found a regular bifurcation for the axial flow around the cylinder at ${\small \text{AR}}=1$ and $Re\approx 278$ (based on the cylinder diameter) without hysteresis, but this bifurcation was not found for the cylinder with ${\small \text{AR}}=3$. Sheard et al. (Reference Sheard, Thompson and Hourigan2008) found that there is a regular bifurcation for the radial flow past a short cylinder with hemispherical ends at $Re=350\pm 2$. Finally, 3-D transitions of the bluff-body cube have been numerically studied by Saha (Reference Saha2004). The cube wake also undergoes a regular bifurcation (losing one symmetric plane but retaining time independence) at $Re=216$ and a Hopf bifurcation at $Re=270$. From these works, it can be hypothesised that the aspect ratio ${\small \text{AR}}$ of the bluff body (including the short cylinder with flat ends as we study here) is related to the existence of a regular bifurcation.

Figure 7.

A typical P2 type wake at $Re=278$ and ${\small \text{AR}}=1$, ($a$) $x$–$y$ plane, ($b$) $x$–$z$ plane. Vorticity ($x$-component) isosurface (60 % transparency) is coloured by values $-0.1$ and $0.1$. The number 22 is the length of the scale ruler. The green object is the cylinder.

Now we present the way we determine $Re_{c1}$. We perform nonlinear DNS for ${\small \text{AR}}=1$, and the critical value $Re_{c1}$ is approached by gradually increasing the Reynolds number from a small value. For each $Re$, we can calculate the growth rate of disturbance development as in figure 8. In this paragraph we focus on the blue curves for $Re=180$ without randomness in the initial condition and will discuss the other curves shortly. The time series of $C_{ly}$, when plotted in a logarithmic scale against the time in a linear scale (as in panel $b$), will display a period of a straight line, which represents the linear phase. The linear growth or decay rate can then be calculated and the critical $Re$ corresponding to the zero growth rate can be interpolated by fitting several data points using a linear function, which is shown in figure 10, to be discussed shortly. For ${\small \text{AR}}$=1, we can see that $Re_{c1}=172.2$, which is consistent with the $Re$ of onset of non-zero $\bar {C}_{ly}$ shown in figure 4($b$). The $Re_{c1}$ values of other ${\small \text{AR}}$ have also been calculated and it is found that when ${\small \text{AR}}$ is small or large, $Re_{c1}$ is relatively large. The ${\small \text{AR}}$ having the minimum $Re_{c1}$ is approximately located around $1.25\sim 1.5$. Besides, the ${\small \text{AR}}$-dependence of the critical Reynolds number will be revealed in § 4.4. In the analysis of Tomboulides & Orszag (Reference Tomboulides and Orszag2000), the authors also performed a logarithmic transformation of the azimuthal velocity at a point located in the wake and near the sphere, and found that the logarithm of the azimuthal velocity is a linear function of time during the initial evolution. The growth rate obtained by this method is consistent with the LSA result of Natarajan & Acrivos (Reference Natarajan and Acrivos1993).

Figure 8.

Time histories of lift coefficient $C_{ly}$ and $\ln (|C_{ly}|)$. $(1+0.2\epsilon _{rand})$ in the legend means that the initial condition is the base flow $(1,0,0)$ plus a white noise of 0.2 times the base flow in the three velocity components. Likewise for the other initial conditions.

Some numerical experiments have also been conducted to perturb the flows in figure 8. We applied white noise (denoted as $\epsilon _{rand}$) to the initial conditions in the DNS at ${\small \text{AR}}=1, Re=165, 168, 170$ (subcritical, as $Re< Re_{c1}=172.2$). The random numbers are shifted to perturb the solution positively and negatively. The amplitude of the white noise is 0.2 times the inlet velocity for the three cases and additionally, for $Re=168$, we also examined the amplitudes of the (white-noise) disturbance being 0 and 0.6 times the inlet velocity. The white noise is applied to all the three velocity components through the following initial conditions ((4.1) and (4.2)):

(4.1)\begin{gather} \left. \begin{array}{ll@{}} U_x = 1.0 +a\epsilon_{rand}, & (a_1=3.0\times10^4, a_2={-}1.5\times10^3, a_3 = 0.5\times10^5),\\ U_y = 0.0 +a\epsilon_{rand}, & (a_1=2.3\times10^4, a_2=2.3\times10^3, a_3 ={-}2.0\times10^5),\\ U_z = 0.0 +a\epsilon_{rand}, & (a_1=2.0\times10^4, a_2=1.0\times10^3, a_3 = 1.0\times10^5), \end{array} \right\} \end{gather}

(4.2)\begin{gather} \left. \begin{array}{l@{}} \epsilon_1 = a_1 (i_g+x \sin y) + a_2 i_x i_y + a_3 i_x, \\ \epsilon_2 = a_1 (i_g + z \sin\epsilon_1) + a_2 i_z i_x + a_3 i_z, \\ \epsilon_{rand} = \cos(10^3\sin(10^3\sin\epsilon_2)), \end{array} \right\} \end{gather}

Here $a$ is the maximum amplitude of the perturbation, $\epsilon _{rand}$ denotes white noise with random numbers between $-1$ and 1 in (4.2). $x,y,z$ are the coordinate values of the element nodes and $i_x,i_y,i_z,i_g$ are the numbers of the element nodes. From figure 8 we can see that the lift coefficients $C_{ly}$ for all the cases aforementioned eventually converge to zero (because the flows are subcritical) and the linear phase survives for a long time. This demonstrates the robustness of the results. A by-product of these numerical experiments is that they seem to indicate that the first bifurcation of the short cylinder around $Re_{c1}$ is probably supercritical for the parameters we consider here, because if the bifurcation was subcritical, the white noise with a large amplitude will probably bring the flow to a stable finite-amplitude nonlinear solution. In order to confirm the supercriticality of the first bifurcation in this flow, we estimate the coefficients of the Landau model from the DNS data (as is well known, the Landau coefficient calculated around the linear critical conditions determines the nature of the bifurcation; see Guckenheimer & Holmes Reference Guckenheimer and Holmes1983). The global variable $C_{ly}$ that can directly reflect the transition from P1 to P2 is used to evaluate the Landau coefficient at $Re=180$ and ${\small \text{AR}}=1$. The original time history of $C_{ly}$ is shown in figure 8($a$), where $|C_{ly}|$ grows and finally saturates. Now, based on the Landau equation ${\textrm {d}A}/{\textrm {d}t}=c_1 A+ c_3 |A|^2$ (where $A$ can be viewed as the amplitude of $C_{ly}$) or, equivalently, $\textrm {d}\log |A|/\textrm {d}t=c_1+ c_3|A|^2$, the Landau coefficients $c_1$ and $c_3$ can be calculated by plotting $\textrm {d}(\log |C_{ly}|)/\textrm {d}t$ vs $|C_{ly}|^2$ (Thompson et al. Reference Thompson, Leweke and Provansal2001), as shown in figure 9. Therefore, the vertical intercept point gives an estimation of $c_1=0.0201$ and the gradient near this point is an approximation of $c_3=-8.78$. The value of growth rate $c_1=0.0201$ is close to the linear growth rate $0.0196$ (at $Re=180$) obtained by calculating the linear slope in figure 8($b$). Furthermore, the negative $c_3=-8.78$ indicates that the (lowest-order) nonlinearity stabilises the flow. In panel ($b$) one can also see that the nonlinearity stabilises the flow after the linear regime. Thus, the bifurcation is supercritical transitioning from P1 to P2. We mention in passing that, theoretically, using this method to infer the bifurcation type of a flow is most accurate when the parameter is close to the critical condition. Nevertheless, when this is satisfied, the evolution time of the flow is very long in DNS. As long as the bifurcation type is concerned, the method can be applied slightly away from the critical condition (Henderson & Barkley Reference Henderson and Barkley1996; Carmo, Meneghini & Sherwin Reference Carmo, Meneghini and Sherwin2010; Gao et al. Reference Gao, Sergent, Podvin, Xin, Le Quéré and Tuckerman2013; Feng et al. Reference Feng, Zhang, Vazquez and Shu2021). Besides, we arrive at the above conclusion using only one instance of $Re$ and admit that a more rigorous analysis would be to apply the weakly nonlinear stability analysis around the linear critical conditions, which can be pursued in a future work.

Figure 9.

Time derivative of log$|A|$ vs $|A|^2$ with the vertical intercept giving the linear growth rate $c_1=-0.0201$ and the gradient near this intercept point giving the Landau coefficient $c_3=-8.78$, which indicates supercritical bifurcation behaviour. Nonlinear DNS data of case ${\small \text{AR}}=1$, $Re=180$.

To sum up, the P2-type steady flow has been found for short cylinders with small ${\small \text{AR}}$. Most of the relevant research on the flow past a short bluff body have focused on the sphere or cylinder with a large ${\small \text{AR}}$, and no previous studies have systematically investigated this P2-type wake mode in the axial flow past a finite cylinder. Through our DNS results in the range of $0.5<{\small \text{AR}}<2$, it is found that the ${\small \text{AR}}$ of the short cylinder has a very significant and important influence on the critical Reynolds number $Re_{c1}$. Only when the ${\small \text{AR}}$ is relatively small (${\small \text{AR}}< 1.75$) will there be a regular bifurcation. As can be seen in figure 10, when ${\small \text{AR}}=1.75$, we cannot find the regular transition by either DNS or LSA methods. In fact, when ${\small \text{AR}}\gtrsim 1.75$, as $Re$ increases, the wake of the radial flow around the short cylinder will not experience the P2-type wake, and directly transition from the P1-type to a periodic shedding of hairpin-shaped vortices P3-0 with $\bar {C}_{ly}=\bar {C}_{lz}=0$, which has been reported many times in the flow past a bluff body with ${\small \text{AR}}>2$, for example, in the flow past a cylinder with two free hemispherical ends (Sheard et al. Reference Sheard, Thompson and Hourigan2005, Reference Sheard, Thompson and Hourigan2008), a cylinder with two free flat ends at moderate $2 \leq {\small \text{AR}} \leq 10$ (named type III by Inoue & Sakuragi Reference Inoue and Sakuragi2008) and at ${\small \text{AR}}=3$ and $Re>125$ (Pierson et al. Reference Pierson, Auguste, Hammouti and Wachs2019's figure 29), a prolate spheroid at $L/D=6$ and $Re=100$ (El Khoury, Andersson & Pettersen Reference El Khoury, Andersson and Pettersen2012) and a freely falling cylinder following rectilinear paths at Archimedes number $Ar=200$ and $L/D=5$ (Toupoint, Ern & Roig Reference Toupoint, Ern and Roig2019's figure 17a). Nevertheless, we will report in the next section that there are two unreported periodic vortex shedding wakes (P3-1 and P3-2) of radial flow around the short cylinder with $\bar {C}_{ly}\neq 0$.

Figure 10.

The growth rate ($G_R$) calculated using the DNS results as a function of the Reynolds number. Unstable flows will transition from P1 pattern to P2 pattern via a regular bifurcation and the critical Reynolds number is $Re_{c1}$.

4.2.2. Vortex shedding

When $Re$ increases further, a second bifurcation appears at $Re\approx 282$ (for ${\small \text{AR}}=1$) with $St\neq 0$, which belongs to a Hopf bifurcation. Figure 11 shows the relationship between the linear growth rate and $Re$ for short cylinders of different ${\small \text{AR}}$ values from $0.5$ to 2 near the Hopf bifurcation point. The relationship between the growth rate and the Reynolds number is approximately linear when the flow is stable and near the critical condition. The solid lines in figure 11 represent linear fitting results which are shown in the legend. The general trend is that the larger the value of ${\small \text{AR}}$ is, the smaller the critical value of $Re_{c2}$. The denoted eigenmodes A and B will be discussed in § 4.3.

Figure 11.

The growth rate $\sigma$ of the leading eigenmode in the global LSA (of the time-mean flow) as a function of $Re$. Unstable flows will transition from P2 pattern to P3 pattern via a Hopf bifurcation and the critical Reynolds number is $Re_{c2}$.

We found that near the Hopf bifurcation point, one of the two periodic wake patterns P3-1 (a transitional state) or P3-2 (a saturated state) may appear and can exist for a long time. The P3-1 wake pattern is a hairpin-like vortex structure shedding from the side surface of the cylinder, as shown in figure 12($a$). The P3-1 type wake is symmetric about the $x$–$y$ plane (note that the green ruler is collinear with the cylinder axis, that is, $z$), and the lift coefficient $C_{lz}$ in the $z$ direction is non-oscillating and approximately zero. Fast Fourier transform (FFT) applied to the drag and the lift coefficients gives the same and single-periodic vortex shedding frequency $St_{Re=283}=0.125$. Interestingly, we note that this value is comparable to the vortex frequency of the sphere $St_{Re=270}=0.1292$, $St_{Re=285}=0.1335$ obtained by Tomboulides & Orszag (Reference Tomboulides and Orszag2000) through nonlinear DNS. The cross-section of the P3-1 hairpin vortex on the $x$–$y$ plane is asymmetric with respect to the $x$-axis, and the wake is offset in the $y$-axis (similar to the P2-type wake). This asymmetric periodic wake pattern is also similar to the asymmetric wakes of bluff bodies, for example, in the axial flow around a short cylinder studied by Pierson et al. (Reference Pierson, Auguste, Hammouti and Wachs2019) (who found that the axial flow becomes unsteady at $Re\approx 355\ {\rm and}\ {\small \text{AR}}=1$, but it still maintains a symmetric plane in the wake until $Re=420$), in the flow past a sphere by Tomboulides & Orszag (Reference Tomboulides and Orszag2000), Johnson & Patel (Reference Johnson and Patel1999) (who showed that the regular bifurcation at $Re=211$, and the asymmetric steady flow becomes unsteady at $Re=275$), and in the flow past a short cylinder with hemispherical ends by Sheard et al. (Reference Sheard, Thompson and Hourigan2008) (who found that the regular bifurcation at $Re=350\pm 2$). The other 3-D wake pattern P3-2 is shown in figure 12($b$). It can be seen that the vortex shedding from the ends begins to be dominant in the wake. The results of FFT applied on the aerodynamic coefficients show that the frequency of $C_d$ and $C_{ly}$ is the same, and the frequency of $C_{lz}$ is 1/2 that of $C_d$ and $C_{ly}$. Besides, the frequency of vortex shedding of the structure P3-2 is larger than that of the structure P3-1; for example, at $Re=290$, we calculated that $St_{P3-2}=0.1395$ whereas $St_{P3-1}=0.1269$. More results of the $St$ data for other values of $Re$ are shown in figure 19 to be discussed shortly.

Figure 12.

Two typical periodic wake patterns for the case $Re=290$, ${\small \text{AR}}=1$. ($a$) wake pattern P3-1 stimulated by the initial condition $1+0.0\epsilon _{rand}$; ($b$) wake pattern P3-2 stimulated by the initial condition $1+0.2\epsilon _{rand}$. The $Q$-criterion isosurfaces $Q=0$ are coloured by the $x$-component of the vorticity ranging from $-10^{-2}$ to $10^{-2}$. The green ruler is collinear with the cylinder axis.

In a subcritical flow (with $Re<282$ for ${\small \text{AR}}=1$), both P3-1 and P3-2 patterns are stable and will converge to the P2 pattern. This is demonstrated in figure 13 using DNS where we have used different initial conditions to trigger the appearance of the two P3 patterns. We found that if one uses initial perturbation with a smaller amplitude at $Re=278$, the flow will more likely develop to the P3-1 structure; whereas, if a large disturbance is adopted, P3-2 is more likely to appear. To explain, the initial condition is the base flow $(U_x, U_y, U_z)$, e.g. ($1, 0, 0)$, plus white noise with a maximum amplitude of $a=0.0, 0.2, 0.5$, denoted as $1+a\epsilon _{rand}$. Let us first claim the results that $1+a\epsilon _{rand}$ leads to the P3-1 structure, and $3+0.2\epsilon _{rand}$ gives rise to the P3-2 structure. The phase trajectory diagram of the lift-drag coefficient is shown in figure 13. For all the initial conditions considered, the $C_d$–$C_{ly}$ curves of P3-1 and P3-2 patterns, as converging spirals, shrink to the same fixed point, that is, the attractor of the subcritical cases is a fixed point, which is the steady flow pattern P2 as shown in figure 7. On the other hand, the oscillation range of $C_{lz}$ corresponding to the initial conditions $1+a\epsilon _{rand}$ is much smaller than $C_{ly}$ and is close to zero, as shown in figure 13($b$). Based on this result, we think that the initial conditions $1+a\epsilon _{rand}$ leads to the P3-1 pattern. The phase trajectory of $C_d$–$C_{lz}$ corresponding to the initial condition $3+0.2\epsilon _{rand}$ is obviously different from that of $1+a\epsilon _{rand}$ (the P3-1 pattern) and corresponds to the P3-2 structure.

Figure 13.

The diagram of $C_d$–$C_{l}$ at $Re=278$ and ${\small \text{AR}}=1$. The expression $1+a \epsilon _{rand}$ in the legends indicates that the initial condition for the velocity is $(1,0,0)$ plus white noise with an amplitude of $a$. The white noise is three dimensional as seen in (4.2).

In a supercritical flow (with $Re>282$ for ${\small \text{AR}}=1$), the P3-1 pattern is transient and will eventually converge to the saturated P3-2 pattern. However, one may be easily fooled to believe that the P3-1 pattern can be a stable mode because when $Re$ is close to the critical condition, the lingering time of the flow around the neighbourhood of P3-1 can be extremely long (before it saturates to the nonlinearly stable P3-2 state). This is shown in figures 14 and 15 for $Re=290$ and $300$, respectively. As one can see from figure 15 for $Re=300$, the two wake patterns and their transition are clear in a relatively short time. When $Re=290$, however, the DNS method becomes quite difficult to differentiate the two because one needs to simulate for a very long time to see the transition. In the above two figures, because a high-$Re$ is more sensitive to numerical error, it makes the $Re=300$ flow transition to the saturated state more quickly. Another way to expedite the convergence to the saturated state is to add perturbation in the initial condition, resulting in a high possibility of throwing the flow out of the neighbourhood of P3-1 and converging to the final saturated state more quickly (results not shown).

Figure 14.

The time histories of aerodynamic coefficients for cylinder ${\small \text{AR}}=1$ at $Re=290$. The initial condition is the wake P3-1 of a lower-$Re$ case.

Figure 15.

The time histories of aerodynamic coefficients for cylinder ${\small \text{AR}}=1$ at $Re=300$. The initial condition is $1+0.0\epsilon _{rand}$.

Some discussions are in order regarding the possible bifurcation type around $Re_{c2}$. In the subcritical flows our numerical simulations with different initial conditions cannot converge to a stable flow state other than P2 (which is a linearly stable state). In the supercritical flows, even though the lingering time around P3-1 can be very long, the final saturation of the flow is the P3-2 pattern; different initial conditions will only change the time spent by the flow around P3-1 before converging to the P3-2 wake. With these results, we infer that the nature of the bifurcation around $Re_{c2}$ should be supercritical. The Landau model has also been used to study the bifurcation type (following a similar analysis presented in figure 9). Again, $c_1=0.0272$ obtained in the Landau model is close to the global stability analysis results based on the SFD base flow (whose linear growth rate is $0.02746$). The negative value of $c_3$ indicates that the flow bifurcation may be supercritical, which can also be seen in the stabilising effect of the (lowest-order) nonlinearity in figure 16($b$). More analyses of the P3-1 and P3-2 patterns and the flow bifurcation will be conducted in § 4.3 on the global modes.

Figure 16.

Time derivative of log$|A|$ vs $|A|^2$ with the vertical intercept giving the linear growth rate $c_1=0.0272$ and the gradient near this intercept point giving the Landau coefficient $c_3=-254.1$, which indicates supercritical bifurcation behaviour. Nonlinear DNS data of case ${\small \text{AR}}=1$, $Re=300$.

4.2.3. Approach to chaos

When $Re$ is further increased, the flow becomes more chaotic. Figure 17 shows $C_d$–$C_{lz}$ phase diagrams and power spectral density (PSD) of the $C_{lz}$ data for four values of $Re$ in $338 \leq Re \leq 1000$. When $Re$ is larger than $332$, a new lower frequency $St=0.072$ starts to emerge in the flow field. This can be seen in panels ($a{,}e$) for $Re=338$ where a small peak begins to form around $St=0.072$ in the PSD figure. This frequency becomes incommensurable when $Re$ further increases and the wake will gradually become chaotic, which is called pattern P4-0 (with $\bar {C}_{ly}=\bar {C}_{lz}=0$) and P4-1 (with $\bar {C}_{ly}\neq 0$, occurs at ${\small \text{AR}}<1.75$) in this article. In Tomboulides & Orszag (Reference Tomboulides and Orszag2000)'s DNS study on the sphere, a similar lower frequency value was detected as $St_{Re=500}=0.045$ at a relatively large $Re$, which was the second highest peak in the PSD of their DNS data. The frequency corresponding to the dominant peak was still $St_{Re=500}=0.167$, which was dominant at all positions of the wake. In the work of the axial flow past a short cylinder, Pierson et al. (Reference Pierson, Auguste, Hammouti and Wachs2019) also found a similar low frequency ($St_{Re=395}=0.03$) when $Re \gtrsim 395$ in their DNS study of the axial flow past a short cylinder.

Figure 17.

The diagrams of $C_{d}$–$C_{lz}$ in (a–d) and the power spectral density for the time history of $C_{lz}$ in (e–f) with ${\small \text{AR}}=1$, at $Re=338$ ($a$,$e$), $Re=344$ ($b$,$f$), $Re=410$ ($c$,$g$) and $Re=1000$ ($d$,$h$).

Back to figure 17, at $Re=344$, a third new frequency signal $St=0.0083$ starts to appear. At this $Re$, the attractor of $C_d$–$C_{ly}$ is no longer a limit cycle, but becomes a limit torus. It means that a certain two frequencies in the flow field are incommensurable, and the phase trajectory is no longer closed. The PSD of these two lower frequencies $St=0.0706$ and $St=0.0083$ is not negligible compared with the dominant frequency $St=0.1528$. At $Re=410$, the phase trajectory of $C_d$–$C_{ly}$ becomes highly disordered, and the PSD curve shows the characteristics of a chaotic signal. It can be seen from figure 17($g$) that the dominant frequency is still around $St=0.1611$, and its PSD curve shows that there are many small peaks around $St=0.033$, $St=0.094$ and $St=0.1611$. Finally, when $Re=1000$, the phase diagram is highly chaotic and a broad spectrum of frequencies appears signalling the disordered state of the flow. In the literature, Sakamoto & Haniu (Reference Sakamoto and Haniu1995)'s experimental work on the flow past a sphere showed that, when $Re>420$, the shedding direction of the hairpin vortex appears intermittent, the oscillation amplitude and waveform caused by the vortex shedding start to become irregular, and the flow field starts to transition from a single-frequency flow to a chaotic state. Pierson et al. (Reference Pierson, Auguste, Hammouti and Wachs2019)'s DNS study (in its supplementary materials) on the axial flow past a finite cylinder (${\small \text{AR}}=1$) shows that the flow exhibits chaotic properties for the $Re$ in $420 \leq Re \leq 460$, and the secondary lower frequency ($St_{Re=460}=0.03$) gradually dominates the entire flow field. These results of the transition from a single frequency to a chaotic state is similar to that of the radial flow past a finite cylinder (${\small \text{AR}}=1$) as we study here.