3.2.1. First bifurcation: stability diagram

We start considering the first bifurcation. Figure 4 summarises the effect of the geometry. Three main regions appear in the $L$ - $W$ plane. The flow past wide bodies (large $W/L$ ) first become unstable to oscillatory $S_y A_z$ perturbations that break the temporal and top/bottom planar symmetries and preserve the left/right planar symmetry, leading to a periodic vortex shedding across the shorter body dimension. This is consistent with the 2-D limit $W\rightarrow \infty$ , and resembles the results for the flow past finite-length circular cylinders (Yang et al. Reference Yang, Feng and Zhang2022). Increasing $L/W$ , the flow first becomes unstable to stationary $S_y A_z$ perturbations that break the top/bottom spatial symmetry and maintain the left/right one. This bifurcation leads to a static vertical deflection of the wake. Finally, further increasing $L/W$ , the flow first becomes unstable to stationary $A_y S_z$ perturbations that break the left/right symmetry while preserving the top/bottom one. This corresponds to a static horizontal deflection of the wake. By symmetry, the $S_y A_z$ and $A_y S_z$ modes become unstable simultaneously in the special case of bodies of square cross-section $W/H=1$ , as observed by Marquet & Larsson (Reference Marquet and Larsson2015), Meng et al. (Reference Meng, An, Cheng and Kimiaei2021) and Zampogna & Boujo (Reference Zampogna and Boujo2023). Interestingly, the oscillatory $A_y S_z$ mode observed by Marquet & Larsson (Reference Marquet and Larsson2015) for thin plates $L=1/6$ of intermediate width $2 \leqslant W \leqslant 2.5$ seems restricted to a narrow region of the $L$ - $W$ plane (see also the neutral curves for $W=2.25$ in figure 5 and for $L=1/6$ and 1 in figure 7).

3.2.2. Eigenmodes and neutral curves

We now proceed to characterise in more detail the linear stability of the steady symmetric base flow for increasing values of $Re$ and a few selected geometries. Figure 5 shows the critical Reynolds number and frequency of the first bifurcations as a function of the body length $L$ for $W=1.2$ , $2.25$ and $5$ . The eigenmodes of the first two bifurcations for these widths and the two body lengths $L=1$ and $5$ are shown in figures 6; see also the supplementary material. We focus on the first few bifurcations, as the eigenmodes that become unstable at significantly larger Reynolds numbers become less relevant.

Figure 5. (a) Neutral curves: critical Reynolds number $Re_c$ as a function of body length $L$ for different body widths $W=1.2$ , 2.25 and 5. Filled symbols and solid lines indicate stationary (pitchfork) bifurcations; open symbols and dashed lines indicate oscillatory (Hopf) bifurcations. (b) Critical frequency as a function of body length $L$ for the first few bifurcations: Strouhal number $St_c=\omega (Re_c)/(2\pi )$ . Different lines correspond to different modes. In both $(a)$ and $(b)$ , thicker lines show the eigenmode that becomes unstable at the lowest Reynolds number.

Figure 6. First and second bifurcating eigenmodes (left and right, respectively), visualised with isosurfaces of streamwise velocity.

For narrow bodies, $W=1.2$ , the first two bifurcations of the steady symmetric base flow occur in short succession (as $W$ is close to 1), and correspond to stationary $S_y A_z$ and $A_y S_z$ modes that break either of the two planar symmetries (Zampogna & Boujo Reference Zampogna and Boujo2023). A pair of streamwise streaks of positive and negative streamwise perturbation velocity arise in the wake (figure 6), leading to a vertical or horizontal displacement of the wake. Other bifurcations occur at larger $Re$ and correspond to oscillatory $S_y A_z$ and $A_y S_z$ modes. For small $L$ , these modes are almost as unstable as their stationary counterparts, but they quickly become much more stable as $L$ increases, i.e. the corresponding $Re_c$ increases faster; accordingly, for long bodies, these oscillatory modes of the symmetric base flow are not directly relevant to the secondary stability of the deflected flow (§ 4.1.2). The associated angular frequency is of the order $\omega \simeq 0.4-0.6$ ( $St = f H / U_\infty \simeq 0.06-0.09$ , where $f = \omega /(2 \pi )$ ) for $L \leqslant 2$ . The increase of $Re_c$ with $L$ is consistent with the stationary bifurcation of the flow past axisymmetric bodies: a disk much thinner than its diameter $L/D \ll 1$ , a sphere $L/D=1$ , bullet-shaped bodies $1 \leqslant L/D \leqslant 3.5$ (Natarajan & Acrivos Reference Natarajan and Acrivos1993; Fabre et al. Reference Fabre, Auguste and Magnaudet2008; Meliga et al. Reference Meliga, Chomaz and Sipp2009; Bohorquez et al. Reference Bohorquez, Sanmiguel-Rojas, Sevilla, Jiménez-González and Martínez-Bazán2011) and with the primary oscillatory bifurcation of the flow past 2-D bodies of different shape (Jackson Reference Jackson1987; Chiarini et al. Reference Chiarini, Quadrio and Auteri2022c ).

For wide bodies, $W=5$ , the first bifurcation corresponds to an oscillatory $S_y A_z$ mode for all $L$ (as already observed in figure 4). The critical Reynolds number increases with $L$ and is rather well separated from the following bifurcations. The angular frequency is $\omega \simeq 0.5-0.6$ ( $St \simeq 0.08-0.10$ ). The unstable eigenmode is stronger in the central region (small $|y|$ ); see figure 6. The second bifurcation corresponds to an oscillatory $S_y S_z$ modes for all $L$ . Like the leading $S_y A_z$ mode, it is associated with vortex shedding in the upper and lower shear layers, with a similar frequency but different inter-layer phasing. Unlike the $S_y A_z$ mode that breaks the top/bottom symmetry, the $S_y S_z$ mode preserves both planar symmetries. In other words, in the vertical plane $y=0$ the two leading modes can be categorised as ‘sinuous’ and ‘varicose’, respectively. To the best of our knowledge, unsteady doubly symmetric $S_y S_z$ modes have not been reported in 3-D bluff body wakes in previous works except for thin plates ( $L=1/6$ , $W\gt 3.1$ , Marquet & Larsson Reference Marquet and Larsson2015); see also § 4.2. At larger $Re$ further bifurcations are observed, including some involving doubly antisymmetric modes ( $A_y A_z$ ). All these bifurcations are oscillatory and the corresponding frequencies are almost independent of $L$ , and differ by one order of magnitude depending on the bifurcation; see figure 5.

For bodies of intermediate width, $W=2.25$ , the increasing trend of $Re_c$ with $L$ is similar, but the leading mode changes with $L$ : stationary for $L \gt 1.5$ (similar to narrower bodies) and oscillatory with $\omega \simeq 0.3-0.6$ ( $St \simeq 0.05-0.10$ ) for $L \lt 1.5$ (similar to wider bodies). As mentioned previously (figure 4), for this specific width, the first bifurcation breaks the left/right symmetry for very short bodies ( $L\lt 0.5$ ), which leads to the unusual vortex shedding across the larger body dimension observed by Marquet & Larsson (Reference Marquet and Larsson2015), and breaks the top/bottom symmetry otherwise, leading to the more usual vortex shedding across the smaller dimension ( $0.5 \lt L \lt 1.75$ ) or to a static vertical wake deflection ( $L\gt 1.75$ ). Interestingly, the stationary $S_y A_z$ mode remains fully stable in the range of $Re$ investigated when $L\lt 1.5$ , while for $L\gt 1.5$ , it is restabilised for large enough $Re$ . Other bifurcations at larger $Re$ draw a more complicated pattern than for $W=1.2$ and $5$ . Their order of appearance depends on $L$ and some modes are unstable only in some interval of $L$ . For almost all values of $L$ , however, doubly symmetric $S_y S_z$ and doubly antisymmetric $A_y A_z$ modes only correspond to the third or higher bifurcations.

Figure 7 offers a complementary picture, with neutral curves shown as a function of the body width $W$ for the three body lengths $L=1/6$ , $L=1$ and $L=5$ . The first critical $Re$ tends to decrease with $W$ , albeit not monotonically: the first bifurcation lies on the envelope of different neutral curves corresponding to different modes, each mode being more unstable in a preferred interval of $W$ .

Figure 7. Same as figure 5, as a function of body width $W$ for different body lengths $L=1/6$ (data from Marquet & Larsson Reference Marquet and Larsson2015), $1$ and $5$ .

4.1.1. Low $Re$ : steady $sS_yA_z$ and $sA_yS_z$ regimes

The nonlinear simulations show that the primary bifurcation breaks the top/bottom planar symmetry leading to the steady $sS_yA_z$ regime, similarly to shorter bodies (Marquet & Larsson Reference Marquet and Larsson2015; Zampogna & Boujo Reference Zampogna and Boujo2023). This bifurcation occurs at $Re \approx 355$ , in agreement with the critical Reynolds number $Re_{c}=353$ found in § 3.2. The simulations show that, for $Re \gtrapprox 365$ , the steady $A_yS_z$ mode dominates and the flow recovers the top/bottom planar symmetry but loses the left/right one, as predicted by the WNL analysis. We observe a small discrepancy between the linear and WNL stability analyses ( $Re_{c} \approx 355$ ) and the fully nonlinear simulations ( $Re_{c} \approx 365$ ).

Figure 11. Steady regimes for $L=5$ and $W=1.2$ . Isosurfaces of streamwise vorticity, with red/blue indicating $\omega _x=\pm 0.075$ . Left: $sS_yA_z$ regime at $Re=355$ . Right: $sA_yS_z$ regime at $Re=370$ . Top: horizontal $x-y$ plane. Centre: vertical $x-z$ plane. Bottom: $y-z$ plane.

To characterise the $sA_yS_z$ and $sS_yA_z$ regimes, figure 11 shows isosurfaces of positive (red) and negative (blue) values of the streamwise vorticity, and illustrates the loss of the planar symmetries. This is similar to the wakes past disks, spheres and bullet-shaped bodies at Reynolds numbers larger than $Re \approx 115, 210$ and $350$ (Johnson & Patel Reference Johnson and Patel1999; Tomboulides & Orszag Reference Tomboulides and Orszag2000; Fabre et al. Reference Fabre, Auguste and Magnaudet2008; Bohorquez et al. Reference Bohorquez, Sanmiguel-Rojas, Sevilla, Jiménez-González and Martínez-Bazán2011). In these steady regimes, the wake features a pair of counter-rotating streamwise vortices that are absent at lower $Re$ . These vortices exhibit an eccentricity that increases with $x$ . The eccentricity is in the $z$ direction in the $sS_yA_z$ regime and in the $y$ direction in the $sA_yS_z$ regime.

In both the $sS_yA_z$ and $sA_yS_z$ regimes we find that the flow asymmetry is confined in the wake, confirming that the $A_yS_z$ and $S_zA_y$ global modes are unstable modes of the wake. Indeed, the recirculating regions that arise along the sides of the prism retain the top/bottom and left/right planar symmetries in both regimes (not shown for brevity). These regions do not play a relevant role in the triggering mechanism of the primary instability, as confirmed by the results of the structural sensitivity (figure 8). A similar conclusion has been drawn for the primary, Hopf instability of the flow past elongated 2-D rectangular cylinders (Chiarini et al. Reference Chiarini, Quadrio and Auteri2021).

Figure 12. Dependence of the aerodynamic forces on $Re$ in the $sS_yS_z$ , $sS_yA_z$ and $sA_yS_z$ regimes for $L=5$ and $W=1.2$ . (a,b) Total forces. (c) Viscous and pressure components of the drag force. (d) Viscous and pressure components of $F_\perp$ , the single non-zero force perpendicular to the incoming flow ( $F_z$ in the $sS_yA_z$ regime and $F_y$ in the $sA_yS_z$ regime). The red line in (a) denotes $8.47 \times Re^{-0.3799}$ .

Figure 12 shows that when the flow bifurcates to the $sS_yA_z$ and $sA_yS_z$ regimes, the drag decreases according to a power law, $F_x \sim Re^{\alpha }$ with $\alpha =-0.3799$ . This is similar to the flow past a sphere (Johnson & Patel Reference Johnson and Patel1999), a short circular cylinder (Yang et al. Reference Yang, Feng and Zhang2022) and a cube (Saha Reference Saha2004; Meng et al. Reference Meng, An, Cheng and Kimiaei2021); for the latter, Saha (Reference Saha2004) report $\alpha =-0.372$ and Meng et al. (Reference Meng, An, Cheng and Kimiaei2021) found a value between $-0.4$ and $-0.3$ . The decrease of $F_x$ is entirely due to a decrease of the friction contribution $F_{x,f}$ (see figure 12(c)), as the pressure contribution $F_{x,p}$ does not change with $Re$ . An increase of $Re$ , indeed, results into a downstream shift of the flow reattachment point $x_r$ over the sides of the prism, leading to a longer recirculating region and to a decrease of $F_{x,f}$ (figure 13). We note that a power function fits well the relation between $x_r$ and $Re$ , being $x_r \sim Re^{1.1651}$ for the left and right sides and $x_r \sim Re^{0.9485}$ for the top and bottom sides. In contrast, the non-zero cross-stream component of the aerodynamic forces increases with $Re$ . Unlike for $F_x$ , this is due to an increase of both the friction and pressure contributions (see the bottom right panel of figure 12), and is associated with the increasing flow asymmetry.

The regular bifurcation towards an asymmetric steady flow is found for prisms with small $W$ only, irrespective of $L$ (§ 3.2). This is the case of many other 3-D bluff bodies: besides the largely studied axisymmetric sphere and disk, the existence of a regular bifurcation in non-axisymmetric flows has been reported by Marquet & Larsson (Reference Marquet and Larsson2015) for thin plates with $W/D\lt 2$ , by Yang et al. (Reference Yang, Feng and Zhang2022) for short circular cylinders with $W/D\lt 1.75$ and by Sheard et al. (Reference Sheard, Thompson and Hourigan2008) for short cylinders with hemispherical ends. The main difference is that the bifurcated wake retains a symmetry plane that is randomly oriented (selected by perturbations; Tomboulides & Orszag Reference Tomboulides and Orszag2000) for axisymmetric bodies, but dictated by the planar symmetries of the geometry for non-axisymmetric bodies.

4.1.2. Large $Re$ : $pA_yS_z$ and $aA_yS_z$ regimes

Figure 13. Position of the reattachment point $x_r$ on the top/bottom and right/left sides (i.e. size of the recirculating regions on the prism walls) as a function of $Re$ for $L=5$ and $W=1.2$ . Red solid line: $0.0023 \times Re^{1.1651}$ . Red dashed line: $0.0095 \times Re^{0.9485}$ . In the unsteady regimes ( $Re\gt 500$ ), $x_r$ is the reattachment point of the time-averaged flow.

A further bifurcation occurs at $Re \approx 510$ . It consists of a Hopf bifurcation of the $sA_yS_z$ deflected wake; the flow becomes unsteady and starts oscillating about the deflected $sA_yS_z$ regime. A similar regime, with the flow oscillating about an asymmetric steady state has been observed for the flow past a sphere (Johnson & Patel Reference Johnson and Patel1999), a cube (Saha Reference Saha2004), a disk (Meliga et al. Reference Meliga, Chomaz and Sipp2009) and a short cylinder (Pierson et al. Reference Pierson, Auguste, Hammouti and Wachs2019). Interestingly, when considering the time-average forces, the decreasing power law that fits $F_x$ and $Re$ in the $sS_yS_z$ , $sS_yA_z$ and $sA_yS_z$ regimes holds also in the unsteady regime for intermediate Reynolds numbers up to $Re=575$ (figure 10). For these $Re$ , indeed, the time-average flow field -- and its dependence on $Re$ -- resembles the steady $sA_yS_z$ regime. The average size of the recirculating regions over the lateral sides increases with the Reynolds number, as shown in figure 13 by means of the reattaching point $x_r$ . For larger $Re$ , the structure of the time-average flow changes and $F_x$ increases. For $Re \geqslant 575$ , indeed, the average $x_r(Re)$ deviates from the power law found at smaller $Re$ : the lateral recirculating regions shrink as $Re$ increases (figure 13).

The dynamics of the flow progressively changes with $Re$ . At $Re=510$ the flow enters the periodic $pA_yS_z$ regime, which is characterised by an alternating shedding of HVs from the top and bottom LE shear layers (figure 14). At this $Re$ , a single peak $St \approx 0.277$ is found in the frequency spectrum $S(F_z)$ , while $F_y$ remains practically constant in time. The attractor draws a limit cycle in the phase space.

For $Re\gtrapprox 515$ , a frequency $St \approx 0.0776$ appears in $S(F_y)$ and the flow becomes aperiodic. Here it is characterised by a superposition of two different modes, (i) the shedding of HVs from the top and bottom LE shear layers (with $St \approx 0.28$ ), and (ii) the oscillating motion of the wake in the $y$ direction (with $St \approx 0.08$ ); see figure 15 and the related discussion. The two frequencies are not commensurate and, as shown in the force diagrams of figure 14, a torus replaces the limit cycle in the phase space. At this $Re$ , the peak at $St \approx 0.28$ is much larger than the other one, indicating that the shedding of HVs dominates.

For $Re \geqslant 550$ , the wake oscillates also in the $z$ direction and a further frequency $St \approx 0.03$ appears in $S(F_z)$ (figure 14). The different $Re$ at which the two oscillating modes arise agree with the results of the LSA, which reveals that, for $L \leqslant 2$ , the unsteady $A_yS_z$ wake mode becomes unstable at smaller $Re$ compared with the unsteady $S_yA_z$ mode. As mentioned above, the presence of this additional mode leads to an increase of the average $F_x$ (figure 10). When $Re$ is further increased, the different modes nonlinearly interact and the flow becomes progressively more chaotic (figure 14). The relative height of the peaks at $St \approx 0.28$ and $St \approx 0.03$ in $S(F_z)$ indicates that the flow dynamics is mainly driven by the LE vortex shedding at low $Re$ , while the vertical wake oscillating motion takes over at large $Re$ (not shown).

Figure 14. Unsteady regimes for $L=5$ and $W=1.2$ . (a) Frequency spectra of $F_y$ (left) and $F_z$ (right), $510 \leqslant Re \leqslant 550$ . (b–c) Structure of the flow for $Re=510$ , 535 and 575 (top to bottom). (b) Instantaneous isosurfaces $\lambda_2 = -0.05$ coloured by streamwise vorticity in the range $-1 \leqslant \omega_x \leqslant 1$ . (c) Force diagrams $F_y-F_x$ (left) and $F_z-F_x$ (right).

Figure 15. The POD analysis for $L=5$ , $W=1.2$ and $Re=515$ . (a) Energy fraction $E_j = \lambda _j/\sum \lambda _j$ of the first eight POD modes. Here $\lambda _j$ denotes the $j$ th eigenvalue of the snapshot covariance matrix, and is related with the $j$ th singular value by $\sigma _j = \lambda _j^2$ (see Appendix C). (b) Frequency associated with the first eight POD modes, ordered by decreasing $\sigma _j$ (i.e. decreasing $E_j$ ). Dashed lines: main DNS frequencies, also shown in figure 10. (c) Visualisation of the POD modes 1 (left) and 3 (right). Isosurfaces of $\lambda _2$ (arbitrary value) coloured by streamwise vorticity (symmetric blue-to-red colour map from negative to positive values).

To examine the spatial structure of the different modes and relate them with the flow frequencies, we use POD; see Appendix C for details. Figure 15 considers $Re=515$ . As shown in Figure 15(a,b), the dominant POD modes are associated with the same frequencies found in the frequency spectra (figure 14) and with their multiples. The POD modes come in pairs (i.e. with the same singular values and frequencies), as typical for oscillating structures. Mode 1 is associated with $St \approx 0.28$ , and a shedding of vortices from the top and bottom sides is clearly visible in its spatial structure. Mode 3, instead, is associated with $St \approx 0.07$ . Its spatial structure is confined in the wake and breaks the left/right symmetry consistently with an oscillating mode of the wake in the $y$ direction about a $A_yS_z$ state. At this $Re$ the energy fraction associated with the LE vortex shedding is approximately $1.5$ times larger than that associated with the lateral wake oscillation (see Figure 15 a), consistently with the peak heights in the frequency spectra.

For elongated prisms with $W=1.2$ and $L=5$ , the characteristic frequency of the wake oscillation ( $St \approx 0.07$ ) is smaller than for a sphere ( $St \approx 0.129$ at $Re=270$ ; Tomboulides & Orszag Reference Tomboulides and Orszag2000), a cube ( $St \approx 0.091-0.095$ at $Re= 270-300$ ; Saha Reference Saha2004), and a short cylinder ( $St \approx 0.125$ at $Re=283$ ; Yang et al. Reference Yang, Feng and Zhang2022). This is consistent with the LSA results (figure 5) that point out a decrease of $St$ with $L$ .

4.2.1. Low $Re$ : steady regimes $sS_yA_z$ and $sA_yS_z$

The nonlinear simulations show that the flow is steady and retains the $S_yS_z$ symmetry for $Re \lessapprox 290$ , in agreement with the results of the LSA. At $Re \approx 290$ the flow undergoes a regular bifurcation towards the steady $sS_yA_z$ regime, qualitatively similar to $W=1.2$ . Further increasing $Re$ , the flow remains steady but recovers the top/bottom symmetry and loses the left/right one, i.e. enters the $sA_yS_z$ regime. Unlike for smaller $W$ , this bifurcation is hysteretic, and both the $sS_yA_Z$ and $sA_yS_z$ regimes are simultaneously stable for a small range of $Re \approx 300$ ; see figure 17(a--d) and the WNL analysis in § 3.4.

By increasing the Reynolds number in the $320 \leqslant Re \leqslant 350$ range, the flow asymmetry in the $y$ direction decreases, and at $Re=350$ the flow recovers the left/right planar symmetry; for $Re \approx 360$ , the flow is steady and $F_y \approx F_z \approx 0$ . This symmetrisation is shown in figure 17. Like for $W=1.2$ (figure 11), at $Re=300$ a pair of counter-rotating streamwise vortices are found in the wake, with eccentricity in the $y$ direction (figure 17 b). As $Re$ increases, a new pair of counter rotating vortices arises, becomes stronger and progressively pushes the other pair away (figure 17 c). Eventually, at $Re \approx 360$ the two pairs of vortices are placed symmetrically with respect to the $z$ axis, and have the same intensity (figure 17 d). At this $Re$ the flow exhibits again both top/bottom and left/right planar symmetries ( $sS_yS_z$ regime). This does not happen for $W=1.2$ , as two pairs of streamwise vortices of characteristic size $H=1$ cannot be simultaneously accommodated at the TE when $W\lt 2$ .

In passing, we note that in the steady $sS_yA_z$ , $sA_yS_z$ and $sS_yS_z$ regimes a decreasing power law fits well the relation between $F_x$ and $Re$ , where $F_x \sim Re^{\alpha }$ with $\alpha =-0.2905$ . However, the agreement does not extend to the unsteady regime, unlike $W=1.2$ . Also, like for smaller $W$ , the decrease of $F_x$ with $Re$ is mainly due to the enlargement of the side recirculating regions (not shown).

4.2.2. Intermediate $Re$ : aperiodic $aS_yA_z$ and periodic $pS_yA_z$ regimes

Figure 18. Forces and frequency content for $L=5$ , $W=2.25$ and $375 \leqslant Re \leqslant 450$ . (a) Frequency spectra for $F_x$ (left) and $F_z$ (right) for $Re=375, 390, 420$ and $450$ . For $Re=420$ , $S(F_z)$ is not visible as fluctuations in $F_z$ are almost null. (b) Main frequencies for $F_x$ (left) and $F_z$ (right). (c) Zoom of figure 16(a,b) in the range $370 \leqslant Re \leqslant 460$ . For $440 \leqslant Re \leqslant 460$ , the bars are not visible as the oscillations of $F_x$ are small.

Figure 19. Structure of the flow for $L=5$ , $W=2.25$ and $Re=375$ , $390$ , $420$ and $450$ (top to bottom). (a) Side view of instantaneous isosurfaces of $\lambda _2=-0.25$ coloured with $-1 \leqslant \omega _x \leqslant 1$ . (b) Force diagrams $F_y-F_x$ and $F_z-F_x$ . The lack of top/bottom symmetry in the top two views of panel (a) is highlighted with arrows.

At $Re \approx 375$ the flow becomes unsteady and enters different regimes in short succession. This is conveniently visualised in figures 18 and 19, where the frequency content and the structure of the flow are shown for $375 \leqslant Re \leqslant 470$ .

For $375 \leqslant Re \leqslant 380$ , the flow approaches the aperiodic $aS_yA_z$ regime: unlike for smaller $W$ , the flow oscillates about a $S_yA_z$ state that retains the left/right symmetry and breaks the top/bottom one. With two non-commensurate frequencies, $St \approx 0.05$ and $0.23$ , the attractor draws a torus in the phase space. As shown in the top panel of figure 19(a), at this $Re$ the flow behaviour is the result of the superposition of two different modes: (i) asymmetric oscillating mode of the wake in the $z$ direction ( $St \approx 0.05$ ) and (ii) symmetric in-phase shedding of HVs from the top and bottom LE shear layers ( $St \approx 0.23$ ). For $Re \approx 385$ , instead, the wake oscillation mode stabilises and the flow approaches the periodic $pS_yA_z$ regime. The wake remains deflected in the $z$ direction and the unsteadiness is only due to the in-phase shedding of HVs, with a frequency $St \approx 0.24$ that slightly increases with $Re$ (see figure 18 c). A limit cycle replaces the torus in the phase space (figure 19).

Figure 20. The POD analysis for $L=5$ , $W=2.25$ and $Re=380$ . (a) Energy fraction $E_j = \lambda _j/\sum \lambda _j$ of the first eight POD modes. (b) Frequency associated with these modes. Dashed lines: main DNS frequencies as in figure 18. (c) The POD modes 1 (left) and 5 (right). Isosurfaces of $\lambda _2$ (arbitrary value) coloured by streamwise vorticity.

We use POD to examine the structure of the flow at $Re=380$ . In figure 20 we separate the modes responsible for the two non-commensurate frequencies detected in the frequency spectra. Here POD mode $1$ is associated with $St \approx 0.23$ and its spatial structure shows a $S_yS_z$ shedding of HVs from the top and bottom sides of the prism. Mode $5$ , instead, matches the $St \approx 0.05$ frequency, with a $S_yA_z$ spatial structure that agrees with an asymmetric oscillating mode of the wake in the $z$ direction. This suggests that the aperiodic $aS_yA_z$ regime found at $Re \approx 380$ is the result of a superposition of an oscillatory $S_yA_z$ mode of the wake and an unsteady $S_yS_z$ mode of the LE shear layers. Interestingly, this interpretation is supported by the LSA, which indeed predicts an unsteady $S_yS_z$ mode to become unstable for $Re \gtrapprox 350$ for this geometry, with a frequency that is compatible with $St \approx 0.2$ (figure 5). Also, the neutral curves (figure 5) show that, for $L\leqslant 4$ , an unsteady $S_yA_z$ mode of the wake with $St \approx 0.06-0.07$ becomes unstable, but for a small range of $Re$ only. For $L \approx 4$ , for example, the unstable range is between $330 \lessapprox Re \lessapprox 350$ . Recalling that these neutral curves refer to the low- $Re$ $sS_yS_z$ steady base flow, it is possible that due to nonlinear effects, an increase of $Re$ extends this curve to $L=5$ , explaining this sudden destabilisation/stabilisation of the wake mode.

Figure 20 shows that at $Re=380$ the energy fraction associated with LE vortex shedding is approximately $25$ times larger than that associated with the asymmetric oscillating mode of the wake; at this $Re$ the flow dynamics is mainly driven by the HVs shed by the LE shear layers. The effect of the Reynolds number on the spatial structure of the POD mode associated with the shedding of HVs from the LE shear layers is further discussed in Appendix C.2 (figure 32).

4.2.3. Intermediate $Re$ : periodic $pS_yS_zl$ regime

As $Re$ is further increased, the flow recovers once again the top/bottom planar symmetry and, for $390 \lt Re \leqslant 470$ , it is in the periodic and symmetric $pS_yS_zl$ regime (figure 19). In this regime, two distinct behaviours can be further distinguished, based on the fluctuations of $F_z$ , characterised by a synchronisation/anti-synchronisation of the vortex shedding from the top and bottom LE shear layers.

For $390 \lt Re \leqslant 420$ , the flow shows an in-phase shedding of HVs from the top and bottom LE shear layers, and the flow instantaneously retains the top/bottom and left/right planar symmetries (third row of figure 19); this is regime $pS_yS_zla$ . Accordingly, the unbalance of the viscous and pressure forces between the top/bottom and left/right sides is null at all times, leading to $F_z \approx F_y \approx 0$ . For this regime, we detect the leading flow frequency by inspecting the $S(F_x)$ spectrum. To the best of our knowledge, such a synchronous vortex shedding that preserves all spatial symmetries at all times and generates neither lift force nor side force has not been reported.

For $420 \lt Re \leqslant 470$ , instead, the flow remains periodic with $St \approx 0.24$ , but the shedding of vortices from the top and bottom LE shear layers is in phase opposition (last row of figure 19); this is regime $pS_yS_zlb$ . In this regime the flow retains the top/bottom planar symmetry in an average sense only, and the oscillations of $F_z$ are not null. Accordingly, unlike for $380 \leqslant Re \leqslant 420$ , at these $Re$ the POD mode associated with the LE vortex shedding is top/down antisymmetric (see Appendix C.2).

We now consider regime $pS_yS_zla$ and detail the dynamics of the HVs shed from the top and bottom LE shear layers at $Re \approx 400$ .

Figure 21. Flow structures of the doubly symmetric $pS_yS_zla$ regime for $L=5$ , $W=2.25$ and $Re=400$ : isosurfaces of $\lambda _2 = -0.05$ coloured by $-1 \leqslant \omega _x \leqslant 1$ . The four snapshots are separated in time by $T/4$ , where $T$ is the period of the shedding of HVs.

Figure 21 shows the flow structures at four different instants equispaced in one shedding period $T=1/St$ . A pair of HVs of opposite vorticity is shed in phase from the top and bottom LE shear layers, with a spanwise wavelength $\lambda _z$ dictated by the width of the prism $\lambda _z \approx W$ . The top/bottom LE shear layers, indeed, periodically release tubes of negative/positive spanwise vorticity that, due to the finite width of the prism, are not exactly aligned with the spanwise direction. As such, they are modulated by the vertical and spanwise velocity gradients and form HVs: the central part of the tubes, farther from the wall, is convected downstream faster and forms the HV heads; the lateral parts, closer to the wall, are convected more slowly and form the HV legs. Accordingly, the HVs induce low-speed velocity (negative velocity fluctuations) between their legs and high-speed velocity (positive velocity fluctuations) in the outer region. Once generated, the two HVs are shed downstream and continue moving as a pair in the wake, where they progressively move apart in the $z$ direction due to their mutual induction, until they are dissipated by viscosity. The shedding of HVs is accompanied by a periodic enlargement and shrinking of the top/bottom recirculation regions (not shown). The recirculating regions enlarge while the LE shear layers accumulate vorticity, and suddenly shrink when the HVs are shed.

The shedding of HVs from the LE shear layers is reminiscent of the flow past 2-D rectangular cylinders: spanwise vortices are periodically shed from the LE shear layers due to the so-called impinging-LE-vortex (ILEV) instability (Naudascher & Rockwell Reference Naudascher and Rockwell1994). The ILEV instability is a resonant oscillation of the fluid. Vortices are shed periodically from the LE shear layer and, when a LE vortex passes over the TE, it triggers the shedding of a new LE vortex (Chiarini et al. Reference Chiarini, Quadrio and Auteri2022d ). The flow frequency thus changes with the length of the body and the dynamics of the LE and TE vortex shedding are synchronised. In the 2-D case, therefore, the shedding of vortices from the LE is not the result of an absolute instability of the LE shear layer, but is interconnected with the dynamics of the TE vortex shedding (Hourigan et al. Reference Hourigan, Thompson and Tan2001; Chiarini et al. Reference Chiarini, Quadrio and Auteri2022d ).

Figure 22. Lateral view of the flow structures for rectangular prisms with (a) $(L,W)=(\infty,2.25)$ and a sharp LE and (b) $(L,W)=(5,2.25)$ and a rounded LE with $R = 1/64$ at $Re=400$ . Isosurfaces of $\lambda _2 = -0.05$ coloured by $-1 \leqslant \omega _x \leqslant 1$ . The flow approaches the regimes $pS_yS_zla$ and $pS_yA_z$ , respectively.

To investigate the role of the LE and TE shear layers in the mechanism that sustains the $pS_yS_zla$ regime, two types of additional simulations have been carried out at $Re=400$ for modified prism geometries. First, to isolate the LE shear layer from the interaction with the TE, we have considered a rectangular prism of infinite length and $W=2.25$ . As shown in figure 22(a), the flow enters the $pS_yS_zla$ regime in this case too, with pairs of HVs shed in phase from the top and bottom LE shear layers. Interestingly, the shedding frequency closely matches that found for $L=5$ , $St \approx 0.24$ . This indicates that, for 3-D prisms, the triggering mechanism of the LE vortex shedding does not require the presence of a TE, as instead observed for 2-D blunt bodies (Chaurasia & Thompson Reference Chaurasia and Thompson2011; Thompson Reference Thompson2012). A possible feedback mechanism that triggers the formation of the HVs may therefore be embedded within the recirculating regions that arise over the top and bottom sides of the prism.

Second, to support this hypothesis, we have considered rectangular prisms with $L=5$ and $W=2.25$ , but with a rounded LE. Five curvature radii $R$ have been considered, ranging from $R=1/2$ (semicircular LE) to $R=1/64$ . For $R=1/2$ , the flow does not separate from the LE, and recirculating regions do not form over the top and bottom sides of the prism. In this case the flow approaches a steady and asymmetric $sS_yA_z$ regime and shedding of HVs from the LE is not detected. When a smaller $R$ is considered, the flow separates from the LE and progressively larger recirculating regions form over the top and bottom sides of the prism. However, the shedding of HVs is only detected for the smallest radius $R=1/64$ . In this case the flow enters a periodic regime around the asymmetric $S_yA_z$ state, characterised by the in-phase shedding of HVs from the top and bottom LE shear layers (figure 22 b). Again, the shedding frequency matches well that found for the sharp configuration, $St \approx 0.238$ .

In summary, these additional simulations support the hypothesis that in the $pS_yS_zl$ regime the shedding of HVs from the LE shear layers is the result of a feedback mechanism that takes place in the recirculating regions over the top/bottom sides of the prism and does not require the presence of a TE.

4.2.4. Large $Re$ : aperiodic regime $aS_yS_z$

Figure 23. Unsteady regimes at larger $Re$ for $L=5$ and $W=2.25$ . (a) Frequency spectra of $F_y$ (left) and $F_z$ (right) for $470 \leqslant Re \leqslant 625$ . (b–c) Structure of the flow for $Re=470$ , $485$ and $625$ (top to bottom). (b) Instantaneous isosurfaces of $\lambda _2=-0.05$ coloured by $-1 \leqslant \omega _x \leqslant 1$ . (c) Force diagrams $F_y-F_x$ (left) and $F_z-F_x$ (right).

For larger Reynolds numbers $Re \geqslant 460$ , the lateral and vertical oscillating modes of the wake become unstable, and the flow progressively enters a chaotic regime (figure 23). Unlike for smaller  $W$ , for $500 \leqslant Re \leqslant 700$ , the flow oscillates around a $S_yS_z$ state, and the time-average value of the cross-stream forces is $F_y=F_z=0$ . This recalls the results of Zdravkovich et al. (Reference Zdravkovich, Brand, Mathew and Weston1989) and Zdravkovich et al. (Reference Zdravkovich, Flaherty, Pahle and Skelhorne1998), who investigated the flow past a finite circular cylinder at $Re \approx 10^5$ , varying the width-to-height ratio between $0.25 \leqslant W \leqslant 10$ . In agreement with our results, they found an asymmetric flow pattern for $1 \leqslant W \leqslant 2$ only.

According to our simulations, the flow loses the instantaneous left/right symmetry at $Re \approx 465$ . For $465 \leqslant Re \leqslant 470$ , $S(F_y)$ (figure 23) shows a single peak at $St \approx 0.134$ , which is half the frequency of the LE vortex shedding (see $S(F_z)$ in the same figure). The flow enters a $pS_yS_z$ regime. In this range of $Re$ the flow remains periodic and the attractor is a limit cycle that draws a closed line in the phase space (see the force diagrams in figure 23). As $Re$ increases, the bifurcated limit cycle becomes unstable. At $Re=485$ , a new incommensurate frequency $St \approx 0.079$ arises in $S(Fy)$ ; the flow is therefore aperiodic and the attractor draws a torus in the phase space.

For $Re \geqslant 500$ , additional frequencies appear and the flow becomes chaotic. In the $aS_yS_z$ regime the flow dynamics is dominated by the nonlinear interaction of three different modes: (i) shedding of HVs from top and bottom LE shear layers; (ii,iii) oscillating modes of the wake in the $z$ and $y$ directions. The frequency of the first two modes are detected with the peaks in $S(F_z)$ , i.e. $St \approx 0.25{-}0.30$ and $St \approx 0.02{-}0.03$ . The frequency $St \approx 0.04{-}0.05$ of the wake oscillating mode in the $y$ direction, instead, is visible in $S(F_y)$ . As $Re$ increases, the frequency of the LE vortex shedding increases and the corresponding peak in the frequency spectrum becomes less evident and more broad-banded (not shown). Like for smaller $W$ , indeed, at larger $Re$ the LE vortex shedding weakens and the flow unsteadiness is mainly driven by the wake oscillating modes. Note that $St \approx 0.03$ is close to what was found for $W=1.2$ , indicating that the frequency of the wake oscillating mode in the $z$ direction is only marginally influenced by the width of the prism for $1.2 \leqslant W \leqslant 2.25$ . In contrast, the frequency of the wake oscillation mode in the $y$ direction clearly decreases with $W$ . Figure 23 shows that, for $Re \gt 500$ , the shedding of HVs arises also over the lateral sides of the prism.

4.3.1. Low $Re$ : periodic regime $pS_yS_zt$

Figure 25. Characterisation of the $pS_yS_zt$ regime for $L=5$ and $W=5$ . (a) Isosurfaces of spanwise vorticity (red and blue for $\omega _y= \pm 0.25$ ) at $Re=275$ . (b) Friction drag for $225 \leqslant Re \leqslant 300$ . (c) Pressure drag for $225 \leqslant Re \leqslant 300$ . (d) Flow frequency for $250 \leqslant Re \leqslant 300$ .

The nonlinear simulations show that the unsteady $S_yA_z$ mode of the wake becomes unstable at $Re \approx 250$ and that, for $250 \lessapprox Re \lessapprox 300$ , the flow experiences a periodic oscillation of the wake in the $z$ direction around the steady $S_yS_z$ state (figure 25 a). Unlike for smaller $W$ , here the flow unsteadiness is driven by the wake oscillation and HVs are not shed by the LE shear layers. The hairpin-like structures observed in the wake are thus the result of an interaction between the top and bottom TE shear layers; this resembles the so-called hairpin shedding regime found for shorter bluff bodies of various shapes (Tomboulides & Orszag Reference Tomboulides and Orszag2000; Saha Reference Saha2004; Yang et al. Reference Yang, Feng and Zhang2022). The flow is periodic with a single frequency close to that found with LSA ( $St=0.09$ ; § 3.2). As expected, the intensity of the wake oscillation increases with $Re$ , as conveniently visualised in figure 24 by means of the root mean square of $F'_z = F_z - \! \left \langle {F_z} \right \rangle \!_t$ ( $ \! \left \langle {\cdot } \right \rangle \!_t$ indicates average in time): $F'_{z,rms} \approx 0.004$ at $Re=250$ and $F'_{z,rms} \approx 0.036$ at $Re=300$ . In this regime, an increase of $Re$ is accompanied by an increase of $F_x$ , which is entirely ascribed to a monotonic increase of the pressure contribution $F_{x,p}$ . In fact, similarly to $W=1.2$ (figure 13), an increase of $Re$ leads to larger side (time-average) recirculating regions and, thus, to a decrease of the friction contribution to $F_x$ (figure 25 b).

4.3.2. Intermediate $Re$ : aperiodic and frequency-locking regimes

Figure 26. First aperiodic regimes for $L=5$ and $W=5$ . (a) Frequency spectra of $F_y$ (left) and $F_z$ (right) for $305 \leqslant Re \leqslant 345$ . (b-c) Structure of the flow for $Re=315$ (top) and $Re=335$ (bottom). (b) Instantaneous isosurfaces $\lambda _2=-0.25$ coloured by $-1 \leqslant \omega _x \leqslant 1$ . (c) Force diagrams $F_y-F_x$ (left) and $F_z-F_x$ (right).

As the Reynolds number increases, the limit cycle loses its stability via a Neimark--Sacker bifurcation, and new frequencies appear in the flow. At $Re \approx 305$ a torus replaces the limit cycle. The unsteady $A_yS_z$ wake mode predicted by the LSA of the steady $S_yS_z$ base flow (§ 3.2) becomes unstable, and the flow experiences an oscillating motion in both $y$ and $z$ directions. The new frequency is $St \approx 0.03$ , in agreement with the value predicted by the LSA. As $Re$ increases, the two modes interact nonlinearly and different frequencies arise in the spectra (figure 26), resulting in a progressive loss of coherency. However, the frequency spectra (and the POD, not shown) indicate that the horizontal oscillations at $St=0.03$ are less intense than the vertical oscillations at $St=0.09$ . As a result, up to $Re \approx 325$ the dynamics is mainly driven by the vertical oscillation of the wake and the resulting hairpin-like structures.

For $Re \gtrapprox 325$ , a new mode arises with a frequency in the range $0.16 \leqslant St \leqslant 0.2$ associated with the shedding of spanwise vorticity tubes from the top/bottom LE shear layers that roll up and generate HVs in the wake, similarly to what was found for smaller  $W$ . The interaction of these vortices with the wake unsteadiness is analysed later in this subsection. The LE vortex shedding frequency for $W=5$ is smaller than that for $W=1.2$ and $2.25$ . This LE vortex shedding progressively breaks the coherency of the flow (see the force diagrams and the emergence of several peaks in the frequency spectra in figure 26). As $Re$ increases, the importance of the LE vortex shedding on the flow dynamics progressively increases (see the peaks at $St \approx 0.09$ and $0.19$ in $S(F_z)$ and the following POD analysis) and, for $Re \gtrapprox 330$ , its contribution is comparable to that of the wake oscillation.

Figure 27. Aperiodic and frequency-locking regimes for $L=5$ and $W=5$ at larger Reynolds numbers $345 \leqslant Re \leqslant 385$ . Same conventions as figure 26, but with the main flow frequencies given in the bottom plot of panel (a). (b,c) Results are shown for $Re=353$ (top) and $Re=375$ (bottom). Note that vortices are shed from the LE shear layer in phase opposition for $Re=353$ (lock-in region I) and in phase for $Re=375$ (lock-in region II).

For two narrow ranges of Reynolds numbers, the flow oscillations influence each other in a way that produces synchronisation into a periodic regime. This phenomenon, known as frequency locking (Iooss & Joseph Reference Iooss and Joseph1990; Kuznetsov Reference Kuznetsov2004), has already been observed by Chiarini et al. (Reference Chiarini, Quadrio and Auteri2022d ) for LE and TE vortex shedding in the flow past 2-D rectangular cylinders. The two sheddings, of periods $T_{LE}$ and $T_{TE}$ , synchronise, and different stable cycles (periodic orbits) arise in the torus, with a long-time period $T_{lp}=p T_{TE}=q T_{LE}$ , where $p,q \in \mathbb {N}$ . In the present case, the frequency locking is visualised in figure 27. The characteristic frequencies of the vertical wake oscillation ( $St \approx 0.09-0.10$ ; squares) and LE vortex shedding ( $St \approx 0.18-0.20$ ; circles) increase weakly with $Re$ . The frequency of the LE vortex shedding is approximately twice that of the wake oscillation, meaning that two LE vortices are shed from the top/bottom LE shear layers during one wake oscillation period. The flow is periodic for $353 \leqslant Re \leqslant 357$ (lock-in region I) and $ 373 \leqslant Re \leqslant 375$ (lock-in region II): the LE vortex shedding synchronises with the vertical wake oscillation, and a limit cycle arises with $T_{lp}=T_{TE} = 2 T_{LE}$ (i.e. $p=1$ and $q=2$ ). Notably, the relative phase between the top and bottom vortex shedding is different in the two synchronised regimes. The LE vortices are indeed shed from the top and bottom LE shear layers in phase opposition in region I and in phase in region II (see figure 27 and the following POD analysis). This explains why the $St = 1/T_{LE}$ frequency is not visible in $S(F_z)$ for the lock-in region II: since the top and bottom LE vortices are shed in phase, they do not create any instantaneous unbalance of vertical force. By contrast, in region I the closed trajectory in the $F_z-F_x$ force diagram is not symmetric with respect to $F_z=0$ , meaning that the time-averaged $F_z$ is non-zero. More generally, the time-averaged top/bottom symmetry is lost for $345 \leqslant Re \leqslant 360$ (figure 24). Regarding the lateral aerodynamic force, the different phase locking in the two synchronised regimes results in a different behaviour: $F_y \approx 0$ at all times in region I, while $F_y$ oscillates with $St = 1/T_{TE} = 1/(2 T_{LE})$ in region II.

Figure 28. Frequency-locking regime for $L=5$ , $W=5$ and $Re=353$ (lock-in region I). Lateral view of isosurfaces $\lambda _2=-0.05$ coloured by streamwise vorticity (blue-to-red colour map for $-1 \leqslant \omega _x \leqslant 1$ ). The snapshots are separated in time by $T/8$ , where $T$ is the period of the wake oscillation. Black/red labels refer to vortices shed in the considered/previous period. Here LEV1, LEV2 and LEV3 refer to LE vortices shed from the top and bottom LE shear layers, respectively; HV1, HV2 and HV3 indicate HVs that arise in the wake once LEV1, LEV2 and LEV3 cross the TE.

We now illustrate the interaction between the LE and TE vortex shedding in region I. Figure 28 shows eight snapshots of the flow at $Re=353$ , equispaced in one period $T_{TE}$ . Black and red labels refer to vortices shed from the prism in the current and previous periods, respectively. Vortices shed from the top and bottom LE shear layers are in phase opposition. At $t = t_1$ a spanwise-aligned vortex, LEV1, is generated from the top LE shear layer and convected downstream (see $t=t_2$ ). At $t=t_3$ , LEV1 crosses the TE corner in phase with the upwards motion of the wake and rolls up ( $t_4$ to $t_6$ ), generating a large head-up hairpin vortex (HV1) that is then shed downstream ( $t_7$ and $t_8$ ). The same dynamics is observed over the bottom side. A spanwise-aligned vortex LEV3 is shed from the LE shear layer at $t=t_3$ . It crosses the bottom TE at $t=t_5$ and rolls up, generating a head-down hairpin vortex HV3 at $t=t_6$ . Moving again to the top side, at $t=t_5$ the second spanwise-aligned vortex LEV2 is shed from the top LE shear layer. It crosses the TE corner at $t=t_5$ in phase with the downwards motion of the wake and generates a hairpin vortex HV2 that is then connected with the HV3 hairpin vortex.

In region II, LE vortices interact with the vertical wake oscillation in a similar way to generate the large HVs that characterise the wake dynamics. However, the top/bottom LE vortices are shed in phase, and thus, cross the top/bottom TE in phase (figure 27 b, bottom row). When a top/bottom LE vortex crosses the top/bottom TE in phase with an upward/downward motion of the wake, a head-up/head-down hairpin-like vortex arises in the wake.

In the aperiodic regimes ( $Re \in [345,350]$ , $[360,365]$ and $[380,385]$ ), a similar interaction between the LE vortex shedding and the wake unsteadiness as in the lock-in region I is observed, although not fully periodic.

Figure 29. The POD analysis for $L=5$ and $W=5$ in the range $305 \leqslant Re \leqslant 385$ . (a) Circles and squares refer to the unsteady wake mode ( $St \approx 0.09$ ) and LE vortex shedding mode ( $St \approx 0.19$ ), respectively. When the two phenomena are distributed over several POD modes, the sum of their energy content is reported in the right panel. (b) Lateral view of the POD mode associated with LE vortex shedding. Isosurfaces of $\lambda _2$ coloured by streamwise vorticity. From left to right: aperiodic regime at $Re=345$ , periodic regimes at $Re=357$ and $Re=375$ .

We use POD to further examine the dependence of the flow structure on $Re$ (figure 29). In agreement with the above discussion, the dominant POD modes are associated with the wake unsteadiness and with the LE vortex shedding, and exhibit a single frequency of $St \approx 0.09$ and $St \approx 0.19$ , respectively. The POD modes associated with the wake unsteadiness mainly evolve in the wake and feature large-scale head-up and head-down HVs downstream of the TE (not shown for brevity). The POD modes associated with the LE vortex shedding, instead, originate over the top and bottom sides where they feature spanwise-aligned structures (figure 29 b). The spatial structure of the LE vortex shedding POD modes changes with $Re$ , as visible along the sides of the prism. In the aperiodic regime ( $Re=345, 365, 385$ ) and in lock-in region I ( $Re=357$ ), the modes are antisymmetric with respect to an inversion of the $z$ axis, with $\omega _x(x,y,z) = \omega _x(x,y,-z)$ . This agrees with the out-of-phase shedding of vortices from the top and bottom LE shear layers. In lock-in region II ( $Re = 375$ ), instead, the modes exhibit a top/down symmetry with $\omega _x(x,y,z) = -\omega _x(x,y,-z)$ , which is consistent with in-phase LE vortex shedding. Also, the POD confirms that in the aperiodic $aS_yS_z$ regime the relative intensity of the LE vortex shedding increases with $Re$ (see right plot of figure 29 a). However, the wake unsteadiness dominates in the periodic regimes.

4.3.3. Large $Re$ : aperiodic regime

Figure 30. Unsteady regimes for $L=5$ and $W=5$ at larger Reynolds numbers $400 \leqslant Re \leqslant 600$ . (a) Frequency spectra of the aerodynamic forces $F_y$ (left) and $F_z$ (right). (b–c) Structure of the flow at $Re=400$ (top) and $Re=500$ (bottom). (b) Isosurfaces of $\lambda_2 = -0.05$ coloured by streamwise vorticity. (c) Force diagrams $F_y-F_x$ (left) and $F_z-F_x$ (right).

For $Re \geqslant 400$ , the synchronisation between the LE and TE vortex shedding is lost, and the flow becomes progressively more chaotic. Like for lower $Re$ , the frequency of the wake oscillation remains approximately constant ( $St \approx 0.095{-}0.1$ for $Re\in [400,600]$ ), while that of the LE vortex shedding keeps increasing ( $St \approx 0.21{-}0.32$ ). The increase in $Re$ is accompanied by a slight increase in the oscillation amplitude in the $y$ direction, as visualised in figure 24. For $Re \geqslant 500$ , a third peak appears in $S(F_y)$ at $St \approx 0.125$ , and a shedding of HVs occurs also along the lateral sides of the prism (figure 30).