Square cylinder in the interface of two different velocity streams

Under consideration for publication in J. Fluid Mech. 1 Banner appropriate to article type will appear here in typeset article Square cylinder in the interface of two1 different-velocity streams2 R. El Mansy1, W. Sarwar2, J.M. Bergadà1and F. Mellibovsky2†3 1Fluid Mechanics Department, Universitat Politècnica de Catalunya, 08034, Barcelona, Spain4 2Department of Physics, Aerospace Engineering Division, Universitat Politècnica de Catalunya, 08034,5 Barcelona, Spain6 (Received 31 May 2022)7 We investigate the incompressible flow past a square cylinder immersed in the wake of an8 upstream nearby splitter plate separating two streams of different velocity. The bottom stream9 Reynolds number, based on the square side, 𝑅𝑒𝐵=56 is kept constant while the top-to-bottom10 Reynolds numbers ratio 𝑅≡𝑅𝑒𝑇/𝑅𝑒𝐵is increased in the range 𝑅∈ [1,6.5], corresponding11 to a coupled variation of the bulk Reynolds number 𝑅𝑒 ≡ (𝑅𝑒𝑇+𝑅𝑒𝐵)/2∈ [56,210]12 and an equivalent nondimensional shear parameter 𝐾≡2(𝑅−1)/(𝑅+1) ∈ [0,1.4667].13 The onset of vortex-sheddingis pushed to higher 𝑅𝑒 as compared to the square cylinder14 in the classic configuration. The advent of three-dimensionality is triggered by a mode-C-15 type instability, much as reported for open circular rings and square cylinders placed at an16 incidence. The domain of minimal spanwise-periodic extension that is capable of sustaining17 spatio-temporally chaotic dynamics, namely one accomodating about twice the wavelength of18 the dominant eigenmode, has been chosen for the analysis of the wake transition regime. The19 path towards spatio-temporal chaos is, in this minimal domain, initiated with a modulational20 period-doubling tertiary bifurcation that also doubles the spanwise periodicity. At slighlty21 higher values of 𝑅, the flow has become spatio-temporally chaotic, but the main features22 of mode C are still clearly distinguishable. Although some of the nonlinear solutions found23 along the wake transition regime employing the minimal domain might indeed be unstable24 to long wavelength disturbances, they still are solutions of the infinite cylinder problem and25 are apt to play a relevant role in the inception of spatio-temporally chaotic dynamics.26 Key words: Authors should not enter keywords on the manuscript, as these must be chosen by27 the author during the online submission process and will then be added during the typesetting28 process (see Keyword PDF for the full list). Other classifications will be added at the same29 time.30 MSC Codes (Optional) Please enter your MSC Codes here31 †Email address for correspondence: fernando.mellibov[email protected] Abstract must not spill onto p.2 2 1. Introduction32 The flow past circular cylinders has been extensively studied as a paradigm of bluff body33 aerodynamics (Williamson 1996c), for it involves several interesting phenomena which34 include, but are not limited to, laminar and turbulent boundary layer separation, vortex35 shedding, or detached shear layer and wake instabilities. Flow unsteadiness, spatio-temporal36 chaos and turbulence are a source of aerodynamic noise and vortex-induced vibration, and37 crucially affect aerodynamic forces, to name but a few issues that are relevant from the38 engineering applications viewpoint. The sole governing parameter, the Reynolds number, is39 defined as 𝑅𝑒 =𝑈𝐷/𝜈, where 𝑈is the free-stream velocity, 𝐷the cylinder diameter and 𝜈40 the kinematic viscosity of the fluid.41 Square (Durao, Heitor & Pereira 1988; Lyn, Einav, Rodi & Park 1995; Luo, Chew & Ng42 2003) and rectangular (Okajima 1982; Norberg 1993) cylinders have also been used, albeit43 to a lesser extent, as an archetype for bluff body aerodynamics. The same definition as for44 the circular cylinder is used for the Reynolds number, except that 𝐷is taken to be the square45 cylinder side length (or the cross-stream side length for a rectangle).46 In real-world applications, bluff bodies are often submerged in boundary layers (Hwang47 & Yao 1997) or wakes and the flow past them is decisively modified by the inhomogeneous48 velocity profiles of the incoming upstream flow. This is the case of a bridge pillar that is close49 to the river shore, long-span bridges immersed in an atmospheric boundary layer (Ahmed50 & Rajaratnam 1998), underwater pipes near the seabed or subject to strong currents, or a51 building (or motor vehicle) in the wake of an upstream building (vehicle) (Bailey & Kwok52 1985; Bhatt & Alam 2018).53 The simplest model for such a situation is the uniform planar shear flow past a circular54 (Kiya, Tamura & Arie 1980; Tamura, Kiya & Arie 1980; Kwon, Sung & Hyun 1992) or55 square (Ayukawa, Ochi, Kawahara & Hirao 1993; Hwang & Sue 1997) cylinder, where the56 homogeneous incoming streamwise velocity is replaced with a linear profile (constant cross-57 stream gradient of streamwise velocity and, therefore, constant non-null shear). The shear58 parameter is defined as 𝐾≡𝐷𝐺/𝑈𝑐, with 𝐷the cylinder characteristic length (diameter59 and side for circular and square cylinders, respectively), and 𝑈𝑐and 𝐺=(𝜕𝑢/𝜕𝑦)𝑦=𝑦𝑐the60 upstream streamwise velocity and dimensional cross-stream gradient of streamwise velocity,61 respectively, at cylinder mid-height.62 Ocasionally, the body of interest lies in the way of a thin shear layer rather than a smooth63 shear profile. This happens, for example, in the near wake of lift-producing devices such as64 airfoils, stator vanes or rotor blades of compressors, turbines or fans. Struts or rods supporting65 a structural casing are examples of objects subject to this type of incoming flows. This is66 precisely the kind of situations we intend to model here by placing a bluff body, the square67 cylinder, in the interface of two streams with different velocity. The wake-body interaction68 problem usually considers a body, streamlined or bluff, placed in the wake of another bluff69 body. The configuration in which a bluff body is placed in the wake of a streamlined body70 such as we intend to address here has very seldom been considered in the literature, and71 then always placing a cylinder or strut in the wake of an airfoil (Zhang, Huang & Zhou72 2005; Niu, Li & Wang 2021). However distant this problem may seem at first from that73 of homogeneous upstream shear, the effects associated to the different velocity seen by the74 upper and lower sides of the bluff body might be expected -and will indeed be shownto75 bear striking resemblance.76 The symmetric and steady flow past a circular cylinder undergoes a supercritical Hopf77 bifurcation, the primary instability, at around 𝑅𝑒𝐻≃47 with dimensionless frequency78 (Strouhal number) 𝑆𝑡𝐻≃0.12 (Provansal et al. 1987; Norberg 1994), resulting in a time-79 periodic two-dimensional solution that consists in the alternated shedding of opposite-signed80 3 vortices from either side of the cylinder – a flow configuration commonly referred to as81 Kármán vortex street (von Kármán 1911, 1912). Although the spatial Z2symmetry associated82 to vertical reflection about a diametral plane aligned with the incoming flow is broken83 (Marques et al. 2004), a spatio-temporal Z2symmetry persists in the form of flow invariance84 upon the combined effect of evolution by a half period followed by reflection about the same85 original reflection-symmetry plane.86 The periodic and space-time-symmetric two-dimensional vortex-shedding solution might87 be observed in experiments all the way up to 𝑅𝑒 ≲190, three-dimensionality consistently88 arising from this point on (Williamson 1996a). Two distinct three-dimensional vortex-89 shedding modes have been reported in the so-called wake transition regime, whose inception90 results in two corresponding discontinuities of the Strouhal number (𝑆𝑡) dependence on 𝑅𝑒91 (Williamson 1988). The first one, mode A, is characterised by the onset of vortex loops that92 are stretched by shear into streamwise vortex pairs of spanwise wavelength around 3∼4𝐷93 and has been shown to persist at flow regimes as low as 𝑅𝑒 ≳180, thus coexisting with94 the two-dimensional solution over a small range of Reynolds numbers (Williamson 1996b).95 Mode A is only regularly patterned at the early stages of inception (as it grows in time on top of96 the two-dimensional solution) and then only transiently, but soon after develops intermintent97 large-scale spot-like wave dislocations that render the spanwise structure rather irregular98 (Williamson 1992). The second, mode B, arises at slightly higher values of the Reynolds99 number 𝑅𝑒 ≳250 and exhibits a fairly regular spanwise pattern with a shorter characteristic100 wavelength of about ∼1𝐷(Williamson 1996b). Mode-B-type vortical structures pervade101 the near wake even at flow regimes where turbulence has already set in at much higher102 Reynolds numbers in excess of 1000 (Mansy et al. 1994). Floquet stability analysis has103 shown that mode A emanates from a secondary instability of the two-dimensional periodic104 vortex-shedding solution at 𝑅𝑒𝐴=188.5±1with wavelength 𝜆𝐴 𝑧=3.96 ±0.02, and105 happens to be subcritical (Henderson & Barkley 1996), hence the hysteretical flow behaviour.106 Meanwhile, mode B seems to be related to a second instability of the same solution that107 occurs supecritcally at 𝑅𝑒 =259 ±2with 𝜆𝐵 𝑧=0.822 ±0.007 (Barkley & Henderson108 1996). The critical values of the parameters at bifurcation have since been further refined to109 (𝑅𝑒𝐴, 𝜆𝐴 𝑧)=(190.2±0.02,3.966±0.002)and (𝑅𝑒𝐵, 𝜆𝐵 𝑧)=(261.0±0.2,0.825±0.002)using110 asymptotically large domains (Posdziech & Grundmann 2001). A third mode, consisting of111 a complex-conjugate pair of eigenvalues and dubbed QP on account of its introducing quasi-112 periodicity into the flow, has been identified as dominant at intermediate wavelengths of113 about ∼2𝐷, in between those characterising modes A and B (Blackburn & Lopez 2003).114 Mode QP bifurcates at 𝑅𝑒𝑄𝑃 ≃377 and generates branches of unstable, and therefore not115 experimentally realisable, quasi-periodic states (Blackburn et al. 2005).116 The spatio-temporal Z2symmetry of the two-dimensional vortex-shedding regime coexists117 with the spanwise invariance of the infinite-cylinder flow problem, represented by the118 orthogonal group O(2)=Z2×SO(2), which includes reflection about any discretionary plane119 that is orthogonal to the spanwise direction (Z2) and every arbitrary translation along the120 span (special orthogonal or rotation group SO(2), which is isomorphic to the circle group121 S1). Notice that translations Δ𝑧in a periodic domain of length 𝐿𝑧may be reinterpreted as122 rotations of angle Δ𝜃=2𝜋Δ𝑧/𝐿𝑧. Systems with Z2×O(2) symmetry, where the Z2refers to123 a spatio-temporal symmetry, rather than simply spatial, and O(2) to space invariance, admit124 two types of synchronous codimension-one bifurcations, one preserving the space-time Z2 125 symmetry and the other one breaking it (Marques et al. 2004). Two-dimensional time-periodic126 vortex-shedding past circular and square cylinders belong to this symmetry class and, among127 the secondary instabilities that three-dimensionalise the flow, mode A is triggered by a Z2-128 preserving bifurcation, while mode B is induced by one that breaks it (Blackburn et al. 2005).129 The symmetries of both modes A and B are exemplified in figure 1through pairs of snapshots130 4 Re α, K, Γ A QP B C 𝑡(𝐷/𝑈)mode A mode B mode QP 0 𝑇/2 Figure 1: Sketch depicting the unfolding in parameter space of the secondary bifurcation curves of two-dimensional vortex shedding past a symmetric bluff body. The abscisa axis represents a top-bottom symmetry-breaking parameter (incidence 𝛼, shear parameter 𝐾or circular ring aspect ratio Γ≡𝑑/𝐷,𝑑and 𝐷being the ring and cross-sectional diameters, respectively). The snapshot pairs, taken half a vortex-shedding cycle apart, correspond to modes A, B and QP for the straight square cylinder at zero incidence subject to homogeneous incoming upstream flow at 𝑅𝑒 =205 (the squares in the bifurcation diagram, corresponding to the symmetric case). Mode C, not depicted, corresponds to a mere frequency locking of mode QP that occurs when breaking the symmetry (bullet), and is therefore barely distinguishable from a symmetry-broken QP mode. Shown are 𝑄=0.05 isosurfaces coloured by 𝜔𝑥∈ [−0.1,0.1]𝑈/𝐷(black to yellow) and 𝜔𝑧∈ [−0.1,0.1]𝑈/𝐷(blue to red). taken a half vortex-shedding period apart for the classic configuration of the homogeneous131 incoming flow past a straight square cylinder at zero incidence and 𝑅𝑒 =205. Notice that the132 signs, and therefore also the colours, of both axial (𝜔𝑥) and spanwise (𝜔𝑧) vorticity switch133 under the space-time symmetry operation for symmetry-preserving modes. Whenever the134 bifurcation in a system with the aforementioned symmetries involves a complex-conjugate135 pair that is non-resonant with the destabilising two-dimensional time-periodic and space-time136 symmetric solution, three quasi-periodic solution branches arise, namely a pair of symmetry-137 conjugate modulated travelling waves and a third of modulated standing waves. Only one of138 the two distinct types of solution branches -either the pair of travelling or the standing waves-139 might be stable at a time (Marques et al. 2004). The third three-dimensionalising secondary140 instability of the two-dimensional time-periodic wake past both circular and square cylinders141 corresponds precisely to a quasi-resonant quasi-periodic subcritical bifurcation. This is mode142 QP, shown in figure 1for the square cylinder in the classic configuration. The fundamental143 frequency is nearly half that of the Kármán frequency, but the flow fields do not exactly repeat144 every two vortex-shedding cycles, the mode being in fact quasi-periodic. The two symmetry-145 conjugate branches of modulated travelling-wave solutions add an unstable eigenmode to146 the count of the already unstable two-dimensional solution, while the modulated standing147 wave adds two (Blackburn et al. 2005). The neighbouring 1:4 resonant case, which would148 correspond to a period-doubling bifurcation, is a codimension-two bifurcation and would149 therefore require the tuning of a second parameter beside the Reynolds number for a complete150 unfolding. The symmetry group of the two-dimensional vortex-shedding regime admits also151 a number of mixed-mode bifurcations and strong 1:1 and 1:2 resonances, all codimension-152 two, none of which seems to bear any relevance to the cylinder wake problem (Marques et al.153 2004).154 Square-cylinder wake dynamics bears compelling resemblance to that past a circular155 cylinder. The symmetries of the problem are the same and the primary instability leads156 to a two-dimensional time-periodic and space-time symmetric vortex-shedding state at the157 slighlty lower 𝑅𝑒𝐻≃45 and 𝑆𝑡𝐻≃0.10 (Norberg 1996; Park & Yang 2016). There are158 however notable differences that concern the location where the boundary layer separates159 Focus on Fluids articles must not exceed this page length 5 from the cylinder surface. While separation points can migrate freely on the surface of a160 circular cylinder, they are bound to coincide with the corners of a square or rectangular161 cylinder. As it happens, separation occurs from the rear corners only at very low Reynolds162 number, and then only after reattachment from an initial separation from the front corners163 (Okajima 1982; Robichaux et al. 1999; Yoon et al. 2010).164 The flow past a square cylinder also exhibits mode Aand B-type structures in the wake165 transition regime (Sohankar et al. 1999; Saha et al. 2003; Luo et al. 2003; Bai & Alam 2018),166 but their respective occurrence starts at lower values of the Reynolds number 𝑅𝑒𝐴≃160 ±2167 and 𝑅𝑒𝐵≃204 ±5and present somewhat larger wavelengths 𝜆𝐴 𝑧/𝐷≃5.1±0.1and168 𝜆𝐵 𝑧/𝐷≃1.3±0.1at onset (Luo et al. 2007). While early experiments failed to detect169 any hysteresis in the inception of mode A and no discontinuity in the Strouhal number170 dependence on Reynolds number was observed (Luo et al. 2003), later experiments that171 strived to accurately resolve variations in the driving parameter produced a small hysteretical172 region (Luo et al. 2007; Tong et al. 2008).173 Linear instabilities akin to modes A and B of the flow past a circular cylinder have174 been identified through Floquet stability analysis also in the wake transition regime of the175 flow past a square cylinder (Robichaux et al. 1999), alongside a third, subharmonic, quasi-176 periodic mode (See the snapshots in figure 1) . As for the circular cylinder, modes A and B177 preserve and break, respectively, the spatio-temporal Z2symmetry of the two-dimensional178 vortex-shedding solution, but occur at lower 𝑅𝑒𝐴≃164 and 𝑅𝑒𝐵≃197, with slightly179 longer 𝜆𝐴 𝑧/𝐷≃5.2and 𝜆𝐵 𝑧/𝐷≃1.1at bifurcation (Sheard et al. 2009; Choi et al. 2012).180 Modulated travellingand standing-wave solution branches also arise in the wake of a square181 cylinder following the bifurcation of a complex-conjugate pair (Blackburn & Lopez 2003),182 which is the counterpart to quasi-periodic mode QP of the circular cylinder. Mode QP, which183 bifurcates at 𝑅𝑒𝑄𝑃 ≃215 with 𝜆𝑧≃2.6for the square cylinder (Sheard et al. 2009), was184 originally mistaken for a subharmonic (period-doubling) mode and dubbed S (Robichaux185 et al. 1999). The blunder followed a shortcoming of the stability analysis method used, which186 was incapable of detecting bifurcations of a quasi-resonant quasi-periodic type. Unlike what187 happens for the flow past a circular cylinder, the symmetry-conjugate branches of modulated188 travelling-wave solutions issued from the QP bifurcation are supercritical and inherit the189 stability properties of the two-dimensional solution, while the modulated standing-wave190 branch, which is also supercritical but with a lesser slope at bifurcation, adds an unstable191 eigenmode (Blackburn et al. 2005).192 Nonlinear analysis using the Landau equation as a model for the secondary instability of the193 flow past a square cyilinder points to a supercritical nature of mode A at bifurcation (Sheard194 et al. 2009), as opposed to what happens for the circular cylinder (Henderson & Barkley195 1996), and in overt contradiction with experimental observation (Luo et al. 2007). This196 dispute as to the subcritical or supercritical nature of mode A between numerical simulation197 and experiment has not yet been settled to the authors knowledge.198 The relation between the instabilities (both primary and secondary) in the wake of square199 and circular cylinders has recently been elucidated by the numerical smooth transformation200 of the former into the latter by gradual rounding of the corners (Park & Yang 2016). The201 primary Hopf bifurcation that brings about two-dimensional vortex shedding is initially202 slightly delayed as the cylinder geometry evolves from circular to square, but the trend203 is reversed halfway and the critical value brought down to 𝑅𝑒𝐻≃44.7. The bifurcation204 remains supercritical all along. Secondary three-dimensionalising instabilities also evolve205 continuously and uneventfully, gradually advancing the occurrence of the bifurcations of all206 three-modes, A, B and QP, to lower critical values of 𝑅𝑒. The order of bifurcation is not207 altered, but the bifurcation points get closely packed, with mode QP strongly promoted for208 6 the square in comparison to the circular shape. Meanwhile, the wavelength at criticality is209 slightly but steadily increased for all three modes in the morphing from circular to square.210 The top-bottom Z2reflection symmetry of the flow past an infinitely long cylinder might211 be broken in several possible ways. For a circular cylinder, the symmetry disruption might212 be achieved by introducing curvature along the spanwise direction turning the cylinder into213 a ring (Monson 1983; Leweke & Provansal 1994, 1995; Sheard et al. 2003), by applying214 rotation about its centerline (Kang et al. 1999; Mittal & Kumar 2003), by subjecting it to a215 non-uniform upstream velocity profile (incoming shear flow) (Jordan & Fromm 1972; Kiya216 et al. 1980; Tamura et al. 1980; Park & Yang 2018), or by combining incoming shear with217 rotation (Yoshino & Hayashi 1984; Sung et al. 1995), among other options. Upstream shear218 is also an option for breaking the symmetry of the square cylinder problem (Hwang & Sue219 1997; Saha et al. 1999; Sohankar et al. 2020), as also is placing the cylinder at an incidence220 with respect to the incoming flow (Norberg 1993; Sohankar et al. 1998; Tong et al. 2008;221 Yoon et al. 2010), or combining both effects together. In most cases, the two-dimensional222 time-periodic vortex-shedding solution persists upon deliberately breaking the spatial Z2 223 symmetry of the problem, but the space-time Z2symmetry is no longer fulfilled (Blackburn224 & Sheard 2010).225 The primary instability of the flow past an open ring leads to an axisymmetric time-226 periodic vortex-shedding regime (Monson 1983; Leweke & Provansal 1994), analogous to227 the two-dimensional vortex-sheding regime past a circular cylinder but obviously lacking its228 spatio-temporal Z2symmetry. Numerical studies have shown that, besides the instabilities229 related to the classic modes A and B, a subharmonic mode C also emerges and is the230 dominant secondary instability for rings of aspect ratio around Γ≡𝑑/𝐷=5, defined as the231 quotient between the diameter of the circle described by the cylinder axis (𝑑) and the cylinder232 diameter itself (𝐷) (Sheard et al. 2003). Both experiments and direct numerical simulations233 show that the secondary linear instability develops nonlinearly into a period-doubled vortex-234 shedding solution with a distinct wavelength somewhere in between those predicted by235 Floquet stability analysis for modes A and B in the ring wake (Sheard et al. 2005a,b). Although236 the solution is period-doubled, aggregate quantities such as aerodynamic forces preserve the237 original period. Two instants exactly one shedding cycle apart are mutually related by an238 appropriate symmetry operation (any of either a spanwise/azimuthal rotation by half the239 angular wavelength or reflection about some collection of appropriately chosen diametral240 planes) and the period-doubling does not initiate a period-doubling cascade. Instead, the241 mode-C structures that characterise the wake after the secondary bifurcation are replaced by242 mode-A-type structures when the Reynolds number is further increased. The scenario that243 follows seems to be analogous to that for the circular cylinder244 A subharmonic instability analogous to mode C is also dominant for a square cylinder245 placed at moderate incidence angles 𝛼(Sheard et al. 2009). Mode A, which was originally246 believed to be dominant for all incidences following experiments that unfortunately failed to247 check the flow topology at intermediate values of 𝛼(Tong et al. 2008), is in fact overtaken248 by mode C for 𝛼≳10.5◦(Yoon et al. 2010; Sheard 2011), which is in turn outdone by249 another mode of characteristics similar to those of mode A for 𝛼≳26◦. This second mode250 𝐴′is distinct from the one evolving from 𝛼=0◦in that it evolves from the space-time251 symmetric mode A corresponding to 𝛼=45◦. Numerical evidence seems to discard any252 smooth connection between modes A and 𝐴′by mere continuous tilting of the cylinder253 from one incidence to the other, although the physical flow mechanisms at play appear to be254 the same. Mode-B-type structures have also been detected in experiments above a critical255 Reynolds number that also evolves smoothly as the incidence angle is changed across the full256 range (Tong et al. 2008). The presence of mode-B structures in the wake of a square cylinder257 at incidence has been confirmed by direct numerical simulation at sufficiently high values of258 7 the Reynolds number 𝑅𝑒 ≳200 (Sheard et al. 2009), which in this case is usually defined259 with the cross-stream projected height 𝐷(cos 𝛼+sin 𝛼)instead of just 𝐷.260 Mode QP of the flow past circular and square cylinders is only the third linear instability,261 after modes A and B, of the periodic and space-time symmetric two-dimensional vortex-262 shedding solution, and the associated growth rate is much smaller. It is therefore not to be263 expected that wake structures related to mode QP might be observed in actual experiments. It264 does however bear strong resemblance to the mode C observed in the wake behind open rings265 (Sheard et al. 2005a) of the right moderate aspect ratio and square cylinders at intermediate266 incidence angles (Sheard et al. 2009), regarding both spanwise wavelength and flow topology.267 As a matter of course, a weak disruption of the spatial Z2symmetry in problems belonging268 to the Z2×O(2) symmetry group will alter the nature and characteristics of the secondary269 bifurcations. Modes A and B may preserve their respective symmetries only approximately,270 but the inconmensurate frequencies of mode QP, which did not retain any remnant of the271 spatio-temporal Z2symmetry, might experience a locking into some strong resonance. As it272 happens, mode QP is quasi-resonant for both circular and square cylinder flows and, as the273 span is curved into a ring or the square tilted into incidence, the associated complex-conjugate274 pair of Floquet multipliers approaches the negative real axis, collides, and separates into a275 couple of negative real eigenvalues, i.e. two subharmonic/period-doubling modes (Blackburn276 & Sheard 2010; Sheard 2011). It is thus that mode QP evolves into mode C, which eventually277 overtakes modes A and B in driving the secondary instability once the reflection symmetry278 across the midplane has been sufficiently broken.279 The bifurcation scenario is clarified by the sketch in figure 1, which shows the schematic280 arrangement of bifurcation curves upon varying a top-bottom-symmetry-breaking parameter281 (for instance incidence 𝛼for a square cylinder, shear parameter 𝐾for either square or circular282 cylinders, or the ring aspect ratio Γfor circular rings).283 Upstream shear has been shown in experiments to delay the onset of periodic vortex284 shedding, i.e. 𝑅𝑒𝐻increases with 𝐾, to the point that shedding can be completely suppressed285 all the way up to 𝑅𝑒 < 220 (Kiya et al. 1980). This effect has also been observed in numerical286 simulation (Tamura et al. 1980). Other computational studies, however, did not detect the287 phenomenon despite exploring similar values of the parameters (Lei et al. 2000; Cao et al.288 2010). No satisfactory explanation for this discrepancy has been offered.289 Stability analysis seems to favour the notion that 𝑅𝑒𝐻is mostly unaffected by 𝐾, and290 that, if anything, it is marginally promoted to slightly lower values, from 46.5 for 𝐾=0291 to 45.5 for 𝐾=0.2(Park & Yang 2018). Floquet stability analysis shows that the wake292 transition regime is instead greatly affected by upstream shear. While modes A, B and QP293 bifurcate at successively large values (𝑅𝑒𝐴≃190)<(𝑅𝑒𝐵≃250)<(𝑅𝑒𝑄𝑃 ≃380)in294 the symmetric problem, mode QP locks into subharmonic mode C as 𝐾is increased and295 gradually overtakes mode B and mode A for 𝐾=0.1and 0.2, respectively (Park & Yang296 2018). As a matter of fact, modes A and B are pushed to higher 𝑅𝑒𝐴≃240 and 𝑅𝑒𝐵≃300,297 while mode C is advanced to as low as 𝑅𝑒𝐶≃150 for 𝐾=0.2. While the critical spanwise298 wavelength of mode A slightly increases to 𝜆𝐴 𝑧≃4.2, mode B remains mostly unaltered at299 𝜆𝐵 𝑧≃0.8and mode C somewhat narrows to 𝜆𝐶 𝑧≃1.6for 𝐾=0.2. Numerical simulation has300 seen mode-A-type structures supressed, and with them three-dimensionality, at 𝑅𝑒 ≃200301 with 𝐾≳0.1, accompanied by the ensuing discontinuity in the 𝑆𝑡 vs 𝑅𝑒 relationship (Cao302 et al. 2010). This phenomenon has not been observed in experiments and the emergence of303 longitudinal vortical structures in the upstream flow may be accounted responsible.304 The onset of time-dependence 𝑅𝑒𝐻is slightly advanced and the mean drag coefficient 𝐶𝑑 305 and Strouhal number 𝑆𝑡 reduced for the flow past a square cylinder subjected to increasing306 𝐾(Cheng et al. 2007; Lankadasu & Vengadesan 2008). At these very low values of 𝑅𝑒, the307 8 lift force coefficient 𝐶𝑙is negative and its modulus increases with increasing 𝐾(Cheng et al.308 2007; Lankadasu & Vengadesan 2008), but the trend is reversed at higher 𝑅𝑒 (Lankadasu &309 Vengadesan 2011).310 The decrease in 𝑅𝑒𝐻has been belied by 2D simulations that explored larger values of 𝐾311 and reported suppression of two-dimensional vortex shedding behind a square cylinder at312 sufficiently low values of 𝑅𝑒 ≲200 whenever 𝐾≳𝐾𝑐exceeds a certain critical value that313 increases with 𝑅𝑒 (Ray & Kumar 2017; Cheng et al. 2007). As for the circular cylinder, an314 increasing 𝐾renders top vortices stronger and rounder, while bottom vortices become weaker315 and elongated and dissipate fast in the wake (Ray & Kumar 2017). As for the circular cylinder,316 the stagnation point systematically rises above the cylinder midplane upon increasing 𝐾at317 any value of 𝑅𝑒 (Cao et al. 2014).318 In the presence of upstream shear, the wake behind a square cylinder experiences a unique319 secondary three-dimensionalising instability characterised by a single mode (Lankadasu &320 Vengadesan 2011), instead of the two modes A and B that are sequentially observed in the321 wake transition regime for both circular and square cylinders in the absence of upstream322 shear. This single mode, which arises at 𝑅𝑒 ∈ (140,150)when 𝐾=0.2, was initially323 mistaken for mode B (Lankadasu & Vengadesan 2011) because of its similar wavelength, but324 the solution seems in fact period-doubled. Its three-dimensional structure appears shifted by325 half a wavelength after every vortex-shedding cycle, so that the instability would plausibly326 correspond to mode C, which would have taken precedence over modes A and B following327 the disruption of the spatial Z2symmetry. At 𝑅𝑒 =200 and 𝐾=0.2, three-dimensional328 simulations have shown mode-Aand mode-Btype structures on the lowand high-velocity329 halves, respectively, of the wake (Lankadasu & Vengadesan 2009), although nothing similar330 has been reported elsewhere.331 Here we choose to submerge the square cylinder in a thin shear layer by replacing the332 classic upstream homogeneous shear by a piecewise-constant velocity profile with the333 discontinuity separating the top and bottom homogeneous streamwise velocities precisely334 located at cylinder mid-height. This same setup has been used twice before, but the flow335 was expressly kept two-dimensional (Mushyam & Bergada 2017; An et al. 2020). In order to336 simulate experimentally reproducible conditions, we choose to separate the top and bottom337 homogeneous velocity streams by a flat plate, such that the shear layer results from the338 reunion, at its trailing edge, of the top and bottom boundary layers and develops smoothly339 downstream before reaching the cylinder. This kind of upstream conditions may be obtained340 experimentally in a wind or water tunnel (Loucks & Wallace 2012), and the development of341 the shear layers thus generated have been thoroughly investigated (Rogers & Moser 1992;342 Moser & Rogers 1993). The length of the plate, the gap left between its trailing edge and343 the cylinder and the Reynolds number below the plate are kept constant. The top-to-bottom344 stream velocity ratio has been varied and the various distinct flow topologies that arise345 classified. The flow mechanisms associated to each one of the solutions occurring along346 the path from steady to chaotic were thoroughly investigated and minutely described in a347 separate publication (El Mansy et al. 2022). Also the effects of upstream shear (as introduced348 through our singular choice of flow configuration) on aerodynamic performance trends was349 thoroughly dissected there. Here we go one step further and analyse and discuss in detail350 the instability mechanism that triggers the onset of three-dimensionality and subsequently351 imprints the ensuing evolution of three-dimensional vortical structures in the cylinder wake.352 Also, we further investigate the extent to which our more realistic and experimentally353 sound configuration, the step-like upstream velocity profile, echoes the case of homogenous354 upstream shear.355 The paper is structured as follows. The mathematical modelling is presented in section356 §2alongside the numerical approach undertaken. §3dissects the flow generated by the357 9 (a) (b) (c) Figure 2: Computational domain. (a) Sketch of the square cylinder. (b) Side and (c) front views of the computational domain.The colours indicate the boundary condition types: velocity inlet (red, with velocities 𝑈𝑇and 𝑈𝐵for the top and bottom streams, respectively), pressure outlet (blue), slip walls (green), non-slip walls (black) and periodic (orange). interaction of the two different-velocity streams as they flow parallel in the splitter plate358 wake towards the square cylinder. A temporal characterisation of the resulting flow past the359 cylinder is then given in section §4, followed by a minute dissection of the wake transition360 regime in section §5. Section §6briefly inspects the aerodynamic performances trends.361 Finally, a raw account of the separate effects of Reynolds number and top-to-bottom velocity362 ratio parameter is given in section§7, and the main results summarised and conclusions363 drawn in section §8.364 2. Mathematical modelling and numerical approach365 Figure 2presents the side and front views of the computational domain employed in the366 simulations. The square cylinder, of side 𝐷, is located at the origin of the domain, of size367 𝐿𝑥×𝐿𝑦=34.5𝐷×16𝐷, at a distance 𝐿𝑢 𝑥=9𝐷downstream from the domain inlet, and368 at mid-height, such that the top and bottom boundaries are at a distance 𝐿𝑦/2=8𝐷above369 and below the centreline (blockage ratio 𝐵=0.0625). All cases were studied with incidence370 𝛼=0◦(all sides are either parallel or orthogonal to the incoming flow direction). The splitter371 plate, of chord 𝐿=6.5𝐷and negligible thickness, extends horizontally from domain inlet at372 mid-height, thus leaving a 𝐺≡𝐿𝑢 𝑥−𝐿−𝐷/2=2𝐷gap between its trailing edge and the front373 face of the cylinder. The small distance left for the development of the shear layer has been374 checked to be sufficiently short to keep it thin and stable while at the same time long enough375 for the flow to freely adapt to the presence of the cylinder (see section 3). Meanwhile, the376 short splitter plate helps smooth the initiation of the shear layer while keeping the boundary377 layer thin and laminar. A downstream extent of 𝐿𝑑 𝑥=𝐿𝑥−𝐿𝑢 𝑥=25.5𝐷has been allowed to378 properly capture the wake and avoid interferences of the domain outlet with the flow around379 the cylinder and in its near-wake.380 The Navier-Stokes equations for incompressible flow, nondimensionalised with the square381 cylinder side length 𝐷and the mean upstream flow velocity 𝑈=(𝑈𝑇+𝑈𝐵)/2(𝑈𝑇and 𝑈𝐵 382 are the top and bottom stream constant velocities), read383 𝜕u 𝜕𝑡 + (u· ∇)u=−∇𝑝+1 𝑅𝑒 ∇2u, ∇ · u=0, (2.1)384 where u(r;𝑡)=(𝑢, 𝑣, 𝑤)and 𝑝(r;𝑡)are the nondimensional velocity and pressure, respec-385 tively, at nondimensional location r=(𝑥, 𝑦, 𝑧)and advective time 𝑡(units of 𝐷/𝑈). Here,386 16 (a) (b) (c) (d) Figure 6: |ˆ 𝐶𝑙|Power spectra of the lift coefficient for different velocity ratios. The insets show the time series of the lift coefficient. (a) 𝑅=3, (b) 𝑅=3.4, (c) 𝑅=3.8, (d) 𝑅=5.357. doubling bifurcation identified here is suggestive of a period-doubling cascade as the origin613 of chaotic dynamics, although this cannot be concluded from the rather coarse discretisation614 of parameter space investigated in the present study.615 5. Three-dimensionalisation of the flow: Wake transition regime616 The reflection symmetry about the horizontal midplane is broken by setting 𝑅≠1. In these617 conditions, the only remaining symmetries of the problem concern the spanwise direction.618 The problem is thus only invariant under the orthogonal group O(2)=SO(2)×Z2symmetry,619 involving both arbitrary spanwise shifts (SO(2)) and mirror reflections about all planes620 orthogonal to the spanwise direction (Z2). As long as the solutions remain two-dimensional,621 spanwise invarariance is preserved, and the onset of time dependence does not affect the only622 remaining symmetry.623 The bifurcation that triggers three-dimensionality, though, necessarily breaks the trans-624 lational invariance SO(2) and restricts the mirror symmetry Z2to a collection of 𝑧-planes625 17 equispaced at intervals 𝜆𝑧/2, where 𝜆𝑧is the fundamental spanwise wavelength of the626 arising three-dimensional solutions. Two-dimensional periodic solutions may still destabilise627 following both synchronous or quasiperiodic bifurcations, as was the case in the presence of628 the space-time Z2symmetry (Marques et al. 2004; Blackburn et al. 2005), but none of the two629 possible types of synchronous bifurcations can preserve a symmetry that was not there in the630 first place. If the quasiperiodic bifurcation becomes resonant and turns into a period-doubling631 bifurcation, the original time-periodicity is broken but still retained in the form of a space-632 time symmetry that recovers the solution after evolution over the original period (half the633 new period) followed by reflection about any 𝑧-plane located halfway between consecutive634 mirror-symmetry planes. In this guise, the period of the three-dimensional solution is double635 that of the destabilising two-dimensional solution, but shall appear to be the same whenever636 aggregate quantities are monitored. The phase maps in figure 5suggest that the advent of637 three-dimensionality follows precisely this path, as has been shown to occur for other systems638 breaking the space-time symmetry of the two-dimensional solution (Blackburn & Sheard639 2010) such as circular rings (Sheard et al. 2005b,a) or inclined cylinders (Sheard et al. 2009;640 Sheard 2011).641 5.1. Stability of two-dimensional vortex-shedding to three-dimensional disturbances642 The precise values of the parameters for which three-dimensional solutions arise in the wake643 transition regime, and the main features that characterise these nonlinear solutions in terms644 of frequency, symmetries and wake vortical structures, may be inferred from linear stability645 analysis of the underlying two-dimensional vortex-shedding solution. Also the requirements646 for the spanwise size of the computational domain at any given value of the Reynolds647 number and velocity ratio can be assessed by determining the range of wavenumbers that648 are unstable. This can be done with a full-fledged Floquet analysis, but given that only the649 dominant eignmode for each spanwise wavenumber, rather than the full spectrum (or even650 a subset of it), is actually required, the stability analysis has been carried out by simply651 time-stepping on a quasi 3D domain, adding to the two-dimensional field a unique spanwise652 Fourier mode of the chosen wavelength, randomly initialised to a very low energy level (See653 appendix A).654 Figure 7a illustrates the time-stepping-based stability analysis for (𝑅, 𝜆𝑧)=(3.4,2.5).655 The two-dimensional vortex-shedding solution whose stability to spanwise disturbances is656 being analysed is shown in the inset. The instantaneous spanwise vorticity 𝜔𝑧∈ [−2,2]657 colour map (blue for negative, yellow for positive) shows the Kármán vortex street, which658 consists of an array of strong clockwise vortices. Only the vortices from the top shear layer659 form due to the severe disruption of the top-bottom symmetry. Consecutive vortices are660 interconnected by weak elongated braids of opposite vorticity. After some initial transients,661 the energy associated to the only spanwise mode considered (𝐸1(𝑡), bottom) starts growing662 exponentially, with a barely perceptible superimposed oscillation that is inherited from the663 underlying two-dimensional vortex-shedding periodicity (𝐶𝑙, top). The exponential growth664 indicates that the 𝑅=3.4solution is already linearly unstable to 𝜆𝑧=2.5perturbations.665 We define a Poincaré section according to {𝐶𝑙=⟨𝐶𝑙⟩𝑡;¤ 𝐶𝑙>0}, where =⟨𝐶𝑙⟩𝑡is the time666 average of the lift coefficient of the two-dimensional periodic solution. An exponential fit667 (dashed line) to the Poincaré crossings in the linear growth regime (red crosses) provides668 an estimation of the modulus of the unstable multiplier |𝜇|. Notice that the modal energy669 oscillation associated to vortex shedding is factored out upon definition of the Poincaré670 section and has therefore no effect whatsoever on the estimation of the multiplier.671 The results of the stability analysis for varying 𝑅and 𝜆𝑧are presented in figure 7b. The672 actual onset of flow three-dimensionality occurs for 𝑅≃3.1. Below this value, all three-673 dimensional perturbations decay (the Floquet multiplier has |𝜇|<1) and the flow remains674 18 (a) (b) Figure 7: Stability analysis of the flow past a square cylinder in the interface of two different-velocity streams with 𝑅𝑒𝐵=56. (a) Time-evolution of the lift coefficient 𝐶𝑙 (top) and of the modal energy associated to the only spanwise-dependent Fourier mode 𝐸1 considered (bottom) for (𝑅, 𝜆𝑧)=(3.4,2.5). The red crosses indicate {𝐶𝑙=⟨𝐶𝑙⟩𝑡;¤ 𝐶𝑙>0}Poincaré crossings in the linear regime used for the exponential fit (red dashed line). The inset depicts the instantaneous (at the Poincaré section) spanwise vorticity 𝜔𝑧∈ [−2,2](blue to yellow) colour map of the (unstable) two-dimensional vortex-shedding solution for 𝑅=3.4. (b) Least stable Floquet multipliers for velocity ratios in the range 𝑅∈ [3,6.5]. The red cross corresponds to the case analysed in panel (a). r3.3 two-dimensional. The largest modulus Floquet multiplier corresponds to a fairly constant675 spanwise wavenumber of about 𝜆𝑧=2𝜋/𝛽𝑧≃2.4all the way up to 𝑅⩽4and then steadily676 increases to 2.8for 𝑅=6.5. Consequently, a spanwise domain size 𝐿𝑧=2.5(and integer677 multiples of it) constitutes a fair choice if the fastest-growing three-dimensional mode is to678 be represented in the simulation. The smooth evolution of the multiplier of the dominant679 mode of the eigenspectrum indicates that, despite its evolution in wavelength, the same mode680 is responsible for the instability over the whole range of 𝑅explored. At 𝑅=6.5, however, a681 second unstable mode, of much shorter associated wavelength 𝜆𝑧≃1.5has destabilised. The682 first mode is clearly dominant, but the presence of this second mode may play some role in683 shaping the small-scale vortical structures that are present in the wake once fully developed684 turbulence has kicked in.685 Not only the multiplier, but also the flow fields of the dominant eigenmode can be recovered686 from the stability analysis. They are simply provided by the sole non-homogeneous spanwise687 mode (or, equivalently, by subtracting the two-dimensional solution from the quasi-three-688 dimensional flow fields) as taken anywhere within the linear regime. The prevailing three-689 dimensional flow structures and the characteristic symmetries of the dominant eigenmodes690 will be discussed presently in section §5.2, alongside the corresponding nonlinear solutions691 at the same values of the parameters, to assist in gauging the extent to which the latter are692 determined by the former.693 5.2. Nonlinear three-dimensional solutions694 Three nonlinear three-dimensional solutions representing the three different types r3.4of temporal695 dynamics found along our coarse exploration of the wake transition regime have been singled696 out for a detailed analysis of the symmetries and the three-dimensional vortical structures697 present in the wake. In particular, a period-doubled, a period-four and a chaotic solution698 have been studied at 𝑅=3.4, 3.8 and 5.357, respectively. All three solutions have been run699 in the domain of minimal spanwise-periodic extent (𝐿𝑧=5) that is capable of sustaining700 19 (a) (b) 𝑤 Figure 8: Space-time properties of the 𝑅=3.4solution. (a) Dynamic evolution of the lift 𝐶𝑙(solid) and drag 𝐶𝑑(dashed) coefficients. Horizontal grey lines bound the absolute maxima and minima of the signals. (b) Space-time diagrams of spanwise velocity 𝑤(𝑥, 𝑦, 𝑧;𝑡)at probe location (𝑥, 𝑦)=(6.0,0.5). Reflection (dash-dotted white lines) and time-shift plus reflection (dashed white) symmetry planes are indicated. spatio-temporally chaotic dynamics, although the latter is presented in the wider 𝐿𝑧=10701 domain.702 Figure 8b contains space-time diagrams of spanwise velocity 𝑤(𝑥, 𝑦, 𝑧;𝑡)along a probe703 array located at (𝑥, 𝑦)=(6.0,0.5)alongside corresponding time series of 𝐶𝑙and 𝐶𝑑(panel a)704 for the three-dimensional periodic solution at 𝑅=3.4. The space-time diagrams correspond705 to 8 vortex-shedding cycles, as clear from the 𝐶𝑙and 𝐶𝑑time series, and their periodicity706 corresponds exactly with that of vortex shedding. The space-time diagrams show instead707 that the actual periodicity of the solution is twice that of vortex shedding, as previously708 unraveled by the corresponding phase maps in figure 5. The nature of the period doubling,709 which does not affect aggregate quantities, is now clearly explained as a result of the way710 in which the spanwise invariance has been disrupted upon three-dimensionalisation of the711 flow. The reflection Z2symmetry is only preserved at all times about planes located at712 𝑧=𝑧0+ (2𝑗)𝜆𝑧/4(white dash-dotted lines), where the origin for the spanwise coordinate has713 been chosen to enforce 𝑧0=0,𝜆𝑧=2.5here and 𝑗∈Z. Additionally, a space-time symmetry714 operation consisting in the evolution by half a period 𝑇/2, where 𝑇is the actual period of715 the solution corresponding to two vortex-shedding cycles, followed by reflection about any716 plane located at 𝑧=𝑧0+ (2𝑗+1)𝜆𝑧/4(white dashed lines) also leaves the solution invariant.717 The appropriate composition of the two symmetries shows that the solution is also invariant718 to evolutions by half a period 𝑇/2, followed by a spanwise shift by a half wavelength 𝜆𝑧/2.719 The ensuing period-doubling bifurcation is of an altogehter different nature. The solution720 at 𝑅=3.8has an actual period of four vortex-shedding cycles, yet aggregate quantities721 repeat every two vortex-shedding cycles. Close inspection of figure 9a shows how the 𝐶𝑙 722 and 𝐶𝑑time series have a period that is about twice that for 𝑅=3.4. The period doubling723 is more clearly evinced by the half frequency peak 𝑓2in figure 6c or the double loop phase724 map trajectory in figure 5a, but it also shows up in the 𝐶𝑙and 𝐶𝑑time series (notice how725 the peaks and valleys come short, every two vortex-shedding cycles, of the horizontal grey726 lines bounding the absolute maxima and minima of the signals.) The space-time diagram727 for 𝑤provide the full picture. The bifurcation that produces the period-doubled solution is728 spatially subharmonic in the sense that the spanwise wavelength is double that at 𝑅=3.4.729 It is a modulational instability to perturbations of a wavelength twice that of the bifurcating730 solution. This breaks the mirror symmetry and the only remaining symmetry leaves the731 20 (a) (b) 𝑤 Figure 9: Space-time properties of the 𝑅=3.8solution. (a) Dynamic evolution of the lift 𝐶𝑙(solid) and drag 𝐶𝑑(dashed) coefficients. Horizontal grey lines bound the absolute maxima and minima of the signals. (b) Space-time diagrams of spanwise velocity 𝑤(𝑥, 𝑦, 𝑧;𝑡)at probe location (𝑥, 𝑦)=(6.0,0.5). Time-shift plus reflection symmetry planes (dashed white) are indicated. solution invariant to evolution for half a period (two vortex-shedding cycles) followed by732 reflection about planes located at 𝑧=𝑧0+𝑗𝜆𝑧/2(white dashed line), where 𝜆𝑧=5now.733 A three-dimensional representation of the solution at 𝑅=3.4is shown in figure 10a734 at two consecutive crossings of the Poincaré section defined by 𝐶𝑙=⟨𝐶𝑙⟩𝑡and ¤ 𝐶𝑙>0.735 Spanwise mode zero, shown with a transparent iso-surface for 𝜔𝑧=±0.1, is taken to736 represent the Kármán vortices. The three-dimensional vortical structures present in the flow737 are exposed by means of the Q-criterion, which identifies vortical structures as regions where738 the second invariant of the velocity gradient tensor becomes positive (𝑄 > 0), i.e. the rotation739 rate overcomes the shear rate (Hunt 1988). Streamwise-cross-stream vortical structures are740 made visible through 𝑄=0.0002 isocontours coloured by streamwise vorticity 𝜔𝑥. A741 first visible effect of three-dimensionalisation is that of introducing spanwise modulation to742 Kármán vortices. Vorticity is no longer merely spanwise but has become slightly tilted and743 alternates positive and negative values of the streamwise component. Elongated streamwise-744 cross-stream vortices extend along the braids connecting consecutive Kármán vortices.745 The three-dimensional flow topology is very different from mode-Aand mode-B-type746 structures observed for circular (Williamson 1996c) and square (Bai & Alam 2018) cylinders747 in the wake transition regime. Instead, the streamwise-cross-stream vortices, in particular748 their wavelength and time periodicity, are highly reminiscent of mode QP in the wake of749 circular (Blackburn et al. 2005), square (Robichaux et al. 1999; Blackburn et al. 2005) and750 rounded-square (Park & Yang 2016) cylinders Some alterations in wake topology are however751 conspicuous: the positive Kármán vortices typically emanating from the bottom shear layer752 are absent and the elongated vortices in the braids that connected them to the preceding and753 following negative Kármán vortices have been spliced. As a matter of fact, the parallel is754 all the more convincing when a comparison is drawn between the present three-dimensional755 wake vortices and mode C structures in the wake of open rings (Sheard et al. 2005a), squares756 at incidence (Sheard et al. 2009) or cylinders submerged in upstream shear (Park & Yang757 2018). In all these problems, mode C results from the frequency locking of mode QP shortly758 after breaking the Z2symmetry about the horizontal mid plane of the classic configuration,759 i.e. the homogeneous incoming flow past an infinite cylinder at zero incidence.760 Comparing two snapshots exactly one Poincaré section crossing apart, the aforementioned761 space-time symmetries become apparent. The solution looks exactly the same but shifted by762 exactly half a wavelength in the spanwise direction or reflected about appropriate symmetry763 21 k (a) (b) 0 1 Figure 10: Instantaneous snapshots of (a) the three-dimensional solution at 𝑅=3.4 (Movie 1) and (b) corresponding dominant eigenmode of the underlying two-dimensional solution (Movie 2), at two consecutive crossings of the Poincaré section. Shown are iso-surfaces of the Q-criterion (𝑄=0.0002 for three-dimensional solution, arbitrarily chosen for the dominant eigenmode to ease comparison), coloured by streamwise vorticity (symmetric arbitrary scale for the eigenmode). Spanwise vortices are shown with transparent iso-surface for 𝜔𝑧=±0.1. The solution is replicated twice in the spanwise direction to 𝐿𝑧=10 only for visualisation purposes. planes as previously observed. The fully developed three-dimensional solution bears strong764 resemblance and shares the same symmetries with the dominant eigenmode of the underlying765 two-dimensional periodic solution at the same value 𝑅=3.4, as clear from figure 10. Here766 the most unstable mode of figure 7b for 𝜆𝑧=2.5has been represented using the Q-criterion767 and coloured by 𝜔𝑥(symmetric arbitrary range). The unstable two-dimensional solution is768 indicated by transparent 𝜔𝑧=±0.1iso-surfaces that clearly show the Kármán vortices. The769 topologies of the dominant eigenmode and of the three-dimensional solution are evidently770 related and so is the space-time symmetry dynamics as illustrated by snapshots taken one771 Poincaré crossing apart. Comparing mode C of figure 10b to mode QP of figure 1some772 intuition as to the morphing from one to the other can be gained. The similarities in the773 arrangement of wake vortices is evident provided that one allows for the suppression of the774 bottom vortices of the Kármán street and for the resulting streamwise splicing of every two775 consecutive braid vortices that follows from the breaking of the top-bottom symmetry.776 The symmetries and wavelength of the dominant eigenmode (and, with it, the nonlinear so-777 lution) make it analogous to the mode C that destabilises the axisymmetric/two-dimensional778 periodic solutions characteristic of the flow past a circular ring (Sheard et al. 2005b,a) or an779 inclined square cylinder (Sheard et al. 2009; Sheard 2011), which evolve from the respective780 modes QP of the top-bottom symmetric cases and take precedence over modes A and B781 when the symmetry has been sufficiently disrupted. r3.5Mode C, its evolution from mode QP782 in a codimension-2 bifurcation, and its overtaking modes A and B, The same bifurcation783 22 scenario has been recently exposed in the somewhat related case of a circular cylinder in784 homogeneous upstream shear (Park & Yang 2018). Here we are only varying one parameter785 𝑅, so that modes A and B are never encountered along the particular 𝑅𝑒 −𝐾path followed.786 The rapid increase of the upstream velocity ratio and its associated mean shear is probably787 suppressing them before any trace can be even found in the Floquet spectrum.788 The dominant eigenmode of the underlying two-dimensional solution is the same for all789 values of 𝑅investigated here. With minor variations, it remains topologicaly identical from790 as low as 𝑅=3, which corresponds to a stable case, to as high as 𝑅=6.5, where the three-791 dimensional solution has evolved into chaotic dynamics. The most unstable (or least stable)792 wavelength remains around 𝜆𝑧≃2.5for low values of 𝑅and then gradually grows as 𝑅is793 increased. In particular, the most unstable mode remains the same for 𝑅=3.8, for which it has794 been seen that a second bifurcation has taken place and the space-time symmetry of 𝑅=3.4is795 no longer present. We surmise that the period doubling bifurcation, that appears to arise from796 a spanwise subharmonic/modulational instability of the 𝑅=3.4solution to perturbations of797 double its spanwise wavelength, may also affect the underlying two-dimensional flow. Since798 this flow is already unstable to the eigenmode of figure 10b, computing the second unstable799 mode is not within the reach of mere time-stepping. Besides, it may also be the case that this800 second instability is related to the interactions of unstable wavelengths across the continuous801 eigenspectrum. For 𝐿𝑧=5the domain admits instability to perturbations of wavelength802 𝜆𝑧=𝐿𝑧/𝑙, with 𝑙∈Z, or, equivalently, wavenumber 𝛽𝑧=2𝜋𝑙/𝐿𝑧. Now, for 𝑅=3.4,803 the unstable dominant mode can only fit twice or thrice within the domain considered and804 the system seems to choose 𝜆𝑧=2.5as the prevailing instability. At 𝑅=3.8, instead, the805 unstable band has grown large enough to allow for the dominant mode to fit either twice,806 thrice, four times or even just once. The three-dimensional solution has 𝜆𝑧=𝐿𝑧=5but it807 retains a 𝜆𝑧≃2.5-dependence to a large extent. This suggests that the modes with 𝜆𝑧=2.5808 and 𝜆𝑧=5, which in fact belong to the same branch and are therefore related by smooth809 continuation in the wavelength parameter, might be competing and that none completely810 prevails over the other.811 Figure 11a shows snapshots of the three-dimensional solution for 𝑅=3.8at four812 consecutive crossings of the Poincaré section. The original spanwise wavelength 𝜆𝑧=2.5813 is preserved to a large extent, but the actual periodicity is now the full 𝜆𝑧=𝐿𝑧=5. The814 similarity with 𝑅=3.4is apparent, but the elongated vortices along the braids reach further815 downstream into the wake. Although crossings 0 and 2 (also 1 and 3) look very much alike,816 careful inspection reveals that very slight differences exist that are related with the period-817 doubled nature of the solution. Also hard to notice, but nevertheless present, is the symmetry818 that leaves the solution invariant after evolution over two consecutive crossings of the Poincaré819 section followed by reflection about appropriately chosen streamwise-cross-stream planes.820 The spanwise periodicity 𝜆𝑧=5of the nonlinear solution is doubtless unrelated to a821 mode A instability despite the compatible wavelength. The typical features of mode A are822 absent from the wake, while mode C vortical structures clearly prevail, albeit with a two-fold823 subharmonic spanwise modulation. In order to discard any involvement of mode A in the824 solution observed at 𝑅=3.8, four consecutive normalised snapshots of the dominant mode825 for wavelengths 𝜆𝑧=2.5and 5, both unstable according to figure 7, are shown in figures 11b826 and c, respectively. The spatial structure of the dominant eigenmode is clearly the same827 at both wavelengths, with minor differences that are perfectly imputable to a continuous828 transformation from one to the other. Moreover, the two modes are definitely subharmonic829 and of the C type. Normalised snaphots taken exactly two vortex-shedding cycles apart are830 identical in all respects, and those taken only one vortex-shedding cycle apart are related by831 the space-time symmetries described earlier. Therefore, the actual frequency and symmetry-832 breaking of the nonlinear solution at 𝑅=3.8cannot be explained by a period-doubling833 23 k (a) (b) (c) 0 1 2 3 Figure 11: Instantaneous snapshots of (a) the three-dimensional solution at 𝑅=3.8 (Movie 3), (b) dominant eigenmode for (𝑅, 𝜆𝑧)=(3.8,2.5)(Movie 4), and (c) dominant eigenmode for (𝑅, 𝜆𝑧)=(3.8,5)(Movie 5), at four consecutive crossings of the Poincaré section. Iso-surfaces and colours as for figure 10, except that 𝑄=0.0001 is used to display nonlinear three-dimensional vortical structures. bifurcation affecting also the already unstable two-dimensional vortex-shedding solution.834 The origin of the period-doubling must probably be sought in some modulational instability835 of the nonlinear solution at 𝑅=3.4.836 Colourmaps of streamwise vorticity 𝜔𝑥on a spanwise-cross-stream plane located at 𝑥=6837 in the cylinder wake -the precise location of the Kármán vortex centre at the instants chosen838 for the snapshotshelp pinpoint the actual spanwise wavelength and space-time symmetry839 of the nonlinear three-dimensional solutions for 𝑅=3.4and 𝑅=3.8in figures 12a and b,840 respectively.841 The exact repetition of the solution every two crossings of the Poincaré section is now842 evinced by the exact matching of crossings 0-2 and 1-3 for 𝑅=3.4. The symmetry planes at843 𝑧=𝑧0+ (2𝑗)𝜆𝑧/4=2.5𝑗are perfectly identifiable, as is also clear that even/odd Poincaré844 crossings are mutually related by a reflection about planes at 𝑧=𝑧0+ (2∗𝑗+1)𝜆𝑧/4=845 1.25 +2.5𝑗. Note that reflection symmetries, as is the case of the two symmetries discussed846 24 𝑘(a) 𝑅=3.4(b) 𝑅=3.8 0 1 2 3 Figure 12: Streamwise vorticity colourmaps 𝜔𝑥∈ [−0.1,0.1](yellow for positive, blue for negative) at 𝑥=6for velocity ratios (a) 𝑅=3.4and (b) 𝑅=3.8. Four consecutive crossings of the Poincaré section have been represented, respectively. here, imply a change of sign of 𝜔𝑥, hence the change of colour. Both symmetries combined847 result in a third symmetry that is detectable as a shift by 𝜆𝑧/2=1.25 from one Poincaré848 crossing to the next.849 For the nonlinear solution at 𝑅=3.8, the planes 𝑧=𝑧0+ (2𝑗)𝜆𝑧/4=2.5𝑗are no850 longer symmetry planes, although the solution keeps the shadow of a reflectional symmetry851 to a certain extent. Remnants of the original spanwise periodicity 𝜆𝑧=2.5are also still852 perceptible, but the actual periodicity has indeed doubled to 𝜆𝑧=5. The only remaining853 symmetry, which is of a spatio-temporal nature, relates even/odd crossings by a reflection854 about planes at 𝑧=𝑧0+ (2𝑗)𝜆𝑧/4=2.5𝑗.855 A single snapshot of a random crossing of the Poincaré section for the chaotic solution856 at 𝑅=5.357 is shown in figure 13a. The Kármán vortices are still clearly identifiable as857 are also the elongated vortical structures along the braids, which are clearly arranged in858 pairs forming horse-shoe vortices on top of the primary spanwise vortices (left arm yellow,859 right arm black, on account of the change of sign in axial vorticity; both legs connect on860 top of the last Kármán vortex shed). The flow remains r3.6fairrily well organised in the near861 wake, but clearly breaks into spatio-temporally chaotic dynamics a little further downstream.862 The typical wavelength of the vortical structures remains of about 𝜆𝑧∼2.5, but there is no863 spanwise periodicity other than that artificially imposed by the fundamental Fourier mode864 employed in the simulation, here 𝐿𝑧=10. A second emerging sub-dominant mode of a865 much shorter spanwise wavelength, already visible at the far end of the Floquet spectrum866 of figure 7, is shown in figure 13b. The mode is still stable at 𝑅=5.357, but bifurcates867 with critical 𝜆𝑧≃10/7for 𝑅 < 6.5. Its characteristically short wavelength and synchronous868 nature identify it as mode-B type (see mode B of figure 1). It is however not detectable in the869 spatio-temporally chaotic solution snapshot and its growth rate once unstable remains well870 below that of the dominant eigenmode, which has shifted to somewhat longer wavelength871 𝜆𝑧≳2.5for these high values of 𝑅. It is therefore improbable that this other mode plays872 any important role in the transition to turbulence of the wake past the square cylinder in873 25 (a) (b) Figure 13: Instantaneous snapshot of (a) the chaotic three-dimensional solution at 𝑅=5.357 at a random crossing of the Poincaré section, and (b) short wavelength dominant eigenmode at (𝑅, 𝜆𝑧)=(5.357,1.429). Iso-surfaces and colours as for figure 11. upstream shear as mode B indeed does for both circular and square cylinders subjected to874 homogeneous incoming flow.875 We have applied both proper orthogonal decomposition and dynamic mode decomposition876 to the spatiotemporally chaotic solution, but results are inconclusive. Besides the average-877 fields mode, the two-dimensional vortex-shedding mode is clearly dominant, followed by878 something akin to mode C, but, other than that, we have been unable to identify any clear879 indication of additional relevant modes that may help understand the actual dynamics.880 A quasi-static variation of 𝑅, which is out of the scope of this study, would be required881 to cast light on the actual path leading to spatio-temporal chaos, but the presence of a882 period-doubling bifurcation suggests that a period-doubling cascade might play a role. There883 is also evidence suggesting that the complexification of the dynamics might arise from884 the competition of an increasing number of unstable wavelengths of the same instability885 type (actually a continuum if an infinite-span cylinder was to be considered), so that an886 instability of the Eckhaus or Benjamin-Feir type might in fact be held responsible, both for887 the period-doubling bifurcation reported here and the route to chaotic dynamics (Leweke888 et al. 1993; Leweke & Provansal 1994, 1995). Confirming or refuting the modulational889 instability hypothesis would not be feasible in a reasonable time scale due to the unjustifiedly890 high computational burden expected, and has therefore been left out of the present analysis.891 5.3. Dependence of the dominant eigenmode on 𝑅and 𝜆𝑧 892 Although the dominant eigenmode curves in the Floquet spectrum of figure 7b are smooth893 in the range of wavelengths observed in nonlinear solutions, evidence that they represent the894 same actual instability requires a close inspection of the eigenfields. Four snapshots taken at895 consecutive crossings within the linear regime of a purposefuly designed Poincaré section,896 well before three-dimensional modal energy reaches nonlinear saturation, illustrate the flow897 topology of the instability for (𝑅, 𝜆𝑧)=(3.4,2.5),(3.8,2.5)and (3.8,5)in figure 14. Shown898 are normalised streamwise vorticity (˜𝜔𝑥≡𝜔𝑥/𝜔max 𝑥)colourmaps on a streamwise-cross-899 stream plane halfway between two consecutive symmetry planes of the eigenmode. 𝜔max 𝑥is900 the maximum instantaneous value of |𝜔𝑥|at any given time. Comparison of all three cases901 points at a common origin of all three instabilities, which in fact are one and the same mode902 just deformed by a continuous change of the parameters. The only apparent effect, besides903 the stretching/compressing associated to changes in 𝜆𝑧which was already obvious from904 32 𝑅𝑒𝑇=56 𝑅𝑒𝐵=56 fixed and vary only 𝑅, such that the closely related shear parameter1092 𝐾≡2(𝑅−1)/(𝑅+1)and 𝑅𝑒 ≡ (𝑅+1)𝑅𝑒𝐵/2=28(𝑅+1)are linked. Nevertheless, we1093 have devoted a section, §7, to explore the decoupled effects of 𝑅𝑒 and 𝑅.1094 Several aerodynamic performance parameters have been assessed along the path from the1095 steady symmetric solution towards the highly asymmetric and spatio-temporally chaotic wake1096 dynamics. Pressure forces, particularly on the upper half of the front wall, have been shown1097 to be the main contributor to the drag coefficient 𝐶𝑑, which increases with 𝑅, particularly1098 after the onset of vortex shedding. The lift coefficient 𝐶𝑙is dominated by downward friction1099 on the front wall at low values of 𝑅, thus producing a net downforce. Time-dependence sets1100 the pressure difference between top and bottom on an increasingly growing trend that ends1101 up yielding positive lift. Force coefficient fluctuations 𝐶′ 𝑙and 𝐶′ 𝑑increase while the solution1102 remains two-dimensional but stagnate to rather constant values once three-dimensionality1103 sets in. The Strouhal number 𝑆𝑡 is rather stable, with a slightly increasing trend mainly pulled1104 by the increase in 𝑅𝑒, while the independent effect of 𝐾is unknown but probably decreasing.1105 Much of the behaviour exhibited by the flow configuration analysed here bears compelling1106 analogy to the somewhat related case of bluff bodies immersed in homogeneous upstream1107 shear. The trends of aerodynamic performance monitors and the qualitative morphing of the1108 wake topology upon varying the bulk Reynolds number 𝑅𝑒 and the equivalent shear parameter1109 𝐾are consistent with those reported by studies dealing with homogeneous upstream shear. It1110 is therefore probable that the velocity jump across the bluff body cross-stream length-scale1111 is in fact to be held responsible for most of the observed phenomena, the actual shape of the1112 velocity profile (linear or step-like) playing only a subsidiary role. 3.12However, Qquantitative1113 comparison is however futile difficult, as the range of 𝐾we have explored is far beyond the1114 reach of computations, or even experiments, dealing with homogeneous upstream shear.1115 The symmetric case 𝑅=1(𝑅𝑒 =56,𝐾=0) is steady, implying that the presence of1116 the splitter plate has a stabilising effect, through the viscous blocakge effect (mass flux1117 and momentum reduction in the near wall and wake regions due to viscosity), on the flow1118 past a square cylinder, which is known to undergo a Hopf bifurcation somewhat earlier at1119 𝑅𝑒𝐻≃45. The first bifurcation upon increasing 𝑅remains the two-dimensional onset of1120 time-periodicity, but this only happens for 𝑅∈ (2.0,2.2)(𝑅𝑒 ∈ (84,89),𝐾∈ (0.67,0.75)).1121 It cannot be concluded from present results whether the delay in the onset of time-dynamics1122 is due to the presence of the splitter plate or whether the upstream velocity ratio also has a1123 stabilising effect.1124 Once vortex-shedding is in place, the Kármán vortex street in the cylinder wake follows1125 the same trends observed in the literature. The vortices on the high velocity side become1126 stronger and rounder with shear, while those on the low velocity side are weak, elongated1127 and dissipate fast in the wake.1128 Stability analysis of the time-periodic two-dimensional vortex-shedding solution to three-1129 dimensional perturbations reveals three distinct modes. A long wavelength mode, possibly1130 related to the mode A that is ubiquitous in the wake transition regime of the classic circular1131 and square cylinders, is dominant but stable for velocity ratios 𝑅⩽3. At the high 𝑅end,1132 a very short wavelength mode arises and bifurcates for 𝑅≳6, but is never even close1133 to dominant. The last mode, of an intermediate wavelegth, prevails for 𝑅 > 3and is the1134 one responsible for wake three-dimensionalisation along the increasing 𝑅path followed1135 here. This mode bifurcates at 𝑅C≃3.1(𝑅𝑒C≃115,𝐾C≃1.02) with critical wavelength1136 𝜆C 𝑧≃2.4, exhibits elongated streamwise-cross-stream vortical structures along the braids1137 connecting consecutive clockwise Kármán vortices, and is subharmonic (period-doubling).1138 While breaking the spanwise translational invariance SO(2) of the two-dimensional time-1139 periodic solution, which belongs to the O(2)=Z2×SO(2) symmetry group class, it still1140 preserves a collection of symmetries. Besides repeating every two vortex-shedding cycles, the1141 33 eigenfields are mirror symmetric with respect to a discrete number of spanwise planes every1142 𝜆𝑧/2, and also invariant under the evolution by half a period (one vortex-shedding cycle)1143 followed by either reflection about spanwise planes located halfway between contiguous1144 reflection-symmetry planes or shift by half a wavelength 𝜆𝑧/2. The wavelength, topology1145 and symmetries of the mode are typical of the mode C that is characterisitc of the wake1146 transition regime of open rings or square cylinders at incidence, and known to evolve from1147 the quasiperiodic mode QP previously reported for circular and square cylinders. Converged1148 three-dimensional nonlinear solutions slightly beyond the bifurcation point exhibit time1149 dynamics and wake structures sharing all properties that are characteristic of mode C.1150 At somewhat higher 𝑅∈ (3.4,3.8), the flow undergoes a tertiary bifurcation that doubles1151 the period yet again (nonlinear solutions repeat every 4 vortex-shedding cycles) and also1152 doubles the spanwise wavelength. The instability is therefore of the modulational type and1153 amplifies perturbations that have double the wavelength of the original spanwise-periodic1154 pattern. The resulting nonlinear solution has been observed in a computational domain1155 with 𝐿𝑧=5, allowing only for a discrete number of periodic patterns with wavelengths1156 𝜆𝑧=𝐿𝑧/𝑗={5,2.5,1.667,1.25, ...}. The spanwise mirror symmetry has finally been1157 broken and the only remaining invariance besides the four-fold time-periodicity is a space-1158 time symmetry that recovers the solution by applying a reflection after the evolution by half a1159 period (two vortex-sheedding cycles). There is no evidence of a secondary bifurcation of the1160 two-dimensional solution exhibiting a 𝜆𝑧=5other than a stretched version of the same mode1161 C, which has also become unstable. As a matter of fact, the range of unstable wavelengths1162 for which mode C is unstable increases with 𝑅but the dominant wavelength remains quite1163 fixed at 𝜆𝑧≃2.5for 𝑅⩽4and only migrates to increasingly wider values beyond this1164 point. Very shortly after the period-doubled three-dimensional pattern, a further increase in1165 𝑅brings spatio-temporal chaos with it.1166 Given that the limited spanwise periodicty 𝐿𝑧=5has been shown already capable of sus-1167 taining spatio-temporally chaotic dynamics, the solutions reported here are bound to play an1168 essential role in the wake transition regime and the inception of turbulence at even higher flow1169 regimes. It might however be the case that these solutions are modulationaly/subharmonically1170 unstable to long wavelength disturbances that hinder their actual observation as stable flows in1171 computations within wider spanwise domains and, therefore, in experiments. We have indeed1172 found evidence that, although stable and unaltered at 𝑅=3.2within an 𝐿𝑧=10 domain,1173 the 𝜆𝑧=2.5solution is already unstable at 𝑅=3.4. Preliminary results show that the stable1174 solution in the 𝐿𝑧=10 domain is still period-doubled periodic, but its wavelength is the full1175 𝜆𝑧=10 and the characteristic space-time symmetry has already been broken. Its appearance1176 suggests a dislocation of a three-replica pattern (the solution with 𝜆𝑧=3.33) trying to1177 accomodate a fourth replica (the 𝜆𝑧=2.5solution) but not quite managing. This brings1178 to mind a mechanism that is responsible for wave dislocation in elongated systems such as1179 channel flows (Mellibovsky & Meseguer 2015). At 𝑅=3.8, the solution seems to be already1180 chaotic for 𝐿𝑧=10 and the three-dimensional vortical structures in the wake incorporate1181 a characteristic spanwise drift. It is therefore possible that the observed period-doubling1182 bifurcation and the ensuing route towards spatio-temporal chaos may be understood in the1183 light of an Eckhaus instability, whereby a periodic pattern becomes modulationally unstable1184 following the bifurcation of a widening spectrum of competing wavelengths. Simulations in1185 much longer domains, allowing for a larger set of unstable wavelengths, would be required in1186 the wake transition regime to elucidate the actual nature of the route to chaotic dynamics and1187 turbulence . An attempt to properly gauge the implications of long-wavelength subharmonic1188 instabilities and their bearing on the actual dynamics of the problem under scrutiny is beyond1189 the scope of the present investigation.1190 We have analysed the wake transition regime along r3.13two different a paths. The first one1191 34 that simultaneously increases the bulk Reynolds number 𝑅𝑒 and the velocity ratio 𝑅(and,1192 with it, the equivalent shear parameter 𝐾), thus masking the independent effects of these1193 two paramenters. r3.13The second path sweeps 𝑅𝑒 at constant 𝑅in a preliminary attempt at1194 decoupling the effects of the two parameters. Some evidence has been reported that 𝑅𝑒1195 is key in engendering chaotic dynamics, while 𝑅is responsible for mode selection upon1196 three-dimensionalisation and, therefore, dictates the specific path that leads towards spatio-1197 temporal complexity. A thorough ongoing analysis decoupling 𝑅𝑒 and 𝑅is expected to1198 provide further insight into their separate contributions to the problem.1199 9. Acknowledgements1200 [Supplementary data]Supplementary material and movies are available at1201 https://doi.org/10.1017/jfm.2019...1202 [Acknowledgements] This work was supported by the Spanish Government grants1203 FIS2016-77849-R and PID2020-114043GB-I00, and by the Catalan Government grant1204 2017-SGR-00785, respectively. Part of the computations were done in the Red Española de1205 Supercomputación (RES), the Spanish supercomputer network, under grants FI-2019-1-00231206 and FI-2018-3-0030. F. Mellibovsky is a Serra-Húnter fellow.1207 [Funding]This work was supported by the Spanish Government (grant numbers FIS2016-1208 77849-R, PID2020-114043GB-I00); and the Catalan Government (grant number 2017-SGR-1209 00785). Part of the computations were done in the Red Española de Supercomputación (RES),1210 the Spanish supercomputer network (grant numbers FI-2019-1-0023, FI-2018-3-0030).1211 [Declaration of interests] The authors report no conflict of interest.1212 [Data availability statement] The data that support the findings of this study may be1213 obtained from the corresponding author upon reasonable request.1214 [Author ORCID] F. Mellibovsky, https://orcid.org/0000-0003-0497-9052; ...1215 [Author contributions]1216 Appendix A. Stability analysis of periodic solutions1217 The spanwise translational invariance of the two-dimensional vortex-shedding flow past1218 an infinite square cylinder breaks as the Reynolds number is increased beyond a critical1219 threshold. Floquet theory provides the framework for analysing the stability of time-periodic1220 base flows and has been employed successfully to shed light on the three-dimensionalisation1221 of the wake behind cylinders (Noack & Eckelmann 1994; Henderson & Barkley 1996; Barkley1222 & Henderson 1996). The instantaneous flow is decomposed into the additive superposition1223 of the periodic solution and a perturbation following1224 (u, 𝑝)=(U, 𝑃)+(˜ u,˜𝑝).(A 1)1225 Formal substitution into the Navier-Stokes equations (2.1) and linearisation yields the1226 governing equations for the perturbation field1227 𝜕˜ u 𝜕𝑡 + ( ˜ u· ∇)U+ (U· ∇) ˜ u=−∇ ˜𝑝+1 𝑅𝑒 ∇2˜ u ∇ · ˜ u=0, (A 2)1228 with homogeneous conditions on all boundaries of the same typology as for the nonlinear1229 problem.1230 The two-dimensionality of the base flow and the translational invariance in the spanwise1231 35 direction allow for the following modal ansatz:1232 ˜ u(r;𝑡)=∫∞ −∞ ˆ u(𝑥, 𝑦, 𝛽;𝑡)ei𝛽𝑧𝑧𝑑𝛽𝑧,(A 3)1233 corresponding to a Fourier decomposition, such that spanwise modes ˆ u(𝑥, 𝑦, 𝛽;𝑡)of different1234 spanwise wavenumber 𝛽𝑧decouple exactly.1235 Since we are not interested in computing the full eigenspectrum nor even a subset of it, we1236 follow here a simple time-stepping approach that provides the most unstable (or least stable)1237 multiplier for any desired Reynolds number 𝑅𝑒 and spanwise wavenumber 𝛽𝑧. We perturb the1238 two-dimensional periodic vortex-shedding solution, duly computed through time stepping1239 in the two-dimensional domain where it remains stable, with a tiny spanwise-dependent1240 random disturbance. The initial condition thus obtained is then evolved in time integrating1241 the full nonlinear Navier-Stokes equations but using a single spanwise Fourier component,1242 besides the homogenous component, in a domain of spanwise size 𝐿𝑧=2𝜋/𝛽𝑧. After the1243 initial transients, the energy contents on the stable manifold vanish and the perturbation1244 aligns with the dominant eigenmode. At this stage, the dominant eigenmode can be extracted1245 directly from the only non-homogeneous Fourier component of the simulation as long as it1246 remains within the linear evolution regime. This can be considered to be the case while the1247 energy does not approach zero-machine (in the stable case) or saturation unto the nonlinear1248 regime (for the unstable case), where it starts feeding energy back into the homogeneous1249 Fourier component via the nonlinear advection term and the simulation becomes utterly1250 under-resolved. Many more modes, providing the standard six orders of magnitude decay,1251 would be required to properly resolve the saturated nonlinear solution, but since we are only1252 interested in the linear regime, we simply dismiss the latter part of the time evolution.1253 While still in the linear regime, but well past the initial transients, the already modal1254 perturbation follows an oscillating behaviour with the periodicity of the underlying two-1255 dimensional solution on top of an exponential decay/growth. The decay/growth rate of the1256 dominant Floquet mode (as represented by the Floquet multiplier 𝜇) can be derived from1257 the energy time-series of the 𝛽𝑧-mode by fitting a power-law to the sequence of crossings1258 (𝐸𝑘 𝛽𝑧 =𝐸𝛽𝑧(𝑡𝑘)) of a purposely defined Poincaré section (in our case, picking 𝐶𝑙=0, and1259 numbering the consecutive crossings at times 𝑡𝑘by an index 𝑘)1260 𝐸𝑘 𝛽𝑧 =𝐸0 𝛽𝑧|𝜇|2𝑘.(A 4)1261 Here 𝐸0 𝛽𝑧is an irrelevant fitting parameter that is simply related to the energy of the initial1262 perturbation we prescribe, while the 2in the exponent arises from the square relation that1263 exists between the velocity field and its corresponding kinetic energy. A more sophisticated1264 fit of the form1265 𝑎𝑘 𝛽𝑧 =𝑎0 𝛽𝑧𝑟𝑘cos (𝑘𝜃 +𝜓),(A 5)1266 where 𝑎is any primitive degree of freedom of the perturbation field (a point velocity1267 component, for example), 𝜓is another fitting parameter representing an initial phase and1268 𝜇≡𝑟e±i𝜃a complex multiplier, allows r3.14for the discrimination between synchronous and1269 quasiperiodic modes (An et al. 2019). We have not used such a fit here, as the only mode1270 present is clearly subharmonic and appears as synchronous when inspecting the modal energy1271 evolution.1272 The envelope of the spectrum |𝜇|(𝛽)for 𝑅𝑒 =200 and 205 has been computed as proof1273 of concept of the aforementioned method, and compared to published results (Robichaux1274 et al. 1999; Blackburn & Lopez 2003; Sheard et al. 2009) in figure 18. The agreement is1275 fair at 𝑅𝑒 =205, for which all three modes, A, B and QP, are duly observed and exhibit the1276 right growth rate dependence on the wavenumber. The most unstable modes at 𝑅𝑒 =2001277 36 Figure 18: Dominant Floquet multipliers at Reynolds numbers 𝑅𝑒 =200 (circles) and 𝑅𝑒 =205 (squares) for the square cylinder without splitter plate. Shown are our numerical results (grey) alongside those by Robichaux et al. (1999) (black). Labels indicate modes A, B and QP. have wavenumbers 𝛽A 𝑧≃1.21 (wavelength 𝜆A 𝑧=5.2) and 𝛽B 𝑧≃5.5(𝜆B 𝑧=1.15), in line with1278 experimental (Luo et al. 2003) and experimental (Luo et al. 2007) results. A third unstable1279 mode arises at 𝑅𝑒QP ≃205 with wavenumber 𝛽QP 𝑧≃2.2(𝜆B 𝑧≃2.8), that was not clearly1280 discernible at 𝑅𝑒 =200. 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