R E M I B O U R G U E T

Vortex-induced vibrations of a flexible cylinder at subcritical Reynolds number

The flow past a fixed, rigid cylinder becomes unsteady beyond a critical Reynolds number close to 47, based on the body diameter and inflow velocity. The present paper explores numerically the vortex-induced vibrations (VIV) that may develop for a flexible cylinder at subcritical Reynolds number (Re), i.e. for Re < 47. Flexible cylinder VIV are found to occur down to Re ≈ 20, as previously reported for elastically mounted, rigid cylinders. A detailed analysis is carried out for Re = 25, in two steps: the system behavior is examined from the emergence of VIV to the excitation of the first structural modes and then, focus is placed on higher mode responses. In all cases, a single vibration frequency is excited in each direction. The cross-flow and in-line responses exhibit contrasted magnitudes (peak amplitudes of 0.35 vs. 0.01 diameters), as well as distinct symmetry properties and evolutions (e.g. standing/traveling waves). The flow, unsteady once the cylinder vibrates, is found to be temporally and spatially locked with body motion. The synchronization with the cross-flow, standing wave responses is accompanied by the formation of cellular wake patterns, regardless the modes involved in the vibrations. Body trajectory varies along the span but dominant orbits can be identified. Despite the low amplitudes of the in-line responses, connections are uncovered between orbit orientation and flow-structure energy transfer, with different trends in each direction.

Introduction

Vortex-induced vibrations (VIV) of flexible bodies with bluff cross-section are encountered in many natural and industrial systems, e.g. plants and trees in wind, cables and mooring lines exposed to ocean currents. Due to their impact on engineering structures and their fundamental interest as a paradigm of fluid-structure interaction involving a synchronization, or lock-in, between body motion and flow unsteadiness, VIV have been the object of a number of studies, as reviewed for example by [START_REF] Williamson | Vortex-induced vibrations[END_REF]. VIV have often been investigated through the canonical problem of a rigid circular cylinder mounted on an elastic support. When the rigid cylinder is fixed, the flow becomes unsteady beyond a critical Reynolds number close to 47, with the formation of the alternating von Kármán vortices [START_REF] Mathis | The Bénard-von Kármán instability : an experimental study near the threshold[END_REF]. The Reynolds number (Re) is based on the body diameter (D) and inflow velocity (U ); the above critical value is denoted by Re c in the following. If the rigid cylinder is elastically mounted, previous works have shown that VIV may occur in the subcritical-Re range, i.e. for Re < Re c , possibly down to Re ≈ 20, depending on the system parameters [START_REF] Cossu | On the instability of a spring-mounted circular cylinder in a viscous flow at low Reynolds numbers[END_REF][START_REF] Mittal | Vortex-induced vibrations at subcritical Re[END_REF][START_REF] Kou | The lowest Reynolds number of vortex-induced vibrations[END_REF][START_REF] Dolci | Bifurcation analysis of the primary instability in the flow around a flexibly mounted circular cylinder[END_REF]. A typical case of rigid cylinder VIV in the subcritical-Re range (Re = 33) is presented in figure 1. The maximum amplitude of cross-flow vibration, normalized by D, is plotted in figure 1(a) as a function of the reduced velocity, U ⋆ , defined as the inverse of the oscillator natural frequency The present work aims at extending the analysis of subcritical-Re VIV to flexible cylinders. Flexible cylinder VIV have been well documented for Re > Re c [START_REF] Chaplin | Laboratory measurements of vortex-induced vibrations of a vertical tension riser in a stepped current[END_REF][START_REF] Bourguet | Vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF][START_REF] Wu | A review of recent studies on vortex-induced vibrations of long slender cylinders[END_REF][START_REF] Huera-Huarte | Towing tank experiments on the vortex-induced vibrations of low mass ratio long flexible cylinders[END_REF][START_REF] Gedikli | Observed mode shape effects on the vortexinduced vibration of bending dominated flexible cylinders simply supported at both ends[END_REF][START_REF] Seyed-Aghazadeh | Effects of boundary conditions on vortex-induced vibration of a fully submerged flexible cylinder[END_REF]. They are still driven by the lock-in mechanism but the flexibility of the body and its distributed interaction with the flow may lead to an increased complexity of the system behavior. For example, several structural wavelengths may be excited, sometimes simultaneously [START_REF] Bourguet | Distributed lock-in drives broadband vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF]. The vibrations often consist of a mixture of standing and traveling wave patterns and so does the wake (Newman & [START_REF] Newman | A direct numerical simulation study of flow past a freely vibrating cable[END_REF]. The phase difference between the streamwise and cross-flow vibrations may vary along the span, leading to diverse trajectory shapes, more or less favorable to body excitation/damping by the flow [START_REF] Fan | Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow[END_REF]. These different aspects remain to be explored in the subcritical-Re range. An attempt is made here on the basis of three-dimensional numerical simulations. The question of the critical value of Re for the onset of flexible cylinder VIV is briefly addressed, then a constant value, Re = 25 (i.e. close to half Re c ), is selected and the investigation is carried out in two steps. First, focus is placed on the evolution of the system behavior with U ⋆ (based on the fundamental natural frequency), from the emergence of VIV to the excitation of the first structural modes; a cylinder of length to diameter aspect ratio equal to 50 is considered and U ⋆ is progressively varied from 1 to 35 ( §3). Second, higher mode vibrations are examined for a cylinder of aspect ratio 400 and two high values of U ⋆ , 80 and 110 ( §4).

Formulation and numerical method

A sketch of the physical system is presented in figure 2. The cylinder has a circular cross-section, is pinned at both ends and aligned with the z axis in the absence of deformation (fluid at rest). Its length and diameter are denoted by L and D, respectively. The cylinder is placed in an incompressible, uniform cross-current, parallel to the x axis. The Reynolds number is based on the inflow velocity (U ) and D, Re = ρ f U D/µ, where

m ζ{x,y} -τ ζ ′′ {x,y} + ξ ζ{x,y} = C {x,y} 2 , (2.1)
where ˙and ′ denote the time and space (z) derivatives. The non-dimensional natural frequency associated with the n th sine Fourier mode (i.e. sin(πnzD/L)) is obtained via the dispersion relation f nat (n) = n τ /m/(2L/D), in vacuum. The reduced velocity is defined as the inverse of the fundamental natural frequency (n = 1), U ⋆ = 1/f nat (1).

The structural damping ratio is defined as γ = ξ(L/D)/(2π √ τ m). Different values of Re, L/D, m, γ and U ⋆ are examined. The numerical method is the same as in previous studies concerning comparable systems at higher Re [START_REF] Bourguet | Vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF][START_REF] Bourguet | Distributed lock-in drives broadband vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF]. It is briefly summarized here and some additional validation results are presented. The coupled flow-structure equations are solved by the parallelized code Nektar, which is based on the spectral/hp element method [START_REF] Karniadakis | Spectral/hp Element Methods for CFD[END_REF]. The computational domain (50D downstream and 20D in front, above, and below the cylinder), boundary conditions (no-slip condition on the cylinder surface, flow periodicity on the side boundaries) and discretization (2175 elements in the (x, y) plane) are the same as in the above mentioned studies. A convergence study was carried out in a typical case of subcritical-Re VIV (Re = 25, L/D = 50, m = 6, γ = 0, U ⋆ = 32) in order to select the non-dimensional time step (0.005), polynomial order in the (x, y) plane ( 5) and number of Fourier modes in the z direction (128). In particular, the relative differences on force/vibration amplitudes and frequencies were lower than 0.05% when increasing the polynomial order from 5 to 6. For the larger aspect ratio configuration (L/D = 400), 1024 Fourier modes are employed. In figure 1(a), a comparison of the CF vibration amplitudes of a rigid cylinder subjected to VIV at Re = 33 (two-degree-of-freedom oscillator), with those reported in prior works, confirms the validity of the present numerical method. Each simulation is initialized with a lowamplitude (10 -4 D), random asymmetrical deformation of the cylinder. The analysis is based on time series of more than 30 oscillation cycles, collected after convergence.

Emergence of VIV and first mode responses

The emergence of VIV and their evolution with U ⋆ are examined for a cylinder of aspect ratio L/D = 50. The maximum amplitude of CF vibration along the span for Re ∈ [20, 25], m = 6 and γ = 0, is plotted at U ⋆ = 8 and U ⋆ = 32 in figure 3(a). Vibrations arise between Re = 20 and 21 at U ⋆ = 8 and between Re = 21 and 22 at U ⋆ = 32; these values of Re are close to those reported for rigid cylinder VIV [START_REF] Dolci | Bifurcation analysis of the primary instability in the flow around a flexibly mounted circular cylinder[END_REF]. Considering the possible influence of the other structural parameters, no attempt was made to determine the value of Re for the onset of vibration for all U ⋆ . The value Re = 25 is selected as a typical value to explore subcritical-Re VIV and it is kept constant in the rest of the paper. To illustrate the existence of vibrations at this Re for different non-dimensional structural masses and when structural damping is added, the maximum amplitudes of CF and IL responses are represented at U ⋆ = 16 for m ∈ {3, 6, 12} and γ = 0, and at U ⋆ = 24 for m = 6 and γ ∈ {0, 0.01, 0.05} (˜denotes the fluctuation about the time-averaged value). In the following, γ is set to 0 to allow maximum amplitude vibrations and m is set to 6.

An overview of the structural responses occurring at Re = 25, for m = 6 and γ = 0, is presented in figure 3. The maximum vibration amplitude exhibits similar trends in each direction as a function of U ⋆ but contrasted magnitudes since the peak amplitudes are close to 0.35 diameters in the CF direction and lower than 0.01 diameters in the IL direction (figure 3(a)). Comparable amplitudes were reported in the rigid cylinder case [START_REF] Mittal | Vortex-induced vibrations at subcritical Re[END_REF]. The successive bell shapes in the evolution of the vibration amplitudes are associated with successive changes in their spatial structures. They are visualized in figure 3(b,c) via the root mean square (RMS) values of the displacement fluctuations along the span. Well-defined sine modes (1 st to 4 th ) can be identified in the CF direction. Less regular patterns develop in the IL direction but it can be noted that the instantaneous response is always symmetrical about the midspan point, while it may be either symmetrical (odd modes) or anti-symmetrical (even modes) in the CF direction. Such phenomenon was also observed at higher Re for symmetrical systems [START_REF] Gedikli | Observed mode shape effects on the vortexinduced vibration of bending dominated flexible cylinders simply supported at both ends[END_REF]. At higher Re, both mono-frequency and multi-frequency vibrations were reported for a flexible cylinder placed in a uniform current (e.g. Seyed-Aghazadeh et al. 2019). Here, a single frequency is excited in each direction, with a ratio of 2 between the IL and CF responses as usually observed for circular bodies. As shown in figure 3(d), the CF vibration frequency (f y ) remains lower than the natural frequency of the corresponding structural mode in vacuum (plain gray lines). It is relatively close to the natural frequency corrected by considering the potential added mass coefficient C m = 1, i.e. f nat m/(m + C m π/4) (gray dashed-dotted lines; [START_REF] Williamson | Vortex-induced vibrations[END_REF]). This is not always the case at higher Re [START_REF] Bourguet | Vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF]. The vibration frequency ranges from 0.1 to 0.13 approximately. This range includes the subcritical Strouhal frequency found by [START_REF] Buffoni | Vortex shedding in subcritical conditions[END_REF] by triggering the flow (St = 0.118, red dashed line).

In the absence of vibration, the flow is steady and similar to that depicted in figure 1(b). When the body vibrates, the frequencies of flow unsteadiness and body CF motion coincide along the entire span, as illustrated in figure 4 by the power spectral densities (PSD) of the CF component of flow velocity sampled at (x, y) = (10, 0), in four typical cases (f y is indicated by a vertical dashed-dotted line); an example is selected for each mode excited in the CF direction (1 st to 4 th ). The lock-in condition is thus established. This justifies the term VIV employed to designate the present vibrations. The absence of response up to U ⋆ = 6 and around U ⋆ = 12 (figure 3(a)) suggests that the admissible range of wake frequencies (i.e. deviation from the subcritical Strouhal frequency) is not compatible with the possible vibration frequencies. In addition, the spanwise structures of the flow and body response are found to coincide, with the same wavelengths and the minima of flow velocity PSD matching CF vibration nodes (horizontal dashed lines in figure 4). This coincidence shows that the flow and the body are not only temporally locked but also spatially locked. In all cases, the wake exhibits a cellular pattern where each cell ends near the nodes of the CF vibration. Instantaneous visualizations of the flow are presented in figure 4. Within each cell, two counter-rotating vortices are formed per oscillation cycle (2S pattern; [START_REF] Williamson | Vortex-induced vibrations[END_REF]) and the flow is comparable to that depicted in figure 1(c).

To further analyze the system behavior, the synchronization of the IL and CF responses and its possible link with flow-structure energy transfer are examined in the following. With a frequency ratio of 2 between the IL and CF responses, the cylinder exhibits figureeight trajectories in the plane perpendicular to the span. The shape and orientation of these trajectories are monitored at each spanwise location, by the phase difference Φ xy = φ x -2φ y , where φ x and φ y are the local phases of the IL and CF responses. A nomenclature is adopted where the cylinder moves downstream when reaching CF oscillation maxima for Φ xy ∈]-180 • , 0 • [ and upstream for Φ xy ∈]0 • , 180 • [; these two types of trajectories are referred to as clockwise and counter-clockwise. The cases Φ xy = 0 • and Φ xy = ±180 • correspond to crescent-shaped orbits, bent downstream and upstream, respectively. A systematic analysis, for each U ⋆ ∈ [1, 35] where vibrations occur, shows that body motion is dominated (73%) by counter-clockwise trajectories and that the most frequent phase difference is Φ xy ≈ 20 • , i.e. counter-clockwise orbit close to a crescent bent downstream. Comparable phasing properties were reported at much higher Re [START_REF] Dahl | Dual resonance in vortex-induced vibrations at subcritical and supercritical Reynolds numbers[END_REF]. The time-averaged power coefficients, C e{x,y} = C {x,y} ζ{x,y} , are used to identify, in each direction, the spanwise regions of excitation (C e{x,y} > 0) and damping (C e{x,y} < 0) of the structure by the flow ( denotes time-averaged values). Histograms of Φ xy and C e{x,y} gathering all the cases are plotted in figure 5. The probability (P ) associated with each histogram quadrant is specified in the plots and the most frequent trajectory is represented in figure 5(a). In spite of the low amplitudes of the IL responses, some connections appear between body trajectory and energy transfer. Counter-clockwise orbits are associated with excitation and damping of the structure in both directions, with a deviation towards damping in the CF direction versus excitation in the IL direction. Clockwise orbits are less frequent but involved in more contrasted energy transfer trends. At higher Re, such orbits were often connected to vibration damping (e.g. [START_REF] Fan | Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow[END_REF].

Here, they are associated with IL vibration damping but CF vibration excitation. Some statistics of fluid forces are reported in figure 6. The structural vibrations are accompanied by a slight amplification of the span-and time-averaged IL force, compared to the fixed, rigid cylinder case (gray dashed-dotted line) and by the emergence of force fluctuations which are quantified by their span-averaged, RMS values ( denotes spanaveraged values). As also observed for the structural responses, force fluctuations present substantially different magnitudes in each direction.

The principal features of the flexible cylinder, subcritical-Re VIV have been described from their emergence to the excitation of the first structural modes. Higher mode vibrations are examined in the next section.

Higher mode responses

In order to extend the above analysis to higher mode vibrations, a flexible cylinder of aspect ratio L/D = 400 is considered in this section, for two high values of U ⋆ , 80 and 110. The values of Re, m and γ are kept equal to 25, 6 and cylinder is found to vibrate for both values of U ⋆ . Figure 7(a-c) an instantaneous visualization of the wake, as well as series of the CF and IL displacement fluctuations along the span, U ⋆ = 110. The vibration amplitudes are similar for both U ⋆ values and are also comparable to those reported in §3, i.e. maxima close to 0.35 and 0.01 diameters in the CF and IL directions. Within the range of vibration frequencies identified in figure 3(d) and based on the frequencies in vacuum, 4 and 7 structural modes could be excited in the CF and IL directions for U ⋆ = 110. it appears that in both studied cases, a single frequency is excited in each direction, with an IL/CF response frequency ratio of 2, as previously observed.

The CF responses remain essentially composed of standing waves. Their spatial struc-tures are close to the 10 th and 13 th sine modes, as shown in figure 7 For both values of U ⋆ , the flow and the body remain temporally and spatially locked, since the frequencies and spanwise structures of the flow and CF vibration coincide. The wakes exhibit cellular patterns comparable to those visualized in figure 4.

A complementary vision of the mixed standing-traveling waves identified in the IL vibrations is proposed in figure 7(e) which represents the response local phases (φ x ) for both U ⋆ values. These plots confirm the transition from standing waves in the central region, with regular jumps near response nodes, to traveling waves moving towards cylinder ends on each side. This orientation of the traveling waves relates to the spanwise distribution of flow-structure energy transfer, as previously observed at higher Re [START_REF] Bourguet | Distributed lock-in drives broadband vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF]. The energy transfer is quantified via C ex in figure 7(e) (for U ⋆ = 110). The structural waves propagate from the main region of excitation (central part) towards the regions where damping dominates (side parts). The global distribution of C ex along the span appears to be determined by the profile of the inflow velocity component locally normal to the body (U ⊥ ). This profile is sheared due to the streamwise deflection of the cylinder (visible in figure 7(a)), which is caused by the mean IL force. The main is found to be located in the area of maximum U ⊥ , i.e. the central part. The present VIV cases exhibit a wider variety of trajectory shapes than those examined in §3. IL/CF response synchronization remains however dominated (60%) by counterclockwise, figure-eight orbits. The values of Φ xy and C e{x,y} obtained for U ⋆ = 80 and U ⋆ = 110 are superimposed on the histograms in figure 5 (gray crosses/red triangles). The general trends previously identified are found to persist and are even more clearly defined for the higher mode responses, in particular the excitation of CF vibrations through clockwise orbits while this orientation is associated with IL vibration damping.

Force statistics are comparable to those reported for lower mode responses, e.g. C x = 1.90 for U ⋆ = 110. The local peaks of C x along the span coincide with CF response antinodes as shown in figure 7(d) (bottom axis); such coincidence is also observed for lower mode responses. The decreasing trend of C x towards the body ends can be connected to a scaling of the force by the inflow velocity component locally normal to the deflected body, i.e. ∝ U 2 ⊥ (red dashed line in figure 7(d)).

Conclusions

The subcritical-Re VIV of a flexible cylinder, pinned at both ends and immersed in a uniform current, have been investigated on the basis of numerical simulations. Vibrations were observed down to Re ≈ 20, as previously reported for rigid cylinders. A constant value, Re = 25, was selected and the analysis was carried out in two steps: a monitoring of the system behavior as a function of U ⋆ , from the emergence of VIV to the excitation of the first structural modes and then, a focus on higher mode responses. The typical features of the subcritical-Re VIV uncovered in this work can be summarized as follows.

A single vibration frequency is excited in each direction, with a ratio of 2 between the IL and CF responses. The flow, which is steady in the absence of vibration, becomes unsteady and its frequency coincides with CF vibration frequency. The peak amplitudes of the CF and IL vibrations are of the order of 0.35 and 0.01 body diameters, respectively. The CF vibrations are essentially composed of standing waves and their spatial structures are close to sinusoidal modes. The spanwise structure of the unsteady flow matches that of the CF vibration, i.e. the flow and the flexible body are not only temporally locked but also spatially locked. This synchronization is accompanied by the formation of cellular wake patterns. The IL responses exhibit less-regular evolutions, in particular with the development of traveling waves propagating from the central part of the cylinder towards its ends; this orientation can be related to the streamwise deflection of the body through the spanwise distribution of the regions of excitation/damping. The instantaneous IL responses are however found to remain symmetrical about the midspan point, which is not necessarily the case in the CF direction. The cylinder describes preferentially counter-clockwise figure-eight orbits, especially for the low mode responses where these orbits are generally close to a crescent-shaped trajectory, bent downstream. In spite of the low amplitudes of the IL responses, the orbit orientation appears to be connected to flow-structure energy transfer and different trends arise in the CF and IL directions.

Figure 1 .

 1 Figure 1. Vortex-induced vibrations of a rigid cylinder at Re = 33: (a) maximum amplitude of cross-flow displacement as a function of the reduced velocity; (b,c) instantaneous iso-contours of spanwise vorticity (ωz ∈ [-0.3, 0.3]), for (b) U ⋆ = 5 and (c) U ⋆ = 7. In (a), the present simulation results are compared to the amplitudes reported by Mittal & Singh (2005) and Dolci & Carmo (2019).

Figure 2 .

 2 Figure 2. Sketch of the physical system.

Figure 3 .

 3 Figure 3. Responses of the flexible cylinder (L/D = 50) at Re = 25, for m = 6 and γ = 0, as functions of U ⋆ : (a) maximum amplitudes of displacement fluctuations in the CF (left axis; black circles) and IL (right axis; blue triangles) directions; (b,c) RMS values of the (b) CF and (c) IL displacement fluctuations along the span; (d) CF vibration frequency (IL vibration frequency is equal to twice this frequency). In (a), the maximum amplitudes of CF displacement for Re ∈[20, 24] are also represented at U ⋆ = 8 and U ⋆ = 32 (grayscale circles); the maximum amplitudes of CF and IL displacement fluctuations are added for Re = 25, at U ⋆ = 16 for m ∈ {3, 12} and γ = 0 (red symbols), and at U ⋆ = 24 for m = 6 and γ ∈ {0.01, 0.05} (green symbols). In (d), the natural frequencies of the first four structural modes in vacuum (fnat) are indicated by plain gray lines; the corrected natural frequencies, with an added mass coefficient of 1, are denoted by gray dashed-dotted lines; the vortex shedding frequency identified by[START_REF] Buffoni | Vortex shedding in subcritical conditions[END_REF] is indicated by a red dashed line.

Figure 4 .

 4 Figure 4. Flow patterns downstream of the flexible cylinder subjected to VIV at Re = 25 (L/D = 50): PSD of the CF component of flow velocity along the span at (x, y) = (10, 0) and instantaneous iso-surfaces of spanwise vorticity (ωz ± 0.07), for (a) U ⋆ = 8, (b) U ⋆ = 16, (c) U ⋆ = 24 and (d) U ⋆ = 32. The PSD are normalized by the peak magnitude; the color code ranges from blue (0) to yellow (1). A vertical dashed-dotted line indicates the CF vibration frequency and horizontal dashed lines denote CF vibration nodes. Arrows represent the oncoming flow. Part of the computational domain is visualized (35D downstream of the cylinder).

Figure 5 .

 5 Figure 5. Flexible cylinder trajectory and flow-structure energy transfer at Re = 25 (L/D = 50): histogram of the IL/CF response phase difference and time-averaged power coefficient, in the (a) CF and (b) IL directions. Each histogram is normalized by the peak magnitude; the color code ranges from blue (0) to yellow (1). A vertical solid line separates the regions of excitation and damping of the structure by the flow (i.e. positive/negative values of C e{x,y} ) while a horizontal dashed line denotes the limit between clockwise and counter-clockwise orbits (Φxy = 0 • ); the probability (P ) associated with each histogram quadrant is specified. The most frequent trajectory shape (Φxy ≈ 20 • ) is represented in (a). The results obtained for L/D = 400 ( §4) are superimposed on the histograms (gray crosses/red triangles for U ⋆ = 80/U ⋆ = 110).

Figure 6 .

 6 Figure 6. Force statistics at Re = 25 (L/D = 50), as functions of U ⋆ : span-averaged values of the time-averaged IL force coefficient (left axis) and RMS values of the force coefficient fluctuations (right axis). A gray dashed-dotted line denotes the value of the IL force coefficient for a fixed rigid cylinder.

Figure 7 .

 7 Figure 7. Flow-structure system behavior for L/D = 400 at Re = 25: (a) instantaneous isosurfaces of spanwise vorticity (ωz ± 0.07) for U ⋆ = 110 (arrows represent the oncoming flow; part of the computational domain is visualized, i.e. 35D downstream of the cylinder); (b,c) time series of the (b) CF and (c) IL displacement fluctuations along the span for U ⋆ = 110; (d) spanwise evolution of the RMS value of the CF displacement for U ⋆ = 80 and U ⋆ = 110 (top axis) and time-averaged IL force coefficient for U ⋆ = 110 (bottom axis; gray area); (e) spanwise evolution of the local phase of the IL response (φx) for U ⋆ = 80 and U ⋆ = 110 (top axis) and time-averaged power coefficient (Cex) for U ⋆ = 110 (bottom axis; regions of excitation/damping of the structure by the flow are represented by yellow/gray areas). In (d), a red dashed line indicates a possible trend of the mean force due to the cylinder streamwise deflection.

  (d) which represents the spanwise evolutions of the displacement RMS values (top axis). In the IL direction, as illustrated in figure7(c), the central region of the cylinder (z ∈ [100, 300]) is characterized by dominant standing wave patterns while traveling waves develop on each side. The excited wavelengths correspond to the 19 th and 25 th modes for U ⋆ = 80 and U ⋆ = 110. The instantaneous IL responses remain symmetrical about the midspan point which is not necessarily the case for the CF responses (anti-symmetrical for U ⋆ = 80). Another persistent trend in the IL responses, also visible in figure3(c), is the presence of amplitude peaks near the body ends. As also reported for lower mode responses, the vibration frequencies (f y = 0.115 and f y = 0.109 for U ⋆ = 80 and U ⋆ = 110) are lower than the corresponding natural frequencies in vacuum (f nat (10) = 0.125 and f nat (13) = 0.118) and close to the corrected natural frequencies with C m = 1 (0.118 and 0.111).

Acknowledgements

This work was performed using HPC resources from CALMIP (grant 2020-P1248).


Patricia Ern

Jérôme Mougel

Sébastien Cazin

Manuel Lorite-Díez

Rémi Bourguet

Bending oscillations of a cylinder freely falling in still fluid

We investigate experimentally the behavior of an elongated flexible cylinder settling at moderate Reynolds number under the effect of buoyancy in a fluid otherwise at rest. The experiments uncover the development of large-amplitude periodic deformations of the cylinder (of the order of its diameter) in specific parameter ranges. Bending oscillations are observed to occur for two base flow situations, involving either a steady or an unsteady wake. In both cases, the sequence of oscillatory deformations emerging when the cylinder length is increased involves the bending modes of an unsupported cylinder with free ends. Comparison of the deformation frequency measured for the falling cylinder with the vortex shedding frequency expected for a non-deformable cylinder at the same Reynolds number indicates that the deformation is coupled to the wake unsteadiness. It also suggests that the cylinder degrees of freedom in deformability allow wake instability to be triggered at Reynolds numbers that would be subcritical for fixed rigid cylinders.

Introduction

The behavior of single bodies freely rising or falling under the effect of buoyancy in a fluid at rest has focused growing interest in the last twenty years, leading to a better understanding of the roles of wake instability and of body anisotropy on the occurrence of non-rectilinear paths (see for instance [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in fluids[END_REF]. However, for a deformable body, the coupling of its deformation with both its path and its wake remains widely unexplored with a few rare exceptions. Among these, the freely rising bubble appears as a paradigmatic case, as it provides a series of advances concerning the impact of mean and oscillatory shape deformations on the onset of path instability [START_REF] Mougin | Path instability of a rising bubble[END_REF][START_REF] Cano-Lozano | Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability[END_REF][START_REF] Bonnefis | Etude des instabilités de sillage, de forme et de trajectoire de bulles par une approche de stabilité linéaire globale[END_REF], on path characteristics [START_REF] Filella | Oscillatory motion and wake of a bubble rising in a thin-gap cell[END_REF], and on wake structure [START_REF] Veldhuis | Shape oscillations on bubbles rising in clean and in tap water[END_REF]; to cite just a few results and references. In the case of solid flexible bodies freely moving in a low-viscosity fluid, only a few studies have dealt with the coupling between path and deformation of the body. [START_REF] Tam | Tumbling dynamics of passive flexible wings[END_REF] investigated the mean reconfiguration to an arched shape of a tumbling rectangular flexible paper strip freely falling in air and the resulting increase in descent rate. They showed that the transition from straight to bent configuration occurs when the destabilizing inertial force induced by the angular velocity of the tumbling strip overbalances the bending resistance of the body. The influence of flexibility on the fluttering motion of a falling strip was investigated by [START_REF] Tam | Flexibility increases lift for passive fluttering wings[END_REF]. They observed localized upward bending deformations, generally not exceeding 5% of the chord length, during short time intervals close to the turning points of the path, the strip remaining flat during intermediate gliding periods. He showed that the observed deformation resulted in turn in increased lift, and therefore in a lower descent rate, and induced shedding of stronger vortices than in the case of stiff plates. At variance with the case of the bubble, literature on freely falling flexible bodies is thus far limited to situations bringing to light flow-induced reconfigurations, in the form of either a steady or a transitory deformation. Such behaviors for freely moving objects are the counterpart of the deformation response observed for partially fixed or locally restrained bodies immersed in a flow. For instance, shape adaptation can be traced in the leaning of an elongated flexible cylinder held on one end under the effect of a steady current [START_REF] Leclercq | Vortex-induced vibrations of cylinders bent by the flow[END_REF], or in the stationary draping featuring multiple lobes observed for a soft disk held in its center in an incoming flow, for both steady and unsteady wakes [START_REF] Schouveiler | Flow-induced draping[END_REF][START_REF] Hua | Dynamics of fluid flow over a circular flexible plate[END_REF]. However, studies also revealed the occurrence of time-periodic deformations of the body associated with vortex shedding, for instance in the case of a flexible plate clutched at its center [START_REF] Pfister | Linear stability analysis of strongly coupled fluid-structure problems with the Arbitrary-Lagrangian-Eulerian method[END_REF]. Also, a flexible cylinder hold from its ends and placed in flowing fluid exhibits vibrations which are driven by the synchronization between body deformation and flow unsteadiness [START_REF] Chaplin | Laboratory measurements of vortex-induced vibrations of a vertical tension riser in a stepped current[END_REF][START_REF] Bourguet | Vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF][START_REF] Gedikli | Observed mode shape effects on the vortexinduced vibration of bending dominated flexible cylinders simply supported at both ends[END_REF][START_REF] Seyed-Aghazadeh | Effects of boundary conditions on vortex-induced vibration of a fully submerged flexible cylinder[END_REF]. These vibrations, often referred to as vortex-induced vibrations, may involve several structural modes, sometimes simultaneously [START_REF] Bourguet | Distributed lock-in drives broadband vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF]. Their amplitudes are typically of the order of one body diameter in the cross-flow direction and one or more orders of magnitude lower in the streamwise direction. Oscillatory deformations may therefore be expected to develop in the case of a freely falling body, as it happens for bubbles or restrained bodies. Following this view, we focused our attention on a flexible cylinder settling under the influence of gravity in still fluid, and investigated experimentally the influence of flexibility on its freely falling motion over a range elongation ratios. The present paper brings to light the onset of oscillatory deformations for two distinct flow situations, both corresponding to rigid-body motions of the undeformed cylinder falling with its axis perpendicular to gravity. In the first one, periodic deformations set in for a cylinder in rectilinear fall with a steady wake. In the second one, they arise for a flexible cylinder undergoing an azimuthal oscillatory motion coupled with an unsteady wake.

Experimental approach

We consider a circular cylinder of diameter d, length L, density ρ c 1160 kg/m 3 and Young modulus E 1 MPa, falling through quiescent water (density ρ f 1000 kg/m 3 , kinematic viscosity ν 10 -6 m/s -2 ) under the influence of gravitational acceleration g. The problem is governed by four dimensionless parameters, including the solid-to-fluid density ratio m * = ρ c /ρ f and the elongation ratio of the cylinder L/d. We next introduce the gravitational velocity V g = (m * -1)gd that was shown by [START_REF] Toupoint | Kinematics and wake of freely falling cylinders at moderate Reynolds numbers[END_REF] to be the relevant velocity scale for the buoyancy-driven fall of cylinders in a comparable range of control parameters. Comparing this free fall velocity with a velocity V c associated with the bending modes of a finite-length cylinder (with a potential added mass coefficient of 1) provides the dimensionless parameter called velocity ratio U * accounting for the body deformability, such that

U * = V g V c with V c = f 0 L and f 0 = d L 2 E ρ c + ρ f . (2.1)
We last introduce the Archimedes number Ar balancing buoyancy and viscosity effects, Ar = V g d/ν. Note that once the mean fall velocity U of the cylinder is determined from the experiments, the Reynolds number Re = U d/ν can be defined, for comparison with the case of fixed bodies embedded in an incoming flow. The procedure followed is to start the series of experiments with a long cylinder (typically L 120 mm for a cylinder with d = 1 mm) and to gradually decrease its length by steps of 5 mm. This results in a joint variation of L/d and U * , while m * and Ar are kept constant. We operated two methods of release of the flexible cylinders, in order to investigate the impact of initial conditions on the cylinder behavior and to test the robustness of the phenomenon uncovered. Both methods used immersed bodies to avoid the presence of bubbles on the body surface. The first one consisted in releasing the cylinder hold straight from both ends. The second one intended to ensure no tension of the cylinder at release, and consisted in delicately pushing with a thin long-enough plate the body lying straight on a diving-board. Both methods provided the same observations, indicating that the origin of the oscillatory deformations is not related to prestressing. In both configurations, we could not avoid a slight initial inclination of the body at release. Inclination angles relative to horizontal are in all cases lower than 7 • , and in most cases lower than 4 • . The resulting lateral drift of the body (lower than 7% of the mean vertical velocity) decreased with the progressive slipping towards horizontality of the body due to the added-mass torque (see for instance [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in fluids[END_REF], for a discussion on the role of this torque). Regardless the release conditions, we observed that the deformations are robust with respect to the presence of a weak drift, as phase locking between the oscillations of the two ends of the cylinder was preserved despite path asymmetry. The inclination angle may however affect the amplitude of oscillation and the drag coefficient, as will be discussed later.

Recordings of the falling cylinders were performed by shadowgraphy with two synchronized sCMOS cameras (2560 × 2160 pixels) at a frequency of 50 Hz and placed in front of two 20 × 20 cm 2 backlight flat panels producing directional illumination. A sketch of the experimental configuration is provided in figure 1. A camera imaged a field of view of 122 × 103 mm 2 in a vertical plane (which limits the length of the cylinders), and the second camera imaged a region of 167× 141 mm 2 in a horizontal plane thanks to a mirror inclined at 45 • . For elongated bodies as those considered here, important distortion effects on the body shape arise with perspective vision, as the body is unavoidably inclined in an angular field of view and off-centre relative to the line of sight of the camera. To disentangle the contributions to the cylinder behavior of the different degrees of freedom is then an awkward procedure, even for rigid bodies, which is often resolved by capturing and reprojecting two generatrices of the cylinder [START_REF] Toupoint | Kinematics and wake of freely falling cylinders at moderate Reynolds numbers[END_REF]). In the case of a deforming body, this routine can no longer be employed straightforwardly.

To solve the problem, we used telecentric lenses, which combine several advantages: magnification and spatial resolution (typically, 15.3 px/mm for the horizontal plane and 21 px/mm for the vertical one) are invariant along the depth of field (telecentricity depth), parallax is avoided and distortion is lower than with standard lenses. Calibration is carried out by recording a fixed object of known dimensions at different locations in the cubic measurement volume, confirming the constant magnification in the field of view (variations below 0.1 px). The analysis of body displacements and deformations in time in the vertical and horizontal planes is performed by image processing, in particular to track the mean centerline of the cylinder (figure 1).

General observations and results

We first consider the case of a cylinder falling with a mean vertical velocity corresponding to Re 42 (d = 1 mm). For sufficiently short cylinders, no deformation is visible. The cylinder maintains a straight conformation, falling with its axis perpendicular to gravity like a rigid cylinder. For L/d 57, the path of the cylinder can be considered as rectilinear, since only very weak non-reproducible irregular horizontal displacements are recorded (lower than 3% of d), as is commonly the case in experiments with freely moving objects (see discussions in [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in fluids[END_REF][START_REF] Toupoint | Kinematics and wake of freely falling cylinders at moderate Reynolds numbers[END_REF]. However, as the elongation ratio of the cylinder increases, a sequence of periodic deformations emerges. For L/d larger than approximately 60, a time-periodic deformation of the cylinder sets in. It is composed of one crest and two nodes, as illustrated in figure 2a. It will be termed M 1 in reference to the corresponding bending mode of a beam. As L/d is further increased, the amplitude of deformation grows, reaches a maximum for L/d 70 (value at the cylinder ends of approximately 0.66 d and span-averaged value of 0.3 d), and decreases beyond. Oscillatory deformation M 1 is observed until L/d 95, featuring a displacement amplitude at the cylinder ends of 0.22 d and a span-averaged value of 0.15 d. For L/d 98, the cylinder switches to M 2 (in reference to mode 2), characterized by two crests and three nodes, as illustrated in figure 2b. The amplitude at the cylinder ends is then 0.37 d (span-averaged amplitude 0.15 d). Behavior M 2 is observed until L/d 110, which is the largest elongation ratio considered for this sequence, featuring a displacement amplitude at the cylinder ends of 0.71 d and a span-averaged value of 0.32 d. The bending deformations uncovered here feature several remarkable properties. First, they are restricted in a very distinctive manner to the horizontal plane, that is per- pendicular to the mean fall velocity of the body. Associated vertical displacements are considerably weaker, one or more orders of magnitude lower than those observed in the horizontal plane, and correspond to vertical velocity fluctuations lower than 10% of U . Such difference in horizontal/vertical oscillation magnitudes is a typical property of freely moving bodies close to onset of oscillatory path (see [START_REF] Ern | Wake-induced oscillatory paths of bodies freely rising or falling in fluids[END_REF] as well as low-Re vortex-induced vibrations observed for rigid [START_REF] Singh | Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes[END_REF] or flexible [START_REF] Bourguet | Vortex-induced vibrations of a long flexible cylinder in shear flow[END_REF] cylinders. Second, only a considerably weaker rigid-body oscillatory motion is observed here in the horizontal plane (displacement of the gravity center lower than 5% d, an order of magnitude comparable to that of fluctuations arising from tiny imperfections in the experiment). The third consideration is that a significant decrease in mean fall velocity occurs when shape oscillations appear. The colored crosses in figure 3a show the evolution of the Reynolds number Re as the elongation ratio and the behavior of the body vary. The periodic deformation of the body results in a loss of mean fall velocity, and therefore Re, of approximately 5 -10% for M 1 . No significant change is then observed for M 2 . The strength of the loss is related to the amplitude of deformation experienced by the body, the lower mean fall velocity being obtained for L/d 70, for which the deformation is most marked. The nearby plotted values for L/d = 72 and 74 correspond to cases presenting a lateral drift of about 7% and interestingly show higher mean vertical velocities than L/d = 70, because of their weaker amplitude of oscillation. The periodic deformation experienced by the cylinder results therefore in additional drag. Drag amplification depends on the amplitude of deformation; a comparable trend was reported for vortex-induced vibrations of rigid cylinders (see for instance [START_REF] Khalak | Motions, forces and mode transitions in vortexinduced vibrations at low mass-damping[END_REF]). This behavior is illustrated in figure 3b, displaying the drag coefficient associated with the mean vertical velocity of the body,

C d = π/2 (V g /U )
2 , obtained by balancing buoyancy and mean drag. Values from [START_REF] Toupoint | Kinematics and wake of freely falling cylinders at moderate Reynolds numbers[END_REF] for freely falling rigid cylinders with m * 1.16 (black symbols) and from [START_REF] Vakil | Drag and lift coefficients of inclined finite circular cylinders at moderate Reynolds numbers[END_REF] for fixed rigid cylinders (dark grey cross) are also displayed, for comparison purpose.

A step forward in the understanding of the flexible body behavior is achieved from the analysis of the frequencies associated with the periodic deformations. Recordings were carried out on long time series (more than 20 oscillation cycles) and frequencies could thus be determined accurately based on Fast Fourier Transform. The oscillatory deformations are dominated by a single frequency, except in some transition regions that will be discussed in the following. Furthermore, both cylinder ends are remarkably synchronized, displaying same frequencies and phase difference over the whole recording. The deformation frequency f of the freely falling cylinder is plotted as a function of the elongation ratio in figure 4a, for M 1 (blue crosses) and M 2 (red crosses). These frequencies can be compared with the natural frequencies of the bending modes i of an unsupported cylinder with free ends immersed in still fluid. Using a linear Euler-Bernoulli beam model and including added mass effects (with an added mass coefficient equal to 1) lead to the bending mode frequencies, f i = α i f 0 , where the constants α i are related to the roots of cosh β i cos β i -1 = 0 by α i = β 2 i /(8π), giving α 1 0.890, α 2 2.454, α 3 4.811, α 4 7.952... At a given time t, the corresponding deformed shape of the cylinder for mode i can be expressed in the form z c sin (2πf i t + φ i ), where φ i is a constant phase and the amplitude z c depends on the longitudinal coordinate x along the cylinder axis as

z c = cos (β i x/L) + cosh (β i x/L) + K [sin (β i x/L) + sinh (β i x/L)] with (3.1a) K = [sin (β i ) + sinh (β i )] / [cos (β i ) -cosh (β i )] .
(3.1b)

Note that for all modes i the integral of z c between 0 and L is zero, so that the position of the center of gravity of the cylinder in this model equation does not evolve in time. The frequencies f i with i ∈ {1, 2} are plotted with thin dashed lines in figure 4a for mode 1 (blue) and mode 2 (red), showing correspondence between the succession with L/d of the vibrational modes f i and the experimental observations. The envelopes of the first two modes provided by equations (3.1) are also superposed (thick black dashed lines) on the deformed states of the cylinder determined experimentally at different times in figure 2, yielding again good agreement. Comparing now the deformation frequency f with the inertial time scale associated with the body free fall d/U , which is also the characteristic time scale for vortex shedding about the body, provides values of f d/U varying in the range 0.105 -0.140. These values are close to the Strouhal number value, St w = 0.105, associated with subcritical vortex shedding downstream of a long cylinder determined by [START_REF] Buffoni | Vortex shedding in subcritical conditions[END_REF] for Re = 42 by slightly vibrating the cylinder to trigger the flow. The wake frequencies f w = St w U/d extracted from [START_REF] Buffoni | Vortex shedding in subcritical conditions[END_REF] using the Reynolds numbers corresponding to each L/d in our experiments are drawn with filled circles in figure 4a; these values are close to the measured deformation frequencies. Figure 4b summarizes the comparison between the frequencies. The data points (blue crosses for M 1 and red ones for M 2 ) indicate the evolution of the body deformation frequency, f , normalized with the proper bending frequency of the cylinder, f 0 , as a function of U * . The wake frequencies f w (also normalized with f 0 ) issued from [START_REF] Buffoni | Vortex shedding in subcritical conditions[END_REF] are drawn with filled circles. We can see that the deformation frequency of the freely moving cylinder is locked to the wake frequency. As L/d varies, the deformation frequency departs from the natural frequencies of the cylinder, indicated by horizontal lines at α i . At a given stage however, the next mode of the cylinder is excited and a frequency jump occurs, as can be seen here between M 1 and M 2 . For all the L/d investigated, the close agreement of the deformation frequency with the wake frequency suggests that the bending deformations observed are linked to the wake unsteadiness. Furthermore, the results indicate that the cylinder degrees of freedom in deformability allow wake instability to be triggered below the threshold for fixed rigid bodies, as the present Reynolds number values (Re 40) would be subcritical in the rigid-body case [START_REF] Inoue | Vortex shedding from a circular cylinder of finite length at low Reynolds numbers[END_REF]. This point will be discussed further in section 4.

The second situation investigated is that of a freely falling cylinder displaying a periodic motion in association with an unsteady wake. For that purpose, a higher Reynolds number is considered, namely Re 120, obtained for cylinders with d = 1.9 mm (figure 3a 4a (orange triangles), the AZI oscillation occurs at a frequency of approximately 4.2 Hz. This corresponds to a Strouhal number of 0.13, which is slightly smaller than that of the Bénard-von Kármán instability about a fixed cylinder at this Re, St w 0.17, obtained from the relation proposed by [START_REF] Williamson | A series in 1/ √ Re to represent the Strouhal-Reynolds number relationship of the cylinder wake[END_REF]. Using dye visualization, [START_REF] Toupoint | Kinematics and wake of freely falling cylinders at moderate Reynolds numbers[END_REF] showed that the AZI motion is coupled with an unsteady wake (see their figure 17c), similar to those for fixed long cylinders referred to as Type II by [START_REF] Inoue | Vortex shedding from a circular cylinder of finite length at low Reynolds numbers[END_REF] and as oblique vortex shedding by [START_REF] Williamson | Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers[END_REF]. However, for the freely-moving cylinder, a periodic beating of the wake was observed near the body ends, at the oscillation frequency of the azimuthal angle, and in phase opposition between the two ends. Now for L/d 39, we observed that mode 1 oscillatory deformations (i.e. M 1 ) sets in. Behavior M 1 is further observed until L/d 57, the largest elongation ratio that could be investigated for this cylinder. The highest amplitude of deformation is reached for L/d 52, with a displacement of the cylinder ends of 0.8 d and a span-averaged value of 0.39 d. Note that in this case also, higher mode responses are expected to exist. Mode 2 deformations were in fact observed for a different cylinder having slightly smaller diameter (d = 1.8 mm) and larger length (L = 131 mm), i.e. L/d 73. This sequence of deformations shares the properties described in the previous case, in particular the significant velocity decrease and associated drop in Re (figure 3a, triangular data points). The resulting increase in drag is conspicuous in figure 3b (triangular data points). For this case also, the proximity between the wake frequency f w determined from the expression St w (Re) proposed by [START_REF] Williamson | A series in 1/ √ Re to represent the Strouhal-Reynolds number relationship of the cylinder wake[END_REF] (open circles), the cylinder natural frequency f i (dotted curves) and the observed deformation frequency f (triangular data points) is manifest in figures 4a and b. In this second situation, the oscillatory deformations supplant an oscillatory rigid-body motion (i.e. AZI). For sufficiently short flexible cylinders, vortex shedding is coupled with a rigid-body vibration of the cylinder. This corresponds to cylinder ends moving in phase opposition, a property shared with M 2 but conflicting with M 1 . Deformation M 1 is nevertheless triggered for a slightly longer cylinder. Closer examination of the main frequency content of the body displacement near the onset of M 1 reveals that the transition features mixed responses, the oscillatory deformation and the rigid-body motion co-exist for both L/d 37 and L/d 39, as can be seen in figure 4c. The rigid-body motion predominates in the former case, whereas it is outweighed by the deformations in the latter, and disappears as the elongation ratio is increased further. Note that co-existence of M 1 and M 2 was not detected here in the measurements for L/d 95 and L/d 98 close to the transition T 1-2 from M 1 to M 2 for Re 42 (figure 4d), while superposition of deformation modes is commonly observed for a flexible cylinder hold from its ends [START_REF] Chaplin | Laboratory measurements of vortex-induced vibrations of a vertical tension riser in a stepped current[END_REF][START_REF] Huera-Huarte | Towing tank experiments on the vortex-induced vibrations of low mass ratio long flexible cylinders[END_REF].

Concluding remarks

We investigated experimentally the behavior of an elongated flexible cylinder settling at moderate Reynolds number under the effect of gravity in a fluid otherwise at rest. For a given cylinder diameter, the experimental approach consisted in gradually decreasing the body length, thereby changing its natural deformation frequency without significantly impacting its mean falling velocity in the absence of deformation. Short enough cylinders behaved like rigid bodies, showing no detectable reconfiguration, and fell with their axis perpendicular to gravity. However, for longer cylinders and therefore higher velocity ratios U * , the experiments brought to light the springing up of periodic oscillatory deformations of the freely falling flexible cylinders in specific parameter ranges. To the best of our knowledge, the only counterpart of this phenomenon in the literature concerns freely rising bubbles, as discussed in the introduction. We further showed that the sequence of oscillatory deformations emerging when the cylinder length is increased involve the bending modes of an unsupported beam with free ends, each mode being associated with a natural frequency f i . Besides, comparison of the deformation frequency with the vortex shedding frequency f w expected for a non-deformable cylinder at the same Reynolds number indicated that the deformations are coupled with wake unsteadiness. Flow-induced bending deformations involving mode i are expected to occur during free fall when f i is close to f w . The simple criterion matching bending mode natural frequency to vortex shedding frequency, f i = f w , corresponds to

L/d = α i /(U St w ) (E/(ρ c + ρ f )) 1/4 . (4.1)
For a cylinder of diameter d, this expression can be used to obtain a prediction of L/d associated with mode i by assuming U = V g (or C d = π/2), which allows to determine

Re and subsequently St w (St w 0.1 and 0.17 for d = 1 and 1.9 mm, respectively). Corresponding estimations for L/d and U * are reported in figures 4a and b for modes 1 and 2 (vertical blue and red lines) and appear consistent with experimental observations. From equation (4.1) we can expect the occurrence of higher order bending modes, for instance in the d = 1 mm case for L/d ≈ 162 (M 3 ) and L/d ≈ 208 (M 4 ).

The experiments also revealed that the oscillatory deformations can develop for distinct flow configurations. Bending oscillations for a freely falling cylinder are here shown to appear for bodies displaying a rectilinear path with a steady wake, and for bodies displaying a rigid-body oscillatory motion coupled with an unsteady wake. In the former case, the close agreement between deformation and wake frequencies suggests that the bending deformations that replace the rectilinear fall are also related to wake unsteadiness, though the Reynolds number would be subcritical in the rigid-body case (Re 40). This indicates that deformability may allow wake instability to be triggered below the threshold corresponding to fixed rigid bodies. The anticipation of wake destabilization due to degrees of freedom of the body has been observed for closely related problems of fluid-body interaction. For instance, [START_REF] Cossu | On the instability of a spring-mounted circular cylinder in a viscous flow at low Reynolds numbers[END_REF] and [START_REF] Meliga | An asymptotic expansion for the vortex-induced vibrations of a circular cylinder[END_REF] investigated the global stability and the nonlinear dynamics close to the threshold of vortex-induced vibrations in the wake of a damped, spring-mounted, circular cylinder. They demonstrated the role of the natural eigenvalue of the cylinder-only system on triggering subcritical vortex shedding at low solid-to-fluid density ratios. In the case of freely moving rigid bodies having different geometries, theoretical and computational studies also revealed that the coupling between the degrees of freedom of the body and the fluid can shift the thresholds and frequencies associated with wake instability about the fixed body or in contrast give rise to different regimes of path instability [START_REF] Tchoufag | Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders[END_REF][START_REF] Mathai | Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders[END_REF]. For instance, in the limit of heavy plates, [START_REF] Assemat | The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study[END_REF] found that the threshold of path instability matches that found for a fixed plate, whereas it decreases when the solid-to-fluid mass ratio decreases, while keeping Strouhal number values comparable with those of the fixed plate. The present observations provide an experimental evidence of the destabilization of a coupled solid-fluid system at a frequency characteristic of vortex shedding, and for Reynolds numbers that are subcritical in the fixed rigid-body case, with the novelty that the degrees of freedom in deformability of the body are involved here. In the second flow configuration, oscillatory bending deformations develop for bodies displaying a periodic motion coupled with an unsteady wake. Except very close to the threshold, the cylinder response to wake forcing becomes essentially a deformation, which takes over the rigid-body displacement of the body. In turn, as the displacements of the cylinder ends shift in this case from phase-opposition to in-phase motion, it leads to a change of wake structure, that still needs to be explored.

A major issue now regards the identification and characterization of the wake structures associated with the different deformation modes. The detailed interaction between the flow and the deforming body remains also to be elucidated, in connection with the different amplitudes of deformation that emerge. We are currently heading in that direction, the challenge being to be able to capture both the body and the flow structure over the two orders of magnitude of length scales involved in the problem (spanning from d to L). At issue here is also a better understanding of the role of the cylinder free ends. As observed for freely falling deformable plates by [START_REF] Tam | Flexibility increases lift for passive fluttering wings[END_REF], deformation may concentrate on the body periphery, where the flow field distribution is likely to trigger it. The degrees of freedom in translation and rotation associated with the cylinder free ends may also play a role in deformation phenomena, in the same way they enable the occurrence of oscillatory paths for finite-length rigid cylinders different from those observed for two-dimensional cylinders. As a first step to address this question, we carried out preliminary experiments with flexible rings in a comparable range of parameters. They brought to light oscillatory deformations of the ring at a nearby frequency, illustrated in figure 5; this indicates that also in a closed configuration (i.e. in the absence of end effects), the degrees of freedom in deformability of the body can be excited by the surrounding flow and can couple with it.

Figure 1 .

 1 Figure1. Sketch of the experimental set-up, showing the positions of the cameras, the measurement volume in the squared-section water tank, along with a qualitative picture of the falling cylinder. Cropped binarized images from both cameras obtained by image processing are also displayed, and show the centerline of the cylinder contour (red line).

Figure 2 .

 2 Figure 2. Illustrations of the bending oscillatory deformations of a cylinder (with d = 1 mm): (a) M1 (L/d = 70); (b) M2 (L/d = 110); left: three-dimensional view, y standing for the vertical coordinate; right: deformation in the horizontal plane (x, z), x describing the undeformed cylinder axis and z the transverse direction. The dashed black lines outline the envelope given by (3.1) with an amplitude adjusted to that of the cylinder ends in the experiments.

Figure 3 .

 3 Figure 3. (a) Reynolds number Re and (b) drag coefficient C d as functions of the elongation ratio L/d; d = 1 mm (crosses); d = 1.9 mm (triangles); rectilinear motion R (green); azimuthal rigid-body oscillatory motion AZI (orange); oscillatory deformations M1 (blue) and M2 (red). Values of C d for freely falling rigid cylinders with L/d = 10 and 20, and m * 1.16 from Toupoint et al. (2019) (black symbols) and for a fixed rigid cylinder with L/d = 20 from Vakil & Green (2009) (dark grey cross).

Figure 4 .

 4 Figure 4. (a) Deformation frequency f of the freely falling cylinder for M1 and M2; natural frequencies fi (dashed line for d = 1 mm, dotted for d = 1.9 mm); frequency of AZI for d = 1.9 mm. Crosses and triangles: same convention as in figure 3a. Open and filled circles: fw from Williamson & Brown (1998) and from Buffoni (2003), respectively. Colored dashed (d = 1 mm) and dotted (d = 1.9 mm) vertical lines: values of L/d corresponding to fi = fw for mode 1 (blue) and 2 (red) according to (4.1). (b) Frequencies f and fw normalized with f0 as functions of the velocity ratio U * . Horizontal lines: values of αi. Vertical lines: values of U * corresponding to those of L/d in (a). Main frequency content (c) near the transition TA-1 from AZI to M1 for Re 120, L/d 37 (orange) and L/d 39 (blue); and (d ) near the transition T1-2 from M1 to M2 for Re 42, L/d 95 (blue) and L/d 98 (red).

Figure 5 .

 5 Figure 5. Deformation at a frequency f 4.5 Hz of a thin flexible ring freely falling at approximately 5 cm/s in water at rest (ring diameter D 7 cm, squared-section width l 1.5 mm), corresponding to Re 75 (based on l). (a) Ring shape in the horizontal plane (x, z) at several times t. The insert figure (i) shows a sketch of the deformation with magnified displacements. The local radial displacement about the time-averaged deformation is denoted r, Ar is the corresponding amplitude and θ is the azimuthal angle. (b) Normalized amplitude Ar/l spanned along θ. (c) Temporal evolutions of r/l for crest points 2 and 4 and for node point 3.

  , triangular data points). At this Re, a rigid-body azimuthal oscillation (termed AZI in the following) of the flexible cylinder is observed from L/d 21 to L/d 37. It corresponds to the azimuthal oscillation reported by Toupoint et al. (2019) for rigid cylinders with L/d = 20, m * 1.16, Re 130 and Re 220. In both cases, this rotational oscillatory motion features reproducible low-amplitude (approximately 0.2 • ) oscillations of the azimuthal angle of the cylinder, on top of irregular weak displacements comparable to those observed for the rectilinear path at Re 42. As shown in figure

† Email address for correspondence: ern@imft.fr

We are grateful to M. V. D'Angelo for her help and advice manufacturing the cylinders. We also thank M. Riodel and J.-D. Barron for technical support. Declaration of Interests. The authors report no conflict of interest.


Timothy Daly
email: timothydaly@paris-sorbonne.fr

Marion Houot

Anouk Barberousse

Yves Agid

Stéphane Epelbaum

Practices of the Amyloid Cascade Hypothesis Between 1992 and 2019

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

There are and have been several aetiological hypotheses of Alzheimer's disease (AD), but the amyloid cascade hypothesis (ACH), which claims that cerebral beta-amyloid deposition is the driving pathogenic factor of AD, has been the most dominant over the last 25 years [START_REF] Hardy | Pathways to Alzheimer's disease[END_REF][START_REF] Liu | History and progress of hypotheses and clinical trials for Alzheimer's disease[END_REF].

Interestingly, despite the dominance of the ACH, there have been no successful clinical trials with anti-amyloid agents (though see promising results from [START_REF] Budd Haeberlein | Clinical Development of Aducanumab, an Anti-Aβ Human Monoclonal Antibody Being Investigated for the Treatment of Early Alzheimer's Disease[END_REF]. The ACH's scientific merit has been drawn into question on multiple occasions [START_REF] Herrup | The case for rejecting the amyloid cascade hypothesis[END_REF], and many experts are skeptical, even cynical about the status it has enjoyed [START_REF] Abbott | Failed Alzheimer's trial does not kill leading theory of disease[END_REF].

The ACH is a biomedical hypothesis in that its evaluation is both biological and medical. In biology, it must adequately explain the etiology of AD. In medicine, it will be evaluated by its ability to help the development of disease-modifying therapies based on its claims. It is because

! 2
of the dual biomedical nature of the hypothesis that not only its confirmation, but also its refutation, is extremely difficult.

The philosopher Sir Karl Popper , famous amongst contemporary biomedical scientists, proposed that testing hypothetical conjectures was the essential activity of science.

Furthermore, he proposed that negative results tell us more about the truth of a hypothesis than confirmatory ones: the accumulation of supposedly confirmatory data can continue ad infinitum, whereas the obtention of a false test result supposedly tells us that something is wrong with the theory now [START_REF] Schilp | The Philosophy of Karl Popper[END_REF].

However, Popper's intuition about falsification can only take us so far, logically speaking. It is extraordinarily difficult with a clinical trial to test the theoretical core of the ACH (that betaamyloid deposition is central to human AD) due to the number of "auxiliary hypotheses" or background assumptions involved in the test of any one anti-amyloid agent [START_REF] Selkoe | The amyloid hypothesis of Alzheimer's disease at 25 years[END_REF]. In short, the ACH predicts that an anti-amyloid agent, in patients correctly diagnosed, given at the appropriate stage of the disease, targeting the correct species of beta-amyloid, with the correct targeting mechanism (see debates around passive vs. active immunisation), at the right dose, will have some important disease-modifying effect. It is only when all of these obstacles have been overcome that we can say that the hypothesis is being truly tested by a clinical trial.

Given these difficulties, we reason that studying confirmation and refutation of the ACH cannot be soundly pursued at this stage by a bibliometric study, but have chosen instead to study the beliefs of scientists towards the ACH. In his famous 1877 article [8] "The Fixation of Belief," the scientist-philosopher Charles Sanders Peirce (1839Peirce ( -1914) ) suggested that there were 4 methods by which we arrive at our beliefs: individual tenacity, collective authority, a priori racionation, ! 3 and science. The scientific method, he claims, is the only method "by which our beliefs may be determined by nothing human, but by some external permanency -by something upon which our thinking has no effect" (CP5.385).

As early as 2001 in AD research, certain researchers have felt that human factors have contributed to the dominance of the ACH: Joseph et al. [START_REF] Joseph | Copernicus revisited: amyloid beta in Alzheimer's disease[END_REF] called this phenomenon "The Church of the Holy Amyloid"; Mudher and Lovestone [START_REF] Mudher | Alzheimer's disease-do tauists and baptists finally shake hands?[END_REF] used religious language to refer to "TAUists" (defending the primacy of tau involvement) and "BAptists" (defenders of the ACH) defending the role of these proteins in AD aetiology.

In light of the difficulties testing the logical confirmation or refutation of the ACH, and the hypothesised existence of divided belief with the possibility of ideology within the AD community, we have chosen to study the acceptance of the ACH and test certain hypotheses about how it has been accepted, using citations as a proxy for acceptance or refusal of the idea by researchers within the field, as a way of measuring their belief in the ACH's validity. The main question we asked about such practices was, "When the authors of scientific papers cite this paper devoted to explicitly defending the ACH, do they tend to do so in a polarised (positive, neutral, or negative) way?" The methodology used in this study was inspired by Greenberg [START_REF] Greenberg | How citation distortions create unfounded authority: analysis of a citation network[END_REF], who likewise used citation practices to measure the "belief in a specific scientific claim by studying the pattern of citations among papers stating it." He found that when the authors of a scientific paper encounter and cite the claim that β-amyloid is associated positively with inclusion body myositis, he could divide the citation into those receiving the idea positively, neutrally, and negatively. He also discovered that amongst citing articles there was an overwhelming preponderance to cite the idea positively.

! 4

We tested four hypotheses about citation practices of articles citing HH92. Firstly, we wished to test Hardy et al. [START_REF] Hardy | Pathways to Alzheimer's disease[END_REF]'s claim of the dominance of the ACH. Secondly, validation of the citation metric: whether HH92 citations could be used to predict how scientists positioned themselves towards the ACH. Thirdly, whether it pays to cite HH92 favourably, leading to higher citation rates of positively-citing articles. Fourthly, whether HH92 was cited earlier within articles with time; for example, as a growing authority on AD.

Finally, recognising the importance of reproducibility of evidence in favour of a hypothesis, as a measure of its scientific support, we measured how many original articles in our sample tested the ACH and how many arrived at a favourable test result. Thus, our study classifies citations according to Greenberg's taxonomy, as a measure of how researchers encounter the idea of "Amyloid deposition as the central event in the aetiology of Alzheimer's disease," and tests hypotheses about such practices and their consequences.

METHODS

HUB SELECTION AND SAMPLING

Four candidates for the ACH hub were identified [START_REF] Hardy | Amyloid deposition as the central event in the aetiology of Alzheimer's disease[END_REF][START_REF] Beyreuther | Amyloid precursor protein (APP) and beta A4 amyloid in the etiology of Alzheimer's disease: precursor-product relationships in the derangement of neuronal function[END_REF][START_REF] Hardy | Alzheimer's disease: the amyloid cascade hypothesis[END_REF][START_REF] Selkoe | The molecular pathology of Alzheimer's disease[END_REF]. "Alzheimer's Disease: The Amyloid Cascade Hypothesis" ("HH92") was chosen, since firstly, as its title suggests, the claims of the ACH are most explicitly laid out, and secondly, this article has the most citations within this wave of "pro-ACH" articles [Figure 1]. A sample of 1 in 10 random articles was chosen (540 / 5400 articles). 95 articles were excluded: those not written in English, not published in peer-reviewed scientific journals, or with 0 citations since their publication (since we wished to measure the transmission of the major idea of the ACH, and an article with 0 citations has no citation-related impact).

DETERMINING HH92 CITATION AND ACH VALENCE ! 6

Inspired by Greenberg's study [START_REF] Greenberg | How citation distortions create unfounded authority: analysis of a citation network[END_REF], who "classified each citation as supportive, neutral, or critical according to how its underlying statement supported the belief" [p. 2], it was hypothesised that articles cite HH92 in a potentially non-neutral way; that is, they may tend to position themselves vis-à-vis the theoretical statement being made explicit within HH92 (that beta-amyloid deposition is the central event in the aetiology of AD). Each citation was therefore classified as "negative" (unconvinced of HH92's ACH's validity) "neutral" (no definitive claim made about the ACH's veracity, or suggesting the need for further evidence before making claims about it) or "positive" (already convinced of the ACH's validity at the time of citing) by Timothy Daly. Furthermore, since it was recognised that the ACH as defended within HH92 is but one version of the ACH, in light of the changes that the field has undergone since 1992 [START_REF] Hardy | The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics[END_REF][START_REF] Selkoe | The amyloid hypothesis of Alzheimer's disease at 25 years[END_REF], the overall conclusion of the article with respect to the suitably of beta-amyloid as a target as compared to other possible therapeutic strategies was also measured ("ACH conclusion"). This conclusion was typically suggested by the study's results, or found explicitly mentioned in the title, abstract, and/or discussion; it was also classified as negative, neutral, or positive in the same way as HH92 citations.

Thereafter, articles were divided into "Original" and "Review" articles, classified by date, citation rate in Google Scholar (measured as citations per year, to account for articles being published during different years). For each article, the place within the bibliography that HH92 occupied was noted (for articles using a quantified bibliography, 303 articles out of 445). The geography of first-author affiliation was also noted as a function of continent (North America: NA, Europe: EU, Asia: AS, Africa: AF, Australasia, AU).

! 7

IDENTIFYING SUPPORT FOR THE ACH FROM ORIGINAL STUDIES WITHIN THE SAMPLE

110 original articles within our sample tested the ACH's basic claim. For each article, we obtained the "post-test result" from a semantic analysis of the article's results and discussion section, divided post-test results into "positive," "inconclusive," or "negative." In "inconclusive" articles, we included those articles whose authors, following their test of the ACH, stressed the need for further testing and interpretation before making claims about the ACH's veracity. We then proceeded to test the reproducibility of pro-ACH test results within the sample.

HYPOTHESES STATISTICAL ANALYSES

Data are presented as counts and percentages for categorical variables and as mean ± standard deviation for normal distribution numerical variables or as median [first quartile Q0.25, third quartile Q0.75] for non-normal distribution numerical variables.

Fisher's exact test were performed to compare category distribution of a categorical variable between several groups. Welch's t-test for normal distribution numerical variables or Wilcoxon rank-sum test were performed to compare distribution of a numerical variable between two groups. To compare more than two groups, ANOVA Fisher test or Kruskal-Wallis test were ! 8 performed when appropriate followed by Tukey's HSD test or pairwise Wilcoxon rank-sum tests with Benjamini Hochberg correction for post hoc comparisons.

In order to test the hypothesis that as time passes, HH92 is mentioned earlier in the articles, a one-sided Spearman's rank test between the year of the publication of the articles and their citations per year was performed with rho < 0 as alternative hypothesis.

HYPOTHESES AND OBJECTIVES: ARTICLE VALENCES

By measuring citation practices, we hoped to gain information regarding the attitudes of practicing scientists within the AD field towards the ACH. In this way, positive citations were used as a proxy for acceptance of the authors' belief that beta-amyloid might indeed have some central role to play in AD; neutral citations were seen as a more reserved attitude in need of more compelling evidence; negative citations were seen to represent authors' mistrust of the ACH's basic claim given the available evidence. The various hypotheses we put forward tested the distribution of citation valences, as well as the consequences of citation valence.

INVESTIGATING HARDY ET AL. (2014)'S CLAIM OF ACH DOMINANCE

We hypothesised that within our sample of articles, given the central place of HH92 in the broader AD literature, there should be a positive majority of both HH92 citations and "pro-ACH" overall article conclusions.

! 9

Furthermore, given the 27-year period over which we measured citation practices, we also measured whether citation valence was associated with three other properties: the article's overall conclusion with respect to the ACH; the Google Scholar citation rate of the citing article itself; and the position of the HH92 within the bibliography of the citing article.

HH92 CITATION AS A PREDICTOR OF ACH VALENCE

When HH92 is cited positively, we hypothesised that it would result in the author's belief "feeding in" to a pro-ACH overall conclusion, whereas neutral and negative citations would not allow for prediction of overall conclusion based on HH92 citation practice due to confounding factors. This allowed us to measure just how much the HH92 served as a positive influencer of the beliefs of citing authors.

THE IMPACT OF ARTICLE VALENCE ON CITATION RATE OF ARTICLES

We wanted to know if articles citing HH92 favourably, or with a favorable conclusion towards the ACH, were themselves cited more frequently than neutral or negative citers, thus contributing to more "pro-ACH" traffic in the literature, which might explain the success of the ACH compared to its competitors.

THE EFFECT OF TIME ON THE POSITION OF HH92 CITATION WITHIN THE BIBLIOGRAPHY ! 10

Finally, we wanted to test whether articles citing HH92 favourably cited it earlier on than articles in previous years, as a proxy for measuring how belief in the AD towards the role of β-amyloid in AD has evolved since 1992, with the idea of its central role becoming more of a foregone conclusion in such positively-citing papers as a function of time.

SUPPORT FOR THE ACH FROM ORIGINAL ARTICLES

Given that the ACH has been dominant, as a necessary scientific condition to justify such dominance, the original articles should provide reproducible support for the ACH in their post-ACH test results. Within each testing article, we studied the nature of the inference made based on the results from the test (as evidence in favour of the ACH; as inconclusive vis-à-vis the ACH; as evidence against the ACH).

RESULTS

GENERAL CHARACTERISTICS AND FINDINGS

Of the 445 articles citing HH92 in the sample, 26 (5.8%) were written between 1992-1999; 97 ( Notes. Data are given as mean ± standard deviation for continuous variables and as count (percentages) for categorical variables. Welch's t-test was used to compare groups for continuous variables and Fisher's exact test for qualitative variables. Abbreviations: MD = data.

THE ASSOCIATION BETWEEN HH92 CITATION AND ACH VALENCE

The ACH conclusion of the articles tended to be different according to HH92 citation practice (Cramer's V=0.33, p<0.001). Indeed, the ACH conclusion was positive in 85.8% of articles citing HH92 positively, in 55.8% citing HH92 neutrally and only in 14.3% citing HH92 negatively.

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HH92 CITATION POSITIVITY AND CITATION RATE OF ARTICLES

Positively-citing articles had a mean ± standard deviation citation rate (CR) of 7.8±10.1 citations per year; neutral articles had a CR of 9.6±14.3; negatively-citing articles had a CR of 12.0±16.4, although these differences were not statistically significant (p=0.208).

Concerning overall ACH conclusion, negative articles had a higher mean CR of 13.6±15.7, neutral 10.1±17.6, and positive 7.9±10.1. Negative articles had a significantly higher CR than positive-ACH articles (Cliff's delta=-0.25, p=0.007).

CHANGES IN HH92'S PLACE WITHIN THE BIBLIOGRAPHY WITHIN POSITIVELY-CITING ARTICLES

303 (68.1%) articles had a numerical bibliography, which we used to measure the place of the HH92 within the bibliography. These articles were not a perfect representation of the greater sample, principally since they tended to cite HH92 more positively than the articles with nonnumerical bibliography (37.6% v 28.9%, p=0.040 and Cramer's V = 0.12).

We found no association between the year of publication and HH92's place in the bibliography for the entire sample (rho=-0.05, p=0.371), but when looking at articles citing HH92 positively, we found an association between place in the bibliography as a function of decade (rho=-0.19, p=0.020) whereas there was no association on articles citing HH92 neutrally (rho=0.03, p=0.670)

! 15

[Figure 2]. Only 6 articles cited HH92 negatively so we did not calculate the Spearman correlation coefficient.

DISCUSSION

The results of this bibliometric study support neither "herd-like behaviour" [17] nor Hunter et al.'s claim of the existence of "two broad groups" [START_REF] Hunter | Time for a change in the research paradigm for Alzheimer's disease: the value of a chaotic matrix modeling approach[END_REF] for and against the ACH, for at least three reasons. Firstly, the majority of authors encountering the idea that beta-amyloid has some central role to play in AD do not cite HH92 positively. Secondly, the majority is actually made up of "neutral" citing authors. Thirdly, any negative citations or anti-ACH traffic were in a small ! 16 minority, and thus such "anti-ACH" opinions are not in a "broad group." In summary, the large amount of "pro-ACH" conclusions and traffic in this network of articles cannot be explained by a bias towards the positivity of HH92 citation practices.

We identified two major trends within citing behaviour: a group of articles citing HH92 positively and tending towards taking it as an established authority with time, versus an unchanging majority of articles with a neutral attitude towards HH92. Negative HH92 citations were exceedingly rare, and anti-ACH traffic in the network represents a 10.6% minority. We did not encounter "herd-like" behaviour: those original articles testing the ACH found an overwhelming majority (89%) of pro-ACH test results (thus justifying their pro-ACH conclusion), and there was no majority of positive HH92 citations or pro-ACH conclusions in review articles. Further studies with actual citation networks, as opposed to individual articles, could shed greater light on the question of the potential "herd-like" behavior of the AD field.

Hardy et al. [START_REF] Hardy | Pathways to Alzheimer's disease[END_REF]'s claim that the ACH has dominated opinion appears true given the majority (67.4%) of pro-ACH conclusions. HH92 can only accurately predict ACH conclusion when HH92 citation is positive (in 86.8% of cases). However, positive HH92 citation did not have a favorable effect on CR, but rather a tendency towards the opposite effect. If anything, our results

suggest that coming to an anti-ACH conclusion is a more favourable way of achieving a higher CR.

The results suggested here do not suggest that one scientific paper can fully "fix belief" in an authoritative manner in a community of researchers. It is clear that ideas around beta-amyloid's

! 17
central role in AD have remained alive in the literature, but the theoretical changes the ACH has undergone have clearly been taken into account by the community.

Furthermore, the empirical support we identified for the ACH was overwhelmingly positive, in terms of the amount of empirical studies with a pro-ACH test result in our sample. In further studies, we will discuss the limits of reproduction as a model of confirmation of a hypothesis, and discuss the idea of the "triangulation" of "different lines of evidence" from different methods [START_REF] Munafò | Robust research needs many lines of evidence[END_REF].

This paper is not the first bibliometric study of the AD field [START_REF] Serrano-Pozo | Four Decades of Research in Alzheimer's Disease (1975-2014): A Bibliometric and Scientometric Analysis[END_REF][START_REF] Dong | Publication Trends for Alzheimer's Disease Worldwide and in China: A 30-Year Bibliometric Analysis[END_REF]. However, previously study have focused on trends, and ours is the first to test hypotheses about the reception of an idea as the central focus of the paper. Furthermore, placing this paper within the context of the intellectual space occupied by the ACH, it is the first time-to the best of our knowledge-that a hypothesis made outside the scientific AD literature (Lovestone's claim that "Our failing in the Alzheimer's field is to have acted like a herd" in a Financial Times article) has been tested by an article within it. Indeed, although the ACH is the object of intense debate within the scientific literature, outside the strict publishing norms of published science such as journalistic interviews, scientists have their say on efforts within the field. Recognising that the scientific literature does not give absolute transparency with regards to scientists' intuitions about research, our group will also publish results from an anonymous, international survey in collaboration with the Alzheimer's Association with scientists and clinicians to have direct access to their opinions on ! 18

the ACH and other theories and treatments of AD without having to pass through the filter of the literature.

Concerning the limitations of the present study, we were well aware when we started this work that GA Higgins, co-author of HH92, is a discredited researcher. The retraction of Kawabata et al. [START_REF] Kawabata | Alzheimer's retraction[END_REF] in which he appeared to have exaggerated the extent of neuronal degeneration in a transgenic mouse model of AD overexpressing APP was rightly seen as a scandal. However, we consider the fact that pro-ACH ideas have survived until 2020 in spite of such unfavourable coverage as a testament to the wide evidence base in favour of them.

Furthermore, we recognise that the number of articles (445) in our sample represents a drop in the ocean of the 35,000 papers studying beta-amyloid in AD. However, defense of the ACH represents a broad church-there is no one amyloid hypothesis-and there are various ways in which authors might defend the ACH alone or integrate it into more complex aetiologies. There is no denying that comparative studies between the fates of the ideas in this hub and those in others would enrich the empirical study of the ACH's place within the literature. One can see our manuscript as a first attempt to grasp the epistemology of the ACH in the vast field of Alzheimer's disease research. Confirmatory studies, including natural language processing analyses of the literature would certainly be important before drawing definitive conclusions about citation practices within the ACH field, but we feel they are outside of the scope of this manuscript.

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Finally, beyond the problem of sample size, the fact that the majority of citations were neutral suggests that nuances within this heterogenous community of researchers are real but not wellcaptured by Greenberg's (2009) methodology. For example, scholars in philosophy of science [START_REF] Van Fraassen | The Scientific Image[END_REF] have previously distinguished between a scientist accepting a theory as a basis for action (i.e. testing its claims and comparing it to other theories within the field in review articles)

versus believing it to be factually true. It is not obvious that the citation polarity used herein allows us to distinguish between these subtleties and further studies could conduct further linguistic analyses to distinguish between such nuances. Furthermore, although the classic distinction between "original" and "review" articles allows us to divide articles in a simple way, there are different kinds of reviews, such as classical "narrative" reviews, meta-analyses, etc., and the lack of distinction between these groups risks glossing over important nuances between article types.

In conclusion, we consider that this objective, empirical study of citation practices and support for the ACH suggests that the ACH has indeed been dominant in the field, has quantitatively good empirical suggest but that there are no problematic citation practices or herd-like behaviour identified. Further studies, using more nuanced measures of scientific belief, as well as more elaborate models of scientific confirmation, could provide further insight into the acceptance of the ACH by the scientific community as well as suggesting ideas on how to improve the ACH's robustness.

! 5 "Figure 1 -

 51 Figure 1 -HH92 is the ACH hub most likely to be influencing the beliefs of citing authors due to

Figure 2 :

 2 Figure 2: The identification of two major trends within citing articles: neutral attitudes were

  7%) tested the claims of the ACH. 98 (89.1%) of 110 articles returned a positive test result, suggesting reproducible support for the ACH from original tests.

	research papers, 110 (50.The majority (343, 77.1%) of papers were produced by European and North American research
	teams, with Asia being the third most productive continent (81, 18.2%).
		all N=445	original (49.21%) N=219	review (50.79%) N=226	p
	Period of publication				0.471
	[1992-2000[	26 (5.84%) 13 (5.94%) 13 (5.75%)
		97	53	44
	[2000-2010[	(21.80%)	(24.20%)	(19.47%)
		322	153	169
	[2010-2019[	(72.36%)	(69.86%)	(74.78%)
		110	110		<0.00
	ACH tested	(24.83%)	(50.69%)	0 (0.00%)	1*
	Result of the ACH test (MD=285)
			98
	ACH positive		(89.09%)
	ACH inconclusive		5 (4.55%)
	ACH negative		7 (6.36%)
					<0.00
	ACH valence				1*
		289	169	120
	positive	(64.94%)	(77.17%)	(53.10%)
	21.8%) between 2000 and 2009, and 322 (77.4%) between 2010 and 2019. 219 articles (49.2%) 109 37 72
	neutral	(24.49%)	(16.89%)	(31.86%)
	were original research articles and 226 (50.8%) were review articles. Within the 219 original 47 34
	negative	(10.56%)	13 (5.94%)	(15.04%)
		! 11

! 12

Table 1 .

 1 Sample characteristics of the 445 articles within the sample citing Hardy & Higgins

	(1992).
	POSITIVITY OF HH92 CITATIONS AND ACH CONCLUSIONS WITHIN THE
	SAMPLE

In overall pro-ACH conclusion articles, there were significantly more original articles compared to negative or neutral ACH conclusion articles (58.5% original articles in pro-ACH, 33.9% in neutral and 27.7% in negative , Cramér's V=0.25, poverall<0.001) ; they tended to test the ACH more than in neutral ACH conclusion articles (32.4% in pro-ACH, vs 8.3% (p<0.001) in neutral and vs 17.0% in negative (p=0.039), Cramér's V=0.24, poverall<0.001); their first author affiliation was more significantly from Asia compared to negative-ACH (20.8% for pro-ACH vs 4.3% for negative-ACH, Cramer's V=0.19, p=0.009).

ACKNOWLEDGEMENTS ! 20

Thanks to Hervé Maisonneuve for showing the Greenberg (2009) paper to Timothy Daly, and to Mathieu Bourdenx for his helpful comments on the manuscript.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest.