Flutter of structurally inhomogeneous cantilevers in laminar channel flow

Journal of Fluids and Structures 90 (2019) 177–189

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Flutter of structurally inhomogeneous cantilevers in laminar channel flow ∗

Julien Cisonni a , , Anthony D. Lucey a , Novak S.J. Elliott a,b a b

Fluid Dynamics Research Group, School of Civil and Mechanical Engineering, Curtin University, Perth, Australia School of Physiotherapy and Exercise Science, Curtin University, Perth, Australia

article

info

Article history: Received 29 November 2018 Received in revised form 13 April 2019 Accepted 13 June 2019 Available online xxxx Keywords: Flexible cantilever Flutter instability Non-prismatic beam Stepped cantilever

a b s t r a c t Flutter instability of flexible cantilevers axially immersed in channel flow has been studied mainly for slender bodies with uniform properties. The present study addresses the stability of one-dimensional stepped cantilevers comprising two sections of different thickness immersed in two-dimensional viscous channel flow. The influence of the relative mass and rigidity of the two sections on the motion of the cantilever is explored through variations of length and thickness ratios. The parametric investigation shows that, for instance, making the free end of the cantilever twice thinner or thicker than the clamped end over a short fraction can produce structures that are either more stable or more unstable, depending on the fluid-to-solid mass ratio. In the case of a heavy and stiff free section and a light and flexible clamped section of comparable length, the excitation of lower structural modes by slower flows is significantly destabilising as compared to a uniform cantilever of same length and total mass. Strong destabilisation and weak stabilisation of the fluid–structure interaction system can result from either thinning or thickening the cantilever free-end which can also lead to changes in the flutter mode shape. These complex variations are quantitatively presented through stability maps. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The vast majority of fundamental studies on flow-induced fluttering bodies has focused on low complexity models including idealised geometries and a limited number of parameters (Shelley and Zhang, 2011). Thus, most analyses conducted for simplified fluid–structure interaction (FSI) systems were based on finite rectangular flags, plates, sheets, or panels of constant thickness. By adopting uniform dimensions and material properties, these investigations were able to provide important insight into FSI mechanisms and showed how irreversible energy transfer from fluid to structure arising from a phase difference between structural motion and fluid pressure (Eloy et al., 2008; Howell et al., 2009) causes flutter. Designs to modify and control the structural response were explored mainly through external alterations and additions to the slender body, including changes of mounts and supports (Howell and Lucey, 2015), and added masses, springs and dampers (Tang et al., 2009; Huang and Zhang, 2013; Tan et al., 2013). This study investigates the flutter instability of flexible cantilevers of non-uniform thickness axially immersed in viscous channel flow. It focuses on the identification of the conditions for which the two sections of stepped cantilevers might sustain each others’ oscillations or behave as mainly separate uncoupled structures. ∗ Corresponding author. E-mail address: [email protected] (J. Cisonni). https://doi.org/10.1016/j.jfluidstructs.2019.06.006 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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Most recent analyses of flutter instability of flexible structures immersed in axial flow provided innovative approaches to energy harvesting (Howell et al., 2009; Michelin and Doaré, 2013; Howell and Lucey, 2015), fluid mixing and cooling (Shoele and Mittal, 2014; Jha et al., 2017), microscale flow-sensors (Liu et al., 2012; Wang et al., 2007), biomechanical systems modelling (Balint and Lucey, 2005; Tetlow and Lucey, 2009; Elliott et al., 2011; Cisonni et al., 2014), and building and vehicle panel-vibration suppression (Tan et al., 2013; Tsigklifis and Lucey, 2017). The solutions investigated for energy harvesting and fluid mixing aimed to extend the parameter ranges over which flutter instability occurs and enhance the structural oscillations. On the other hand, for most other applications, the objectives are usually delay and mitigation of flutter to avoid structural instabilities and reduce their adverse effects, such as destructive failures. Both promoting and preventing flutter oscillations require reliable prediction and efficient control of the FSI instabilities and their desired or undesired effects, such as material fatigue, power extraction (Elvin and Erturk, 2013; Howell and Lucey, 2015), unsteady wakes (Michelin et al., 2008; Tang and Païdoussis, 2008), variations of hydro-/aero-dynamic efficiency (Carruthers and Filippone, 2005; Alben and Shelley, 2008) and sound generation (Bae et al., 2008; Manela and Howe, 2009). In many fields of engineering, compliant structures interacting with aerodynamic forces can have non-uniform geometric and mechanical properties (Dowell, 2015). In biomechanics, for instance, the vibrations of the soft-palate and its conical extension, the uvula, induced by inspiratory airflow is an FSI system studied extensively in recent years for its implications in breathing disorders such as snoring and obstructive sleep apnoea. The complex shape and composite structure of the uvulopalatal system originate from the layered soft tissues with viscoelastic properties varying anisotropically (Berry et al., 1999; Birch and Srodon, 2009; Cheng et al., 2011). Numerical models of this particular FSI system have been proposed to account for the intricate geometric configuration of the airway and the tissue (Zhu et al., 2012; Pirnar et al., 2015) as well as the local variations in mechanical properties (Xu et al., 2009). Most of these studies have provided detailed accounts of the passive motion of the soft palate and its effect on the flow properties, but only for a limited number of specific cases. These more elaborate and comprehensive models can be challenging to control and computationally demanding so that they are often used for particular problems and conditions. Therefore, this approach usually only provides limited understanding of the physical phenomena and the critical parameters underlying FSI instabilities in laminar flow which can be found, for instance, during slow breathing and in blood flow. In this study, the coupling of one-dimensional stepped cantilevers, comprising two sections with different mass and flexural rigidity, to two-dimensional Navier–Stokes equations is numerically simulated for a Reynolds number of 200, based on the channel height, to ascertain the nature of the cantilever’s flow-induced motion. The instability boundaries of the FSI system are determined through a comprehensive parametric analysis of its linear stability. Thus, over the wide ranges of parameters explored, results demonstrate the complex dynamics of the inhomogeneous flexible structure interacting with the surrounding fluid. This investigation ultimately provides details to elucidate whether a stepped cantilever with a thinner or thicker free-end immersed in an axial flow can be more unstable than its uniform counterpart.

2. Method 2.1. Model and parameters An infinitely-thin flexible cantilever axially immersed in two-dimensional viscous channel flow constitutes the FSI system represented schematically in Fig. 1(a) in dimensional form (all dimensional quantities are denoted by ∗ ). The ∗ ∗3 cantilever beam of length L∗C and thickness h∗C is characterised by its density ρC∗ and flexural rigidity B∗ = Eeff hC /12, ∗ ∗ where Eeff is the effective Young’s modulus defined as Eeff = E ∗ /(1 − ν 2 ) with E ∗ and ν being, respectively, the Young’s modulus and Poisson’s ratio of the solid material. The fluid, characterised by its density ρF∗ and dynamic viscosity µ∗F , is flowing in a channel of length L∗ and height H ∗ . The flexible cantilever is clamped to a rigid wall of length L∗inlet positioned along the centreline of the channel and parallel to the channel walls. The upstream end of the channel is thus divided into two inlets of identical height H ∗ /2 at which identical steady Poiseuille velocity profiles with average velocity U ∗ are imposed. The downstream end of the channel, located at a distance L∗outlet from the downstream free end of the flexible cantilever, is set as the outlet where the flow is assumed to be parallel and axially traction-free. The shape of the flexible cantilever is represented solely by its centreline, parameterised by the one-dimensional Lagrangian coordinate ξ ∗ . The non-uniform flexural rigidity and mass are varied locally through a thickness function h∗C (ξ ∗ ) dividing the cantilever into two sections of length L∗clamp and L∗free , and thickness h∗clamp and h∗free , corresponding respectively to the clamped-end and free-end as shown in Fig. 1(b). To avoid discontinuities in the thickness profile, a smoothstep-2 polynomial function (see Eq. (1)) is used over the relatively short length L∗trans to smooth the transition between the two sections. The shape of the thickness profile is based on the constant reference thickness h∗0 (corresponding to the uniform cantilever for which h∗C (ξ ∗ ) = h∗0 ) and parameterised by the length ratio ϑL = L∗free /L∗C and thickness ratio ϑh = h∗free /h∗clamp

J. Cisonni, A.D. Lucey and N.S.J. Elliott / Journal of Fluids and Structures 90 (2019) 177–189

179

Fig. 1. Description of the FSI system: (a) physical quantities of the model of flexible cantilever axially immersed in viscous channel flow and (b) flexible-cantilever thickness profile defined in Eq. (1).

so that L∗free = ϑL L∗C , L∗clamp = (1 − ϑL )L∗C and

⎧ h∗0 L∗ ⎪ , ξ ∗ < ξA∗ = L∗clamp − trans h∗clamp = ⎪ ⎪ ⎪ ϑL)ϑh 2 ⎪ ((1∗− ϑL ) + ⎪ ∗ ∗ ⎪ h + h − h ⎪ clamp free clamp ⎨ ( ( ) ( ∗ ) ( ∗ ) ) ξ − ξA∗ 4 ξ − ξA∗ 3 ξ ∗ − ξA∗ 5 h∗C (ξ ∗ ) = ⎪ − 15 + 10 , ξA∗ ≤ ξ ∗ ≤ ξB∗ × 6 ⎪ ⎪ L∗trans L∗trans L∗trans ⎪ ⎪ ⎪ ⎪ ϑh h∗0 L∗ ⎪ ⎩ h∗ = , ξ ∗ > ξB∗ = L∗clamp + trans . free (1 − ϑL ) + ϑL ϑh 2

(1)

This piecewise function allows keeping the average thickness h∗C and total mass of the stepped cantilever beams the same as those of the uniform cantilever beam, so that h∗C = h∗0 . To analyse the problem in non-dimensional form, all geometric dimensions and spatial coordinates are scaled with the channel height H ∗ , flow velocity components with the average inlet velocity U ∗ , the fluid stresses with the viscous ∗ , and time with H ∗ /U ∗ . This leads to five scale µ∗F U ∗ /H ∗ , solid stresses and loads with effective Young’s modulus Eeff non-dimensional parameters, in addition to the length and thickness ratios ϑL and ϑh , characterising the FSI dynamics:

ρF∗ L∗C , = M= ρC∗ h∗0 ρC∗ h∗C ρF∗ L∗C

√ ∗ ∗

U = U LC

ρC∗ h∗C B∗

√ ∗ ∗

= U LC

ρC∗ h∗0 B∗

,

Re =

ρF∗ U ∗ H ∗ , µ∗F

LH =

L∗C H∗

,

hL =

h∗0 L∗C

.

(2)

The stability of the FSI system depends primarily on the average mass ratio M and average reduced velocity U. For different stepped cantilevers obtained from variations of ϑL and ϑh , the average mass ratio remains constant as the thickness profile function is set to keep h∗C = h∗0 and the total solid mass constant. However, for a particular value of average reduced velocity, the behaviour of the FSI can differ significantly depending on ϑL and ϑh , as these two ratios affect the local mass and rigidity of the beam. The Reynolds number Re, the ratio LH between cantilever-length and channel-height and the cantilever thickness-to-length ratio hL are independent of the thickness profile and set constant for all the configurations investigated (see Section 2.5). The open-source finite-element library oomph-lib is used to carry out time-marching numerical simulations of the FSI system. An overview of the approach employed and its implementation is provided in the following sections. Further details can be found in the extensive tutorials available on the oomph-lib webpages (Heil and Hazel, 2006). Rigorous validation of the FSI methods and modelling used in the present study can be found in the study conducted by Cisonni et al. (2017) for a uniform cantilever. 2.2. Fluid mechanics The two-dimensional fluid domain shown in Fig. 1(a) is discretised using nine-node quadrilateral Taylor–Hood elements implemented with adaptive mesh-refinement capabilities. The body-fitted fluid-mesh is updated using an algebraic node update procedure, based on a generalisation of Kistler and Scriven’s ‘‘method of splines’’ (Kistler and Scriven, 1983). A second-order backward differentiation formula scheme is used for the fluid time-stepping. The incompressible Navier–Stokes equations,

( Re

) ∂u + u · ∇ u = −∇ p + ∇ 2 u , ∂t

(3)

and the continuity equation,

∇ ·u=0 ,

(4)

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are solved in the Eulerian coordinate system x = ⟨x1 , x2 ⟩ for the non-dimensional fluid velocity u and pressure p. At both inlets, a parabolic velocity profile is imposed for the axial velocity so that

{ u=

12x2 (1 − 2x2 ) e1 ,

0 ≤ x2 ≤ H /2 (upper inlet) −12x2 (1 + 2x2 ) e1 , −H /2 ≤ x2 ≤ 0 (lower inlet),

(5)

where e1 and e2 are the unit vectors of the Eulerian coordinate system. At the outlet, where the flow is assumed parallel, the transverse velocity is set to zero while the axial velocity is determined using the axially-traction-free outflow condition. The no-slip condition is applied on all stationary (u = 0) and flexible-cantilever walls. 2.3. Beam mechanics The motion of the flexible cantilever is solved using a one-dimensional, elastic Kirchhoff–Love beam discretised using two-node Hermite finite elements. A Newmark scheme is used for the solid time-stepping. The position vector to a material point is given by r(ξ ) in the undeformed cantilever and by R(ξ ) in the cantilever deformed by a resultant load Teff combining the tractions acting on the top and bottom of the beam. The principle of virtual displacements that governs the cantilever motion is given by LH

∫

{ γ δγ +

0

h2C 12

( κ δκ −

1

√

hC

A a

( Teff −

UhL 12

)2

∂ 2R ∂t2

)

} · δR

√

a dξ = 0 ,

(6)

where a=

∂r ∂r · , ∂ξ ∂ξ

A=

∂R ∂R · ∂ξ ∂ξ

(7a,b)

are the squares of the lengths of√infinitesimal material line elements in the undeformed and deformed configurations, respectively. Therefore, the ratio A/a represents the stretch of the cantilever centreline while the strain γ and bending κ are given by

γ =

1

(A − a) ,

2

κ = − (C − c ) ,

(8a,b)

with c =n·

∂ 2r , ∂ξ 2

C =N·

∂ 2R ∂ξ 2

(9a,b)

representing the curvature of the cantilever centreline, respectively, before and after the deformation. n and N denote the unit normals (pointing into the fluid, as shown in Fig. 1(a)) to the top face of the undeformed and deformed cantilever centreline, respectively. The downstream end (ξ = 1) of the flexible cantilever is free while its upstream end (ξ = 0) is clamped to the rigid wall dividing the channel inlet, so that R(ξ = 0) = r(ξ = 0) ,

⏐ ⏐ ⏐

d(R · e2 ) ⏐ dξ

ξ =0

=0.

(10)

2.4. Fluid–structure coupling Eqs. (3), (4) and (6) are discretised monolithically (Heil et al., 2008) and the Newton–Raphson method is used to solve the non-linear system of equations for the FSI system, employing the SuperLU direct linear solver within the Newton iteration. The no-slip condition on the flexible walls (top and bottom of the cantilever) is given by u=

∂ Rˆ , ∂t

(11)

ˆ is the position vector of the FSI interface. The fluid load acting on the beam in Eq. (6) combines the fluid tractions where R on the top and bottom of the cantilever so that

( Teff =

UhL 12

)2

MhL {( Re

)⏐ )

}

p|top I − ∇ u + (∇ u)T ⏐top · N − p|bottom I − ∇ u + (∇ u)T ⏐bottom · N

(

(

(

)⏐

)

.

(12)

J. Cisonni, A.D. Lucey and N.S.J. Elliott / Journal of Fluids and Structures 90 (2019) 177–189

181

Fig. 2. Example of flow-domain mesh adaptation: (a) initial mesh, (b) mesh at the end of steady initialisation and (c) mesh during large-amplitude oscillation of the flexible cantilever. The inlet rigid wall and flexible cantilever are indicated in cyan and magenta, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.5. Numerical simulations The primary objective of this study is to characterise the critical( instability, i.e. that which first appears with increasing ) mean flow velocity, of the immersed stepped cantilever in the M , U parameter space when the relative length and thickness of the two beam sections, clamped and free, change through variations of the length ratio ϑL between 1/8 and 7/8, and of the thickness ratio ϑh between 0.1 and 10. The base configuration is kept constant for all the cases considered. To ensure that the slender beam approximation holds for all cases, the base cantilever thickness-to-length ratio hL is set to 0.01 (h0 = 0.02). Also, the Reynolds number Re and the cantilever-length-to-channel-height ratio LH are both fixed to 200 and 2, respectively. While the fluid relative viscosity, corresponding to different values of Re, and the cantilever confinement, corresponding to different values of LH , can play a significant role in the FSI instability mechanisms (Cisonni et al., 2017), these effects are not explored in the present study. As regards the non-dimensional channel dimensions, the channel height is set to H = 1, the length of the wall dividing the inlet to Linlet = 1 and the distance between the outlet and the free-end of the flexible cantilever to Loutlet = 3 (Linlet + LH ) = 9. The latter is prescribed to ensure that the effect of the outlet boundary condition on FSI stability characteristics remains marginal while restricting excessive computational cost.) ( Each simulation corresponding to a set of non-dimensional parameters M , U , ϑL , ϑh is initialised as follows: (I) the inlet flow velocity is gradually increased in a sequence of nine steady solutions while the flexible cantilever is constrained in its undeformed shape, and (II) the flexible cantilever is then given a small amplitude deformation and the system is again solved as a steady problem. This initial shape corresponds to the uniform-cantilever in vacuo Mode 2 (n = 2, β2 ≈ 4.6941) transverse deflection with amplitude ηT0 = h0 = 0.02 at the free-end tip

η (ξ ) =

ηT0

[(

2

(

cosh βn

ξ

)

LC

( )) ( ( ) ( ))] ξ Cn ξ ξ − cos βn − sinh βn − sin βn , LC

Sn

LC

LC

(13)

where Cn = cos(βn ) + cosh(βn ), Sn = sin(βn ) + sinh(βn ) and βn satisfies the dispersion relation cos(βn ) cosh(βn ) = −1. The cantilever is then released and the unsteady FSI problem solved with a time step ∆t = 0.02 until 20 periods of the cantilever-tip oscillation are obtained. The flow domain is meshed initially using a 10 × 90 grid. As shown in Fig. 2, all elements are allowed to be refined or coarsened during the simulation (three times per steady solve during the initialisation and once per time step during the unsteady run) depending on local normalised error. The flexible cantilever is discretised with 80 elements and the transition in the thickness profile (see Fig. 1(b)) is prescribed over 4 of these elements so that the transition length Ltrans represents 5% of the flexible cantilever length LC . This total number of solid elements ensures that enough elements are present in each section of the flexible cantilever to capture accurately the kinematics of the beam deformation, even when Lfree = LC /8 or Lclamp = LC /8. The linear stability analysis of the FSI system is based on the time trace of the transverse deflection of the flexiblecantilever tip ηT . The stability or instability of the system is characterised by the exponential decay or growth rate αT of the tip oscillations normalised by the frequency of oscillation fT . In this analysis, all frequencies are scaled by the characteristic in vacuo frequency of the uniform cantilever fC∗ =

1 L∗C2

√

B∗

ρC∗ h∗C

=

1 L∗C2

√

B∗

ρC∗ h∗0

.

(14)

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Fig. 3. Stability map for the immersed uniform cantilever: (a) Exponential growth rate αT and (b) cantilever-tip oscillation frequency fT as functions of the mass ratio M and reduced flow velocity U. The uniform cantilever eigenfrequencies (in vacuo linear theory) associated with Modes 2 to 4 are indicated with dashed lines in (b).

The frequency fT is determined by performing a peak detection on a 8192-FFT of the signal ηT while the growth rate αT is estimated through a linear interpolation of the log envelope detected on the same signal. For particular cases, the cantilever oscillations can very quickly be damped out or leave the linear small-amplitude regime. Therefore, when the FSI system is very stable or unstable, αT and fT are quantified using fewer than 20 periods of the cantilever-tip oscillation. This process allows the mapping of the FSI stability characteristics over 41 values of M, logarithmically spaced between 0.1 and 10, and 51 values of U, logarithmically spaced between 1 and (50, for variations of the two non-dimensional parameters ) ϑL and ϑh , to extract the critical combinations of parameters M , U , ϑL , ϑh for which the immersed stepped cantilever become unstable. 3. Stability of immersed uniform cantilever Detailed characterisation of(the dynamics of the uniform cantilever (ϑh = 1) for fixed LH = 2 and Re = 200 is obtained ) from the consolidation in the M , U parameter space of the data simulated with both sections of the cantilever having the same mass and rigidity as hclamp = hfree = h0 . The resulting topography shown in Fig. 3(a) indicates the strength of stability (αT < 0) or instability (αT > 0) represented by the coloured contours of the normalised exponential growth rate αT of the tip oscillations. The thick solid line denotes the neutral stability (αT = 0) demarcating the boundary between stable and unstable motions of the cantilever. The shape of this curve shows that the critical flow velocity U crit required to initiate self-sustained oscillations is strongly linked to the structural mode most excited by the flow, hence to the relative properties of the flow and solid. The sequence of lobes representing the cascade from lower to higher order modes with increasing M has been reported in previous studies based on viscous flow modelling (Shoele and Mittal, 2016; Cisonni et al., 2017) or using the inviscid-flow assumption (Eloy et al., 2008). When mounted axially in a channel, flexible cantilevers have been shown to exhibit flow-induced single-mode flapping motions for which the onset is significantly influenced by the transverse and span-wise confinement (Wu and Kaneko, 2005; Howell et al., 2009; Doaré et al., 2011; Shoele and Mittal, 2016). As the channel height is reduced, the critical flow velocity required to destabilise the cantilever decreases because of the greater pressure difference across the cantilever associated with the strengthened axial momentum of the flow. For relatively low Reynolds numbers, the flow characteristics downstream of the cantilever can change considerably (Jha et al., 2017; Shoele and Mittal, 2014) and the inertial effects on the critical velocity are, in relative terms, diminished by the increased viscous contribution to the hydrodynamic forces (Cisonni et al., 2017). Thus, stabilisation of the immersed-cantilever FSI system as the Reynolds number decreases has been shown to be stronger for high fluid-to-solid mass ratios, usually associated with higher-mode structural oscillations. The influence of the Reynolds number Re and the cantilever-length-to-channel-height ratio LH on the cantilever-tip oscillation frequency, and more generally on the stability boundary, can become significant as these two control parameters alter the interactions between the viscous flow and the individual structural modes. However, the neutral-stability curves globally retain their main features with variations of Re and LH so that the critical values U crit and f crit of the immersed uniform cantilever for LH = 2 and Re = 200 can reliably represent the phenomenology described below. For M < 0.7, Mode 2 is the dominant structural mode and the higher order modes remain unexcited by the flow. Over this range of mass ratio, the frequency f T of the oscillations varies relatively smoothly with increasing reduced flow velocity as shown in Fig. 3(b). For M > 0.8, Modes 3, 4 and 5 successively become the dominant structural mode excited by the flow as the mass ratio increases to 10. For reduced flow velocities lower than the critical velocity, the mode-cascade structure can be observed on the stability map. This translates into abrupt changes in oscillation frequency with increasing reduced flow velocity. For all modal branches of the neutral-stability curve, the critical cantilever-tip oscillation frequency f crit is lower than the corresponding in vacuo linear eigenfrequency of the cantilever. In comparison to in vacuo condition, the decrease in frequency at which immersed cantilevers oscillate is caused by the fluid-inertia loading that primarily depends upon the mass ratio (Fu and Price, 1987; Yadykin et al., 2003; Shabani et al., 2013).

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183

Fig. 4. Comparison of FSI stability boundary for variations of thickness ratio with Lfree = Lclamp = LC /2: (a) Critical flow velocity U crit and (b) critical cantilever-tip oscillation frequency f crit as functions of average mass ratio M for different values of ϑh .

Fig. 5. Stability map for the immersed stepped cantilever with M = 1 and Lfree = Lclamp = LC /2: (a) Exponential growth rate αT as a function of the thickness ratio ϑh and reduced flow velocity U, and (b) critical cantilever-tip oscillation frequency f crit as a function of the thickness ratio ϑh . Labels (i) to (ix) indicate cases for which the mode shape is presented in Fig. 6.

4. Stability of immersed stepped cantilevers with two sections of the same length The stability of immersed stepped cantilevers divided in half (ϑL = 1/2) is analysed for variations of the relative thickness ϑh = hfree /hclamp of the free and clamped sections. The thickness profile of the cantilever with both sections of the same length is given by Eq. (1) (in dimensional form) with Lfree = Lclamp = LC /2. The results for this particular configuration are presented first to illustrate the range of phenomena associated with the stepped-cantilever instabilities. As shown in Fig. 4(a), when the free section is thinner than the clamped section (ϑh < 1), the neutral-stability curve is in general gradually shifted to lower average fluid-to-solid mass ratio M and lower critical velocity U crit . Thus, compared to the uniform case, higher structural modes are triggered at lower mass ratios and become more easily excited by the flow as the free section becomes lighter and more flexible. Conversely, when the free section is thicker than the clamped section (ϑh > 1), the neutral-stability curve is in general only slightly shifted to higher average fluid-to-solid mass ratio M. However, as the free section becomes heavier and stiffer, the critical velocity U crit initially increases for 1 < ϑh ≲ 2 before significantly decreasing for ϑh ≳ 2. The influence of the relative thickness ϑh on the stability boundary does not produce the same pattern over the higher range of mass ratio for which higher structural modes dominate the motion of the stepped cantilever. This is an indication that the complexity of the three-way interactions between the flow and the two cantilever sections is increased by higher structural modes coming into play with both thinner and thicker free sections. However, at high mass ratio, uniform and stepped cantilevers remain very stable since the reduced flow velocities required to initiate the oscillations become relatively high. It must also be noted that the resolution over the higher range of U is coarser in this study. Thus, overall, for behaviours corresponding to Mode 2, Mode 3 and Mode 4 of the immersed uniform cantilever, an increase in thickness of the free or clamped section of the stepped cantilever tends to destabilise the FSI system except over a narrow range of thickness ratio, 1 < ϑh ≲ 2. As shown in Fig. 4(b), the decrease in critical velocity through most of the range of thickness ratio considered is accompanied by a decrease of the critical frequency at which the stepped-cantilever tip oscillates when the system is neutrally stable. The relative thinning or thickening of the free part of the stepped cantilever induces non-linear variations in the FSI system behaviour. The influence of the thickness ratio on the flutter instability thresholds and on the pre- and postcritical cantilever-tip oscillations is illustrated in Fig. 5 for M = 1. In this figure, the instability of the reference uniform

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Fig. 6. Mode shapes associated with the data points indicated in Fig. 5(a) (M = 1 and ϑL = 1/2). Axial coordinates x1 are scaled with the length of the flexible cantilever LC and transverse coordinates x2 are scaled with the maximum cantilever deflection ηmax . Clamped and free parts of the stepped cantilever are represented in blue and red, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

cantilever (ϑh = 1) corresponds to point (v), which is demonstrated to be a Mode 3 oscillation in Fig. 6 (case (v)), and which is also confirmed in Fig. 3 at M = 1. For comparison, the critical velocity and frequency curves of the uniform cantilever equivalent (LH = 1) to the free and clamped sections of the stepped cantilever are included in Fig. 5. These curves intersect for ϑh = 1 when the two sections are identical and the equivalent uniform cantilever flutters in a Mode 2 motion. When the free part of the stepped cantilever is thinner than its clamped part (ϑh < 1), the critical velocity decreases with decreasing thickness ratio, as shown in Fig. 5(a). Thus, the stepped cantilever becomes increasingly more unstable than the uniform cantilever as its free section becomes lighter and more flexible. While the modal motions of the uniform and stepped cantilevers are comparable, the deflection and bending of the clamped part are greatly reduced with decreasing ϑh , as shown in Fig. 6 (cases (iv) to (ii)). However, for ϑh < 0.2, other modal branches emerge on the stability map because of higher modes of the free section being excited by the flow, as illustrated in Fig. 6 (case (i)). Indeed, when the free section becomes much thinner than the clamped section, the latter is much heavier and stiffer than the former and the reference uniform cantilever. Consequently, the free part of the stepped cantilever mainly interacts with the flow while the clamped part remains very stable and contributes only marginally to the motion of its fluttering counterpart. Over this range of thickness ratio, the neutral-stability curve for the stepped cantilever gets closer to that of the uniform cantilever equivalent to the free part, showing that the behaviour of the free part is more independent from the mostly undeformed thick clamped part. It must be noted that the neutral-stability curve for the stepped cantilever does not strictly converge to that of the uniform cantilever equivalent to the free part for very low thickness ratio since the stepped cantilever’s free section is not clamped (by imposed boundary condition) to its clamped part like the equivalent uniform cantilever is to the fixed wall. When the free part of the stepped cantilever is thicker than the clamped part (ϑh > 1), its deformation is reduced with increasing thickness ratio but its large displacement is driven by the highly deformed clamped section, as shown in Fig. 6 (cases (vi) to (viii)). This results in an initial increase of the critical velocity when the thickness ratio is increased from 1. For a similar modal motion, the thick free section makes the stepped cantilever more stable than the uniform cantilever up to ϑh ≈ 2.2 but strongly destabilises it beyond this value, as indicated in Fig. 5 by the rapidly decreasing critical velocity with increasing thickness ratio. However, for ϑh > 1.8, a lower mode, as illustrated by case (ix) in Fig. 6, is also excited by the flow at lower reduced velocities, causing the stepped cantilever to be significantly more unstable than the uniform cantilever. The emergence of this lower-mode instability lobe in the stability map for very low values of U results from the effect of an ‘elongated’ added mass, that constitutes the heavier and stiffer free-end section, connected to the lighter

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Fig. 7. Comparison of FSI stability boundary for variations of length ratio with (a) hfree = hclamp /2 and (b) hfree = 2 hclamp : Critical flow velocity U crit as a function of average mass ratio M for different values of ϑL .

and more flexible clamped section. Indeed, Huang and Zhang (2013) have shown that a concentrated mass had the most destabilising effect on the flow-induced flutter of a plate when it was added to the trailing edge. For the stepped cantilever, the relatively thicker free section destabilises the clamped section in a similar way but, unlike a concentrated mass, also interacts with the flow even if it does not deform considerably. Therefore, it has a more extensive effect on the motion of the clamped section and on the whole FSI system behaviour. This is clearly indicated by the stepped cantilever’s lower mode being increasingly unstable with increasing thickness ratio while only modes of the clamped section’s equivalent uniform cantilever higher than Mode 3 can become unstable for ϑh > 4.5. The modifications of the stepped cantilever’s stability boundary caused by variations of the thickness ratio for different average mass ratios follow similar trends to that observed for M = 1. However, for some average mass ratios, e.g. M = 0.25 or M = 2, the neutral stability of the reference uniform cantilever is clearly associated with a single modal motion. In these cases where the regions of transition to lower or higher mode instabilities are sufficiently separated, lower structural modes of the stepped cantilever cannot become unstable with a thicker free section so that new modal branches do not appear on the stepped cantilever’s stability map as the thickness ratio is increased beyond ϑh = 1. At the other end of the stepped cantilever’s stability map, for decreasing thickness ratio below ϑh = 1, higher modes of the thinner free section can become unstable and dominate the stepped cantilever motion for any average mass ratio. This can result in the stepped cantilever being more stable than the reference uniform cantilever. Thus, with a free part significantly thicker than the clamped part, the stepped cantilever is always more unstable than the reference uniform cantilever, as well as more unstable than the uniform cantilever equivalents to both the free and clamped sections. Otherwise, the changes in thickness ratio can make the stepped cantilever either more stable or more unstable than the reference and equivalent uniform cantilevers. 5. Stability of immersed stepped cantilevers with two sections of varying length In this section, the influence of the relative length of the free section on the stability of stepped cantilevers is examined. The analysis focuses on the two particular cases in which the free section is either twice as thin (ϑh = 1/2 or hfree = hclamp /2) or twice as thick (ϑh = 2 or hfree = 2 hclamp ) as the clamped section. As defined in Eq. (1), when the stepped cantilever consists almost exclusively of its clamped part (ϑL → 0+ ) or of its free part (ϑL → 1− ), it becomes identical to the reference uniform cantilever. Therefore, for any thickness ratio ϑh , the total volume of solid material remains constant and the stability characteristics of the stepped cantilever evolve cyclically from and to those of the reference uniform cantilever as the length ratio ϑL increases from 0 to 1. For the case when the free section is thinner than the clamped section, it can be seen from Fig. 7(a) that the neutralstability curve is generally shifted to lower average fluid-to-solid mass ratio M in comparison to the reference uniform cantilever, except when the thin free part occupies more than three quarters of the stepped cantilever. For ϑL = 7/8, the critical velocity curves for the uniform and stepped cantilever are almost superimposed, indicating that the stepped cantilever with a long and thin free part, and a short and thick clamped part behaves almost like its uniform counterpart. This is confirmed by the mode shape for case (xi) shown in Fig. 9 which is similar to that of the reference uniform cantilever (case (v)) shown in Fig. 6. For ϑL < 3/4, the shift to lower mass ratio of Mode 2, Mode 3 and Mode 4 branches is accompanied by a general decrease of the critical velocity except when the free section becomes relatively short. Indeed, the neutral-stability curve corresponding to ϑL = 1/8 is the only curve with higher U crit than that of the uniform cantilever throughout almost the whole range of M. In this case, the short and thin free section makes the stepped cantilever more stable than the reference uniform cantilever for all mass ratios considered. Thus, the long and stiffer clamped section does not bend significantly and most of the deformation of the stepped cantilever occurs in the more flexible free section, as shown in Fig. 9 (case (x)). Also, the modal transitions are smoother for ϑL = 1/8, and to a lesser extent for ϑL = 1/4,

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Fig. 8. Critical flow velocity U crit as a function of length ratio ϑL for different average mass ratio, and with (a) hfree = hclamp /2 and (b) hfree = 2 hclamp . −⊕ → symbols indicate the region bounded by the curves in which the FSI system is unstable. Labels (x) to (xiv) indicate cases for which the mode shape and flow field are presented in Figs. 9 and 10, respectively.

denoting a damping effect of a shorter thin free section on the motion of the stepped cantilever. As shown in Fig. 8(a) for different average mass ratios, relatively short and long free sections tend to stabilise stepped cantilevers with a free part twice as thin as their clamped part, and this stabilisation can be more pronounced for ϑL < 3/8. On the other hand, these stepped cantilevers tend to be more prone to instability for 3/8 < ϑL < 5/8. Over this range of length ratio, free and clamped sections are of comparable length and thickness, so that they can more easily interact to reciprocally sustain their flow induced motion, while the bandwidth of destabilising flow velocity is increased for the whole stepped cantilever system. When the free section is thicker than the clamped section, the neutral-stability curve is generally shifted to higher average fluid-to-solid mass ratio M in comparison to the reference uniform cantilever, as shown in Fig. 7(b), except when the thick free part occupies more than three quarters of the stepped cantilever. Similar to the case hfree = hclamp /2, the critical velocity curves for the uniform cantilever and the stepped cantilever with ϑL = 3/4 and ϑL = 7/8 are almost superimposed when hfree = 2 hclamp . Thus, e.g. for M = 1, the mode shape for case (xiv) shown in Fig. 9 is comparable to that of the reference uniform cantilever (case (v)) shown in Fig. 6. The shift to higher mass ratios of the different modal branches is more pronounced for relatively short and thick free section (ϑL < 3/8) and all the higher modes become unstable at lower reduced flow velocity. However, for M < 3, the critical velocities associated with Mode 2 instability for all the values of length ratio considered almost coincide with that of the reference uniform cantilever. Thus, a free section twice as thick the clamped section, and of any length, appears to have very little influence on the stepped cantilever’s stability for the lower range of mass ratio. More generally, for mass ratios at which the neutral stability of the reference uniform cantilever is clearly associated with a single modal motion, e.g. M = 0.25 or M = 2, variations of ϑL appears to have limited effect on the stability boundary, as shown on Fig. 8(b). On the contrary, for mass ratios corresponding to regions of transition to lower or higher mode instabilities, e.g. M = 1 or M = 4, lower structural modes can become unstable at significantly lower reduced velocities when the thick free part occupies less than half of the length of the stepped cantilever, as illustrated in Fig. 9 (case (xiii)). For oscillations of the stepped cantilevers in the linear regime, the influence of the modal shapes would have overall very limited impact on the flow in the channel. Indeed, for small-amplitude deflection of the cantilever, flow separation at the trailing edge and vertex shedding downstream are nearly non-existent or remain negligible. However, as shown in Fig. 10, for large-amplitude oscillations, the modal shapes resulting from placing the step at different locations along the length of the cantilever can significantly influence the flow patterns generated by the cantilever motions. Thus, with a relatively short and light free section (e.g. case (x)), most of the stepped cantilever’s deflection is concentrated around this free part, so that the flow directly above and below the clamped section remains mainly undisturbed. When the clamped section undergoes larger-amplitude oscillation (e.g. case (xii)), the flow is disturbed further upstream from the trailing edge. This appears to alter flow separation at the trailing edge and the shape of the vortices shedding in the wake region immediately downstream of the trailing edge. The influence of the modal shape on the vortical structures is less prominent further downstream of the trailing edge (region not shown in Fig. 10). Shoele and Mittal (2014) showed the rapid dissipation of the flow disturbances generated by oscillating cantilevers at low Reynolds numbers. Moreover, Shoele and Mittal (2014) also showed that this attenuation effect was amplified with an increasing confinement of the cantilever. Therefore, for the configuration considered in this study (Re = 200 and LH = 2), an early attainment of plane Poiseuille flow can be expected in the downstream part of the channel. Configurations based on higher Reynolds number and wider channels to limit interference of the channel walls with the system dynamics might be more relevant to determine the influence of the stepped cantilevers’ modal shapes on the vortical patterns in the wake and the fluid–structure interactions mechanisms in the non-linear regime of oscillation.

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Fig. 9. Mode shapes associated with the data points indicated in Fig. 8(a) (M = 1 and ϑh = 1/2) and Fig. 8(b) (M = 1 and ϑh = 2). Axial coordinates x1 are scaled with the length of the flexible cantilever LC and transverse coordinates x2 are scaled with the maximum cantilever deflection ηmax . Clamped and free parts of the stepped cantilever are represented in blue and red, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Flow fields associated with the data points indicated in Fig. 8(a) (M = 1 and ϑh = 1/2) and Fig. 8(b) (M = 1 and ϑh = 2): (left) velocity magnitude averaged over a period of large-amplitude oscillation of the flexible cantilever and (right) velocity magnitude at an instant corresponding to a flexible-cantilever maximum deflection in a large-amplitude oscillation period. The inlet rigid wall and flexible cantilever are indicated in grey and white, respectively.

6. Conclusions Variations of the relative thickness of two connected flexible sections forming a stepped cantilever immersed in axial channel flow can have contrasting effects on the flutter instability thresholds and induce non-linear variations of the preand post-critical behaviours of the FSI system. Thinning or thickening the free end of a flexible cantilever is shown to influence greatly the bending motions of the structure. By altering the cantilever dynamics, this can result in a relatively weak stabilisation or a strong destabilisation of the FSI system and can change the shape of the mode that first becomes unstable with increasing flow speed. While the neutral-stability curves corresponding to different stepped cantilevers retain a similar shape to that of the uniform cantilever when the overall fluid-to-solid mass ratio is kept consistent between all cases, they are shifted along both the mass ratio and reduced flow velocity axes of the stability map when a non-uniform thickness is introduced. The figures presented in this paper can thus be used as lookup charts to evaluate

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Table 1 Semiquantitative summary of the effects on the stability characteristics of stepped cantilevers caused by variations of the thickness and length ratios for an average fluid-to-solid mass ratio is kept constant.

quantitatively the overall consequences of different degrees of non-uniformity in immersed cantilevers on the neutral stability of the FSI system. In general, predominance of the higher mode instabilities is triggered for lower mass ratios for the flexible cantilevers with a thinner free-end, in comparison to the uniform case. Conversely, for the flexible cantilevers with a thicker freeend, the predominance of the higher mode instabilities is triggered only for higher mass ratios. An overview of the effects on the stability characteristics expected at any particular average fluid-to-solid mass ratio for variations of the relative thickness and length of the free section of the stepped cantilever is presented in Table 1. In this table, it can be noted that the flutter instability thresholds for most modes, from 2 to 5, remain comparable (within 5%) to those of the equivalent uniform cantilever when the thickness of the free section of the stepped cantilever is kept between 0.7 and 2 times that of the clamped section. With two sections of the same length and a free part more than twice thicker than the clamped part, a stepped cantilever is expected to be significantly more unstable than the corresponding uniform cantilever, as well as uniform cantilevers equivalent to the free and clamped sections, particularly if the thickness ratio is sufficiently high to destabilise a lower structural mode excited by much slower axial flows. However, results show that a relatively long free section generally reduces the effects introduced by its thickening or thinning, and yields stability characteristics for stepped cantilevers that are similar to those of the uniform cantilever. Over the wide of range of thickness and length ratios considered in this study, it is found that stepped cantilevers can become significantly more unstable in many cases but can only be more stable, slightly or moderately, for just a few particular conditions. Therefore the introduction of a non-uniform thickness in a cantilever immersed in axial channel flow appears to be more appropriate for applications aiming to promote FSI instability and strengthening fluttering motions, such as those in energy harvesting. Acknowledgements The authors gratefully acknowledge the support of the Australian Research Council, Australia through project DP0559408 and the WA State Centre of Excellence in eMedicine, Australia through project ‘Airway Tomography Instrumentation’. This work was supported by resources provided by The Pawsey Supercomputing Centre, Australia with funding from the Australian Government and the Government of Western Australia, Australia. References Alben, S., Shelley, M.J., 2008. Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100, 074301. Bae, Y., Jang, J.Y., Moon, Y.J., 2008. Effects of fluid-structure interaction on trailing-edge noise. J. Mech. Sci. Technol. 22, 1426–1435. Balint, T.S., Lucey, A.D., 2005. Instability of a cantilevered flexible plate in viscous channel flow. J. Fluid Struct. 20, 893–912. Berry, D.A., Moon, J.B., Kuehn, D.P., 1999. A finite element model of the soft palate. Cleft Palate Craniofac J. 36 (3), 217–223. Birch, M.J., Srodon, P.D., 2009. Biomechanical properties of the human soft palate. Cleft Palate Craniofac J. 46 (3), 268–274. Carruthers, A., Filippone, A., 2005. On the aerodynamic drag of streamers and flags. J. Aircr. 42, 976–982. Cheng, S., Gandevia, S.C., Green, M., Sinkus, R., Bilston, L.E., 2011. Viscoelastic properties of the tongue and soft palate using MR elastography. J. Biomech. 44 (3), 450–454. Cisonni, J., Elliott, N.S.J., Lucey, A.D., Heil, M., 2014. A compound cantilevered plate model of the palate-uvula system during snoring. In: 19th Australasian Fluid Mechanics Conference.

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