Gautier Verhille

Deformability of discs in turbulence

Keywords: MSC Codes 76F99, 76F25, 76-05

The aim of this study is to investigate experimentally the transition from a rigid regime to a deformed regime for flexible discs freely advected in turbulent flows. For a given disc, the amplitude of the deformation is expected to increase when its bending modulus decreases or when the turbulent kinetic energy increases. To quantify this qualitative argument, experiments are performed where the deformation of flexible discs is measured using three cameras. The amplitude of the deformation has been characterised by the eigenvalues of the moment of inertia tensor. Experimental results exhibit a transition from a rigid regime to a deformed regime that depends on the size, the density and the flexibility of the disc and the turbulent kinetic energy. The modeling of this transition is a generalisation and an extension of the previous models used to characterise the deformation of flexible fibres in turbulent flows.

Introduction

Modeling the advection of particles in turbulent flows is a fundamental problem with various applications going from the advection of plankton in the ocean (Guasto et al. 2012) to the formation of planet (Pumir & Wilkinson 2016). The particles considered in these problems have many different sizes, shapes, densities and rheologies. The role of size (Qureshi et al. 2007;Cisse et al. 2013;Klein et al. 2013), of the shape (Parsa et al. 2012;Parsa & Voth 2014;Byron et al. 2015;Pujara et al. 2018) and of the particle to carrying fluid density ratio (Bec et al. 2007;Volk et al. 2008) on the particle advection have been addressed in several studies since the 2000's. In turbulence, the first investigations on deformable particles were partly motivated by drag reduction and focused on bubbles or polymers (Vanapalli et al. 2006;Ravelet et al. 2011;van Gils et al. 2013;Loisy & Naso 2017;Lohse 2018;Vincenzi et al. 2021). The question of the role of flexibility on the advection of fibres in turbulent flows has been raised recently. The transition from straight to bent fibres is controlled by the ratio of the fibre length to persistence length. This persistence length depends on the mechanical properties of the fibre, on those of the fluid and on the turbulent kinetic energy ε (Brouzet † Email address for correspondence: gautier.verhille@irphe.univ-mrs.fr Abstract must not spill onto p.2 et al. 2014). More recently, it has been shown that, in the flexible regime, the statistics of the fibre deformations depend on the fibre length (Gay et al. 2018;Sulaiman et al. 2019) and that the dynamics of the deformation are given by the coherent structures of the flow (Rosti et al. 2018;Allende et al. 2018). Moreover, increasing fibre flexibility leads to an increase of preferential accumulation in high vortical regions (Picardo et al. 2020). One direct application of the research on deformable objects is the formation of microplastics in the oceans, a global environmental threat (Andrady 2017). Recent investigations on the fragmentation of brittle fibres in turbulence highlight the relation between the deformation of these fibres and the fragment size distribution (Allende et al. 2020;Brouzet et al. 2021). A relevant next step would be to extend these results to 2D and 3D objects which constitute the large majority of plastic debris in the ocean (Morét-Ferguson et al. 2010;Cózar et al. 2017). For slender bodies, such as fibres, fluid and particle inertia can generally be neglected due to the slenderness of the particle even for particles longer than the Kolmogorov length (Batchelor 1970;Shin & Koch 2005;Bounoua et al. 2018). However, this assumption does not hold for 2D or 3D objects larger than the Kolmogorov length. Therefore, if for fibres the main contribution of the hydrodynamic stress tensor is due to the viscous stress, for 3D objects, such as spheres, larger than the Kolmogorov scale the hydrodynamic stress tensor is dominated by the pressure term (Volk et al. 2011). For 2D objects, such as discs, the modeling of the hydrodynamic stress is still an open question at moderate to high Reynolds numbers.

In turbulent flows, the dynamics of rigid discs have been mainly investigated numerically by considering rigid oblate spheroids with a major axis smaller than the Kolmogorov length (Voth & Soldati 2017). Disc-like particles tend to have their axis of symmetry perpendicular to the local vorticity leading to a predominance of tumbling over spinning (Chevillard & Meneveau 2013;Byron et al. 2015;Pujara et al. 2018). In these works, the question of disc deformability has never been addressed. Conversely, several studies investigate the role of the deformation on the motion of a settling disc. Most of these studies aim to understand the different oscillatory motions of a disc falling in a fluid at rest and relate the motions to the disc's wake (Jenny et al. 2004;Fernandes et al. 2008;Auguste et al. 2013;Heisinger et al. 2014). The influence of deformability on the settling of 2D objects has been investigated both theoretically (Alben 2010) and experimentally (Tam et al. 2010;Vincent et al. 2020). They all show that deformability increases the settling speed due to a modification of the shape of the object. This phenomenon is well known by the fluid structure interaction community where the object is held fixed in a flow. In that case, the drag reduction is due to the bending of a plate by the flow (Schouveiler & Boudaoud 2006;Gosselin et al. 2010). The transition from flat to bent plate is governed by a Cauchy number defined by the ratio of the hydrodynamic pressure to the plate rigidity C Y ∼ ρL 3 U 2 /B where ρ is the fluid density, U its mean streamwise velocity and L and B are the typical length and the bending modulus of the plate. In these studies, the hydrodynamical constraint is mainly due to the mean streamwise flow as the turbulent fluctuations are negligible compared to the mean flow. On the contrary, the settling of rigid discs is still poorly understood when turbulent fluctuations are important. Some recent results show that the settling speed increases due to turbulence when the turbulent fluctuations are smaller than the settling speed of the disc in a fluid at rest (Esteban et al. 2019) and decreases when the turbulent fluctuation are higher (Byron et al. 2019). However a model for this phenomenon is still missing.

The present study is focused on the deformation of freely advected flexible discs in turbulent flow. We limit ourselves to the case of low settling rate where the turbulent fluctuations are of the order of or larger than the settling speed of the disc. In that respect, this study is closer to the one of Byron et al. (2019)

[cm] ε [m 2 .s -3 ] λ T [mm] η K [µm] τ K [ms] R λ 5-7
3.8-1.4 1.5-2.5 29-72 0.8 -5.2 300 -710

Table 1: Ranges of variation in the main parameters of turbulence in the measurement volume.

velocity, generally used in fluid-structure interaction community (Tam et al. 2010;Vincent et al. 2020).

In the next section, the experimental setup, the turbulence properties and the manufacturing of the particles are presented. The main results are presented in the third section and their interpretation in the fourth. The final section is a discussion of the results and a conclusion.

Experimental setup

The experiments are performed in a 60 cm cubic tank filled with water. The turbulent flow is generated by the rotation of eight rotors with a diameter D r of 17 cm and fitted with six straight blades 5 mm in height. Each impeller is located at a vertex and points towards the center of the cube. The rotation rate F r ot of each impeller can be set independently between 5 and 19 Hz. In the present study, they all rotate at the same frequency but with an opposite direction to their three closest neighbours. An image of the tank can be seen in Oehmke et al. (2021). The flow properties have been determined by Particle Image Velocimetry (PIV) (Xu & Chen 2013). The measurements have been performed in a cubic volume of nearly 10×10 × 10 cm 3 at the center of the tank where the mean flow is negligible ( Ū2 /U 2 r ms ∼ 10 -2 , where U r ms is the root mean square velocity and Ū the magnitude of the mean flow). The turbulent dissipation rate ε is given by the structure function relation Frisch 1995). It scales as ε ∼ F 3 mot as expected in von Kármán like devices (Labbé et al. 1995;Pinton et al. 1999;Monchaux et al. 2009). The comparison of the component of the U r ms in different locations of the volume of measurement shows that the turbulence is fairly homogenous and isotropic. The relative homogeneity and the isotropy is confirmed by our previous studies mainly on the rotation of fibres (Bounoua et al. 2018;Bordoloi et al. 2020;Oehmke et al. 2021). The variance of the tumbling rate and the correlation times do not depend on the location of the measurements and are equal, up to the statistical error bars, for the three components of the tumbling vector for all the particles we have tested. From the measurements of U r ms and ε one can deduce the main timescales and lengthscales of the turbulence (considering a kinematic viscosity ν = 10 -6 m 2 /s for pure water): the Kolmogorov length η K = (ν 3 /ε) 1/4 , the integral length L I = U 3 r ms /ε, the Taylor length λ T = (15νU 2 r ms /ε) 1/2 and the Reynolds number based on the Taylor length R λ = λU r ms /ν. The ranges of variation of these quantities are listed in Table 1. These measurements have been validated by our previous studies on the rotational dynamics of fibres where our measurements were in agreement with different experiments (Bounoua et al. 2018;Oehmke et al. 2021) and numerical simulations of the literature (Bordoloi et al. 2020) .

D ll (r) = (u( ) -u( + r)) 2 = C 2 ε 2/3 r 2/3 (
The aspect ratio, the Young's modulus and the density of the discs have been varied to investigate their influence on the transition from the rigid regime to the flexible one. The elastic discs are manufactured in the lab with two different silicones from Esprit Composite, EC00 and R1, having different Young's moduli. First, silicone is poured on a glass plate Table 2: Mechanical properties of the different discs used in this study. Note that the concentration of copper varies between the two disc made of EC00+Cu. The standard deviation of the thickness was of the order of 25 µm for all the cases. The symbols correspond to the symbols used for each disc throughout this article.

and levelled at the desired thickness e using tape as spacers. The final sheet thickness varies between 190 and 540 µm and was controlled with a 2D laser displacement sensor LJ-V7080 from Keyence with a resolution of 1 µm. The discs are then cut with various hollow punches having different radii R. To vary the density of the discs, some of them have been made with silicone loaded with copper or tungsten powder having a grain size of ∼ 45 µm and of ∼ 25 µm respectively. The final density was measured by weighing 20 discs of known radius and thickness. Finally, the stress-strain curve of a silicone stripe (∼ 10 × 50 mm 2 ) cut in the same sheet as the discs was measured using a ZwikiLine testing machine from Zwick/Roell company in order to estimate the Young's modulus. The mechanical properties of the discs used in this study are summarised in Table 2. The influence of buoyancy can be estimated by comparing the typical settling velocity u s = (( ρ/ρ f )eg) 1/2 , where ρ = ρρ f is the density difference between the particle and the fluid (Jenny et al. 2004;Esteban et al. 2019), to the turbulent fluctuation U r ms . The ratio u s /U r ms is always small for lighter discs (ρ 1200 kg.m -3 ) so buoyancy is always negligible in these cases. For heavy discs this settling velocity might be higher than U r ms at very low rotation rates F < F min . To minimise buoyancy effects for these discs, the minimal rotation rate F min investigated here has been chosen so that the root mean square of the turbulent flow U r ms = αD r F rot , where α is a constant, is equal to the settling speed u s : F rot > F min = u s /αD r . Finally, the disc concentration c d is always very low, c d = N d πR 2 e/V tot 2.10 -5 . The interactions between discs and their feedback on the flow can then be neglected Elgobashi (1994).

From rigid to flexible discs

Determining if an object is bent or not does not require measuring its exact shape. For instance, Brouzet et al. (2014) quantified the deformability of fibres in turbulence using the distance between the two extremities of the fibre which is smaller than the fiber length for a bent fibre. Here, a similar approach is used by looking at the global shape of flexible discs advected in a turbulent flow. Three 1 MP cameras are used to simultaneously image the volume of measurement. Only the discs seen by the three cameras are reconstructed. We are not interested in the dynamics of the deformation so the acquisition rate F acq is purposely low, F acq = F r ot . This minimises the correlation between images and the quantity of data needed to ensure a good convergence of the deformation statistics. The cameras are modelled by the pinhole model whose 11 parameters (position and orientation Here, the volume of reconstruction is divided into cubic voxels of 500 µm in length. Each voxel is projected onto each image plane. The voxels for which their projections belong to a disc in all images are stored. At the end of this stage the deformed disc is made of a group of voxels.

Focus on Fluids articles must not exceed this page length

As we aim to reconstruct a surface, voxels which are not at the boundary of the object are discarded. Moreover, if two voxels are projected identically onto an image, the voxel closer to the camera is stored and the farther is discarded. An illustration of the voxel selction is sketched in Figure 1.d, and an exemple of a reconstructed disc from 3 images is shown in Figure 1.e. The convex hull volume method gives only access to the convex envelope of the object and not to its real shape. However, this information is sufficient for determining if a disc is

(a) (b) (c)
Figure 2: a) Ratio of the shape factor λ 2 /λ 3 as a function of λ 1 /λ 3 for all of the reconstructed discs made from silicone R1 (R = 8 mm, e = 190 µm) at F rot = 19 Hz. The shaded area is an unreachable zone as λ 2 > λ 1 by definition. b) Evolution of the mean shape factor for the same disc at different frequencies. The color codes the frequency. The plain line and the dashed line correspond to theoretical prediction, see text for more detail. c) Evolution of the mean shape factor as function of the turbulent dissipation rate for all the discs (for symbols, see table 2).

bent. Here, disc deformations are quantified by the moment of inertia tensor I defined by:

I i j = N k=1 m k |r k | 2 δ i j -x (k) i x (k) j (3.1)
where

r k = (x (k) 1 , x (k) 2 , x (k)
3 ) is the vector connecting the voxel k to the center of mass of the group of voxels and m k is the mass of point k which is constant here, so m k = m. The shape of the object can be characterised by the three eigenvalues λ 1 λ 2 λ 3 . These quantities are extensive, as they depend on the mass of the particle. This prevents an easy comparison of discs having different sizes. To work with intensive quantities, we define two shape factors λ 1 /λ 3 and λ 2 /λ 3 independent of the disc size. These shape factors are not shape specific, meaning that two different objects can have the same shape factors. For instance, a cuboid of dimension d 1 = d 2 > d 3 and a cylinder of length L f and radius r f have the same shape factors

if 3d 2 3 /d 2 1 = L 2 f /r 2 f -2. A cube (d 1 = d 2 = d 3
) and a cylinder of aspect ratio L f /r f = √ 5 also have the same shape factors. In the plane (λ 1 /λ 3 ,λ 2 /λ 3 ), three points are remarkable: the point (1/2,1/2) which corresponds to a thin disc, (1,0) to a thin cylinder and (1,1) to a sphere. In Figure 2a the different shape factors measured for discs made with the silicone R1 (e = 190 µm and R = 8 mm) at F r ot = 19 Hz are shown. Most of the points stay in the vicinity of the point (0.5,0.5) corresponding to a flat disc. This shows that in this case most of the discs are undeformed. There is also a cluster of points in the neighbourhood of the point (0,1) corresponding to a cylinder. This suggests that at the onset of the transition, deformable discs tend to roll up more than to crumple like a packed paper sheet.

To characterize the transition from rigid to flexible disc, we focus on the mean shape factors λ 1 / λ 3 and λ 2 / λ 3 , where • is a time and an ensemble average. The evolution of these parameters when the rotation rate F r ot is varied is shown in Figure 2b for the same discs as in Figure 2a. One can see that these shape factors are good proxies for the disc deformability as the evolution of λ 2 / λ 3 and λ 1 / λ 3 is monotonic when the rotation rate of the impellers increases, and so when the turbulent dissipation rate ε increases. In the following the disc deformability will be quantified by λ 2 / λ 3 which increases when the deformation of the disc increases. The evolution of this parameter as a function of ε is shown on Figure 2c. As expected the deformability increases with the turbulent dissipation rate and is higher for discs having a smaller Young's modulus or a smaller aspect ratio Λ = e/R.

Modeling

Disc shape

We will first model the evolution of the disc shape. As shown on Figures 2a and2b, at the onset of the transition, discs tend to wrap and form cylinder like particles. The exact shape of the disc is given by the equation of elasticity:

σ∂ tt r + B∇ 2 ∇ 2 r = ξ, (4.1)
where r is the position of a point on the disc, ∂ tt the second temporal derivative, σ = ρe the surface density and B = Ee 3 /12(1 -ν) the bending modulus of the disc, with ν 0.5 the Poisson coefficient of silicone, and ξ the hydrodynamic stress. The disc being freely advected and as no external torque is applied at the boundary, the boundary condition is κ(r = R) = 0.

To model the evolution of the shape, we consider a simple solution where a flat disc lies in the xz plane and can only deform in the y direction according to:

y d (x) = K 0 2 x 2 1 - x 2 6x 2 R , (4.2)
where K 0 is the curvature at the center of the disc and x R the lateral extension of the disc. For an inextensible disc, x R is defined by the arc length which should be equal to the disc radius:

R = ∫ x R 0 ds = ∫ x R 0 dx 2 + dy 2 = ∫ x R 0 1 + K 2 0 x 2 1 - x 2 3x 2 R 2 dx, (4.3)
Representations of a flat and a bent disc are shown in Figure 3. The (x,y,z) axes being the principal axes of the object, the eigenvalues of the moment of inertia of the bent disc are given by:

I x = ∫ x R x R dx ∫ ∞ 0 dy ∫ z m -z m dz((y -y G ) 2 + z 2 )δ (y -y d (x)) (4.4) I y = ∫ x R x R dx ∫ ∞ 0 dy ∫ z m -z m dz(x 2 + z 2 )δ (y -y d (x)) (4.5) I z = ∫ x R x R dx ∫ ∞ 0 dy ∫ z m -z m dz((y -y G ) 2 + x 2 )δ (y -y d (x)) (4.6)
where z m (x) is the maximum height of the disc at the position x and is given by:

z 2 m (x) = R 2 - ∫ x 0 ds 2 , (4.7)
and y G = 3K 0 x 2 R /20 is the y-coordinate of the center of gravity. The exact expressions of the moments of inertia are relatively tricky to compute analytically. We then compute the evolution of the shape factors numerically for different K 0 , cf. Figure 4a. The theoretical evolution of the bent disc in the (λ 1 /λ 3 , λ 2 /λ 3 ) plane is represented by the plain line in Figure 2b. As one can see the general trend is well captured by this model, but there is an offset between the measurements and the theoretical prediction. Two sources of discrepancy can be identified. First, the number of voxels forming the disc is much smaller than the number of voxels used to compute the theoretical moment of inertia. A second source of error is related to the convex hull volume method. As detailed previously, for voxels projecting onto the same pixel only the voxel closest to the camera is stored. The reconstructed disc is then, in general, not a disc but a partially filled cylinder, as illustrated by Figure 1d. To estimate the order of magnitude of this last source of error, we can compute the shape factor of an object made of touching two coaxial cylinders: one filled of radius R and thickness h, and one empty cylinder of inner/outer radii R 1 and R with the same height h. Here h is given by the voxel size and we take R 1 = Rh. The evolution of λ 2 /λ 3 for this object is shown on Figure 4b. The value of the shape factor for h = 0 is equal to the one of a flat disc and increases when h is increased. As mentioned previously the exact value of the offset cannot be determined rigorously and has been considered as a constant free parameter to fit the data. The dashed line on Figure 2b is obtained by shifting the theoretical prediction for a disc by 0.1 in the vertical direction ( λ 2 / λ 3 ) and 0.01 in the horizontal direction ( λ 1 / λ 3 ). These coefficients will remain fixed in the following sections.

Here, the agreement between our model and our measurements suggests that near the transition the disc bend and adopt a U-shape which can be described by equation 4.2. All the physics of the deformation are then hidden in the parameter K 0 which depends on the disc deformability and the turbulent intensity. The following sections aim to model these dependencies.

A power budget argument

As shown in Figure 4a, if K 0 R 10 -2 the deformations are weak and the disc can be seen as rigid, whereas if K 0 R > 10 -2 deformations are important and the disc is indeed flexible. The curvature K 0 then plays a role similar to the persistence length p for the deformation of wormlike chain polymers. The stiffness of polymers is characterised by the ratio of their length L p to the persistence length p , defined by t(s) • t(s + ) = e -/ p , where t(s) is the tangent vector at the position s (Yamakawa 1971). If L p p deformations are important. On the contrary if L p p the polymer remains straight and is considered stiff. In polymer theory, the order of magnitude of the persistence length is given by the balance of the thermal energy k B T with the elastic energy E el ∼ E I/ p , where k B is the Boltzmann constant, T the temperature and I the area moment of inertia. Brouzet et al. (2014) draw an analogy between this system and flexible fibres distorted by a turbulent flow. They showed that the energy balance was not able to capture the transition from rigid to flexible fibres. The reason is that the correlation time of turbulence is, in general, of the same order of magnitude as the dynamical timescale of the deformations. Therefore, contrary to polymers, the forcing cannot be modeled by delta correlated noise. The energy budget has then to be replaced by a balance of power. Indeed, the deformation of a fibre in a turbulent flow is an out-of-equilibrium stationary process. The mean elastic energy stored by the fibre E el is then constant and:

dE el dt = P in j -P dis = 0, (4.8)
where P in j is the power injected by the turbulence and P dis is the power dissipated by viscosity. For fibres whose lengths are in the inertial range, deformations are due to eddies of similar sizes. The turbulent power P tur b is then given by P tur b ∼ ρ f L 3 p ε. The dissipative term should be proportional to the elastic energy E el stored in the fibre and should depend on the typical timescale of the deformation τ de f . The ratio of these two terms defines an elastic power P el . For fibres, they showed that the timescale of the deformation is given by the relaxation time defined by the ratio of the viscous stress to the bending stress.

Following a similar argument, one can assume that the transition from a rigid to a flexible disc is given by the balance of the turbulent power:

P tur b ∼ ρ f R 3 ε (4.9)
and the elastic power: (4.10) where τ B is the relaxation timescale of the deformation given by the balance of the viscous stress ξ v with the bending stress B∇ 2 ∇ 2 r in equation 4.1. From dimensional analysis, at small Reynolds number, ξ v scales as:

P el,v = Bκ 2 R 2 /τ B ,
ξ v ∼ µ R (u f -∂ t r), (4.11)
where u f is the fluid velocity and µ the dynamical viscosity of the fluid. The relaxation time is then:

τ B ∼ µR 3 /B, (4.12)
and, assuming that at the transition κ can be replaced by 1/R in equation 4.10, the elastic power scales as:

P el,v ∼ B 2 µρ f R 3 , (4.13)
The balance of power P tur b = P el,v defines a critical turbulent dissipation rate ε v :

ε v = B 2 ρ f µR 6 . (4.14)
If this model was able to capture the physics of the deformation, the evolution of the mean shape factors should only depend on the ratio ε/ε v . To compare the theoretical evolution of the shape factor with this prediction one needs to relate the curvature K 0 to ε. Using equation 4.9, 4.10 and 4.12, which define the turbulent and elastic power, one can show that near the threshold the typical curvature κ, and therefore K 0 , should be proportional to ε 1/2 . The comparison of the theoretical and measured evolution of the mean shape factors as a function of ε/ε v is shown in Figures 5a and5b. The dashed line is the best fit of the experimental measurements considering K 0 as a free parameter. Here, the global trend is well captured by the theoretical prediction. However, the estimation of the threshold with this model is not valid as evidenced by the scattering of the experimental points around the theoretical prediction. In particular, the threshold of the discs having the highest densities (symbols and •) seems to be at least one order of magnitude lower than the model prediction.

In this model, inertia is always neglected. It is then not surprising that the dependency of the transition on the disc density is not captured for high densities. Another model, derived by Rosti et al. (2018), considers the density of fibres. The transposition of their model to discs is presented in the following section.

A temporal argument

The second model used to quantify fibre deformation is based on a temporal argument. If the forcing timescale is smaller than the deformation timescale, deformations are weak as fibres do not have the time to adapt their shapes to the flow. On the contrary, if the forcing timescale is larger than the deformation timescale, the fibre is distorted by the coherent structures of the flow.

As for the model presented in section 4.2, the transposition of this idea to the current study requires an estimation of the deformation timescale of the discs. Two timescales are relevant depending on the influence of particle inertia. The first one is the relaxation timescale, as defined by equation 4.12. It is used to define the Weissenberg number which generally quantifies polymer deformations in turbulent flows (Vincenzi et al. 2021). The second one is the resonant frequency which has been used by Rosti et al. (2018) to quantify fibre deformations in the underdamped regime, i.e. when viscous stress is negligible as compared to particle inertia. As shown previously, the relaxation time is independent of the particle density. Therefore a model solely based on this timescale should not be able to describe our measurements.

The resonant frequency of the disc ω d is given by the equation of elasticity 4.1 by balancing the inertial term σ∂ tt r and the bending term B∇ 2 ∇ 2 r:

ω 2 d = c ω B σR 4 , (4.15)
where c ω is a constant which depends on the excited mode. In the following we will consider the case where c ω = 1. When the disc radius R is in the inertial range of turbulence, the relevant timescale for the forcing τ tur b is the typical time of eddies of similar size: (4.16) where u R ∼ (εR) 1/3 is the typical velocity at scale R. From these two timescales, one can define an inertial Weissenberg number Wi = ω d τ tur b which should be of order unity at the transition. This defines a critical turbulent dissipation rate ε r above which a disc can be bent:

τ tur b ∼ R/u R ∼ R 2/3 ε -1/3 ,
ε r = B 3/2 σ 3/2 R 4 (4.17)
Contrary to the previous model, this model cannot predict the evolution of the mean curvature κ as function of ε. There was relatively good agreement between the shape of the theoretical prediction and the global trend of the measurement in the previous scaling argument, so we will assume that κ ∼ ε 1/2 . Once again K 0 will be considered as a free parameter in order to fit the experimental data.

The measurements and the theoretical predictions are compared on Figures 6a and6b. As for the previous case, there is a scattering of the experimental points around the theoretical prediction for both shape factors. The failure of this approach is not surprising as it was derived to describe the transition between two modes of deformation of fibres: the excitation of the first bending mode if ω d τ tur b 1 and the deformation dependent upon the coherent structures of the flow if ω d τ tur b 1 (Rosti et al. 2018). On the contrary, we focus here on the transition from rigid to flexible discs which should occur at smaller ε.

An inertial power balance argument

In the first model presented in section 4.2, the transition from rigid to flexible discs was modeled using a balance between the turbulent power P tur b and the elastic power P el = Bκ 2 R 2 /τ el assuming that the timescale of the deformation τ el was the relaxation timescale, cf. equation 4.12. In the second model (section 4.3), we introduce a second timescale for the deformation: the frequency of the first bending mode, cf. equation 4.15. The relevance of each timescale depends on the importance of the disc inertia in the equation of elasticity 4.1. This is quantified by the ratio of the inertial term σ∂ tt r ∼ σω 2 f ζ, where ω f is the forcing frequency and ζ the typical displacement, to the viscous term µ∂ t r/R ∼ µω f ζ/R. This ratio defines an elastic Stokes number St d :

St d ∼ ρ ρ f e η K 4/3 R e 1/3 . (4.18)
Here we have assumed that the forcing timescale is given by the eddies at the scale of the disc ω f ∼ u R /R. In this study, St d varies between 14 and 320. Inertia is then larger than the viscous dissipation and the timescales of the deformation should be given by ω d . This regime corresponds to the underdamped regime investigated by Rosti et al. (2018) for fibres.

Replacing the relaxation time scale by ω -1 d in the definition of the elastic power leads to:

P el,w ∼ B 3/2 κ 2 σ 1/2 . (4.19)
As for the first model, balancing this elastic power with the turbulent power P tur b = ρR 3 ε, we find that κ ∼ ε 1/2 near the threshold. Moreover, the critical turbulent dissipation rate ε w can be estimated by defining the threshold of the transition by κ ∼ 1/R:

ε w = B 3/2 ρ f σ 1/2 R 5 (4.20)
The evolution of the shape factors as a function of the normalised turbulent dissipation rate ε/ε w is shown in Figures 7a and7b. There is now a good agreement between the measurements and the theoretical predictions for all the discs except for the most flexible ones. In that case, deformations probably involved several modes of bending. This cannot be captured by our model and will be discussed in the following section.

Discussion and conclusion

We have presented here an experimental investigation on the deformation of discs within a turbulent flow. The disc deformations have been measured using the convex hull volume method and quantified by the evolution of the shape factors λ 1,2 / λ 3 , where λ i are the eigenvalues of the moment of inertia tensor. We showed that near the threshold of the transition, the discs adopt a ∪ shape with a maximum curvature κ located at the center and which scales as κ ∼ ε 1/2 . The threshold of the transition is given by an equilibrium between the turbulent power P tur b and the elastic power P el defined by the ratio of the elastic energy to the timescale of the deformation which is here given by the frequency of the first bending mode. This timescale depends on the elastic Stokes number St d which is relatively large in this study. When St d 1, i.e. when e/η K (R/e) 1/4 , viscous stress is dominant and the timescale should then be given by the relaxation time τ B , cf. equation 4.12, as it was for fibres (Brouzet et al. 2014). More studies are needed to understand the influence of the Stokes number on the timescale of the deformations and, hence, on the onset of the deformation.

In our modeling we compare the disc deformation to a folded disc shown in Figure 3. For this simple shape the variation of the curvature and of the deflection at the extremity, given by y d (x R ) = 5K 0 x 2 R /12, evolves linearly with K 0 for small deformation, as shown in Figure 8a. The agreement between the experiments and this simple model calls into question about the term "transition" we use throughout this paper which suggests that two different states exist: the rigid and the flexible regime. In fact, the existence of two different regimes can be seen in the radial extension x R of the disc, cf. Figure 8b, and in the evolution of the shape factors, cf. Figure 7. For K 0 R 1, x R ∼ R so the shape of the disc is well approximated by a flat disc. For K 0 R 1, x R < R meaning that the radial extension in the x direction is smaller than in the z direction, cf. Figure 3. In this regime, deformations cannot be neglected and should impact the disc dynamics. Moreover, investigation of the fragmentation of brittle fibres shows evidence that the critical length p , defined by the balance of power, plays a major role in the fragmentation process (Brouzet et al. 2021). Extrapolating this result to brittle discs allows us to define a critical size above which deformations, and the following fragmentation, are important for 2D objects: R c = B 3/10 ρ f σ 1/2 ε 1/5 .

(5.1)

This length scale could be relevant to determine the size of microplastic fragmented in the ocean. Considering a plastic bag made of polyethylene with a Young's modulus of 500 kPa and a thickness of 80 µm within a turbulent flow with ε ∈ [10 -1 ; 10 2 ] m 2 .s -3 , typical during a storm (Gemmrich & Farmer 2004), the critical size R c varies between 3.2 mm and 790 µm. These values show that the plastic bag can easily be deformed within the turbulent ocean. Moreover, these lengthscales are compatible with field measurements where the fragment size distribution exhibits a maximum for microplastic particles around 1 mm (Cózar et al.

2014).

Finally in section 4.3, we claimed that the critical value ε r derived from the temporal argument (Wi = 1) should be larger than ε w . In fact, ε r characterises a transition where the topology and the dynamics of the deformation are given by the coherent structures of the flow and not by the eigenmode of the disc. To fully validate this claim more measurements are needed. However, we can compare the two thresholds ε r and ε w :

ε w ε r = σ ρ f R = ρ ρ f e R .
(5.2)

In the case of the most flexible discs investigated here ( ), this ratio is of the order of 1/30. As the threshold of the transition from flat to bent disc occurs for ε/ε w ∼ 3, cf. Figure 7, the second regime of deformation where coherent structures of the flow are responsible of the disc deformation occur then at ε/ε w 90, as it is the case here, cf. Figure 7. In general, when ρ ∼ ρ f , for 2D objects where e R, the transition derived from the temporal argument occurs at higher turbulent dissipation rate ε than the one corresponding to the excitation of the bending mode. When the particle and the carrying fluid have very different densities, like a plastic bag advected in air, this ratio can be of order unity (for a plastic bag with a thickness e = 80 µm and of typical size 20 cm ε w /ε r ∼ 2). In that case the deformation should directly be driven by the coherent structures of the flow. Further investigation is needed to validate this point.

Figure 1

 1 Figure 1: a,b and c) Typical images with a superimposition of the reconstructed disc shown in red dots. d) Illustration of the reconstruction process with the voxel selection. Here for simplicity, the axis of the voxel array is aligned with the axes of the cameras. e) Reconstructed object from the convex hull volume method with the voxel selection detailed in the text. The color codes the z coordinate of the voxel

Figure 3 :

 3 Figure 3: 3D representation of an undistorted disc K 0 = 0, on the left, and of a bent disc, K 0 R=1, on the right.

  Figure 4: a) Theoretical evolution of the shape factors of a bent disc represented on Figure 3 as a function of the normalised curvature K 0 R. b) Evolution of the mean shape factor λ 2 / λ 3 for a partially empty cylinder.

Figure 5 :

 5 Figure 5: Evolution of the mean shape factors λ 1 / λ 3 (a) and λ 2 / λ 3 (b) as a function of the normalised turbulent dissipation rate ε/ε v . The dashed line represents the best fit of an ideal bent disc, see text for more details.

Figure 6 :

 6 Figure 6: Evolution of the mean shape factors λ 1 / λ 3 (a) and λ 2 / λ 3 (b) as a function of the normalised turbulent dissipation rate ε/ε r . The dashed line represents the best fit of an ideal bent disc, cf. text for more details.

Figure 7 :

 7 Figure 7: Evolution of the mean shape factors λ 1 / λ 3 (a) and λ 2 / λ 3 (b) as a function of the normalised turbulent dissipation rate ε/ε r . The dashed line represents the best fit of an ideal bent disc, cf. text for more details.

Figure 8 :

 8 Figure 8: Evolution of the deflection y(x R ) = δ (a) and the radial extension x R (b), see equation 4.3, as function of the curvature K 0 .

  on the settling of hydrogel particles than the one ofEsteban et al. (2019) on the settling of thin rigid discs. The governing non dimensional parameter is then expected to be different than the classical Cauchy number, based on the streamwise

	Integral	Turbulent	Taylor Kolmogorov Kolmogorov Reynolds
	length dissipation rate length	length	time	number
	L I

Acknowledgements. This work was carried out in the framework of FlexFiT Project (ANR-17-CE30-0005-01) funded by the French National Research Agency (ANR).