Jaromir Havlica
email: havlica@icpf.cas.cz

David Kramolis

Florian Huchet

A revisit of the electro-diffusional theory for the wall shear stress measurement

Keywords: Boundary layer, Lévêque, Electrochemical method, Mass transfer, Wall shear rate

This article intends to revisit the electro-diffusional theory for the wall shear stress measurement from mass transfer probes of rectangular shape by considering the existence of two components of the wall shear rate (i.e., axial and transversal). General analytical formulas for the effective transfer length and the dimensionless mass transport coefficient were derived as a function of only two parameters: a dimensionless angle of the flow direction, relative to the leading edge of the probe, and the aspect ratio between the width and the length of the strip probe. The correctness of the analytical relations for arbitrary flow direction and the aspect ratio was confirmed by numerical solutions of the transport equation in the convective-diffusive regime. It has also been proved that

the differences between the Lévêque solution and the general analytical formula exhibit a significant deviation for a specific range of parameters. In the case of the three-dimensional boundary layers, in addition to the magnitude of the wall shear stress, the direction of the fluid flow in the vicinity of the probe's surface is of paramount importance. Accordingly, a measurement methodology is proposed using two strip probes with different aspect ratios. The resulting equations required to quantify the magnitude of the wall shear rate vector and the dimensionless angle are also derived.

Introduction

The efficiency of mass and energy transfer between a solid wall and the surrounding fluid depends on the nature of the hydrodynamics boundary layer [START_REF] Bu | Experimental study of flow transitions in structured packed beds of spheres with electrochemical technique[END_REF][START_REF] Gelderblom | Analytical and experimental characterization of a miniature calorimetric sensor in a pulsatile flow[END_REF], and the wall shear stress fundamentally influences these transport phenomena. However, to obtain information about the spatial changes of velocity using experimental measurements in the viscous sub-layers is not a trivial matter. The inability of the classical techniques to describe fluid dynamics in the near-wall region has led to the development of a group of non-invasive methods based on a heating element or a mass transfer probe flush-mounted to the wall [START_REF] Stone | Heat mass-transfer from surface-films to shear flows at arbitrary peclet numbers[END_REF]. The Electro-Diffusion (ED) technique is one of the essential representatives of these methods.

The ED method measures the wall shear rate in the liquid phase from the mass transfer probes. A potential difference between an anode of a broad surface (usually located far from the measuring area) and the small working electrode acting as a cathode leads to a fast redox reaction. The ion disappearance occurs only on the microprobe active surface, causing the development of the diffusion boundary layer, in which the fast electro-diffusion reaction rate is controlled only by diffusion in the case of the sufficiently large Schmidt number. The sufficiently large Schmidt number guarantees that the diffusion boundary layer thickness is smaller than the width of the viscous sub-layer, and it allows the simplification of a description of the hydrodynamics near the wall by using the assumption of simple shear flow [START_REF] Hanratty | Measurement of wall shear stress[END_REF]. The Schmidt number is given by the ratio of the liquid viscosity to the diffusion coefficient of the electrolyte ions. Since in the most measurement cases water and ions of ferricyanide are used, the assumption of sufficiently large Schmidt number can be easily fulfilled for small size probes.

The main drawback of the ED method is posttreatment of measured data to find out the wall shear stress from the limiting diffusion current measurements on the base of individual assumptions, which may not always be fulfilled. The fundamentals of ED have been addressed in several works [START_REF] Hanratty | Measurement of wall shear stress[END_REF][START_REF] Mitchell | A study of turbulence at a wall using an electrochemical wall shear-stress meter[END_REF]. They conclude that ED measurement has to respect the quasi-steady regime of simple shear flow in the viscous sub-layer perpendicular to the strip probe leading edge. In this case, an analytical solution of the convection-diffusion equation, initially derived by Lévêque [START_REF] Lévêque | Les lois de la transmission de chaleur par convection[END_REF], can be applied. In 1963 Reiss and Hanratty extended the Lévêque solution also to a circular probe [START_REF] Reiss | An experimental study of the unsteady nature of the viscous sublayer[END_REF].

Nonetheless, the Lévêque solution has limited validity for sufficiently large Péclet numbers, where the convection has to outweigh the diffusion in the axial direction significantly. Then axial diffusion flux can be neglected. However, this assumption cannot be used for smaller values of Péclet number [START_REF] Geshev | Angular and transient characteristics of circular electrochemical friction probes[END_REF][START_REF] Ling | Heat transfer from a small isothermal spanwise strip on an insulated boundary[END_REF][START_REF] Phillips | Heat and mass-transfer from a film into steady shear-flow[END_REF].

Such cases have been studied by the matched asymptotic expansions [START_REF] Ackerberg | Heat-mass transfer to a finite strip at small peclet numbers[END_REF] and integral boundary method [START_REF] Geshev | Angular and transient characteristics of circular electrochemical friction probes[END_REF] with some simplifications for limiting cases or by a combination of numerical solution of convection-diffusion equation and the inverse method with the necessity of an iterative process [START_REF] Lamarche-Gagnon | An inverse problem to assess the twocomponent unsteady wall shear rate[END_REF][START_REF] Rehimi | Inverse method for electrodiffusional diagnostics of flows[END_REF]. The presented results for small Péclet numbers are pointing at a significant deviation to the Lévêque solution [START_REF] Phillips | Heat and mass-transfer from a film into steady shear-flow[END_REF].

Whenever substantial inertial effect is combined with unsteady phenomenon governs the transport processes, wall shear rate fluctuations are occurring [START_REF] Mathis | Estimating wallshear-stress fluctuations given an outer region input[END_REF][START_REF] Martemianov | Statistical theory of turbulent mass transfer in electrochemical systems[END_REF], and the frequency response of the mass transfer probes must be taken into account [START_REF] Deslouis | Frequency-response of electrochemical sensors to hydrodynamic fluctuations[END_REF]. If the amplitude of the flow perturbations is relatively small compared to the time-averaged value, the time-variation of the measured electric current respects the temporal fluctuations of the wall shear rate, and a quasisteady state is generally assumed. At more elevated fluctuation rates, it is necessary to take into consideration the frequency response of the mass transfer probes due to the sufficiently large Schmidt number. In these cases, the capacitive effect of the concentration boundary layer acts as a low-pass filter upon the current fluctuations [START_REF] Fortuna | Frequency response of boundary layer on wall transfer probes[END_REF], and the assumption of quasi-steady state is erroneous.

The correction between wall shear stress fluctuations and measured fluctuations of limiting diffusion current is restored by transfer functions [START_REF] Nakoryakov | Electrodiffusion method of investigation into the local structure of turbulent flows[END_REF], and their uses have been found relevant, especially for turbulent flow measurements [START_REF] Deslouis | Frequency-response of electrochemical sensors to hydrodynamic fluctuations[END_REF]. By using the transfer function, it is possible to evaluate the phase shift of the measured current signal concerning the imposed wall shear rate variation of a given frequency, the amplitude attenuation, and correct the power spectra obtained from the measurements of near-wall turbulence [START_REF] Tihon | Electrodiffusion detection of the near-wall flow reversal in liquid films at the regime of solitary waves[END_REF]. The transfer functions can provide information on the frequency-domain from a signal analysis procedure at the cost of the loss of the time-domain information. Therefore, some authors, similarly to the small Péclet number cases, have considered using other methods as, for example, the inverse mass transfer method explored firstly by Mao and Hanratty [START_REF] Mao | Application of an inverse mass-transfer method to the measurement of turbulent fluctuations in the velocity-gradient at the wall[END_REF], to deal with the probe inertia while preserving the time-resolved ED measurements. This approach has improved the measurements of the amplitude probability distribution and the frequency spectrum of the streamwise component of the fluctuating velocity gradient. Another procedure for sufficiently high frequency fluctuating flow was introduced by Sobolik et al. [START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear-stress in wavy flow of non-newtonian liquids[END_REF].

Their method corrects the quasi-steady solution by adding a term, derived from the transient response of the ED probe extended by the time derivative of the wall shear rate. It is based on the assumption of similarity of the concentration fields over the probe surface under steady and unsteady flow conditions. This method was verified numerically for small amplitudes of the fluctuation rate [START_REF] Wang | An approximate solution for the dynamicresponse of wall transfer probes[END_REF] and experimentally at higher amplitudes of the fluctuation rate [START_REF] Huchet | Unsteady flows in milli-and microsystems: analysis of wall shear rate fluctuations[END_REF] showing its relevancy in turbulent flows compared to the transfer function.

Another problem that may interfere with the correct posttreatment of measured data by ED is the direction of fluid flow. For a circular electrode, due to its symmetry, there is no influence of the angle of the liquid flow on the relation between measured limiting current and wall shear stress. This feature simplifies the interpretation of experimental data, but on the other hand, it does not allow more work with information about the direction of the wall shear stress vector. Unlike the circular probes, rectangular probes are sensitive to the flow orientation in both the axial and transverse directions. The expression for the evaluation of the mass transport coefficient for the slanted probe was firstly presented by Sirkar & Hanratty [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF]. For that evaluation, knowledge of the direction of flow around the surface probe is required. However, this information is often not available. It has to be determined by other means than solely measurements with one strip probe because from obtained data it is possible to evaluate only the magnitude of the wall shear rate vector. Therefore, if additional information about the flow direction is needed, the more-segmented probes have been proposed. Sirkar & Hanratty [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF] developed the chevron-electrode arrangement formed by a pair of slanted electrodes. Py & Gosse [START_REF] Py | Sur la réalisation d'une sonde polarographique sensible à la vitesse et à la direction de l'écoulement[END_REF] used semi-circular active surfaces with an isolated gap, and Tournier & Py [START_REF] Tournier | Behavior of naturally oscillating 3-dimensional flow around a cylinder[END_REF] suggested utilizing the pair of twin rectangular probes. The most commonly used two-segmented probes are, so-called 'sandwich probes', are formed by two strips separated in the mean flow direction by a gap created by an insulator [START_REF] Son | Numerical solution for flow around a cylinder at reynolds numbers of 40,200 and 500[END_REF]. The main problem of this type of two-segmented probes lies in poor sensitivity to transverse flow, and their usage is, therefore, often reduced only to detection of reversal flow in low-frequency oscillating flows [START_REF] Lamarche-Gagnon | An inverse problem to assess the twocomponent unsteady wall shear rate[END_REF]. Due to unsatisfactory results with two-segmented probes in terms of flow direction evaluation, the three-segment sensors were developed [START_REF] Wein | Theory of direction sensitive probes for electrodiffusion measurement of wall velocity-gradients[END_REF]. The three-segment probes can detect the magnitude of wall shear stress vector and flow direction with a precision better than 10 • with appropriate calibration [START_REF] Sobolik | Tribollet, 3-segment electrodiffusion probes for measuring velocity-fields close to a wall[END_REF]. However, they exhibit problems with stagnation point flow evaluation and are not suitable for the measurement of high frequency fluctuating flow due to the probe inertia [START_REF] Lamarche-Gagnon | An inverse problem to assess the twocomponent unsteady wall shear rate[END_REF]. Such latter complex micro-sensors also suffer from manufacturing constraints for implementation, and the diffusion layer approximation in the trailing edge has to be carefully identified.

In the early stages in the 1960s, the ED method was considered as the sole experimental method able to provide relevant insight into the near-wall flow behavior. Measurements in simple geometries such as straight channel were firstly considered for a better understanding of the boundary layer nature in fully developed turbulent flow [START_REF] Mitchell | A study of turbulence at a wall using an electrochemical wall shear-stress meter[END_REF][START_REF] Reiss | An experimental study of the unsteady nature of the viscous sublayer[END_REF][START_REF] Reiss | Measurement of instantaneous rates of mass transfer to a small sink on a wall[END_REF]. Afterward, ED technique usage was expanded to more complex flows such as flows around bypassing bodies [START_REF] Fadla | Electrochemical measurements for real-time stochastic reconstruction of large-scale dynamics of a separated flow[END_REF], creating of stagnation points e.g. flow-through suddenly changing cross-section [START_REF] Tihon | The transitional backwardfacing step flow in a water channel with variable expansion geometry[END_REF], or the mixing of fluid streams [START_REF] Huchet | Use of electrochemical microsensors for hydrodynamics study in crossing microchannels[END_REF][START_REF] Mouheb | Flow characterization in t-shaped and cross-shaped micromixers[END_REF]. Since the last two decades, the electro-diffusion method has also been extended to the multiphase systems as flows through microporous systems [START_REF] Lesage | New local measurements of hydrodynamics in porous media[END_REF]. This sudden transition from fundamental to advanced flow characterization may have led to a misunderstanding in the interpretation of the results while considering the hypothesis of the ED method initially introduced sixty years ago.

If the simple geometry as a straight channel is used, the strip probe can be uniquely positioned on the wall so that its leading edge is perpendicular to the flow direction. In many applications [START_REF] Altheimer | Flow regime and liquid solid mass transfer investigation in a designed porous structure using electrochemical micro-probes[END_REF][START_REF] Huchet | Comments on the paper "flow regime and liquid-solid mass transfer investigation in a designed porous structure using electrochemical micro-probes[END_REF], the situation is less clear, and the flow direction is also dependent on the input conditions. In such cases, the probe cannot be placed universally in the space so that the liquid flow is always perpendicular to the leading edge, and the existence of transverses flow directions must be taken into account in addition to the main axial component.

In such flow conditions, the analytical solution of Lévêque [START_REF] Lévêque | Les lois de la transmission de chaleur par convection[END_REF], used in most of the research works, is no longer apply.

In that context, this work intends to revisit the ED technique and show the benefit it may have for the scientist community to take into account the existence of two components of the wall shear rate (i.e., axial and transversal) for the interpretation of measured data. For the Lévêque solution users, pitfalls and sources of errors are outlined. In the present paper, we start from the diffusion-convection equation assuming a homogeneous two-dimensional simple shear flow without the existence of stagnation points in the measured area (u x , u y = 0; u z ≈ 0). We reduced the general parametric space of convectiondiffusion transport equation into solely two parameters; one corresponding to the aspect ratio between the width and the length of the strip probe, while the second is related to the flow direction relative to the leading edge of the probe.

We derived general analytical formulas for the effective transfer length and the dimensionless mass transport coefficient for fluid flow of arbitrary orientation and any aspect ratio of the strip probe. This relation is a generalization of the solution presented by Sirkar & Hanratty [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF] for a slanted strip probe. The results of the analytical solution for the dimensionless mass transport coefficient were confirmed by a numerical solution of the transport equation with 2D convective flux. Finally, we presented the new methodology of measurement with a strip probe concerning the axial and transversal flow directions.

Strip mass transfer probes: measurement of wall shear stress

The principle of the electro-diffusion method is based on the measurement of the current under limiting diffusion conditions, I lim . In this case, the mass transfer coefficient averaged through a surface probe, kz , can be calculated using the Faraday relation:

kz = J z c ∞ -c w = I lim ν e F S e (c ∞ -c w ) (1)
where S e is the surface of the electrode, F is Faraday constant, ν e is the number of electrons involved in the redox reaction, c ∞ is bulk concentration, and c w is equilibrium concentration on the probe surface. From the knowledge of the average mass transfer coefficient, the wall shear rate can be obtained based on the relationship derived from the local description of the mass transport over the probe surface.

The mass transport in the vicinity of the strip probe flush-mounted to the wall can be described by the governing equation:

∂c ∂t + u x ∂c ∂x + u y ∂c ∂y + u z ∂c ∂z = D ∂ 2 c ∂x 2 + D ∂ 2 c ∂y 2 + D ∂ 2 c ∂z 2 , (2)
with the following boundary conditions:

x → -∞ ∀y z > 0 c = c ∞ x ∈ 0, L x y ∈ 0, L y z = 0 c = c w x < 0 ∪ x > L x y < 0 ∪ y > L y z = 0 ∂c ∂z = 0 ∀x ∀y z → ∞ c = c ∞ (3)
where x, y, z are Cartesian coordinates, L x is the width of the probe in the xdirection, L y is the length of the probe in the y-direction, and u ≡ (u x , u y , u z )

is the velocity vector.

This work is focused on the simple shear flow flowing under an arbitrary angle α over the mass transfer probe of rectangular shape with any aspect ratio R. In such cases, in addition to the axial flow (u x = 0), the existence of a transverse direction of the velocity must be taken into account (u y = 0).

Since the flow near the probe is supposed to be parallel with the wall, and the existence of stagnation points in the measured area is not expected, the convective transport in the z-direction is negligible (u z ≈ 0). It is also assumed that the flow is homogeneous over the surface of the probe and the components of the velocity vector (u x , u y ) vary only in the normal direction to the electrode surface as a linear function in the viscous sub-layer:

u x = ∂u x ∂z z = S x z, (4)
u y = ∂u y ∂z z = S y z, (5)
where S x and S y are components of the wall shear rate vector. This assumption can be made since the Schmidt number is sufficiently large. It is also expected that the flow regime is in steady state ( ∂c ∂t = 0). Based on the expectations of sufficiently large Péclet numbers, it can also be assumed that mass transport by diffusion mechanism close to the wall is essential only in the normal direction.

Applying the above assumptions to mass transport governing equation, the eq. ( 2) is reduced to the form:

S x z ∂c ∂x + S y z ∂c ∂y = D ∂ 2 c ∂z 2 . ( 6
)
For the sake of simplification, the following discussion of mass transport on the strip probe surface is divided into two parts based on spatial orientations between the probe's leading edge and fluid flow. The examined situation is schematically illustrated in figure 1a. The first conventional case assumes the perpendicular fluid flow across the leading edge of the probe (figure 1b). The second case assumes fluid flows under the general angle α considering the leading probe edge (figure 1c). The first case leads to the Lévêque solution [START_REF] Lévêque | Les lois de la transmission de chaleur par convection[END_REF] and the second one to the generalized Sirkar and Harantty solution for the slanted strip probe [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF].

Perpendicular fluid flow: Lévêque solution for strip probe

If the fluid flow is perpendicular to the leading edge of the probe (see figure 1b), equation ( 6) is simplified into the following form:

S x z ∂c ∂x = D ∂ 2 c ∂z 2 . (7)
This partial differential equation with boundary conditions (3) has an analytical 200 solution [START_REF] Reiss | An experimental study of the unsteady nature of the viscous sublayer[END_REF] such as:

c -c w c ∞ -c w = 1 Γ 4 3 η 0 e -η 2 dη, ( 8
)
where Γ is the gamma function, and variable η is defined as:

η = z S x 9Dx 1 3
.

Based on the knowledge of the analytical expression for the concentration field, eq. ( 8), the relation for the mass transport coefficient on the strip probe was derived by Lévêque [START_REF] Lévêque | Les lois de la transmission de chaleur par convection[END_REF]:

205 kz = 3 2Γ 4 3 9 1 3 D 2 S x L x 1 3 = 0.8075 D 2 S x L x 1 3 . ( 10
)
In its normalized form, eq. ( 10) can be rewritten as:

Sh * Lx = Sh Lx Pe 1 3 s,Lx = kz L x D 2 S x 1 3 = 0.8075, (11)
where Sh * Lx is normalized Sherwood number evaluated for the characteristic length L x . For the perpendicular transfer surface case, non-dimensional Sherwood number, Sh Lx , and shear Péclet number, Pe s,Lx , are defined as:

Sh Lx = kz L x D ; ( 12
)
Pe s,Lx = L 2 x S x D . (13)
Both non-dimensional numbers are also related to the characteristic length corresponding to the width of the probe L x .

Using the Faraday relation [START_REF] Bu | Experimental study of flow transitions in structured packed beds of spheres with electrochemical technique[END_REF] and the definition of the normalized Sherwood number [START_REF] Ackerberg | Heat-mass transfer to a finite strip at small peclet numbers[END_REF], it is possible to evaluate the wall shear rate S x from measuring current under the limiting diffusion conditions.

The arbitrary orientation of simple shear flow

For the homogeneous fluid flow, the inlet angle α is related to the components of the wall shear rate vector S over the whole electrode surface. Therefore, the angle α can be defined in the form of the non-dimensional β parameter that is introduced as a square of the ratio of the transverse shear rate to its magnitude, S :

β = S 2 y S 2 x + S 2 y = S 2 y S 2 , (14)
and the individual components of the wall shear rate on the wall surface could be re-written as:

S x = 1 -β S ; (15)
S y = β S . (16)
The exact relation between the β parameter and inlet angle α is presented in the Appendix A by equations (A.18) and (A. [START_REF] Tihon | Electrodiffusion detection of the near-wall flow reversal in liquid films at the regime of solitary waves[END_REF].

By substituting equations ( 15) and ( 16) into equation ( 6) one can obtain:

1 -β S z ∂c ∂x + β S z ∂c ∂y = D ∂ 2 c ∂z 2 . ( 17
)
Equation ( 17) can be converted into a dimensionless form to simplify the analysis of its solution within the parametric space on the base of the following formulas:

c + = c -c w c ∞ -c w ; x + = x L x ; y + = y L y ; z + = z δ D ; δ D = Dl S 1 3 ; R = L x L y , ( 18
)
where l is characteristic length on which the thickness of the diffusion boundary layer, δ D , is calculated. Substituting formulas (18) into eq. ( 17) and subsequent simple modifications, the transformed non-dimensional form of eq. ( 17)

is obtained:

z + ∂c + ∂x + + z + √ β √ 1 -β ∂c + ∂y + = 1 √ 1 -β L x l ∂ 2 c + (∂z + ) 2 . ( 19
)
Due to the introduction of bidirectional fluid flow, the transport eq. ( 19) has no analytical solution. It fundamentally complicates the wall shear rate evaluation by using the ED method. Two ways of solving this problem are possible: (i) determine an analytical solution of the normalized Sherwood number or (ii) compute the concentration profiles numerically from the equation [START_REF] Tihon | Electrodiffusion detection of the near-wall flow reversal in liquid films at the regime of solitary waves[END_REF] to assess the normalized Sherwood number.

Analytical approach: extension Sirkar and Harantty solution for strip probe

General analytical relation for normalized Sherwood number for arbitrary orientation β ∈ 0, 1 and any aspect ratio of strip mass transfer probe, R ≥ 0, can be derived on the base of the method initially used by Reiss & Hanratty [START_REF] Reiss | An experimental study of the unsteady nature of the viscous sublayer[END_REF] for a circular probe. Later, Sirkar & Hanratty [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF] applied this approach on a slanted strip probe. This method is based on dividing the probe into narrow strips oriented in the direction of flow. Since each strip is infinitesimally wide, they can be considered as a separate rectangular probe of length l(v) and width dv. On the base of the Lévêque solution, the mass transfer coefficient for a given narrow strip can be defined according to the equation (10) as follows:

kz = 0.8075 D 2 S l(v) 1 3
.

The mean integral value of the mass transfer coefficient, kz , over the whole probe surface is given by: kz

= 1 L x L y S k z dS = 1 L x L y v k z l(v)dv = 0.8075 D 2 S 1 3 L x L y v l(v) 2 3 dv. ( 21
)
The eq. ( 21) is possible rewrite into the form:

kz = 0.8075 D 2 S l e 1 3 , (22)
where the effective transfer length l e is defined as:

l 1 3 e = L x L y v l(v) 2 3 dv . ( 23
)
The integral in the denominator has to be evaluated for two basic situations:

β ≤ β c and β > β c
, where β c is the critical parameter β. The critical parameter β c corresponds to a condition where the aspect ratio of the probe equals the ratio of the axial to the transversal velocity components of the flowing fluid over the probe:

L x L y = u x u y = S x z S y z = √ 1 -β c S z √ β c S z = √ 1 -β c √ β c . ( 24
)
It is possible to express β c by a simple modification of eq. ( 24):

β c = L 2 y L 2 x + L 2 y = 1 R 2 + 1 . ( 25
)
For the evaluation of the effective transfer length, the integral in eq. ( 23) must be divided into three parts, as displayed in figure 2. In both cases (β ≤ β c and β > β c ), the rectangular probe is divided into two triangles A, C, and one parallelogram B. The triangles A and C are identical from the integration point of view. Based on this division, eq. ( 23) could be rewritten into the form:

l 1 3 e = L x L y 2 vs 0 l t (v) 2 3 dv + vc 0 l 2 3 p dv , ( 26
)
where v s is the altitude of triangles A and C, v c is the altitude of parallelogram B, and l t (v) is the length of strips in both triangles (A and C), and l p is the length of strips in the parallelogram. As shown in figure 2, the length l p is independent from the parameter v.

By solving the integrals in eq. ( 26) and subsequent simple adjustments, the equations describing the effective transfer length l e as a function of probe width 270 L x , aspect ratio R and parameter β can be obtained:

l 1 3 e = L x L y v l(v) 2 3 dv =                        L 1 3
x (1 -β)

1 6 1 + R √ β 5 √ 1 -β for β ≤ β c L 1 3 x β 1 6 R 1 3 1 + √ 1 -β 5R √ β for β > β c . (27)
A detailed derivation of the equation ( 27) is given in the Appendix A.

In the case of multidimensional flow, the dimensionless Sherwood and Péclet numbers are defined as follows:

Sh l = kz l D ; ( 28
)
Pe s,l = l 2 S D , ( 29
)
where l is the characteristic length, and its choice will be discussed below. Now the normalized Sherwood number can be expressed as:

Sh * l = Sh l Pe 1 3 s,l = kz l D 2 S 1 3 , (30)
By substituting a combination of eqs. ( 22) and ( 27) into equation [START_REF] Reiss | Measurement of instantaneous rates of mass transfer to a small sink on a wall[END_REF], one can obtain a relation for evaluation of Sh * l :

Sh * l = 0.8075

D 2 S l e 1 3 l D 2 S 1 3 = 0.8075 l l e 1 3 , (31)
By a combination of the expression ( 27) and the equation ( 31) the final relation of normalized Sherwood number is obtained:

Sh * l =                0.8075 (1 -β) 1 6 1 + R √ β 5 √ 1 -β l L x 1 3 for β ≤ β c 0.8075β 1 6 R 1 3 1 + √ 1 -β 5R √ β l L x 1 3 for β > β c . ( 32
)
The resulting value of the normalized Sherwood number depends on the selection of the characteristic length l. It is clear from the relation (31) that if the substitution of the characteristic length l by the effective transfer length l e is used (see eq. ( 27)), the equation ( 32) is reduced to the form:

Sh * le = Sh le Pe 1 3 s,le = kz l e D 2 S 1 3 = 0.8075, (33)
In this case, Sh * le is constant, and it is independent from both parameters R and β. The value of the normalized Sherwood number is identical to the value of the normalized Sherwood number obtained by the Lévêque solution for the perpendicular probe, compare eqs. ( 11) and [START_REF] Huchet | Use of electrochemical microsensors for hydrodynamics study in crossing microchannels[END_REF]. If the effective transfer length l e is known, the magnitude of wall shear rate S is evaluated directly from eq. ( 33) on the base of the evaluated mass transfer coefficient.

Another possibility is to assume that the characteristic length is equal to the width of the probe. Using substitution l = L x , the relation (32) has the 34). For practical reasons, the aspect ratio is limited to an interval R ∈ 0, 1 . The black curve describes the value of the critical βc defined by eq. ( 25).

following form:

Sh * Lx = kz L x D 2 S 1 3 =              0.8075 (1 -β) 1 6 1 + R √ β 5 √ 1 -β for β ≤ β c 0.8075β 1 6 R 1 3 1 + √ 1 -β 5R √ β for β > β c . ( 34
)
295

In this situation, the value of the normalized Sherwood number is generally a function of two parameters: probe aspect ratio R and parameter β. The resulting values of normalized Sherwood number defined by eq. ( 34) are presented in figure 3. In general, the more the effective transfer length differs from the probe width, the more Sh * Lx will vary from the derived Lévêque solution, see eq. [START_REF] Fadla | Electrochemical measurements for real-time stochastic reconstruction of large-scale dynamics of a separated flow[END_REF].

Figure 3 shows that the minimum value of the normalized Sherwood number can be found for β = 1, and the electrode width is close to zero (R → 0). In this situation, normalized Sherwood number is also approaching to zero. Conversely, the maximum normalized Sherwood number can be found for the following parameters: β = 0.5 (β = β c ) and R = 1 (square probe). In this case, Sh * Lx = 0.8627.

The presented value of Sh * Lx corresponds to the normalized Sherwood number of a circular probe relative to its diameter derived by Reiss & Hanratty [START_REF] Reiss | An experimental study of the unsteady nature of the viscous sublayer[END_REF]. It is clear from equation ( 34) that the Lévêque solution (Sh * Lx = 0.8075) applies only to the perpendicular transfer surface (β = 0). In such a case, the effective transfer length is equal to the electrode width (l e = L x ).

During experimental measurements, the direction of fluid flow can be unknown. This fact fundamentally complicates a posttreatment process with the measured data. In the literature, the effective transfer and characteristic length are often substituted by the electrode width L x without taking into account the fluid flow direction. In such cases, it is assumed that the difference between the real value of normalized Sherwood number and the normalized Sherwood number obtained for the perpendicular flow from the Lévêque solution is negligible.

But, as shown in figure 3, the difference between both values of the normalized Sherwood numbers may vary considerably. It applies that the more Sh * Lx differs from the value equal to 0.8075, more significant errors occur whenever the calculation of the wall shear rate is solely based on the Lévêque solution assuming the case of the perpendicular fluid flow over the surface of the probe.

Numerical approach

A second approach, on how to get the appropriate value of the normalized Sherwood number, is based on the calculation of the concentration profiles close to the probe surface by numerical solution of the transport equation [START_REF] Tihon | Electrodiffusion detection of the near-wall flow reversal in liquid films at the regime of solitary waves[END_REF]. In our case, the numerical solution was used, especially to verify the validity of the derived analytical relation for Sh * Lx , see eq. [START_REF] Mouheb | Flow characterization in t-shaped and cross-shaped micromixers[END_REF]. Therefore, the characteristic length l in eq. ( 19) is defined as the width of the probe l = L x . In this case, the eq. ( 19) is changed to:

z + ∂c + ∂x + + z + √ β √ 1 -β ∂c + ∂y + = 1 √ 1 -β ∂ 2 c (∂z + ) 2 . (35)
The eq. ( 35) is dependent only on two parameters: R and β. Due to the transformation into the non-dimensional form, the individual velocity components and the diffusion coefficient have changed, see eq. ( 35):

u + x = S + x z + = 1.z + ; (36)
u + y = S + y z + = √ β √ 1 -β .z + ; (37)
D + = 1 √ 1 -β .z + . (38)
The mass transport in the viscous boundary layer is modeled with a simple shear flow over the wall of the square shape (L + x = 1, L + y = 1). A probe is placed on the bottom wall near the inlet. The chosen size of the domain did not affect the shape of the diffusion boundary layer over the probe while minimizing the calculation time. Domain dimensions are in dimensionless quantities. The height of the domain is H = 6. The length and width are 9 × 9. The square electrode has dimensions 1 × 1. The following boundary conditions were set for dimensionless concentration: on the probe surface: c + = 0; inlet: c + = 1; top wall: c + = 1; outlet: ∇ + c + = 0; bottom wall (except the probe): ∂c + ∂z + = 0. The finite volume method was used for discretization. A numerical grid was containing 10 7 elements and was non-uniform. Since the greatest change in concentration occurs at the inflow edges of the probe, the numerical grid at the given locations had the highest density to refine the numerical solution. Convection terms were discretized by the UPWIND scheme and diffusion term by the central differencing scheme. For the solution of a system of linear algebraic equations was used smoothSolver with preconditioner method DILU (Simplified diagonal-based incomplete LU smoother for asymmetric matrices). Individual simulations were performed within the OpenFOAM software in which the solver for the numerical solution of equation [START_REF] Lesage | New local measurements of hydrodynamics in porous media[END_REF] was programmed.

Examples of concentration profiles for different probe aspect ratios R and different parameters β are shown in figure 4. In figures 4a, c, and e, the concentration profiles are presented directly on the bottom wall with the placed probe having a square shape after transformation. In figures 4b, d, and f, the concentration profiles are shown in a plane perpendicular to the bottom wall, passing through the centre of the probe and its orientation is in the direction of fluid flow.

As can be seen in figure 4, the individual concentration profiles depend on the β and R parameters. The resulting shape of the concentration profiles depends on the mutual competition between convective and diffuse transport, which could be characterized by the Péclet number:

Pe s,L + x = (L + x ) 2 S D + ; ( 39
)
whereas the characteristic length corresponds to the width of transformed probe

L + x = 1.
By substitution of eqs. ( 36) -(38) into eq. ( 39), the Péclet number is obtained as a function of β and R parameters:

Pe s,L + x = 1 -β 1 -β (1 -R 2 ) 1 -β = 1 -β (1 -R 2 ). (40)
In general, if the Péclet number increases, the importance of convective transport increases compared to diffusion. Therefore, the higher Péclet number results in the reduction of the thickness of the diffusion boundary layer. The increasing the ratio R causes an increase of the Péclet number and a decrease of diffusion boundary layer thickness (compare figure 4d; Pe s,L + x = 0.456 and 4f; Pe s,L + x = 0.921). The increasing the parameter β causes a reduction of the Péclet number and, at the same time, the increase of diffusion boundary layer thickness (compare figure 4b; Pe s,L + x = 0.777 and 4d; Pe s,L + x = 0.456). The evaluation of the normalized Sherwood number is computed from the numerical concentration fields around the probe surface, such as:

Sh * Lx = Sh Lx Pe s,Lx = L 2 x S D 1 3 L x δ D 1 0 1 0 ∂c + ∂z + dx + dy + = = 1 0 1 0 ∂c + ∂z + dx + dy + = Sh L + x Pe s,L + x = Sh * L + x . (41)
As follows from eq. ( 41) for calculating the normalized Sherwood number, knowledge of the concentration derivative according to the z-direction on the 34), was found, on average, around 10 -4 . A further reduction of the error could be achieved by using a higher density numerical grid, especially in the area of liquid inflow on the inner edges of the probe. The agreement between the results proves the correctness of eqs. ( 31)- [START_REF] Mouheb | Flow characterization in t-shaped and cross-shaped micromixers[END_REF] for the interpretation of the ED method measurements on the strip probe with any aspect ratio R for a simple shear flow of arbitrary spatial orientation.

Measurement of wall shear stress with the strip probe for the arbitrary orientation of simple shear flow

Based on Faraday's relation, see eq. ( 1), the mass transfer coefficient can be obtained from the measured electrical current under limiting diffusion conditions. Unlike the perpendicular case, where the wall shear rate can be calculated directly from eq. ( 11), in the case of the arbitrary orientation of simple shear flow, eq. ( 34) cannot be used in this straightforward manner. In addition to the magnitude of the wall shear rate vector, an unknown β parameter also occurs in eq. [START_REF] Mouheb | Flow characterization in t-shaped and cross-shaped micromixers[END_REF]. It is necessary to obtain one more independent equation for dimensionless mass transport coefficient to obtain an unambiguous solution (β, S ). This equation can be defined based on a new measurement with a strip probe of other aspect ratios.

It is necessary to make measurements with two probes, namely called A and B, with different aspect ratios in such a way that R A < R B . During the experiment, it must be guaranteed that the concentration wakes of both probes do not interact with each other, which can be overcome by alternative timeacquiring synchronization of the measurements from both probes or a suitable inter-probe distance. Because of the small dimension of the probe surfaces, the flow field in the near-electrode surfaces is assumed to be homogeneous and, so, the magnitude of the wall shear rate vector and parameter β are identical in both cases under a given hydrodynamical flow condition. In that situation, the relation (34) can be written as follows:

Sh * L x,A (β, R A ) = kz,A L x,A D 2 S 1 3
;

(42)

Sh * L x,B (β, R B ) = kz,B L x,B D 2 S 1 3 . ( 43
)
By combining the equations ( 42) and ( 43), the unknown S is eliminated, and the new relation is only dependent on one unknown, namely the parameter β:

Sh * L x,A (β, R A ) kz,A L 1 3 x,A = Sh * L x,B (β, R B ) kz,B L 1 3 x,B . (44)
As follows from equation [START_REF] Mouheb | Flow characterization in t-shaped and cross-shaped micromixers[END_REF], the exact relation describing the normalized Sherwood number is dependent on the critical parameter β c . Because of the difference in β c values for both measurement sets, the equation (44) can occur in the following three forms:

f (β) =                                    R A f 1 + (1 -β) 1 6 L 1 3
x,A kz,A x,A kz,A

- R B f 1 + (1 -β)
- f 2 + β 1 6 R B L 1 3
x,B kz,B R x,A kz,A R

2 3 A - f 2 + β 1 6 R B L 1 3
x,B kz,B R

where

f 1 = √ β 5 (1 -β) 1 3
;

f 2 = √ 1 -β 5β 1 3 . ( 46
)
The solution of equation (45) corresponds to looking for the root of the function f (β). It is a nonlinear continuous monotonic decreasing function with only one real β root, which can be obtained by any iterative numerical method for finding roots such as the Newton's method. In general, the bigger the difference between the aspect ratio R A and R B , the steeper the function f (β) decreases. After obtaining the appropriate value of β parameter, the last stage is to calculate the magnitude of the wall shear rate vector S from eq. ( 42) or (43).

Conclusion

The purpose of this paper is to revisit the electro-diffusional theory for the measurement of wall shear stress from strip mass transfer probes based on the consideration of the existence of the two components of the wall shear rate (i.e., axial and transversal). Therefore, the two-dimensional simple shear flow flowing under an arbitrary angle over the probe of rectangular shape with any aspect ratio was considered. For this type of uniform flow without stagnation points in the measured area, two general analytical formulas were derived for the effective transfer length and the dimensionless mass transport coefficient. These equations are a generalized solution presented by Sirkar & Hanratty [START_REF] Sirkar | Limiting behaviour of turbulent transverse velocity component close to a wall[END_REF] for a slanted strip probe. The correctness of the electro-diffusional theory revision for interpreting measurement data from the ED method was confirmed by numerical solutions of the non-dimensional convection-diffusion equation. It has been shown that if the characteristic length is substituted by the effective transfer length, the dimensionless mass transport coefficient is constant, and the value of the normalized Sherwood number is identical to that obtained with the Lévêque solution for the perpendicular probe. On the other hand, if the characteristic length is defined as the width of the probe, as commonly assumed, the value of the normalized Sherwood number is a function of two parameters: the aspect ratio R and the non-dimensional angle β. By using the electro-diffusion method for the arbitrary fluid flow direction, it is also often necessary to evaluate, in addition to the magnitude of the wall shear rate vector, the unknown angle of the fluid flow. Therefore, a new measurement methodology based on the proposed analytical relations is applied for two measuring probes of different aspect ratios.

Figure 1 :

 1 Figure 1: Scheme of the investigated system: a) flow and mass transfer in the near-wall region of the strip probe in XZ plane; b) perpendicular fluid flow over probe surface in XY plane; c) arbitrary orientation fluid flow over probe surface in XY plane.

Figure 2 :

 2 Figure 2: Scheme and description of the rectangular probe to derive the effective transfer length le for cases: a) β ≤ βc and b) β > βc.

Figure 3 :

 3 Figure 3: Normalized Sherwood number for length l = Lx as function of the parameter β and aspect ratio R evaluated from eq. (34). For practical reasons, the aspect ratio is limited to an

Figure 4 :

 4 Figure 4: 2D slices of concentration profiles for different probe aspect ratios R and parameters β: a), b) R = 0.1, β = 0.4; c), d) R = 0.1, β = 0.8; e), f) R = 0.9, β = 0.8. The first column -the bottom wall with the electrode; the second column -the plane perpendicular to the bottom wall, passing through the center of the electrode and its orientation is in the direction of fluid flow. The black dashed lines in the subplots in the first column represent the position of planes in the second column. Variable l + is a relative length of the black dashed line. The black lines are contours.

Figure 5 :

 5 Figure 5: The derivative of concentration in the z-direction on the probe surface (logarithmic color bar): the first row -β = 0; the second row -β = 0.4; the third row -β = 0.8; the first column -R = 0.1; the second column -R = 0.4; the third column -R = 0.8.

  β ≤ β c,A , β c,B R A f 1 + (1 -β)

  β > β c,A , β c,B ,

Acknowledgments

This work was supported by the Ministry of Health of the Czech Republic (project No. 16-29738A) and by Internal Grant Agency of Jan Evangelista

Appendix A.

The calculation of the effective transfer length is based in particular on the quantification of integrals in the denominator of the equation [START_REF] Tournier | Behavior of naturally oscillating 3-dimensional flow around a cylinder[END_REF] for two cases:

For the case β ≤ β c , the following equations for altitudes and lengths of strips are applied, see figure 2:

where α is the angle between the leading edge of the rectangular probe, L y , and the velocity vector of fluid flow over the surface of the probe. The integrals in the denominator in relation [START_REF] Tournier | Behavior of naturally oscillating 3-dimensional flow around a cylinder[END_REF] after applying terms (A.1) -(A.4) can be solved as follows:

x cos (α) sin (α)

x L y sin (α)

Substituting the solutions of the integrals (A.5) and (A.6) into the equation for calculating the effective transfer length [START_REF] Tournier | Behavior of naturally oscillating 3-dimensional flow around a cylinder[END_REF], can be get the following formula:

x L y sin (α) x sin(α)

For the case β > β c , the altitude of triangles v s , the altitude of the parallelogram v c , the length of strips in both triangles l t (v) and the length of strips in the parallelogram l p are defined by relations:

The integrals in the denominator in relation ( 26) after applying terms (A.9) -(A.12) can be solved as follows: It follows from the definition of beta by eq. ( 14):

From the equation (A.17), the relation between the goniometric functions and β parameter can be obtained:

x

(1 -β)


Richard Dearden
email: richard.dearden@mnhn.fr

Rohan Mansuit

Antoine Cuckovic

Anthony Herrel

Dominique Didier

Paul Tafforeau

Alan Pradel

a digital dissection of the elephantfish Callorhinchus milii and the catshark Scyliorhinus canicula

Keywords: chondrichthyan, cranial muscles, digital dissection, elasmobranch, holocephalan

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Inv nr.

Taxon Description M3#708 Callorhinchus milii cranial skeleton and muscles M3#709 Scyliorhinus canicula cranial skeleton and muscles

Table 1. List of 3D models.

INTRODUCTION

As the sister-group to bony fishes, the anatomy of sharks, rays and chimaeras (=chondrichthyans) plays a crucial role in our understanding of jawed vertebrate evolution. The two constituent chondrichthyan groups -holocephalans (chimaeras) and elasmobranchs (sharks and rays) -display wildly different cranial architectures, the origins of which have important implications for the origins of jawed vertebrate anatomy. In the associated manuscript (Dearden et al. 2021), we carried out a digital dissection of two chondrichthyans widely used as model species, the elephantfish Callorhinchus milii and the catshark Scyliorhinus canicula, which we used to review evidence for the evolution of chondrichthyan cranial muscles in the fossil record. Resulting from this digital dissection are detailed models of the cranial and pharyngeal skeleton and associated muscles, which we present here (Fig. 1 andtable 1).

METHODS

The 3D surfaces were extracted in Mimics 21.0 (Materialise), using the segmentation threshold tool. This was done with a combination of manual and interpolatory segmenting. Models were imported into Blender (blender.org), which was used to capture images. The models are provided here in .ply format, which are openable in many free software packages.

BIBLIOGRAPHY

Dearden, R.P., Mansuit, R., Cuckovic, A., Herrel, A., Didier, D., Tafforeau, P., Pradel, A. 2021. The morphology and evolution of chondrichthyan cranial muscles: a digital dissection of the elephantfish Callorhinchus milii and the catshark Scyliorhinus canicula. Journal of Anatomy. https://doi.org/10.1111/joa.133 62

Figure 1 .

 1 Figure 1. The cranial skeleton and select muscles in two chondrichthyans, based on the data published herein. A, the holocephalan Callorhinchus milii. B, the elasmobranch Scyliorhinus canicula. Colours: cream, cartilage; beige, pinks, muscles; greys, deeper muscles. Abbreviations: m.: muscle; mm.: muscles; mm. con. branchiales: mm. constrictors branchiales; m. coracoarc.: m. coracoarcualis; mm. coracobr.: mm. coracobranchiales; m. cuc. prof.: m. cucullaris profundus; m. epax.: m. epaxialis; m. lev. ang. oris ant.: m. levator anguli oris anterior; m. lev. ang. oris ant. pars. rost.: m. levator anguli oris anterior pars rostralis; m. lev. ang. oris post.: levator anguli oris posterior; m. lev. palp. nict.: m. levator palpebrae nictitantis; m. prot. dors. pect.: m. protractor dorsalis pectoralis; m. ret. dors. pect.: m. retractor dorsalis pectoralis; m. ret. lat.-vent. pect. l: m. retractor latero-ventralis pectoralis lateral; m. ret. palp. sup.: m. retractor palpebrae superioris; m. subspin: m. subspinalis.

ACKNOWLEDGEMENTS

The main work was supported by the Paris Ile-de-France Region -DIM "Matériaux anciens et patrimoniaux"-DIM PHARE projet, the ESRF (beamline ID19, proposal ec361) and the H.R. & E. Axelrod Research Chair in paleoichthyology at the AMNH.


Zhong-Sen Li

Post-Doctor

Matthieu Blanc
email: matthieu.blanc@ifsttar.fr

Luc Thorel

Using Fibre Bragg Grating sensors to estimate the horizontal response of a monopile in geotechnical centrifuge

Keywords: Centrifuge, Monopile, Transversal loading, Fibre Bragg Grating sensors, Sand

A 50-mm-diameter circular aluminium tube was instrumented with two optical fibres that consist of 13 Fibre Bragg Grating sensors (FBGS) for each. The performance of the FBGS was evaluated by applying a series of increasing transversal loads at 1×g level and comparing the strains measured by FBGS with those calculated from the Euler-Bernoulli beam theory. Centrifuge test was then conducted at 100×g to estimate the transversal response of the calibrated model pile that had been jacked 450 mm into saturated sand and horizontally loaded at 500 mm above the ground. The profiles of the normal strain, bending moment, soil reaction and pile deflection were measured or determined, allowing to construct the soil reaction -pile deflection (P-y) curves. The results confirmed the reliability of the FBGS at 100×g by giving satisfactory measurements on bending moments and coherent measurements on shear force at the ground level.

List of notation

Introduction

Monopile is a typical foundation for offshore wind farms. It takes over 80% of recently installed wind turbines (Wind Europe, 2017) and represent about 15-20% of the total investment. The lifecycle of the monopile is influenced by the horizontal service loads resulting from winds, currents and waves.

The generation of horizontal deflection may degrade the foundation soil and as a result trigger the instability problems. Faced with the increasing capacity of the next-generation wind turbines, the dimension of the future monopiles is expected to increase, for example, to as large as 10 m in diameter. For such large-diameter monopiles, current design codes for long and slender piles are deemed inappropriate and new standards need to be developed.

In the literature, three experimental methods exist to characterise the response of pile or monopile under transversal loads: i) the full-scale field tests (e.g. [START_REF] Davisson | Lateral load tests on drilled piers. Performance of Deep Foundations[END_REF][START_REF] Baguelin | Etude expérimentale du comportement de pieux sollicités horizontalement[END_REF], Cox et al., 1974;[START_REF] Reese | Analysis of laterally loaded piles in sand[END_REF][START_REF] Ting | Full-scale cyclic dynamic lateral pile responses[END_REF][START_REF] Little | Full scale cyclic lateral load tests on six single piles in sand[END_REF][START_REF] Koukoura | Identification of support structure damping of a full scale offshore wind turbine in normal operation[END_REF][START_REF] Barbosa | Large Scale Offshore Static Pile Tests-Practicality and Benefits[END_REF][START_REF] Jardine | Behaviour of piles driven in chalk[END_REF][START_REF] Mcadam | Monotonic lateral loaded pile testing in a dense marine sand at Dunkirk[END_REF]; ii) reduced-scale laboratory tests (e.g. [START_REF] Leblanc | Response of stiff piles in sand to long-term cyclic lateral loading[END_REF][START_REF] Abadie | Rigid pile response to cyclic lateral loading: laboratory tests[END_REF] and iii) reduced-scale centrifuge tests (e.g. [START_REF] Georgiadis | Centrifugal testing of laterally loaded piles in sand[END_REF][START_REF] Verdure | Lateral cyclic loading of single piles in sand[END_REF][START_REF] Rosquoet | Lateral cyclic loading of sand-installed piles[END_REF][START_REF] Li | Centrifuge modelling of mono-pile under cyclic lateral loads[END_REF]Klinkvort andHededal, 2013, 2014;Klinkvort et al., 2013[START_REF] Klinkvort | Centrifuge modelling considerations of laterally loaded monopiles in sand[END_REF][START_REF] Choo | Experimental Development of the p-y Relationship for Large-Diameter Offshore Monopiles in Sands: Centrifuge Tests[END_REF][START_REF] Truong | Empirical approach based on centrifuge testing for cyclic deformations of laterally loaded piles in sand[END_REF].

Each above-mentioned method presents its own inherent advantages and disadvantages. For performing parametric studies in a first step, reduced-scale laboratory and centrifuge tests are more feasible due to the consideration of testing budget. Comparing to laboratory tests, centrifuge tests at elevated gravity have the advantage of reproducing the field stress condition for monopiles. Through carrying out centrifuge tests, researchers were able to test monopiles as large as 50 mm in diameter (e.g. [START_REF] Li | Centrifuge modelling of mono-pile under cyclic lateral loads[END_REF][START_REF] Klinkvort | Centrifuge modelling considerations of laterally loaded monopiles in sand[END_REF]. The pile local strains were measured and then used to determine the soil reaction -pile deflection (P-y) relationship. These results, together with the full-scale field tests and reduced-scale laboratory tests, are of interest for the design of next generation monopiles.

The present study is part of the SOLCYP+ project (France Energies Marines, 2017), which is an extension of the SOLCYP project [START_REF] Puech | Design of piles under cyclic loading: SOLCYP recommendations[END_REF]. The main objective was to experimentally investigate the response of the model piles under horizontal load in geotechnical centrifuge and then to extrapolate the prototype pile behaviour. Fibre Bragg Grating sensors (FBGS)

were used to measure the local strains and to determine the moment profile of the pile. Experimental results are presented following the local pile analysis [START_REF] Garnier | Advances in lateral cyclic pile design: contribution of the SOLCYP project[END_REF].

Principle and geotechnical application of FBGS

Exposing the Germanium-doped optical fibre to the spatial pattern of ultraviolet light (e.g. [START_REF] Hill | Fiber Bragg grating technology fundamentals and overview[END_REF][START_REF] Kreuzer | Strain measurement with fiber Bragg grating sensors[END_REF] can artificially alter the refractive index of the fibre and form a periodic modulation of the refractive index (i.e., grating). When a beam of light passes a grating, part of the light will be reflected and the other be transmitted. The reflected lights accumulate and form a peak if an integer of the light wavelength fits into two times of the fringe spacing, as shown in Figure 1.

Under the effect of the external stress or temperature, the optical fibre shrinks or swells. This shrinkage-swelling results in a change in grating period and as a result a corresponding shift of the reflected peak wavelength in the recorded spectra. Under temperature-controlled condition, the strain (e) can be calculated with the peak wavelength shift (Dl) by the following formula:

! = #$ $% (1)
where & (nm) is the initial peak wavelength under null external stress (i.e., reference scan), '& (nm)

is the peak wavelength shift and k, the gage factor, is equal to 0.78 for the fibre used in this study.

As the FBGS has very small dimension (e.g., diameter of several hundred micro-meters), it can be adhered on or embedded into the piles or beams without significantly changing their geometry and mechanical property. For such reason, it has been increasingly used. In geotechnical laboratory and field testing, FBGS have been used to measure the load transfer of the model pile (e.g., [START_REF] Lee | Measurement of pile load transfer using the Fiber Bragg Grating sensor system[END_REF], the strain distribution along the soil nails during pull-out test (e.g., [START_REF] Zhu | Soil nail monitoring using Fiber Bragg Grating sensors during pullout tests[END_REF]Zhu et al., , 2011;;[START_REF] Hong | New LGFBG-Based structural integrity evaluation method for cement-Grouted soil nails[END_REF] and the transversal displacement of ballast under cyclic loading conditions (e.g., [START_REF] Hussaini | Application of optical-fiber Bragg grating sensors in monitoring the rail track deformations[END_REF]. Besides, FBGS have been used to monitor the dynamic process of the wind turbine blades (e.g., [START_REF] Park | Real-time monitoring of composite wind turbine blades using fiber Bragg grating sensors[END_REF], large pile under vertical driving conditions (e.g., [START_REF] Doherty | Field validation of fibre Bragg grating sensors for measuring strain on driven steel piles[END_REF] and strains of buried pipelines under different external loads (e.g. [START_REF] Glisic | Fiber optic method for health assessment of pipelines subjected to earthquake-induced ground movement[END_REF][START_REF] Simpson | Distributed sensing of circumferential strain using fiber optics during full-scale buried pipe experiments[END_REF][START_REF] Ni | Distributed fibre optic sensing of strains on buried fullscale PVC pipelines crossing a normal fault[END_REF].

However, the report of the FBGS in geotechnical centrifuge is limited. [START_REF] Kapogianni | The use of optical fibre sensors in a geotechnical centrifuge for reinforced slopes[END_REF] investigated the performance of FBGS in a drum centrifuge at ETH Zurich. The FBGS gave coherent measurements at different g-level (0-100 g) even though the unstable measuring signal was observed due to the movements of the FBGS cables. Da [START_REF] Da Silva | Fibre optic instrumentation and calibration in the geotechnical centrifuge[END_REF] reported their experience on FBGS instrumentation and calibration in centrifuge test at the University of Cambridge. A first issue that they considered was the location and orientation of the fibre optic interrogator. In their study, the interrogator was placed on the centrifuge beam as close as possible to the centre to minimise the gravitational acceleration experienced by the interrogator. In addition, the interrogator box was orientated such that the axis of the fans aligned with the direction of the gravitational acceleration to minimise impact of g-level on the operation of the interrogator. After these efforts, the authors found that the interrogator remained securely in place with g-level up to 40 g (i.e., 6 g for the interrogator).

Excellent correlation was achieved between the strains measured by FBGS and those by LVDT, with the difference generally smaller than 5%. To avoid the high g-forces within the interrogator, [START_REF] Correia | Interrogation of fibre Bragg gratings through a fibre optic rotary joint on a geotechnical centrifuge[END_REF] reported a new method to place the FBGS interrogator. This method consists of using a fibre optic rotary joint and placing the interrogator outside of the centrifuge. At Chang'an University (China), [START_REF] Weng | Application of Fiber Bragg Grating Strain Sensors to a Centrifuge Model of a Jacked Pile in Collapsible Loess[END_REF] conducted a series of centrifuge tests, using both FBGS and strain gauge to measure the vertical strains of a jacked steel pile previously soaked by water. The difference between the strains measured by FBGS and the strains measured by strain gauges was, in most cases, within 10%.

In conclusion, the FBGS i) provides a precise, reliable and convenient method to monitor strain change of piles and pipes; ii) allows multiple strain measurements with a single fibre; iii) is easy to be installed and suitable under both static and dynamic load conditions and iv) is rarely used in geotechnical centrifuge. Compared to the strain measuring methods that the authors previously adopted (e.g., gluing the strain gages outside or even inside the tube in Rosquoët et al., 2007 and[START_REF] El Haffar | Monotonic lateral loading on single piles in sand: parametric centrifuge modelling in eccentricity, saturation and installation mode[END_REF], the choice of the optical fibres in this study was mainly to maintain the wall thickness of the tube while working on an open-ended model pile.

Instrumentation and calibration of the model pile

The geometric and mechanical characteristics of the model pile are described in Figure 2 and Table 1. The choice of an aluminium tube instead of a steel bar allows to respect the key scale factor of the rigidity (EI) while maintaining a suitable tube thickness for embedding FBG sensors. Besides, aluminium has been commonly used in geotechnical centrifuge to reproduce the behaviour of steel structures (e.g. [START_REF] Ng | Centrifuge modelling of heating effects on energy pile performance in saturated sand[END_REF][START_REF] Saiyar | Response of pipelines of differing flexural stiffness to normal faulting[END_REF]. The pile was chiselled with two semi-cylindrical grooves that are diametrically opposed. The radius of the grooves is 0.5 mm, i.e., 1/5 the wall thickness of the pile. Two 200-micrometer-diameter optical fibres were laid straight in the grooves and then sealed by a bi component cold curing epoxy, i.e., X120 adhesive. X120 adhesive was chosen mainly due to its easy usage and good strain transfer property (HBM, 2019b). Another consideration was that the stiffness of the adhesiveness is smaller than that of the pile so that the local gauge hardening effect can be avoided. Placing two optical fibres in an extension-compression configuration, i.e., diametrically opposed as shown in Figure 2a, aimed to compensate the temperature effect on the measurement. This effect can be compensated by averaging the difference of the two measured strains (i.e., εav. = (εext.εcomp.)/2) as a temperature change has identical effect on both fibres in terms of the peak wavelength shift, e.g., one fibre being compressed and features a negative strain, whereas the other being stretched and has a positive strain. Within each optical fibre, 13 FBGS were integrated and distributed every 25 mm for the first eight FBGS (#1 -#8) and every 50 mm for the rest (#9 -#13). Such a configuration allows to measure the pile strains at 13 different positions ranging from 0 -425 mm away from the FBGS #1, as shown in Figures 2 and3.

The calibration of the model pile is intended to compare the the strain measured by FBGS and that calculated from the Euler-Bernoulli beam theory. Two calibrating methods are proposed: i) the pile clamped at one end (i.e., cantilever beam) and ii) the pile put on two rigid supports (e.g., simply supported beam). The apparatus and the corresponding schematic drawing are presented in Figure 3.

To avoid the possible damage on the pile end in particular for the cantilever beam configuration where the pile was clamped only 10 mm at the end, the maximum applied weight F was limited, for example 100 N and 500 N for the cantilever and simply supported beam, respectively. For this reason, the calibration strains were within 350 microstrain (10 -6 ), which cannot cover the full range of the following centrifuge tests. Transversal loads were applied at 1×g level with an incremental ratio of about 2, i.e., 20, 50, 100 N for the cantilever beam and 20, 50, 100, 200 and 500 N for the simply supported beam. In order to evaluate the performance of the FBGS under both tensile and compressive conditions, the pile was calibrated at 0° (initial position) and 180° (turning the pile over).

As the transversal load increases, the model pile deflects and generates tensile or compressive strains.

These strains can be on the one hand calculated based on the direct measurements of the FBGS as shown in Equation 1, and on the other hand determined by the Euler-Bernoulli beam theory using the following formula:

! = () *+, ( 2
)
where M is the moment at different sections, F, the applied transversal force, D, the outside diameter of the pile, E, the Young modulus of the pile and I, the moment of inertia of the section..

For the cantilever beam configuration, the moment M is:

-= / • (500 + 5 6789 ) (3)
and for the simply supported beam, the moment M is:

-= 6•*;<•(==>?@ ABCD ) E<< (4)
where zFBGS is the position of the FGBS in mm, as illustrated in Figure 3. When the pile is supported by two tips (i.e., simply supported beam) and the transversal loads are applied in between, strains are in tension on the lower side and in compression on the upper side.

Similarly, the experimental points, excluding those measured by FBGS #13, are located around the correlation lines (Figure 4b). However, for the maximum transversal load, the measured strains are deviated from those calculated according to Euler-Bernoulli beam theory. The FBGS measurements derivate from -11.99 to 4.89% from the theoretical predictions for FBGS #1-8. The relative difference increases significantly for FBGS #9-13. This may result from many aspects in such a free system where both the stress concentration and pile movement are possible. As concerning the moment profiles, results in the third diagram of Figure 4b shows better accordance between the experiment and theoretic correlations, with their difference generally smaller than 5%.

Pile behaviour under transversal loads in centrifuge test

Preparation of the centrifuge test

The experimental study was carried out at the IFSTTAR large beam centrifuge (D = 11 m) inaugurated in 1985 [START_REF] Corté | Une centrifugeuse pour la recherche en géotechnique[END_REF].The centrifuge can generate a maximum gravitational acceleration of 100 g in the bracket for a sample mass up to 2 tons. The FS22DI Industrial

BraggMETERInterrogator box from HBM was placed in the data acquisition chamber that is attached to the rotation centre. The longitude direction of the interrogator was in line with the centrifuge arm.

The gravitational acceleration experienced by the interrogator is estimated about 1/15 of the centrifuge basket. The wavelengths of the 13 FBGS at reference scan range from 1520-1580 nm and increase every 5 mm. To prepare for the centrifuge test, three steps were employed: i) model sand preparation, ii) pile installation and iii) load application.

The model soil was the Fontainebleau NE34 poorly graded sand (Table 2). Using the automated raining technique [START_REF] Garnier | Modèles physiques en géotechnique-I-Évolution des techniques expérimentales et des domaines d'application[END_REF], the sand was dropped from a constant height (70 cm in this study)

and then pluviated into a 1200 mm (length) by 800 mm (width) by 720 mm (height) rectangular strongbox. The depth of the sand mass was 560 mm and the relative density was 81%. To simulate the prototype condition where the foundation is under water, the sand was saturated by slowly (about 6 hours) injecting water through four draining channels located at the bottom of the strongbox. The attained water table was about 30 mm above the sand surface. The effective volumetric weight of the saturated sand was 10.3 kN/m 3 .

Concerning the pile installation, the instrumented model pile (Figure 2) was pushed into the sand at 1×g level by a hydraulic jack. The jacking speed was 1 mm/s and the attained depth is 450 mm, i.e.,

9D. The sand level inside the pile was checked and there was no plug generated. After pile installation, an electric actuator was mounted on two supporting beams that perpendicularly placed on the longitudinal edges of the strongbox. The actuator has a stroke of 150 mm and can measure the load with a precision of 0.1 N for a range from 0 -5 kN. At the end of the piston, a force transducer and a loading fork were integrated, aiming to measure and control the applied horizontal load. In front of the actuator, a laser displacement transducer, with measuring range of 120 mm and resolution of 20 µm, was fixed on a supporting beam. This allows to monitor the pile deflection at 410 mm above the ground level. The established experimental set-up is schematically shown in Figure 5.

Centrifuge test was performed at 100´g level. Horizontal load was applied through the centre of the cross section by pushing the steel rod that crosses the monopile perpendicularly to the instrumentation plan (Figures 2 and3). The loading point situates at 500 mm (i.e. 10D) above the ground level. The loading was displacement-controlled with a rate of 1 mm/s at the actuator level and terminates at 50 mm (i.e. 1D). The measurements of the FBGS, the laser and force transducer were recorded and registered every 0.02 second (i.e., 50 Hz).

Horizontal response of the model pile

In Figure 6a, the normal strains measured at each incremental force (every 0.25 kN until 1.75 kN)

were plotted versus the pile depth As the applied horizontal load H increases, the normal strains within the pile increase as well. Under the same H, normal strains first slightly increase with depth, reach to the maximum at the depth of 100 mm and then decrease almost to zero at 425 mm below the ground level. The average value of the tensile and compressive strains was used to determine the bending moments and the used equation was shown in Figure 6b. At the ground level, the moments measured by FBGS (empty circular points) are very close to the moments calculated from H (solid square points), confirming the reliability of the FBGS under flight condition. At the pile end (FBGS #13, z = 425), the bending moments are close to zero. The discrete moments along the depth need to be fitted into continuous so that further calculation can be proceeded. In [START_REF] Haiderali | Evaluation of curve fitting techniques in deriving p-y curves for laterally loaded piles[END_REF], the authors compared different curve fitting techniques and found that the cubic and cubic Bsplines gave more consistent and accurate P-y curves. In this study, the cubic splines function was used to interpolate the experimental results. According to the beam theory, the shear force V and the transversal distributed loads along the pile P (equal to the soil reaction) can be obtained by solving the first and second derivatives of the moment profiles (i.e. cubic splines), respectively. Due to the lack of the experimental results above the ground surface and below the pile tip, the algorithm of the cubic spline forces the spline approximate the end points (i.e., FBGS #1 and FBGS #13) in a default way, considering the second derivative of the spline to be zero. Such algorithm is normally reasonable for FBGS #1 as the second derivative of the bending moment (i.e., soil reaction) should be theoretically null at the ground level, however, probably incorrect for FBGS #13 because the soil reaction at the pile tip should not be zero. As a result, the first and second derivatives of FBGS #13 (i.e. pile depth from 375-425) are not necessarily significant and were not determined. In Figures 6c and6d, the V and P profiles are presented. For the former, V at the ground level are coherent with H applied by the actuator. Slight difference is due to uncertainties in finding the first derivative accurately as it lacks experimental points above the ground surface. Concerning the P profile, as H increases, the pile rotation centre defined here as the point where the soil reaction is nil, P = 0 increases from 210 -275 mm (i.e., 4.2 -5.5 D) below the ground level.

Finding the second-order definite integral of the moment profiles, one may obtain the pile transversal deflection y. This was usually achieved by solving a simultaneous equation that contains two integration constants. Two limit conditions are introduced in this study. The first one is the pile deflection measured by the electric actuator at the pile head. The second is the null pile deflection (y = 0) at the rotation centre, which was determined by finding the depths of the null reaction (P = 0), as presented in El [START_REF] El Haffar | Monotonic lateral loading on single piles in sand: parametric centrifuge modelling in eccentricity, saturation and installation mode[END_REF]. Figure 7 presents the pile horizontal deflection at different H values. Due to the large embedment depth (i.e., 450 mm or 9D), pile deflections are close to zero below the rotation centre (e.g., 300 -450 mm below the ground surface). Presented also are the pile deflections measured by the laser transducer. As H increases, the laser provides extra measurements that are comparable with the measurements provided by the electric actuator.

P-y curves are obtained by plotting the pile deflection y versus the soil reaction P at different depths (e.g., the depth of the FBGS). In Figure 8, each curve demonstrates the P-y relationship as H increases from 0.25 -1.75 kN. P is positive above the pile rotation centre and negative under. The initial soil stiffness, i.e., the initial slope of the P-y curve, increases as the embedment depth increases (from the ground surface to the pile tip). For the soil at a certain depth (e.g., z = 75 mm), its stiffness decreases as the horizontal load H increases. In this study, a plateau is reached only for the second level, z = 25 mm. For deeper levels (e.g. z = 50 -225 mm), the P-y curves show a tendency to be stabilized.

However, much more lateral displacement is needed to reach such limit.

Conclusions

A model pile was instrumented with two diametrically-opposed optical fibres. The instrumented pile was calibrated at 1×g and then subjected to lateral loading in centrifuge at 100×g. The experimental results show that:

• All the 26 FBGS survived in the 100×g centrifuge test;

• At the ground level, the moments determined by FBGS are in accordance with the moments calculated from the transversal force;

• Shear forces at the ground level are slightly larger than the applied transversal forces. Such difference mainly results from the difficulty of the accurate derivation operation at the ground level.

• The pile local behaviour is characterized: under the effect of the increasing horizontal load, pile rotates at the depth 210 -275 mm (i.e., 4.2 -5.5D) below the ground level. Due to the large embedment depth, the pile deflection below the rotation centre close to zero. The P-y curves at different depths were also determined.

FIGURE CAPTIONS

  diameter at which 10, 50 or 60% of the sample's mass is comprised of smaller particles (µm) ρd-min minimum dry density (g/cm 3 ) ρd-max maximum dry density (g/cm 3 ) Λ distance between two adjacent fringe (nm)

Figure 4

 4 Figure 4 presents the calibration results of the model pile. When the pile is clamped on one tip (i.e.,

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 1 Figure 5 Figure 6 Figure 7 Figure 8

 123456781234567815678 Figure 1 Working principle of the Fibre Bragg Grating sensors

Table 1 .

 1 Geometric and mechanical characteristics of the pile

	Material	Prototype	Model
	L (m)	50	0.5
	D (m)	5	0.05
	d (m)	4.5	0.045
	E (GPa)	74 GPa	74 GPa
	EI (N.m 2 )	7.8×10 11	7.8×10 3

Acknowledgement

This work benefited from France Energies Marines and State finance managed by the National Research Agency under the Investments for the Future program bearing the reference SOLCYP+ ANR-10-IEED-0006-18. Assistances from the technical staffs of the centrifuge group and discussion with Dr. Thierry Dubreucq are acknowledged.