Yoga Satria Putra

Anthony Beaudoin
email: anthony.beaudoin@univ-poitiers.fr

Germain Rousseaux

Lionel Thomas

Serge Huberson

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

Keywords: Sediment transport, Maxey-Riley equations, tracker method, tidal bore, Froude number, Ratio of water depths, OpenFoam

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The tidal bore is unique phenomenon in the estuaries having a convergent river mouth with a uterine section shape (Fig. 1 (a) and(b)). This phenomenon has an influence on the transport of sediment particles in the estuaries as studied by [START_REF] Donnelly | Environmental impact of undular tidal bores in tropical rivers[END_REF] in [START_REF] Donnelly | Environmental impact of undular tidal bores in tropical rivers[END_REF]. They observed that the high bed shear stresses induce erosion beneath a tidal bore. There is a lot of studies in the 5 scientific literature that discussed this phenomenon. In field studies, the tidal bore phenomenon has been studied recently by [START_REF] Chanson | Tidal bores, aegir, eagre, mascaret, pororoca: theory and observations[END_REF], [START_REF] Furgerot | Sediment transport induced by tidal bores. an estimation from suspended matter measurements in the see river (mont saint-michel bay, northwestern france)[END_REF] and [START_REF] Bonneton | Conditions for tidal bore formation in convergent alluvial estuaries[END_REF] [START_REF] Chanson | Tidal bores, aegir, eagre, mascaret, pororoca: theory and observations[END_REF][START_REF] Furgerot | Sediment transport induced by tidal bores. an estimation from suspended matter measurements in the see river (mont saint-michel bay, northwestern france)[END_REF][START_REF] Bonneton | Conditions for tidal bore formation in convergent alluvial estuaries[END_REF]. In these studies, they have characterized the tidal bore with respect to the Froude number F r from the field data. Analytically, the tidal bore phenomenon has been studied by [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF][START_REF] Andersen | Undular hydraulic jump[END_REF][START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF]. An analytical study of conjugate depth ratio 10 r = h 2 /h 1 , wavelength L w and wave amplitude a w has been presented in these three studies. In the laboratories, the tidal bore phenomenon has been studied by [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF], [START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF], [START_REF] Hornung | The flow field downstream of a hydraulic jump[END_REF], [START_REF] Koch | Turbulent mixing beneath an undular bore front[END_REF] and [START_REF] Rousseaux | A novel method to generate tidal-like bores in the laboratory[END_REF] [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF][START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF][START_REF] Hornung | The flow field downstream of a hydraulic jump[END_REF][START_REF] Koch | Turbulent mixing beneath an undular bore front[END_REF][START_REF] Rousseaux | A novel method to generate tidal-like bores in the laboratory[END_REF]. In these studies, two types of tidal bores, undular and breaking, have been generated. The analysis of free surface, velocity and pressure of tidal bores has been performed. And numerically, the tidal bore phenomenon has been studied by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF], [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] and Berchet et al. (2018) [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF][START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Berchet | Flow fields and particle trajectories beneath a tidal bore: A numerical study[END_REF]. They have generated numerical simulations of tidal bores using the CFD software Thétis developed in the I2M laboratory of Bordeaux University, France. This Thétis software has been developed by applying the LES turbulence model to solve the Navier-Stokes equations [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF][START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Berchet | Flow fields and particle trajectories beneath a tidal bore: A numerical study[END_REF]. The flow pattern of velocity beneath a tidal bore classified into 4 types (A, B, C and D) has been performed in the study by Berchet et al. (2018). Figure 1: (a) Bono on Kampar river, Indonesia (photograph by Lawrence in [START_REF] Lenstraffic | Bono Tidal Bore Surfing in Sumatera[END_REF]) and (b) Kampar river map in Sumatera, Indonesia where the tidal bore bono appears (https://maps.google.com/).

There is a similitude between a hydraulic jump and a tidal bore. These two phenomena can be characterized by the Froude number F r, a dimensionless number representing the ratio between the flow inertia to the external field as gravity (see in detail the part 2.1.4). A hydraulic jump is a transition from a supercritical flow with a Froude number F r < 1 to a subcritical flow with a Froude number F r > 1. The tidal bore is also called (improperly) a moving hydraulic jump or a hydraulic bore [START_REF] Chanson | Tidal bores, aegir, eagre, mascaret, pororoca: theory and observations[END_REF][START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF]. The tidal bore is similar to a hydraulic jump because it is generated by the opposite flow of the tide with respect to the river flow. The criteria to generate the tidal bore in nature are well-known: a large tidal range, a shallow and convergent channel, and a low freshwater discharge [START_REF] Bartsch-Winkler | Catalog of worldwide tidal bore occurrences and characteristics[END_REF][START_REF] Bonneton | Conditions for tidal bore formation in convergent alluvial estuaries[END_REF].

To study the impact of tidal bores on the hydrodynamics of river flows, we have referred to some studies of the scientific literature that are correlated with our study. The latter were realized to define the transition between no tidal bore and the undular tidal bore, and the transition between the undular and breaking tidal bores. In their studies, [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] and [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] in [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF][START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] generated experimentally and numerically two types of tidal bores, undular and breaking. They used the CFD software Thétis to generate the 2D numerical simulations of tidal bores by applying the LES turbulence model. They have also estimated the wave amplitude a w and the wavelength L w from the form of free surface. From their numerical results, the wavelength L w decreases when the Froude number F r increases. The wave amplitude a w increases when the Froude number F r increases in the regime of undular tidal bore. Otherwise, the wave amplitude a w decreases when the Froude number F r increases in the regime of breaking tidal bore. In our study, the numerical study of two tidal bores types, undular and breaking, was done using OpenFOAM (Open Source Field Operation And Manipulation) CFD software by applying a LES turbulence model to complete the studies by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] and [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF]. Seventeen 2D numerical simulations were generated to identify the transition from undular tidal bores to breaking tidal bores. A study has been done by [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF] in [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF] that classified the tidal bores with the Froude number F r only (not based on r ). She has performed a classification of tidal bores based on the field data where a small wave appears on the tidal bore with a Froude number F r < 1.25. She has also observed that the undular and breaking tidal bores appear with a Froude number 1.25 < F r < 1.50 and F r > 1.50 respectively. The low Froude number (F r < 1.52) brings up a low turbulence in the river flow. Then it generates an undular tidal bore in the river channel. For a Froude number F r > 1.50, the high turbulence induces a breaking tidal bore. This classification by [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF] in [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF] is confirmed and refined in this work using the 2D numerical simulations obtained with OpenFOAM.

To study the impact of tidal bores on the transport of non-cohesive sediment particles, the studies by [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF] and [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF] have been referred in our study [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF][START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF]. [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF] in [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF] have studied analytically the trajectory of non-cohesive particles in a two dimensional wave-current field using the fully Lagrangian framework. They have derived analytical solutions that can be used to describe the trajectory of non-cohesive sediment particles beneath an undular tidal bore. [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF] in [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF] has developed a tracker model based on the Maxey-Riley equations to estimate the trajectory of non-cohesive sediment particles under an undular tidal bore. From 2D numerical simulations of undular tidal bores using the CFD software Thétis, he has obtained that the numerical trajectories are similar to those given by the analytical model proposed by [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF] in [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF]. In our study, we complete the studies by Chen et al. (2012) and [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF] taking into account the effects of gravity, elevation and attenuation on the trajectory of non cohesive sediment particles. Three parameters of the modified Chen's model, β 1 , β 2 and β 3 , are identified as functions of the Froude number F r.

In order to reach the two objectives of our study, the study of the impact of tidal bores on the hydrodynamics of river flows and the transport of non cohesive sediment particles, a methodology based on different tools is proposed in the section 2. For the analysis of tidal bore flows, we perform a comparison between the numerical simulations obtained with OpenFOAM [START_REF] Team | The Open Source CFD Toolbox: Programmer's Guide[END_REF] and the theory proposed by [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF][START_REF] Andersen | Undular hydraulic jump[END_REF][START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] for various values of the Froude number F r, and the ratio of water depths r for Berry only. For the analysis of trajectories of non-cohesive sediment particles, we performe a comparison between the numerical results obtained with a tracker model based on the modified Maxey and Riley equations [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] and the modified Chen's model [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF]. The results of these two comparisons are given in section 3. A validation of 2D numerical simulations obtained with OpenFOAM and a classification of tidal bores are presented in this section. The relationships between the three parameters β 1 , β 2 and β 3 of the modified Chen's model and the Froude number F r are given in section 3. In our study, 2D numerical simulations of tidal bores were generated using OpenFOAM (Open Source Field Operation And Manipulation) CFD software. OpenFOAM is an open source CFD software released and developed primarily by OpenCFD Ltd since 2004 [START_REF]About OpenFOAM[END_REF]. This software uses the Finite Volume discretization and free surface capturing technique based on the Volume of Fluid (VoF) method to solve the Navier-Stokes equations [START_REF] Team | The Open Source CFD Toolbox: Programmer's Guide[END_REF]. We have used the LES turbulence model of OpenFOAM, as it has been done by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] and [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF], to generate seventeen 2D numerical simulations of tidal bore. These numerical simulations cover two types of tidal bore, undular and breaking tidal bores. In the LES model, we have used the Smagorinsky model as the turbulence model with a turbulent kinetic energy k = 0.0001 m 2 /s 2 . The viscosity of air and water is set to µ air = 1.85 × 10 -5 kg/ms and µ water = 10 -3 kg/ms respectively. The densities of air and water are set to ρ air = 1.1768 kg/m 3 and ρ water = 1000 kg/m 3 respectively, under a gravity acceleration equal to g = 9.80 m/s 2 . The numerical results have been compared with experimental and numerical results by [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] for undular tidal bores and Khezri (2013) for breaking tidal bores [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF].

Methodology

The 2D computational domain of tidal bores is derived from the experimental setup in an open channel given in Figure 2. The transition between the undular and breaking tidal bores is studied by using the configurations performed by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF]. To obtain seventeen 2D numerical simulations of tidal bore that vary in the value of the Froude number F r, we have set the different gate height h g for each simulation, between 0 and 0.135 m. The initial flow velocity V 0 and the initial water depth h 1 were fixed to 0.74 m/s and 0.14 m respectively. In these configurations, the size of the 2D computational domain is fixed to 10 m for the length and 0.5 m for the height. The mesh grid of computational domain is generated by the mesh generation utility, blockM esh of OpenFOAM.

The geometry of computational domain given in Figure 2 was built from a dictionary file, named blockM eshDict, located in the system directory of OpenFOAM. In the blockM eshDict file, we could set the cell number of computational domain Cn. In Figure 2, we divided the computational domain into two blocks. The block 1 is water zone and the block 2 is air zone. The cell number of computational domain could be increased in the block 1 (water zone) and reduced in the block 2 (air zone). The cell number in the block 1 is smoother than the block 2 because in the block 1, the parameters of velocity, pressure and free surface of tidal bore will be calculated. The time step of numerical simulations δt was set in a dictionary file, named controlDict, located in the system directory of OpenFOAM. These two numerical parameters, cell number of computational domain Cn and time step of numerical simulations δt, will be fixed by a parameter study in subsection 3.1.

Flow model

Navier-Stokes equations

The flow model of tidal bores is governed by the Navier-Stokes (NS) equations expressed in a velocity-pressure formulation as given in equations 1 and 2:

∇ • (ρu) = 0, (1)
∂ρu ∂t + ∇ • (ρuφ) -∇ • (µ ef f ∇(ρu)) = -∇ + ρg + F s (2)
where u, P , ρ, φ and g are the flow velocity (m/s), the pressure (P a), the fluid density (kg/m 3 ), the flux field of fluid and the gravity acceleration (m/s 2 ) respectively. µ ef f = µ + µ t is the effective dynamic viscosity (kg/m.s) where µ is the kinematic viscosity and µ t is the turbulent viscosity. F s (N/m) is the source of momentum which corresponds to the surface tension σ that will be described in the following section. The density ρ in the equation 2 is the density of water and air.

The dynamic viscosity and density of fluids, µ and ρ, can be defined as:

µ = αµ water + (1 -α)µ air , (3)
ρ = αρ water + (1 -α)ρ air (4)
where α is the volume fraction of fluid. To express the transport of α was formulated by [START_REF] Weller | Derivation modelling and solution of the conditionally averaged two-phase flow equations[END_REF] in [START_REF] Weller | Derivation modelling and solution of the conditionally averaged two-phase flow equations[END_REF] by introducing an extra term in the phase fraction function. It is called the artificial compression term:

∂α ∂t + ∇ • (αū) + ∇ • [u Γ α(1 -α)] artificial compression term = 0 (5)
where u Γ is the vector of the relative velocity between water and air, also called the compression velocity. ū is the mean velocity calculated by a weighted average of water and air velocities (u w and u a ):

ū = αu w + (1 -α)u a . (6)
The Finite Volume method is used to discretize the Navier-Stokes equations [START_REF] Team | The Open Source CFD Toolbox: Programmer's Guide[END_REF]. The pressure P is estimated at the cells vertices. The components of the flow velocity u are obtained on the cells sides. The Navier-Stokes equations are integrated on each control volume and discretised in terms 120 of physical fluxes through the control volume faces.

In OpenFOAM, the Volume Of Fluid (VOF) method by [START_REF] Hirt | Volume of fluid (vof) method for the dynamics of free boundaries[END_REF] is applied [START_REF] Hirt | Volume of fluid (vof) method for the dynamics of free boundaries[END_REF]. It uses the volume fraction α as an indicator function (in OpenFOAM called alpha) to define the portion of the cell occupied by water. It is defined as:

125    α(x, y, z, t) = 1
for a position (x,y,z) occupied by water at a time t, 0 < α(x, y, z, t) < 1 for a position (x,y,z) in the interface at a time t, α(x, y, z, t) = 0 for a position (x,y,z) occupied by air at a time t.

Surface tension force

The numerical simulation of tidal bores is a multiphase problem where a surface tension appears between air and water. This surface tension is taken into account in the NS equations by means of a surface tension force per unit volume F s (see Eq. 2). Using a volumetric formulation, the volumetric surface tension force is given by [START_REF] Popinet | Numerical models of surface tension[END_REF]:

F s = σκ∇α (8)
where σ and κ are the surface tension (N/m) and the curvature, respectively. The difficulty in estimating the surface tension force F s lies in estimating the curvature κ. In the literature, two families of methods are distinguished to estimate κ, direct derivation from the implicit representation of the interface and discrete differential geometry operators applied to an explicit description of the location of interface [START_REF] Popinet | Numerical models of surface tension[END_REF]. In the first family, the Levelset (LS) methods allow for an accurate estimation of the curvature κ. However the LS methods are found to be non conservative [START_REF] Cassidy | An investigation of interface-sharpening schemes for multiphase mixture flows[END_REF] [START_REF] Tsui | Flux-blending schemes for interface capture in two-fluid flows[END_REF]. Still in this first family, the VOF methods tend to provide an inaccurate estmation of the curvature. But they are conservative [START_REF] Raessi | A volume-of-fluid interfacial flow solver with advected normals[END_REF]. Heyns et al. have implemented in OpenFOAM a method mixed between LS and VOF methods [START_REF] Heyns | Modelling surface tension dominated multiphase flows using the vof approach[END_REF]:

κ = -∇n = -∇ • ∇α |∇α| . ( 9
)
This mixed method is similar to the CSFVOF method of Sussman and Pucket [START_REF] Sussman | A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows[END_REF].

Taking into account the volumetric form of surface tension force (Eq. 8), the final form of NS equations in equations 1 and 2 can be written as:

∇ • (ρu) = 0, ( 10
)
∂ρu ∂t + ∇ • (ρuφ) -∇ • (µ ef f (∇ρu)) = -∇p * + σκ∇α. (11)
The NS equations could be solved together with the constitutive relations for the dynamic viscosity (Eq. 3) and the density of fluid (Eq. 4) by using the Finite Volume Method [START_REF] Versteeg | An Introduction to Computational Fluid Dynamics: the Finite Volume Method[END_REF].

LES turbulence model

Based on previous works of the scientic literature [START_REF] Furuyama | A numerical study of open channel flow hydrodynamics and turbulence of the tidal bore and dam-break flows[END_REF][START_REF] Furuyama | A numerical study of a tidal bore flow[END_REF][START_REF] Lubin | Numerical simulation of a weak breaking tidal bore[END_REF][START_REF] Lubin | Large eddy simulation of turbulence generated by a weak breaking tidal bore[END_REF], we have taken the LES model to simulate the turbulent flows of two tidal bore types, undular and breaking. The first 2D numerical simulations of tidal bores, using the Large Eddy Simulation (LES) model, have been realized by Furuyama andChanson (2008 and[START_REF] Chanson | Tidal bores, aegir and pororoca: the geophysical wonders[END_REF] [START_REF] Furuyama | A numerical study of open channel flow hydrodynamics and turbulence of the tidal bore and dam-break flows[END_REF][START_REF] Furuyama | A numerical study of a tidal bore flow[END_REF]. The lack of mesh grid resolution did not allow to simulate the complete flow dynamic of tidal bores. In 2010, Lubin et al. showed that in the 2D numerical simulations the turbulent LES model is able to simulate correctly the main features of breaking tidal bores in accordance with the experimental results, as the time evolution of the free surface [START_REF] Lubin | Numerical simulation of a weak breaking tidal bore[END_REF][START_REF] Lubin | Large eddy simulation of turbulence generated by a weak breaking tidal bore[END_REF]. However the agreement between numerical and experimental results of velocity profile is only showed at a selective range of vertical elevation beneath the free-surface [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF][START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF]. In 2018, Leng et al. have performed 3D numerical simulations where the numerical boundary layer presents a thickness close to that of experimental results [START_REF] Leng | Cfd modeling of tidal bores: development and validation challenges[END_REF].

Instead of time-averaging, the LES model resolves the large scale motions of flows by solving the spatial-filtered NS equations. In this model, the smaller universal scales, the sub-grid scales (SGS), are modeled. The spatial filtered velocity u of LES model is given by:

u(x) = Ω u(x , t)G(x, x , )ds (12)
where G(x, x , ) is the spatial filter with as the filter size, and Ω is the entire space. In LES, the spatial filtering controls the large eddies to be resolved and removes the small eddies to be modelled. To achieve this, spatial filters should be able to smooth away the small eddies [START_REF] Gong | Comparison of rans and les models in the laminar limit for a flow over a backward-facing step using openfoam[END_REF] [START_REF] Versteeg | An Introduction to Computational Fluid Dynamics: the Finite Volume Method[END_REF].

The three filter types are defined in OpenFOAM, i.e. tophat G 1 , Gaussian G 2 and sharp Fourier cut-off G 3 . They are expressed as:

G 1 = 1/ 3 if |x -x | ≤ /2, 0 otherwise, (13)
G 2 = 6 π 2 3/2 exp -6||x -x || 2 2 , (14)
G 3 = 3 i=1 sin π ||x -x || π||x -x || (15)
where = ( 1 2 3 ) 1/3 with i being the filter width in the i -th spatial coordinate [38] [32]. The tophat G 1 filter (Eq. 13) was used in this work. This is because the tophat G 1 filter is applicable for the finite volume implementation of LES [START_REF] Versteeg | An Introduction to Computational Fluid Dynamics: the Finite Volume Method[END_REF]. While for the filters of Gaussian G 2 (Eq. 14) and sharp Fourier cut-off G 3 (Eq. 15) are implemented for the finite difference implementation of LES. Thus, the spatial filtering defines a new term, the SGS stress tensor τ S ij . Then, the fluctuation velocity could be expressed as:

u i u j = u i u j + u i u j = u i u j -τ S ij . (16)
Furthermore, the spatially-filtered NS equations is written as:

∂u i ∂t + u j ∂u i ∂x j = - 1 ρ ∂p ∂x i + 1 ρ ∂τ ij ∂x j - ∂τ S ij ∂x j . (17)
2.1.4. Froude number 150

The strength and free surface shape of tidal bores are linked to the Froude number F r [START_REF] Chanson | The Hydraulics of Open Channel Flow: An Introduction[END_REF]. Originally, for a channel with an irregular cross-sectional shape and a river flow condition, the Froude number F r is usually defined as [START_REF] Chanson | The Hydraulics of Open Channel Flow: An Introduction[END_REF]:

F r = v 0 g(A/B) (18)
where v 0 is a river flow velocity (m/s), A is the cross-sectional area (m 2 ) and B is the free-surface width (m). For open channel flows, the ratio A/B in equation 18 is relevant with the water depth of river h 1 [START_REF] Chanson | The Hydraulics of Open Channel Flow: An Introduction[END_REF]. So, for a rectangular channel, the Froude number F r is defined as:

F r = v 0 √ gh 1 . ( 19
)
For the tidal bores where the flow velocities of river and tidal bore have opposite directions, the definition of Froude number F r in equation 19 is written as:

F r = v 0 + U b √ gh 1 (20)
with U b is the front velocity of tidal bore (m/s).

Lemoine's, Andersen's and Berry's theories

In our study, the analysis of tidal bore flows is assessed with theories developed by [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF][START_REF] Andersen | Undular hydraulic jump[END_REF][START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF]. These theories are based on approximations of linear wave theory. [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] proposed formulas giving the wavelength L w (m) and the wave amplitude a w (m) for an undular tidal bore that are expressed in equations 21, 22 and 23 respectively:

L w h 1 = π √ 3 (F r -1) -1/2 and a w h 1 = 8 3 √ 3 (F r -1) (Lemoine's theory), (21)
L w h 1 = 1.79 (1 -F r) 0.614 and a w h 1 = 0.741(F r -1) 1.028 (Andersen's theory), (22)
L w h 1 = 0.3426 (r + 2) 3/2 (r -1) and a w h 1 = 0.2331(r -1) (Berry's theory) ( 23
)
where r = h 2 /h 1 is the ratio of water depth of river h 1 (m) to water depth of tidal bore h 2 (m).

The ratio r can be written as a function of the Froude number F r given as [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF]:

r = 1 2 ( 1 + 8F r 2 -1). ( 24
)
Because of this equation, the classification in term of the Froude number is equivalent to the classification in term of r on the assumption of Bélanger hypotheses (hydrostaticity, no friction and 155 rectangular geometry). If the friction is included for example, then r will possibly be a function of the Reynolds number Re in addition to the Froude number F r.

The numerical results of the wave amplitude a w and wavelength L w obtained with OpenFOAM are given by the following equations:

L w = X 1 -X 2 and a w = Y 1 -Y min 2 ( 25
)
where Y min (m) is the lowest height between the front and the first stubble (secondary wave), Y 1 (m) and X 1 (m) are associated to the front elevation, Y 2 (m) and X 2 (m) are associated to the first stubble elevation. All these quantities are shown in Figure 3.

Transport model 2.2.1. Modified Maxey-Riley equations

A transport model was used to simulate the trajectory of non-cohesive sediment particles. The trajectory of each particle can be described by the fundamental dynamics principle proposed by [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] given in equations 26 and 27:

dx i dt = v i (t), (26)
m i dv i dt = V i (ρ p -ρ f ) g + m f D u i Dt - 1 2 m f d dt v i -u i - 1 10 r 2 i ∇ 2 u i -6πr i µ v i -u i - 1 6 r 2 i ∇ 2 u i -6πr 2 i µ t t0 d dτ v i (τ ) -u i (τ ) - 1 6 r 2 i ∇ 2 u i dτ πν(t -τ ) (27)
where x i , v i , m i , V i , ρ i and r i are the position (m), the velocity (m/s), the mass (kg), the volume (m 3 ), the density (kg/m 3 ) and the radius of each particle i respectively at time t (s) to indicates the initial time t 0 (s). Next, m f , u i , ρ f , µ and ν are the mass (kg), the flow velocity (m/s), the density (kg/m 3 ), the dynamic (kg/m.s) and kinematic (m 2 /s) viscosity of fluid respectively. The time discretization of the previous system of equations uses the fourth order Runge-Kutta scheme.

To obtain the trajectory of non-cohesive sediment particles, it is necessary to have the velocity field during the passage of tidal bores. The flow velocity field is estimated with OpenFOAM.

Modified Chen's model

Chen et al. (2012) [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF] have computed the particle trajectories in wave-current interaction for a two-dimensional field. In our work, we have proposed three modifications of Chen's model to obtain an analytical solution allowing to analyse the trajectory of non-cohesive sediment particles under undular tidal bores [START_REF] Berchet | Modélisation par des méthodes lagrangiennes du transport sédimentaire induit par les mascarets[END_REF][START_REF] Berchet | Flow fields and particle trajectories beneath a tidal bore: A numerical study[END_REF]:

1 Gravity: Considering a constant settling velocity w s (m/s), the contribution of gravity in the sediment particle trajectories is given by:

y(t) = -w s (t -t 0 ). ( 28
)
2 Elevation: During the passage of an undular tidal bore, an elevation of the free surface appears simultaneously to a slowing down of the river current. Their effects on the sediment particle trajectories can be collected in the same equation with the changing of the water height from h 1 (m) to h 2 (m) and the current velocity from V 1 to V 2 :

x(t) = 1 2β 1 ((U 1 -U 2 ) log(cosh(-β 1 )t B ) -log(cosh(β 1 (t -t B )))) + β 1 (U 1 + U 2 )t, (29)
y(t) = h 2 -h 1 2 tanh(β 2 (t -t B )) ( 30
)
where t B is the time taken by the sediment particles to go through the jump front (s), β 1 is the parameter controlling the flow velocity during the slowing down between the two states before and after the front, and β 2 is the parameter controlling the velocity of elevation between the two states.

3 Attenuation: The secondary waves (stubbles) amplitude decreases with the distance from the front (see Berry, 2018 [7]). It is probably that the attenuation is a combination of dispersive and dissipative effects to be elucidated. This effect on the sediment particle trajectories can be described by:

x(t) = x chen (t) exp(-β 3 (t)), (31)
y(t) = y chen (t) exp(-β 3 (t)) ( 32
)
where β 3 is the attenuation parameter of the related wave-current interaction problem where each stubble is considered as a water wave on its own. x chen and y chen are the coordinates given by the Chen's model [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF].

Numerical Results

Convergence study

In this convergence study, the undular tidal bore simulation with Froude number F r = 1.10 has been chosen by using the LES turbulence model. The initial velocity and water depth of undular tidal bore were chosen as V 0 = 0.189 m/s and d 0 = 0.2 m respectively. While for the breaking tidal bore simulation, the initial velocity and water depth are such that V 0 = 0.74 m/s and d 0 = 0.14 m respectively with Froude number F r = 1.48. The convergence study of numerical simulations with the time step δt = 0.1, 0.01, 0.001 and 0.0001 s has been studied in this section. We have drawn the time evolution of horizontal component V x of flow velocity at observation point X = (5 m, 0.1 m) in Figure 5. To achieve temporal accuracy and numerical stability, the time step δt must be constrained by the Courant-Friedrichs-Lewy (CFL) condition given by [START_REF] Xu | An improved weakly compressible sph method for simulating free surface flows of viscous and viscoelastic fluids[END_REF]:

Cells number Cn

Time step δt

δt CF L = min δx U , δx 2 ν w ( 33
)
where U is the magnitude of flow velocity and δx is the cell size in the direction x. As the water viscosity ν w is very small, the first criterion on the advection time imposes the limit of time step δt CF L to ensure the stability of numerical simulations. In our test case, the flow velocity of the river v 0 being equal to 0.189 m/s, the limit of time step δt CF L is equal to 0.05 s. For the rest of numerical simulations, the time step δt will be fixed to 0.001 s.

Validation study

To validate our numerical simulations, we have generated two types of tidal bores, undular and breaking. For the case of undular tidal bores in Figures 6 and7, the validation of numerical results obtained with OpenFOAM was achieved with analytical, numerical and experimental results obtained in [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Chanson | Characteristics of undular hydraulic jumps: Experimental apparatus and flow patterns[END_REF] with the Froude number F r = 1.10 compared with the results proposed by [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF] [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF]. For the case of a partially breaking tidal bore, the dimensionless time evolution of the free surface, horizontal and vertical components v x /v 0 and v y /v 0 of flow velocity have been drawn in Figures 8 and9 respectively with Froude number F r = 1.48 compared with the results obtained by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF].

Undular tidal bore case

We analyzed the numerical results of the free surface shape for an undular tidal bore at an abscissa x = 5.15 m given in Figure 6. At this abscissa, we obtained that the free surface shape has an identical trend with the results of the scientific literature [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Chanson | Characteristics of undular hydraulic jumps: Experimental apparatus and flow patterns[END_REF]. The wave train generated using OpenFOAM has a similar trend with the numerical results of the free surface using the Thétis code. The free surface shape in Figure 6 shows the similar pattern to the undular tidal bore of type A given by Berchet et al. (2018) [START_REF] Berchet | Flow fields and particle trajectories beneath a tidal bore: A numerical study[END_REF] where the free surface shape is dominated by undulations without global flow reversal but with local flow reversals. In Figure 7, we also plotted the dimensionless time evolution of horizontal and vertical components, v x /v 0 and v y /v 0 , of flow velocity at a position X =(7.15 m, 0.036 m) for a Froude number F r = 1.10. In this figure, we can observe that the dimensionless time evolution of horizontal and vertical components, v x /v 0 and v y /v 0 , of flow velocity obtained with OpenFOAM shows similar trends with the results of the scientific literature in [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Chanson | Characteristics of undular hydraulic jumps: Experimental apparatus and flow patterns[END_REF]. According to our numerical results in Figure 7, we can observe that the horizontal component v x /v 0 of flow velocity beneath the crest and trough has positive and negative values respectively. The greatest velocity value occurs beneath the first crest. After that, the velocity value decreases and reaches to the lowest velocity value beneath the first trough. The velocity value re-increases beneath the second crest. However it is smaller than the velocity value beneath the first crest. These numerical results are validated by the numerical and experimental results of the scientific literature [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF][START_REF] Chanson | Characteristics of undular hydraulic jumps: Experimental apparatus and flow patterns[END_REF].

Partially breaking tidal bore case

For the case of breaking tidal bores in Figures 8 and9, the validation of numerical results obtained with OpenFOAM was achieved with numerical and experimental results obtained in [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF].

The numerical results of the free surface shape for a breaking tidal bore have been observed to follow the results obtained by Khezri (2013) at an abscissa x = 5 m given in Figure 8. At this abscissa, we obtained that the free surface shape given by OpenFOAM are closer with the results of scientific literature [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF].

(a)

Figure 8: Dimensionless time evolution of the free surface of a partially breaking tidal bore with a Froude number F r = 1.48 at an abscissa x = 5 m using OpenFOAM compared with the results reported by Khezri (2013) [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF]. Physically, for a partially breaking tidal bore, the wave amplitude of the front wave is smaller than an undular tidal bore [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF]. Based on the experimental photograph by Khezri (2013) in [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF], the free surface shape of the front wave for partially a breaking tidal bore is broken and not smooth but still features secondary waves. The free surface shape doesn't have a wave train as in an undular tidal bore. We have also observed that the free surface shape of a partially breaking tidal bore obtained with the OpenFOAM is similar to the numerical results obtained by Lubin et al. (2010) in [START_REF] Lubin | Large eddy simulation of turbulence generated by a weak breaking tidal bore[END_REF].

For a partially breaking tidal bore, the dimensionless horizontal and vertical components, v x /v 0 and v y /v 0 , of flow velocity were plotted in Figure 9 at a position X = (5m, 0.04m) (in the water column) by following the work by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] with a Froude number F r = 1.48 using Open-FOAM. According to Figure 9, we can observe that OpenFOAM is close to experimental and numerical given by [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF]. As with experimental and numerical given by Khezri, OpenFOAM produces negative-positive velocity values beneath a partially breaking tidal bore. OpenFOAM gives a numerical simulation similar to the type C of Berchet's paper (2018) where the flow velocity is positive v > 0 beneath the crest and negative v < 0 beneath the trough.

Classification

Figure 10 shows the dimensionless values of conjugate water depth r = h 2 /h 1 as a function of the Froude number F r. In this figure, some reference data are taken from the various papers [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF][START_REF] Benet | Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels[END_REF][START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF][START_REF] Koch | An experimental study of tidal bores and positive surges: hydrodynamics and turbulence of the bore front[END_REF][START_REF] Chanson | Flow field in a tidal bore: a physical model[END_REF][START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF]. Equation 24 as a momentum principle was plotted in this figure. The numerical results of OpenFOAM are in good agreement with the momentum principle and some reference data of the scientific literature. From Figure 10, we confirm that the conjugate depth ratio r = h 2 /h 1 increases when the Froude number F r increases. The dimensionless wavelength L w /h 1 was plotted as a function of the Froude number F r given in Figure 11 (a). The numerical results of OpenFOAM were compared with the theoretical results given by [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and Berry (2018) in equations 21, 22 and 23 respectively [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF][START_REF] Andersen | Undular hydraulic jump[END_REF][START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF]. We also compared the numerical results of OpenFOAM with the experimental results given by Favre (1935), [START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF] and Tissier [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF][START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF][START_REF] Tissier | étude numérique de la transformation des vagues en zone littorale, de la zone de levée aux zones de surf et de jet de river[END_REF]. In Figure 11 (a), we can observe that the numerical results of OpenFOAM follow the trend described in the scientific literature. According to this figure, we obtain that the wavelength L w decreases when the Froude number F r increases.

The dimensionless wave amplitude A w /h 1 was plotted as a function of the Froude number F r given in Figure 11 (b). The numerical results of OpenFOAM were compared with the theoretical results given by [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF], [START_REF] Andersen | Undular hydraulic jump[END_REF] and [START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF] in equations 21, 22 and 23 respectively [START_REF] Lemoine | Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude[END_REF][START_REF] Andersen | Undular hydraulic jump[END_REF][START_REF] Berry | Minimal analytical model for undular tidal bore profile; quantum and hawking effect analogies[END_REF]. We also compared the numerical results of OpenFOAM with the experimental results given by [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF], [START_REF] Benet | Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels[END_REF], [START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF], [START_REF] Koch | Turbulence measurements in positive surges and bores[END_REF], [START_REF] Chanson | Tidal bores, aegir and pororoca: the geophysical wonders[END_REF], [START_REF] Tissier | étude numérique de la transformation des vagues en zone littorale, de la zone de levée aux zones de surf et de jet de river[END_REF] and [START_REF] Docherty | Physical modeling of unsteady turbulence in breaking tidal bores[END_REF] [START_REF] Favre | Etude Théorique et Expérimentale des Ondes de Translation dans les Canaux Découverts (Theoretical and Experimental Study of Travelling Surges in Open Channels)[END_REF][START_REF] Benet | Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels[END_REF][START_REF] Treske | Undular bores (favre-waves) in open channels-experimental studies[END_REF][START_REF] Koch | Turbulence measurements in positive surges and bores[END_REF][START_REF] Chanson | Tidal bores, aegir and pororoca: the geophysical wonders[END_REF][START_REF] Tissier | étude numérique de la transformation des vagues en zone littorale, de la zone de levée aux zones de surf et de jet de river[END_REF][START_REF] Docherty | Physical modeling of unsteady turbulence in breaking tidal bores[END_REF]. The numerical results of OpenFOAM were also compared to the field results given by Simpson at al.

(2004) at Dee river, United Kingdom [START_REF] Simpson | Reynolds stress and tke production in an estuary with a tidal bore[END_REF]. The numerical results of OpenFOAM were compared to the numerical results given by Simon (2013) using the Thétis code [START_REF] Simon | Effects of tidal bores on turbulent mixing : a numerical and physical study in positive surges[END_REF]. According to these comparisons, we obtain that the numerical results of OpenFOAM follow the trend of the scientific literature. We observed that the ratio of wave amplitude a w /h 1 is even greater when the tidal bore is undular. Then, when the tidal bore turns into a breaking one, the wave amplitude ratio a w /h 1 of the tidal bore decreases. Figure 11 (b) shows that the transition between a total undular tidal bore and a partial breaking tidal bore is a Froude number F r 2 = 1.43, and the transition between a partial breaking tidal bore and a total breaking tidal bore is a Froude number F r 3 = 1.57. For Froude number F r = 1, the wave amplitude a w of the tidal bore is difficult to be observed as the free surface of the tidal bore has similarities with the surface of river flow. To confirm the values of F r 1 , F r 2 and F r 3 in Figure 11 (b), the variation rate of the elevation of free surface over the time, dh/dt, is studied. In Figure 12, the variation rate dh/dt was drawn for the four conditions of tidal bore: the small wave of tidal bore with the Froude number F r = 1 (Fig. 12 (a)), the total undular tidal bore with Froude number F r = 1.35 (Fig. 12 (b)), the partial breaking tidal bore with F r = 1.43 (Fig. 12 (c)) and the total breaking tidal bore with F r = 1.57 300 (Fig. 12 (d)). The variation rate dh/dt increases between the Froude number F r = 1 to 1.43 and when the Froude number F r = 1.57, the variation rate dh/dt decreases with the appearance of strong oscillations. To confirm this analysis, we have referred to the work provided by Furgerot (2014) in [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF] that defined the type of tidal bore based on the Froude number F r. She has observed that for F r < 1 there is no tidal bore, 1 < F r < 1.5 is the undular tidal bore, 1.5 < F r < 1.6 is the transtition of the tidal bore and F r > 1.6 is a breaking tidal bore. Based on the observation of the variation rate of the elevation of free surface dh/dt and Furgerot's classification, we have proposed numerically that for Froude number 1 < F r < 1.43 the tidal bore is undular, 1.43 < F r < 1.57 the tidal bore is partially breaking that is similar with the tidal bore transtition and F r > 1.57 the tidal bore is totally breaking. In the case of undular tidal bores, the trajectory of non-cohesive sediment particles were estimated to etablish the relationship between the parameters of modified Chen's model, β 1 , β 2 and β 3 , and the Froude number F r. Figure 13 (a) shows the trajectory of a non-cohesive sediment particle with or without the effect of gravity. By subtracting the contribution of the settling velocity w s to the complete trajectory, the new trajectory is not affected by gravity. Figure 13 (b) shows the trajectory of a non-cohesive sediment particle obtained with the modified Chen's model considering the effect of elevation. Figure 13 (c) shows the trajectory of a non-cohesive sediment particle obtained with the modified Chen's model considering all the mechanisms, gravity, elevation and attenuation. According to Figure 13, we observe that the three effects of gravity, elevation and attenuation influence the trajectory of non-cohesive sediment particles benath an undular tidal bore. From Figure 13 (a), we can observe that a trajectory of non-cohesive sediment particles influenced by gravity (blue curve) decreases with increasing the time t. While for the trajectory of non-cohesive sediment particles without gravity (red curve) has a constant value with increasing the time t.

y = 17.44x -17.49 with R 2 = 0.99 for β 2 , (34)
y = 2.74x -3.13 with R 2 = 0.99 for β 3 . (35)
The least squares fits were plotted in Figures 14 (a), (b) and (c). The relationships between the three parameters, β 1 , β 2 and β 3 , and the Froude number F r are linear. It may be explained by the low level of turbulence beneath an undular tidal bores. The flow induced by undular tidal bores is not complex. This physical phenomenon is quasi-linear. Thus the theory of wave-current interactions is applicable. Then, the Chen's model can be adapted to the study of undular tidal 330 bores. The parameter β 1 decreases when the Froude number F r increases (see Fig. 14 (a)). The decrease of β 1 means a decrease of the velocity of the tidal bore front. The passage from undular tidal bores to breaking tidal bores induces a slowing of the passage of tidal bores. The two other parameters, β 2 and β 3 , increase with the Froude number F r (see Fig. 14 (b) and (c)). The parameter β 2 simulates the free surface elevation generated by the undular tidal bores passage. The increasing 335 of Froude number F r induces an increase of this elevation. The flow of tidal bores, being more turbulent, has more energy for producing a more important elevation of free surface. The parameter β 3 simulates the attenuation of the wave train following the tidal bore front. The increase of Froude number F r induces an increase of this attenuation. The wave train disappears progressively with the Froude number F r. The breaking tidal bores don't feature this wave train.

Conclusions

Using the 2D numerical simulations of tidal bores generated by OpenFOAM, we studied the hydrodynamics of two tidal bores types: undular and breaking. The undular tidal bore type of this study is similar to the tidal bore of type A proposed by Berchet et al. (2018) [START_REF] Berchet | Flow fields and particle trajectories beneath a tidal bore: A numerical study[END_REF]. We defined three transitions of tidal bores based on the Froude number F r. For a Froude number 1 < F r < 1.43

(1 < r < 1.57), the tidal bore is undular. For a Froude number 1.43 < F r < 1.57 (1.57 < r < 1.75), the tidal bore is partially breaking, that is similar to the tidal bore transition defined by Furgerot (2014) [START_REF] Furgerot | Propriétés hydrodynamiques du mascaret et de son influence sur la dynamique sé dimentaire. une approche couplée en canal et in situ (estuaire de la sée, baie du mont saint-michel)[END_REF]. And for a Froude number F r > 1.57 (r > 1.75), the tidal bore is totally breaking. By using LES turbulence model of OpenFOAM, we have access to the velocity in the water column and the free surface to classify the tidal bore in relation to the free surface and parameterize the modified Chen's model.

The impact of tidal bores on the transport of non-cohesive sediment particles was studied in our work. For the non-cohesive sediment particles, we observed that the trajectory using the flow generated by OpenFOAM is similar to the typical trajectory proposed by Chen et al. (2010) [START_REF] Chen | Particle trajectories beneath wave-current interaction in a two-dimensional field[END_REF].

The modifications of Chen's model were done by including the effects of gravity, elevation and attenuation to reproduce non-cohesive particle trajectories under an undular tidal bore. We obtained that the relationships between the parameters of modified Chen's model, β 1 , β 2 and β 3 , and the Froude number F r are linear. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter β 1 related to the front celerity of tidal bores, decreases when the Froude number F r increases. We can say that the passage from undular to breaking tidal bores induces a deceleration of tidal bores. The parameter β 2 related to the elevation of tidal bores, increases when the Froude number F r increases. The tidal bore flow, being more turbulent, has more energy to produce a more important elevation of the free surface. The parameter β 3 , related to the tidal bores attenuation, increases when the Froude number F r increases. The wave train of undular tidal bores disappears progressively while the Froude number F r increases. The breaking tidal bores don't present this wave train. The energy associated to this wave train is dissipated by the flow turbulence.

As a final remark, we have chosen following the literature to express our results as a function of the Froude number F r only. For the sake of completeness, we add the range of the r values. Since both parameters are related by the Bélanger equation 24, to choose r or Fr is just a matter of convenience. It seems to us that r is more logical when speaking of the transition to wave breaking instead of the Fr. Bélanger equation should be generalized in order to take into account the potential non-hydrostaticity as well as the structure of the boundary layer as it was done to integrate the effect of the river geometry [START_REF] Chanson | The Hydraulics of Open Channel Flow: An Introduction[END_REF]. Hence, we guess that a future and broader classification of bores taking into account both the free surface (this work) and what happens beneath it (Berchet et al., 2018 [15]), should feature not only the Froude number F r but others dimensionless numbers like r (encoding wave breaking), the Reynolds number Re (encoding the friction), geometrical parameters like the slope of the banks (encoding curved bore front), etc. This will be our future line of investigation.

Figure 2 :

 2 Figure 2: Sketch of computational domain used for the 2D numerical simulations of tidal bores.

Figure 3 :

 3 Figure 3: Definition of variables Y 1 , Y 2 , X 1 , X 2 and Y min used in equation 25 to estimate the wave amplitude aw and the wavelength Lw.

Figure 4 :

 4 Figure 4: Horizontal component Vx of flow velocity beneath an undular tidal bore with F r = 1.10 at X = (5 m, 0.1 m) using the LES model of OpenFOAM for various values of cell number Cn.

Figure 5 :

 5 Figure 5: Horizontal component Vx of flow velocity beneath an undular tidal bore with F r = 1.10 at X = (5 m, 0.1 m) using the LES model of OpenFOAM for various values of time step δt.

Figure 6 :Figure 7 :

 67 Figure 6: Dimensionless time evolution of the free surface of an undular tidal bore with a Froude number F r = 1.10 at an abscissa x = 5.15 m compared with the results proposed by Simon (2013) [14].

Figure 9 :

 9 Figure 9: Dimensionless time evolution of the horizontal and vertical components vx/v 0 and vy/v 0 of flow velocity for a partially breaking tidal bore with a Froude number F r = 1.48 at a position X = (5 m, 0.04 m)compared with the results reported by[START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF] [START_REF] Khezri | Modelling turbulent mixing and sediment process beneath tidal bores: physical and numerical investigations[END_REF].

Figure 10 :

 10 Figure 10: Dimensionless variable h 2 /h1 as a function of the Froude number F r.

Figure 11 :Figure 12 :

 1112 Figure 11: Dimensionless wavelength Lw/h 1 (a) and wave amplitude aw/h 1 (b) as a function of the Froude number F r.

Figure 13 :

 13 Figure 13: Trajectories of a non-cohesive sediment particle induced by three effects: (a) gravity, (b) elevation and (c) attenuation.

Figure 14 :

 14 Figure 14: Parameters of modified Chen's model, β 1 (a), β 2 (b) and β 3 (c), as functions of the Froude number F r in the case of undular tidal bores (the solid line is a fitting linear function).

Figures 14 (

 14 Figures 14 (a), (b) and (c) show the numerical values of the three parameters of the modified Chens's model, β 1 , β 2 and β 3 , as functions of Froude number F r. The standard method of least squares was used in order to obtain the best fitting of curves of three parameters, β 1 , β 2 and β 3 : y = -0.41x + 0.78 with R 2 = 0.98 for β 1 ,(34)

Acknowlegments

This work is a collaboration between the University of Tanjungpura (Untan), West Kalimantan, Indonesia and the Institute Pprime, University of Poitiers, France with the financing fully covered by the Ministry of Research, Technology and Higher Education of the Republic of Indonesia (Ristekdikti). We would like to thank Michael Berry for enligthering discussions on his model.


N Zory

Roche

Maxime Geiger
email: maxime.geiger@gmail.com

Minimal detectable change of kinematic and spatiotemporal parameters in patients with chronic stroke across three sessions of gait analysis

Keywords: stroke, gait analysis, reproducibility, minimum detectable change

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Stroke-related hemiparesis alters the gait pattern [START_REF] Pélissier | Analyse instrumentale de la marche de l'hémiplégique adulte: revue de la littérature[END_REF][START_REF] Pinzur | Gait changes in adult onset hemiplegia[END_REF], and medical and surgical treatments and rehabilitation often focus on improving gait [START_REF] Bleyenheuft | The effect of botulinum toxin injections on gait control in spastic stroke patients presenting with a stiff-knee gait[END_REF][START_REF] Boudarham | Effects of quadriceps muscle fatigue on stiff-knee gait in patients with hemiparesis[END_REF][START_REF] Flansbjer | Knee muscle strength, gait performance, and perceived participation after stroke[END_REF][START_REF] Roche | Effect of rehabilitation and botulinum toxin injection on gait in chronic stroke patients: a randomized controlled study[END_REF]. The effects of treatment can be measured using threedimensional gait analysis (3D-gait analysis) [START_REF] Bleyenheuft | The effect of botulinum toxin injections on gait control in spastic stroke patients presenting with a stiff-knee gait[END_REF][START_REF] Pittock | A double-blind randomised placebo-controlled evaluation of three doses of botulinum toxin type A (Dysport) in the treatment of spastic equinovarus deformity after stroke[END_REF][START_REF] Roche | Effect of rehabilitation and botulinum toxin injection on gait in chronic stroke patients: a randomized controlled study[END_REF], the gold standard for gait assessment. 3D-gait analysis systems provide precise measurements of spatiotemporal, kinematic and kinetic gait parameters [START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF]. The reliability of these system has been shown to be good in patients with stroke.

Reliability is high for kinematic parameters in the sagittal plane [START_REF] Kadaba | Repeatability of kinematic, kinetic, and electromyographic data in normal adult gait[END_REF][START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF][START_REF] Schwartz | Measurement and management of errors in quantitative gait data[END_REF]) and for spatiotemporal parameters [START_REF] Oken | Repeatability and variation of quantitative gait data in subgroups of patients with stroke[END_REF][START_REF] Yavuzer | Repeatability of lower limb three-dimensional kinematics in patients with stroke[END_REF], the reliability has been mainly assessed using the intraclass coefficient correlation (ICC) and the standard error measurement (SEM).

However, another very important psychometric parameter to consider is the minimal detectable change (MDC). MDC relates to measurement bias. If the change following treatment is greater than the MDC, it is therefore likely due to the treatment. [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF].

MDC values have been calculated for certain parameters evaluated during 3D-gait analysis in patients with stroke, including gait profile score and gait deviation index [START_REF] Correa | Reliability and Minimum Detectable Change of the Gait Deviation Index (GDI) in poststroke patients[END_REF][START_REF] Devetak | Reliability and minimum detectable change of the gait profile score for post-stroke patients[END_REF], ground reaction forces [START_REF] Campanini | Reliability, smallest real difference and concurrent validity of indices computed from GRF components in gait of stroke patients[END_REF] and kinematic (such as peak ankle angle during swing or peak knee flexion during swing) and spatiotemporal (such as step length) parameters and ground reaction forces during treadmill gait [START_REF] Kesar | Minimal detectable change for gait variables collected during treadmill walking in individuals post-stroke[END_REF].

However, treatment often aims to improve kinematic parameters during a specific phase of the gait cycle to improve locomotion of patients with stroke (i.e botulinum toxin injection (BTI) in the rectus femoris muscle to increase peak knee flexion in swing phase, or surgery to the triceps surae to improve ankle dorsiflexion in swing phase). It is thus important to determine the MDC for specific kinematic parameters during stance and swing in patients with chronic stroke-related hemiparesis to better analyses results from studied using gait analysis to assess effect of a given treatment in a specific gait phase.. Furthermore, hemiparetic gait is characterized by an asymmetry between duration of swing and stance phase, knowing the MDC of these parameters could also be relevant to better analyses results of studied aiming to restore the symmetry of phases [START_REF] Bohannon | Gait performance of hemiparetic stroke patients: selected variables[END_REF][START_REF] Pinzur | Gait changes in adult onset hemiplegia[END_REF][START_REF] Sheffler | Hemiparetic Gait[END_REF].

The effect of treatment is often evaluated over three 3D-gait analyses: baseline, mid-treatment and end of treatment, or baseline, end of treatment and after a wash out period to determine if the effect is maintained. It is therefore important to assess the MDC over three sessions.

The aim of this study was thus to define the MDC for hip, knee and ankle angles in the sagittal plane during both stance and swing and the MDC of spatiotemporal parameters and to assess changes in MDC across three repeated 3D-gait analyses. To that end, the kinematic and spatiotemporal parameters of patients with chronic stroke-related hemiparesis were compared over three 3D-gait analyses performed at 7-day intervals. We hypothesized that performance would be most variable during the first 3D-gait analysis, inducing a larger MDC than during the second and third analyses. If this was confirmed, it would imply that an initial 3D-gait analysis should be carried out simply for familiarization purposes prior to the data collection.

Methods

Sample

Twenty-six -patients with chronic stroke were included in the study (n=19 men; mean age 58.2±13.1 years; mean time since stroke 9.7±7.1 years; n=14 right hemiparesis). They were recruited during routine follow-up visits in the physical medicine and rehabilitation department of a university teaching hospital. Subjects were eligible for inclusion if they: i) were over eighteen years old, ii) had hemiparesis due to a single stroke more than six month previously, and iii) were able to walk 10 minutes independently with or without walking aids. Subjects were excluded if they had: i) bilateral cortical lesions, ii) cerebellar syndrome, iii) severe comprehensive deficit or severe aphasia, iv) apraxia or v) musculoskeletal surgery less than six months ago. All subjects gave written informed consent before participation. The study was performed in accordance with the ethical codes of the World Medical Association (Declaration of Helsinki) and was approved by the local Ethics Committee.

Study design

Each subject participated in 3 visits at 7-day intervals (V1, V2 and V3). One 3D-gait analysis was carried out at each visit.

Gait analysis

At least three trials were performed before the beginning of the recording in order to familiarise the patient with the experimental conditions, since it has been shown that the first three trials of a gait analysis session differ from subsequent trials (Boudarham, Roche, et al., 2013). A minimum of 4 trials were then recorded at a spontaneous walking speed in a 10-meter-long corridor. To balance the influence of each visit, the same number of gait cycles was analysed for each subject (based on the visit with the least number of gait cycles). We manually deleted the gait cycles in excess. Therefore, the number of gait cycles analyzed varied across subjects (from 5 to 18). Seven optoelectronic cameras (Motion Analysis Corporation, Santa Rosa, CA, USA, 100 Hz sampling frequency) recorded the trajectories of 30 reflective markers, positioned on the skin of the subjects according to the Helene Hayes model. The trajectories were manually processed using Cortex 1.3 and OrthoTrack 6.5 software (Motion Analysis Corporation, Santa Rosa, CA, USA) to extrapolate joint kinematics and spatiotemporal parameters.

The same operator positioned all the markers for each patient at each visit and carried out the whole 3D-gait analysis in order to limit extrinsic variability [START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF].

Data Analysis

3D gait analysis

Kinematic data in the sagittal plane (peak hip and knee flexion and extension, ankle dorsiflexion and plantarflexion and total range of motion (RoM) during the stance and swing phases) were calculated for the paretic lower limb. Spatiotemporal gait parameters (gait speed, cadence, stride length, step length, step width and duration of stance phase) were also calculated. Values for each parameter were averaged across all the gait cycles for each patient and the mean was used for the analysis.

Analysis of raw differences, reliability and minimal detectable change

Mean values of the kinematic and spatiotemporal parameters were compared to determine if there were significant differences between each visit. Reproducibility of measures is often evaluated by comparing two datasets with different indices. The most common and relevant indices are the ICC and the SEM, the MDC is then derived from the SEM. ICC is useful to observe the correspondence between two measures, however it is dependent on the standard deviation in the sample [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF].

To give more weight to the reliability of the data assessed by the ICC, it is necessary to concomitantly evaluate the SEM and the MDC. Low SEM and MDC values signify that changes observed are likely due to the treatment and not the measurement [START_REF] Correa | Reliability and Minimum Detectable Change of the Gait Deviation Index (GDI) in poststroke patients[END_REF][START_REF] Devetak | Reliability and minimum detectable change of the gait profile score for post-stroke patients[END_REF][START_REF] Flansbjer | Reliability of gait performance tests in men and women with hemiparesis after stroke[END_REF][START_REF] Kadaba | Repeatability of kinematic, kinetic, and electromyographic data in normal adult gait[END_REF][START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF][START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF].

Reliability was thus evaluated using two methods, the ICC and the SEM. ICC was calculated using custom software written by Arash Salarian (Copyright 2016) in MATLAB (Mathworks). The ICC 3,k , was used because the data were acquired over three sessions by the same operator and the trials within each session were averaged [START_REF] Shrout | Intraclass correlations: uses in assessing rater reliability[END_REF][START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF]. The formulae used to calculate the ICC was:

-Where MS S is the subjects mean square and MS E is the error mean square of the 2-way ANOVA used to compare the data [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF]. The ICC varies between 0 and 1; the higher the ICC 3,k the higher the reproducibility. There is no real consensus regarding "cut-off" values, however, it has been suggested that values <0.59 reflect "poor reproducibility", 0.60-0.79 "moderate reproducibility" and > 0.80 "high reproducibility" [START_REF] Bushnell | Retrospective assessment of initial stroke severity: comparison of the NIH Stroke Scale and the Canadian Neurological Scale[END_REF]. The SEM can be calculated using two formulae : = √1 -, or SEM= , MS E being the square root of the within-subjects error of the repeated measures ANOVA [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF]. was chosen because the ICC depends on the standard deviation of the data, thus for two sets of data with the same means and two different standard deviations, data with a low standard deviation will have a lower ICC than data with a large standard deviation. takes this into account [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF].

The MDC was calculated using the following equation: = * 2.056 * √2. The value 2.056 corresponds to the student-t distribution with a 95% confidence interval for the study sample size (n=26) [START_REF] Beckerman | Smallest real difference, a link between reproducibility and responsiveness[END_REF]. SEM and MDC are expressed in raw units and not in percentages to facilitate future comparisons with other studies. However, the mean MDC and SEM values for the spatiotemporal parameters are presented as percentages in order to compare parameters with different units [START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF]. The MDC% was calculated using the following formula: % = * 100, being the mean value of the parameter compared between 2 sessions (either V1/V2 or V1/V3 or V2/V3) [START_REF] Flansbjer | Reliability of gait performance tests in men and women with hemiparesis after stroke[END_REF]. The SEM% was calculated using the same technique.

Statistical analysis

The statistical analysis was based on the recommendations of [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF] [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF]. If the data followed a normal distribution (according to the Shapiro-Wilk test), a one-factor (time), repeated measures ANOVA was performed (sessions V1, V2 and V3). If the results of the repeated measures ANOVA were significant, a post hoc Tukey HSD test was performed to determine which visits differed: (V1vsV2 or V1vsV3 or V2vsV3). If the data did not follow a normal distribution, a Friedman test was performed. Significant differences were analysed using a Wilcoxon test to identify which visits differed significantly from each other. Simultaneously, the ICC 3,k , SEM and MDC were calculated for each parameter between V1 and V2, V1 and V3; V2 and V3.

3 Results

Kinematics

Table 1 shows the kinematic values (mean and SD) for each parameter studied during each visit. The details of the repeated measures ANOVAs are available in S1 appendix Table A. --Table 1--

The only parameter that changed significantly was hip RoM in swing, which increased significantly (p=0.045) from V1 to V2 (Tables 1 andtable A).

ICC and SEM

The mean ICC values were high for all parameters between all visits (from 0.96 to 0.97). The reliability was higher between V2 and V3 than between the other visits as showed by higher ICC (0.97) and lower SEM (2.01°) in V2vsV3 than in V1vsV2 and V1vsV3 (see Table 2).

MDC Values

The mean MDC was lowest between V2vsV3 (5.86°) than in V1vsV2 (7.20°) and V1vsV3 (6.48°) (Table 2).

Spatiotemporal parameters

Table 3 presents the values (means and SD) of spatiotemporal parameters during the three visits. All the results of the ANOVAs are available in appendix S1 Table A.

Table 3: Spatiotemporal parameters with mean and SD at each visit.

--Table3--There were no significant differences between the values of the spatiotemporal parameters for the three visits.

ICC and SEM

Mean ICC values were high for all parameters between all visits (0.96 to 0.97). The reliability was higher between V2 and V3 than between the other visits as showed by higher ICC (0.97) and lower %SEM (3.90%) in V2vsV3 than in V1vsV2 and V1vsV3 (see Table 4).

MDC values

The mean %MDC was lowest between V2vsV3 (11.45%) than in V1vsV2 (13.75%) and V1vsV3

(13.10%) (Table 4). --Table 4--

Discussion

The purpose of this study was to determine the MDC for paretic-limb kinematic and spatiotemporal parameters in both swing and stance phase, evaluated during 3D gait analysis, in order to improve interpretation of the results of treatments in patients with hemiparesis following stroke. Reliability was higher between the second and third visits than between the first and second and first and third visits:

ICC values were higher and mean SEM and MDC values were lower. However, the differences in reliability across sessions were small, likely because of the lack of differences in kinematic values across sessions that resulted in high ICC values. This result is in accordance with previous studies that have also found a high reliability of raw kinematic data in a similar population of patients [START_REF] Correa | Reliability and Minimum Detectable Change of the Gait Deviation Index (GDI) in poststroke patients[END_REF][START_REF] Devetak | Reliability and minimum detectable change of the gait profile score for post-stroke patients[END_REF].

The aim of treatment is often to improve peak hip extension in stance, and peak knee and ankle flexion in the swing phase of the gait cycle [START_REF] Pomeroy | Electrostimulation for promoting recovery of movement or functional ability after stroke[END_REF]. The MDC values for these three parameters were lowest between V2 and V3. There was a difference of 1° to 3° in the MDCs depending on the visits compared. This is highly relevant considering the small, but statistically significant changes reported in the literature following various treatments [START_REF] Bonnyaud | Effects of gait training using a robotic constraint (Lokomat®) on gait kinematics and kinetics in chronic stroke patients[END_REF][START_REF] Boudarham | Changes in electromyographic activity after botulinum toxin injection of the rectus femoris in patients with hemiparesis walking with a stiff-knee gait[END_REF][START_REF] Novak | Gait changes following botulinum toxin A treatment in stroke[END_REF][START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF][START_REF] Robertson | Relevance of botulinum toxin injection and nerve block of rectus femoris to kinematic and functional parameters of stiff knee gait in hemiplegic adults[END_REF][START_REF] Roche | Use of gait parameters to predict the effectiveness of botulinum toxin injection in the spastic rectus femoris muscle of stroke patients with stiff knee gait[END_REF]. Some changes reported in studies of patients with stroke are close to, or sometimes lower than, the smallest MDC defined in the present study. Roche et al, (2015) found a significant improvement in peak hip flexion during swing of 4.2° (MDC=9.1°) after BTI in the rectus femoris [START_REF] Roche | Use of gait parameters to predict the effectiveness of botulinum toxin injection in the spastic rectus femoris muscle of stroke patients with stiff knee gait[END_REF]; Roche et al, (2015), [START_REF] Robertson | Relevance of botulinum toxin injection and nerve block of rectus femoris to kinematic and functional parameters of stiff knee gait in hemiplegic adults[END_REF] and Boudarham et al, (2013) found significant improvements in peak knee flexion in swing of respectively 6.5°, 8° and 3.4° (MDC=6.5°) after BTI in the rectus femoris [START_REF] Boudarham | Changes in electromyographic activity after botulinum toxin injection of the rectus femoris in patients with hemiparesis walking with a stiff-knee gait[END_REF][START_REF] Robertson | Relevance of botulinum toxin injection and nerve block of rectus femoris to kinematic and functional parameters of stiff knee gait in hemiplegic adults[END_REF][START_REF] Roche | Use of gait parameters to predict the effectiveness of botulinum toxin injection in the spastic rectus femoris muscle of stroke patients with stiff knee gait[END_REF]. [START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF] found a significant increase in peak ankle dorsiflexion in stance of 4° (MDC=5.4°) and a significant increase in peak knee flexion in swing of 6° to 10° (MDC=6.5°) after BTI in the triceps surae [START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF]. [START_REF] Novak | Gait changes following botulinum toxin A treatment in stroke[END_REF] found a significant decrease in plantarflexion at the end of the swing phase of 2.5° (MDC=5.8°) after BTI in the triceps surae [START_REF] Novak | Gait changes following botulinum toxin A treatment in stroke[END_REF]. [START_REF] Bonnyaud | Effects of gait training using a robotic constraint (Lokomat®) on gait kinematics and kinetics in chronic stroke patients[END_REF] found a significant increase in peak knee flexion during swing of 2.5° (MDC=6.5°) after a single Lokomat session [START_REF] Bonnyaud | Effects of gait training using a robotic constraint (Lokomat®) on gait kinematics and kinetics in chronic stroke patients[END_REF]. Thus, it is not possible to know how much the change relates to the treatment and how much to the repetition of the measurements. Treatment also aims to improve gait speed, stride length and step length. The MDCs for the former two parameters were lowest between the second and third visits, while the MDC for step length was lowest between the first and third visits. The changes in spatiotemporal parameters reported in patients with stroke following treatments are often close to or below the MDC values found in this study. [START_REF] Wallard | Effects of robotic gait rehabilitation on biomechanical parameters in the chronic hemiplegic patients[END_REF] found an increase in gait speed of 35 cm/s (MDC=14.6 cm/s) and a 2 cm increase step length (MDC=6.3 cm) after 20 intensive sessions of lokomat training [START_REF] Wallard | Effects of robotic gait rehabilitation on biomechanical parameters in the chronic hemiplegic patients[END_REF]. After a single session of Lokomat training, Bonnyaud et al, (2014) found a significant increase in gait speed of 5.4 cm/s (MDC=14.6 cm/s), of 3.1 step/min in cadence (MDC=8.6 step/min) and 3.6cm in step length (MDC=6.3 cm). [START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF] found a significant increase in gait velocity of 17 cm/s (MDC=14.6 cm/s), and a 7 cm increase in step length (MDC=6.3 cm) and 15 cm in stride length (MDC=11.9 cm) after BTI in the triceps surae [START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF].

Thus, statistically significant improvements in kinematic and spatiotemporal parameters are often close to, or even below, the lowest MDC values found in the present study. The purpose of these comparisons is not to diminish the scientific impact of these studies, far from it. We believe that our study supports the need to decrease CMD values in studies using the 3D gait-analysis to evaluate the effects of a treatment. This familiarization visit seems important to reduce as possible the MDC values, but comparisons are only possible between populations with similar characteristics such as walking speed, thus this does not apply to all studies, so these comparisons should be used sparingly.

Indeed, in our study the average gait speed of patients was 0.77 m/s; in Roche et al., (2015) it was 0.58 m/s, in [START_REF] Robertson | Relevance of botulinum toxin injection and nerve block of rectus femoris to kinematic and functional parameters of stiff knee gait in hemiplegic adults[END_REF] it was 0.52 m/s, in Boudarham et al., (2013) it was 0.61 m/s, in [START_REF] Pradon | A pilot study to investigate the combined use of Botulinum toxin type-a and ankle foot orthosis for the treatment of spastic foot in chronic hemiplegic patients[END_REF] it was 0.55 m/s, in [START_REF] Novak | Gait changes following botulinum toxin A treatment in stroke[END_REF] it was 0.50 m/s, in [START_REF] Bonnyaud | Effects of gait training using a robotic constraint (Lokomat®) on gait kinematics and kinetics in chronic stroke patients[END_REF] it was 0.76 m/s, in [START_REF] Wallard | Effects of robotic gait rehabilitation on biomechanical parameters in the chronic hemiplegic patients[END_REF] it was 0.84m/s.

In contrast with our hypothesis, kinematic data were very reliable across visits as shown by the

ANOVAs results (only one parameter shown a significant modification between V1 and V2) and ICC values (0.95 to 0.97). This is in accordance with the literature [START_REF] Awad | Effects of repeated treadmill testing and electrical stimulation on post-stroke gait kinematics[END_REF][START_REF] Correa | Reliability and Minimum Detectable Change of the Gait Deviation Index (GDI) in poststroke patients[END_REF][START_REF] Devetak | Reliability and minimum detectable change of the gait profile score for post-stroke patients[END_REF][START_REF] Kadaba | Repeatability of kinematic, kinetic, and electromyographic data in normal adult gait[END_REF][START_REF] Mcginley | The reliability of three-dimensional kinematic gait measurements: a systematic review[END_REF].

Only hip RoM during swing differed between visits one and two. Spatiotemporal data were also highly reliable across visits as shown by the ANOVAs results (none significant ANOVAs) and ICC values (0.96 to 0.97). This is also in agreement with the literature [START_REF] Cho | Test-retest reliability of the GAITRite walkway system for the spatio-temporal gait parameters while dual-tasking in post-stroke patients[END_REF][START_REF] Kesar | Minimal detectable change for gait variables collected during treadmill walking in individuals post-stroke[END_REF].

A study of five 3D-analyses of treadmill gait found no changes in mean peak knee and ankle angles or standard deviations across sessions. The authors therefore did not recommended the use of a familiarization session [START_REF] Awad | Effects of repeated treadmill testing and electrical stimulation on post-stroke gait kinematics[END_REF]. However, gait on a treadmill is different from gait overground [START_REF] Bayat | Speed and temporal-distance adaptations during treadmill and overground walking following stroke[END_REF][START_REF] Brouwer | A comparison of gait biomechanics and metabolic requirements of overground and treadmill walking in people with stroke[END_REF][START_REF] Kautz | Comparison of motor control deficits during treadmill and overground walking poststroke[END_REF]. The results of the present study indicate that for over-ground gait analysis, a familiarization session would increase the reliability of the data, particularly for hip extension in swing and stance, as well as ankle RoM in swing which showed the greater decreased of MDC value because of the familiarization visit.

Conclusion

In conclusion, kinematic and spatiotemporal data recorded during three 3D-gait analyses at intervals of 7 days were reliable. However, the MDC was lowest between the second and third visits, suggesting that patients should attend a familiarization session prior to carrying out the actual evaluations. This would ensure changes measured are related to the treatment and are not an effect of the repeated evaluations.

Limitations

The results of this study may not be generalizable to the whole population of patients with stroke since the patients included all had moderate to good functional recovery (based on mean gait speed) [START_REF] Beyaert | Gait post-stroke: Pathophysiology and rehabilitation strategies[END_REF]. In clinical practice, gait analyses may be carried out at intervals greater than 7 days. It is possible that the reliability and MDC may differ for intervals of 1 month for example. Further studies are required to test this.

Table 1 :

 1 Kinematic parameters during the swing (SwP) and stance phases (StP) of gait cycle. Positive values denote flexion and negative values denote extension.

Table 2 :

 2

ICC, SEM and MDC of kinematic parameters of the paretic lower limb during the swing (SwP) and stance phases (StP) of gait cycle. The lowest MDC values are in bold --Table2--

Table 4 :

 4 ICC, SEM and MDC of spatiotemporal parameters in paretic lower limb. The lowest MDC values are in bold. The mean values of SEM and MDC are presented as percentages in order to rationalize and compare the different units.

Table 2 :

 2 ICC, SEM and MDC of kinematic parameters of the paretic lower limb during the swing (SwP) and stance phases (StP) of gait cycle. The lowest MDC values are in bold.

	Kinematic Parameters	Gait Cycle		ICC			SEM (degrees)			MDC (degrees)
		Phase
			V1vsV2	V1vsV3	V2vsV3	V1vsV2	V1vsV3	V2vsV3	V1vsV2	V1vsV3	V2vsV3
	Hip Maximum Angle	SwP StP	0.93 0.95	0.94 0.95	0.94 0.96	3.21 2.92	3.09 2.58	3.12 2.50	9.36 8.54	9.01 7.52	9.12 7.28
	Hip Minimum Angle	SwP StP	0.93 0.94	0.94 0.95	0.97 0.98	3.58 3.26	3.22 2.96	2.59 2.22	10.44 9.53	9.39 8.63	7.56 6.48
	Hip RoM	SwP StP	0.98 0.99	0.99 0.99	0.99 0.99	1.87 1.55	1.61 1.41	1.66 1.37	5.47 4.53	4.69 4.13	4.85 4.01
	Knee Maximum Angle	SwP StP	0.98 0.96	0.99 0.97	0.99 0.98	2.61 2.46	2.43 2.40	2.24 1.69	7.61 7.19	7.10 7.01	6.54 4.93
	Knee Minimum Angle	SwP StP	0.96 0.97	0.97 0.98	0.98 0.98	2.61 2.17	2.24 2.12	2.02 1.87	7.62 6.34	6.53 6.18	5.90 5.47
	Knee RoM	SwP StP	0.98 0.96	0.99 0.98	0.98 0.96	2.80 2.44	2.20 1.80	2.91 2.23	8.16 7.12	6.43 5.25	8.49 6.52
	Ankle Maximum Angle	SwP StP	0.92 0.91	0.93 0.91	0.95 0.92	2.38 1.94	2.22 1.83	1.87 1.71	6.93 5.65	6.49 5.33	5.47 4.99
	Ankle Minimum Angle	SwP StP	0.92 0.9	0.95 0.95	0.95 0.94	2.43 2.57	2.00 2.01	1.95 2.14	7.10 7.50	5.84 5.86	5.69 6.24
	Ankle RoM	SwP StP	0.92 0.93	0.89 0.93	0.99 0.94	2.08 1.51	2.33 1.53	0.70 1.35	6.08 4.40	6.80 4.48	2.05 3.95
	Mean		0.95	0.96	0.97	2.47	2.22	2.01	7.20	6.48	5.86

Table 4 :

 4 ICC, SEM and MDC of spatiotemporal parameters in paretic lower limb. The lowest MDC values are in bold. The mean values of SEM and MDC are presented as percentages to rationalize and compare the different units.

	Spatiotemporal Parameters		ICC			SEM			MDC
		V1vsV2 V1vsV3 V2vsV3 V1vsV2 V1vsV3 V2vsV3 V1vsV2 V1vsV3 V2vsV3
	Gait speed (cm/s)	0.96	0.96	0.98	6.38	6.08	5.03	18.54	17.67	14.61
	Stride length (cm)	0.98	0.98	0.98	4.38	4.13	4.11	12.73	12.02	11.96
	Cadence (step/min)	0.95	0.95	0.98	4.06	4.09	2.95	11.81	11.9	8.58
	Step length (cm)	0.96	0.97	0.96	2.42	2.18	2.42	7.02	6.33	7.02
	StP (%)	0.96	0.97	0.97	1.52	1.30	1.24	4.42	3.79	3.60
	SwP (%)	0.96	0.97	0.97	1.52	1.30	1.24	4.42	3.79	3.60
	Step width (cm)	0.97	0.96	0.98	1.03	1.12	0.85	3.00	3.25	2.47
	Mean (% for SEM and MDC)	0.96	0.97	0.97	4.66	4.32	3.90	13.75	13.10	11.45

Acknowledgments

The authors would like to thank all the study participants and « Assistance Publique -Hôpitaux de Paris ». We also thank Johanna Robertson for her English writing correction and assistance.