V Afentoulis

A Papadimitriou

V Tsoukala

M Benoit

Numerical Approaches for the Evaluation of Sediment Transport Mechanisms on a Shallow Sloping Sea Bottom

Keywords: Sediment transport, Numerical simulation, Morphodynamics, Coastal hydrodynamics

The seabed evolution due to wave-current interaction is examined in the case of a gently sloping bottom in a shallow water coastal area. In this study, an attempt has been made to estimate bed level changes over a time period of 96 hours, under accretive and erosive wave sequences. A numerical model based on the non-linear shallow water equations (NSWE) is utilized in order to simulate the wave propagation and hydrodynamic conditions. Two different approaches are used for the detailed investigation of sediment transport and seabed-fluid interactions. It is shown that the numerical approximation performs reasonably well, as the erosive sequence is characterized by an offshore sediment movement while the sandbar is driven shoreward under accretive wave conditions. The simulation is carried out on a variable coastal bathymetry corresponding to the wave basin experiments of Michallet et al. (2013) at LHF facility (Grenoble INP, France). The obtained results are qualitatively compared with laboratory observations.

INTRODUCTION

The evolution of seabed under the combined action of waves and currents has been extensively studied, but until nowadays is still not fully understood. Historically, many studies have been carried out on the sediment transport pathway investigation, generation and migration of different bed forms, such as ripples and dunes (Sleath 1984), (Nielsen 1992), (Fredsøe and Deigaard 1992), (Soulsby 1997), and (Van Rijn 1993, 2007). Due to the complex nature of these phenomena, it is difficult to strictly interpret their mechanisms with a rational theory. Sand fluxes can be generated and driven by the combination of steady flows (currents) and oscillatory flows (waves). However, several other processes need to be taken into consideration, such as the mean water level variations (tide, wave setup and set-down), breaking effects, and influence of the bed slope and form (Camenen and Larroudé 2003).

The main objective of the present study is to provide a comprehensive understanding of seabed-fluid interactions, using numerical modeling approaches. Hence, a numerical model is employed to simulate the hydrodynamic conditions and morphological seabed changes under erosive and accretive wave sequences. In order to assess the interaction between hydrodynamic and morphodynamic processes, the wave-current-bed evolution system is evaluated in an interactive mode, by considering the feedback on the former from the latter (Wang et al. 2012). Thus, a significant difference of the initial wave height distribution is observed due to the bathymetry changes, while topographically controlled currents are generated as a result of this interaction. It is worth pointing out that the effects of swash and groundwater dynamics in the unsaturated zone of beach is beyond the scope of the present work. Furthermore, measurements and observations from laboratory experiments have been used in order to verify the accuracy and credibility of the numerical approach.

General model setup

The simulation is carried out on a variable coastal bathymetry corresponding to the wave basin experiments of Michallet et al. (2013) at LHF facility (Grenoble INP, France). Figure 1 depicts the 3D bottom topography colored map and the alongshore-averaged beach profiles at the three main stages of the experiments. The applied initial wave climates, defined by their significant wave height and peak period, are divided into two main categories: energetic conditions (A: Hm0 = 23 cm / Tp = 2.3 s) and moderate conditions with a large period (B: Hm0 = 18 cm / Tp = 3.5 s). These wave characteristics correspond to the so-called down-state and up-state wave transitions respectively (i.e. the change from high to low wave energy in the case of the accretive sequence and the change from low to high wave energy in the case of the erosive sequence). The still water depth is set to h0= 0.765 m and the length and time scales are 1/10 and ≈1/√10 respectively, using Froude similarity law. Fine sand is considered, with specific gravity (Gs) of 2.65 and a median diameter (d50) of 0.166 mm which corresponds to a settling velocity (ws) = 2 cm/s. The total simulation time is 96.4 hours and it is divided into two parts, 62 hours duration of B wave conditions and 34.4 hours duration of A wave conditions. In order to create alongshore nonuniformity in the wave breaking spatial pattern, the wave directional spreading at the inlet varies from 20 to 30 degrees.

HYDRODYNAMICS

For the assessment of hydrodynamic conditions, a model that solves the two-dimensional non-linear shallow water equations (NSWE) in conservative form is applied (Marche and Bonneton 2006), (Marche et al. 2007). These equations are discretized by a finite volume formulation, which preserves steady state solutions on non-flat sea beds in the absence of perturbations. The main advantage of this numerical approach is that breaking events can be modeled as the development of a free-surface and current discontinuity (Bonneton et al. 2010). It is noted, however, that dispersion effects are not considered, which is a limitation of this approach. A wave-maker is implemented for the generation of irregular waves using directional spectral data. These irregular waves are obtained by superposing a series of wave components with different frequencies and directions, using random phases. In Figure 2 the evolution of wave height distribution during the erosive sequence is shown. The change of the computed significant wave height is due to the wave-bed evolution joint actions.

MORPHODYNAMICS

In the present work, the mode of sediment movement is investigated using two different transport approaches. The first one is based on a quasi-steady approximation suggested by Ribberink (1998) for the estimation of bed load transport, while the suspended transport is calculated from Camenen and Larson's (2005, 2007, 2008) formula. The second approach is based on Soulsby-Van Rijn approximation (Soulsby 1997). This formula applies to total (bed and suspended load) sediment transport in combined waves and currents on sloping beds. Averaged sediment fluxes are obtained, by integrating the transport load over 50 wave periods. Furthermore, a slope limiting methodology is applied to control the maximum bottom slope, which is considered equal to the sediment repose angle (≈32 deg) (Bailard and Inman 1981).

Ribberink's and Camenen -Larson's methods

Bed load transport

Ribberink suggested a model of bed load transport where the solid flux is proportional to a function of the difference between the actual time-dependent bed shear stress and the critical bed shear stress. This formula is described by the following expression for the sand transport rate:

𝑞 𝑆𝐵 = 𝑚 𝑅𝑖𝑏 √(𝑠 -1)𝑔𝑑 3 ⟨(| 𝜃(𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗⃗ | -𝜃 𝑐𝑟 ) 𝑛 𝑅𝑖𝑏 𝜃(𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | 𝜃(𝑡)| ⟩ (1)
where, θ(𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 0.5𝑓 𝑐𝑤 |𝑢(𝑡)|𝑢(𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗ /[(𝑠 -1)𝑔𝑑] is the time-dependent Shields parameter with the instantaneous velocity 𝑢(𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑈 𝑐 ⃗⃗⃗⃗ + 𝑢 𝑊 (𝑡) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , 𝑠 the relative density of sediment and 𝑓 𝑐𝑤 the wavecurrent friction factor. ⟨ ⟩: time-averaged over several wave periods; and 𝑚 𝑅𝑖𝑏 = 11, 𝑛 𝑅𝑖𝑏 = 1.65: adjusted coefficients.

Suspended load transport

The Camenen and Larson (2005, 2007, 2008) transport rate formula is adopted for estimating suspended load. This formula is described by the following expression for the suspended load transport:

𝑞 𝑆𝐵 = 𝑈 𝐶 𝐶 𝑅 𝜀 𝑊 𝑠 [1 -𝑒 -𝑊 𝑠 ℎ 𝜀 ] (2)
where, 𝑐 𝑅 = 3.51 -3 𝑒 -0.3𝑑 * 𝜃 𝑐𝑤,𝑚 𝑒 -4.5

𝜃 𝑐𝑟 𝜃 𝑐𝑤 is the reference concentration at the bottom with 𝑑 * = √(𝑠 -1)𝑔/𝑣 2 𝑑 50 3 the dimensionless grain size, 𝑣 the kinematic viscosity of water, 𝜃 𝑐𝑤,𝑚 and 𝜃 𝑐𝑤 the mean and maximum Shields parameters due to wave-current interaction and ε the sediment diffusivity.

Soulsby-Van Rijn's method

Soulsby-Van Rijn's formula (Soulsby, 1997) was derived for total (suspended and bed load) sediment transport in combined wave and currents on horizontal and sloping beds. This approach is described by the following expression for the sand transport rate:

𝑞 𝑡 = 𝐴 𝑠 𝑈 ̅ [(𝑈 ̅ 2 + 0.018 𝐶 𝐷 𝑈 𝑅𝑀𝑆 2 ) 1 2 -𝑈 ̅ 𝐶𝑅 ] 2.4 (1 -1.6𝑡𝑎𝑛𝛽) (3)
where, 𝐴 𝑆 = 𝐴 𝑆𝐵 + 𝐴 𝑆𝑠 with 𝐴 𝑆𝐵 = 0.005ℎ( 𝑑50 ℎ

) 1.2

[(𝑠-1)𝑔𝑑 50 ] 1.2 and 𝐴 𝑆𝑆 = 0.012𝑑 50 𝑑 * -0.6

[(𝑠-1)𝑔𝑑 50 ] 1.2 . 𝑈 ̅ is the depth averaged velocity, 𝑈 𝑅𝑀𝑆 the root-mean-square wave orbital velocity, 𝐶 𝐷 the drag coefficient due to current alone, 𝑈 ̅ 𝐶𝑅 the threshold current velocity and 𝛽 the bed slope.

RESULTS AND DISCUSSION

The sea bed evolution at the end of the accretive and erosive sequence using Ribberink's and Camenen -Larson's formulas is presented in Figure 3, while the results corresponding to the Soulsby's approximation are depicted in Figure 4. In Figure 5 alongshore-averaged beach profiles are demonstrated, as obtained using the two methods. It is observed that the numerical approximation performs reasonably well, as the erosive sequence is characterized by an offshore sediment movement while the sandbar is driven shoreward under accretive wave conditions. The obtained results show a good agreement between the two different applied methods, as the identified accretive and erosion areas in Figure 3 and in Figure 4 look similar. However, small discrepancies around the offshore zone are noticed during the accretion sequence. The intensity and direction of the wave-induced currents are determinant for the morphological changes of the sea bottom. The flow is characterized by a rip current, which has a shore-normal orientation and a magnitude of about 0.05 m/s and 0.1 m/s during the accretion and erosive sequence respectively. The intensity of the seaward current causes an offshore sandbar migration under energetic wave conditions. The mean circulations obtained during the experiments of Michallet et al. ( 2013) have a same mean flow magnitude but their directions are highly variable during each sequence. This is to some extent due to the imposed wave conditions of the experiment, where the wavemaker is designed to have a varying damping at the center of the wave front. Thus, the seabed alongshore nonuniformity obtained by the numerical simulation is more intense. Seabed changes at the end of each numerical simulation (Figure 5) are in the same order of magnitude with the measured ones (Figure 3), between -0.1 m and +0.1 m. Overall, a good agreement between both experimental and numerical results is achieved, as the seabed adapts correctly to the wave climate changes. Further numerical investigations are needed to investigate the transition from the dissipative to the reflective beach state and back again, considering the intermediate sub-states according to the work of Wright and Short (1984).

Figure 1 .

 1 Figure 1. 3D bathymetry colored elevation map (left). Alongshore-averaged beach profiles at the three main stages of the experiments (Michallet et al. 2013) (right).

Figure 2 .

 2 Figure 2. Map of computed significant wave height at the beginning (left) and at the end (right) of the erosive sequence.

Figure 3

 3 Figure 3 Map of bottom changes and associated mean flow field (vectors) at the end of the accretive (left) and at the end of the erosive sequence (right) using Ribberink's and Camenen -Larson's formulas.

Figure 4

 4 Figure 4 Map of bottom changes and associated mean flow field (vectors) at the end of the accretive (left) and at the end of the erosive sequence (right) using Soulsby's formula.

Figure 5

 5 Figure 5 Alongshore-averaged beach profiles at the two main stages of the simulation using Soulsby's formula (left) and Ribberink's and Camenen -Larson's formulas (right).


Maria João Teles
email: maria.teles@edf.fr

Thierry Fouquet
email: thierry.fouquet@edf.fr

Antonio Pires-Silva

Michel Benoit

Computation of 3D coastal hydrodynamics through the vortex force formalism implemented

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

INTRODUCTION

The coastal domain is a complex hydrodynamic system where physical phenomena with different time and space scales interact. This is the case of waves and currents interaction.

The wave breaking process together with wave-induced currents can create a dangerous environment for swimmers and have a great impact on morphodynamics in the nearshore areas. Depending on the bottom morphology and on the incident wave field, currents can have different characteristics. If obliquely incident waves break on a alongshore uniform planar beach, a longshore current is generated. If the beach has, for example, sand bars or cusps, rip currents can be generated.

The wave-current environment is well described in a 2DH framework by the work of [1], through the radiation stress concept, already implemented by coupling the hydrodynamic model TELEMAC-2D [START_REF] Hervouet | Hydrodynamics of free surface flows, modelling with the finite element method[END_REF] and spectral wave modelling TOMAWAC [START_REF] Benoit | Development of a third generation shallow water wave model with unstructured spatial meshing[END_REF]. A couple of years ago, this approach was extended to TELEMAC-3D code assuming a uniform distribution of the radiation stresses over depth. Therefore the vertical structure of the flow was not properly assessed.

Nevertheless the three-dimensional (3D) structure of the flow can give relevant information about sediment transport, linked to the value of the velocities near the bottom, or to assess wave power availability at a certain location.

During the PhD thesis of Teles [START_REF] Teles | Wave-current modelling at local and regional scales[END_REF], [START_REF] Teles | Computation of 3D coastal hydrodynamics through the vortex force formalism implemented by coupling[END_REF] a new TELEMAC-3D -TOMAWAC coupling system was developed, based on a vortex force formalism.

The theoretical framework chosen was the glm2z-RANS [START_REF] Ardhuin | Explicit wave-averaged primitive equations using a generalized Lagrangian mean[END_REF] equations. To do so, the three-dimensional TELEMAC-3D equations were modified together with new boundary conditions and new parametrizations were included in TOMAWAC to calculate the wave forcing terms.

The purpose of this paper is to present the implementation of the developed TELEMAC-3D -TOMAWAC coupled system [START_REF] Teles | Wave-current modelling at local and regional scales[END_REF], [START_REF] Teles | Computation of 3D coastal hydrodynamics through the vortex force formalism implemented by coupling[END_REF] on the latest TELEMAC-MASCARET version (v8p1). The improvement of wave-current interaction effects on the 3D hydrodynamics description is evidenced.

In the following section a brief description of the new wave forcing terms included in TOMAWAC and the modified equations implemented on TELEMAC-3D is given together with the new key-words created in the steering file to activate this coupling mode.

In section III, a barred beach test case is used to analyse the vertical structure of the flow, followed by section IV, where some concluding remarks and perspectives are given.

II. COUPLING SYSTEM

A. Governing equations

To take into account the 3D effects of the combined environment the mathematical framework proposed by [START_REF] Ardhuin | Explicit wave-averaged primitive equations using a generalized Lagrangian mean[END_REF], the glm2z-RANS equations, was implemented. Following [START_REF] Bennis | On the coupling of wave and three-dimensional circulation models: Choice of theoretical framework, practical implementation and adiabatic tests[END_REF] the vertical current shear was neglected in the wave forcing terms. (u , w ) represent the quasi-Eulerian velocities ( = 1, 2 corresponding to horizontal coordinates), defined in a second order theory approach, by the difference between the Lagrangian mean velocities and the Stokes drift (U αs , W s ).

S α represents the hydrodynamic model horizontal source or sink terms of momentum, for instance, the Coriolis force The acceleration due to gravity is given by g, and S xα represent the hydrodynamic model horizontal source terms, for instance, the Coriolis force.  H and  z are, respectively, the horizontal and vertical turbulence viscosities. The viscosity values can either be prescribed by the user or computed by a turbulence closure model. Furthermore, within the wave-current environment and due to wave breaking there is an enhancement of the vertical mixing. To take account of this effect, the approach proposed by [START_REF] Battjes | Energy loss and set-up due to breaking of random waves[END_REF] was followed by adding the wave-enhanced vertical mixing,  wz to the vertical turbulence viscosity  z :

  wz (z)= c b Q br 1 3 H s √2 Df wb (z) (3)
with : The vortex force in x α direction, which is represented in ∈ α3β (f 3 +𝜔 3 )U βs , is defined by the vectorial product between the mean flow vertical vorticity 𝜔 3 and the horizontal Stokes drift U βs . The Stokes-Coriolis force is represented by ∈ α3β f 3 U βs .

 f wb (z)= 1-tanh k b ( -z) 2 ∫ (1-tanh k b  -z 2 )dz  -h  (4)
Moreover, the bottom shear stress in the hydrodynamic model was modified in order to take into account the wavecurrent interaction effects on the bottom roughness (CFW), following [START_REF] Christoffersen | Bed friction and dissipation in a combined current and wave motion[END_REF]'s theoretical framework.

To guarantee mass conservation, the mass transport induced by the Stokes drift in the depth-integrated continuity equation ( 5) was included. The overbar symbol denotes a depth-integrated variable.

 ∂h ∂t + ∂hu α ∂x α = - ∂hU αs ∂x α  (5)
At the offshore open boundary, in the case of no other external forcing terms (such as tidal forces), two conditions are imposed for the phase-averaged elevation [START_REF] Andrews | An exact theory of non linear waves on a Lagrangian mean flow[END_REF] and the horizontal velocities [START_REF] Ardhuin | Explicit wave-averaged primitive equations using a generalized Lagrangian mean[END_REF] [START_REF] Rascle | Impact of waves on the ocean circulation[END_REF]:

  = - J g  (6)
 u α = -U  (7)
The momentum lost by waves due to depth-induced wave breaking (FDX, FDY) and bottom friction (FBX, FBY) is imposed in the hydrodynamic model as free surface and bottom stresses, respectively. The momentum lost by waves du to white-capping together with the input momentum transferred from wind to the wave field (FWX, FWY) is also imposed as a surface stress in the hydrodynamic model.

B. Implementation

In the following, we give a brief explanation on the way the coupled system works (Figure 1) [START_REF] Teles | Computation of 3D coastal hydrodynamics through the vortex force formalism implemented by coupling[END_REF]. Both TOMAWAC and TELEMAC-3D models run with the same horizontal mesh.

TELEMAC-3D starts the calculation. The Nikuradse roughness, the z-levels, and the computed depth-integrated velocities and mean surface elevation are communicated to TOMAWAC. In its turn, TOMAWAC computes, over a time step, the wave forcing terms: the Stokes drift components, the wave-induced pressure, the wave breaking and the bottom-induced dissipation momentum contributions. The last two terms are imposed as surface and bottom stresses, respectively, in the hydrodynamic model. Furthermore, the wave model calculates the wave-enhanced vertical mixing coefficient  wz that is added to the vertical turbulence viscosity in TELEMAC-3D. This process is repeated each time step or made within a coupling period defined by the user. The coupling period between TELEMAC-3D and TOMAWAC can be larger than the time step of the models. The time step of each of the models does not have to be the same, just a multiple of each other.

C. Steering file: new key words

In order to compute a wave-current environment in a 3D framework, with the coupling system based in the vortex force formalism described above, the user must include the key word COUPLING WITH: TOMAWACT3D in the steering file. The existing coupling, where the radiation stresses are distributed uniformly over the depth, is then distinguished from the new developments.

If the user wants to include the lost by waves due to bottom friction the key word BOTTOM FRICTION DUE TO WAVES must be set to TRUE in the steering file. Please be aware that this term shall be included only when the vertical resolution near the bottom in TELEMAC-3D is sufficiently refined.

Moreover the wave enhanced vertical mixing formulation follows [START_REF] Battjes | Energy loss and set-up due to breaking of random waves[END_REF] approach with the same parameters used in that paper. A sensitivity analysis would be advised if the wave enhanced vertical mixing is expected to have a predominant role in the 3D flow dynamics.

III. TEST CASE

In the present section the capability of the new coupled system to model rip currents is tested and the vertical structure of the flow is evidenced. The comparison of numerical results between the existing TELEMAC-3D -TOMAWAC coupling, based on the radiation stresses uniformly distributed over depth, and the new one, based on a vortex-force formalism is made.

A. Rip current

The present test-case is based on laboratory wave-basin experiment [START_REF] Haas | Laboratory measurements of the vertical structure of rip currents[END_REF]. The bathymetry is made of two bars which induce wave breaking and the generation of a rip current system. The wave basin is 17.2 m in the cross-shore direction and 18.7 m in the longshore direction. The slope is 1:5 from offshore up to three meters from the wave maker and then 1:30 till the end of the beach. The generated waves are monochromatic and perpendicular to the beach.

The same wave basin configuration was defined on the numerical model. The computational domain was discretized equally for both models with 0.2 m ix x-and y-directions (Figure 2). In TELEMAC-3D, 8 horizontal planes were distributed over depth.

The time step was set to Δt= 0.03 s for both hydrodynamic and wave models. Conditions (4) and ( 5) were assigned at the offshore boundary and walls were defined at lateral and shoreward boundaries. The Nikuradse roughness was set to ks= 0.01 m. The k- model was the chosen turbulence model to compute the vertical turbulence viscosity. To take into account the wave-enhanced mixing in the hydrodynamic model due to wave breaking, the parameters cb = 0.03 and ab = 1.2 were used [START_REF] Battjes | Energy loss and set-up due to breaking of random waves[END_REF].

A value of  H = 1.10 -2 m 2 s -1 was set for the horizontal turbulence viscosity and  z = 1.10 -6 m 2 s -1 . The numerical simulations with TOMAWAC were performed with spectral parameters that match the monochromatic laboratory experiments. This way, a significant wave height was set to Hs= 0.067 m. The minimum frequency was set to 0.187 Hz, the number of frequencies to 7 and the frequential ratio to 1.4. The directional discretization was made through 24 direction bins. For the depth-induced breaking the model proposed by [START_REF] Battjes | Energy loss and set-up due to breaking of random waves[END_REF] was chosen with  = 1.0.

B. Results

The wave height evolution over the domain is of major importance to correctly obtain the flow pattern of a rip current system. In its turn the current field is going to have a significant role on the wave height evolution [START_REF] Haas | Laboratory measurements of the vertical structure of rip currents[END_REF]. Therefore a great influence is also noticed on the rip currents from the interaction with waves. Due to bathymetry features, the wave pattern in the domain is quite different in the channel areas and over the bars (upper panel of Figure 3). The variation of the mean surface elevation (lower panel of Figure 3) alongshore induces longshore pressure gradients. The latter generate offshore oriented currents that converge into the rip channel and onshore oriented currents over the bar.

The breaking pattern reveals a quite distinct behaviour between the bar and the rip channel. While over the bars the waves break suddenly and then near the shoreline a second less intense breaking is observed, in the channel, due to the greater water depth, the waves break more progressively and penetrate further into the inshore zone of the channel, inducing a longshore current near the shoreline that flows away from the channel (Figure 6).

The comparison between numerical results and measurements of the significant wave height and mean water level over the bar (at y = 9.2 m) and through the rip channel (y = 13.6 m) are shown on Figure 4 and Figure 5 respectively. Over the bar the comparisons between the numerical simulations and data are similar. The waves break suddenly when encountering the bar and a sudden and significant rise of the water level occurs due to the strong wave breaking in this region. Through the rip channel, the coupled system reproduces quite well the progressive evolution of the wave set-up up to the shoreline but it shows some difficulties in reproducing the exact location where waves start to break through the rip channel. Possibly, the numerical opposing currents were to strong and induce the waves to break earlier than observed in the experiments. Within the rip channels, the vertical profile of the crossshore velocities is not so sheared, but it shows relatively high negative velocities approximately in the middle of the water column. Then it starts to slightly decrease in magnitude towards the free surface and the bottom.

The longshore components of the quasi-Eulerian velocity are one order of magnitude lower than the quasi-Eulerian cross-shore velocities. This is caused by the weak longshore component of the Stokes drift (lower panel of Figure 8). Even so, they present high variability along the longshore bars and channels. At least three inflexion points for the velocities can be clearly identified: two in each channel and one over the bar. These inflexion points are generated by the variability of the mean surface elevation in the longshore direction which in its turn exists due to the spatial variability of the dissipation rate of the incident wave field. In the figures below we plot the evolution of cross-shore and longshore current components along two cross-shore directions of the domain (y = 13.6 m in the rip channel and y = 9.2 m over the bar). It can be seen that in both wave breaking locations there is a strong vertical shear of the cross-shore quasi-Eulerian velocity: near the free surface it is directed onshore while near the bottom an undertow is verified. When analysing the rip current through the rip channel the velocities oriented offshore are stronger than the ones observed over the bar.

One of the great advantages of the vortex force formalism to describe waves and current interactions is that the vortex force is clearly distinguished. Therefore it is possible to study the circulations and water motions in the mean flow. Moreover, within this approach, it is possible to show the distribution of the vortex force over the domain that in its turn contributes to the vortices located between the bar and the channel and near the shoreline.

If the radiation stress is assumed to be uniformly distributed over the depth, the vertical structure of the flow cannot be assessed. As it is shown above, there is an important vertical shear of the velocity at certain locations of the domain, which can strongly impact the morphodynamics in similar beach configurations.

IV. CONCLUDING REMARKS

The implementation of the coupled system between the three-dimensional flow model TELEMAC-3D and the spectral wave model TOMAWAC [START_REF] Teles | Wave-current modelling at local and regional scales[END_REF], [START_REF] Teles | Computation of 3D coastal hydrodynamics through the vortex force formalism implemented by coupling[END_REF] in the next official TELEMAC-MASCARET v8p1 version is presented. The capability of the new coupled system, based on a vortex force formalism, to represent rip currents on a barred beach is assessed and the vertical structure of the flow is analysed.

Other parametrizations and vertical distributions of the non-conservative wave forcing terms should be the subject of further research.

Moreover in the present work, the vertical current shear is ignored within the wave forcing terms, and the horizontal diffusion velocity coefficient was set by using a simple approach (imposing a constant value along the domain). The former feature should be included and different and more complex turbulence modelling approaches should be tested.

Q

  br represents the wave breaking sink term, D the total depth and k b = √2 a b H s , with a b = 1.2 and c b = 0.03.

Figure 1

 1 Figure 1 Scheme of the different terms computed and exchanged by TELEMAC-3D and TOMAWAC

Figure 2

 2 Figure 2 Wave basin bathymetry of the rip current test case

Figure 3 Figure 4

 34 Figure 3 Wave height (m) and mean free surface elevation (m) patterns over the domain

Figure 5

 5 Figure 5 Cross-shore evolution of significant wave height (above) and mean water level (below) through the rip channel at y=13.6 m. Comparison between numerical results (lines) and data (dots) from [14].

Figure 6

 6 Figure 6 Depth integrated rip current (m/s) patterns over the domain The vertical structure of the cross-shore and longshore currents is presented below in the longshore direction of the domain at x = 11.6 m.

Figure 7

 7 Figure 7 Alongshore variation of cross-shore and longshore current (m/s) components at x = 11.6 m It can be seen a strong vertical shear of the quasi-Eulerian cross-shore velocity component above the bar crest. It increases from bottom to the free surface, being oriented towards the shoreline. When encountering the bars, waves break, and a strong mass flux occurs. This can be confirmed by the vertical distribution of the cross-shore component of the Stokes drift (upper panel of Figure 8).

Figure 8

 8 Figure 8 Alongshore variation of crosshore and longshore Stokes drift (m/s) components at x = 11.6 m

Figure 9

 9 Figure 9 Crosshore variation of cross-shore and longshore current (m/s) components in the rip-channel at y = 13.6 m.

Figure 10

 10 Figure 10 Crosshore variation of cross-shore and longshore current (m/s) components over the bar at y = 9.2 m

ACKNOWLEDGEMENT

Maria João Teles would like to acknowledge the support of a PhD grant (SFRH/BD/61269/2009) from FCT (Fundação para a Ciência e Tecnologia), Portugal. The authors would also like to acknowledge all the support given by the TELEMAC-MASCARET development team.


Marco Licari
email: marco.licari@mail.polimi.it

Michel Benoit
email: benoit@irphe.univ-mrs.fr

Fabien Anselmet
email: anselmet@irphe.univ-mrs.fr

Vincent Kocher

Simon Clément

Pierre Le Faucheux

Study of low-head hydrostatic pressure water wheels for harnessing hydropower on small streams

Keywords: experimental hydraulics, hydroelectricity, hydraulic wheel, hydraulic energy, small hydropower, hydrostatic pressure machine

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

Nomenclature

1.Introduction: context and concept presentation

A renewed interest in hydroelectric plants of medium and small size has been demonstrated by the scientific and international community in the recent years (see, e.g., [START_REF]HYLOW project: Development of hydropower converters with very low head differences[END_REF][START_REF] Paish | Small hydro power: technology and current status[END_REF][START_REF] Senior | New hydropower converters for very low-head differences[END_REF]), as these technologies are not only regarded as promising answers to cope with climate change impacts, but they could also unlock a huge hydraulic potential, currently unexploited. Considering European countries only, the estimated available sites account for 13 GW of untapped hydraulic energy potential [START_REF] Paish | Small hydro power: technology and current status[END_REF].

Various concepts of small hydraulic turbines are under research and optimisation process, and the "Hydrostatic Pressure Machine" (HPM) is the one considered in the present study [START_REF] Senior | New hydropower converters for very low-head differences[END_REF], operating on sites with ultra-low hydraulic heads between 0.5 m and 2 m. The HPM concept, shown in Figure 1, derives from the hydraulic wheel, but has one peculiarity compared to the other hydraulic turbines. This machine, in fact, is not only an energy converter, but it could also be used as a water level regulation system: it can be installed in any river, but also in any irrigational canal, acting as a regulatory weir. Due to its design, the HPM converts the hydrostatic head created by the turbine itself: the difference between the upstream and downstream water levels applies a net pressure force on the machine blade in the same direction as the water flow, resulting in an applied torque on the machine shaft, thus converting hydraulic power into mechanical power. In recent studies [START_REF] Senior | New hydropower converters for very low-head differences[END_REF][START_REF] Senior | Hydrostatic pressure converters for the exploitation of very low head hydropower potential[END_REF][START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF], various models have been proposed to explain the power conversion mechanism, but still no optimised HPM in terms of shape and dimensions has been proposed yet. The collaboration between the research institute IRPHE, specialized in fluid mechanics, and the PYTHEAS Technology company, developing energy conversion solutions for slow and intermittent energy sources, aims to deepen the knowledge of the power conversion mechanism and to study the effect of the geometry and shape of the HPM on its performances. Two experimental scale models have been designed, built and tested in the "Herode channel", a testing facility available at the IRPHE, in Marseille. In this paper, the experimental results obtained are presented in terms of power, efficiency and performance predictions of the HPM. Thanks to these experimental campaigns, it can be assessed that the HPM is effectively a promising device both for power production and water level regulation purposes, while the collaboration between the two institutions is still undergoing, in order to produce more results soon. The physical models developed are presented in Section 2, then the experimental set-up in the Herode flume is described in Section 3, as well as the design of the two tested scale models of the HPM, along with their respective mechanical and electrical conversion chains. Their different conversion chains were designed to impose various working conditions on the wheels by using either an electromagnetic brake or a mechanical brake, simulating the presence of a power take-off system. Experimental results are presented and discussed in Section 4, before conclusions and perspectives are summarized in Section 5.

Physics and modelling of the Hydrostatic Pressure Machine

The most advanced model in the scientific literature seems to be the one proposed by Linton [START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF] following earlier works of Senior [START_REF] Senior | Hydrostatic pressure converters for the exploitation of very low head hydropower potential[END_REF] and Senior et al. [START_REF] Senior | New hydropower converters for very low-head differences[END_REF]. This model relies on several simplifying assumptions:

 steady state conditions (for the flow and motion of the wheel),  the physics is bi-dimensional (and described in the vertical plane),  there are no interactions between blades,  blades are perpendicular to the water flux (as represented in Figure 2).  upstream and downstream water levels 𝑑 1 and 𝑑 2 are constant, creating a nominal static head equal to the hub diameter 2𝑟 0 .

Figure 2. 2D scheme of an HPM under nominal operating conditions (reprinted from [START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF]).

HPM Ideal Model

The so-called "ideal model" does not consider any dissipative effect acting on the machine blades. Since the HPM is designed to convert mainly pressure energy, in this ideal approach, only pressure forces are considered. The net pressure force acting on the blade is expressed as:

𝐹 𝑝 = 𝐹 1 -𝐹 2 = ρ𝑔(𝑑 1 -𝑑 2 -∆ℎ)𝑆 𝑏𝑙 (1)
where 𝐹 1 and 𝐹 2 are the resultant pressure forces on the blade on the upstream and downstream sides respectively. 𝑆 𝑏𝑙 is the surface of the blade and, given that 𝐿 is its width and 𝑏𝑙 is its radial length, it is estimated as follows:

𝑆 𝑏𝑙 = 𝐿 𝑏𝑙 (2)
∆ℎ is the drop in hydrostatic head due to the acceleration of the water flow, and it can be computed thanks to Bernoulli's theorem:

∆ℎ = 𝑣 2 2 -𝑣 1 2 2𝑔 (3)
The acceleration of the water flow can be estimated thanks to the continuity equation:

𝑄 = 𝐿𝑑 1 𝑣 1 = 𝐿𝑑 2 𝑣 2 (4)
As the upstream water level is higher than the downstream one, the water flow is faster downstream than upstream (as a result of eq. ( 4)).

Another assumption made by Linton needs to be highlighted. Thanks to his experimental campaign, Linton observed a linear relationship between the wheel rotation velocity , and the water discharge passing through the HPM. Moreover, considering the operating conditions depicted in Figure 2, where the downstream water level rises up to the blade attach, Linton hypothesizes that the blades move at the same velocity as the downstream water velocity [START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF].

It is then possible to compute the ideal power as:

𝑃 𝑖𝑑 = 𝐹 𝑝 𝑣 𝑏𝑙 = 𝜌𝑔(𝑑 1 -𝑑 2 -∆ℎ)𝑆 𝑏𝑙 𝑣 2 (5)
To calculate the efficiency of the machine, Linton only takes into account the hydrostatic head [START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF]:

𝜂 𝑖𝑑 = 𝑃 𝑖𝑑 𝜌𝑔𝑄𝐻 = 𝑃 𝑖𝑑 𝜌𝑔𝑄(𝑑 1 -𝑑 2 ) (6)
The normalized ideal results obtained using Linton's model are plotted in Figure 3 and they are independent of the machine geometry. In Figure 3, both the power and the water flow rate are normalized by their respective maximum values. While the maximum ideal power can be directly computed by maximising the power function relatively to the blade velocity, more considerations are needed for the definition of the maximum water flow dischargeable by the HPM. The operational limit of the machine occurs when all the hydrostatic head available is converted into dynamic head:

𝛥ℎ 𝑚𝑎𝑥 = 𝑑 1 -𝑑 2 (7)
In this condition, the net pressure force acting on the blade disappears, and no power is produced, as it can be verified applying eq. ( 7) to eq. ( 1) and to eq. ( 5). When this operational boundary is reached, the HPM rotates at its maximum angular speed, and the maximum discharged flow rate can be computed considering eq. ( 3), eq. ( 4) and eq. ( 7):

𝑣 2,𝑚𝑎𝑥 = √ 2𝑔(𝑑 1 -𝑑 2 ) 1 -( 𝑑 2 𝑑 1 ) 2 (8)
𝑄 𝑚𝑎𝑥 = 𝐿𝑑 2 𝑣 2,𝑚𝑎𝑥 (9)

More realistic (non-ideal) models

Although the ideal model allows a first assessment of the HPM operation and performances, various phenomena occur under real operating conditions, reducing the power effectively obtainable by the hydraulic machine. Some of these effects are discussed below, and expressions are proposed to be incorporated in more realistic versions of the HPM model.

Blade-fluid interaction losses

First, power losses due to viscosity are an inevitable consequence of the blade-fluid turbulent interaction. The following semi-empirical function is proposed for the estimation of these losses:

𝑃 𝑘,1 = 𝑘 1 𝜌𝑔𝑄𝐻 𝑙𝑜𝑠𝑡 ( 10
)
𝐻 𝑙𝑜𝑠𝑡 = (𝑣 2 -𝑣 1 ) 2 2𝑔 (11)
Alternatively, a second formulation can be used to estimate the losses due to the blade-fluid interaction, expressed as follows:

𝑃 𝑘,2 = 1 2 𝜌𝑘 2 𝐿𝑏𝑙𝑣 𝑏𝑙 3 (12)
The dimensionless coefficients 𝑘 𝑖 in eq. ( 10) and eq. ( 12) are determined experimentally, allowing to express a first definition for the non-ideal power:

𝑃 𝑛𝑖 = 𝑃 𝑖𝑑 -𝑃 𝑘,1 (13)
The eq. ( 13) computes the maximum hydraulic power convertible by the HPM under the non-ideal condition of power losses due to the blade-fluid turbulent interaction.

Ventilation losses

This phenomenon occurs at the exit of the wheel blade from water during the emptying mechanism of the water volume delimited between two blades (the so called "cells").

Assuming cells are fully filled with water until reaching the atmosphere, power at the exit of the wheel is estimated by applying the momentum equation at the exit of the wheel:

𝑃 𝑒𝑥𝑖𝑡 = 𝑄 3 𝜌 𝑏𝑙 𝑑 2 -1 (𝐿𝑏𝑙) 2 + 1 2 𝜌𝑔𝑄 𝑑 2 2 -𝑏𝑙 2 𝑏𝑙 (14)
However, this involves that an important volume of water is lifted until the water free surface. In reality, air starts replacing the water long before the surface. Assuming that air can freely fill the cell from the angle φ, the power is adjusted as follows:

𝑃 𝑢𝑛𝑣𝑒𝑛𝑡 = 𝑃 𝑒𝑥𝑖𝑡 + 𝜌𝑔𝑄 2𝐿 𝑍 𝑣𝑒𝑛𝑡 (𝑍 𝑣𝑒𝑛𝑡 -2𝐿) (15)
Z vent = r mean (1 -cos φ) (16)
Currently, eq. ( 15) largely over-estimates losses as air filling is restrained. A coefficient α is then introduced to estimate the rate of the cell filling. Finally, it is possible to compute the power lost at the exit section of the machine due to the cell ventilation losses as follows:

𝑃 𝑒𝑥𝑖𝑡 = (1 -α φ ω )𝑃 𝑢𝑛𝑣𝑒𝑛𝑡 (17)
For the sake of clarity, it is here highlighted that the empirical coefficient α is not dimensionless but, has a unit of 𝑠 -1 . Therefore, scaling laws need to be applied if it is needed to compare α obtained by different experiments.

Acceleration losses

Acceleration losses are calculated by applying the momentum equation on the upstream part of the fluid domain:

𝑃 𝑎𝑐𝑐 = 𝑟 𝑚𝑒𝑎𝑛 𝜔 {𝐹 ℎ𝑢𝑏 + 𝐹 𝑏𝑙 + 𝐹 𝑠𝑖𝑑𝑒 -𝐹 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 + 𝑄𝜌𝑣 𝑏𝑙 (1 - 𝑏𝑙𝐿 𝑑1𝐿 𝑐ℎ )} (18)
where F hub , F bl , F side and F upstream are the respective pressure forces.

Flow leakage

Another negative effect concerning the power production is the presence of by-passed water flow during operation. The totality of the by-passed flow is due to the gap between the fixed and rotating components of the HPM. In order to avoid mechanical friction, these gaps are needed between the hydraulic wheel and the lateral walls, as well as between the wheel and the channel floor. While the lateral gaps have constant dimensions, a bigger portion of the flow is able to by-pass the wheel underneath the blade whenever the blade is not perpendicular to the channel floor.

The processed flow 𝑄 𝑤𝑜𝑟𝑘 , considering the operating conditions depicted in Figure 2, can be analytically estimated given that the wheel angular velocity and the velocity profile of the water flow at section T are known:

𝑄 𝑤𝑜𝑟𝑘 = ∫ 𝑣(𝑟) 𝑑𝑟 = ∫ 𝜔𝑟𝐿 𝑑𝑟 𝑟 𝑡𝑖𝑝 𝑟 0 = 𝜔𝑟 𝑚𝑒𝑎𝑛 𝐿𝑏𝑙 𝑟 𝑡𝑖𝑝 𝑟 0 (19)
The geometrical constant 𝑟 𝑚𝑒𝑎𝑛 is the radius of the rotor at the mean blade height:

𝑟 𝑚𝑒𝑎𝑛 = 𝑟 0 + 𝑏𝑙 2 (20)
As discussed in the experimental results presentation (see Section 4 hereafter), flow leakage appears to be mainly driven by the increase of the water level difference and, as for the blade-fluid interaction losses (see Sub-section 2.2.1), leakage losses will be experimentally estimated.

Definition of the machine efficiency

Various efficiencies are defined to properly characterise the HPM ability to convert hydraulic power. While the total efficiency 𝜂 𝑡𝑜𝑡 is computed on the total water flow rate and evaluates the global performance, the available efficiency 𝜂 𝑎𝑣 considering only the net processed flow rate only accounts for the hydrodynamics quality of the HPM:

𝜂 𝑡𝑜𝑡 = 𝑃 𝑛𝑖 𝜌𝑔𝑄 𝑡𝑜𝑡 𝐻 (21) 𝜂 𝑎𝑣 = 𝑃 𝑛𝑖 𝜌𝑔𝑄 𝑤𝑜𝑟𝑘 𝐻 (22)
Furthermore, it is convenient to define the volumetric efficiency as the ratio between the effectively processed water flow rate and the total water flow rate:

𝜂 𝑣𝑜𝑙 = 𝑄 𝑤𝑜𝑟𝑘 𝑄 𝑡𝑜𝑡 (23)
Doing so, the volumetric efficiency allows to better evaluate the leakage's influence on the HPM performance, being the volumetric efficiency related to the hydrodynamic quality and global quality of the machine as follows:

𝜂 𝑡𝑜𝑡 = 𝜂 𝑣𝑜𝑙 𝜂 𝑎𝑣 (24)
It is finally possible to define the technical efficiency, as the efficiency considering the full set of power losses and computing the full technical performance of the machine:

𝜂 𝑡ℎ = 𝑃 𝑖𝑑 -𝑃 𝑘,2 -𝑃 𝑎𝑐𝑐 -𝑃 𝑒𝑥𝑖𝑡 𝜌𝑔𝑄 𝑡𝑜𝑡 𝐻 (25)

Experimental set-up

Two different scale models of HPM have been tested in the "Herode Channel", a pre-existing experimental facility located in the research institute IRPHE, in Marseille. The two experimental scale models have different geometries and they have been designed to study the HPM under various operating conditions. The concepts behind these two machines are discussed in Subsections 3.2 and 3.3, while the Herode Channel is first presented in Sub-section 3.1.

The Herode hydraulic channel

In Figure 4, a schematic view of the Herode channel is given. This testing facility is divided in two main areas: the free-surface zone with a rectangular cross-section, that corresponds to the actual testing environment, and the recirculation pipe located below. The total water volume present in the channel can be regulated by a secondary system of charging and discharging conducts, allowing to directly control the free-surface level. Whenever a physical barrier of any kind is installed in the channel and the pump is running, the upstream and downstream free-surface levels cannot be directly fixed nor modified due to the lack of regulatory weir (as the total volume of water circulating in the flume is fixed for a given test).

In the testing section, the free-surface level can reach up to 0.55 m and the internal width of the channel is of 0.62 m, while the total length of the testing section is 8 m.

The water flow rate is measured using an electromagnetic flow meter, while a hydraulic pump allows the operator to adapt the water discharge to the chosen testing conditions.

The IRPHE experimental HPM scale model

A first HPM scale model was built at IRPHE using a simple geometry for the blades (Figure 5). It consists in a 12 blades rotor, where a transmission belt conveys the mechanical power to the measure and control ("M&C") system, located well above the water surface.

The 12 blades of the hydraulic wheel have a 2 mm thickness, a radial length of 185 mm and they span the hub axial length of 488 mm. The central hub shape is a dodecagonal prism, 488 mm of width and a dodecagon circumferential radius of 145 mm. The transmission system is composed of a toothed transmission belt, 16 mm wide, which connects the "motor" gear mounted on the wheel shaft, to the "transmission" gear, mounted on the M&C system. The motor gear has 84 teeth and the transmission gear 14, so the angular speed of the M&C shaft is 6 times the motor shaft angular speed. This transmission ratio has been chosen to respect various electrical and mechanical limits concerning the M&C system and is related to the expected HPM operating conditions in terms of torque and angular speed. Regarding the M&C system represented in Figure 6, the measurement system features a dynamic torque transducer (T22/10Nm made by HBM) and a RVDT angular position sensor (R30D by TE Connectivity ltd.). Two brakes have been selected for applying the mechanical load to the transmission shaft, a mechanical band brake and an electro-magnetic (EM) brake. The mechanical brake has been designed, built and assembled by IRPHE and it is adjustable manually thanks to a regulation screw. The EM brake was bought (model FRAT 120 by Merobel), it is remotely controllable by altering the incoming excitation current and it applies a constant resistive torque independent from the angular speed.

The hydraulic wheel, the transmission system and the M&C system have been mounted in a PMMA supporting structure, so that it is possible to position and fix the HPM without modifying the preexisting testing facility.

PYTHEAS Technology experimental HPM scale model

For the PYTHEAS Technology experimental HPM model, two parts were designed: the HPM wheel and a shroud. The hydraulic wheel is designed with 12 blades attached to a central hub. Blades are as long as the diameter of the central hub. To minimize turbulence each time a blade enters the free surface, they present an entry angle of 30°, and the twisted shape not only improves the blade cell emptying but also prevents air to be sucked in at the entrance. The second main element of the system is the shroud. It consists of an adaptation of the channel floor to ensure efficient compression during the wheel cycle and to reduce the by-passed flow underneath the wheel blades, as depicted in Figure 8. Various shroud configurations have been tested: the covered angle between the vertical and the entrance or exit section could be of 40° or 56°, as shown in Figure 8. To optimise the filling and emptying of the cells, the wheel should be smaller than the complete inner width of the channel where the HPM is installed. According to [START_REF] Paudel | Experimental and numerical study of Dethridge Wheel for pico-scale hydropower generation[END_REF], the gap between the wheel sides and the channel lateral walls must be at least equal to half of the wheel's width, especially on the downstream part where the water needs to be quickly discharged. This is the reason why the wheel axial length is 238 mm, the blade radial length is 150 mm while the hub diameter is 150 mm. Like for the IRPHE HPM model, rotational speed and torque are measured on a shaft above the free surface: an encoder ERN 1020 from Heidenhain is used to measure the angular velocity and a torque transducerT40S2 (range 0-100N.m) from HBM for the torque. The resistive torque is applied with a mechanical brake, manually adjustable thanks to two plates tightened by screws on the transmission shaft. Basically, the measure and control system set-up of the PYTHEAS Technology and IRPHE HPM wheels share the same concept, with the only difference of a second electromagnetic brake installed on the IRPHE transmission shaft.

Analysis of experimental results

General considerations on experimental protocol

The results in terms of converted power and related efficiency are disclosed hereafter, both for the IRPHE and the PYTHEAS HPM wheels. Furthermore, it has to be highlighted that the PYTHEAS HPM operated under Linton's nominal conditions of constant downstream and upstream water levels: the downstream surface rises up to the lowest point on the hub, while the upstream water level reaches the highest point on the hub circumference (as depicted in Figure 2). Since the hydraulic system response to a higher mechanical load imposed to the HPM is a wider difference between the upstream and downstream levels, both Senior's [START_REF] Senior | Hydrostatic pressure converters for the exploitation of very low head hydropower potential[END_REF] and PYTHEAS experiments relied on a system of either adjustable or fixed weirs and variation of the total water volume available in the testing channel for exogenously fixing the free-surface levels at the protocol values. In order to study a wider range of operating conditions, the IRPHE experimental campaign did not operate under constant water levels: for a higher mechanical load imposed on the HPM, the upstream and downstream water levels were free to adapt themselves at the hydraulic system equilibrium. These two approaches have been chosen for better understanding the HPM physics, since this hydraulic machine is studied not only as an energy converter but also as a water level regulation system. Previous experiments in the scientific literature [START_REF] Linton | Field trials and development of a hydrostatic pressure machine[END_REF] already demonstrated how, during real field tests, the assumption of constant water levels is not always granted, so further investigation on the relation between the machine mechanical load and the free-surface levels was considered useful and justified the two chosen complementary approaches.

Five distinct parameters are measured during the experimental campaigns:

 Two hydrostatic probes measure the upstream and downstream water levels. Incertitude on these measurements is about 1 mm.  As explained in the scale models descriptions, encoders measure the angular velocity and torque transducers measure the applied resistive torque.  The total water flow discharged from the testing section is measured on the exit pipe. Doing so, the measured flow rate corresponds to the total flow rate streaming in the free-surface section, accounting for both the water flux effectively processed by the HPM and the bypassed water flow.

Experimental results obtained with the IRPHE wheel

Since the water heights are not constant for the experiments with IRPHE wheel, the number of free variables increases compared to Linton's model. More specifically, there are three variable parameters for the IRPHE experiments: the resulting torque and the angular velocity of the HPM, which can be studied imposing different mechanical loads and water flow rates, while the water levels difference represents the third variable. Thanks to the conducted experimental campaign, empirical observations suggested that the difference between the upstream and downstream water levels is directly related to the total water volume available in the testing channel: setting the HPM torque and angular speed at the same values produced the same upstream and downstream water heights and difference only if the same water volume was present. From this observation, the following experimental protocol was developed: 4 different initial water volumes were tested, and for each water volume 4 angular velocities were studied, increasing progressively the mechanical load. 82 different equilibrium points have been collected, and it is possible to observe the related power and efficiency curves in Figures 9, 10 and 11.

All the variables are plotted as functions of the hydrostatic head difference 𝑑 1 -𝑑 2 , while the curves are parameterized for a fixed motor-shaft RPM. Considering the plotted results, the amount of water volume available in the channel is conveniently expressed by the term ℎ 𝑧 , the homogenous water height when the channel is completely at rest. Moreover, for the sake of clarity, only the results for ℎ 𝑧 at 25 cm are plotted here, while the qualitative comments on this case hold across all the IRPHE results. Starting with the results in Figure 9, the experimental ideal power 𝑃 𝑚𝑒𝑎𝑠 + 𝑃 𝑘 increases for higher static head and higher RPM, in accordance with the power formalisation expressed in eq. ( 5). Given the relatively low values of RPM studied, the nonlinear effects of static head drop affecting the obtained power and expressed by eq. ( 3) were slightly appreciable, explaining the linear slope of the curves at 3.7 𝑟𝑎𝑑. 𝑠 -1 , 5.2 𝑟𝑎𝑑. 𝑠 -1 and 6.8 𝑟𝑎𝑑. 𝑠 -1 , while for the fastest RPM curve, due to faster flow velocities involved, nonlinear effects were more appreciable. Furthermore, the power curves show already that, for an ideal machine, the HPM can convert up to 172 W per meter of width, given a very low hydrostatic head of 25 cm. The results in terms of efficiency seem promising as well.

As shown in Figure 10, unlike the power curves, the efficiency curves drop for higher RPM: the effect is due to the viscous effects for the blade-fluid interactions. Eq. ( 10) and eq. ( 11) show the relation between the power losses and the blade velocity: as the HPM is rotating faster, higher blade velocities cause higher turbulent power dissipation. Furthermore, it can be noticed that the gap between total and available efficiency increases as the static head difference increases. This effect can be explained considering eq. ( 24): the water leakage increases with the upstream and downstream water level difference. Consequently the volumetric efficiency drops, widening the gap between available and total efficiency.

Experimental results obtained with the PYTHEAS Technology wheel

For the experimental protocol adopted for the PYTHEAS experiments, it must be considered that, as explained in Sub-sections 3.1 and 3.2, the testing channel operates as a closed loop. At the beginning of the tests, with the pump running, the operator sets the upstream water level at the chosen nominal value of 31.5 cm while, at the same time, the resistive torque is set at its maximum. Then, the operator increases the pump power and reduces the breaking torque to keep the upstream water level constant and record data for various flow rate and mechanical load conditions. Given the initial presence of enough water in the channel, it was possible to fix the downstream water level at the nominal value of 16.7 cm. The nominal head difference is then 14.8 cm and is constant all along the studied power curve conditions.

Figure 12 shows the power curves obtained with two shroud conditions. The red curve corresponds to a 40° downstream shroud, the blue one is set with a downstream shroud angle of 56°, while the upstream shroud angle is kept constant at 56° for both tests. The gap between the two curves can be explained by the ventilation phenomenon, as formalized by eq. ( 14) and eq. ( 15). With the tallest shroud, the water remains in the cell for a longer time: an effort from the wheel is then required to rise this volume before air fills it, creating additional losses.

However, an optimum design should be reached. Due to the twisted and oblique shape of the blades, the downstream shroud should be designed containing at least an entire blade, but a shorter shroud could bring new other losses. During the compression phase, in fact, a shorter shroud would cause the water to be depressurized before the whole volume crosses the vertical line. For the current experiment, the optimal trade-off angle is estimated around 30°. Figure 13 shows the corresponding technical efficiencies. In both configurations, efficiency is nearly linear with the flow rate. It can be seen that efficiency is lower with the 56° shroud, which matches the observations made from the power curves in Figure 12. In this configuration, both data sets are close. The maximum difference (12%) appears for small flow rates. As ventilation power largely over-estimates losses for low rotation velocities, it is neglected in this area. In the range where the turbine is used, power differences are below 4.5%.

The analytical model's efficiency results are within the incertitude of the experiments. Less than 2.5% deviation is observed for flow rates above 0.0056 m 3 .s -1 . This accurate correlation between the analytical and experimental datasets will allow to extrapolate data for different geometries. As expected, the peak of power and efficiency does not always match, but the model still appears to explain reasonably well the power prediction and the main sources of losses. Finally, Figure 15 shows the flow rate as a function of the rotational velocity. As expected, the flow rate through the turbine is nearly linear with its rotation speed. Color on test data is function of the head loss. A small variation occurs at high flow rates where fluid is not homogenous anymore. A significant amount of air is observed in the cell, reducing the flow rate through the wheel.

Conclusions and perspectives

The tests performed in this study show a very good agreement between the analytical models and the experimental results, by adding just a few semi-empirical models describing non-ideal effects, with empirical coefficients determined from experimental measurements. Based on these reliable results, estimation of the power production and efficiency for a site of any scale can be assessed. These tests also permitted to identify the hydrodynamic behavior of the wheel, especially how changing the breaking torque (representing the power take-off system) and the water level impacts the behavior of the HPM. Possible improvements of the wheel (geometry, dimensions, shape of the blades, etc.) and the shroud were identified, which will be tested in the next steps of the project. The optimal operating point in flow rate, or equivalently in rotation speed, confirms the relevance and benefit of the HPM wheel. This type of turbine appears as one of the most suitable solutions to efficiently harvest energy from such small hydrostatic head difference.

The next steps of the project will address: i) a dedicated study to improve the performances of the wheel, ii) the adaptation of the generator that converts the slow motion of the wheel into electricity, and iii) the elaboration of the control law of the wheel. In fact, another advantage of this wheel is the linear relation between flow rate and rotational speed. This relation makes the wheel also useful for flow rate control. By adjusting the rotation speed of the wheel, we can control the flow rate and produce power simultaneously: this is no less than an hydraulic regulation work harvesting energy.

Acknowledgements

M. Benoit and F. Anselmet thank the board of IRPHE lab. for funding the experimental program on the IRPHE wheel and the 6-month internship scolarship of M. Licari.

1 𝑣 2 ,

 12 , measured from the blade-hub attach to the blade tip the fluid pressure on the upstream side of the blade N 𝐹 2 force due to the fluid pressure on the downstream side of the blade N 𝐹 𝑏𝑙 force due to the fluid pressure on the blade N 𝐹 ℎ𝑢𝑏 force due to the fluid pressure on the wheel hub N 𝐹 𝑠𝑖𝑑𝑒 force due to the fluid pressure on the machine lateral side N 𝐹 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 force due to the fluid pressure on the frontal upstream section N 𝑔 gravity acceleration m.s -2 𝐻 hydrostatic head m 𝐻 𝑙𝑜𝑠𝑡 energy loss expressed as a loss of the available hydrostatic head m 𝑘 loss coefficient due to viscous power dissipation -𝐿 axial length of the hydrostatic pressure machine m L ch width of the channel m 𝑃 𝑎𝑐𝑐 power loss due to the fluid acceleration W 𝑃 𝑒𝑥𝑖𝑡 power loss at the exit section due to partial ventilation W 𝑃 𝑖𝑑 power of an ideal machine, with no energy loss W 𝑃 𝑘 viscous power loss due to machine-fluid interactions W 𝑃 𝑚𝑒𝑎𝑠 power measured at the shaft by the instruments W 𝑃 𝑛𝑖 power of a non-ideal machine, with energy losses considered W 𝑃 𝑢𝑛𝑣𝑒𝑛𝑡 power loss at the exit section due to perfectly unvented water cell W 𝑄 volume flow rate m 3 .s -1 𝑄 𝑚𝑎𝑥 maximum volume flow rate dischargeable m 3 .s -1 𝑄 𝑡𝑜𝑡 total volume flow accounting for both the processed water flow and leakage m 3 .s -1 𝑄 𝑤𝑜𝑟𝑘 working volume flow processed by the machine, without the leakage removed m 3 .s -1 𝑟 0 radius of the machine measured from the rotation center to the blade-hub attach m 𝑟 𝑚𝑒𝑎𝑛 radius of the machine measured from the rotation center to the blade mean height m 𝑟 𝑡𝑖𝑝 radius of the machine measured from the rotation center to the blade tip m 𝑆 𝑏𝑙 radial-axial surface of the blade 𝑚𝑎𝑥 maximum velocity of the downstream water flow m.s -1 𝑍 𝑣𝑒𝑛𝑡 increase in elevation occurring in the cell before the fluid evacuation m 𝛼 empirical coefficient of partial ventilation s -1 𝛥ℎ free surface level drop due to the water flow acceleration m 𝛥ℎ 𝑚𝑎𝑥 maximum free surface level drop due to the water flow acceleration m 𝜂 𝑎𝑣 efficiency of the machine considering the available working volume flow rate -𝜂 𝑡𝑜𝑡 efficiency of the machine considering the total volume flow rate -𝜂 𝑣𝑜𝑙 volumetric efficiency -𝜌 density of the fluid kg.m -3 𝜑 ventilation angle that determines the start of the cell emptying rad 𝜔 angular velocity of the machine rad.

Figure 1 .

 1 Figure1. HPM under real field trial, developed by the HYLOW project, Bulgary (pictures taken from[START_REF]HYLOW project: Development of hydropower converters with very low head differences[END_REF]).

Figure 3 .

 3 Figure 3. Power and efficiency of an HPM wheel based on the ideal model results, under nominal operating conditions.

Figure 4 .

 4 Figure 4. Schematic view of the Herode channel at IRPHE.

Figure 5 .

 5 Figure 5. IRPHE HPM wheel operating in the Herode Channel. The water flows from the left and the wheel is rotating counterclockwise.

Figure 6 .

 6 Figure 6. Measurement and control system of the IRPHE experimental HPM model.

Figure 7 .

 7 Figure 7. Model views of the PYTHEAS experimental system (left: drawing of the wheel and M&C system ; right: picture of the wheel operating in the flume)

Figure 8 .

 8 Figure 8. Shroud design of PYTHEAS HPM scale model

Figure 9 .

 9 Figure 9. Experimental ideal power for various fixed angular velocities (IRPHE wheel)

Figure 10 .Figure 11 .

 1011 Figure 10. Available and total efficiency compared for different fixed angular velocities (IRPHE wheel)

Figure 12 .

 12 Figure 12. Converted power for two downstream shroud configurations (PYTHEAS wheel)

Figure 13 .

 13 Figure 13. Technical efficiency  th (eq. (25)) for two downstream shroud configurations(PYTHEAS wheel)

Figure 14 .

 14 Figure 14. Comparison of experimental and analytical power and efficiency (PYTHEAS wheel)

Figure 15 .

 15 Figure 15. Variation of flow rate as a function of the rotation velocity (PYTHEAS wheel).