© L. Cassan and P. Laurens, published by EDP Sciences 2016

1 Introduction

Over the last twenty years, there have been many plans to restore the populations of migratory fish species (e.g. salmon, sea-trout, shad, lamprey) in France's waterways. More recently, holobiotic species (e.g. barbel, riverine trout, nase) are also more and more taken into account. One of the necessary measures involves re-establishing the connectivity along these waterways and in particular the passage of fish at obstacles (weirs and dams). In general, according to all the fishway design guidelines and taking into account the specific biological constrains, it is possible to design any type of fishway for most species and life stages presented in a river reach (FAO, 2002; Larinier et al., 2006a). However, engineering and economic constraints make only possible to design some types, such as technical fishways, for species with good swimming abilities. In addition, technical fishways (e.g. pool and weir or vertical slot fishways) are usually built with more frequency that nature-like fishways (bypass channels or rock-ramps) due its shorter topographic development. Nevertheless, for small weirs (height mostly lower than 2–3 m), rock-ramp passes are being developed (Baki et al., 2014) and can have some advantages: possibility of high discharges interesting for the attractiveness of the facility and a lower sensitivity than technical fishways to clogging by floating debris and sediments. Three types of rock ramps can be encountered: (i) rough rock-ramp, (ii) rock-ramp with perturbation boulders and (iii) step-pool rock-ramp. In this paper the second type is studied for both emergent and submerged condition of perturbation boulders. Indeed, fishways have to be functional over a wide range of river flow and thus have to be adapted to the variations of upstream and downstream water levels. This is the reason why rock ramps usually have a half V-profile section. Within a ramp, there may be sub-sections where the blocks are submerged and others sub-sections where the blocks are emergent, depending on the upstream water level. In practice, blocks can be submerged with heights of water up to twice their height at the higher river flow of the functionality range (Larinier et al., 2006b). The submergence of some sub-sections of a ramp results in a rapid increase in their discharge and is interesting for the attractiveness of the facility, while more gentle hydraulic conditions are maintained in emergent sub-sections. The submerged sub-sections may also remain passable at least at low submergence and for species with high swimming capacities, but up to now, there was no model to compute the flow velocities in each flow layer (between blocks and above blocks). In Cassan et al. (2014), an analytical model was firstly developed for emergent rock-ramp fish pass, where the contribution of drag and bed on energy dissipation was quantified. Compared to previous methods (FAO, 2002; Heimerl et al., 2008), the evolution of the boulder drag coefficient can be estimated as a function of hydraulic parameters (Froude number, block shape and slope).

The first objective of this study is to propose a simpler version of the model proposed by Cassan et al. (2014) for emergent ramps, and to extend its relevance to blocks’ arrangements with transverse and longitudinal spaces between blocks that are uneven. This is based on new experiments results obtained on a down-scale physical model and on the comparison with other models from bibliography.

The second and main objective is to develop an analytical model for submerged ramps to estimate the stage-discharge relationship and the velocities in the different flow layers. This model is adapted from one-dimensional vertical models developed for vegetation (Klopstra et al., 1997; Huthoff et al., 2007; Murphy and Nepf, 2007; King et al., 2012), and by analyzing experiments results obtained on a down-scale physical model. Vegetation models usually study the turbulent flow as a function of the geometry, being comparable with submerged rock-ramps. The difficulty in these models arises in simulating the total turbulence intensity based on the configuration of the flow (arrangement of obstacles, slope and flow rate). Here, a turbulence closure model is proposed to estimate vertical velocity profiles and the turbulent viscosity for a large range of blocks arrangements. The proposed model is also compared to existing experimental stage-discharge correlations (Larinier et al., 2006b; Heimerl et al., 2008; Pagliara et al., 2008).

2 Method

In this part, the experimental device is firstly presented. Secondly, the method described in Figure 1 to design a rock-ramp fish pass, is proposed and each step is detailed in the following paragraphs. After the choice of water depth, slope and block (steps 1 and 2), experiments were performed to establish relationships for the velocity computation (step 3). The last steps consist in checking the passability and adjusting the design if necessary (steps 4 and 5). Some theoretical aspects are included in the corresponding step whereas the experiments analysis leading to the design formula is presented in the results part.

Fig. 1 Flow chart of the design method.

2.1 Experimental device

The fish pass is modeled as an arrangement of blocks (or macro-roughness) spaced regularly in the transverse (a_x) and longitudinal (a_y) directions (Cassan et al., 2014). The arrangement is expressed with the concentration C = D²/a_xa_y), where D is the characteristic width facing the flow. The blocks are defined by D, by their height k, and by the minimum distance between them, s = D(1/C^1/2 − 1) (Fig. 2). The averaged water depth is denoted h.

The experiments were carried out on a rectangular channel (0.4 m wide and 4.0 m long) with a variable slope. The macro-roughness consisted of plastic cylinders 0.035 m in diameter with a height (k) of either 0.03 m, 0.07 m or 0.1 m. The blocks were arranged in a staggered pattern with several densities (see Tab. 1 and Fig. 2). For experiments, the bed is horizontal in the transverse direction even if it can be sloped for some real scale fishways. The bed was covered by Polyvinyl chloride plate. A camera (1024 × 1280 pixels) was used to view the free surface of a pattern, using shadowscopy and a LED lighting system to differentiate the air from the water (Fig. 3). The image-acquisition frequency was 3 Hz and a series of 50 images were taken for each flow rate. The time-averaged water depth in the transverse direction (compared to the water's direction) was provided by the position of the minimum signal. The mean water depth (h) on the pattern was then deduced by integrating the free surface in the longitudinal direction. Flow rates were measured using KROHNE electromagnetic flow meters, accurate to 0.5%. Tests were carried out with slopes (S) of 1, 2, 3, 4 and 5% for all arrangements (Tab. 1). The flow rates for each slope were between 0.001 m³ s⁻¹ and 0.018 m³ s⁻¹ with a 0.002 m³ s⁻¹ step. The performance of flow (emergent or submerged) depends on the discharge and the slope (see Supplementary data).

Fig. 2 Definition of geometric variables for experiments (left) and view of the transverse section for real scale fishway (right). The water flows in the y-direction.

Fig. 3 Photography of one block (a) and instantaneous picture of the flow for k = 0.1 m, S = 0.01 and C = 0.1 (b).

2.2 Step 1 and 2: Geometrical characteristics

The block arrangements are depicted in Figure 2. They are characterized by D, k and C. The ratio between water depth and the characteristic width is denoted by h_* = h/D. Cassan et al. (2014) emphasized that the flow pattern depends on the Froude number (g is the gravitational acceleration) based on the averaged velocity between blocks V_g (Eq. (1)).$$\frac{V_g}{V}=\frac{1}{1-\sqrt{(a_x/a_y)C}},$$(1)where V is the bulk velocity, i.e. the total discharge divided by the ramp width and by h. The cross section is rectangular for experiments and for this theoretical approach. However some fish passes have a half V-profile section. They can be approximated as several rectangular sub-sections juxtaposed in the transverse direction. The method is applied for each sub-section and the water depth is modified as a function of the bed level. The design relationships remain relevant if the influence of the lateral slope on the transverse transfers is neglected (Fig. 2). The validation of this assumption is given by in situ measurement available in Tran (2015) for emergent condition.

2.3 Step 3: Computation of velocity

2.3.1 Step 3a and 3b

To compute the stage-discharge relationship for emergent performance, the flow analysis is based on the momentum balance applied on a cell (a_x × a_y) around one block where resistance forces are equal to the gravity force. In Cassan et al. (2014), as the flow around a block is influenced by other blocks and the bed, the drag coefficient was decomposed by three functions f_CC), f_F(F) and ( where C_d0 is the drag coefficient of a single, infinitely long block with F ≪ 1, S is the bed slope) which allow taking into account the concentration, Froude number and aspect ratio influences. As a consequence the momentum balance can be written in a dimensionless form as follows:$$C_{d0}{f}_C(C)f_F(F)f_{h_{*}}(h_{*})\frac{\mathrm{C\; h}}{D}\left(\frac{1+N}{1-\sigma C}\right)F^2=2S,$$(2)with N = (αC_f)/(C_dCh_*) and . N is the ratio between bed friction force and drag force, α is the ratio of the area where the bed friction occurs on a_x × a_y and σ is the ratio between the block area in the x, y plane and D² (for a cylinder σ = π/4), C_f is the bed friction coefficient from Rice et al. (1998) (Cassan et al., 2014). C_f is calculated by:$$C_f=\frac{2}{{(5.1\text{log}(h/k_s)+6)}^2}.$$(3)

The roughness parameter (k_s) is assumed to be equal to the mean diameter of pebbles on the bed. A common value for real scale fish pass is k_s = 0.1 m (Tran, 2015).

The influence of concentration on drag coefficient is estimated with a model based on the interaction between two cylinders (Nepf, 1999). The correction function f_C(C) proposed in Cassan et al. (2014) is only valid for a_x/a_y ≈ 1 which does not correspond to all the present arrangements. A solution is to assume that f_C = (V/V_g)², the validity of this hypothesis is discussed in the results part. Then the momentum can be expressed as follows:$$C_{d0}{f}_F(F)f_{h_{*}}(h_{*})\frac{\mathrm{C\; h}}{D}\left(\frac{1+N}{1-\sigma C}\right)F_0^2=2S,$$(4)where F₀ is the Froude number based on h and V.

The function f_F(F) is based on the fact that velocity increases because of the vertical contraction and that it is fixed to the critical velocity when a transition occurs. The analytical expression selected to reproduce these phenomena are the following (Cassan et al., 2014):$$f_F(F)=min{\left(\frac{1}{1-(F^2/4)},\frac{1}{F^{\frac{2}{3}}}\right)}^2,$$(5)$$f_{h_{*}}(h_{*})=1+\frac{0.4}{h_{*}^2}.$$(6)

The bulk velocity of rock-ramp fish pass is done by applying equation (4) with the correction function and f_F(F). The relationship between F and F₀ is deduced from equation (1) and the friction coefficient (C_f) from Rice formula (Eq. (3)).

2.3.2 Step 3c

The model is based on the spatially double-average method developed for atmospheric or aquatic boundary layers (Klopstra et al., 1997; Lopez and Garcia, 2001; Nikora et al., 2001; Katul et al., 2011) for which the submergence ratio are similar (h/k ∈ [1, 3]). First, the velocity at the bed is computed. As the discharge continuity between emergent and submerged rock-ramp is assumed, the correction is also applied in the C_d calculation whereas f_F could be neglected. Indeed, when the blocks are submerged the correction function due to the vertical contraction of the flow becomes non significant. Like for emergent conditions, the drag force within the block layer is expressed as a function of the spatially averaged velocity, then the function f_C is also neglected. At the top of the canopy, the total stress τ is computed with the shear velocity ( where ρ is the water density).

2.3.3 Step 3d

Within the canopy, an analytical formulation for the velocity u is obtained by modeling τ with the following equation:$$\tau =\rho {\nu }_t\frac{\mathrm{d\; u}}{\mathrm{d\; z}}=\rho {\alpha }_{t}u\frac{\mathrm{d\; u}}{\mathrm{d\; z}},$$(7)where z is the vertical coordinate, ν_t is the turbulent viscosity, and α_t is a turbulent length scale (Meijer and Velzen, 1999; Poggi et al., 2009).

The momentum balance in dimensionless form can be written as (Defina and Bixio, 2005):$$\frac{1}{{\beta }^2}\frac{{\partial }^2\xi }{\partial {\tilde{z}}^2}+1-\xi =0,$$(8)with β² = (k/α_t)(C_dCk/D)/(1 − σC) is the force ratio between drag and turbulent stress, ξ = (u/u₀)² is the dimensionless square of the velocity, and is the dimensionless vertical position. Viscous terms are neglected because the Reynolds number Re = u₀k/ν (ν is the water kinematic viscosity) is considerably larger than the values used for studies with vegetation (Meijer and Velzen, 1999; Defina and Bixio, 2005). The drag coefficient and diameter are assumed to be constant vertically. Finally, the velocity profile between the blocks can then be expressed by solving equation (8) with the boundary condition ξ(0) = 1:$$u(\tilde{z})=u_0\sqrt{\beta \left(\frac{h}{k}-1\right)\frac{\sinh (\beta \tilde{z})}{\cosh (\beta )}+1}.$$(9)

The continuity of the eddy viscosity at the canopy provides the relationship between the turbulent length scale at the top of blocks (l₀) and α_t:$$l_0{u}_{*}={\alpha }_t{u}_k.$$(10)In this step, equations (9) and (10) have to be solved simultaneously since u_k (velocity at the top of the canopy) depends on α_t. Using the experimental results (see further), the value of l₀ is given by the following equation:$$l_0=min(s,0.15k).$$(11)

2.3.4 Step 3e

The velocity above the canopy is assumed to be logarithmic (Eq. (12)).$$\frac{u}{u_{*}}=\frac{1}{\kappa }ln\left(\frac{z-d}{z_0}\right),$$(12)where u is the velocity above the canopy, κ the von Karman constant (κ = 0.41), d the displacement height of the logarithmic velocity profile, and z₀ the hydraulic roughness. A velocity defect law is not used because of low confinements (h/k < 3). The continuity of the velocity and the derivative at the top of the canopy can be used to obtain an expression of the coefficients d and z₀ of logarithmic law by applying equations (9) and (12) (Defina and Bixio, 2005).$$\frac{d}{k}=1-\frac{1}{\kappa }\frac{{\alpha }_t}{k}\frac{u_k}{u_{*}},$$(13)$$\frac{z_0}{k}=\left(1-\frac{d}{k}\right)\times exp\left(-\kappa \frac{u_k}{u_{*}}\right).$$(14)

2.4 Step 4 and 5: Implication for fish passage

For emergent blocks, the bulk velocity is directly deduced from equation (4) and the velocity between block is calculated with equation (1) to verify that it is lower than the fish swimming ability (criterion for fish passability). For submerged flows, the validity of l₀, d/k and z₀/k is shown by the good agreement of the velocity profiles calculated and measured within and above the canopy (Fig. 4). The advantage of the proposed equations (Eqs. (10) and (11)) is that it establishes the conditions on the bed u₀ but also that it provides an exponential profile near the canopy whose coefficients are determined by the way the obstacles are arranged. With equation (1), the maximal velocity within the block layer can be deduced from the vertical profile. Then, the location where the velocity is lower than swimming abilities is estimated.

The total discharge by unit width is obtained by the integration of the modeled velocity profiles. It must be sufficient to create an attracting current. Otherwise the same method has to be applied with lower concentration, steeper slope or considering submerged blocks.

Fig. 4 Spatially double-averaged velocity profiles of experiments compared to the model, according to Ghisalberti and Nepf (2006) (R2 = 0.98, 0.92, 0.98), Lopez and Garcia (2001) (R2 = 0.93), Meijer and Velzen (1999) (R2 = 0.98), Shimizu and Tsujimoto (1994) (R2 = 0.97, 0.95). The model presented is in solid lines. The abbreviations correspond to experimental series.

3 Results and discussion

3.1 Emergent condition

In Figure 5, the assumption on f_C(C) is confirmed by the comparison between this formula and those of Nepf (1999) and Idelcick (1986) for a set of vertical tubes. Then the stage-discharge relationship can be deduced from equation (4) where f_C(C) is omitted but with a Froude number based on the bulk velocity V. This method avoids a complex function for f_C(C) depending on a_y/a_x. The experimental results are analysed considering the bulk velocity both for the drag force and the bed friction. It is worth mentioning that knowing V_g remains important because it is the criterion for the fish passability and it fix the flow pattern by the Froude number.

Assuming f_C = (V/V_g)², the functions f_F(F) and are experimentally deduced. Their experimental values are obtained by considering the measured discharge and waterdepth (F₀ and h_*) and equation (4).

We found that the correction function defined by Cassan et al. (2014) are still valid when a_x ≠ a_y. When h_* < 0.5, the drag force and friction force have the same magnitude. As a consequence, the measurement error increases and the determination coefficients (R²) are low (Fig. 6). This inaccuracy on C_d provides a weak variation of the total discharge because the bed friction is strong. In comparison with results obtained in Cassan et al. (2014) with a more accurate definition of f_C, the uncertainty of the model is slightly increased (around 10% in the range 0.1 < C < 0.25) but the parameter C is now sufficient to characterize the geometry regardless of the ratio a_x/a_y. This remark is particularly important when the drag resistance is computed for an irregular arrangement of blocks or when the submerged flows model is applied.

Nevertheless, the maximal velocity is dependent on the ratio f = a_x/a_y. To quantify the influence of f, the model for emergent block is applied to a real scale fishway (S = 0.05, D = 0.4 m, k_s = 0.1 m, C_d0 = 1). In Figure 7, it appears that reducing this ratio does not involve a significant increase of maximal velocity but it can lengthen the resting zone since a_y is higher than a_x. But, the velocity between two blocks becomes faster because the frontal area of blocks is larger at a given water depth. As it is shown that the stage-discharge relationship only changes with C the curves are plotted for a constant total discharge in the fish pass. A limitation to f > 0.5 can be proposed. Anyway for very low f value, the function f_C cannot be pertinent.

Fig. 5 Measured corrective function as a function of concentration compared with formula of Nepf (1999) (a) and Idelcick (1986) (b).

Fig. 6 Measured corrective function as a function of Froude number (a) and dimensionless water depth (b).

Fig. 7 Velocity between blocks as a function of the ratio ax/ay. Computation for a real scale fishway (S = 0.05, D = 0.4 m, ks = 0.1 m, Cd0 = 1).

3.2 Submerged conditions

The experiments were used to determine the relationship between the geometric characteristic and l₀. The l₀ value can be expressed experimentally using several approaches (Huthoff et al., 2007; Konings et al., 2012; Luhar et al., 2008; Nepf, 2012; Poggi et al., 2009). The present approach is a combination of these formulas in order to use a formula available for a large range of macro-roughness arrangement. Unlike cases involving vegetation, k/D for macro-roughness is close to one. As a result, the influence of the bed is greater when the obstacles are shallow. Figure 8 shows the different possible configurations for each of the three length scales: s, k and D. To determine experimentally l₀, the flow rate by integrating the calculated vertical velocity profile (steps 3c, 3d and 3e) is compared to the measured flow rate with an optimization method (simplex algorithm from Matlab). For all experiments from literature (Kouwen and Unny, 1969; Meijer and Velzen, 1999; Lopez and Garcia, 2001; Righetti and Armanini, 2002; Poggi et al., 2004; Jarvela, 2005; Ghisalberti and Nepf, 2006; Murphy and Nepf, 2007; Kubrak et al., 2008; Nezu and Sanjou, 2008; Huai et al., 2009; Yang and Choi, 2009; Florens et al., 2013), C_d is considered equal to 1 if the block is circular and C_d = 2 otherwise. The results performed by Poggi et al. (2009), Konings et al. (2012), Nezu and Sanjou (2008), Yang and Choi (2009), Huai et al. (2009), and Kubrak et al. (2008) are reused, together with those obtained specifically for the present study. As indicated by Konings et al. (2012), the experiments carried out with leafy vegetation behaved in a particular way because of viscosity terms. The interpretation of l₀ is based on the assumptions of Belcher et al. (2003) and King et al. (2012). As expected, the experimental values of l₀ (Fig. 9) are similar to s when s/k ≪ 1, and proportional to k when s/k > 0.15 which yields to equation (11).

Equation (11) is consistent with literature for shallow cases (Coceal and Belcher, 2004; Nikora et al., 2013) or for deep cases (Ghisalberti and Nepf, 2006; Luhar et al., 2008; Huai et al., 2009; Poggi et al., 2009). For all experiments considered, the averaged error between the experimental and computed (with equation (11)) discharge is about 20% as indicated by the dashlines in Figure 9b. For the experiments performed in this study, the averaged error is 15.8%.

Fig. 8 Definition of lengths and turbulent length scale (l0) as a function of blocks arrangement.

Fig. 9 Vegetation studies used to evaluate the model. Data from Lopez and Garcia (2001), Poggi et al. (2004), Meijer and Velzen (1999), Ghisalberti and Nepf (2006), Murphy and Nepf (2007), Nezu and Sanjou (2008), Yang and Choi (2009), Kubrak et al. (2008), Jarvela (2005), Kouwen and Unny (1969), Florens et al. (2013), Huai et al. (2009), Righetti and Armanini (2002).

3.3 Model validation

Lastly, the model is compared to the experimental correlation proposed by Larinier et al. (2006b) for rock-ramp fish passes (Eqs. (15) and (16)). This correlation is deduced from a statistical study of a large number of experiments in the laboratory, on cylindrical macro-roughness with 8 % < C < 16%, 1 % < S < 9%, k = 0.07 or 0.1 m and D = 0.035 m, the maximum ratio for h/k is 3.6.

For emergent conditions:$$q=0.815\sqrt{\mathrm{gS}}{\left(\frac{h}{D}\right)}^{1.45}{C}^{-0.456}{D}^{1.5}.$$(15)

For submerged conditions:$$q=1.12\sqrt{\mathrm{gS}}{\left(\frac{h}{D}\right)}^{2.282}{C}^{-0.255}{\left(\frac{k}{D}\right)}^{-0.799}{D}^{1.5}.$$(16)

The experimental correlation of Pagliara et al. (2008) is also used (for emergent and submerged conditions):$$q=\mathrm{Vh}=\sqrt{\frac{8\mathrm{ghS}}{(-7.82S+3.04)(1.4exp(-2.98C)+ln(h/k))}}h.$$(17)

The model results are consistent with the formula of Larinier et al. (2006b), including high concentrations (Fig. 10) superior to C = 0.2. However the model differs from statistical formulation for low value of k/D. Experimental data with k/D < 1 are used to calibrate the turbulence model whereas no such of experiments were used to establish the experimental correlation in Larinier et al.'s (2006) study. Similarly, equation (16) indicates that D has no influence except for C because only one diameter was used. In the presented model, D modifies the values of C_d, u₀, s and then l₀ for deep cases. In Figure 10, the stage-discharge relationship is depicted for a real scale fishway. Comparison with other guidelines (Larinier et al., 2006b; Heimerl et al., 2008) indicates that the model allows to reproducing experimental correlation between h_* and q_* = q/D^5/2. The same dependence on C is found between the present study and results of Pagliara et al. (2008). For emergent performance, the model with high concentration C = 0.3 provides the same stage-discharge relationship than method from Heimerl et al. (2008). Therefore, the model is validated by other studies. But, as mentioned before, some advantages are added like the applicability to different shapes and the validity for a large range of geometry.

Moreover, the model allows estimating the double averaged velocity profile (Fig. 4) and turbulent shear stress (Eq. (7)) within and above the block layer whereas equations (16) and (17) only provide the total discharge. It is possible to know if hydrodynamic parameters are suitable with the swimming ability of fishes within the block layer even if the velocity is higher in the surface layer.

Fig. 10 Comparison of the stage-discharge relationship between the model and the empirical formula of Larinier et al. (2006b) (a) as a function of concentration and formula of Larinier et al. (2006b), Heimerl et al. (2008) and Pagliara et al. (2008) with k/D = 1 (b). Computation for a real scale fishway (S = 0.05, D = 0.4 m, ks = 0.1 m, Cd0 = 1).

4 Conclusion

This paper presents an analytical model for calculating the stage-discharge relationship for emergent and submerged rock-ramp fish passes. New experiments are conducted to prove that results obtained by Cassan et al. (2014) are available whatever the block arrangement. For submerged blocks, canopy vegetation models are improved to take mixing length into account by linking it directly to the geometric characteristics of the arrangement. The model has been adjusted on the basis of a large number of experiments found in the literature as well as the presented fish passes configurations. The model seems to offer a good trade-off between its validity for a large range of geometrical arrangements and simplicity of use (number of parameters, calculation time, etc.). Although the mixing-length model provides few explanations about the structure of the turbulence, it can be used to estimate mean flow rate between and above the blocks fairly accurately. It is thus possible to predict the velocities that the different species of fish will need to overcome. It can also help to design effective and durable passes. The implementation of the model could be difficult but all equations can be solved with numerical tools. Software is developed currently and its ergonomy has been designed to help to use the presented flow chart.

Supplementary Material

Access here

The Supplementary Material is available at http://www.kmae-journal.org/10.1051/kmae/2016032/olm.

Acknowledgments

The research was supported by the ONEMA under the grant “Caractérisation des conditions hydrodynamiques dans les écoulements à fortes rugosités émergentes ou peu submergées”. This financial support is greatly appreciated.

Notations

α : ratio of the area where the bed friction occurs on a_x × a_y

α _t : turbulent length scale (m) within the blocks layer

β : force ratio between drag and turbulent stress

κ : von Karman's constant

λ : frontal density

σ : ratio between the block area in the x, y plane and D²

ξ : dimensionless square of the velocity

a _x : width of a cell (perpendicular to flow) (m)

a _y : length of a cell (parallel to flow) (m)

C : blocks concentration

C _d0 : drag coefficient of a block considering a single block infinitely high with F ≪ 1

C _d : drag coefficient of a block under the actual flow conditions

C _f : bed-friction coefficient

d : zero-plane displacement of the logarithmic profile (m)

D : characteristic width facing the flow (m)

F : Froude number based on h and V_g

F ₀ : Froude number based on h and V

g : gravitational constant (m s⁻²)

h : mean water depth in a cell (m)

h _* : dimensionless water depth (h/D)

k : height of blocks (m)

k _s : height of roughness (m)

l ₀ : turbulent length scale (m) at the top of blocks (m)

N : ratio between bed friction force and drag force

q : specific discharge per unit width (m² s⁻¹)

q _* : specific discharge per unit width (m^−0.5 s⁻¹)

Re : Reynolds number based on k and u₀

R ² : determination coefficient

s : minimum distance between blocks

S : bed slope

u : averaged velocity at a given vertical position (m s⁻¹)

u ₀ : averaged velocity at the bed (m s⁻¹)

u _k : averaged velocity at the top of blocks (m s⁻¹)

u _* : shear velocity (m s⁻¹)

V _g : averaged velocity in the section between two blocks (m s⁻¹)

V : bulk velocity (m s⁻¹)

z : vertical position (m)

z ₀ : hydraulic roughness (m)

: dimensionless vertical position

References

All Tables

All Figures

	Fig. 1 Flow chart of the design method.
In the text

	Fig. 2 Definition of geometric variables for experiments (left) and view of the transverse section for real scale fishway (right). The water flows in the y-direction.
In the text

	Fig. 3 Photography of one block (a) and instantaneous picture of the flow for k = 0.1 m, S = 0.01 and C = 0.1 (b).
In the text

	Fig. 4 Spatially double-averaged velocity profiles of experiments compared to the model, according to Ghisalberti and Nepf (2006) (R2 = 0.98, 0.92, 0.98), Lopez and Garcia (2001) (R2 = 0.93), Meijer and Velzen (1999) (R2 = 0.98), Shimizu and Tsujimoto (1994) (R2 = 0.97, 0.95). The model presented is in solid lines. The abbreviations correspond to experimental series.
In the text

	Fig. 5 Measured corrective function as a function of concentration compared with formula of Nepf (1999) (a) and Idelcick (1986) (b).
In the text

	Fig. 6 Measured corrective function as a function of Froude number (a) and dimensionless water depth (b).
In the text

	Fig. 7 Velocity between blocks as a function of the ratio ax/ay. Computation for a real scale fishway (S = 0.05, D = 0.4 m, ks = 0.1 m, Cd0 = 1).
In the text

	Fig. 8 Definition of lengths and turbulent length scale (l0) as a function of blocks arrangement.
In the text

	Fig. 9 Vegetation studies used to evaluate the model. Data from Lopez and Garcia (2001), Poggi et al. (2004), Meijer and Velzen (1999), Ghisalberti and Nepf (2006), Murphy and Nepf (2007), Nezu and Sanjou (2008), Yang and Choi (2009), Kubrak et al. (2008), Jarvela (2005), Kouwen and Unny (1969), Florens et al. (2013), Huai et al. (2009), Righetti and Armanini (2002).
In the text

	Fig. 10 Comparison of the stage-discharge relationship between the model and the empirical formula of Larinier et al. (2006b) (a) as a function of concentration and formula of Larinier et al. (2006b), Heimerl et al. (2008) and Pagliara et al. (2008) with k/D = 1 (b). Computation for a real scale fishway (S = 0.05, D = 0.4 m, ks = 0.1 m, Cd0 = 1).
In the text