Calculation of the flow rate of a rockramp pass
The calculation of the flow rate of a rockramp pass corresponds to the implementation of the algorithm and the equations present in Cassan et al. (2016)^{1}.
General calculation principle
After Cassan et al., 2016^{1}
There are three possibilities:
 the submerged case when \(h \ge 1.1 \times k\)
 the emergent case when \(h \le k\)
 the quasiemergent case when \(k < h < 1.1 \times k\)
In the quasiemergent case, the calculation of the flow corresponds to a transition between emergent and submerged case formulas:
with \(a = \dfrac{h / k  1}{1.1  1}\)
Submerged case
The calculation is done by integrating the velocity profile in and above the macroroughnesses. The calculated velocities are the temporal and spatial averages per plane parallel to the bottom.
In macroroughnesses, velocities are obtained by double averaging the NavierStokes equations in uniform regime with a mixing length model for turbulence.
Above the macroroughnesses, the classical turbulent boundary layer analysis is maintained. The velocity profile is continuous at the top of the macroroughnesses and is dependent on the boundary conditions set by the hydraulics:
 velocity at the bottom (without turbulence) in m/s:
 total shear stress at the top of the roughnesses in m/s:
The average bed velocity is given by integrating the flows between and above the blocks:
with respectively \(Q_{inf}\) and \(Q_{sup}\) the unit flows for the part in the canopy and the part above the canopy.
Calculation of the unit flow rate Q_{inf} in the canopy
The flow in the canopy is obtained by integrating the velocity profile (Eq. 9, Cassan et al., 2016):
with
with
with
and \(\alpha_t\) obtained by solving the following equation:
with
with
Calculation of the unit flow Q_{sup} above the canopy
with (Eq. 12, Cassan et al., 2016)
with (Eq. 14, Cassan et al., 2016)
and (Eq. 13, Cassan et al., 2016)
which gives
Emerging case
The calculation of the flow rate is done by successive iterations which consist in finding the flow rate value allowing to obtain the equality between the flow velocity \(V\) and the average velocity of the bed given by the equilibrium of the friction forces (bottom + drag) with gravity:
with
with
Formulas used
Bulk velocity V
Average speed between blocks V_{g}
From Eq. 1 Cassan et al (2016)^{1} and Eq. 1 Cassan et al (2014)^{2}:
Drag coefficient of a single block C_{d0}
\(C_{d0}\) is the drag coefficient of a block considering a single block infinitely high with \(F << 1\) (Cassan et al, 2014^{2}).
Block shape  Cylinder  "Rounded face" shape  Squarebased parallelepiped  "Flat face" shape 

Value of \(C_{d0}\)  1.0  1.21.3  2.0  2.2 
When establishing the statistical formulae for the 2006 technical guide (Larinier et al. 2006^{4}), the definition of the block shapes to be tested was based on the use of quarry blocks with neither completely round nor completely square faces. The socalled "rounded face" shape was thus not completely cylindrical, but had a trapezoidal bottom face (seen in plan). Similarly, the "flat face" shape was not square in crosssection, but also had a trapezoidal bottom face. These differences in shape between the "rounded face" and a true cylinder on the one hand, and the "flat face" and a true parallelepiped with a square base on the other hand, result in slight differences between them in the shape coefficients \(C_{d0}\).
Block shape coefficient σ
Cassan et al. (2014)^{2}, et Cassan et al. (2016)^{1} define \(\sigma\) as the ratio between the block area in the \(x,y\) plane and \(D^2\). For the cylindrical form of the blocks, \(\sigma\) is equal to \(\pi / 4\) and for a square block, \(\sigma = 1\).
Ratio between the average speed downstream of a block and the maximum speed r
The values of (\r) depends on the block shapes (Cassan et al., 2014^{2} et Tran et al. 2016 ^{3}):
 round : \(r_Q=1.1\)
 "rounded face" shape : \(r=1.2\)
 squarebased parallelepiped : \(r=1.5\)
 "flat face" shape : \(r=1.6\)
Cassiopée implements a formula depending on \(C{d0}\):
Froude F
Frouderelated drag coefficient correction function f_{F}(F)
If \(F < 1\) (Eq. 19, Cassan et al., 2014^{2}):
otherwise \(f_F(F) = 1\) because a torrential flow upstream of the blocks is theoretically impossible because of the hydraulic jump caused by the downstream block.
Maximum speed u_{max}
According to equation 19 of Cassan et al, 2014^{2} :
Drag coefficient correction function linked to relative depth f_{h*}(h_{*})
The equation used in Cassiopeia differs slightly from equation 20 of Cassan et al. 2014^{2} and equation 6 of Cassan et al. 2016^{1}. This formula is a fit to the experimental measurements on circular blocks used in Cassan et al. 2016^{1}:
Coefficient of friction of the bed Cf
If \(k_s < 10^{6} \mathrm{m}\) then we use Blasius' formula
with
Else (Eq. 3, Cassan et al., 2016 d'après Rice et al., 1998^{5})
Notations
 \(\alpha\): ratio of the area affected by the bed friction to \(a_x \times a_y\)
 \(\alpha_t\): length scale of turbulence in the block layer (m)
 \(\beta\): ratio between drag stress and turbulence stress
 \(\kappa\): Von Karman constant = 0.41
 \(\sigma\): ratio between the block area in the plane X,y et \(D^2\)
 \(a_x\): cell width (perpendicular to the flow) (m)
 \(a_y\): length of a cell (parallel to the flow) (m)
 \(B\): pass width (m)
 \(C\): blocks concentration
 \(C_d\): drag coefficient of a block under current flow conditions
 \(C_{d0}\): drag coefficient of a block considering an infinitely high block with \(F \ll 1\)
 \(C_f)\): bed friction coefficient
 \(d\): displacement in the zero plane of the logarithmic profile (m)
 \(D\): width of the block facing the flow (m)
 \(F\): Froude number based on \(h\) and \(V_g\)
 \(g\): acceleration of gravity = 9.81 m.s^{2}
 \(h\): average depth (m)
 \(h_*\): dimensionless depth (\(h / D\))
 \(k\): useful block height (m)
 \(k_s\): roughness height (m)
 \(l_0\): length scale of turbulence at the top of the blocks (m)
 \(N\): ratio between bed friction and drag force
 \(Q\): flow (m^{3}/s)
 \(S\): pass slope (m/m)
 \(u_0\): average bed speed (m/s)
 \(u_*\): shear velocity (m/s)
 \(V\): flow velocity (m/s)
 \(V_g\): velocity between blocks (m/s)
 \(s\): minimum distance between blocks (m)
 \(z\): vertical position (m)
 \(z_0\): hydraulic roughness (m)
 \(\tilde{z}\): dimensionless stand \(\tilde{z} = z / k\)

Cassan L, Laurens P. 2016. Design of emergent and submerged rockramp fish passes. Knowl. Manag. Aquat. Ecosyst., 417, 45. https://doi.org/10.1051/kmae/2016032 ↩↩↩↩↩↩

Cassan, L., Tien, T.D., Courret, D., Laurens, P., Dartus, D., 2014. Hydraulic Resistance of Emergent Macroroughness at Large Froude Numbers: Design of NatureLike Fishpasses. Journal of Hydraulic Engineering 140, 04014043. https://doi.org/10.1061/(ASCE)HY.19437900.0000910 ↩↩↩↩↩↩↩

Tran, T.D., Chorda, J., Laurens, P., Cassan, L., 2016. Modelling naturelike fishway flow around unsubmerged obstacles using a 2D shallow water model. Environmental Fluid Mechanics 16, 413–428. https://doi.org/10.1007/s1065201594303 ↩

Larinier, Michel, Courret, D., Gomes, P., 2006. Guide technique pour la conception des passes à poissons “naturelles,” Rapport GHAPPE RA. Compagnie Nationale du Rhône / Agence de l’Eau Adour Garonne. http://dx.doi.org/10.13140/RG.2.1.1834.8562 ↩

Rice C. E., Kadavy K. C., et Robinson K. M., 1998. Roughness of Loose Rock Riprap on Steep Slopes. Journal of Hydraulic Engineering 124, 179‑85. https://doi.org/10.1061/(ASCE)07339429(1998)124:2(179) ↩