# Calculation of the flow rate of a rock-ramp pass

The calculation of the flow rate of a rock-ramp 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., 20161

There are three possibilities:

• the submerged case when $$h \ge 1.1 \times k$$
• the emergent case when $$h \le k$$
• the quasi-emergent case when $$k < h < 1.1 \times k$$

In the quasi-emergent case, the calculation of the flow corresponds to a transition between emergent and submerged case formulas:

$Q = a \times Q_{submerge} + (1 - a) \times Q_{emergent}$

with $$a = \dfrac{h / k - 1}{1.1 - 1}$$

## Submerged case

The calculation is done by integrating the velocity profile in and above the macro-roughnesses. The calculated velocities are the temporal and spatial averages per plane parallel to the bottom.

In macro-roughnesses, velocities are obtained by double averaging the Navier-Stokes equations in uniform regime with a mixing length model for turbulence.

Above the macro-roughnesses, the classical turbulent boundary layer analysis is maintained. The velocity profile is continuous at the top of the macro-roughnesses and is dependent on the boundary conditions set by the hydraulics:

• velocity at the bottom (without turbulence) in m/s:
$u_0 = \sqrt{2 g S D (1 - \sigma C)/(C_d C)}$
• total shear stress at the top of the roughnesses in m/s:
$u_* = \sqrt{gS(h-k)}$

The average bed velocity is given by integrating the flows between and above the blocks:

$\bar{u} = \frac{Q_{inf} + Q_{sup}}{h}$

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 Qinf in the canopy

The flow in the canopy is obtained by integrating the velocity profile (Eq. 9, Cassan et al., 2016):

$Q_{inf} = \int_{0}^1 u(\tilde{z}) d \tilde{z}$

with

$u(\tilde{z}) = u_0 \sqrt{\beta \left( \frac{h}{k} -1 \right) \frac{\sinh(\beta \tilde{z})}{\cosh(\beta)} + 1}$

with

$\beta = \sqrt{(k / \alpha_t)(C_d C k / D)/(1 - \sigma C)}$

with

$C_d = C_{x} f_{h_*}(h_*)$

and $$\alpha_t$$ obtained by solving the following equation:

$\alpha_t u(1) - l_0 u_* = 0$

with

$l_0 = \min \left( s, 0.15 k \right)$

with

$s = D \left( \frac{1}{\sqrt{C}} - 1 \right)$

### Calculation of the unit flow Qsup above the canopy

$Q_{sup} = \int_k^h u(z) dz$

with (Eq. 12, Cassan et al., 2016)

$u(z) = \frac{u_*}{\kappa} \ln \left( \frac{z - d}{z_0} \right)$

with (Eq. 14, Cassan et al., 2016)

$z_0 = (k - d) \exp \left( {\frac{-\kappa u_k}{u_*}} \right)$

and (Eq. 13, Cassan et al., 2016)

$d = k - \frac{\alpha_t u_k}{\kappa u_*}$

which gives

$Q_{sup} = \frac{u_*}{\kappa} \left( (h - d) \left( \ln \left( \frac{h-d}{z_0} \right) - 1\right) - \left( (k - d) \left( \ln \left( \frac{k-d}{z_0} \right) - 1 \right) \right) \right)$

## 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:

$u_0 = \sqrt{\frac{2 g S D (1 - \sigma C)}{C_d f_F(F) C (1 + N)}}$

with

$N = \frac{\alpha C_f}{C_d f_F(F) C h_*}$

with

$\alpha = 1 - (a_y / a_x \times C)$

## Formulas used

### Bulk velocity V

$V = \frac{Q}{B \times h}$

### Average speed between blocks Vg

From Eq. 1 Cassan et al (2016)1 and Eq. 1 Cassan et al (2014)2:

$V_g = \frac{V}{1 - \sqrt{(a_x/a_y)C}}$

### Drag coefficient of a single block Cd0

$$C_{d0}$$ is the drag coefficient of a block considering a single block infinitely high with $$F << 1$$ (Cassan et al, 20142).

Block shape Cylinder "Rounded face" shape Square-based parallelepiped "Flat face" shape
Value of $$C_{d0}$$ 1.0 1.2-1.3 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 so-called "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 cross-section, 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., 20142 et Tran et al. 2016 [^3]):

• round : $$r_Q=1.1$$
• "rounded face" shape : $$r=1.2$$
• square-based parallelepiped : $$r=1.5$$
• "flat face" shape : $$r=1.6$$

Cassiopée implements a formula depending on $$C{d0}$$:

$r = 0.4 C_{d0} + 0.7$

### Froude F

$F = \frac{V_g}{\sqrt{gh}}$

If $$F < 1$$ (Eq. 19, Cassan et al., 20142):

$f_F(F) = \min \left( \frac{r}{1- \frac{F_{g}^{2}}{4}}, \frac{1}{F^{\frac{2}{3}}} \right)^2$

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 umax

According to equation 19 of Cassan et al, 20142 :

$u_{max} = V_g \sqrt{f_F(F)}$

### Drag coefficient correction function linked to relative depth fh*(h*)

The equation used in Cassiopeia differs slightly from equation 20 of Cassan et al. 20142 and equation 6 of Cassan et al. 20161. This formula is a fit to the experimental measurements on circular blocks used in Cassan et al. 20161:

$f_{h_*}(h_*) = (1 + 1 / h_*^{2})$

### Coefficient of friction of the bed Cf

If $$k_s < 10^{-6} \mathrm{m}$$ then we use Blasius' formula

$C_f = 0.3164 / 4 * Re^{-0.25}$

with

$Re = u_0 \times h / \nu$

Else (Eq. 3, Cassan et al., 2016 d'après Rice et al., 1998)

$C_f = \frac{2}{(5.1 \mathrm{log} (h/k_s)+6)^2}$

## 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 (m3/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 $$\tilde{z} = z / k$$

1. Cassan L, Laurens P. 2016. Design of emergent and submerged rock-ramp fish passes. Knowl. Manag. Aquat. Ecosyst., 417, 45

2. Cassan, L., Tien, T.D., Courret, D., Laurens, P., Dartus, D., 2014. Hydraulic Resistance of Emergent Macroroughness at Large Froude Numbers: Design of Nature-Like Fishpasses. Journal of Hydraulic Engineering 140, 04014043. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000910