Commit c5e7790f authored by Dorchies David's avatar Dorchies David
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doc(macrorugo): English corrections

Refs #477, #488
Showing with 30 additions and 9 deletions
+30 -9
...@@ -24,4 +24,4 @@ It requires the following values to be entered: ...@@ -24,4 +24,4 @@ It requires the following values to be entered:
- The background roughness (m); - The background roughness (m);
- The width of the blocks \(D\) facing the flow (m); - The width of the blocks \(D\) facing the flow (m);
- The useful height of the blocks \(k\) (m); - The useful height of the blocks \(k\) (m);
- The shape parameter of the blocks (1 for round, 2 for square) - The drag coefficient of a single block (1 for round, 2 for square).
# Calculation of the flow rate of a rock-ramp pass # 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 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].
*Cassan L, Laurens P. 2016. Design of emergent and submerged rock-ramp fish passes. Knowl. Manag. Aquat. Ecosyst. 417, 45, <https://doi.org/10.1051/kmae/2016032>*.
## General calculation principle ## General calculation principle
...@@ -55,7 +54,9 @@ with ...@@ -55,7 +54,9 @@ with
$$\beta = \sqrt{(k / \alpha_t)(C_d C k / D)/(1 - \sigma C)}$$ $$\beta = \sqrt{(k / \alpha_t)(C_d C k / D)/(1 - \sigma C)}$$
with \(\sigma = 1\) for \(C_{d0} = 2\) (square blocks), \(\sigma = \pi/4\) otherwise (circular blocks) with
$$C_d = C_{d0} (1 + 1 / h_*^2)$$
and \(\alpha_t\) obtained by solving the following equation: and \(\alpha_t\) obtained by solving the following equation:
...@@ -93,13 +94,11 @@ $$Q_{sup} = \frac{u_*}{\kappa} \left( (h - d) \left( \ln \left( \frac{h-d}{z_0} ...@@ -93,13 +94,11 @@ $$Q_{sup} = \frac{u_*}{\kappa} \left( (h - d) \left( \ln \left( \frac{h-d}{z_0}
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: 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:
$$u_0 = \sqrt{\frac{2 g S D (1 - \sigma C)}{C_d C (1 + N)}}$$ $$u_0 = \sqrt{\frac{2 g S D (1 - \sigma C)}{C_d f_F(F) C (1 + N)}}$$
with with
$$C_d = C_{d0} (1 + 0.4 / h_*^2) f_F(F)$$ $$N = \frac{\alpha C_f}{C_d f_F(F) C h_*}$$
$$N = \frac{\alpha C_f}{C_d C h_*}$$
with with
...@@ -115,6 +114,24 @@ $$V = \frac{Q}{B \times h}$$ ...@@ -115,6 +114,24 @@ $$V = \frac{Q}{B \times h}$$
$$V_g = \frac{V}{1 - \sqrt{(a_x/a_y)C}}$$ $$V_g = \frac{V}{1 - \sqrt{(a_x/a_y)C}}$$
<div style="position: relative"><a id="coefficient-de-trainee-dun-bloc-cd0" style="position: absolute; top: -60px;"></a></div>
### Drag coefficient of a single block *C<sub>d0</sub>*
\(C_{d0}\) is the drag coefficient of a block considering a single block
infinitely high with \(F << 1\) (Cassan et al, 2016[^1]).
### 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, \(\sigma = 1\).
The formula used in Cassiopée has been revised to better correspond to the experimental measurements and to take into account the intermediate block shapes between circular and square.:
$$ \sigma = 0.4 C_{d0} + 0.7 $$
We now have \(\sigma = 1.1\) for a circular block and \(\sigma = 1.5\) for a square block.
### Froude *F* ### Froude *F*
$$F = \frac{V_g}{\sqrt{gh}}$$ $$F = \frac{V_g}{\sqrt{gh}}$$
...@@ -123,7 +140,7 @@ $$F = \frac{V_g}{\sqrt{gh}}$$ ...@@ -123,7 +140,7 @@ $$F = \frac{V_g}{\sqrt{gh}}$$
If \(F < 1.3\) (Eq. 5, Cassan et al., 2016) If \(F < 1.3\) (Eq. 5, Cassan et al., 2016)
$$f_F(F) = \mathrm{min} \left( \frac{0.4 C_{d0} + 0.7}{1- (F^2 / 4)}, \frac{1}{F^{\frac{2}{3}}} \right)^2$$ $$f_F(F) = \mathrm{min} \left( \frac{\sigma}{1- (F^2 / 4)}, \frac{1}{F^{\frac{2}{3}}} \right)^2$$
else else
...@@ -180,3 +197,7 @@ $$C_f = \frac{2}{(5.1 \mathrm{log} (h/k_s)+6)^2}$$ ...@@ -180,3 +197,7 @@ $$C_f = \frac{2}{(5.1 \mathrm{log} (h/k_s)+6)^2}$$
- \(z\): vertical position (m) - \(z\): vertical position (m)
- \(z_0\): hydraulic roughness (m) - \(z_0\): hydraulic roughness (m)
- \(\tilde{z}\): dimensionless stand \(\tilde{z} = z / k\) - \(\tilde{z}\): 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
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