theorie_plans.md

# Plane baffles (Denil) fishway

## Geometrical characteristics

![Characteristics of a plane baffles (Denil) fishway](theorie_plans_schema.png)

*Excerpt from Larinier, 2002[^1]*

## Hydraulic laws given by abacuses

Experiments conducted by Larinier, 2002[^1] allowed to establish abacuses that link adimensional flow \(Q^*\):

$$ Q^* = \dfrac{Q}{\sqrt{g}L^{2,5}} $$

 to upstream head \(ha\) and the average water level in the pass \(h\) :

![Abacuses of a plane baffles (Denil) fishway for a slope of 10%](baffle_fishway_plans_slope_10_.png)

*Abacuses of a plane baffles (Denil) fishway for a slope of 10% (Excerpt from Larinier, 2002[^1])*

![Abacuses of a plane baffles (Denil) fishway for a slope of 15%](baffle_fishway_plans_slope_15_.png)

*Abacuses of a plane baffles (Denil) fishway for a slope of 15% (Excerpt from Larinier, 2002[^1])*

![Abacuses of a plane baffles (Denil) fishway for a slope of 20%](baffle_fishway_plans_slope_20_.png)

*Abacuses of a plane baffles (Denil) fishway for a slope of 20% (Excerpt from Larinier, 2002[^1])*

To run calculations for all slopes between 8% and 22%, polynomes coefficients of abacuses above are themelves adjusted in the form of slope \(S\) depending polynomes.

We thus have:

$$ ha/L = a_2(S) Q^{*2} + a_1(S) Q^* + a_0(S) $$

$$a_2(S) = 315.110S^2 - 115.164S + 6.85371$$

$$a_1(S) = - 184.043S^2 + 59.7073S - 0.530737$$

$$a_0(S) = 15.2115S^2 - 5.22606S + 0.633654$$

And:

$$ h/L = b_2(S) Q^{*2} + b_1(S) Q^* + b_0 $$

$$b_2(S) = 347.368S^2 - 130.698S + 8.14521$$

$$b_1(S) = - 139.382S^2 + 47.2186S + 0.0547598$$

$$b_0(S) = 16.7218S^2 - 6.09624S + 0.834851$$

## Calculation of \(ha\), \(h\) and \(Q\)

We can then use those coefficients to calculate \(ha\), \(h\) and \(Q^*\):

$$ ha = L \left( a_2 (Q^*)^2 + a_1 Q^* + a_0 \right)$$

$$ h = L \left( b_2 (Q^*)^2 + b_1 Q^* + b_0 \right)$$

Using the positive inverse function, depending on \(ha/L\), we get:

$$ Q^* = \dfrac{-a_1 + \sqrt{a_1^2 - 4 a_2 (a_0 - h_a/L)}}{2 a_2}$$

And we finally have:

$$ Q = Q^* \sqrt{g} L^{2,5} $$

Calculation limitations of \(Q^*\), \(ha/L\) and \(h/L\) are determined based on the extremities of the abacuses curves.

## Flow velocity

Flow velocity \(V\) corresponds to the minimum flow speed given the flow section \(A_w\) at the perpendicular of the baffle :

$$ V = \dfrac{Q}{A_w} $$

for plane baffles fishways using the notation of the schema above, we have:

$$ A_w = B \times \left( h - \dfrac{C+D}{2} \sin(45°) \right)$$

Which gives with standard proportions:

$$ A_w = L \left(0.583 h - 0.146L \right) $$

## Upstream apron elevation \(Z_{r1}\)

$$ Z_{r1} = Z_{d1} - D \sin(45° + \arctan(S)) $$

## Minimal rake height of upstream side walls \(Z_m\)

$$ Z_m = Z_{r1} + - H_{min} \sin(45° + \arctan(S)) $$

[^1]: Larinier, M. 2002. “BAFFLE FISHWAYS.” Bulletin Français de La Pêche et de La Pisciculture, no. 364: 83–101. doi:[10.1051/kmae/2002109](https://doi.org/10.1051/kmae/2002109).