theorie_mixte.md

# Mixed / chevron baffles fishway

![Characteristics of a mixed / chevron baffles fishway](theorie_mixte_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 \(q^*\) :

$$ q^* = \dfrac{Q/L}{\sqrt{2g}a^{1,5}} $$

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

![Abacuses of a mixed / chevron baffles fishway for a slope of 10%](baffle_fishway_mixte_slope_10_.png)

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

![Abacuses of a mixed / chevron baffles fishway for a slope of 16%](baffle_fishway_mixte_slope_16_.png)

*Abacuses of a mixed / chevron baffles fishway for a slope of 15% (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:

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

$$a_2(S) = 0.188324S - 0.0427461$$

$$a_1(S) =  - 2.47998S + 1.25363$$

$$a_0(S) = 5.02138S + 0.709434$$

And:

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

$$b_2(S) = - 0.0733832S - 0.00839864$$

$$b_1(S) = 0.176261S + 0.661656$$

$$b_0(S) =  - 4.97686S + 1.30546$$

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

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

$$ ha = a \left( a_2 (q^*)^2 + a_1 q^* + a_0 \right)$$

$$ h = a \left( b_2 (q^*)^2 + b_1 q^* + b_0 \right)$$

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

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

And we finally have:

$$ Q = L q^* \sqrt{g} a^{1,5} $$

Calculation limitations of \(q^*\), \(ha/a\) and \(h/a\) 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 mixed / chevron baffles fishways using the notation of the schema above, we have:

$$ A_w = h \times L$$

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

$$ Z_{r1} = Z_{d1} + \frac{3 a S - a}{\sqrt{1 + S^2}} $$

[^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).