Plane baffles (Denil) fishway

Geometrical characteristics

Excerpt from Larinier, 2002¹

Hydraulic laws given by abacuses

Experiments conducted by Larinier, 2002¹ 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% (Excerpt from Larinier, 2002¹)

Abacuses of a plane baffles (Denil) fishway for a slope of 15% (Excerpt from Larinier, 2002¹)

Abacuses of a plane baffles (Denil) fishway for a slope of 20% (Excerpt from Larinier, 2002¹)

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 \(Z_m\)

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

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1. Larinier, M. 2002. “BAFFLE FISHWAYS.” Bulletin Français de La Pêche et de La Pisciculture, no. 364: 83–101. doi:10.1051/kmae/2002109. ↩↩↩↩↩