fente_noyee.md

# Submerged slot formula

![Submerged slot diagram](fente_noyee_schema.png)

*Excerpt from Larinier, M., Travade, F., Porcher, J.-P., Gosset, C., 1992. Passes à poissons : expertise et conception des ouvrages de franchissement. CSP. (page 94)*

Larinier (1992) suggests the following equation:

$$Q = C_d b H_1\sqrt{2g \Delta H}$$

With:

* *b* the slot width in m
* *H<sub>1</sub>* the head on the slot m
* *C<sub>d</sub>* the discharge coefficient.

# Discharge coefficient **C<sub>d</sub>** for the submerged slot formula (vertical slot fish ladder)

The discharge coefficient **C<sub>d</sub>** is an important parameter for the design of vertical slot weirs. This term is integrated in the formula of conveyance of a submerged slot, which connects the transited flow with the fall between basins, the upstream head on the slot and its width.

**For single vertical slot weirs**, model-reduced studies and numerical simulations have determined average values of discharge coefficients for different configurations, for a typical slot and baffle geometry defined from vertical slot weirs built in France (Figure 1). A different slot and baffle geometry can induce deviations from the average values of the discharge coefficients according to the configurations indicated below.

![Figure 1](fente_noyee_fig1.png)

*Figure 1: typical slot and baffle geometry from which the discharge coefficient values were determined (Ballu, 2017)*

* **Under smooth apron conditions (no bottom roughness) and in the absence of a weir in the slot**, Wang *et al.* (2010) and Ballu *et al.* (2017) showed that the value of **C<sub>d</sub>** is not significantly influenced by discharge and depends mainly on the slope **S** (%) and the ratio of the basin width **B** to the slot width **b** (**B/b**).

* **Under smooth invert conditions (no bottom roughness) with the presence of a weir in the slot**, under 3 configurations of weir heights **h<sub>s</sub>/b** related to slot width **b** (**h<sub>s</sub>/b** = 0.5, 1, and 1.5), Ballu *et al.* (2015) and Ballu (2017) showed that the value of **C<sub>d</sub>** is not significantly influenced by discharge, and is mainly influenced by slope **S**, aspect ratio **B/b**, and weir height **h<sub>s</sub>/b**.

* **Under apron conditions with precast bottom roughnesses and in the absence of a weir in the slot**, with evenly distributed roughnesses of height **h<sub>r</sub>/b** = 2/3 and diameter **Ø<sub>r</sub>/b** = 1/2, under 2 configurations of densities **d<sub>r</sub>** of 10% and 15%, Ballu *et al.* (2017) and Ballu (2017) showed that the value of **C<sub>d</sub>** is not significantly influenced by discharge, and is mainly influenced by slope **S**, aspect ratio **B/b**, and the presence of the bottom roughness **d<sub>r</sub>**.

Depending on these different configurations, the average values of the discharge coefficient range from about 0.62 to nearly 0.88 (Figure 2).

![Figure 2](fente_noyee_fig2.png)

*Figure 2: average values and uncertainties (95% confidence intervals, k = 2) of the discharge coefficients according to the slope S and the shape ratio B/b, for different configurations: (A) smooth raft without weir in the slot, (B) bottom roughness d<sub>r</sub> = 10% without weir in the slot, (C) bottom roughness d<sub>r</sub> = 15% without weir in the slot, (D) weir in the slot h<sub>s</sub>/b = 0.5 and smooth raft, (E) weir in the slot h<sub>s</sub>/b = 1 and smooth raft, (F) weir in the slot h<sub>s</sub>/b = 1.5 and smooth raft*

**For passes with double vertical slots**, **C<sub>d</sub>** values between 0.75 and 0.80 are commonly used for recently constructed devices with longitudinal slopes between 4% and 5.5%. Studies are under way to clarify the influence of different configurations on discharge coefficients and their corresponding values.

**Bibliography:**

Ballu A., Pineau G., Calluaud D., David L. (2015). Experimental study of the influence of sills on vertical slot fishway flow. *36<sup>th</sup> IAHR World Congress*, 7p.

Ballu A. (2017). Étude numérique et expérimentale de l’écoulement turbulent au sein des passes à poissons à fentes verticales. Analyse de l’écoulement tridimensionnel et instationnaire. *Thèse de l’Université de Poitiers*, 223p.

Ballu A., Calluaud D., Pineau G., david L. (2017). Experimental study of the influence of macro-roughnesses on vertical slot fishway flows. *La Houille Blanche*, 2: 9-14.

Wang R.W., David L., Larinier M. (2010). Contribution of experimental fluid mechanics to the design of vertical slot ﬁsh passes. *Knowledge and Management of Aquatic Ecosystems*, 396(2).