# CEM88(V) : Weir / Undershot gate (small sill elevation)

**Figure 18. Device schematic view**

### Weir - Free-flow

### Weir - Submerged

**[17]**

with coefficient of reduction for submerged flow.

The flow reduction coefficient is a function of and of the value of this ratio at the instant of the free-flow/submerged transition. The submerged conditions are obtained when . The law of variation of the coefficient has been derived from experimental results ().

Let :

If :

If :

With

One calculates an equivalent coefficient for free-flow conditions as before.

### Undershot gate - Free-flow

**[18]**

It has been established experimentally that the undershot gate discharge coefficient increases with . A law of variation of of the following form is adopted:

avec :

Hence,

In order to ensure the continuity with the open channel free-flow conditions for , we must have:

Hence, for

### Undershot gate - Submerged

#### Partially submerged flow

**[19]**

being the same as for open channel flow.

The following free-flow/submerged transition law has been derived on the basis of experimental results:

In order to ensure continuity with the open channel flow conditions, the free-flow/submerged transition under open channel conditions has to be realized for instead of in the weir/orifice formulation.

#### Totally submerged flow

**[20]**

The equation is the same as the one for where is replaced by (and by ) for the calculation of the coefficient and (and therefore for the calculation of ).

The transition to totally submerged flow occurs for:

with:

()

The functioning of the weir / undershot gate device is represented by the above equations and displayed in figure 20. Whatever the conditions of the pipe flow, one calculates an equivalent free-flow discharge coefficient, corresponding to the classical equation for the free-flow undershot gate.

The reference coefficient introduced for the device is the classic coefficient of the free-flow undershot gate, usually close to . It is then transformed to which allows to compute and from equation **[18]** for the free-flow undershot gate.

Remark: it is possible to get , even under free flow conditions, since the discharge coefficient increases with the ratio.

(12): Weir - Free flow

(19): Undershot gate - Partially submerged

(17): Weir - Submerged

(20): Undershot gate - Totally submerged

(18): Undershot gate - Free flow

**Figure 20. Weir - Undershot gate**

Equations are also available in a Matlab script file (function Qouvrage) here.