# Froude number and critical discharge

Top: behaviour of the discharge depth h of an open-channel fl ow with permanent disturbance, bottom: propagation of a surface wave after a momentary disturbance (red dot, blue lines = disturbance fronts)

**1** subcritical discharge, **2** critical discharge, **3** supercritical discharge

**Subcritical discharge **

Disturbances in the discharge behaviour are noticeable upstream. The flow velocity **v** is less than the propagation velocity **c** of a surface wave. Subcritical discharge usually has a large discharge depth** h** at low flow velocity **v**.

**Critical discharge **

Disturbances in the discharge behaviour are not noticeable upstream. The flow velocity **v** is equal to the propagation velocity **c **of a surface wave.

**Supercritical discharge**

Disturbances in the discharge behaviour are not noticeable upstream. The flow velocity** v** is greater than the propagation velocity **c** of a surface wave.

The** Froude number **describes the ratio of flow velocity **v** to propagation velocity **c** of a surface wave and therefore serves as a measure of subcritical or supercritical discharge. The same Froude number means a dynamically similar open-channel flow.

Fr < 1: subcritical

Fr = 1: critical

Fr > 1: supercritical

Specific energy diagram with Froude number

**h** discharge depth, **E** specific energy, **Fr** Froude number

Open-channel flow has many similarities with compressible flow. In both cases there is a dimensionless number (Froude or Mach) that characterises the flow. Many of the differences between subcritical and supercritical discharge have analogies in subsonic and supersonic flow.

### Critical discharge (Froude number = 1)

At the minimum specific energy **Emin**, the discharge depth **h** corresponds to the critical depth **hc**. At this point, the Froude number is Fr = 1, there is a prevailing critical discharge and the propagation velocity **c **is equal to the flow velocity **v**. Also, at this point the specific force **F** in the flume is minimal.

Examples where critical depths (critical discharge) may occur

**1** critical depth near free overfall,

**2** change in the bottom slope,

**3** flow over a broad-crested weir,

**4** hydraulic jump