# Uniform discharge in a rectangular flume

**I** non-uniform discharge, **II** uniform discharge;

**h** depth of discharge, **Js** uniform bottom slope,** Jw** slope of water surface profi le,** L0** length of the flume with bottom slope **Js** and constant width, **v** flow velocity,** red frame** control volume

In uniform open-channel flow the discharge depth **h** remains equal, i.e. parallel to the bottom. This also means that the flow velocity **v** remains constant. The discharge depth h can also be described as a pressure head (a component of the specifi c energy). These energy heads are often applied in the form of what are known as grade lines. In the energy grade line** J** the most significant component in many cases is the discharge depth **h**. In uniform open-channel flow the energy grade line **J** is equal to the bottom slope **Js** and thus equal to the discharge depth **h**. In uniform open-channel flow the **normal discharge** prevails, i.e. the bottom slope** Js** balances out the friction losses in the discharge **Q**. The energy grade line, water surface profile and bottom slope are all parallel.

**blue** energy grade line **J:** hv/L=(E1-E2)/L

**green** slope of water surface profi le **Jw:** [(h1+z1)-(h2+z2)]/L

**red** bottom slope **Js:** (z1-z2)/L

According to Bernoulli, the total energy Etot is composed of three components:

velocity head (v²/2g)

pressure head (h=p/ρg)

elevation (z)

### Flow formulae

Flow formulae describe the relationship between the discharge **Q** and the discharge depth **h** at a given shape of cross-section and roughness characteristic. The shape of cross-section is taken into account in the hydraulic radius; the discharge depth **h** comes into play via the energy grade line **J**.

Commonly used formulae for general flumes are

Darcy-Weisbach

Manning-Strickler (also Gauckler-Manning-Strickler).

Flow formulae are based on empirical values.