Energy Variation in Rivers
Water above sea level has potential energy (PE). Its quantity is proportional to the mass of the water, and the vertical distance to sea level.
Kinetic energy (KE) is caused by movement, and is derived from potential energy.
Energy is lost to overcome the internal friction of the water (viscosity) and the friction with the bed of the channel.
KE represented by the velocity of the water in the channel does work in eroding and transporting sediment.
If the channel gradient is steep, then the change from PE to KE is rapid and the velocity is higher.
Conversely, on gentle gradients the velocity is lower.
The amount of work done depends not only on the velocity, but the mass and thus volume. The volume passing a given point in one second is the discharge. The greater the discharge, the greater the total energy of the river.
KE is represented by velocity. It is determined by a combination of internal friction, bed friction, slope of the channel, discharge and the size of the channel.

Types of Flow
Turbulent Flow
When a water particle flows along a open channel, it does not follow a direct path, nor a smoothly curving trajectory. It moves vertically and laterally in eddies as well as its overall downstream direction. The true velocity of this particle is much higher than the average velocity. It predominates in natural river channels and accounts for their efficiency in eroding and transporting sediment. Turbulence varies with the velocity of the river, which in turn, depends upon the amount of energy available after friction has been overcome. It is estimated that under ‘normal’ conditions, about 95% of a river’s energy is expended in order to overcome friction.
Laminar flow
This is so rarely encountered except for under laboratory conditions that it is usually discounted. It can be found at lower flow velocities. It is recognised by nearly linear trajectories of water particles. Such a method of flow would travel over sediment on the river bed without disturbing it.

Velocity and Discharge
Velocity varies vertically and laterally across a river so we refer to mean velocity as ‘velocity’.
Frequently velocity is measured only at the surface. We know that this is the fastest point in the river profile, thus this value is often multiplied by 0.8 to give an approximation to mean velocity. The best depth for measuring velocity is at 60-70% depth.
The line at right angles from bank to bank is the width, and the depth is the vertical distance from the water surface to the bed of the channel. If the depth is measured at equally spaced points along a width line, a reasonably accurate cross section can be drawn. Either by reading off the diagram, or by taking the average depth and multiplying by the width, the Cross Sectional Area (CSA)of the river can be found. The length of bed in contact with the water can be measured, and is known as the wetted perimeter.
The volume of water passing a point in a given time is the discharge. It depends on velocity and CSA at the point. Usually, the equation is written as
Q         =          A          X          V
Discharge = Cross sectional area x Velocity

Friction of the water with the bed
The effect of the bed on flow is determined by how much water comes in to contact with it. The relationship between the wetted perimeter and the CSA is termed the hydraulic radius, and forms the equation;
Hydraulic Radius = CSA / Wetted Perimeter
This value indicates the amount of water that each metre of cross section affects in the channel. A large hydraulic radius means that there is a small amount of water in contact with the wetted perimeter. This creates less friction, which in turn reduces energy loss and allows greater velocity. This creates higher efficiency.

The shape of the X-section controls the point of maximum velocity in a river’s channel. The point of max. velocity in a river with a straight course which is likely to have a symmetrical channel is different to that of a meandering chanel which is likely to have an asymmetrical X-section.
The friction of the bed varies with how irregular the bed of the river is. If the bed is silt, it has a lower frictional effect than a bed of rocks and boulders.

Manning’s Equation (Manning’s n)
Linking the majority of the variables discussed previously, Manning (1889) devised a formula which states;
Q = A x ((R0.67 x S0.5) / n)
where  Q = Discharge
            A = Cross sectional area
            R = Hydraulic Radius
            S = Slope (Gradient)
            n = Manning’s coefficient of bed roughness
The slope is measured as a fraction, the vertical fall of water in the vicinity of a section divided by the horizontal distance over which the fall takes place. For example, if a river falls 1m in 1000m, the slope will be 0.001.

Bed Profile

Sand and Gravel

Coarse Gravel










Highly irregular




Table 1: Manning's Frictional Coefficiency Values

By rearranging this formula, we can calculate n in a field situation and thus classify a channel. Manning’s formula is useful in estimating the discharge in flood conditions. The height of the water can be assessed from debris stranded in trees or high on the bank. All that need be measured is the cross section and the slope.

Results of Previous Research
In 1953, Leopold and Maddock published the results of their research in to the hydraulic geometry of streams. They found that in both normal and flood conditions, stream width, depth and velocity increase as simple power functions of discharge. From records of streams and rivers in central and south west USA they obtained the direct relationships.

W is proportional to Q0.5
            where W is width and Q is discharge
            - the width of the channel increases as the square root of discharge
D is proportional to Q0.4
            where D is mean depth
            - the depth of the channel increases nearly as the square root of discharge
V is proportional to Q0.1
            where V is mean velocity
-the mean velocity increases as the tenth root of discharge

These relationships are shown graphically. As the volume of water flow, and hence discharge increase downstream, it follows that width, mean depth and mean velocity all increase with increasing distance from the source. It was accepted that width and depth increased downstream, but more surprising that velocity increases. However, a river flows faster and more efficiently with increased depth, and this accounts for increased velocity.