An investigation of surge wave profiles in open channel flow

When a sudden increase in the discharge occurs in an open channel, a surge wave is formed. This body of water appears to move along the initial surface. Depending on the discharge this surge can be undular, breaking undular, or steep fronted as the discharge increases.

A theoretical expression has been derived for the undular form, but no allowances have been made in the theory for the effect of forces that cause the waves to break. To simplify matters the surge was assumed to have been arrested, by superimposing on it a velocity equal and opposite to that of the mean velocity of the head of the surge. Then it appears as an undular hydraulic jump, with moving boundaries. The expression for the profile is derived for permanent flow,^if solved alone, with no allowance for friction gives a solitary wave profile. Hence two further expressions have been derived for the changes in energy and momentum. After simplifying and assuming that the channel bed is horizontal and the channel cross section is rectangular, the resulting non dimensional equations are:-

1/6 (dY/dX)² = EY² - Y/2³ +1/2 - $Y (41)

dY/dX is the slope of the water surface. E refers to energy, $ to momentum, and Y to depth (y = ycY. yc - critical depth).

d$/dX = g/E²[1/Y₀ - 1/Y]²[1 + 2y/l] (71) C - coefficient of friction. Suffix ₀ - initial conditions. l - width of channel. y - depth of water.

dE/dX = g/YC²[1/Y₀ - 1/Y]²[1 + 2y/l] (72)

With the small channel used in the experiments, allowances had to be made for wall friction (1 + 2Y/l).

Benjamin and Lighthill show that this undular form of surge is not possible unless losses in energy and momentum occur.

The waves are termed 'cnoidal' waves because the profile can be represented, to a very close approximation by, the graph of the square of the Jacobean elliptic function cn x. The term 'cnoidal' was coined by Korteweg and de Vries.

The Equations 41, 71 and 72 were then obtained in a form suitable for computation, and a number of numerical examples were solved. The resulting profiles were checked by experiment, and the agreement between the results was considered to be good.

It is believed that if the calculations were made for greater initial depths then those possible in the model channel, that there would be greater agreement with recorded values. This is because of the uncertainty of the determination of the value of the coefficient of friction at low Reynolds numbers. At high values of R the friction coefficient can be determined more accurately. It is thought that probably the values derived from the Bazin, Manning, or Gauckler-Strickler formulae would then be suitable.

A considerable number of experimental determinations of wave profiles were made, and the results listed in graphical and tabular form. The curves show that until breaking occurs there is a definite dependence between wave length and amplitude of the waves.

Probably the most significant result of this study of the undular surge, is the realisation of the importance of the effect of friction on the shape of the waves constituting the surge. In a rough-sided channel for a given Y₀, the crest height is greater and increases more rapidly from wave to wave, and the wave length is shorter than in a channel with a smoother surface.