Conditions of non-uniform flow exist at the outlet of all sewers, except under the unusual conditions where the depth of flow in the sewer under conditions of steady, uniform flow with the given rate of discharge would raise the surface of water in the sewer, at the point of discharge, to the same elevation as the surface of the body of water into which discharge is taking place. By an application of the principles of non-uniform flow to the design of outfall sewers, smaller sewers, steeper grades, greater depth of cover, and other advantages can be obtained.
The backwater curve is caused by an obstruction in the sewer, by a flattening of the slope of the invert, or by allowing the sewer to discharge into a body of water whose surface elevation would be above the surface of the water in the sewer, at the point of discharge, under conditions of steady, uniform flow with the given rate of discharge.
The drop-down curve is caused by a sudden steepening of the slope of the invert; by allowing a free discharge; or by allowing a discharge into a body of water whose surface elevation would be below the surface of the water in the sewer, at the point of discharge, under conditions of steady, uniform flow with the given rate of discharge. The last described condition is common at the outlet of many sewers, hence the common occurrence of the drop-down curve.
The hydraulic jump is a phenomenon which is seldom considered in sewer design. If not guarded against it may cause trouble at overflow weirs and at other control devices, in grit chambers, and at unexpected places. The causes of the hydraulic jump are sufficiently well understood to permit designs that will avoid its occurrence, but if it is allowed to occur the exact place of the occurrence of the jump and its height are difficult, if not impossible, to determine under the present state of knowledge concerning them. The hydraulic jump will occur when a high velocity of flow is interrupted by an obstruction in the channel, by a change in grade of the invert, or the approach of the velocity to the “critical” velocity. The “critical” velocity is equal to √(gh), where h is the depth of flow and g is the acceleration due to gravity. The velocity in the channel above the jump must be greater than √(gh1), where h1 is the depth of flow in the channel above the jump. The velocity in the channel below the jump must be greater than √(gh2), where h2 is the depth of flow below the jump. The jump will not take place unless the slope of the invert of the channel is greater than g
C2,in which C is the coefficient in the Chezy formula. With this information it is possible to avoid the jump by slowing down the velocity by the installation of drop manholes, flight sewers, or by other expedients.
The shape of the drop-down curve can be expressed, in some cases, by mathematical formulas of more or less simplicity, dependent on the shape of the conduit. The formula for a circular conduit is complicated. Due to the assumptions which must be made in the deduction of these formulas, the results obtained by their use are of no greater value than those obtained by approximate methods. A method for the determination of the drop-down curve is given by C. D. Hill.[[32]] In this method it is necessary that the rate of flow past all sections shall be the same; that the depth of submergence at the outlet shall be known; and that the depth of flow at some unknown distance up the stream shall be assumed. The shape and material of construction of the sewer and the slope of the invert should also be known. The problem is then to determine the distance between cross-sections, one where the depth of flow is known, and the other where the depth of flow has been assumed. This distance can be expressed as follows:
L = (d2 − d1) − (H1 − H2)
S − S1 = d′ − H′
S′,
in which L = the distance between cross-sections; d1 = the depth of flow at the lower section; d2 = the depth of flow at the upper section; H1 = the velocity head at the lower section; H2 = the velocity head at the upper section; S = the hydraulic slope of the stream surface; S1 = the slope of the invert of the sewer.
In order to solve such problems with a satisfactory degree of accuracy the difference between d1 and d2 should be taken sufficiently small to divide the entire length of the sewer to be investigated into a large number of sections. The solution of the problem requires the determination of the wetted area, the hydraulic radius, and other hydraulic elements at many sections. The labor involved can be simplified by the use of diagrams, such as Fig. 19, or by specially prepared diagrams such as those accompanying the original article by C. D. Hill. The solution of the problem can be simplified by tabulating the computations as follows:
| Drop-down Curve Computation Sheet | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Uniform discharge. Varying depth | ||||||||||||
| D = Q = A = V = Q A = S1 = L = d1 − H1) S1 | ||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Depth | R | H | H1 | d1 − H1 | V | S | S1 | L | Elevation | |||
| D | d | d1 | Sewer | W. L. | ||||||||
At the head of the computation sheet should be recorded the diameter of the sewer in feet, the assumed volume of flow, the area of the full cross-section of the sewer, the velocity of the assumed volume flowing through the full bore of the sewer, and the gradient or slope of the invert. In the 1st column enter the assumed depth in decimal parts of the diameter for each cross-section; in the 2nd column enter the same depth in feet; in the 3rd column enter the difference in feet between the successive cross-sections; in the 4th column enter the hydraulic radius corresponding to the depth at each cross-section; in the 8th column enter the velocity, equal to the volume divided by the wetted area, for each cross-section; in the 5th column enter the corresponding velocity head; in the 6th column enter the difference between the velocity heads at successive cross-sections; in the 7th column enter the difference between the quantities in the third and in the sixth columns; in the 9th column enter the hydraulic slope corresponding to the velocity and hydraulic radius of each cross-section; in the 10th column enter the difference between the hydraulic slope and the slope or gradient of the sewer; in the 11th column enter the computed distance between successive cross-sections; in the 12th column enter the elevation of the bottom of the sewer at each cross-section; and in the 13th column enter the corresponding elevation of the surface of the water.