Flow in Open Channels

FLOW IN OPEN CHANNELS As in the case of pipe flow, the earlier experimenters assumec the loss of head during steady flow in an open channel to be pro portional to the square of the velocity, and adopted one or other modification of the Chezy formula v=CV(mi), where in is the hydraulic mean depth (=cross-sectional area [A1-2.--wetted perimeter [P] ), and i is the gradient of the channel The best-known of these formulae are due to Ganguillet one butter. and to Bazin.

form of the section, while the resistance to flow increases as A=P diminishes, it becomes important to determine what form of channel will give the maximum value of A=P for a given value of A, since this will give the maximum discharge for a given slope. Further, as the sectional area of this channel is a minimum, the cost of construction is a minimum, and since in general the perimeter P increases with the area, the cost of lining the channel is also a minimum.

Theoretically the best form of channel is the semicircular section, and for steel and wooden flumes this section is often adopted. For earthen channels the zoidal section (fig. 9) with sides sloping at S horizontal to I vertical is common, and, for rock channels, the rectangular section. It may be shown that the most economical proportions for such sections are obtained when a circle, with its centre in the water surface, touches the sides and bottom. In a where h is the bottom breadth and d the depth.

Such a channel has a hydraulic mean radius equal to d=a. Of the trapezoidal sections, that having side slopes of .5 to I is the most efficient. The section to be adopted, however, depends also on other considerations. The minimum permissible side slope depends on the character of the soil and varies from o to i in rock to 1.5 to 1 in ordinary loamy soil, and 2 to I in loose sandy soil. In loose soil a concrete lining enables side slopes of I to 1 to be used, and, by preventing erosion of the banks, en ables higher velocities of flow to be adapted, while the increased smoothness of the channel enables these velocities to be attained without any greater loss of head. In such a case a concrete lined channel may be cheaper than one which is unlined.

Velocity of Flow in Open Channels.

The permissible ve locity of flow depends on the tendency to erosion of the sides and bed. Experiment shows that the safe velocity increases with the depth. For medium depths in light soil a mean velocity of from 1.2 to 1.8 f.s. is safe, while in firm loamy soil the safe velocity is from 3.0 to 3.5 f.s. On firm well-rammed gravel this may be increased to between 5 and 7 f.s. In a concrete-lined channel faced with cement, the maximum safe velocity with water which carries solid material in suspension is about 9 f.s. A higher velocity wears and roughens the bottom until this roughness reduces the velocity sufficiently to prevent further erosion. With a brick or dry-laid heavy rubble channel the ve locity should not exceed 15 f.s. Any higher velocity necessitates a carefully-laid facing of heavy masonry with cemented joints.

Distribution of Velocity in an Open Channel.—The dis tribution of velocity in a straight channel depends somewhat upon the mean velocity. The maximum velocity is found near the centre and in general below the surface, even with a down stream wind. Its depth varies from • I h to .4h, where h is the depth of the stream. The curves of fig. Io show typical contours of equal velocity, and the distribution of velocity in a series of verticals in a rectangular channel. The curves of fig. II show the variations of velocity in a vertical plane. The effect of an increase in mean velocity in raising the filament of maximum velocity is well shown by these curves. It is found that the depth of the point of mean velocity in any vertical is sensibly inde pendent of the direction of the wind. It varies from about .55h to .70/i, depending on the depth and roughness of the channel as indicated below.

Generally speaking, the velocity at six-tenths depth in any vertical gives the mean velocity in that vertical within 5% ex cept in abnormal cases, while the mean of the velocities at one fifth and four-fifths of the depth also gives the mean velocity within narrow limits. While the surface velocity should only be used for gauging purposes when other measurements are Im practicable, its value, on a still day, is between 8o and i 00% of the mean velocity in its own vertical. This factor increases with the depth of the stream and with the smoothness of the channel.

Gauging of Stream Flow.

The method to be adopted in stream gauging depends on the size of the stream, its state, and on the degree of accuracy required. Where the installation of a weir capable of taking the whole flow is feasible, this forms the most accurate method. For a stream of medium size the rectan gular weir is most suitable. For small flows the triangular notch has advantages. For large streams the weir becomes too costly as a temporary measuring device, and if no permanent weir is available the only way of obtaining the discharge is to measure the mean velocity of the stream and to multiply this by the cross-sectional area. The mean velocity may be obtained in a number of ways. (a) by current meter; (b) by floats; (c) by colour or chemical methods.

Current Meters.

Various types of current meter are in use. In one common type the meter is suspended from a rod or cable, and is provided with a guide vane which keeps its axis per pendicular to the direction of the current. The wheel makes and breaks the contact in an electrical circuit at each fifth and tenth revolution, thus enabling the number of revolutions to be indi cated by means of a buzzer or telephone carried by the observer. The instrument is previously calibrated by towing at known ve locities through still water, the number of revolutions corre sponding to these velocities being recorded. It has the disad vantages that it cannot be used where floating grass or weed is prevalent and that it requires rating at frequent intervals. Further, it is unsuitable for very low ve locities. The minimum permis sible velocity depends on the type of meter, but in general varies from 3 to 6 in. per second.

Meter Observations.—The most usual method of using the meter is the "point" method, in which it is held successively at certain points in a cross section.

In a shallow stream this may be done by mounting it on a staff which is carried by an observer in waders. In deeper streams it is attached to a heavy sinker, and is suspended from a convenient bridge or from a car carried by a cable across the stream, or from an outrigger fixed to an anchored boat.

In this method, the meter may either be held (I) at several equidistant points in a number of equidistant verticals, the mean velocity being deduced from these readings; (2) at six-tenths, or at mid-depth in a series of equidistant verticals, the mean ve locity in each of these verticals then being found by applying a factor; (3) at the surface and bottom only, or at two-tenths and eight-tenths of the depth in a series of verticals; (4) at the surface only. While the first method gives the most accurate re sults in a steady stream, the length of time necessary to obtain the many observations is a serious drawback, and renders it un suitable in a stream which is rising or falling.

Generally speaking, the velocity at .6 of the depth will give the mean velocity in that vertical within 5%, while the velocity at mid-depth multiplied by .96 will give the mean velocity within about 3 per cent. Method (3) in which the surface and bottom velocities are measured, is only suitable for shallow streams. Ex periments show that the results are fairly accurate if the bed is smooth or gravelly, the depth from .4 to i •o ft., and the veloc ity from .5 to 1•5 f t. per second. For deeper streams the mean of readings at .2h and .8h is in close agreement with the mean ve locity in the vertical, and this method is often adopted for gen eral stream gauging.

While it is usually inadvisable to use the surface velocity alone for computing the discharge, it is sometimes impossible in times of flood to make any other measurements. The meter should then be sufficiently submerged to eliminate any disturbance of the surface. Except as affected by the wind, the surface velocity multiplied by a constant which varies from about .85 in a shal low stream to •95 in a deep stream gives the mean velocity in a vertical with a fair degree of approximation.

Soundings.—Simultaneously with the meter observations, soundings should be made from which the cross section of the stream may be obtained.

Having recorded the observations for a series of verticals over the cross section of the stream, these are plotted and a smooth curve is drawn through the plotted points. From this curve ve locities are read off at the top and bottom, and at equal intervals of, say, each 0.5 feet and are set down in order.

Assuming that in a particular case there are six intervals of depth, giving seven such recorded velocities, including the sur face velocity and the bottom velocity the mean velocity in the vertical is then computed from the prismoidal formula for seven abscissae.

vm vi v7+4(v2+v4+v6) + 2 (v3+vb) } • The cross section having been plotted, the areas of the various compartments, having such verticals as their centre lines, may be measured and the discharge through each compartment cal culated. The sum of these gives the total discharge.

Float Measurements.

Floats may be divided into three classes : (I) surface floats; (2) sub-surface floats; (3) rod floats.

Surface floats are liberated at a series of points across the stream at the head of a long straight reach, whose length should be not less than about 200 ft., and the time occupied in covering a measured distance is noted. The surface velocity in each of a number of vertical sections is obtained by repeated observations, and the mean velocity in each vertical is then obtained by multiplying the surface velocity by a factor varying from .85 to -95, depending on the depth and condition of the channel. The stream sections may be marked, in a channel of moderate width, by ropes hanging from a bridge or tempo rary support and trailing in the stream. In a large river this method is impracticable, and observations with the theodolite are necessary to determine the path of the float. The effect of the wind on the surface velocity renders this method of measurement very unsatisfactory.

Sub-surface floats consist of bodies having surfaces of large area, attached to small surface floats for ease of observation, the length of connection being adjusted so as to allow the true float to remain at any given depth. The velocity of the float will then be approximately that of the current at the required depth. A series of such floats liberated at different points in the cross sec tion of a stream, the depth of each being .6 that of the stream at the point of introduction, may be taken as giving the mean veloc ities in their respective sections. This type is more reliable than the surface float. Experiments show that the errors involved by the use of such floats may be between 5 per cent and 25 per cent.

The "rod float" consists of a light wooden rod or tin tube about I in. in diameter, and made in adjustable lengths. The lower end of the bottom section is weighted, and the length adjusted until the rod floats vertically with its lower end clearing the bottom by a few inches. In a large river where these are not likely to inter fere with navigation, logs of wood having their lower ends weighted, may be used. The velocity of the rod gives the mean velocity over the vertical in which it floats. The difficulty in us ing the rod lies in its tendency to drag over shoals and weeds, and to obviate this its lower end may be arranged to float at a height above the bed of the stream.

For such a case Francis gives the empirical formula If 0 be the angle through which the jet is diverted (fig. 13), and if the relative velocity of the water and the vane is unaffected by the impact, the final velocity in the original direction will be v cose, and the final momentum in this direction will be giving the mean velocity in the vertical containing the rod in terms of the velocity of the rod and of and h the depth of the stream. Here should be less than .25h.

In channels of moderate and uniform depth, the rod float is capable of giving results in close agreement with weir gaugings.

Measurement of Velocity by Colour Injection.—The ve locity may be determined by injecting colouring matter into the stream, and noting the time this takes to traverse a measured dis tance. For successful results the colour must be injected in a single burst. In clear water a solution of permanganate of potash may be used. In waters discoloured by organic matter or vege table stains, red or green aniline dye gives good results.

Gauging by Chemical or Electrical Methods.—By adding a strong solution of some chemical, for which sensitive reagents are available, at a uniform and known rate into a stream, and by collecting and analyzing a sample taken from the stream at some point below, where admixture is complete, the volume of flow can readily be computed.

Electrical Method.—This is based on the fact that salt in solution increases the electrical conductivity of water. Two pairs of insulated electrodes (fig. 12) are mounted in the conduit at a measured distance apart, and are coupled to a battery with a voltmeter or ammeter in the circuit. Salt in solution is injected at a single burst at a point above the upper pair of electrodes and the passage of this over the electrodes is indicated by the deflec tion of the needle of the record ing instrument. This method which is also applicable to pipe flow, is quicker, cheaper and probably more accurate than the chemical method.

These two methods are best adapted to rapid and irregular streams in which the admixture is most thorough and which, inci dentally, are most difficult to gauge by other means.

Impact of Jets.—In the case of the impact of a jet on a sta tionary or moving surface, the force exerted in any direction is equal to the rate of change of momentum per second in that direction.

Impact on a Fixed Surface.— Let a= sectional area of jet in square feet, v=initial velocity in feet per second.

Then the weight of water impinging on the surface per second =wav lb.

The initial momentum of this in the original direction of motion Impact on a Series of Moving Vanes.—If the vanes are moving in the original direction of motion of the jet, with velocity u, f.s., the relative velocity of the water and the vane is v—u, and the final absolute velocity of the water in the original direction of motion is of Motion Makes an Angle with That of the Jet.---This problem is one of much importance in the design of impulse turbines. Let a be the angle between the directions of v and of u, and let 8 be the total angle through which the vane is recurved (fig. id). Then if, as is usual, the incidence is tangential, the relative velocity of jet and vane at impact is given by yr2 = — 2 V u cosy (triangle abc) and, neglecting friction and eddy losses, the relative velocity at discharge will be the same as this.

Also, for tangential incidence, the direction of the vane at incidence must be parallel to the direction of the jet relative to the vane, and must therefore make an angle 0 with the direction of motion of the vane where 0 is obtained from the relationship (triangle abc).

v Sin0 = — • sina.

The initial velocity of jet in the direction of motion of vane = v coca• The final velocity of jet in the direction of motion of vane ... Change of momentum per sec. in direction of motion of vane = WQ{ v cosa —u—v, cos(0-}-j3)}1b.

g Force on vane in this WQ direction = { v Cosa — u — cos (9 + j3) } lb.

g BIBLIOGRAPHY. See P. A. M. Parker, Control of Water (1925) ; Bibliography. See P. A. M. Parker, Control of Water (1925) ; Mechanical Properties of Fluids (Blackie, 1923) ; D. Spataro, Trattato di Idraulica (1924) ; A. A. Barnes, Hydraulic Flow Reviewed (1916) ; F. C. Lea, Hydraulics (5923) ; W. C. Unwin, Hydraulics (1918) ; A. H. Gibson, Hydraulics and Its Applications (1924) ; and Le Conte, Hydraulics (59 26) . (A. H. G.)