472 HYDROMECHANICS [HYDRAULICS. tained as in the preceding section. Thus for a rectangular notch, put H^Oin (8). Then 5 Q = cBV2# H (11), where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 48 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small. Since the area of the notch opening is BH, the above formula is of the form where & is a factor depending on the form of the notch and express ing the ratio of the mean velocity of discharge to the velocity due to the depth H. 38. Francis s Formula for Rectangular Notches. The jet dis charged through a rectangular notch has a section smaller than BH, (a) because of the fall of the water surface from the point where H is measured towards the weir, (b) in con sequence of the crest contraction, (c) in consequence of the end contrac tions. It may be pointed out that while the- diminu tion of the section of the jet due to the surface fall and to the crest contrac tion is proportional to the length of the weir, the end con tractions have near ly the same effect whether the weir is wide or narrow. Mr Francis s ex periments showed that a perfect end contraction, when the heads varied from 3 to 24 inches, and the length of the weir was not less than three times the head, dimin ished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if I is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be Z-0 2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Hence, generally, the width of the jet is I - O lnH, where n is the number of end con tractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(l - lwH)H and ( 37) the mean velocity will be f /2<yH. Consequently the discharge will be given by an equation of the form Q = lc(l - -1 iH)H V 20H = 5-35c(/-0-l?iH)H ? . This is Francis s formula, in which the coefficient of discharge c is much more nearly constant for different values of I and h than in Fig. 49. the ordinary formula. Francis found for c the mean value 622, tliu weir being sharp-edged. 39. Triangular Notch (fig. 49). Consider a lamina issuing be tween the depths h and h + dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is /2gh. Hence the dis charge for this lamina is I V2f//t dh . But B II b H - h Hence discharge of lamina and total dischare of notch j// or, introducing a coefficient to allow for contraction, When a notch is used to gauge a stream of varying flow, the ratio varies if the notch is rectangular, but is constant if the notch is
H triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with differ ent values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 617. It will be seen, as in 37, that since ^BH is the area of section of the stream through the notch, the formula is again of the form where & = W i g the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form. 40. Weir with a Broad Sloping Crest. Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a move ment sensibly rectilinear and uniform (fig. 50). Let the inner edge be Fig. 50. so rounded as to prevent a crest contraction. Consider a filament aa , the point a being so far back from the weir that the velocity of approach is negligible. Let 00 be the surface level in the reservoir, and let a be at a height h" below 00, and h above a . Let h be the distance from 00 to the weir crest and c the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh"; at it is Gz. If v be the velocity at a , Theory does not furnish a value for c, but Q = for c = and for c = h. Q has therefore a maximum for a value of c between and fi, obtained by equating ^ to zero. This gives c = h, and, inserting this value, as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately ap plicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction or* the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent
to the ordinary weir formula with c = 577.
