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CROSS SECTION AND
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See also:CROSS See also:SECTION AND SLOPE § 115. In every stream the See also:discharge of which is See also:constant, or may be regarded as constant for the See also:time considered, the velocity at different places depends on the slope of the See also:bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the cross sections normal to the direction of flow. Suppose the See also:mass of See also:water AoBoA1B, comes in a See also:short time 0 to C0DoCID1, and let the See also:work done on the mass be equated to its See also:change of kinetic See also:energy during that See also:period. Let l be the length AoA1 of the portion of the stream considered, and z the fall, of See also:surface level in that distance. Let Q be the discharge of the stream per second. Change of Kinetic Energy.—At the end of the time 0 there are as many particles possessing the same velocities in the space CoDoAIB1 as at the beginning. The change of kinetic energy is therefore the difference of the kinetic energies of AoBoCoDo and AlBiCID1. Let fig. 119 represent the cross section AoBo, and let w be a small See also:element of its See also:area at a point where the velocity is v. Let S2o be the whole area of the cross section and uo the mean velocity for the whole cross section. From the See also:definition of mean velocity we have uo =Ewv/52o. Let v=uo+w, where w is the difference between the velocity at the small element w and the mean velocity. For the whole cross section, now = O. The mass of fluid passing through the element of section w, in 0 seconds, is (G/g)wvo, and its kinetic energy is (G%2g)wv2O. For the whole section, the kinetic energy of the mass AoBoCoDo passing in 9 seconds is (GB/2g)Ewv3 = (GO/2g)Ew(uo3+3uo2W+3uow2+w'), _ (GO/2g){uo312+Eww2(3uo+ze ) The See also:factor 3uo+w is equal to 2uo+v, a quantity necessarily See also:positive. Consequently Ewv3> flouo3, and consequently the kinetic energy of AoBoCoDo is greater than (GO/2g)12ouo3 or (GO/2g)Quo2, which would be its value if all the particles passing the section had the same velocity uo. Let the kinetic energy be taken at a(GO/2g)12ouo3 = a(GO/2g)Quo2, where a is a corrective factor, the value of which was estimated by J. B. C. J. Belanger at 1.1.1 Its precise value is not of See also:great importance. In a similar way we should obtain for the kinetic energy of AIBICIDI the expression a(Go/2g)S2,uk3 = a(G6/2g)Qui', where f21, ul are the section and mean velocity at A1BI, and where a may be taken to have the same value as before without any important See also:error. Hence the change of kinetic energy in the whole mass AoBoAIB1 in 0 seconds is a(G0/2g)Q(uk2-uo2). (I) See also:Motive Work of the See also:Weight and Pressures.—Consider a small filament aoa1 which comes in 0 seconds to See also:cock. The work done by gravity during that See also:movement is the same as if the portion aoco were carried to See also:ale'. Let dQo be the See also:volume of aoco or alcl, and yo, yk the depths of ao, al from the surface of the stream. Then the volume Boussinesq has shown that this mode of determining the corrective factor a is not satisfactory. C: See also:Scale A- Inc1t=1 See also:Foot. dQB or GdQB pounds falls through a See also:vertical height z+y,—yo, and the work done by gravity is GdQB (z+yl —yo) . Putting pa for atmospheric pressure, the whole pressure per unit of area at as is Gyo+pa, and that at a, is — (Gyi+pa). The work of these pressures is G(yo+pu/G—yi—pu/G)dQO =G(yo—yi)dQC. Adding this to the work of gravity, the whole work is GzdQO; or, for the whole cross section, GzQB. (2) Work expended in Overcoming the See also:Friction of the Stream Bed.—Let A'B', A"B" be two cross sections at distances s and s+ds from AOB0. Between these sections the velocity may be treated as See also:uniform, because by See also:hypothesis the changes of velocity from section to section are See also:gradual. Hence, to this short length of stream the See also:equation for uniform See also:motion is applicable. But in that See also:case the work in overcoming the friction of the stream bed between A'B' and A"B" is GQBI-(u2/2g) (x/S2)ds, where u, x,12 are the mean velocity, wetted perimeter, and section at A'B'. Hence the whole work lost in friction from AoB0 to See also:AlBi will be ,GQB f ol.(u2/2g)(x/Sc)ds. (3) Equating the work given in (2) and (3) to the change of kinetic energy given in (1), a(GQB/2g) (u12—uo2) =GQzO—GQBfo' (u2/2g)(x/t2)ds; z = a(u12 —uo2)12g+ f -(n2/2g) (x/SI)ds. § i i6. Fundamental See also:Differential Equation of Steady VariedMotion.—Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, aibi, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of ------------------------- ;See also:fir ,d ' mod h I u+du --------------- a and a,, may be written dz. Also, if we write u for uo, and u+du for the See also:term (uo2—u,2)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is dz = udu/g+(x/St)l-(u2/2g)ds. (I) From this equation some See also:general conclusions may be arrived at as to the See also:form of the See also:longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem. Modification of the See also:Formula for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope.—Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac See also:horizontal. Then dz is sensibly equal to a'c. The depths of the stream, h and h+dh, are sensibly equal to ab and a'b', and therefore dh=a'd. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence dz = a'c = cd —a'd = ids —dh. Since the motion is steady Q =12u = constant. Sldu+udtt = o ; ...du= —udit/S2. Let x be the width of the stream, then dtl=xdh very nearly. Inserting this value, du = — (ux(St)dh. (3) Putting the values of du and dz found in (2) and (3) in equation (i), ids —dh = — (u2x/g1t)dh+(x/0) (u2/2g)ds. dh/ds ={i—(x/St)l(u2/2g)}/}i—(u2/g) (x/S2)•} (4) Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width.—The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, xh=9, and if x is large compared with h, S2/x = xh/x ---h nearly. Then equation (4) becomes dhlds=i(i-g-u2/2gih)/(I—u2/gh). (5) § 117. General Indications as to the Form of Water Surface furnished by Equation (5).—Let AoA, (fig. 121) be the water surface, BoB1 the bed in a longitudinal section of the stream, and ab any section at a distance s from Bo, the See also:depth ab being h. Suppose BoB1, BoAo taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the See also:angle which the surface of the stream at a makes with the See also:axis BoB1_ This tangent dh/ds will be positive, if the stream is increasing in depth in the direction BoB1; negative, A if the stream is diminishing in depth from Bo towards Bi. If dh/ds =o. the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4) dh/ds=o, if i—(x/12)l'(u2/2g)=o; -(u2/2g) = (flx)i =mi, which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion. Consider the possible changes of value of the fraction (I —l"u2/2gih)/(I —u2/gh)• As h tends towards the limit o, and consequently is is large, the numerator tends to the limit—co. On the other See also:hand if h = co , in which case u is small, the numerator becomes equal to 1. For a value H of h given by the equation I — l"u2/2gi H =o, H = I u2/2gi, we fall upon the case of uniform motion. The results just stated may be tabulated thus: For h=o,H,>H,co, the numerator has the value —co, o, > o, 1. Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit — co. As h becomes very large and u consequently very small, the de-nominator tends to the limit 1. For h=u2/g, or u=s/ (gh), the denominator becomes zero. Hence, tabulating these results as before: For h =o, u2/g, >u2/g, co, the denominator becomes — co, o, > o, i. § 118. Case 1.-Suppose h>u2/g, and also h> H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let BoB1 be the bed, CoCt a See also:line parallel to the bed and at a height above it equal to H. By hypothesis, the surface AoAI of the stream is above CoC1, and it has just been shown that the depth of the stream increases from Bo towards B1. But going up stream h approaches more and more nearly the value H, and there-fore dh/ds approaches the limit o, or the surface of the stream is asymptotic to C5C1. Going down stream h increases and u diminishes, thenumeratorand denominator of thefraction(' — See also:rue/2gih)/(1—u2jgh) both tend towards the limit i, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line DoD1. The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a See also:weir. The raising of the water surface above the level CoCI is termed the backwater due to the weir. § 119. Case 2.-Suppose h> u2/g, and also h <H. Then dh/ds is ---------------- (2) Differentiating, negative, and the stream is diminishing in depth in the direction of flow. In fig. 123 let BoB0 be the stream bed as before; CoC1 a line See also:drawn parallel to BoB1 at a height above it equal to H. By hypo- thesis the surface A0 Al of the stream is below CoC1, and the depth has just been shown to diminish from Bo towards B1. Going up stream h ap- proaches the limit T H, and dh/ds tends to the limit zero. Q That is, up stream 1 i Bi AoAI is asymptotic ~~~, ~~~ to CoC,. Going down On the See also:Ganges See also:canal, as orig- inally See also:con- structed, there were abrupt falls precisely of this See also:kind, and it appears that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some See also:miles up stream, was not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and considerable damage was done to the See also:works. " When the canal was first opened the water was allowed to pass freely over the crests of the overfalls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles up rapidly followed, and it soon became apparent that means must be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were See also:accord- ingly fixed in the grooves above the See also:bridge See also:arches, or temporary weirs were formed over which the water was allowed to fall; in some cases the surface of the water was thus raised above its normal height, causing a backwater in the channel above " (Crofton's See also:Report on the Ganges Canal, p. 14). Fig. 125 represents in an ex- aggerated form what probably occurred, the See also:diagram being intended to represent some miles' length of the canal bed above the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the See also:ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the See also:crest of the fall, as shown by dotted lines, a new water surfae CC corresponding to Case i would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA. § I20. Case 3.—Suppose a stream flowing uniformly with a depth -'h<u2/g. For a stream in uniform motion 'u2/2g=mi, or if thestream is of indefinitely great width, so that m= H, then 'u2/2g =See also:ill, and H ='u2/2gi. Consequently the See also:condition stated above involves that i u2/2gi <u2/g, or that i> i-/2. If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes hi> u2; g, then for a portion of the stream the depth h will satisfy the conditions h <u2/g and h> H, which are not the same as those assumed in the two previous cases. At some point of the stream above the weir the depth h becomes equal to u2/g, and at that point dh/ds becomes See also:infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the See also:influence of See also:internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the See also:free surface of the stream, or a See also:standing See also:wave is formed, the conditions of motion in which will be examined presently. It appears that the condition necessary to give rise to a standing wave is that i> /2. Now I depends for different channels on the roughness of the channel and its See also:hydraulic mean depth. See also:Bazin calculated the values of for channels of different degrees of roughness and different depths given in the following table, and the corresponding minimum values of i for which the exceptional case of the See also:production of a standing wave may occur. Nature of Bed of Stream. Slope below Standing Wave Formed. which a Stand- See also:ing Wave is "-- impossible in Slope in feet Least Depth feet per foot. per foot. in feet. (0.002 0.262 Very smooth cemented surface 0.00147 < 0.003 •098 10.004 •065 0.003 '394 See also:Ashlar or See also:brickwork . . 0.00186 0.004 .197 0.006 •098 (0.004 1.181 See also:Rubble See also:masonry . . . Additional information and CommentsThere are no comments yet for this article.
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