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Chapter 3Uniform Flow3.1 INTRODUCTION A flow is said to be uniform if its properties remain constant with respect to distance. As mentioned earlier, the term uniform flow…
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Chapter 3Uniform Flow3.1 INTRODUCTION A flow is said to be uniform if its properties remain constant with respect to distance. As mentioned earlier, the term uniform flow in open channels is understood to mean steady uniform flow. The depth of flow remains constant at all sections in a uniform flow (Fig. 3.1). Considering two sections 1 and 2, the depths and hence Since , it follows that in uniform flow . Thus in a uniform flow, the depth of flow, area of cross-section and velocity of flow remain constant along the channel. The trace of the water surface and channel bottom slope are parallel in uniform flow (Fig.3.1) 3.2 CHEZY EQUATION By definition there is no acceleration in uniform flow. By applying the momentum equation to a control volume encompassing sections 1 and 2, distance L apart, as shown in Fig. 3.1, (3.1) where and are the pressure forces and and are the momentum fluxes at section 1 and 2 respectively = weight of fluid in the control volume and = shear force at the boundary. Since the flow is uniform, Also, where = average shear stress on the wetted perimeter of length and = unit weight of water. Replacing by (= bottom slope), Eq. (3.1) can be written as or (3.2) where is defined as the hydraulic radius. is a length parameter accounting for the shape of the channel. It plays a very important role in developing flow equations which are common to all shapes of channels. Expressing the average shear stress as , where =a coefficient which depends on the nature of the surface and flow parameters, Eq. (3.2) is written as leading to (3.3) where = a coefficient which depends on the nature of the surface and the flow. Equation (3.3) is known as the Chezy formula after the French engineer Antoine Chezy, who is credited with developing this basic simple relationship in 1769. The dimensions of are and it can be made dimensionless by dividing it by . The coefficient is known as the Chezy coefficient.3.3 DARCY-WEISBACH FRICTION FACTOR f Incompressible, turbulent flow over plates, in pipes and ducts has been extensively studied in the fluid mechanics discipline. From the time of Prandtl (1875- 1953) and Von karman (1881 一 1963) research by numerous eminent investigators has enabled considerable understanding of turbulent flow and associated useful practical applications. The basics of velocity distribution and shear resistance in a turbulent flow are available in any good text on fluid mechanics . Only relevant information necessary for our study is summed up in this section.Pipe Flow A surface can be termed hydraulically smooth, rough or in transition depending on the relative thickness of the roughness magnitude to the thickness of the laminar sub-layer. The classification is as follows: where =sand grain roughness, = shear velocity and = kinematic viscosity. For pipe flow, the Darcy-Weisbach equation is (3.4) where = head loss due to friction in a pipe of diameter and length ; = Darcy-Weisbach friction factor. For smooth pipes, is found to be a function of the Reynolds number only. For rough turbulent flows, is a function of the relative roughness and type of roughness and is independent of the Reynolds number. In the transition regime, both the Reynolds number and relative roughness play important roles. The roughness magnitudes for commercial pipes are expressed as equivalent sand-grain roughness . The extensive experimental investigations of pipe flow have yielded the following generally accepted relations for the variation of in various regimes of flow: 1. For smooth walls and (Blasius formula) (3.5) 2. For smooth walls and (karman-Prandtl equation) (3.6) 3.For rough boundaries and (Karman-Prandtl equation) (3.7) 4. For the transition zone (Colebrook-White equation) (3.8) It is usual to show the variation of with and by a three-parameter graph known as the Moody chart. Studies on non-circular conduits, such as rectangular, oval and triangular shapes have shown that by introducing the hydraulic radius ,the formulae developed for pipes are applicable for non-circular ducts also. Since for a circular shape , by replacing by , Eqs. (3.5) through (3.8) can be used for any duct shape provided the conduit areas are close enough to the area of a circumscribing circle or semicircle.Open channels For purposes of flow resistance which essentially takes place in a thin layer adjacent to the wall, an open channel can be considered to be a conduit cut into two. The appropriate hydraulic radius would then be a length parameter and a prediction of the friction factor can be done by using Eqs. (3.5) through (3.8). It should be remembered that and the relative roughness is . Equation (3.4) can then be written for an open channel flow as which on rearranging gives (3.9) Noting that for uniform flow in an open channel = slope of the energy line = = , it may be seen that Eq. (3.9) is the same as Eq. (3.3) with (3.10) For convenience of use, Eq (3.10) along with Eqs (3.5) through (3.8) can be used to prepare a modified Moody chart showing the variation of C with If is to be calculated by using one of the Eqs (3.5) through (3.8), Eqs (3.6) and (3.8) are inconvenient to use as is involved on both sides of the equations. Simplified empirical forms of Eqs (3.6) and (3.8), which are accurate enough for all practical purposes, are given by Jain as follows: (3.6a) and (3.8a) Equation (3.8a) is valid for These two equations are very useful for obtaining explicit solutions of many flow-resistance problems. Generally, the open channels that are encountered in the field are very large in size and also in the magnitude of roughness elements. 3.4 MANNING’S FORMULA A resistance formula proposed by Robert Manning, an Irish engineer, for uniform flow in open channels, is (3.11) where = a roughness coefficient known as Manning’s . This coefficient is essentially a function of the nature of boundary surface. It may be noted that the dimensions of dimensions of are . Equation (3.11) is popularly known as the Manning's formula. Owing to its simplicity and acceptable degree of accuracy in a variety of practical applications, the Manning’s formula is probably the most widely used uniform-flow formula in the world. Comparing Eq. (3.11) with the Chezy formula, Eq. (3.3), we have (3.12) From Eq. (3.10) i.e. (3.13) since Eq. (3.13) does not contain any velocity term (and hence the Reynolds number), we can compare Eq. (3.13) with Eq. (3.7), i.e. the Pranal-Karman relationship for rough turbulent flow. If Eq. (3.7) is plotted as vs. on a log-log paper, a smooth curve that can be approximated to a straight line with a slope of is obtained (Fig. 3.2). From this the term can be expressed as Since from Eq. (3.13), , it follow that . Conversely, if , the Manning’s formula and Dracy-Weisbach formula both represent rough turbulent flow 3.5 OTHER RESISTANCE FORMULAE Several forms of expressions for the Chezy coefficient have been proposed by different investigators in the past. Many of these are archaic and are of historic interest only. A few selected ones are listed below: 1. Pavlovski Formula (3.14) in which and = Manning’s coefficient. This formula appears to be in use in Russia. 2. Ganguillet and Kutter Formula (3.15) in which = Manning’s coefficient 3. Bazin’s Formula in which = a coefficient dependent on the surface roughness.3.6 VELOCITY DISTRIBUTIONWide Channels (i) Velocity-defect Law: In channels with large aspect ratio , as for example in rivers and very large canals, the flow can be considered to be essentially two dimensional. The fully developed velocity distributions are similar to the logarithmic form of velocity-defect law found in turbulent flow in pipes. The maximum velocity occurs essentially at the water surface, (Fig.3.3). The velocity at a height above the bed in a channel having uniform flow at a depth is given by the velocity-defect law for as (3.17) where = shear velocity = = , = hydraulic radius, = longitudinal slope, and = Karman constant = 0.41 for open channel flow . This equation is applicable to both rough and smooth boundaries alike. Assuming the velocity distribution of Eq. (3.17) is applicable to the entire depth , the velocity can be expressed in terms of the average velocity (3.18) From Eq (3.18), it follows that (3.19) (ii) Law of the wall: For smooth boundaries, the flow of the wall as (3.20) is found applicable in the inner wall region ( < 0.20). The values of the constants are found to be = 0.41 and = 5.29 regardless of the Froude number and Reynolds number of the flow . Further, there is an overlap zone between the law of the wall region and the velocity-defect law region. For completely rough turbulent flows, the velocity distribution in the wall region ( < 2.0) is given by (3.21) where = equivalent sand grain roughness. It has been found that is a universal constant irrespective of the roughness size . Values of = 0.41 and = 8.5 are appropriate. For further details of the velocity distributions Ref. [5] can be consulted.(b) Channels with Small Aspect Ratio In channels which are not wide enough to have two dimensional flow, the resistance of the sides will be significant to alter the two-dimensional nature of the velocity distribution given by Eq.(3.17). The most important feature of the velocity distributions in such channels is the occurrence of velocity-dip, where the maximum velocity occurs not at the free surface but rather some distance below it, (Fig. 3.4). Typical velocity distributions in rectangular channels with = 1.0 and 6.0 are shown in Fig. 3.5(a) and (b) respectively.3.7 SHEAR STRESS DISTRIBUTION The average shear stress on the boundary of a channel is, by Eq. (3.2), given as . However, this shear stress is not uniformly distributed over the boundary. It is zero at tile intersection of the water surface with the boundary and also at the corner in the boundary. As such, the boundary shear stress will have certain local maxima on the side as well as on the bed. The turbulence of the flow and the presence of secondary currents in the channel also contribute to the non-uniformity of the shear stress distribution. A knowledge of the shear stress distribution in a channel is of interest not only in the understanding of the mechanics of flow but also in certain problems involving sediment transport and design of stable channels in non-cohesive material, (Chapter 11). Preston tube is a very convenient device for the boundary shear stress measurements in a laboratory channel. Distributions of boundary shear stress by using Preston tube in rectangular , trapezoidal and compound channels have been reported. Is sacs and Macintosh report the use of a modified Preston tube to measure shear stresses in open channels. Lane obtained the shear stress distributions on the sides and bed of trapezoidal and rectangular channels by the use of membrane analogy. A typical distribution of the boundary shear stress on the side and bed in a trapezoidal channel of =4.0 and side slope =1.5 obtained by Lane is shown in Fig.(3.6). The variation of the maximum shear stress on the bed and on the sides in rectangular and trapezoidal channels is shown in Fig. (3.7). It is noted from this figure that for trapezoidal sections approximately and when .3.8 RESISTANCE FORMULA FOR PRACTICAL USE Since a majority of the open channel flows are in the rough turbulent range, the Manning's formula (Eq. 3.11) is the most convenient one for practical use. Since it is simple in form and is also backed by considerable amount of experience, it is the most preferred choice of hydraulic engineers. However, it has a limitation in that it cannot adequately represent the resistance in situations where the Reynolds number effect is predominant and this must be borne in mind. In this book, the Manning's formula is used as the resistance equation. The Darcy-Weisbach coefficient used with the Chezy formula is also an equally effective way of representing the resistance in uniform flow. However, field-engineers generally do not prefer this approach, partly because of the inadequate information to assist in the estimation of and partly because it is not sufficiently backed by experimental or field observational data. It should be realised that for open channel flows with hydrodynamically smooth boundaries, it is perhaps the only approach available to estimate the resistance.3.9 MANNING’S ROUGHNESS COEFFICIENT In the Manning's formula, all the terms except are capable of direct measurement. The roughness coefficient, being a parameter representing the integrated effects of the channel cross-sectional resistance, is to be estimated. The selection of a value for n is subjective, based on one's own experience and engineering judgement. However, a few aids are available which reduce to a certain extent the subjectiveness in the selection of an appropriate value of n for a given channel. These include: 1. Photographs of selected typical reaches of canals, their description and measured values of . These act as type values and by comparing the channel under question with a figure and description set that resembles it most, one can estimate the value of fairly well. Movies, sterioscopic colour photographs and video recordings of selected typical reaches are other possible effective aids under this category. 2. A comprehensive list of various types of channels, their descriptions with the associated range of values of . Some typical values of for various normally encountered channel surfaces prepared from information gathered from various sources are presented in Table 3.2.EXAMPLE 3.1 A rectangular channel 2.0m wide carries water at at a depth of 0.5m.The channel is laid on a slope of 0.0004. Find the hydrody- namic nature of the surface if the channel is made of (a) very smooth concrete and (b) rough concrete.Solution Hydraulic radius (a) For a Smooth Concrete Surface Form Table 3.1, Since this value is slightly greater than 4.0, the boundary is hydrodynamically in the early transition from smooth to rough surface. (b) For a Rough Concrete Surface From Table 3.1, Since this value is greater than 60, the boundary is hydrodynamically rough.EXAMPLE 3.2 For the two cases in Example 3.1, estimate the discharge in the channel using (i) the Chezy formula with Darcr-Weisbach and (ii) the Manning's formula.Solution Case (a) : Smooth Concrete Channel (i) Since the boundary is in the transitional stage, Eq. (3.8a) would be used. Here Re is not known to start with and hence a trial and error method has to be adopted. By trial (ii) Referring to Table 3.2, the value of for smooth trowel-finished concrete can be taken as 0.012. By the Manning’s formula (Eq. 3.11), Case (b): Rough Concrete Channel (i) Since the flow is in the rough-turbulent state, by Eq. (3.7), (ii) By the Manning’s Formula Form Table 3.2, for rough concrete, = 0.015 is appropriate.Empirical Formulae for n Many empirical formulae have been presented for estimating Manning's coefficient in natural streams. These relate to the bed-particle size. The most popular form under this type is the Strickler formula: (3.22) Where is in meters and represents the particle size in which 50 per cent of the bed material is her. For mixtures of bed materials with considerable coarse-grained sizes, Eq. (3.17) has been modified by Meyer . As (3.23) where = size in metres and in which 90 per cent of the particles are finer than .This equation is reported to be useful in predicting in mountain streams paved with coarse gravel and cobbles.Factors Affecting n The Manning's is essentially a coefficient representing the integrated effect of a large number of factors contributing to the energy loss in a reach. Some important factors are: (a) surface roughness, (b) vegetation, (c) cross-section irregularity and (d) irregularity alignment of channel. The chief among these are the characteristics of the surface. The dependence of the value of n on the surface roughness in indicated in Tables 3.1 and 3.2. Since n is proportional to ,a large variation in the absolute roughness magnitude of a surface causes correspondingly a small change in the value of n. The vegetation on the channel perimeter acts as a flexible roughness element. At low velocities and small depths vegetation, such as grass and weeds, can act as a rigid roughness element which bends and deforms at higher velocities and depths of flow to yield lower resistance. For grass-covered channels, the value of n is known to decrease as the product VR increases. The type of grass and density of coverage also influence the value of n. For other types of vegetation, such as brush, trees in Rood plains, etc. the only recourse is to account for their presence by suitably increasing the values of n given in Table 3.2, which of course is highly subjective.
  • Channel irregularities and curvature, especially in natural streams, produce energy losses which are difficult to evaluate separately. As such, they are combined with the boundary resistance by suitably increasing the value of n. The procedure is sometimes also applied to account for other types of form losses, such as obstructions that may occur in a reach of channel.
  • 3.10 EQUIVALENT ROUGHNESS
  • In some channels different parts of the channel perimeter may have different roughnesses. Canals in which only the sides are lined, laboratory flumes with glass walls and rough beds, rivers with a sand bed in deepwater portion and flood plains covered with vegetation, are some typical examples. This equivalent roughness, also called the composite roughness, represents a weighted average value for the roughness coefficient. Several formulae exist for calculating the equivalent roughness. All are based on certain assumptions and are approximately effective to the same degree. One such method of calculation of equivalent roughness is given below.
  • Consider a channel having its perimeter composed of types of roughnesses. are the lengths of these parts and are the respective roughness coefficients (Fig. 3.8). Let each port be associated with a partial area such that It is assumed that the mean velocity in each partial area is the mean velocity for the entire area of flow, i.e. By the Mannning’s formula (3.24) where = equivalent roughness From Eq. (3.24) (3.25) i.e. (3.26) This equation affords a means of estimating the equivalent roughness of a channel having multiple roughness types in its perimeter. If the Darcy-Weisbach friction formula is used under the same assumption of (i) velocity being equal in all the partial areas and (ii) slope is common to all partial areas, then Hence Thus and on summation i.e. or (3.27)EXAMPLE 3.3 An earthen trapezoidal channel (n = 0.025) has a bottom width of 5.0 m, side slopes of 1.5 horizontal:1 vertical and a uniform flow depth of 1.1m. In an economic study to remedy excessive seepage from the canal two proposals, viz. (a) to line the sides only and (b) to line the bed only are considered. If the lining is of smooth concrete (n=0.012), determine the equivalent roughness in the above two cases.Solution Case (a) : Lining on the side only Here for the bed For the sides: Equivalent roughness, by Eq. (3.26) Case b: Lining on the bottom only Equivalent roughness3.11 UNIFORM FLOW COMPUTATIONS The Manning's formula (Eq. 3.11) and the continuity equation, Q =AV form the basic equations for uniform-flow computations. The discharge Q is then given by (3.28) (3.28a) where, is called the conveyance if the channel and expresses the discharge capacity of the channel per unit longitudinal slope. The term is sometimes called the section factor for uniform-flow computations. For a given channel, is a function of the depth of flow. For example, consider a trapezoidal section of bottom width =B and side slope m horizontal: 1 vertical. Then, (3.29) For a given channel, and are fixed and = . Figure 3.9 shows the relationship of Eq (3.29) in a non-dimensional manner by plotting for different values of . It may be seen that for , the
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