 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Chapter 2.1 2.2 2.3 2.4 2.5 UNIFORM FLOW Introduction Basic equations in uniform open-channel flow Most economical cross-section Channel with compound cross-section Permissible velocity against erosion and sedimentation Summary The chapter on uniform flow in open channels is basic knowledge required for all hydraulics students In this chapter, we shall assume the flow to be uniform, unless specified otherwise This chapter guides students how to determine the rate of discharge, the depth of flow, and the velocity The slope of the bed and the cross-sectional area remain constant over the given length of the channel under the uniform-flow conditions The same holds for the computation of the most economical cross section when designing the channel The concept of permissible velocity against erosion and sedimentation is introduced Key words Uniform flow; most economical cross-section; discharge; velocity; erosion; sedimentation 2.1 INTRODUCTION 2.1.1 Definition Uniform flow relates to a flow condition over a certain length or reach of a stream and can occur only during steady flow conditions Uniform flow may be also defined as the flow occurring in a channel in which equilibrium has been reached between gravitational force and shear force Many irrigation and drainage canals and other artificial channels are designed to carry water at uniform depth and cross section all along their lengths Natural channels as rivers and creeks are seldom of uniform shape The design discharge is set by considerations of acceptable risk and frequency analysis, whereas the channel slope and the cross-sectional shape are determined by topography, and soil and land conditions Uniform equilibrium open-channel flows are characterized by a constant depth and a constant mean flow velocity: h V 0 and 0 (2-1) s s where s is the coordinate in the flow direction, h the flow depth and V the flow velocity Uniform equilibrium open-channel flows are commonly called “uniform flows” or “normal flows” Note: The velocity distribution in fully-developed turbulent open channel flows is given approximately by Prandtl’s power law (Fig 2.1): V  y N   (2-2) Vmax  h  where the exponent 1/N varies from ¼ down to ½ depending on the boundary friction and the cross-section shape The most commonly-used power law formulae are the one-sixth Chapter 2: UNIFORM FLOW 25 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - power (1/6) and the one-seventh power (1/7) formulas It should be noted that the velocity in open-channel flow is assumed constant over the entire cross-section Vmax V velocity distribution h v y Fig 2.1 Velocity distribution profile in turbulent flow Such flow conditions are represented schematically in Fig 2.2 Considering Bernoulli’s theorem of the conservation of energy, between cross-sections and 2, leads to the expression:   p V2  p V2  (2-3) E1   z1   1   E  h L   z    2   h L  2g   2g    where 1 and 2 are the Corriolis-coefficients corresponding to the velocities V1 and V2, respectively They are also called the kinetic-energy correction coefficients  is equal to or larger than but rarely exceeds 1.1 (Li and Hager, 1991) For a uniform velocity distribution,  = The slope of the energy gradient line S is equal to the bed slope i of the channel, or: h S= i L (2-4) L   energy-gradient line hL V12 2g V 22 2g S E1 p h1  g hydraulic-gradient line h2  i z1 E2 z2 L  p2 g  Datum Fig 2.2 Energy and hydraulic gradient in uniform-flow channel If the flow is uniform, the cross sections at points and must be constant Consequently, the velocity and the depth of flow must also remain constant, or: V1 = V2 and h1 = h2 (2-5) The flow resistance in an open channel is more difficult to quantify The importance of the resistance coefficient goes beyond its use in channel design for uniform flow Chapter 2: UNIFORM FLOW 26 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 2.1.2 Momentum analysis Consider a control volume of length L in uniform flow, as shown in Fig 2.3 L FP1 A Wsin h FP2 = FP1 oPL  W = gAL P Fig 2.3 Force balance in uniform flow By definition, the hydrostatic forces, Fp1 and Fp2, are equal and opposite In addition, the mean velocity is invariant in the flow direction, so that the change in momentum flux is zero Thus, the momentum equation reduces to a balance between the gravity force component in the flow direction and the resisting shear force: A L sin = o P L (2-6) in which  = g = specific weight of the fluid, A = cross-sectional area of flow, o = mean boundary shear stress, and P = wetted perimeter of the boundary on which the shear stress acts If Eq (2-6) is divided by PL, the hydraulic radius R = A/P appears as an intrinsic variable Physically, Eq (2-6) represents the ratio of flow volume to boundary surface area, or shear stress to unit weight, in the flow direction Eq (2-6) can be written as: o = R sin   RS (2-7) if we replace sin with S = tan for small values of  Furthermore, if we solve Eq (2-7) for the bed slope, which equals the slope of the energy grade line, hL/L, and express the shear stress in terms of the friction factor f for uniform pipe flow according DarcyWeisbach: o V  f (2-8) we have the Darcy-Weisbach equation (for uniform pipe flow): i=S= h f  o f .V f V2    L R R 4R 2g (2-9) from which it is evident that the appropriate length scale, when applied to open-channel flow, is 4R It seems reasonable to use 4R as the length scale in the Reynolds-number and the relative roughness as well Before applying uniform flow formulas to the design of open channels, the background of Chezy’s as well as Manning’s formulas for steady, uniform in open channels are presented in the next section Chapter 2: UNIFORM FLOW 27 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 2.2 Basic equations in uniform open-channel flow 2.2.1 Chezy’s formula Consider an open channel of uniform cross-section and bed slope as shown in Fig 2.4: L Q Let and L A V P f i = = = = = = i VA Q Fig.2.4 Sloping bed of a channel length of the channel; cross-sectional area of flow; velocity of water; wetted perimeter of the cross-section; friction coefficient according to Darcy-Weisbach; uniform slope of the bed It has been experimentally found, that the total frictional resistance along the length L of the channel, follows the law: f Frictional resistance =    contact area  (velocity)2 f =    P.L  Vn (2-10) The exponent n has been experimentally found to be nearly equal to But for all practical purposes, its value is taken to be Therefore, Frictional resistance = f    P.L  V2 (2-11) Since the water moves over a distance V in second, therefore, the work done in overcoming the friction reads as: Frictional resistance  distance V in second = f f PLV2V =  PL V3 (2-12) 8 The weight of the water, W, in the channel over a length of L is: W = .A.L (2-13) This water “falls” vertically down over a distance V.i in second, so Loss of potential energy = Weight of water  Height = .A.L.V.i We know that work done in overcoming friction = Loss of potential energy f i.e    P.L  V3 = .A.L.V.i (2-14) (2-15) Chapter 2: UNIFORM FLOW 28 OPEN CHANNEL HYDRAULICS FOR ENGINEERS -  V2 =8 or V = where C =  A.i f  P 8 A  i  f P (2-16) A 8 8g is known as Chezy’s coefficeint and R = as hydraulic radius   f f P The discharge of flow then is Q = A  V = AC Ri (2-17) Note: Unlike the Darcy-Weisbach coefficient f , which is dimensionless, the Chezy coefficient C has the dimension, [L1/2T-1], as mentioned in Chapter Chezy’s coefficient C depends on the mean velocity V, the hydraulic radius R, the kinematic viscosity  and the relative roughness There is experimental evidence that the value of the resistance coefficient does vary with the shape of the channel and therefore with R and possibly also with the bed slope i, which for uniform flow will be equal to the slope of the energy-head line io, yielding a relationship for the velocity of the form: V = K Rx.ioy where K, x and y are constants (2-18) Example 2.1: A rectangular channel is m deep and m wide Find the discharge through the channel, when it runs full Take the slope of the bed as 1:1000 and Chezy’s coefficient as 50 m1/2s-1 Solution: Given: Depth h = m, Width b = m, h Bed slope i = 1/1000 = 0.001, Chezy’s coefficient C = 50 m1/2s-1 b Q = ? (m3/s) A=hb P = b + 2h Area of the rectangular channel: Perimeter of the rectangular channel: Hydraulic radius of the flow: Discharge through the channel: = 24 m2 = 14 m A = 1.71 m P Q = AC Ri = 49.62 m3s-1 R = Ans Example 2.2 : Water is flowing at the rate of 8.5 m3s-1 in an earthen trapezoidal channel with a bed width m, a water depth 1.2 m and side slope 2:1 Calculate the bed slope, if the value of C in Chezy’s formula be 49.5 m1/2s-1 Solution: Given: Discharge Q = 8,5 m3/s, Bed width b = m, Depth h = 1.2 m, Side slope m = 2, h Chezy’s coefficient C = 49.5 m1/2s-1, Bed slope = ? Surface width of the trapezoidal channel B h/2 h B = b + 2( ) b = 10.2 m Chapter 2: UNIFORM FLOW 29 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Area of the trapezoidal channel: A =  b  B h   Wetted perimeter: Hydraulic radius: Now using the relation:  P = b   h  h    A R = P Q = AC Ri Q2 i = R(AC)2 = 11.52 m2 = 11.68 m = 0.986 m = 4440 Ans 2.2.2 Manning’s formula Manning, after carrying out a series of experiments, deduced the following relation for the value of C in Chezy’s formula: 1 C = R (2-19) n where n is the Manning constant in metric units, n = [m-1/3s] n is expressing the channel’s relative roughness properties and values are given in Table 2.1 Now we see that the velocity: 1 1 1 V = C Ri =  R  Ri =  R  R  i n n 1  V =  R i (2-20) n 1 Now, the discharge is: Q = AV = A   R  i (2-21) n Table 2.1: Values of Manning coefficient n [m-1/3s] Wetted perimeter n Wetted perimeter n A Natural channel Clean and straight Sluggish with deep pools Major rivers B Flood plain Pasture, farmland Light brush Heavy brush Trees C Excavated earth channels Clean Gravelly Weedy Stony, cobbles 0.030 0.040 0.035 0.035 0.050 0.075 0.150 0.022 0.025 0.030 0.035 D Artificially lined channel Glass Brass Steel, smooth Steel, painted Steel, riveted Cast iron Concrete, finished Concrete, unfinished Planned wood Clay tile Brickwork Asphalt Corrugated metal Rubble masonry 0.010 0.011 0.012 0.014 0.015 0.013 0.012 0.014 0.012 0.014 0.015 0.016 0.022 0.025 For a more detailed description, we can take the value of Manning’s n from the Table at the end of this chapter Chapter 2: UNIFORM FLOW 30 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Example 2.3: An earthen trapezoidal channel with a m wide base and side slopes 1:1 carries water with a depth of m The bed slope is 1/1600 Estimate the discharge Take the value of n in Manning’s formula as 0.04 m-1/3s Solution: Given: Base width b = m, Side slope = 1:1, Water depth h = m, Bed slope 1/1600, Manning’s coefficient n = 0.04 m-1/3s Discharge Q = ? (m3/s) Surface width of the trapezoidal channel Area of the trapezoidal channel: Wetted perimeter: Hydraulic radius: Now using the relation: B h h b B = b + 2h = m  b  B A= = m2 h P = b  h2  h2 A R = P 1 Q = A  R i n = 5.828 m = 0.686 m = 1.94 m3/s Ans Example 2.4 : Water at the rate of 0.1 m3/s flows through a vitrified sewer with a diameter of m with the sewer pipe half full Find the slope of the water surface, if Manning’s n is 0.013 m-1/3s Solution: Given: Discharge Q = 0.1 m3/s, Diameter of pipe D = m, Manning’s n = 0.013 m-1/3s Sewer slope i = ? Area of the flow: A= Wetted perimeter: P = Hydraulic radius: Using Manning’s formula:  Water surface slope:   D2    = 0.393 m 2  D A R= P D = 1.57 m = 0.25 m 1 Q = A  R i n  Qn  i = =   A  R  1430 Ans Chapter 2: UNIFORM FLOW 31 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 2.2.3 Discussion of factors affecting f and n The dependence of f on the relative roughness in open channel flow is not as well known as in pipe flow, because it is difficult to assign an equivalent sand-grain roughness to the large values of the absolute roughness height typically found in open channels The dependence of flow resistance on the cross-sectional shape occurs as a result of changes of both the channel hydraulic radius, R, and the cross-sectional distribution of velocity and shear There is no substitute for experience in the selection of Manning’s n for natural channels Table 2.2 (at the end of this chapter) from Ven Te Chow (1959) gives an idea of the variability to be expected in Manning‘s n 2.3 MOST ECONOMICAL CROSS-SECTION 2.3.1 Concept A typical uniform flow problem in the design of an artificial canal is the economical proportioning of the cross-section A canal, having a given Manning coefficient n and a slope i, is to carry a certain discharge Q, and the designer’s aim is to minimize the cross-sectional area Clearly, if A is to be a minimum, the velocity V is to be a maximum The Chezy and Manning formulas indicate, therefore, that the hydraulic radius R = A/P must be a maximum It can be shown that the problem is equivalent to that of minimizing P for a given constant value of A This concept has a practical application in estimating the cost for a canal excavation and /or lining From economic considerations of minimizing the flow cross-sectional area for a given design discharge, a theoretically optimum cross-section will be introduced 2.3.2 Conditions for maximum discharge The conditions for maximum discharge for the following cross-sections will be dealt with: (a) Rectangular cross-section, and (b) Trapezoidal cross-section (a) Channel with rectangular cross-section Consider a channel of rectangular cross-section as shown in Fig 2.5 Let h b = breadth of the channel, and h = depth of the channel Area of flow: A=b h b= Discharge: Q=AV b A h (2-20) Fig.2.5 A rectangular channel = AC Ri  AC A i P (2-22) Keeping A, C and i constant in the above equation, the discharge will be maximum when A/P is maximum or the wetted perimeter P is minimum Or in other words, when: dP 0 (2-23) dh A We know that P = b + 2h =  2h (2-24) h Chapter 2: UNIFORM FLOW 32 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Differentiating the above equation with respect to h and equating to zero yields: dP A  20 dh h  A = 2h = b  h or b = 2h i.e the breadth is two times the depth In this case, the hydraulic radius is: A bh 2h  h 2h h R     P b  2h 2h  2h 4h (2-25) (2-26) (2-27) Hence, we can say that for the maximum discharge or the maximum velocity, these two conditions (i.e b = 2h and R = h/2) should be used for solving the problem of optimizing channels of rectangular cross-section 1.00 0.95 Q Q max 0.90 h b A = bh = constant 0.85 Fig 2.6 Experimental relationship between b h Q b and h Qmax As can be seen in Fig 2.6, the maximum represented by this optimal configuration is a b rather weak one For example, for aspect ratios, , between and 4, the flow rate is within h 96% of the maximum flow rate obtained with the same area and by b/h = Example 2.5.: Find the most economical cross-section of a rectangular channel to carry 0.3 m3/s of water, when the bed slope is 1/1000 Assume Chezy’s C = 60 m-1/3s-1 Solution: Given: Discharge Q = 0.3 m3/s, Bed slope i = 1/1000, Chezy coefficient C = 60 m-1/3s-1 Breadth of channel b = ? (m) and depth of the channel h = ? (m) We know that for the most economical rectangular section: b = 2h Area: A = b  h = 2h  h = 2h2 and hydraulic radius: R = h/2 = 0.5 h Using the relation: Q = AC Ri Chapter 2: UNIFORM FLOW 33 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - and squaring both sides yields: 0.09 m5  7.2h or h  0.0125  h  0.0125 m Depth of the channel: h = 0.416 m And breadth: b = 2d = 0.832 m Ans Ans (a) Channel with trapezoidal cross-section Consider a channel of trapezoidal cross-section ABCD as shown in Fig 2.7 B = b + 2nh D C  B nh n A b h Fig.2.7 A trapezoidal channel Let b = breadth of the channel at the bottom, h = depth of the channel, and = side slope (i.e vertical to n horizontal) n Area of flow: or Discharge: A = h(b + nh) A A = b + nh  b = - nh h h Q = A  V  AC Ri  AC (2-28) A i P (2-29) Keeping A, C and i constant in the above equation, the discharge will be maximum, when A/P is maximum or the wetted perimeter P is minimum Or in other words: dP 0 dh We know that: P = b  n h  h  b  2h n  (2-30) Substituting the value of b from equation (2-28) yields: A P= (2-31)  nh  2h n  h Differentiating the above equation with respect to d and equating to zero results into: dP A    n  n2    (2-32) dh h A  n  n2 1 h2 Chapter 2: UNIFORM FLOW 34 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - h(b  nh)  n  n2 1 h [A = h(b + nh)] b  nh h  n    n2 1 h h b  2nh  h n2 1 or (2-33) We see that b + 2nh = B is the top width of the channel and h n  is the length of the sloping side, i.e the length of the sloping side is equal to half the top width In this case, the hydraulic radius: A h(b  nh) h(b  nh) h(b  nh) h R=     P b  2h n  b  (b  2nh) 2(b  nh) (2-34) Hence, we can say that for the maximum discharge or the maximum velocity, these two b  2nh h conditions (i.e  h n  and/or R = ) should be used for solving the problems 2 in the case of channels of trapezoidal cross-section Example 2.6: A canal of trapezoidal cross-section has to be excavated through hard clay at the least cost Determine the dimensions of the channel for a discharge equal to 14 m3/s, a side slope for hard clay n = 1:1, a bed slope 1:2500 and Manning’s n = 0.02 m-1/3s Solution: Given: Discharge Q = 14 m3/s; Bed slope i = 1/2500; Side slope n = 1:1 Manning’s n = 0.02 m-1/3s Breath b = ? (m) Depth h = ? (m) h h b We know that for the least cost: half of the top width = length of sloping side b  2nh  h n 1 with n =1  b  2h  2h  2.83h b = 0.83 h Area of flow: A = h(b + nh) = 1.83 h2 Using Manning’s formula: 1 Q = A  R i n yields: And h  12.142 m8/3 h  2.55 m Ans b = 0.83 h = 2.12 m Ans Chapter 2: UNIFORM FLOW 35 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Note: The semi-circular section (the semi-circle having its center in the surface) is the best hydraulic section The best hydraulic cross-section for other shapes can be drawn as presented in Fig 2.8 D R D R D R D R 60 90 trapezoidal triangular channel channel Fig 2.8 Cross-sections of maximum flow rate: i.e “optimum design” circular channel rectangular channel Students should try to proof the conditions for circular and triangular channels for the best hydraulic cross-section based on the below relationship: Q 1 1 A V     R i    i A n n P  A   Q.n = constant   P  i  3 (2-25) Table 2.2.: The “best” design summary for several channel cross-sections Cross-section Optimum width B Optimum wetted perimeter P  D2 D D D2 2D 2D D Semi-circle D Rectangular Trapezoidal Triangular Optimum cross-section A 2D D2 Optimum hydraulic radius R D D D 3D D 2D 2.3.3 Problems of uniform-flow computation The computation of uniform flow may be performed by the use of two equations: the continuity equation and a uniform-flow formula When the Manning formula is used as the uniform-flow formula, the computation will involve the following six variables: (1) the normal discharge Q (2) the normal depth h (3) the channel slope i (4) the mean velocity of flow V (5) the coefficient of roughness (6) the geometric elements that depend on the shape of the channel section, such as n A,R, etc, … Chapter 2: UNIFORM FLOW 36 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - When any four of the above six variables are given, the remaining two unknowns can be determined by the two equations The following represents some types of problems of uniform flow computation: A to compute the normal discharge: In practical applications, this computation is required for the determination of the capacity of a given channel or for the construction of a sysnthetic rating curve of the channel B to determine the velocity of the flow: This computation has many applications For example, it is often required for the study of scouring and silting effects in a given channel C to compute the normal depth: This computation is required for the determination of the stage of flow in a given channel D to determine the channel roughness: This computation is used to ascertain the roughness coefficient in a given channel; the coefficient thus determined may be used in other similar channels E to compute the channel slope: This computation is required for adjusting the slope of a given channel F to determine the dimensions of the channel section: This computation is required mainly for design purposes Table 2.3 lists the known and unknown variables involved in each of the six types of problems mentioned above Table 2.3: Problems of uniform-flow computation Type of problem Discharge Q Velocity V Depth d Roughness n Slope i Geometric elements A B C D E F ?     ? -   ?       ?       ?       ? Notes:  The known variables are indicated by the check mark () and the unknown required in the problems by the question mark (?) The unknown variable(s) that can be determined from the known variables is(are) indicated by a dash (-)  Table 2.3 does not include all types of problems By varying combinations of various known and unknown variables, more types of problems can be formed In design problems, the use of the best hydraulic section and of empirical rules is generally introduced and thus new types of problems are created Chapter 2: UNIFORM FLOW 37 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - CHANNEL WITH COMPOUND CROSS-SECTION A compound channel consists of a main channel, which carries the base flow (frequently running off up to bank-full conditions), and a floodplain on one or both sides that carries over-bank flow during the time of flooding as sketched in Fig 2.9 The compound cross-section of a channel may be composed of several distinct subsections with each subsection different in roughness form the others canal bank high water level dyke low water level (1): A1, P1, n1 (2): A2, P2, n2 (3): A3, P3, n3 Fig.2.9 Over-bank flow in a compound channel The roughness of the side channels will be different (generally rougher) from that of the main channel and the method of analysis is to consider the total discharge to be the sum of the component discharges computed by the Manning equation The mean velocity for the whole channel section is equal to the total discharge divided by the total water area The classical method of computation of discharge, as presented by Chow in 1959, consisted in subdividing the composite cross-section into sub-areas with vertical interfaces in which the shear stresses are neglected The discharge for each sub-area is calculated by assuming a common friction slope i for the whole channel Thus in the channel, as shown in Fig 2.9, assuming that the bed slope is the same for the three sub-areas, it holds: AR Q   Qi  i  i i ni i 1 i 1 m m (2-26) The division of the channel by these artificial vertical boundaries assumes implicity that the shear stress on these interfaces is relatively small with respect to the shear stress acting on the wetted perimeter of the channel Note: It has been found by more recent experimentation that this hypothesis is incorrect and that it leads to a considerable over-estimation of the discharge in the compound channel Example 2.7: Water flows along a drainage canal having the properties shown in the figure below If the bottom slope i = 1/500=0.002, estimate the discharge 3m 0.6 m n1 = 0.020 (1) 2m (2) n2 = 0.015 3m n3 = 0.030 0.8 m (3) [ni] = m-1/3s Chapter 2: UNIFORM FLOW 38 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Solution: We divide the cross-section into three subsections as is indicated in the figure and write the discharge as Q = Q1 + Q2 + Q3, where for each section, it holds: Qi =AiVi = A i  12 Ri i ni The appropriate values of Ai, Pi, Ri and ni are listed in the table below: i (1) (2)* (3) Ai (m2) 1.8 2.8 1.8 Pi (m) 3.6 3.6 3.6 Ri (m) 0.500 0.778 0.500 ni (m-1/3s) 0.020 0.015 0.020 Note that the imaginary portions of the wetted perimeter between the sections (denoted by the dashed lines in the figure) are not implemented in Pi That is, for section (2): So that A2 =  (0.8 + 0.6) m2 P2 = {2 + 2(0.8)} m A 2.8 R2   m P2 3.6 = 2.8 m2 = 3.6 m = 0.778 m Thus the total discharge is: Q = Q1 + Q2 + Q3 = 2  3 1.8 0.500 2.8 0.778 1.8 0.500           Q =  0.002     0.020 0.015 0.030    m3s-1   Q = 11.275 m3/s Ans If the entire channel cross-section were considered as one flow area, then : A = A1 + A2 + A3 = 6.4 m2 P = P1 + P2 + P3 = 10.8 m A Then R= = 0.593 m P The total discharge can be written as 1 Q= A  R 3i n eff n where neff is the effective value of n for the whole compound channel With Q = 11.275 m3/s, as determined above, the value of neff is found to be: A R i n eff  = 0.01681 m-1/3s Q As expected, the effective roughness (Manning’s n) is between the minimum (n2 = 0.015 m-1/3s) and maximum (n3 = 0.030 m-1/3s) values for the individual subsections Chapter 2: UNIFORM FLOW 39 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 2.5 PERMISSIBLE VELOCITY AGAINST EROSION AND SEDIMENTATION The excavation and lining cost of open channels or conduits varies with their size With respect to water-resources-system economics, erosion of and sedimentation in channels are problems in hydraulic engineering Erosion and sedimentation must be predicted because they can change the bed slope, the channel width and therefore the flow conditions So, if the available slope permits, the cost of the initial construction may be reduced by using the highest velocity However, if the velocity is becoming too high, the channel may be damaged or destroyed by erosion This must be avoided by limiting the velocities according to the boundary materials For clear water in hard-surfaced water conductors, the limiting velocity is beyond practical requirements Velocities above 10 m/s for clear water in concrete channels have been found to no harm If the water carries abrasive material, damage may occur at lower velocities No definite relation has been established between the nature of abrasive materials, the material of channel bank and bed, and a permissible velocity In unlined earthen channels, the limiting velocity involves many factors Generally, a fine soil is eroded more easily than a coarse one, but the effect of the grain size may be obscured by the presence or absence of a cementing or binding material The tendency to erode is reduced by seasoning Groundwater conditions can exert an important influence Seepage out of the channel, particularly if the water is turbid, tends to toughen the banks, whereas infiltration reduces the resistance to erosion Erosion can be reduced or avoided by designing for low velocities If the water carries an appreciable amount of silt in suspension, too low a velocity will cause the canal to fill up until its capacity is impaired It is necessary to choose a velocity that will keep the silt in motion but that will not erode the bank or bottom of the canal The margin of permissible velocities depends on the amount and nature of the silt in the water, the nature of the bank material, the size and shape of the canal, and many other factors The silt content of most turbid water varies with the season, as does also the demand for water and the resultant velocity of the flow The determination of non-scouring, non-silting velocities for earthen canals has attracted the attention of many investigators over a long period of time, and a considerable mass of data and formulas have been accumulated However, for preliminary purposes, and for design in many cases, use may be made of the approximate values purposed by Fortie and Scobey, in 1926, as shown in Table 2.4 Where the silt is important, it is better to make the slope a little too steep rather than a little too flat A gradient that proves to be too steep can be controlled by checks In hard-surfaced channels, silting is easily controlled if fall for scouring velocity is available Chapter 2: UNIFORM FLOW 40 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Table 2.4: Permissible canal velocities (Fortier and Scobey, 1926) Original material excavated for canal (1) Fine sand (non-colloidal) Sandy loam (non-colloidal) Sandy loam (non-colloidal) Alluvial silts when non-colloidal Ordinary firm loam Volcanic ash Fine gravel Stiff clay (very colloidal) Graded, loam to cobbles, when non-colloidal Alluvial silts when colloidal Graded, loam to cobbles, when colloidal Coarse gravel (non-colloidal) Cobbles and shingles Shales and hardpans Velocity, m/s, after aging, of canal carrying: Clear Water Water transporting water, no transporting non-colloidal silts, detritus colloidal silts sands, gravels, or rock fragments (2) (3) (4) 0.46 0.76 0.46 0.53 0.76 0.61 0.61 0.91 0.61 0.61 1.07 0.61 0.76 1.07 0.69 0.76 1.07 0.61 0.76 1.52 1.14 1.14 1.52 0.91 1.14 1.52 1.52 1.14 1.52 0.91 1.22 1.68 1.52 1.22 1.83 1.98 1.52 1.68 1.98 1.83 1.98 1.52 Chapter 2: UNIFORM FLOW 41 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Table 2.5: Value of Manning’s Roughness Coefficient n [m-1/3s] (Ven Te Chow, 1973) Type of channel and description A Closed Conduits Flowing Party Full A.1 Metal a Brass, smooth b Steel Lock bar and welded Riveted and spiral c Cast iron Coated Uncoated d Wrought iron Black Galvanized e Corrugated metal Subdrain Storm drain A.2 Nonmetal a Lucite b Glass c Cement Neat surface Mortar d Concrete Culvert, straight and free of debris Culvert with bends, connections, and some debris Finished Sewer with manholes, inlet, etc., straight Unfinished, steel form Unfinished, smooth wood form Unfinished, rough wood form e Wood Stave Laminated, treated f Clay Common drainage tile Vitrified sewer Vitrified sewer with manholes, inlet, etc Vitrified subdrain with open joint g Brickwork Glazed Lined with cement mortar h Sanitary sewers coated with sewage slimes, with bends and connections i Paved invert, sewer, smooth bottom j Rubble masonry, cemented Minimum Normal Maximum 0.009 0.010 0.013 0.010 0.013 0.012 0.016 0.014 0.017 0.010 0.011 0.013 0.014 0.014 0.016 0.012 0.013 0.014 0.016 0.015 0.017 0.017 0.021 0.019 0.024 0.021 0.030 0.008 0.009 0.009 0.010 0.010 0.013 0.010 0.011 0.011 0.013 0.013 0.015 0.010 0.011 0.011 0.013 0.013 0.014 0.011 0.013 0.012 0.012 0.015 0.012 0.015 0.013 0.014 0.017 0.014 0.017 0.014 0.016 0.020 0.010 0.015 0.012 0.017 0.014 0.020 0.011 0.011 0.013 0.014 0.013 0.014 0.015 0.016 0.017 0.017 0.017 0.018 0.011 0.012 0.012 0.013 0.015 0.013 0.015 0.017 0.016 0.016 0.018 0.019 0.025 0.020 0.030 Chapter 2: UNIFORM FLOW 42 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Type of channel and description B Lined or built-up channels B.1 Metal a Smooth steel surface Unpainted Painted b Corrugated B.2 Nonmetal a Cement Neat surface Mortar b Wood Planed, untreated Planed, creosoted Unplanted Plank with battens Lined with roofing paper c Concrete Trowel finish Float finish Finished, with gravel on bottom Unfinished Gunite, good section Gunite, navy section On good excavated rock On irregular excavated rock d Concrete bottom float finished with sides of Dressed stone in mortar Random stone in mortar Cement rubble masonry, plastered Cement rubble masonry Dry rubble or riprap e Gravel bottom with sides of Formed concrete Random stone in mortar Dry rubble or riprap f Brick Glazed In cement mortar g Masonry Cemented rubble Dry rubble h Dressed ashlar i Asphalt Smooth Rough j Vegetal lining Minimum Normal Maximum 0.011 0.012 0.021 0.012 0.013 0.025 0.014 0.017 0.030 0.010 0.011 0.011 0.013 0.013 0.015 0.010 0.011 0.011 0.012 0.010 0.012 0.012 0.013 0.015 0.014 0.014 0.015 0.015 0.018 0.017 0.011 0.013 0.015 0.014 0.016 0.018 0.017 0.022 0.013 0.015 0.017 0.017 0.019 0.022 0.020 0.027 0.015 0.016 0.020 0.020 0.023 0.025 0.015 0.017 0.016 0.020 0.020 0.017 0.020 0.020 0.025 0.030 0.020 0.024 0.024 0.030 0.035 0.017 0.020 0.023 0.020 0.023 0.033 0.025 0.026 0.036 0.011 0.012 0.013 0.015 0.015 0.018 0.017 0.023 0.013 0.025 0.032 0.015 0.030 0.035 0.017 0.013 0.016 0.030 0.013 0.016 … 0.500 Chapter 2: UNIFORM FLOW 43 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Type of channel and description C Excavated or dredged a Earth, straight and uniform Clean, recently completed Clean, after weathering Gravel, uniform section, clean With short grass, few weeds b Earth, winding and sluggish No vegetation Grass, some weeds Dense weeds or aquatic plants in deep channels Earth bottom and rubble sides Stony bottom and weedy banks Cobble bottom and clean sides c Dragline-excavated or dredged No vegetation Light brush on banks d Rock cuts Smooth and uniform Jagged and irregular e Channels not maintained, weeds and brush uncut Dense weeds, high as flow depth Clean bottom, brush on sides Same, highest stage of flow Dense brush, high stage D Natural streams D.1 Minor stream (top width at flood stage < 100 ft) a Streams on plain Clean, straight, full stage, no rifts or deep pools Same as above, but no more stones and weeds Clean, winding, some pools and shoals Same as above, but some weeds and stones Same as above, lower stages, more ineffective slopes and sections Same as 4, but more stones Sluggish reaches, weedy, deep pools Very weedy reaches, deep pools, or floodways with heavy stand of timber and underbrush b Mountain stream, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages Bottom: gravels, cobbles, and few boulders Bottom: cobbles with large boulders Minimum Normal Maximum 0.016 0.018 0.022 0.022 0.018 0.022 0.025 0.027 0.020 0.025 0.030 0.033 0.023 0.025 0.030 0.028 0.025 0.030 0.025 0.030 0.035 0.030 0.035 0.040 0.030 0.033 0.040 0.035 0.040 0.050 0.025 0.035 0.028 0.050 0.033 0.060 0.025 0.035 0.035 0.040 0.040 0.050 0.050 0.040 0.045 0.080 0.080 0.050 0.070 0.100 0.120 0.080 0.110 0.140 0.025 0.030 0.033 0.035 0.040 0.030 0.035 0.040 0.045 0.048 0.033 0.040 0.045 0.050 0.055 0.045 0.050 0.075 0.050 0.070 0.100 0.060 0.080 0.150 0.030 0.040 0.040 0.050 0.050 0.070 Chapter 2: UNIFORM FLOW 44 ... Chapter 2: UNIFORM FLOW 27 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 2. 2 Basic equations in uniform open- channel flow 2. 2.1... 0. 022 0.013 0.015 0.017 0.017 0.019 0. 022 0. 020 0. 027 0.015 0.016 0. 020 0. 020 0. 023 0. 025 0.015 0.017 0.016 0. 020 0. 020 0.017 0. 020 0. 020 0. 025 0.030 0. 020 0. 024 0. 024 0.030 0.035 0.017 0. 020 ... 0.018 0. 022 0. 022 0.018 0. 022 0. 025 0. 027 0. 020 0. 025 0.030 0.033 0. 023 0. 025 0.030 0. 028 0. 025 0.030 0. 025 0.030 0.035 0.030 0.035 0.040 0.030 0.033 0.040 0.035 0.040 0.050 0. 025 0.035 0. 028 0.050