WATER FLOW PROPERTIES OF FLUIDS The kinematic viscosity, v, is defined as the ratio of the coefficient of viscosity to the density and is expressed in The fluid properties most commonly encountered in water flow problems are presented in the following paragraphs The International System of units is used throughout the discussion unless otherwise stated to the contrary The unit of mass, m, is the kilogram (kg) A mass of one kg will be accelerated by a force of one newton at the rate of m per sec2 The density, r, of a fluid is its mass per unit volume and is expressed in kilograms per cubic meter The specific weight, g, is the weight per unit volume and denotes the gravitational force on a unit volume of fluid and is expressed in newtons per cubic meter Fluid density and specific weight are related by the expression: r= g g (1) in which g is the acceleration due to gravity The specific gravity of a fluid is found by dividing its density by the density of pure water at 4ЊC The relative shearing force required to deform a fluid gives a measure of the viscosity of the fluid An increase in temperature causes a decrease in viscosity of a liquid and vice versa Consider the space between two parallel plates (Figure 1) which is filled with fluid; the bottom plate remains at rest while the upper plate moves with velocity V under an applied force The velocity of the fluid particles will range from V at the top boundary to zero at the bottom as they will assume the same velocity as the boundary in which they are in contact Experiments have demonstrated that the shear stress, t, is directly proportional to the rate of deformation, du/dy Mathematically, this can be written as: tϭ du dy ⋅ (2) Equation (2) is known as Newton’s equation of viscosity The constant of proportionality, µ, in newton-second per square meter (N-s/m2), is termed the coefficient of viscosity, the dynamic viscosity or the absolute viscosity vϭ m r⋅ (3) A more proper term for surface tension, s, would be surface energy Surface tension is a liquid surface phenomenon and is caused by the relative forces of cohesion, the attraction of liquid molecules for each other, and adhesion, the attraction of liquid molecules for the molecules of another liquid or solid Surface tension has the units of newtons per meter (N/ m) When a liquid surface is in contact with a solid, a contact angel u, greater than 90Њ results with depression of the liquid surface if the liquid does not “wet” the tube such as mercury and glass If the solid boundary has a greater attraction for a liquid molecule than the surrounding liquid molecules, then the contact angle is less than 90ЊC and the liquid is said to “wet” the wall leading to a capillary rise as in the case of water and glass Table gives the values of the fluid properties discussed in the preceding paragraphs for a few common fluids PRESSURE FLOW Friction Formulae Darcy-Weishbach’s Equation The Darcy-Weishbach formula was first proposed empirically but later found by dimensional reasoning to have a rational basis: hf ϭ f LV 2gD (4) in which f ϭ friction factor, L ϭ pipe length, V ϭ mean velocity, D ϭ diameter, h ϭ head loss, g ϭ acceleration due to gravity Equation (4) was derived for circulation sections flowing full and the equation itself is dimensionally homogeneous It can be extended to other cross-sections provided these shapes are not too different from circular; in this case, the equation has to be transformed by using the hydraulic radius, R, instead of the diameter, D: hf ϭ f LV , 8gR (5) 1275 © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1275 11/18/2005 11:12:11 AM 1276 WATER FLOW where r ϭ D/4 for flow at full bore The use of Darcy’s equation in the form given by Eq (5) is sometimes extended to open channel flow The determination of the friction factor, f, depends on the flow regime, that is, whether the flow is laminar, critical, transitional, smooth, turbulent or rough fully turbulent Laminar Flow Consider the mean pipe velocity, V, as given by Hagen-Poiseuille’s equation for laminar flow: Vϭ gSD 32 m 64 fϭ , Re V Moving plate u+dy u Applied force (8) At the centre line, the velocity is a maximum: umax ϭ gSD 16 m (9) The mean velocity is: umean ϭ (umax ) / ϭ gSD 32 m (10) Critical Flow From a Reynolds number of about 2000 and extending to 4000 lies a critical zone where the flow may be either laminar or turbulent The flow regime is unstable and no equation adequately describes it Smooth Turbulent For pipes fabricated from hydraulically smooth materials such as copper, plexiglass and glass, the flow is smooth turbulent for a Reynolds number exceeding 4000 The von Karman-Nikuradse smooth pipe equation is: ϭ log10 ( Re f ) Ϫ 0.80 f (11) Equation (11) indicates that the friction factor depends on the fluid properties and deceases with increasing Reynolds number Rough Fully Turbulent Nikuradse experimented with pipes artificially roughened with uniform sand grains The results were fitted to the theory of Prandtl–Karman to give the well known rough-pipe equation: du u=0 gS ⎛ D 2⎞ ⎜ Ϫr ⎟ 4m ⎝ ⎠ (7) where Re ϭ nDրg is the Reynolds number Eq (7) can be used for all pipe roughness as the friction factor in laminar flow is independent of the wall protuberances and is inversely proportional to the Reynolds number The energy loss varies directly as the mean pipe velocity in laminar flow which persists up to a Reynolds number of about 2000 The velocity profile, which has a parabolic distribution, can be obtained from Hagen-Poiseuille’s equation The y uϭ (6) in which S ϭ energy slope Combining Eq (4) with Eq (6) and noting that S ϭ Hf /L, g ϭ rg, and nmրr, the friction factor is given by: dy velocity, u, at any radius, r, of the pipe of diameter, D, is given by: Stationary plate f FIGURE Fluid shear ϭ log10 ( D/) ϩ1.14 (12) TABLE Fluid properties Fluid Temperature ЊC Mass density kg/m3 Specific weight kN/m3 Dynamic viscosity N-s/m3 Kinematic viscosity m2/s Surface tension N/m Water 1000 9.81 1.75 ϫ 10Ϫ3 1.75 ϫ 10Ϫ6 0.0756 — 1000 9.81 1.52 ϫ 10Ϫ3 1.52 ϫ 10Ϫ6 0.0754 9.81 1.30 ϫ 10 — 10 1000 1.30 ϫ 10 0.0742 Mercury 20 13,570 133.1 1.56 ϫ 10Ϫ3 1.15 ϫ 10Ϫ7 0.514 Sea water 20 1028 10.1 1.07 ϫ 10Ϫ3 1.04ϫ 10Ϫ6 0.073 Ϫ3 Ϫ3 © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1276 11/18/2005 11:12:11 AM WATER FLOW Equation (12) states that beyond a certain Reynolds number, when the flow is fully turbulent, the friction factor is influenced only by the relative roughness,րD and independent of the Reynolds number Transition Flow Most commercial pipe flows not follow either the smooth pipe or rough pipe equations Colebrook and White proposed a transitional flow equation which would be asymptotic to both: ⎛ e/D 2.51 ϭϪ2 log10 ⎜ ϩ f ⎝ 3.7 Re f ⎞ ⎟ ⎠ Re 200 D (14) / 1/ R S , N (15) in which N ϭ roughness coefficient Equation (15) can also be transformed to: 3.05 0.31 Steel vϭ 9.14 Smooth concrete ϭ Mannning’s Equation The Manning equation, although originally developed for open channel flow, has often been extended for use in pressure conduits The equation is usually favored for rough textured material (rough concrete, unlike rock tunnels) and cross-sections that are not circular (rectangular, horseshoe) It is most commonly given in the form: (mm) Rough concrete (13) TABLE Equivalent sand-grain diameter Riveted steel Moody Diagram (Moody, 1944.) The Moody Diagram (Figure 2) summarises and solves graphically the four friction factor equations Eqs (7), (11), (12), (13) as well as delineating the zones of the various flow regimes The line separating transitional and fully turbulent flow is given by Rouse’s equation: f Equation (13) approaches the smooth pipe equation for low and the rough pipe equation for high values of the Reynolds number respectively Unlike Nikuradse’s , which represents the actual height of the sand grains, the of Colebrook— White’s equation is not an actual roughness dimension but a representative height describing the roughness projections It is referred to as the equivalent sand-grain diameter since the friction loss it represents is the same as the equivalent sand-grain diameter; Table gives experimentally observed values: Pipe material 1277 0.05 h f ϭ19.6 N Sϭ L V2 , R / 2g (16) Q2 N A2 R / (17) 0.06 Critical ⑀/D = 0.010 0.03 0.004 0.002 0.001 0.02 Smooth pipes 0.004 Relative roughness ⑀/D Friction factor f 0.04 turbulent Fully 0.05 0.0001 0.01 103 104 105 106 107 Reynolds number VD/ν FIGURE Pipe friction factors © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1277 11/18/2005 11:12:12 AM 1278 WATER FLOW The dimension of the roughness coefficient, N, is frequently taken as L1/6 as the equation by itself is not dimensionally homogenous Table provides values of Manning’s N for the more widely used pipe materials Hazen-William’s Equation This equation is used mainly in sanitary engineering Contraction The contraction loss equation can be expressed in terms of the downstream velocity, V2, as: V ϭ 0.36CR0.63S0.54, Typical values of the coefficient of contraction, Cc, are given in Table Entrance Loss The head loss at the entrance of a conduit can be compared to that of a short tube: (18) where C ϭ roughness coefficient Typical values of the Hazen-William’s C is given in Table ⎞ V2 ⎛ H ϭ ⎜ Ϫ1⎟ ⎠ 2g ⎝ Cc ⎞V ⎛ hent ϭ ⎜ Ϫ1⎟ ⎠ 2g ⎝C Energy Losses Due to Cross-Sectional Changes, Bends and Valves (20) (21) Cross-Sectional Changes hent ϭ K end Expansion Energy loss in an expansion is principally a form loss: ⎡ ⎛ D ⎞2⎤ V2 hexp ϭ ⎢1Ϫ ⎜ ⎟ ⎥ , ⎢ ⎝ D2 ⎠ ⎥ 2g ⎣ ⎦ (19) in which D ϭ diameter of the conduit, and subscripts and denote upstream and downstream values From Eq (19), if D2 is very large compared to D1, such as the discharge into a reservoir, the entire velocity head is lost TABLE Normal values of Manning’s N Material Brass N 0.010 V2 , 2g (22) in which C ϭ coefficient of discharge, K ϭ entrance loss coefficient Typical values of C and K are given in Table Transition In gradual contractions and expansions, the lead losses are calculated in terms of the difference of velocity heads in the upstream and downstream pipes: ⎛V2 V2 ⎞ Gradual contraction: htc ϭ K te ⎜ Ϫ ⎟ , ⎝ 2g 2g ⎠ (23) ⎛V2 V2 ⎞ Gradual expansion: htc ϭ K te ⎜ Ϫ ⎟ ⎝ 2g 2g ⎠ (24) Ktc values vary from 0.1 to 0.5 for gradual to sudden contractions Values of Ktc range from 0.03 to 0.80 for flare angles of 2Њ to 60Њ Corrugated metal 0.024 Glass 0.010 Concrete, unfinished 0.014 Vitrified clay 0.014 Steel 0.012 A2/A1 Cc Cement 0.012 0.25 0.64 Brick 0.013 0.50 0.68 0.75 0.78 1.00 1.00 TABLE Coefficient of contraction TABLE Hazen-William’s C Material (new) TABLE Values of C and Kent C C Kent Circular bellmouth 0.98 0.05 Square bellmouth 0.93 0.16 130 Fully rounded 0.95 0.10 120 Moderately rounded 0.89 0.25 Sharp cornered 0.82 0.50 Cast iron 130 Welded steel 119 Riveted steel 110 Concrete Wood-stave Vitrified clay 110 Type of entrance © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1278 11/18/2005 11:12:13 AM WATER FLOW Bends The effect of the presence of bends is to induce secondary flow currents which are responsible for the additional energy dissipation: hb ϭ K b V2 2g (25) The bend loss coefficient, Kb, depends on the ratio of the bend radius, r, to the pipe diameter, d, as well as the bend angel For a 90Њ bend and r/d ratio varying from to 12, values of Kb range from 0.20 to 0.07 Gates and Gate Valves The gate and gate valve loss can be expressed as: V2 hg ϭ K g 2g (26) 1279 Energy-discharge Relation In pressure conduit flow, the water is transmitted through a closed boundary conveying structure without a free surface Figure illustrates graphically the various forms of energy losses which could take place within the conduit The following energy relation can be written: hl ϭ hent ϩ htc ϩ h f (28) in which hend ϭ entrance loss, htc ϭ transition loss, hf ϭ skin friction loss If H denotes the total head required to produce the discharge and hv represents the existing velocity head, H ϭ hl ϩ hv (29) Writing Eq (29) in terms of the velocity heads and their respective loss coefficients, The value of the loss coefficient, Kg, for gates depends on a variety of factors The value of Kg for the case having the bottom and sides of the jet suppressed ranges from 0.5 to 1.0 for typical values of Kg for gate valves see Table H ϭ Cl ⎛V2 V22 ⎡ V2 V2 ⎞ ϭ ⎢ K ent ϩ K tc ⎜ − ⎟ 2g ⎣ 2g 2g ⎠ ⎝ 2g ⎢ LV22 V2 ⎤ ϩf ϩ Kv ⎥ , 2g ⎦ 2gD2 TABLE Kg for gate values (30) 0.2 where Kv ϭ combined velocity head and exit loss coefficient By the continuity equation: open 1.3 A1V1 ϭ A2V2 open 5.5 open 24.0 Fully open (31) and V12 A2 V ϭ 2g A1 2g Equation (30) could be expressed as, Exit Loss In general the entire velocity head is lost at exit and the exit loss coefficient, Ke is unity in the equation: he ϭ K e V2 2g (27) H ϭ Cl V22 2g ⎤ ⎛ ⎛ A2 ⎞ V22 ⎡ A2 ⎞ fL ⎢ K ent ⎜ ⎟ ϩ K tc ⎜ − ⎟ ϩ ϭ ϩ Kv ⎥ 2g ⎢ A1 ⎠ gD2 ⎝ A1 ⎠ ⎥ ⎝ ⎣ ⎦ (32) hent in which htc TEL –V 2/2g p/y hl hf H hv ⎡ ⎤ ⎛ A2 ⎞ ⎛ A2 ⎞ fL Cl ϭ ⎢ K ent ⎜ ⎟ ϩ K tc ⎜ − ⎟ ϩ ϩ Kv ⎥ A1 ⎠ 2gD2 ⎝ A1 ⎠ ⎝ ⎢ ⎥ ⎣ ⎦ V2 ϭ gH Cl1/ Transition Q ϭ A2V2 ϭ FIGURE Energy relations A2 gH C1 / (33) (34) (35) ϭ CA2 2gH , © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1279 11/18/2005 11:12:13 AM 1280 WATER FLOW in which C ϭ1/Cl1/ is the discharge coefficient Equation (35) can be readily extended to multiple conduits in parallel Q Q + + Pipe Networks Introduction The Hardy Cross method is most suitably adapted to the resolution of pipe networks The statement of the problem resolves itself into: 1) the method of balancing heads is directly applicable if the discharges at inlets and outlets are known, 2) the method of balancing flows is very suitable if the heads at inlets and outlets are known It is assumed that: a) sizes, lengths and roughness of pipes in the system are given, b) law governing friction loss and flow for each pipe is known, c) equations for losses in junctions, bends, and other minor losses are known These relations are most conveniently expressed in terms of equivalent lengths of pipes FIGURE Pipe network and Method of Balancing Heads Based on the condition required by Eq (36), the following equations for any closed pipe loop results (Figure 4): ∑ H ϭ ∑ r (Q ϩ ⌬Q) ⌬Q ϭ a) to determine the flow distribution in the individual pipes of the network, b) to compute the pressure elevation heads at the junctions In applying the Hardy Cross Method, two sets of conditions have to be satisfied: ∑ Q ϭ Using Darcy’s formula: Hϭ f LV ⎛ f L ⎞ ϭ Q 2gD ⎜ gp D ⎟ ⎝ ⎠ Ϫ∑ rQ ∑ 2rQ ⎛ H⎞ ⌬H ϭ (36) (39) (40) ∑⎜ r ⎟ ⎝ ⎠ 1/ ⎛ H⎞ ∑ 2H ⎜ r ⎟ ⎝ ⎠ 1/ (41) In both Eq (40) and (41), the proper sign conventions must be used in the numerators (37) For the pressure head change in any closed path, the clockwise positive sign convention is used For the discharge continuity requirement at a nodal point, the inward flow positive sign convention is adopted The friction head loss equation is used in the form: H ϭ rQ ϭ 0, Method of Balancing Flows Utilising the continuity requirement at a pipe junction as given by Eq (37), the head correction, ⌬H, at anodal point is given by the equation: a) the total change in pressure head along any closed circuit is zero: b) the total discharge arriving at any nodal point equals the total flow leaving it: where Q0 ϭ assumed flow in the circuit for any one pipe, ⌬Q ϭ required flow correction Expanding Eq (39) and approximating by retaining only the first two terms, the flow correction ⌬Q, can be expressed as: The objectives of the analysis are: ∑ H ϭ 0, 8fL gp D rϭ (38) OPEN CHANNEL FLOW Introduction Open channel flow refers to that class of water discharge in which the water flows with a free surface The stream flow is said to be steady if the discharge does not vary with time If the discharge is time dependent, the water flow is termed unsteady Uniform flow refers to the case in which the mean velocities at any cross-section of the stream are identical; if these mean velocities vary from one cross-section to another, the flow is considered non-uniform Steady uniform flow requires the conveyance section of the stream channel to be prismatic Where the water surface profile is controlled © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1280 11/18/2005 11:12:13 AM WATER FLOW principally by channel friction, this phenomenon is known as gradually varied flow For the type of flow in which the water surface changes substantially within a very short channel length due to a sudden variation in bed slope or cross-section, this category is referred to as rapidly varied flow Open channels fabricated from concrete are often rectangular or trapezoidal in shape Canals excavated in erodible material have trapezoidal cross-sections Although sewer pipes are closed sections, they are still considered as open channels so long as they are not flowing full; these crosssections are usually circular Normal values of Manning’s N for straight channels and various modifying values are given in Table The total mean N value for the channel is obtained from the relation: N ϭ Ns ϩ ∑ Nm In deriving the energy relationships for open channel flow, the following assumptions are normally used: 1) a uniform velocity distribution over the crosssection is assumed, that is, the velocity coefficient, a, in the velocity head term, aV2/2g, is taken as unity In practice, the value of ␣ depends on the shape of the stream channel and has an average value of about 1.02 which makes this assumption sufficiently valid 2) streamlines are essentially parallel, 3) channel slopes are small The most widely used open channel friction formula is the Manning equation as mentioned earlier in pressure flow: AR / S 1/ N (42) Q2 N A2 R / (17) Sϭ Manning’s equation in hydraulic engineering is used for fully turbulent flow and, as such, the values of Manning’s N apply to this flow regime In a natural tortuous stream channel, the mean value of Manning’s N can be obtained from the following considerations: 1) estimate an equivalent basic Ns, for a straight channel of that material, 2) select modifying values of Nm for non-uniform roughness, irregularity, variation in shape of crosssection, vegetation, and meandering, 3) sum the basic, Ns together with the modifying values to obtain the total mean N (43) Energy Principles Channel Friction Equation Qϭ 1281 Consider the water particle of mass, m, and of weight, W (Figure 5) The elevation and pressure energies of the particle are Wh1; and Wh2 respectively Thus, the potential energy of the water particles is, W(h1 ϩ h2) and is independent of its elevation over the flow cross-section As the kinetic energy is WV2/2g the total energy of the water particles, e is: ⎛ V2 ⎞ e ϭ W ⎜ h1 ϩ h2 ϩ ⎟ 2g ⎠ ⎝ (44) Z ϩ D ϭ h1 ϩ h2 (45) and noting that the total flow passing the cross-section is gQ the total energy of the water passing the cross-section per second, Et is given by: ⎛ V2 ⎞ Et ϭ g Q ⎜ Z ϩ D ϩ ⎟ 2g ⎠ ⎝ TABLE Values of Manning’s N (46) Basic NS for straight channels Type of channel Ns Earth 0.010 Sand 0.012 Fine gravel 0.014 Rock 0.015 Coarse gravel 0.028 Cobbles and boulders 0.040 Modifying values of Nm (1) (2) hf V1/2g V1 D1 0.005 to 0.020 Vegetation 0.005 to 0.100 Meander V2 i 0.005 to 0.020 Changes in shape V2/2g D h2 Ns Irregularity TEL D2 0.10 Ns to 0.40 Ns Z1 Z h1 L Z2 FIGURE Energy principles © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1281 11/18/2005 11:12:14 AM 1282 WATER FLOW Thus, the energy per unit weight of water passing the crosssection per second, H is: H ϩZ ϩ Dϩ V2 2g (47) The term, z ϩ D ϩ V2ր2g is known as the total head or total energy level (TEL); the latter name is used here The slope of the total energy level line is the energy gradient or friction slope and gives the rate of energy dissipation in the flow The energies at Sections and are related by the expression: z1 ϩ D1 ϩ If the mean depth of the flow section is defined as Dm ϭ A/T, substitution of this relation into Eq (49) would give the significant expressions: V12 V2 Q2 N ϭ Z ϩ D2 ϩ ϩ / 2g 2g A R (48) in which the Manning equation is used to calculate the friction slope and the mean values for the flow area, A, and the hydraulic radius, R, are to be used Vc2 Dm ϭ 2g and Vc gDm Flow Regimes E ϭ Dϩ Q1 2gA2 ϭ1 (53) At critical flow, Eq (52) demonstrates that the velocity head equals one-half the mean depth and Eq (53) indicates that the Froude number equals unity Specific Energy Diagram for Rectangular Channel For a rectangular channel, Q ϭ qB in which q ϭ discharge per unit width, B ϭ channel width, and Eq (49) becomes, EϭD ϩ Critical Flow The specific energy, E, is defined as the total head referred to the channel bottom (Figure 6): (52) q2 2gD (54) A plot of Eq (54) for any given constant unit discharge gives Figure 7, which is known as the specific energy diagram The (49) Differentiating Eq (47) with respect to D and equating the derivative to zero to obtain its minimum value, dE Q dA ϭ 1Ϫ ϭ g A dD dD (50) 45° line E=D Subcritical Q A3 ϭ g T (51) Equation (51) is the fundamental equation for critical flow and is applicable to all shapes of cross-sections TEL Flow depth, D Noting that dA ϭ TdD, Eq (50) becomes: Ec = Dc Supercritical q2 Critical line T q1 V2/2g D dD dA FIGURE Derivation of critical flow E D FIGURE Specific energy diagram © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1282 11/18/2005 11:12:14 AM 1283 WATER FLOW following flow regimes could be defined with reference to the specific energy diagram Subcritical flow denotes tranquil flow in which the Froude number and the mean velocity are less than unity and the celerity of the gravity wave respectively Critical flow represents the discharge phenomenon where: (1) for a constant specific energy, the discharge is a maximum; (2) the specific energy is a minimum for a constant discharge; (3) the critical velocity equals the celerity of a small gravity wave; (4) the Froude number equals unity; (5) the critical depth is also the depth of minimum pressuremomentum force Supercritical flow which is also known as shooting or rapid flow, is that state of water flow where the Froude number and mean velocity exceed unity and the speed of transmission of a surface wave respectively Based on the equations developed earlier for critical flow, particular formulae can be derived for a rectangular section: ⎛q ⎞ Dc ϭ ⎜ c ⎟ ⎝ g⎠ Vc gDc 1) the bed slope is considered small and neglected, 2) frictional resistance along the bed and sides of the channel are omitted Consider the control volume between Sections (1) and (2) Applying the impulse-momentum principle: g F1 Ϫ F2 ϭ Q(V2 Ϫ V1 ) g g g F1 ϩ QV1 ϭ F2 ϩ QV2 g g (55) (56) Vc2 g (57) ϭ1 (60) in which F1 and F1 denote the hydrostatic forces at sections and respectively The term (F ϩ g/gQV) is given the name – pressure-momentum force Let A ϭ flow area, y ϭ distance of centroid of flow area from surface; then they hydrostatic – force ϭ gAy and Eq (60) can be written as: (58) Flow Transition The concept of the normal depth is an important parameter in the study of flow transition For a given channel and any fixed discharge, uniform flow will occur at one unique depth It is the depth attained in a long channel when the component of gravity force is just balanced by the frictional resistance of the channel When the normal and critical depth are equal, the flow is critical and the bed and energy slopes are the same The channel bed then has a critical slope The bed slope is termed mild when the normal depth exceeds the critical depth; the bed slope is then less than the critical energy slope and the flow regime is subcritical When the normal depth lies below the critical depth and, hence, the bed slope is greater than the critical energy slope, the channel slope is considered to be steep and the supercritical flow regime prevails When water makes a transition from a channel with a mild slope to another with a steep slope, or vice versa, the flow passes through the critical depth close to the junction of the two channels The section in which the water depth is critical defines a channel control The weir acts as a control when water flows over it as critical depth is attained there Hydraulic Jump A hydraulic jump occurs when supercritical flow makes a transition to subcritical flow A common occurrence of a hydraulic jump takes place at the base of a chute (59) or 1/ Dc ϭ ( Ec ) Dc ϭ spillway Figure shows the energy momentum and depth relations for a hydraulic jump and also defines the symbols to be used In developing the equations for the hydraulic jump, the following assumptions are used: A1 y1 ϩ Q2 Q2 ϭ A2 y1 ϩ gA1 gA2 (61) Equation (60) states the condition for the formation of a hydraulic jump and suggests a graphical solution A plot of Eq (60) for any fixed discharge is shown in Figure For any given up-stream supercritical water depth, which is usually known such as at the toe of a spillway, the subcritical hydraulic jump depth or sequent depth can be obtained from the graph For a rectangular cross-section channel and utilising the continuity relation: Q ϭ V1 A1 ϭ V2 A2 P.M.F S.E (62) FLOW (2) (1) D D D Ej TEL V2/2g V1/2g F E Dc F2 P.M.F FIGURE Hydraulic jump relations © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1283 11/18/2005 11:12:14 AM 1284 WATER FLOW Equation (6) can be written as: D2 ϭ ( 1ϩ F12 Ϫ1), D1 (63) where F1 ϭ V1 / (gD1 ) is the upstream Froude number The energy loss Ej across the hydraulic jump on a horizontal floor can be obtained by coming Eqs (61), (62) and the energy equation: D1 ϩ V12 V2 ϭ D2 ϩ ϩ E j 2g 2g (64) where the Manning equation is used to calculate the energy slope Flow computation must start at a control section where all the flow parameters are known The calculation proceeds upstream for subcritical and downstream for supercritical flow In Eq (69), the solution of the reach length, ⌬L, is direct if the immediately upstream depth, D2, is given a value If ⌬L is given a value, D2 has to be solved by trial This method of computing surface water profiles is suitable for regular channels Classification of Flow Profiles Twelve distinct types of non-uniform profiles have been systematically classified (Figure 10) 1) Firstly, the curves are identified according to bed slopes as mild (M), steep (S), horizontal (H), critical (C) and adverse (A) 2) Secondly, numbers are assigned to flow regions The numerical refers to actual flow depths exceeding both critical (Dc) and normal (D) depths For flow depths less than both critical and normal, the number is affixed to it The numeral is for depths intermediate between critical and normal to give: Ej ϭ ( D2 Ϫ D1 )3 D1 D2 (65) The head loss Ej is graphically shown in the specific energy and flow diagrams (Figure 8) Surface Water Profiles Non-uniform Differential Equation Using the notation given in Figure 9, the energy relations can be expressed as: iL ϩ D ϩ ⎛V2 V2 V2 ⎞ ϭ SdL ϩ ( D ϩ dD) ϩ ⎜ ϩ d ⎟ 2g 2g ⎠ ⎝ 2g dL ϭ (66) d( D ϩV / 2g) (i Ϫ S ) Water Profiles in Irregular Channels The river channel has to be divided into panels (Figure 11) with the side panels conveying overbank flow Let Q ϭ total flow, Qc ϭ central channel discharge, Ql ϭ left overbank flow, Qr ϭ right overbank flow The continuity condition requires that: Q ϭ Qc ϩ Ql ϩ Qr , (67) dE ϭ (i Ϫ S ) dL (68) By Manning’s Equation: Qϭ For a finite length, ⌬L, Eq (67) becomes: ⌬L ϭ ( L2 Ϫ L1 ) ⎛ ( D ϩV / 2g ) Ϫ ( D2 ϩV / 2g ) ⎞ ϭ⎜ ⎟, i Ϫ Q N /A R / ⎠ ⎝ 2 (69) 1 Ac Rc2 / S 1/ ϩ Al R / S 1/ ϩ A R / S 1/ (71) Nc Nl Nr r r The energy slope, S, has been taken as the same for Qc, Ql, Qr; this assumption seemed to be justified in practice Due to different channel roughness, vegetative and other obstructions, Manning’s N for the three flow panels would (2) (1) V2 S TEL 2g (70) D S dL V2 + d 2g i idL (V ) 2g (D + dD) dL FIGURE Non-uniform flow derivation © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1284 11/18/2005 11:12:14 AM WATER FLOW M1 Left panel S1 M2 1285 Right panel Center panel S2 S3 M3 Mild slope Steep slope FIGURE 11 River channel division A2 C1 C3 It is assumed that constant maintenance of canals suppresses the canal’s tendency to meander The water-sediment flow is usually regarded as the independent parameter In the concept of flow in mobile channels where the transport of sediment is an integral part of the system, two philosophies have emerged Based on the work of Lindley (1919), Lacey (1952), Inglis (1949), and Blench (1953, 1966) in India and Pakistan, the regime theory has evolved On the other hand, the United States Bureau of Reclamation under the direction of Lane (1952, 1953) developed the tractive force method A3 Adverse slope Critical slope Legend Normal depth Critical depth H2 H3 Regime Theory for Canals FIGURE 10 Surface water profiles not have the same values As the energy gradient, S, is common to the panels in Eq (71), water level versus energy slope curves can be plotted for any selected discharge for water level computations The energy equation for the step method of surface profile calculation can conveniently be written in the form: WL2 ϩ Q2 Q2 ϭ WL1 ϩ ϩ SL1,2 2g A2 2g A12 (72) If in practice, the change in kinetic head, (Q / 2gA2 Ϫ Q / 2g A1 ) is small and could be removed from Eq (72), this would greatly simplify the work In determining the wetted perimeter for the calculation of the hydraulic radius, only the water-channel contact lines are relevant and the water-water contact lines between panels are excluded 2 2 FLOW IN ERODIBLE CHANNELS The regime theory postulates that for given water-sediment flow and bed material, there exists a regime channel which determines uniquely the flow area, cross-sectional shape and bed slope The regime channel is considered a stable channel which on the average will neither silt nor scour The flow occupying the regime channel is the dominant discharge and it is also variously referred to as the formative, regime, or bank-full discharge Lacey’s Equations Based on extensive flow observations of the canals in India, Lacey (1952) proposed a set of formulae for alluvial channels with sandy mobile beds with the discharge ranging from 25 cfs to 2500 cfs, the bed material size varying from 0.2 mm to 0.6 mm and with the quantity of solids conveyed being less than 50 ppm: Wetted perimeter: P ϭ 2.67 Q1/ (73) Flow area: Aϭ 1.25 Q / f 1/ (74) 0.0054 f / Q1/ (75) Bed slope: Introduction Flow in erodible channels can be divided into two types, namely, canal and river flows Blench describes canal flow as possessing there degrees of freedom due to its ability to adjust itself with respect to its flow depth, bed slope, and side widths which are taken to be the dependent variables River flow, in addition to having the three degrees of freedom of canals, has a fourth degree by virtue of its ability to meander Silt factor: Sϭ 1/2 f ϭ 8dinch (76) From the above equations, the following two equations can be derived: Mean velocity: V ϭ 0.8951/ Q1/ r © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1285 11/18/2005 11:12:15 AM 1286 WATER FLOW TABLE Values of the bed factor 1.0 Remarks Very sandy loam banks 0.1 Erosion if Ͼ (Fs)max Silty clay loam 0.2 Erosion if Ͼ (Fs)max Very cohesive banks 0.3 Erosion if Ͼ (Fs)max 0.473Q1/ Dϭ f 1/ (77) Blench’s Equations Blench (1953, 1966), using the concept that canals possessing three degrees of freedom must, therefore, have three basic equations to describe their motion, presented the stream equations for small bed load as: V D ϭ Fb SS 1.5:1, sides 0.6 SS 0:1, sides Rectangular 0.4 SS 0:1, bottom 0.2 (78) V2 ϭ Fs B SS 2:1, sides 0.8 Maximum unit tractive force Ϭ γDS (Fs)max Flow depth: bottom SS 2:1 and 1.5:1 Trapezoidal Material (79) 0 V2 ϭ 3.63(VB/v)1/ gDS Width/depth ratio (80) In Eq 78, Fb is bed factor and the equation itself expresses the statement that channels with similar water-sediment flows tend towards the same Froude number in relation to a suitable depth Equation (79) describes the scouring action on the hydraulically smooth sides and defines the side Factor, Fs The dissipation of energy per unit mass of water per unit time in the channel is given by Eq (80) in which S is the energy gradient For appreciable bed load, Eq (80) becomes: V2 C ⎞ ⎛ VB ⎞ ⎛ ϭ 3.63 ⎜ 1ϩ ⎟⎜ ⎟ ⎝ gDS 233 ⎠ ⎝ v ⎠ 1/ , (81) where C is the bed load charge in parts per hundred thousand by weight of fluid discharge The bed factor, Fb, for sand of subcritical flow is given by the empirical equations: Fb ϭ Fb (1ϩ 0.12C ) (82) Fb ϭ 1.9 dmm (83) in which Fb0 ϭ zero bed factor and is the value of Fb when C tends to zero, dmm ϭ median bed material size by weight in millimeter As a guide to the value of the side factor, Fs, the following table has been suggested by Blench (1966) FIGURE 12 Maximum tractive force Tractive Force Method for Canals Unit Tractive Force The stability of an erodible channel depends on (a) the resistance of the material lining the bottom and sides against the erosive force of the stream and (b) the ability of the stream to transport the sediment load without giving rise to significant deposition The shear or drag force exerted by the water on the bed and sides of the channel is termed the tractive force The average unit tractive force, t, in uniform flow is the component of the gravity force acting on the water parallel to the channel bottom per unit area, thus: t ϭ gRS (84) For wide channels, the flow depth can replace the hydraulic radius: t ϭ gDS (85) The distribution of tractive force has been investigated by the United States Bureau of Reclamation (Lane, 1952; 1953, Olsen and Florey, 1951, 1952) The maximum values of the unit tractive force for the bottom and sides of rectangular and trapezoidal cross-sections are given in Figure 12 Tractive Force Ratio A soil particle of effective area, Ae, resting on the side of a channel is acted on by the tractive © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1286 11/18/2005 11:12:15 AM WATER FLOW The tractive force ratio, K, is defined as, ts/tl is obtained by dividing Eq (87) to Eq (88) and simplifying: 1.0 Critical tractive force, 1bs./ft2 1287 Coarse noncohesive material 25% larger Large amount of fine sediment K ϭ (1Ϫ sin w/ sin u)1/ From Eq (89) it can be seen that the tractive force ratio, K is a function of the side slope and angle of repose of the material only Critical Tractive Force The permissible tractive force is the maximum unit tractive force that will not cause significant scour of the material lining the channel bed on a level surface It is often found from laboratory observations and is known as the critical tractive force It is influenced by the amount of organic matter and fine suspended sediment in the water The effect of the fine sediment is to increase the allowable critical tractive force Figure 13 shows curves of permissible tractive forces as recommended by the United States Bureau of Reclamation 0.1 Clear water Small amount of fine sediment 0.01 1.0 0.1 (90) 10 100 Mean diameter, mm River Engineering FIGURE 13 Critical tractive force for canals force, Aets, in the direction of the flow and the gravity force component, Ws sinw which attempts to cause the particle to roll down the side slope, where ts ϭ unit tractive force on the side of the channel, Ws ϭ submerged weight of the particle, w ϭ angle of the channel side The resultant of these two forces, F, is: F ϭ (Ws2 sin w ϩ Ac2 t s2 )1 / (86) The motion of the article is resisted by its frictional force (R): R ϭ Ws cos w tan u, (87) where tan u is the coefficient of friction and u is the angle of repose of the material Equating Eq (86) and (87) for the condition of impending motion and solving for ts: ⎛ W tan w ⎞ t s ϭ r cos w tan u ⎜ 1Ϫ Ae tan u ⎟ ⎝ ⎠ (88) A similar equation can be written for the case of a particle on a level bed when motion is impending, thus: tl ϭ Wr tan q, Ae where tl denotes the unit tractive force on the level bed (89) In river flows, a greater number and range of factors have to be considered in addition to those parameters used in the analysis of canals These variables include bigger size bed materials, large suspended and bed sediment loads, unsteady and a wide variation of flood flows, meandering and braiding, large changes in stream channel cross-sections, obstructions to flow, and other factors involved An analysis of river engineering is, therefore, beyond the scope of this chapter Readers are recommended to consult the works of Blench (1966), Shen (1971, 1972), Inglis (1949) and Leopold, Wolman and Miller (1964) More specialized treatment of sediment transport, bedforms and stream geometry can be found in the publications of Einstein (1972), Leopold and Maddock (1953), Richardson and Simons (1967), Yalin (1971), Kennedy (1963), Christensen (1972), and Ackers (1964) Standard texts which cover the subject more formally include those of Graf (1971), Henderson (1966), Raudkivi (1967) and Leliavsky (1955) FLOW WITH AN ICE COVER A river flowing with an ice cover has, in addition to the bed and side frictional forces, the shear resistance imposed by a buoyant boundary represented by the floating ice cover Chee and Haggag (1984) have developed equations concerning floating boundary stream flow which are reproduced here The essential concepts and assumptions are first discussed A channel with a buoyant cover can be divided into two subsections as shown in Figure 14 The flow in subsection (1) is influenced by the bed and sides while subsection (2) is controlled by the cover The two subsections are divided by a separation surface which represents the locus of no shear and maximum velocity The equations of energy, continuity, © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1287 11/18/2005 11:12:15 AM 1288 WATER FLOW P2 P1 FIGURE 14 Ice covered channel and momentum are applicable to each subsection individually as well as to the entire channel cross-section Composite Roughness Equation The Reynolds form of the Navier-Stokes equation was used to develop the shear distribution and the velocity profile was obtained using the Prandtl-von Karman mixing length theory In addition, a channel momentum equation and the flow resistance formula of Manning were utilised to derive the relationship for the division surface separating the two flow subsections as (0.66) ( l1/ − 1) R1/ ϭ [aϩ (1Ϫ a) l]1/ 1/ 1Ϫ N1 /N l2 / Nqg (91) in which R ϭ hydraulic radius of entire channel, N1, N2 ϭ Manning’s roughness for the bed and cover respectively, g ϭ acceleration due to gravity, l ϭ R1/R2 is hydraulic radius ratio of the bed subsection to the cover subsection, a ϭ P1/P is the wetted perimeter ratio of the entire channel to the bed subsection The division surface is found by solving for l using Eq (91) The complete roughness, N, of an ice-covered channel is given by ⎤ N1 N Ϫ5 / ⎡ ϭ [1ϩ (1Ϫ a) l] ⎢a ϩ (1Ϫ a) l5 / ⎥ N N2 ⎣ ⎦ (92) REFERENCES Ackers, P., Experiments on small streams in alluvium, proceedings, American Society of Civil Engineers Journal, Hydraulics Division, 90, no HY4, pp 1–37, July 1964 Albertson, M.L., J.R Barton, and D.B Simons, Fluid Mechanics for Engineers, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961 Allen, J and S.P Chee, The resistance to the flow of water round a smooth circular bend in an open channel proceedings, Institution of Civil Engineers, London, 23, pp 423–424, November 1962 Blench, T., Regime theory equations applied to a tidal river estuary, proceedings, Minnesota International Hydraulics Convention I.A.H.R IAHR and ASCE Joint Meeting, pp 77–83, September 1953 Blench, T., Mobile-bed fluviology, University of Alberta, 1966 Chee, S.P and M.R.I Haggag, Flow resistance of ice-covered streams, Canadian Journal of Civil Engineering, Vol II, No 4, pp 815–823, December 1984 Chow, Ven Te, Open-Channel Hydraulics, McGraw-Hill Book Company, Inc., New York, 1959 Christensen, B.A., Incipient motion on cohesionless banks, Chapter in Shen, Hsieh Wen, Sedimentation, Colorado, 1972 Einstein, H.A., The bed load function for sediment transportation in open channel flows, technical bulletin No 1026, US Department of Agriculture, Soil Conservation Service, September 1950 Also reprinted in Appendix B in Shen, Hsieh Wen, Sedimentation, Colorado, 1972 10 Graf, W.H., Hydraulics of Sediment Transport, McGraw-Hill Book Company, New York, 1971 11 Henderson, F.M., Open Channel Flow, The Macmillan Company, New York, 1966 12 Inglis, C.C., The behavior and control of river and canals, research publication No 13, Central Waterpower Irrigation and Navigation Research Staton, Poona, 1949 13 Kennedy, J.F., The mechanics of dunes and antidunes in erodible bed channels, J Fluid Mechanics, 16, part 4, 1963 14 King, H.W., Handbook of Hydraulics, McGraw-Hill Book Company, Inc., New York, 1954 15 Lacey, G., Flow in alluvial channels with sandy mobile beds, Proceedings, Institution of Civil Engineers, London Vol 9, pp 145–164, 1952 16 Lane, E.W., Progress report on studies on the design of channels by the Bureau of Reclamation, Proceedings, American Society of Civil Engineers, Irrigation and Drainage Division, Vol 79, pp 280-1–280-31, September 1953 17 Lane, E.W., Progress report on results on design of stable channels, US Bureau of Reclamation, Hydraulic Laboratory Report No Hyd-352, June 1952 18 Leliavsky, Serge, An Introduction to Fluvial Hydraulics, Constable and Co Ltd., London, 1955 19 Leopold, L.B and T Maddock, Jr., The hydraulic geometry of stream channels and some physiographic implications, US Geological Survey, Professional paper 252, 1953 20 Leopold, L.B., M.G Wolman, and J.P Miller, Fluvial Process in Geomorphology, W.H Freeman and San Francisco, 1964 21 Lindley, E.S., Regime channels, Minutes or Proceedings, Punjab Engineering Congress, Lahore, India, Vol 7, pp 63–74, 1919 22 Moody, L.F., Friction factors for pipe flow, Trans ASME., 66, No 8, 1944 23 Morris, H.M., Applied Hydraulics in Engineering, The Ronald Press Company, New York, 1963 24 Olsen, O.J and Q.L Florey (compilers), Stable Channel Profiles, Hydraulic Laboratory Report No Hyd-325, US Bureau of Reclamation, 1951 25 Olsen, O.J and Q.L Florey (compilers), Sedimentation studies in open channels: boundary shear and velocity distribution by membrane analogy, analytical and finite difference methods, US Bureau of Reclamation, Laboratory Report No Sp-34, 1952 26 Raudkivi, A.J., Loose Boundary Hydraulics, Pergamon Press, Oxford, 1967 27 Richardson, E.V and D.B Simons, Resistance to flow in sand channels, Twelfth Congress, IAHR, 1, pp 141–150, 1967 28 Rouse, H., editor, Engineering Hydraulics, John Wiley and Sons, Inc., New York, 1950 29 Shen, Hsieh Wen, Editor, River Mechanics, I and II, Colorado, 1971 30 Shen, Hsieh Wen, Editor, Sedimentation: Symposium to honour Professor H.A Einstein, Colorado, 1972 31 Streeter, S.L., Fluid Mechanics, McGraw-Hill Book Company, New York, 1966 32 United States Bureau of Reclamation, Design of small dams, US Printing Office, 1961 33 Yalin, M.S., On the formation of dunes and meanders, paper C-13, proceedings Fourteenth Congress, IAHR, 3, 1971 S.P CHEE University of Windsor © 2006 by Taylor & Francis Group, LLC C023_003_r03.indd 1288 11/18/2005 11:12:15 AM ... Consider the water particle of mass, m, and of weight, W (Figure 5) The elevation and pressure energies of the particle are Wh1; and Wh2 respectively Thus, the potential energy of the water particles... solved by trial This method of computing surface water profiles is suitable for regular channels Classification of Flow Profiles Twelve distinct types of non-uniform profiles have been systematically... lines are relevant and the water- water contact lines between panels are excluded 2 2 FLOW IN ERODIBLE CHANNELS The regime theory postulates that for given water- sediment flow and bed material,