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Open channel flow, 1. Introduction, a) Characteristic of Open channel flow, • Flow in open channel is defined as the flow of a liquid with a free surface. A free surface is a, surface having constant pressure., • In case of open channel flow, as the pressure is atmospheric, the flow takes place under the force, of gravity which means the flow takes place due to the slope of the bed of the channel only. HGL, coincides with top surface of water. (pressure head + datum head), • The flow channel can be of any shape, such as rectangular, circular, trapezoidal. And Froud, number is used for analysis., b) Types of channels, • Artificial and natural channel, • Prismatic and non-prismatic, ¨ If cross-section shape size and bed slope remain constant in the direction of flow, then, channel is considered to be prismatic., ¨ All artificial channels are considered as prismatic for longer length., • Rigid and Mobile boundary channel, ¨ In rigid boundaries are non-deformable, no silting or scouring takes place, water is having, only one degree of freedom., ¨ In mobile boundary, water is having four degree of freedom, such as depth of flow, width, of channel, bed slope and layout. In OCF only rigid boundary channel are considered., c) Type of flow in open channel, • Steady and unsteady, • Uniform and non-uniform, • Laminar, transition and turbulent, Pipe flow, OCF, Laminar, Re < 2000, < 500, Transition, 2000 < Re < 4000 500 < Re < 2000, Turbulent, 4000 < Re, 2000 < Re, ¨ In OCF flow is generally considered to be turbulent., • Critical, sub critical, super critical flow., 𝐹𝑟 = 1 Critical, ¨ Froud Number, 𝐹𝑟 > 1 Super critical, 𝐹%, 𝑣, 𝐹𝑟 < 1 Sub critical, 𝐹𝑟 = $ =, 𝐹& (𝑔𝐿+, o, , Characteristic length, 𝐿+ =, , 𝐴, 𝑇, , 𝑄/ 𝑇, =1, 𝑔𝐴0, d) General parameters of channel section, • Depth of flow: it is vertical distance measured from bed of the channel to top surface of water., • Depth of flow section: it is normal distance measured from bed of the channel to top surface of, water., ¨ Since in OCF channel bed slope is considered small hence vertical depth and normal depth, is almost same (HGL coincides with top surface of water). For larger bed slope due to, difference in vertical depth and normal depth, HGL lies below the top surface of water., • Hydraulic radius, 𝐴, area, 𝑅= =, 𝑃 wetted perimeter, , 1

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e) Possible combination of flow, • Steady uniform, • Steady non-uniform, • Unsteady uniform, • Unsteady non-uniform, • If flow is uniform it is always steady, and if flow in non-uniform it can either be steady or, unsteady. Non-uniform flow is also known as varied flow. Which can further be classified as, Gradually varied flow and Rapidly varied flow., • GVF: if change in depth is gradual in the direction of flow, such that curvature of water surface is, very less then flow is considered GVF. Friction plays important role in GVF analysis., • If change in depth is sudden such that curvature of water surface profile is very large then flow is, considered to be RVF. Friction is insignificant in RVF analysis., • Spatially varied flow: it is to be noted that any change in depth in case of GVF and RVF shall not, be due to some addition or subtraction of water. If depth changes due to some addition and, subtraction of water then flow is considered as spatially varied flow., f) Velocity distribution in Open Channel, • Average velocity, ¨ 𝑣@A& =, , AB.D EAB.F, /, , deep river, , ¨ 𝑣@A& = 𝑣G.H, shallow river, ¨ 𝑣@A& = (0.8 𝑡𝑜 0.95)𝑣QRST@+U, ¨ 𝑣@A& =, •, , •, , ∫ AWX, X, , Non-uniform velocity distribution at any section is due to presence of free water surface and, resistance from the channel boundaries. Presence of corners and boundaries causes the velocity, vector of flow to have component not only in longitudinal and lateral direction but also in normal, direction of flow. Current in normal direction is known as secondary current., Dip of maximum velocity: it is depth from the surface from which maximum velocity occurs., Surface velocity is less because of secondary currents, and it is a function of aspect ratio., ¨ Aspect ratio =, , YUZ[\ ]T +\@^^U_, `%W[\ ]T +\@^^U_, , ¨ Dip of maximum is at larger depth for the channel having larger aspect ratio (Narrow and, deep channel), • Contours of equal velocity are known as Isovels., • For artificial channel as we move from centre line toward boundaries, depth of flow increases and, velocity reduces. (slightly curved surface profile), g) Correction factor, • Kinetic energy correction factor, ¨ Coriolis coefficient 𝛼, KE actual velocity ∫ 𝑣 0 𝑑𝐴, 𝛼=, = 0, KE avg velocity, 𝑣@A 𝐴, • Momentum correction factor, ¨ Boussinesq coefficient 𝛽, Momentum actual velocity ∫ 𝑣 / 𝑑𝐴, 𝛽=, = /, momentum avg velocity, 𝑣@A 𝐴, h) Equation in use, • Continuity equation: based on law of conservation of mass, ¨ Steady state, o 𝐴o 𝑣o = 𝐴/ 𝑣/, o, , 2, , Wp, Wq, , =0

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•, , At conjugate or sequent depths Specific Force remains same (hydraulic jump), pD, , ¨ 𝐹Q = 𝐴𝑧̅ + &X, b) At critical depth 𝑆§ is minimum, also discharge is maximum for that 𝑆§ and 𝑆¨ ., • As discharge is increased keeping specific energy as constant, 𝑆§ curve shifts right and upwards, and difference in alternate depth reduces. It is possible to imagine a discharge 𝑄 = 𝑄•@q at which, specific energy curve would just be tangential to specific energy line at critical depth., c) Specific force diagram, • For rectangular section, ¨ 𝐹Q =, , ©u D, , +, , /, , ©¤ D, &u, , ¨ Depth values having same specific force are called, conjugate/sequent depth., d) Critical depth and Energy, • Rectangular channel, o/0, , 𝑞/, 𝑦+ = ª «, 𝑔, , ¨ tan 𝛼 =, , •, , u¬, 0//u¬, , solving 𝛼 = 33.7], , 3, 𝐸+ = 𝑦+, 2, , ¨ When alternate depths are given then critical depth for that discharge can be computed, directly as, 𝑞/, 2𝑦o/ 𝑦//, 𝑦+0 =, =, 𝑔, 𝑦o + 𝑦/, ¨ Specific Energy at the given alternate depth, 𝑦o/ + 𝑦o 𝑦/ + 𝑦//, 𝐸=, 𝑦o + 𝑦/, ¨ If critical depth at this energy is required 𝐸 = 1.5𝑦+Triangular section, /p D, , o/Ÿ, , ¨ 𝑦+ = w&•D z, Ÿ, , ¨ 𝐸+ = ® 𝑦+, e) Channel transition, • Flow through hump, ¨ Assumption: channel is horizontal, frictionless, rectangular, ¨ Since channel is horizontal and frictionless, total, energy and specific energy remains constant., Construction of hump at downstream reduces the, specific energy by an amount equal to height of, hump. Due to decrease in specific energy in the, direction of flow,, o Depth of flow reduces if flow is, subcritical, o Depth increases if flow is supercritical., ¨ Δ𝑍• is minimum height of hump for critical flow, or maximum height of hump for which upstream, flow is not affected., ¨ When ΔZ < ΔZ² (no chocking), , 5

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¨ Δ𝑍 > Δ𝑍• flow is not possible for the given specific energy E, to make flow possible, upstream depth 𝑦 has to increase to 𝑦′ to create a new specific energy 𝐸 - (subcritical flow), so that critical depth corresponding to minimum specific energy is always maintained over, the hump., ¨ Calculating Δ𝑍•, Δ𝑍• = 𝐸o − 𝐸+, 𝑣o/ 3, Δ𝑍• = 𝑦o +, − 𝑦, 2𝑔 2 +, ¤D, , •, , o/0, , Put 𝑦+ = w & z, , o, , Width contraction, ¨ Since channel is horizontal and frictionless and also there is no hump provided, specific, energy and total energy remains constant., ¨ Width contraction at downstream increases discharge per unit width due to which specific, energy curve shifts right and upwards. Since specific energy remains constant hence due to, increase in 𝑞 depth of flow reduces in subcritical condition, and increases for supercritical, condition. (difference in alternate depth decreases), ¨ Here 𝐵/• is greatest allowable contraction at which flow is critical and upstream flow is, not affected., ¨ Calculation of 𝐵•, 𝐵• = ’, , o, , /³p D, , “&§…´, , ¨ If 𝐵/ < 𝐵•, o Chocking will occur, flow is not possible for given specific energy, 𝐸o to make, flow possible upstream depth 𝑦o has to increase to maintain 𝑦o- for subcritical flow, to create a new specific energy which will just be sufficient to maintain critical, depth over the contracted section. It is to be noted here that the new critical depth, 𝑦+/, > 𝑦+•, 4. Gradually varied flow, a) Assumptions, • Region of high curvature is excluded from the analysis and pressure distribution is assumed to be, hydrostatic., • Resistance to flow at any depth is given by uniform flow equation such as Chezy’s and manning’s, equation by replacing bed slope with friction slope., • Slope of channel is small and channel is considered to be prismatic., • Roughness coefficient such as 𝐶, 𝑛 and kinetic energy correction factor are independent of depth, of flow., b) Differential equation for GVF, •, , W§, , •, , Dynamic equation for GVF, , Wq, , = 𝑆] − 𝑆T, 𝑆] − 𝑆T, 𝑑𝑦, =, 𝑑𝑥 1 − 𝛼𝐹𝑟 /, ¨ As 𝑦 is the depth of flow and x is the distance measured along the bottom of the channel, hence, , Wu, Wq, Wu, , represents the variation of the water depth along the bottom of the channel., , ¨ When Wq = 0, 𝑦 is constant i.e. depth of the water above the bottom of channel is constant., It means the free surface of water is parallel to the bed of the channel., ¨ When, , Wu, Wq, , > 0, It means that depth of the water increases in the direction of flow. The, , profile of water so obtained is called back water curve., , 6

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¨ When, •, , Wu, Wq, , = 0, It means that the depth of water decreases in the direction of flow. The, , profile of the water so obtained is called drop down curve., For wide channel we can derive that, ¨ From manning’s equation, Q¶, , o, , Q·, , u, , oG/0, , = w u·z, , ¨ Chezy’s equation, Q¶, , o, , Q·, , u, , 0, , = w u·z, , •, , As per manning’s, , •, •, , 𝑦] oG/0, ¹, 𝑑𝑦 𝑠] ¸1 − w 𝑦 z, =, 𝑦 0, 𝑑𝑥, ¸1 − w 𝑦+ z ¹, Where 𝑦] is normal depth, 𝑦 is depth of flow, 𝑦+ is critical depth, General equation, ¨, , Wu, Wq, , =, , º », Q· ¸o–w ·z ¹, º, , º ¼, ¸o–w ¬z ¹, º, , ¨ N and M are hydraulic exponent corresponding to normal depth and critical depth., c) Classification of channel, Characteristic, Channel category, Symbol, Remark, condition, Mild slope, M, Subcritical flow at normal depth, 𝑦] > 𝑦+, Steep slope, Supercritical flow at normal depth, S, 𝑦] < 𝑦+, Critical slope, Critical flow at normal depth, C, 𝑦] = 𝑦+, Horizontal bed, Cannot sustain uniform flow, H, 𝑠] = 0, Adverse slope, Cannot sustain uniform flow, A, 𝑠] < 0, • Critical slope can be calculated by (using manning’s equation for wide channel), ¨ 𝑠+ =, , •, , ^D &…B/ž, ¤ D/ž, , ¨ A given channel may be classified as mild for one discharge, critical for another discharge,, and steep for yet another discharge., Regions of flow profile, , Types of GVF possible (12), M1 M2 M3, S1 S2 S3, C1 C3, H2 H3, A2 A3, d) Various GVF profile, , 7

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Profile with subscript 1 and 3 are rising and, profile with subscript 2 are falling., Critical slope is highly unstable as it does, not have falling water profile., , e) Control section, • A control section is defined as a section in which fixed relationship exists between depth and, discharge of flow., • Weir, sluice gate, dam, change in bed slope, change in channel geometry, change in channel, roughness characteristic etc. acts as a control section., • Subcritical section has control section in downstream and supercritical flow has control section in, upstream., • When flow changes from subcritical to supercritical, critical depth is control point., • When flow changes from supercritical to subcritical, hydraulic jump is formed, vice versa is not, true., f) Analysis of flow profile for break in grade, • First NDL (normal depth line) and CDL (critical depth line) for both channels are drawn. As CDL, depends on Q only there is no change., , 8

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a) Assumptions, • Depth of flow before and after the jump is uniform and pressure distribution is assumed to be, hydrostatic., • Length of jump is so small such that frictional resistance is negligible., • Channel bed is horizontal or slope is so gentle such that weight component of water mass, comprising the jump is very small and hence neglected., b) Analysis, • The two depths before and after the jump are conjugate or sequent depths at which specific force, is constant, due to high turbulence and eddies formation, significant amount of energy is lost, which is initially unknown to us, hence momentum equation is used for analysis., c) Hydraulic jump, 𝑦/ 1, = Ä−1 + ’1 + 8Fro/ Æ, 𝑦o 2, ¤D, , ¨ 𝐹𝑟 = &u ´, …, , •, , u…, , •, , Energy loss (head), , uD, , o, , = / Ç−1 + (1 + 8Fr// È, 𝐸É =, , •, •, , (𝑦/ − 𝑦o )0, 4𝑦o 𝑦/, , Power dissipated, ¨ 𝑃 = 𝑚̇𝑔𝐸É, Discharge, 𝑦+0 =, , 𝑞/ 𝑦o 𝑦/ (𝑦o + 𝑦/ ), =, 𝑔, 2, , d) Standard terms, • Height of jump = 𝑦/ − 𝑦o, u –u, • Relative height of jump = D§ …, …, , •, •, •, , Length of jump is 5 to 7 times of height., Relative initial depth = 𝑦o /𝐸o, Relative sequent depth = 𝑦/ /𝐸o, , •, , Relative energy loss = § Ë, , •, , Efficiency of jump = §D = 1 − § Ë, , §, , §, , …, , …, , §, , …, , ¨ Efficiency is only a function of 𝐹𝑟o, e) Classification of jump, Fr, Type, Undular jump (or standing wave). Small rise in surface level. Low energy, 1–1.7, dissipation. Surface rollers develop near Fr =1.7., 1.7–2.5, weak jump. Surface rising smoothly, with small rollers. Low energy dissipation., Oscillating jump. Pulsations caused by jets entering at the bottom generate large, 2.5–4.5, waves that can travel for miles and damage earth banks. Should be avoided in the, design of stilling basins., Steady jump Stable, well-balanced, and insensitive to downstream conditions., 4.5–9, Intense eddy motion and high level of energy dissipation within the jump., Recommended range for design., Strong jump. Rough and intermittent. Very effective energy dissipation, but may, >9, be uneconomical compared to other designs because of the larger water heights, involved., f) Location of jump, , 10

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•, •, , 𝑦[ = 𝑦/ free hydraulic jump will form at Venna contracta., 𝑦[ < 𝑦/ free repelled jump, in this case formation of jump is delayed, it is formed at some, distance downstream of venna contracta at any height 𝑦o > 𝑦@ such that it will end at 𝑦[ . In this, case calculation will proceed in backward direction., • 𝑦[ > 𝑦/ submerged or drowned jump, in this case energy loss is very less and jump will form at, venna contracta only., g) Application, • Energy dissipaters, • Mixing of chemicals in water treatment plants, • Increase water load on apron to counteract uplift., 6. Rapidly varied unsteady flow, a) Positive surge, • The rapidly-varied transient phenomenon in an open channel, commonly known under the general, term surge, occurs whenever there is a sudden change in the discharge or depth or both., • Such situations occur, during the sudden closure of a gate. A surge producing an increase in depth, is called positive surge and the one which causes a decrease in depth is known as negative surge., • Further a surge can travel either in the upstream or downstream direction, thus giving rise to four, basic types., • Positive surges moving downstream and moving upstream are often termed moving hydraulic, jumps in view of their similarity to a steady state hydraulic jump in horizontal channels., b) Celerity, • The velocity of the surge relative to the initial flow velocity in the canal is known as the celerity, of the surge (𝐶Q )., • Thus, for the surge moving downstream 𝐶Q = 𝑣` − 𝑣o and for surge moving upstream 𝐶Q = 𝑣` +, 𝑣o, 𝐶Q = $, , •, •, , •, , 𝑔 𝑦/, (𝑦 + 𝑦/ ), 2 𝑦o o, , ¨ Suffix 1 and 2 refers to the conditions before and after the passage of the surge., When height of wave is very small then 𝑦/ → 𝑦o, 𝐶Q = (𝑔𝑦, Consider a surge moving downstream. If the surge is considered to be made up of large number of, elementary surges of very small height piled one over the other, for each of these 𝑣` = 𝑣o + (𝑔𝑦., The top point of the surge moves faster than bottom of the surge. This causes the top to overtake, the lower portions and in this process the flow tumbles down on to the wave front to form a roller, of stable shape. Thus, the profile of a positive surge is stable and its shape is preserved., In a negative surge, by a similar argument, a point on the top of the surge moves faster than a, point on the lower water surface. This results in the stretching of the wave profile. The shape of, the negative surge at various time intervals will be different and as such the analysis used in, connection with positive surges will not be applicable., , 11