OPEN CHANNEL FLOW Open channel is a conduit in which the liquid flows with a free surface subjected to atmospheric pressure. The flow is caused by the slope of the channel and of liquid surface. ‘Types of Channels 1, Natural Channels. These are irregular in shape and their size varies from hilly rivulets to large rivers. 2, Artificial Channels. These are constructed and developed by human efforts. Their shape and size depend upon the requirement of the project. Geometric Elements of Open Channel 1. Prismatic and Non-prismatic Channel . A channel of constant slope and cross-section for sufficient length is called prismatic channel. Artificial channels are generally prismatic channel. In natural streams, slope at various cross-sections do not remain same throughout a reach. Such channels are called non- prismatic channels, 2. Depth of flow (y). It is.vertical distance from free surface to the bed of the channel. 3. Depth of flow-section (d). It is the depth of flow normal to the channel bed. d=ycos8 where, @ = angle for bed slope of the channel. 4. Channel top width (T). It is width of water surface exposed to the atmospheric pressure. dA tS, dy 5. Free board. It is vertical distance from the top of channel to the water surface. It is provided to prevent overflowing of water due to waves and fluctuations. 6. Cross-sectional area of flow (A). It is the area of flow through which flow takes place. 1. Wetted perimeter (P). It is length of boundary in contact with the liquid flowing in this cross-section plane x-x. 8, Hydraulic radius (R). Itis the ratio of cross-sectional area of flow to the wetted perimeter, ie. R= A/P Hydraulic radius is also called Hydraulic mean depth (H.M.D), 9. Hydraulic depth (D). It is the ratio of cross-sectional area (A) to the top width (T), : A ie, D=z 10. Bed slope (S,). It is the slope of the bed of the channel. It is given by S, = sin 0 or tan UL. Hydraulie slope (S,). It is the slope of energy line. head lost due to friction (h,) in length (L) = Tength (L) 12. Water surface slope (S.,). [tis the slope of water surface line at any section, Ina channel it represents the hydraulic graident.Classification of open Channel Flow Open Channel flow Unsteady flow (time dependent flow) Steady flow (time independent flow) ae ss Uniform flow Non-uniform flow Uniform flow (Flow depth __Non-uniform flow or varied remainsconstanteverywhere _flow (flow depth changes (Practically impossible es along the channel) along the channel) to exist) varied flow a ———— Gradually varied flow Rapidly varied (flow depth Gradullay varied Rapidly varied (depth variation alongthe _variation along the channel flow flow is sudden) channel is gradualy) 1. Based on Change in Flow depth with respect to Time and Space (i) Steady and Unsteady Flows (Time basis): In steady flow, conditions of flow e.g. depth of flow and velocity does not vary with time and if conditions vary with time, itis called unsteady flow. ev oy For stead: S20 S20 ‘or steady flows, ae and FF 8V 29 and 2% 20 For unsteady flows, et Flow (Space basis) : In uniform flow, conditions of flow, e.g. depth, mn length of channel et (ii) Uniform and Non-uniform slope, velocity and cross-section remains constant over givel ie, 2Y 20, 2% cote. éL oL In non-uniform or varied flow, depth of flow changes along length of the open channel, ie. 29 40, 2% =octe. oL” ’ OL A uniform flow may be steady or unsteady depending on whether depth changes with time or not. (a) Steady uniform flow : In this flow, depth of flow does not change along length of channel during given interval if time, i.e. depth of flow remains same at all cross-section of the channel. (6) Unsteady uniform flow : In this flow, water surface fluctuates from time while remaining parallel to the channel bottom. Obviously it is practically impossible condition. (c) Steady non-uniform flow : In this flow, depth of flow at a section remains constant with time, but varies from section to section. (d) Unsteady non-uniform flow : In this flow, depth of flow must vary from section to section as well as with time also e.g. flood waves travelling in a natural stream. Types of Varied or Non-uniform Flow. (i) Rapidly Varied Flow (RVF) : In this flow, flow condition changes significantly in a relatively shor distance of the channel eg. hydraulic jump. (ii) Gradually Varied Flow (GVF) : In this flow, flow conditions changes gradually over a long distanc the channel e.g. flow behind a dam and at a channel transition. 2. Based on Forces Acting in Open Channel Flow (@ Effects of Viscous forces : In open channels, Reynold ich is Effect of Viscous fo Pp is, Reynolds number of flow which is a measure of effect Reem vwhere, v = kinematic viscosity R = hydraulic radius, which is ratio of flow area to wetted perimeter. Depending on value of Reynolds number, flow can be classified as follows : (a) Laminar flow, R < 500 (®) Transition, 500 < R < 2000 (©) Turbulent, R < 2000 a Gi) Effect of Gravity : It is represented by Froude number, which is the ratio of square root of 2 (i a at) inertial forces to gravity forces v Froude number, F = veD where, V = average velocity of flow D = hydraulic depth (= A/P) Depending on Froude number, flow can be classified as follows : (a) Critical flow, F = (®) Subcritical or tranquil flow, F <1 (©) Super critical or rapid flow, F > 1 Celerity (C) In shallow water, momentary change in the local depth of water causes small gra these small waves is called eelerity, C= gD Celerily can be more than the velocity of flow in subcritical flow; thus a gravity wave can travel upstream. In case of supercritical flow, the wave cannot travel upstream. wity waves. The velocity of Velocity Distribution Over Open Channel Cross-section Itis not uniform because of effect of friction at the bed, banks and free surface. ‘Theoretically velocity of flow should be maximum at the topmost point on the vertical center-line. However due to surface tension effect and resistance offered by the air, velocity is reduced at the water surface. Maximum velocity occurs at a distance of 0.05 to 0.25 times the flow depth y from the free surface. ‘Average velocity in an open channel cross-section is usually the local velocity occuring at 0.6 y from the free surface. Another practice is to measure local velocities at 0.2 y and 0.8y from the free surface velocity whereas that of average velocity is about 0.7 to 0.8 times. velocity Best Hydraulic Section or Most Economical Section distribution Best hydraulic section is the most efficient section for conveying a given discharge for given slope and the channel roughness factor, when area of cross-section is minimum. Thus for best hydraulic section, cost of eel lining and excavation is minimum and itis the most economical section, 1. Trapezoidal channel section. Wetted perimeter, P= b +2y ym? +1 Cross-sectional area, A= by + my? —<>» 4, @ pe (2-my)+ 2x Vim? y . ' Now for a given cross-sectional area and slope, the rate of flow through a° Se ee eae will be maximum, when hydraulic radius will be maximum, i.e. wetted perimel dP First codition of minimum P: T~ =0 ly Differentiating P with respect to y keeping m constant, we get betny yao 2 ie, Half the top width = sloping side ‘Second condition for minimum P : Tecan be obtained by differentiating P with respect to m keeping y constant [> +m -my]y 2 [ay +m? ~2my] Thus for maximum hydraulic efficiency, the channel should be designet (i) Hydraulic radius is equal to half the depth of flow 1 i 0° W) mee ‘These two conditions show that hydraulically economical trapezoi R .d as follows : dal section is half of a regular hexagon, Rectangular Channel, “ For a rectangular channel section to be most economical, put m = 0. Thus from condition 0 bt2my _y faqs => 48 Wor 2 > b=2y Hence width of channel should be equal to twice the flow depth. Triangular Section. For maximum R, ae dé 7 cos=0, 0245" = 4 Most economical triangular section have the angle between two sides as 90°. Each side making an angle of 45° with the vertical, i.e. m = 1 7 R=, 22 Semicircular Section. For circular pipes, area of flow corresponding to depth y is, 2 A= > (0—sin 6) where, r is radius of the conduit and 0 is given by, j Wetted perimeter, For maximum discharge based on Chezy’s formula : 1 7 O-sin®) _ (0~sing)*? : 2 Discharge, Q = AV. = A.C. RSp = F@-sind).C. Keeping r, C and S, constant, for maximum. discharge 2Q PT) a (AR™? d0 which for a channel reduces toThis on solving gives, 6=308 For this value of 6 and diameter d, we have y = 0.05 d, A= -sin) P=r0=5.876randR=0.573r (i) Based on Manning’s formula : For maximum discharge, Qe Lar?" x we have @ = 302° 20, y = 0.938 d, A = 3.061 72, P = 5.277 r and R= 0.587 Steady Uniform Flow (Resistance Equation) In steady uniform flow in a prismatic channel, channel slope S,, water energy line S, are equal. In this case, average velocity V, hydraulic radius Resistance laws of Manning, Chezy or Darcy - Weisbach. 1. Chezy’ formula v= CRS, where Cis called Chezy’s coefficient and is a measure of roughness of the boundary. It depends upon hydraulic radius of the section, channel slope and the boundary roughness. surface slope S,, and slope of the total Rand channel slope S, are related by Chezy's is related to Darey's friction factor fa, c= ee a3, womss 2. Kutter’s formula =] aaa 1+(23+ 200 where n is called Kutter’s coefficient. 8. Bazin’s formula. According to Bazin, value of C depends only on the hydraulic radius. c= 8 14 vR where m is called Bazin’s coefficient. Rue 4. Manning formulas o- = : 1 and Average velocity, V = +R®? 8,12 n where n is called Manning's rugosity coefficient. This formula is used most widely for open channel flow due to its simplicity. (a) Determination of : Value ofn can be obtained by studying factors affecting itor from following tabl S.No. | Name of surface ‘Average value ofa 1. _| Closed conduits flowing partly fully 0.013-0.114 2. | Lined channels 0.011-0.027 3, | Excavated earth channel 0.018-0.040 4,_| Natural streams 0.03-0.15 (®) Factors affecting n (@® Surface roughness (i) Channel irregularity (ii) Channel alignment (iv) Size and shape () Vegetation (i) Silting and scouring (vii) Obstruction (viii) Stage and discharge5. Stickler’s formula. When surface roughness can be represented by an equivalent roughness size k, (ag defined by Nikuradse), then n can be determined for hydrodynamically rough channel by KS 256 This is valid for R/K, varying from 5 to 700 Critical Flow in a Channel Specific Energy (E) Itis defined as the energy per unit weight of liquid, using channel bottom as the datum. 2 2g A? E = Depth + Velocity head = y + 2 For rectangular channel, E=y+ = gy where q = g is discharge per unit width. Specific energy is a function of y and Q both. Therefore three different curves can be plotted : () EVsy (i) QVsy Gii) EVs Qorq Specific Energy Curve For a given discharge Q, a curve between E and y is plotted which is called specific energy curve. Specific energy has following two components : @® Depth, y v (i) Velocity head, Y—. 2g ‘There are separately plotted as curves I and II. By adding ordinates of these two curves, a composite curve is obtained, which is required specific energy curve. Characteristics of Curve (1) Curve is asymptotic to energy line at one end to a datum energy line at other end (i.e. a line making an angle of 45° with X-axis) (2) Static energy line makes an angle of 45° with X-axis for small slope channel and for steeper slope the static energy line will be steeper. (3) For any value of specific energy E > E,j., there are two flow depths satisfying energy equation, y, andy, 85 shown in the figure are respectively called subcritical and supercritical depths and are jointly called alternate depths. (4) Specific energy is minimum at critical point C and the depth corresponding to this point y, is called critical depth. (5) Ifdepth of y >y,, flow is subcritical or tranquil flow. If depth y Q,, curve will lie above it and Q, > Q,, curve will lie below it, Condition for Critical Flow For any shape of cross-section of channel, let top width at water surface be T. Then QT ZA?2@ = Froude number, F, = 0 for critical flow; F, < 1 for subcritical flow; and F, > 1 for supercritical flow. 2 i D From equation (), we get > = 2 ‘Therefore in critical flow, velocity head is equal to half the hydraulic depth. From the curve it is observed that, there are two depths of flow y, and y, for given discharge Q and given specific energy E. These are alternate depths. The discharge is maximum at critical depth y,. Relation between Discharge and Depth of flow Discharge, Q = A /2g (=) Formaximum discharge, “2 <9 | 4 vd ar bent ga u ale where Z, is section factor for critical flow. Therefore conditions for critical flow are : @ Eis minimum for given discharge. i) Qis maximum for given E. (iii) F,=1 Relation between E and Q From the curve E Vs Q, if E @ _ 2x93 € nity 2 ¢ But © zys gn e yi = 208 A But since Application of Specific Energy 1, Measurement of flow in Open channel. Critical 1 1 hannels- The devices are called critical flow meters, '°°TY C4” be used for measurement of low nope ©. venturiflume, standing wave flume ete : 2. Flow through Rectangular channel transiti, i ema siet ‘ansition. Transition ina channel is done by following : (i) Changing width of section (iii) Combination of both,Hydraulic Jump fdepth of flow in a channel is supereritical i THETE chan flow changes fr rm and there is adquate depth available in the down stream ¢l shen ow changes from superriial to subcritical through a sudden rise in depth. ‘This phenomenon is called Conjugate Depth Relations (Characteristics of Hydraulic Jump) I represents @ discontinuty in liquid surface and causes air entrainmen tohigh turbulence resulting in loss of energy. Hydraulic jump is therefore frequently used as an. For the free body between sections (1) and (2) considering a unit width of channel and unit flow q nel, it and violent mixing due energy dissipater. 1 1 ZP ati and P= > Pw? From principle of impulse and momentum, we get 1 1 © Fyn Ory; and x8m 5 92 O14) 2 Wy, =- B+ [e282] we -14/i+8R , Go) yy Froude Number Relations 8 yh _ 1+ 2F De i) = iy - Wt2Fe @ Ry = Bon @ B= Galery Git) = oe 2a F, (iv) Relation between pre and post - jump Froude number : 2 (®) —14 1+ 8F? Length of Hydraulic Jump Itis approximately given by, It is generally found to vary between AB Yo Energy loss in Hydraulic Jump ve L,=8 to5 times (2-9) and 5.39, From energy equation, 1 + 3 #) o-oo ff (ft) PE 29" \2e yt 98 WAY | - Qr-y* Wy Also, sp-o.-v( remul w( 28 an Energy loss, AE = ¥i ~ 92 Therefore, hydraulic jump is used as an energy dissipater.Efficiency of Hydraulic Jump mp. Itis given by It's the ratio of specifi energy after the jump to that before the jump. Ey (SFP +1)? -4Fy +1 Eh 8F? (2+ FP Open Channel Surges If flow in a channel is increased by sudden increasing opening of @ sluice gate, a wave is formed which travel downstream and if flow is decreased by sudden partial closure of the gate, wave so formed travels upstream. Such waves are called surges. : A surge is called a positive surge, when it causes an increase in depth in the direction of ts travel and negative surge, when it causes decrease in depth. Consider a positive surge in a rectangular channel with horizontal bottom and eae ear vary om " to y2, By partially closing a gate, the surge formed travels upstream with a ean Tha me rine aoe e channel bottom producing the condition of unsteady flow. The velocity of wave Vi called Celeriy ofthe ve For analysis, moving surge can be brought to rest by superimposing an equal and opposite velocity Vn theo, 9 that velocity, V, change to (V, + V,) and V, to (V,+V,,). Conditions of flow now are steady. Following Relations are obtained : (1) Negative surge moving downstream (V,+V,)9,=(V,+V,)yand V, (2) Positive surge moving downstream W,-Viyy=(V,+Vy)¥, and + Fron +y) (8) Positive surge moving upstream (Wy +V)91= Vy +V_)¥y and sae (4) Negative surge moving downstream WV, ~- Ve) ¥2=(V- Vy) 94 and V,=V,+ 2 On + y2) 7 ‘Trick : For upstream surges, (y.~y, 1 ) and—V, and for downstream surges (y, ~y,) and + V, +)-V, yp OV YD= Vi . N-H Thus for negative surge moving upstream, y, +y)- Yaa 7 Yay O14 = Mi and for positive surge moving downstream, V,, = a = e jee Oty) +V, 2M FA Gradually Varied Flow (GVF) Asteady, non-uniform flow in which depth of flow v: gradually varied flow. +y)-M g ‘aries gradually along the length of the channel is called To analyse GVF, following assumptions are made : {2 Channel is prismatic channel section and alignment remains same (2) Bed slope is small, ie. S =S, ? () Energy loss is same for a uniform flow at a section (4) Energy correction factor a = 1Dynamic equation for gradually varied flow is given by dy, So-8p _ 8-8 aL = 2 Fee ry For wide rectangular channel, 4” . (9) _ a z ) j eZ 5 Exponent used in the above equation is 3 when Chezy’s equation is used while 10/3 is used, when Manning’s ‘equation is used. ‘ Classification of Channel Bed Slopes 1. Various Slopes Bed slope (S,) of the channel as compared to the critical slope (S,) are classified as : @ Mildslope: _ $, y, ii) Steep slope: 8. >S, and therefore y, 0, i.e. channel is rising in flow direction 3) = 0 and therefore y, 2, Various Zones Zone 1 : y>y, andy, Zone 2 : y lies between y, and y, Zone 3 : yIn>¥e + Back water M, _ | Subcritical Mn? I>Ye ~ Dropdown M, _ | Suberitical ale + Back water mM, _ | Supercritical Steep S)>S.>0[¥ >.> In + Back water 8, _ | Suberitical Yer IP In - drop down Sy ‘Supercritical penne + Back water g, _ | Supercritical Critical I> Ie“ In su Bock water ©, | Suberitical Ss, <8, +paralleltobed| ¢, | Uniform 9, + Back water c, _ | Supereritical ‘Adverse y>Ye Soran A, — | Suberitical 8) <0 rd + Back water A, — | Supercritical yee Horizontal >. ~ Dropdown H, | Subcritical y<%e + Back water H, _ | Supercritical