Chapter 6. GraduallyVaried Flow in Open Channels


 Abner West
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1 Chapte 6 GaduallyVaied Flow in Open Channels 6.. Intoduction A stea nonunifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gaduallyvaied flow (GVF). The backwate poduced by a dam o wei acoss a ive and dawdown poduced at a sudden dop in a channel ae few typical examples of GVF. In a GVF, the velocity vaies along the channel and consequently the bed slope, wate suface slope, and enegy line slope will all diffe fom each othe. Regions of high cuvatue ae excluded in the analysis of this flow. The two basic assumptions involved in the analysis of GVF ae:. The pessue distibution at any section is assumed to be hydostatic. This follows fom the definition of the flow to have a gadually vaied wate suface. As gadual changes in the suface cuvatue give ise to negligible nomal acceleations, the depatue fom the hydostatic pessue distibution is negligible.. The esistance to flow at any depth is assumed to be given by the coesponding unifom flow equation, such as the Manning equation, with the condition that the slope tem to be used in the equation is the enegy line slope, S e and not the bed slope, S. Thus, if in a GVF the depth of flow at any section is y, the enegy line slope S e is given by, n V S e (6.) 4 3 R whee R hydaulic adius of the section at depth y. 6.. Basic Diffeential Equation fo the GaduallyVaied Flow Wate Suface Pofile Since, S Channel bed slope fo unifom flow depth y, Wate depth vaiation fo the canal each, d (V /g) Velocity head vaiation fo the each. Witing the enegy equation between the cosssections and (Fig. 6..),
2 V S + y + g + + ( y + ) V S e + d g S S S e e e d V g d V g V V + + d + Se g g S e (6.) d V + g y V E.L S e d V /g+d(v /g S d y+d d V+d S Figue 6.. V g Q ga ( y) da T ( y) d Q ga V A g T Substituting this to Equ. (6.), ( y) ga( y) ( y) ( y) Q A F e F da 4 Q T ga S (6.3) ( y) 3 ( y)
3 Equ. (6.3) is the geneal diffeential equation of the wate suface pofile fo the gadually vaied flows. / gives the vaiation of wate depth along the channel in the flow diection Classification of Flow Suface Pofiles Fo a given channel with a known Q Dischage, n Manning coefficient, and S Channel bed slope, y c citical wate depth and y Unifom flow depth can be computed. Thee ae thee possible elations between y and y c as ) y > y c, ) y < y c, 3) y y c. Fo hoizontal (S ), and advese slope ( S < ) channels, Q A R n 3 S Hoizontal channel, S Q, Advese channel, S <, Q cannot be computed, Fo hoizontal and advese slope channels, unifom flow depth y does not exist. Based on the infomation given above, the channels ae classified into five categoies as indicated in Table (6.). Table 6.. Classification of channels Numbe Channel categoy Symbol Chaacteistic condition Remak Mild slope M y > y c Subcitical flow at nomal depth Steep slope S y c > y Supecitical flow at nomal depth 3 Citical slope C y c y Citical flow at nomal depth 4 Hoizontal H S Cannot sustain unifom flow bed 5 Advese slope A S < Cannot sustain unifom flow Fo each of the five categoies of channels, lines epesenting the citical depth (y c ) and nomal depth (y ) (if it exists) can be dawn in the longitudinal section. These would divide the whole flow space into thee egions as: Region : Space above the topmost line, Region : Space between top line and the next lowe line, Region 3: Space between the second line and the bed. 3
4 Figue (6.) shows these egions in the vaious categoies of channels. Figue 6.. Regions of flow pofiles Depending upon the channel categoy and egion of flow, the wate suface pofiles will have chaacteistics shapes. Whethe a given GVF pofile will have an inceasing o deceasing wate depth in the diection of flow will depend upon the tem / in Equ. (6.3) being positive o negative. S (6.3) e F 4
5 Fo a given Q, n, and S at a channel, y Unifom flow depth, y c Citical flow depth, y Nonunifom flow depth. The depth y is measued vetically fom the channel bottom, the slope of the wate suface / is elative to this channel bottom. Fig. (6.3) is basic to the pediction of suface pofiles fom analysis of Equ. (6.3). Figue 6.3 To assist in the detemination of flow pofiles in vaious egions, the behavio of / at cetain key depths is noted by stuing Equ. (6.3) as follows: y > y y < y y y c c c F F F < > And also, y > y y < y y y S S S e e e < S > S S. As y y, V V, S e S S lim y y F cons The wate suface appoaches the nomal depth asymptotically. 5
6 . As y yc, F, F, S e S lim y y c F e The wate suface meets the citical depth line vetically. 3. As y, V F S e lim y S F e S S The wate suface meets a vey lage depth as a hoizontal asymptote. Based on this infomation, the vaious possible gadually vaied flow pofiles ae gouped into twelve types (Table 6.). Table 6.. Gadually Vaied Flow pofiles Channel Region Condition Type Mild slope 3 Steep slope 3 Citical slope 3 Hoizontal bed 3 Advese slope 3 y > y > y c y > y > y c y > y c > y y> y c > y y c > y > y y c > y > y y > y y y < y y c y > y c y < y c y > y c y < y c M M M 3 S S S 3 C C 3 H H 3 A A 3 6
7 6.4. Wate Suface Pofiles M Cuves Figue 6.4 Geneal shapes of M cuves ae given in Fig. (6.4). Asymptotic behavios of each cuve will be examined mathematically. a) M Cuve Wate suface will be in Region fo a mild slope channel and the flow is obviously subcitical. S e < S Mild slope channel y > y c Subcitical flow S e F F < Subcitical flow ( F ) > y > y S e < S + + > (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; Wate depth can be between ( > y > y ) fo Region. The asymptotic behavios of the wate suface fo the limit values (, y ) ae; 7
8 a) y, V, F, ( F ) y, V, S e lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. b) y y, V V, S e S S lim y y F The wate suface appoaches the nomal depth asymptotically. The most common of all GVF pofiles is the M type, which is a subcitical flow condition. Obstuctions to flow, such as weis, dams, contol stuctues and natual featues, such as bends, poduce M backwate cuves (Fig. 6.5). These extend to seveal kilometes upsteam befoe meging with the nomal depth. b) M Cuve Figue 6.5. M Pofile Wate suface will be in Region fo a mild slope channel and the flow is obviously subcitical. (Fig. 6.4). y > y > y c y < y V > V S e > S (S S e ) < y > y c F < ( F ) > S F e + (Wate depth decease in the flow diection) 8
9 Asymptotic behavio of the wate suface is; y > y > y c y y, S e S, (S S e ) S lim y y F The wate suface appoaches the nomal depth asymptotically. y y c, F, ( F ) S lim y yc e The wate suface meets the citical depth line vetically. The M pofiles occu at a sudden dop of the channel, at constiction type of tansitions and at the canal outlet into pools (Fig. 6.6). c) M 3 Cuve Figue 6.6 Wate suface will be in Region 3 fo a mild slope channel and the flow is obviously subcitical. (Fig. 6.4). y > y c > y < y V > V S e > S (S S e ) < y < y c Supecitical flow F > ( F ) < S F e + (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; 9
10 y y c F ( F ) S e lim y yc The wate suface meets the citical depth line vetically. y, S e, (S S e ) y, F V gy lim (Unknown) y y The angle of the wate suface with the channel bed may be taken as S. y c 3 Figue 6.7. M 3 Pofile Whee a supecitical steam entes a mild slope channel, M 3 type of pofile occus. The flow leading fom a spillway o a sluice gate to a mild slope foms a typical example (Fig. 6.7). The beginning of the M 3 cuve is usually followed by a small stetch of apidly vaied flow and the downsteam is geneally teminated by a hydaulic jump. Compaed to M and M pofiles, M 3 cuves ae of elatively shot length.
11 Example 6. : A ectangula channel with a bottom width of 4. m and a bottom slope of.8 has a dischage of.5 m 3 /sec. In a gadually vaied flow in this channel, the depth at a cetain location is found to be.3 m. assuming n.6, detemine the type of GVF pofile. Solution: a) To find the nomal depth y, A By 4y P B + y R A 4y P 4 + y 4 + y b) Citical depth y c, c) Type of pofile, A Q VA A n P.5 q S 5 3 ( 4y ) ( 4 + y ) 3 y 5 3 ( 4 + y ) 3.8 By tial and eo, y.43 m. Q B.5.375m sec q.375 yc. 4m g 9.8 y.43 m > y c.4 m (Mild slope channel, M pofile) Wate suface pofile is of the M type. y.3 m y > y > y c (Region ).5
12 6.4.. S Cuves Figue 6.8 Geneal shapes of S cuves ae given in Fig. (6.8). Asymptotic behavios of each cuve will be examined mathematically. a) S Cuve Wate suface will be in Region fo a steep slope channel and the flow is obviously supecitical. y > y c > y y > y V < V S e < S (S S e ) > y > y c F < ( F ) > S F e (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; > y > y c y, V, F, ( F ) y, V, S e lim y S F e S S
13 The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F, ( F ) S e S lim y y c F e The wate suface meets the citical depth line vetically. Figue 6.9. S Pofile The S pofile is poduced when the flow fom a steep channel is teminated by a deep pool ceated by an obstuction, such as a wei o dam (Fig. 6.9). At the beginning of the cuve, the flow changes fom the nomal depth (supecitical flow) to subcitical flow though a hydaulic jump. The pofiles extend downsteam with a positive wate slope to each a hoizontal asymptote at the pool elevation. b) S Cuve Wate suface will be in Region fo a steep slope channel and the flow is supecitical. y < y < y c y > y S e < S (S S e ) > y < y c Supecitical flow F > ( F ) < S F e + (Wate depth will decease in the flow diection) 3
14 Asymptotic behavio of the wate suface is; y < y < y c y y c, F, ( F ) S e S lim y y c F e The wate suface meets the citical depth line vetically. y y, V V, S e S, (S S e ) lim y y F The wate suface appoaches the nomal depth asymptotically. Figue 6.. S Pofile Pofiles of the S type occu at the entance egion of a steep channel leading fom a esevoi and at a bake of gade fom mild slopes to steep slope (Fig. 6.). Geneally S pofiles ae shot of length. c) S 3 Cuve Wate suface will be in Region 3 fo a steep slope channel and the flow is supecitical. < y < y y < y S e > S (S S e ) < y < y c Supecitical flow F > ( F ) < S e + F (Wate depth will incease in the flow diection) 4
15 Asymptotic behavio of the wate suface is; < y < y y, S e, (S S e )  y, F V gy lim (Unknown) y y The angle of the wate suface with the channel bed may be taken as S. y c y y, S e S, (S S e ) 3 lim y y F The wate suface appoaches the nomal depth y asymptotically. Figue 6.. S 3 Pofile Figue 6.. S 3 Pofile Fee flow fom a sluice gate with a steep slope on its downsteam is of the S 3 type (Fig. 6.). The S 3 cuve also esults when a flow exists fom a steepe slope to a less steep slope (Fig. 6.). 5
16 C Cuves Figue 6.3 Geneal shapes of C cuves ae given in Fig. (6.3). Asymptotic behavios of each cuve will be examined mathematically. Since the flow is at citical stage, y y c, thee is no Region. a) C Cuve Wate suface will be in Region fo a citical slope channel. y y c < y < y > y c S e < S (S S e ) > y > y c Subcitical flow F < ( F ) > S F e (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, ( S S e ) S y, V F ( F ) lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F ( F ) y y c y, S e S S c (S S e ) lim y y y c (Unknown) C and C 3 pofiles ae vey ae and highly unstable 6
17 H Cuves Figue 6.4 Geneal shapes of H cuves ae given in Fig. (6.4). Fo hoizontal slope channels, unifom flow depth y does not exist. Citical wate depth can be computed fo a given dischage Q and theefoe citical wate depth line can be dawn. Since thee is no unifom wate depth y, Region does not exist. a) H Cuve Wate suface will be in Region fo a hoizontal slope channel. > y > y c y > y c S e < S (S S e ) < y > y c subcitical flow F < ( F ) > + (Wate depth will decease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, ( S S e ) S y, V F ( F ) lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F ( F ) y y c, S e S c S, (S S e ) S e S e e lim y y c F The wate suface meets the citical depth line vetically. 7
18 b) H 3 Cuve > y > y c y < y c S e > S (S S e ) S e y y c, supecitical flow, F > ( F ) < S F e + (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, (S S e )  y, F V gy lim (Unknown) y y y c F ( F ) y y c, S e S c S, (S S e ) S e S lim y y c F e e The wate suface meets the citical depth line vetically. Figue 6.5 A hoizontal channel can be consideed as the lowe limit eached by a mild slope as its bed slope becomes flatte. The H and H 3 pofiles ae simila to M and M 3 pofiles espectively (Fig. 6.5). Howeve, the H cuve has a hoizontal asymptote. 8
19 A Cuves Figue 6.6 Geneal shapes of A cuves ae given in Fig. (6.6). Fo advese slope channels, unifom flow depth y does not exist. Citical wate depth can be computed fo a given dischage Q and theefoe citical wate depth line can be dawn. Since thee is no unifom wate depth y, Region does not exist as well as in A cuves. A and A 3 cuves ae simila to H and H 3 cuves espectively. Figue 6.7 Advese slopes ae athe ae (Fig. 6.7). These pofiles ae of vey shot length Contol Sections A contol section is defined as a section in which a fixed elationship exists between the dischage and depth of flow. Weis, spillways, sluice gates ae some typical examples of stuctues which give ise to contol sections. The citical depth is also a contol point. Howeve, it is effective in a flow pofile which changes fom subcitical to supecitical flow. In the evese case of tansition fom supecitical flow to subcitical flow, a hydaulic jump is usually fomed bypassing the citical depth as a contol point. Any GVF pofile will have at least one contol section. 9
20 (d) Figue 6.8 In the synthesis of GVF pofiles occuing in a seially connected channel elements, the contol sections povide a key to the identification of pope pofile shapes. A few typical contol sections ae shown in Fig. (6.8, ae). It may be noted that subcitical flows have contols in the downsteam end, while supecitical flows ae govened by contol sections existing at the upsteam end of the channel section. In Figs. (6.8 a and b) fo the M pofile, the contol section (indicated by a dak dot in the figues) is just at the upsteam of the spillway and sluice gate espectively. In Figs. (6.8 b and d) fo M 3 and S 3 pofiles espectively, the contol point is at the vena contacta of the sluice gate flow. In subcitical flow esevoi offtakes (Fig. 6.8 c), even though the dischage is govened
21 by the esevoi elevation, the channel enty section is not stictly a contol section. The wate suface elevation in the channel, will be lowe than the esevoi elevation by a headloss amount equivalent to ( + ζ )V /g whee ζ is the entance loss coefficient. The tue contol section will be at a downsteam location in the channel. Fo the situation shown in Fig. (6.8 c) the citical depth at the fee oveflow at the channel end acts as the downsteam contol. Fo a sudden dop (fee oveflow) due to cuvatue of the steamlines the citical depth usually occus at distance of about 4y c upsteam of the dop. This distance, being small compaed to GVF lengths, is neglected and it is usual to pefom calculations by assuming y c to occu at the dop. (e) Fo a supecitical canal intake (Fig. 6.8 e), the esevoi wate suface falls to the citical depth at the head of the canal and then onwads the wate suface follows the S cuve. The citical depth occuing at the upsteam end of the channel is the contol fo this flow. Figue 6.9
22 6.5.. Feeding a Pool fo Subcitical Flow A mild slope channel dischaging into a pool of vaiable suface elevation is indicated in Fig. (6.9). Fou cases ae shown. ) The pool elevation is highe than the elevation of the nomal (unifom) wate depth line at B. This gives ise to a downing of the channel end. A pofile of the M type is poduced with the pool level at B as contol. ) The pool elevation is lowe than the elevation of the nomal depth line but highe than the citical depth line at B. The pool elevation acts as a contol fo the M cuve. 3) The pool elevation has dopped down to that of the citical depth line at B and the contol is still at the pool elevation. 4) The pool elevation has dopped lowe than the elevation of the citical depth line at B. The wate suface cannot pass though a citical depth at any location othe than B and hence a sudden dop in the wate suface at B is obseved. The citical depth at B is the contol of this flow Feeding a Pool fo Supecitical Flow When the slope of the channel is steep, that is geate than the citical slope, the flow in the channel becomes supecitical. In pactical applications, steep channels ae usually shot, such as aft and log chutes that ae used as spillways. Figue 6. As the contol section in a channel of supecitical flow is at the upsteam end, the delivey of the channel is fully govened by the citical dischage at section.
23 ) When the tailwate level B is less than the outlet depth (nomal depth) at section, the flow in the canal is unaffected by the tailwate. The flow pofile passes though the citical wate depth nea C and appoaches the nomal depth by means of a smooth dawdown cuve of the S type. ) When the tailwate level B is geate than the outlet depth, the tailwate will aise the wate level in the upsteam potion of the canal to fom an S pofile between j and b`, poducing a hydaulic jump at the end j of the pofile. Howeve, the flow upsteam fom the jump will not be affected by the tailwate. Such a hydaulic jump is objectionable and dangeous, paticulaly when the canal is aft chute o some othe stuctue intended to tanspot a floating aft fom the upsteam esevoi to a downsteam pool. A neutalizing each may be suggested fo the solution of this poblem. (Chow, 959). (Fig. 6.) Figue 6.. Elimination of hydaulic jump by neutalizing each. In this each the bottom slope of the channel is made equal to the citical slope. Accoding to a coesponding case of C pofile in Fig. (6.), the tailwate levels will be appoximately hoizontal lines which intesect the suface of flow in the canal without causing any distubance. At the point of intesection, theoetically, thee is a jump of zeo height. 3) As the tailwate ises futhe, the jump will move upsteam, maintaining its height and fom in the unifom flow zone nb, until it eaches point n. The height of the jump becomes zeo when it eaches the citical depth at c. In the meantime, the flow pofile eaches its theoetical limit cb`` of the S pofile. Beyond this limit the incoming flow will be diectly affected by the tailwate. In pactical applications, the hoizontal line cb``` may be taken as the pactical limit of the tailwate stage. 3
24 6.6. Analysis of Flow Pofile A channel caying a gadually vaied flow can in geneal contain diffeent pismoidal channel cosssections of vaying hydaulic popeties. Thee can be a numbe of contol sections at vaious locations. To detemine the esulting wate suface pofile in a given case, one should be in a position to analyze the effects of vaious channel sections and contols connected in seies. Simple cases ae illustated to povide infomation and expeience to handle moe complex cases Beak in Gades Simple situations of a seies combination of two channel sections with diffeing bed slopes ae consideed. In Fig. (6..a), a beak in gade fom a mild channel to a milde channel is shown. It is necessay to fist daw the citicaldepth line (CDL) and the nomaldepth line (NDL) fo both slopes. Since y c does not depend upon the slope fo a taken Q dischage, the CDL is at a constant height above the channel bed in both slopes. The nomal depth y fo the mild slope is lowe than that of the of the milde slope (y ). In this case, y acts as a contol, simila to the wei o spillway case and an M backwate cuve is poduced in the mild slope channel. Vaious combinations of slopes and the esulting GVF pofiles ae pesented in Fig. ( 6., ah). It may be noted that in some situations thee can be moe than one possible pofiles. Fo example, in Fig. (6. e), a jump and S pofile o an M 3 pofile and a jump possible. The paticula cuve in this case depends on the channel and its flow popeties. In the examples indicated in Fig. (6.), the section whee the gade changes acts a contol section and this can be classified as a natual contol. It should be noted that even though the bed slope is consideed as the only vaiable in the above examples, the same type of analysis would hold good fo channel sections in which thee is a maked change in the oughness chaacteistics with o without change in the bed slope. A long each of unlined canal followed by a line each seves as a typical example fo the same. A change in the channel geomety (the bed width o side slope) beyond a section while etaining the pismoidal natue in each each also leads to a natual contol section. 4
25 5
26 6
27 Figue Seial Combination of Channel Sections To analyze a geneal poblem of many channel sections and contols, the following steps ae to be applied.. Daw the longitudinal section of the system.. Calculate the citical depth and nomal depths of vaious eaches and daw the CDL and NDL in all eaches. 3. Mak all the contols, both the imposed as well as natual contols. 4. Identify the possible pofiles. Example 6. : Identify and sketch the GVF pofiles in thee mild slopes which could be descibed as mild, steepe mild and milde. The thee slopes ae in seies. The last slope has a sluice gate in the middle of the each and the downsteam end of the channel has a fee ovefall. Solution: The longitudinal pofile of the channel, citical depth line and nomal depth lines fo the vaious eaches ae shown in the Fig. (6.3). The fee ovefall at E is obviously a contol. The vena contacta downsteam of the sluice gate at D is anothe contol. Since fo subcitical flow the contol is at the downsteam end of the channel, the highe of the two nomal depths at C acts as a contol fo the each CB, giving ise to an M pofile ove CB. At B, the nomal depth of the channel CB acts as a contol giving ise to an M pofile ove AB. The contols ae maked distinctly in Fig. (6.3). With these contols the possible flow pofiles ae: an M pofile on channel AB, M pofile on channel BC, M 3 pofile and M pofile though a jump on the stetch DE. 7
28 Figue 6.3 Example 6.3: A tapezoidal channel has thee eaches A, B, and C connected in seies with the following physical chaacteistics. Reach 3 Bed width (B) 4. m 4. m 4. m Side slope (m)... Bed slope S n.5..5 Fo a dischage Q.5 m 3 /sec though this channel, sketch the esulting wate suface pofiles. The length of the eaches can be assumed to be sufficiently long fo the GVF pofiles to develop fully. Solution: The nomal depths and citical wate depths in the vaious eaches ae calculated as: T y m B4m A ( B + my) P B + y T B + my y + m 8
29 A (4 + y) y P 4 + y T 4 + y y Q AV.5 A A n P [( 4 + y ) y ] n 3 S S.5 ( y ) 3 Unifom flow depths fo the given data fo evey each ae calculated by tial and eo method; Reach A: Reach B: Reach C: S A.4, n A.5 y A.6 m S B.9, n B. y B.8 m S C.4, n C.5 y C.7 m Since the channel is pismoidal (the geomety does not change in the eaches), thee will be only one citical wate depth. Q T ga c 3 c ( 4 + yc ) [( 4 + y ) y ] c c 3 y c.3 m Reach A is a mild slope channel as y A.6 m > y c.3 m and the flow is subcitical. Reach B and C ae steep slope channels as y B, y C < y c and the flow is supecitical on both eaches. Reach B is steepe than each C. The vaious eaches ae schematically shown in Fig. (6.4). The CDL is dawn at a height of.3 m above the bed level and NDLs ae dawn at the calculated y values. The contols ae maked in the figue. Reach A will have an M dawdown cuve, each B an S dawdown cuve and each C a ising cuve as shown in the figue. It may be noted that the esulting pofile as above is a seial combination of Fig. (6. d and f). 9
30 Figue 6.3 3
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