Lecture Note for Open Channel Hydraulics. V F =, gl

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1 Leture Note for Open Chnnel Hdrulis CPR HR CRIICL FLOW.. Criterion for ritil stte of flow he Froude number of ritil flow is unit F Critil Flow F, L Where men veloit of flow F< Sub Critil Flow L Chrteristi lenth hdruli rdius (D for open hnnel F > Super ritil Flow he speifi ner nd speifi fore re minimum he dishre is mximum t ritil flow for iven hnnel setion he veloit hed is equl to hlf the hdruli depth t ritil flow o If this riterion is to be used in n problem, the followin onditions must be stisfied Flow prllel or rdull vried Chnnel of smll slope ner oeffiient ssumed to be unit he veloit of flow in hnnel of smll slope with uniform veloit distribution is equl to the elerit of smll rvit wves (C h in shllow wter used b lol distribution. Flow t the ritil stte is unstble o minor hne in speifi ener t or lose to ritil stte will use mjor hne in depth Note: If the ritil ste of flow hve referred minl to prtiulr setion of hnnel, known s the ritil setion If the ritil ste of flow exists throuhout the entire lenth of the hnnel or over reh of the hnnel the flow in the hnnel is ritil flow..he setion Ftor for ritil flow he setion ftor for ritil flow omputtion ( is the produt of the wter re nd the squre root of the hdruli depth. D D D For ritil flow D, b substitutin D z herefore (. Where represent the dishre tht would mke the depth ritil nd know s the ritil dishre. When the ener oeffiient is not ssumed to be unit α (. q. (. nd. re ver useful tool for the omputtion nd nlsis of ritil flow in open hnnel, when the dishre is ive the equtions will ive the ritil setion ftor (. nd hene the ritil Depth. Setion ftor ( for different hnnel setion shpe is ive s Retnulr:.5 b rinulr: m.5 rpezoidl: [ * ( b m ] b m o simplif the omputtion of ritil flow, dimensionless urves showin the reltion between the depth nd the setion ftor hve been prepred for different tpe of hnnels setions. hese self-explntor urves will help to determine the depth for iven setion ftor nd vie vers..5 elete. i Deprtment of Civil nineerin Otober 0

2 Leture Note for Open Chnnel Hdrulis xmple.. Compute the ritil depth nd veloit of the trpezoidl hnnel its setion iven in the fiure nd rrin dishre of 45m /se. Given b4.0m m.5 45m /se.5 4.0m ( [ *( b m ] [ *(4.5] ( m 85 b m 4 *.5* 4 ( b m (4 (.5*.85 *.85.5m.59m / s First Hdruli xponent ( 45.5 In mn omputtions involvin wide rne of depths in hnnel, suh s in the GF omputtions, it is onvenient to express the vrition of with in n exponentil form. he (- reltionship C is found to be ver dvnteous. In this eqution C oeffiient nd n exponent lled first hdruli exponent. It is found tht enerll is slowl vrin funtion of the spet rtio for most of the hnnel shpes. n estimte of n lso be obtined b the reltion xmple.: Obtin the vlue of the first hdruli exponent ( for Retnulr hnnel b xponentil hnnel where the re K d d.5 b b C b, b k k d d k { k ( } [ k k ( ] ( ( d d k k k k. rnsitions he onepts of speifi ener nd ritil ener re useful in the nlsis of trnsition problems. rnsitions in retnulr hnnels re presented here. he priniples re equll pplible to hnnels of n shpe nd other tpes of trnsitions... Chnnel with Hump Subritil Flow Consider horizontl, fritionless retnulr hnnel of width rrin dishre t depth. Let the flow be subritil. t setion (Fi.. smooth hump of heihts is built on the floor. Sine there re no ener losses between setions nd, onstrution of hump uses the speifi ener t setion to derese b. hus the speifi eneries t setions nd re, Fiure:.. Chnnel trnsition with hump elete. i Deprtment of Civil nineerin Otober 0

3 Leture Note for Open Chnnel Hdrulis Sine the flow is subritil, the wter surfe will drop due to derese in the speifi ener. In Fi. (., the wter surfe whih ws t P t setion will ome down to point R t setion. he depth will be iven b, Fiure.. Speifi ener dirm for Fi. (. It is es to see from Fi. (. tht s the vlue of is inresed, the depth t setion, or, will derese. he minimum depth is rehed when the point R oinides with C, the ritil depth. t this point the hump heiht will be mximum, mx, ritil depth, nd minimum ener for the flowin dishre. he ondition t mx is iven b the reltion, mx he question m rise s to wht hppens when > mx. From Fi. (. it is seen tht the flow is not possible with the iven onditions (iven dishre. he upstrem depth hs to inrese to use nd inrese in the speifi ener t setion. If this modified depth is represented b, ( with > nd > t setion the flow will ontinue t the minimum speifi ener level, i.e. t the ritil ondition. t this ondition,, nd, mx Reolletin the vrious sequenes, when 0 < < mx the upstrem wter level remins sttionr t while the depth of flow t setion dereses with rehin minimum vlue of t mx. (Fi... with further inrese in the vlue of, (i.e., for > mx, will hne to while will ontinue to remin. he vrition of nd with in the subritil reime n be lerl seen in Fi.. Fiure.: rition of nd in subritil flow over hump Fiure.4. rition of nd in superritil flow over hump b Superritil Flow If is in the superritil flow reime Fi (. shows tht the depth of flow inreses due to the redution of speifi ener. In Fi (. point P` orresponds to nd point R` to depth t the setion. Up to the ritil depth, inreses to reh t mx For > mx, the depth over the hump will remin onstnt nd the mx upstrem depth will hne. It will derese to hve hiher speifi ener ` b inresin veloit. he vrition of the depths nd with in the superritil flow is shown in Fi. (.4. xmple.: retnulr hnnel hs width of.0 m nd rries dishre of 4.80m /se with depth of.60 m. t ertin ross-setion smll, smooth hump with flt top nd heiht 0.0 m is proposed to be built.. Clulte the likel hne in the wter surfe. Nelet the ener loss. b. If the heiht of the hump is 0.50 m, estimte the wter surfe elevtion on the hump nd t setion upstrem of the hump :.5 F r 9.8* , it show the upstrem flow is subritil nd hump will use drop in the wter surfe elevtion m Let the suffixes nd refer to the upstrem nd downstrem setions respetivel s shown in the fi. 4.8 q.40 q.40m /se/ m.50m /se m *9.8 t setion q / * m m m.48, elete. i Deprtment of Civil nineerin Otober 0

4 Leture Note for Open Chnnel Hdrulis he drop t wter surfe elevtion is b. If z * m m Solvin b tril nd error nd seletin the positive root ives >,.648m he minimum speifi ener required t setion is reter thn, (.6 >.5, the vilble speifi ener t tht setion. Hene, the depth t setion will be t the ritil depth nd.6m. he upstrem depth will inrese to depth suh tht the new speifi ener t the upstrem setion is xmple.4: Wter flow in wide hnnel pprohes 0 m hih hump t.50 m/se veloit nd depth of m. stimte he wter depth over the hump nd b he hump heiht tht will use the rest flow to be ritil.. Froude number t upstrem of the hump is.5 F r 0.48 < ( subritil 9.8* m m b. For ritil flow in wide hnnel q v.5*.5m /se/m min.5(q / /.5*(.5 /9.8 / 0.98m herefore the mximum heiht of the hump for fritionless flow over the hump is mx - min m q / here re three roots 0.859m, 0.45m nd -0.96m. Sine the third (netive solution is phsill impossible nd the seond (smller one is for superritil ondition, the re rejeted for this subritil hump. hus the depth ( 0.859m. he wter surfe level dropped b h m.. rnsition with Chne in Width. Subritil Flow in Width Constrition Consider fritionless horizontl hnnel of width rrin dishre t depth s in Fi. (.5. t setion hnnel width hs been onstrited to b smooth trnsition. Sine there re no losses involved nd sine the bed elevtions t setions nd re the sme, the speifi ener t setion is equl to the speifi ener t setion. Fiure.5. rnsition with width onstrition It is onvenient to nlze the flow in terms of the dishre intensit q /. t setion, q / nd t setion, q /. sine <, q > q. In the speifi ener dirm (Fi..6 drwn with the dishre intensit, point P on the urve q orresponds to depth nd speifi ener. Sine t setion, nd q q, point P will move vertill downwrd to point R on the urve q to reh the depth. hus, in subritil flow the depth is <. If is mde smller, then q will inrese nd will derese. he limit of the ontrted width min is rehed when orrespondin to, the dishre intensit q q mx, i.e. the mximum dishre intensit for iven speifi ener (ritil flow ondition will previl. elete. i Deprtment of Civil nineerin Otober 0

5 Fiure.6: Speifi ener dirm for Fi. (.5 Leture Note for Open Chnnel Hdrulis t the minimum width, m Critil depth C min m ( min m For retnulr hnnel t ritil flow Sine min If < min, the dishre intensit q will be lrer thn q mx, the mximum dishre intensit onsistent. he flow will not, therefore, be possible with the iven upstrem onditions. he upstrem depth will hve to inrese to. he new speifi ener will min m min X C / min min 7 8 m ( be formed whih will be suffiient to use ritil flow t setion. It m be noted tht the new ritil depth t setion for retnulr hnnel is, / q.5 / Sine < min, will be lrer thn m, > m. hus even thouh ritil flow previls for ll < min, the depth setion is not onstnt s in the hump se but inreses s `nd hene ` rises. he vrition of, nd with / is shown shemtill in Fi..7. b. Superritil Flow in Width Constrition If the upstrem depth is in the superritil flow reime, redution of the flow width nd hene n inrese in the dishre intensit use rise in depth. In Fi. (.6, point P` orresponds to nd point R` to. s the width is deresed, R` moves up till it beomes ritil t min. n further redution in uses the upstrem depth to derese to ` so tht rises to `. t setion, ritil depth ` orrespondin to the new speifi ener ` will previl. he vrition of, nd with / in superritil flow reime is indited in Fi. (5.. Fiure.7. rition of nd in subritil flow in width onstrition.. Chokin Fiure.8: rition of nd onstrition in superritil flow in width In the se of hnnel with hump, nd lso in the se of width onstrition, it is observed tht the upstrem wter surfe elevtion is not ffeted b the onditions t setion till ritil ste is first hieved. hus in the se of hump for ll mx, the upstrem wter depth is onstnt nd for ll > mx the upstrem depth is different from. Similrl, in the se of the width onstrition, for min, the upstrem depth is onstnt; while for ll < min, the upstrem depth underoes hne. his onset of ritil ondition t setion is prerequisite to hokin. hus ll ses with > mx or < min re known s hoked onditions. Obviousl, hoked onditions re undesirble nd need to be wthed in the desin of ulverts nd other surfe drine fetures involvin hnnel trnsitions. xmple.5: retnulr hnnel is.50 m wide onves dishre of 5.0 m /se t depth of.0 m. It is proposed to redue the width of the hnnel t hdruli struture. ssumin the trnsition to be horizontl nd the flow to be fritionless determine the wter surfe elevtions upstrem nd downstrem of the onstrition when the onstrited width is.50 m nd b.0 m...4. Generl rnsition trnsition in enerl form m hve hne of hnnel shpe, provision of hump or depression, ontrtion or expnsion of hnnel width, in n ombintion. In ddition, there m be vrious derees of loss of ener t vrious omponents. However, the bsi dependene of the depths of flow on the hnnel eometr nd speifi ener of flow will remin the sme. n omplited trnsition situtions n be nlzed b usin the priniples of speifi ener nd ritil depth. In subritil flow trnsitions the emphsis is essentill to provide smooth nd rdul hnes in the boundr to prevent flow seprtion nd onsequent ener losses. he trnsitions in superritil flow re different nd involve suppression of shok wves relted disturbnes. xmple.6: dishre of 6.0 m /se flows with depth of.0 m in retnulr hnnel 4.0 m wide. t downstrem setion the width is redued to.50 m nd the hnnel bed is rised b. nlze the wter surfe elevtions in the trnsitions when 0.0 m nd b 0.5 m. elete. i Deprtment of Civil nineerin Otober 0