NOTES ON OPEN CHANNEL FLOW

Transcription

1 NOTES ON OPEN CANNEL FLOW Prof. Marco Plott Facoltà d Ingegnera, Unverstà degl Stud d Bresca Profl d moto permanente n un canale e n una sere d due canal - Boudne, 86

2 OPEN CANNEL FLOW: quanttatve profles n complex cannels Intally, n order to understand wat s te actual applcablty scope of unform moton we asked ow muc far away s far? Wen an M profle s consdered a frst guess can be provded by te followng table, tat s vald for nfntely wde rectangular cannel, accordng to Bresse s soluton

3 OPEN CANNEL FLOW: quanttatve profles n complex cannels Drect step metod dstance calculated from dept S b S f dx de y at te poston x along te cannel s known as a boundary condton. From te qualtatve dscusson of te profle, one knows wat s te asymptotc dept e.g., f M, t wll tend to 0. Accordngly one selects n an adaptve way a dept value y < y < 0 for te secton at te unknown staton x, computng te correspondng values E and S f tat depend only on y n prsmatc cannels. Te unknown staton x s obtaned by dscretzaton of te dynamc equaton. If Fr < ten x grows from downstream to upstream [ ] f f b S S S E E x x [ ] f f b x x S S x x S E E Let us frst consder a metod wc s very convenent but s vald only n prsmatc cannel, were, ndependently from x, one knows Ax and S b x If Fr > ten x grows upstream to downstream and te equaton s [ ] f f b S S S E E x x

4 OPEN CANNEL FLOW: quanttatve profles n complex cannels Let us now consder a general metod wc can be as convenent as te drect step f solved explctly or just a bt more complex f solved mplctly. Its scope s not lmted to prsmatc cannels. We know te boundary condton y at te poston x along te cannel. By a frst order approxmaton of te energy balance equaton, we obtan : d dx S f [ S f S f ] x x Te unknown value y at te known poston x s suc tat F0 Standard step metod dept calculated from dstance F y 0, were F y [ S f S f ] x x Ts equaton s non lnear and must be solved by, e.g., a Newton Rapson metod. y * y * * F y F y As a frst guess for te teraton, a frst estmate y* of y can be obtaned as y Q z S f x ga y α x tat can also be used as an explct approxmaton of te energy balance equaton / IMPLICIT EXPLICIT

5 OPEN CANNEL FLOW: quanttatve profles n complex cannels If te cannel s a prsmatc cannels see classwork 3 were we know te boundary condton y at te poston x along te cannel, te standard step metod can be gven as: dy Sb S f x x y y Sb S f S f dx Fr [ Fr ] Fr Tat s an mplct equaton tat provdes te unknown value y at te known poston x. Alternatvely, te metod can be wrtten n a fully explct way as: y b Fr [ S S ] x x y f [ ] Wc works equally well provded te space ncrement x - x are adequately small. Implct Standard step metod Explct Standard step metod As sown before for te drect step metod, f Fr > ten te ntegraton moves from te boundary condton, gven at a pont x 0 upstream, to downstream, so tat x grows n te same drecton of te man flow. If Fr < ten te boundary condton x 0 s gven downstream and te ntegraton moves upstream. In ts case f x grows from te poston of te boundary condton to upstream, y [ S S ] x x y Sb f f [ Fr ] Fr y b Fr [ S S ] x x y f

6 OPEN CANNEL FLOW: nlet of a mld/steep cannel Let us consder te outlet of a lake n a cannel wose geometry s known. ow can we fnd out te dscarge Q tat flows out from te lake? To solve te problem, as a frst step we ave to make an educated guess on S b : s t mld or steep? To provde an answer, let us refne our sensblty on ts pont. Let us consder an nfntely wde cannel wt crtcal slope. It must be Q k 3 c S c /6 0 Q gb B 0 0 o S b From ts system, we may obtan S b eter 0/9 g as a functon of qq/b, k s : Sb _ crtcal / 9 k q g or as a functon of, k s : Sb _ crtcal / 3 k S S Conclusons:. Te rouger te cannel, te ger te crtcal slope S b_crtcal : e.g., a mountan creek could ave a mld slope f t s very roug.. Te ger te Q or, te smaller te crtcal slope S b_crtcal : e.g., a cannel could go from mld to steep f Q ncreases

7 OPEN CANNEL FLOW: nlet of a mld/steep cannel

8 OPEN CANNEL FLOW: nlet of a sort and wde cannel Real case can be a bt more complcate but can be easly solved f one understands te rules of ts teratve game. Let us consder ts case were te geometry of te cannel, te water dept n te lake and te elevaton of te gate are known As a frst step, we ave to make an educated guess on S b : s t mld or steep?

9 OPEN CANNEL FLOW: nlet of a sort and wde cannel - mld slope case. Accordng to our guess, S b s mld. Solve Energy balance at to fnd bot 0 and Q under te assumpton true or false? of nfntely long cannel. 3. Make an energy balance between and 3 to fnd Compute te M and verfy f te specfc energy on te sll wt possble entry loss comples wt te water stage n te lake. If not satsfed, decrease Q and go back to pont 3 5. Wen Q as been found, compute te M3 profle from staton and te M from staton 4. Compute also te specfc force of tese profles and locate staton Is te sluce gate submerged.e., te M3 profle s not present? If no, end of te game; oterwse go back to 3 and go on playng untl convergence of Q.

10 OPEN CANNEL FLOW: nlet of a sort and wde cannel - steep slope case If our guess s wrong and te cannel s steep. Compute Q under te assumpton true or false? of nfntely long cannel.. Make an energy balance between and 3 to fnd Compute te S and S profles. Compute also te specfc force of tese profles and locate te staton 4 of te ydraulc jump. If te S profles exst, tan go on to te followng step, oterwse te outlet of te lake s submerged by te S profle, te cannel s not nfntely long and te dscarge must be decreased, gong back to pont 4. Compute te S3 profle from downstream to staton 5.

11 OPEN CANNEL FLOW: nlet of a an nfntely long wde cannel Let us consder a wde cannel Y/B << wc s orgnated from a reservor were water s motonless Let us suppose tat te cannel s nfntely long so tat we can dsregard te nfluence of boundary condtons. In general term we can wrte an energy balance between te reservor and te flow at te nlet of te cannel, also consderng te presence of a local dsspaton tat s proportonal to te knetc energy. Q s unknown If te slope S b of te cannel s mld, ten we sould ave normal dept up to te cannel nlet, so tat we ave to solve E Q Q ξ o gao gao Q χa R S o o b Wc s solved for te normal dept and Q Accordngly, by ncreasng S b Q ncreases as well untl te crtcal condton s obtaned at te nlet If te cannel s steep, ten te cannel nlet s a Q Q transton troug te crtcal dept between mld E ξ c gac ga and steep cannel. Te system s solved for te crtcal dept and Q, tat s ndpendent from S b Q da 3 ga d c c

12 OPEN CANNEL FLOW: nlet of a an nfntely long wde cannel In bot cases, provded tat te water stage E n te lake s known, one as to solve a non lnear equaton to fnd and ten Q E E o c ξ ξ k s R 4/3 o g gac da g d S b In rectangular cross secton te second equaton greatly smplfes, n te form 3 E ξ c

13 OPEN CANNEL FLOW: nlet of a sort and wde cannel - mld slope case If te cannel as a lengt l tat s not nfntely long, ten we may ave a backwater rgurgto or drawdown camata effect caused by te boundary condton located at te cannel outlet. In ts case te dscarge must be computed troug an teratve procedure. Let us suppose tat te cannel outlet s nto anoter reservor, tat constrans te level of water at te cannel end, e. If te cannel s mld, te dscarge Q computed from te equaton seen before s only an ntal guess Usng Q, compute te crtcal dept c If e < c compute a M profle startng from c, oterwse eter a M profle c < e < o or a M one e > o 3 Compare te computed water dept at te cannel sll nlet wt te normal dept. If t s ger, tan Q must be decreased; oterwse t must be ncreased. If e E S b l, ten Q 0; If e > E S b l, te flow s reversed from downstream to upstream. Te mld slope cannel turns nto an adverse slope one

14 OPEN CANNEL FLOW: nlet of a sort and wde cannel - steep slope case If te cannel s steep, tere mgt be a backwater effect caused by te boundary condton at te cannel outlet, tat could cause an ydraulc jump wtn te cannel stretc. If te specfc force of te S profle s larger tan te specfc force of te accelerated supercrtcal S profle, te ydraulc jump moves backward locatng closer and closer to te cannel nlet were eventually tere mgt be a submerged ydraulc jump. Ts appens wen e s close to te upstream energy level E If e E S b l, ten Q 0 as before, l s te cannel lengt. If e > E S b l, te flow s reversed from downstream to upstream. Te orgnal cannel turns nto an adverse slope one Now you know wat Infntely Long means

15 OPEN CANNEL FLOW: passage over a sll ump, bump: sogla Wen te flow passes over an ump, several stuatons may appen, dependng on te Froude number and on Energy content. Locally tere s a sudden curvature of te flow, te cannel s not prsmatc and te teory on water surface profles s of no use. owever an energy balance can be accomplsed to study ts transton. Let us frst suppose tat. no ead loss s present 0 E a E0; E E0. te sll egt a s small wt respect to te energy upstream. 3. Te cannel s nfntely long downstream and upstream, so tat te dept of te flow approacng te sll and downstream of t s te normal dept If te slope s mld, water dept on te sll lowers more tan te sll egt. If te slope s steep, te effect of rse of te sll bed prevals

16 OPEN CANNEL FLOW: passage over a sll ump, bump: sogla Sometmes te egt of te sll s suc tat te specfc energy of te normal flow of te approacng current s not suffcent to pass over t. In suc a case te flow upstream must gan energy and we ave to dstngus between mld and steep cannel

17 AN IMPORTANT EXAMPLE: Broad crested, round nose, orzontal crest wer Upstream corner well rounded to prevent separaton Geometrcal requrements as n fgure above and n te specfc publcatons

18 WEIRS: Broad crested orzontal crest wer

19 OPEN CANNEL FLOW: passage over a sll ump, bump: sogla But an ead loss s almost nevtable so tat 0: normal flow : on te sll; : downstream; 0m: upstream Makng an energy balance startng downstream, one sees tat n a mld cannel te level upstream s ger M. Dependng on Te lengt of te cannel, ts could affect Q And n a steep cannel, Startng upstream, one sees tat te rse on te ump s stronger and Te level downstream Is greater tan te Normal dept S 0; ; 0m ; 0 m 0; 0m ; ; E 0m 0m E 0 0 E 0 0 E

20 OPEN CANNEL FLOW: passage troug a contracton Te same stuaton occurrng wen a flow passes over an ump can be observed n te passage troug a contracton. Usually a contracton can be caused by te pers or abutments of a brdge If no localzed losses are Present, ten te specfc energy s constant

21 OPEN CANNEL FLOW: passage troug a contracton Sometmes te Energy upstream sn t enoug

22 OPEN CANNEL FLOW: passage troug a contracton Altoug one can suppose tat no ead loss s present,ts s not generally true. Accordngly, te flow must gan energy to compensate for te localzed ead loss. Ts appens upstream f Fr < M and downstream f Fr > S f f 0m Fr < Fr > V 0m V 0 M 0 S Te process s smlar to te one consdered for te passage over a bump

23 OPEN CANNEL FLOW: Transtons As a frst approxmaton one can dsregard te energy losses mpled n a transton. In suc a case te followng stuatons arse for a sudden rse/fall of te bed or contracton/expanson

24 OPEN CANNEL FLOW: Transtons n subcrtcal flow wt ead loss Let us consder an abrupt drop n te cannel bed. If we ave an ead loss we cannot drectly use an energy balance and we ave to revert to a momentum balance, under te same assumptons usually used to derve Borda s ead loss n a ppe. γ Q β ga γ Q gb E a Π b γ γ Q β ga a E ; Π γ Q gb b γ < a If we now consder an energy balance Q ga a Q ga we get under reasonable assumptons V V g Accordngly, provded tat s E < E a E0 a E0 te drawdown effect s dmnsed by te localzed loss

25 OPEN CANNEL FLOW: Transtons n subcrtcal flow b V V b Q g V a g V b QV a b QV γ ρ γ ρ Let us solve for, neglectng te meanngless negatve root 0 a V V V g a g V V V V V V g a V V V g gb b V a gb b V V V V g a V V V g a V V g a V V g Tat s a reasonable approxmaton

26 OPEN CANNEL FLOW: Varable dscarge due to lateral nflow/outflow Man ypotess: Steady moton n a rectangular cannel base s B wt a small and constant slope; gradually vared flow Neglgble wegt component n te drecton of moton and of sear along te wall; α and β Let us consder te equaton of momentum balance r r r r r r ρvdw ρv V n ds ρgdw σ nds t W S W S and ts component along te man flow drecton M Π ρq dsv * M s ds Π s ds ρq dsv d M Π ds ρds QV* QoV ds d B ρq γ ρ QV* QoV ds B Were we suppose tat te outflow velocty s V. Te LS vares wt s because bot and Q are a functon of s d ds ρq ρq dq γb ρ Q V * B B ds Q V Let us now consder te mass balance equaton Q s Q ds Q s ds Q ds dq ds Q Q o o o o

27 OPEN CANNEL FLOW: lateral outflow - Q decreasng along te flow drecton Case A: Q 0; dscarge decreasng along te flow drecton d ρq γb ds B dq Qo ds ρq B dq ds ρq V Wc can be combned to obtan d ρq dq ρq d ρq γb ρv γb ds B ds B ds B If we now consder te flow specfc energy E Q E gb It vares wt s as a functon of and Q de ds E Q gb E Q E gb d ds Q E Q 3 dq ds o ρq B Drop Intake of a small ydropower plant dq ds 0 Lateral outflow on te left and nflow on te rgt

28 OPEN CANNEL FLOW: lateral outflow - Q decreasng along te flow drecton If one consder tat E ρq γb γb B E ρq γb Q B Te momentum balance equaton can be wrtten as E d E dq γb γb 0 ds Q ds or, more smply water overflow from te cannel appens wtout decreasng te energy per unt de 0 wegt of te water flowng n te cannel. Its value wll be determned on te ds bass of te boundary condton And alternatvely d dq g E dq n an alternatve way, ts equaton provdes te ds g B Q ds gb3 E ds dfferental equaton tat governs te water surface Q profle. It can be ntegrated numercally. Bot equatons requre an addtonal equaton for water overflowng out of te cannel. Usually t s n te form dq Q g c 3/ o µ ds Altoug an analytcal soluton s possble f µ s constant, a numercal soluton provdes a more general approac

29 OPEN CANNEL FLOW: lateral outflow - Q decreasng along te flow drecton E constant and Q decreasng along te flow: use of te Specfc dscarge curve Q B q g α E Two dfferent classes of problem: L and c are gven; fnd out Q 0 qds,.e., QL : FUNCTIONAL VERIFICATION problem If Fr <, starts downstream staton A wt a temptatve QL and a correspondng QL and compute profle n a backward fason. Cange QL untl Q0 s found. If Fr >, starts upstream B knowng and Q0 and ntegrate te eqauton movng downward. Q0 and qds are gven; fnd out L wt c beng usually constraned : DESIGN problem If Fr <, starts downstream staton A wt te known value QL, and compute profle n a backward fason. Wen Qs Q0, ten L s. If Fr >, starts upstream B wt te known value Qs, and compute profle untl QsQ0 - qds. ten L s.

30 OPEN CANNEL FLOW: lateral outflow - Q decreasng along te flow drecton Te effcency of te lateral wer can be ncreased by operatng downstream on te boundary condton.

31 OPEN CANNEL FLOW: lateral nflow - Q ncreasng along te flow drecton Case B: dscarge ncreasng along te flow drecton d ρq ρq dq γ B ρq V * ds B B ds ere we need te velocty component V * of te enterng dscarge along te flow drecton. Often ts quantty can be set 0, so tat ρq d B dq ds ρq ds γb B Wc s an equaton statng te conservaton of te specfc force SF d M Π ds 0 Accordngly, te SF s constant wlst E s not. Te constant value of te Specfc Force, S, must be determned on te bass of te boundary condton. Te SF equaton must be consdered along wt te mass conservaton equaton dq ds Q were te enterng dscarge Q s a known functon.

32 OPEN CANNEL FLOW: lateral nflow - Q ncreasng along te flow drecton In order to nvestgate te possble profles, we consder ρq γ B B γ B S ; Q S B ρ wose maxmum s te crtcal dept. As one can see, wlst Q ncreases wt s, n a subcrtcal flow te dept decreases. te contrary appens n a supercrtcal flow. In bot cases te secton were te crtcal dept occurs can only be located downstream. In bot cases, E decreases movng from upstream to downstream, due to te enterng dscarge tat as no momentum n te average flow drecton Q SB 3 ρg /3 /3 3/ 4

33 OPEN CANNEL FLOW: lateral nflow - Q ncreasng along te flow drecton In ts case, only an S profle s possble. Actually E, wc s a specfc quantty, keeps decreasng along te flow entrance flow stretc, because dq enters wt 0 momentum n te flow drecton. Accordngly, at te end te flow must gan energy to attan a fnal downstream normal flow tat s more energetc te te one upstream If Fr>, t mgt appen tat te overall nflow cannot be supported by te specfc force of te normal flow upstream. In suc a case ts stuaton may occur. Beng a mld profle, one must start downstream from te crtcal dept and compute te profle movng upstream

34 OPEN CANNEL FLOW: Brdge and culvert Wen flows nteract wt te nvert of a brdge, a sudden reducton of te ydraulc radus appens, so tat also te stage-dscarge relatonsp of te brdge s modfed. Te upstream propagatng M profle s strongly condtoned by te boundary condton exerted by te brdge Frenze, 966, Ponte Vecco

35 OPEN CANNEL FLOW: Culvert tombno o botte a sfone Often a small cannel s use to convey water from one sde to te oter of a levee often a road. Te ydraulc beavour can be qute complex and, apart from te geometry, depends on te level upstream m and downstream v and on te culvert lengt L. a Intally, wen bot m and v are small: open cannel flow troug a contracton b Ten, wen m grows but bot L and v are small: orfce flow c Eventually, pressure flow Te transton between and 3 mples a reducton of R. Accordngly, a strongly backwater effect may occur

36 OPEN CANNEL FLOW: Brdge Brdges are te most common obstructon and tey can strongly condton te upstream water surface profle. On te oter and, a wde varety of stuatons s possble and ts must be treated on a case by case bass In general terms, passage troug a brdge usually mples a contracton, due to pers and abutments.

37 OPEN CANNEL FLOW: Specfc Dscarge Specfc dscarge for E constant E g q B Q k E d dq E g A Q α α 0 In a rectangular cross secton