Original Research Comparison of Analytical and Numerical Solutions for Steady, Gradually Varied Open-Channel Flow

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1 Polis J. of Eviro. Stud. Vol., No. 4 (), 95-9 Origial Researc Compariso of Aalytical ad Numerical Solutios for Steady, Gradually Varied Ope-Cael Flow Jacek Kuratowski* Departmet of Hydroegieerig, West Pomeraia Uiversity of Tecology, Piastów 5a, 7- Szczeci, Polad Received: 9 August Accepted: Jauary Abstract Te depts for a steady, gradually varied flow i a rectagular ope cael wit ifiite widt obtaied aalytically by te Bresse formula ad umerically usig trapezoidal rule for itegratio ave bee compared. It as bee sow tat umerical itegratio may geerate sigificat errors of flow values for looped river etworks. Te metod for coosig te optimum itegratio step tat miimizes umerical itegratio errors as bee preseted. Keywords: ope-cael flow, river looped etworks, gradually varied flow, Bresse formula, umerical itegratio Itroductio * jkuratowski@zut.edu.pl Amog various ope-cael systems, a special positio is occupied by looped etworks, i.e. etworks wit cyclic sequeces of reaces. Looped structure ofte appes at irrigatio systems; i te case of rivers tis tis results from atural bifurcatios, i particular at deltaic mouts, like te Red River i Vietam, te Gages ad te Mekog [] as from ydrotecical works (bypasses). Rivers of braided ad aastomosig types [] ca be perceived as looped etworks as well. Te Lower Oder River is a example of suc a system i Polad []. Looped etworks are worty of special regard amog oters due to te fact tat ay local cage of flow coditios, e.g. te arrowig or deepeig of a river bed i oe place, may affect te flow values at te major part of te etwork. Hece, tese etworks require computatioal metods tat allow us to determie flows ad water levels wit ig accuracy. Te preset paper aims to sow circumstaces of sigificat errors at flow determiatio i looped etworks usig stadard umerical metods. Simultaeously, some possibilities of computatioal accuracy improvemet by relevat umerical itegratio are sow. Gradually Varied Flow i Ope Caels Aalytical Solutios Te eergy equatio is te base for all metods of steady, gradually varied flow computatios i ope caels. Assumig abscissa x directed opposite te flow (Fig. ), tis equatio ca be preseted i a differetial form: v z g z dz g v dv Sdx...were z water surface elevatio [m], α Coriolis coefficiet [-], v mea velocity i a cross-sectio [m s - ], g gravity [m s - ], S eergy lie slope (ydraulic slope) [-], or i a itegral oe: x z z v v Sdx () g x...were z, z are elevatios ad v, v are mea velocities i cross-sectios positioed at x ad x, respectively. ()

2 96 Kuratowski J. Cosiderig te rectagular bed cross-sectio, Eq. () ca be writte as follows: d S S () dx c...were is dept [m], c critical dept [m], S bed slope [-]. Solutio of Eq. () requires specificatio for te eergy loss fuctio. Assumig te cocept of te ifiitely wide cael wit a costat bed slope as a base for furter cosideratios, at costat value of Cézy coefficiet Eq. () is give by:...were is te ormal dept [m], wile Maig s approac leads to te equatio: Te aalytical solutio of Eq. (4) is kow as te Bresse formula [4], wic is for te give iitial coditio (x, ) ad at te followig deotatios:...were Fr is a Froude umber for uiform flow ad may be writte as follows:...were: Sdx v g z I Z S l v y, d dx x x F d dx S y y Fr S c y y, F y Fy / c Fr 6 arcta y c eergy lie water surface bottom v+dv II (4) (5) (6) (7) (8) (v dv) g z+dz Aalytical solutio for equatio (5) was give by Veutelli [5]. Gradually Varied Flow i Ope Caels Compariso of Numerical ad Aalytical Solutios Te itegral form () of a gradually varied flow equatio is basically applied i atural caels, wile eergy losses beig a itegral of te ydraulic slope fuctio are calculated usig te trapezoidal rule metod [6]: xx x S Sdx x x Sx x Tis metod is widely accepted as a sufficietly accurate oe [4, 7], so te applicatios of oter metods of umerical itegratio, like Ruge-Kutta [4, 8] or Kutta- Merso [4], are rare. Some researcers [, 9], wile discussig metods of flow calculatios i looped etworks, assume arbitrarily te acceptable error of water elevatios i a sigle itegratio step as. m, wic creates a impressio of accuracy sufficiet for practical purposes. Adlul Islam et al. [], comparig te effectiveess of various algoritms for river etwork calculatios, assume (also arbitrarily) te acceptable error te times smaller, i.e.. m. However, cosiderig te looped etwork as te trapezoidal metod as te metioed error, values may appear isufficietly accurate, as i te example discussed below. Example: Let us cosider a river etwork (Fig. ) cosistig of two mai caels, - ad -, wit costat bed slope joied i ode No. ad coected additioally by trasverse cael -. Te followig assumptios are made: - cross-sectios for eac cael are rectagular wit ifiite widt, - te Cézy coefficiet for eac cael is a costat idepedet of te dept ad calculated as for uiform flow due to te Maig formula for te give rougess coefficiet. Tese assumptios make te Bresse formula (7, 8) te exact solutio for te problem of water surface level determiatio for eac reac of te etwork. Altoug Maig formula is basically redudat ere due to te ivariability of te Cézy coefficiet, it was used to demostrate te proximity of assumptios to a possible real situatio. Usig idepedetly te Bresse formula ad te itegral form of Eq. () at differet itegratio steps Δx by (9) datum Fig.. Gradually varied ope-cael flow. dx X Fig.. Example of cyclic looped river etwork.

3 Compariso of Aalytical ad Numerical Table. Determiatio of flows i a example of a cyclic looped river etwork. Value Assumptios Reac - - Legts [m],, Bed slopes [-].. Uit flows [m s - ]. 5. Rougess coefficiets [m -/ s].. Dept i ode No. [m]. Reac - Legt [m] Bed slope [-] Rougess coefficiet [m -/ s].6 Values commo for bot metods Calculatio results Reac - - Normal depts [m].95. Critical depts [m]..4 Cézy coefficiets [m / s - ] Values for particular metods Bresse (exact) Δx=5 m Δx=, m Δx=, m Depts i ode No. [m] Errors of dept i ode No. i relatio to exact value [mm] Depts i ode No. [m] Errors of dept i ode No. i relatio to exact value [mm] Uit flow at reac - [m s - ] trapezoidal rule (9), uit flow at reac - sould be calculated. Te followig values ave bee assumed for reaces - ad -: - uit flows q [m s - ], - Maig rougess coefficiet [m -/ s]. Additioally te followig values are give: - dept i te ode No. commo for bot mai reaces [m], - costat Coriolis coefficiet α=.. Te sequece of calculatios is preseted below:. Determiatio of critical dept, ormal dept, ad Cézy coefficiets for reaces - ad - due to te relatios: / 5 q q S, c, g C / 6 (). Calculatio of depts i odes ad usig adopted metods.. Determiatio of uit flow at reac - usig te Cézy-Maig formula. All assumptios ad results of computatios performed due to te preseted algoritm are sow i Table. Aalysis of te calculatio results sows tat differeces betwee exact water level elevatios ad te elevatios obtaied by itegratio of eergy losses usig trapezoidal rule metod are seemigly small ad egligible (a few millimeters); owever, tese differeces are te source of ig cages of calculated uit flow values at reac -. Tis may produce biased results of matematical modelig of flows i a complicated river etwork wit looped structure ad give eve etirely false flow coditios i suc a etwork as a cosequece. Additioally, te elevatio differeces are ot a mootoe fuctio of te itegratio step legt Δx. Terefore, te problem of accuracy for water level determiatio is essetial we gradually varied flow as

4 98 Kuratowski J. bee cosidered by a itegral form of te eergy equatio (). I particular, te proper coice of te itegratio steps miimizig te errors of cael depts ad elevatios determiatio becomes importat. Earlier, tis problem was reported by Taveer []. Next, Družeta et al. [], wile aalyzig te ifluece of fiite elemet size o te accuracy of te solutio for te -D ope-flow problem, sowed te existece of a optimal elemet size tat, we decreased, may produce te worse quality of te model. Oe of te possible aalysis optios is te applicatio of te modified trapezoidal rule due to te formula: ()...were λ is te weigt coefficiet, ad determiatio of te λ value tat results i te umerical itegral of te eergy ead losses equal to te relevat aalytical value. For rectagular cael Eq. () yields: ()...were Δx = x x. Assumig coditios adequate to te Bresse solutio te ydraulic slope ca be expressed as: () Terefore, usig deotatios (6), relatios (), ad formula (), Eq. () ca be writte as: Rewritig Eq. (7) i a form: (4) (5)...te set of equatios is obtaied tat allows us to determie relevat λ value for ay itegratio step as a fuctio:...were: xx x q g S Sdx S x x x Sx S q C q x g S x F S (6) (7) For practical reasos it is more coveiet to sow te relatio (6) i a logaritmic scale, i.e.:,, y (8)...were: y log y (9),, y S x y y S x y y y y y Fy x xx x Sdx y Calculatios of te λ values were performed for te followig rages: β,, γ ;.5, ad y * ;. Tis variability covers practically te wole rage of flow parameters at backwater computatios. Te graps for some β values are sow i Fig.. It results tat for te give values of β ad γ tere exists oly oe value y * =y *, were λ=.5 ad umerical itegratio by trapezoidal rule gives a ubiased result. Aalogically, at give values β ad y * tere exists te sigle (at te outmost) value γ=γ fulfillig tis coditio. Tis meas tat wile performig oe itegratio step for te gradual flow equatio (4) by trapezoidal rule, te dept is equal to te exact aalytical value at oe, if ay, legt of te spatial itegratio step Δx=Δx. Te graps sow tat at sufficietly small values y * (oe may estimate y * <, so at depts ) te value Δx > does ot exist; terefore, every itegratio step must be biased. Worty of otice is tat at y * y *, plots λ=λ(β,γ,y * ) ave poit of iflexio ad ig gradiets tat cause eac iexactess of y * determiatio affectig λ value relatively strogly. Substitutig λ=.5 ad puttig γ=γ to Eq. (8) te followig iverse fuctio ca be foud: () Te graps for relatio () are sow at Fig. 4, wilst coefficiets of te approximatig triomial fuctio y y y * *, a a * a () are give i Table togeter wit relevat determiatio coefficiets R. Table. Approximatio of te relatio y * =y * (β, γ ). β a a a R

5 Compariso of Aalytical ad Numerical... 99,,9 =.,,9,8 =.5,7 =. =.7,6 =.4 =.,5,4,,,, -, -,5,,5,,5, y *,,9 =.,8,7,6,5 * y =.95,4 =.9,,,, =.8 =.7 =.6 =.5 =.,,,,,4,5,6,7,8,9,,,,,4,5 Fig. 4. Values y * at differet β ad γ.,8,7,6,5,4,, =. =.4 =.5 =.7 =. Applicatio of liear iterpolatio to te triomial approximatio coefficiets give i Table for te first step of te eergy equatio itegratio i te mai reaces of te example river etwork results i te followig values: for te reac -: β=.5, y * =.69, γ =.6, Δx =,7 m (exact value,76 m), for te reac -: β=.75, y * =., γ =.99, Δx =,88 m (exact value,847 m).,, -, -,5,,5 y *,,5,, =.6,9 =.5,8,7 =. =.7,6 =.4 =.,5,4,,,, -, -,5,,5,,5, y *, =.9 =.5,9 =. =.7,8,7 =.4,6 =.,5,4,,,, -, -,5,,5,,5, y * Fig.. Sample values of λ at differet β, γ, ad y *. Coclusios Te simulatios performed ave revealed tat umerical itegratio of te steady, gradually varied flow i looped etworks of ifiitely wide rectagular ope caels may lead to sigificat errors of water level slopes ad be te source of errors of flow values at particular reaces of te etwork. It as bee sow tat umerical itegratio of tis equatio is exact oly for at most oe itegratio step wit its legt beig a fuctio of te flow parameters. Terefore, i order to obtai te ubiased solutio, oe may eiter apply specified icostat step legt or modify trapezoidal rule by applicatio of variable weigt coefficiets accordig to Eq. (). Oe ca aticipate tat similar relatios exist for a rectagular cael of fiite widt; owever, weigt coefficiets as optimum itegratio steps require furter researc i tis case. Refereces. PRASHANTH REDDY H., MURTY BHALLAMUDI S. Gradually Varied Flow Computatio i Cyclic Looped Cael Networks. Joural of Irrigatio ad Draiage Egieerig, ASCE,, (5), 44, 4.. SMITH D.G. Aastomosed fluvial deposits: moder examples from Wester Caada. Special Publicatios, Iteratioal Associatio of Sedimetologists, 6, 55, 98.. KURNATOWSKI J. River Rougess Coefficiet as a Fuctio of Vertical Referece System. Polis Joural of Evirometal Studies, 8, (5A), 79, 9.

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