4.1 Introduction. 4. Uniform Flow and its Computations

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1 4. Uiform Flow ad its Computatios 4. Itroductio A flow is said to be uiform if its properties remai costat with respect to distace. As metioed i chapter oe of the hadout, the term ope chael flow i ope chael is to mea steady uiform flow. Sice QAV, it follows that i uiform flow V V V. thus i uiform flow, depth of flow, area of cross sectio ad velocity of flow remai costat alog the chael. It is obvious; therefore, that uiform flow is possible oly i prismatic (artificial) chaels. 4. Expressig the velocity of a uiform flow For hydraulic computatio the mea velocity of a turbulet uiform flow i ope chaels is usually expressed approximately by a so-called uiform flow formula. Most practical uiform flow formula ca be expressed i the followig geeral form: V CR x S y Where V is the mea velocity, R is the hydraulic gradiet, S is the eergy slope, x ad y are expoets ad C is a factor of flow resistace. Chezy equatio By defiitio there is o acceleratio i uiform flow. By applyig the mometum equatio to a cotrol volume, distace L apart. F x W siθ + F F F Fa Qρ( β v vβ) Where F ad F are pressure forces ad M ad M are the mometum fluxes at sectio ad respectively W weight of the fluid i the cotrol volume ad F is the shear force at the boudary. Sice the flow is uiform: F F ad M M Also WγAL ad F τ o PL Where τ o average shear stress o the wetted perimeter of legth P ad γ is uit weight of water. Replacig siθ by S o (bottom slope) ca be writte as γ ALS τ opl o A τ o γ So γrs P Where o R A P is defied as hydraulic radius. Expressig the average shear stress τ o as τ o kρv ature of the surface ad flow parameters. A τ o γ S o γrs o kρv P Leadig to V C RS is kow as Chezy formula o, where k is a coefficiet which depeds o the g Where C a coefficiet which deped o the ature of the surface ad the flow ad kow as k chezy coefficiet.

2 Determiatio of Chezy resistace factor Several forms of expressios for the Chezy coefficiet C have bee proposed by differet ivestigatios i the past. A few selected oes are selected below:. Pavlovski formula x C R I which x R ( 0.). Gaguiller ad Kutter formula So C So R. Bazi s formula C M R Maig s formula ad is Maig s coefficiet I which M is a coefficiet depedat o the surface roughess A resistace formula proposed by Robert Maig, owig to its simplicity ad acceptable degree of accuracy i a variety of practical applicatios, Maig s formula is probably the most widely used uiform flow formula i the world. V R S o Where is roughess coefficiet kow as Maig s [L -/ T]. This coefficiet is essetially a fuctio of the ature of the boudary surface. Comparig Chezy ad Maigs: R 6 C I the Maig s formula, all the terms except are capable of direct measuremet. The roughess coefficiet, beig a parameter represetig the itegrated effect of the chael cross sectioal resistace, is to be estimated. The selectio of a value for is subjective, based o oe s experiece ad egieerig judgemet. However, a few aids are available which reduce to a certai extet the subjectiveess i the selectio of a appropriate value of for a give chael. These icludes: A compressive list of various types of chaels, there descriptios with the associated rage of values of ad photographs of selected typical reaches of caals, there descriptio ad measured value of.

3 Table 4. values of roughess coefficiet Factors affectig It is ot commo for egieers to thik of a chael as havig a sigle value of for all occasios. I reality, the value of is highly variable ad depeds o a umber of factors. I selectig a proper value of for various desig coditios, a basic kowledge of these factors should be foud useful. The maig s is essetially a coefficiet represetig the itegrated effect of a large umber of factors cotributig to the eergy loss i a reach. Some importat factors are: a) surface roughess b) vegetatio c) chael irregularity ad d) Chael aligmet. The chief amog these are the characteristics of the surface. Surface roughess: represeted by the size ad shape of the grais of the material formig the weighted perimeter ad producig a retardig effect o the flow. Fie grais result i relatively low value of ad course grais, i a high value of. Vegetatio: also markedly reduce the capacity of the chael ad retard the flow. Chael irregularity: comprises irregularity i wetted perimeter ad variatio i crosssectio ad shape alog the chael legth. Chael aligmet: smooth curves with larger radius will give a relatively low value of, whereas sharp curvatures with severe meaderig will icrease. Empirical formula for May empirical formulas have bee preseted for estimatig Maig s coefficiet i atural streams. These relate to the bed particle size. The most popular form uder this type is the Strickler formula: 6 d 50. Where d 50 is i meters represets the particle size i which 50% of the bed material is fier. For mixtures of bed material with cosiderable course graied size the above equatio is modified by Mayer as:

4 6 d 90. Where d 90 is size i meters with 90% of the particle are fier tha d Composite roughess of chaels I some chaels differet parts of the chael perimeter may have differet roughesses. Caals i which oly the sides are lied, laboratory flumes with glass walls ad rough beds, rivers with a sad bed i deepwater portio ad flood plais covered with vegetatio, are some typical examples. For such chaels it is ecessary to determie a equivalet roughess, a coefficiet that ca be applied to the etire cross-sectioal perimeter i usig the Maig s formula. This equivalet roughess, also called the composite roughess, represets a weighted average value for the roughess coefficiet. Oe such method of calculatios of equivalet roughess is give below. Cosider a chael havig its perimeter composed of N types of roughess P, P, P P N are the legths of these N parts ad, N are the respective roughess coefficiets. Let each part P i be associated with a partial area A i such that: N i Ai A + A + A... AN A Total area Figure 4. multi-roughess type sectio It is assumed that the mea velocity i each partial area is the mea velocity V for the etire flow of the area, i.e V + V + V... V N V By maig s formula V V V V S O... R R R R V R i i i N N N Where is equivalet roughess Ai A i P P i A i i A P i P 4

5 A i A i P P i P i P i This equatio affords a meas of estimatig the equivalet roughess of a chael havig multiple roughess types i its perimeter. 4.4 uiform flow computatios The Maig s formula ad the cotiuity equatio QAV form the basic equatios for uiform flow computatios. The discharge Q is the give by: Q AR S o Where, K AR is called the coveyace of the chael ad expresses the discharge capacity of the chael per uit logitudial slope. The term K AR is sometimes called the sectio factor for uiform flow computatios. For a give chael, AR is a fuctio of depth of flow. Q AR the write side of the equatio cotais the value of, Q ad S; but the left sides S o depeds oly o the geometry of the water area. For example cosider a trapezoidal chael with bottom width B ad side slope m the: A ( B + my)y ( B + y m ) P + AR ( B + my) 5 y 5 ( B + y m +) ( + mη) AR Q 8 8 B So B η 5 η 5 ( + m + ) y where η Eq A B 5

6 Figure 4. Variatio of φ with y/b i trapezoidal chael For a give chael, B ad m are fixed ad sectio factor is a fuctio of depth oly. The above / figure shows the relatioship of equatio A i a o-dimesioal maer by plottig AR vs y/b for differet values of m. it may be see that for m 0. There is oly oe value y/b for each value of φ, idicatig that for m 0, AR is a sigle valued fuctio of y. this is true for ay other shape of chaels provided that the top width is either costat or icreasig with depth these are called chaels of the first kid. Sice Q AR ad if ad S o are fixed for a chael, the chael of the first kid have uique S o depth i uiform flow associated with each discharge. This depth is called ormal depth. Thus the ormal depth is defied as the depth of flow at which a give discharge flows as uiform flow i a give chael. The ormal depth is desigated as y o, the chaels of the first kid have oe ormal depth oly. While a majority of the chaels belog to the first kid, sometimes oe ecouters chaels with closig top width. Circular ad void sewers are typical examples of this category. Chaels with a closig top-width ca be desigated as chaels of the secod kid. φ B 8 / 6

7 Figure 4. variatio of AR i chaels of the secod kid The variatio of AR with depth of flow for a few chaels of this secod kid is show i the above figure. It may be see that i some rages of depth, AR is ot a sigle valued fuctio of depth. Types problems i uiform flow Uiform flow computatio problems are relatively simple. The available relatios are: Maig s formula Cotiuity equatio Geometry of the cross-sectio The basic variables i uiform flow situatio ca be the discharge Q, velocity of flow V, ormal depth y o, roughess coefficiet, chael slope S o ad the geometric elemets (e.g. B ad m for trapezoidal chael). The followig five types of basic problems are recogized i ope chael flows. No Give Required y o,, S o, geometric elemets Q ad V y o,, Q, geometric elemets S o y o, S o, Q, geometric elemets 4, S o, Q, geometric elemets y o 5, S o, Q, y o geometric Geometric elemets 7

8 Problems of type, ad ormally have explicit solutios ad hece do ot preset ay difficulty i there calculatios. Problems of type 4 ad 5 usually do ot have explicit solutios ad as such may ivolve trial-ad-error solutio procedure. 4.5 Computatio of ormal depth It is evidet from the above table that the calculatio of the ormal depth for may chaels ivolves a trial-ad-error solutio. Sice all ope chael problems ivolve ormal depth, special attetio towards providig aids for quicker calculatios of ormal depth is warrated. A few aids for computig ormal depth i some commo chael sectios are give below. a) Rectagular chael 8

9 9

10 0

11

12 4.6 Computatio of critical ad ormal slopes Usig the Froude umber formula it is possible to develop a geeral flow relatio that is: V F ga/ T Q F A g T If the discharge Q occurs as a uiform flow, the slope S o required to sustai this discharge is, by the Maig s formula; Q S o Substitutig the Froude umber equatio 4 A R F S o TA gp 4 4 S o P f ( y) (eq A) F g TA For a trapezoidal chael of side slope m, S o ( B + m + y) f ( y) F g ( B + my)(( B + my) y) / S o B ( + m + η) F g ( + mη)(( B + mη) η) 4 4 f ( y) Desigatig S / B o S geeralized slope * F g S * f ( m, η) The above equatio represets the relatioship betwee the various elemets of uiform flow i a chael sectio i a geeralized maer. The limitig value of S is obtaied by puttig, ds y 0 whereη o. dη B The slope of the chael which carries a give discharge as a uiform flow at the critical depth is kow as the critical slope, S c. the coditio goverig the critical slope i ay chael ca be easily obtaied from eq A by puttig F. Whe the discharge ad roughess are give, the Maig s formula ca be used to determie the slope of a prismatic chael i which the flow is uiform at a give ormal depth y. The slope thus determied is sometimes called specifically the ormal slope S o

13 4.7 Hydraulically efficiet chael sectio The coveyace of a chael sectio of a give area icreases with a decrease i its perimeter. Hece a chael sectio havig the miimum perimeter for a give area of flow provides the maximum value of the coveyace. With the slope, roughess coefficiets ad area of flow fixed, a miimum perimeter sectio will represet the hydraulically efficiet sectio as it coveys the maximum discharge. The chael sectio is also called the best sectio. Of all the various possible ope chael sectios, the semicircular shape has the least amout of perimeter of a give area. A) Rectagular sectio Bottom width B ad depth of flow y Area of flow A BY costat Wetted perimeter P B + Y A/Y +Y

14 4

15 5

16 6

17 4.7 Compoud sectios Some chael sectios may be formed as a combiatio of elemetary sectios. Typical atural chaels, such as rivers, have flood plais which are wide ad shallow compared to the deep mai chael. The figure below represets a simplified sectio of a stream with flood baks. Chael of this kid is kow as compoud sectios. Figure 4.4 compoud sectio Cosider the compoud sectio to be divided ito subsectios by arbitrary lies. These ca be either extesios for the deep chael boudaries as i the figure above or vertical lies draw at the edge of the deep chaels. Assumig the logitudial slope to be the same for all subsectios, it is easy to see that the subsectios will have differet mea velocities depedig up o the depth ad roughess of the boudaries. Geerally, overbaks have larger size roughess tha the deeper mai chael. If the mea velocities V i i the various subsectios are kow the the total discharge is V i A. i If the depth of flow is cofied to the deep chael oly (i.e y < h), calculatio of discharge by usig the Maig s is very simple. However, whe the flow spills over ito the flood plai (i.e y>h), the problem of the discharge calculatio is complicated as the calculatio may give a smaller hydraulic radius for the whole stream sectio ad hece the discharge may be uder estimated. This uderestimatio of discharge happes i all rages i small rages y, say h < y <y m, where y m is maximum value of y beyod which the uderestimatio of the discharge as above doest occur. For the value of y > y m, the calculatio of the discharge by cosiderig the whole sectio as oe uit would be adequate. For the value of y i the rage of h < y <y m, the chael has to be cosidered to be made up of sub-areas ad the discharge i each sub-area determied separately. The total discharge is obtaied as a sum of discharge through all such sub-areas. The value of y m would deped upo the chael geometry. However, for practical purpose the followig method of discharge estimatio ca be adopted. i.) The discharge is calculated as the sum of the partial discharge i the sub-areas; as, ad as the above figure. Qp Qi AiVi ii.) The discharge is also calculated by cosiderig the whole sectio as oe uit, (portio ABCDEFGH as oe uit) say Q w. 7

18 iii.) The larger of the above two discharges Q p ad Q w, is adopted as the discharge at the depth y. For determiig the partial discharge Q i ad hece Q p i step oe above, two methods are available. Posey s method: i this method while calculatig the wetted perimeter for the sub-areas, the imagiary divisios (FJ ad CD refer the above figure) are cosidered as boudaries for the deeper sectios oly ad eglected completely i the calculatio relatig to the shallower portio. Zero shear method: some ivestigators mostly i computatioal work treat the iterface as purely a hypothetical iterface with zero shear stress. As such, the iterfaces are ot couted as perimeter either for the deep portio or for the shallow portio. Problems:. a trapezoidal chael of bed-width 4.0m ad side slope.5: has a sad bed (0.05). at a certai reach the sides are lied by smooth cocrete (0.0). Calculate the equivalet roughess of this reach if the depth of flow is.5m. As a earthe trapezoidal chael ( 0.05) has a bottom width of 5.0m, side slope of.5: ad a uiform flow depth of.m. I a ecoomic study to remedy excessive seepage from the chael two proposals are plaed a.) to lie the sides oly ad b.) to lie the bed oly are cosidered. If the liig is of smooth cocrete (0.0), determie the equivalet roughess i the above two cases. As 0.0 ad A rectagular chael.6m wide hade badly-damaged surfaces ad with Maig s coefficiet 0.0. As a first phase of repair, its bed was lied with cocrete If the depth of flow remais the same at.m before ad after the repair, What is the percetage icrease i discharge as a result of the maiteace? As 8.9% 4. a circular chael.5m i diameter is made of cocrete (0.04) ad is laid o a slope of i 00. a.) Calculate the discharge if the ormal depth is.5m. b.) Calculate the depth of flow for a discharge of 5m /s. As.76m /s.75m 5. A rectagular chael is to be laid o a slope of The sides will be of smooth cocrete (0.0). What width of chael is ecessary to carry a discharge of 9m /s with a ormal depth of.6m? As B.656m 6. a trapezoidal chael of bed width.0m ad side slope of.5: carries a full supply of 0m /s at a depth of.5m. What would be the discharge at half of full supply depth (i.e. at 0.75m)? What would be the depth of half of full supply discharge? As.7m/s, 0.09m 7. A trapezoidal chael havig a side slope of.5: carries a discharge of 00m/s with a depth of flow equal to 0.75 width. If S o ad 0.05 fid the bed width ad depth of flow. As 4.85m &.68m 8. a trapezoidal chael is 5.0m wide ad has a side slope of 0.5:. Fid the depth of flow which ca make the chael a efficiet sectio. If S o ad 0.0, fid the correspodig discharge. As 4.045m &.m /s 8