NOTES ON OPEN CHANNEL FLOW

Transcription

1 NOTES ON OPEN CHNNEL FLOW Prof. Marco Pilotti DICTM, Università degli Studi di Brescia Profili di moto permanente in un canale e in una serie di due canali - Boudine, 86

2 OPEN CHNNEL FLOW: basic assumptions Free surface flow: the upper surface is limited by a gas (tipically, the atmosphere) so that its pressure is constant Tipical cases: channel (irrigation, hydropower supply, water supply, land reclamation ) river, sewer conduits, lakes Particular cases: groundwater flow, free surface flow in a syphon Main hypothesis in these lectures D approach in steady conditions Single-phase flow in unerodible, fixed bed Newtonian, constant density fluid Mostly, linear flow in rough turbulent conditions Bed slope i f < 0. m/m Minimum slope Maximum slope Irrigation or land reclamation channel Sewer free surface pipe Floodplain river creek

3 OPEN CHNNEL FLOW: typical slopes Mississippi river between St. Louis and Minneapolis (U.S. Corps of Engineers) 60 Typical mountain creek in Italian alps (T. Rossiga, 3.7 Km ) 500 Pendenza media z [m] Profilo altimetrico pendenza [%] s [m]

4 OPEN CHNNEL FLOW: relevance and applicability of basic hypothesis Unerodible and fixed bed: the area surrounding Isola Pescaroli (from braided river to a single bed river)

5 OPEN CHNNEL FLOW: a remark on the applicability of the fixed bed hypothesis The hypothesis of unerodible and fixed bed is true whenever the flow lies below the Shields diagram : Y < Y c Y v γ D ρv D µ X * s γ s τ D ρ * 0 Where mobility number [-], with γ s sediment submerged specific weight [N/m 3 ] grain Reynolds number [-] For a better definition of the left side of the Shields diagram see Pilotti M., Menduni G., Beginning of sediment transport of incoherent grains in shallow shear flows, Journal of Hydraulic Research, IHR, 39, 5-4, 00.

6 OPEN CHNNEL FLOW: relevance and applicability of basic hypothesis Roll waves in steep channels Debris flow (colata detritica) (see USGS movie, by Costa and Garret) HOMEWORK: see USGS movie on debris-flow

7 OPEN CHNNEL FLOW: mass balance r r ρ( V n) d Q QM U verage velocity on ρ D Dt M Mass discharge through, the area of the cross section r r ( ρdw ) ( ) ( ) 0 ρdw ρ V n d t W W S ρd S ρ r r V nd Q V verage density on Q Volumetric discharge through Mass balance for a control volume dw dt dy r r S(Y ) (V n )d Q dt in Q out (G.) Mass balance for a basin, under constant density assumption ( ρ ) dx Q( x ) t M ( ρ) ( ρu) + 0 t x ( ) ( U + ) t x q Q( x + dx ) M ( ρu ) dx x Mass balance for D flow (G.a) Mass balance for D flow when density varies (e.g.,turbiditic flow or debris flow) (G.b) Mass balance for D flow when density is constant (typically, in open channel flow), and with net discharge q per unit length HOMEWORK: see exercises on Sarnico dam and on dam breach

8 OPEN CHNNEL FLOW: energy balance First and second Coriolis coefficient (G.) Energy balance equation for D gradually unsteady varied flow (G.a) Total head (G.b) Specific Energy with respect to the thalweg Relationship between (G.a) and (G.b) (G.c) Energy balance in terms of E S f t U g s H β g U h z g U p z H α α γ U d u U d u 3 3 ; β α t U g S S s E s E S s E s z s H E z g U h z H f b b β α ) ( h g Q h g U h E α α + + S b S f ds de (G.d) Energy balance in terms of E in steady state conditions

9 OPEN CHNNEL FLOW: uniform motion Let us consider a D flow in steady state condition with no lateral influx. ccordingly Q is constant u s U s s h 0 s de ds dh ds S S b f (U.) The definition of uniform motion in a weak sense, imply a cilindric boundary and can be immediately rewritten in term of average velocity Q α g S w That, if one consider (G.b), implies i.e, the wetted cross section does not vary. But in a prismatic boundary (h), so that If now we consider (G.b), 3 d ds 0 that implies (U.) Energy head loss bed slope water surface slope The assumption (U.) at the basis is never fully verified but it is often verified in an approximate way. The implication (U.) is of paramount importance because it implies a steady energy content of the flow ccordingly, uniform flow is considered the reference state for all the other flow conditions

10 OPEN CHNNEL FLOW: uniform motion ccordingly, from the kinematic point of view uniform flow is characterized by du ds dh ds d dh 0 ; 0; 0 ds ds Whilst, from the energetic point of view S f de ; Sb S f ; 0 ds nd finally, from the momentum point of view γ Q γ Q β + Π + W sinθ β + Π + Tf g g W sinθ T γ LS b τ PL τ γ RS where Π: pressure force acting on the given cross section; W: weight of the water enclosed between the sections; T f : total external force of friction acting along the wetted boundary. 0 0 b γ RS f HOMEWORK: Find D of equilibrium for the Po river at Isola Pescaroli

11 OPEN CHNNEL FLOW: uniform motion In order to have a uniform flow, a prismatic channel is a necessary condition. This channel, of trapezoidal cross section (b 6m, B7 m), is used to convey Q 5 mc/s of drinkable water to a large american town. Its length is 300 kms. However, this is not a sufficient condition because many man-made structures can interact with the flow causing departure from uniform flow (e.g., the gate on the left) In these situations uniform motion still holds but one has to be sufficiently far away from the disturbance How much far away is far? We have to compute the profiles

12 OPEN CHNNEL FLOW: uniform motion Let us consider the problem of finding the relationship between h and Q in uniform flow U ε Q Sb S f λ λ(re,, Fr, f ) 8gR R 8gR / 6 / 6 U χ RSb ksr RSb R RSb n / 6 / 6 Q χ RSb ksr RSb R RS n χ 8g λ b (U.3) Darcy-Weisbach relationship, with friction coefficient l (U.3a) Chezy equation with Gauckler Strickler and Manning s coefficient (U.3b) Chezy equation with Gauckler Strickler and Manning s coefficient By comparing (U.3) and (U.4a-U.4b) one sees that the friction coefficient and the Chezy coefficient have the same informative content. ctually, if one compare a logaritmic law for hydraulically rough flow for l and admits that k s is proportional to e -/6 k s R / 6 8g λ ε R / 6 ε C log( ) R Conclusion: law valid for hydraulically rough turbulent motion with k s being a conveyance coefficient proportional to e -/6 k s ccording to Manning, n is a resistence n coefficient proportional to e +/6

13 OPEN CHNNEL FLOW: cross-sections geometry From W. H. Graf and M. S. ltinakar, 998

14 OPEN CHNNEL FLOW: different formulation for friction - gravel bed rivers - Q χ RS 6 6 χ K s R K s / D R 6 χ n χ n b 8g χ ; χ ; λ g For natural channels with sediments of diameter D (DD 50 [m], Strickler, and DD 75, Lane). Manning δ R δ.5 n R ( n 0.) χ χ ε 5.75log + g g Re f 3. 3Rf χ ar 5.6 log g 3.5D χ 5.6 log g S n 0.3 R χ g 0.38 b 0.6 y D log λ χ.4 0. g + 4 y D R D ln λ Pavloskii, 95, took into account the exponent variation with relative roughness (0.m <R< 3m ; 0.0<n <0.04) Marchi (96), for situations where a logarithmic profile holds. f is a shape factor varying between 0.8 (wide rectangular cross section) and.3 (triangular equilateral cs) Hey (979), for gravel bed rivers, where a varies with bed slope between. and χ g y D 50 Bathurst (978) for rivers where slope is > 0.4% Jarret (99), for mountain creeks log λ R D χ g y D 50 Ferro e Giordano for gravel bed rivers.06 ln.73 y D 50 Butera e Sordo (984), for beds with medium and high relative roughness

15 OPEN CHNNEL FLOW: different formulation for friction How do these formulas compare? Let us consider an infinitely wide bed with y m, DD m: / 6 R χ 7.5 [ m / s / 6 D / χ 9.6 [ m s χ 3.5 [ m / s χ 8. [ m / s χ.7 [ m / s χ 4.8 [ m / s ] ] ] ] ] ] Gauckler-Strickler: Ferro e Giordano: first equation Butera e Sordo: first and second equation Hey s equation with a. Bathurst χ 7.6 [ m / s ] Marchi: with εd and we suppose hydraulically rough regime with f 0.8 Bed of gravel and well-rounded small boulders. Right bank is fairly steep and lined with trees and brush. Left bank slopes gently and has tree and brush cover.

16 OPEN CHNNEL FLOW: uniform motion and selection of roughness coefficient Tables of roughness coefficient for channels (Marchi e Rubatta, UTET)

17 OPEN CHNNEL FLOW: uniform motion and selection of roughness coefficient Tables of n values for channels of various kinds can be found in the literature (e.g. Chow, 964)

18 OPEN CHNNEL FLOW: uniform motion and selection of roughness coefficient There is no substitute for experience in the selection of Manning's for natural channel

19 OPEN CHNNEL FLOW: uniform motion In river or streams the uniform flow is a mere approximation, still extremely usefull Sometime the approximation is very good, as in this case (plan sketch and cross sections, Columbia River at Vernita, Wash)

20 OPEN CHNNEL FLOW: uniform motion In other cases it is a crude approximation, but still very usefull (plan sketch and cross sections of some creeks in the US, from Barnes, USGS)

21 OPEN CHNNEL FLOW: uniform motion Computation of Manning s n according to Barnes, USGS L g U g U k H H Z Z n Q S Z Z n Q S Z n S R n Q L g U g U k H H S g U g U k LS H H f f f / f f K is a coefficient taken to be 0 for contracting reaches and 0.5 for expanding ones Conveyance Geometric average

22 OPEN CHNNEL FLOW: uniform motion Stage-discharge (scala delle portate) relationship in uniform flow (also, normal rating curve) ( h 5/ 3 / 6 0 Q ksr( h0 ) ( h0 ) R( h0 ) Sb ks / 3 P( h0 ) ) S b The encircled expression is known as conveyance, being a function of h and representing a measure of capacity of water transport h h 0 is the so called NORML DEPTH (profondità di moto uniforme

23 OPEN CHNNEL FLOW: uniform motion Stage-discharge relationship for closed conduits in uniform flow ( h) P( h) 5/3 / 6 Q ksr( h) ( h) R( h) Sb ks / 3 r ( ϕ sen( ϕ)) P rϕ R r( sen( ϕ) ) ϕ S b HOMEWORK: Compute these relationship by using a spreadsheet

24 OPEN CHNNEL FLOW: uniform motion Stage-discharge relationship for irregular cross section in uniform flow P pi a R n P n j j i Q k R( h) ( h) R( h) / 6 s S b

25 OPEN CHNNEL FLOW: uniform motion Zona golenale of the Po river at Isola Pescaroli (floodplain) lveo di magra (main bed or channel)

26 OPEN CHNNEL FLOW: uniform motion Often natural or man-made cross section are composite, i.e., composed of different subsections, maybe with different roughness and local slope, due to different lengths of the thalweg. The lower subsection (alveo di magra) conveys water during drought or low flows. The overflow sections (alvei di piena e golenale) are activated during floods Without a proper decomposition the D assumption is violated and the hydraulic radius shows sudden reductions that have unrealistic effects on the other hydraulic quantities HOMEWORK: Give a careful look to the spreadsheet Compound_Cross_Section.xls

27 OPEN CHNNEL FLOW: uniform motion in channels of compound section The cross section of a channel may be composed of several subsections, e.g. a main channel and two side channels (flood plains). In this case the Chézy equation has to be applied to each subsection i to compute the corresponding discharge Q i. The total discharge is obtained as QΣQ i For the evaluation of the wetted perimeter P i of each subsection only the solid boundaries are considered. This criterion would require to subdivide the section along the lines orthogonal to the isotachs; actually, along these lines no internal shear stress takes place; however, vertical lines are generally used. The hydraulic radius R i of each subsection is calculated as R i i /P i. Q 3 Qi Ri i Sb KSb, K i i n i i n i R 3 i i conveyance

28 OPEN CHNNEL FLOW: equivalent roughness In case of compact sections in which the roughness may be different from part to part of the perimeter the discharge can be computed without actually subdividing the section. To this purpose, an equivalent roughness coefficient can be introduced dividing the water area into N parts of which the wetted perimeter P i (calculated taking into account only the solid boundaries) and roughness coefficients n i are known. ssuming the same mean velocity and slope for each partial area, in uniform flow (according to Horton and Einstein) V 3 V R Pi ni n P from which 3 n k s P n 3 i P i V i i Sb S n P P b ni i S n b i 5/3 ( Pk i siri ) PR 5/3 3 3 P i ni i n ( P Pi ) P In a similar way, Pavloskii and Einstein, considering the tractive force along the boundary as the sum of the single contributions k s P 0.5 P k i si 0.5 nd Lotter, regarding the overall discharge as the sum of the single contributions HOMEWORK: Compare these relationship by using a spreadsheet i

29 OPEN CHNNEL FLOW: uniform motion Section of Maximum discharge Q k ( h) P( h) 5/3 s S /3 b ( h) C P( h) 5/3 /3 Let us suppose that Q, ks and S b are kept constant. Let us explore the condition for which Q will be maximixed, under the constraint that is constant. R must be maximized, that is for given, P must be minimum ccordingly, by minimizing the wetted perimeter dp dh h dp h + ds By substituting () in () s + + s 0; hs 0; + s + s Let us consider a trapezoidal cross-section, that is a function of b, h and s. If is kept constant, these three variables are constrained ( b + sh) h; b sh h P h b + h + s + + h ( ) sh h s + s s s + h (3) P ( h) R( h) h P h () () Which is the condition that maximizes R indipendently from s. If s 0 we have an optimal rectangular cross section for bh

30 OPEN CHNNEL FLOW: uniform motion If one can minimize also with respect to s one obtains (3) that corresponds to stg30 + s s; sen( α) / ; α 30 ; s tan(30 ) ccordingly, if everything can be chosen, an half exagonal cross section seems to be the most reasonable. On the other hand, if the choice is constrained to a rectangular cross-section, bh provides the best choice. Whether this choice is practicable or not depends on other constraints, such as, for instance, the type of lining used to cover the channel surface, the actual availability of space around the channel or the maximum allowable velocity.