NOTES ON OPEN CHANNEL FLOW

Transcription

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

2 OPEN CANNEL FLOW: passage over a sill ump, bump: soglia Wen te flow passes over an ump, several situations may appen, depending on te Froude number and on Energy content. Locally tere is a sudden curvature of te flow, te cannel is not prismatic and te teory on water surface profiles is of no use. owever an energy balance can be accomplised to study tis transition. Let us first suppose tat. no ead loss is present 0 E a E0; E E0. te sill eigt a is small wit respect to te energy upstream. 3. Te cannel is infinitely long downstream and upstream, so tat te dept of te flow approacing te sill and downstream of it is te normal dept If te slope is mild, water dept on te sill lowers more tan te sill eigt. If te slope is steep, te effect of rise of te sill bed prevails M. Pilotti - lectures of Environmental ydraulics

3 OPEN CANNEL FLOW: passage over a sill ump, bump: soglia Sometimes te eigt of te sill is suc tat te specific energy of te normal flow of te approacing current is not sufficient to pass over it. In suc a case te flow upstream must gain energy and we ave to distinguis between mild and steep cannel M. Pilotti - lectures of Environmental ydraulics

4 AN IMPORTANT EXAMPLE: Broad crested, round nose, orizontal crest weir Upstream corner well rounded to prevent separation Geometrical requirements as in figure above and in te specific publications M. Pilotti - lectures of Environmental ydraulics

5 WEIRS: Broad crested orizontal crest weir M. Pilotti - lectures of Environmental ydraulics

6 OPEN CANNEL FLOW: passage over a sill ump, bump: soglia But an ead loss is almost inevitable so tat 0: normal flow : on te sill; : downstream; 0m: upstream Making an energy balance starting downstream, one sees tat in a mild cannel te level upstream is iger M. Depending on Te lengt of te cannel, tis could affect Q And in a steep cannel, Starting upstream, one sees tat te rise on te ump is 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 M. Pilotti - lectures of Environmental ydraulics

7 OPEN CANNEL FLOW: passage troug a contraction Te same situation occurring wen a flow passes over an ump can be observed in te passage troug a contraction. Usually a contraction can be caused by te piers or abutments of a bridge If no localized losses are Present, ten te specific energy is constant M. Pilotti - lectures of Environmental ydraulics

8 OPEN CANNEL FLOW: passage troug a contraction Sometimes te Energy upstream isn t enoug M. Pilotti - lectures of Environmental ydraulics

9 OPEN CANNEL FLOW: passage troug a contraction Altoug one can suppose tat no ead loss is present,tis is not generally true. Accordingly, te flow must gain energy to compensate for te localized ead loss. Tis appens upstream if Fr < M and downstream if Fr > S if if 0m Fr < Fr > 0m 0 M 0 S Te process is similar to te one considered for te passage over a bump M. Pilotti - lectures of Environmental ydraulics

10 OPEN CANNEL FLOW: Transitions As a first approximation one can disregard te energy losses implied in a transition. In suc a case te following situations arise for a sudden rise/fall of te bed or contraction/expansion M. Pilotti - lectures of Environmental ydraulics

11 OPEN CANNEL FLOW: Transitions in subcritical flow wit ead loss Let us consider an abrupt drop in te cannel bed. If we ave an ead loss we cannot directly use an energy balance and we ave to revert to a momentum balance, under te same assumptions usually used to derive Borda s ead loss in a pipe. 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 consider an energy balance Q ga a Q ga we get under reasonable assumptions g Accordingly, provided tat is E < E a E0 a E0 te drawdown effect is diminised by te localized loss M. Pilotti - lectures of Environmental ydraulics

12 OPEN CANNEL FLOW: Transitions in subcritical flow M. Pilotti - lectures of Environmental ydraulics b b Q g a g b Q a b Q γ ρ γ ρ Let us solve for, neglecting te meaningless negative root 0 a g a g g a g gb b a gb b g a g a g a g Tat is a reasonable approximation

13 OPEN CANNEL FLOW: ariable discarge due to lateral inflow/outflow Main ypotesis: Steady motion in a rectangular cannel base is B wit a small and constant slope; gradually varied flow Negligible weigt component in te direction of motion and of sear along te wall; α and β Let us consider te equation of momentum balance r r r r r r ρdw ρ n ds ρgdw σ nds t W S W S and its component along te main flow direction M Π ρq i * M s Π s ρq d M Π ρ Qi* Qo d B ρq γ ρ Qi* Qo B Were we suppose tat te outflow velocity is. Te LS varies wit s because bot and Q are a function of s d ρq ρq dq γb ρ Q i * B B Q Let us now consider te mass balance equation Q s Q Q s Q dq i Q i Q o o o o M. Pilotti - lectures of Environmental ydraulics

14 OPEN CANNEL FLOW: lateral outflow - Q decreasing along te flow direction Case A: Q i 0; discarge decreasing along te flow direction d ρq γb B dq Qo ρq B dq ρq Wic can be combined to obtain d ρq dq ρq d ρq γb ρ γb B B B If we now consider te flow specific energy E Q E gb It varies wit s as a function of and Q de E Q gb E Q E gb d Q E Q 3 dq o ρq B dq 0 Lateral outflow on te left and inflow on te rigt Drop Tyrolean Intake of a small ydropower plant M. Pilotti - lectures of Environmental ydraulics

15 OPEN CANNEL FLOW: lateral outflow - Q decreasing along te flow direction If one consider tat E ρq γb γb B E ρq γb Q B Te momentum balance equation can be written as E d E dq γb γb 0 Q or, more simply water overflow from te cannel appens witout decreasing te energy per unit de 0 weigt of te water flowing in te cannel. Its value will be determined on te basis of te boundary condition And alternatively d dq g E dq in an alternative way, tis equation provides te g B Q gb3 E differential equation tat governs te water surface Q profile. It can be integrated numerically. Bot equations require an additional equation for water overflowing out of te cannel. Usually it is in te form dq Q g c 3/ o µ Altoug an analytical solution is possible if µ is constant, a numerical solution provides a more general approac M. Pilotti - lectures of Environmental ydraulics

16 OPEN CANNEL FLOW: lateral outflow - Q decreasing along te flow direction Te efficiency of te lateral weir can be increased by operating downstream on te boundary condition. For instance, By placing a sluice gate one can raise te water level and greatly increase te amount of discarge released by te weir. M. Pilotti - lectures of Environmental ydraulics

17 OPEN CANNEL FLOW: lateral outflow - Q decreasing along te flow direction E constant and Q decreasing along te flow: use of te Specific discarge curve Q B q g α E Structural variables: L, c usually constrained ydraulic variables: QL, η Q 0 -QL/ Q 0 efficiency Two different problems: Q 0,L and c are given; find out Q 0 q, i.e., QL : FUNCTIONAL ERIFICATION problem If Fr <, start downstream station A wit a temptative value of QL and a corresponding i QL and compute te corresponding profile in a backward fasion. Cange QL until Q 0 is found. If Fr >, start upstream B knowing i and Q 0 and integrate te equation moving downward. In te former case te procedure is iterative, not in te latter. Q 0, c and q are given; find out L: DESIGN problem If Fr <, start downstream station A wit te known values of [QL, i ] and compute profile in a backward fasion. Wen Qs Q 0, ten L s. If Fr >, start upstream B wit te known value Qs, i and compute te profile until Qs Q 0 - q. ten L s. M. Pilotti - lectures of Environmental ydraulics

18 OPEN CANNEL FLOW: lateral inflow - Q increasing along te flow direction Case B: discarge increasing along te flow direction d ρq ρq dq γ B ρq i * B B ere we need te velocity component * of te entering discarge along te flow directon. Often tis quantity can be set 0, so tat ρq d B dq ρq γb B Wic is an equation stating te conservation of te specific force SF d M Π 0 Accordingly, te SF is constant wilst E is not. Te constant value of te Specific Force, S, must be determined on te basis of te boundary condition. Te SF equation must be considered along wit te mass conservation equation dq Q i were te entering discarge Q i is a known function. M. Pilotti - lectures of Environmental ydraulics

19 OPEN CANNEL FLOW: lateral inflow - Q increasing along te flow direction In order to investigate te possible profiles, we consider ρq γ B B γ B S ; Q S B ρ wose maximum is te critical dept. As one can see, wilst Q increases wit s, in a subcritical flow te dept decreases. te contrary appens in a supercritical flow. In bot cases te section were te critical dept occurs can only be located downstream. In bot cases, E decreases moving from upstream to downstream, due to te entering discarge tat as no momentum in te average flow direction Q SB 3 ρg /3 /3 3/ 4 M. Pilotti - lectures of Environmental ydraulics

20 OPEN CANNEL FLOW: lateral inflow - Q increasing along te flow direction In tis case, only an S profile is possible. Actually E, wic is a specific quantity, keeps decreasing along te stretc were flow is entering, because dq enters wit 0 momentum in te flow direction. Accordingly, at te end te flow must gain energy to attain a final downstream normal flow tat is more energetic te te one upstream If Fr>, it migt appen tat te overall inflow cannot be supported by te specific force of te normal flow upstream. In suc a case tis situation may occur. Being a mild profile, one must start downstream from te critical dept and compute te profile moving upstream M. Pilotti - lectures of Environmental ydraulics

21 OPEN CANNEL FLOW: Bridge and culvert Wen flows interact wit te invert of a bridge, a sudden reduction of te ydraulic radius appens, so tat also te stage-discarge relationsip of te bridge is modified. Te upstream propagating M profile is strongly conditioned by te boundary condition exerted by te bridge Firenze, 966, Ponte eccio M. Pilotti - lectures of Environmental ydraulics

22 OPEN CANNEL FLOW: Culvert tombino o botte a sifone Often a small cannel is use to convey water from one side to te oter of a levee often a road. Te ydraulic beaviour can be quite complex and, apart from te geometry, depen on te level upstream m and downstream v and on te culvert lengt L. a Initially, wen bot m and v are small: open cannel flow troug a contraction b Ten, wen m grows but bot L and v are small: orifice flow c Eventually, pressure flow Te transition between and 3 implies a reduction of R. Accordingly, a strongly backwater effect may occur M. Pilotti - lectures of Environmental ydraulics