NOTES ON OPEN CHANNEL FLOW

Transcription

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

2 OPEN CHANNEL FLOW: uniform motion In order to have a uniform flow, a prismatic channel is a necessar condition. This channel, of trapezoidal cross section b 6m, B7 m), is used to conve 5 mc/s of drinkable water to a large american town. Its length is 3 kms. However, this is not a sufficient condition because man 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 sufficientl far awa from the disturbance How much far awa is far? We have to compute the profiles M. Pilotti - lectures of Environmental Hdraulics

3 OPEN CHANNEL FLOW: Specific Energ E h α α h g ga h) G.b) Specific Energ with respect to the thalweg, with constant de dh h k E k k A h) B h) α ga k ) 3 3 α gb k E k ; α ga h) 3 k B k ) k da dh E k ; α gh k 3 5 E k α E.) Equivalent average) Hdraulic Depth E.3) General expression for critical depth E.4) Critical depth in rectangular channel General espression for Specific Energ in critical Condition E.) Minimum of Eh) Ek) in Rectangular, Triangular and Parabolic cross-section; M. Pilotti - lectures of Environmental Hdraulics

4 OPEN CHANNEL FLOW: Specific Energ For a given channel section and a given discharge the critical depth c depends onl on the geometr of the section, while the normal depth h h depends on the slope of the channel and the roughness coefficient. Depending on the relative position between h and k, the bottom slope is defined as Mild slope: h k see figure above); Steep slope: h k; Critical slope: h k M. Pilotti - lectures of Environmental Hdraulics

5 OPEN CHANNEL FLOW: Specific Energ EY) for different cross sections Triang.; θ 9 5 mc/s E [m] Circular; R m R; B4 m Triang.; θ Rect.; B 8 m Trap; B 4m, side slope / Y [m] M. Pilotti - lectures of Environmental Hdraulics

6 OPEN CHANNEL FLOW: oude number Let us underline the meaning of the oude number. Let us consider an infinitel wide channel where water flows in uniform motion with depth h and velocit. If perturbation affects the whole water column tsunami like), we have a wave of positive height dh that ma travel upstream and downstream with absolute celerit ±a. Due to its passage is modified, as -d. Given that the motion is an unstead one, it is convenient to stud the process as seen from astride the wave. This is a inertial frame of reference so that both energ and mass balance can be written in terms of relative velocit. We can write r r r v v v a de ds d ds a r r m t a; h gh a) g h dh ) a) h d a) h dh ) r m c a gh m ) d a) a) g dh d d dh g a) h energ balance mass balance this is the wave relative celerit with respect to the flow M. Pilotti - lectures of Environmental Hdraulics

7 OPEN CHANNEL FLOW: oude number Let us observe our final result a m gh m c gh m ) Where c is the wave relative celerit according to Lagrange 788) see following part of the course on nstead motion). Accordingl: if then there is a positive value of a and a negative one and ever perturbation can move both upstream and downstream. if both values of a are positive, so that the wave cannot propagate upstream. if, c see previous slides) and a Note that a is generall different from. The infinitel small wave propagates with a celerit that is different from the average mass velocit,. M. Pilotti - lectures of Environmental Hdraulics

8 OPEN CHANNEL FLOW: overall significance of the oude number; ρv vv ρgv ρ ρ ) g p γ r* Dv Dτ r* v p ρv vv gv * g γh γ * z p * f x, τ,,re); f x * ρ L ρg ρ L ρ ρg ρ gh gl, τ,,re) * ρ gl ρ r v Re * oude number can be introduced when studing jets, as the ratio between inertial and gravitational Forces d In general terms it should take into account the densit of the fluid where the jet is taking place densimetric oude number) But also in open channel flow as the semi-ratio between the kinetic energ per unit weight over the energ related to pressure after Bakhmeteff, 9). Finall it arises when Navier Stokes equations are made dimensionless. This result is particularl important because it dictates the oude similarit criterion. Within geometricall similar boundaries, if oude and Renolds numbers are the same, also the solution are the same. M. Pilotti - lectures of Environmental Hdraulics

9 OPEN CHANNEL FLOW: Phsical models and the oude similarit criterion When Re is sufficientl large, dnamic similitude for fixed bed models is obtained b imposing oude similarit onl. This is shown taking as an example the Cancano Test case. In 943, in the middle of War World II, under the effect of the beginning of the bombing of Milan and the precedent of the attacks on 6 Ma 943 to the Ruhr dams, where mines, factories and houses where flooded for 8 Km and over people drowned, it was considered that Cancano dam could be regarded as a militar target. The arched gravit Cancano dam had been built in 9 in val aele, Valtellina, north Ital, to the purpose of the hdropower suppl of Milan. Accordingl, it was asked to Prof. De Marchi, one of the leading hdraulicians of the time, to stud the effects of a possible bombing of Cancano I dam. Let us define the ratios between model m) and prototpe p) λ l If oude similarit is imposed, then the constraints hold p gh p h m m ; λv hp l p l And eventuall m gh m ; m p m p tm λ t t p h h m p ; λ ; l λ v λ l M. Pilotti - lectures of Environmental Hdraulics

10 OPEN CHANNEL FLOW: Phsical models and the oude similarit criterion A phsical model of the first 6 km stretch of the alpine valle; total and partial collapses of the dam were considered and discharge hdrographs were measured in three locations l λl l m p λ λ v m p t λ 5 / l 5 λl See: De Marchi, G. Sull onda di piena che seguirebbe al crollo della diga di Cancano. L Energia Elettrica, 945,, Pilotti M., Maranzoni A., Milanesi L., Tomirotti M., Valerio G., Hdraulic hazard mapping in alpine dam break prone areas: the Cancano dam case, IAHR Congress, 3, Chengdu, China, Tsinghua niversit Press, Beijing, ISBN , on SB), 7 pp. M. Pilotti - lectures of Environmental Hdraulics

11 OPEN CHANNEL FLOW: stead flow profiles in prismatic channels Let us consider a graduall varied flow, i.e. one in which vertical acceleration on the cross section are negligible, and, accordingl, an hdrostatic pressure distribution is present. This happens if the slope of the channel is small and the geometr of the boundar is such that the streamlines are practicall parallel. Let us consider constant. nder the above hpothesis,starting from energ equation of graduall varied flow. dh dh d d d S f d z ga E da Sb S f A E S E d E da A b S f Sb S f de d b, 3 de d d d ga ga S S H x x ) H x ) S x x ) z x x ) x ) z x ) S f x )) x ga x x )) ga x )) b f K Sb ) S S b b f x This equation could be approximated and solved for but this would not tell us anthing about the theor of profiles This equation provides x) for a general channel. Let us now consider a prismatic channels, so that AAx)) and S b constant where the Chez equation has been used to compute S f and is the normal flow discharge for a given value of M. Pilotti - lectures of Environmental Hdraulics

12 OPEN CHANNEL FLOW: stead flow profiles in prismatic channels om the equation of graduall varied flow in a prismatic channel, the following general properties of the flow profile x) are easil obtained:. d N ; ), N Sb, D D ; ) c N D N D N D d Boundar conditions known): N D S b d d asmptotic to a horizontal line) asmptotic to normal-depth line) asmptotic to a vertical line) d/ if Manning eq. Is used for ) If, Y downstream and the computation proceeds in the upstream direction along the channel. If, Y upstream and the computation proceeds in the downstream direction along the channel. M. Pilotti - lectures of Environmental Hdraulics

13 OPEN CHANNEL FLOW: stead flow profiles in prismatic channels M. Pilotti - lectures of Environmental Hdraulics d D N c d D N c d D N c ),, D S N D N d b Mild slope prismatic channel M profile M profile M3 profile

14 OPEN CHANNEL FLOW: stead flow profiles in mild slope prismatic channels om W. H. Graf and M. S. Altinakar, 998 M. Pilotti - lectures of Environmental Hdraulics

15 OPEN CHANNEL FLOW: stead flow profiles in prismatic channels M. Pilotti - lectures of Environmental Hdraulics Steep slope prismatic channel S profile S profile S3 profile d D N c d D N c d D N c ),, D S N D N d

16 OPEN CHANNEL FLOW: stead flow profiles in steep slope prismatic channels om W. H. Graf and M. S. Altinakar, 998 M. Pilotti - lectures of Environmental Hdraulics

17 OPEN CHANNEL FLOW: stead flow profiles in prismatic channels Horizontal slope prismatic channel Adverse slope prismatic channel Critical slope prismatic channel M. Pilotti - lectures of Environmental Hdraulics

18 OPEN CHANNEL FLOW: stead flow profiles in various slope prismatic channels Horizontal slope prismatic channel Adverse slope prismatic channel om W. H. Graf and M. S. Altinakar, 998 Critical slope prismatic channel M. Pilotti - lectures of Environmental Hdraulics

19 OPEN CHANNEL FLOW: hdraulic jump Supercritical flow in mild slope prismatic channels M3 profile) and subcritical flow in steep slope prismatic channels S profile) are limited downstream and upstream respectivel at the critical depth. In these cases it ma happen that supercritical flow has to be followed b subcritical flow to cover the whole channel length. The change from supercritical to subcritical flow takes place abruptl through a vortex known as the hdraulic jump, characterized b considerable turbulence and energ loss. The flow depths upstream and downsteam of the jump are called sequent depths or conjugate depths. M. Pilotti - lectures of Environmental Hdraulics

20 OPEN CHANNEL FLOW: hdraulic jump M. Pilotti - lectures of Environmental Hdraulics

21 OPEN CHANNEL FLOW: hdraulic jump M. Pilotti - lectures of Environmental Hdraulics Due to the loss of linearit and to the unknown energ loss we have to revert to the Momentum balance where Π: pressure force acting on the given section, computed as Πγ g A where g is the depth of the centroid of flow area A W: weight of the water enclosed between the sections; T f : total external force of friction acting along the boundar. If the tractive force on the boundar and the component of the weight compensate each other, we can write According to which Specific Force S in conserved across the hdraulic jump. This function has some interesting properties. For instance T f ga W ga Π Π sin γ β θ γ β A ga A ga S g g γ γ γ γ ) ) A b g A A A A ga b A A ga S g Is zero when, I.e., in critical conditions Y Y Y Y g Y g A d b d Y f dy d Y f Y Y f d Y f dy d d b Y dy d dy A d A d b Y ) ),,) ), ), ) ) ) ; ) ) ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

22 OPEN CHANNEL FLOW: hdraulic jump om the momentum equation in the form S S it turn out that since E E an energ loss EE -E takes place across the jump is the initial depth depth before the jump) and the sequent depth. Both are coniugate depths M. Pilotti - lectures of Environmental Hdraulics

23 OPEN CHANNEL FLOW: hdraulic jump M. Pilotti - lectures of Environmental Hdraulics For rectangular sections the condition of momentum conservation between sections and can be written as Whose solution, due to the smmetr of the equation, can be put in one of the followig forms that can be used to calculate downstream or upstream) depth once upstream or downstream) conditions are known: The energ loss across the jump can be calculated as: ) 8 ) ) gb gb E E ) b g b gb b gb

24 OPEN CHANNEL FLOW: hdraulic jump M. Pilotti - lectures of Environmental Hdraulics

25 OPEN CHANNEL FLOW: qualitative profiles in complex channels Real cases can be obtained b combining the simple profiles seen before. As a first step control section must be identified, where the depth is known as a function of. om there one starts computing the profile moving in the direction dictated b the oude number, as far as the critical depth is reached. At this stage, in some stretch of the channel, more than a single profile is potentiall present. The final choice will be the one whose Specific Force prevails. M. Pilotti - lectures of Environmental Hdraulics