Open Channel Hydraulic

Transcription

1 Open Cannel Hydraulic Julien Caucat Associate Professor - Grenoble INP / ENSE3 - LEGI UMR 5519 julien.caucat@grenoble-inp.fr Winter session /2016 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 1/ 79

2 Introduction 1 Introduction Cannel types and geometries Flow in cannels Velocity, pressure and turbulent stress distributions Hydrodynamic considerations 2 Uniform flows Momentum balance and friction coefficients Discarge calculation 3 Non-Uniform Flows Gradually Varied Flows Rapidly Varied Flows 4 Bibliograpy julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 2/ 79

3 Introduction Cannel types and geometries Natural / artificial cannels River Irrigation cannel Wastewater treatment plant Storm water overflow sewer julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 3/ 79

4 Introduction Cannel types and geometries Cannels geometries Water surface slope piezometric slope Bed slope Definitions: Cross-section (CS): plane normal to te flow direction A: Wetted surface = portion of te cross section occupied by te liquid P : Wetted perimeter = lengt of contact line between liquid and bed and banks R = A : Hydraulic radius = reference lengt of te CS P B: Widt or top widt = widt of te cannel at te free surface D = A : Hydraulic dept = water dept of an equivalent rectangular CS B : Water dept = maximum dept in te CS julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 4/ 79

5 Introduction Cannel types and geometries Cannels geometries From Graf and Altinakar (1998) Open Cannel Hydraulic 5/ 79

6 Introduction Flow in cannels Type of flows Flow wit a free surface (air/water interface) on wic te pressure is at te atmosperic value Te flow is mostly driven by gravity due to te inclination of te bed (and not due to a pressure drop like in closed conduit flows) Classification: julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 6/ 79

7 Introduction Flow in cannels Type of steady flows Uniform flows: x = 0 and S f = S w Non-uniform flows: D (x) varies and S f S w Gradually varied flows: slow variation / quasi-uniform flow Rapidly varied flows: abrupt cange of D (x) like at a singularity (weir, gate,...) or a Hydraulic jump julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 7/ 79

8 Introduction Flow in cannels Navier-Stokes equations Navier-Stokes equation (incompressible) Mass conservation. ( u ) = 0 Momentum balance equation ρ u t + ρ. ( u u ) = p + η 2 u + ρ g }{{}}{{}}{{}}{{} Inertia P ressure V iscous stress Gravity julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 8/ 79

9 Introduction Flow in cannels Dimensional analysis (1/5) Let s consider a lengt scale: L 0 a velocity scale: U 0 based on wic we can deduce a time scale: T 0 = L0 U 0 a pressure scale: P 0 = ρ (U 0) 2 Wit tese scales we can make te variables dimensionless: position : x x = vitesse : u ū = L 0 U 0 temps : t = t U0 pression : p = p L 0 ρu 2 0 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 9/ 79

10 Introduction Flow in cannels Dimensional analysis (2/5) We now substitute tese dimensionless variables into te Navier-Stokes equations: Mass conservation U ( ) 0 ū. = 0. ( ) ū = 0 L 0 Momentum balance ρ U 2 0 L 0 [ ū t +. ( ) ] ū ū = ρ U 0 2 L 0 p + U 0 L 2 0 η 2 ū + ρ g g g dividing by ρ U 0 2 : L 0 [ ū t +. ( ) ] ū ū = η U 0 L 0 p + ρ L 2 0 U 0 2 }{{} η ρ U 0 L 0 2 ū + L 0 g U 2 0 }{{} g L 0 U0 2 g g julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 10/ 79

11 Introduction Flow in cannels Dimensional analysis (3/5) Tese two numbers are noting else but: te Reynolds number: Re = ρ U0 L0 η te Froude number: F r 2 = L 0 U 2 0 g Te dimensionless momentum balance equation can be rewritten as: [ ū t +. ( ) ] ū ū = 1 p + Re ū + 1 ḡ F r 2 wit ḡ = g g julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 11/ 79

12 Introduction Flow in cannels Dimensional analysis (4/5) Now let s take some caracteristic scales of te problem: Lengt scale = water dept : L 0 = m Velocity scale = mean velocity : U 0 = Q m/s A and compute te order of magnitude of tese two dimensionless numbers: Reynolds: Re = ρ U0 L0 η = 103 ( ) ( ) 10 3 = Froude: F r = U 0 L0 g = ( ) ( ) 10 = Conclusion: In natural systems, te flow is always fully turbulent (Re > ) and can be sub- or supercritical. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 12/ 79

13 Introduction Flow in cannels Flow regimes From Graf and Altinakar (1998) Open Cannel Hydraulic 13/ 79

14 Introduction Flow in cannels Dimensional analysis (5/5) Te order of magnitude of te different terms in te momentum equation can be estimated as follows: [ ū t +. ( ) ] ū ū = 1 p + Re 1 ḡ ū + }{{}} F r {{ 2 } O( ) O( ) Te viscous stress tensor is negligible but due to turbulence te inertial term give rise to a turbulent stress, te so-called Reynolds stress. Very quickly (you will see tat in more details in te mixing part of tis lecture) te Reynolds stress is obtained by introducing te Reynolds decomposition < u >= 1 τ t+τ t u =< u > + u u dt and applying it to te inertia term: u ū = (< u > + ) u (< u > + ) u < u >=<< u >> < u >= 0 u ū =< u > < u u > + u }{{} Reynolds stress tensor julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 14/ 79

15 Introduction Velocity, pressure and turbulent stress distributions Transverse velocity profile In Hydraulic engineering we assume tat te flow one-dimensional: or U = 1 u (x,y,z). x da A A U = 1 0 u (x,z). x d Looking back at te measurements tis assumption is reasonable and anyway it is te only way to acieve calculations by and. Te aspect ratio B as an influence on te velocity profile due to te tickness of te lateral boundary layers. For B > 5 te flow can be considered as 2D. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 15/ 79

16 Introduction Velocity, pressure and turbulent stress distributions Velocity profile Te profiles presented ere corresponds to wat you actually measure in an open cannel flow under uniform flow conditions. Te velocity vanises at te bed and exibits a logaritmic profile for z/ < 0.2 Te Reynolds stress exists! < u 2 > and < w 2 > are te diagonal terms and < u w > is te off-diagonal term (=sear stress). julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 16/ 79

17 Introduction Velocity, pressure and turbulent stress distributions Velocity and Reynolds stress profiles Te following formula can be used to estimate te average velocity in a given cross section: Prony formula: U = ( )U surface USGS formula : U = 0.5 (U U 0.8) simple formula : U = U 0.4 were U α represents te value of U(z) at te elevation z/ = α. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 17/ 79

18 Introduction Velocity, pressure and turbulent stress distributions Pressure distribution in uniform flows Hydrostatic pressure Navier-Stokes equation on (0z): 0 = p z + ρ gcosα p = ρ g ( z)cosα ρ g ( z) for small α (α < 6 ) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 18/ 79

19 Introduction Hydrodynamic considerations Continuity equation (steady state) S w Q 1 A 1 S f U 1 A 2 Q 2 α U 2 Consider a 2D flow for simplicity: u = (u,0,w) Te conservation of mass for an incompressible fluid reads: u x + w z = 0 ( u Integrating over te water dept: 0 x + w ) dz = 0 (w=0 at z = 0; ) z ( ) so tat x 0 u dz Q 1 = Q 2 = Q or U 1 A 1 = U 2 A 2 = U x = 0. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 19/ 79

20 Introduction Hydrodynamic considerations Total Energy Total ead at elevation z: p t(z p) ρ g = p(zp) + z p + u(zp)2 ρ g }{{} 2 g }{{} Potential }{{} Pressure Velocity expressed in meter of water column. Total ead in a cross section: α = 1 U 3 H = αu 2 2 g + + z f 0 u 3 dz: Energy coefficient in general, for a turbulent flow α 1 Piezometric or reduced pressure: p = p + ρ g z wit ydrostatic pressure distribution p = ρ g ( z) p = ρ g ( z) + ρ g z p = ρ g = constant julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 20/ 79

21 Introduction Hydrodynamic considerations Specific energy (1/2) Tis concept as been first introduced by Bakmeteff in 1932 and corresponds to te total energy in a cross section wit reference to te bed elevation: H = αu 2 2 g + }{{} +z f H s = αu 2 2 g + H s: Specific Energy Using continuity equation: Q = U A H s = αq2 2 g A 2 + For a rectangular cross section cannel: A = b : H s = αq2 2 g b curve example for b = 10 m julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 21/ 79

22 Introduction Hydrodynamic considerations Specific energy (2/2) Supercritical Subcritical Te minimum of H s versus corresponds to te critical dept c: H min s dhs d dh s d dh s d = 0 =c = αq2 d 2 =c 2 g b 2 d + 1 =c H min s αq2 g b 2 3 c = αq g b 2 3 c = αq2 + 1 g b 2 3 c = F 2 = 1 ( αq 2 Critical dept: c = g b 2 ) 1/3 rectangular cannel julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 22/ 79

23 Introduction Hydrodynamic considerations Waves propagation and flow regimes Long waves propagates wit celerity: C 2 = g (rectangular cannel in quiescent water). long wave i.e. L >> 1 If tere is a flow in te cannel, te absolute wave celerity is given by: C w = U ± C Critical flow: single wave (downstream) Subcritical flow: two waves (up- and downstream) Supercritical flow: two waves (downstream only) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 23/ 79

24 Introduction Hydrodynamic considerations From Navier-Stokes to Sallow water equations Streamwise component of te Navier-Stokes equation (2D flow v = 0) u t }{{} unst. + u u x + w u = 1 p + ρ g sinα }{{ z } ρ x }{{}}{{} gravity advection pressure + 1 τ xx ρ x + 1 τ xz ρ z } {{ } total st. Assuming: 1 steady flows i.e. t = 0 2 negligible normal stresses τ xx 0; τ zz 0 u u x = 1 p ρ x + g S f + 1 ρ τ xz z 3 weak slopes: sinα tanα S f Using te ydrostatic pressure distribution p = ρ g ( z): u u = g x x + g S f + 1 τ xz ρ z julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 24/ 79

25 Introduction Hydrodynamic considerations From Navier-Stokes to Sallow water equations Integrating along te vertical direction (Oz): 0 u u }{{ x } 1 u 2 = 2 x dz = for a non-uniform velocity profile: wit 0 0 g x dz τ xz g S f dz + 0 ρ z dz u 2 dz = β U 2 (β = O(1): Boussinesq coefficient) 1 β U 2 = g 2 x x + g S f + τ xz s τxz b ρ τ s xz: sear stress acting at te free surface (by te wind for example ; will be neglected) τxz b : sear stress acting at te bed 1 β U 2 } 2 {{ x } inertia = g x }{{} pressure + g S f }{{} gravity τ b xz ρ }{{} bed stress Dept averaged momentum balance for a steady non-uniform gradually varied flow julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 25/ 79

26 Introduction Hydrodynamic considerations Exercices exercice 1 Compute and estimate te average velocity Te table below gives te values of a vertical velocity profile corresponding to a flow aving a water dept of = 9.9 m in a prismatic cannel. z (m) u (m/s) Give a metod to compute te average velocity U 2 Estimate te average velocity U and compare it wit te metod proposed in tis lecture. exercice 2 Specific Energy curve for a rectangular cross section cannel of widt B = 5 m 1 Give te relationsip for te critical dept c as a function of B and Q. 2 Compute te value of c for a discarge Q = 40 m 3 /s. 3 Find te expression of H c and give its value for Q = 40 m 3 /s. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 26/ 79

27 Uniform flows 1 Introduction Cannel types and geometries Flow in cannels Velocity, pressure and turbulent stress distributions Hydrodynamic considerations 2 Uniform flows Momentum balance and friction coefficients Discarge calculation 3 Non-Uniform Flows Gradually Varied Flows Rapidly Varied Flows 4 Bibliograpy julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 27/ 79

28 Uniform flows Uniform Flows Te flow in a cannel is considered as uniform and steady, if te flow remains constant in te flow direction and in time. Tis configuration is taken as te base configuration for all te oter ones. In reality uniform flows are rarely encountered. It is only possible in very long prismatic cannels far from upstream and downstream boundaries. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 28/ 79

29 Uniform flows Momentum balance and friction coefficients Momentum balance Rectangular cannel (from momentum equation): 1 β U 2 2 x = g }{{} x }{{} inertia pressure + g S f }{{} gravity τ b xz ρ }{{} bed stress = g S f }{{} = gravity τ b xz ρ }{{} bed stress General case Friction force: F F = τ 0 P dx Gravity force: F G = ρ g A sinα dx Gravity force = Friction force ρ g A sinα dx = τ 0 P dx τ 0 = ρ g A P sinα = ρ g R S f (*) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 29/ 79

30 Uniform flows Momentum balance and friction coefficients Friction coefficients: Darcy-Weisbac equation Te wall sear stress can be modeled according to: τ 0 = f 8 ρ U 2 (**) ( ) ks were f = f,re is te Darcy-Weisbac friction coefficient tat depends on te R Reynolds number Re and te relative rougness ks R. Moody diagram (1944) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 30/ 79

31 Uniform flows Momentum balance and friction coefficients Friction coefficients: Cézy formula Identifying (*) and (**) gives: f 8 ρ U 2 = ρ g R S f tat can be recasted into: U = 8 g f R1/2 S 1/2 f or U = C R 1/2 S 1/2 f wic is known as te Cézy equation were C = 8 g/f is te Cézy coefficient (m 1/2 /s). Cézy was a Frenc civil and ydraulic engineer wo establises tis first law for uniform flows in 1775 (one year before US declaration of independence). It is usually more convenient to use te discarge in place of te average velocity: Q = U A Q = C A R 1/2 S 1/2 f Table of typical Cézy coefficient values julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 31/ 79

32 Uniform flows Momentum balance and friction coefficients Friction coefficients: Manning-Strickler Te major limitation of te Cézy formula resides in te lack of dependency of C to te relative rougness k s/r. Manning (1881), Iris engineer, and Strickler ( ), Autrician engineer, independently proposed te following parametrization for C: C = R1/6 n Introducing tis parametrization into te Cézy equation U = C R 1/2 S 1/2 f leads to te Manning-Strickler equation: = Ks R1/6 n: Manning coefficient (s/m 1/3 ) K s: Strickler coefficient (m 1/3 /s) Henderson (1966) U = K s R 2/3 S 1/2 f = 1 n R2/3 S 1/2 f or Q = K s A R 2/3 S 1/2 f = 1 n A R2/3 S 1/2 f Tis parametrization made te unanimity. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 32/ 79

33 Uniform flows Momentum balance and friction coefficients Friction coefficients: Manning-Strickler Te values of K s or n as been tabulated by many autors, tey mostly depends on te nature of te cannel s bed and banks (rougness). Te coice of te values remains someow subjectives and needs to be calibrated using measurements. Te oter problem is tat tese friction coefficients are dimensional. However, te values are te same in te metric and Imperial/US systems. Te correction is made on te definition of te Cézy coefficient for te US system: C = 1.48 R1/6 n = 1.48 K s R 1/6 See HECRAS 4.1 Hydraulic reference table 3-1 p.80 (from Cow, 1959) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 33/ 79

34 Uniform flows Momentum balance and friction coefficients Friction coefficients: Composite rougness In case te rougness is not omogeneous over te entire cross section it becomes necessary to compute an equivalent friction coefficient. Following Einstein (see cow, 1959) te cross section A can be divided into N parts of almost omogeneous rougness n i aving a wetted area A i and wetted perimeter P i. Assuming tat te average velocity is identical in all te sub cross sections: U 1 = U 2 =... = U N = U one can write: U = 1 ( A n e P ) 2/3 S 1/2 f = 1 ( ) 2/3 A1 S 1/2 f =... = 1 n 1 P 1 n N ( AN P N ) 2/3 S 1/2 f using te equality A = i Ai leads to: n e = ( n 3/2 i P P i ) 2/3 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 34/ 79

35 Uniform flows Discarge calculation Conveyance To determine te discarge Q in a cannel one needs to know te cannel s geometry (A, P ), rougness (n) and longitudinal slope (S f ). Q = 1 n A R2/3 S 1/2 f = K() S 1/2 f were K() is te cannel conveyance tat depends only on te water dept. Tis dept is usually called te normal dept and is denoted as n. Te conveyance caracterizes te water transport capacity of a given cross section. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 35/ 79

36 Uniform flows Normal dept for a rectangular cannel Manning-Strickler: Q = 1 n A R2/3 S 1/2 f Discarge calculation Wetted area: A = B Wetted perimeter: P = B + 2 B wit ydraulic radius: R = A P Rectangular cross section: R = B B + 2 Q = 1 (B ) 5/3 S1/2 n 2/3 f (B + 2 ) Normal dept: ( ) 3/5 ( n Q n = ) 2/5 B S 1/2 B f Wide rectangular cannel: B >> B + 2 B R = B B + 2 Q = 1 n B 5/3 S 1/2 f Normal dept: W C n = ( n Q B S 1/2 f Wide Cannel assumption (WC) ) 3/5 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 36/ 79

37 Uniform flows Discarge calculation Application to te Delaware river Geometric data: Lengt: L = 484 km Average slope: S f = Discarges at Trenton: Q min = 122 m 3 /s Q ave = 371 m 3 /s Q max = 9,316 m 3 /s Widt at Trenton: B 300 m Manning coefficient : n = s/m 1/3 Compute te critical and normal depts at Trenton for te tree given disarge values. For te computations, assume a rectangular cross section for te river. source: Wikipedia julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 37/ 79

38 Uniform flows Discarge calculation Composite cross section During a flood, te flow takes place in two very different part of te cannel: te main cannel and te floodplain (overbanks). Te rougness and te slope of tese two parts can be very different and tis situation can be dealt wit te following formula: Q = Q c + Q o = 1 n c A c R 2/3 c 1/2 + 1 A o R 2/3 L c n o o 1/2 L o were P c and P o sould only include fluid-bed or banks contact lines i.e no fluid-fluid contact lines. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 38/ 79

39 Non-Uniform Flows 1 Introduction Cannel types and geometries Flow in cannels Velocity, pressure and turbulent stress distributions Hydrodynamic considerations 2 Uniform flows Momentum balance and friction coefficients Discarge calculation 3 Non-Uniform Flows Gradually Varied Flows Rapidly Varied Flows 4 Bibliograpy julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 39/ 79

40 Non-Uniform Flows Non-Uniform Flows: Introduction Non uniform flows are encountered wen te water dept varies along te streamwise direction. Two different cases will be considered, if te free surface curvature is small compared wit te cannel slope te flow is called Gradually Varied Flow (GVF) and if te free surface curvature is important te flow will be called Rapidly Varied Flow (RVF). Te ypoteses and equations tat govern tese two non-uniform flow regimes are different and will be detailed ereafter. Open Cannel Hydraulic 40/ 79

41 Non-Uniform Flows Gradually Varied Flows Water dept equation S e Cannel - prismatic: B; S f x = 0 - rectangular of widt B S w Q U 2 (x) 2g (x) energy gradeline dh Energy equation: H = U 2 2 g + + z f = dh dx = d dx ( S f Q 2 z f (x) 2 g B z f Q 2 ) 2 g B d dx + dz f dx dh dx = Q2 d g B 2 3 dx + d dx + dz f dx S e = ( 1 F r2) d dx S f τ 0 derivation d dx d 2 z f (x+dx) ( x+dx) d dx = S f S e 1 F r 2 Tis equation remains true for an arbitrary cross section dx = 2 3 d dx and F r2 = dh dx = Se and dz f dx = S f Q 2 g B 2 3 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 41/ 79

42 Non-Uniform Flows Gradually Varied Flows Water dept equation S e Water dept equation: S w U 2 (x) 2g energy gradeline dh d dx = S f S e 1 F r 2 Q (x) Energy slope: S e = dh dx S f z f (x) τ 0 z f (x+dx) corresponds to te slope of te energy grade line = energy dissipated by friction ( x+dx) for a uniform flow: S e = S f and Q = 1 n A R2/3 S 1/2 f (assuming a Manning-Strickler law) for a non uniform GVF: S e S f and Q = 1 n A R2/3 Se 1/2 S e = ( n Q A R 2/3 S e is te slope tat an equivalent uniform flow would ave in a cannel aving a bed slope equal to te energy gradeline one. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 42/ 79 ) 2

43 Non-Uniform Flows Gradually Varied Flows Water dept equation prismatic rectangular cannel and Manning-Strickler law d dx = S f S e d = 1 F r2 dx = S f 1 S e/s f 1 F r 2 ( ) n Q 2 wit S e = B 5/3 we can write Se = S f and F r 2 = Q2 g B 2 3 = ( ) 3 c Te water dept equation can be written as: 2 n Q B S 1/2 f 5/3 ( ) 10/3 n = ( ) 10/3 n 1 d dx = S f ( ) 3 c 1 Tis equation is valid only for a wide rectangular cannel assuming a Manning-Strickler law Remarks: 1 If = n ten d = 0 and remains constant: te uniform flow is stable. dx 2 If = c ten d and te free surface is vertical! Incompatible wit slow dx variation in te streamwise direction. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 43/ 79

44 Non-Uniform Flows Gradually Varied Flows Water dept variation prismatic rectangular cannel and Manning-Strickler law Subcritical flow: n > c 3 cases 1 > n > c : = d dx = > 0 > 0 > 0 1 d dx = S f ( ) 10/3 n ( c 1 ) 3 (wide rectangular cannel assuming a Manning-Strickler law) n < 1 and c ( ) n 10/3 ( ) c 3 < 1 = 1 > 0 and 1 > 0 2 n > > c : = d dx = < 0 > 0 < 0 n > 1 and c ( ) n 10/3 ( ) c 3 < 1 = 1 < 0 and 1 > 0 3 n > c > := d dx = < 0 < 0 > 0 n > 1 and c ( ) n 10/3 ( ) c 3 > 1 = 1 < 0 and 1 < 0 Q S f n c julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 44/ 79

45 Non-Uniform Flows Gradually Varied Flows Water dept variation prismatic rectangular cannel and Manning-Strickler law Supercritical flow: c > n 3 cases 1 > c > n : = d dx = > 0 > 0 > 0 1 d dx = S f ( ) 10/3 n ( c 1 ) 3 (wide rectangular cannel assuming a Manning-Strickler law) n < 1 and c ( ) n 10/3 ( ) c 3 < 1 = 1 > 0 and 1 > 0 2 c > > n : = d dx = > 0 < 0 < 0 n < 1 and c ( ) n 10/3 ( ) c 3 > 1 = 1 > 0 and 1 < 0 3 c > n > := d dx = < 0 < 0 > 0 n > 1 and c ( ) n 10/3 ( ) c 3 > 1 = 1 < 0 and 1 < 0 Q S f c n julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 45/ 79

46 Non-Uniform Flows Gradually Varied Flows Water dept variation: syntesis Subcritical flow Supercritical flow n c Q c Q n S f S f julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 46/ 79

47 Non-Uniform Flows Gradually Varied Flows Water dept equation For a prismatic wide rectangular cannel and Cézy law, te water dept equation is: 1 d dx = S f 1 ( ) 3 n ( c ) 3 In te general case, te water dept equation is: 1 (Q/A)2 d dx = S C 2 R S f f ( ) Q 2 2 or B 1 g A 3 Cézy law d dx = S f 1 n2 (Q/A) 2 1 Manning law R 4/3 S f ) 2 ( Q 2 B g A 3 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 47/ 79

48 Non-Uniform Flows Gradually Varied Flows Te notion of critical slope If te uniform flow is in te critical regime i.e: n = c Ten te cannel is at te critical slope S c Demo: Q = 1 n A R2/3 Sc 1/2 Q 2 B g A 3 = 1 Q 2 = 1 n 2 A2 R 4/3 S c Q 2 = g A3 B 1 n 2 R4/3 S c = g A B S c = n2 g A B R 4/3 S c = n2 g 1/3 wide rectangular cannel and Manning-Strickler law julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 48/ 79

49 Non-Uniform Flows Gradually Varied Flows Te different sapes of water surface profiles Te classification of water surface profiles is usually done according to te bed slope S f (Graf and Altinakar, 1998): S f < S c cannels on Mild slope : M S f > 0 S f > S c cannels on Steep slope : S S f = S c cannels on Critical slope : C S f = 0 cannels on Horizontal slope : H S f < 0 cannels on Adverse slope : A Eac curve is composed of different brances depending on te actual value of te water dept compared wit te normal and critical depts. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 49/ 79

50 Non-Uniform Flows Gradually Varied Flows Te different sapes of water surface profiles Open Cannel Hydraulic 50/ 79

51 Non-Uniform Flows Gradually Varied Flows Te different sapes of water surface profiles Open Cannel Hydraulic 51/ 79

52 Non-Uniform Flows Gradually Varied Flows Metods for computing water surface profiles S e S w i energy gradeline dh Q = i+ i+1 2 i+1 S f x i Δ x x i+1 For a prismatic wide rectangular cannel: d dx = S 1 ( n/) 10/3 f 1 ( c/) 3 wic can also be written as: d = S f 1 ( n/) 10/3 1 ( c/) 3 dx integrating between x i and x i+1 gives: i+1 i = S f 1 ( n/ ) 10/3 1 ( c/ ) 3 (x i+1 x i) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 52/ 79

53 Non-Uniform Flows Gradually Varied Flows Metods for computing water surface profiles Te discrete equation: i+1 i = S f 1 ( n/ ) 10/3 1 ( c/ ) 3 (x i+1 x i) can be used to compute te water surface profile in te following way: 1 For a reac of a small distance x one can compute te water dept variation ; tis metod is know as te standard step metod. 2 For a small water dept difference one can compute te distance x between tese two water depts; tis metod is know as te dept variation or direct metod. 3 Before starting te computation one as to establis te control sections were a known relationsip between te water dept and te discarge exists (could be a an outlet/inlet of a cannel, a weir, a gate or a ydraulic drop). 4 Computations proceed upstream for subcritical flows (F r < 1) and downstream for supercritical flows (F r > 1). 5 Te standard step metod is more time consuming but usually more precise. Tis is te metod used in HEC-RAS. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 53/ 79

54 Non-Uniform Flows Gradually Varied Flows Flow profiles examples: orizontal cannel Free overfall or drop: no longer a gradually varied flow but in a subcritical regime te free overfall controls te flow dept upstream. If te sluice gate forces te flow below te critical dept a supercritical regime is observed downstream te gate so te water surface profile as to be computed in te downstream direction. Te downstream gate imposes a water level above te critical dept upstream so te flow as to be subcritical tere. Te water surface profile as to be computed in te upstream direction. Te matcing of tis two brances a ydraulic jumps takes place to ensure te transition from a supercritical to a subcritical flow. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 54/ 79

55 Non-Uniform Flows Gradually Varied Flows Flow profiles examples Open Cannel Hydraulic 55/ 79

56 Non-Uniform Flows Gradually Varied Flows Standard step metod ( x fixed) Solve: i+1 i = S f 1 ( n/ ) 10/3 1 ( c/ ) 3 }{{} = d dx (x i+1 x i) S e S w Q S f i energy gradeline = i+ i+1 2 x i Δ x x i+1 In tis example we assume tat te flow is supercritical and so we compute te water dept profile in te downstream direction. Te value of Q and i at te upstream cross section are given. Te question is: find te value of i+1 at te position x i+1. Te space step x is given by: x = x i+1 x i. Te solution procedure is as follows: 1 Coose a value for k i+1 (extrapolation, zero gradient, secant metod,...) 2 Compute k = k i+1 + i 2 3 Compute te new k+1 i+1 4 if k+1 i+1 k i+1 and d dx ( k ) d value using: k+1 i+1 = i + dx ( k ) x (non linear equation iterative solution) > ɛ ten step 1 to 3 are repeated wit a new guess for i+1 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 56/ 79 dh i+1

57 Non-Uniform Flows Gradually Varied Flows Detp variation metod ( fixed) Solve: x i+1 x i = S 1 1 ( ) 3 c/ f 1 ( ) ( 10/3 i+1 i) n/ }{{} dx = d S e S w Q S f i energy gradeline = i+ i+1 2 x i Δ x x i+1 Again we assume tat te flow is supercritical and te value of Q and i at te upstream cross section are given. Te question is: find te value of x i+1 at wic te water dept is i+1. Te dept variation is given by: = i+1 i. Te solution procedure is as follows: 1 Compute k = i + and dx d ( ) 2 Compute te value of x using: x = dx d ( ) 3 Compute te value of x i+1 = x i + x ten move on to te next step i+2 by using te known value at i+1 at x i+1. dh i+1 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 57/ 79

58 Non-Uniform Flows Gradually Varied Flows Water surface profile computations: some remarks 1 Te two metods presented above can also be used for an arbitrary cross section geometry. Te equation to be solved is ten: d dx = S f 1 n2 (Q/A) 2 1 R 4/3 S f ) 2 ( Q 2 B g A 3 more complicated but it s not a big deal wit a computer. 2 Te dept variation metod is te easiest metod for computation by and (no iterations) 3 Te standard step metod is more accurate and more convenient for numerical computations especially for non prismatic cannel. Tis is te metod used in HECRAS. 4 Oter metods exist (see Cow 2009 or Graf and Altinakar 1998 for details), some of tem are semi analytic but sligtly more complicated to use. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 58/ 79

59 Non-Uniform Flows Rapidly Varied Flows Rapidly Varied Flows: introduction Upstream and downstream of te critical dept c te flow is rapidly varied. Te assumptions of te Gradually Varied Flow are no longer valid: slow canges wit x, small curvature of te streamlines and ydrostatic pressure distribution. Te passage of te critical dept is usually associated wit a sudden cange in water dept like a ydraulic jump wen te flow dept increases or a ydraulic drop wen te flow dept decreases. Rapidly varied flows usually corresponds to control sections of te flow profile. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 59/ 79

60 Non-Uniform Flows Rapidly Varied Flows Weirs and spillways A weir or a spillway is a device or structure tat allows to measure or control te discarge Q in cannel. Te flow passes over te weir towards te downstream. It is caracterized by its eigt H D and its lengt in te cross section L D. Only unsubmerged weirs will be studied ere but remember tat a weir can be submerged. ttps://en.wikipedia.org/wiki/radyr ttp:// dobbs weir julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 60/ 79

61 Non-Uniform Flows Rapidly Varied Flows Broad crested weir Q 1 2 c D Assumptions: Te flow is supercritical downstream. Te flow is subcritical upstream. Negligible ead loss between te upstream cross section and te weir. Horizontal bed In tis condition te total ead is conserved: Upstream: H = D + + Q 2 2 g L 2 (subcritical flow upstream negligible) D (D + )2 Q 2 Critical section: H = D + c + = 2 g L 2 D + 3 ( ) Q 2 1/3 D 2 c 2 g L 2 D julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 61/ 79

62 Non-Uniform Flows Rapidly Varied Flows Broad crested weir Q 1 2 c D D + = D + 3 ( ) Q 2 1/3 ( ) 3/2 2 Q = L 2 g L 2 D g 3/2 2 = L D 2 g 3/2 D 3 3 3/2 Q = L D g 3/2 In te general case, te coefficient is replaced by a discarge coefficient K D tat depends on te weir geometry: 2 Q = L D K D 2 g 3/2 3 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 62/ 79

63 Sarp crested weir Non-Uniform Flows Rapidly Varied Flows Warning: notation cange H 2 Weir equation: Q = L D K D 2 g H 3/2 3 wit K D ( H H D ) [0.6; 1.2] Wen H H D increases te kinetic energy contribution is no more negligible and one sould use te following weir equation: Q = L D K D ( ) H wit K D [0.58; 0.67] H D 2 2 g H 3/2 were H = H + U g julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 63/ 79

64 Spillways Non-Uniform Flows Rapidly Varied Flows Warning: notation cange H Weir equation: Q = L D K D g H 3/2 wit K D ( H H D ) [0.65; 1.2] Mobile spillways julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 64/ 79

65 Non-Uniform Flows Rapidly Varied Flows Underflow gates Assumptions: Te flow is subcritical upstream. Te flow can be supercritical downstream (non-submerged) or subcritical (submerged). Negligible ead loss between cross section 1 and 2. Horizontal bed In tis condition te specific ead is conserved: Q Upstream: H = g B 2 2 (subcritical flow upstream negligible) 1 Q 2 Downstream: H = g B julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 65/ 79

66 Non-Uniform Flows Rapidly Varied Flows Underflow gates Q 2 1 = g B Q = 2 B 2 g ( 1 2) In te non-submerged case, 2 << 1 and te vena contracta is parametrized using a contraction coefficient: C c = 2 0 wit C c [0.5; 1]: Q = C c 0 B 2 g 1 Te general gate equation is given by: Q = K v A 0 2 g 1 and te gate coefficient K v = C q C c/ 1 + C c(a 0 /A 1 ) wit C q [0.95; 0.99] and C c = A 2 /A 0. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 66/ 79

67 Non-Uniform Flows Rapidly Varied Flows Underflow gates Many underflow gates geometry exists, te basic equation remains te same te difference is in te K v coefficient tat depends on geometrical parameters (sould be given by te manufacturer). Underflow gates are very sensitive devices tat needs to be carefully calibrated! julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 67/ 79

68 Non-Uniform Flows Rapidly Varied Flows Hydraulic drop and free overfall Wen a subcritical open cannel flow freely discarge into te atmospere, tis is a ydraulic drop or free overfall. Tis situation is te limit case of Mild slope cannel flowing in a steep slope cannel (wit infinite slope). Te flow as to experience a transition from subcritical to supercritical so te water dept as to be critical somewere. Te critical dept is observed sligtly before te outlet: b = c In fact te flow can not be assumed gradually varied anymore and so te pressure is not ydrostatic at te outlet. Tis can be demonstrated using te momentum balance between a and te free fall. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 68/ 79

69 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump Te ydraulic jump is encountered wen te flow passes abruptly from supercritical to subcritical in te downstream direction. sudden water surface elevation (discontinuity) or stationary wave very turbulent motion: undulation and air entrainment form of a aerated wave breaking in roller causes strong dissipation of energy: HJ julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 69/ 79

70 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: classification From Mosin Siddique Open Cannel Hydraulic 70/ 79

71 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: conjugate dept Applying te momentum balance on te control volume: ρ( V. ) V dω = p dω + τ dω + ρ g dω Ω Ω Ω Ω neglecting te gravity and te friction terms and applying te divergence teorem: ρ V ( V. n ) dγ = p. n dγ (1) Γ Γ julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 71/ 79

72 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: conjugate dept Momentum balance in te steam wise direction: ρ u ( u. n ). e x dγ = Γ Γ On Γ 2 and Γ 1: n = ± e x: ρ u ( u. n ). 1 e xdγ = B Γ 1 0 and ρ Γ2 u ( u. n ). e xdγ = ρ Q2 B 2 p. n. e x dγ ρu 2 dz = ρbu = ρ Q2 B 1 julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 72/ 79

73 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: conjugate dept On Γ 2 and Γ 1: n = ± e x: p n. e xdγ = B Γ ρ g( 1 z)dz = Bρ g and p Γ2 n. e xdγ = Bρ g Te momentum balance is terefore: ρ Q2 B ( ) = Bρ g ( ) julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 73/ 79

74 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: conjugate dept ( Momentum balance: ρ Q2 1 1 ) B g B Q2 = Bρ g Q 2 2 g B 2 3 = F r 2 1 = 2 1 ( ) = ( 1 2 ) ( ) ( ( Tis is a second order equation in Y = 2 1 : Y 2 + Y 2F r 2 1 = 0 of wic te solutions are: Y 1/2 = 1 ± F1 2 ( F 1 > F1 2 2 > 1) Only te plus solution is pysically admissible 2 = F ) ) fully symmetric: 1 = F Tis equation is known as te Bélanger equation julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 74/ 79

75 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: Head loss ) ) Head loss definition: H = H 2 H 1 = ( 2 + Q2 ( 2 gb Q2 2 2 gb ( H = ( 2 1) + Q2 1 2 gb ) H = ( 2 1) + 4 ( 1 + 2) H = ( 2 1) + 1 (1 2)(1 + 2) (1 + 2) ( H = ( 1 2) ) ( 1 + 2) H = ( 1 2) H = ( 1 2) (1 2)2 H = ( 1 2) Head loss across a Hydraulic Jump: H = (1 2) valid for a rectangular cross section and an orizontal cannel julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 75/ 79

76 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: conclusion Te ydraulic jump lengt can be estimated as: L HJ for 4.5 < F r 1 < 13 (Henderson, 1966) Hydraulic Jumps are usually used in ydraulic engineering as an energy dissipator (for example after a Dam). To prevent erosion in case of mobile beds, a still basin is build. Tis is te work of te ydraulic engineer to determine wen and were te Hydrualic Jumps will form and to design te adequate still basin. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 76/ 79

77 Non-Uniform Flows Rapidly Varied Flows Hydraulic jump: Numerical Application An ydraulic jump is used to determine te flow rate flowing in a rectangular cross-section cannel of widt b = 10 m wit downstream water dept 1 = 0.5 m and downstream water dept 2 = 1.5 m. 1 Compute te flow rate Q 2 Compute te values of te upstream and downstream Froude numbers. 3 Compute te power dissipated by tis ydraulic jump. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 77/ 79

78 Bibliograpy 1 Introduction Cannel types and geometries Flow in cannels Velocity, pressure and turbulent stress distributions Hydrodynamic considerations 2 Uniform flows Momentum balance and friction coefficients Discarge calculation 3 Non-Uniform Flows Gradually Varied Flows Rapidly Varied Flows 4 Bibliograpy julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 78/ 79

79 Bibliograpy Bibliograpie 1 Walter Hans Graf, M. S. Altinakar, Fluvial ydraulics: flow and transport processes in cannels of simple geometry, Wiley, 1998, 681 pages. 2 Ven Te Cow, Open Cannel Hydraulics, Blackburn Press, 2009, 650 pages. 3 Henderson, Open Cannel Hydraulics, Macmillan, 1966, 522 pages. 4 H. Canson, Hydraulics of Open Cannel Flow: An Introduction - Basic Principles, Sediment Motion, Hydraulic Modeling, Design of Hydraulic Structures (Second Edition), Butterwort Heinemann, 2004, 650 pages. julien.caucat@grenoble-inp.fr Open Cannel Hydraulic 79/ 79