Open Channel Flow - General. Hydromechanics VVR090

Transcription

1 Open Channel Flow - General Hydromechanics VVR090 ppt by Magnus Larson; revised by Rolf L Jan 2014, Feb 2015

2 SYNOPSIS 1. Introduction and Applications 2. The History of Open Channel Flow 3. Flow Classification 4. Gravity Wave 5. Channel Types 6. Definitions of Channel and Flow Properties 7. Governing Equations

3 1. Introduction and Applications Open channel: a conduit for flow which has a free surface Free surface: interface between two fluids of different density Characteristics of open channel flow: pressure constant along water surface gravity drives the motion pressure is approximately hydrostatic flow is turbulent and unaffected by surface tension Note, dual function of free surface and the extra degree of freedom Example: Constriction. Compare closed conduit flow with open channel

4 Water Supply Aqueduct, Pont du Gard, France

5 Water Power Cross-section of power plant ITAIPU power plant (Brazil/Paraguay)

6 Zola dam, Aix-en-Provence Spillway, ITAIPU dam

7 Transportation Panama Canal

8 Flow Control and Measurement

9 Flow Phenomena Tidal bore, Hangzou, China

10 Flooding Yellow River, China

11 2. The History of Open Channel Flow Main periods of development: ancient times (river cultures) roman times (aqueducts) renaissance (first theory) 17th century (experimental techniques + theory) 18th century (rise of hydrodynamics) 19th century (split between hydraulics and hydrodynamics) 20th century (boundary layer theory)

12 Ancient Times Centers of early civilization around the large rivers

13 The Nile River The Nile Delta The nilometer on the Island of Rhoda

14 Indus civilization Public bath Drainage pipe

15 Yellow River Levee construction Sediment-laden river water

16 Roman aqueducts Aqua Claudia Construction of an aqueduct

17 Characteristics of Aqueducts

18 Top of aqueduct Reservoar A Roman fountain Frontinus ( A.D.) Vitruvius (55 B.C. 14 A.D.)

19 Renaissance Leonardo da Vinci ( ) Water flow When you put together the science of the motion of water, remember to include in each proposition its application and use, in order that these sciences may not be useless.

20 Experimental Techniques (17 th century) Galileo Galilei ( ) Evangelista Torricelli ( ) barometer

21 Rapid developments in mathematics Blaise Pascal ( ) Isaac Newton ( ) Gottfried Leibniz ( )

22 Daniel Bernoulli ( ) The Rise of Hydrodynamics

23 Experimental Hydraulics (18 th Century) Italy: Poleni, Venturi France: Pitot, Chezy, Borda England: Smeaton Pitot tube on an airplane wing

24 19 th Century Developments Main efforts: collect experimental data formulate empirical relationships derive general physical principles Split into hydraulics and hydrodynamics

25 Hydraulics: Germany: Hagen, Weisbach France: Poiseuille, Darcy England: Manning, Froude Hydrodynamics: France: Navier, Cauchy, Poisson, Boussinesq England: Stokes, Reynolds Germany: Helmholtz, Kirchoff

26 Navier-Stokes Equations u v w P t x y z x x y z u u u u 1 p u u u u v w Q t x y z y x y z v v v v 1 p v v v u v w R t x y z z x y z w w w w 1 p w w w 2 2 2

27 Increased gap between hydraulics and hydrodynamics Bridged by the introduction of boundary layer theory by Ludwig Prandtl, the father of modern fluid mechanics. Ludwig Prandtl ( )

28 3 Flow Classification I steady unsteady uniform non-uniform varied flow (= non-uniform): gradually varied rapidly varied

29 Flow Classification II

30 Flow Classification III Laminar, transitional, and turbulent flow Characterized by Reynolds number: Re UL L taken to be the hydraulic radius R=A/P Re < 500 laminar 500 < Re < 12,500 transitional 12,500 < Re turbulent

31 Flow Classification IV homogeneous stratified flow depends on the density variation subcritical supercritical flow characterized by the Froude number Fr U gl L taken to be the hydraulic depth D=A/T Fr < 1 subcritical flow Fr = 1 critical flow Fr > 1 supercritical flow

32 4 Gravity Wave I Celerity of gravity wave: c gl (denominator in Froude number) Movement of impermeable plate

33 Gravity Wave II Continuity equation: cy y yc u (coordinate system moving with velocity c) Simplifying: u c y y

34 Gravity Wave III Momentum equation: y y y cy c u c Simplifying: u y g c c gy

35 Gravity Wave IV Interpretation: 1. Subcritical flow (Fr < 1): Velocity of flow is less than the celerity of a gravity wave. Gravity wave can propagate upstream. Upstream areas in hydraulic communication with downstream areas. 2. Supercritical flow (Fr > 1): Velocity of flow is greater than the celerity of a gravity wave. Gravity wave cannot propagate upstream. Upstream areas not in hydraulic communication with downstream areas.

36 5. Channel Types Natural channels: developed by natural processes (e.g., creeks, small and large rivers, estuaries) Artificial channels: channels developed by human efforts (e.g., navigation channels, power and irrigation channels, drainage ditches) Easier to treat artificial channels.

37 Artificial Channels 1. Prismatic (constant shape and bottom slope) 2. Canal (long channel of mild slope) 3. Flume (channel built above ground) 4. Chute and drop (channel with a steep slope) 5. Culvert (pipe flowing only partially full)

38 6. Definitions of Channel and Flow Properties I Depth of flow (y): vertical distance from channel section to water surface y d cos (d = depth of flow measured perpendicular to the channel bottom; = slope angle of channel bottom) Small slopes: y d

39 Definition of Channel and Flow Properties II Stage: elevation of the water surface relative to a datum Top width (T): width of channel section at water surface Flow area (A): cross-sectional area of the flow taken perpendicular to the flow direction Wetted perimeter (P): length of the line which is the interface between the fluid and the channel boundary

40 Definition of Channel and Flow Properties III Hydraulic radius (R): ratio of flow area to wetted perimeter R A P Hydraulic depth (D): ratio of flow area to top width D A T For irregular channels: integrate and use representative values for above-discussed quantities

41 Definition of Channel and Flow Properties IV

42 7. Governing Equations Flow is turbulent in situation of practical importance (Re > 12,500) => Laminar flow is not discussed. Description of turbulent flow: u = u + u v = v + v w = w + w Average in time: Average in space: u T = 1 T න 0 T u dt u = 1 A න A u da

43 Statistical Quantities Root-mean-square (rms) value of velocity fluctuation: rms u = 1 T න 0 T u 2 dt 1/2 Average kinetic energy (KE) of the turbulence per unit mass: KE 1 mass = ( u' ) 2 ( v' ) 2 ( w' ) 2 ) Reynolds stresses: u v = 1 T න 0 T u v dt

44 Energy Equation Bernoulli equation (along a streamline): H z d cos A A 2 ua 2g Small values of : H z y 2 u 2g

45 Fundamental Equations Conservation of mass: Q ua Conservation of momentum: F Q( u u ) 2 1 Conservation of energy: H z y 2 u 2g

46 Correction of Momentum Flux True transfer of momentum: 2 u da A Transfer based on average velocity: Qu Momentum correction coefficient: A u da 2 2 Qu A u da 2 u A

47 Correction of Energy Flux True transfer of energy: A u da Transfer based on average velocity: 1 2 Qu 2 Energy correction coefficient: A u da Qu u da 3 2 A 2 3 u A

48 Properties of and equal to unity for flow with uniform velocity distribution (otherwise greater than 1) is more sensititve to velocity variations than and used only for complex cross-sectional shapes (e.g., compound sections)

49 Boundary Layers Consider flat surface: boundary layer depends on U, ρ,, and x. Laminar boundary layer thickness (Blasius): 5x u at 0.99 Re x U Transition to turbulent boundary layer: 500,000 Re 1,000,000 x Turbulent boundary layer thickness: 0.37 x 0.2 Re x u at 0.99 U

50 Observations Regarding Boundary Layers I The following relationships exist: x x,,, U,,, U Boundary layers can grow within boundary layers (e.g., change in channel shape or roughness)

51 Observations Regarding Boundary Layers II

52 Observations Regarding Boundary Layers III Boundary layers classified as hydraulically smooth or rough. Hydraulically smooth: laminar sublayer cover the roughness elements Hydraulically rough: roughness elements project through the laminar sublayer ku s * 0 5 smooth ku s * 5 70 transition ku s * 70 rough

53 Resistance Estimate Chezy equation: u C RS u * grs ksu g 0 5 C smooth ksu g 5 70 C transition ksu g 70 C rough