Open Channel Flow - General Hydromechanics VVR090 Open Channel Flow 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 1
Water Supply Aqueduct, Pont du Gard, France Water Power Cross-section of power plant ITAIPU power plant 2
Zola dam, Aix-en-Provence Spillway, ITAIPU dam Transportation Panama Canal 3
Flow Control and Measurement Flow Phenomena Tidal bore, Hangzou, China 4
Flooding Yellow River, China 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) 5
Ancient Times Centers of early civilization around the large rivers The Nile River The Nile Delta The nilometer on the Island of Rhoda 6
Indus civilization Public bath Drainage pipe Yellow River Levee construction Sediment-laden river water 7
Roman aqueducts Aqua Claudia Construction of an aqueduct Characteristics of Aqueducts 8
Top of aqueduct Reservoar Frontinus (40-103 A.D.) Vitruvius (55 B.C. 14 A.D.) A Roman fountain Renaissance Leonardo da Vinci (1452-1519) 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. 9
Experimental Techniques (17 th century) Galileo Galilei (1564-1642) Evangelista Torricelli (1608-1647) barometer Rapid developments in mathematics Blaise Pascal (1623-1662) Isaac Newton (1642-1727) Gottfried Leibniz (1646-1716) 10
The Rise of Hydrodynamics Daniel Bernoulli (1700-1782) Experimental Hydraulics (18 th Century) Italy: Poleni, Venturi France: Pitot, Chezy, Borda England: Smeaton Pitot tube on an airplane wing 11
19 th Century Developments Main efforts: collect experimental data formulate empirical relationships derive general physical principles Split into hydraulics and hydrodynamics Hydraulics: Germany: Hagen, Weisbach France: Poiseuille, Darcy England: Manning, Froude Hydrodynamics: France: Navier, Cauchy, Poisson, Boussinesq England: Stokes, Reynolds Germany: Helmholtz, Kirchoff 12
Navier-Stokes Equations 1 μ + u + v + w = + P+ + + t x y z ρ x ρ x y z 2 2 2 u u u u p u u u 2 2 2 1 μ + u + v + w = + Q+ + + t x y z ρ y ρ x y z 2 2 2 v v v v p v v v 2 2 2 1 μ + u + v + w = + R+ + + t x y z ρ z ρ x y z 2 2 2 w w w w p w w w 2 2 2 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 (1875-1953) 13
Flow Classification I steady unsteady uniform non-uniform varied flow (= non-uniform): gradually varied rapidly varied Flow Classification II 14
Flow Classification III Laminar, transitional, and turbulent flow Characterized by Reynolds number: Re UL = ν L taken to be the hydraulic radius R=A/P Re < 500 laminar 500 < Re < 12,500 transitional 12,500 < Re turbulent 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 15
Gravity Wave I Celerity of gravity wave: c= gl (denominator in Froude number) Movement of impermeable plate Gravity Wave II Continuity equation: cy = ( y +Δy)( c Δu) (coordinate system moving with velocity c) Simplifying: Δu c= y Δy 16
Gravity Wave III Momentum equation: 1 1 2 γ γ ( +Δ ) =ρ ( Δ ) 2 2 ( ) 2 y y y cy c u c Simplifying: Δ u = Δy g c c= gy 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. 17
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. 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) 18
Definition of Channel and Flow Properties I Depth of flow (y): vertical distance from channel section to water surface y = d cos θ (d = depth of flow measured perpendicular to the channel bottom; q = slope angle of channel bottom) Small slopes: y d 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 19
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 A D = T For irregular channels: integrate and use representative values for above-discussed quantities Definition of Channel and Flow Properties IV 20
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: ut 1 T = udt T 0 1 u= uda A A Statistical Quantities Root-mean-square (rms) value of velocity fluctuation: T 1 2 rms( u') ( u' ) dt = T 0 1/2 Average kinetic energy (KE) of the turbulence per unit mass: KE 1 (( u' ) 2 ( v' ) 2 ( w' ) 2 ) mass = 2 + + Reynolds stresses: 1 uv ' ' = uvdt ' ' T T 0 21
Energy Equation Bernoulli equation (along a streamline): 2 ua H = za + dacosθ+ 2g Small values of q: 2 u H = z+ y+ 2g Fundamental Equations Conservation of mass: Q = ua Conservation of momentum: F =ρq( u u ) 2 1 Conservation of energy: 2 u H = z+ y+ 2g 22
Correction of Momentum Flux True transfer of momentum: 2 ρuda A Average transfer: ρqu Momentum correction coefficient: ρuda 2 2 A A β= = ρqu ρuda ρ 2 u A Correction of Energy Flux True transfer of energy: A 1 2 3 ρu da Average transfer: 1 2 ρqu 2 Energy correction coefficient: ρuda ρuda 3 2 A A α= = 2 3 ρqu ρu A 23
Properties of a and b equal to unity for uniform flow (otherwise greater than 1) a is more sensititve to velocity variations than b a and b used only for complex cross-sectional shapes (e.g., compound sections) Boundary Layers Consider flat surface: boundary layer depends on U, r, m, and x. Laminar boundary layer thickness (Blasius): 5x u δ= at = 0.99 Re x U Transition to turbulent boundary layer: 500,000 < Rex < 1,000,000 Turbulent boundary layer thickness: 0.37 u δ= at = 0.99 U x 0.2 Re x 24
Observations Regarding Boundary Layers I The following relationships exist: x μ ρ,,, U δ x μ ρ,,, U δ Boundary layers can grow within boundary layers (e.g., change in channel shape or roughness) Observations Regarding Boundary Layers II 25
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 ν ku ν ku ν s * 0 5 smooth s * 5 70 transition s * 70 rough Resistance Estimate Chezy equation: u = C RS u * = grs ku s g 0 5 Cν smooth ku s g 5 70 Cν transition ku s g 70 Cν rough 26