Watershed Sciences 6900 FLUVIAL HYDRAULICS & ECOHYDRAULICS
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1 Watershed Sciences 6900 FLUVIAL HYDRAULICS & ECOHYDRAULICS WEEK Four Lecture 6 VELOCITY DISTRIBUTION Joe Wheaton FOR TODAY, YOU SHOULD HAVE READ 1
2 LET S GET ON WITH IT TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) 2
3 VELOCITY DISTRIBUTIONS Most of discussion limited to vertical velocity profiles WHY DO WE CARE? Basis for formulating expressions of flow resistance 3
4 WHAT IS A VELOCITY PROFILE? A mathematical function for velocity as a function of depth (or height above bed):? 1 1 Is this a velocity profile? - Local average vertical velocity U - Average cross section velocity - Local Depth A - Cross Sectional Area w - Water surface width HOW DO I MEASURE A VELOCITY PROFILE? You take a bunch of points Say every few centimeters? 4
5 BOUNDARY SHEAR STRESS DISTRIBUTION Depth-Slope Product: Boundary Shear Stress: Linear shear stress relationship for both laminar & turbulent flow: TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) 5
6 LAMINAR FLOW - PROFILE 2 WHAT S HE TELLING US? Average vertical velocity 6
7 TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) SO IN TURBULENT FLOWS. It s most commonly treated with the Law of the Wall (i.e. Prandtl von Karman) For practical purposes it is very difficult to measure the viscous sublayer in natural streams 7
8 WHAT DOES IT MEAN? In log-log space, a straight line means? The profile is logarimthmic TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) 8
9 DERIVATON OF Prandtl-von Kármán (P-vK) Starting with equations 3.40a & 5.6 (in sections you will have read but we did not cover Bunch of math magic 1 Where you know everything except In practice = 0.4 Can range from 0.2 to 0.4 SHEAR VELOCITY (FRICTION VELOCITY) 1 1 9
10 RELATIVE VELOCITY PROFILE (DIMENSIONLESS) ROUGNESS REYNOLDS NUMBER Recall: = Let thickness of the sublayer be the characteristic length Then, the boundary Reynolds number (or roughness Re) is: Experiments have shown that : Smooth: > 5 Transitional: 5 70 Rough: >70 10
11 WHAT IT ALL BOILS DOWN TO AVERAGE VERTICAL VELOCITY (TURBULENT FLOW) Starting with a definition of the average vertical velocity, derived by integration P-vK law over its range of validity (i.e. above the top of the buffer zone:
12 PRACTICAL UTILITY OF P-VK LAW The six-tenths rule If velocity profile follows P-vK (i.e. logarithmic) then I can just take one measurement! TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) 12
13 THE VELOCITY-DEFECT LAW P-vK states is constant through flow boundary (wall); this is known as the law of the wall Far from the bed the velocity gradient ( ) does not depend on viscosity ( ) or bed roughness, but only on distance from bed In this region, the velocity profile is the difference between velocity at the surface ( ) and the velocity at an arbitrary level ( ) and is only a function of VELOCITY-DEFECT REARRANGED & APPLIED The function is determined by experiment Daily & Harleman (1966), for > 0.15: 3.74 But: 13
14 TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) POWER-LAWS AS AN ALTERNATIVE What is a power law? A polynomial relationship that exhibits scale invariance and of the form: Can t do it for, but can do it for. When integrated over depth 1 14
15 TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) A BORING RECTANGULAR FLUME 15
16 SOMETHING MORE INTERESTING HOW GOOD IS Pv-K? 16
17 SOME EMPIRICAL DATA TODAY S PLAN VELOCITY DISTRIBUTIONS I. Velocity Profile in Laminar Flows II. Velocity Profile in Turbulent Flows I. Prandtl von Karman Velocity Profile II. Velocity Defect Law III. Power Law Profiles III. Observed Velocity Distributions From Chanson (2004) 17
18 READING/LAB I ll post this tonight (Chapter 6 Uniform Flow & Flow Resistance) This week s lab more time to work through Chapter 6 Next week, get back to something more fun in lab flume experiment 18
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