# ch65 - Chapter 6: Viscous Flow in Ducts 6.1 Reynolds Number...

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Chapter 6: Viscous Flow in Ducts 6.1 Reynolds Number Regimes 6.2 Internal vs External Fluid Flows 6.3 Head Loss – The Friction Factor 6.4 Laminar Fully Developed Pipe Flow 6.5 Turbulence Modeling 6.6 Turbulent Pipe Flow 6.7 Four Types of Pipe Flow 6.8 Flow in Noncircular Ducts* 6.9 Minor Losses Pipe Systems* 6.10 Multiple Pipe Systems* 6.11 Diffuser Performance* 6.12 Fluid Meters* * skim

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6.5 TURBULENT MODELLING (applies to turbulent pipe flow but generally true for all turbulent flows)
“ I am an old man, and when I die and go to heaven, there are two matters on which I hope for enlightenment: one is quantum electrodynamics and the other is the turbulent motion of fluids. About the former I am rather optimistic” Sir Horace Lamb, 1932

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Shaft of water issuing from a square hole into a pool, forming bubbles and eddies (1509 by Leonardo da Vinci) , drawn while working on a hydraulic project in Milan
Smoke Plume Laminar Transition Turbulent

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Laminar Flow Transitional Flow Turbulent Flow
MYO Pipe Flow hot-wire anemometer

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MYO u(r) = u avg (r) + u(r)’
Typical u x fluctuations at the center of narrow pipe, u avg = 12 m/s Turbulence Intensity = u’ rms /u avg % ~ 0.1 for atmosphere and rivers ~ 0.01 for typical wind tunnel

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Turbulence Phenomena ~ Davies Prandtl 1924
° ° ° ° ° ° ° ° ° ° ° ° ° U avg ~ 2 m/s y = 0.15 and 1.0 radius ----------- U avg ~ 6 m/s Turbulence Phenomena - Davies “eddy frequency”

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Mixing Layer Bottom picture is twice the Re number of the top. An Album of Fluid Motion –Van Dyke
LAMINAR FLOW (& incompressible) τ xy = μ ( u/ y + v/ x); τ xx = 2 μ ( u/ x); … (4.37) (additional term for turbulent shear stress)

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τ ~ μ du/dy + ρ u’v’ for boundary layer and pipe/duct flow Osborne Reynolds, 1895