Lecture 17 Turbulent Flow - Thermal and Fluids Engineering...

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Thermal and Fluids Engineering I Prof. Deborah A. Kaminski Lecture 17 – Turbulent Flow Reynolds Number For fully-developed laminar flow in a horizontal pipe: 2 8 m L P R µ ∆= V The pressure drop, P , depends on ,, , a n d . m LR In turbulent flow, P also depends on these four parameters, but the relationship is more complex. To reduce the number of variables, we non-dimensionalize. 2 2 83 2 2 mm LL P D D = ⎛⎞ ⎜⎟ ⎝⎠ To find a representative pressure, consider Bernoulli’s equation along a streamline in stagnation flow: 22 11 2 2 12 PP gz gz ρρ ++= ++ VV 2 2 2 P P ρ += Lecture 17 Page 1
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Thermal and Fluids Engineering I Prof. Deborah A. Kaminski 2 1 12 2 P P ρ + = V static pressure + dynamic pressure = stagnation pressure We now use dynamic pressure to define a non-dimensional pressure as: * 2 1 2 m P P ∆= Also define a non-dimensional length * L L D = ( ) * *2 2 32 1 2 m m DL P D µ ⎛⎞ ⎜⎟ ⎝⎠ * * 64 m P LD = Define the Reynolds number as char L Re = where L char is a characteristic length for the geometry under consideration. In a tube flow, L char is D . * * 64 D P LR e = laminar, fully developed This is a non-dimensional pressure drop per unit length of pipe. It depends only on the Reynolds number. By experiment in pipe flow: Re < 2100 laminar Lecture 17 Page 2
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Thermal and Fluids Engineering I Prof. Deborah A. Kaminski 2100 < Re < 4000 transitional 4000 < Re turbulent Friction Factor Define the Darcy friction factor as * * P f L = Since * 2 1 2 m P P ρ ∆= V and * L L D = 2 1 2 m P f L D = ⎛⎞ ⎜⎟ ⎝⎠ 2 2 m L Pf D laminar or turbulent
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This note was uploaded on 04/06/2009 for the course ENGR 2250 taught by Professor Borca-tasciuc during the Spring '08 term at Rensselaer Polytechnic Institute.

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Lecture 17 Turbulent Flow - Thermal and Fluids Engineering...

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