Notes 1 - Flows in pipes

Notes 1 - Flows in pipes - MECH-414 Momentum & Energy...

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1 MECH-414 Momentum & Energy Transport ANALYSIS OF FLOW IN PIPES AND OVER SURFACES 1. Introduction The Navier-Stokes equations govern the motion of a Newtonian fluid; see page 214, Fluid Mechanics Text by Fox, McDonald and Pritchard, 6 th Edition. When non- dimensionalized, the parameter Re appears in these equations. The Reynolds number Re is used to distinguish between laminar flow and turbulent flow, and accounts for viscous effects. In the first course in fluid mechanics, the effects of friction were considered only in an overall manner, using friction factor f. The objective of the present study is to understand the basis of the relationship between friction factor and Reynolds number. Some effort was also made earlier to study laminar flow in a pipe of circular cross section. We will now review that briefly and then examine the corresponding turbulent flow. Similarly, we will then study flow over flat plates. 2. Laminar and Turbulent Flow Recall: For Fully Developed Flow in a Circular Pipe friction coefficient f varies with Re.
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2 At Re 3000, the f vs. Re curve shows a sudden change in slope. Reynolds Experiment: Water flows through a transparent pipe. Far downstream, dye is injected along the pipe center. At low Re, (i.e., low flow rate), the dye follows a straight line parallel to the pipe axis. As Re increased, the dye streak becomes wavy. At still higher Re, the dye line disappears completely and the dye spreads throughout the pipe. Low Re Moderate Re High Re (laminar flow) (transitional flow) (turbulent flow) Transition occurs at the values of Re where f experiences a sudden change in slope. Transition is associated also with hydrodynamic instability . Transition depends on Re as well as other factors (disturbances) such as surface roughness, temperature, etc. Turbulent flows occur in pipes (internal flow) as well as in flows over surfaces (external flow). Turbulence occurs in many flows encountered in daily situations, e.g., atmosphere, oceans, jets, smoke stacks, etc. Turbulent flows are not truly steady. At best, the short-time average of the flow can be
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3 considered steady. Therefore, when we speak of steady flow, we imply either laminar flow, or time- averaged steady turbulent flow. 3. Shear Distribution in Circular Pipes Consider fully developed flow in a straight pipe of circular cross section of radius r o carrying a fluid of constant density ρ . p/ l = constant for fully developed pipe flow p: pressure drop over length l ( p is negative) For a cylindrical element of radius r and length l , equilibrium requires that Here τ is negative. For r = r o , 2 o o oo 2 o o 2r p2 p( r ) = (2 r ) or = = r r π τ ∆π π ττ π A A (2) so that i. e., τ varies linearly with r. 2 p ( r ) = (2 r ) . τπ A (1) τ τ r r = o o (2a)
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4 Now, friction factor f is defined as non-dimensional pressure drop per unit non-dimensional length .
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Notes 1 - Flows in pipes - MECH-414 Momentum & Energy...

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