Chapter9 - 57:020 Mechanics of Fluids and Transport...

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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2010 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes, turbomachinery, open channel/river, which are bounded by walls or fluid interfaces: Chapter 8. 2. External flows such as flow around vehicles and structures, which are characterized by unbounded or partially bounded domains and flow field decomposition into viscous and inviscid regions: Chapter 9. a. Boundary layer flow: high Reynolds number flow around streamlines bodies without flow separation. b. Bluff body flow: flow around bluff bodies with flow separation.
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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2010 2 3. Free Shear flows such as jets, wakes, and mixing layers, which are also characterized by absence of walls and development and spreading in an unbounded or partially bounded ambient domain: advanced topic, which also uses boundary layer theory. Basic Considerations Drag is decomposed into form and skin-friction contributions: ( ) τ + ρ = S w S 2 D dA i ˆ t dA i ˆ n p p A V 2 1 1 C C Dp C f
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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2010 3 ( ) ρ = dA j ˆ n p p A V 2 1 1 C S 2 L c t << 1 C f > > C Dp streamlined body c t 1 C Dp > > C f bluff body Streamlining: One way to reduce the drag reduce the flow separation reduce the pressure drag increase the surface area increase the friction drag Trade-off relationship between pressure drag and friction drag Trade-off relationship between pressure drag and friction drag Benefit of streamlining : reducing vibration and noise
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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2010 4 Qualitative Description of the Boundary Layer Flow-field regions for high Re flow about slender bodies:
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57:020 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 2010 5 τ w = shear stress τ w rate of strain (velocity gradient) = 0 y y u = µ large near the surface where fluid undergoes large changes to satisfy the no-slip condition Boundary layer theory and equations are a simplified form of the complete NS equations and provides τ w as well as a means of estimating C form . Formally, boundary-layer theory represents the asymptotic form of the Navier-Stokes equations for high Re flow about slender bodies. The NS equations are 2 nd order nonlinear PDE and their solutions represent a formidable challenge. Thus, simplified forms have proven to be very useful. Near the turn of the last century (1904), Prandtl put forth boundary-layer theory, which resolved D’Alembert’s paradox: for inviscid flow drag is zero. The theory is restricted to unseparated flow. The boundary-layer equations are singular at separation, and thus, provide no information at or beyond separation. However, the requirements of the theory are met in many practical situations and the theory has many times over proven to be invaluable to modern engineering.
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This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

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Chapter9 - 57:020 Mechanics of Fluids and Transport...

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