Ch09_2_PB - MAE130B Ch 09(2 Prandtl/Blasius Boundary Layer...

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MAE130B – Ch 09 (2) Prandtl/Blasius Boundary Layer Solution Changzheng Huang
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Ch 09 (2) Changzheng Huang 2 Prandtl/Blasius Boundary Layer Solution 1. Derivation of boundary layer equations 2. Seeking similarity solution 3. Boundary layer thickness
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Ch 09 (2) Changzheng Huang 3 1. Derivation of boundary layer equations x x= 0 Laminar boundary layer Turbulent boundary layer Inviscid outer flow Viscous boundary layer Incompressible flow past a thin flat plate. Note the distortion of a fluid particle as it flows past within the boundary layer. U UU U Re Ux ρ µ = 56 Re 2 10 3 10 xcr = ×× y P
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Ch 09 (2) Changzheng Huang 4 1. Derivation of boundary layer equations 2D Navier-Stokes equations for incompressible and steady laminar flow past a flat plate, 22 0 1 1 uv xy uu p u u x x y vv p v v y x y ν ρ ∂∂ += + + ⎛⎞ + + ⎜⎟ ⎝⎠ U U P x y Boundary layer Outer flow x= 0 δ ( x )
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Ch 09 (2) Changzheng Huang 5 1. Derivation of boundary layer equations Order of magnitude analysis within the boundary layer U x y Outer flow L δ 0 Denote L as a distance from the Leading edge in flow direction. Denote δ 0 as the average boundary Layer thickness within this range. Introduce boundary layer assumptions, (1) Boundary layer is very thin. (2) Velocity changes much faster in y-direction than in x-direction. 0 L δ ± 22 xy ± ±
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Ch 09 (2) Changzheng Huang 6 1. Derivation of boundary layer equations Boundary layer assumptions, (1) Boundary layer is very thin. (2) Velocity changes much faster in y-direction than in x-direction. 0 L δ ± 22 xy ± ± 0 1 1 uv uu p u u x x y vv p v v y x y ν ρ ∂∂ += + + ⎛⎞ + + ⎜⎟ ⎝⎠ 2 2 1 p u x y x y + With assumption (2)
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Ch 09 (2) Changzheng Huang 7 1. Derivation of boundary layer equations 2 2 0 1 uv xy uu p u ν ρ ∂∂ += + Now consider the continuity equation and x-momentum equation, v u y These two terms have same order. So keep them in the analysis.
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Ch 09 (2) Changzheng Huang 8 1. Derivation of boundary layer equations 2 2 0 1 uv xy uu p u ν ρ ∂∂ += + Note that to transition from the boundary layer flow to outer flow, the inertia term must balance the viscous term. This can be used to estimate the order of the boundary layer thickness and pressure, 2 U L 2 0 U δ Same order 1/2 0 Re LU L = 2 2 0 1 p u
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This note was uploaded on 03/10/2009 for the course MAE 130B taught by Professor Huang during the Spring '09 term at UC Irvine.

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Ch09_2_PB - MAE130B Ch 09(2 Prandtl/Blasius Boundary Layer...

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