lecture17 - 2.20 - Marine Hydrodynamics, Spring 2005...

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Lecture 17 - Marine Hydrodynamics Lecture 17 4.6 Laminar Boundary Layers U o L u, v viscous flow δ y x U potential flow 4.6.1 Assumptions 2D flow: w, 0 and u ( x, y ) ,v ( x, y ) ,p ( x, y ) ,U ( x, y ). ∂z Steady flow: 0. ∂t For δ<< L , use local (body) coordinates x, y , with x tangential to the body and y normal to the body. u tangential and v normal to the body, viscous flow velocities (used inside the boundary layer). U, V potential flow velocities (used outside the boundary layer). 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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± ² ± ² 4.6.2 Governing Equations Continuity ∂u ∂v + = 0 (1) ∂x ∂y Navier-Stokes: 1 ∂p 2 u 2 u u + v = + ν + (2) ρ∂x 2 2 1 2 v 2 v u + v = + ν + (3) ρ∂y 2 2 4.6.3 Boundary Conditions KBC Inside the boundary layer: No-slip u ( x, y =0)=0 No-flux v ( x, y Outside the boundary layer the velocity has to match the P-Flow solution. Let y ± y/δ , y y/L , and x x/L . Outside the boundary layer y ± →∞ but y 0. We can write for the tangential and normal velocities u ( x ,y ± )= U ( x 0) u ( x ± U ( x , 0), and v ( x ± V ( x 0) v ( x ± V ( x , 0) = 0 No-flux P-Flow In short: u ( x, y ± U ( x, 0) v ( x, y ± )=0 DBC As y ± , the pressure has to match the P-Flow solution. The x -momentum equation at y = 0 gives ∂U 1 dp 2 U d p U U + V = + ν = ρU ρdx 2 dx 0 0 2
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±²³´ ± ²³ ´ µ · 4.6.4 Boundary Layer Approximation Assume that R e L >> 1, then ( u, v ) is confined to a thin layer of thickness δ ( x ) << L . For flows within this boundary layer, the appropriate order-of-magnitude scaling / normalization is: Variable Scale Normalization u U u = U u x L x = Lx y δ y = δy v V =? v = V v Non-dimensionalize the continuity, Equation (1), to relate V to U · · · U ∂u V ∂v δ + =0= ⇒V =O U L∂ x δ∂ y L Non-dimensionalize the x -momentum, Equation (2), to compare δ with L U 2 · UV · 1 ∂p ν U δ 2 2 u · 2 u · u + v = + + L x δ y ρ∂x δ 2 L 2 ∂x 2 ∂y 2 O ( U 2 /L ) ignore The inertial effects are of comparable magnitude to the viscous effects when: U 2 ν U δ ν 1 = = << 1 L δ 2 L U L R e L The pressure gradient must be of comparable magnitude to the inertial effects U 2 = O ρ L 3
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± ² ± ² ± ² Non-dimensionalize the y -momentum, Equation (3), to compare ∂p to ∂y ∂x ± ² ± ² ± ² ± ² UV ∂v V 2 1 ν V 2 v ν V 2 v u + v = + + L x δ y ρ y L 2 2 δ 2 2 ³´µ¶ ³´µ¶ ³´µ¶ ³´µ¶ O ( U 2 δ O ( U 2 δ O ( U 2 δ 3 U 2 δ L L ) L L ) L L 3 ) O ( L L ) The pressure gradient must be of comparable magnitude to the inertial effects U 2 δ = O ρ L L Comparing the magnitude of to we observe U 2 δ U 2 = O ρ while = O ρ = L << = 0 = p = p ( x ) Note: - From continuity it was shown that V / U∼ O ( δ/L ) v<< u , inside the boundary layer.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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lecture17 - 2.20 - Marine Hydrodynamics, Spring 2005...

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