lecture17

# lecture17 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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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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4.6.2 Governing Equations Continuity ∂u ∂v + = 0 (1) ∂x ∂y Navier-Stokes: ∂u ∂u 1 ∂p 2 u 2 u u + v = + ν + (2) ∂x ∂y ρ ∂x ∂x 2 ∂y 2 ∂v ∂v 1 ∂p 2 v 2 v u + v = + ν + (3) ∂x ∂y ρ ∂y ∂x 2 ∂y 2 4.6.3 Boundary Conditions KBC Inside the boundary layer: No-slip u ( x, y = 0) = 0 No-ﬂux v ( x, y = 0) = 0 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 , y 0) u ( x , y → ∞ ) = U ( x , 0), and v ( x , y → ∞ ) = V ( x , y 0) v ( x , y → ∞ ) = 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 ∂U 1 dp 2 U dp ∂U U + V = + ν = ρU ∂x ∂y ρ dx ∂y 2 dx ∂x 0 0 2
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 ∂u UV ∂u 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 ∂p must be of comparable magnitude to the inertial effects ∂x ∂p U 2 = O ρ ∂x L 3

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Non-dimensionalize the y -momentum, Equation (3), to compare ∂p to ∂p ∂y ∂x UV ∂v V 2 ∂v 1 ∂p ν V 2 v ν V 2 v u + v = + + L ∂x δ ∂y ρ ∂y L 2 ∂x 2 δ 2 ∂y 2 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 ∂p must be of comparable magnitude to the inertial effects ∂y ∂p U 2 δ = O ρ ∂y L L Comparing the magnitude of ∂p to ∂p
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