Notes 6_Boundary Layer Equations

Notes 6_Boundary - Boundary-Layer Thickness This is a very thin layer in the immediate neighborhood of the body where the fluid viscosity exerts an

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1 Boundary-Layer Thickness, δ This is a very thin layer in the immediate neighborhood of the body where the fluid viscosity exerts an essential influence, in the sense that the shearing stresses are important here. Displacement Thickness, δ * This is a more meaningful measure for boundary-layer thickness. It is the distance by which the external flow streamlines are displaced outwards as a consequence of the formation of the boundary layer, i.e., due to the decrease in velocity in the boundary layer. Decrease in volume flow rate is U o δ * which is also equal to and replaced by: Momentum Thickness, δ m The loss of momentum in the boundary layer is expressed as: Note: The upper limit in the above integrals can be replaced by , since the contribution to the integrand, and hence to the resulting integral, from the portion beyond δ is negligible. e U 99 . o u | y = = δ ∫∫ δδ δ = δ = δ = δ 00 o o o 0 o o dy U u 1 * dy U ) u U ( * dy ) u U ( * U δ δ = δ ρ = δ ρ 0 2 o 2 o m 0 o m 2 o dy U u U u dy ) u U ( u U
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2 Given : Velocity profiles for laminar boundary layer. Find : * δ δ for each profile Definition : 00 uu *1d y1d y UU ∞δ ⎛⎞ δ= ⎜⎟ ⎝⎠ ∫∫ ± Solution : () 1 *1 u 1d y 1 u d y U δ δ =−= δδ ²² or 1 0 * 1ud δ =− η δ ² where yu y, u U η δ ³± ³ (1) Laminar: u y == η 1 2 1 0 0 1 d 0.500 22 ⎡⎤ δη η η = η = = ⎢⎥ δ ⎣⎦ (2) Parabolic: 2 u2 −η ² 1 3 1 0 0 12 d 0 . 3 3 3 33 η + η η = η η + = = δ (3) Cubic: 3 31 u ² 1 1 32 4 0 0 *3 1 3 1 3 1 d 0.375 4 8 8 δ η + η η = η η + η = = δ (4) Sinusoidal: us i n 2 π ² 1 1 0 0 *2 2 1 sin d cos 1 0.363 δπ η π η
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This note was uploaded on 10/13/2010 for the course MECH 414 taught by Professor Ghia during the Winter '06 term at University of Cincinnati.

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Notes 6_Boundary - Boundary-Layer Thickness This is a very thin layer in the immediate neighborhood of the body where the fluid viscosity exerts an

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