lecture12

# lecture12 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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Lecture 12 - Marine Hydrodynamics Lecture 12 3.14 Lifting Surfaces 3.14.1 2D Symmetric Streamlined Body No separation, even for large Reynolds numbers. U stream line Viscous eﬀects only in a thin boundary layer. Small Drag (only skin friction). No Lift. 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

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3.14.2 Asymmetric Body (a) Angle of attack α , chord line U α (b) or camber η ( x ), chord line mean camber line U (c) or both amount of camber U α angle of attack mean camber line chord line Lift to U ± and Drag ± to U ± 2
3.15 Potential Flow and Kutta Condition From the P-Flow solution for ﬂow past a body we obtain P-Flow solution, inﬁnite velocity at trailing edge. Note that (a) the solution is not unique - we can always superimpose a circulatory ﬂow without violating the boundary conditions, and (b) the velocity at the trailing edge . We must therefore, impose the Kutta condition, which states that the ‘ﬂow leaves tangentially the trailing edge, i.e., the velocity at the trailing edge is ﬁnite’ . To satisfy the Kutta condition we need to add circulation. Circulatory ﬂow only. Superimposing the P-Flow solution plus circulatory ﬂow, we obtain Figure 1: P-Flow solution plus circulatory ﬂow. 3

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3.15.1 Why Kutta condition ? Consider a control volume as illustrated below. At t = 0, the foil is at rest (top control volume). It starts moving impulsively with speed U (middle control volume). At t =0 + , a starting vortex is created due to ﬂow separation at the trailing edge. As the foil moves, viscous eﬀects streamline the ﬂow at the trailing edge (no separation for later t ), and the starting vortex is left in the wake (bottom control volume). t = 0 + t = 0 for later t Kelvin’s theorem: After a while the Γ S Γ Γ Γ Γ S S S S U U no Γ starting vortex left in wake Γ = 0 starting vortex due to separation (a real fluid effect, no infinite vel of potetial flow) d Γ Γ = 0 for t 0i fΓ( t =0)=0 dt in the wake is far behind and we recover Figure 1.
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lecture12 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

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