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Unformatted text preview: Lecture 16 D (drag) x F (lift) y F U Marine Hydrodynamics Lecture 16 4.1.4 Vortex Shedding and Vortex Induced Vibrations Consider a steady ﬂow U o over a bluff body with diameter D . We would expect the average forces to be: F y F x t F However, the measured oscillatory forces are: Average Average t F F x F y • The measured drag F x is found to oscillate about a nonzero mean value with frequency 2 f . • The measured lift F y is found to oscillate about a zero mean value with frequency f . • f = ω /2 π is the frequency of vortex shedding or Strouhal frequency. 1 2.20  Marine Hydrodynamics, Spring 2005 2.20 U o D Von Karman Street F y F x Reason: Flow separation leads to vortex shedding. The vortices are shed in a staggered array, within an unsteady nonsymmetric wake called von Karman Street . The frequency of vortex shedding is the Strouhal frequency and is a function of U o , D , and ν . i) Strouhal Number We define the (dimensionless) Strouhal number S ≡ Strouhal frequency f D U . The Strouhal number S has a regime dependence on the R e number S = S ( R e ). 10 5 10 6 10 7 0.22 0.3 S(Re) Re For a cylinder : Laminar ﬂow S ∼ . 22 • Turbulent ﬂow S ∼ . 3 • ii) Drag and Lift The drag and lift coeﬃcients C D and C L are functions of the correlation length . For ‘ ∞ ’ correlation length: If the cylinder is fixed, C L ∼ O (1) comparable to C D . • • If the cylinder is free to move, as the Strouhal frequency f S approaches one of the cylinder’s natural frequencies f n , ‘lockin’ occurs. Therefore, if one natural frequency is close to the Strouhal Frequency f n ∼ f S , we have large amplitude motions Vortex Induced Vibration (VIV). ⇒ 2 L b D U o 4.2 Drag on a Very Streamlined Body UL R e L ≡ ν C f ≡ D 1 2 ρU 2 ( Lb ) S = wetted area one side of plate C f = C f ( R e L , L/b ) C f 0.01 Laminar 10 5 Re 0.001 Turbulent 10 6 = ν ∂u Unlike a bluff body, C f is a strong function of R e L since D is proportional to ν τ ∂y . See an example of C f versus R e L for a ﬂat plate in the figure below. Skin friction coeﬃcient as a function of the R e for a ﬂat plate • R e L depends on plate smoothness, ambient turbulence, . . . In general, C f ’s are much smaller than C D ’s ( C f /C D ∼ O (0 . 1) to O (0 . 01)). Therefore, • designing streamlined bodies allows minimal separation and smaller form drag at the expense of friction drag. • In general, for streamlined bodies C Total Drag is a combination of C D ( R e ) and C f ( R e L ), 1 and the total drag is D = 2 ρU 2 C D S + C f A w , where C D has a regime frontal area wetted area dependence on R e and C f is a continuous function R e L . 3 4.3 Known Solutions of the NavierStokes Equations 4.3.1 Boundary Value Problem NavierStokes’: • ∂v 1 1 ∂t + ( v · ) v = − ρ p + ν 2 v + ρ f Conservation of mass: • · v = 0 • Boundary conditions on solid boundaries “noslip”: v = U Equations very diﬃcult to solve, analytic solution only for a few very special cases (usually when v = 0...) v · 4 4.3.24....
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 Spring '05
 DickK.P.Yue
 Fluid Dynamics, UO, skin friction, boundary layer thickness, dp dx

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