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lecture9

# lecture9 - 2.20 Marine Hydrodynamics Spring 2005 Lecture 9...

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( ) Lecture 9 - Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full Navier-Stokes equations (unsteady, viscous momentum equations) to deduce the vorticity equation and study some additional properties of vorticity. In paragraph 3.6 we introduce the concept of potential ﬂow and velocity potential. We formulate the governing equations and boundary conditions for potential ﬂow and finally introduce the stream function. 3.5 Vorticity Equation Return to viscous incompressible ﬂow. The Navier-Stokes equations in vector form ∂v + v · ∇ v = −∇ p + gy + ν 2 v ∂t ρ By taking the curl of the Navier-Stokes equations we obtain the vorticity equation. In detail and taking into account ∇ × u ω we have ∇ × (Navier-Stokes) → ∇ × ∂v + ∇ × ( v · ∇ v ) = −∇ × ∇ p + gy + ∇ × ( ν 2 v ) ∂t ρ The first term on the left side, for fixed reference frames, becomes ∂v ∂ω ∇ × = ( ∇ × v ) = ∂t ∂t ∂t In the same manner the last term on the right side becomes ∇ × ν 2 v = ν 2 ω Applying the identity ∇ × ∇ · scalar = 0 the pressure term vanishes, provided that the density is uniform ∇ × ( p + gy ) = 0 ρ 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

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The inertia term v · ∇ v , as shown in Lecture 8, § 3.4, can be rewritten as 1 v 2 v · ∇ v = ( v · v ) v × ( ∇ × v ) = v × ω where v 2 ≡ | v | 2 = v · v 2 2 and then the second term on the left side can be rewritten as 2 v ∇ × ( v · ∇ ) v = ∇ × ∇ − ∇ × ( v × ω ) = ∇ × ( ω × v ) 2 = ( v · ∇ ) ω ( ω · ∇ ) v + ω ( ∇ · v ) + v ( ∇ · ω ) =0 =0 since incompressible ∇· ( ∇× v )=0 ﬂuid Putting everything together, we obtain the vorticity equation = ( ω · ∇ ) v + ν 2 ω Dt Comments-results obtained from the vorticity equation Kelvin’s Theorem revisited - from vorticity equation: If ν 0 , then = ( ω · ∇ ) v , so if ω 0 everywhere at one time, ω 0 always. Dt ν can be thought of as diffusivity of vorticity (and momentum), i.e., ω once generated (on boundaries only) will spread/diffuse in space if ν is present. ω v ω v Dv v v = υ∇ 2 v v + ... D ... ω v = υ∇ 2 ω v + ... Dt Dt 2
∂T Diffusion of vorticity is analogous to the heat equation: = K 2 T , where K is the ∂t heat diffusivity. Numerical example for ν 1 mm 2 /s. For diffusion time t = 1 second, diffusion ( ) distance L O νt O ( mm ). For diffusion distance L = 1cm, the necessary diffusion time is t O ( L 2 ) O (10)sec.

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