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Unformatted text preview: ( ) 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 ow and velocity potential. We formulate the governing equations and boundary conditions for potential ow and finally introduce the stream function. 3.5 Vorticity Equation Return to viscous incompressible ow. The Navier-Stokes equations in vector form v + v v = p + gy + 2 v t By taking the curl of the Navier-Stokes equations we obtain the vorticity equation. In detail and taking into account u we have (Navier-Stokes) v + ( v v ) = p + gy + ( 2 v ) t The first term on the left side, for fixed reference frames, becomes v = ( v ) = t t t In the same manner the last term on the right side becomes 2 v = 2 Applying the identity scalar = the pressure term vanishes, provided that the density is uniform ( p + gy ) = 0 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20 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 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 ) = ( v ) 2 = ( v ) ( ) v + ( v ) + v ( ) =0 =0 since incompressible ( v )=0 uid Putting everything together, we obtain the vorticity equation D = ( ) v + 2 Dt Comments-results obtained from the vorticity equation Kelvins Theorem revisited- from vorticity equation: If , then D = ( ) v , so if everywhere at one time, always. Dt can be thought of as diffusivity of vorticity (and momentum), i.e., once generated (on boundaries only) will spread/diffuse in space if is present. v v v v Dv v v v = 2 v 2 v v v + ... D ... D v v = 2 2 v v + ... ......
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.
- Spring '05