{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
( ) 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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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.
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern