chpfour - Chapter 4 Potential ow Potential or irrotational...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 4 Potential flow Potential or irrotational flow theory is a cornerstone of fluid dynamics, for two reasons. Historically, its importance grew from the developments made possible by the theory of harmonic functions, and the many fluids problems thus made accessible within the theory. But a second, more important point is that po- tential flow is actually realized in nature, or at least approximated, in many situations of practical importance. Water waves provide an example. Here fluid initially at rest is set in motion by the passage of a wave. Kelvin’s theorem insures that the resulting flow will be irrotational whenever the viscous stresses are negligible. We shall see in a later chapter that viscous stresses cannot in gen- eral be neglected near rigid boundaries. But often potential flow theory applies away from boundaries, as in effects on distant points of the rapid movements of a body through a fluid. An example of potential flow in a barotropic fluid is provided by the theory of sound. There the potential is not harmonic, but the irrotational property is acquired by the smallness of the nonlinear term u · ∇ u in the momentum equation. The latter thus reduces to u ∂t + 1 ρ p 0 . (4.1) Since sound produces very small changes of density, here we may take ρ to be will approximated by the constant ambient density. Thus u = φ with ∂φ ∂t = - p/ρ . 4.1 Harmonic flows In a potential flow we have u = φ. (4.2) We also have the Bernoulli relation (for body force f = - ρ Φ) φ t + 1 2 ( φ ) 2 + integraldisplay dp ρ + Φ = 0 . (4.3) 47
Image of page 1

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

View Full Document Right Arrow Icon
48 CHAPTER 4. POTENTIAL FLOW Figure 4.1: A domain V , bounded by surfaces S i,o where ∂φ ∂n is prescribed. Finally, we have conservation of mass ρ t + ∇ · ( ρ φ ) = 0 . (4.4) The most extensive use of potential flow theory is to the case of constant density, where ∇ · u = 2 φ = 0. These harmonic flows can thus make use of the highly developed mathematical theory of harmonic functions. in the problems we study here we shall usually consider explicit examples where existence is not an issue. On the other hand the question of uniqueness of harmonic flows is an important issue we discuss now. A typical problem is shown in figure 4.1. A harmonic function φ has prescribed normal derivatives on inner and outer boundaries S i ,S o of an annular region V . The difference u d = φ d of two solutions of this problem will have zero normal derivatives on these boundaries. That the difference must in fact be zero throughout V can be established by noting that ∇ · ( φ d φ d ) = ( φ d ) 2 + φ d 2 φ d = ( φ d ) 2 . (4.5) The left-hand side of (4 . 5) integrates to zero over V to zero by Gauss’ theo- rem and the homogeneous boundary conditions of ∂φ d ∂n . Thus integraltext V ( φ d ) 2 dV = 0, implying u d = 0.
Image of page 2
Image of page 3
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