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Unformatted text preview: 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 ≈ . (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 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....
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This note was uploaded on 01/05/2012 for the course MECHANICAL 101 taught by Professor Jose during the Spring '11 term at Rensselaer Polytechnic Institute.
- Spring '11