fluidsCh5 - Chapter 5 Inviscid Flows and Irrotational Flows...

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Unformatted text preview: Chapter 5 Inviscid Flows and Irrotational Flows This chapter is divided between two topics that are related but must be dis- tinguished carefully: inviscid flows and irrotational flows. Even though there is considerable overlap between the two topics, inviscid flows (Euler equa- tion) can contain vorticity. The mechanisms by which viscous forces forces introduce vorticity in a flow will be treated in Ch. 8 and 9. And another distinction must be made right between inviscid flows, which have unique solutions (if irrotational), and very-large Reynolds number flows, which are generally turbulent. 5.1 Inviscid flow In the absence of viscosity, Euler’s equations are the expression of Newton’s second law for incompressible flow. As seen above ∂ t u + ω × u =-∇ ( p ρ + u 2 2 + gz ) =-∇ B (5.1) and in index notations ∂ t u i + u j ∂ j u i =- gδ i 3- 1 ρ ∂ i p (5.2) 119 120 CHAPTER 5. INVISCID FLOWS AND IRROTATIONAL FLOWS 5.2 Bernoulli’s equation There are several forms of Bernoulli’s equations, relying on more or less restrictive assumptions. The student should make a clear distinction between them. In all cases, we start from the Navier-Stokes equations: ∂ t u + ω × u =-∇ B- ν ∇ × ω , (5.3) where B = p ρ + u 2 2 + gz, (5.4) and look for ways to isolate the gradient in the r.h.s. (The Bernoulli term B is a scalar and cannot be confused with the body force per unit volume, which is a vector.) 5.2.1 The strong form The simplest way is to assume that the flow is irrotational. A necessary and sufficient condition for this to hold is that u = ∇ φ ↔ ω = 0 . (5.5) (see Helmholtz decomposition). Then, the Magnus and viscous terms vanish, and the time derivative can be brought inside the gradient, to give ∇ ( ∂ t φ + p ρ + u 2 2 + gz ) = 0 (5.6) implying that ∂ t φ + B = 0 (5.7) has a constant value in the entire field. The assumptions made here seem to hold in simple wave motion, where the time-dependent potential is needed (see Ch. 11 for simple examples). If, furthermore, the flow is steady, there is also a useful relation known as Euler’s normal equation. The normal pressure gradient is only balanced by the centripetal acceleration. So, 1 ρ ∂ n p = u 2 R (5.8) where R is the local radius of curvature of the streamline. 5.2. BERNOULLI’S EQUATION 121 5.2.2 The weaker form A separate version of Bernoulli’s equation is not as restrictive: allowing vor- ticity, the initial assumption is that viscous effects can be neglected. Thus, we work with Euler’s equation ∂ t u + ω × u =-∇ B (5.9) Then, project the equation on the direction of the velocity vector itself, i.e. along the streamlines. This cancels out the Magnus term, and we have ∂ t u 2 2 =- u · ∇ B (5.10) In words, the local rate of change of kinetic energy (per unit mass) comes at the expense of the convective (directional) derivative of the Bernoulli ex- pression. If we restrict ourselves to steady flows, we see that B = p ρ + u 2 2 + gz...
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This note was uploaded on 10/16/2010 for the course L.C.SMITH MAE643 taught by Professor Xx during the Spring '10 term at Syracuse.

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fluidsCh5 - Chapter 5 Inviscid Flows and Irrotational Flows...

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