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Unformatted text preview: Chapter 10 Rotating Flows Flows in rotating frames of reference include atmospheric flows and flows in turbomachinery. Fictitious inertia forces are added to the NavierStokes equations: the centrifugal force is easily included the conventional framework, but the Coriolis force introduces some new physics. Relevant dimensionless parameters are the Ekman and Rossby numbers. The case of dominant Coriolis force corresponds to a small Rossby num ber. See Fig. 10.1. 10.1 Equations of motion The equations of motion in a rotating frame of reference can be found in most dynamics textbooks. The derivation relies on the observation that, as the base vectors rotate at constant angular velocity Ω (Fig. 10.2), we have ∂ t e i = Ω × e i (10.1) Note that Ω is the rotation vector (Poisson vector) of the frame of reference as seen from an assumed inertial reference. Then, the second derivative of the vector position r = x i e i is easily obtained ∂ 2 tt r = ∂ 2 tt x i e i + 2 ∂ t x i Ω × e i + x i Ω × (Ω × e i ) (10.2) In addition to the apparent acceleration, two new terms appear: the Coriolis acceleration which depend on velocity, and the centripetal acceleration which depends on position relative to the axis of rotation. 201 202 CHAPTER 10. ROTATING FLOWS D. Fultz’s movie: Rotating flows • Coriolis force: motion of vortex ring (selfpropelled) or similar object • Rossby number, geostrophic flow – linear equations – 2D: TaylorProudman theorem – Taylor columns – nonlocal effects: ∇ 2 – Rossby waves • atmospheric motion: large scale only • Ekman layers: secondary flows Figure 10.1: Small Rossby number flows, movie by D. Fultz Figure 10.2: Inertial and rotating frames of reference 10.2. SCALING 203 Moving these two accelerations to the other side of Newton’s second law, they become fictitious forces (with signs reversed), and the NavierStokes equations become ∂ t u + ω × u =∇ ( p ρ + u 2 2 + gz ) ν ∇ × ω Ω × (Ω × r ) 2Ω × u (10.3) The first added term is the centrifugal force, quadratic in Ω. Because it can be rewritten in the form Ω × (Ω × r ) =∇ ( 1 2 Ω 2 r 2 ) (10.4) where r denotes the distance from the axis of rotation, the centrifugal term is easily combined with the pressure term. The term linear in Ω is the Coriolis force. It is essential in explaining our largescale weather systems, among other features. This chapter revolves around it. Questions for discussion: Is your lab an inertial lab? Rotor? 10.1.1 Coriolis force Effects of the Coriolis force are illustrated in the movie shown in class. The trajectory of a vortex ring in the rotating tank is a good example (Fig. 10.3). Note that the direction of rotation is actually for tank rotation as seen from (inertial) lab....
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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.
 Spring '10
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