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p1002 - flow necessarily their cross product is zero...

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1318 CHAPTER 10. VORTICITY, VISCOSITY, LIFT AND DRAG 10.2 Chapter 10, Problem 2 Problem: Beltrami flow is an idealized type of flow sometimes used in turbomachinery analysis. In this type of flow, the velocity vector, u , is parallel to the vorticity vector, ω , everywhere. (a) What is the vortex force in Beltrami flow? (b) If the body force is conservative with f = −∇ V , determine the pressure, p , as a function of V , u and density, ρ , for steady, incompressible Beltrami flow. Solution: (a) By definition, the vortex force is f vortex = ρ u × ω But, since u and ω are parallel for Beltrami
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Unformatted text preview: flow, necessarily their cross product is zero. Therefore, f vortex = for Beltrami flow (b) For steady Beltrami flow, ∂ u / ∂ t = 0 , and the vortex force is zero, so that the Euler equation simplifies to ρ ∇ w 1 2 u · u W = −∇ p − ρ ∇ V If the flow is also incompressible, ρ is constant so that all terms can be grouped together in parentheses as follows. ∇ w p + 1 2 ρ u · u + ρ V W = Therefore, the quantity in parentheses must be constant, viz., p + 1 2 ρ u · u + ρ V = constant...
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