Chapter10

Chapter10 - Vorticity and Viscosity We have seen that there...

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Vorticity and Viscosity Vorticity and Viscosity z We have seen that there is direct connection between vorticity and the force that develops on an object moving through a fluid z In this chapter, we will prove that there is no physical mechanism for developing vorticity in an inviscid fluid for most practical applications Helmholtz’s Theorem d’Alembert’s Paradox z We will see how viscosity provides a mechanism for developing vorticity Navier-Stokes Equation Boundary Layers, Separation z We will discuss lift and drag on common objects
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The Vortex Force The Vortex Force - - 1 1 z To understand the connection between vorticity and the force on an object moving through a fluid, recall that we can write Euler’s Equation as z The term proportional to u × ω acts like a Coriolis acceleration…moving it to the right-hand side… z As indicated, we call this the vortex force ω = vorticity = ρ ( u ) u
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The Vortex Force The Vortex Force - - 2 2 z For example, in a 2-D flow the vorticity has only a z component z So, the vortex force acts normal to the direction of flow z Because of the vortex force, powerful aerodynamic forces develop at 90 o to the direction of motion when A propeller spins An airplane’s wing moves A tacking sailboat’s sail advances into the wind
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Helmholtz’s Theorem Helmholtz’s Theorem - - 1 1 z Since vorticity is essential for developing forces in a fluid flow, it would be helpful to determine the conditions needed to create vorticity z Helmholtz’s Theorem was a major stumbling block in developing theoretical fluid mechanics z This means that when a flow starts from rest, for example, there is no physical mechanism for developing vorticity in an unbounded, inviscid, incompressible fluid acted on by gravity (e.g., the atmosphere) “If a fluid particle moving in an unbounded frictionless, incompressible fluid under conservative body forces has zero vorticity initially, it always has zero vorticity”
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Helmholtz’s Theorem Helmholtz’s Theorem - - 2 2 z Proof: Recall that, for incompressible flow with a conservative body force, Euler’s Equation is z We can derive a differential equation for vorticity by taking the curl of this equation z Making use of several vector identities (see text) such as ∇×∇ p = 0 for any scalar p , we find that z In words, the rate of change of vorticity following a fluid particle equals the “vortex-stretching” process = d ω /dt Vortex stretching
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.

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Chapter10 - Vorticity and Viscosity We have seen that there...

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