Chapter11

Chapter11 - Potential Flow Potential Potential-flow theory...

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Potential Flow Potential Flow z Potential-flow theory applies in the limiting case of flows that are both incompressible and irrotational, which is not a pathological case… Incompressible flows are very common Vorticity is generally confined to thin “boundary layers” for slender bodies (like wings) so that many flows of engineering interest are irrotational except in small regions z All of the nonlinearity of the basic equations of motion is confined to Bernoulli’s Equation z We will see the direct connection between vorticity and forces that develop on an object
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Differential Equations of Motion Differential Equations of Motion z Potential-flow theory is based on the mass- and momentum-conservation equations… and z Making use of the vector identity for u u in Appendix C, the acceleration is and we can rewrite the momentum equation as We assume the body force is conservative Æ Irrotational ∇× u = 0
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Mathematical Foundation Mathematical Foundation - - 1 1 z We will build a theory that begins with and z Irrotationality: For any scalar function φ z Thus, if a function exists such that the irrotationality condition is automatically satisfied z We call the velocity potential
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Mathematical Foundation Mathematical Foundation - - 2 2 z Incompressibility: For any vector Ψ z Thus, if a vector Ψ exists such that the incompressibility condition is automatically satisfied z In the special case of 2-D flow… z We call ψ the streamfunction
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Velocity Velocity - - Potential Representation Potential Representation z In terms of velocity potential, we have z Substituting into the continuity equation yields z We will make use of cylindrical coordinates throughout this chapter… and Laplace’s equation Æ Laplace’s equation in cylindrical coordinates Circumferential velocity Radial velocity
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Streamfunction Representation Streamfunction Representation z In terms of streamfunction, we have z Substituting into ∇× u = 0 yields z In terms of cylindrical coordinates… and Laplace’s equation Æ Laplace’s equation in cylindrical coordinates Circumferential velocity Radial velocity
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Mathematical Mathematical - - Representation Summary Representation Summary z In terms of velocity potential, φ , and streamfunction, ψ , the mathematical representations of 2-D potential- flow theory are as follows
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Streamlines and Equipotential Lines Streamlines and Equipotential Lines - - 1 1 z One of the most useful features of the streamfunction is its relation to streamlines…consider the curve defined by z Differentiating, we have z Therefore, on the contour for which ψ ( x,y ) = constant which is the differential equation of a streamline z Thus, ( x,y ) = constant defines a streamline Æ
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Streamlines and Equipotential Lines Streamlines and Equipotential Lines - - 2 2 z Another useful feature of the streamfunction is its relation to flow rate z The mass-flow rate across an arc of length s is z Using the indicated control volume, mass conservation for a steady flow yields… z Therefore, we conclude that Æ
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.

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Chapter11 - Potential Flow Potential Potential-flow theory...

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