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Unformatted text preview: ow for ψ we have ∆ψ Velocity potential Beginnings of potential flow theory (applies to irrotational flow) There is a vector identity to be found in any multivariable calculus book 0 for any scalar function φ. For irrotational flow, we know that 0 So there must be a φ such that ϕ The function , , is called the velocity potential. Expressing this in component form ̂ ̂
̂ ̂ We can write this as three component equations So, if φ is known we can find u, v and w or anywhere by differentiating Relationship between φ and ψ in 2D A curve along which is called an equipotential line Along this line, applying the chain rule ϕ 0 or 0 So the slope of the equipotential line is 1 Likewise for a streamline, applying the chain rule we have 0 Or 0 So the slope of the streamline is If we combine the two slope equations we find that 1 So the slopes are negative reciprocals of one another. The curves are normal to one another. This is true everywhere. Linearity of velocity potential The gradient operator is linear, so if we can build up multiple simple solutions in order to make a more complicated solution ϕ
ϕ Elementary examples: Source/sink Vortex Uniform flow Can be used to construct Doublet Flow over a cylinder Lifting flow over a rotating cylinder With some further math tricks Flow over some wing shapes We divide the flow around an aircraft between a boundary layer and an irrotational outer flow. For very many cases we can solve the outer flow using potential flow equations Analytically or numerically Homework on chapter 2 due Friday Exam one week from friday, on 31 May Will cover first two chapters in the book Please look at homework problems, book example problems, example problems worked in class on board, or ones you worked...
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This note was uploaded on 01/19/2014 for the course AEEM 2042C taught by Professor Munday during the Fall '10 term at University of Cincinnati.
- Fall '10