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MAE 150B - 03B - Fundamentals of Invicid Incompressible Flow - part B

# MAE 150B - 03B - Fundamentals of Invicid Incompressible Flow - part B

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MAE 150B - Chap 3 part B Fundamentals of Inviscid  Incompressible Flow

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Complex Number and Complex Potential  Functions  (supplement materials) In some circumstances, it may be more convenient to use complex function and complex number to express the potential and stream functions ( 29 z f ressed be may flows some iy x z i = Φ + = + = Φ exp ψ φ
Complex Potential  (supplement  materials) dy y dx x d + = φ φ φ dy y dx x d + = ψ ψ ψ + + + = Φ + = Φ dy y dx x i dy y dx x d id d d ψ ψ φ φ ψ φ ( 29 ( 29 ( 29 ivdz udz d idy dx iv idy dx u d udy vdx i vdy udx d - = Φ + - + = Φ + - + + = Φ ) ( vi u dz d - = Φ α i Ve dz d - = Φ

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Stream and Potential functions of Basic  Flows
Complex Potential of Uniform Flow x V = φ y V = ψ z V yi V x V z F = + = ) ( = V dz dF We may replace V with a general uniform flow of arbitrary direction. The above expression can be extended for any uniform flow. α i Ve dz dF - =

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Complex Potential of a Line Source r M ln 2 π φ = θ π ψ 2 M = ( 29 ( 29 θ θ π π π π π θ π θ θ θ sin cos 2 2 2 2 ln 2 ln 2 ) ( i r M e r M re M z M dz dF re iy x z z M i r M z F i i i - = = = = = + = = + = - ( 29 z M i r M z F ln 2 ln 2 ) ( π θ π = + =
Complex Potential of a Vortex θ π φ 2 Γ - = r ln 2 π ψ Γ = z i F ln 2 π Γ =

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Complex Potential of Doublet If we have a source at x = -a and a sink of equal strength at x = a for small x If we have a approaches to zero by keeping 2aM as a constant ( 29 ( 29 - + = - + = - - + = z a z a M a z a z M F a z M a z M F 1 1 ln 2 ln 2 ln 2 ln 2 π π π π x x x 2 1 1 ln - + z F π κ 2 =
Point Source (just for your information  will not be included in the test) We may use a spherical coordinate system to develop a point source (sink) stream function and potential function) Question: The radial velocity will decrease inversely proportional to what power of r? Combining a point source and a point sink, we may develop a point doublet They can be combined with a uniform flow to form some useful flow over a 3-D object (with the methodology we are going to discuss next)

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Superposition (Combination) of  Elementary Flows
Method of images (not in the  textbook) Method of images is a special form of method of superposition.

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