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Unformatted text preview: AE 2020 Chapter III. Part III Flow over a Circular Cylinder In the previous handout we derived the following equation for the doublet: r Doublet θ π µ ψ sin 2 − = = (1) The quantity µ is called the doublet strength. When this doublet is superposed over a uniform flow parallel to the x- axis, we get: r y u θ π µ ψ sin 2 − = ∞ (2) The above equation contains some terms in the Cartesian coordinate system, while the second term is in the polar coordinate system. To avoid unnecessary confusion, we express y as rsin θ . Then, equation (2) becomes: θ π µ θ π µ θ ψ sin 1 2 1 sin 2 sin 2 r r u u r r u − = − = ∞ ∞ ∞ (3) If we replace the constant ∞ u π µ 2 by a new constant R 2 , the above equation becomes: θ ψ sin 1 2 2 r r R u − = ∞ (4) Let us examine this equation. At all points where r=R (i.e. at all points on a circle of radius R), the stream function is zero. Secondly, taking the derivative of the above stream radius R), the stream function is zero....
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- Summer '07
- Cartesian Coordinate System, Velocity, µ, Polar coordinate system, Ψ