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Unformatted text preview: Fluids Lecture 15 Notes 1. Uniform ow, Sources, Sinks, Doublets Reading: Anderson 3.9 3.12 Uniform Flow Definition V A uniform ow consists of a velocity field where vector = u + v is a constant. In 2-D, this velocity field is specified either by the freestream velocity components u , v , or by the freestream speed V and ow angle . u = u = V cos v = v = V sin 2 Note also that V 2 = u + v 2 . The corresponding potential and stream functions are ( x, y ) = u x + v y = V ( x cos + y sin ) ( x, y ) = u y v x = V ( y cos x sin ) V v u Zero Divergence A uniform ow is easily shown to have zero divergence vector u v V = + = 0 x y since both u and v are constants. The equivalent statement is that ( x, y ) satisfies Laplaces equation. 2 ( u x + v y ) 2 ( u x + v y ) 2 = + = 0 x 2 y 2 Therefore, the uniform ow satisfies mass conservation. Zero Curl A uniform ow is also easily shown to be irrotational, or to have zero vorticity . vector u V vector = v k = 0 x y 1 The equivalent irrotationality condition is that ( x, y ) satisfies Laplaces equation. 2 ( u y v x ) 2 ( u y v x ) 2 = + = 0 x 2 y 2 Source and Sink Definition A 2-D source is most clearly specified in polar coordinates. The radial and tangential velocity components are defined to be V r = , V = 0 2 r where is a scaling constant called the...
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- Fall '05