Transcription of Fluids – Lecture 15 Notes - MIT
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Fluids – Lecture 15 Notes - MIT
Fluids Lecture 15 Notes1. Uniform flow, Sources, Sinks, DoubletsReading: Anderson FlowDefinitionAuniform flowconsists of a velocity field where~V=u +v is a constant. In 2-D, thisvelocity field is specified either by the freestream velocitycomponentsu ,v , or by thefreestream speedV and flow angle .u=u =V cos v=v =V sin Note also thatV2 =u2 +v2 . The corresponding potential and stream functions are (x,y) =u x+v y=V (xcos +ysin ) (x,y) =u y v x=V (ycos xsin )Vvu Zero DivergenceA uniform flow is easily shown to havezero divergence ~V= u x+ v y= 0since bothu andv are constants. The equivalent statement is that (x,y) satisfiesLaplace s equation. 2 = 2(u x+v y) x2+ 2(u x+v y) y2= 0 Therefore, the uniform flow satisfies mass CurlA uniform flow is also easily shown to be irrotational, or to havezero vorticity.
The origin location (0,0) is called a singular point of the source flow. As we approach this point, the magnitude of the radial velocity tends to infinity as Vr ∼ 1 r Hence the flow at the singular point is not physical, although this does not prevent us from using the source to represent actual flows. We will simply need to ensure that ...
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