f12_fall

f12_fall - Fluids Lecture 12 Notes 1. Stream Function 2....

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Unformatted text preview: Fluids Lecture 12 Notes 1. Stream Function 2. Velocity Potential Reading: Anderson 2.14, 2.15 Stream Function Definition Consider defining the components of the 2-D mass ux vector vector V as the partial derivatives of a scalar stream function , denoted by ( x, y ): u = , v = y x For low speed ows, is just a known constant, and it is more convenient to work with a scaled stream function ( x, y ) = which then gives the components of the velocity vector vector V . u = , v = y x Example Suppose we specify the constant-density streamfunction to be 1 2 ( x, y ) = ln x 2 + y 2 = ln( x 2 + y ) 2 which has a circular funnel shape as shown in the figure. The implied velocity components are then y x u = y = x 2 + y 2 , v = = x x 2 + y 2 which corresponds to a vortex ow around the origin. x y vortex flow example 1 Streamline interpretation The stream function can be interpreted in a number of ways. First we determine the differ- ential of as follows. d = dx + dy x y d = u dy v dx Now consider a line along which is some constant 1 ....
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f12_fall - Fluids Lecture 12 Notes 1. Stream Function 2....

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