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

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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 flux vector ρ vector V as the partial derivatives ¯ of a scalar stream function , denoted by ψ ( x, y ): ¯ ¯ ∂ψ ∂ψ ρu = , ρv = ∂y ∂x For low speed flows, ρ is just a known constant, and it is more convenient to work with a scaled stream function ¯ ψ ψ ( x, y ) = ρ which then gives the components of the velocity vector vector V . ∂ψ ∂ψ u = , v = ∂y ∂x Example Suppose we specify the constant-density streamfunction to be 1 2 ψ ( 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 ∂ψ y ∂ψ x u = ∂y = x 2 + y 2 , v = = ∂x x 2 + y 2 which corresponds to a vortex flow around the origin. ψ x y vortex flow example 1
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Streamline interpretation The stream function can be interpreted in a number of ways. First we determine the differ- ¯ ential of ψ as follows. ¯ ¯ ∂ψ ∂ψ ¯ = dx + dy ∂x ∂y ¯ = ρu dy ρv dx ¯ ¯ Now consider a line along which ψ is some constant ψ 1 . ¯ ¯ ψ ( x, y ) = ψ 1 ¯ ¯ Along this line, we can state that = 1 = d (constant) = 0, or dy v ρu dy ρv dx = 0 = dx u ¯ which is recognized as the equation for a streamline.
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