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f15_fall - Fluids – Lecture 15 Notes 1 Uniform flow...

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Unformatted text preview: Fluids – Lecture 15 Notes 1. Uniform flow, Sources, Sinks, Doublets Reading: Anderson 3.9 – 3.12 Uniform Flow Definition V A uniform flow consists of a velocity field where vector = u ˆ ı + v ˆ 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 flow angle α . u = u ∞ = V ∞ cos α v = v ∞ = V ∞ sin α 2 Note also that V 2 = u + v 2 . The corresponding potential and stream functions are ∞ ∞ ∞ φ ( x, y ) = u ∞ x + v ∞ y = V ∞ ( x cos α + y sin α ) ψ ( x, y ) = u ∞ y − v ∞ x = V ∞ ( y cos α − x sin α ) V v u α Zero Divergence A uniform flow 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 φ ( x, y ) satisfies Laplace’s equation. ∂ 2 ( u ∞ x + v ∞ y ) ∂ 2 ( u ∞ x + v ∞ y ) ∇ 2 φ = + = 0 ∂x 2 ∂y 2 Therefore, the uniform flow satisfies mass conservation. Zero Curl A uniform flow is also easily shown to be irrotational, or to have zero vorticity . ∇ × vector ∂u ∞ ˆ V ≡ ξ vector = ∂v ∞ − k = 0 ∂x ∂y 1 The equivalent irrotationality condition is that ψ ( x, y ) satisfies Laplace’s equation. ∂ 2 ( u ∞ y − v ∞ x ) ∂ 2 ( u ∞ y − v ∞ x ) ∇ 2 ψ = + = 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 Λ V r = , V θ = 0 2 π r where Λ is a scaling constant called the...
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f15_fall - Fluids – Lecture 15 Notes 1 Uniform flow...

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