f15_fall

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online