Chapter_Four_Part_Three_web

# Chapter_Four_Part_Three_web - ENU 4133 – Chapter 4, Part...

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Unformatted text preview: ENU 4133 – Chapter 4, Part 3 February 10, 2010 Chapter 4 – Differential Relations for Fluid Flow (Advanced Topics) In scope of Exam #2 I 4.7 – stream function I 4.8 – voriticity and irrotationality I 4.9 – frictionless, irrotational flows Stream Function When only two terms of the continuity equation are non-zero, it is possible to develop an equation for a single variable (the “stream function”, ψ ). Ignore cases where one of the non-zero terms is ∂ρ/∂ t . Incompressible, Cartesian: ∂ u ∂ x + ∂ v ∂ y = 0 (1) u = ∂ψ ∂ y (2) v =- ∂ψ ∂ x (3) ∂ ∂ x ∂ψ ∂ y + ∂ ∂ y- ∂ψ ∂ x = 0 (4) Steam Function (2) ~ V = ~ i ∂ψ ∂ y- ~ j ∂ψ ∂ x (5) ∇ × ~ V =- ~ k ∇ 2 ψ (6) ∇ × ~ V =- ~ k ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 (7) Stream Function along Streamline Along streamline in 2-D flow: dx u = dy v (8) udy- vdx = 0 (9) ∂ψ ∂ x dx + ∂ψ ∂ y dy = 0 (10) d ψ = 0 (11) Therefore, ψ is constant along a streamline. Other Cases Compressible, Cartesian:...
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## This note was uploaded on 07/14/2011 for the course ENU 4133 taught by Professor Schubring during the Spring '11 term at University of Florida.

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Chapter_Four_Part_Three_web - ENU 4133 – Chapter 4, Part...

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