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Technical Brief

Exact Solutions for Starting and Oscillatory Flows in an Equilateral Triangular Duct

[+] Author and Article Information
C. Y. Wang

Department of Mathematics;Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: cywang@mth.msu.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 18, 2015; final manuscript received February 4, 2016; published online May 19, 2016. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 138(8), 084503 (May 19, 2016) (4 pages) Paper No: FE-15-1674; doi: 10.1115/1.4032936 History: Received September 18, 2015; Revised February 04, 2016

Exact series solutions, some in closed-form, for starting flow and oscillatory flow in an equilateral triangular duct are presented. The complete set of eigenvalues and eigenfunctions of the Helmholtz equation is derived, and the method of eigenfunction superposition is used. Exact solutions are rare, fundamental, and serve as accuracy standards for approximate methods.

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Figures

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Fig. 1

The equilateral triangular duct

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Fig. 2

Rotational symmetric modes of the equilateral triangle. Darker curves are nodal lines.

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Fig. 3

Velocity distributions for starting flow in an equilateral triangular duct. Left t = 0.01, center t = 0.1, and right t = 1. The magnitude of the velocity is represented by the height from the triangular base.

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Fig. 4

The flow rate Q for starting flow as a function of time t

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Fig. 5

Instantaneous velocity profiles for oscillatory flow, s = 20. From top: st = 0, st = 0.25π, st = 0.75π, st = 0.75π, and st = π. We see that the magnitude of the instantaneous velocity is maximum near the triangular boundary, especially evident for st = 0 or π.

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Fig. 6

Left: the amplitude M and right: the phase lag β versus normalized frequency s for oscillatory flow

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