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Fundamental Issues and Canonical Flows

Optimized Laminar Axisymmetrical Nozzle Design Using a Numerically Validated Thwaites Method

[+] Author and Article Information
Philippe Versailles1

Department of Mechanical Engineering,  McGill University, Montréal, QC, H3A 0C3, Canadaphilippe.versailles@mail.mcgill.ca

Jeffrey M. Bergthorson

Department of Mechanical Engineering,  McGill University, Montréal, QC, H3A 0C3, Canadajeff.bergthorson@mcgill.ca

1

Corresponding author.

J. Fluids Eng 134(10), 101203 (Sep 28, 2012) (9 pages) doi:10.1115/1.4007155 History: Received August 22, 2011; Revised July 10, 2012; Published September 27, 2012; Online September 28, 2012

This paper presents the Thwaites method as an accurate and efficient design tool for laminar, axisymmetrical nozzles. Based on historical developments, it is improved to describe internal flows with highly favorable pressure gradients in cylindrical coordinates. The calculation of the core flow velocity distribution based on the continuity equation is proposed as a replacement to other sophisticated numerical methods. A remarkably good agreement is obtained when comparing the results of the current Thwaites method against those of computational fluid dynamics (CFD) simulations, for which the integral boundary layer thicknesses are calculated with equations developed from first principles in the course of the work. This consistency among the results and the low time and resource costs of the Thwaites method confirm its applicability and usefulness as an engineering design and optimization tool.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Correlations of the shape factor, H, against the Holstein–Bohlen parameter, λ

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Figure 2

Nozzle contour, first and second-order derivatives of the radius as a function of x

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Figure 3

Computational domain

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Figure 4

Control volume for the calculation of the integral parameters in internal cylindrical coordinates

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Figure 5

Momentum thickness versus axial location obtained from the Thwaites method and CFD simulation

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Figure 6

Displacement thickness versus axial location obtained from the Thwaites method and CFD simulation

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Figure 7

Free-stream velocity gradients versus axial location obtained from the Thwaites method

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Figure 8

Negative of the pressure gradient versus axial location obtained from the Thwaites method and CFD simulation

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Figure 9

Axial velocity versus axial location obtained from the Thwaites method and CFD simulation at different radial positions

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Figure 10

Shape factor versus λ obtained from the Thwaites method and CFD simulation

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Figure 11

Interaction between the parameters of the Thwaites method

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Figure 12

F(λ) correlation versus λ obtained from the Thwaites method and CFD simulation

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Figure 13

Displacement-thickness-based critical Reynolds number for transition as a function of the shape factor H

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Figure 14

Reδ * /Reδ * crit ratio versus axial location obtained from the Thwaites method and CFD simulation

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Figure 15

Görtler parameter versus axial location obtained from the Thwaites method and CFD simulation

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