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12 - Finite Element Analysis in Curvilinear Coordinate

Published online by Cambridge University Press:  05 June 2014

Erian A. Baskharone
Erian A. Baskharone
Affiliation:
Texas A & M University
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Summary

In this chapter we apply Galerkin's weighted-residual finite element approach to a special category of flow problems. This is where only the through-flow and tangential momentum equations (beside the continuity equations, of course) suffice as the flow-governing equations. This problem is perhaps best represented by the socalled quasi-three-dimensional flow field in analyzing airfoil cascades. Some terms within the finite element formulation are presented and modeled as “source” terms, in analogy with a special problem category in heat conduction. Also, implicit means are used in enforcing the cascade periodicity conditions.

Introduction

In the cascade theory discipline, the basic problem is that of a three-dimensional periodic flow in the blade-to-blade hub-to-casing passage (Figure 12.1). In modeling this flow type, it is crucial to account for such real-flow effects as boundary layer separation, flow recirculation, and trailing-edge mixing losses. Existing numerical models in this area vary in complexity from the potential flow category [1–3] to that of the fully three-dimensional viscous flow field [4, 5]. Compared with the strictly two-dimensional and three-dimensional flow models, the quasi-three-dimensional approach (which is the topic in this chapter) to the cascade flow problem has been recognized as a sensible compromise in terms of both economy and precision. It is, however, the viscous flow version of the problem, under this approach, that is in need of further enhancement, particularly in the area of simulating the hubto-casing flow interaction effects on the blade-to-blade flow field.

Type
Chapter
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The Finite Element Method with Heat Transfer and Fluid Mechanics Applications , pp. 149 - 175
Publisher: Cambridge University Press
Print publication year: 2013

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References

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