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# Quasi Two-Dimensional Flows Through Cascades

## Citation

Mani, Ramani (1967) Quasi Two-Dimensional Flows Through Cascades. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/01sg-we85. https://resolver.caltech.edu/CaltechETD:etd-10012002-093723

## Abstract

The present thesis is an attempt to develop a thin airfoil theory for an airfoil which spans the gap between a pair of stream surfaces which are slowly diverging or converging, the motivation being to predict, theoretically, the effect of varying axial velocity on cascade performance of axial flow compressor rows.

The procedure involves, firstly, derivation of approximate equations satisfied by suitably defined average potentials and stream functions in such quasi two-dimensional flows. The flow is assumed to be inviscid, irrotational, and incompressible, but as will be argued later, the quasi two-dimensional type equations also result from less restrictive assumptions. Next, fundamental solutions to these equations, corresponding to bound, line sources and vortices, are found. A distribution of such solutions is used to formulate the airfoil problem, using the condition that the flow be tangential to the airfoil contour. The vorticity distribution appears as the solution to a singular integral equation, which is solved by an approximate method. Simple yet physically realistic assumptions are made concerning the gap width as a function of the streamwise length, to obtain numerical results for the effect of contraction of the stream surfaces. Varying degrees of approximation, later discussed, are used in the calculation procedures. A wide variety of the location and the extent of the contraction, with respect to the airfoil, is investigated.

In all cascade calculations the contraction of the stream surfaces was assumed to be in the same direction as the cascade axis. The main conclusions of the thesis can be summarized as below:

1. The theory predicts a lesser circulation round an airfoil in a contracting flow as compared to the circulation round the same airfoil in a plane flow. There is a similar reduction of circulation for a cascade of airfoils. The percentage reduction of circulation is greater for the cascade case as compared to the isolated case, assuming the contractions to be geometrically similar in both cases. The effect on the circulation of contractions, considered physically reasonable in extent and magnitude, either fully upstream or fully downstream of the airfoil, is quite small.

2. As a very rough rule of thumb, it may be stated that the reduction of circulation as compared to the two-dimensional theory, in the range of parameters applicable to compressors, has about the same magnitude as the reduction of gap between the stream surfaces taking place across the airfoil chords.

3. In a comparison with fixed mean angle of attack, the change in flow turning and deviation angles of the flow are much smaller than changes of circulation and may be stated to be of the order of one degree or less for contraction extents and magnitudes considered realistic for compressor cascades.

Item Type: Thesis (Dissertation (Ph.D.)) Mechanical Engineering California Institute of Technology Engineering and Applied Science Mechanical Engineering Public (worldwide access) Acosta, Allan J. Unknown, Unknown 16 December 1966 CaltechETD:etd-10012002-093723 https://resolver.caltech.edu/CaltechETD:etd-10012002-093723 10.7907/01sg-we85 No commercial reproduction, distribution, display or performance rights in this work are provided. 3843 CaltechTHESIS Imported from ETD-db 02 Oct 2002 16 Apr 2021 23:10

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