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# Topics in 2-D Separated Vortex Flows

## Citation

Tanveer, Saleh Ahmed (1984) Topics in 2-D Separated Vortex Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BR0B-QH56. https://resolver.caltech.edu/CaltechETD:etd-02012007-133412

## Abstract

This thesis is concerned with vortices in steady two dimensional inviscid incompressible flow. In the first three chapters, separated vortex flows are considered in the context of inviscid flow past two dimensional airfoils for which the action of the vortex is to induce large lift. In the fourth and last chapter, we consider vortices in uniform flow in the absence of any physical bodies.

In chapter I, we consider two configurations of vortices for flow past a flat plate with a forward facing flap attached to its rear edge. In the first case, case (a), we consider a potential vortex in the vicinity of the airfoil, while for case (b), we consider a vortex sheet coming off the leading edge of the plate and reattaching at the leading edge of the flap such that the region between the vortex sheet and the airfoil is stagnant. For case (a), the Schwarz-Christoffel transformation is used to find exact solutions to the flow problem. It is found that by suitably placing a potential vortex of appropriate strength it is possible to satisfy the Kutta condition of finite velocity at both the leading edges of the plate and the flap in addition to satisfying it at the trailing edge, provided the plate flap combination satisfies a geometric constraint. The action of the potential vortex is to create a large circulatory region bounded by the airfoil and the streamline that separates smoothly at the leading edge of the plate (due to the Kutta condition) and reattaches smoothly at the leading edge of the flap (from the Kutta condition again). The circulation induced at infinity for such a flow and hence the lift on the airfoil is found to be very large. For case (b), where the vortex sheet location is unknown, a hodograph method is used to find exact solutions. It is found that once a geometric constraint is satisfied, flows exist for which the Kutta condition is satisfied at the trailing edge of the plate-flap combination. As in (a), large values of lift are obtained. However, in both cases (a) and (b), the adverse pressure gradient of top of the flap is recognized as a source of potential difficulty in the experimental realization of the calculated flow.

In chapter II, successive modifications are made to the airfoil considered in chapter I. Exact solutions are once again obtained by a variation of the hodograph method of chapter I. The lift for these airfoils is found to be significantly larger than the one in chapter I. Because the trailing edge is no longer a stagnation point, it is felt that these flows may be easier to realize experimentally.

Chapter III is concerned with the so-called Prandtl-Batchelor flow past the plate-flap geometry of chapter I. The flow consists of an inner region which has a constant vorticity. The region outside of the airfoil and the vortex sheet coming off the leading edge of plate and reattaching at the leading edge of the flap (as in chapter I) is once again irrotational. The common boundary between the exterior flow and the inner flow, i.e. the vortex sheet, is unknown a priori and is determined by continuity of pressure, which translates into a nonlinear boundary condition on an unknown boundary. By extending the function theoretic approach of complex variables to this problem, we reduce the entire problem into one of determining one unknown function of one variable on a fixed domain from which everything else can be calculated. This is then solved numerically. Our calculations provide what we believe to be the first such calculation of a Prandtl-Batchelor flow. The calculations also provide a more realistic model for the vortex sheet flow considered in chapter I.

Chapter IV deals with a steadily translating pair of equal but opposite vortices with uniform cores and vortex sheets on their boundaries, moving without the presence of any physical boundary. The solutions were found for such flows using the function theoretic approach introduced earlier in chapter III for flows where the velocity on the vortex sheet is not a constant. The solutions form a continuum between the hollow vortex case of Pocklington (1898) and those of Deem and Zabusky (1978) and Pierrehumbert (1980) who consider uniform core with no vortex sheet. The iterative scheme for numerical calculation, however, turns out to have severe limitations, as it fails to converge for the cases with no vortex sheet or when the vortex sheet strength is small. In the last section of the chapter, a more traditional approach due to Deem and Zabusky is taken to calculate a pair of touching vortices with uniform core and no vortex sheet on the boundary and an error in Pierrehumbert's (1980) calculations is pointed out.

In appendix I, we point out some errors in Pocklington's paper on the motion of a hollow vortex pair. The errors are corrected and the results are found to be then in agreement with results using the method in chapter IV.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:2-D separated vortex flows
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Applied Mathematics
Thesis Availability:Public (worldwide access)
• Saffman, Philip G.
Thesis Committee:
• Keller, Herbert Bishop (chair)
• Roshko, Anatol
• Fornberg, Bengt
• Saffman, Philip G.
Defense Date:26 October 1983
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
Record Number:CaltechETD:etd-02012007-133412
Persistent URL:https://resolver.caltech.edu/CaltechETD:etd-02012007-133412
DOI:10.7907/BR0B-QH56
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:442
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:13 Feb 2007