Date of Award

Spring 1999

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

John Kenneth Bechtold

Second Advisor

Moshe Matalon

Third Advisor

Demetrius T. Papageorgiou

Fourth Advisor

Michael R. Booty

Fifth Advisor

Jonathan H.C. Luke

Abstract

In this dissertation, the response of a premixed flame to time-dependent strained flow fields is investigated. Because of the potential application to turbulent combustion modeling, the main focus is on the particular case of a flame in stagnation point flow with an imposed oscillatory strain rate. The flame is modeled as a hydrodynamic discontinuity separating burned from unburned gasses. To complete the formulation of the problem, conditions relating the fluid variables across the flame front are needed, as is a flame speed equation that determines the evolution of the discontinuity. These conditions are derived through asymptotic analysis of the flame structure.

In the first part of this dissertation, an existing hydrodynamic model is employed to assess flame response to oscillating stagnation point flow. The model is valid for near-equidiffusional conditions, i.e. for near-unity Lewis numbers. Under these conditions, the flame speed varies linearly with strain. Unlike previous theoretical investigations, the present formulation places no restrictions on the amplitude of the oscillations, and we account for the full interaction between the flame and the flow. Solutions are constructed by a combination of asymptotic and numerical methods. Results regarding flame response are in agreement with previous experiments and studies. We also obtain the following results as a consequence of the underlying time-periodic flow: (a) the mean flame position is shifted upstream from the steady state location, (b) a region of reverse flow appears immediately ahead of the flame front during part of each cycle, and (c) there is a maximum amplitude of oscillation beyond which the flame fails to exist. These results are most pronounced at high frequencies and agree with the asymptotic solution constructed in that regime.

In the second part of this dissertation, a new model is derived which exhibits a nonlinear dependence of flame speed on strain. The model is valid for arbitrary Lewis number, and unlike previous models, it allows for an unsteady flame structure. Asymptotic methods are used to construct solutions across the narrow flame zone (and reaction zone), and asymptotic matching then yields the nonlinear flame speed equation. The new model is then employed to investigate flame response to unsteady strained flows. Our results predict that the flame becomes most sentive to fluctuations in the flow as steady state extinction conditions are approached. Also, at high frequency the flame response is the same, regardless of the mixture properties. These results are in good agreement with experiments.

Included in

Mathematics Commons

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