LINEAR STABILITY OF SLOWLY VARYING UNSTEADY FLOWS IN A CURVED CHANNEL

G. SEMINARA, P. HALL

Research output: Contribution to journalArticleResearchpeer-review

12 Citations (Scopus)

Abstract

THE LINEAR STABILITY OF VISCOUS FLOWS IN A CURVED CHANNEL DUE TO TIME DEPENDENT SLOWLY VARYING PRESSURE GRADIENTS IS CONSIDERED BY AN APPROACH OF THE W.K.B.TYPE.THE ASYMPTOTIC BEHAVIOUR OF SMALL PERTURBATION WAVES IS DETERMINED, ALLOWING THEIR CHARACTERISTICS (AMPLITUDE, TRANSVERSE STRUCTURE, AMPLICATION RATE) TO BE SLOWLY VARYING WITH TIME.THE EVOLUTION OF SUCH DISTURBANCES IS FOLLOWED AND THE INSTANTANEOUS MARGINAL STATES ARE DETERMINED ACCORDING TO A MINUTES MOMENTARY MINUTES CRITERION FOR STABILITY WHICH IS BASED ON THE DEFINITION OF A MINUTES GROWTH RATE MINUTES OF THE DISTURBANCE.AN ASYMPTOTIC REPRESENTATION FOR THE GROWTH RATE IS FOUND WHICH, AT LOWEST ORDER, REDUCES TO THE QUAS-STEADY AMPLIFICATION RATE OF THE DISTURBANCE VELOCITY FIELD ASSOCIATED WITH THE LOWEST ORDER COMPONENT OF THE INSTANTANEOUS BASIC FLOW.A QUASI-STEADY CORRECTION TERM APPEARS AT THE NEXT ORDER TOGETHER WITH TERMS WHICH ARISE FROM THE SLOW VARIATION WITH TIME OF THE ACTUAL DISTURBANCE STRUCTURE.SLOWLY ACCELERATED BASIC FLOWS ARE ALSO INVESTIGATED.THE EFFECT OF THE QUASI-STEADY CORRECTION IS FOUND TO BE STABILIZING WHEREAS PART OF THE MINUTES SLOWLY VARYING MINUTES TERMS EXHIBIT THE OPPOSITE BEHAVIOUR.HOWEVER THE INFLUENCE OF THE CORRECTION TERMS APPEARS TO BE SMALL, AS EXPECTED SINCE THE CHARACTERISTIC SCALE OF THE INSTABILITY IS MUCH BIGGER THA N THE MINUTES SLOW MINUTES SCALE OF THE BASIC STATE.UNDER SUCH CIRCUMSTANCES THE QUASI-STEADY APPROXIMATION MAY BE CONSIDERED A FAIRLY ACCURATE APPROACH.LOW FREQUENCY MODULATED BASIC FLOWS ARE ALSO INVESTIGATED BY USING THE PERIODICITY CRITERION TO DEFINE THE MARGINAL STATE.THE MODULATION IS ALWAYS FOUND TO DESTABILIZE THE MEAN FLOW AND THE MINUTES CRITICAL MINUTES WAVENUMBER IS FOUND TO DECREASE FROM ITS UNMODULATED VALUE WHEN THE AMPLITUDE OF THE OSCILLATION INCREASES.SOME DISCUSSION FOLLOWS.(A).

Original languageEnglish
Pages (from-to)279-303
Number of pages25
JournalProceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences
Volume346
Issue number(NOVEMBER 4, 1975)
Publication statusPublished - 1 Jan 1975
Externally publishedYes

Cite this

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title = "LINEAR STABILITY OF SLOWLY VARYING UNSTEADY FLOWS IN A CURVED CHANNEL",
abstract = "THE LINEAR STABILITY OF VISCOUS FLOWS IN A CURVED CHANNEL DUE TO TIME DEPENDENT SLOWLY VARYING PRESSURE GRADIENTS IS CONSIDERED BY AN APPROACH OF THE W.K.B.TYPE.THE ASYMPTOTIC BEHAVIOUR OF SMALL PERTURBATION WAVES IS DETERMINED, ALLOWING THEIR CHARACTERISTICS (AMPLITUDE, TRANSVERSE STRUCTURE, AMPLICATION RATE) TO BE SLOWLY VARYING WITH TIME.THE EVOLUTION OF SUCH DISTURBANCES IS FOLLOWED AND THE INSTANTANEOUS MARGINAL STATES ARE DETERMINED ACCORDING TO A MINUTES MOMENTARY MINUTES CRITERION FOR STABILITY WHICH IS BASED ON THE DEFINITION OF A MINUTES GROWTH RATE MINUTES OF THE DISTURBANCE.AN ASYMPTOTIC REPRESENTATION FOR THE GROWTH RATE IS FOUND WHICH, AT LOWEST ORDER, REDUCES TO THE QUAS-STEADY AMPLIFICATION RATE OF THE DISTURBANCE VELOCITY FIELD ASSOCIATED WITH THE LOWEST ORDER COMPONENT OF THE INSTANTANEOUS BASIC FLOW.A QUASI-STEADY CORRECTION TERM APPEARS AT THE NEXT ORDER TOGETHER WITH TERMS WHICH ARISE FROM THE SLOW VARIATION WITH TIME OF THE ACTUAL DISTURBANCE STRUCTURE.SLOWLY ACCELERATED BASIC FLOWS ARE ALSO INVESTIGATED.THE EFFECT OF THE QUASI-STEADY CORRECTION IS FOUND TO BE STABILIZING WHEREAS PART OF THE MINUTES SLOWLY VARYING MINUTES TERMS EXHIBIT THE OPPOSITE BEHAVIOUR.HOWEVER THE INFLUENCE OF THE CORRECTION TERMS APPEARS TO BE SMALL, AS EXPECTED SINCE THE CHARACTERISTIC SCALE OF THE INSTABILITY IS MUCH BIGGER THA N THE MINUTES SLOW MINUTES SCALE OF THE BASIC STATE.UNDER SUCH CIRCUMSTANCES THE QUASI-STEADY APPROXIMATION MAY BE CONSIDERED A FAIRLY ACCURATE APPROACH.LOW FREQUENCY MODULATED BASIC FLOWS ARE ALSO INVESTIGATED BY USING THE PERIODICITY CRITERION TO DEFINE THE MARGINAL STATE.THE MODULATION IS ALWAYS FOUND TO DESTABILIZE THE MEAN FLOW AND THE MINUTES CRITICAL MINUTES WAVENUMBER IS FOUND TO DECREASE FROM ITS UNMODULATED VALUE WHEN THE AMPLITUDE OF THE OSCILLATION INCREASES.SOME DISCUSSION FOLLOWS.(A).",
author = "G. SEMINARA and P. HALL",
year = "1975",
month = "1",
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language = "English",
volume = "346",
pages = "279--303",
journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
issn = "1364-5021",
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number = "(NOVEMBER 4, 1975)",

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LINEAR STABILITY OF SLOWLY VARYING UNSTEADY FLOWS IN A CURVED CHANNEL. / SEMINARA, G.; HALL, P.

In: Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 346, No. (NOVEMBER 4, 1975), 01.01.1975, p. 279-303.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - LINEAR STABILITY OF SLOWLY VARYING UNSTEADY FLOWS IN A CURVED CHANNEL

AU - SEMINARA, G.

AU - HALL, P.

PY - 1975/1/1

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N2 - THE LINEAR STABILITY OF VISCOUS FLOWS IN A CURVED CHANNEL DUE TO TIME DEPENDENT SLOWLY VARYING PRESSURE GRADIENTS IS CONSIDERED BY AN APPROACH OF THE W.K.B.TYPE.THE ASYMPTOTIC BEHAVIOUR OF SMALL PERTURBATION WAVES IS DETERMINED, ALLOWING THEIR CHARACTERISTICS (AMPLITUDE, TRANSVERSE STRUCTURE, AMPLICATION RATE) TO BE SLOWLY VARYING WITH TIME.THE EVOLUTION OF SUCH DISTURBANCES IS FOLLOWED AND THE INSTANTANEOUS MARGINAL STATES ARE DETERMINED ACCORDING TO A MINUTES MOMENTARY MINUTES CRITERION FOR STABILITY WHICH IS BASED ON THE DEFINITION OF A MINUTES GROWTH RATE MINUTES OF THE DISTURBANCE.AN ASYMPTOTIC REPRESENTATION FOR THE GROWTH RATE IS FOUND WHICH, AT LOWEST ORDER, REDUCES TO THE QUAS-STEADY AMPLIFICATION RATE OF THE DISTURBANCE VELOCITY FIELD ASSOCIATED WITH THE LOWEST ORDER COMPONENT OF THE INSTANTANEOUS BASIC FLOW.A QUASI-STEADY CORRECTION TERM APPEARS AT THE NEXT ORDER TOGETHER WITH TERMS WHICH ARISE FROM THE SLOW VARIATION WITH TIME OF THE ACTUAL DISTURBANCE STRUCTURE.SLOWLY ACCELERATED BASIC FLOWS ARE ALSO INVESTIGATED.THE EFFECT OF THE QUASI-STEADY CORRECTION IS FOUND TO BE STABILIZING WHEREAS PART OF THE MINUTES SLOWLY VARYING MINUTES TERMS EXHIBIT THE OPPOSITE BEHAVIOUR.HOWEVER THE INFLUENCE OF THE CORRECTION TERMS APPEARS TO BE SMALL, AS EXPECTED SINCE THE CHARACTERISTIC SCALE OF THE INSTABILITY IS MUCH BIGGER THA N THE MINUTES SLOW MINUTES SCALE OF THE BASIC STATE.UNDER SUCH CIRCUMSTANCES THE QUASI-STEADY APPROXIMATION MAY BE CONSIDERED A FAIRLY ACCURATE APPROACH.LOW FREQUENCY MODULATED BASIC FLOWS ARE ALSO INVESTIGATED BY USING THE PERIODICITY CRITERION TO DEFINE THE MARGINAL STATE.THE MODULATION IS ALWAYS FOUND TO DESTABILIZE THE MEAN FLOW AND THE MINUTES CRITICAL MINUTES WAVENUMBER IS FOUND TO DECREASE FROM ITS UNMODULATED VALUE WHEN THE AMPLITUDE OF THE OSCILLATION INCREASES.SOME DISCUSSION FOLLOWS.(A).

AB - THE LINEAR STABILITY OF VISCOUS FLOWS IN A CURVED CHANNEL DUE TO TIME DEPENDENT SLOWLY VARYING PRESSURE GRADIENTS IS CONSIDERED BY AN APPROACH OF THE W.K.B.TYPE.THE ASYMPTOTIC BEHAVIOUR OF SMALL PERTURBATION WAVES IS DETERMINED, ALLOWING THEIR CHARACTERISTICS (AMPLITUDE, TRANSVERSE STRUCTURE, AMPLICATION RATE) TO BE SLOWLY VARYING WITH TIME.THE EVOLUTION OF SUCH DISTURBANCES IS FOLLOWED AND THE INSTANTANEOUS MARGINAL STATES ARE DETERMINED ACCORDING TO A MINUTES MOMENTARY MINUTES CRITERION FOR STABILITY WHICH IS BASED ON THE DEFINITION OF A MINUTES GROWTH RATE MINUTES OF THE DISTURBANCE.AN ASYMPTOTIC REPRESENTATION FOR THE GROWTH RATE IS FOUND WHICH, AT LOWEST ORDER, REDUCES TO THE QUAS-STEADY AMPLIFICATION RATE OF THE DISTURBANCE VELOCITY FIELD ASSOCIATED WITH THE LOWEST ORDER COMPONENT OF THE INSTANTANEOUS BASIC FLOW.A QUASI-STEADY CORRECTION TERM APPEARS AT THE NEXT ORDER TOGETHER WITH TERMS WHICH ARISE FROM THE SLOW VARIATION WITH TIME OF THE ACTUAL DISTURBANCE STRUCTURE.SLOWLY ACCELERATED BASIC FLOWS ARE ALSO INVESTIGATED.THE EFFECT OF THE QUASI-STEADY CORRECTION IS FOUND TO BE STABILIZING WHEREAS PART OF THE MINUTES SLOWLY VARYING MINUTES TERMS EXHIBIT THE OPPOSITE BEHAVIOUR.HOWEVER THE INFLUENCE OF THE CORRECTION TERMS APPEARS TO BE SMALL, AS EXPECTED SINCE THE CHARACTERISTIC SCALE OF THE INSTABILITY IS MUCH BIGGER THA N THE MINUTES SLOW MINUTES SCALE OF THE BASIC STATE.UNDER SUCH CIRCUMSTANCES THE QUASI-STEADY APPROXIMATION MAY BE CONSIDERED A FAIRLY ACCURATE APPROACH.LOW FREQUENCY MODULATED BASIC FLOWS ARE ALSO INVESTIGATED BY USING THE PERIODICITY CRITERION TO DEFINE THE MARGINAL STATE.THE MODULATION IS ALWAYS FOUND TO DESTABILIZE THE MEAN FLOW AND THE MINUTES CRITICAL MINUTES WAVENUMBER IS FOUND TO DECREASE FROM ITS UNMODULATED VALUE WHEN THE AMPLITUDE OF THE OSCILLATION INCREASES.SOME DISCUSSION FOLLOWS.(A).

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M3 - Article

VL - 346

SP - 279

EP - 303

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - (NOVEMBER 4, 1975)

ER -