Temporal stability of Jeffery–Hamel flow

Hamadiche, Mahmoud; Scott, Julian F.; Jeandel, D.

doi:10.1017/s0022112094001266

Cited by 67 publications

(38 citation statements)

References 17 publications

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“…Hamadiche et al [21] considered the flow in a symmetric diverging channel over a finite domain (i.e. a ≤ r ≤ b) and indeed found 219 the bifurcation to be supercritical.…”

Section: Note On Weakly Nonlinear Stability Theorymentioning

confidence: 99%

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

McAlpine

Nash

Lowson

1999

Journal of Sound and Vibration

126

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Tonal noise, the self-induced discrete frequency noise generated by aerofoils, is investigated. It is heard from an aerofoil placed in streams at low Mach number flows when inclined at a small angle to the stream. The tones are heard as a piercing whistle, commonly up to 30 dB above the background noise level. The work is motivated by the occurrence of tonal noise from rotors, fans and recently wind-turbines. Previous authors have attributed tonal noise to a feedback loop consisting of a coupling between laminar boundary-layer instability waves and sound waves propagating in the free stream. The frequency has been predicted by use of various methods based on this model.In this thesis a review of wind-tunnel results obtained by Dr. E.C. Nash at the University of Bristol is presented. Boundary-layer measurements show the presence of tonal noise is closely related to the existence of a region of separated flow close to the trailing edge of the aerofoil. Highly amplified boundary-layer instability waves were observed close to the trailing edge of the aerofoil at the frequency of the tone.A comprehensive analysis of the linear stability of the boundary-layer flow over the aerofoil is presented. The growth of boundary-layer instability waves over the aerofoil is calculated. The growth rates of the waves were obtained by solving the OrrSommerfeld problem at several stations on the aerofoil. The Falkner-Skan boundary layers were found to be a suitable form of velocity profiles to incorporate the adverse pressure gradients experienced by the flow over an aerofoil. The amplification of the instability waves is shown to be controlled almost entirely by the region of separated flow close to the trailing edge. The calculated frequency of the linear modes with maximum amplification over the aerofoil is found to be close to the observed frequency of the acoustic tone.A weakly nonlinear stability analysis was also performed and this appears to be a suitable description of the boundary-layer instability waves. The results indicate that the frequency of the tones may commonly be predicted to within 10% by using weakly nonlinear stability theory.The generation of sound by diffraction of the boundary-layer instability waves at the trailing edge of the aerofoil is also discussed as well as the proposed feedback models. A modified feedback model is proposed, being based on the experimental and theoretical results.i

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“…Hamadiche et al [21] considered the flow in a symmetric diverging channel over a finite domain (i.e. a ≤ r ≤ b) and indeed found 219 the bifurcation to be supercritical.…”

Section: Note On Weakly Nonlinear Stability Theorymentioning

confidence: 99%

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

McAlpine

Nash

Lowson

1999

Journal of Sound and Vibration

126

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“…However, their work did not compute the bifurcation directly and no critical parameter values were provided. Hamadiche, Scott & Jeandel (1994) presented the results of unsteady calculations in a wedge of finite length, finding a neutral curve for temporal stability in good agreement with B 2 for small α. However, they observed time-periodic nonlinear states for Reynolds numbers greater than the critical value perhaps indicating that they believed the loss of stability to arise via a Hopf bifurcation, although this is not stated explicitly.…”

Section: Introductionmentioning

confidence: 71%

“…Although we have computed stable periodic orbits we have not been able to do so for the parameter values (α = 0.1, 0.3) cited by Hamadiche et al (1994). We do not believe that stable periodic orbits exist at such small values of α but can offer no explanation for this discrepancy.…”

mentioning

confidence: 79%

“…In particular, there are contradictions regarding the bifurcation sequence obtained as the flow rate is increased, as highlighted by the supercritical pitchfork inferred by Sobey & Drazin (1986) and Dennis et al (1997), the supercritical Hopf bifurcation observed by Hamadiche et al (1994) and the symmetry-broken 'spatial wave' found by Tutty (1996) and Kerswell et al (2004). Whilst the available experimental evidence (Sobey & Drazin 1986;Putkaradze & Vorobieff 2006) qualitatively supports a supercritical pitchfork bifurcation, neither Sobey & Drazin nor Dennis et al gave values for the critical Reynolds number of this bifurcation.…”

Section: Introductionmentioning

confidence: 97%

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The Jeffery–Hamel similarity solution and its relation to flow in a diverging channel

2011

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We explore the relevance of the idealized Jeffery-Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle 2α, bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the (Re, α) plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where Re is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for α 1, superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231-250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.

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“…Except a limited number of these problems, most of them do not have precise analytical solutions so that they have to be solved using other methods. Many different new methods have recently presented some techniques to eliminate the small parameter; for example, the variational iteration method (VIM) [6][7][8][9] and the Exact solutions (ADM) [10,11], HPM [12][13][14][15][16][17], and others [18,19]. In [25], a new perturbation algorithm combining the method of multiple scales and lindstedt-Poincare techniques is proposed for first time.…”

Section: Introductionmentioning

confidence: 99%

Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM)

Saedodin

Shahbabaei

2013

Arab J Sci Eng

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In this paper, porous fin has been studied and its nonlinear ordinary differential equation has been solved through homotopy perturbation method. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary and initial conditions. To study the thermal performance, one type case is considered: finite-length fin with insulated tip.

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Temporal stability of Jeffery–Hamel flow

Cited by 67 publications

References 17 publications

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

On the Generation of Discrete Frequency Tones by the Flow around an Aerofoil

The Jeffery–Hamel similarity solution and its relation to flow in a diverging channel

Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM)

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