Please use this identifier to cite or link to this item: http://hdl.handle.net/1783.1/1934
A quasisteady approach to the instability of timedependent flows in pipes
Authors  Ghidaoui, Mohamed Salah
Kolyshkin, Andrei A. 


Issue Date  2002  
Source  Journal of fluid mechanics, v. 465, 2002, AUG 25, p. 301330  
Summary  Asymptotic solutions for unsteady onedimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady onedimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a threedimensional perturbation is investigated and the results are used to reinterpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. delta(1)R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is nonaxisymmetric for delta(1)R = 0.55 and 0.85. In addition, when delta(1)R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasisteady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasisteady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex GinzburgLandau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finiteamplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).  
Subjects  
ISSN  00221120  
Rights  © Cambridge University Press 2002. This paper was published in Journal of Fluid Mechanics, v. 465, 2002, p. 301330 and is reprinted with permission.  
Language  English 

Format  Article  
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