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Perturbation dynamics in unsteady pipe flows
- M. Zhao, M. S. Ghidaoui, A. A. Kolyshkin
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- Journal:
- Journal of Fluid Mechanics / Volume 570 / 10 January 2007
- Published online by Cambridge University Press:
- 14 October 2021, pp. 129-154
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This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.
Double conductor line above a two-layered cylinder with varying properties
- A. A. Kolyshkin, Rémi Vaillancourt
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- Journal:
- Journal of the Australian Mathematical Society. Series B. Applied Mathematics / Volume 38 / Issue 1 / July 1996
- Published online by Cambridge University Press:
- 17 February 2009, pp. 1-15
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The change of impedance per unit length in a single or double conductor line situated parallel to an infinitely long two-layered metallic circular cylinder is found (within the quasistatic approximation) in the form of an infinite series. The cylinder consists of an inner core and an outer annulus. The properties of the inner core are assumed to be constant. The relative magnetic permeability, μ(r) = rα and the conductivity, σ(r) = σ(0)rκ, of the outer annulus vary with respect to the radial coordinate, r, and α and κ are arbitrary real numbers. Numerical results are presented in the form of figures and tables.
Linear and nonlinear analysis of shallow wakes
- M. S. GHIDAOUI, A. A. KOLYSHKIN, J. H. LIANG, F. C. CHAN, Q. LI, K. XU
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- Journal:
- Journal of Fluid Mechanics / Volume 548 / 10 February 2006
- Published online by Cambridge University Press:
- 01 February 2006, pp. 309-340
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The bottom friction and the limited vertical extent of the water depth play a significant role in the dynamics of shallow wakes. These effects along with the effect of the strength of the shear layer define the wake parameter $S$. A nonlinear model, based on a second-order explicit finite volume solution of the depth-averaged shallow water equation in which the fluxes are obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, is developed and applied to shallow wake flows for which laboratory data are available. The velocity profiles, size of the recirculating wake, oscillation frequency, and wake centreline velocity are studied. The computed and measured results are in reasonable agreement for the vortex street (VS) and unsteady bubble (UB) regimes, but not for the steady bubble (SB). The computed length of the recirculation region is about 60% shorter than the measured value when $S$ belongs to the SB regime. As a result, the stability investigation performed in this paper is restricted to $S$ values away from the transition between SB and UB. Linear analysis of the VS time-averaged velocity profiles reveals a region of absolute instability in the vicinity of the cylinder associated with large velocity deficit, followed by a region of convective instability, which is in turn followed by a stable region. The frequency obtained from Koch's criterion is in good agreement with the shedding frequency of the fully developed VS. However, this analysis does not reveal the mechanism that sets the global shedding frequency of the VS regime because the basic state is obtained from the VS regime itself. The mechanism responsible for VS shedding is sought by investigating the stability behaviour of velocity profiles in the UB regime as $S$ is decreased towards the critical value which defines the transition from the UB to the VS. The results show that the near wake consists of a region of absolute instability sandwiched between two convectively unstable regions. The frequency of the VS appears to be predicted well by the selection criteria given in Pier & Huerre (2001) and Pier (2002), suggesting that the ‘wave-maker’ mechanism proposed in Pier & Huerre (2001) in the context of deep wakes remains valid for shallow wakes. The amplitude spectra produced by the nonlinear model are characterized by a narrow band of large-amplitude frequencies and a wide band of small-amplitude frequencies. Weakly nonlinear analysis indicates that the small amplitude frequencies are due to secondary instabilities. Both the UB and VS regimes are found to be insensitive to random forcing at the inflow boundary. The insensitivity to random noise is consistent with the linear results which show that the UB and VS flows contain regions of absolute instabilities in the near wake where the velocity deficit is large.
Stability analysis of shallow wake flows
- A. A. KOLYSHKIN, M. S. GHIDAOUI
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- Journal:
- Journal of Fluid Mechanics / Volume 494 / 10 November 2003
- Published online by Cambridge University Press:
- 22 October 2003, pp. 355-377
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Experimentally observed periodic structures in shallow (i.e. bounded) wake flows are believed to appear as a result of hydrodynamic instability. Previously published studies used linear stability analysis under the rigid-lid assumption to investigate the onset of instability of wakes in shallow water flows. The objectives of this paper are: (i) to provide a preliminary assessment of the accuracy of the rigid-lid assumption; (ii) to investigate the influence of the shape of the base flow profile on the stability characteristics; (iii) to formulate the weakly nonlinear stability problem for shallow wake flows and show that the evolution of the instability is governed by the Ginzburg–Landau equation; and (iv) to establish the connection between weakly nonlinear analysis and the observed flow patterns in shallow wake flows which are reported in the literature. It is found that the relative error in determining the critical value of the shallow wake stability parameter induced by the rigid-lid assumption is below 10% for the practical range of Froude number. In addition, it is shown that the shape of the velocity profile has a large influence on the stability characteristics of shallow wakes. Starting from the rigid-lid shallow-water equations and using the method of multiple scales, an amplitude evolution equation for the most unstable mode is derived. The resulting equation has complex coefficients and is of Ginzburg–Landau type. An example calculation of the complex coefficients of the Ginzburg–Landau equation confirms the existence of a finite equilibrium amplitude, where the unstable mode evolves with time into a limit-cycle oscillation. This is consistent with flow patterns observed by Ingram & Chu (1987), Chen & Jirka (1995), Balachandar et al. (1999), and Balachandar & Tachie (2001). Reasonable agreement is found between the saturation amplitude obtained from the Ginzburg–Landau equation under some simplifying assumptions and the numerical data of Grubĭsić et al. (1995). Such consistency provides further evidence that experimentally observed structures in shallow wake flows may be described by the nonlinear Ginzburg–Landau equation. Previous works have found similar consistency between the Ginzburg–Landau model and experimental data for the case of deep (i.e. unbounded) wake flows. However, it must be emphasized that much more information is required to confirm the appropriateness of the Ginzburg–Landau equation in describing shallow wake flows.
A quasi-steady approach to the instability of time-dependent flows in pipes
- M. S. GHIDAOUI, A. A. KOLYSHKIN
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- Journal:
- Journal of Fluid Mechanics / Volume 465 / 25 August 2002
- Published online by Cambridge University Press:
- 02 September 2002, pp. 301-330
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Asymptotic solutions for unsteady one-dimensional 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 one-dimensional 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 three-dimensional perturbation is investigated and the results are used to re-interpret 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. δ1/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 non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/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 quasi-steady 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 quasi-steady 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 Ginzburg–Landau 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, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).