Spatial and temporal instability of slightly curved particle-laden shallow mixing layers

In the present paper we present linear and weakly nonlinear models for the analysis of stability of particle-laden slightly curved shallow mixing layers. The corresponding linear stability problem is solved using spatial stability analysis. Growth rates of the most unstable mode are calculated for different values of the parameters of the problem. The accuracy of Gaster’s transformation away from the marginal stability curve is analyzed. Two weakly nonlinear methods are suggested in order to analyze the development of instability analytically above the threshold. One method uses parallel flow assumption. If a bed-friction number is slightly smaller than the critical value then it is shown that the evolution of the most unstable mode is governed by the complex Ginzburg-Landau equation. The second method assumes that the base flow is slightly changing downstream. Applying the WKB method we derive the first-order amplitude evolution equation for the amplitude

Stability analysis of velocity profiles in water-hammer flows

This paper performs linear stability analysis of base flow velocity profiles for laminar and turbulent water-hammer flows. These base flow velocity profiles are determined analytically, where the transient is generated by an instantaneous reduction in flow rate at the downstream end of a simple pipe system. The presence of inflection points in the base flow velocity profile and the large velocity gradient near the pipe wall are the sources of flow instability. The main parameters that govern the stability behavior of transient flows are the Reynolds number and dimensionless timescale. The stability of the base flow velocity profiles with respect to axisymmetric and asymmetric modes is studied and its results are plotted in the Reynolds number/timescale parameter space. It is found that the asymmetric mode with azimuthal wave number 1 is the least stable. In addition, the results indicate that the decrease of the velocity gradient at the inflection point with time is a stabilizing mechanism whereas the migration of the inflection point from the pipe wall with time is a destabilizing mechanism. Moreover, it is shown that a higher reduction in flow rate, which results in a larger velocity gradient at the inflection point, promotes flow instability. Furthermore, it is found that the stability results of the laminar and the turbulent velocity profiles are consistent with published experimental data and successfully explain controversial conclusions in the literature. The consistency between stability analysis and experiments provide further confirmation that (I) water-hammer flows can become unstable; (2) the instability is asymmetric; (3) instabilities develop in a short (water-hammer) timescale; and (4) the Reynolds number and the wave timescale are important in the characterization of the stability of water-hammer flows. Physically, flow instabilities change the structure and strength of the turbulence in a pipe, result in strong flow asymmetry, and induce significant fluctuations in wall shear stress. These effects of flow instability are not represented in existing water-hammer models

Linear stability analysis of lateral motions in compound open channels

A linear stability analysis of lateral motions in shallow flow open channels with a free surface is presented. Unlike previous studies, this work does not employ the rigid-lid assumption. Computations revealed that the rigid-lid assumption works well for weak shear flows and/or small Froude numbers. However, the validity of the rigid-lid assumption is reduced as the Froude number increases. Also, when Reynolds numbers are larger than 1,000, the stability domain is insensitive to the Reynolds number for all Froude numbers. The size of the stability domain is found strongly affected by the velocity and length scales of base velocity and the shape of the velocity profile. The analysis shows that the stability domains obtained by the Manning formula are larger than those obtained by the Chezy formula for the same Froude number. The analysis shows that a compound channel, whose floodplain has larger flow resistance than the main channel does, has larger lateral transport of flow mass, momentum, and energy

Stability analysis of shallow wake flows

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 Grubisic 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

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. 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 non-axisymmetric 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 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)