Plenary Lecture
Amplitude Equations for the Analysis of Stability of Fluid Flows
Professor Andrei Kolyshkin
Department of Engineering Mathematics
Riga Technical University
LATVIA
E-mail: andrejs.koliskins@rbs.lv
Abstract: Linear theory of hydrodynamic stability is used in practice in order to determine conditions where a particular base flow becomes unstable. Linear stability analysis provides a marginal stability curve which separates regions of linear stability and instability in the parameter space. In addition, critical value of the parameter S (where S is the critical Reynolds number for viscous flows or critical bed-friction number for shallow flows) can be obtained numerically. However, the linear stability theory cannot predict what happens with the most unstable mode when the base flow becomes unstable. Weakly nonlinear theories (based on the method of multiple scales) are used in such cases in order to analyze the development of instability above the threshold.
In the present talk we use weakly nonlinear theory to derive amplitude evolution equations for the most unstable mode. It is shown that evolution equations are obtained using the Fredholm’s alternative. Two main approaches are used in the talk to illustrate the theory: (a) weakly nonlinear analysis based on a parallel flow assumption and (b) weakly nonlinear analysis based on the assumption that the base flow is slightly changing downstream. In case (a) the base flow is fixed at a particular station downstream (thus, the base flow is a function of one transverse coordinate) and weakly nonlinear expansion is constructed in the neighbourhood of the critical point where the parameter S is slightly above the threshold. Examples presented in the talk include single-phase and two-pahse shallow flows and transient flows in pipes. It is shown that in all the above mentioned cases the amplitude evolution equation is the complex Ginzburg-Landau (GL) equation. The coefficeints of the GL equation are expressed in terms of integrals containing characteristics of the linear stability problems. In case (b) the small parameter is the ratio of the length scale of the unstable mode and the length scale of the evolution of the base flow. First-order evolution equation is obtained in the case of single-phase and two-phase shallow flows where the base flow is slightly changing downstream. Numerical values of the coefficients of evolution equations can be used in order to describe possible scenarios of the development of instability. For example, the sign of the real part of the Landau constant in the GL equation can be used to describe whether a finite amplitude saturation of the most unstable mode will take place. Numerical examples are discussed in detail.
Brief Biography of the Speaker: Andrei Kolyshkin received his undergraduate degree in Applied Mathematics in 1976 at the Riga Technical University. In 1981 he received a Ph.D in differential equations and mathematical physics at the University of St. Petersburg (Russia). Andrei Kolyshkin is currently a full professor at the Department of Engineering Mathematics at the Riga Technical University. His current research interests include investigation of stability problems in fluid mechanics with applications to open- channel flows, transient flows in hydraulic systems and mathematical models for eddy current testing. He is the co-author of three monographs published by Academic Press and CRM. Andrei Kolyshkin has participated in more than 40 international conferences and has published more than 70 papers in refereed journals since 1980. As a visiting professor and visiting researcher he spent a few years at the University of Ottawa and Hong Kong University of Science and Technology.