SVD ANALYSIS OF COMPLEX FLOW PATTERNS IN TWO-PHASE FLOWS
Luciano Cantelli, Alberto Fichera
This paper describes a novel approach for the analysis of air-water two phase flows in vertical channels, aiming at exploiting the chaotic nature of experimental flow patterns.
An experimental apparatus, consisting of a vertical pipes connected through a mixing section to the air and water lines, has been tested for a wide range of air and water mass flow rates, corresponding to several kinds of flow patterns, ranging from bubbly flow to churn flow. During the experimental tests the nonlinear void fraction oscillations have been detected by means of a fast response resistive void fraction sensor [1]. Preliminary application of linear tools, such as the analysis of power spectral density distributions and of autocorrelation functions, have pointed out the inadequacy of such approaches to deal with the complexity of the system dynamics [2].
In this study phase space analysis has been preferred in order to allow for the observation of the complex dynamics of the system, with the aim of searching for hints of its chaotic nature.
At first, an n-dimensional representation state space has been reconstructed on the basis of Takens’ theorem, i.e. by means of a set of n delayed versions of the experimental void fraction time series [3]. The analysis of the attractors points out the existence of a complex but regular structure in phase space, which constitutes a first hint of deterministic behaviour. Nonetheless, attractors obtained in this way are affected by a relatively high amount of noise, mostly due to the superposition of the high order dynamics associated to small diameter dispersed bubbles to the dominant dynamics characterizing the flow pattern.
In order to address this problem, Singular Value Decomposition (SVD) has been applied to the n-dimensional state space in order to determine its eigenvalues and, in particular, the attractor projection onto the eigenvectors state space [4-5]. This has allowed substantial separation of the dominant features of the system dynamics from the noise-like dynamics. In fact, the projection of the n-dimensional attractors in the three-dimensional representations space formed by the dominant eigenvectors are satisfactorily unfolded. This can be observed by comparing the projection of the attractor in the representation space spanned by the three principal eigenvectors. From this kind of analysis it is possible to observe the existence of a well defined and unfolded structure, in various cases characterised by an evident fractal nature, which is a clear proof of the chaotic nature of void fraction dynamics in two-phase flow patterns.
Moreover, comparative analyses have shown that each type of flow pattern is characterised by a specific morphology, sufficiently different from that of other flow patterns. In particular, different regions of the phase space are occupied by the attractors of the various flow patterns.
Further analyses will be developed in order to define an appropriate methodology for two phase flow patterns classification based on the results of SVD analysis.
References
[1] Cantelli L., Fichera A., Guglielmino I.D., Pagano A., Preliminary Analyses of the Experimental Dynamics of an Air-Water Two-Phase Flow in a Vertical Pipe, XXIII Congr. UIT 2005, Parma, 20-22 Giugno 2005.
[2] Cantelli L., Fichera A., Guglielmino I.D., Pagano A., Non-Linear Dynamics of Air-Water Mixtures in Vertical Pipes: Experimental Trends, Int. J. Bifurc. nd Chaos, Vol. 16-9, 2006.
[3] Takens F., Lecture Notes in Mathematics, Dynamical System and Turbulence, D.A. Rand & L.S. Young Editions, Springer, New York, 1981.
[4] Broomhead D.S., King G.P., Extracting Qualitative Dynamics from Experimental Data, Physica D, Vol. 20, 217-236 1986.
[5] Drahos J., Zahradnik J., Puncochar M., Fialova M., Chen K., Bradka F., Effect of operating conditions of the characteristics of pressure fluctuations in a bubble column, Chemical Eng. Processing, Vol. 29, pp. 107-105, 1991.