Hessian-based topology of two-phase slug flow

Abstract

Experimental slug flow is analyzed through a topological method and proper orthogonal decomposition (POD). Local phase fractions of a pipe cross-section, acquired through X-ray tomographic reconstruction, are analyzed by extracting critical points via eigenvalues associated with the Hessian matrix. Based on the sign of the eigenvalues, three types of critical points are classified: local maxima, minima and saddle points. Reduced order descriptions (ROD), obtained as results of the POD, are examined to investigate the flow dynamics associated with the primary eigenfunctions with respect to critical point placement and frequency. Voronoï mapping is employed for visualization of the critical points as a function of cross-sectional location, and quantification of the spatial distribution of the critical points. For each classification, the sum of the respective critical point over the cross-section correlates to the temporal holdup of liquid within the pipe. The local maxima and minima display an increase in their occurrence that relates to the liquid slug passage while the local saddle points follow the profile of the liquid holdup as the bubble is forming after the passage of the liquid slug. The total number of local maxima per snapshot as a function of time peaks prior to the liquid holdup peak, displaying the instabilities present in the flow field preceding the liquid slug reaching the top of the pipe. Applying critical point theory to the RODs reveals that the first two modes capture the liquid-dominated slug body region and the Taylor bubble/liquid film region. The observed time lag between the total maxima and the liquid holdup provides opportunity for implementation of predictive control monitoring in fields where instabilities influence system mechanics.