Distortion of line and surface elements in model turbulent flows

Abstract

Material lines and surfaces transported in a random velocity field undergo bending and stretching. In this paper we investigate the time evolution of curvature in line and surface elements both analytically and by numerical simulation for a simple model turbulence. Our analysis is close to that of Pope (1988) for the evolution of curvature in surface elements. We show that the equation governing the evolution of curvature in a line element is very similar to that governing the evolution of the principal curvature in a surface patch. We investigate the circumstances in which the effect of straining fluctuations is to cause the exponential rate of growth of curvature discovered by Pope et al. (1989). Our simulation confirms that the presence of helicity in the turbulent flow results in the development of a non-vanishing mean torsion in a line element. The results of the simulation also suggest that the generation of curvature tends to occur in regions different from those associated with rapid stretching. The generation of torsion, however, is found not to be correlated with either bending or stretching.