The measurement and interpretation of fractal dimensions of the scalar interface in turbulent flows

Abstract

One of the recently established results concerns the fractal-like properties of surfaces such as the turbulent/nonturbulent interface. Although several confirmations have been reported in recent literature, enough discussion does not exist on how various flow features as well as measurement techniques affect the fractal dimension obtained; nor, in one place, is there a full discussion of the physical interpretation of such measurements. This paper serves these two purposes by examining in detail the specific case of the interface of scalar-marked regions (scalar interface) in turbulent shear flows. Dimension measurements have been made in two separate scaling regimes, one of which spans roughly between the integral and Kolmogorov scales (the K range), and the other between the Kolmogorov and Batchelor scales (the B range). In the K range, the fractal dimension is 2.36±0.05 to high degree of reliability. This is also the dimension of the vorticity interface. The dimension in the B range approaches (logarithmically) the value 3 in the limit of infinite Schmidt number, and is 2.7±0.03 when the diffusing scalar in water is sodium fluorescein (Schmidt number of the order 1000). Among the effects considered are those of (a) the flow Reynolds number, (b) developing regions such as the vicinity of a jet nozzle or a wake generator, (c) the free-stream and other noise effects, (d) the validity of the method of intersections usually invoked to relate the dimension of a fractal object to that of its intersections, (e) the effect of intersections by ‘‘slabs’’ of finite thickness and ‘‘lines’’ of finite width, and (f) the computational algorithm used for fractal dimension measurement, etc. The authors’ previous arguments concerning the physical meaning of the fractal dimension of surfaces in turbulent flows are recapitulated and amplified. In so doing, turbulent mixing is examined, and by invoking Reynolds and Schmidt numbers similarities, the fractal dimensions of scalar interfaces are deduced when the Schmidt number is small, unity, and large.