Fractal dimensions and spectra of interfaces with application to turbulence

Abstract

This paper is concerned with the analysis of any convoluted surface in two or three dimensions which has a self-similar structure, and which may simply be defined as a mathematical surface or as an interface where there is a sharp change in the value of a scalar field F(x) (say from 0 to 1). The different methods of analysis that are related to each other here are based on spectra of the scalar which have the form Γ(k) ∝ k - p , the Kolmogorov capacity D K of the interface or D' K of the intersections of the interface with a plane or a line (both being defined by algorithms for counting the minimum number of boxes of sizes ∊ either covering the surface or its intersection), and the Hausdorff dimensions D H , D' H (which are defined differently). It is demonstrated that interfaces with a localized self-similar structure around accumulation points, such as spirals, may have non-integer capacities D K and D' K even though their Hausdorff dimension is integer and equal to the topological dimension of the surface. It is explained how the same surface can have different values of D K and D' K over different asymptotic ranges of ∊ . There are other (fractal) surfaces where both D H and D K are non-integer which are convoluted on a wide range of scales with the same form of self-similarity everywhere on the surface. Distinctions are drawn between these two kinds of interface which have local and global self­- similarity respectively. If the intersections of the interface with any line form a set of points that is statistically homogeneous and independent of the location and orientation of the line and also that is self-similar over a sufficiently wide range of spacing that a capacity D' K can be defined, it is shown that the scalar F has a power spectrum of the form of (1) and that the exponent p is related to D' K by p + D' K = 2. This quite general result for interfaces is verified analytically and computationally for spirals. In experiments with scalar interfaces in different turbulent flows at high Reynolds and Prandtl number, K. R. Sreenivasan, R. Ramshankar and C. Meneveau’s measurements showed that D' K = 0.33 for values of ∊ within the inertial range and D' K = 0 for smaller values (in the microscale range). The values of p derived from (2) are consistent with the theory of G. K. Batchelor and many measurements of scalar spectra. For fractal interfaces with global self-similarity, the values of D H of an interface and D' H of the intersections of the interface with a line, have been shown previously to be related simply to each other by the topological dimension E of the interface so that D H = D' H + E . No such theorem exists in general for the Kolmogorov capacities D K and D' K . But it is shown analytically and computationally that for the case of spirals, over a certain range of resolutions ∊ , D K = D' K + E . This corresponds in practice to the measurable range of typical experimental spirals with fewer than about five turns. Over a range of smaller length scales ∊ , where a larger number of turns is resolved and which is experimentally difficult to measure, (3) is not correct. The result (3) has been previously suggested based on experimental results. Finally we demonstrate that interfaces for which there is a well-defined value of capacity (which is indeed the case for spirals of three turns) are only found to have self-similar spectra if there is a much wider range of length scales (e. g. more than 50 turns of the spiral) than is needed for the capacity D K to be measurable. As well as demonstrating this computationally, this is proved mathematically for interfaces having a particular class of accumulation points whose intersections with straight lines form a self-similar sequence of points x n ∝ n -α ; the power spectrum of F only tends to the self-similar form (1) if ( ∊ min / ∊ max ) 1- D' K ≪ 1, whereas the capacity measure simply requires that ∊ min / ∊ max ≪ 1. So when D' K > 0, the criteria for the spectra requires a wider range of scales in the convolutions of the interface. This is consistent with the finding that reliable measurements of D K can be computed from measurements of interfaces in laboratory experiments, but in the same experiments computations of spectra are often not of the form (1). Therefore, despite this apparent discrepancy with the general result (2) (which implies a value of p given a value of D' K ), the above theoretical argument supports the deduction from such experiments that if a non-integer value of D K is measured, the interface does indeed have a self-similar structure; but it is not self-similar over a wide enough range of length scales to satisfy (2). Consequently, for turbulent flows, stating that the capacity should have its asymptotic value rather than (as is usual) the spectrum should be equal to its asymptotic form (e. g. in (1) p = 5/3) may be the correct necessary condition for deciding whether, in a given flow, the interfaces have the characteristic structure found at very high Reynolds number. Many of the results here may be of value in other scientific fields, where convoluted interfaces are also studied.