Understanding fully developed turbulence: Some new ideas

Abstract

The present paper is a contribution within the E(∞) Cantorian space theory. After a brief review of the historical development of length and time measurements, we consider conventional definitions and methods of calculation of vector and fractal dimensions. A definition of emptiness as zero entropy is discussed.Treating the `fabric' of space-time as non-permanent, we consider other possible ways of filling emptiness. This leads to a definition of range dimension (p), the properties of which are then discussed. p=1/2 and p=1/4 range dimensions are linked to p=1 (conventional) range dimensions in fluid environments using differential velocities and accelerations. Applications of p=2 range dimensions to quantum events are also considered, and eclectic evidence for the existence of range dimensions is presented.The bijection relationship, defining space-filling fractal dimension (dc,n) is extended to cover the range 1<n<4 vector dimensions, and a physical interpretation in terms of a characteristic length ratio (ln/l0) is developed. The n-dimensional term (ln/l0)n is then closely correlated to the ratio (l1/l0)dc,n. The bijection relationship is extended to other range dimensions.An approximate empirical relationship is now presented, which provides a basis for defining conventional (p=1) range dimensional space, by introducing Nn, the number of elements needed to set up 1–4 vector dimensions. Two other ways of obtaining approximately the same number of elements are also given, one of which leads to the concept of a p=2 range dimensional substrate for conventional (p=1) range dimensions. It is found that a single track in p=1 range dimensions needs nearly p=2 range dimensions to support it. The required value then falls as n increases, equaling approximately unity for n=3 vector dimensions. A second empirical equation defining the same value of Nn for 3<n<4 vector dimensions, based on space-filling fractal dimensions for p=2 range dimensions is also presented.A fractal form for conventional space is then defined, characterised by tiny elements of negative entropy (in log space). This definition is then readily amalgamated with dynamically generated space, and the latter is shown to correlate with the classical gas equation and Hooke's law for solids. Based on the new definition of space, a proof of both the well-known inverse square law and Hubble's astronomical law is given.A relationship between the number of events in p=1andp=2 range dimensions is next established, and the effect of `information loss' corresponding to the reduction in the required value of range dimension as vector dimension (n) increases is tentatively linked to wave functions, for n=4 vector dimensions. The corresponding `useful' information is then shown to be contained in the exponent field, which is of the form {2f(e−1)pdc,4}, where f≈1,p=2 range dimensions, and e (within the exponent!) equals the numerical value of the exponent. This field is then sub-divided to various extents, and the corresponding entropies are used to determine sets of distinguishable arrangements N. These sets are successfully compared with `magic' numbers in atomic and nuclear physics, and with preferred sodium atom groupings. A prediction of the number of occupied energy levels in atomic electron shell theory is successfully carried out.Finally, combining `useful' and `hidden' information the p=2 range dimension substrate is re-created by allowing matter to collapse and `drain away' from its p=2 range dimensional space base. Light propagation through space having p=2 range dimensions is then discussed.The paper concludes with a discussion of various ideas implicit in the text, though not necessarily directly obvious. The interdependence of fractal, range and vector dimensions is first considered, followed by a review of the universal application of range dimensions. The paper ends with a study of the fractal nature of space and its justification.