ASPECTS OF LOITSIANSKY’S TYPE OF INVARIANT
*,1Ghosh, N. C., 2Pijush Basak and 3Abhijit Bhattacharya
1S.N. Bose Institute for Mathematics & Mathematical Sciences
2Narula Institute of Technology
3B.P. Poddar Institute of Management & Technology
ABSTRACT
Turbulence is seen as one of the last outstanding unsolved problems in classical physics. In the last century, great minds viz, Heisenberg, von Weizs"acker, Kolmogorov, Prandtl and G.I. Taylor had worked on it. Einstein put his last postdoc Bob Kraichnan on the subject of Turbulence. Despite the fact that isotropic turbulence constitutes the simplest type of turbulent flow, it is still not possible to render the problem analytically traceable without introducing the two point double and triple longitudinal velocity correlations to admit self-similarity solution with respect to a single length-scale, which has served as a useful hypothesis since its inception by von Karman and Howarth (1938). Rapid development of experimental and numerical techniques in this area and the growth of computing power created a lot of activities on turbulence research. Here authors have elaborated a debated concept Loitsiansky’s type of invariant associated with turbulent study from analytical point.
Copyright©2016 Ghosh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.INTRODUCTION
Homogeneous isotropic turbulence is a kind of idealization for real turbulent motion, under the assumption that the motion is governed by a statistical law invariant for arbitrary translation (homogeneity), rotation or reflection (isotropy) of the coordinate system. This idealization was first introduced by Taylor (1935) and used to reduce the formidable complexity of statistical expression of turbulence and thus made the subject feasible for theoretical treatment. Up to now, a large amount of theoretical work has been devoted to this rather restricted kind of turbulence. However, turbulence observed either in nature or in laboratory has much more complicated structure.
Although remarkable progress has been achieved so far in discovering various characteristics of turbulence, our understanding of the fundamental mechanism of turbulence is still partial and unsatisfactory. The assumption of similarity and self-preservation, which permits an analytical determination of the energy decay in isotropic turbulence, has played an important role in the development of turbulence theory for more than half a century. In the traditional approach to search for similarity solutions for turbulence, the existence
of a single length and velocity scale has been assumed, and then the conditions for the appearance of such solutions have been examined. Excellent contributions had been made to this direction by von Karman and Howarth (1938), who firstly deduced the basic equation and presented a particular set of its solutions for the final decaying turbulence. Later on, two Russian scholars, Loitsiansky (1939) and Millionshtchikov (1941), separately obtained the solutions for the Karman-Howarth equation after the term related to the effect of the
triple velocity correlation has been neglected (Ghosh and Ghosh, 1982; Ghosh, 2001). Their work was an extension of the “small Reynolds number” solution first given by von Karman and Howarth. Dryden gave a comprehensive review on this subject (Dryden, 1943). Detailed research on the
solutions of the Karman-Howarth equation was conducted by Sedov, who showed that one could use the separability constraint to obtain the analytical solution of the Karman-Howarth equation (Sedov, 1944). Sedov’s solution could be expressed in terms of the confluent hypergeometric function. Batchelor (1948) readdressed this problem under the assumption that the Loitsiansky integral is a dynamic invariant, which was a widely accepted assumption, but was later found to be invalid. Batchelor concluded that the only complete self-preserving solution which was intrinsically consistent existed at low turbulence Reynolds number, for which the turbulent kinetic energy is accordant with the final period of turbulent decay. Batchelor also found a self-preserving solution to the Karman-Howarth equation in the limit of infinite Reynolds number, for which the Loitsiansky integral is an invariant. Objections were later raised against using the Loitsiansky integral as a dynamic invariant. In fact, at high Reynolds number this integral can be proved to be a weak function of time Proudman and Reid (1954) and Batchelor and Proudman (1956). Saffman proposed an alternative dynamic invariant which yielded another powerlaw decay in the limit of infinite Reynolds number (Hinze,1975). While the results of Batchelor and Saffman formally constitute complete self-preserving solutions to the inviscid
Karman-Howarth equation, it must be kept in mind that they only exhibit partial self-preservation with respect to the full viscous equation. Later on, George (1992) revived this issue concerning the existence of complete self-preserving solutions in isotropic turbulence. In an interesting paper he claimed to find a complete self-preserving solution, valid for all Reynolds numbers. George’s analysis was based on the dynamic equation for the energy spectrum rather than on the Karman-Howarth equation. Strictly speaking, the solution presented by George was an alternative self-preserving solution to the equations of Karman-Howarth and Batchelor since George relaxed the constraint that the triple longitudinal velocity correlation is self-similar in the classical sense. Speziale and Bernard (1992) reexamined this issue from a basic theoretical
and computational standpoint. Several interesting conclusions have been drawn from their analysis. From the development of turbulence theory, it is known that the research on decaying homogeneous isotropic turbulence is one of the most important and extensively explored topics. Despite all the efforts, a general theory describing the decay of turbulence based on the first principles has not yet been developed (Skrbek and Stalp,2000). It seems that the theory of self-preservation in homogeneous turbulence has lots of interesting features which have not yet been fully understood and are worth of further study (Speziale and Bernard, 1992). This paper offers a short, but interesting feature of Loitsiansky invariant unified investigation of isotropic turbulence, based on the exact solutions of the Karman-Howarth equation. The statistical procedures of random fluctuation in a turbulent flow has been investigated by Taylor (1935), Robertson (Robertson, 1940) and many others. Karman and Howarth (1938) proposed the equation of translation for a second order velocity correlation of a homogeneous isotropic flow field. The equation constituting the relation of second order velocity correlation with the third order velocity correlation in collaboration with
viscous damping denoted by Laplacian of second order velocity correlations.*Corresponding author: Ghosh, N. C.
S.N. Bose Institute for Mathematics & Mathematical Sciences
ISSN: 0976-3376 Asian Journal of Science and Technology
Vol. 07, Issue, 11, pp.3857-3861, November, 2016
