A New Look at the Final Period Decay of. Homogeneous Isotropic Turbulence

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1 International Mathematical Forum Vol. 8 1 no HIKARI Ltd A New Look at the Final Period Decay of Homogeneous Isotropic Tbulence C. Mamaloukas Department of Statistics Athens University of Economics and Business Greece mamkris@aueb.gr Copyright 1 C. Mamaloukas. 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. Abstract Final-period decay law in homogeneous and isotropic tbulence is reexamined here from the concept of a simple point-merging technique. Keywords: Homogeneous and isotropic tbulence simple point-merging technique 1 Introduction Ghosh (197) investigated the early-period decay process of a general type of tbulence using quasi-normality hypothesis. He established two lemmas concerning merger of points in the physical space. First lemma: This lemma is concerned with the behavior of correlation tensors in the energy space when two or more points under reference coincide. u κκ t κ κκ κ t are spectrum functions which Let ( u ) and ( ) F x x t uu i j and F ( x x x κ uu i ju κ pertaining to points say x x' x at the same instant of time t in the tbulent medium. When the third point x merges with the first point x then we have: correspond respectively to correlation function ( )

2 496 C. Mamaloukas uuuuu κ λ uu κ κ κ κ u λ κ where λ κ + κ ( d ik j( (1.1) Second lemma: The second lemma is concerned with the Millionschikov s quasi-normality hypothesis. We consider additional fluctuation velocity component u l at the uu κ l κκ κ κ t which correspond to point x and the spectrum tensor ( ) F l x x x x t uu κ i ju κu l. When the foth point x coincides with the first point x we derive the relation u uuuuuu uu il j κ ( κκ κ ( κ κ κ κ l ( κ κ dκ uuuuuu uu + ( κ κ κ j l( κ κ dκ (1.) u + κ t κ κ t correlation function ( ) il ( ) j κ ( ) If the third point x merges with the second point x we derive the relation uuuuu uuuuu u il jκ ( κκ ( κ κ κ κ κ l ( κ κ t ) uuuuu uuuuu u +i κ ( κ κ κ j l( κ κ κ dκ dκ (1.) + κ t κ t il ( ) j κ ( ) The relation (1.) is derived from Millionschikov s quasi-normality hypothesis e.g. ' '' ''' ' '' ''' '' ' ''' ''' ' '' uu i jukul uu i j ukul + uu i k ujul + uu i l ujuk (1.4) Ghosh (197) derived the early period decay equation e.g. u u ( κκ I1( κκ + I( κκ (1.5) for the general type of tbulence where I1 ( κκ and I ( κκ have the requisite expressions (see the original paper of Ghosh (197)). Fther Ghosh showed that for homogeneous and isotropic tbulence the equation (1.5) reduces to: u ( κκ I1( κκ (1.6) Expressing the second rank spectrum tensor for homogeneous and isotropic mn κ κ t as tbulence ( )

3 Final period decay of homogeneous isotropic tbulence 497 ( F κ κ mκ n mn ( κ κ δ mn 4πκ κ (1.7) he obtained finally the Proudman-Reid equation e.g. 4 F ( d F( d κ κ κ κ κ κ (1.8) In the present note we however solve the final period decay equation of homogeneous isotropic tbulence using the concept of point-merging technique of Ghosh (197) as explained above. Final-period decay equation of homogeneous and isotropic tbulence In this case we would use the Ghosh s lemma (1.1) as applied here to the merging of second point to the first point in a two point correlation tensor as uuuuu ( κ u κ κ dκ ij( κ (.1) The decay equation for the homogeneous and isotropic tbulence may be read as (Hinze 1959) u u ( ) ( ) ( κκ t T κκ t v κ + κ ) i j( κκ (.) T κκ t represents the non-linear transfer of homogeneous and where i j( ) isotropic tbulence energy (i.e. through a cascade of energy transfer process). For the final-period decay process the equation (.) reduces to the simple case e.g. u ( ) ( κκ t v κ + κ ) i j( κκ (.) Solution of this equation is given by u v ( κ + κ ) t κκ t κκ t e (.4) ( ) i j( ) where t is an initial instant of time. We express the above equation in the form: uuuuu uuuuu v ( κ κ ) + k t κ κ κ tdκdκ κ κ κ t e dκdκ or ij ( ) i j( ) ( ) ij ( ) vκ t κ tdκ κ t e dκ (.5)

4 498 C. Mamaloukas Let us introduce now the transformations: d dκ1dκdκ ; κ1 sinθcosφ κ sinθsinφ κ cosθ where κ Therefore ij ππ v t ij sin v t π ij ( t ) e d 1 4π ps ij ( s t ) s e ds where vt p s 1 ps ps 4 ( ) ( ) td t e θ dθ dφd ii κ tdκ π A se ds πa se ds ( ) where A is constant. Now 1 1 ii κ tdκ 5 Γ 1 π 4π 4π π 4 p ( v ( v ( ) ui say (.6) and in isotropic tbulence Thus π 1 5 ( ) 1 u + u + u u say 5 t ( vt ) u (.7) Hinze (1959) discussed that the inverse five-halves decay law for u is obtained on the assumption of analytic behavior at k of the energy-spectrum tensor E. In his text Davidson (5) also discussed elaborately that in the case of ii final-period decay of isotropic tbulence the non-linear term becomes unimportant and we have the longitudinal correlation function as

5 Final period decay of homogeneous isotropic tbulence 499 and r f exp ( 8 v 5 t u Conclusions One may conclude that the point-merging technique may be treated as an important concept and applied in deriving relations in homogeneous tbulence. References [1] Davidson P.A. Tbulence: An Introduction for Scientists and Engineers Oxford University Oxford England (5). [] Ghosh K.M. Some Concequences of Millionschikov s Hypothesis in the Early-Period Decay Process of Tbulence Indian Jonal of Pe and Appl. Math.Vol. No.1 (197) 157. [] Hinze J.O. Tbulence McGraw-Hill New York (1959). [4] Mazumdar H.P. On the decay process of tbulence at large Reynolds and Peclet numbers App. Sci. Res. Vol. (1976) 571. [5] Mazumdar H.P. On Incompressible Hydromagnetic Tbulence (Research monograph) Calcutta Mathematical Society (1). [6] Millionschikov M. On the theory of homogeneous isotropic tbulence Dokl. Akad. Nauk SSSR (1941) [7] Monin A.S. and Yaglom A.M. Statistical Fluid Mechanics Vol. II MIT Press Cambridge MA (1975). [8] Proudman I. Reid W.H. On the Theory of a Normally Distributed and Homogeneous Tbulent Velocity Field. Phil. Trans. R. Soc. 47A (1954) Received: November 1