Deduction of the Reynolds Equations of Turbulence from the Gyarmati Principle [antikvár]

Acta Chimica Academiae Scientiarum Hungaricae, Tomus 75 (1), pp. 33—43 (1972) DEDUCTION OF THE REYNOLDS EQUATIONS OF TURBULENCE FROM THE GYARMATI PRINCIPLE Gy. ViNczE (Institute for Physics, University of Gödöllő, Hungary) Received October 15, 1971 It is shown that, similarly to the Navier - Stokes equation of viscous flow, also the Reynolds equations of turbulent flow are deducible from the Gyakmati integral principle of non-equilibrium thermodynamics. In the course of the deduction Lobentz's dissipation potential is somewhat generalized, and it is demonstrated that the Lagran-gian density in the Gyarmati principle can be split into two separate parts belonging to the mean motion and the fluctuating motion, respectively. The validity of Gyar-matl's supplementary theorem is discussed in the case of quasi-linear constitutive equations, the variational principle is proved to remain operative in such cases, too. 1. Preliminaries It is well known that the Navier—Stokes equation of viscous flow was not deducible from a variational principle for a long time. Although great efforts were made to draw up the problem within the scope of a suitable variational task [1—4], the researches — excluding some very special cases — did not lead to any positive result. The negative results were characterized by Serrín in his excellent article [5]: "Other negative results concerning variational principles yielding Navier—Stokes equation are due to Gerber [6] and Bateman [7]." Of course, the possibility of deducing the Reynolds equations of turbulent flow from a variational principle could not arise until the problem of derivation of the Navier—Stokes equation was solved. In 1965 the tide was turned; starting from an alternative form of Onsager's principle of minimum energy dissipation [8], Gyarmati formulated the integral principle of dissipative processes [9, 10] within the framework of non-equilibrium thermodynamics. From this principle Verhas [11] first deduced the simple Navier—Stokes equation, then Borocz [12] derived also the generalized hydrodynamic equation of motion belonging to the antisymmetric part of the pressure tensor and describing the internal rotation too. After these results Gyarmati wrote already in 1967 [13]: "It is evident that the Reynolds equations of turbulent flow can also be derived from the integral principle, and the situation is similar with respect to the fundamental equations of magnetohydrodynamics, plasma physics, etc." 3 Acta aim. (Budapest) 75, 1972