Modeling turbulent transfer in a channel by means of point vortices

Abstract

i0. Yu. B. Kolesnikov and A. B. Tsinober, "Experimental study of two-dimensional turbulence behind a grid," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 4 (1974). ii. J. Sommeria and R. Moreau, "Why, how, and when MHD-turbulence becomes two-dimensional," J. Fluid Mech., 118 (1982). 12. Y. Couder, "Two-dimensional grid turbulence in a thin liquid film," J. Phys. (Paris) Lett., 45, No. 8 (1984). 13. D. V. Lyubimov, G. F. Putin, and V. I. Chernatynskii, "Convective motions in a Hele- Shaw cell," Dokl. Akad. Nauk SSSR, 235, No. 3 (1977). 14. A. L. Tseskis, "Two-dimensional turbulence," Zh. Eksp. Teor. Fiz., 83, No. i (1982). 15. G. P. Bogatyrev, V. G. Gilev, and V. D. Zimin, "Space-time spectra of stochastic oscil- lations in a convection cell," Pis'ma Zh. Eksp. Teor. Fiz., 32, No. 3 (1980). 16. V. A. Barannikov, G. P. Bogatyrev, V. D. Zimin, A. I. Ketov, and V. G. Shaidurov, "Laws of alternation of peaks in spectra of stochastic oscillations of hydrodynamic systems," Preprint Inst. Mekh. Sploshnykh Sred UNTs Akad. Nauk SSSR (1982). MODELING TURBULENT TRANSFER IN A CHANNEL BY MEANS OF POINT VORTICES P. I. Geshev and B. S. Ezdin UDC 532.527+532.4 Much attention has recently been paid to the direct numerical modeling of turbulence. Some studies have examined three-dimensional turbulent flow in a channel at moderate Reynolds numbers Re by numerically solving the complete system of Navier-Stokes equations [i]. The main difficulty in such calculations is that motions on a scale much smaller than the dis- tance between the nodes of the finest computing grids used in practice are important in tur- bulence at sufficiently large Re. Despite the increasing capacity of modern computers, the limitation on Re remains. There are other approaches to the numerical modeling of wall tur- bulence, such as the method of large vortices [2]. In this method, the scales of motion are divided into a calculable part (by means of "filtered" Navier-Stokes equations for large scales) and a closable, small-scale part (a one-parameter closing relation is generally used), i.e., the hypothesis of the independence of small-scale motions from large-scale motions is employed. In accordance with the principle of the similarity of turbulent flows with respect to the Reynolds number [3], the large-scale motion of a continuum away from the walls is slight- ly dependent on Re. Thus, it can be described by the equations of an ideal swirled fluid. In the proposed computational scheme, transverse motion is modeled by the inviscid two-dimen- sional motion of point vortices, while the complete Navier-Stokes equation, with a constant pressure gradient, is calculated in the mean direction of motion. Two-dimensional point vor- tices have been used to study mainly free flows - jets and wakes in flow about different re- cesses and projections. It was shown in [4] that the spectral energy flux is constant in a system of point vortices and the flow spectrum is close to a Kolmogorov spectrum. The "5/3" law follows from similarity theory in the case of isotropic turbulence. In wall turbulence, this theory leads to logarithmic velocity profiles in the region where the flow of the longi- tudinal component of momentum to the wall is constant [5]. Considering the successful model- ing of isotropic turbulence in [4], there is hope for obtaining interesting results in wall turbulence by modeling turbulent transfer by the method of longitudinal point vortices. It was shown in the present study that such calculations give results which agree qualitatively with experimental findings; the logarithmic profiles of velocity and temperature are calcu- lated, profiles of the Reynolds stresses and turbulent heat fluxes are obtained, and the amp- litudes of fluctuating quantities are investigated. A model of turbulence based on point vortices should be considered a direct numerical model. Such an approach has an undoubted advantage, since it does not require any closing assumptions. Novosibirsk. Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 79-84, March-April, 1986. Original article submitted February 6, 1985. 0021-8944/86/2702-0228512..50 9 1986 Plenum Publishing Corporation 228