Информация о публикации

Просмотр записей
Инд. авторы: Grebenev V.N., Oberlack M.
Заглавие: Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor
Библ. ссылка: Grebenev V.N., Oberlack M. Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor // Journal of Nonlinear Mathematical Physics. - 2018. - Vol.25. - Iss. 4. - P.650-672. - ISSN 1402-9251. - EISSN 1776-0852.
Внешние системы: DOI: 10.1080/14029251.2018.1503447; РИНЦ: 35760386; SCOPUS: 2-s2.0-85050815845; WoS: 000440363400010;
Реферат: eng: The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B-ij (x,x',t) and isometries of the correlation space K-3 equipped with a (pseudo-) Riemannian metrics ds(2)(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds(2)(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M-t, ds(2)(t)), M-t subset of K-3. Here the metric ds(2)(t), whose components are the correlation functions, evolves due to the von Karman-Howarth equation. We review the explicit geometric realization of ds(2)(t) in K-3 and present symmetries (or isometric motions in K-3) of the metric ds(2)(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds(2)(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale lambda(g) is a second-order differential invariant and show how lambda(g) can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.
Ключевые слова: minimal set of differential invariants; infinite-dimensional Lie algebra; two-point correlation tensor; homogeneous isotropic turbulence;
Издано: 2018
Физ. характеристика: с.650-672
Цитирование:
1. A.A., Belavin, A.M., Polyakov, A.B., Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333–380. doi: 10.1016/0550-3213(84)90052-X
2. P.A., Davidson, Turbulence. An introduction for scientists and engineers (Oxford Univiersity Press, 2004).
3. V.N., Grebenev, M., Oberlack, Geometric realization of the two-point correlation tensor for isotropic turbulence, J. Nonl. Math. Phys. 18(1) (2011) 109–120. doi: 10.1142/S1402925111001192
4. V.N., Grebenev, M., Oberlack, A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turulence. ZAMM. Z. Angew. Math. Mech. 92(3) (2012) 179–195. doi: 10.1002/zamm.201100021
5. V.N., Grebenev, M., Oberlack, A.N., Grishkov, Infinite dimensional Lie algebra associated with the two- point velocity correlation tensor from isotropic, ZAMP, Z. Angew. Math. Phys. 64 (2013) 599–620. doi: 10.1007/s00033-012-0251-7
6. V.N., Grebenev, A.N., Grishkov, M., Oberlack, The extended symmetry Lie algebra and the asymptotic expansion of the transversal correlation function for isotropic turbulence, Adv. Math. Phys. 2013 article ID 46954 (2013) 11 pp.
7. K., Hasselmann, Zur Deutung der dreifachen Geschwindigkeits korrelationen der isotropen Turbulenz, Dtsh. Hydrogr. Zs. 11 (1958) 207–211. doi: 10.1007/BF02020016
8. N.R., Kamyshanskij and A.S., Solodovnikov, Semireducible analytic spaces “in the large”, Russ. Math. Surv. 35(5) (1980) 1–56. doi: 10.1070/RM1980v035n05ABEH001921
9. Th., von Kármán, L., Howarth, On the statistical theory of isotropic turbulence, Proc. Roy. Soc. A164 (1938) 192–215. doi: 10.1098/rspa.1938.0013
10. A.M., Lavrentjev and B.V., Shabat, Problems of Hydrodynamics (Nauka, Moscow, 1973).
11. A.S., Monin and A.M., Yaglom, Statistical Hydromechanics (Gidrometeoizdat, St.-Petersburg, 1994).
12. A.G., Megrabov, Group spliting and Lax representation, Dokl. Math. 67 (3) (2003) 335–349.
13. S.V., Meleshko, Homogeneous autonomic systems with three independent variables, J. Appl. Math. Mech. 58 (1994) 857–863. doi: 10.1016/0021-8928(94)90011-6
14. L.V., Ovsynnikov, Group analysis of differential equations (Nauka, Moscow, 1978).
15. S., Oughton, K.-H., Rädler and W.H., Matthaeus, General second-rank correlation tensors for homogeneous magnetohydrodynamics, Phys. Rev. E 56 (1977) 2875–2888. doi: 10.1103/PhysRevE.56.2875
16. J.H., Rubinstein, R., Sinclair, Visualizing Ricci flow of manifolds of revolution, Exp. Math. 14 (2005) 285–298. doi: 10.1080/10586458.2005.10128930
17. M., Schottenloher, A mathematical introduction to conformal field theory. Lectures Notes in Physics. 759 (Springer, Berlin Heiderberg, 2008).
18. I.N., Vekua, Generalized analytical functions (Nauka, Moscow, 1988).