Non-Gaussianity, bounds on turbulent scaling parameter and conformal transformations - analyzing the Lundgren and Hopf functional equation of turbulence using Lie symmetries
Final Report Abstract
In the present project, the symmetry-based turbulence theory was significantly advanced. The main goal of this theory is, on the one hand, to determine the symmetries of different mathematical descriptions of turbulence with the help of the methods of Lie’s group theory, and on the other hand, to use such symmetries for the construction of special basis solutions, so-called turbulent scaling laws. In this project, the main focus was concretely on the symmetry analysis of the statistical description of turbulence based on three different hierarchies of probability density functions (PDFs), which determine the statistics of essential field quantities. Of particular interest here in the incompressible case were, first, the PDFs for the (absolute) flow velocity, second, the PDFs of the vorticity, both in the two- and three-dimensional cases, and, third, the PDFs of the velocity increments. The transfer of the analysis strategy used here to the case of the PDFs of the velocity increments led to identical results as for those for the absolute velocities. The symmetry expressions obtained corresponded to those for the hierarchy of the PDFs of the velocities after application of a coordinate transformation, which transforms the two hierarchies into each other. The analysis of the equation hierarchy of PDFs of vorticity in 2D focused on the question of the possible existence of the group of conformal, i.e., angle-preserving, transformations as a subgroup of the full symmetry group of the hierarchy. This conjecture could not be confirmed for the hierarchy, but as a partial result the conformal invariance of the equation along isolines of vorticity could be shown. Various attempts to use extended symmetry concepts or to develop an extension of the previously used analysis method for integro-differential equations were not successful. The question raised in the course of the project as to the transferability of the previous methods to the analogous equation hierarchies for compressible flows was dealt with. For this purpose, a different mathematical formulation of the turbulence via so-called characteristic functions (CF) came to the fore. For the compressible case, therefore, first both the corresponding equation hierarchies for the PDFs as well as those for the corresponding CFs were derived. In a second step, the symmetry analysis of the obtained hierarchy was performed for the CFs. Here, however, again only the a priori expected symmetries, the symmetries of the underlying physical equations as well as the symmetries due to the naturally resulting linearity of the statistical equations of motion were obtained. With respect to the second central question of the project, the derivation of invariant solutions of the PDF hierarchies by means of symmetry methods, the so-called turbulent scaling laws, such solutions could be explicitly obtained here. Using scaling symmetries in time and space as well as statistical symmetries inherent to the statistical description, a complete hierarchy of invariant PDFs was constructed, which is of particular interest for the description of wall-parallel flows. As a first test of these scaling laws, the theoretical expressions were compared with the data of a direct numerical simulation (DNS). For this purpose, the free parameters contained in the model were determined by the DNS data using a fitting algorithm. A first comparison shows a good agreement of the theoretical PDF expressions with the simulation data, at least in a limited spatial range. However, this spatial range, in which the PDF expressions are a good approximation, is less pronounced than in the comparable case of the analogously derived scaling laws for the downstream velocity moments. An analogous comparison of the theory with data from simulations or experiments of flows in other configurations has not yet been considered.
Publications
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Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence. Journal of Physics A: Mathematical and Theoretical, 52(33), 335501.
Grebenev, V. N.; Wacławczyk, M. & Oberlack, M.
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Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models. Symmetry, 12(9), 1482.
Wacławczyk, Marta; Grebenev, Vladimir N. & Oberlack, Martin
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The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence. Physics of Fluids, 32(6).
Praturi, Divya Sri; Plümacher, Dominik & Oberlack, Martin
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Second-order invariants of the inviscid Lundgren–Monin–Novikov equations for 2d vorticity fields. Zeitschrift für angewandte Mathematik und Physik, 72(3).
Grebenev, V. N.; Grichkov, A. N.; Oberlack, M. & Wacławczyk, M.
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On the Lie symmetries of characteristic function hierarchy in compressible turbulence. European Journal of Applied Mathematics, 34(5), 913-935.
PRATURI, D.S.; PLÜMACHER, D. & OBERLACK, M.
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Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach. Physical Review Letters, 128(2).
Oberlack, Martin; Hoyas, Sergio; Kraheberger, Stefanie V.; Alcántara-Ávila, Francisco & Laux, Jonathan
