Approximate solution to one model of axisymmetric turbulent wake

Shmidt A.V.
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In this paper we consider the problem of modeling of flow in the far regions of an axisymmetric momentumless turbulent wake that occurs when a body is flowed. The main features of the studied flow are its almost shear-free nature and its proximity to the self-similar regime, which is confirmed by the available experimental data. A semi-empirical $e-\varepsilon$ model of turbulence based on the Rodi algebraic model of Reynolds stresses is invoked to describe a flow in the far wake. Using group-theoretic analysis, similarity reduction of the invoked model of turbulence to a system of ordinary differential equations is obtained. An approximate self-similar solution of the second kind of the corresponding boundary value problem for the reduced system of equations was constructed using a method based on matching the truncations of asymptotic expansions of its solution found in neighborhoods of the boundary points. The involved approach also allowed for the extraction of an approximate value of the self-similarity exponent and other unknown parameters of the problem during the procedure of matching of the asymptotic expansion. Comparison of the approximate solution of the given boundary value problem constructed in the form of Puiseux series, which includes 16 approximating terms, with the numerical solution found using the shooting method, showed that the maximum relative error between the solutions does not exceed 2%. The approximate value of the self-similarity exponent of the problem established in this work is in a good agreement with the exact value obtained earlier by other authors as a result of the analytical solution of the corresponding nonlinear eigenvalue problem. The approximate solution was compared with the available experimental data on the axisymmetric momentumless turbulent wake obtained at the Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, and a satisfactory agreement was found. The approximate formulas developed in this work have practical significance and can be used for quick estimation of the values of hydrodynamic quantities in the far regions of the axisymmetric momentumless turbulent wake. The developed method for constructing approximate solutions to the boundary value problems of free turbulence can also be used in other applied fields that are not related to turbulence theory.

Keywords: momentumless turbulent wake, e-ε model of turbulence, similarity of the second kind, boundary value problem, approximate solution, asymptotic expansion
Citation in English: Shmidt A.V. Approximate solution to one model of axisymmetric turbulent wake // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 823-836
Citation in English: Shmidt A.V. Approximate solution to one model of axisymmetric turbulent wake // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 823-836
DOI: 10.20537/2076-7633-2026-18-4-823-836
URL: http://crm-en.ics.org.ru/journal/article/3747/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Shmidt A.V.

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