Calculation of the dissipation coefficient of the mechanical energy of moving fluid in the smooth cylindrical tube in a
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ORIGINAL RESEARCH
Calculation of the dissipation coefficient of the mechanical energy of moving fluid in the smooth cylindrical tube in a turbulent flow taking into account the turbulent viscosity Alla Yovchenko1 · Sergii Bespalko1 · Sviatoslav Poliakov1 · Tymofii Veretilnyk2 · Ruslan Kapitan2 Received: 18 January 2020 / Accepted: 27 May 2020 © Islamic Azad University 2020
Abstract Using a three-layer turbulence model for a cylindrical tube, an analytical calculation of the dissipation coefficient of the mechanical energy of flow in a smooth-walled cylindrical tube was performed, taking into account the turbulent viscosity. To take into account the turbulent viscosity, the turbulence model developed by Y. V. Lapin, O. A. Nekhamkin and M. Kh. Strelets was applied, considering Van Driest’s damping multiplier. Analytical formulas were obtained describing the dependence of the dissipation coefficient of the mechanical energy of the flow on the number Re in the degree and logarithmic velocity distribution in a smooth cylindrical tube. Based on the performed calculations, a plot of the coefficient of dissipation on the number Re is constructed. The obtained results are consistent with the results obtained by other authors and are not inferior to modern, more complex mathematical models and can be used in engineering calculations. This analysis provides simple analytical expressions to determine the coefficient of dissipation concerning turbulent viscosity using a universal velocity profile in a smooth cylindrical tube. Keywords Turbulent viscosity · Mechanical energy dissipation coefficient · Weighted average dissipation value · Laminar layer · Turbulent nucleus
Introduction Most of the flows occurring in nature and man-made processes are turbulent and characterized by high values of the Reynolds number Re ≥ 2300. In turn, the turbulent flows are used in various thermal processes to intensify heat and mass transfer. Thus, knowledge of the velocity distribution in the flow is very important since it significantly affects mixing, emulsification, heat transfer, etc., determines the shear stresses and dissipation rate. Turbulent flow is complicated by unsteadiness, threedimensionality, dissipativity, a vortex formation. Therefore, exact analytical solutions for turbulent flows in a pipe do * Ruslan Kapitan [email protected] 1
Department of Energy Technologies, Cherkasy State Technological University, Cherkasy, Ukraine
Department of Mechanics, Printing Machines and Technology, Cherkasy State Technological University, Cherkasy, Ukraine
2
not exist, and all the formulas either derived directly from experience or using the semiempirical approach. The first fundamental studies of turbulent flows were carried out in XIX century by G. Hagen, O. Reynolds, J. Boussinesque, G. Lorenz and in the early twentieth century were developed by many scientists (L. Prandtl, T. Karman, V. Ekman, I. Burgers, A. Kolmogorov, H. Dryden, G. Clauser, J. Taylor, G. Schlichting, L. Loitsyanskiy, et al.). Various semiempirical models
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