The Navier – Stokes equation describes not only the laminar regime, but also the turbulent one. but the turbulent solution is complex. In addition, there is a connection between the solution of the Navier – Stokes equation and the solution of the <PERSON> equation see file 3. See attached files where there is a solution for the sphere in file 2
file 1 Values of large dimensionless unknown functions (for example, a large <PERSON> number) can be found out as solutions of non-linear partial differential equation. In this case these equations can be brought to some number of non-linear ordinary differential equations. Turbulent solutions corresponding to large values of unknown function are complex. Transition from real solution to complex turbulent solution is realized through infi nity of the right parts of ordinary differential equation system to which Navier – Stokes equations are brought. Thus, real solution of Navier – Stokes equation for turbulent mode yields function going to infi nity. At the same time, complex solution for the turbulent mode is fi nite. Fluid fl ow resistance coeffi cient is calculated for round pipeline with different pipeline walls roughness.
file 2 The problem of turbulent fluid motion description has not been solved yet. It creates difficulties when oil, gas pipelines design calculation is performed. Besides, there are no theoretical methods for description of bodies motion in turbulent environment. These methods would be necessary for description of motion of aircrafts, submarines or above-water ships in the turbulent mode. 614
Without simulation of bodies motion in wind tunnels or water basins, design of the bodies moving in the viscous environment is impossible.
file 3 In this part of the article is described the physical meaning of complex velocity. The relationship between the <PERSON> equation and the Navier – Stokes equation obtain. <PERSON> equation and the Navier – Stokes equations are, in general, have a countable number of turbulent energy and solutions.
1. <PERSON>, <PERSON> "STUDY OF NAVIER-STOKES EQUATION SOLUTION I. THE GENERAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 60-66. https://world-science.ru/pdf/2016/3/14.pdf
2. YAKUBOVSKIY, <PERSON> "STUDY OF NAVIER-STOKES EQUATION SOLUTION II. THE USE OF LAMINAR SOLUTIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 67-83.https://world-science.ru/pdf/2016/3/15.pdf
3. YAKUBOVSKIY, E. G. "STUDY OF NAVIER–STOKES EQUATION SOLUTION III. THE PHYSICAL SENSE OF THE COMPLEX VELOCITY AND CONCLUSIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 84-87. https://www.world-science.ru/pdf/2016/3/16.pdf 614