Turbulent motion described by three-dimensional Navier-Stokes equations Yu Chen For more than 130 years, scientists have believed that the three-dimensional Navier-Stokes equations can fully describe the movement of viscous fluids, including laminar flow, transition flow and turbulence, but the mathematical difficulties in solving these equations hinder the development of its theory. The International Towing Tank Conference (ITTC, the authoritative academic organization of the international hydrodynamics community) began to clarify CFD (Computational Fluid Dynamics) in 1960, that is, how to quickly approximate the Navier-Stokes equation, and after working hard for more than half a century, it has not been solved quantitatively yet, and can only be qualitatively analyzed. Recently, chen sir used a completely different algorithm from traditional methods such as finite difference, finite element and Galerkin to quickly approximate the four-dimensional space-time solution of three-dimensional Navier-Stokes equations,and directly find the velocity u（x,y,z,t） along the x direction, the velocity v（x,y,z,t） along the y direction, the velocity w（x,y,z,t） along the z direction, and the four-dimensional spatiotemporal numerical solution of the pressure p（x,y,z,t） in the equations,and by depicting the image of the p=f(u,v,w) function,the motion of the incompressible viscous fluid described by the three-dimensional Navier-Stokes equations:visualize the evolution process of “laminar flow → transfer turbulence → turbulence”. This numerical simulation (see Figure 1) is consistent with the classic flow experiments done by British scientist Osborne Reynolds in 1880. In fact, Osborne Reynolds pointed out in the An Experimental Investigation of the Circumstances which Determine the the Motion of Water Shall be Direct Or Sinuous: And of the Law of Resistance in Parallel Channels the vortex interprets the resistance as the square of the velocity, the change in resistance (a function of speed) is related to the generation and development of the vortex. It is proved that not only is there a critical velocity at which eddy currents enter, but it is also proportional to the viscosity and inversely proportional to the diameter of the tube. In other words, chen sir takes three-dimensional Navier-Stokes by depicting the image of the p=f(u,v,w) function relationship formed by four four-dimensional space-time solutions in the three-dimensional Navier-Stokes equations,the turbulent motion visualization scheme described by the equation is correct. Figure 1: Turbulent motion described by three-dimensional Navier-Stokes equations On February 22, 1880, Reynolds injected the aniline dye solution into a long horizontal pipe stream to make a tracer, so that the flow of water in the pipe can be seen. When the flow rate is small, the aniline dye solution forms a slender straight line parallel to the tube axis, indicating that the flow is a stable and regular flow, called laminar flow; when the flow rate slowly increases, reaching a certain value, the flow form suddenly The change occurred, the fine line of the aniline dyeing liquid was severely disturbed, and the aniline dye solution was quickly dispersed throughout the tube, indicating that the flow was very disordered, called turbulence. (See Figure 2-3) This test clearly suggests two different flow states (laminar and turbulence) and their concept of transition (transfer turbulence). The history of turbulent research has generally been recognized from the classic mobile experiment of Osborne Reynolds in 1880. Figure 2: The device of the flow experiment done by Reynolds in 1880 and the phenomenon he observed (1) Figure ３: The device of the flow experiment done by Reynolds in 1880 and the phenomenon he observed (２) References: Osborne Reynolds (1883). An Experimental Investigation of the Circumstances which Determines the Motion of Water Shall be Direct Or Sinuous: And of the Law of Resistance in Parallel Channels [J] . The Royal Society Journal Philosophical transactions Vol. 174, No. 3 Article. 顯示郵件原件
