Fundamental equations for the turbulent motion of an incompressible viscous fluid

Drawing upon the fundamental principle that turbulence is inherently contingent upon the interaction between fluid viscosity and the velocity gradient, this paper embarks on a profound and thorough reexamination of the establishment of fluid motion equations from their very origins. Through meticulous analysis, it is discerned that within the process of formulating fluid motion equations, the sole aspect influenced by human judgment lies in the determination of the fluid's constitutive equation. This paper posits that, owing to the exceedingly high-velocity gradient characteristic of turbulent flow, the second-order term associated with the deformation rate in the fluid's constitutive equation is of such significance that it cannot be disregarded under any circumstances. Instead, it is imperative to retain this term to ensure a comprehensive and precise formulation of the equation. By adhering to this principle, a complete and accurate constitutive equation for viscous fluids is successfully derived. From this refined constitutive equation, hydrodynamic equations that are applicable to turbulent motion can be deduced, and these equations are free from any adjustable parameters, thereby enhancing their predictive accuracy and theoretical rigor. As a practical demonstration of the utility of this new theoretical framework, the paper extends Prandtl's boundary layer theory and applies it to solve the boundary layer problem of wedge flow. To further solidify the theoretical foundation, in order to ascertain the newly introduced viscosity coefficient in the constitutive equation, the paper leverages the exact solution of the pressure field of the free fall flow within a circular tube. Based on this exact solution, an explicit expression for determining the new viscosity coefficient is put forward, providing a crucial quantitative tool for practical applications and further theoretical exploration in the field of fluid dynamics.