Fundamental Equations for the Turbulent Motion of an Incompressible Viscous Fluid

In the realm of fluid dynamics, turbulence inherently necessitates the interactions between fluid viscosity and velocity gradients. This study delves into the foundation of fluid motion equations, identifying the determination of the fluid's constitutive equation as the sole artificial element in their derivation. The paper posits that in turbulent flows, characterized by intense velocity gradients, the second-order terms associated with the deformation rate in the constitutive equation must be retained, rather than dismissed. This revelation yields an accurate constitutive equation for viscous fluids, enabling the derivation of fluid dynamics equations tailored for turbulent motion, devoid of adjustable parameters, and thus refining the Navier-Stokes equations. As a practical application, we present an extended version of Prandtl's boundary layer equation and provide a numerical solution for the wedge flow boundary layer.