  In this study, we propose a computational method for solving the turbulence
problem of incompressible viscous Newtonian fluids based on the extended
Navier-Stokes (N-S) equations. With some phenomenological observations and H.
J. Kreuer's interpretation of the origin of viscosity, we make a hypothesis in
the fluid flow that the shear process is accompanied by eddy formation.
Considering the influence of eddy on convection and diffusion, the classical
N-S equations have been improved to obtain the extended N-S equations. The
extended equations are closed and the source of velocity fluctuations is
explicitly considered as additional convection and diffusion. The extended
equations are compatible with the classical N-S equations and are able to
describe the laminar and turbulent flow in a unified way. In fluid flow
simulations, the equations describing the mean flow quantities can be obtained
directly from the extended N-S equations without any additional turbulence
model. The flow over a cube placed in a channel was numerically investigated to
verify the extended equation, these simulation results were in good agreement
with the Large Eddy Simulation (LES) and experimental results.

Fluid Dynamics