Turbulence calculation based on the extended Navier-Stokes equations

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.