On analytical solutions to the three-dimensional incompressible Navier-Stokes equations with general forcing functions and their relation to turbulence

The three-dimensional incompressible viscous flows are solved in this work with the application of the transformed coordinate which defines as a set of functionals, h_i(ζ) = k_ix + l_iy + m_iz-c_it. The solution is proposed from the base of general Riccati equation, which is firstly substituted into the Navier-Stokes equations to produce the polynomial equation with variable coefficients. The resultant solutions from the system of Riccati and polynomial are then evaluated by the proposed method of integral evaluation. The simulation shows the fluctuation of the decaying velocity through time due to energy dissipation. The rapid velocity accumulations are also detected providing that the solutions may produce singularity in the small scale of turbulent flows. The bifurcation is then detected which revealed the strong nonlinearity in the small scale of turbulent flows.