Singular solution of the Navier–Stokes equation for plane Poiseuille flow

In previous studies, experiments and numerical simulations have shown that the transition to turbulence depends on both the Reynolds number and the magnitude of disturbances. However, the underlying physical mechanisms driving this phenomenon are still poorly understood. In this study, we analyzed the solution of the Navier–Stokes equation for plane Poiseuille flow, with a particular focus on the behavior of instantaneous velocity under disturbance effects. For the first time, we demonstrate how the interaction between the base flow and disturbance evolves to produce a point in instantaneous velocity where the Laplace term (viscous term) equals zero. Subsequently, a rigorous analysis is conducted on the solution of the Navier–Stokes equation at the critical condition of zero Laplace term, revealing that it constitutes a physical singularity of the equation. The instability of flow at singularity leads to the formation of negative velocity spikes in the instantaneous velocity. The theoretical results are compared against experimental data, yielding qualitative agreement that aligns with the simulation results reported in existing literature. The appearance of singularities and the generation of turbulence lie on the streamwise extension and compression of fluid elements. The research results indicate that although the Navier–Stokes equation may have a unique solution in theory for transitional and turbulent flows, this solution lacks global smoothness due to the non-differentiability at singularities. In addition, based on the vorticity transport equation, we have demonstrated that singularity is necessary within the flow field when turbulence is generated. Finally, it is elucidated that the conserved soliton-like coherent structure observed in transitional flows arises from the negative velocity spike.