A class of exact solutions to the three-dimensional incompressible Navier–Stokes equations

An exact solution of the three-dimensional incompressible Navier–Stokes equations with the continuity equation is produced in this work. The solution is proposed to be in the form V=∇Φ+∇×ΦV=∇Φ+∇×Φ where ΦΦ is a potential function that is defined as Φ=P(x,y,ξ)R(y)S(ξ)Φ=P(x,y,ξ)R(y)S(ξ), with the application of the coordinate transform ξ=kz−ς(t)ξ=kz−ς(t). The potential function is firstly substituted into the continuity equation to produce the solution for RR and SS. The resultant expression is used sequentially in the Navier–Stokes equations to reduce the problem to a class of nonlinear ordinary differential equations in PP terms, in which the pressure term is presented in a general functional form. General solutions are obtained based on the particular solutions of PP where the equation is reduced to the form of a linear differential equation. A method for finding closed form solutions for general linear differential equations is also proposed. The uniqueness of the solution is ensured because the proposed method reduces the original problem to a linear differential equation. Moreover, the solution is regularised for blow up cases with a controllable error. Further analysis shows that the energy rate is not zero for any nontrivial solution with respect to initial and boundary conditions. The solution being nontrivial represents the qualitative nature of turbulent flows.