Purely dissipative solutions of Navier-Stokes equations for three-dimensional incompressible flows without wall

J Chai (1) , T Wu (1) , L Fang (1)

1 LMP, Ecole Centrale de Pékin, Beihang University, Beijing 100191, China

Existing analytical solutions of Navier-Stokes equations (NSE) are rare. Starting from the he-lical decomposition, we derive analytically a series of purely dissipative solutions of NSE for three-dimensional incompressible flows without wall. The quasi-two-dimensional solutions are generalized Beltrami flows, and the three-dimensional solutions are find to be Beltrami flows. The two-dimensional Taylor-Green (TG) vortex and the Arnold-Beltrami-Childress (ABC) flows are our particular solutions. By choosing N different wave vectors at the same wave length, our solutions have 2N + 2 degrees of freedom for the quasi-two-dimensional cases and 2N degrees of freedom for the three-dimensional cases, indicating that the solution space can be at any high dimensions.

Domains: Fluid mechanics [physics.class-ph]

J Chai, T Wu, L Fang. Purely dissipative solutions of Navier-Stokes equations for three-dimensional incompressible flows without wall. 2019. ⟨hal-02407364⟩