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Viscous three-dimensional periodic solutions

Two examples of periodic fully-three-dimensional viscous solutions are described in. These solutions are defined on a three-dimensional torus T 3 = 3 {\displaystyle \mathbb {T} ^{3}=^{3}} and are characterized by positive and negative helicity respectively.The solution with positive helicity is given by: u x = 4 2 3 3 U 0 e − 3 ν k 2 t u y = 4 2 3 3 U 0 e − 3 ν k 2 t u z = 4 2 3 3 U 0 e − 3 ν k 2 t {\displaystyle {\begin{aligned}u_{x}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\lefte^{-3\nu k^{2}t}\\u_{y}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\lefte^{-3\nu k^{2}t}\\u_{z}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\lefte^{-3\nu k^{2}t}\end{aligned}}} where k = 2 π / L {\displaystyle k=2\pi /L} is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is U 0 2 / 2 {\displaystyle U_{0}^{2}/2} at t = 0 {\displaystyle t=0}.The pressure field is obtained from the velocity field as p = p 0 − ρ 0 ‖ u ‖ 2 / 2 {\displaystyle p=p_{0}-\rho _{0}\|{\boldsymbol {u}}\|^{2}/2} (where p 0 {\displaystyle p_{0}} and ρ 0 {\displaystyle \rho _{0}} are reference values for the pressure and density fields respectively).Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by ω = 3 k u {\displaystyle \omega ={\sqrt {3}}\,k\,{\boldsymbol {u}}}. These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green Taylor–Green vortex.