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We study the dynamics and indications of the flows with all the eigenvaluesof the velocity gradients being real, thus `lone', \textit{i.e.}, withoutforming the complex conjugate pairs associated to the swirls. A genericprototype is the `lone Schur flow (LSF)' whose velocity gradient tensor isuniformly of Schur form but free of complex eigenvalues. A (partial)integral-differential equation governing such LSF is established, and asemi-analytical algorithm is accordingly designed for computation. Simulatedevolutions of example LSFs in 2- and 3-spaces show rich dynamics and vorticalstructures, but no obvious swirls (nor even the homoclinic loops in whateverdistorted forms) could be found. We discovered the flux loop scenario and theanisotropic analogy of the incompressible turbulence at or close to thecritical dimension $D_c =4/3$ decimated from 2-space.

Structures and Dynamics of Lone Schur Flows with Vorticity but no Swirls