Lagrangian dynamical theory of the velocity gradient tensor: real Schur form, Schur frame, characteristic vorticity modes, and commutative vorticity operators

Abstract

The present work proposes a novel Lagrangian dynamical theory for the velocity gradient tensor (VGT) A ≡ ∇ u , formulated on the basis of its canonical real Schur form. Starting from the Navier–Stokes equations for compressible Newtonian fluids, we derive the general evolution equations for the six rotational invariants in the real Schur form, in which the angular velocity of the Schur frame along any Lagrangian trajectory is explicitly determined. These results then enable us to obtain the evolution equations for the derived VGT invariants, which also encompass the three principal VGT invariants and their Schur representations. Possibly for the first time, the evolution equations for the characteristic vorticity modes ( R _N , S _N ) are derived in both intrinsic and component forms within the Schur frame. Beyond the conventional vortex stretching term in the Helmholtz vorticity transport equation, the theory reveals all stretching and tilting mechanisms of the distinct vorticity modes due to their interactions with the straining and shear strain-rate tensors. Moreover, in the present formulation, it unveils a new physical role of the enthalpy Hessian tensor (replaced by the pressure Hessian tensor in the incompressible limit) in the mutual transformation and redistribution of the two vorticity modes, as embodied in the longitudinal-transverse dynamical transfer term ϒ in the evolution equations. Physically, this term elucidates the mechanism of mutual transformation between axial vortices and surrounding shear layers coexisting within a swirling flow system, a process occurring in both two- and three-dimensional flows. Furthermore, the viscous source pair ( J [ R _N ], J [ S _N ]) in the evolution equations is shown to arise from the spatial derivatives of the Schur frame, the dilatation, and the external force. As an illustrative example, we analyze the kinematics and dynamics of the vorticity modes for the Lamb–Oseen vortex, which validates the proposed theory. Using a similar approach, we discuss the Lagrangian dynamics of the commutative vorticity-operator pair ( Ψ _R , Γ _S ). Finally, neglecting the nonlocal pressure Hessian contribution, we examine the restricted Euler dynamics for incompressible inviscid flow as a reduced case, which yields the Lagrangian invariance of the Schur frame along a pathline and the well-known Vieillefosse–Cantwell closed system of equations for the second and third principal VGT invariants. The proposed theory is expected to be useful for understanding the evolution dynamics of the VGT rotational invariants and intrinsic vorticity modes in a variety of vortical flows.