Lagrangian Dynamical Theory of the Velocity Gradient Tensor: Real Schur Form, Schur Frame, Characteristic Vorticity Modes, and Commutative Vorticity Operators

RESUMO

The present work proposes a novel Lagrangian dynamical theory for the velocity gradient tensor (VGT) $\bm{A}\equiv\bm{\nabla}\bm{u}$, formulated on the basis of its canonical real Schur form in the traditional vortex region with positive discriminant. 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 is explicitly determined along any Lagrangian trajectory. 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. Notably, and perhaps for the first time, the evolution equations for the characteristic vorticity modes $(\bm{R}_{N},\bm{S}_{N})$ are derived in both intrinsic and component forms within the Schur frame, and are subsequently extended to the recently proposed commutative vorticity-operator pair $(\bm{\Psi}_{R},\bm{\Gamma}_{S})$. We find that the straining strain-rate tensor $\bm{D}_{EL}$ acts solely to stretch or contract the integral lines of the rigid-rotation vorticity $\bm{R}_{N}$, whereas the interaction between the shear vorticity $\bm{S}_{N}$ and the shear strain-rate tensor $\bm{D}_{SH}$ alters $\bm{S}_{N}$ within the rotation-axis-normal plane. The theory further unveils a new physical role of the pressure Hessian tensor (more precisely, the enthalpy Hessian tensor for compressible flow) in the mutual transformation and redistribution of the two vorticity modes/operators. The proposed theory could be useful for understanding the dynamics of VGT constituents and intrinsic vorticity modes in a variety of vortical flows.