Speculation of fluid dynamics equations based on Liutex theory and constitutive relation of symmetric shearing deformation

Abstract

The fluid kinematics of Liutex decomposes a velocity gradient tensor (VGT) of ∇v into four components, including rotation (R), stretching/compressing (SC), anti-symmetric shear (S_anti-sym) and symmetric shear (S_sym), as oppose to the traditional Cauchy-Stokes decomposition where a VGT was decomposed into the strain rate and vorticity tensors. The current study limpidly clarified the physical meanings of these deformations in the newly-proposed decomposition from the perspectives of both fluid kinematics and dynamics. With in-depth understanding the physical connotations of these deformations, the present study further suggests that the S_sym be the only deformation appropriately correlated to the stress tensor, leading to the establishment of a new constitutive relation for Newtonian fluids with the modified model assumptions originated from Stokes in 1845. Moreover, the present research finds that the “principal decomposition” proposed by Liu is not mathematically unique when a VGT has three real eigenvalues (TR). Within the context, a new decomposition method is introduced to avoid the non-uniqueness issue arising from using the principal decomposition to establish fluid dynamics equations. Based on the modified Stokes assumptions and the novel VGT decomposition method, a set of new fluid dynamics momentum equations are obtained for Newtonian fluid. The added stress tensor of F^add is identified as the key difference between the newly-derived governing equations and the conventional Navier-Stokes (N-S) equations, which is caused by excluding the SC correlation to the stress tensor in the new constitutive equation. Finally, a preliminary analysis of F^add is conducted using the existing channel turbulence direct numerical simulations (DNS) data based on the traditional N-S equations. The F^add is found widely existing in turbulence and is of the same order of magnitude with the other force terms. Therefore, the F^add is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations, which will be further verified by performing the “DNS” simulation using the newly-derived fluid dynamics equations in near future.