Fluid Dynamics Equations based on Constitutive Relation of Symmetric Shearing Deformation

Abstract

The fluid kinematics of Liutex decomposes a velocity gradient tensor (VGT) of \(\nabla \vec {v}\) into four components, including rotation (\(\varvec{R}\)), stretching/compressing (\(\varvec{SC}\)), anti-symmetric shear (\({\varvec{S}_{anti - sym}}\)) and symmetric shear (\({\varvec{S}_{sym}}\)), 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 an in-depth understanding the physical connotations of these deformations, the present study further suggests that the \({\varvec{S}_{sym}}\) 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 possesses three real eigenvalues (TR). Within the context, a new decomposition method is introduced to resolve the non-uniqueness issue. Based on the modified Stokes assumptions and the associated VGT decomposition method, a set of new fluid dynamics momentum equations are obtained for Newtonian fluid. The added stress tensor of \({\varvec{F}^{add}}\) is identified as the key difference between the newly-derived governing equations and the conventional N-S equations, which is caused by excluding the \(\varvec{SC}\) correlation to the stress tensor in the new constitutive equation. Finally, a preliminary analysis of \({\varvec{F}^{add}}\) is conducted using the existing channel turbulence DNS data based on the traditional N-S equations. The \({\varvec{F}^{add}}\) is found widely existing in the flow field and is at the same order of magnitude with the other force terms in these equations. Therefore, the \({\varvec{F}^{add}}\) is expected to have some tangible 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 dynamic equations in near future.