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Uniform decomposition of velocity gradient tensor

Journal of Hydrodynamics 2024年第1期36卷 24-34页

作者： Chenxi Ma Chaoqun Liu Department of Mathematics University of Texas at ArlingtonArlingtonTXUSA

In this paper,the principal decomposition of the velocity gradient tensor[∇v]is discussed in 3 cases based on the discriminant∆:∆0 with 3 distinct real eigen values,and∆=0 with 1 or 2 distinct real eigen values.The velocity gradient tensor can also be classified as rotation point,which can be decomposed into three parts,i.e.,rotation[R],shear[S]and stretching/compression[SC],and non-rotation point,we defined a new resistance term[L],and the tensor can be decomposed into three parts,i.e.,resistance[L],shear[S]and stretching/compression[SC].Example matric are also displayed to demonstrate the new decomposition.Connections of principal decomposition between 3 different cases,and between Resistance and Liutex will also be discussed.

关键词： Velocity gradient Liutex resistance fluid kinematics

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

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Journal of Hydrodynamics 2023年第4期35卷 607-629页

作者： Shuai-chen Zhu Duo Wang Yang Liu Hongyi Xu Department of Aeronautics and Astronautics Fudan UniversityShanghai200433China

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(Santi-sym)and symmetric shear(Ssym),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 Ssym 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 Fadd 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 Fadd is conducted using the existing channel turbulence direct numerical simulations(DNS)data based on the traditional N-S equations.The Fadd is found widely existing in turbulence and is of the same order of magnitude with the other force terms.Therefore,the Fadd is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations,which will be furt

关键词： Constitutive relation Liutex principal decomposition Newtonian fluid Navier-Stokes equations fluid kinematics fluid dynamics

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