arXiv subjects

Chaoqun Liu

Publications and source records attributed to Chaoqun Liu.

At least 37 records · Page 2Linked to original sources

arXiv2022

An objective vortex identification method

Generally, the vortex structures should be independent of the observers who are moving, especially when their coordinates are non-inertial, which may result in confusions in communications between researchers. The property that not being influenced by the choice of coordinate is called objective. Mostly, researchers would like to gather data in the inertial coordinate because most physical laws are valid in the inertial coordinate. As a result, how to transfer the observations in a non-inertial coordinate to those in an inertial coordinate raises interest. In this paper, methods are provided to obtain velocity and velocity gradient tensor in an inertial coordinate from the non-inertial observations based on one known zero-vorticity point which can be chosen from physical fact such as the point is located in the inviscid region.Further, objective vortex structure can be calculated from the objective velocity or velocity gradient tensor. With Liutex chosen as the indicator of the vortex, two numerical examples are used to test the proposed methods. The results show that the methods perform very well and the relative error is extremely small.

physics.flu-dyn↗

arXiv2021

New Governing Equations for Fluid Dynamics

There are several questions with NS, which include: 1. Both symmetric shear terms and stretching terms in strain and stress are coordinate-dependent and thus not Galilean invariant; 2. The physical meaning of both diagonal and off-diagonal elements are not clear, which is coordinate-dependent; 3. It is hard to measure the strain and stress quantitatively, and viscosity is really measured by vorticity not by symmetric strain; 4. There is no vorticity tensor in NS, which plays important role in fluid flow especially for turbulent flow. The new proposed governing equations for fluid dynamics use vorticity tensor only, which is anti-symmetric. The advantages include: 1. Both shear and stress are anti-symmetric, which are Galilean invariant and independent of coordinate rotation; 2. The physical meaning of off diagonal elements is clear, which is anti-symmetric shear stress, 3. Viscosity coefficients are obtained by experiment which uses vorticity, 4. The vorticity term can be further decomposed to rigid rotation and anti-symmetric shear, which are important to turbulence research, 5. The computation cost for viscous term is reduced to half as the diagonal terms are all zero as six elements are reduced to three. Several computational results are made, which clearly demonstrate both NS and new governing equations have exactly same results. As shown below, the new governing equation is identical to NS in mathematics, but the former has lower cost and several advantages mentioned above including possibility to better study turbulent flow. There are reasons to use the new governing equations to replace Navier-Stokes equations. The unique definition and operation of vector and tensor by matrix and matrix operation are also discussed in this paper.

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arXiv2021

Principal Decomposition of Velocity Gradient Tensor in the Cartesian Coordinates

Traditional Cauchy-Stokes decomposition of velocity gradient tensor gives a symmetric and an anti-symmetric subtensors which are called the strain-rate and vorticity tensors. There are two problems with Cauchy-Stokes decomposition. The first one is that the anti-symmetric or vorticity tensor cannot represent the fluid rotation or vortex. The second one is that the symmetric (strain-rate) tensor cannot distinguish the stretching (compression) and shear. The stretching and shear are dependent on the coordinate or are not Galilean invariant. Since vorticity cannot distinguish between the non-rotational shear and the rigid rotation, vorticity has been decomposed to a rigid rotation called Liutex and anti-symmetric shear in our previous work. A Liutex-based principal coordinate was developed and the velocity gradient tensor was decomposed in the principal coordinate as a rigid rotation (Liutex tensor), a pure shear tensor and a stretching (compression) tensor, which is called the principal decomposition. However, the principal decomposition is made at each point which has own principal coordinate different from other points. This paper derives the principal decomposition in the original xyz coordinate system, and, therefore, provides a new tool for fluid kinematics to conduct the velocity gradient tensor decomposition to rigid rotation, pure shear, and stretching (compression) which is Galilean invariant and has clear physical meanings. The new velocity gradient tensor decomposition could become a foundation for new fluid kinematics.

physics.flu-dyn↗

arXiv2021

A mathematical study on the local fluid rotation axis

As widely recognized, vortex represents flow rotation. Vortex should have a local rotation axis as its direction and angular speed as its strength. Vorticity vector has been considered the rotation axis, and vorticity magnitude the rotational strength for a long time in classical fluid kinematics. However, this concept cannot stand in viscous flow. This study demonstrates by rigorous mathematical proof that the vorticity vector is not the fluid rotation axis, and vorticity is not the rotation strength. On the other hand, the Liutex vector is mathematically proved as the fluid rotation axis, and the Liutex magnitude is twice the fluid angular speed.

physics.flu-dyn↗

arXiv2020

Stretching and Shearing Contamination Analysis for Liutex/Rortex and Other Vortex Identification Methods

Although traditional vortex identification methods such as Q, Delta, Lambda2, Lambdaci remain popular in the identification and visualization of vortices, these methods count on shearing and stretching as a part of vortex strength. However, shearing and stretching do not contribute to fluid rotation. In this paper, the contamination effects of stretching and shearing of these methods are investigated and compared with Liutex method. From our investigation, the Liutex is an exact definition of fluid rotation or vortex, while other vortex identification methods are contaminated by stretching and shearing at different levels. The decomposition of the velocity gradient tensor can only be conducted in a so-called Principal Coordinate for uniqueness. The mathematical relation between Liutex and other vortex identification function are derived in this paper and then the effects of shearing and stretching on different vortex identification methods are studied. The mathematical formula and computation of the stretching and shearing effects on different schemes clearly show that the Liutex method has superiority over the other vortex identification methods as it only counts on local fluid rigid rotation while other methods count on stretching/compression and shearing as a part of the fluid rotation or vortex.

physics.flu-dyn↗

arXiv2020

Correlation Analysis among Vorticity, Q method and Liutex

Influenced by the fact that vorticity represents rotation for rigid body, people believe it also works for fluid flow. However, the theoretical predictions by vorticity do not match experiment results, which drove scientists to look for better methods to describe vortex. According to Dr. Liu classification, all methods applied to detect vortex can be categorized into three generations. The vorticity-based method is classified as the first generation. Methods relying on eigenvalues of velocity gradient tensor are considered as the second generation. Although so many methods appeared, people still believe vorticity is vortex since vorticity theory looks perfect in math, and all other methods are only scalars and unable to indicate swirl direction. Recently, Dr. Liu innovated a new vortex identification method called Liutex. Liutex, a vector quantity, which is regarded as the third-generation method, not only overcomes all previous methods drawbacks, but also has a clear physical meaning. The direction of Liutex represents the swirl axis of rotation, and its strength is equal to twice of angular speed. In this paper, we did a correlation analysis between vorticity, Q, Lambda ci, Lambda 2 methods and Liutex based on a DNS case of boundary layer transition. The results show that the correlation between vorticity and Liutex is minimal in strong shear region, which demonstrates the idea that using vorticity to detect vortex lacks a scientific foundation; in other words, vorticity is not vortex.

physics.flu-dyn↗

arXiv2020

Liutex-based Modified Navier-Stokes Equation

The Navier-Stokes (NS) partial differential equations, as the governing equation of fluid dynamics, are based on particles with zero volume. In such a condition, conservation law of moment of momentum is automatically satisfied, and thus NS equations only contain conservation of mass, momentum and energy. Because of the difficulty to get an analytical solution of NS equations, scientists develop finite element method (FEM) and finite volume method (FVM) to obtain approximate solutions. However, these methods are dependent on the size of finite volumes, which is not zero. Considering the size of the finite volume, conservation law of moment of momentum is no longer automatically satisfied and it is necessary to develop a new control equation to take effect of momentum into account. In this paper, new relations of reciprocal shear stresses are derived from conservation law of moment of momentum, and so are new constitutive relations and modified NS equations. The Liutex based NS equation may be the universal governing equations for both laminar and turbulent flow which is dominated by vortices and conservation of moment of momentum must be satisfied.

physics.flu-dyn↗

arXiv2019

Vortex Core and POD Analysis on Hairpin Vortex Formation in Flow Transition

In this study, the new concept of vortex core line based on Liutex definition is applied to demonstrate that vortex ring is not part of the Lambda-vortex and the generation of ring like vortex is formed separately from the Lambda-vortex. The proper orthogonal decomposition is also applied to analyze the Kelvin Helmholtz instability happening in hairpin ring areas of flow transition on the flat plate to understand the ring like vortex formation. The new identification method named modified Omega-Liutex method is efficiently used to visualize and observe the shapes of vortex structures in 3D. The streamwise vortex structure characteristics can be found in POD Mode1 as the mean flow. The other POD modes are in spanwise structures and have the fluctuation motions, which are induced by KH instability. Moreover, the result shows that fluctuated POD modes are in pairs and share the same characteristics such as amplitudes, mode shapes and time evolutions. The vortex core and POD results confirm that the Lambda-vortex is not self deformed to a hairpin vortex, but it is formed by the KH instability during the formation of Lambda vortex and hairpin vortex in boundary layer flow transition.

physics.flu-dyn↗

arXiv2019

Letter: Mathematical Foundation of Turbulence Generation-Symmetric to Asymmetric Liutex/Rortex

Vortex has been considered as the building block and muscle of turbulence for long time. A new physical quantity called Liutex (previously named Rortex) has been defined as the rigid rotation part of fluid motion. From DNS and experiment, forest of hairpin vortices has been found in flow transition and low Reynolds turbulence, but more and more one leg vortices appear in fully developed turbulence. This letter shows hairpin vortex is unstable. Hairpin vortex will weaken or lose one leg by the shear and vortex interaction. This conclusion is made by Liutex definition and mathematical analysis without any physical assumptions. The asymmetry of vortex is caused by the interaction of symmetric shear and symmetric Liutex due to the definition of Liutex, which considers the smaller element of a pair of vorticity elements as the rotational strength. For a 2-D fluid rotation, if a disturbance shear effects the larger element, the rotation strength will not be changed, but if the disturbance shear effects the smaller element, the rotation strength will be immediate changed due to the Liutex strength definition. For a rigid rotation, if both vorticity of the shear and Liutex have the same directions, e.g., clockwise, the Liutex strength will not be changed. If the vorticity of the shear and Liutex have different directions, e.g., one clockwise and one counterclockwise, the Liutex strength will be weakened. Then, the hairpin vortex could lose the symmetry and even deform to a one leg vortex. The one leg vortex cannot keep balanced, and the chaotic motion and flow fluctuation are doomed. This is considered as the mathematical foundation of turbulence formation. This theory has been checked and proved by our DNS results of boundary layer transition.

physics.flu-dyn↗

arXiv2019

Letter: Modified normalized Rortex/vortex identification method

In this paper, a modified normalized Rortex/vortex identification method named Omega_R is presented to improve the original Omega_R method and cure the bulging phenomenon on the iso-surfaces caused by the original Omega_R method. Detailed mathematical explanations and the relationship with the Q criterion are described. In addition, the new developed formula does not require two original coordinate rotations, and the calculation is greatly simplified. The numerical results are demonstrated to show the effectiveness of the new modified normalized Rortex/vortex identification method.

physics.flu-dyn↗

arXiv2019

A Liutex based definition of vortex axis line

Eulerian local region-type vortex identification criteria, including the criterion, the criterion and the criterion et al., are widely used for vortex identification due to the simplicity in applications. However, most of these criteria are based on a scalar quantity, unable to identify vortex axis (core) lines. On the other hand, the current line-type methods, which seek to extract line-type features such as vortex core lines, are not entirely satisfactory since most of these methods are based on vorticity or pressure minimum which will fail in many cases. To address this issue, a novel Liutex (previously named Rortex) based definition of vortex axis line is proposed in this paper. Mathematically, the vortex axis lines are defined by points where the gradient of Liutex magnitude is aligned with the direction of the Liutex vector, which implies that the cross product of the gradient of Liutex magnitude and the Liutex vector is equal to zero. A preliminary manual process for extracting vortex axis lines is presented. Two test cases, namely Burgers vortex and hairpin vortex, are examined to verify the proposed method.

physics.flu-dyn↗

arXiv2018

Physical meaning of vorticity based on the RS decomposition and an explicit formula for the Liutex vector

In the present study, the physical meaning of vorticity is revisited based on the RS decomposition proposed by Liu et al. in the framework of Liutex (previously named Rortex), a vortex vector field with information of both rotation axis and swirling strength [ C. Liu et al., "Rortex-A new vortex vector definition and vorticity tensor and vector decomposition", Phys. Fluids 30, 035103 (2018)]. It is demonstrated that the vorticity in the direction of rotational axis is twice the spatial mean angular velocity in the small neighborhood around the considered point while the imaginary part of the complex eigenvalue (λ_ci) of the velocity gradient tensor (if exist) is the pseudo-time average angular velocity of a trajectory moving circularly or spirally around the axis. In addition, an explicit expression of the Liutex vector in terms of the eigenvalues and eigenvectors of velocity gradient is obtained for the first time from above understanding, which can further, though mildly, accelerate the calculation and give more physical comprehension of the Liutex vector.

physics.flu-dyn↗

arXiv2018

DNS Study on Vorticity Structures in Late Flow Transition

Vorticity and vortex are two different but related concepts. This paper focuses on the investigation of vorticity generation and development, and vorticity structure inside/ outside the vortex. Vortex is a region where the vorticity overtakes deformation. Vortex cannot be directly represented by the vorticity. Except for those vorticity lines which come from and end at side boundaries, another type of vorticity, self-closed vorticity lines named vorticity rings, is numerously generated inside the domain during flow transition. These new vorticity rings are found around the hairpin vortex heads and legs. The generation and growth of vorticity rings are produced by the buildup of the vortices according to the vorticity transport equation. On the other hand, vortex buildup is a consequence of vorticity line stretching, tilting and twisting. Both new vorticity and new vortices are generated during the flow transition. According to the Helmholtz vorticity flux conservation law, vorticity line cannot be interrupted, started, or ended inside the flow field, the newly produced vorticity has only one form which is the vorticity rings. In addition, an interesting finding is that a single hairpin vortex consists of several types of vorticity lines which could come from the side boundaries, whole vorticity rings and part of vorticity rings.

physics.flu-dyn↗

arXiv2018

Rortex and comparison with eigenvalue-based vortex identification criteria

Most of the currently popular Eulerian vortex identification criteria, including the Q criterion, the Delta criterion and the Lambda_ci criterion, are based on the analysis of the velocity gradient tensor. More specifically, these criteria are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria have been found to be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to contamination by shearing. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The real eigenvector of the velocity gradient tensor is used to define the direction of Rortex as the possible axis of the local fluid rotation, and the rotational strength obtained in the plane perpendicular to the possible local axis is defined as the magnitude of Rortex. This alternative definition is mathematically equivalent to our previous one but allows a much more efficient implementation. Furthermore, a complete and systematic interpretation of scalar, vector and tensor versions of Rortex is presented to provide a unified and clear characterization of the instantaneous local rigidly rotation. By relying on the tensor interpretation of Rortex, a new decomposition of the velocity gradient tensor is proposed to shed light on the analytical relations between Rortex and eigenvalue-based criteria. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength.

physics.flu-dyn↗

arXiv2018

Rortex A New Vortex Vector Definition and Vorticity Tensor and Vector Decompositions

A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally-accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity which is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and an anti-symmetric deformation part, and the vorticity vector is decomposed to a rigidly rotational vector and a non-rotational vector. Several cases, including 2D Couette flow, 2D rigid rotational flow and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.

physics.flu-dyn↗

arXiv2017

A Definition of Vortex Vector and Vortex

Vortex is ubiquitous in nature. However, there is not a consensus on the vortex definition in fluid dynamics. Lack of mathematical definition has caused considerable confusions in visualizing and understanding the coherent vortical structures in turbulence. According to previous study, it is realized that vortex is not the vorticity tube and vorticity should be decomposed into a rotational part which is the vortex vector and a non-rotational part which is the shear. In this paper, several new concepts such as fluid rotation of local point, the direction of fluid rotation axis and the strength of fluid rotation are proposed by investigating the kinematics of fluid element in the 2D and 3D flows. A definition of a new vector quantity called vortex vector is proposed to describe the local fluid rotation. The direction of the vortex vector is defined as the direction of local fluid rotation axis. The velocity components in the plane orthogonal to the vortex vector have zero derivatives along the vortex vector direction. The magnitude of the vortex vector is defined as the rotational part of vorticity in the direction of the vortex vector, which is the twice of the minimum angular velocity of fluid around the point among all azimuth in the plane perpendicular to vortex vector. According to the definition of the vortex vector, vortex is defined as a connected flow region where the magnitude of the vortex vector at each point is larger than zero. The new definition for the vortex vector and vortex follows three principles: 1. Local in quantity, 2. Galilean invariant, 3. Unique. The definitions are carefully checked by DNS and LES examples which clearly show that the new defined vortex vector and vortex can fully represent the complex structures of vortices in turbulence.

physics.flu-dyn↗

arXiv2014

DNS Study for the origin of the flow Randomization in Late Boundary Layer Transition

This paper is devoted to the investigation of the origin and mechanism of randomization in late boundary layer transition over a flat plate without pressure gradient. The flow randomization is a crucial phase before flow transition to the turbulent state. According to existing literatures, the randomization was caused by the big background noises and non-periodic spanwise boundary conditions. It was assumed that the large ring structure is affected by background noises first, and then the change of large ring structure affects the small length scales quickly, which directly leads to randomization and formation of turbulence. However, by careful analysis of our high order DNS results, we believe that the internal instability of multiple ring cycles structure is the main reason. What we observed is that randomization begins when the third cycle overlaps the first and second cycles. A significant asymmetric phenomenon is originated from the second cycle in the middle of both streamwise and spanwise directions. More technically, a visible asymmetric phenomenon in the middle vortex ring cycle starts at time step t=16.25T and x=838.9δin where the top and bottom level rings are still completely symmetric. The non-symmetric structure of middle level ring affects the small length scale in boundary layer bottom quickly. The randomization phenomenon spreads to top level through ejections. Finally, the whole flow domain becomes randomized. A hypothesis of C- and K-types shift is given as a possible mechanism of flow randomization.

nlin.CD↗

arXiv2014

DNS Study on Physics of Late Boundary Layer Transition

This paper serves as a review of our recent new DNS study on physics of late boundary layer transition. This includes mechanism of the large coherent vortex structure formation, small length scale generation and flow randomization. The widely spread concept vortex breakdown to turbulence,which was considered as the last stage of flow transition, is not observed and is found theoretically incorrect. The classical theory on boundary layer transition is challenged and we proposed a new theory with five steps, i.e. receptivity, linear instability, large vortex formation, small length scale generation, loss of symmetry and randomization to turbulence. We have also proposed a new theory about turbulence generation. The new theory shows that all small length scales (turbulence) are generated by shear layer instability which is produced by large vortex structure with multiple level vortex rings, multiple level sweeps and ejections, and multiple level negative and positive spikes near the laminar sub-layers.Therefore, turbulence is not generated by vortex breakdown but rather positive and negative spikes and consequent high shear layers. Shear layer instability is considered as the mother of turbulence. This new theory may give a universal mechanism for turbulence generation and sustenance which shows that the energy is brought by large vortex structure through multiple level sweeps.

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