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.

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

A vortex is intuitively recognized as the rotational/swirling motion of fluids, but a rigorous and universally-accepted definition is still not available. Vorticity tube/filament has been regarded equivalent to a vortex since Helmholtz proposed the concepts of vorticity tube/filament in 1858 and the vorticity-based methods can be categorized as the first generation of vortex identification methods. During the last three decades, a lot of vortex identification methods, including Q, Δ, λ2 and λci criteria, have been proposed to overcome the problems associated with the vorticity-based methods. Most of these criteria are based on the Cauchy-Stokes decomposition and/or eigenvalues of the velocity gradient tensor and can be considered as the second generation of vortex identification methods. Starting from 2014, the Vortex and Turbulence Research Team at the University of Texas at Arlington (the UTA team) focus on the development of a new generation of vortex identification methods. The first fruit of this effort, a new omega (Ω) vortex identification method, which defined a vortex as a connected region where the vorticity overtakes the deformation, was published in 2016. In 2017 and 2018, a Liutex (previously called Rortex) vector was proposed to provide a mathematical definition of the local rigid rotation part of the fluid motion, including both the local rotational axis and the rotational strength. Liutex/Rortex is a new physical quantity with scalar, vector and tensor forms exactly representing the local rigid rotation of fluids. Meanwhile, a decomposition of the vorticity to a rotational part namely Liutex/Rortex and an anti-symmetric shear part (RS decomposition) was introduced in 2018, and a universal decomposition of the velocity gradient tensor to a rotation part (R) and a non-rotation part (NR) was also given in 2018 as a counterpart of the traditional Cauchy-Stokes decomposition. Later in early 2019, a Liutex/Rortex based Omega method called Omega-Liutex, which combines the respective advantages of both Liutex/Rortex and Omega methods, was developed. And a latest objective Omega method, which is still under development, is also briefly introduced. These advances are classified as the third generation of vortex identification methods in the current paper. To elaborate the advantages of the third-generation methods, six core issues for vortex definition and identification have been raised, including: (1) the absolute strength, (2) the relative strength, (3) the rotational axis, (4) the vortex core center location, (5) the vortex core size, (6) the vortex boundary. The new third generation of vortex identification methods can provide reasonable answers to these questions, while other vortex identification methods fail to answer all questions except for the approximation of vortex boundaries. The purpose of the current paper is to summarize the main ideas and methods of the third generation of vortex identification methods rather than to conduct a comprehensive review on the historical development of vortex identification methods.

Third generation of vortex identification methods: Omega and Liutex/Rortex based systems

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 that 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 possible 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 a non-rotationally anti-symmetric part, and the vorticity vector is decomposed to a rigidly rotational vector which is called the Rortex vector and a non-rotational vector which is called the shear vector. Several cases, including the 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.

Rortex—A new vortex vector definition and vorticity tensor and vector decompositions

The relationship between the upstream boundary layer and the low-frequency, large-scale unsteadiness of the separated flow in a Mach 2 compression ramp interaction is investigated by performing wide-field particle image velocimetry (PIV) and planar laser scattering (PLS) measurements in streamwise–spanwise planes. Planar laser scattering measurements in the upstream boundary layer indicate the presence of spanwise strips of elongated regions of uniform momentum with lengths greater than 40?. These long coherent structures have been observed in a Mach 2 supersonic boundary layer (Ganapathisubramani, Clemens & Dolling 2006) and they exhibit strong similarities to those that have been found in incompressible boundary layers (Tomkins & Adrian 2003; Ganapathisubramani, Longmire & Marusic 2003). At a wall-normal location of y/?=0.2, the inferred instantaneous separation line of the separation region is found to oscillate between x/?=?3 and ?1 (where x/?=0 is the ramp corner). The instantaneous spanwise separation line is found to respond to the elongated regions of uniform momentum. It is shown that high- and low-momentum regions are correlated with smaller and larger size of the separation region, respectively. Furthermore, the instantaneous separation line exhibits large-scale undulations that conform to the low- and high-speed regions in the upstream boundary layer. The low-frequency unsteadiness of the separation region/shock foot observed in numerous previous studies can be explained by a turbulent mechanism that includes these elongated regions of uniform momentum

Effects of upstream boundary layer on the unsteadiness of shock induced separation

Most of the current Eulerian vortex identification criteria, including the Q criterion and the λci criterion, 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 will be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to severe contamination by shearing. To address these issues, a new vector named Rortex which represents the local fluid rotation was proposed in our previous work. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then the rotational strength obtained in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. This new equivalent definition allows a much more efficient implementation. Furthermore, a systematic interpretation of scalar, vector, and tensor versions of Rortex is presented. By relying on the tensor interpretation, the velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition. 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. In addition, in contrast to eigenvalue-based criteria, not only the iso-surface of Rortex but also the Rortex vectors and the Rortex lines can be applied to investigate vortical structures. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Rortex.

Rortex and comparison with eigenvalue-based vortex identification criteria

A new vortex identification criterion called W -method is proposed based on the ideas that vorticity overtakes deformation in vortex. The comparison with other vortex identification methods like Q -criterion and l 2-method is conducted and the advantages of the new method can be summarized as follows: (1) the method is able to capture vortex well and very easy to perform; (2) the physical meaning of W is clear while the interpretations of iso-surface values of Q and l 2 chosen to visualize vortices are obscure; (3) being different from Q and l 2 iso-surface visualization which requires wildly various thresholds to capture the vortex structure properly, W is pretty universal and does not need much adjustment in different cases and the iso-surfaces of W =0.52 can always capture the vortices properly in all the cases at different time steps, which we investigated; (4) both strong and weak vortices can be captured well simultaneously while improper Q and l 2 threshold may lead to strong vortex capture while weak vortices are lost or weak vortices are captured but strong vortices are smeared; (5) W =0.52 is a quantity to approximately define the vortex boundary. Note that, to calculate W , the length and velocity must be used in the non-dimensional form. From our direct numerical simulation, it is found that the vorticity direction is very different from the vortex rotation direction in general 3-D vortical flow, the Helmholtz velocity decomposition is reviewed and vorticity is proposed to be further decomposed to vortical vorticity and non-vortical vorticity.

New omega vortex identification method

Vortex and vorticity are two correlated but fundamentally different concepts which have been the central issues in fluid mechanics research. Vorticity has rigorous mathematical definition (curl of velocity), but no clear physical meaning. Vortex has clear physical meaning (rotation) but no rigorous mathematical definition. For a long time, many people treat them as a same thing. However, based on our high-order direct numerical simulation (DNS), we found that first, “vortex” is not “vorticity tube” or “vortex tube” which is widely defined as a bundle of vorticity lines without any vorticity line leak. Actually, vortex is an open area for vorticity line penetration. Second, vortex is not necessarily congregation of vorticity lines, but dispersion in many 3-dimensional cases. Some textbooks say that vortex cannot end inside the flow field but must end on the solid wall (and/or boundaries). Our DNS observation and many other numerical results show almost all vortices are ended inside the flow field. Finally, a more theoretical study shows that neither vortex nor vorticity line can attach to the solid wall and they must be detached from the wall.

DNS Study on Vortex and Vorticity in Late Boundary Layer Transition

Shock oscillation with low-frequency unsteadiness commonly occurs in supersonic flows and is a top priority for the control of flow separation caused by shock wave and boundary layer interaction. In this paper, the interaction of the shock caused by the compression ramp and the vortex rings generated by a micro-vortex generator (MVG) in a supersonic flow at Ma = 2.5 is simulated by the implicit large eddy simulation method. The analysis of observation and the frequency of both the vortex ring motion and the shock oscillation is carried out. The results show that the shock produced by a compression ramp flow at Ma = 2.5 has a dominant non-dimensional low frequency, which is around St = 0.002, while the vortex rings behind the MVG have a dominant high frequency which is around St = 0.038. The dominant low frequency of the shock, which is harmful, can be removed or weakened through the shock-vortex ring interaction by the vortex rings which generate high frequency fluctuations. In the shock and vortex ring interaction region, a dominant high frequency St = 0.037–0.038 has been detected rather than the low frequency St = 0.002, which indicates that the vortex ring is stiff enough to break or weaken the shock. This analysis could provide an effective tool to remove or weaken the low frequency pressure fluctuation below 500 Hz, which has a negative effect on the flight vehicle structures and the environmental protection, through the high frequency vortex generation.

Spectrum study on unsteadiness of shock wave–vortex ring interaction

A high order implicit large eddy simulation (ILES) is carried out to study the mechanism of shock induced flow separation reduction under ramp-type MVG control The mechanism was originally considered as that MVG can generate streamwise vortices which strongly mix boundary layer and the boundary layer becomes more capable to resist strong adverse pressure gradient caused by shock and to keep the boundary layer attached However, according to our ILES, a chain of ring-like vortices is generated behind the ramp-type MVG and goes further to interact with the shock When the ring-like vortices pass through the shock, the shock wave is weakened and altered while the vortex structures are quite stable The instantaneous simulation shows that the spanwise ring-like vortex, not the streamwise vortex, plays a key role to weaken the shock and reduce the shock-induced separation Detailed investigation on ring-like vortices and shock interaction will be given in this paper

Yong Yang

Papers

New omega vortex identification method

DNS Study on Vortex and Vorticity in Late Boundary Layer Transition

Spectrum study on unsteadiness of shock wave–vortex ring interaction

ILES for mechanism of ramp-type MVG reducing shock induced flow separation

DNS study on three vortex identification methods