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.

Although the vortex is ubiquitous in nature, its definition is somewhat ambiguous in the field of fluid dynamics. In this absence of a rigorous mathematical definition, considerable confusion appears to exist in visualizing and understanding the coherent vortical structures in turbulence. Cited in the previous studies, a vortex cannot be fully described by vorticity, and vorticity should be further decomposed into a rotational and a non-rotational part to represent the rotation and the shear, respectively. In this paper, we introduce several new concepts, including local fluid rotation at a point and the direction of the local fluid rotation axis. The direction and the strength of local fluid rotation are examined by investigating the kinematics of the fluid element in two- and three-dimensional flows. A new vector quantity, which is called the vortex vector in this paper, is defined to describe the local fluid rotation and it is the rotational part of the vorticity. This can be understood as that the direction of the vortex vector is equivalent to the direction of the local fluid rotation axis, and the magnitude of vortex vector is the strength of the location fluid rotation. With these new revelations, a vortex is defined as a connected region where the vortex vector is not zero. In addition, through direct numerical simulation (DNS) and large eddy simulation (LES) examples, it is demonstrated that the newly defined vortex vector can fully describe the complex vertical structures of turbulence.

In this paper, two functions are introduced to describe the turbulence generation in late flow transition. One is called the volume omega bar (volume $\overline{\mathrm{\Omega }}$ ), which represents the flow rotation or vortex strength. The other is called the volume vorticity, which shows the flow statistical fluctuations. Although they have very different definitions, one is for fluctuation and the other is for rotation, volume $\overline{\mathrm{\Omega }}$ and volume vorticity are found highly correlated with a correlation factor greater than 0.9, which means that there is a very close correlation between flow fluctuation and flow rotation (vortex). While the vorticity flux keeps constant in the late flow transition through the integration over any sections either parallel or perpendicular to the flow direction, the volume $\overline{\mathrm{\Omega }}$ is greatly increased along the flow direction during the flow transition process. This means that the vortex structures are greatly built up and rotation becomes more and more dominant. On the other hand, the total volume vorticity is also quickly increased, which shows that the fluctuation is stronger. The flow transition is a process with significant volume vorticity increase, which is attributed to two of the following crucial factors: the first one is the lengthening of the vorticity tubes from side boundary due to the vorticity line stretching, distortion, and tangling; the second one is the generation of countless self-closed vorticity rings within the domain. Both the increase of the volume vorticity and the volume $\overline{\mathrm{\Omega }}$ can be a significant symbol of the flow transition from the laminar flow to turbulent flow. It also shows that vorticity (tubes or lines) cannot directly represent vortex and should not be simply used as a signal of the turbulent transition process.

The overset grid method for simulation of unsteady flow with moving bodies faces several issues, including low assembly efficiency for a large number of bodies, difficult parallel implementation, and the requirement of manual intervention. To address these issues, in this paper, we develop an efficient, automatic, and robust parallel dynamic overset-unstructured-grid assembly method. It is achieved by parallelizing its two main steps: hole cutting, and identification of interpolation stencils. The hole cutting is simply implemented in parallel by identifying the active zone of each grid with a criterion of the global minimum wall distance. The global minimum wall distance is efficiently calculated on each process with the oriented-bounding-box-based [ ]-dimensional ([ ]-D) trees of the wall surface grids. For identification of the interpolation stencil, we use the results of the first step, optimally define the interpolation boundary nodes, and determine their candidate donor partitions in each process. This efficiently minimizes the number of query nodes and their donor cell candidates. The donor cell search is efficiently performed in parallel in each candidate donor partition by using the oriented-bounding-box-based [ ]-D tree of the field volume grid. Several cases are adopted to test the efficiency and capability of the proposed parallel dynamic overset-unstructured-grid assembly approach.

In this work, a vortex identification method is developed by analyzing the physical meaning of the local rotation of fluid elements. It is shown that a point is locally rotational when the velocity gradient tensor at the point has a pair of complex eigenvalues. The local rotation can be represented by a so called vortex vector. The direction of the vortex vector is defined as that of the local fluid rotation axis, which is parallel to the eigenvector of velocity gradient tensor corresponding to the real eigenvalue. The magnitude is evaluated as the twice of the minimum angular velocity around the point among all azimuth in the plane perpendicular to the vortex vector. Based on the local fluid rotation, a vortex is identified as a connected region where the vortex vector at each point is not equal to zero. The vortex identification method is validated by applying it to Reynolds-averaged Navier-Stokes data and direct numerical simulation data. The results reveal that the method can fully describe the complex structures of vortices.