An easy-to-interpret kinematic quantity measuring the average corotation of material line segments near a point is introduced and applied to vortex identification. At a given point, the vector of average corotation of line segments is defined as the average of the instantaneous local rigid-body rotation over 'all planar cross-sections' passing through the examined point. The vortex identification method based on average corotation is a one-parameter, region-type local method sensitive to the axial stretching rate as well as to the inner configuration of the velocity gradient tensor. The method is derived from a well-defined interpretation of the local flow kinematics to determine the 'plane of swirling' and is also applicable to compressible and variable-density flows. Practical application to DNS data sets includes a hairpin vortex of boundary-layer transition, the reconnection process of two Burgers vortices, a flow around an inclined flat plate, and a flow around a revolving insect wing. The results agree well with some popular local methods and perform better in regions of strong shearing.
Nomenclature
Ma= Mach number Q = vortex-identification criterion Q D = measure based on the deviatoric part of ∇u Q M = modification of the vortex-identification Q-criterion Re = Reynolds number S = strain-rate tensor, symmetric part of ∇u S D = deviatoric part of the strain-rate tensor S u, u i = velocity vector, components of u u i;j = spatial partial derivatives of components of u II = second invariant of an arbitrary second-order tensor A is defined by II A equals trA 2 − trA 2 =2 II S = second invariant of (the strain-rate tensor) S II SD = second invariant of (the deviatoric part of the strain-rate tensor) S D II ∇u = second invariant (see the definition of II above) of ∇u λ 2 = second-largest eigenvalue of S 2 Ω 2 , vortexidentification criterion σ = principal strain-rate difference vector,= deviatoric principal strain rates Ω = vorticity tensor, antisymmetric part of ∇u ω, ω 1 , ω 2 , ω 3 = vorticity vector, vorticity vector components ∇u = velocity-gradient tensor ∇ × u = curl of velocity vector
This paper attempts to answer Lyman's question (1990) on the nonuniqueness in de ning the 3D measure of the boundary vorticity-creation rate. Firstly, a straightforward analysis of the vorticity equation introduces a de nition of a general vorticity ®ux-density tensor and its`e¬ective' part. The approach is strictly based on classical eld theory and is independent of the constitutive structure of continuous medium. Secondly, the fundamental question posed by Lyman dealing with the ambiguity of the 3D measure of the boundary vorticity-creation rate for incompressible ®ow is discussed. It is shown that the original 3D measure (for an incompressible Newtonian ®uid de ned by Panton 1984), which is reminiscent of an analogy to Fourier's law, is in its character`e¬ective' and plays a crucial role in the prognostic vorticity transport equation. The alternative 3D measure proposed by Lyman includes, on the other hand, à non-e¬ective' part, which plays a role in the local determination of the`e¬ective' measure as well as in a certain diagnostic integral boundary condition.
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