Invariants of the reduced velocity gradient tensor in turbulent flows. Cardesa, J., A., I., Mistry, D., Gan, L., & Dawson, J., A., R. Journal of Fluid Mechanics, 716:597-615, 4, 2013. ![Invariants of the reduced velocity gradient tensor in turbulent flows [link]](https://bibbase.org/img/filetypes/link.svg "link")
Website doi abstract bibtex In this paper we examine the invariants p and q of the reduced 2 × 2 velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of p and q exhibit a common characteristic asymmetric shape consistent with < 0. An explanation for this inequality is proposed. Assuming local homogeneity we derive = 0 and = 0. With the addition of local isotropy the sign of is proved to be the same as that of the skewness of ∂u 1 /∂x 1 , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of p–q stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of Q–R obtained from the full 3 × 3 VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full 3 × 3 strain-rate tensor. 1. Introduction With the arrival of data sets providing access to the full three-dimensional velocity gradient tensor (3D VGT), new ways of analysing this wealth of information have been introduced. The 3D VGT can be written as A ij = ∂u i /∂x j = S ij + W ij , where S ij is the symmetric rate-of-strain tensor and W ij is the antisymmetric rate-of-rotation tensor. One approach is to examine the eigenvalues of S ij . They indicate if the strain-rate tensor is compressing or extending along its principal axes, which in turn determines whether or not the overall strain-rate configuration at a point is sheet forming or tube forming. Another approach is to analyse the local flow topology using the invariants of A ij , as set out by the work of Chong, Perry & Cantwell (1990). In an incompressible flow the number of non-vanishing similarity invariants of the second-order tensor A ij reduces to two, namely Q and R. These two quantities can fully characterize the category to which the local flow topology belongs. The local flow topology refers to the streamline pattern of a region of the flow in the immediate vicinity of the point
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title = {Invariants of the reduced velocity gradient tensor in turbulent flows},
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year = {2013},
keywords = {isotropic turbulence,turbulence theory,turbulent flows},
pages = {597-615},
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abstract = {In this paper we examine the invariants p and q of the reduced 2 × 2 velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of p and q exhibit a common characteristic asymmetric shape consistent with < 0. An explanation for this inequality is proposed. Assuming local homogeneity we derive = 0 and = 0. With the addition of local isotropy the sign of is proved to be the same as that of the skewness of ∂u 1 /∂x 1 , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of p–q stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of Q–R obtained from the full 3 × 3 VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full 3 × 3 strain-rate tensor. 1. Introduction With the arrival of data sets providing access to the full three-dimensional velocity gradient tensor (3D VGT), new ways of analysing this wealth of information have been introduced. The 3D VGT can be written as A ij = ∂u i /∂x j = S ij + W ij , where S ij is the symmetric rate-of-strain tensor and W ij is the antisymmetric rate-of-rotation tensor. One approach is to examine the eigenvalues of S ij . They indicate if the strain-rate tensor is compressing or extending along its principal axes, which in turn determines whether or not the overall strain-rate configuration at a point is sheet forming or tube forming. Another approach is to analyse the local flow topology using the invariants of A ij , as set out by the work of Chong, Perry & Cantwell (1990). In an incompressible flow the number of non-vanishing similarity invariants of the second-order tensor A ij reduces to two, namely Q and R. These two quantities can fully characterize the category to which the local flow topology belongs. The local flow topology refers to the streamline pattern of a region of the flow in the immediate vicinity of the point},
bibtype = {article},
author = {Cardesa, J A I and Mistry, D and Gan, L and Dawson, J A R},
doi = {10.1017/jfm.2012.558},
journal = {Journal of Fluid Mechanics}
}
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Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of p and q exhibit a common characteristic asymmetric shape consistent with < 0. An explanation for this inequality is proposed. Assuming local homogeneity we derive = 0 and = 0. With the addition of local isotropy the sign of is proved to be the same as that of the skewness of ∂u 1 /∂x 1 , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of p–q stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of Q–R obtained from the full 3 × 3 VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full 3 × 3 strain-rate tensor. 1. Introduction With the arrival of data sets providing access to the full three-dimensional velocity gradient tensor (3D VGT), new ways of analysing this wealth of information have been introduced. The 3D VGT can be written as A ij = ∂u i /∂x j = S ij + W ij , where S ij is the symmetric rate-of-strain tensor and W ij is the antisymmetric rate-of-rotation tensor. One approach is to examine the eigenvalues of S ij . They indicate if the strain-rate tensor is compressing or extending along its principal axes, which in turn determines whether or not the overall strain-rate configuration at a point is sheet forming or tube forming. Another approach is to analyse the local flow topology using the invariants of A ij , as set out by the work of Chong, Perry & Cantwell (1990). In an incompressible flow the number of non-vanishing similarity invariants of the second-order tensor A ij reduces to two, namely Q and R. These two quantities can fully characterize the category to which the local flow topology belongs. 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