Abstract
AbstractThis paper examines the kinematic behaviour of the reduced velocity gradient tensor (VGT),$\tilde{\unicode[STIX]{x1D608}}_{ij}$, which is defined as a$2\times 2$block, from a single interrogation plane, of the full VGT$\unicode[STIX]{x1D608}_{ij}=\partial u_{i}/\partial x_{j}$. Direct numerical simulation data from the fully developed turbulent region of a nominally two-dimensional mixing layer are used in order to examine the extent to which information on the full VGT can be derived from the reduced VGT. It is shown that the reduced VGT is able to reveal significantly more information about regions of the flow in which strain rate is dominant over rotation. It is thus possible to use the assumptions of homogeneity and isotropy to place bounds on the first two statistical moments (and their covariance) of the eigenvalues of the reduced strain-rate tensor (the symmetric part of the reduced VGT) which in turn relate to the turbulent strain rates. These bounds are shown to be dependent upon the kurtosis of$\partial u_{1}/\partial x_{1}$and another variable defined from the constituents of the reduced VGT. The kurtosis is observed to be minimised on the centreline of the mixing layer and thus tighter bounds are possible at the centre of the mixing layer than at the periphery. Nevertheless, these bounds are observed to hold for the entirety of the mixing layer, despite departures from local isotropy. The interrogation plane from which the reduced VGT is formed is observed not to affect the joint probability density functions (p.d.f.s) between the strain-rate eigenvalues and the reduced strain-rate eigenvalues despite the fact that this shear flow has a significant mean shear in the cross-stream direction. Further, it is found that the projection of the eigenframe of the strain-rate tensor onto the interrogation plane of the reduced VGT is also independent of the plane that is chosen, validating the approach of bounding the full VGT using the assumption of local isotropy.
Highlights
The fine scales of turbulence are primarily characterised by the velocity gradient tensor (VGT), which can be split into a symmetric and skew-symmetric tensor as follows: Aij = ∂ ui ∂ xj Sij + Ωij ∂ uj ∂ xi − (1.1)P
The reduced VGT (Aij) is defined as a 2 × 2 block, from a single interrogation plane, of the full VGT. It has subsequently been shown in the study of Cardesa et al (2013) that the joint p.d.f. of the invariants, p and q, for the characteristic equation for Aij displays a characteristic ‘teapot’ shape for a number of turbulent flows
This distribution is confirmed in this study, lending credence to the notion that this shape is a ‘universal’ feature of turbulent flows similar to the ‘teardrop’ shape for the joint p.d.f. between the second and third invariants of the full VGT
Summary
The fine scales of turbulence are primarily characterised by the velocity gradient tensor (VGT), which can be split into a symmetric and skew-symmetric tensor as follows: Aij. They made use of microscopic, planar PIV experiments (purely two-dimensional data) to show the tendency for regions of high dissipation to be distributed as sheet-like structures surrounding the perimeter of high-enstrophy tubes at a characteristic offset that scales with the Kolmogorov length scale These results are in agreement with the scalings derived in much lower Reynolds number 3D3C simulations/experiments (e.g. Jiménez et al 1993 and Mullin & Dahm 2006) and demonstrate the capability to infer fully three-dimensional topological velocity gradient information from the reduced VGT. R, on the other hand, is a sum of triple velocity gradient products and reveals the local excess of inviscid strain-rate amplification over enstrophy amplification These terms are not present in the dynamics of two-dimensional turbulence, with the only mechanism by which enstrophy changes in time being via the direct action of viscosity (Batchelor 1969). This is reflected in (2.4) in which no decoupling of the terms of the reduced VGT and those not captured in the reduced VGT is possible with the exception of the first term, −pq, the mean of which is shown to predict ωiSijωj under the assumption of isotropy in (2.2)
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