Invariants of the reduced velocity gradient tensor in turbulent flows

Cardesa, JI; Mistry, D; Gan, L; Dawson, JR

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 ⟨pq⟩<0. An explanation for this inequality is proposed. Assuming local homogeneity we derive ⟨p⟩=0 and ⟨q⟩=0. With the addition of local isotropy the sign of ⟨pq⟩ is proved to be the same as that of the skewness of ∂u1/∂x1, hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of p\ndashq 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\ndashR 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.

Citation

Cardesa, J., Mistry, D., Gan, L., & Dawson, J. (2013). Invariants of the reduced velocity gradient tensor in turbulent flows. Journal of Fluid Mechanics, 716, 597-615. https://doi.org/10.1017/jfm.2012.558

• Journal Article Type: Article
• Acceptance Date: Nov 9, 2012
• Online Publication Date: Jan 28, 2013
• Publication Date: Feb 10, 2013
• Deposit Date: May 16, 2016
• Journal: Journal of Fluid Mechanics
• Print ISSN: 0022-1120
• Electronic ISSN: 1469-7645
• Publisher: Cambridge University Press
• Volume: 716
• Pages: 597-615
• DOI: https://doi.org/10.1017/jfm.2012.558
• Public URL: https://durham-repository.worktribe.com/output/1381790