Journal of Fluid Mechanics


Invariants of the reduced velocity gradient tensor in turbulent flows

J. I. Cardesaa1 c1, D. Mistrya2, L. Gana2 and J. R. Dawsona2

a1 School of Aeronautics, Universidad Politécnica de Madrid, 28040 Madrid, Spain

a2 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK


In this paper we examine the invariants and of the reduced 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 and exhibit a common characteristic asymmetric shape consistent with . An explanation for this inequality is proposed. Assuming local homogeneity we derive and . With the addition of local isotropy the sign of is proved to be the same as that of the skewness of , hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of obtained from the full 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 strain-rate tensor.

(Received July 25 2012)

(Revised October 10 2012)

(Accepted November 09 2012)

Key words

  • isotropic turbulence;
  • turbulence theory;
  • turbulent flows


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