FP7 Marie Curie Re-Integration Grant 228389
structure tensors
Homogeneous turbulence is the random field in which, all statistical properties are invariant under a fixed translation of the coordinate system. A convenient method to describe the morphology of homogeneous turbulent fields is by using the one point turbulence tensors, introduced by Reynolds (1989) and Kassinos and Reynolds (1994, 2001). Starting from the Reynolds stress tensor, which is defined by Rij= uiuj =εipqεjtsΨq,pΨs,t, introducing the isotropic tensor identityand assuming homogeneity one finds Rij+Ψk,iΨk,j+Ψi,kΨj,k=dijq^2, where q^2=2k=Rkk, is twice the turbulent kinetic energy (TKE). This constitutive equation shows that for a proper characterization of non-equilibrium turbulence, the componentality information found in Rij must be supplemented by structure information found in the one-point turbulent structure tensors defined by Dij=Ψk,iΨk,j, Fij=Ψi,kΨj,k. In addition to the above basic definitions of the structure tensors, one can use equivalent representations for homogeneous turbulence in terms of the velocity spectrum tensor E(k) and the vorticity spectrum tensor W(k). These tensors help to distinguish between the componentality of the turbulence, which is described by the Reynolds stress tensor and its dimensionality, which has to do with the dimensional structure of the turbulenceeddies, and is described by the structure dimensionality tensor. For homogeneous turbulence, the structure dimensionality tensor, which describes the elongation and orientation of energy-containing eddies, takes the form Dij=Ψk,iΨk,j. Furthermore, the structure circulicity tensor, which describes the distribution of large-scale circulation in the turbulence field, can be written as Fij=Ψi,kΨj,k. From the non-dimensional, normalized form of equation rij+dij+fij=dij, the two-linear independence of the structure tensors in homogeneous cases becomes evident. Thus, for homogeneous cases any pair of the three tensors defines the turbulent structure. For isotropic turbulence, rij=dij=fij=dij/3.
Fig. Schematic diagram showing idealized 2D structures in
homogeneous turbulence and the associated componentality
and dimensionality for: (a) vortical, (b) jetal, (c) helical eddy.
Fig. Schematic diagram showing idealized 2D structures in
homogeneous turbulence and the associated componentality
and dimensionality for: (a) vortical, (b) jetal, (c) helical eddy.