Abstract
Tomographic particle image velocimetry experiments were performed in the near field of the turbulent flow past a square cylinder. A classical Reynolds decomposition was performed on the resulting velocity fields into a time invariant mean flow and a fluctuating velocity field. This fluctuating velocity field was then further decomposed into coherent and residual/stochastic fluctuations. The statistical distributions of the second and third invariants of the velocity-gradient tensor were then computed at various streamwise locations, along the centreline of the flow and within the shear layers. These invariants were calculated from both the Reynolds-decomposed fluctuating velocity fields and the coherent and stochastic fluctuating velocity fields. The range of spatial locations probed incorporates regions of contrasting flow physics, including a mean recirculation region and separated shear layers, both upstream and downstream of the location of peak turbulence intensity along the centreline. These different flow physics are also reflected in the velocity gradients themselves with different topologies, as characterised by the statistical distributions of the constituent enstrophy and strain-rate invariants, for the three different fluctuating velocity fields. Despite these differing flow physics the ubiquitous self-similar ‘tear drop’-shaped joint probability density function between the second and third invariants of the velocity-gradient tensor is observed along the centreline and shear layer when calculated from both the Reynolds decomposed and the stochastic velocity fluctuations. These ‘tear drop’-shaped joint probability density functions are not, however, observed when calculated from the coherent velocity fluctuations. This ‘tear drop’ shape is classically associated with the statistical distribution of the velocity-gradient tensor invariants in fully developed turbulent flows in which there is no coherent dynamics present, and hence spectral peaks at low wavenumbers. The results presented in this manuscript, however, show that such ‘tear drops’ also exist in spatially developing inhomogeneous turbulent flows. This suggests that the ‘tear drop’ shape may not just be a universal feature of fully developed turbulence but of turbulent flows in general.
Highlights
The smallest scales present in a turbulent flow have long been thought to be well approximated as statistically homogeneous and isotropic (Kolmogorov 1941; Batchelor 1953)
A sensitivity study was conducted to assess the size of this window on the ability to faithfully reproduce the Q–R joint p.d.f. and it was found to be optimal in the sense that it was the minimal window size that generated relatively noise-free statistics that were insensitive to modest increases/decreases in window size
Tomographic particle image velocimetry (PIV) experiments were performed in the near field of the turbulent flow around a square cylinder generating a 3D3C data set
Summary
The smallest scales present in a turbulent flow have long been thought to be well approximated as statistically homogeneous and isotropic (Kolmogorov 1941; Batchelor 1953). The general topology of these fine scales may be shown to depend on the invariants of the velocity-gradient tensor (VGT) which may be split up into a symmetric and skew-symmetric component, respectively the strain-rate and rotation tensors, aij = ∂ ui ∂ xj sij + ωij ∂ uj ∂ xi − (1.1). The first invariant, P, is the negative trace of the VGT (P = −aii) and is identically zero for an incompressible flow. The generalised topology of the flow may be described by the invariants Q and R, defined as
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