Since the introduction of the Richardson-Kolmogorov cascade in which the flow in wind tunnels, pipes, jets, wakes or along moving surfaces is injecting energy into the turbulent fluid motion at large scales which break down to smaller and smaller scales due to entrainment effects, shear instability mechanisms and the dynamics of turbulent scale interaction a picture of turbulence has been created that intrinsically connects this (in general) directional down-scaling process featuring vortical flow structures with the overall energy transfer finally ending into viscous dissipation at the smallest scales of the cascade. The energy transfer mechanism over many scales of turbulent structures is governed by the (local) energy dissipation rate ε and plays a significant role for the understanding of turbulence and its self-sustaining spatial and temporal organization as it delivers the knowledge about how much energy of the turbulent motion is transferred globally from scale-to-scale and locally into heat in a statistical sense. After Pope (2000) the picture of the cascade indicates that ε at sufficiently high Reynolds numbers has to scale with ~U³/L, with U² representing the turbulent kinetic energy and L the integral length scale of the turbulent motion. If the range of spatial scales found in the turbulent structures is large enough, while the range is increasing with Reynolds number approximately with L/h ~ ReL9/4 (Vassilicos 2015) with h being the Kolmogorov length scale, a third range of scales develops, where neither the peculiarities of energy injection (at scales ~L), nor viscous dissipation at small scales ~ influence the spatial scale-to scale energy transfer, which is called inertial range. In this intermediate range following a -5/3 power law in the corresponding spectrum statistical properties can be described by a balance of the scale-to-scale transfer of kinetic energy and the kinetic energy dissipation ε (dissipated power per unit mass). These unique statistical properties in the inertial range are used by several methods allowing to approximate ε without the need to resolve the full (fluctuating) velocity gradient tensor Aij at the smallest scales. In the inertial range of flow structures estimates of ε can be calculated using the energy spectra, the second- or third order structure function (the latter known from the famous -4/5’s-law (Kolmogorov 1941)) e.g. of the longitudinal component (see e.g. Mann et al 1999 or Xu and Chen 2012) or the velocity-acceleration structure function -2ε = <du∙da> (Falkovich et al. 2012), while all methods assuming homogeneity. It has been shown that for many turbulent flows some anisotropy remains at the inertial subrange (Davidson 2004) or even down to dissipative scales even at moderate and higher Reynolds numbers and that large kinetic energy dissipation events happened intermittently in time and space depending on specific flow structure interactions. Therefore a method for a direct 3D measurement of the full Aij resolving all relevant scales at once is still required, which would allow to statistically estimating all energy dissipation rate elements εij without assumptions of symmetry, homogeneity or (local) isotropy. According to Komolgorov’s similarity hypothesis the turbulence statistics of the velocity gradients at dissipative scales are governed only by the kinematic viscosity nu and the energy dissipation rate ε.
The first work presenting a (moderately accurate) direct estimation of the (low-pass filtered) VGT in low Reynolds number homogenous turbulence was done by Lüthi et al 2005 using classical 3D-PTV data with linear interpolation between tetrahedrons of particles in closer proximity. Due to the tremendous problems associated with the classical measurement techniques described above in estimating all elements of εij because of low pass filtering, (systematic) noise and/or intrusiveness issues when aiming at resolving the smallest scales of the velocity gradients at sufficiently high Reynolds numbers there have been only few reports on experimental investigations aiming at a direct determination of all elements of εij, mostly with relatively large error bars and uncertainties. Therefore, PIV related measurements of velocity gradients typically can only claim resolving the full inertial range, sometimes down to the larger dissipative scales, but need to rely on proper correction schemes reflecting the various error sources (Tanaka and Eaton 2007) when the smallest scales are considered. A more recent study in a von Kármán flow (Knutsen et al. 2019) used scanning PIV with a two-camera MART processing for volumetric reconstruction in order achieve the full local Aij (slightly low pass-filtered with correlation-window sizes of ~ 3h) and investigated anisotropy and directionality effects of the inter-scale energy transfer based on the generalized version of the Kármán-Horwarth equation, extended by Monin and Hill.
Recent developments in dense Lagrangian Particle Tracking with Shake-The-Box (Schanz et al. 2016) and data assimilation techniques, like FlowFit (Gesemann et al. 2016), raises hope that for moderate Reynolds numbers the full (time-resolved) velocity gradient tensor might be captured experimentally at high-spatial resolution more accurately than in previous generic turbulence measurements based e.g. on Tomo- or scanning stereo PIV. In (Schröder et al. 2015) it was shown that the wall-shear stress and instantaneous friction velocity vectors in a turbulent boundary layer flow could be estimated accurately by STB from particle tracks in close vicinity of the wall. Here we attempt to increase the spatial range and resolution down to ~h at moderate image magnification by detecting a sufficient amount of events with close tetrahedron particle track constellations in order to reach statistical convergence for determining all elements of the dissipation rate tensor in homogeneous turbulence directly without losing the global picture of the surrounding energy containing flow structures in the inertial regime. At the same time we are reducing the noise level for determining the related instantaneous velocity gradients from such particle constellations in close proximity by making use of an optimal temporal track fit algorithm (Gesemann et al. 2016) and of analytical derivatives along the cubic 3rd order B-splines which are the base functions for the continuous interpolation of the flow field by the FlowFit data assimilation scheme.