Ali Rahimi Khojasteh INRAE, OPAALE 17 avenue de Cucillé, 35044, Rennes, France ali.rahimi-khojasteh@inrae.fr Dominique Heitz INRAE, OPAALE dominique.heitz@inrae.fr Lionel Fiabane INRAE, OPAALE lionel.fiabane@inrae.fr
ABSTRACT
In the present study, we investigate the computation of the Lagrangian second-order structure-function to characterise the multiscale dynamics of turbulence from measured particle tra- jectories. We performed time-resolved three-dimensional par- ticle tracking velocimetry (4D-PTV) to study the anisotropic and inhomogeneous flow field of the wake behind a cylinder at a Reynolds number equal to 3900.  We performed Lagrangian statistical analysis on nearly 12000 trajectories for 4000 time steps.
LAGRANGIAN DIFFUSION PROPERTIES
Taylor's turbulent diffusion theory (Taylor, 1922) has been used widely to study homogeneous isotropic turbulent (HIT) flows.  In a given time τ, Taylor's theory computes the Lagrangian two-point correlation function R uy ( τ ) for an en- semble of particle trajectories based on the mean square dis- placements of particles σ 2 (τ) that can be written as,
The turbulent diffusion process to Lagrangian statistical properties of particle trajectories can be linked if we assume that particles in the present study act as a tracer (Viggiano et al. , 2021).  This means that all the inertial effects are ne- glected, and particles perfectly follow the flow motion. There- fore, we can compute the Lagrangian second-order structure- function as,
where the Lagrangian trajectories were obtained from three- dimensional particle tracking velocimetry (4D-PTV). There- fore, we can compute the Lagrangian universal constant C 0 for the Lagrangian second-order structure function. This constant was found to be strongly sensitive to the Reynolds number, large-scale anisotropy and inhomogeneity of flow (Viggiano et al. , 2021). This means that computation of the C 0 constant is complicated in anisotropic and inhomogeneous turbulent cases. In the wake behind a cylinder, turbulent length and time scales evolve as flow goes downstream, which creates non- stationary anisotropic and inhomogeneous dynamics.  Batche- lor's diffusion theory (Batchelor, 1957) as an extension of Tay- lor's theory, proposed using the Lagrangian stationarisation idea for inhomogeneous cases such as the wake flow. Station- arisation is a process based on Eulerian self-similarity prop- erties that stationarises the Lagrangian dynamics.  Recently, Viggiano et al. (2021) investigated the highly anisotropic and inhomogeneous case in a free shear turbulent jet.  Viggiano et al. (2021) characterised the inertial-range dynamics and the Lagrangian universal constant C 0 by the Lagrangian stationar- isation idea. Two Lagrangian second-order structure function and two-point correlation function statistics needs to be com- puted. The Lagrangian second-order structure function can be written as,
where ε is the turbulent energy dissipation.  To stationarise the Lagrangian instationarity of the wake flow, as proposed by Viggiano et al. (2021), we compute the Eulerian mean ve- locity.  Then the deviation between the instantaneous and the mean components non-dimensionalised by the Reynolds stress terms as follows,
We can achieve a stationarised flow field with non- dimensionalised fluctuations through the entire spatial domain for evert time step.  With the same derivation spirit discussed in Viggiano et al. (2021); Ouellette (2021), we are interested in exploring Lagrangian properties for the wake behind a cylin- der. We will characterise the Eulerian turbulent properties as well as self-similarity behaviour, followed by estimation of the Lagrangian universal constant C 0 . 1
4D-PTV EXPERIMENTAL STUDY
Recent advances in time-resolved particle tracking ve- locimetry (4D-PTV) have led to accurate and long trajectory reconstruction in turbulent flows (Schanz et al. (2016); Kho- jasteh et al. (2021 b )).  We performed an experimental study of the cylinder wake flow at Reynolds number equal to 3900. We designed an experimental setup with four CMOS Speed- Sense DANTEC cameras and spatial resolutions of 1280 × 800 pixels.  We used the camera’s maximum frequency at 3 kHz in the present study to achieve the maximum reachable tem- poral resolution.  Four Nikon 105 mm lenses were installed on the cameras.  Considering the illumination direction, the first two cameras received particle signals in backward light scattering, while the second two cameras received maximum intensity signal in forward scattering.  The calibration error reduced down to 0.04 pixel after the volume self-calibration. We captured roughly volume of 280 mm × 160 mm × 46 mm starting from roughly 4D downstream of the cylinder, knowing that the vortex formation zone ends at 4D where the cylinder diameter was D = 12 mm.  Details of the experiment setup can be found in Khojasteh et al. (2021 a ).  The velocity com- ponent of each trajectory was computed after fitting a curve over noisy reconstructed positions.  The Eulerian fields were computed using fine-scale reconstruction (VIC#, Jeon et al. (2018)) between trajectories to achieve gridded velocity fields. Thereafter, time-averaged mean ūj(x(τ)) and Reynolds stress σιι ,(x(τ)) terms can be computed by averaging all instanta- neous Eulerian velocity fields to start the process in equa- tion (4).
WAKE FLOW STATISTICS
Due to the loss of momentum cases by the cylinder, the wake velocities are smaller than the free stream region.  The wake thickness increases as the flow travels downstream of the cylinder.  We can compute the Eulerian statistics within the area of the wake.  Ensemble of all trajectories passing a vir- tual volume inside the wake at a certain x/D is considered to achieve statistically converged Eulerian properties.  The Eu- lerian volume has the dimension of δ(x/D) × 2D in y and z directions with 0.5mm depth in x direction, as suggested by Viggiano et al.  Viggiano et al. (2021).  Therefore, we can compute the Eulerian second order structure function over the ensemble of spatial velocity increments of each pair trajecto- ries passing the Eulerian volume.  Velocity components were stationarised with equation (4). We computed the Eulerian tur-
bulent properties in seven downstream positions varying from x/D = 9 to 25 as listed in table 1.  Nearly constant values of the Taylor microscale Re λ  shows that the stationarisation pro- cess suggested by Batchelor Batchelor (1957) is valid for the self similar wake flow far downstream of the wake flow. De- cay of the dissipation rate ε π , toward downstream is also in with power-law decay in self-similar flows. Both Kolmogorov η x , and integral L E  length scales are growing as flow goes far downstream. The evolution of Eulerian dissipation rate ε u x , Kol- mogorov length scale η x , and integral scale T E z will be used to compute Lagrangian statistical properties.  To compute the Lagrangian second order statistics, we assume a small cube volume inside the wake with the length of δ(x/D)/3 (sug- gested by Viggiano et al.  Viggiano et al. (2021)) and index all trajectories passing the volume.  Following equation (3), we compute temporal velocity increments on each individual trajectory. The ensemble of computed temporal increments is then averaged to compute the Lagrangian second order struc- ture function. By solving the left side of equation (3), we can compute the C 0 constant where ε ; and σ 2 , are the Eulerian dis- sipation rate and the averaged velocity standard deviation over the Lagrangian volume. The Lagrangian second order structure function at four downstream locations is plotted in figure 1.  The result of non-dimensional S 2 x in far downstream shows that the La- grangian statistics becomes independent of the downstream position (x/D > 10).  In the present study, we were unable to reach converged statistics for x/D < 10. We also observe that the non-dimensional S 2 x has linear relation with τ where η) x < τ < T E .  These findings are in agreement with the free jet self-similar case Viggiano et al. (2021).
CONCLUSION AND OUTLOOK
In the present study, we are interested in computing the Lagrangian universal constant C 0 for the wake behind a cylinder at subcritical Reynold number.  This constant in the Lagrangian framework is in a similar role as the Kolmogorov constant in the Eulerian framework (Viggiano et al. , 2021). We acquired 4000 time steps to achieve the converged statis- tics using in-house time-resolved three-dimensional particle tracking velocimetry (4D-PTV) code.  Lagrangian statistics of the wake flow experiment showed that the Lagrangian statistics becomes independent of the downstream position in far downstream x/D > 10 (see figure 2).  These findings are in agreement with the free jet self-similar case (Viggiano et al. (2021)). Figure 2.b suggests the C 0 value, which should stay nearly constant in the inertial range.  C 0 is found to be between 2 — 4 for the selected downstream locations.  Based on the modelling suggested by Sawford Sawford (1991), C 0 should be around 2.6 for the corresponding Taylor microscale Reynolds number. Therefore, the estimated C 0 of the present study is in the same order as Sawford’s findings.