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Abstract

Regions of negative turbulent kinetic energy (TKE) production are observed and studied in two different flows,namely in turbulent thermal convection in a cubic Rayleigh-Bénard convection cell, and in a pure shear flow in absence of buoyancy. The experimental investigation is performed using three-dimensional particle tracking velocimetry (3D-PTV) which allows for measuring the three velocity components and the full tensor of velocity derivatives in a finite 3D volume. The capability to compute the production term in its complete form is crucial due to the three dimensionality of the flow. A comparative analysis of four different flow situations is performed in regions with positive and negative TKE production with and without buoyancy effects. In both, convective flow and pure shear flow, negative TKE production is associated with the contribution of the positive (i.e. stretching) eigenvalue and eigenvector of the tensor of the mean rate of strain, as contrasted to the positive TKE production associated with the contribution of the negative (i.e. compressing) eigenvalue and eigenvector of the tensor of the mean rate of strain. In the negative TKE production regions of convective flow we find i) increased values for mean strain, ii) persistent alignment of the fluctuating velocity vector u to the stretching eigenvector , iii) stronger anisotropy of , iv) higher levels of fluctuating strain and enstrophy as well as higher rates of their production, , , compared to the respective values in positive TKE production regions. In the pure shear flow all the mentioned quantities are lower in the negative TKE production regions than in the positive TKE production regions. From this we conclude that the inverse energy transfer in the pure shear flow is depleting the field of velocity derivatives. This does not occur in the Rayleigh-Bénard cell. In this flow, buoyancy is observed to be effective on the field of velocity derivatives, velocity fluctuations as well as on the mean flow field. It is inferred that buoyancy is able to maintain a region with substantial negative TKE production by acting on all these levels simultaneously.

Experimental setup

Fig. 1. (A) Sketch of the Rayleigh-Bénard experimental facility with the geometry, coordinates and the location of the measurement volume. (B) Schematic view of the pure shear flow experiment, including the geometry, the location of the observation volume and the forcing scheme of rotating disks.

Experiment A: Rayleigh-Bénard experimental facility

The detailed scheme of the experimental facility, shown in Figure 1 is given in Ref. [2] and here only a brief description is presented. The experiments are performed in water in a cubic aquarium with a side length of 200 mm. The observation volume of 18 × 12 × 10 mm³ is located 40 mm from the left wall and 1 mm from the bottom wall and it is shown as a shaded volume in Fig. 1A. Due to optical limitations, the observation volume is located 80 mm from the observational wall (i.e. pointing to the camera system), which is not precisely the mid-plane of the cubic box. The location of the observational volume is chosen to be exactly where Burr et al. (2001) have reported a region negative turbulent kinetic energy production. All precautions were taken to reproduce the same experimental conditions as in Ref. [2]: i) the temperature difference between lower and upper wall was maintained at , ii) mean temperature of the fluid of T_m = 23.5^oC, iii) Rayleigh and Prandtl numbers of 1.61 × 10⁹ and 6.1, respectively, and iv) the sense of rotation was fixed by a slight tilting of the box around the z-axis by an angle of 1.5^o. The flow was left to develop for at least 24 hours, then small polystyrene particles of were injected through a small hole in the corner of the top wall and additional time was allowed for the flow to evenly distribute the particles before the measurements were started.

Experiment B: pure shear flow forced by rotating disks

In order to rule out the effect of buoyancy on the turbulence in the negative TKE production region, we conducted the second experiment in the setup of four-counter rotating disks⁹. The schematic view is shown in Fig. 1B. The experiment was performed in a glass tank, 120 × 120 × 140 mm³, in water. The turbulent flow field is maintained by four counter-rotating disks of 40 mm in diameter, as it is shown in Fig 1B. A controlled servo-motor rotates the disks with a constant angular speed of 250 rpm, such as to produce a three-dimensional quasi-isotropic turbulent flow in the center of the tank with a weak mean flow on the order of 1 mm^.s^-1, and a velocity fluctuation on the order of 10 mm^.s^-1 (see Liberzon et al. 2005). Close to the disc the flow is the result of the complex interaction between the boundary layers of four rotating disks and the large scale circulation. The center of the observation volume is located 10 mm from the right wall, 55 mm from the bottom wall, and 53 mm from the front wall, respectively.

References

Liberzon A., Lüthi B., Guala M., Kinzelbach W. and Tsinober A. (2005) Experimental study of the structure of flow regions with negative turbulent kinetic energy production in confined three dimensional shear flows with and without buoyancy, Phys. Fluids 17, 095110.