We present high-resolution direct numerical simulations of turbulent three-dimensional Rayleigh-Benard convection with a focus on the Lagrangian properties of the flow. The volume is a Cartesian slab with an aspect ratio of four bounded by free-slip planes at the top and bottom and with periodic side walls. The turbulence is inhomogeneous with respect to the vertical direction. This manifests in different lateral and vertical two-particle dispersion and in a dependence of the dispersion on the initial tracer position for short and intermediate times. Similar to homogeneous isotropic turbulence, the dispersion properties depend in addition on the initial pair separation and yield a short-range Richardson-like scaling regime of two-particle dispersion for initial separations close to the Kolmogorov dissipation length. The Richardson constant is about half the value of homogeneous isotropic turbulence. The multiparticle statistics is very close to the homogeneous isotropic case. Clusters of four Lagrangian tracers show a clear trend to form flat, almost coplanar objects in the long-time limit and deviate from the Gaussian prediction. Significant efforts have been taken to resolve the statistics of the acceleration components up to order four correctly. We find that the vertical acceleration is less intermittent than the lateral one. The joint statistics of the vertical acceleration with the local convective and conductive heat flux suggests that rising and falling thermal plumes are not associated with the largest acceleration magnitudes. It turns out also that the Nusselt number which is calculated in the Lagrangian frame converges slowly in time to the standard Eulerian one.
Deep Dive into Lagrangian studies in convective turbulence.
We present high-resolution direct numerical simulations of turbulent three-dimensional Rayleigh-Benard convection with a focus on the Lagrangian properties of the flow. The volume is a Cartesian slab with an aspect ratio of four bounded by free-slip planes at the top and bottom and with periodic side walls. The turbulence is inhomogeneous with respect to the vertical direction. This manifests in different lateral and vertical two-particle dispersion and in a dependence of the dispersion on the initial tracer position for short and intermediate times. Similar to homogeneous isotropic turbulence, the dispersion properties depend in addition on the initial pair separation and yield a short-range Richardson-like scaling regime of two-particle dispersion for initial separations close to the Kolmogorov dissipation length. The Richardson constant is about half the value of homogeneous isotropic turbulence. The multiparticle statistics is very close to the homogeneous isotropic case. Clusters
Turbulent convection is one of the best studied fundamental flows in fluid dynamics research [1,2]. One reason is the large range of examples and applications in nature and technology for which a turbulent motion is initiated and sustained by heating a fluid from below and cooling from above. Almost all of these studies have been conducted in the Eulerian frame of reference. They were primarily focussed to the mechanisms of local [3,4,5,6] and global [7,8,9,10] turbulent heat transfer.
The Lagrangian perspective of turbulence, in which the fields are monitored along the trajectories of infinitesimal fluid parcels, has recently produced new insights into the local topology of fluid parcel tracks, the local strength of accelerations, and the statistics of time increments of turbulent fields [11,12]. The progress is caused on the one hand by significant innovations in the experimental techniques, such as three-dimensional particle tracking [13,14,15] or acoustic methods [16]. On the other hand, direct numerical simulations of turbulence become now feasible that resolve three-dimensional Lagrangian turbulence at moderate and higher Reynolds numbers [17,18,19,20]. Both, experiments and simulations, made a deeper understanding of the small-scale intermittency and its connection with large accelerations of fluid parcels possible.
Lagrangian investigations in convective turbulence are however rare. Several reasons can be given for this cir-cumstance. First, on the experimental side it is desirable to monitor the temperature along the particle tracks beside the velocity components and the accelerations. Only recently, Gasteuil et al. [21] constructed therefore a smart particle, that monitors velocity, temperature and orientation while moving through the cell. Due to the integrated power supply the particle diameter remained however larger than the thermal boundary layer thickness, such that the large-scale bulk motion can be monitored only. Second, it is also clear that the complexity of direct numerical simulations increases since the temperature field has to be advected in addition to the velocity. Temperature tracking along the tracer positions requires additional interpolations. Furthermore, one cannot return to simulations in a fully periodic cube, the so-called homogeneous Rayleigh-Bénard convection setup, since the periodicity in the direction of the mean temperature gradient causes a self-amplifying fluid motion. This was discussed in detail by Calzavarini et al. [22,23]. Third, the turbulence is inhomogeneous -at least in the vertical direction as in the following setup-and it is thus not clear which of findings from the homogeneous isotropic purely hydrodynamic turbulence pertain. For example, the height dependence of the statistics has to be considered additionally.
First numerical attempts have been made recently to study some aspects of the heat transfer and tracer dispersion in the Lagrangian framework of convective turbulence [24]. The motivation of the study can be condensed in one question: Which new insight into the nature of turbulent convection provides the complementary Lagrangian view? One result of [24] was to determine a mixing zone which is dominated by rising and falling thermal plumes. This is done by combining acceleration and local convective heat flux statistics. The mixing zone starts right above the thermal boundary layer and extends several tens of the boundary layer thickness into the bulk of the cell. Thermal plumes are fragments of the thermal boundary layer that detach in the vicinity of the top and bottom isothermal planes. The existence of a mixing zone has been suggested in several Eulerian studies on the basis of other criteria, e.g. [25,26] and was thus confirmed in the complementary Lagrangian frame of reference [24].
The present work extends the previous study [24] into several directions. Beside the local convective, the local conductive heat flux is studied along the tracer tracks. It requires to monitor temperature gradient components. Furthermore, the analysis of the Lagrangian tracer dispersion is extended. In addition to the hydrodynamic case [20], we study the dependence of pair dispersion on the initial separation and the initial seeding position. As discussed in Refs. [19,27,28] for the pure hydrodynamic case, higher order particle statistics requires to track little clusters of tracers. We provide here an analysis of the four-particle-statistics, where the tracers start out of groups of tetrahedra of different sidelengths and initial vertical positions.
The outline of the manuscript is as follows. In the next section the equations of motion, the numerical scheme, the Lagrangian tracer tracking and the turbulent heat transfer. In section III, some results of the Eulerian statistics of the temperature field are presented. This section is followed by sections on the Lagrangian particle dispersion, the acceleration statistics and the conductive and convective he
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