Janet Scheel

Small-scale universality in fluid turbulence

Significance Since the time Kolmogorov postulated the universality of small-scale turbulence, an important research topic has been to experimentally establish it beyond doubt. The likelihood of small-scale universality increases with increasing distance (say, in wave number space) from the nonuniversal large scales. This distance increases as some power of the flow Reynolds number, and so a great deal of emphasis has been put on creating and quantifying very high Reynolds number flows under controlled conditions. The present paper shows that the universal properties of inertial range turbulence (thought to exist only at very high Reynolds numbers) are already present in an incipient way even at modest Reynolds numbers and hence changes the paradigm of research in this field. Turbulent flows in nature and technology possess a range of scales. The largest scales carry the memory of the physical system in which a flow is embedded. One challenge is to unravel the universal statistical properties that all turbulent flows share despite their different large-scale driving mechanisms or their particular flow geometries. In the present work, we study three turbulent flows of systematically increasing complexity. These are homogeneous and isotropic turbulence in a periodic box, turbulent shear flow between two parallel walls, and thermal convection in a closed cylindrical container. They are computed by highly resolved direct numerical simulations of the governing dynamical equations. We use these simulation data to establish two fundamental results: (i) at Reynolds numbers Re ∼ 102 the fluctuations of the velocity derivatives pass through a transition from nearly Gaussian (or slightly sub-Gaussian) to intermittent behavior that is characteristic of fully developed high Reynolds number turbulence, and (ii) beyond the transition point, the statistics of the rate of energy dissipation in all three flows obey the same Reynolds number power laws derived for homogeneous turbulence. These results allow us to claim universality of small scales even at low Reynolds numbers. Our results shed new light on the notion of when the turbulence is fully developed at the small scales without relying on the existence of an extended inertial range.

Local boundary layer scales in turbulent Rayleigh-Benard convection

We compute fully local boundary layer scales in three-dimensional turbulent Rayleigh-Benard convection. These scales are directly connected to the highly intermittent fluctuations of the fluxes of momentum and heat at the isothermal top and bottom walls and are statistically distributed around the corresponding mean thickness scales. The local boundary layer scales also reflect the strong spatial inhomogeneities of both boundary layers due to the large-scale, but complex and intermittent, circulation that builds up in closed convection cells. Similar to turbulent boundary layers, we define inner scales based on local shear stress which can be consistently extended to the classical viscous scales in bulk turbulence, e.g. the Kolmogorov scale, and outer scales based on slopes at the wall. We discuss the consequences of our generalization, in particular the scaling of our inner and outer boundary layer thicknesses and the resulting shear Reynolds number with respect to Rayleigh number. The mean outer thickness scale for the temperature field is close to the standard definition of a thermal boundary layer thickness. In the case of the velocity field, under certain conditions the outer scale follows a similar scaling as the Prandtl-Blasius type definition with respect to Rayleigh number, but differs quantitatively. The friction coefficient c_epsilon scaling is found to fall right between the laminar and turbulent limits which indicates that the boundary layer exhibits transitional behavior. Additionally, we conduct an analysis of the recently suggested dissipation layer thickness scales versus Rayleigh number and find a transition in the scaling. We also performed one study of aspect ratio equal to three in the case of Ra=1e+8.

Resolving the fine-scale structure in turbulent Rayleigh-Benard convection

We present high-resolution direct numerical simulation studies of turbulent Rayleigh–Benard convection in a closed cylindrical cell with an aspect ratio of one. The focus of our analysis is on the finest scales of convective turbulence, in particular the statistics of the kinetic energy and thermal dissipation rates in the bulk and the whole cell. The fluctuations of the energy dissipation field can directly be translated into a fluctuating local dissipation scale which is found to develop ever finer fluctuations with increasing Rayleigh number. The range of these scales as well as the probability of high-amplitude dissipation events decreases with increasing Prandtl number. In addition, we examine the joint statistics of the two dissipation fields and the consequences of high-amplitude events. We have also investigated the convergence properties of our spectral element method and have found that both dissipation fields are very sensitive to insufficient resolution. We demonstrate that global transport properties, such as the Nusselt number, and the energy balances are partly insensitive to insufficient resolution and yield correct results even when the dissipation fields are under-resolved. Our present numerical framework is also compared with high-resolution simulations which use a finite difference method. For most of the compared quantities the agreement is found to be satisfactory.

Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Significance Low-Prandtl-number thermal convection flows in liquid metals for which the temperature diffusivity is much larger than the fluid viscosity have been studied much less frequently than convective flows in air or water, despite many important applications reaching from astrophysics to energy conversion. Currently, the turbulence in low-Prandtl-number flows is fully accessible only by three-dimensional simulations. Our numerical studies reveal why the small-scale turbulence is much more vigorous compared with convection in air. We also find that the generation of small-scale vorticity in the bulk of convection follows the same mechanisms and statistics as in idealized isotropic turbulence, especially for the low-Prandtl-number flow. This opens new perspectives for necessary turbulence parameterizations in applications. Turbulent convection is often present in liquids with a kinematic viscosity much smaller than the diffusivity of the temperature. Here we reveal why these convection flows obey a much stronger level of fluid turbulence than those in which kinematic viscosity and thermal diffusivity are the same; i.e., the Prandtl number Pr is unity. We compare turbulent convection in air at Pr=0.7 and in liquid mercury at Pr=0.021. In this comparison the Prandtl number at constant Grashof number Gr is varied, rather than at constant Rayleigh number Ra as usually done. Our simulations demonstrate that the turbulent Kolmogorov-like cascade is extended both at the large- and small-scale ends with decreasing Pr. The kinetic energy injection into the flow takes place over the whole cascade range. In contrast to convection in air, the kinetic energy injection rate is particularly enhanced for liquid mercury for all scales larger than the characteristic width of thermal plumes. As a consequence, mean values and fluctuations of the local strain rates are increased, which in turn results in significantly enhanced enstrophy production by vortex stretching. The normalized distributions of enstrophy production in the bulk and the ratio of the principal strain rates are found to agree for both Prs. Despite the different energy injection mechanisms, the principal strain rates also agree with those in homogeneous isotropic turbulence conducted at the same Reynolds numbers as for the convection flows. Our results have thus interesting implications for small-scale turbulence modeling of liquid metal convection in astrophysical and technological applications.

Transitional boundary layers in low-Prandtl-number convection

The boundary layer structure of the velocity and temperature fields in turbulent Rayleigh-Benard flows in closed cylindrical cells of unit aspect ratio is revisited from a transitional and turbulent viscous boundary layer perspective. When the Rayleigh number is large enough, the dynamics at the bottom and top plates can be separated into an impact region of downwelling plumes, an ejection region of upwelling plumes and an interior region away from the side walls. The latter is dominated by the shear of the large-scale circulation (LSC) roll which fills the whole cell and continuously varies its orientation. The working fluid is liquid mercury or gallium at a Prandtl number Pr=0.021 for Rayleigh numbers between Ra=3e+5 and 4e+8. The generated turbulent momentum transfer corresponds to macroscopic flow Reynolds numbers with values between 1800 and 46000. It is shown that the viscous boundary layers for the largest Rayleigh numbers are highly transitional and obey properties that are directly comparable to transitional channel flows at friction Reynolds numbers Re_tau slightly below 100. The transitional character of the viscous boundary layer is also underlined by the strong enhancement of the fluctuations of the wall stress components with increasing Rayleigh number. An extrapolation of our analysis data suggests that the friction Reynolds number Re_tau in the velocity boundary layer can reach values of 200 for Ra beyond 1e+11. Thus the viscous boundary layer in a liquid metal flow would become turbulent at a much lower Rayleigh number than for turbulent convection in gases and gas mixtures.

The amplitude equation for rotating Rayleigh–Bénard convection

The amplitude equation for rotating Rayleigh–Benard convection is derived from the Boussinesq equations with the Coriolis force included. The vertical boundary conditions are no-slip, and the lateral boundary conditions are either periodic or rigid. In order to keep track of the mean flow, the full system of equations is considered, instead of a potential formulation. A multiple scales perturbation expansion in the control parameter ϵ is performed, and appropriate solvability conditions are imposed. This leads to the usual amplitude equation at order ϵ3∕2, but a new rotation term enters at order ϵ7∕4. This rotation term will cause a change of phase with respect to time, whenever there is a gradient in the amplitude in the direction parallel to the rolls. As a result, rolls terminating perpendicularly to a wall will precess in the direction of rotation. The new rotation term will also cause stationary dislocations to glide perpendicular to the rolls. Amplitude equation results for a specific set of paramet...

Patterns in rotating Rayleigh–Bénard convection at high rotation rates

We present the results from numerical and theoretical investigations of rotating Rayleigh-Benard convection for relatively large dimensionless rotation rates, 170 < Ω < 274, and a Prandtl number of 6.4. Unexpected square patterns were found experimentally by Bajaj et al. (Phys. Rev. Lett., vol. 81, 1998, p. 806) in this parameter regime and near threshold for instability in the bulk. These square patterns have not yet been understood theoretically. Sanchez-Alvarez et al. (Phys. Rev. E, vol. 72, 2005, p. 036307) have found square patterns in numerical simulations for similar parameters when only the Coriolis force is included. We performed detailed numerical studies of rotating Rayleigh-Benard convection for the same parameters as the experiments and simulations. To better understand these patterns, we compared the effects of the Coriolis force as well as the centrifugal force. We also computed the coefficients of the amplitude equation describing one-, two- and three-mode bulk solutions to rotating Rayleigh-Benard convection. We find that squares are unstable, but we do find stable limit cycles consisting of three coupled oscillating amplitudes, which can superficially resemble squares, since one of the three amplitudes is rather small.

Turbulent superstructures in Rayleigh-Bénard convection

Turbulent Rayleigh-Bénard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. Here we report numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to 107. We identify characteristic scales and times that separate the fast, small-scale turbulent fluctuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale patterns, which change with Prandtl and Rayleigh number, are also correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo- and astrophysical settings.Turbulent fluids in nature, counter-intuitively, can exhibit large-scale order that persists for long times. Pandey et al. numerically characterize the formation of these superstructures in turbulent convection by separating the fast motions at small-scales from those that gradually vary at large scales.

Onset of Rayleigh-Bénard convection for intermediate aspect ratio cylindrical containers

The convection patterns that occur at and slightly above the onset of convection in cylindrical containers were determined as a function of aspect ratio, using simulations of Rayleigh-Benard convection and linear stability analysis. The study focused primarily on aspect ratios 6≤Γ≤20, where Γ = diameter/depth, with conducting or insulating, and no-slip boundary conditions and Prandtl numbers Pr = 0.7 and 28.9. Simulations demonstrate azimuthally pure Fourier mode patterns at onset consistent with what is expected from bifurcation theory, with an m = 1 mode, for even values of Γ, and a concentric roll pattern, or m = 0 mode, for odd values of Γ. For Rayleigh numbers slightly higher than onset other pure or mixed mode patterns were found and then for even higher Rayleigh numbers, straight parallel rolls were found. A linear stability analysis was used to determine the critical Rayleigh number, Rac, and flow pattern for a large range of aspect ratios and was found to agree with the simulation results.

Extreme dissipation event due to plume collision in a turbulent convection cell

An extreme dissipation event in the bulk of a closed three-dimensional turbulent convection cell is found to be correlated with a strong reduction of the large-scale circulation flow in the system that happens at the same time as a plume emission event from the bottom plate. The reduction in the large-scale circulation opens the possibility for a nearly frontal collision of down- and upwelling plumes and the generation of a high-amplitude thermal dissipation layer in the bulk. This collision is locally connected to a subsequent high-amplitude energy dissipation event in the form of a strong shear layer. Our analysis illustrates the impact of transitions in the large-scale structures on extreme events at the smallest scales of the turbulence, a direct link that is observed in a flow with boundary layers. We also show that detection of extreme dissipation events which determine the far-tail statistics of the dissipation fields in the bulk requires long-time integrations of the equations of motion over at least a hundred convective time units.