Mean velocity and temperature profiles in turbulent Rayleigh-Bénard convection at low Prandtl numbers. - In: Journal of fluid mechanics, ISSN 1469-7645, Bd. 918 (2021), A1, S. A1-1-A1-20
We report a direct numerical simulation (DNS) study of the mean velocity and temperature profiles in turbulent Rayleigh-Bénard convection (RBC) at low Prandtl numbers (Pr). The numerical study is conducted in a vertical thin disk with Pr varied in the range 0.17 ≤ Pr ≤ 4.4 and the Rayleigh number (Ra) varied in the range 5 × 10^8 ≤ Ra ≤ 1 × 10^10. By varying Pr from 4.4 to 0.17, we find a sharp change of flow patterns for the large-scale circulation (LSC) from a rigid-body rotation to a near-wall turbulent jet. We numerically examine the mean velocity equation in the bulk region and find that the mean horizontal velocity profile u(z) can be determined by a balance equation between the mean convection and turbulent diffusion with a constant turbulent viscosity νt. This balance equation admits a self-similarity jet solution, which fits the DNS data well. In the boundary-layer region, we find that both the mean temperature profile T(z) and u(z) can be determined by a balance equation between the molecular diffusion and turbulent diffusion. Within the viscous boundary layer, both u(z) and T(z) can be solved analytically and the analytical results agree well with the DNS data. Our careful characterisation of the mean velocity and temperature profiles in low-Pr RBC provides a further understanding of the intricate interplay between the LSC, plume emission and boundary-layer dynamics, and pinpoints the physical mechanism for the emergence of a pronounced LSC in low-Pr RBC.
https://doi.org/10.1017/jfm.2021.255
Electromagnetic interaction between a permanent magnet and a sphere moving in liquid metal. - In: Experiments in fluids, ISSN 1432-1114, Bd. 62 (2021), 5, 109, S. 1-16
We present a series of model experiments where an electrically non-conductive solid sphere moves inside a vertical column of liquid alloy GaInSn. The experimental setup consists of the liquid metal container, the sphere driving system and the permanent magnet with the attached force sensor. The sphere moves at a controllable constant velocity U0 and follows a straight route, which in turn generates a liquid metal flow around the sphere. This flow interacts with the localized magnetic field of the permanent magnet, and thus a weak reaction force on the magnet is generated. The force sensor attached on the magnet has a resolution of the order 10^-6. Upon elimination of high frequency noise, reproducible time-dependent signals for the forces on the magnet are obtained in the experiments for several Reynolds numbers Re between 160 and 2000. The force component Fz on the magnet parallel to the direction of particle motion exhibits a typical two-peak structure with different peak heights, whereas the transverse force component Fx resembles an antisymmetric pulse. The results demonstrate that the force sensor can detect the presence of a moving particle in a quiescent conducting liquid. They also show that the structure of the Fx signal can be reproduced with less variation and is less sensitive to the Reynolds number than the Fz signal. Moreover, the structure and magnitude of time-dependent Lorentz force signals can be reasonably predicted by a numerical model.
https://doi.org/10.1007/s00348-021-03209-4
Electrical voltage by electron spin-vorticity coupling in laminar ducts. - In: Physical review fluids, ISSN 2469-990X, Bd. 6 (2021), 4, 043703, insges. 14 S.
We report a linear scaling law for an electrical voltage as a function of the pressure drop in capillary pipes and ducts. This voltage is generated by a process which is termed spin hydrodynamic generation (SHDG), a result of the collective electron spin-coupling to the vorticity field in the laminar flow in combination with an inverse spin-Hall effect. We study this phenomenon in laminar duct flows with different width-to-height aspect ratios ranging from 1 (square ducts) to infinite (two dimensional channels). First, we analytically solve the governing Valet-Fert spin diffusion equations for the SHDG by means of the method of small parameters together with proper boundary conditions for the set of inhomogeneous elliptic partial differential equations. Second, the proposed linear scaling law is validated through a series of experiments using capillary tubes with rectangular and square cross sections. The experimental results show very good agreement to the analytically found scaling law. A subsequent substitution of the bulk velocity of the laminar wall-bounded flows by the pressure drop reveals a universal scaling law for the electrical voltage that incorporates all pipe and duct geometries which we could study in our experiments. Finally, the efficiency of the system is estimated for circular pipes, rectangular and square ducts. This study shows that the efficiency of a spin hydrodynamic generator is the same for a circular pipe and a square duct with the same diameter and height, respectively. Hence, due to the ease of manufacturing and the possibility to scale the experiments up to parallel settings in a compact form, micro-channels with a square cross section seem to be the optimum for a spin hydrodynamic generator.
https://doi.org/10.1103/PhysRevFluids.6.043703
Lagrangian heat transport in turbulent three-dimensional convection. - In: Physical review fluids, ISSN 2469-990X, Bd. 6 (2021), 4, L041501, insges. 9 S.
Spatial regions that do not mix effectively with their surroundings and, thus, contribute less to the heat transport in fully turbulent three-dimensional Rayleigh-Bénard flows are identified by Lagrangian trajectories that stay together for a longer time. These trajectories probe Lagrangian coherent sets (CSs) which we investigate here in direct numerical simulations in convection cells with a square cross section of aspect ratio [Gamma]=16, Rayleigh number Ra=10^5, and Prandtl numbers Pr=0.1,0.7, and 7. The analysis is based on N=524288 Lagrangian tracer particles which are advected in the time-dependent flow. Clusters of trajectories are identified by a graph Laplacian with a diffusion kernel, which quantifies the connectivity of trajectory segments, and a subsequent sparse eigenbasis approximation (SEBA) for cluster detection. The combination of graph Laplacian and SEBA leads to a significantly improved cluster identification that is compared with the large-scale patterns in the Eulerian frame of reference. We show that the detected CSs contribute by a third less to the global turbulent heat transport for all investigated Prandtl numbers compared to the trajectories in the spatial complement. This is realized by monitoring Nusselt numbers along the tracer trajectory ensembles, a dimensionless local measure of heat transfer.
https://doi.org/10.1103/PhysRevFluids.6.L041501
Supergranule aggregation for constant heat flux-driven turbulent convection. - In: Physical review research, ISSN 2643-1564, Bd. 3 (2021), 1, S. 013231-1-013231-14
Turbulent convection processes in nature are often found to be organized in a hierarchy of plume structures and flow patterns. The gradual aggregation of convection cells or granules to a supergranule which eventually fills the whole horizontal layer is reported and analyzed in spectral element direct numerical simulations of three-dimensional turbulent Rayleigh-Bénard convection at an aspect ratio of 60. The formation proceeds over a time span of more than 104 convective time units for the largest accessible Rayleigh number and occurs only when the turbulence is driven by a constant heat flux which is imposed at the bottom and top planes enclosing the convection layer. The resulting gradual inverse cascade process is observed for both temperature variance and turbulent kinetic energy. An additional analysis of the leading Lyapunov vector field for the full turbulent flow trajectory in its high-dimensional phase space demonstrates that turbulent flow modes at a certain scale continue to give rise locally to modes with a longer wavelength in the turbulent case. As a consequence, successively larger convection patterns grow until the horizontal extension of the layer is reached. This instability mechanism, which is known to exist near the onset of constant heat flux-driven convection, is shown here to persist into the fully developed turbulent flow regime, thus connecting weakly nonlinear pattern formation with the one in fully developed turbulence. We discuss possible implications of our study for observed, but not yet consistently numerically reproducible, solar supergranulation which could lead to improved simulation models of surface convection in the Sun.
https://doi.org/10.1103/PhysRevResearch.3.013231
Suppression of free convection effects for spherical 1 kg mass prototype. - In: International journal of heat and mass transfer, ISSN 1879-2189, Bd. 170 (2021), 121037, insges. 13 S.
We investigate the free convection processes in the vicinity of a spherical 1 kg mass standard by two- and three-dimensional direct numerical simulations using a spectral element method. Our focus is on the determination and suppression of updraft forces in a high-precision mass comparator which are caused by temperature differences between mass standard and its environment in the millikelvin range - a source of systematic uncertainties in the high-precison mass determination. A two-dimensional model is presented first, which obtains a good agreement with previous laboratory measurements for the smaller temperature differences up to 15 mK. The influence of different boundary conditions and side lengths of the square domain is discussed for the mass standard positioned in the center of the chamber. The complexity is increased subsequently in configurations with additional built-ins for counter heating in form of planar plates or hemispherical shells above the mass standard. The latter ones lead to a full compensation of the updraft force. Three-dimensional simulations in a closed cubic chamber confirm the two-dimensional findings and additionally reveal complex secondary flow patterns in the vicinity of the mass standard. The reduction of the heat transfer due to the built-ins is also demonstrated by a comparison of the Nusselt numbers as a function of the Rayleigh number in the chosen parameter range. Our simulations suggest that such additional constructive measures can enhance the precision of the mass determination by suppression of free convection and related systematic uncertainties.
https://doi.org/10.1016/j.ijheatmasstransfer.2021.121037
Non-Boussinesq low-Prandtl-number convection with a temperature-dependent thermal diffusivity. - In: The astrophysical journal, ISSN 1538-4357, Volume 907 (2021), number 1, 56
In an attempt to understand the role of the strong radial dependence of thermal diffusivity on the properties of convection in Sun-like stars, we mimic that effect in non-Oberbeck-Boussinesq convection in a horizontally extended rectangular domain (aspect ratio 16) by allowing the thermal diffusivity to increase with the temperature (as in the case of stars). Direct numerical simulations (i.e., numerical solutions of the governing equations by resolving up to the smallest scales without requiring any modeling) show that, in comparison with Oberbeck-Boussinesq simulations (two of which we perform for comparison purposes), the symmetry of the temperature field about the mid-horizontal plane is broken, whereas the velocity and heat flux profiles remain essentially symmetric. Our choice of (T), which resembles the variation in stars, results in a temperature field that loses its fine structures toward the hotter part of the computational domain, but the characteristic large scale of the turbulent thermal superstructures, which are structures whose size is typically larger than the depth of the convection domain, continues to be largely independent of the depth.
https://doi.org/10.3847/1538-4357/abd1d8
Lagrangian analysis of long-term dynamics of turbulent superstructures. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 20 (2021), 1, e202000197, insges. 4 S.
https://doi.org/10.1002/pamm.202000197
Dynamics of a magnetic pendulum in the presence of an oscillating conducting plate. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 20 (2021), 1, e202000083, insges. 2 S.
A pendulum with an attached permanent magnet moving near a conductor is a typical experiment for the demonstration of electromagnetic braking. When the conductor itself moves, it can transfer energy to the pendulum. We study a simple but exact analytical model where the conductor is a horizontally unbounded flat plate. For this geometry, eddy currents and induced Lorentz force due to the motion of a magnetic dipole are known analytically in the quasistatic limit. A vertical oscillation of such a horizontal plate located beneath the magnet is considered. In this setup, the vertical position of the pendulum is an equilibrium point when the magnetic moment of the magnet is perpendicular to its plane of motion. Depending on the strength of the magnetic dipole moment, the frequency and amplitude of the plate as well as the distance between plate and magnet, the plate oscillation can destabilize the equilibrium. The stability limits for weak electromagnetic coupling are computed analytically using the harmonic balancing method. For stronger coupling, the stability limits are obtained numerically using Floquet analysis. Chaotic motions with finite amplitudes are also found.
https://doi.org/10.1002/pamm.202000083
Turbulent convection for different thermal boundary conditions at the plates. - In: Journal of fluid mechanics, ISSN 1469-7645, Bd. 907 (2021), A27, S. A27-1-A27-22
https://doi.org/10.1017/jfm.2020.830