Publikationen

The impact of turbulent mixing on an ensemble of initially monodisperse water droplets is studied in a turbulent bulk that serves as a simplified setup for the interior of a turbulent ice-free cloud. A mixing model was implemented that summarizes the balance equations of water vapor mixing ratio and temperature to an effective advection-diffusion equation for the supersaturation field s(x,t). Our three-dimensional direct numerical simulations connect the velocity and scalar supersaturation fields in the Eulerian frame of reference to an ensemble of cloud droplets in the Lagrangian frame of reference. The droplets are modeled as point particles with and without effects due to inertia. The droplet radius is subject to growth by vapor diffusion. We report the dependence of the droplet size distribution on the box size, initial droplet radius, and the strength of the updraft, with and without gravitational settling. In addition, the three finite-time Lyapunov exponents λ1 ≥ λ2 ≥ λ3 are monitored which probe the local stretching properties along the particle tracks. In this way, we can relate regions of higher compressive strain to those of high local supersaturation amplitudes. For the present parameter range, the mixing process in terms of the droplet evaporation is always homogeneous, while it is inhomogeneous with respect to the relaxation of the supersaturation field. The probability density function of the third finite-time Lyapunov exponent, λ3 < 0, is related to the one of the supersaturation s by a simple one-dimensional aggregation model. The probability density function (PDF) of λ3 and the droplet radius r are found to be Gaussian, while the PDF of the supersaturation field shows sub-Gaussian tails.

Supergranule aggregation, i.e. the gradual aggregation of convection cells to horizontally extended networks of flow structures, is a unique feature of constant heat flux-driven turbulent convection. In the present study, we address the question if this mechanism of self-organisation of the flow is present for any fluid. Therefore, we analyse three-dimensional Rayleigh-Bénard convection at a fixed Rayleigh number Ra ≈ 2.0 × 10^^ 5 across 4 orders of Prandtl numbers Pr ∈ [10^−2, 10^2] by means of direct numerical simulations in horizontally extended periodic domains with aspect ratio Γ = 60. Our study confirms the omnipresence of the mechanism of supergranule aggregation for the entire range of investigated fluids. Moreover, we analyse the effect of Pr on the global heat and momentum transport, and clarify the role of a potential stable stratification in the bulk of the fluid layer. The ubiquity of the investigated mechanism of flow self-organisation underlines its relevance for pattern formation in geophysical and astrophysical convection flows, the latter of which are often driven by prescribed heat fluxes.

We employ a linear stability analysis and direct numerical simulations to study the characteristics of wall modes in thermal convection in a rectangular box under strong and inclined magnetic fields. The walls of the convection cell are electrically insulated. The stability analysis assumes periodicity in the spanwise direction perpendicular to the plane of a homogeneous magnetic field. Our study shows that for a fixed vertical magnetic field, the imposition of horizontal magnetic fields results in an increase of the critical Rayleigh number along with a decrease in the wavelength of the wall modes. The wall modes become tilted along the direction of the resulting magnetic fields and therefore extend further into the bulk as the horizontal magnetic field is increased. Once the modes localized on the opposite walls interact, the critical Rayleigh number decreases again and eventually drops below the value for onset with a purely vertical field. We find that for sufficiently strong horizontal magnetic fields, the steady wall modes occupy the entire bulk and therefore convection is no longer restricted to the sidewalls. The aforementioned results are confirmed by direct numerical simulations of the nonlinear evolution of magnetoconvection. The direct numerical simulation results also reveal that at least for large values of horizontal magnetic field, the wall-mode structures and the resulting heat transfer are dependent on the initial conditions.

We detail our developments in the high-fidelity spectral-element code Neko that are essential for unprecedented large-scale direct numerical simulations of fully developed turbulence. Major innovations are modular multi-backend design enabling performance portability across a wide range of GPUs and CPUs, a GPU-optimized preconditioner with task overlapping for the pressure-Poisson equation and in-situ data compression. We carry out initial runs of Rayleigh-Bénard Convection (RBC) at extreme scale on the LUMI and Leonardo supercomputers. We show how Neko is able to strongly scale to 16,384 GPUs and obtain results that are not possible without careful consideration and optimization of the entire simulation workflow. These developments in Neko will help resolving the long-standing question regarding the ultimate regime in RBC.

The spatial prediction of the turbulent flow of the unsteady von Kármán vortex street behind a cylinder at Re = 1000 is studied. For this, an echo state network (ESN) with 6000 neurons was trained on the raw, low-spatial resolution data from particle image velocimetry. During prediction, the ESN is provided one half of the spatial domain of the fluid flow. The task is to infer the missing other half. Four different decompositions termed forward, backward, forward-backward, and vertical were examined to show whether there exists a favorable region of the flow for which the ESN performs best. Also, it was checked whether the flow direction has an influence on the network's performance. In order to measure the quality of the predictions, we choose the vertical velocity prediction of direction (VVPD). Furthermore, the ESN's two main hyperparameters, leaking rate (LR) and spectral radius (SR), were optimized according to the VVPD values of the corresponding network output. Moreover, each hyperparameter combination was run for 24 random reservoir realizations. Our results show that VVPD values are highest for LR ≈ 0.6, and quite independent of SR values for all four prediction approaches. Furthermore, maximum VVPD values of ≈ 0.83 were achieved for backward, forward-backward, and vertical predictions while for the forward case VVPDmax = 0.74 was achieved. We found that the predicted vertical velocity fields predominantly align with their respective ground truth. The best overall accordance was found for backward and forward-backward scenarios. In summary, we conclude that the stable quality of the reconstructed fields over a long period of time, along with the simplicity of the machine learning algorithm (ESN), which relied on coarse experimental data only, demonstrates the viability of spatial prediction as a suitable method for machine learning application in turbulence.

Fully compressible turbulent convection beyond the Oberbeck-Boussinesq limit and anelastic regime is studied in three-dimensional numerical simulations. Superadiabaticity ε and dissipation number D, which measures the strength of stratification of adiabatic equilibria, cause two limits of compressible convection - nearly top-down-symmetric, strongly superadiabatic convection and highly top-down-asymmetric, strongly stratified convection. The highest turbulent Mach numbers Mt follow for a symmetric blend of these two limits, which we term the fully compressible case. Particularly, the strongly stratified convection case leads to a fluctuation-reduced top layer in the convection zone, a strongly reduced global heat transfer, and differing boundary layer dynamics between top and bottom. We detect this asymmetry for growing dissipation number D also in the phase plane, which is spanned by the turbulent Mach number Mt and the dilatation parameter δ, which relates the dilatational velocity fluctuations to the solenoidal ones. A detailed analysis of the different transport currents in the fully compressible energy budget relates the low-D convection cases to the standard definition of the dimensionless Nusselt number in the Oberbeck-Boussinesq limit.

The present study investigates the non-Oberbeck-Boussinesq (NOB) effects which arise due to the temperature dependence of material properties in cryogenic helium experiments of turbulent Rayleigh-Bénard convection. Here we quantify these effects solely by the difference of the measured mean temperature at the center of the closed cell, Tc, from the arithmetic mean temperature obtained from the prescribed fixed and uniform temperatures at the top and bottom copper plates of the apparatus, Tm = (Tbot + Ttop) /2. To this end, the material properties such as specific heat at constant pressure, dynamic viscosity, thermal conductivity, the isobaric expansivity, and the mass density are expanded into power series with respect to temperature up to the quadratic order with coefficients obtained from the software package HEPAK. A subsequent nonlinear regression that uses deep convolutional networks delivers a dependence of the strength of non-Oberbeck-Boussinesq effects in the pressure-temperature parameter plane. Strength of the NOB effects is evaluated via the deviation of the mean temperature profile ξNOB ≡ Tm − Tc from the top-bottom-symmetric Oberbeck-Boussinesq case ξNOB = 0. Training data for the regression task are obtained from 236 individual long-term laboratory measurements at different Rayleigh numbers which span eight orders of magnitude.

The threshold for the onset of thermocapillary flow in a planar liquid layer heated from below is increased by a vertical magnetic field when the liquid is a good electric conductor. The magnetic damping effect is reduced when the induced eddy currents are blocked by insulating side walls. Neutral conditions for this specific Bénard-Marangoni stability problem with a vertical field and side walls are obtained numerically for three-dimensional perturbations assumed periodic in one horizontal direction. The domain is bounded by a free-slip wall at the bottom, a free surface at the top and two free-slip lateral walls in the other horizontal direction. Buoyancy forces and surface deformations are neglected and a constant heat flux is imposed on the free surface. Upon increasing the magnetic induction, the least stable modes become localized near the side walls and the convective threshold increases at a lower rate than for the least stable bulk mode.

The dynamics in a confined turbulent convection flow is dominated by multiple long-lived macroscopic circulation states that are visited subsequently by the system in a Markov-type hopping process. In the present work, we analyze the short transition paths between these subsequent macroscopic system states by a data-driven learning algorithm that extracts the low-dimensional transition manifold and the related new coordinates, which we term collective variables, in the state space of the complex turbulent flow. We therefore transfer and extend concepts for conformation transitions in stochastic microscopic systems, such as in the dynamics of macromolecules, to a deterministic macroscopic flow. Our analysis is based on long-term direct numerical simulation trajectories of turbulent convection in a closed cubic cell at a Prandtl number Pr=0.7 and Rayleigh numbers Ra=10^6 and 10^7 for a time lag of 10^5 convective free-fall time units. The simulations resolve vortices and plumes of all physically relevant scales, resulting in a state space spanned by more than 3.5 million degrees of freedom. The transition dynamics between the large-scale circulation states can be captured by the transition manifold analysis with only two collective variables, which implies a reduction of the data dimension by a factor of more than a million. Our method demonstrates that cessations and subsequent reversals of the large-scale flow are unlikely in the present setup, and thus it paves the way for the development of efficient reduced-order models of the macroscopic complex nonlinear dynamical system.

We study the influence of fringing magnetic fields on turbulent thermal convection in a horizontally extended rectangular domain. The magnetic field is created in the gap between two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap. We employ direct numerical simulations in this set-up for fixed Rayleigh and small Prandtl numbers, but vary the fringe width by controlling the gap between the magnetic poles and the convection cell. The magnetic field generated by the magnets is strong enough to cease the flow in the high magnetic flux region of the convection cell. We observe that as the local vertical magnetic field strength increases, the large-scale structures become thinner and align themselves perpendicular to the longitudinal sidewalls. We determine the local Nusselt and Reynolds numbers as functions of the local Hartmann number (based on the vertical component of the magnetic field), and estimate the global heat and momentum transport. We show that the global heat transport decreases with increasing fringe width for strong magnetic fields but increases with increasing fringe width for weak magnetic fields. In the regions of large vertical magnetic fields, the convective motion becomes confined to the vicinity of the sidewalls. The amplitudes of these wall modes show a non-monotonic dependence on the fringe width.

The working conditions of the Scaled Convective Airflow Laboratory Experiment (SCALEX) at Technische Universität Ilmenau and sample experiments are reported. The SCALEX facility is a pressure vessel which allows for downscaling of laboratory experiments up to a factor of 20 by compression of gaseous working fluids, air or sulfur hexafluoride, to change the material properties of the fluid. The requirements and conditions for downscaling of fluid dynamical problems are discussed in detail. Long-term high and low pressure tests are conducted to screen the stability of the experimental environment inside the vessel against pressure and temperature fluctuations. Finally, a Rayleigh-Bénard convection experiment at an aspect ratio 10 is performed inside the SCALEX facility as a proof of concept. The reference experiment was conducted under 4.5 bar pressure for Ra = 1.9 × 10^5. However, the Rayleigh number could be varied in a wide range of Ra = 10^4 … 10^8. The flow investigation was pursued with stereoscopic particle image velocimetry in horizontal mid-plane through the convection cell. To improve the image quality the cameras were placed inside the pressure cell and tested up to 6 bar. Thus the feasibility of optical flow measurements at elevated pressures is shown.

A liquid metal flow in the form of a submerged round jet entering a square duct in the presence of a transverse magnetic field is studied experimentally. A range of high Reynolds and Hartmann numbers is considered. Flow velocity is measured using electric potential difference probes. A detailed study of the flow in the duct's cross-section about seven jet's diameters downstream of the inlet reveals the dynamics, which is unsteady and dominated by high-amplitude fluctuations resulting from the instability of the jet. The flow structure and fluctuation properties are largely determined by the value of the Stuart number N. At moderate N, the mean velocity profile retains a central jet with three-dimensional perturbations increasingly suppressed by the magnetic field as N grows. At higher values of N, the flow becomes quasi-two-dimensional and acquires the form of an asymmetric macrovortex, with high-amplitude velocity fluctuations reemerging.

We explore the generalization properties of an echo state network applied as a reduced-order model to predict flux-driven two-dimensional turbulent convection. To this end, we consider a convection domain with constant height with a variable ratio of buoyancy fluxes at the top and bottom boundaries, which break the top-down symmetry in comparison to the standard Rayleigh-Bénard case, thus leading to highly asymmetric mean and fluctuation profiles across the layer. Our direct numerical simulation model describes a convective boundary layer in a simple way. The data are used to train and test a recurrent neural network in the form of an echo state network. The input of the echo state network is obtained in two different ways, either by a proper orthogonal decomposition or by a convolutional autoencoder. In both cases, the echo state network reproduces the turbulence dynamics and the statistical properties of the buoyancy flux, and is able to model unseen data records with different flux ratios.

The study of instabilities in the convection of rotating fluids is one of the classical topics of research. However, in spite of more than five decades of research, the instabilities and related transition scenarios near the onset of rotating convection of low Prandtl number fluids are not well understood. Here, we investigate the transition scenario in rotating Rayleigh–Bénard convection with no-slip boundary conditions by performing 3D direct numerical simulations (DNS) and low-dimensional modeling. The governing parameters, namely, the Taylor number (Ta), Rayleigh number (Ra), and Prandtl number (Pr), are varied in the ranges 0 < Ta ≤ 8 × 10^3, 0 < Ra < 1 × 10^4, and 0 < Pr ≤ 0.35, where convection appears as a stationary cellular pattern. In DNS, for Pr < 0.31, the supercritical or subcritical onset of convection appears, according as Ta > Tac(Pr) or Ta < Tac(Pr), where Tac(Pr) is a Pr dependent threshold of Ta. On the other hand, only supercritical onset of convection is observed for Pr ≥ 0.31. At the subcritical onset, both finite amplitude stationary and time dependent solutions are manifested. The origin of these solutions are explained using a low dimensional model. DNS show that as Ra is increased beyond the onset of convection, the system becomes time dependent and depending on Pr, standing and traveling wave solutions are observed. For very small Pr (≤ 0.045), interestingly, finite amplitude time dependent solutions are manifested at the onset for higher Ta.

Inverse cascades of kinetic energy and thermal variance in the subset of vertically homogeneous modes in spectral space are found to cause a slow aggregation to a pair of convective supergranules that eventually fill the whole horizontally extended, three-dimensional, turbulent Rayleigh-Bénard convection layer when a heat flux is prescribed at the top and bottom. An additional weak rotation of the layer around the vertical axis stops this aggregation at a scale that is smaller than the lateral domain extension and ceases the inverse cascade for the thermal variance. The inverse cascade for the kinetic energy remains intact, even for times at which the root-mean-square values of temperature and velocity have reached the statistically steady state. This kinetic energy inverse cascade sustains the horizontally extended convection patterns which are best visible in the temperature field. The resulting characteristic length of the aggregated convection patterns depends on the thermal driving and linearly on the strength of rotation. Our study demonstrates the importance of inverse energy cascades beyond the two-dimensional turbulence case in a three-dimensional convection flow that is subject to a multiscale energy injection by thermal plumes and driven by boundary heat fluxes as typically present in natural geo- and astrophysical systems, such as solar convection.

Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers, such as occur in stellar convection and in the Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in square boxes of side length L and height H with the aspect ratio Γ = L/H of 25, for Prandtl numbers that span almost 4 orders of magnitude, 10^−3 ≤ Pr ≤ 7, and Rayleigh numbers 10^5 ≤ Ra ≤ 10^7, obtained by massively parallel computations on grids of up to 5.36 × 10^11 points. The low end of this Pr-range cannot be accessed in controlled laboratory measurements. We report the essential properties of the flow and their trends with the Rayleigh and Prandtl numbers, in particular, the global transport of momentum and heat - the latter decomposed into convective and diffusive contributions - across the convection layer, mean vertical profiles of the temperature and temperature fluctuations and the kinetic energy and thermal dissipation rates. We also explore the degree to which the turbulence in the bulk of the convection layer resembles classical homogeneous and isotropic turbulence in terms of spectra, increment moments and dissipative anomaly, and find close similarities. Finally, we show that a characteristic scale of the order of the mesoscale seems to saturate to a wavelength of λ ≳ 3H for Pr ≲ 0.005. We briefly discuss possible implications of these results for the development of subgrid-scale parameterization of turbulent convection.

Turbulent convection at low Prandtl numbers is in many aspects still terra incognita on the parameter map. One reason for this fact is that laboratory experiments on turbulent convection in this parameter regime are notoriously challenging as they require the use of opaque liquid metals. These working fluids prevent the application of typical optical imaging techniques such as particle image velocimetry. Recent experiments by Grannan et al. (J. Fluid Mech., vol. 939, 2022, R1) shed new light on the variety of regimes in liquid metal flows which include rotating convection, magnetoconvection and rotating magnetoconvection next to the classical Rayleigh-Bénard case. More importantly, the authors manage the seamless crossover from one regime into another. They were thus able to study low-Prandtl-number convection at different levels of complexity in a single experimental set-up. Their work provides new insights into the tight connections between characteristic large-scale flow behaviours and the resulting global heat transfer magnitudes. This has implications for convection in planetary cores and stellar convection zones and connected dynamo action.

The vertically driven pendulum is one of the classical systems where parametric instability occurs. We study its behavior with an additional electromagnetic interaction caused by eddy currents in a nearby thick conducting plate that are induced when the bob is a magnetic dipole. The known analytical expressions of the induced electromagnetic force and torque acting on the dipole are valid in the quasistatic limit, i.e., when magnetic diffusivity of the plate is sufficiently high to ensure an equilibrium between magnetic field advection and diffusion. The equation of motion of the vertically driven pendulum is derived assuming that its magnetic dipole moment is aligned with the axis of rotation and that the conducting plate is horizontal. The vertical position of the pendulum remains an equilibrium with the electromagnetic interaction. Conditions for instability of this equilibrium are derived analytically by the harmonic balance method for the subharmonic and harmonic resonances in the limit of weak electromagnetic interaction. The analytical stability boundaries agree with the results of numerical Floquet analysis for these conditions but differ substantially when the electromagnetic interaction is strong. The numerical analysis demonstrates that the area of harmonic instability can become doubly connected. Bifurcation diagrams obtained numerically show the co-existence of stable periodic orbits in such conditions. For moderately strong driving, chaotic motions can be maintained for the subharmonic instability.

Continuous wetting of a surface with liquid metal is indispensable in many applications, such as in fusion reactors. In the present study, we provide data on the suppression of free-surface instabilities of liquid metal film flows under the action of strong streamwise magnetic fields in analogy to the poloidal fields used in application. We have designed and built up an experimental test setup which allows studying the influence of magnetohydrodynamics on the dynamic behaviour of liquid metal GaInSn film flows in laminar, transient, and turbulent regimes. While the width and the length of the film are adjusted at w = 23 mm and l = 120 mm, respectively, we are able to apply strong uniform magnetic fields up to B = 5 T over the entire fluid-flow volume. Moreover, the setup allows to vary the Reynolds number within the range 200 ≤ Re ≤ 1700. The corresponding Hartmann and Stuart numbers are Ha ≤ 180 and N ≤ 40, respectively. This study shows that a streamwise magnetic field is capable of suppressing free-surface instabilities even in the turbulent regime of the film flow by dampening any motion perpendicular to the applied magnetic field. Plans for future studies include the quantitative investigation of the parameter space.

A combined convolutional autoencoder-recurrent neural network machine learning model is presented to directly analyze and forecast the dynamics and low-order statistics of the local convective heat flux field in a two-dimensional turbulent Rayleigh-Bénard convection flow at Prandtl number Pr=7 and Rayleigh number Ra=10^7. Two recurrent neural networks are applied for the temporal advancement of turbulent heat transfer data in the reduced latent data space, an echo state network, and a recurrent gated unit. Thereby, our work exploits the modular combination of three different machine learning algorithms to build a fully data-driven and reduced model for the dynamics of the turbulent heat transfer in a complex thermally driven flow. The convolutional autoencoder with 12 hidden layers is able to reduce the dimensionality of the turbulence data to about 0.2% of their original size. Our results indicate a fairly good accuracy in the first- and second-order statistics of the convective heat flux. The algorithm is also able to reproduce the intermittent plume-mixing dynamics at the upper edges of the thermal boundary layers with some deviations. The same holds for the probability density function of the local convective heat flux with differences in the far tails. Furthermore, we demonstrate the noise resilience of the framework. This suggests that the present model might be applicable as a reduced dynamical model that delivers transport fluxes and their variations to coarse grids of larger-scale computational models, such as global circulation models for atmosphere and ocean.

We study the connection between extreme events of thermal and kinetic energy dissipation rates in the bulk of three-dimensional Rayleigh-Bénard convection and the wall shear stress patterns at the top and the bottom plates that enclose the layer. Zero points of this two-dimensional vector field stand for detachments of strong thermal plumes. If their position at the opposite plates and a given time is close then they can be considered as precursors for high-amplitude bulk dissipation events triggered by plume collisions or close passings. This scenario requires a breaking of the synchronicity of the boundary layer dynamics at both plates which is found to be in line with a transition of the bulk derivative statistics from Gaussian to intermittent. Our studies are based on three-dimensional high-resolution direct numerical simulations for moderate Rayleigh numbers between and .

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.

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.

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.

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.

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.

This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.

The article is devoted to an experimental study of a submerged flat jet flow in a transverse magnetic field. Two different approaches to the experimental study of jet flows are described. Detailed information about the experimental program and measuring methods presented here. The flow of a flat jet 6 mm high in a square channel with a side of 56 mm is considered. The channel is positioned so that the plane of the jet is perpendicular to the magnetic field induction. The results of measuring velocity profiles and waveforms by swivel-type probe with potential sensor are presented. Effects that can be interpreted in different ways are found: strongly unstationary flow regimes, mean flow reorganization, and development of near-wall jets. Additional experiments are prepared to obtain more detailed information about the restructuring and development of the jet. In particular, continuous measurements along the channel will be made in the presence of a slight main flow.

Reservoir computing is an efficient implementation of a recurrent neural network that can describe the evolution of a dynamical system by supervised machine learning with- out solving the underlying mathematical equations. In this work, reservoir computing is applied to model the large-scale evolution and the resulting low-order turbulence statistics of a two-dimensional turbulent Rayleigh-Bénard convection flow at a Rayleigh number Ra = 10^7 and a Prandtl number Pr = 7 in an extended spatial domain with an aspect ratio of 6. Our data-driven approach, which is based on a long-term direct numerical simulation of the convection flow, comprises a two-step procedure: (1) reduction of the original simulation data by a proper orthogonal decomposition (POD) snapshot analysis and subsequent truncation to the first 150 POD modes which are associated with the largest total energy amplitudes; (2) setup and optimization of a reservoir computing model to describe the dynamical evolution of these 150 degrees of freedom and thus the large-scale evolution of the convection flow. The quality of the prediction of the reservoir computing model is comprehensively tested by a direct comparison of the results of the original direct numerical simulations and the fields that are reconstructed by means of the POD modes. We find a good agreement of the vertical profiles of mean temperature, mean convective heat flux, and root-mean-square temperature fluctuations. In addition, we discuss temperature variance spectra and joint probability density functions of the turbulent vertical velocity component and temperature fluctuation, the latter of which is essential for the turbulent heat transport across the layer. At the core of the model is the reservoir, a very large sparse random network characterized by the spectral radius of the corresponding adjacency matrix and a few further hyperparameters which are varied to investigate the quality of the prediction. Our work demonstrates that the reservoir computing model is capable of modeling the large-scale structure and low-order statistics of turbulent convection, which can open new avenues for modeling mesoscale convection processes in larger circulation models.

We report on Rayleigh-Bénard convection with strongly varying fluid properties experimentally and theoretically. Using pressurized sulfur-hexafluoride (SF6) above its critical point, we are able to make measurements at mean temperatures (Tm) and pressures (Pm) along Prandtl-number isolines in the (T,P) parameter space. This allows us to keep the mean Rayleigh- (Ram) and Prandtl number (Prm) constant while changing the temperature dependences of the fluid properties independently, e.g., probing the liquidlike or gaslike region that are left and right of the supercritical isochore. Hence, non-Oberbeck-Boussinesq (NOB) effects can be measured and analyzed cleanly. We measure the temperature at midheight (Tc) as well as the global vertical heat flux. We observe a significant heat transport enhancement of up to 112% under strong NOB conditions. Furthermore, we develop a theoretical model for the global vertical heat flux based on ideas of Grossmann and Lohse (GL) in OB systems, adjusted for nonconstant fluid properties. In this model, the NOB effects influence the boundary layer and hence Tc, but the change of the heat flux is predominantly due to a change of the fluid properties in the bulk, in particular the heat capacity cp and density p. Predictions from our model are consistent with our experimental results as well as with previous measurements carried out in pressurized ethane and cryogenic helium.

In this work we study numerically liquid metal flow in a squareduct under the influence ofa transverse magnetic field applied in a spanwise direction (coplanar). The key interest of thepresent study is an attempt of passive control of flow regimesdeveloped under magnetic fieldand thermal loads by applying specially shaped conditions,such as swirling, at the duct inlet.In this paper, we report results of numerical simulations ofthe interaction of swirling flow andtransverse magnetic field in a square duct flow. Analysis of the obtained regimes might beimportant for the development of an experimental setup, in order to design corresponding inletsections.

Turbulence in a wall-bounded flow of supercritical fluid can be significantly modulated by nonuniform body forces. This study presents direct numerical simulations performed with a nonuniform streamwise body force varying in the wall-normal direction in a fully developed channel flow. A quasilaminar state and reorganized turbulence were obtained by changing the amplitude of the nonuniform body force which distorts the parabolic mean velocity profile and thus alters the turbulence production due to mean shear. Weak production of the flattened mean velocity profile leads to a relaminarization. In the quasilaminar state, all components of the Reynolds stress tensor are fairly weak except the streamwise fluctuations distant from the wall, which indicates the collapse of the near-wall turbulence self-sustaining cycle. The remaining streamwise fluctuations away from the wall exhibit elongated streaks, which is shown by premultiplied spectra. In the recovered turbulence regime, the Reynolds shear stress has a negative range in the bulk connected with a positive range near the wall. This corresponds to the nonmonotonic shear stress resulting from an M-shaped velocity profile. In a subsequent quadrant analysis of the Reynolds shear stress, the sweep and ejection events are found to dominate near the wall only while inward Q1 and outward Q3 motions are significant in the bulk. Both kinds of high-speed events can be attributed to the velocity maxima of the M-shaped mean velocity profile and are found to penetrate towards the wall and the channel center. Interestingly, the flow topology shows the typical teardrop shape once turbulence recovered. Our study contributes to the understanding of flow relaminarization in mixed convection and assists in developing further flow control techniques.

The physical complexity and the large number of degrees of freedom that can be resolved today by direct numerical simulations of turbulent flows, and by the most sophisticated experimental techniques, require new strategies to reduce and analyse the data so generated, and to model the turbulent behaviour. We discuss a few concrete examples for which the turbulence data have been analysed by machine learning tools. We also comment on work in neighbouring fields of physics, particularly astrophysical (and astronomical) work, where Big Data has been the paradigm for some time. We discuss unsupervised, semi-supervised and supervised machine learning methods to direct numerical simulations data of homogeneous isotropic turbulence, Rayleigh-Bénard convection, and the minimal flow unit of a turbulent channel flow; for the last case, we discuss in some detail the application of echo state networks, this being one implementation of reservoir computing. The paper also provides a brief perspective on machine learning applications more broadly.

Magnetohydrodynamic convection in a downward flow of liquid metal in a vertical duct is investigated experimentally and numerically. It is known from earlier studies that in a certain range of parameters, the flow exhibits high-amplitude pulsations of temperature in the form of isolated bursts or quasi-regular fluctuations. This study extends the analysis while focusing on the effects of symmetry introduced by two-sided rather than one-sided wall heating. It is found that the temperature pulsations are robust physical phenomena appearing for both types of heating and various inlet conditions. At the same time, the properties, typical amplitude, and range of existence in the parametric space are very different at the symmetric and asymmetric heating. The obtained data show good agreement between computations and experiments and allow us to explain the physical mechanisms causing the pulsation behavior.

Direct numerical simulations are carried out to study the flow structure and transport properties in turbulent Rayleigh-Bénard convection in a vertical cylindrical cell of aspect ratio one with an imposed axial magnetic field. Flows at the Prandtl number 0.025 and Rayleigh and Hartmann numbers up to 10^9 and 1400, respectively, are considered. The results are consistent with those of earlier experimental and numerical data. As anticipated, the heat transfer rate and kinetic energy are suppressed by a strong magnetic field. At the same time, their growth with Rayleigh number is found to be faster in flows at high Hartmann numbers. This behaviour is attributed to the newly discovered flow regime characterized by prominent quasi-two-dimensional structures reminiscent of vortex sheets observed earlier in simulations of magnetohydrodynamic turbulence. Rotating wall modes similar to those in Rayleigh-Bénard convection with rotation are found in flows near the Chandrasekhar linear stability limit. A detailed analysis of the spatial structure of the flows and its effect on global transport properties is reported.

The interactions between turbulence and cloud microphysical processes have been investigated primarily through numerical simulation and field measurements over the last 10 years. However, only in the laboratory we can be confident in our knowledge of initial and boundary conditions and are able to measure under statistically stationary and repeatable conditions. In the scope of this paper, we present a unique turbulent moist-air wind tunnel, called the Turbulent Leipzig Aerosol Cloud Interaction Simulator (LACIS-T) which has been developed at TROPOS in order to study cloud physical processes in general and interactions between turbulence and cloud microphysical processes in particular. The investigations take place under well-defined and reproducible turbulent and thermodynamic conditions covering the temperature range of warm, mixed-phase and cold clouds (25 &ring;C > T > 40 &ring;C). The continuous-flow design of the facility allows for the investigation of processes occurring on small temporal (up to a few seconds) and spatial scales (micrometer to meter scale) and with a Lagrangian perspective. The here-presented experimental studies using LACIS-T are accompanied and complemented by computational fluid dynamics (CFD) simulations which help us to design experiments as well as to interpret experimental results. In this paper, we will present the fundamental operating principle of LACIS-T, the numerical model, and results concerning the thermodynamic and flow conditions prevailing inside the wind tunnel, combining both characterization measurements and numerical simulations. Finally, the first results are depicted from deliquescence and hygroscopic growth as well as droplet activation and growth experiments. We observe clear indications of the effect of turbulence on the investigated microphysical processes.

Decay of honeycomb-generated turbulence in a duct with a static transverse magnetic field is studied via high-resolution direct numerical simulations. The simulations follow the experimental study [1], in particular the paradoxical observation of high-amplitude velocity fluctuations, which exist in the downstream portion of the flow when the strong transverse magnetic field is imposed in the entire duct including the honeycomb exit, but not in other configurations. It is shown that the fluctuations are caused by the large-scale quasi-two-dimensional structures forming in the flow at the initial stages of the decay and surviving the magnetic suppression. The study demonstrates that turbulence decay in the presence of a magnetic field is a complex phenomenon critically depending on the state of the flow at the moment the field is introduced.

Purpose - This paper aims to address the performance of different subgrid-scale models (SGS) for hydro- (HD) and magnetohydrodynamic (MHD) channel flows within a collocated finite-volume scheme. Design/methodology/approach - First, the SGS energy transfer is analyzed by a priori tests using fully resolved DNS data. Here, the focus lies on the influence of the magnetic field on the SGS energy transport. Second, the authors performed a series of 18 a posteriori model tests, using different grid resolutions and SGS models for HD and MHD channel flows. Findings - From the a priori analysis, the authors observe a quantitative reduction of the SGS energy transport because of the action of the magnetic field depending on its orientation. The a posteriori model tests show a clear improvement because of the use of mixed-models within the numerical scheme. Originality/value - This study demonstrates the necessity of improved SGS modeling strategies for magnetohydrodynamic channel flows within a collocated finite-volume scheme.

Combined measurements of velocity components and temperature in a turbulent Rayleigh-Bénard convection flow at a low Prandtl number of Pr = 0.029 and Rayleigh numbers of 10^6 ≤ Ra ≤ 6 × 10^7 are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation roll in a cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on a time scale up to a hundred times slower. The coherent large-scale circulation drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam-line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.

Decay of honeycomb-generated turbulence in a duct with a static transverse magnetic field is studied via direct numerical simulations. The simulations follow the revealing experimental study of Sukoriansky et al. (Exp. Fluids, vol. 4 (1), 1986, pp. 11-16), in particular the paradoxical observation of high-amplitude velocity fluctuations, which exist in the downstream portion of the flow when the strong transverse magnetic field is imposed in the entire duct including the honeycomb exit, but not in other configurations. It is shown that the fluctuations are caused by the large-scale quasi-two-dimensional structures forming in the flow at the initial stages of the decay and surviving the magnetic suppression. Statistical turbulence properties, such as the energy decay curves, two-point correlations and typical length scales are computed. The study demonstrates that turbulence decay in the presence of a magnetic field is a complex phenomenon critically depending on the state of the flow at the moment the field is introduced.

Large-eddy simulations of thermal convection are presented and discussed for a cube with rough horizontal surfaces. Two types of roughness are considered: uniformly placed pyramids, and grooves aligned parallel to one set of sidewalls. The Rayleigh number is 10^8, the Prandtl number 0.7, and the aspect ratio 1, as in a previous study [N. Foroozani, J. J. Niemela, V. Armenio, and K. R. Sreenivasan, Phys. Rev. E 95, 033107 (2017)], except that the meshes here are finer. When the thermal boundary layers are sufficiently large relative to the characteristic roughness height, i.e., for hydrodynamically smooth conditions, the mean properties of the large scale circulation (LSC) are qualitatively similar to the case of smooth surfaces. In particular, the LSC is always aligned along one of the diagonals of the cube. When the boundaries are hydrodynamically rough, the same result holds true only for the case of pyramidal structures; for grooved surfaces, the LSC is forced to be parallel to the sidewalls on average, alternating rapidly between the two diagonals of the cube with a mean period of the order 10 turnover times. Our analysis suggests that the difference from the pyramidal case is due to the breaking of the horizontal x-z symmetry under conditions of hydrodynamical roughness, and the corresponding directional concentration of plume emission along the grooves, from which the LSC is generated, providing a strong restoring force. Furthermore, in this study we observed a small reduction in heat transport for both roughness configurations which is in good agreement with past studies.

Spatial evolution of electrically conducting jet flow at a supercritical Reynolds number, entering a duct filled with the liquid of the same physical properties, is studied by direct numerical simulations for the case of a streamwise uniform magnetic field. In contrast to the case of a transverse field, here the jet mean velocity does not interact with the streamwise field, and only the turbulent fluctuations of the flow are influenced and suppressed by the field. In this case, the jet saves its energy and has a tendency to spread at much larger distances. Therefore, one interesting and important property of this setup is the flow stabilization, i.e., transition to turbulence can be largely delayed due to the stabilizing effect of the magnetic field. This occurs in the presence of moderate magnetic fields. At strong magnetic fields, the second instability evolves-the jet profile becomes unsteady due to the traveling waves, which propagate along the jet while not interacting with the field. These traveling waves are generated by the interaction of secondary radial flows and magnetic field.

The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of 650 is studied using direct numerical simulations on a 4096^3 grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to 1.2 for moment orders beyond about 12, indicating that scalar intermittency is dominated by the most singular shocklike cliffs in the scalar field. We show that the fractal dimension of the spatial support of steep cliffs is about 1.8, whose sum with the saturation exponent value of 1.2 adds up to the space dimension of 3, thus demonstrating a deep connection between the geometry and statistics in turbulent scalar mixing. The anomaly for the fourth and sixth order moments is comparable to that in the Kraichnan model for the roughness exponent of 4/3.

The dynamics and lifetime of atmospheric clouds are tightly coupled to entrainment and turbulent mixing. This paper presents direct numerical simulations of turbulent mixing followed by droplet evaporation at the cloud-clear air interface in a meter-sized volume, with an ensemble of up to almost half a billion individual cloud water droplets. The dependence of the mixing process on domain size reveals that inhomogeneous mixing becomes increasingly important as the domain size is increased. The shape of the droplet size distribution varies strongly with spatial scale, with the appearance of a pronounced negative exponential tail. The increase of relative dispersion during the transient mixing process is strongly dependent on the scale of the mixing and therefore on the Damköhler number, defined as the turbulence large-eddy time scale divided by the cloud supersaturation relaxation time.

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by Yakhot and Donzis [Phys. Rev. Lett. 119, 044501 (2017)] for homogeneous and isotropic turbulence. In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, is much more complex. In this paper, we discuss the Reynolds-number dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-Bénard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by Pandey et al. [Nat. Commun. 9, 2118 (2018)]. Different Reynolds numbers 1[less-than or equivalent to]Rel[less-than or equivalent to]1000 which are based on the thickness of the bulk region l and the corresponding root-mean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number Ra=10^5. A few specific features of the data agree with the theory. The normalized moments of the kinetic energy dissipation rate En show a nonmonotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.

Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Bénard convection for infinite Prandtl number. We find that the large-scale circulation reverses irregularly, with the waiting time between two consecutive genuine reversals exhibiting a Poisson distribution on long timescales, while the interval between successive crossings on short timescales shows a power-law distribution. We observe that the vertical velocities near the sidewall and at the center show different statistical properties. The velocity near the sidewall shows a longer autocorrelation and 1/f2 power spectrum for a wide range of frequencies, compared to shorter autocorrelation and a narrower scaling range for the velocity at the center. The probability distribution of the velocity near the sidewall is bimodal, indicating a reversing velocity field. We also find that the dominant Fourier modes capture the dynamics at the sidewall and at the center very well. Moreover, we show a signature of weak intermittency in the fluctuations of velocity near the sidewall by computing temporal structure functions.

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as 1000 convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity-field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.

Liquid metal flows in the presence of a uniform magnetic field experience electromagnetic induction. Eddy currents and associated Lorentz force density modify the flow and give rise to thin electromagnetic boundary Jayerson the walls of the channel or duct. We describe features of transition to turbulence in the sidewall jets that occur in a duct flow when the Hartmann walls perpendicular to the magnetic field are electrically conducting. We also examine the modification of the magnetic field and the turbulence in a duct with insulatingwalls in the case when the magnetic diffusion time is not small compared to the flow time scale.

The effect of an axial homogeneous magnetic field on the turbulence in the Taylor-Couette flow confined between two infinitely long conducting cylinders is studied by the direct numerical simulation using a periodic boundary condition in the axial direction. The inner cylinder is rotating, and the outer one is fixed. We consider the case when the magnetic Reynolds number Rem 1, i.e., the influence of the induced magnetic field on the flow is negligible that is typical for industry and laboratory study of liquid metals. Relevance of the present study is based on the similarity of flow characteristics at moderate and high magnetic field for the cases with periodic and end-wall conditions at the large flow aspect ratio, as proven in the earlier studies. Two sets of Reynolds numbers 4000 and 8000 with several Hartmann numbers varying from 0 to 120 are employed. The results show that the mean radial induced electrical current, resulting from the interaction of axial magnetic field with the mean flow, leads to the transformation of the mean flow and the modification of the turbulent structure. The effect of turbulence suppression is dominating at a strong magnetic field, but before reaching the complete laminarization, we capture the appearance of the hairpin-like structures in the flow.

Lorentz force velocimetry (LFV) is a contactless velocity measurement technique for electrically conducting fluids. When a liquid metal or a molten glass flows through an externally applied magnetic field, eddy currents and a flow-braking force are generated inside the liquid. This force is proportional to the velocity or flow rate of the fluid and, due to Newton's third law, a force of the same magnitude but in opposite direction acts on the source of the applied magnetic field which in our case are permanent magnets. According to Ohm's law for moving conductors at low magnetic Reynolds numbers, an electric potential is induced which ensures charge conservation. In this paper, we analyze the contribution of the induced electric potential to the total Lorentz force by considering two different scenarios: conducting walls of finite thickness and aspect ratio variation of the cross-section of the flow. In both the cases, the force component generated by the electric potential is always in the opposite direction to the total Lorentz force. This force component is sensitive to the electric boundary conditions of the flow of which insulating and perfectly conducting walls are the two limiting cases. In the latter case, the overall electric resistance of the system is minimized, resulting in a considerable increase in the measured Lorentz force. Additionally, this force originating from the electric potential also decays when the aspect ratio of the cross-section of the flow is changed. Hence, the sensitivity of the measurement technique is enhanced by either increasing wall conductivity or optimizing the aspect ratio of the cross-section of the flow.

We report a semianalytic and numerical investigation of the maximal induced Lorentz force on an electrically conducting cylinder in translation along its axis that is caused by the presence of multiple distant magnetic dipoles. The problem is motivated by Lorentz force velocimetry, where induction creates a drag force on a magnet system placed next to a conducting flow. The magnetic field should maximize this drag force, which is usually quite small. Our approach is based on a long-wave theory developed for a single distant magnetic dipole. We determine the optimal orientations of the dipole moments providing the strongest Lorentz force for different dipole configurations using numerical optimization methods. Different constraints are considered for dipoles arranged on a concentric circle in a plane perpendicular to the cylinder axis. In this case, the quadratic form for the force in terms of the dipole moments can be obtained analytically, and it resembles the expression of the energy in a classical spin model. When all dipoles are equal and their positions on the circle are not constrained, the maximal force results when all dipoles are gathered in one point with all dipole moments pointing in radial direction. When the dipoles are equal and have equidistant spacing on the circle, we find that the optimal orientations of the dipole moments approach a limiting distribution. It differs from the so-called Halbach distribution that provides a uniform magnetic field in the cross section of the cylinder. The corresponding force is about 10% larger than that for the Halbach distribution but 60% smaller than for the unconstrained dipole positions. With the so-called spherical constraint for a classical spin model, the maximal force can be found from the eigenvalues of the coefficient matrix. It is typically 10% larger than the maximal force for equal dipoles because the constraint is weaker. We also study equal and evenly spaced dipoles along one or two lines parallel to the cylinder axis. The patterns of optimal magnetic moment orientations are fairly similar for different dipole numbers when the inter-dipole distance is within a certain interval. This behavior can be explained by reference to the magnetic field distribution of a single distant dipole on the cylinder axis.

Liquid metal flows in the presence of a uniform magnetic field experience electromagnetic induction. The eddy currents and associated Lorentz force density modify the flow and give rise to thin electromagnetic boundary layers on the walls of the channel or duct. Hartmann layers develop on the walls perpendicular to the magnetic field whereas side layers develop on the parallel walls. The structure of the laminar flow depends on the conductivity of the walls. The side layers play a critical role in the transition to turbulence and are also strongly affected by the anisotropic character of the Lorentz force. We focus on duct flows with conducting Hartmann walls that give rise side-layers jets and report numerical studies of the transitional and turbulent regimes. We also examine one-point statistics and describe specific transitional patterns.

Lorentz force velocimetry (LFV) is a non-contact electromagnetic flow measurement technique for electrically conducting liquids. It is based on measuring the flow-induced force acting on an externally arranged permanent magnet. Motivated by extending LFV to liquid metal two-phase flow measurement, in a previous test we considered the free rising of non-conductive bubbles/particles in a thin tube of liquid metal (GaInSn) initially at rest. We observed that the Lorentz force signals strongly depend on the size of the bubble/particle and on the position, where it is released. Moreover, the force signals cannot be reproduced in detail, which necessitates a statistical analysis. This is caused by chaotic trajectories due to the rising velocities of about 200 mm/s. Therefore, in this paper, we use an improved setup for controlled particle motions in liquid metal. In this experiment, the particle is attached to a straight fishing line, which suppresses any lateral motion, and is pulled by a linear driver at a controllable velocity (0-200 mm/s). For comparison, we solve the induction problem numerically using Oseen's analytical solution of the flow around a translating sphere that is valid for small but finite Reynolds numbers. This simplification is made since the precise hydrodynamic flow is difficult to measure or to compute. The aim of the present work is to check if our simple numerical model can provide Lorentz forces comparable to the experiments. Although Oseen's solution becomes inaccurate near the sphere for finite Reynolds numbers, it provides a fore-aft asymmetry of the flow and is globally well-behaved. It provides an upper limit to the measurement results. We recover the peak-delay of the Lorentz force signals as well.

We discuss two aspects of turbulent Rayleigh-Bénard convection (RBC) on the basis of high-resolution direct numerical simulations in a unique setting: a closed cylindrical cell of aspect ratio of one. First, we present a comprehensive comparison of statistical quantities such as energy dissipation rates and boundary layer thickness scales. Data are used from three simulation run series at Prandtl numbers Pr that cover two orders of magnitude. In contrast to most previous studies in RBC the focus of the present work is on convective turbulence at very low Prandtl numbers including Pr 0.021 for liquid mercury or gallium and Pr 0.005 for liquid sodium. In this parameter range of RBC, inertial effects cause a dominating turbulent momentum transport that is in line with highly intermittent fluid turbulence both in the bulk and in the boundary layers and thus should be able to trigger a transition to the fully turbulent boundary layers of the ultimate regime of convection for higher Rayleigh number. Second, we predict the ranges of Rayleigh numbers for which the viscous boundary layer will transition to turbulence and the flow as a whole will cross over into the ultimate regime. These transition ranges are obtained by extrapolation from our simulation data. The extrapolation methods are based on the large-scale properties of the velocity profile. Two of the three methods predict similar ranges for the transition to ultimate convection when their uncertainties are taken into account. All three extrapolation methods indicate that the range of critical Rayleigh numbers Rac is shifted to smaller magnitudes as the Prandtl number becomes smaller.

Lorentz force velocimetry (LFV) is a non-contact electromagnetic flow measurement technique for electrically conductive liquids. It is based on measuring the flow-induced force acting on an external permanent magnet. Motivated by extending LFV to liquid metal two-phase flow measurement, in a first test we consider the free rising of a non-conductive spherical particle in a thin tube of liquid metal (GaInSn) initially at rest. Here the measured force is due to the displacement flow induced by the rising particle. In this paper, numerical results are presented for three different analytical solutions of flows around a moving sphere under a localized magnetic field. This simplification is made since the hydrodynamic flow is difficult to measure or to compute. The Lorentz forces are compared to experiments. The aim of the present work is to check if our simple numerical model can provide Lorentz forces comparable to the experiments. The results show that the peak values of the Lorentz force from the analytical velocity fields provide us an upper limit to the measurement results. In the case of viscous flow around a moving sphere we recover the typical time-scale of Lorentz force signals.

The transfer of a solute between two liquid layers is susceptible to convective instabilities of the time-dependent diffusive concentration profile that may be caused by the Marangoni effect or buoyancy. Marangoni instabilities depend on the change of interfacial tension and Rayleigh instabilities on the change of liquid densities with solute concentration. Such flows develop increasingly complex cellular or wavy patterns with very fine structures in the concentration field due to the low solute diffusivity. They are important in several applications such as extraction or coating processes. A detailed understanding of the patterns is lacking although a general phenomenological classification has been developed based on previous experiments. We use both highly resolved numerical simulations and controlled experiments to examine two exemplary systems. In the first case, a stationary Marangoni instability is counteracted by a stable density stratification producing a hierarchical cellular pattern. In the second case, Rayleigh instability is opposed by the Marangoni effect causing solutal plumes and eruptive events with short-lived Marangoni cells on the interface. A good qualitative and acceptable quantitative agreement between the experimental visualizations and measurements and the corresponding numerical results is achieved in simulations with a planar interface, and a simple linear model for the interface properties, i.e. no highly specific properties of the interface are required for the complex patterns. Simulation results are also used to characterize the mechanisms involved in the pattern formation.

The combined effects of buoyancy-driven Rayleigh-Bénard convection (RC) and surface tension-driven Marangoni convection (MC) are studied in a triple-layer configuration which serves as a simplified model for a liquid metal battery (LMB). The three-layer model consists of a liquid metal alloy cathode, a molten salt separation layer, and a liquid metal anode at the top. Convection is triggered by the temperature gradient between the hot electrolyte and the colder electrodes, which is a consequence of the release of resistive heat during operation. We present a linear stability analysis of the state of pure thermal conduction in combination with three-dimensional direct numerical simulations of the nonlinear turbulent evolution on the basis of a pseudospectral method. Five different modes of convection are identified in the configuration, which are partly coupled to each other: RC in the upper electrode, RC with internal heating in the molten salt layer, and MC at both interfaces between molten salt and electrode as well as anticonvection in the middle layer and lower electrode. The linear stability analysis confirms that the additional Marangoni effect in the present setup increases the growth rates of the linearly unstable modes, i.e., Marangoni and Rayleigh-Bénard instability act together in the molten salt layer. The critical Grashof and Marangoni numbers decrease with increasing middle layer thickness. The calculated thresholds for the onset of convection are found for realistic current densities of laboratory-sized LMBs. The global turbulent heat transfer follows scaling predictions for internally heated RC. The global turbulent momentum transfer is comparable with turbulent convection in the classical Rayleigh-Bénard case. In summary, our studies show that incorporating Marangoni effects generates smaller flow structures, alters the velocity magnitudes, and enhances the turbulent heat transfer across the triple-layer configuration.

The spatially evolving flow of a liquid-metal jet in a duct with a streamwise uniform magnetic field is studied by direct numerical simulation. In contrast to the case of imposed transverse field, only the turbulent fluctuations of the flow are affected in this setup. They tend to form structures elongated along the applied magnetic field. In that case, turbulence becomes strongly anisotropic and, therefore, may completely change its properties. One interesting and important effect is the flow stabilization due to the magnetic field, whereby transition to turbulence can be delayed significantly. This occurs in the presence of moderate magnetic fields.

Three-dimensional direct numerical simulations of a shearless mixing layer in a small fraction of the cloud-clear air interface are performed to study the response of an ensemble of cloud water droplets to the turbulent entrainment of clear air into a cloud filament. The main goal of this work is to understand how mixing of cloudy and clear air evolves as turbulence and thermodynamics interact through phase changes, and how the cloud droplets respond. In the main simulation case, mixing proceeds between a higher level of turbulence in the cloudy filament and a lower level of turbulence in the clear air environment - the typical shearless mixing layer set-up. Fluid turbulence is driven solely by buoyancy, which incorporates feedbacks from the temperature, the vapour content and the liquid water content fields. Two different variations on the core set of shearless mixing layer simulations are discussed, a simulation in a larger domain and a simulation with the same turbulence level inside the filament and its environment. Overall, it is found that, as evaporation occurs for the droplets that enter subsaturated clear air regions, buoyancy comes to dominate the subsequent evolution of the mixing layer. The buoyancy feedback leads initially to downdraughts at the cloudy-clear air interface and to updraughts in the bulk regions. The strength of the turbulence after initial transients depends on the domain size, showing that the range of scales is an important parameter in the shearless mixing layer set-up. In contrast, the level of turbulence in the clear air is found to have little effect on the evolution of the mixing process. The distributions of cloud water droplet size, supersaturation at the droplet positions and vertical velocity are more sensitive to domain size than to the details of the turbulence profile, suggesting that the evolution of cloud microphysics is more sensitive to large-scale as opposed to small-scale properties of the flow.

The scaling theory of Grossmann and Lohse [J. FluidMech. 407, 27 (2000)] for turbulent heat and momentum transfer is extended to the magnetoconvection case in the presence of a (strong) vertical magnetic field. A comparison with existing laboratory experiments and direct numerical simulations in the quasistatic limit allows us to restrict the parameter space to very low Prandtl and magnetic Prandtl numbers and thus to reduce the number of unknown parameters in the model. Also included is the Chandrasekhar limit, for which the outer magnetic induction field B is large enough such that convective motion is suppressed and heat is transported by diffusion. Our theory identifies four distinct regimes of magnetoconvection that are distinguished by the strength of the outer magnetic field and the level of turbulence in the flow, respectively.

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.

We present results from numerical reconstructions of magnetic obstacle experiments performed in liquid metal flows. The experimental setup consists of an open rectangular container filled with a thin layer of liquid metal (GaInSn). A permanent magnet is installed on a rail beneath the container and is moved with a constant velocity U0, which in turn induces a flow inside the liquid metal due to Lorentz forces. The setup allows experiments in a parameter range that is accessible by direct numerical simulations (DNS). We present results from realizations with four different parameter sets, covering flows with stable stationary vortex structures in the reference system of the moving magnet as well as time-dependent flow regimes. Although the liquid metal layer is very thin, the flow shows a highly three-dimensional character in the near and in the far wake of the magnetic obstacle. We conclude that the streamline visualization in the experiment (using gas bubbles at the surface of the liquid metal layer) is insufficient to picture the flow structure occurring in the liquid metal. To underpin our conclusions, we introduce a modified numerical model which aims to mimic the movement of these gas bubbles. Although this model is a strong simplification of the highly complicated behavior of bubbles at a fluid-fluid interface, it captures the main effects and provides a good reproduction of the experimental results. Furthermore, transient effects are investigated when the flow is initiated, i.e., when the magnet approaches the container and crosses its front wall.We conclude that the process of vortex formation is accompanied by a decrease of the streamwise component of the Lorentz force compared to the time when the fluid is still quiescent. This decrease occurs only for flows with stable vortex structures, which might be of interest for practical applications like Lorentz force velocimetry. The Lorentz forces obtained from our DNS are in good agreement with the values measured in experiment.

Transition in a pressure driven magnetohydrodynamic duct flow, subjected to a uniform transverse magnetic field is studied with direct numerical simulations. The electric boundary conditions correspond to a Hunt's flow regime, with perfectly insulating side walls and perfectly conducting Hartmann walls. The flow forms strong side wall jets at already moderate fields, represented here by Hartmann number Ha = 100. While increasing Reynolds number Re, various unsteady regimes are identified: Ting-Walker (TW) vortices at the side walls, elongated vortices, jet detachment and finally, fully turbulent side-wall jets. We have also found hysteresis behaviour in a broad range of Re.

We present results from numerical reconstructions of magnetic obstacle experiments in which a moving permanent magnet drives a free-surface flow in a container filled with liquid metal (GaInSn). The results cover different flow regimes, such as stationary (in the reference system of the moving magnet) so-called six vortex structures, as well as time-dependent flows. The numerical results show that - although the liquid metal layer is thin - the flow structure is highly three dimensional. We conclude that the experimental technique for streamline visualization (gas bubbles at the surface of the liquid metal) is insufficient to picture the occurring flow structure. To underpin our conclusion, we introduce a modified numerical model that aims to mimic the movement of the gas bubbles. The results of the modified model and the experiments are in excellent agreement. Furthermore, Lorentz forces obtained from simulations match well with those from the experiments.

A streamwise magnetic field leads to turbulent drag reduction in channel flow of a conducting liquid due to the selective Joule damping of certain flow structures. Near the walls, the turbulent mean velocity profile retains the logarithmic layer but the von Kármán constant decreases with increasing magnetic field strength. In the outer region, the flow is characterized by persistent streaky structures of large streamwise extent, which lead to a rather flat mean velocity profile. In addition, the streamwise velocity fluctuation profiles develop a pronounced second peak upon increasing the magnetic induction as well as a second logarithmic layer that increases in steepness.

Boundary layers of the velocity and temperature fields are essential for the transport of heat and momentum in turbulent Rayleigh-Bénard convection. Here we study the combined effects of a prescribed pressure gradient and buoyancy for a boundary layer flow by an extension of the two-dimensional, steady Falkner-Skan model. The resulting model, which couples the evolution of the velocity field to that of the temperature field, is used for a comparison with the dynamics in the vicinity of the isothermal bottom plate in a fully turbulent Rayleigh-Bénard convection flow in a closed cylindrical cell. The obtained set of boundary layer equations is solved numerically and the results are compared with direct numerical simulation data at a Rayleigh number Ra = 3 × 10^9 and for Prandtl numbers Pr = 0.7 and Pr = 7. The additional buoyancy effects improve the agreement with the data for the lower of the two Prandtl numbers. The improvements remain however very small for the larger Prandtl number. In the latter case, the simulation data show a much weaker coherence and strength of the large-scale circulation. Pressure gradients are then generated by local impacts of colder fluid at the bottom rather than by a coherent circulation which fills the whole convection cell.

Many technological processes, such as extraction, evaporation, or absorption are accompanied by solutal Marangoni instability. This type of convection, which is driven by gradients in interfacial tension, can significantly influence process performance in industrial production. The patterns originating from this convection are of a complex and unsteady nature. One particularly remarkable feature is the formation of hierarchical structures, that is, larger patterns emerging on a background of smaller flow patterns. For the example of liquid-liquid extraction, hierarchical roll cells were already observed in the early experiments on solutal Marangoni convection [1-5]. Despite the continuing research [6-13], the details of the multiscale patterns and the mechanisms underlying the hierarchy formation remained unresolved for a long time. Only recently, a thorough characterization of hierarchical Marangoni roll cells could be achieved by a combination of highly resolved three-dimensional (3D) simulations and specifically designed validation experiments [14,15]. However, the hierarchical nature requires the use of a well adapted numerical technique and of large parallel computers to simultaneously resolve both the very fine and the large-scale features of solutal Marangoni convection. To reduce computational cost compared to full 3D simulations, two-dimensional (2D) models are frequently employed [9,16]. Experimentally, such a reduction can be realized to a certain degree in the Hele-Shaw (HS) cell. Here, the liquids are placed between two parallel plates that are sufficiently close together such that the fluid motion becomes mainly 2D [17]. This can be represented by gap-averaged equations [6,7] which - contrary to pure 2D models - take into account the influence of wall friction. Despite the benefits of the HS cell, differences may arise by reducing 3D dynamics to a 2D situation. In this chapter, we present mathematical models, which are able to reproduce the hierarchical patterns observed in the experiments. Therefore, particular focus is laid on providing sufficiently comparable situations in experiments and simulations. The methods are applied to an exemplary two-layer system where hierarchical Marangoni roll cells develop due to the mass transfer of a weakly surface-active solute. In combination with the validation experiments, the results of the simulations provide new insights into the hierarchical nature of the patterns. On this basis, we discuss the applicability of the simplified theoretical models and point out limitations when comparing experimental and numerical results.

Stofftransport über die Grenzfläche zwischen nicht mischbaren Flüssigkeiten ist in der Lage Konvektion durch Dichtegradienten (Rayleigh-Konvektion) oder Gradienten in der Grenzflächenspannung (Marangoni-Konvektion) zu erzeugen. Direkte numerische Simulationen eines Zweischichtsystems wurden durchgeführt, um zwei klassische Experimente aus diesem Bereich zu reproduzieren und zu erklären. Dazu wurden die Navier-Stokes-Boussinesq- und die Transportgleichung für einen gelösten Stoff in zwei, durch eine ebene Grenzfläche gekoppelten Schichten, für all drei Raumdimensionen gelöst. Eine Pseudo-Spektral-Methode wurde zur numerischen Lösung der Gleichungen eingesetzt, wobei Fourier-Moden in beiden horizontalen Richtungen und Chebyshev-Moden in der vertikalen Richtung eingesetzt wurden. Der anfänglich nur in einer Phase gelöste Stoff diffundiert in die andere Phase, welches im Laufe des Stofftransportes Konvektion auslöst. Zwei unterschiedliche Stoffsysteme wurden simuliert, zuerst das ternäre Gemisch aus Cyclohexanol, Wasser und Butanol. Dabei ist Butanol zu Beginn nur in der oberen organischen Phase gelöst. Da Butanol die Grenzflächenspannung sowie Dichte verringert, entsteht Marangoni-Konvektion mit einer stabilisierenden Dichteschichtung. Die durchgeführten Simulationen reproduzierten erfolgreich die experimentell bekannten mehrskaligen Strömungsmuster. Eine zweistufige Hierarchie von Konvektionszellen wurde beobachtet: große, langsam wachsende Zellen, welche kleinere, stetig bewegte Zellen einschließen. Die Ursache für den Musteraufbau wurde durch zwei Mechanismen, Vergröberung und eine lokale Instabilität, erklärt. Die zeitliche Entwicklung der Muster wurde mit zwei unabhängigen Experimenten aus der Literatur verglichen. Dazu wurden Längenskalen und der optische Fluss aus Schlierenbildern abgeleitet. Neben einer guten qualitativen Übereinstimmung erschienen jedoch Simulationen verlangsamt im Vergleich mit den Experimenten. Parameterstudien zeigten, dass Konzentrationsänderungen von Butanol teilweise durch eine Reskalierung von Länge und Zeit berücksichtigt werden können. Bei dem zweiten Stoffsystem wurde die Übergangskomponente durch Isopropanol ersetzt (ähnliche Eigenschaften wie Butanol) und nun in der unteren wässrigen Phase gelöst. Hierfür konnten Simulationen die experimentell beobachteten Strukturen (Eruptionen) reproduzieren und deren Ursprung durch die Wechselwirkung von Rayleigh- und Marangoni-Konvektion erklären. Ein Vergleich mit experimentellen Ergebnissen zeigte eine gute qualitative Übereinstimmung, jedoch waren auch hier die experimentell ermittelten Geschwindigkeiten höher. Parameterstudien ergaben, dass Variationen in der Ausgangskonzentration teilweise durch eine Reskalierung der Zeit berücksichtigt werden können.

We report numerical simulations of a submerged round jet of conducting liquid issuing into a square duct in the presence of a uniform axial magnetic field. Starting from a quiescent initial condition, the spatial development of the jet is simulated until the transition or breakup zone is no longer evolving in space or the jet has reached the outlet while eventually be completely suppressed within the computational domain. For even stronger fields the laminar jet becomes unsteady again, which is presumably caused by a MHD pinching mechanism.

We study the effects of electrically conducting walls on the interaction between a permanent magnet and a liquid-metal flow in a cylindrical pipe using experiments and numerical simulation. The problem is motivated by Lorentz force velocimetry, where the drag force on the magnet due to the induced eddy currents in the flow is used for flow measurement. Compared with insulating walls, the conducting walls lead to an increased drag force on the magnet. Except for low distances, the experimental results are satisfactorily reproduced in simulations using two different approximations of the magnetic field distribution.

We present results of a numerical analysis of transition to turbulence and laminarization processes in magnetohydrodynamic duct flows with a transverse magnetic field. The simulations continue our earlier work [1, 2], where flow regimes with localized turbulent spots near the side walls parallel to the imposed magnetic field have been discovered. The new results extend the analysis to the case of large Reynolds and Hartmann numbers and to ducts of various aspect ratios. The results show good agreement with experiments and confirm that the states with localized turbulent spots are a robust feature of transitional magnetohydrodynamic duct flows.

We study Marangoni roll cell convection in a ternary mass transfer system composed of cyclohexanol, water and butanol (transferred species) placed in a Hele-Shaw cell. A detailed comparison of experimental and numerical results is carried out for four different cases including horizontal and vertical orientation as well as small and large gap width of the Hele-Shaw cell. The numerical simulations are based on a common gap-averaged model. For the small gap width, the numerical results qualitatively represent the experimental results. However, by comparison with the experiments, the time evolution in the simulation appears generally retarded. The results show that three-dimensional flow effects have to be expected in particular for a horizontal orientation and a large gap width.

We present results of a numerical analysis of relaminarization processes in MHD duct and pipe flows. It is motivated by Julius Hartmann's classical experiments on flows of mercury in pipes and ducts under the influence of magnetic fields. The simulations, conducted both in periodic and non-periodic settings, provide a first detailed view of flow structures that have not been experimentally accessible. The main novelty of the analysis is very long (tens to hundreds of hydraulic diameters) computational domains that allows to discover new flow regimes with localized turbulent spots near the side walls parallel tonthe magnetic field. The computed critical parameters for transition as well as the friction coefficients are in good agreement with Hartmann's data.

This study combines observations, large-eddy simulations (LES), and direct numerical simulations (DNS) in order to analyze entrainment and mixing in shallow cumulus clouds at all relevant spatial scales and, additionally, to verify the results by the multiple methods used. The observations are based on three flights of the CARRIBA campaign which are similar to the classical BOMEX case used for LES. Virtual flights in the LES data are used to validate the observational method of line measurements. It is shown that line measurements overrepresent the cloud core, and it is quantified how derived statistics depend on small perturbations of the flight track, which has to be taken in account for the interpretation of airborne observations. A linear relation between fluctuations of temperature and liquid water content has been found in both LES and observations in a good quantitative agreement. However, the constant of proportionality deviates from purely adiabatic estimates, which can be attributed to cloud edge mixing. The cloud edge is compared in detail in observations and LES, which agree qualitatively although the LES cloud edge is smoother due to the model's resolution. The resulting typical amplitudes of the turbulence fields from this comparison are compared with the large-scale forcing model which is used in a series of DNS which study the mixing below the meter scale, which show that LES does not resolve the intermittency of small-scale turbulence.

We compute fully local boundary layer scales in three-dimensional turbulent Rayleigh-Bénard 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 that 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 the 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 scaling similar to that of the Prandtl-Blasius type definition with respect to the 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 behaviour. Additionally, we conduct an analysis of the recently suggested dissipation layer thickness scales versus the Rayleigh number and find a transition in the scaling. All our investigations are based on highly accurate spectral element simulations that reproduce gradients and their fluctuations reliably. The study is done for a Prandtl number of Pr=0.7 and for Rayleigh numbers that extend over almost five orders of magnitude, 3 × 10^5 ≤ Ra ≤ 10^10, in cells with an aspect ratio of one. We also performed one study with an aspect ratio equal to three in the case of Ra=108. For both aspect ratios, we find that the scale distributions depend on the position at the plates where the analysis is conducted.

We study the effects of electrically conducting walls on the interaction between a small cubic permanent magnet and liquid-metal flow in a cylindrical pipe using experiments and electromagnetic simulation. The problem is motivated by Lorentz force velocimetry, where the drag force on the magnet due to the induced eddy currents in the flow is used for flow measurement. Compared with insulating walls, the conducting walls lead to an increased drag force on the magnet. Except for low distances, the experimental results are satisfactorily reproduced in simulations using a point dipole approximation of the magnetic field.

We present results of a numerical analysis of relaminarization processes in MHD duct and pipe flows. It is motivated by Julius Hartmann's classical experiments on flows of mercury in pipes and ducts under the influence of magnetic fields. The computed critical parameters for transition as well as the friction coefficients are in good agreement with Hartmann's data. The simulations provide a first detailed view of flow structures that are experimentally inaccessible. Novel flow regimes with localized turbulent spots near the side walls parallel to the magnetic field are observed.

We study the linear stability of annular MHD channel flow with a uniform axial magnetic field in order to determine when two-dimensional instabilities of Orr type can appear. They are a prerequisite for intermittent turbulent behavior known from plane MHD channel flow with a spanwise field. The annular flow is driven by Lorentz forces caused by a radial electric current and the imposed axial field. Stability of this MHD Dean flow is investigated for axially uniform Orr modes and axisymmetric Dean modes. Orr mode instability dominates only for small gap width and in strong magnetic fields.

A magnetic field imposed on a flow of an electrically conducting fluid can profoundly change flow behavior. We consider this effect for the situation of laminar-turbulent transition in magnetohydrodynamic duct, pipe, and channel flows with homogeneous magnetic field and electrically insulating walls. Experimental and recent computational results obtained for flows in pipes, ducts and channels are reviewed.

According to the seminal theory by Sternling and Scriven [1], solutal Marangoni convection during mass transfer of surface-active solutes may occur as either oscillatory or stationary instability. With strong support of Manuel G. Velarde, a combined initiative of experimental works, in particular to mention those of Linde, Wierschem and coworkers, and theory has enabled a classification of dominant wave types of the oscillatory mode and their interactions. In this way a rather comprehensive understanding of the nonlinear evolution of the oscillatory instability could be achieved. A comparably advanced state-of-the-art with respect to the stationary counterpart seemed to be out of reach a short time ago. Recent developments on both the numerical and experimental side, in combination with assessing an extensive number of older experiments, now allow one to draw a more unified picture. By reviewing these works, we show that three main building blocks exist during the nonlinear evolution: roll cells, relaxation oscillations and relaxation oscillations waves. What is frequently called interfacial turbulence results from the interaction between these partly coexisting basic patterns which may additionally occur in different hierarchy levels. The second focus of this review lies on the practical importance of such convection patterns concerning their influence on mass transfer characteristics. Particular attention is paid here to the interaction between Marangoni and buoyancy effects which frequently complicates the pattern formation even more. To shed more light on these dependencies, new simulations regarding the limiting case of stabilizing density stratification and vanishing buoyancy are incorporated.

In this paper, we report an analytical study of the forces and torques acting upon a magnetic dipole interacting with a moving electrically conducting sphere. The work is motivated by the question whether Lorentz force velocimetry [Thess et al., Phys. Rev. Lett., vol. 96, 2006, 164501] - a noncontact flow measurement technique for liquid metals and electrolytes - can be applied to granular materials as well. We derive explicit expressions for all forces and torques for the case of low magnetic Reynolds number and small particle size. After a discussion of symmetry and reciprocity relations among the forces and torques, we apply the general theory to the particular cases of a translating (nonrotating) and rotating (nontranslating) sphere. The analysis for the purely translating sphere leads to the conclusion that the force is proportional to where is the radius of the sphere and is its minimum distance to the magnetic dipole. This result indicates that Lorentz force velocimetry can indeed be applied to granular metallic materials. The analysis for the purely rotating sphere leads to the result that the torque is proportional to. This result can be applied to derive a rigorous solution for a rotary Lorentz force flowmeter interacting with a rotating sphere. This solution implies that, contrary to intuitive expectation, a frictionless rotary Lorentz force flowmeter rotates with only 4/5 of the angular velocity of the sphere with which it interacts rather than undergoing synchronous rotation.

The present work analyzes the impacts of radiative cooling in three-dimensional high-resolution direct numerical simulations of moist Rayleigh-Bénard convection. An atmospheric slab is destabilized by imposing a warm, moist lower boundary and a colder, dryer upper boundary. These boundary conditions are chosen such that the atmosphere is relaxed toward a conditionally unstable state in which unsaturated air parcels experience a stable stratification and unsaturated parcels experience an unstable one. Conditionally unstable moist Rayleigh-Bénard convection in the absence of radiative transfer produces self-aggregated convectively active cloudy regions separated by a quiescent unsaturated environment. Such convection is strongly limited by diffusion and is unable to transportmuch energy. As radiative cooling partially compensates for the adiabatic warming in the unsaturated environment and destabilizes the lower unsaturated boundary, its inclusion results in a significant enhancement of convective activity and cloud cover. A dry convectively unstable region develops at the lower boundary in a way that is reminiscent of the planetary boundary layer. Convective transport increases through the entire layer, leading to a significant enhancement of the upward transport of energy and water.

We consider liquid metal flow in a square duct with electrically insulating walls under the influence of a magnetic point dipole using three-dimensional direct numerical simulations with a finite-difference method. The dipole acts as a magnetic obstacle. The Lorentz force on the magnet is sensitive to the velocity distribution that is influenced by the magnetic field. The flow transformation by an inhomogeneous local magnetic field is essential for obtaining velocity information from the measured forces. In this paper we present a numerical simulation of a spatially developing flow in a duct with laminar inflow and periodic boundary conditions.

The statistical properties of the thermal dissipation rate in turbulent Rayleigh-Bénard convection in a cylindrical cell are studied by means of three-dimensional direct numerical simulations for a fixed Prandtl number Pr = 0.7 and aspect ratio = 1. The Rayleigh numbers Ra are between 107 and 3×1010. We apply a criterion that decomposes the cell volume into two disjoint subsets: the plume-dominated part and the turbulent background part. The plume-dominated set extends over the whole cell volume and is not confined to the boundary layers. It forms a complex spatial skeleton on which the heat is transported in the convection cell and its volume fraction decreases with increasing Rayleigh number. The latter finding holds also when the threshold, which separates both subvolumes, is varied. The Rayleigh number dependence of the mean moments and probability density functions of the thermal dissipation are analyzed on the subvolumes and related to other possible divisions of the convection volume, such as into boundary layer and bulk. The largest thermal dissipation events are always found in the plume-dominated subset.

We report results of studies of the boundary layer structure in turbulent Rayleigh-Bénard convection. The geometry is a closed cylindrical cell. The results are based on three-dimensional direct numerical simulations (DNS) of the Boussinesq equations at Rayleigh numbers Ra = 3 109, Ra = 3 1010, at an aspect ratio = 1 and at fixed Prandtl number Pr = 0.7. Similar to experiments in the Barrel of Ilmenau, the velocity and temperature profiles are analysed at different positions in the cell in the vicinity of the cooling and heating plates. The profiles differ from the classical Prandtl-Blasius-Pohlhausen theory, even when a so-called dynamic rescaling is performed. We show that in the present flow none of the assumptions made to derive the Blasius solution is satisfied. Furthermore, fluctuations of the local transport currents are analysed.

Gegenstand der vorliegenden Arbeit ist die Untersuchung der thermischen Konvektion hinsichtlich der Rayleigh-Abhängigkeit charakteristischer Bereiche des Strömungsfeldes wie den Grenzschichten, den thermischen Plumes und dem Kernvolumen mithilfe von DNS und LES. Die Analyse der Wahrscheinlichkeitsdichtefunktion (PDF) der thermischen Dissipationsraten zeigt drei markante Bereiche, die dem turbulenten Hintergrund (Bulk), den Plumes / Mischungsschichten und der konduktiven Unterschicht zugeordnet werden können. Mit steigender Rayleigh-Zahl (Ra) wird die Dynamik des Kernvolumens zunehmend von den Grenzschichten entkoppelt, was sich u.a. in einer schwindenden Korrelation der turbulenten Fluktuationen von Temperatur und Geschwindigkeit niederschlägt. Damit wird das von Castaing et al. [J. Fluid Mech., 204, 1989] skizzierte Verhalten weicher und harter Turbulenz bestätigt. - Es wird außerdem gezeigt, dass die thermische Dissipation in der konduktiven Unterschicht zunehmend von den turbulenten Fluktuationen bestimmt wird und somit eine von den Plumes/Mischungsschichten grundverschiedene Dynamik aufweist. Es wird daher vorgeschlagen, den Ansatz von Grossmann & Lohse [Phys. Fluids, 16(12), 2004] zur Aufteilung der thermischen Dissipationsraten so zu erweitern, dass diesem Verhalten Rechnung getragen wird.Der Vergleich dreier unterschiedlicher Geometrien (periodische, quaderförmige und kubische Zelle) ergibt, dass die Strömungsfelder bei geringen Rayleigh-Zahlen signifikante Unterschiede aufweisen, sich mit steigender Rayleigh-Zahl aber einander annähern, was sich sowohl in den Beiträgen der charakteristischen Bestandteile des Strömungsfeldes zur Wärmeübertragung als auch in der Skalierung der Nusselt-Rayleigh-Beziehung widerspiegelt. Abschließend wird die turbulente Rayleigh-Benard-Konvektion (Ra = 3,5 mal 10 5) mit der thermischen Mischkonvektion (Ar = 1) verglichen. Hieraus folgt, dass sich die Strukturbildung bei der Mischkonvektion signifikant ändert. Die Plumes sind bei gleicher Rayleigh-Zahl kleiner und zahlreicher. Zudem wird die Position der Ablösung des Zuluftstrahls stark von der Wechselwirkung mit den Plumes beeinflusst und der konvektive Wärmestrom weist im Auslass eine Oszillation auf. Diese Beobachtung ist in qualitativer Übereinstimmung mit experimentellen Ergebnissen. Ein Vergleich der thermischen Dissipationsraten von Rayleigh-Benard- und Mischkonvektion zeigt, dass sich der Einfluss des Zuluftstrahls vorwiegend auf die groß-skaligen Strukturen, nicht jedoch auf die klein-skaligen Mischungsvorgänge auswirkt.

Direct numerical simulations (DNS) of a turbulent channel flow with 2D wedges of random height on the bottom wall have been performed. In addition, two other simulations have been carried out to assess the effect of the geometry on the overlying flow. In the first simulation, the four smallest elements were removed while in the other, a uniform distribution of wedges with the same area was used. Two Reynolds numbers were studied, Reb = 2500 and Reb = 5000 which correspond in case of smooth walls to Rer = 180 and 300, respectively. Roughness on the wall induces separated regions, the reattachment occurring on the walls of the wedges or on the bottom wall. The pressure gradients on the walls increase the ejections and inrushes towards the wall. As a consequence the flow is more isotropic. The mechanism inducing an improved isotropy has been explained in term of the spectra and budgets of Reynolds stress. The comparison of the 3 surfaces has shown that near the wall, the uniformly distributed roughness represents only a poor approximation of the surface with wedges of random height. The Reynolds stresses, pressure distribution and spectra on the modified wall agree well with those on the random surface. Energy spectra show the pitch to height ratio of the largest elements to be the more appropriate geometrical parameter to describe the geometry.