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Krasnov, Dmitry; Akhtari, Ali; Zikanov, Oleg; Schumacher, Jörg
Tensor-product-Thomas elliptic solver for liquid-metal magnetohydrodynamics. - In: Journal of computational physics, ISSN 1090-2716, Bd. 474 (2023), 111784, S. 1-23

A new approach to numerical simulation of magnetohydrodynamic flows of liquid metals is presented. It combines the conservative finite-difference discretization with a tensor-product-Thomas solution of the elliptic problems for pressure, electric potential, velocity, and temperature. The method is realizable on an arbitrarily clustered structured grid. The main novelty of the approach is the efficient solution of the practically important and computationally challenging elliptic problems for electric potential in flow domains with thin electrically conducting walls. The method is verified via solution of benchmark problems for streamwise-uniform and nonuniform, steady and unsteady magnetohydrodynamic flows in ducts, and for thermal convection in boxes of various aspect ratios. Computational efficiency of the method in comparison to the existing ones is demonstrated.
Heyder, Florian; Mellado, Juan Pedro; Schumacher, Jörg
Generalizability of reservoir computing for flux-driven two-dimensional convection. - In: Physical review, ISSN 2470-0053, Bd. 106 (2022), 5, S. 055303-1-055303-21

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.
Bhattacharya, Shashwat; Verma, Mahendra K.; Bhattacharya, Arnab
Predictions of Reynolds and Nusselt numbers in turbulent convection using machine-learning models. - In: Physics of fluids, ISSN 1089-7666, Bd. 34 (2022), 2, S. 025102-1-025102-10

In this paper, we develop a multivariate regression model and a neural network model to predict the Reynolds number (Re) and Nusselt number in turbulent thermal convection. We compare their predictions with those of earlier models of convection: Grossmann-Lohse [Phys. Rev. Lett. 86, 3316 (2001)], revised Grossmann-Lohse [Phys. Fluids 33, 015113 (2021)], and Pandey-Verma [Phys. Rev. E 94, 053106 (2016)] models. We observe that although the predictions of all the models are quite close to each other, the machine-learning models developed in this work provide the best match with the experimental and numerical results.
Mandal, Sutapa; Ghosh, Manojit; Maity, Priyanka; Banerjee, Ankan; Pal, Pinaki
Supercritical and subcritical rotating convection in a horizontally periodic box with no-slip walls at the top and bottom. - In: Physics of fluids, ISSN 1089-7666, Bd. 34 (2022), 10, 104117, insges. 14 S.

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.
Sharifi Ghazijahani, Mohammad; Heyder, Florian; Schumacher, Jörg; Cierpka, Christian
On the benefits and limitations of Echo State Networks for turbulent flow prediction. - In: Measurement science and technology, ISSN 1361-6501, Bd. 34 (2022), 1, 014002, S. 1-18

The prediction of turbulent flow by the application of machine learning (ML) algorithms to big data is a concept currently in its infancy which requires further development. It is of special importance if the aim is a prediction that is good in a statistical sense or if the vector fields should be predicted as good as possible. For this purpose, the statistical and deterministic prediction of the unsteady but periodic flow of the von Kármán Vortex Street (KVS) was examined using an Echo State Network (ESN) which is well suited for learning from time series due to its recurrent connections. The experimental data of the velocity field of the KVS were collected by Particle Image Velocimetry (PIV). Then, the data were reduced by Proper Orthogonal Decomposition (POD) and the flow was reconstructed by the first hundred most energetic modes. An ESN with 3000 neurons was optimized with respect to its three main hyperparameters to predict the time coefficients of the POD modes. For the deterministic prediction, the aim was to maximize the correct direction of the vertical velocities. The results indicate that the ESN can mimic the periodicity and the unsteadiness of the flow. It is also able to predict the sequence of the upward and downward directed velocities for longer time spans. For the statistical prediction, the similarity of the probability density functions of the vertical velocity fields between the predicted and actual flow was achieved. The leaking rate of the ESN played a key role in the transition from deterministic to statistical predictions.
Vieweg, Philipp; Scheel, Janet D.; Stepanov, Rodion; Schumacher, Jörg
Inverse cascades of kinetic energy and thermal variance in three-dimensional horizontally extended turbulent convection. - In: Physical review research, ISSN 2643-1564, Bd. 4 (2022), 4, S. 043098-1-043098-14

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.
Pandey, Ambrish; Krasnov, Dmitry; Schumacher, Jörg; Samtaney, Ravi; Sreenivasan, Katepalli R.
Similarities between characteristics of convective turbulence in confined and extended domains. - In: Physica, ISSN 1872-8022, Bd. 442 (2022), 133537, S. 1-11

To understand turbulent convection at very high Rayleigh numbers typical of natural phenomena, computational studies in slender cells are an option if the needed resources have to be optimized within available limits. However, the accompanying horizontal confinement affects some properties of the flow. Here, we explore the characteristics of turbulent fluctuations in the velocity and temperature fields in a cylindrical convection cell of aspect ratio 0.1 by varying the Prandtl number Pr between 0.1 and 200 at a fixed Rayleigh number Ra = 3 × 10^10, and find that the fluctuations weaken with increasing Pr, quantitatively as in aspect ratio 25. The probability density function (PDF) of temperature fluctuations in the bulk region of the slender cell remains mostly Gaussian, but increasing departures occur as Pr increases beyond unity. We assess the intermittency of the velocity field by computing the PDFs of velocity derivatives and of the kinetic energy dissipation rate, and find increasing intermittency as Pr decreases. In the bulk region of convection, a common result applicable to the slender cell, large aspect ratio cells, as well as in 2D convection, is that the turbulent Prandtl number decreases as Pr^−1/3.
Pandey, Ambrish; Krasnov, Dmitry; Sreenivasan, Katepalli R.; Schumacher, Jörg
Convective mesoscale turbulence at very low Prandtl numbers. - In: Journal of fluid mechanics, ISSN 1469-7645, Bd. 948 (2022), S. A23-1-A23-27

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.
Pfeffer, Philipp; Heyder, Florian; Schumacher, Jörg
Hybrid quantum-classical reservoir computing of thermal convection flow. - In: Physical review research, ISSN 2643-1564, Bd. 4 (2022), 3, S. 033176-1-033176-14

We simulate the nonlinear chaotic dynamics of Lorenz-type models for a classical two-dimensional thermal convection flow with three and eight degrees of freedom by a hybrid quantum-classical reservoir computing model. The high-dimensional quantum reservoir dynamics are established by universal quantum gates that rotate and entangle the individual qubits of the tensor product quantum state. A comparison of the quantum reservoir computing model with its classical counterpart shows that the same prediction and reconstruction capabilities of classical reservoirs with thousands of perceptrons can be obtained by a few strongly entangled qubits. We demonstrate that the mean squared error between model output and ground truth in the test phase of the quantum reservoir computing algorithm increases when the reservoir is decomposed into separable subsets of qubits. Furthermore, the quantum reservoir computing model is implemented on a real noisy IBM quantum computer for up to seven qubits. Our work thus opens the door to model the dynamics of classical complex systems in a high-dimensional phase space effectively with an algorithm that requires a small number of qubits.
Schumacher, Jörg;
The various facets of liquid metal convection. - In: Journal of fluid mechanics, ISSN 1469-7645, Bd. 946 (2022), S. F1-1-F1-5

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.