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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.
Moller, Sebastian; Käufer, Theo; Pandey, Ambrish; Schumacher, Jörg; Cierpka, Christian;
Combined particle image velocimetry and thermometry of turbulent superstructures in thermal convection. - In: Journal of fluid mechanics, ISSN 1469-7645, Bd. 945 (2022), S. A22-1-A22-25

Turbulent superstructures in horizontally extended three-dimensional Rayleigh-Bénard convection flows are investigated in controlled laboratory experiments in water at Prandtl number Pr = 7. A Rayleigh-Bénard cell with square cross-section, aspect ratio Γ = l/h = 25, side length l and height h is used. Three different Rayleigh numbers in the range 10^5 < Ra < 10^6 are considered. The cell is accessible optically, such that thermochromic liquid crystals can be seeded as tracer particles to monitor simultaneously temperature and velocity fields in a large section of the horizontal mid-plane for long time periods of up to 6 h, corresponding to approximately 10^4 convective free-fall time units. The joint application of stereoscopic particle image velocimetry and thermometry opens the possibility to assess the local convective heat flux fields in the bulk of the convection cell and thus to analyse the characteristic large-scale transport patterns in the flow. A direct comparison with existing direct numerical simulation data in the same parameter range of Pr, Ra and Γ reveals the same superstructure patterns and global turbulent heat transfer scaling Nu(Ra). Slight quantitative differences can be traced back to violations of the isothermal boundary condition at the extended water-cooled glass plate at the top. The characteristic scales of the patterns fall into the same size range, but are systematically larger. It is confirmed experimentally that the superstructure patterns are an important backbone of the heat transfer. The present experiments enable, furthermore, the study of the gradual evolution of the large-scale patterns in time, which is challenging in simulations of large-aspect-ratio turbulent convection.
Boeck, Thomas; Sanjari, Seyed Loghman; Becker, Tatiana;
Parametric instability of a vertically driven magnetic pendulum with eddy-current braking by a flat plate. - In: Nonlinear dynamics, ISSN 1573-269X, Bd. 109 (2022), 2, S. 509-529

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.
Valori, Valentina; Kräuter, Robert; Schumacher, Jörg;
Extreme vorticity events in turbulent Rayleigh-Bénard convection from stereoscopic measurements and reservoir computing. - In: Physical review research, ISSN 2643-1564, Bd. 4 (2022), 2, S. 023180-1-023180-14

High-amplitude events of the out-of-plane vorticity component ωz are analyzed by stereoscopic particle image velocimetry (PIV) in the bulk region of turbulent Rayleigh-Bénard convection in air. The Rayleigh numbers Ra vary from 1.7×104 to 5.1×105. The experimental investigation is connected with a comprehensive statistical analysis of long-term time series of ωz and individual velocity derivatives ∂ui/∂xj. A statistical convergence for derivative moments up to an order of 6 is demonstrated. Our results are found to agree well with existing high-resolution direct numerical simulation data in the same range of parameters, including the extreme vorticity events that appear in the far exponential tails of the corresponding probability density functions. The transition from Gaussian to non-Gaussian velocity derivative statistics in the bulk of a convection flow is confirmed experimentally. The experimental data are used to train a reservoir computing model, one implementation of a recurrent neural network, to reproduce highly intermittent experimental time series of the vorticity and thus reconstruct extreme out-of-plane vorticity events. After training the model with high-resolution PIV data, the machine learning model is run with sparsely seeded, continually available, and unseen measurement data in the reconstruction phase. The dependence of the reconstruction quality on the sparsity of the partial observations is also documented. Our latter result paves the way to machine-learning-assisted experimental analyses of small-scale turbulence for which time series of missing velocity derivatives can be provided by generative algorithms.
Maity, Priyanka; Koltai, Péter; Schumacher, Jörg;
Large-scale flow in a cubic Rayleigh-Bénard cell: long-term turbulence statistics and Markovianity of macrostate transitions. - In: Philosophical transactions of the Royal Society, ISSN 1471-2962, Bd. 380 (2022), 2225, 20210042, S. 1-18

We investigate the large-scale circulation (LSC) in a turbulent Rayleigh-Bénard convection flow in a cubic closed convection cell by means of direct numerical simulations at a Rayleigh number Ra = 106. The numerical studies are conducted for single flow trajectories up to 105 convective free-fall times to obtain a sufficient sampling of the four discrete LSC states, which can be summarized to one macrostate, and the two crossover configurations which are taken by the flow in between for short periods. We find that large-scale dynamics depends strongly on the Prandtl number Pr of the fluid which has values of 0.1, 0.7, and 10. Alternatively, we run an ensemble of 3600 short-term direct numerical simulations to study the transition probabilities between the discrete LSC states. This second approach is also used to probe the Markov property of the dynamics. Our ensemble analysis gave strong indication of Markovianity of the transition process from one LSC state to another, even though the data are still accompanied by considerable noise. It is based on the eigenvalue spectrum of the transition probability matrix, further on the distribution of persistence times and the joint distribution of two successive microstate persistence times.
Schindler, Felix; Eckert, Sven; Zürner, Till; Schumacher, Jörg; Vogt, Tobias;
Collapse of coherent large scale flow in strongly turbulent liquid metal convection. - In: Physical review letters, ISSN 1079-7114, Bd. 128 (2022), 16, S. 164501-1-164501-6
Giannakis, Dimitrios; Ourmazd, Abbas; Pfeffer, Philipp; Schumacher, Jörg; Slawinska, Joanna;
Embedding classical dynamics in a quantum computer. - In: Physical review, ISSN 2469-9934, Bd. 105 (2022), 5, 052404, insges. 47 S.

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, H. Furthermore, an embedding of classical observables into self-adjoint operators on H is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of H, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a 2n-dimensional tensor product Hilbert space associated with n qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an n-channel quantum circuit of size O(n) and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size O(n), and a quantum Fourier transform stage of size O(n2), which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, n&flech;∞. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension 2n using circuits of size O(n2). We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.
Pandey, Sandeep; Teutsch, Philipp; Mäder, Patrick; Schumacher, Jörg;
Direct data-driven forecast of local turbulent heat flux in Rayleigh-Bénard convection. - In: Physics of fluids, ISSN 1089-7666, Bd. 34 (2022), 4, S. 045106-1-045106-14

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.
Schneide, Christiane; Vieweg, Philipp P.; Schumacher, Jörg; Padberg-Gehle, Kathrin;
Evolutionary clustering of Lagrangian trajectories in turbulent Rayleigh-Bénard convection flows. - In: Chaos, ISSN 1089-7682, Bd. 32 (2022), 1, S. 013123-1-013123-11

We explore the transport mechanisms of heat in two- and three-dimensional turbulent convection flows by means of the long-term evolution of Lagrangian coherent sets. They are obtained from the spectral clustering of trajectories of massless fluid tracers that are advected in the flow. Coherent sets result from trajectories that stay closely together under the dynamics of the turbulent flow. For longer times, they are always destroyed by the intrinsic turbulent dispersion of material transport. Here, this constraint is overcome by the application of evolutionary clustering algorithms that add a time memory to the coherent set detection and allows individual trajectories to leak in or out of evolving clusters. Evolutionary clustering thus also opens the possibility to monitor the splits and mergers of coherent sets. These rare dynamic events leave clear footprints in the evolving eigenvalue spectrum of the Laplacian matrix of the trajectory network in both convection flows. The Lagrangian trajectories reveal the individual pathways of convective heat transfer across the fluid layer. We identify the long-term coherent sets as those fluid flow regions that contribute least to heat transfer. Thus, our evolutionary framework defines a complementary perspective on the slow dynamics of turbulent superstructure patterns in convection flows that were recently discussed in the Eulerian frame of reference. The presented framework might be well suited for studies in natural flows, which are typically based on sparse information from drifters and probes.