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Register a new userPerformance of parallel-in-time integration for Rayleigh Benard Convection
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Rayleigh-Benard convection (RBC) is a fundamental problem of fluid dynamics, with many applications to geophysical, astrophysical, and industrial flows. Understanding RBC at parameter regimes of interest requires complex physical or numerical experiments. Numerical simulations require large amounts of computational resources; in order to more efficiently use the large numbers of processors now available in large high performance computing clusters, novel parallelisation strategies are required. To this end, we investigate the performance of the parallel-in-time algorithm Parareal when used in numerical simulations of RBC. We present the first parallel-in-time speedups for RBC simulations at finite Prandtl number. We also investigate the problem of convergence of Parareal with respect to to statistical numerical quantities, such as the Nusselt number, and discuss the importance of reliable online stopping criteria in these cases.
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The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and are carried out in parameter regimes many orders of magnitude away from real conditions. Parallelization in space is a common strategy to speed up simulations on high performance computers, but eventually hits a scaling limit. Additional directions of parallelization are desirable to utilise the high number of processor cores now available. Parallel-in-time methods can deliver speed up in addition to that offered by spatial partitioning but have not yet been applied to dynamo simulations. This paper investigates the feasibility of using the parallel-in-time algorithm Parareal to speed up initial value problem simulations of the kinematic dynamo, using the open source Dedalus spectral solver. Both the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos are investigated over a range of magnetic Reynolds numbers. Speed ups beyond those possible from spatial parallelization are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3. Roberts flow results are less efficient, but Parareal still shows some speed up over spatial parallelization alone. Parallel in space and time speed ups of $sim300$ were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of $sim0.16$.
Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-Benard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Rasim 10^9$. Power law scalings of $Nusim Ra^{gamma}$ are observed with $gammaapprox 0.31$, where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $Pr$ is found to be extremely weak. On the other hand, the presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $Pr lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid which spawns left/right horizontal arms near the top of the channel, leading to downdrafts adjacent to the central updraft. For $Pr > 7$ at high-enough Ra, the optimal structure is a single updraft absent significant horizontal structure, and characterized by the larger maximal wavenumber.
Using direct numerical simulations, we study the statistical properties of reversals in two-dimensional Rayleigh-Benard 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 time scales, while the interval between successive crossings on short time scales 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 $1/f^2$ 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 study numerically the melting of a horizontal layer of a pure solid above a convecting layer of its fluid rotating about the vertical axis. In the rotating regime studied here, with Rayleigh numbers of order $10^7$, convection takes the form of columnar vortices, the number and size of which depend upon the Ekman and Prandtl numbers, as well as the geometry -- periodic or confined. As the Ekman and Rayleigh numbers vary, the number and average area of vortices vary in inverse proportion, becoming thinner and more numerous with decreasing Ekman number. The vortices transport heat to the phase boundary thereby controlling its morphology, characterized by the number and size of the voids formed in the solid, and the overall melt rate, which increases when the lower boundary is governed by a no-slip rather than a stress-free velocity boundary condition. Moreover, the number and size of voids formed are relatively insensitive to the Stefan number, here inversely proportional to the latent heat of fusion. For small values of the Stefan number, the convection in the fluid reaches a slowly evolving geostrophic state wherein columnar vortices transport nearly all the heat from the lower boundary to melt the solid at an approximately constant rate. In this quasi-steady state, we find that the Nusselt number, characterizing the heat flux, co-varies with the interfacial roughness, for all the flow parameters and Stefan numbers considered here. This confluence of processes should influence the treatment of moving boundary problems, particularly those in astrophysical and geophysical problems where rotational effects are important.
We have developed novel instrumentation for making Lagrangian measurements of temperature in diverse fluid flows. A small neutrally buoyant capsule is equipped with on-board electronics which measure temperature and transmit the data via a wireless radio frequency link to a desktop computer. The device has 80 dB dynamic range, resolving milli-Kelvin changes in temperature with up to 100 ms sampling time. The capabilities of these smart particles are demonstrated in turbulent thermal convection in water. We measure temperature variations as the particle is advected by the convective motion, and analyse its statistics. Additional use of cameras allow us to track the particle position and to report here the first direct measurement of Lagrangian heat flux transfer in Rayleigh-B{e}nard convection. The device shows promise for opening new research in a broad variety of fluid systems.
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