Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

Farhat, Aseel; Glatt-Holtz, Nathan; Martinez, Vincent R.; McQuarrie, Shane A.; Whitehead, Jared P.

doi:10.1137/19m1248327

“…In the context of NWP, different formulations of nudging have been used to study the state estimation problem using finite-dimensional dynamical systems and weather models [25,[29][30][31], and for boundary condition matching [32][33][34]. In the context of turbulence, for the cases of a two-dimensional Navier-Stokes equation (NSE) [35][36][37][38], the three-dimensional Navier-Stokes α model [39], and Rayleigh-Bénard convection [40,41], it has been rigorously proven that given a sufficient amount of input data, a nudged field will eventually synchronize with its nudging field. Indeed, both DA [42] and nudging can be framed as a synchronization problem; see Ref.…”

Section: Introductionmentioning

confidence: 99%

Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Leoni

¹

,

Mazzino²,

Biferale

³

2020

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Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it to the canonical problem of fluid dynamics: three-dimensional homogeneous and isotropic turbulence. By doing numerical experiments we perform a systematic assessment of how well the technique reconstructs large-and small-scale features of the flow with respect to the quantity and the quality or type of data supplied to it. The types of data used are (i) field values on a fixed number of spatial locations (Eulerian nudging), (ii) Fourier coefficients of the fields on a fixed range of wave numbers (Fourier nudging), or (iii) field values along a set of moving probes inside the flow (Lagrangian nudging). We present state-of-the-art quantitative measurements of the scale-by-scale transition to synchronization and a detailed discussion of the probability distribution function of the reconstruction error, by comparing the nudged field and the truth point by point. Furthermore, we show that for more complex flow configurations, like the case of anisotropic rotating turbulence, the presence of cyclonic and anticyclonic structures leads to unexpectedly better performances of the algorithm. We discuss potential further applications of nudging to a series of applied flow configurations, including the problem of field reconstruction in thermal Rayleigh-Bénard convection and in magnetohydrodynamics, and to the determination of optimal parametrization for small-scale turbulent modeling. Our study fixes the standard requirements for future applications of nudging to complex turbulent flows.

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“…By adapting a data assimilation scheme for Bénard convection from Farhat et al (2016Farhat et al ( , 2020 present a data assimilation scheme for noisy temperature measurements of large to infinite Prandtl number RBC flows based on Newtonian relaxation. Evaluating their data assimilation scheme analytically and by means of two-dimensional direct numerical simulations of moderately turbulent flow, Farhat et al (2020) establish that their scheme successfully assimilates temperature fields if the number of projected modes and the relaxation parameter relative to the Rayleigh number are large enough. More often than not, only the measured velocity fields are available, whereas pressure, temperature, or other scaler fields are also required to fully characterise the flow phenomenology.…”

Section: Introductionmentioning

confidence: 99%

Assimilation and extension of particle image velocimetry data of turbulent Rayleigh–Bénard convection using direct numerical simulations

Bauer

¹

,

Schiepel

²

,

Wagner

³

2022

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A novel method for assimilating and extending measured turbulent Rayleigh–Bénard convection data is presented, which relies on the fractional step method also used to solve the incompressible Navier–Stokes equation in direct numerical simulations. Our approach is used to make measured tomographic particle image velocimetry (tomo PIV) fields divergence-free and to extract temperature fields. Comparing the time average of the extracted temperature fields with the temporally averaged temperature field, measured using particle image thermometry in a subdomain of the flow geometry, shows that extracted fields correlate well with measured fields with a correlation coefficient of $$C_{T\tilde{T}}=0.84$$ C T T ~ = 0.84 . Additionally, extracted temperature fields as well as divergence-free velocity fields serve as initial fields for subsequent direct numerical simulations with and without feedback which generate small-scale turbulence initially absent in the experimental data. Although the tomo PIV data set was spatially under-resolved and did not include any information on the boundary layers, the here-proposed method successfully generates velocity and temperature fields featuring small-scale turbulence and thermal as well as kinetic boundary layers, without disturbing the large-scale circulation contained in the original experimental data significantly. The latter is underpinned by high vertical and horizontal velocity correlation coefficients—computed from velocity fields averaged in time and horizontal x-direction obtained from the measurement and from the simulation without feedback—of $$C_{v\tilde{v}}=0.92$$ C v v ~ = 0.92 and $$C_{w\tilde{w}}=0.91$$ C w w ~ = 0.91 representing the large-scale structure. For simulations with feedback, the generated velocity fields resemble the experimental data increasingly well for higher feedback gain values, whereas the temperature fluctuation intensity deviates noticeably from the values obtained from a direct numerical simulation without feedback for gain values $$\alpha \ge 1$$ α ≥ 1 . Thus, a feedback gain of $$\alpha =0.1$$ α = 0.1 was found optimal with correlation coefficients of $$C_{v\tilde{v}}=0.96$$ C v v ~ = 0.96 and $$C_{w\tilde{w}}=0.95$$ C w w ~ = 0.95 as well as a realistic temperature fluctuation intensity profile. The xt-averaged temperature fields obtained from the direct numerical simulations with and without feedback correlate somewhat less with the extracted temperature field ($$C_{T\tilde{T}}\approx 0.6$$ C T T ~ ≈ 0.6 ), which is presumably caused by spatially under-resolved and temporally oscillating initial tomo PIV fields reflected by the extracted temperature field. Graphical abstract

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“…Early studies 13,14 showed that the conjectures is true in simple systems, but also presented numerical evidence to the contrary. In the past few years the conjecture has been studied in different models, such as in a 3D Planetary Geostrophic model 15 , a Rayleigh-Bénard flow in the infinite Prandtl number limit 16 , and a Rayleigh-Bénard flow in the non-turbulent regime 17 . The first two studies were successful in reconstructing the whole state out of thermal measurements, but the third study struggled to do so when the prior on the velocity field was not set correctly.…”

Section: Introductionmentioning

confidence: 99%

Reconstructing Rayleigh-Bénard flows out of temperature-only measurements using nudging

Agasthya,

Di Leoni,

Biferale

2022

Preprint

0

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Nudging is a data assimilation technique that has proved to be capable of reconstructing several highly turbulent flows from a set of partial spatiotemporal measurements. In this study we apply the nudging protocol on the temperature field in a Rayleigh-Bénard Convection system at varying levels of turbulence. We assess the global, as well as scale by scale, success in reconstructing the flow and the transition to full synchronization while varying both the quantity and quality of the information provided by the sparse measurements either on the Eulerian or Lagrangian domain. We asses the statistical reproduction of the dynamic behaviour of the system by studying the spectra of the nudged fields as well as the correct prediction of the heat transfer properties as measured by the Nusselt number. Further, we analyze the results in terms of the complexity of the solutions at various Rayleigh numbers and discuss the more general problem of predicting all state variables of a system given partial or full measurements of only one subset of the fields, in particular temperature. This study sheds new light on the correlation between velocity and temperature in thermally driven flows and on the possibility to control them by acting on the temperature only.

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Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

Cited by 41 publications

References 71 publications

Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Assimilation and extension of particle image velocimetry data of turbulent Rayleigh–Bénard convection using direct numerical simulations

Reconstructing Rayleigh-Bénard flows out of temperature-only measurements using nudging

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