[Paper Review] Direct data-driven forecast of local turbulent heat flux in Rayleigh-B\'{e}nard convection

Sandeep Pandey, Philipp Teutsch+2 more|arXiv (Cornell University)|Feb 26, 2022

Fluid Dynamics and Turbulent Flows76 references32 citations

TL;DR

This study presents a data-driven, reduced-order model using a convolutional autoencoder (CAE) combined with recurrent neural networks (RNNs) to forecast local turbulent heat flux in two-dimensional Rayleigh-Bénard convection at Ra = 10⁷ and Pr = 7. The framework achieves 99.7% data compression while accurately reproducing first- and second-order statistics, intermittent plume dynamics, and the probability density function of the heat flux, demonstrating robustness to noise and potential for use in large-scale climate models.

ABSTRACT

A combined convolutional autoencoder-recurrent neural network machine learning model is presented to analyse and forecast the dynamics and low-order statistics of the local convective heat flux field in a two-dimensional turbulent Rayleigh-B\'{e}nard convection flow at Prandtl number ${ m Pr}=7$ and Rayleigh number ${ m Ra}=10^7$. Two recurrent neural networks are applied for the temporal advancement of flow data in the reduced latent data space, a reservoir computing model in the form of an echo state network and a recurrent gated unit. Thereby, the present 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 which suggests the present model might be applicable as a reduced dynamical model that delivers transport fluxes and their variations to the coarse grid cells of larger-scale computational models, such as global circulation models for the atmosphere and ocean.

Motivation & Objective

• To develop a fully data-driven, reduced-order model for local turbulent heat flux in Rayleigh-Bénard convection without solving the underlying Navier-Stokes and energy equations.
• To reduce high-dimensional simulation data (up to 181 GB per snapshot) to a low-dimensional latent space while preserving key statistical and dynamical features of the heat flux field.
• To evaluate the performance of two RNN architectures—echo state networks (ESNs) and gated recurrent units (GRUs)—in forecasting temporal dynamics of the heat flux in the latent space.
• To assess the model's ability to reproduce intermittent plume-mixing dynamics and the full probability density function (PDF) of the local convective heat flux.
• To test the noise resilience of the framework for potential application in coarse-grid models like global atmospheric and oceanic circulation models.

Proposed method

• A 12-layer convolutional autoencoder (CAE) is used to compress high-dimensional heat flux snapshots into a low-dimensional latent space, reducing data size to 0.2% of original.
• Two RNN architectures—echo state networks (ESNs) and gated recurrent units (GRUs) with an encoder-decoder structure—are trained on the latent space data to model temporal evolution.
• The trained RNNs autonomously advance the dynamics in the latent space, and a decoder network reconstructs the high-dimensional heat flux fields from the latent states.
• The model is trained and validated on direct numerical simulation (DNS) data of 2D Rayleigh-Bénard convection at Ra = 10⁷ and Pr = 7.
• Hyperparameters of the RNNs are tuned via random search, and the framework is tested for noise resilience by injecting perturbations into input data.
• The model's performance is evaluated using statistical metrics such as mean and fluctuation profiles, PDFs, and qualitative analysis of plume dynamics.

Experimental results

Research questions

• RQ1 Can a combined CAE-RNN framework accurately forecast the temporal dynamics of local turbulent heat flux in a 2D Rayleigh-Bénard convection system?
• RQ2 To what extent can the model preserve the first- and second-order statistics of the heat flux field, including mean and variance profiles?
• RQ3 How well does the model reproduce intermittent plume-mixing dynamics at the thermal boundary layers?
• RQ4 Can the model accurately reconstruct the full probability density function (PDF) of the local convective heat flux, especially in the far tails?
• RQ5 How resilient is the model to noise in input data, and what does this imply for its use in large-scale climate modeling?

Key findings

• The CAE reduces the dimensionality of the heat flux data by 99.7%, retaining approximately 95% of the original variance.
• The model accurately reproduces the mean and fluctuation profiles of the local convective heat flux, with good agreement between predicted and reference data.
• The model captures intermittent plume-mixing dynamics at the upper thermal boundary layers, though with some deviations in timing and intensity.
• The probability density function (PDF) of the local heat flux is well reproduced, with minor discrepancies observed in the far tails of the distribution.
• The framework demonstrates strong noise resilience, maintaining predictive accuracy even when perturbations are added to the input data.
• The combination of CAE with ESN and GRU RNNs enables a fully data-driven, equation-free model that can serve as a reduced-order dynamical model for transport fluxes in large-scale simulations.