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Pre-processing DNS data to improve statistical convergence and accuracy of mean velocity fields in invariant data-driven turbulence models

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Abstract

Machine learning (ML) techniques for turbulence modeling are becoming an important tool to build the bridge between low-cost-low-accurate turbulence models (like RANS) and high-cost-high-accurate procedures to represent turbulence (like DNS). In recent studies, however, it was observed that the DNS data for the Reynolds stress tensor (RST) do not satisfactorily recover the mean velocity field. This fact has two rooting sources, the lack of convergence of statistical fields, and the ill-conditioning of the RANS equations. To address these two aspects, we employ two remedies in the turbulent flow through a square duct (SD). On the one side, we applied symmetry filters on the flow data to provide more converged statistical quantities. On the other side, we contrast the traditional approach where the model target is the Reynolds stress tensor with a recent approach where the Reynolds force vector (RFV) is the target. We also provide a comparison between two ML techniques commonly used in the literature, neural network and random forest, in an invariant formulation recently proposed. The results have shown that there is a direct relation between the convergence of DNS data and the performance of data-driven turbulence models. The models obtained from symmetrical data presented lower error propagation to the mean velocity field. The Reynolds force vector is shown to be a target that can produce more accurate results, corroborating recent findings of the literature. The performance of the two ML techniques was equivalent, with small differences depending on the target quantity (RST or RFV) and the velocity component considered (main flow or secondary flow).

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Acknowledgements

We would like to acknowledge also CNPq (No. 304095/2018-4) and CAPES (No. PROEX 803/2018) for financial support.

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Authors and Affiliations

  1. Department of Mechanical Engineering, COPPE, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, Rio de Janeiro, RJ, 24210-240, Brazil

    Eduardo F. Fonseca, Victor B. Rangel, Bernardo P. Brener, Matheus A. Cruz & Roney L. Thompson

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  1. Eduardo F. Fonseca
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  2. Victor B. Rangel
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  3. Bernardo P. Brener
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  4. Matheus A. Cruz
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Correspondence to Roney L. Thompson.

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Communicated by Vassilios Theofilis.

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Appendix: \(R^2\) scores

Appendix: \(R^2\) scores

Since NN has a random starting point for weights and bias, even with the same set of hyper-parameters, the accuracy of a NN can vary. In this work, a set of 20 NN were built for each model and the one with highest \(R^{2}\) score was used to predict the desired fields. Each of these 20 NN is called a fold, so for each model 20 folds will be made and the evaluation of the model will be expressed as the mean of the \(R^{2}\) score for each fold. As well as for NN models, a set of 20 RF were built for each model and the one with highest \(R^{2}\) score was used to predict the desired fields (Tables 6 and 7).

Table 6 \(R^2\) score of NN quadrant sets
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Table 7 \(R^2\) score of RF quadrant sets
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Fonseca, E.F., Rangel, V.B., Brener, B.P. et al. Pre-processing DNS data to improve statistical convergence and accuracy of mean velocity fields in invariant data-driven turbulence models. Theor. Comput. Fluid Dyn. 36, 435–463 (2022). https://doi.org/10.1007/s00162-022-00603-4

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