Application of Gene Expression Programming to a-posteriori LES modeling of a Taylor Green Vortex

https://doi.org/10.1016/j.jcp.2020.109859Get rights and content

Highlights

  • In-the-loop a-posteriori Large Eddy Simulation subgridscale model development.

  • Models have been trained against Direct Numerical Simulation data of a Taylor Green Vortex.

  • Gene Expression Programming provides an interpretable mathematical expression.

  • Optimal model performs better than existing models even on coarse meshes.

  • Best model is robust in response to changes of resolution and Reynolds number.

Abstract

Gene Expression Programming (GEP), a branch of machine learning, is based on the idea to iteratively improve a population of candidate solutions using an evolutionary process built on the survival-of-the-fittest concept. The GEP approach was initially applied with encouraging results to the modeling of the unclosed tensors in the context of RANS (Reynolds Averaged Navier–Stokes) turbulence modeling. In a subsequent study it was demonstrated that the GEP concept can also be successfully used for modeling the unknown Sub-Grid Stress (SGS) tensor in the context of Large Eddy Simulations (LES). This was done in an a-priori analysis, where an existing Direct Numerical Simulation (DNS) database was explicitly filtered to evaluate the unknown stresses and to assess the performance of model candidates suggested by GEP. This paper presents the next logical step, i.e. the application of GEP to a-posteriori LES model development. Because a-posteriori analysis, using in-the-loop optimization, is considered the ultimate way to test SGS models, this can be considered an important milestone for the application of machine learning to LES based turbulence modeling. GEP is here used to train LES models for simulating a Taylor Green Vortex (TGV) and results are compared with existing standard models. It is shown that GEP finds a model that outperforms known models from literature as well as the no-model LES. Although the performance of this best model is maintained for resolutions and Reynolds numbers different from the training data, this is not automatically guaranteed for all other models suggested by the algorithm.

Introduction

Data from experiments and DNS have historically been used to calibrate turbulence models. The increasing size and quality of the datasets, together with algorithmic innovations and advances in computer hardware have allowed the use of more complex algorithms that infer not just closure constants but also functional forms. As a result machine-learning algorithms have become a popular tool for improving turbulence models. For a recent overview the reader is referred to [1]. Machine learning includes a wide range of techniques, essentially sophisticated curve-fitting algorithms, within the broader field of artificial intelligence [1]. This concept is by no means new in turbulence modeling (e.g. [2], [3]) but it has recently attracted considerable attention. Typically, the algorithms have been applied to closing the RANS equations, e.g. [4], [5], [6], [7], although the scalar flux [8], [9] equations and hybrid RANS/LES approaches [10] have also been tackled. Activities in LES include the use of neural networks to model subgrid-scale stresses [11], [12], to estimate turbulent sub-grid scale reaction rates [13], [14], sub-grid curvature effects in two phase flow simulations [15] or LES wall modeling [16]. GEP has been used recently [17] for modeling the unknown sub-grid stress tensor in the context of a-priori analysis. It has been also demonstrated in [18] that the sparse regression technique STRidge and the evolutionary optimization algorithm GEP are effective tools for identifying hidden physical laws from observed data. The aforementioned studies can be split into two categories: those that are transparent and those that are not. Essentially, this dictates whether the non-linearity of the machine-learnt model exists on a level of description interpretable by a human [17] or not. For methodologies such as random forests and neural networks, the non-linearity is built over hundreds, if not thousands of interactions. The existence of complexity at the lowest level forces the user to treat the model as a non-transparent black box. Diagnosing problems and sharing models with the community are thus not straightforward. For symbolic regression, such as in GEP, the model inferred is a mathematical expression. Therefore, using GEP for turbulence modeling is an emerging field of research. This method has the advantage, relative to other machine learning models such as artificial neural networks, that it produces a turbulence model as a function of key physical parameters. The obtained function can be readily implemented, and is also repeatable and provides insight into the phenomenon of interest [19]. For these reasons the GEP approach is adopted in this study as the model can be interpreted and easily implemented in existing LES solvers. The work presented in [17] was based on a-priori analysis, i.e. filtering of DNS data. Such analysis, however, has limitations [20], and it cannot be guaranteed that findings based on a-priori analysis necessarily result in improved performance in an actual LES simulation. The ultimate test of an LES must therefore consist of comparing actual simulation results to measurements or DNS; this is known as a posteriori testing [21]. Hence, the present optimization strategy is based on a-posteriori analysis, i.e. it uses tens of thousands in-the-loop LES runs to find a good LES representation of the DNS based reference data. It is worth mentioning that model development based on a-posteriori LES is not only more expensive, but also much more intricate than in the case of a-priori LES, because of the close interaction of modelling and numerical errors. Furthermore, there is no guarantee that every candidate model suggested by GEP will result in a stable LES. This paper is structured as follows: first the methodology is presented, outlining the overall algorithm for in-the-loop optimization and providing a short summary of the GEP approach. Then the performance of the trained models is presented on the training case, assessing their generalizability to other grid resolutions and flow conditions.

Section snippets

Methodology

This section outlines the numerical methodology and the computational configuration as well as details regarding the GEP framework used for the optimization.

Results

In this section the results of the optimization process will be presented and the models suggested by GEP will be compared to standard models from literature. In the following subsection a sensitivity study on the model parameters will be presented. Finally, the robustness of the models with respect to filter size (i.e. grid size) and Reynolds number variations will be discussed.

Conclusions

The concept of Gene Expression Programming was applied to model the unknown subgrid stresses in the context of Large Eddy Simulation of a Taylor Green vortex. To the best knowledge of the authors this is the first application of GEP to in-the-loop, a-posteriori LES model development. A DNS of the same configuration has been performed, using the same code, to provide the reference data such that the cost function could be chosen as the weighted mean absolute error between DNS and LES of kinetic

CRediT authorship contribution statement

Maximilian Reissmann: Data curation, Investigation, Software. Josef Hasslberger: Conceptualization, Formal analysis, Supervision, Validation, Writing - review & editing. Richard D. Sandberg: Methodology, Software, Supervision, Writing - review & editing. Markus Klein: Conceptualization, Formal analysis, Funding acquisition, Supervision, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The visit to University of Melbourne of the first author was supported by the German Federal Ministry of Defence.

References (34)

  • B.A. Younis et al.

    A rational model for the turbulent scalar fluxes

    Proc. R. Soc. Lond. A, Math. Phys. Eng. Sci.

    (2005)
  • J. Ling et al.

    Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

    J. Fluid Mech.

    (2016)
  • J. Wu et al.

    Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework

    Phys. Rev. Fluids

    (2018)
  • P.M. Milani et al.

    A machine learning approach for determining the turbulent diffusivity in film cooling flows

  • J. Weatheritt et al.

    Hybrid Reynolds-averaged/large-eddy simulation methodology from symbolic regression: formulation and application

    AIAA J.

    (2017)
  • A. Vollant et al.

    Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures

    J. Turbul.

    (2017)
  • M. Gamahara et al.

    Searching for turbulence models by artificial neural network

    Phys. Rev. Fluids

    (2017)
  • Cited by (0)

    View full text