Reynolds-averaged Navier–Stokes simulations represent a cost-effective option for practical engineering applications, but are facing ever-growing demands for more accurate turbulence models. Recently, emerging machine learning techniques have had a promising impact on turbulence modeling, but are still in their infancy regarding widespread industrial adoption. Toward their extensive uptake, this paper presents a universally interpretable machine learning (UIML) framework for turbulence modeling, which consists of two parallel machine learning-based modules to directly infer the structural and parametric representations of turbulence physics, respectively. At each phase of model development, data reflecting the evolution dynamics of turbulence and domain knowledge representing prior physical considerations are converted into modeling knowledge. The data- and knowledge-driven UIML is investigated with a deep residual network. The following three aspects are demonstrated in detail: (i) a compact input feature parameterizing a new turbulent timescale is introduced to prevent nonunique mappings between conventional input arguments and output Reynolds stress; (ii) a realizability limiter is developed to overcome the under-constrained state of modeled stress; and (iii) fairness and noise-insensitivity constraints are included in the training procedure. Consequently, an invariant, realizable, unbiased, and robust data-driven turbulence model is achieved. The influences of the training dataset size, activation function, and network hyperparameter on the performance are also investigated. The resulting model exhibits good generalization across two- and three-dimensional flows, and captures the effects of the Reynolds number and aspect ratio. Finally, the underlying rationale behind prediction is explored.

Reynolds-stress closure modeling is critical to Reynolds-averaged Navier-Stokes (RANS) analysis, and it remains a challenging issue in reducing both structural and parametric inaccuracies. This study first proposes a novel algebraic stress model named as tensorial quadratic eddy-viscosity model (TQEVM), in which nonlinear terms improve previous model-form failure due to neglection of nonlocal effects. Then a data-driven regression model based on a fully-connected deep neural network is designed to determine the TQEVM coefficients. The well-trained data-driven model using high-fidelity direct numerical simulation (DNS) data successfully learned the underlying input-output relationships, further obtaining spatial-dependent optimal values of these coefficients. Finally, detailed validations are made in wall-bounded flows where nonlocal effects are expected to be significant. Comparative results indicate that TQEVM provides improvements both for the stress-strain misalignment and stress anisotropy, which are clear advantages over linear and quadratic eddy-viscosity models. TQEVM extends to the scope of resolution to the wall distance y + ≈ 9 as well as provides a realizable solution. RANS simulations with TQEVM are also carried out and the obtained mean-flow quantities of interest agree well with DNS. This work, therefore, results in a high-fidelity representation of Reynolds stresses and contributes to further understanding of machine-learning-assisted turbulence modeling and regression analysis.Energies 2020, 13, 258 2 of 21 improving sensitivity to curvature and/or rotation effects, such as introducing corrections into the eddy viscosity equation [10], the dissipation rate equation [11], the specific dissipation rate equation [12], and introducing corrections into eddy-viscosity coefficient [12,13]. However, none of the essential changes occur in the baseline framework of closure modeling. Obviously, the variant closure coefficients cannot improve the inherent limitations due to the underlying model-form inaccuracy rooted in Boussinesq hypothesis.Linear stress-strain relationship origins from the Boussinesq hypothesis that turbulence react locally to the changes of mean-flow behavior in equilibrium flows. However, that is not really the case. In most turbulent flows, there exists a large misalignment in stress-strain response which has been repeatedly confirmed with high-fidelity DNS investigation [14][15][16]. In view of pressure-strain correlations in the exact Reynolds-stress transport equations, the stress-strain misalignment is theoretically generated from the nonlocal response of the pressure to the entire flow field through Green's functions [17]. As a result, the in-phase type EVMs certainly experience failure, especially in flows with strong spatial inhomogeneity (e.g., rotation/curvature [10,18], three-dimensionality [19], secondary-flow [20], etc.). These flow effects make changes in the mean shear (i.e., spatial inhomogeneity) and turbulence structures [21][22][23], further enhancing nonlocality on the...

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