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.

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