Data from experiments and direct simulations of turbulence have historically been used to calibrate simple engineering models such as those based on the Reynolds-averaged Navier–Stokes (RANS) equations. In the past few years, with the availability of large and diverse data sets, researchers have begun to explore methods to systematically inform turbulence models with data, with the goal of quantifying and reducing model uncertainties. This review surveys recent developments in bounding uncertainties in RANS models via physical constraints, in adopting statistical inference to characterize model coefficients and estimate discrepancy, and in using machine learning to improve turbulence models. Key principles, achievements, and challenges are discussed. A central perspective advocated in this review is that by exploiting foundational knowledge in turbulence modeling and physical constraints, researchers can use data-driven approaches to yield useful predictive models.

Turbulence modeling is a critical component in numerical simulations of industrial flows based on Reynolds-averaged Navier-Stokes (RANS) equations. However, after decades of efforts in the turbulence modeling community, universally applicable RANS models with predictive capabilities are still lacking. Large discrepancies in the RANS-modeled Reynolds stresses are the main source that limits the predictive accuracy of RANS models. Identifying these discrepancies is of significance to possibly improve the RANS modeling. In this work, we propose a data-driven, physics-informed machine learning approach for reconstructing discrepancies in RANS modeled Reynolds stresses. The discrepancies are formulated as functions of the mean flow features. By using a modern machine learning technique based on random forests, the discrepancy functions are trained by existing direct numerical simulation (DNS) databases and then used to predict Reynolds stress discrepancies in different flows where data are not available. The proposed method is evaluated by two classes of flows: (1) fully developed turbulent flows in a square duct at various Reynolds numbers and (2) flows with massive separations. In separated flows, two training flow scenarios of increasing difficulties are considered: (1) the flow in the same periodic hills geometry yet at a lower Reynolds number and (2) the flow in a different hill geometry with a similar recirculation zone. Excellent predictive performances were observed in both scenarios, demonstrating the merits of the proposed method.

Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high-fidelity simulations [J.-X. Wang et al., Phys. Rev. Fluids 2, 034603 (2017)]. However, solving for mean flows from the improved Reynolds stresses still poses significant challenges due to potential ill-conditioning of RANS equations with Reynolds stress closures. Enabling improved predictions of mean velocities is of profound practical importance, because often the velocity and its derived quantities (quantities of interest, e.g., drag, lift, and surface friction), and not the Reynolds stress itself, are of ultimate interest in RANS simulations. To this end, we present a comprehensive framework for augmenting turbulence models with physics-informed machine learning, illustrating a complete workflow from identification of input features to final prediction of mean velocities. This work has two innovations. First, we demonstrate a systematic procedure to generate mean flow features based on the integrity basis for mean flow tensors. Second, we propose using machine learning to predict linear and nonlinear parts of the Reynolds stress tensor separately. Inspired by the finite polynomial representation of tensors in classical turbulence modeling, such a decomposition is instrumental in overcoming the ill-conditioning of RANS equations. Numerical tests demonstrate the merits of the proposed framework.

While data-based approaches were found to be useful for subgrid scale (SGS) modeling in Reynolds-averaged Navier-Stokes (RANS) simulations, there have not been many attempts at using machine learning techniques for wall modeling in large-eddy simulations (LESs). Large-eddy simulation differs from RANS simulation in many aspects. For one thing, LES is scale resolving. For another, LES is in and of itself a high-fidelity tool. Because data sets of higher fidelity are in general not frequently accessible or available, this poses additional challenges to data-based modeling in LES. Further, SGS modeling usually needs flow information at only large scales, in contrast with wall modeling, which needs to account for both near-wall small scales and large scales above the wall. In this work we discuss how the above-noted challenges may be addressed when taking a data-based approach for wall modeling. We also show the necessity of incorporating physical insights in model inputs, i.e., using inputs that are inspired by the vertically integrated thin-boundary-layer equations and the eddy population density scalings. We show that the inclusion of the above physics-based considerations would enhance extrapolation capabilities of a neural network to flow conditions that are not within the train data. Being cheap to evaluate and using only channel flow data at ${\mathrm{Re}}_{\tau }=1000$, the trained networks are found to capture the law of the wall at arbitrary Reynolds numbers and outperform the conventional equilibrium model in a nonequilibrium flow.