A CFD-driven deterministic symbolic identification algorithm for learning explicit algebraic Reynolds-stress models (EARSM) from high-fidelity data is developed building on the frozen-training SpaRTA algorithm of [1]. Corrections for the Reynolds stress tensor and the production of transported turbulent quantities of a baseline linear eddy viscosity model (LEVM) are expressed as functions of tensor polynomials selected from a library of candidate functions. The CFD-driven training consists in solving a blackbox optimization problem in which the fitness of candidate EARSM models is evaluated by running RANS simulations. The procedure enables training models against any target quantity of interest, computable as an output of the CFD model. Unlike the frozen-training approach, the proposed methodology is not restricted to data sets for which full fields of high-fidelity data, including second flow order statistics, are available. However, the solution of a high-dimensional expensive blackbox function optimization problem is required. Several steps are then undertaken to reduce the associated computational burden. First, a sensitivity analysis is used to identify the most influential terms and to reduce the dimensionality of the search space. Afterwards, the Constrained Optimization using Response Surface (CORS) algorithm, which approximates the black-box cost function using a response surface constructed from a limited number of CFD solves, is used to find the optimal model parameters. Model discovery and cross-validation is performed for three configurations of 2D turbulent separated flows in channels of variable section using different sets of training data to show the flexibility of the method. The discovered models are then applied to the prediction of an unseen 2D separated flow with higher Reynolds number and different geometry. The predictions of the discovered models for the new case are shown to be not only more accurate than the baseline LEVM, but also of a multi-purpose EARSM model derived from purely physical arguments. The proposed deterministic symbolic identification approach constitutes a promising candidate for building accurate and robust RANS models customized for a given class of flows at moderate computational cost.@en

A novel deterministic symbolic regression method SpaRTA (Sparse Regression of Turbulent Stress Anisotropy) is introduced to infer algebraic stress models for the closure of RANS equations directly from high-fidelity LES or DNS data. The models are written as tensor polynomials and are built from a library of candidate functions. The machine-learning method is based on elastic net regularisation which promotes sparsity of the inferred models. By being data-driven the method relaxes assumptions commonly made in the process of model development. Model-discovery and cross-validation is performed for three cases of separating flows, i.e. periodic hills (Re=10595), converging-diverging channel (Re=12600) and curved backward-facing step (Re=13700). The predictions of the discovered models are significantly improved over the k-ω SST also for a true prediction of the flow over periodic hills at Re=37000. This study shows a systematic assessment of SpaRTA for rapid machine-learning of robust corrections for standard RANS turbulence models.@en

In this work recent advancements are presented in utilising deterministic symbolic regression to infer algebraic models for turbulent stress-strain relation with sparsity-promoting regression techniques. The goal is to build a functional expression from a set of candidate functions in order to represent the target data most accurately. Targets are the coefficients of a polynomial tensor basis, which are identified from high-fidelity data using regularised least-square regression. The method successfully identified a correction term for the benchmark test case of flow over periodic hills in 2D at Reh = 10595.@en

In direct and large eddy simulations, very small space steps are used close to the solid walls in order to resolve the boundary-layer structures. Due to the restrictive CFL stability criteria of explicit time-stepping schemes, the maximum allowable time step is also very small, leading to high computational costs, notably for converging flow statistics. The use of an implicit integration scheme may overcome this limitation at the price of an increased computational cost per step. Moreover, the most commonly used fully implicit schemes induce higher errors due to the necessary approximations and poor dispersion and dissipation properties. As a compromise, a fourth-order implicit residual smoothing scheme (IRS4), successfully validated for a finite volume solver in Cinnella and Content (2016); Hoarau and Cinnella (2020), has been introduced in a multiblock high-order finite-difference solver. Several improvements are proposed to ensure better dissipation properties, a more efficient treatment of physical boundaries and an accurate and stable parallel multiblock implementation. For moderate CFL numbers, a similar accuracy as the explicit method is obtained with substantial savings in terms of computational time.@en

In direct and large eddy simulations, very small space steps are used close to the solid walls in order to resolve the boundary-layer structures. Due to the restrictive CFL stability criteria of explicit time-stepping schemes, the maximum allowable time step is also very small, leading to high computational costs, notably for converging flow statistics. The use of an implicit integration scheme may overcome this limitation at the price of an increased computational cost per step. Moreover, the most commonly used fully implicit schemes induce higher errors due to the necessary approximations and poor dispersion and dissipation properties. As a compromise, a fourth-order implicit residual smoothing scheme (IRS4), successfully validated for a finite volume solver in Cinnella and Content (2016); Hoarau and Cinnella (2020), has been introduced in a multiblock high-order finite-difference solver. Several improvements are proposed to ensure better dissipation properties, a more efficient treatment of physical boundaries and an accurate and stable parallel multiblock implementation. For moderate CFL numbers, a similar accuracy as the explicit method is obtained with substantial savings in terms of computational time.@en

The lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at Re= 44,000 over 2D periodic hills at ReH= 5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project).@en

The lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at Re= 44,000 over 2D periodic hills at ReH= 5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project).@en

This work presents developments towards a deterministic symbolic regression method to derive algebraic Reynolds-stress models for the Reynolds-Averaged Navier-Stokes (RANS) equations. The models are written as tensor polynomials, for which optimal coefficients are found using Bayesian inversion. These coefficient fields are the targets for the symbolic regression. A method is presented based on a regularisation strategy in order to promote sparsity of the inferred models and is applied to high-fidelity data. By being data-driven the method reduces the assumptions commonly made in the process of model development in order to increase the predictive fidelity of algebraic models.@en

Computational fluid dynamics analyses of high-Reynolds-number flows mostly rely on the Reynolds-averaged Navier–Stokes equations. The associated closure models are based on multiple simplifying assumptions and involve numerous empirical closure coefficients, which are calibrated on a set of simple reference flows. Predicting new flows using a single closure model with nominal values for the closure coefficients may lead to biased predictions. Bayesian model-scenario averaging is a statistical technique providing an optimal way to combine the predictions of several competing models calibrated on various sets of data (scenarios). The method allows a stochastic estimate of a quantity of interest in an unmeasured prediction scenario to be obtain by 1) propagating posterior probability distributions of the parameters obtained for multiple calibration scenarios, and 2) computing a weighted posterior predictive distribution. Although step 2 has a negligible computational cost, step 1 requires a large number of samples of the solver. To enable the application of the proposed approach to computationally expensive flow configurations, a modified formulation is used where a maximum posterior probability approximation is used to drastically reduce the computational burden. The predictive capability of the proposed simplified approach is assessed for two-dimensional separated and three-dimensional compressible flows.@en