Periodic Hills Research Articles

Development of a Generalizable Data-Driven Turbulence Model: Conditioned Field Inversion and Symbolic Regression

This paper addresses the issue of predicting separated flows with Reynolds-averaged Navier–Stokes (RANS) turbulence models, which are essential for many engineering tasks. Traditional RANS models usually struggle with this task, so recent efforts have focused on data-driven methods such as field inversion and machine learning (FIML) to correct this issue by adjusting the baseline equations. However, these FIML methods often reduce accuracy in attached boundary layers. To address this issue, we developed a “conditioned field inversion” technique. This method adjusts the corrective factor β (used by FIML) in the shear-stress transport (SST) model. It multiplies β with a shield function fd that is off in the boundary layer and on elsewhere. This maintains the accuracy of the baseline model for the attached flows. We applied both conditioned and classic field inversion to the NASA hump and a curved backward-facing step, creating two datasets. These datasets were used to train two models: SR-CND (symbolic regression-conditioned, from our new method) and SR-CLS (symbolic regression-classic, from the traditional method). The SR-CND model matches the SR-CLS model in predicting separated flows in various scenarios, such as periodic hills, the NLR7301 airfoil, the 3D SAE (Society of Automotive Engineers) car model, and the 3D Ahmed body, and outperforms the baseline SST model in the cases presented in the paper. Importantly, the SR-CND model maintains accuracy in the attached boundary layers, whereas the SR-CLS model does not. Therefore, the proposed method improves separated flow predictions while maintaining the accuracy of the original model for attached flows, offering a better way to create data-driven turbulence models.

Data-Guided Low-Reynolds-Number Corrections for Two-Equation Models

Abstract The baseline Launder–Spalding k−ε model cannot be integrated to the wall. This paper seeks to incorporate the entire law of the wall into the model while preserving the original k−ε framework structure. Our approach involves modifying the unclosed dissipation terms in the k and ε equations specifically within the wall layer according to direct numerical simulation (DNS) data. The resulting model effectively captures the mean flow characteristics in both the buffer layer and the logarithmic layer, resulting in robust predictions of skin friction for zero-pressure-gradient (ZPG) flat-plate boundary layers and plane channels. To further validate our formulation, we apply our model to boundary layers under varying pressure gradients, channels experiencing sudden deceleration, and flow over periodic hills, with highly favorable results. Although not the focus of this study, the methodology here applies equally to the k–ω formulation and yields improved predictions of the mean flow in the viscous sublayer and buffer layer.

Non-unique machine learning mapping in data-driven Reynolds-averaged turbulence models

Recent growing interest in using machine learning for turbulence modeling has led to many proposed data-driven turbulence models in the literature. However, most of these models have not been developed with overcoming non-unique mapping (NUM) in mind, which is a significant source of training and prediction error. Only NUM caused by one-dimensional channel flow data has been well studied in the literature, despite most data-driven models having been trained on two-dimensional flow data. The present work aims to be the first detailed investigation on NUM caused by two-dimensional flows. A method for quantifying NUM is proposed and demonstrated on data from a flow over periodic hills and an impinging jet. The former is a wall-bounded separated flow, and the latter is a shear flow containing stagnation and recirculation. This work confirms that data from two-dimensional flows can cause NUM in data-driven turbulence models with the commonly used invariant inputs. This finding was verified with both cases, which contain different flow phenomena, hence showing that NUM is not limited to specific flow physics. Furthermore, the proposed method revealed that regions containing low strain and rotation or near pure shear cause the majority of NUM in both cases—approximately 76% and 89% in the flow over periodic hills and impinging jet, respectively. These results led to viscosity ratio being selected as a supplementary input variable (SIV), demonstrating that SIVs can reduce NUM caused by data from two-dimensional flows and subsequently improve the accuracy of tensor-basis machine learning models for turbulence modeling.

Framework of Physically Consistent Homogenization and Multiscale Modeling of Shell Structures**

A framework of physically consistent homogenization and multiscale modeling (PCHMM) for reduced-order analysis of plate/shell structures is developed in this paper. To address the inapplicability of conventional periodic boundary conditions and Hill’s condition involved in homogenization of shear-deformable shell structures, the paper proposes physically consistent boundary conditions and modified Hill’s condition for plate/shells. Unlike the PCHMM method for beams, considering the contradiction between high-order displacement fields induced by shear forces and low-order kinematic assumptions, additional constraints are applied to the plate/shell structure sectional strains during the solution of perturbation fields. The correctness and effectiveness of the proposed plate/shell PCHMM framework and method are verified by typical numerical examples. The proposed theory can also be conveniently embedded into commercial finite element software for homogenization and multiscale analysis of structures such as microscale metamaterials like lattice plates and large complex structures like aircraft fuselage sections.

An implicit large-eddy simulation perspective on the flow over periodic hills

The periodic hills simulation case is a well-established benchmark for computational fluid dynamics solvers due to its complex features derived from the separation of a turbulent flow from a curved surface. We study the case with the open-source implicit large-eddy simulation (ILES) software Lethe. Lethe solves the incompressible Navier–Stokes equations by applying a stabilized continuous finite element discretization. The results are validated by comparison to experimental and computational data available in the literature for Re = 5600. We study the effect of the time step, averaging time, and global mesh refinement. The ILES approach shows good accuracy for average velocities and Reynolds stresses using less degrees of freedom than the reference numerical solution. The time step has a greater effect on the accuracy when using coarser meshes, while for fine meshes the results are rapidly time-step independent when using an implicit time-stepping approach. A good prediction of the reattachment point is obtained with several meshes and this value approaches the experimental benchmark value as the mesh is refined. We also run simulations at Reynolds equal to 10600 and 37000 and observe promising results for the ILES approach.

Wall Modeling of Turbulent Flows with Varying Pressure Gradients Using Multi-Agent Reinforcement Learning

We propose a framework for developing wall models for large-eddy simulation that is able to capture pressure-gradient effects using multi-agent reinforcement learning. Within this framework, the distributed reinforcement learning agents receive off-wall environmental states, including pressure gradient and turbulence strain rate, ensuring adaptability to a wide range of flows characterized by pressure-gradient effects and separations. Based on these states, the agents determine an action to adjust the wall eddy viscosity and, consequently, the wall-shear stress. The model training is in situ with wall-modeled large-eddy simulation grid resolutions and does not rely on the instantaneous velocity fields from high-fidelity simulations. Throughout the training, the agents compute rewards from the relative error in the estimated wall-shear stress, which allows them to refine an optimal control policy that minimizes prediction errors. Employing this framework, wall models are trained for two distinct subgrid-scale models using low-Reynolds-number flow over periodic hills. These models are validated through simulations of flows over periodic hills at higher Reynolds numbers and flows over the Boeing Gaussian bump. The developed wall models successfully capture the acceleration and deceleration of wall-bounded turbulent flows under pressure gradients and outperform the equilibrium wall model in predicting skin friction.

Reconstruction of the turbulent flow field with sparse measurements using physics-informed neural network

Accurate high-resolution flow field prediction based on limited experimental data is a complex task. This research introduces an innovative framework leveraging physics-informed neural network (PINN) to reconstruct high-resolution flow fields using sparse particle image velocimetry measurements for flow over a periodic hill and high-fidelity computational fluid dynamics data for flow over a curved backward-facing step. Model training utilized mean flow measurements, with increased measurement sparsity achieved through various curation strategies. The resulting flow field reconstruction demonstrated marginal error in both test cases, showcasing the ability of the framework to reconstruct the flow field with limited measurement data accurately. Additionally, the study successfully predicted flow fields under two different noise levels, closely aligning with experimental and high-fidelity results (experimental, direct numerical simulation, or large eddy simulation) for both cases. Hyperparameter tuning conducted on the periodic hill case has been applied to the curved backward-facing step case. This research underscores the potential of PINN as an emerging method for turbulent flow field prediction via data assimilation, offering reduced computational costs even with sparse, noisy measurements.

Bayesian interface technique-based inverse estimation of closure coefficients of standard k−ϵ turbulence model by limiting the number of DNS data points for flow over a periodic hill

Among different numerical methods for modeling turbulent flow, Reynolds-averaged Navier–Stokes (RANS) is the most commonly used and computationally reasonable. However, the accuracy of RANS is lower than that of other high-fidelity numerical methods. In this work, the uncertainties associated with the coefficients of the standard k−ϵ RANS turbulence model are estimated and calibrated to improve the accuracy. The calibration is performed by considering the coefficients individually as well as collectively. The first three coefficients of the standard k−ϵ turbulence model are calibrated among the five coefficients ( Cμ,Cϵ1,Cϵ2,σϵ and σ k ). The Bayesian inference technique using the Metropolis–Hastings algorithm is applied to quantify uncertainties and calibration. Flow over a periodic hill is selected as a test case. The separation height of the bubble at x/h=2 and x/h=4 , along with the streamwise velocity at various locations, has been chosen as the quantities of interest for comparing the results with DNS. The calibration is performed using known high-fidelity data (direct numerical simulation) from the available data set. The velocity field is re-calculated from the calibrated closure coefficients and compared with the same calculated with the standard coefficients of k−ϵ turbulence model (baseline). The deviation of calibrated C µ is almost 50%–60% from baseline and for Cϵ1 and Cϵ2 it is 3%–12% and 6%–9% respectively. The algorithm is tested for different Reynold numbers and data points. A sensitivity analysis is also performed.

Implementation and validation of a generalized wall stress function

The generalized wall function by Shih et al. [Report No. M-1999-209398 (1999)], which accounts for non-equilibrium effects by the presence of favorable and adverse pressure gradients in turbulent flows, is addressed with the aim of performing high Reynolds number large-eddy simulations of the wall-bounded flow. The model uses a corrected law of the wall with a pressure gradient contribution to approximate the wall stress and applies to the entire viscous layer, buffer layer, and inertial region. A fully developed channel flow is first tested to validate the solver and model implementation, and then the wall function is assessed for the flow over a periodic hill. Wall-resolved simulations are in good agreement with reference results. A priori investigation with own experimental results corroborates the mathematical form of the model and suggests using different coefficients. The wall-modeled simulations show that the implemented wall model is able to improve the wall shear stress predictions compared to a standard equilibrium wall model. It corrects the underestimation of wall shear stresses by equilibrium models in the favorable pressure gradient region and the overestimation of wall shear stresses in the adverse pressure gradient region. The positions of the separation and reattachment points are also in good agreement with reference results. Furthermore, the prediction of the wall shear stress maximum in the favorable pressure gradient zone at the windward side of the hill is quite robust against coarsening the wall-normal grid spacing.

Optimal sensor placement for ensemble-based data assimilation using gradient-weighted class activation mapping

We introduce an optimal sensor placement method using convolutional neural networks for ensemble-based data assimilation. The proposed method utilizes the gradient-weighted class activation mapping of the convolutional neural networks to identify important regions for assimilation processes. It is achieved by using the initial ensemble of samples for data assimilation as training data to construct a convolutional neural network-based surrogate model. In doing so, the method can estimate optimal sensor locations in an a priori manner, allowing for sensor placement before conducting data assimilation processing. Moreover, the gradient-weighted class activation mapping is used to alleviate the effect of error accumulation during the backpropagation process through global averaging. Further, these observation sensors are incorporated to reconstruct mean turbulent flow fields based on the ensemble Kalman method. The proposed optimal sensor placement method is applied to two flow applications with complex geometries, i.e., flows around periodic hills and an axisymmetric body of revolution. Both cases demonstrate that the proposed method can significantly reduce the number of sensors without sacrificing the accuracy of the reconstructed flow field.

The effect of variations in experimental and computational fidelity on data assimilation approaches

AbstractWe conduct a comprehensive analysis of two data assimilation methods: the first utilizes the discrete adjoint approach with a correction applied to the production term of the turbulence transport equation, preserving the Boussinesq approximation. The second is a state observer method that implements a correction in the momentum equations alongside a turbulence model, both applied to fluid dynamics simulations. We investigate the impact of varying computational mesh resolutions and experimental data resolutions on the performance of these methods within the context of a periodic hill test case. Our findings reveal the distinct strengths and limitations of both methods, which successfully assimilate data to improve the accuracy of a RANS simulation. The performance of the variational model correction method is independent of input data and computational mesh resolutions. The state observer method, on the other hand, is sensitive to the resolution of the input data and CFD mesh.

Scale-resolving simulations of turbulent flows with coherent structures: Toward cut-off dependent data-driven closure modeling

Complex turbulent flows with large-scale instabilities and coherent structures pose challenges to both traditional and data-driven Reynolds-averaged Navier–Stokes methods. The difficulty arises due to the strong flow-dependence (the non-universality) of the unsteady coherent structures, which translates to poor generalizability of data-driven models. It is well-accepted that the dynamically active coherent structures reside in the larger scales, while the smaller scales of turbulence exhibit more “universal” (generalizable) characteristics. In such flows, it is prudent to separate the treatment of the flow-dependent aspects from the universal features of the turbulence field. Scale resolving simulations (SRS), such as the partially averaged Navier–Stokes (PANS) method, seek to resolve the flow-dependent coherent scales of motion and model only the universal stochastic features. Such an approach requires the development of scale-sensitive turbulence closures that not only allow for generalizability but also exhibit appropriate dependence on the cut-off length scale. The objectives of this work are to (i) establish the physical characteristics of cut-off dependent closures in stochastic turbulence; (ii) develop a procedure for subfilter stress neural network development at different cut-offs using high-fidelity data; and (iii) examine the optimal approach for the incorporation of the unsteady features in the network for consistent a posteriori use. The scale-dependent closure physics analysis is performed in the context of the PANS approach, but the technique can be extended to other SRS methods. The benchmark “flow past periodic hills” case is considered for proof of concept. The appropriate self-similarity parameters for incorporating unsteady features are identified. The study demonstrates that when the subfilter data are suitably normalized, the machine learning based SRS model is indeed insensitive to the cut-off scale.

Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation.

Time-Resolved Local Loss Analysis of Single- and Two-Blade Pump Flow

Abstract A method for the evaluation of time-resolved entropy production in isothermal and incompressible flow is presented. It is applied as a postprocessing of the three-dimensional (3D) flow field obtained by time-resolved computational fluid dynamics (CFD) with scale adaptive turbulence modeling. Wall functions for direct and turbulent entropy production are presented for a cell-centered finite volume method, implemented in the open-source software OpenFOAM and validated on channel, asymmetric diffuser, and periodic hill flow. Single- and two-blade centrifugal pump flow is considered for a wide range of load conditions. Results are compared to experimental data. Time-averaged analysis shows essentially the same loss density distribution among pump components for both pumps, with the impeller and volute region contributing the most, especially in off-design conditions. For both pumps, the losses exhibit significant fluctuations due to impeller–volute interactions. The fluctuation magnitude of loss density is in the same range as flowrate fluctuations and much smaller than pressure fluctuation magnitude. For the two-blade pump (2BP), loss fluctuation magnitude is smaller than for the single-blade pump (1BP). Distinct loss mechanisms are identified for different load conditions. Upon blade passage, a promoted or attenuated volute tongue separation is imposed at part or overload, respectively. In between blade passages, a direct connection from pump inlet to the discharge leads to enhanced flowrate and loss density fluctuations. Future work aims at extending this analysis to stronger off-design conditions in multiblade pumps, where stochastic cycle fluctuations occur.

Data-driven RANS closures for improving mean field calculation of separated flows

Reynolds-averaged Navier-Stokes (RANS) simulations have found widespread use in engineering applications, yet their accuracy is compromised, especially in complex flows, due to imprecise closure term estimations. Machine learning advancements have opened new avenues for turbulence modeling by extracting features from high-fidelity data to correct RANS closure terms. This method entails establishing a mapping relationship between the mean flow field and the closure term through a designated algorithm. In this study, the k-ω SST model serves as the correction template. Leveraging a neural network algorithm, we enhance the predictive precision in separated flows by forecasting the desired learning target. We formulate linear terms by approximating the high-fidelity closure (from Direct Numerical Simulation) based on the Boussinesq assumption, while residual errors (referred to as nonlinear terms) are introduced into the momentum equation via an appropriate scaling factor. Utilizing data from periodic hills flows encompassing diverse geometries, we train two neural networks, each possessing comparable structures, to predict the linear and nonlinear terms. These networks incorporate features from the minimal integrity basis and mean flow. Through generalization performance tests, the proposed data-driven model demonstrates effective closure term predictions, mitigating significant overfitting concerns. Furthermore, the propagation of the predicted closure term to the mean velocity field exhibits remarkable alignment with the high-fidelity data, thus affirming the validity of the current framework. In contrast to prior studies, we notably trim down the total count of input features to 12, thereby simplifying the task for neural networks and broadening its applications to more intricate scenarios involving separated flows.

Large Eddy Simulation of Separated Flows on Unconventionally Coarse Grids

Abstract We examine and benchmark the emerging idea of applying the large eddy simulation (LES) formalism to unconventionally coarse grids where Reynolds-averaged Navier–Stokes (RANS) would be considered more appropriate at first glance. We distinguish this idea from very large eddy simulation and detached eddy simulation, which require switching between RANS and LES formalism. LES on RANS grid is appealing because first, it requires minimal changes to a production code; second, it is more cost-effective than LES; third, it converges to LES; and most importantly, it accurately predicts flows with separation. This work quantifies the benefit of LES on RANS-like grids as compared to RANS on the same grids. Three canonical cases are considered: periodic hill, backward-facing step, and jet in cross flow. We conduct direct numerical simulation (DNS), proper LES on LES grids, LES on RANS-quality grids, and RANS. We show that while the LES solutions on the RANS-quality grids are not grid converged, they are twice as accurate as the RANS on the same grids.

On robust boundary treatments for wall‐modeled LES with high‐order discontinuous finite element methods

SummaryTo robustly and accurately simulate wall‐bounded turbulent flows at high Reynolds numbers, we propose suitable boundary treatments for wall‐modeled large‐eddy simulation (WMLES) coupled with a high‐order flux reconstruction (FR) method. First, we show the need to impose an auxiliary boundary condition on auxiliary variables (solution gradients) that are commonly introduced in high‐order discontinuous finite element methods (DFEMs). Auxiliary boundary conditions are introduced in WMLES, where the grid resolution is too coarse to resolve the inner layer of a turbulent boundary layer. Another boundary treatment to further enhance stability with under‐resolved grids, is the use of a modal filter only in the wall‐normal direction of wall‐adjacent cells to remove the oscillations. A grid convergence study of turbulent channel flow with a high Reynolds number () shows that the present WMLES framework accurately predicts velocity profiles, Reynolds shear stress, and skin friction coefficients at the grid resolutions recommended in the literature. It was confirmed that a small amount of filtering is sufficient to stabilize computation, with negligible influence on prediction accuracy. In addition, non‐equilibrium periodic hill flow with a curved wall, including flow separation, reattachment, and acceleration at a high Reynolds number (), is reported. Considering stability and the prediction accuracy, we recommend a loose auxiliary wall boundary conditions with a less steep velocity gradient for WMLES using high‐order DFEMs.

Turbulence model augmented physics-informed neural networks for mean-flow reconstruction

Experimental measurements and numerical simulations of turbulent flows are characterized by a tradeoff between accuracy and resolution. In this study, we combine accurate sparse pointwise mean velocity measurements with the Reynolds-averaged Navier-Stokes (RANS) equations using data assimilation methods. Importantly, we bridge the gap between data assimilation (DA) using physics-informed neural networks (PINNs) and variational methods based on classical spatial discretization of the flow equations, by comparing both approaches on the same turbulent flow case. First, by constraining the PINN with sparse data and the underdetermined RANS equations without closure, we show that the mean flow is reconstructed to a higher accuracy than a RANS solver using the Spalart-Allmaras (SA) turbulence model. Second, we propose the SA turbulence model augmented PINN (PINN-DA-SA), which outperforms the former approach by up to 73% reduction in mean velocity reconstruction error with coarse measurements. The additional SA physics constraints improve flow reconstructions in regions with high velocity and pressure gradients and separation. Third, we compare the PINN-DA-SA approach to a variational data assimilation using the same sparse velocity measurements and physics constraints. The PINN-DA-SA achieves lower reconstruction error across a range of data resolutions. This is attributed to discretization errors in the variational methodology that are avoided by PINNs. We demonstrate the method using high-fidelity measurements from direct numerical simulation of the turbulent periodic hill at Re=5600. Published by the American Physical Society 2024

Ensemble variational method with adaptive covariance inflation for learning neural network-based turbulence models

This work introduces an ensemble variational method with adaptive covariance inflation for learning nonlinear eddy viscosity turbulence models where the Reynolds stress anisotropy is represented with tensor-basis neural networks. The ensemble-based method has emerged as an important alternative to data-driven turbulence modeling due to its merit of non-derivativeness. However, the training accuracy of the ensemble method can be affected by the linearization assumption and sample collapse issue. Given these difficulties, we introduce the hybrid ensemble variational method, which inherits the merits of the ensemble method in non-derivativeness and the variational method in nonlinear analysis. Moreover, a covariance inflation scheme is proposed based on convergence states to alleviate the detrimental effects of sample collapse. The capability of the ensemble variational method in model learning is tested for flows in a square duct, flows over periodic hills, and flows around the S809 airfoil, with increasing complexity in the training data from direct observation to sparse indirect observation. Our results show that the ensemble variational method can learn relatively accurate neural network-based turbulence models in scenarios of small ensemble size and sample variances, compared to the ensemble Kalman method. It highlights the superiority of the ensemble variational method in practical applications, since small ensemble sizes can reduce computational costs, and small sample variance can ensure the training robustness by avoiding nonphysical samples of Reynolds stresses.

Data-driven discovery of turbulent flow equations using physics-informed neural networks

In the field of fluid mechanics, traditional turbulence models such as those based on Reynolds-averaged Navier–Stokes (RANS) equations play a crucial role in solving numerous problems. However, their accuracy in complex scenarios is often limited due to inherent assumptions and approximations, as well as imprecise coefficients in the turbulence model equations. Addressing these challenges, our research introduces an innovative approach employing physics-informed neural networks (PINNs) to optimize the parameters of the standard k−ω turbulence model. PINNs integrate physical loss functions into the model, enabling the adaptation of all coefficients in the standard k−ω model as trainable parameters. This novel methodology significantly enhances the accuracy and efficiency of turbulent flow simulations, as demonstrated by our application to the flow over periodic hills. The two coefficients that have been modified considerably are σω and α, which correspond to the diffusion and production terms in the specific dissipation rate equation. The results indicate that the RANS simulation with PINNs coefficients (k−ω−PINNs simulation) improves the prediction of separation in the near-wall region and mitigates the overestimation of turbulent kinetic energy compared to the base RANS simulation. This research marks a significant advancement in turbulence modeling, showcasing the potential of PINNs in parameter identification and optimization in fluid mechanics.