Non-linear Eddy-viscosity Models Research Articles

Generalized field inversion strategies for data-driven turbulence closure modeling

Most data-driven turbulence closures are based on the general structure of nonlinear eddy viscosity models. Although this structure can be embedded into the machine learning algorithm and the Reynolds stress tensor itself can be fit as a function of scalar- and tensor-valued inputs, there exists an alternative two-step approach. First, the spatial distributions of the optimal closure coefficients are computed by solving an inverse problem. Subsequently, these are expressed as functions of solely scalar-valued invariants of the flow field by virtue of an arbitrary regression algorithm. In this paper, we present two general inversion strategies that overcome the limitation of being applicable only when all closure tensors are linearly independent. We propose to either cast the inversion into a constrained and regularized optimization problem or project the anisotropy tensor onto a set of previously orthogonalized closure tensors. Using the two-step approach together with either of these strategies then enables us to quantify the model-form error associated with the closure structure independent of a particular regression algorithm. Eventually, this allows for the selection of the a priori optimal set of closure tensors for a given, arbitrary complex test case.

Numerical analyses of water swirls in aquaculture tanks

Water swirls were observed in both model and prototype aquaculture tanks with bottom drains. This phenomenon was seldom replicated in studies of flow fields of water in aquaculture tanks with computational fluid dynamics. A commonality of these studies is that the linear eddy-viscosity turbulence models were employed. This study delves into the water swirls from the aspect of the generation of the turbulence kinetic energy; both the linear and nonlinear eddy-viscosity turbulence models are used, as well as a Reynolds stress turbulence model. A novel identification method is utilized to pinpoint vortex structures. Numerical results affirm the outstanding performance of the Reynolds stress model to estimate the flow fields of water in aquaculture tanks. Both the linear and nonlinear eddy-viscosity models overestimate the kinetic turbulence energy in the stagnation zone of swirls. The overestimation can be restrained by including the effect of the streamline curvature correction (SCC) to the turbulence production term. Even so, the SCC does not help to improve the estimating accuracy between the Reynolds stress tensor and the mean rate-of-strain tensor. While the nonlinear model offers some improvements, the Reynolds stress model provides a more effectvie solution. After that, we calculate the flow fields of water in aquaculture tanks with different inlet–outlet configurations with the Reynolds stress turbulence model. Both the vertical and horizontal scales of water swirls increase with inlet velocities and Froude numbers, where the Froude number is based on the bottom draining velocity and the water depth above the bottom drain. The vertical distributions of the water residence time in aquaculture tanks are shown to improve the design of tanks.

Integrating NSGA-II and CFD for enhanced urban airflow prediction: Recalibration of closure coefficients for a nonlinear eddy viscosity model

Accurately predicting urban environmental airflows using the Reynolds-averaged Navier‒Stokes (RANS) approach presents significant challenges due to unsolved closure issues. Previous attempts to address these limitations through coefficient adjustments were constrained by isotropic assumptions, resulting in limited generalizability and reliability, especially for complex building configurations. In this study, we proposed a recalibration approach within a multi-objective optimization framework. Nondominant sorting genetic algorithm-II (NSGA-II) and computational fluid dynamics (CFD) numerical computations were utilized to optimize the closure coefficients of a nonlinear eddy viscosity (NLEV) model, with a focus on improving the accuracy of the mean velocity, turbulent kinetic energy, and airflow characteristic predictions. To ensure the robustness of the model, appropriate objective functions were carefully defined. The proposed approach was evaluated using three-dimensional (3D) building cases related to urban prototypes, and the results demonstrated the necessity of multi-objective optimization. Our findings indicated that the trade-off solution on the Pareto front consistently outperformed the baseline and single-objective optimal versions across different types of urban airflow, demonstrating superior generalizability. The simulation results of this solution exhibited closer agreement with the wind tunnel experimental data and provided more accurate distributions of the time-averaged values and airflow characteristics. The normalized root mean square errors are reduced by about 33 %, 52 %, 35 % and 9 % in velocity, and 31 %, 25 %, 24 % and 11 % in turbulent kinetic energy, respectively. These results underscore the effectiveness of the multi-objective optimization framework in addressing closure issues and improving the prediction accuracy of complex urban environmental airflows.

A tensor basis neural network-based turbulence model for transonic axial compressor flows

Traditional turbulence models encounter limitations when simulating intricate flows within transonic axial compressors. In contrast, recent advancements in machine learning turbulence models have demonstrated enhanced potential in refining the precision of turbulence modeling. Notably, the tensor basis neural network (TBNN) methodology has successfully developed non-linear eddy viscosity turbulence models. These models possess the capability to capture the anisotropic characteristic of Reynolds stress. However, applying machine learning non-linear eddy viscosity models to aircraft turbomachinery remains relatively infrequent. Current research mainly focuses on linear eddy viscosity turbulence models based on the Boussinesq hypothesis that presupposes isotropy in Reynolds stress. In this work, we introduce a non-linear eddy viscosity turbulence model, denoted as the k-ω-SST-TBNN model based on the TBNN framework. This model has been employed to simulate the transonic axial compressor NASA Rotor 37. The TBNN is trained using large eddy simulation (LES) results datasets. The importance of input scalar features is analyzed using the random forest method, from which significant and low-degree variables are selected as inputs for the TBNN. Furthermore, this study proposes including the turbulent Mach number as one of the extra features, representing fluid compressibility, thereby extending the computational mass flow rate range of compressor. Additionally, this paper proposes a method involving the weighted average of Reynolds stress, combining the high-precision but less robust TBNN predictions with results based on the Boussinesq assumption to enhance the turbulence model's robustness. The trained k-ω-SST-TBNN model undergoes validation on Rotor 37, where it exhibits a marked improvement in the prediction of overall performance, tip-gap vortex, and radial distribution of flow parameters. The model also displays a commendable capacity for generalization.

Physical interpretation of neural network-based nonlinear eddy viscosity models

Neural network-based turbulence modeling has gained significant success in improving turbulence predictions by incorporating high–fidelity data. However, the interpretability of the learned model is often not fully analyzed, which has been one of the main criticisms of neural network-based turbulence modeling. Therefore, it is increasingly demanding to provide physical interpretation of the trained model, which is of significant interest for guiding the development of interpretable and unified turbulence models. The present work aims to interpret the predictive improvement of turbulence flows based on the behavior of the learned model, represented with tensor basis neural networks. The ensemble Kalman method is used for model learning from sparse observation data due to its ease of implementation and high training efficiency. Two cases, i.e., flow over the S809 airfoil and flow in a square duct, are used to demonstrate the physical interpretation of the ensemble-based turbulence modeling. For the flow over the S809 airfoil, our results show that the ensemble Kalman method learns an optimal linear eddy viscosity model, which improves the prediction of the aerodynamic lift by reducing the eddy viscosity in the upstream boundary layer and promoting the early onset of flow separation. For the square duct case, the method provides a nonlinear eddy viscosity model, which predicts well secondary flows by capturing the imbalance of the Reynolds normal stresses. The flexibility of the ensemble-based method is highlighted to capture characteristics of the flow separation and secondary flow by adjusting the nonlinearity of the turbulence model.

Testing and performance analysis of an integrated electromagnetic and hydraulic retarder for heavy-duty vehicles

Hydraulic retarder and eddy current retarder are important auxiliary braking component, which installed on heavy-duty vehicles. However, the hydraulic retarder has poor braking performance at low speeds, while the eddy-current retarder presents serious thermal recession at high speed. To overcome these issues and ensure safe driving during downhill conditions, a novel Integrated Electromagnetic and Hydraulic Retarder (IEHR) is proposed. The design includes a rotor with blades, double salient poles, and a coil built in a C-type stator. The electromagnetic field model of the IEHR is established and analyzed using the Finite Element Method. Moreover, the nonlinear eddy viscosity model of the flow field is generated and analyzed using the Computational Fluid Dynamics. The Blend braking control strategy is adopted to distribute the braking torque at high speed and low speed The IEHR performance was tested at a test bench. The test results show that the calculated braking characteristics of the electromagnetic and hydraulic subsystems were reasonable agreement with the measured results. The road test was carried out on a vehicle with a total mass of 49 tons. The deceleration and torque peaked at 0.75 m/s2 and 3620 N m, respectively. These results indicated that the IEHR has better performance than conventional hydraulic retarders and eddy current retarder, and is more suitable for heavy vehicles.

Investigation on performance of a marine centrifugal pump with broken impeller

As an important auxiliary equipment for ships, the stable operation of centrifugal pumps has a significant impact on the safe operation of ships, so monitoring the operation of marine centrifugal pumps is particularly urgent. In this paper, a marine centrifugal pump is studied for four cases: the normal state, the initial stages of broken state, the intermediate stages of broken state, and the late stages of broken state. The Delayed Detached Eddy Simulation (DDES) with a nonlinear eddy viscosity model is used for the nonstationary calculation of the whole channel, aiming to study the performance of the centrifugal pump in the broken state. A comprehensive analysis of the hydrodynamic characteristics of the centrifugal pump, such as external characteristics, flow situation, pressure pulsation, and flow analysis based on the proper orthogonal decomposition (POD) method, was carried out. Taken together, these results suggest that as the number of broken blades increases, the corresponding head and efficiency gradually decrease. Blade loss mainly affects the uniformity and stability of the flow field, so that the centrifugal pump performance loss increases, which in turn causes the centrifugal pump head and efficiency to decline. It also leads to an increase in radial force during operation, which in turn causes an increase in the amplitude of pump vibration. The results of the study provide some references for the fault diagnosis of centrifugal pumps.

Ensemble Kalman method for learning turbulence models from indirect observation data

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.

A realizable and scale-consistent data-driven non-linear eddy viscosity modeling framework for arbitrary regression algorithms

A data-driven modeling framework for non-linear eddy viscosity models is presented. In contrast to the majority of similar approaches, it splits the multivariate regression problem into a set of univariate problems and is, furthermore, independent of a specific machine learning algorithm. Instead of inferring the closure equation from high-fidelity data as a whole, the coefficients of the tensor polynomial are learned individually. The target variables are obtained from an efficient field inversion procedure by virtue of successive tensor projections. The hypotheses for each closure coefficient are then fitted separately by employing an arbitrary regression technique. Ordinary curve fitting, neural networks and gene expression programming are considered as examples. The robustness of the model against an extrapolation and the stability of the solver are promoted by coefficient limiters and a full barycentric realizability correction. Additionally, turbulent scale consistency is guaranteed by a k-corrective frozen-RANS method, which is the subject of a companion paper (Mandler and Weigand, 2022). The proposed strategy leads to models which are well suited for the application to the same class of flows they were inferred from, namely separated channel flows. As proven by an extensive extrapolation study, the resulting neuralSST model is robust against geometry and Reynolds number modifications provided the type of flow does not drastically change. It consistently outperforms both the shear stress transport (SST) and a more complex elliptic-blending model and agrees well with the reference data.

A Delayed Detached Eddy Simulation Model for the Simulation of Complex Turbulent Flow

A new turbulent model based on Delayed Detached Eddy Simulation (DDES) with non-linear eddy viscosity model (NLEVM) was developed to predict the complex turbulent flow. The numerical simulation of the triangular cylinder and the centrifugal pump was carried out to investigate the ability and applicability of the DDES model based on NLEVM (DDES_NL). Compared to the turbulent model based on the eddy viscosity model, the computational results of the triangular cylinder showed the advantage of the non-linear eddy viscosity modification in the DDES_NL model which can improve the accuracy of the prediction in the flow phenomenon with a relatively simple turbulence structure. Regrettably, some small-scale turbulent structures among those still cannot be captured accurately. The numerical simulation of the centrifugal pump predicted by the DDES_NL model shows more abundant flow structures and gets close to the realistic statistical characteristics. It also proves the good applicability of the DDES_NL model in complex flow. This study aims to contribute to the growing area of turbulence modeling by exploring it.

Thermal, thermodynamic and exergoeconomic investigation of a parabolic trough collector utilizing nanofluids

The exploitation of solar energy facilitates the renewable energy paradigm. In this regard, parabolic trough collectors (PTC) are considered as a useful set-up to absorb solar energy. Simultaneous study of thermal, thermodynamic, and exergoeconomic performance of PTC systems paves the way for designers and manufacturers to not only have a better insight into understanding the underlying concepts about the operation of PTC systems but also to find the most effective and cost-effective circumstances. This study aims at analyzing a practical PTC system by considering an evacuated absorber tube with glass cover, non-uniform heat flux, and taking into account the convective and radiative heat losses. Obtained results demonstrate that employing the glass cover, especially in the low Reynolds numbers regime could remarkably reduce the heat losses up to 22%. The present analysis indicated that considering oil temperature-dependent properties reduced the friction factor around 94 % in the low Reynolds numbers regime compared to that of the cases based on constant properties. In this work a CFD code in the OpenFOAM software was developed to simulate both laminar and turbulent regimes with Lien cubic k-∊ model (non-linear eddy viscosity model) by adding three types of nanoparticles (Al2O3,Cu,andSWCNT) individually into the synthetic oil. Moreover, the Buongiorno’s model (BGM) which considers Brownian, thermophoresis, and turbulence diffusion phenomena was implemented to model the nanoparticles and base fluid interactions. It was found that adding nanoparticles and increasing the Reynolds number have no substantial impact on thermal efficiency of the system, whereas an optimum Reynolds number was found for exergy efficiency and net profit per unit transferred heat load (ηp) of the system. Although injecting SWCNT into pure oil led to augmentation of average convective heat transfer coefficient, PEC, and exergy efficiency, it was found not to be cost-effective. It was established that the obtained results by homogenous model (single-phase model) and BGM were roughly the same, with relative difference of less than 3.1 and 2.4% for the average Nusselt number and friction factor, respectively.

Turbulence model assessment and heat transfer phenomena inside a rectangular channel under forced and mixed convection

A very high-temperature gas-cooled reactor (VHTR) is an attractive option for hydrogen production due to its high operating temperature. The reactor cavity cooling system (RCCS), a passive safety system of the VHTR, surrounds the reactor vessel with rectangular riser channels and removes decay heat from it. Under emergency operation conditions such as a decrease in the flow rate of the RCCS, the flow condition inside the RCCS riser can become a mixed convection condition accompanying heat transfer deterioration. To design and operate the VHTR with reliable safety, accurate predictions for various convective heat transfer conditions in RCCS riser channels are required. In this study, by comparing the results of an air flow visualization experiment, the predictability of the turbulence models was assessed for the convective heat transfer and complex flow characteristics near the corner region of the rectangular duct confirmed from the experimental study. In this paper, the heat removal rate through the test section of the visualization experiment was quantified under various convective heat transfer conditions, and local temperature measurement data from the experiment were presented to describe the heat transfer deterioration phenomena inside a rectangular riser in turbulent mixed convection. Linear and nonlinear eddy viscosity models were evaluated by comparing the heat transfer coefficient, local temperature, local flow structure including secondary flow, and turbulence quantities obtained from the experiment. Furthermore, improvements to turbulence models for enhancing the prediction of mixed convection heat transfer in rectangular riser ducts were proposed.

Uncertainty quantification for data-driven turbulence modelling with Mondrian forests

Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. Such approaches generally aim to improve the modelled Reynolds stresses by leveraging data from high fidelity turbulence resolving simulations. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is essential since the predictive capability of a data-driven model diminishes when predicting physics not seen during training. In this work, we explore the suitability of Mondrian forests (MF's) for data-driven turbulence modelling. MF's are claimed to possess many of the advantages of the commonly used random forest (RF) machine learning algorithm, whilst offering principled uncertainty estimates. An example test case is constructed, with a turbulence anisotropy constant derived from high fidelity turbulence resolving simulations. A number of flows at several Reynolds numbers are used for training and testing. MF predictions are found to be superior to those obtained from a linear and non-linear eddy viscosity model. Shapley values, borrowed from game theory, are used to interpret the MF predictions. Predictive uncertainty is found to be large in regions where the training data is not representative. Additionally, the MF predictive uncertainty is found to exhibit stronger correlation with predictive errors compared to an a priori statistical distance measure, which indicates it is a better measure of prediction confidence. The MF predictive uncertainty is also found to be better calibrated and less computationally costly than the uncertainty estimated from applying jackknifing to random forest predictions. Finally, Mondrian forests are used to predict the Reynolds discrepancies in a convergent-divergent channel, which are subsequently propagated through a modified CFD solver. The resulting flowfield predictions are in close agreement with the high fidelity data. A procedure for sampling the Mondrian forests' uncertainties is introduced. Propagating these samples enables quantification of the uncertainty in quantities of interest such as velocity or a drag coefficient, due to the uncertainty in the Mondrian forests' predictions. This work suggests that uncertainty quantification can be incorporated into existing data-driven turbulence modelling frameworks by replacing random forests with Mondrian forests. This would also open up the possibility of online learning, whereby new training data could be added without having to retrain the Mondrian forests.

Predicting turbulent flows in butterfly valves with the nonlinear eddy viscosity and explicit algebraic Reynolds stress models

The development of turbulence modeling is crucial for the numerical prediction of the flow behavior, especially for separation, stagnation, reattachment, recirculation, and streamline curvature of the flow through complex structures. In this study, the capability of turbulence models was estimated for predicting the flow in a butterfly valve. The explicit algebraic Reynolds stress model (EARSM) and nonlinear eddy viscosity model (NLEVM) were evaluated in terms of the velocity profile, turbulence intensity, and Reynolds stress, and their results were compared with those of the standard k–ε and renormalization group (RNG) models. A numerical validation was conducted with the flow past a backward-facing step as the benchmark test. Comparison with the validation test showed that the NLEVM accurately predicted the reattachment length. For the flow in a butterfly valve, the NLEVM and EARSM indicated a smaller velocity increase than the standard k–ε and RNG models in the recirculation area near the valve region. The NLEVM and EARSM demonstrated an ability to predict anisotropic stresses with a dominant stress value near the valve region.

A priori tests of eddy viscosity models in square duct flow

We carry out a priori tests of linear and nonlinear eddy viscosity models using direct numerical simulation (DNS) data of square duct flow up to friction Reynolds number {text {Re}}_tau =1055. We focus on the ability of eddy viscosity models to reproduce the anisotropic Reynolds stress tensor components a_{ij} responsible for turbulent secondary flows, namely the normal stress a_{22} and the secondary shear stress a_{23}. A priori tests on constitutive relations for a_{ij} are performed using the tensor polynomial expansion of Pope (J Fluid Mech 72:331–340, 1975), whereby one tensor base corresponds to the linear eddy viscosity hypothesis and five bases return exact representation of a_{ij}. We show that the bases subset has an important effect on the accuracy of the stresses and the best results are obtained when using tensor bases which contain both the strain rate and the rotation rate. Models performance is quantified using the mean correlation coefficient with respect to DNS data {widetilde{C}}_{ij}, which shows that the linear eddy viscosity hypothesis always returns very accurate values of the primary shear stress a_{12} ({widetilde{C}}_{12}>0.99), whereas two bases are sufficient to achieve good accuracy of the normal stress and secondary shear stress ({widetilde{C}}_{22}=0.911, {widetilde{C}}_{23}=0.743). Unfortunately, RANS models rely on additional assumptions and a priori analysis carried out on popular models, including k–varepsilon and v^2–f, reveals that none of them achieves ideal accuracy. The only model based on Pope’s expansion which approaches ideal performance is the quadratic correction of Spalart (Int J Heat Fluid Flow 21:252–263, 2000), which has similar accuracy to models using four or more tensor bases. Nevertheless, the best results are obtained when using the linear correction to the v^2–f model developed by Pecnik and Iaccarino (AIAA Paper 2008-3852, 2008), although this is not built on the canonical tensor polynomial as the other models.

Analysis and modelling of the relation between the shear rate and Reynolds stress tensors in transitional boundary layers

A non-linear eddy-viscosity transition model is presented, tuned by a large experimental data set describing transitional boundary layers. Data have been acquired by TR-PIV on a flat plate placed in a 2D converging-diverging channel with variable opening angle, allowing variation of the adverse pressure gradient, the free-stream turbulence intensity and the flow Reynolds number. Overall, 48 different combinations of these flow parameters encompass different modes of transition from bypass to separated-flow mechanisms, thus allowing fine tuning of the model, spanning significantly different conditions. The model is tuned locally as a function of the turbulent kinetic energy, a Reynolds number based on the wall distance and the ℓ2-norm of the shear rate tensor. A first correlation determines the rotation for alignment of the principal axes of the shear and stress tensors. By a second correlation, the eigenvalues of the stress tensor are obtained. The non-linear eddy-viscosity relation reproduces the anisotropy of the turbulence field observed for both bypass and separated-flow transitional cases. The relation has been applied to another experimental data set that did not participate to the fitting of the model and that is characterized by a different range of Reynolds number and turbulence intensity and a significantly stronger adverse pressure gradient with respect to the tuning dataset. Such application further strengthens the capability of the proposed correlations, that can easily be implemented in existing CFD solvers.

Assessment of RANS models in predicting mixing flow induced by split-type vanes in rod bundle

Turbulence model assessment is carried out based on particle image velocimetry (PIV) measurement data in a 5 × 5 rod bundle with split mixing vane for the nonlinear eddy viscosity models (NLEVMs) and the Reynolds stress transport (RST) model. The NLEVMs are applied with the two-layer wall treatment. The RST model is closed with the two-layer linear pressure strain model. The validation shows that the RST model is able to predict the mean flow vortices in the subchannel. The quadratic NLEVM predicts the similar, but smaller, vortices, while the cubic NLEVM fails to show the vortices. Comparing with the cross-flow velocities, prediction of axial velocity is relatively unsatisfactory. In general, the lateral normal Reynolds stresses are overpredicted by the models, while the axial normal Reynolds stress is underpredicted. In the near-wake region the lateral normal Reynolds stress can be reasonably predicted with the models, but the axial normal Reynolds stress cannot be predicted in a good accuracy. With the cubic term, the NLEVM predicts smaller normal Reynolds stresses, and especially small value of lateral component in the subchannel center. The NLEVMs which depend on local parameters, show relatively good prediction of the Reynolds stress in the subchannel center where the Reynolds stress is less influenced by the convection and diffusion effect. Invariant analysis of Reynolds stress anisotropy obtained with the models are carried out based on the Lumley triangle. Comparison of invariant analysis with large eddy simulation results of (Busco et al., 2019) indicates that the RST model can predict the far-wake turbulence anisotropy. The cubic NLEVM predicts non-realizable Reynolds stress close to the mixing vanes, while the quadratic model predicts inadequate turbulence anisotropy. All the models cannot predict the disk-like turbulence near the mixing vanes, which can be main reason of inaccurate prediction in the near-wake region.

The use of a general convected time derivative to compute the Reynolds stress tensor for a compressible turbulent flow

This paper deals with the turbulence closure problem associated with the Reynolds-averaged Navier–Stokes equations for compressible flows. Due to the known limitations faced by the turbulent closure model based on the Boussinesq hypothesis, additional tensors are needed in order to capture the complete Reynolds stress tensor. Models such as this are in the group of the so-called nonlinear eddy viscosity models. The methodology here presented consists mainly in translating the coefficients obtained via orthogonal projection of the Reynolds stresses that are calculated from a large eddy simulation onto a given basis of tensors. In addition, these coefficients can be seen as turbulent parameters which have analogous interpretation in a class of non-Newtonian fluids. The proposed closure, expressed in terms of the mean strain-rate and its Gordon–Schowalter convected derivative, has a concise and clear physical interpretation. In addition to the turbulent viscosity, parameters associated with the Gordon–Schowalter convected derivative are incorporated to help the interpretation of the nonlinear turbulent model. The methodology is applied to a compressible flow through a narrow channel with a series of holes of cylindrical shape. This configuration is representative of what is generally found in a Hole-Pattern turbomachinery seal. Different number of holes are tested in order to evaluate the influence of neighborhood holes in the analysis. The comparison between the Boussinesq hypothesis and the extended model shows how the latter one outperforms the former. Even though the nonlinear model has shown some limitations, there are still tensorial subspaces of the Reynolds stress tensor obtained from the LES approach that the present nonlinear model could not reach. In addition, the physical coefficients did not present strong similarities for the different configurations tested, what can be a challenge for constructing a complete closure model.

A new non-linear RANS model with enhanced near-wall treatment of turbulence anisotropy

A new ω-based non-linear eddy-viscosity model is proposed. It was developed based on the original k−ω model and formulated using a quadratic stress-strain relation for the Reynolds stress tensor, with an added realisability condition. For enhanced treatment of near-wall turbulence anisotropy, a formulation that scales only with the turbulent Reynolds number is proposed for the first time. The new model has been implemented in the open-source Computational Fluid Dynamics (CFD) package OpenFOAM and validated against plane channel flow, a zero-pressure-gradient flat plate, and a U-bend curved channel configuration. To further assess the performance of the model for more complex geometries, it has been tested on configurations relevant to automotive applications. Overall, the new model outperforms the standard k−ω model. For example, on a curved channel, improved predictions for the minimum pressure and maximum skin friction of approximately 50% are obtained. Improved predictions are also obtained for quantities of practical engineering relevance, such as the pressure distribution along the wall of a sudden expansion diffuser, a configuration used to inform the design of automotive exhaust systems. This demonstrates that the proposed model has important practical applications for internal flows where anisotropic turbulence effects dominate.

Towards a general data-driven explicit algebraic Reynolds stress prediction framework

We propose a method to extend linear eddy-viscosity models up to a second order formulation using machine learning algorithms. The method is applicable to any turbulence model equipped with a transport equation for the wall-normal component of the Reynolds stress tensor v2 and is based on our ITALO framework (Sotgiu et al., 2018). We implement and test our extension using as base the elliptic-blending k−ϵ−ϕ−α linear eddy-viscosity model of Billiard and Laurence (2011). Neural networks are used to derive scalar functional forms for the nonlinear part of the Reynolds stress constitutive equation. The networks have been trained with data from a planar channel at Reτ=1020 and a two-dimensional periodic channel perturbed with corrugations at Reb=32,156. The test cases comprise channel flows at Reynolds numbers ranging from Reτ=180 to Reτ=1020, a periodic square duct at Reb=10,320, and a periodic ribbed channel at Reb=37,200. A comparison with other nonlinear eddy-viscosity models is reported. The new proposed model shows very good agreement with the reference data.