Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

Purpose In recent years, the partially averaged Navier–Stokes (PANS) methodology has earned acceptability as a viable scale-resolving bridging method of turbulence. To further enhance its capabilities, especially for simulating separated flows past bluff bodies, this paper aims to combine PANS with a non-linear eddy viscosity model (NLEVM). Design/methodology/approach The authors first extract a PANS closure model using the Shih’s quadratic eddy viscosity closure model [originally proposed for Reynolds-averaged Navier–Stokes (RANS) paradigm (Shih et al., 1993)]. Subsequently, they perform an extensive evaluation of the combination (PANS + NLEVM). Findings The NLEVM + PANS combination shows promising result in terms of reduction of the anisotropy tensor when the filter parameter (fk) is reduced. Further, the influence of PANS filter parameter f on the magnitude and orientation of the non-linear part of the stress tensor is closely scrutinized. Evaluation of the NLEVM + PANS combination is subsequently performed for flow past a square cylinder at Reynolds number of 22,000. The results show that for the same level of reduction in fk, the PANS + NLEVM methodology releases significantly more scales of motion and unsteadiness as compared to the traditional linear eddy viscosity model (LEVM) of Boussinesq (PANS + LEVM). The authors further demonstrate that with this enhanced ability the NLEVM + PANS combination shows much-improved predictions of almost all the mean quantities compared to those observed in simulations using LEVM + PANS. Research limitations/implications Based on these results, the authors propose the NLEVM + PANS combination as a more potent methodology for reliable prediction of highly separated flow fields. Originality/value Combination of a quadratic eddy viscosity closure model with PANS framework for simulating flow past bluff bodies.

Liquid waste discharged from industrial outfalls is categorized into two major classes based on their density. One type is the effluent that has a higher density than that of the ambient water body. In this case, the discharged effluent has a tendency to sink as a negatively buoyant jet. The second type is the effluent that has a lower density than that of the ambient water body and is hence defined as a (positively) buoyant jet that causes the effluent to rise. Negatively/Positively buoyant jets are found in various civil and environmental engineering projects: discharges of desalination plants, discharges of cooling water from nuclear power plants turbines, mixing chambers, etc. This thesis investigated the mixing and dispersion characteristics of such jets numerically. In this thesis, mixing behavior of these jets is studied using a finite volume model (OpenFOAM). Various turbulence models have been applied in the numerical model to assess the accuracy of turbulence models in predicting the effluent discharges in submerged outfalls. Four Linear Eddy Viscosity Models (LEVMs) are used in the positively buoyant wall jet model for discharging of heated waste including: standard k-e, RNG k-e, realizable k-e and SST k-ω turbulence models. It was found that RNG k-e, and realizable k-e turbulence models performed better among the four models chosen. Then, in the next step, numerical simulations of 30˚ and 45˚ inclined dense turbulent jets in stationary ambient water have been conducted. These two angles are examined in this study due to lower terminal rise height for 30˚ and 45˚, which is very important for discharges of effluent in shallow waters compared to higher angles. Five Reynolds-Averaged Navier-Stokes (RANS) turbulence models are applied to evaluate the accuracy of CFD predictions. These models include two LEVMs: RNG k-e, and realizable k-e; one Nonlinear Eddy Viscosity Model (NLEVM): Nonlinear k-e; and two Reynolds Stress Models (RSMs): LRR and Launder-Gibson. It has been observed that the LRR turbulence model as well as the realizable k-e model predict the flow more accurately among the various turbulence models studied herein.

Reynolds-averaged Navier-Stokes (RANS) equations are presently one of the most popular models for simulating turbulence. Performing RANS simulation requires additional modelling for the anisotropic Reynolds stress tensor, but traditional Reynolds stress closure models lead to only partially reliable predictions. Recently, data-driven turbulence models for the Reynolds anisotropy tensor involving novel machine learning techniques have garnered considerable attention and have been rapidly developed. Focusing on modelling the Reynolds stress closure for the specific case of turbulent channel flow, this paper proposes three modifications to a standard neural network to account for the no-slip boundary condition of the anisotropy tensor, the Reynolds number dependence, and spatial non-locality. The modified models are shown to provide increased predicative accuracy compared to the standard neural network when they are trained and tested on channel flow at different Reynolds numbers. The best performance is yielded by the model combining the boundary condition enforcement and Reynolds number injection. This model also outperforms the Tensor Basis Neural Network in Ling et al. [Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J Fluid Mech. 2016;807:155–166] on the turbulent channel flow dataset.

Steady-state and scale-resolving simulations have been performed for flow in and around a model scramjet combustor flameholder. The cases simulated corresponded to those used to examine this flowfield experimentally using particle image velocimetry. A variety of turbulence models were used for the steady-state Reynolds-averaged simulations which included both linear and non-linear eddy viscosity models. The scale-resolving simulations used a hybrid Reynolds-averaged / large eddy simulation strategy that is designed to be a large eddy simulation everywhere except in the inner portion (log layer and below) of the boundary layer. Hence, this formulation can be regarded as a wall-modeled large eddy simulation. This effort was undertaken to formally assess the performance of the hybrid Reynolds-averaged / large eddy simulation modeling approach in a flowfield of interest to the scramjet research community. The numerical errors were quantified for both the steady-state and scale-resolving simulations prior to making any claims of predictive accuracy relative to the measurements. The steady-state Reynolds-averaged results showed a high degree of variability when comparing the predictions obtained from each turbulence model, with the non-linear eddy viscosity model (an explicit algebraic stress model) providing the most accurate prediction of the measured values. The hybrid Reynolds-averaged/large eddy simulation results were carefully scrutinized to ensure that even the coarsest grid had an acceptable level of resolution for large eddy simulation, and that the time-averaged statistics were acceptably accurate. The autocorrelation and its Fourier transform were the primary tools used for this assessment. The statistics extracted from the hybrid simulation strategy proved to be more accurate than the Reynolds-averaged results obtained using the linear eddy viscosity models. However, there was no predictive improvement noted over the results obtained from the explicit Reynolds stress model. Fortunately, the numerical error assessment at most of the axial stations used to compare with measurements clearly indicated that the scale-resolving simulations were improving (i.e. approaching the measured values) as the grid was refined. Hence, unlike a Reynolds-averaged simulation, the hybrid approach provides a mechanism to the end-user for reducing model-form errors.

This article reviews explicit algebraic Reynolds stress models and other non-linear eddy-viscosity turbulence models utilised in the context of Reynolds-Averaged Navier-Stokes simulations of turbulent flows. Since the 1990s, these modelling classes have become important intermediate classes between the linear eddy viscosity models and full Reynolds stress transport models. The derivation of explicit algebraic Reynolds stress models from the Reynolds stress transport models, and the required simplifications are discussed. The most important simplification is the weak-equilibrium assumption. Properties of the basic weak-equilibrium assumption and its extended form are discussed. Differences between explicit solutions based on complete tensor representation and those based on reduced representation as well as other non-linear eddy-viscosity models are pointed out. The scale-determining models are also discussed briefly.

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.

This paper is interested in the mathematical modeling and approximation of the turbulent flow with strong anisotropy for the case of flow in a rack station. Turbulent models based on Boussinesq eddy viscosity assumption are traditionally employed for the solution of these types of flows. Linear eddy viscosity model and the assumption of isotropic Reynolds stress tensor are typically used in many cases. This leads to certain limitations in the cases of either separated flow or flow with significant curvature of the mean flow. The application of full Reynolds stress model is suitable for this case but it introduces at least seven additional equations and the model has much higher demand for computing time, grid parameters and discretization schemes. Therefore the Explicit Algebraic Reynolds Stress Model (EARSM) is used here. This model is based on two equations k – omega turbulence model with the nonlinear eddy viscosity model and explicit terms for anisotropic parts of the Reynolds stress tensor. The computational requirements are similar to the standard k – omega model. The numerical and experimental results from PIV measurement are compared.

Nonlinear eddy viscosity models have received significant interest because of the shortcomings of linear eddy viscosity models used in the numerical simulation of flow around building structures. There are many kinds of nonlinear eddy viscosity models. This paper adopts one of them to simulate backward facing step flow, which aims to evaluate the properties of this turbulence model. Compared with linear eddy viscosity model and experimental measurements, the simulation of nonlinear eddy viscosity model gives better results, especially for the reattachment length and pressure coefficient values. For this reason, this nonlinear eddy viscosity model is more suitable for numerical simulation of complex flow around building structures than linear eddy viscosity model.

Dispersion of turbulent jets in shallow coastal waters has numerous engineering applications. The accurate forecasting of the complex interaction of these jets with the ambient fluid presents significant challenge and has yet to be fully elucidated. In this paper, numerical simulation of $$30{^\circ }$$ and $$45{^\circ }$$ inclined dense turbulent jets in stationary water have been conducted. These two angles are examined in this study due to lower terminal rise heights for $$30{^\circ }$$ and $$45{^\circ }$$ , this is critically important for discharges of effluent in shallow waters compared to higher angles. Mixing behavior of dense jets is studied using a finite volume model (OpenFOAM). Five Reynolds-Averaged Navier–Stokes turbulence models are applied to evaluate the accuracy of CFD predictions. These models include two Linear Eddy Viscosity Models: RNG $$ k-\varepsilon $$ , and realizable $$k-\varepsilon $$ ; one Nonlinear Eddy Viscosity Model: nonlinear $$k-\varepsilon $$ ; and two Reynolds Stress Models: LRR and Launder–Gibson. Based on the numerical results, the geometrical characteristics of the dense jets, such as the terminal rise height, the location of centerline peak, and the return point are investigated. The mixing and dilution characteristics have also been studied through the analysis of cross-sectional concentration and velocity profiles. The results of this study are compared to various advanced experimental and analytical investigations, and comparative figures and tables are discussed. It has been observed that the LRR turbulence model as well as the realizable $$k-\varepsilon $$ model predicts the flow more accurately among the various turbulence models studied herein.

Numerical simulations based on Reynolds-averaged Navier Stokes (RANS) models are still the work-horse tool in engineering design involving turbulent flows. Two decades ago, when LES started gaining popularity thanks to the increasing availability of computational resources, it was widely expected that it would have gradually replaced RANS methods in industrial CFD for decades to come. In the past two decades, however, while LES-based methods gained widespread applications and the earlier hope did not diminish, the predicted time when LES would replace RANS has been significantly delayed. Most industrial users are probably decades away from any routine use of scale resolving simulations, not to mention the cost, time and user skill it take to run these computations. In brief, RANS solvers, particularly those based on standard eddy viscosity models (e.g k-e, k-ω, S-A and k-ω SST) are expected to remain the workhorse in the CFD of high Reynolds number flows for decades. However, predictions from RANS simulations are known to have large discrepancies in many flows of engineering relevance, including those with swirl, pressure gradients, or mean streamline curvature. It is a consensus that the dominant cause for such discrepancies is the RANS-modeled Reynolds stresses. In light of the long stagnation in traditional turbulence modeling, researchers explored machine learning as an alternative to improve RANS modeling by leveraging data from highfidelity simulations. The goal is to make use of vast amounts of turbulent flows data, machine learning techniques and current understanding of turbulence physics to develop models with better predictive capabilities in the context of RANS simulations. Recently, in a seminar work, Ling et al (2016) developed a neural network architecture capable of embedding invariance properties into the Reynolds stress tensor predicted in output. Such a network, named the tensor basis neural network (TBNN), was applied to a variety of flow fields with encouraging results compared to both classical turbulence models and neural networks that do not preserve Galilean invariance. Yet, as in most data driven turbulence modelling approaches, the TBNN was used as a post-processing tool to correct the Reynolds stress tensor field predicted by a RANS simulation run with standard closure models. This means that, theoretically, the network can be applied only to correct the Reynolds stress tensor for the same RANS model on which it has been trained since, in general, different turbulence models yield different results depending on the flow type. Moreover, there is no physisical insight that suggests a relation between the RANS velocity gradients - used as inputs of the machine learning model - and the true Reynolds stress tensor. Differently, in this work a network with a similar architecture to the Ling’s one was trained and tested on a database of high-fidelity data of eight different flows to learn a functional mapping between the inputs of Pope’s General Eddy Viscosity Model and the anisotropic part of the Reynolds stress tensor. Then the network was embedded into a CFD RANS solver as a replacement of the standard closure model - and therefore called at every solver’s iteration. Lasty, the RANS solver with embedded TBNN was be tested on a canonical flow case - turbulent channel flow - to evaluate its performances. As for the organization of this work: in Chapter 1 further details on the data driven turbulence modelling will be given, RANS models and equations will be introduced and also an introduction to Neural Networks will be presented. In Chapter 2, it will be given a detailedexplanation of the RANS CFD solver and the neural network’s implementation. In Chapter 3, the method will be tested on a turbulent channel flow case and the results will be discussed. Lastly, in Chapter 4, some meaningful conclusions will be drawn.

Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

A linear eddy viscosity model and a non-linear eddy viscosity model were used to simulate a centrifugal pump at an off-design point. Compared with the inner flow of experimental results, turbulence model was accurate in the calculation of the rotating stall phenomenon in the runner of low flow conditions. The turbulence model and three dimensional (3-D), unsteady flow in a pump-turbine was used to study the instability of a high-head pump-turbine at no-load opening. Fluid coupling and dynamic mesh were used to simulate the change of runner's rotational speed. Stall phenomena in the runner caused by a large incidence angle of flow at the region between stay vanes and guide vanes were analyzed. Calculations based on different moments of inertia were accomplished in order to investigate the influence of moment of inertia to the stall phenomena in the runner. The Rayleigh criterion was introduced to analyze the instability of the stall phenomena in the runner. The explicit characteristics such as the flow-rate, rotational speed, torque of the runner etc. were analyzed. Results show that the moment of inertia has great influence on the stall phenomena in the runner. The Rayleigh criterion can be used to evaluate the instability of the pump-turbine at no-load opening. The flow in the pump-turbine at runaway speed is more stable when the moment of inertia increases. The study of a pump-turbine at no-load opening can provide a basic foundation for the improvement of S characteristics.

Kinetic theory offers a promising alternative to conventional turbulence modelling by providing a mesoscopic perspective that naturally captures non-equilibrium physics such as non-Newtonian effects. In this work, we present an extension and theoretical analysis of the kinetic model for incompressible turbulent flows developed by Chen et al. ( Atmosphere , 2023, vol. 14(7), p. 1109), constructed for unbounded flows. The first extension is to reselect a relaxation time such that the turbulent transport coefficients are obtained consistently and better align with well-established turbulence theory. The Chapman–Enskog (CE) analysis of the kinetic model reproduces the linear eddy-viscosity and gradient diffusion models for Reynolds stress and turbulent kinetic energy flux at the first order, and yields nonlinear eddy-viscosity and closure models at the second order. In particular, a previously unreported CE solution for turbulent kinetic energy flux is obtained. The second extension is to enable the model for wall-bounded turbulent flows with preserved near-wall asymptotic behaviours. This involves developing a low-Reynolds-number model incorporating wall damping effects and viscous diffusion, with boundary conditions enabling both viscous sublayer resolution and wall function application. Comprehensive validation against experimental and direct numerical simulation data for turbulent Couette flow demonstrates excellent agreement in predicting mean velocity profiles, skin friction coefficients and Reynolds shear-stress distributions, although the near-wall-normal stress anisotropy is underestimated. The results show that averaged turbulent flow behaves similarly to rarefied-gas flow at finite Knudsen number, capturing non-Newtonian effects beyond linear eddy-viscosity models. This kinetic model provides a physics-based foundation for turbulence modelling with reduced empirical dependence.

In the present work, the numerical simulation was adopted to resolve the problem of the turbulent forced convection coupled to heat conduction in a square cross-section duct. The governing equations for turbulent convection are the continuity, momentum, and energy equations. These equations are coupled to heat conduction comings through of four plates situated around of the channel flow. The plates among itself with thermal resistance of ideal contact are coupled. Two turbulence models to resolve the momentum equations and one model to resolve the energy equation were used. To determine the profiles of velocity, the models of turbulence, k-ε Non Linear Eddy Viscosity Model (NLEVM) and Reynolds Stress Model (RSM) were adopted. The fluid temperature field was determined from the model Simple Eddy Diffusivity (SED). The dimensionless energy equation was developed in a code of programming FORTRAN. The models have been validated in base the experimental and numerical results of literature. Finally, the results of this investigation allow evaluating the fluid temperature field for different square cross-sectional sections throughout of the main flow direction, which is influenced mainly by the temperature distribution at the wall.

In this study, the deep multi-layered Group Method of Data Handling (GMDH)-type neural network algorithm using principal component-regression analysis is applied to recognition problems of the right and left kidney regions. The deep multi-layered GMDH-type neural network algorithm can automatically organize the deep neural network architectures which have many hidden layers and these deep neural networks can identify the characteristics of very complex nonlinear systems. The architecture of the deep neural network with many hidden layers is automatically organized using the heuristic self-organization method, so as to minimize the prediction error criterion defined as Akaike's information criterion (AIC) or Prediction Sum of Squares (PSS). The heuristic self-organization method is a type of the evolutional computation. In this deep GMDH-type neural network, principal component-regression analysis is used as the learning algorithm of the weights in the deep GMDH-type neural network, and multi-colinearity does not occur and stable and accurate prediction values are obtained. This new algorithm is applied to the medical image recognitions of the right and left kidney regions. The optimum neural network architectures, which fit the complexity of the right and left kidney regions, are automatically organized and the right and left kidney regions are automatically recognized and extracted by the organized deep GMDH-type neural networks. The recognition results are compared with the conventional sigmoid function neural network trained using back propagation method and it is shown that this deep GMDH-type neural networks are useful for the medical image recognition problems of the right and left kidney regions.