Reynolds Stress Transport Research Articles

Particle Image Velocimetry And Scaling Of Mach 0.3 And 1.25 Axisymmetric Turbulent Jets

The influence of compressibility on the scaling of the Reynolds stress transport (RST) budgets is investigated for Mach 0.3 and Mach 1.25 free axisymmetric turbulent jets using particle image velocimetry (PIV). It is proposed that the RST equations can be shown in self-preserving form scaled by the local streamwise evolving spreading rate, b’. It is shown that b’ is a function of the local convective Mach number, M_c. Compressibility effects on the suppression of the Reynolds stresses and higher turbulence moments are detailed. From self-preservation principles, the budgets scaled by powers of b’. In this form, most terms in the budgets are scaled appropriately indicating self-preservation. Terms which are not properly scaled are isolated then which allows for insight on effects of compressibility on turbulence.

A validated numerical methodology for flow-induced vibration of a semi-spherical end cantilever rod in axial flow

This study presents a simulation method for turbulent flow-induced vibrations of cantilever rods with a semi-spherical end exposed to axial flow, a configuration investigated for the first time. This simulation strategy has been developed using solids4Foam, a toolkit for the open-source package OpenFOAM, which uses the finite-volume approach. The fluid and solid domain equations are solved separately. Coupling is achieved with the Interface Quasi-Newton Inverse Least-Squares (IQN-ILS) algorithm. The mean flow is described by the unsteady Reynolds-averaged Navier–Stokes equations. Turbulence is modeled through either the stress-transport model of Launder, Reece, and Rodi or the effective-viscosity k–ω shear stress transport model, both with the wall-function approach accounting for near-wall turbulence. The methodology is validated using experimental data produced during this study. The simulations show good agreement with the measured values of the oscillation amplitude and frequency for both flow directions (toward rod free-end and away from it). Turbulence model comparisons show that (a) Reynolds stress transport models are necessary to reproduce the vibration amplitude and (b) wall functions enable the simulations to be completed in realistic time scales. The significance to the fluid–solid-interaction (FSI) process of a so far overlooked (with the exception of a couple of recent studies) dimensionless number, the ratio of the flow dynamic pressure to the rod's Young's modulus of elasticity, is also explored. Simulations, which decouple the variation of this dimensionless number from that of the Reynolds number, demonstrate this number's strong effect on the vibration amplitude. This finding is important to the contact of further FSI studies and the scaling of FSI data.

Interplay of scales during the spatial evolution of energy-containing motions in wall-bounded turbulent flows

This article extends the previous investigation of the spatial evolution of energy-containing motions in wall-bounded turbulent flows (Kannadasan et al., J. Fluid Mech., vol. 955, 2023, R1) by examining their scale-interactions through spectral analysis based on the spanwise scale decomposition of turbulent kinetic energy and the Reynolds stress transport equation. The energy-containing motions located at the inflow of a turbulent channel flow are artificially removed and the interscale transport mechanisms involved in their spatial evolution are studied. This scale interaction analysis reveals the presence of a significant inverse transfer of streamwise Reynolds stress from the near-wall streaks to larger scales in the spatial evolution of energy-containing motions. This transfer is due to the spanwise variation of streamwise velocity fluctuations, represented by $\partial u'/\partial z$ , which is the primary mechanism of streak instability. The analysis presented in this study also shows that the inverse cascade of spanwise energy may correspond to the regeneration of streamwise vortices in the process of reactivating the self-sustaining mechanism in the spatial evolution of energy-containing motions.

A data‐driven turbulence modeling for the Reynolds stress tensor transport equation

AbstractThe long lasting demand for better turbulence models and the still prohibitively computational cost of high‐fidelity fluid dynamics simulations, like direct numerical simulations and large eddy simulations, have led to a rising interest in coupling available high‐fidelity datasets and popular, yet limited, Reynolds averaged Navier–Stokes simulations through machine learning (ML) techniques. Many of the recent advances used the Reynolds stress tensor or, less frequently, the Reynolds force vector as the target for these corrections. In the present work, we considered an unexplored strategy, namely to use the modeled terms of the Reynolds stress transport equation as the target for the ML predictions, employing a neural network approach. After that, we solve the coupled set of governing equations to obtain the mean velocity field. We apply this strategy to solve the flow through a square duct. The obtained results consistently recover the secondary flow, which is not present in the baseline simulations that used the model. The results were compared with other approaches of the literature, showing a path that can be useful in the seek of more universal models in turbulence.

Comparative predictions of turbulent non-isothermal flow of a viscoplastic fluid with yield stress

RANS simulation of turbulent non-isothermal flow in a pipe by transition of Newtonian fluid into a viscoplastic non-Newtonian fluid is carried out. Carrier phase turbulence was modeled using two-equation isotropic k‒ε˜ – model and Reynolds stress transport model. Results of calculations of Newtonian and non-Newtonian power-law fluids were compared with data of DNS calculations of other authors. Comparisons with data from other works for isothermal Schwedoff-Bingham fluid were performed in this work using k‒ε˜ – model. Satisfactory agreement was obtained with data from other studies for the axial averaged velocity and turbulent kinetic energy radial profiles (the difference is up to 15 %). Reynolds stress transport model showed significant anisotropy between streamwise and transverse velocity fluctuations (up to several times) and good agreement with DNS results of other authors. Averaged and pulsation profiles express the indicated transformation of the non-isothermal turbulent flow.

Large eddy simulation of particle hydrodynamic characteristics in a dense gas-particle bubbling fluidized bed

A subgrid scale particle kinetic turbulent energy model coupled with a second order moment model is developed to describe the particle hydrodynamic characteristics in dense gas-particle fluidized bed. Anisotropic behaviors of particle dispersions and interactions between gas and particle are fully revealed by means of two-phase Reynolds stresses transport eqs. A four-way coupling strategy for the consideration of particle-particle collisions is modeled by frictional granular temperature transport equation. Simulation results by large eddy simulation are validated by experimental measurement with acceptable agreement. Heterogenous flow structures, anisotropic characteristics of particle dispersions and wider-size distribution behaviors of bubbles are indicated. At the height of 0.2 m, the largest dominant frequency of porosity is 25.0 Hz at lateral 0.375 cm away from center, it is approximately 5.0 times larger than that of lateral 9.375 cm. Compared to 0.45 m, the standard deviations of 82.5 cm/s at lateral region is approximately 1.5 times larger than that of center region.

Two self-similar Reynolds-stress transport models with anisotropic eddy viscosity.

Two Reynolds-averaged Navier-Stokes models with full Reynolds-stress transport (RST) and tensor eddy viscosity are presented. These new models represent RST extensions of the k-2L-a-C and k-ϕ-L-a-C models by Morgan [Phys. Rev. E 103, 053108 (2021)10.1103/PhysRevE.103.053108; Phys. Rev. E 105, 045104 (2022)10.1103/PhysRevE.105.045104]. Self-similarity analysis is used to derive constraints on model coefficients required to reproduce expected growth parameters for a variety of canonical flows, including Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) mixing layers. Both models are then applied in one-dimensional simulation of RT and KH mixing layers, and the expected self-similar growth rates and anisotropy are obtained. Next, models are applied in two-dimensional simulation of the so-called "tilted rocket rig" inclined RT experiment [J. Fluids Eng. 136, 091212 (2014)10.1115/1.4027587] and in simulation of a shock-accelerated localized patch of turbulence. It is found that RST is required to capture the qualitative growth of the shock-accelerated patch, and an anisotropic eddy viscosity provides substantial improvement over a Boussinesq treatment for the tilted rocket rig problem.

Analysis of improved digital filter inflow generation methods for compressible turbulent boundary layers

We propose several enhancements to improve the accuracy and performance of the digital filter turbulent inflow generation technique and assess their efficacy in the context of wall-resolved large-eddy simulations of a compressible turbulent boundary layer. Improvements of accuracy include a more realistic correlation function for the transversal directions, target length scales that vary with wall-distance, and a counter-intuitive approach that involves the suppression of streamwise velocity fluctuations at the inflow. For improving the computational performance, we propose to generate the inflow data in parallel in single precision and at a prescribed time interval based on the turbulence time scale, and not at every time-step of the simulation. Based on the results of 7 wall-resolved large-eddy simulations, we find that the new correlation functions and the considered performance improvements are beneficial and therefore desired. Suppressing streamwise velocity fluctuations at the inflow leads to the fastest relaxation of the pressure fluctuations; however, this approach increases the adaptation length defined in terms of compliance with the von Kármán integral equation. The adaptation length can be shortened by artificially increasing the wall-normal Reynolds stresses, thereby preserving the desired turbulence kinetic energy level. A detailed inspection of the Reynolds stress transport budgets reveals that the observed spurious spatial transients are largely driven by pressure-related terms. For instance, increased values of u′p′¯ are found throughout the computational domain when a physical Reynolds stress distribution is prescribed at the inflow. Therefore, efforts to enhance digital filter techniques should aim at modeling pressure fluctuations as well as their correlation with the velocity components.

Validation of RANS-Based Turbulence Models Against High-Resolution Experiments and DNS for Buoyancy-Driven Flow with Stratified Fronts

To provide computational fluid dynamics (CFD)–grade experimental data for studying stratification, measurements on the High-Resolution Jet (HiRJet) facility at the University of Michigan have been conducted with density differences of 1.5 % and − 1.5 % , respectively. Fluid with a density different from the fluid initially present in the HiRJet tank was injected, and the propagation of the time-dependent density stratification was captured on a two-dimensional plane with the aid of the wire-mesh sensor technique for Reynolds numbers near 5000 and Richardson numbers near 0.29. Direct numerical simulations (DNSs) of the two cases have also been conducted to expand the multifidelity database. The novel experimental and DNS data were then used to assess the predictive capabilities of the Standard k − ε (SKE) model and the Reynolds Stress Transport (RST) model. In particular, the propagation speed and thickness of the stratification fronts were assessed by comparing the CFD results against the experimental and DNS data. It was found that the general trends of the stratified density fronts were well predicted by the CFD simulations; however, slight overprediction of the thickness of the stratification layer was found with the SKE model while the RST model gave a larger overprediction of the mixing. Examination of the turbulent statistics showed that the turbulent viscosity was largely overpredicted by the RST model compared to the SKE model.

An equivariant neural operator for developing nonlocal tensorial constitutive models

Developing robust constitutive models is a fundamental and longstanding problem for accelerating the simulation of multiscale physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. However, with either traditional or machine learning techniques, the constitutive model for tensorial quantities has been much less studied than scalar quantities due to more complicated physics, even though the former plays an important role in many scientific and engineering applications. In this work, we propose a neural operator to develop nonlocal constitutive models for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models, faithfully reflects the region of influence in physics, and is applicable to different spatial resolutions. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We evaluate the VCNN-e by using it to emulate the Reynolds stress transport model for turbulent flows, which directly computes the Reynolds stress tensor to close the Reynolds-averaged Navier–Stokes (RANS) equations. The evaluation is performed in two situations: (1) emulating the Reynolds stress model through synthetic data generated from the Reynolds stress transport equations with closure models, and (2) predicting the Reynolds stress by learning from data generated from direct numerical simulations. Such a priori evaluations of the proposed network on realistic and challenging datasets pave the way for developing and calibrating robust and nonlocal, non-equilibrium constitutive models for the RANS equations and other mechanical problems.

Experimental and computational investigation of the vortical structures generated from a blended-wing-body UAV model

The current study presents a combination of experimental and computational investigations of a scaled-down Blended Wing Body (BWB) Unmanned Aerial Vehicle (UAV) model. The BWB model features highly swept wings. The experiments are conducted in a subsonic windtunnel in the Laboratory of Fluid Mechanics and Turbomachinery, in the Aristotle University of Thessaloniki. More specifically, surface oil flow visualization and Laser-Doppler Anemometry (LDA) are employed to investigate the vortical structures emanating from the leading edge and the wingtips of the model and provide the corresponding velocity and vorticity profiles. Using the windtunnel measurement data as a reference point, both steady-state and transient computational fluid dynamics (CFD) analyses are also conducted. The flow around the BWB scaled-down model is modeled by adopting four turbulence models, three of them based on the concept of the eddy-viscosity and one, on the Reynolds-stress transport equation modeling. The modeling results are compared to the experimental measurements, emphasizing on the streamwise and spanwise velocity profiles, axial vorticity profiles and separation patterns. The results of this study provide insight on the vortical phenomena dominating the flowfield over and around BWB configurations and conclusions concerning the tradeoffs between accuracy and computational efficiency are drawn.

Flow Field Investigation of a Single Engine Valve Using PIV, POD, and LES

Due to stringent emission regulations, it is of practical significance to understand cycle-to-cycle variations in the combustion of fossil or renewable fuels to reach future emission regulations. The present study aims to conduct a parametric investigation to analyse the influence of the valve lift and different mass flows of an inlet valve of the test engine “Flex-OeCoS” on the flow structures. To gain a deeper understanding of the flow behaviour, an optical test bench for 2D Particle Image Velocimetry (PIV) and a Large Eddy Simulation (LES) are used. Turbulence phenomena are investigated using Proper Orthogonal Decomposition (POD) with a quadruple decomposition and the Reynolds stress transport equation. The results show good agreement between the PIV and LES. Moreover, the main flow structures are primarily affected by valve lift while being unaffected by mass flow variation. The turbulent kinetic energy within the flow field increases quadratically to the mass flow and to the decreasing valve lift, where large high-energetic flow structures are observed in the vicinity of the jet and small low-energetic structures are homogeneously distributed within the flow field. Furthermore, the convective flux, the turbulent diffusive flux, the rate of change, and the production of specific Reynolds stress are the dominant terms within the specific Reynolds stress transport equation.

A turbulence modeling study of the precessing vortex core in a pipe flow

The phenomenon of precessing vortex core is observed experimentally when swirl is imparted on an axial flow in a pipe. It manifests as a coherent structure in the form of a helical vortex of regular wavelength whose axis is coincident with the pipe’s axis. The most striking consequence of this pattern of flow is the generation of periodic fluctuations in the streamwise distribution of the wall static pressure and skin friction. While the prediction of the precessing vortex has proved possible with large-eddy simulations, there is no record of this phenomenon being captured in great detail by Reynolds-averaged Navier-Stokes methods utilizing turbulence models to close the time-averaged equations. The purpose of the research reported here was to determine whether the precessing vortex core and its impact on conditions at the wall can be captured using this approach. The turbulence model used was of the Reynolds-stress transport type which involves the solution of a differential transport equation for each of the six non-zero components of the Reynolds stress tensor. Previous studies in which such models were used in the prediction of rotating and swirling flows have shown that their performance is largely determined by the way in which the difficult fluctuating pressure-strain correlations that appear in these equations are modeled. To this end, four very different alternative models for these correlations were assessed by comparisons with experimental data for a swirling pipe flow at relatively high swirl number. It was found that the models do indeed capture the precessing vortex revealing it to be in the form of an exceptionally well-defined double vortex. It was also found that the expected periodic fluctuations in static pressure and wall skin friction, not previously obtained in turbulence model studies, are captured by the present closures.

Effect of sudden constriction of a flat duct on forced convection in a turbulent droplet-laden mist flow

Numerical modeling of the flow structure and heat transfer in a gas-droplet turbulent flow in a duct with forward-facing step is carried out. The two-dimensional RANS equations are used in the numerical solution. The Eulerian two-fluid approach is used for describing the flow dynamics and heat transfer in the gaseous and dispersed phases. The turbulence of the carrier phase is described using an elliptical Reynolds stress model with taking the presence of dispersed phase. It is shown that finely-dispersed droplets are involved in the separation recirculation motion of the gas phase. The addition of evaporating droplets to a single-phase turbulent flow in the forward-facing step leads to a significant intensification of heat transfer (more than 2 times) compared to a single-phase air flow, all other parameters being equal. This effect is enhanced with an increase in the initial mass fraction of the water droplets. Keywords: Numerical simulation, Reynolds stress transport model, forward-facing step, droplet evaporation, turbulence, heat transfer enhancement.

High-Order Accurate Numerical Simulation of Supersonic Flow Using RANS and LES Guided by Turbulence Anisotropy

This paper discusses accuracy improvements to Reynolds-Averaged Navier–Stokes (RANS) modeling of supersonic flow by assessing a wide range of factors for physics capture. Numerical simulations reveal complex flow behavior resulting from shock and expansion waves and so, a supersonic jet emanating from rectangular nozzle is considered. PIV based experimental data for the jet is available from literature and is used for validation purposes. Effect of various boundary conditions and turbulence modeling approaches is assessed qualitatively and quantitatively. Of particular interest are the inlet conditions considering the turbulence intensity and the effect of upstream air supply duct, the effect of nozzle wall surface roughness on nozzle internal flow and downstream, wall y+ sensitivity for boundary layer resolution and laminar to turbulent transition modeling. In addition to mesh sensitivity, domain dependency is conducted to evaluate the appropriate domain size to capture the kinetic energy dissipation downstream of the nozzle. To further improve the flow characteristics, accounting for the anisotropy of Reynolds stresses is also one of the focuses. Therefore, non-linear eddy viscosity-based two-equation model and Reynolds stress transport model are also investigated. Additionally, the results of baseline linear (Boussinesq) RANS are compared. Corresponding comparisons with high-fidelity LES are presented. Jet self-similar behavior resulting from all simulation fidelities is assessed and it appears that turbulent flow in LES becomes self-similar, but not in RANS. Finally, various factors such as the nozzle geometry and numerical modeling choices influencing the anisotropy in jet turbulence are discussed.

Suppression of the turbulent kinetic energy and enhancement of the flame-normal Reynolds stress in premixed jet flames at small Lewis numbers

We report the effects of the Lewis number on the turbulent kinetic energy, Reynolds stresses, and scalar fluxes in premixed jet flames with strong mean shear. The direct numerical simulation (DNS) of piloted turbulent premixed jet flames is performed under the same condition except for different Lewis numbers to isolate the Lewis number effects. The turbulence intensity within the mean flame brush is found to decay more rapidly along the axial distance with a smaller Lewis number. This Lewis number dependence results from the coupling of the mean shear and the combustion heat release, where the mean shear is the primary source of turbulence production. We show that, under the strain imposed by the mean shear and turbulence fluctuation, the heat release of premixed jet flames is significantly enhanced at the Lewis number smaller than unity, which is represented by the rising flamelet consumption speed and flame surface area. As a consequence, the mean shear within the flame brush is suppressed at small Lewis numbers via the dilatation effect, and the weakened mean shear production causes the fast turbulence suppression along the axial direction. The enhanced heat release at small Lewis numbers also increases the flame-normal Reynolds stress, and leads to the counter-Boussinesq behavior of the shear Reynolds stress and the counter-gradient transport of scalar fluxes. We model the Lewis number effects on local turbulence and scalar transport using the stretched laminar flame speed and thickness, and achieve overall good estimations of the flame-normal Reynolds stress and scalar transport at small Lewis numbers.

A PDE-free, neural network-based eddy viscosity model coupled with RANS equations

In fluid dynamics, constitutive models are often used to describe the unresolved turbulence and to close the Reynolds averaged Navier–Stokes (RANS) equations. Traditional PDE-based constitutive models are usually too rigid to calibrate with a large set of high-fidelity data. Moreover, commonly used turbulence models are based on the weak equilibrium assumption, which cannot adequately capture the nonlocal physics of turbulence. In this work, we propose using a vector-cloud neural network (VCNN) to learn the nonlocal constitutive model, which maps a regional mean flow field to the local turbulence quantities without solving the transport PDEs. The network is strictly invariant to coordinate translation, rotation, and uniform motion, as well as ordering of the input points. The VCNN-based nonlocal constitutive model is trained and evaluated on flows over a family of parameterized periodic hills. Numerical results demonstrate its predictive capability on target turbulence quantities of turbulent kinetic energy k and dissipation ɛ. More importantly, we investigate the robustness and stability of the method by coupling the trained model back to RANS solver. The solver shows good convergence with the simulated velocity field comparable to that based on k–ɛ model when starting from a reasonable initial condition. This study, as a proof of concept, highlights the feasibility of using a nonlocal, frame-independent, neural network-based constitutive model to close the RANS equations, paving the way for the further emulation of the Reynolds stress transport models.

Benchmark simulation of the flow-induced vibrations for nuclear applications

This study investigates the computational modelling of flow-induced vibrations of cantilever rods subjected to turbulent axial flow at operating conditions relevant to those of fuel rods of pressurized-water-cooled (PWR) nuclear reactors. The aim is to assemble all the modelling elements needed for a cost-effective and thus URANS-based (Unsteady Reynolds Averaged Navier–Stokes) modelling strategy, employing high-Reynolds-number models of turbulence. This objective is pursued through three stages. First, we investigate the numerical FSI (Fluid–Structure Interaction) strategy adopted, through the computation of flow-induced vibration of an elastic plate subjected to axial laminar flow. We subsequently assess the suitability of URANS models through computations of turbulent flow over a forward-facing step, for which measurements of the fluctuating wall pressure are available. On the numerical side, these explorations led to adopting a two-way FSI strategy, using a single finite-volume solver, with the Arbitrary Lagrangian–Eulerian (ALE) approach, high order convective discretization schemes and Laplacian smoothing for the displacement of the mesh in the fluid domain. On the physical modelling side, they resulted in the use of high-Reynolds-number Reynolds stress transport models. The resulting modelling strategy is subsequently validated against the experimentally investigated case of a steel cantilever rod caused to vibrate through exposure to turbulent axial flow. The resulting comparisons show that for the first time, to our knowledge, both the frequency and the amplitude of the flow induced vibrations of this case, have been reproduced with good accuracy.

Data-Driven Prediction of Complex Flow Field Over an Axisymmetric Body of Revolution Using Machine Learning

Abstract Computationally efficient and accurate simulations of the flow over axisymmetric bodies of revolution (ABR) have been an important desideratum for engineering design. In this article, the flow field over an ABR is predicted using machine learning (ML) algorithms (e.g., random forest (RF), artificial neural network (ANN), and convolutional neural network (CNN)) using trained ML models as surrogates for classical computational fluid dynamics (CFD) approaches. The data required for the development of the ML models were obtained from high fidelity Reynolds stress transport model (RSTM)-based simulations. The flow field is approximated as functions of x and y coordinates of locations in the flow field and the velocity at the inlet of the computational domain. The optimal hyperparameters of the trained ML models are determined using validation. The trained ML models can predict the flow field rapidly and exhibit orders of magnitude speedup over conventional CFD approaches. The predicted results of pressure, velocity, and turbulence kinetic energy are compared with the baseline CFD data. It is found that the ML-based surrogate model predictions are as accurate as CFD results. This investigation offers a framework for fast and accurate predictions for a flow scenario that is critically important in engineering design.

On the explainability of machine-learning-assisted turbulence modeling for transonic flows

Machine learning (ML) is a rising and promising tool for Reynolds-Averaged Navier–Stokes (RANS) turbulence model developments, but its application to industrial flows is hindered by the lack of explainability of the ML model. In this paper, two types of methods to improve the explainability are presented, namely the intrinsic methods that reduce the model complexity and the post-hoc methods that explain the correlation between the model inputs and outputs. The investigated ML-assisted turbulence model framework aims to improve the prediction accuracy of the Spalart–Allmaras (SA) turbulence model in transonic bump flows. A random forest model is trained to construct a mapping between the input flow features and the output eddy viscosity difference. Results show that the intrinsic methods, including the hyperparameter study and the input feature selection, can reduce the model complexity at a limited cost of accuracy. The post-hoc Shapley additive explanations (SHAP) method not only provides a ranked list of input flow features based on their global significance, but also unveils the local causal link between the input flow features and the output eddy viscosity difference. Based on the SHAP analysis, the ML model is found to discover: (1) the well-known scaling between eddy viscosity and its source term, which was originally found from dimensional analysis; (2) the well-known rotation and shear effects on the eddy viscosity source term, which was explicitly written in the Reynolds stress transport equations; and (3) the pressure normal stress and normal shear stress effect on the eddy viscosity source term, which has not attracted much attention in previous research. The methods and the knowledge obtained from this work provide useful guidance for data-driven turbulence model developers, and they are transferable to future ML turbulence model developments.