Reynolds Stress Transport Model Research Articles

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.

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.

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.

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.

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.

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.

Flow Non-Uniformity and Secondary Flow Characteristics Within a Serpentine Cooling Channel of a Realistic Gas Turbine Blade

Abstract Unlike the ideal conditions considered in most previous studies, the actual cooling flow passage inside gas turbine blades is extremely complex. This complexity is due to the geometrical restrictions resulting from the external cooling holes and blade shape, which change the secondary flow and flow non-uniformity of the internal cooling flow. This study conducted an experimental and numerical analysis to characterize the secondary flow and flow non-uniformity in a realistic internal cooling serpentine passage. Magnetic resonance velocimetry was utilized to measure the average three-dimensional–three-components of the mean velocity. By integrating the flow field, parameters indicating the flow non-uniformity and secondary flow strength were obtained. Reynolds-averaged Navier–Stokes simulations were also conducted, and the Reynolds stress transport model showed relatively good performance when predicting the separation bubble in the U-bend. The secondary flow intensity exponentially decreases after the U-bend, but the rib turbulators maintain the secondary flow at a certain level. Additionally, the high-velocity regions in the inlet zone and beyond the separation bubble create significant flow non-uniformity and inherent shear. At the same time, the turbulence intensity becomes strong at the low-velocity region, which is key for heat transfer enhancement. Therefore, high flow non-uniformity has the potential to enhance heat transfer.

Reynolds Stress Turbulence Modelling with γ Transition Model

Implementation of the SSG/LRR-ω-γ model is carried out in the current study to predict transition to turbulence. The framework uses a Reynolds stress transport model, SSG/LRR-ω, as the base turbulence formulation and is coupled with Menter's γ transition model. To extend its applicability to different transition mechanisms, the production terms in the Reynolds stress and specific dissipation rate transport equations are modified. SSG/LRR-ω-γ uses simplified correlations that do not depend on freestream quantities, making it coordinate independent and rendering it Galilean invariant. Additionally, using a second-order closure turbulence model makes it suitable for complex flow fields. The influence of different grid parameters on the model is also discussed. A validation study is performed using multiple benchmark flat plate cases as well as 2D and 3D geometries to demonstrate the model's capability in predicting different transition mechanisms. When compared to state-of-the-art transition models, the proposed model shows equivalent or higher predictive capability.

CFD analysis of a solid pin-fueled small modular fluoride salt-cooled reactor

A computational fluid dynamics (CFD) study is performed for the pre-conceptual solid pin-fueled small modular fluoride salt-cooled reactor developed by the Oak Ridge National Laboratory (ORNL). The solid and fluid regions in a 1/12th section of a fuel assembly are modeled. The solid region includes fuel and non-fuel pins, with heat generation in the active region of the fuel pins. Two different power profiles, uniform and center-peaked, are considered. The FLiBe coolant flows from the bottom to the top of the core, parallel to the bank of fuel and non-fuel pins. The k-ω shear-stress transport (SST) model is chosen as the baseline turbulence model. The effects of grid refinement, inlet turbulence specification, and turbulence models on the temperature and pressure drop predictions are studied. Turbulence model sensitivity is investigated by comparing the results from the k-ω SST model with other two-equation models (k-ω baseline or BSL and k-ε realizable) as well as with anisotropic Reynolds stress transport models (linear pressure-strain and stress-BSL). The results show that the choice of turbulence model has a significant impact on the pin temperature and bundle pressure drop predictions. The results from the baseline turbulence model show good agreement between the fuel pin temperatures and bundle pressure drop values predicted using a subchannel model previously developed by the authors of the present study.

A Validation Study for RANS Based Modelling of Swirling Pulverized Fuel Flames

A swirling pulverized coal flame is computationally investigated. A Eulerian–Lagrangian formulation is used to describe the two-phase flow. Turbulence is modelled within a RANS (Reynolds averaged numerical simulation) framework. Four turbulence viscosity- (TV) based models, namely the standard k-ε model, realizable k-ε model, renormalization group theory k-ε model, and the shear stress transport k-ω model are used. In addition, a Reynolds stress transport model (RSM) is employed. The models are assessed by comparing the predicted velocity fields with the measurements of other authors. In terms of overall average values, the agreement of the predictions to the measurements is observed to be within the range 20–40%. A better performance of the RSM compared to the TV models is observed, with a nearly twice as better overall agreement to the experiments, particularly for the swirl velocity. In the second part of the investigation, the resolution of the discrete particle phase in modelling the turbulent particle dispersion (TPD) and particle size distribution (SD) is investigated. Using the discrete random walk model for the TPD, it is shown that even five random walks are sufficient for an accuracy that is quite high, with a less than 1% mean deviation from the solution obtained by thirty random walks. The approximation of the measured SD is determined by a continuous Rosin–Rammler distribution function, and inaccuracies that can occur in its subsequent discretization are demonstrated and discussed. An investigation on the resolution of the SD by discrete particle size classes (SC) indicates that 12 SC are required for an accuracy with a less than 1% mean deviation from the solution with 18 SC. Although these numbers may not necessarily be claimed to be sufficiently universal, they may serve as guidance, at least for SD with similar characteristics.

Computational fluid dynamics study of the impact of surface roughness on cyclone performance and erosion

In this study, the effect of roughness of cyclone wall on the rate of erosion and cyclone performance was studied using the CFD technique with the aid of the Ansys-Fluent 19.2 software. The Reynolds stress transport model was used to simulate the airflow turbulence in the rotating flows in the cyclone. The validated computational model was used, and the variations of axial velocity, tangential velocity, collection efficiency, cyclone pressure drop, and rate of erosion for different wall roughness were evaluated. For simulating surface erosion, the Oka model was used. The result of this study revealed that the wall roughness significantly affected airflow behavior in the cyclone and the resulting cyclone performance. In addition, when the wall roughness increased, the wear rate decreased. It was also found that the solid loading and inlet gas velocity significantly affect the erosion rate and even more than that of particle size.

Performance improvement of a cyclone separator using different shapes of vortex finder under high-temperature operating condition

In the present study, the effect of inlet temperature on cyclone performance was numerically investigated based on computational fluid dynamics (CFD) approach. The Eulerian–Lagrangian approach in conjunction with the Reynolds stress transport model was employed to solve the unsteady Reynolds-averaged Navier–Stokes equations. A series of numerical simulations were carried out at a wide temperature range of 293 K to 700 K. Conclusive results indicated that the separation efficiency of cyclone significantly decreased with any increment of inlet temperature due to weaker swirling flow across the cyclone at a higher temperature. In this condition, it is essential to find an impressive way to overcome this negative effect. Hence, four proposed shapes, namely divergent vortex finder and convergent vortex finder (CVF) were designed to use instead of base one under high-temperature operating conditions. The results extracted from CFD simulations demonstrated that the maximum tangential velocity was obtained about 2.2 times of the inlet velocity (v = 11.99 m/s) at T = 700 K for cyclone with CVF 1, while it was predicted 1.9 times of the inlet velocity for a base cyclone. Moreover, other cyclones generated tangential velocities even less than the base cyclone. Higher tangential velocity led to increase the centrifugal force and cyclone separation efficiency. Using CVF 1 instead of a base vortex finder significantly helped the cyclone to collect finer particles. Whereas the separation efficiency enhanced 9.5% for the particle size of 2 µm at T = 700 K and v = 20.18 m/s compared to the base cyclone.

Particle Concentration Distribution in a Gas–Droplet Confined Swirling Flow: Euler and Lagrange Approaches

This paper considers the problem of the numerical simulation of the propagation dynamics of a dispersed admixture and heat transfer in a swirling turbulent gas–droplet flow behind a sudden tube expansion. The gas phase is described by a system of three-dimensional Reynolds-Averaged Navier-Stokes equations with taking into account the effect of particles on the transport processes in the carrier phase. The gas-phase turbulence is calculated with the Reynolds stress transport model with allowance for the effect of the dispersed phase. The Euler and Lagrangian descriptions give qualitatively similar results for small droplet sizes of up to d1 ≤ 30 μm; it is only for the largest particles studied in this paper (with an initial diameter of d1 = 100 μm) did the difference in the calculation results exceed 15%.

Digital Design of Batch Cooling Crystallization Processes: Computational Fluid Dynamics Methodology for Modeling Free-Surface Hydrodynamics in Agitated Crystallizers

A framework for the digital design of batch cooling crystallization processes is presented comprising three stages, which are based on different levels of process complexity, integrating crystallizer hydrodynamics with crystallization kinetics and consequently with expected crystal size distribution. In the first stage of the framework, a computational fluid dynamics methodology is developed to accurately assess hydrodynamics in a typical batch crystallizer configuration, comprising a 20 L scale dish-bottom vessel with a single beavertail baffle agitated by a retreat curve impeller, used in the pharmaceutical as well as in the fine chemical industries. The hydrodynamics of crystallizers with such configurations is characterized by vortex formation on the free liquid surface. It is therefore important to model the free surface using the Volume-of-Fluid (VoF) method. Comparison of the predicted mean velocity components with experimental measurements using laser Doppler anemometry reveals that improved predictions are obtained using a differential Reynolds-stress transport model for turbulence coupled with the VoF for modeling the gas-liquid interface compared with those using the Shear-stress transport model and with a flat liquid surface. This study demonstrates that an accurate treatment of the liquid free surface for capturing vortex formation is essential for reliable predictions of the crystallizer’s flow field. While the vortex depth is predicted to increase with increasing impeller Reynolds number, the dependence of hydrodynamic macroparameters, including power number, impeller flow number, and secondary circulation flow number, on Reynolds number reveals that they are essentially constant within the turbulent regime but fluctuate when the flow is in the transitional and laminar regimes as fluid viscosity increases.

Structure of a turbulent bubbly flow and heat transfer in a vertical tube

The results of numerical simulation of the flow structure and heat transfer in a vertical polydispersed bubbly flow are presented. The mathematical model is based on the Euler approach taking into account the effect of bubbles on the mean characteristics and turbulence of the carrier phase. The polydispersed distribution of bubbles size in a two-phase flow is modeled by the method of delta approximation considering the process of bubble break-up and coalescence. Carrier phase (fluid) turbulence is predicted using the Reynolds stress transport model. The simulation results showed good agreement with the experimental data presented in the literature. The measured and predicted thermal-hydraulic parameter distribution indicates that in a turbulent bubbly flow, the wall friction increase is greater than heat transfer enhancement.

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.