Quadratic Constitutive Relation Research Articles

Validation and analyses of QCR correction turbulence model in sub-/super-sonic inner flows

The inlet, isolation section, and other internal flow components are important parts of the aircraft propulsion system. Their performances affect the stability of the entire propulsion system. According to those components, complicated shock wave/boundary layer interaction (SBLI), flow separation, and secondary flow phenomena would occur. The commonly used turbulence models, SA and SST, cannot predict the anisotropy of turbulence. This deficiency makes the calculated results differ significantly from the experimental results and cannot accurately predict their aerodynamic performance. This paper validates the feasibility and effectiveness of the turbulence models based on quadratic constitutive relation (QCR) correction applied to the flow of square duct, compression corners, diffusing 3D S-Duct, and axisymmetric cylindrical isolator. This can support future calculations of complex flow fields with flow separation and secondary flow phenomena in the subsonic or supersonic inlet. The results show that the turbulence model with QCR correction is better than the original turbulence model. Among them, the SA-QCR2020 turbulence model is the best, which is able to predict the presence of secondary flows and large boundary layer separated flows well.

DPW-7: Steady and Unsteady Computations of the Common Research Model at Different Reynolds Numbers

This article presents the numerical computations performed at ONERA for the Seventh AIAA Drag Prediction Workshop. By introducing Reynolds numbers up to 30 million closer to the flight conditions, greater lift levels beyond the design point, and time-accurate simulations, this new session has allowed the previous studies to be extended. The Common Research Model aircraft configuration has been considered in its academic wing-body version and calculated in this work with point-matched structured grids. The ONERA Cassiopee software as well as the elsA solver and the FFDπ far-field drag code have been used. The grid convergence study has shown larger pressure drag variations than what was obtained at the cruise lift coefficient, but increasing the Reynolds number seems to reduce this trend. Then, the angle-of-attack sweep study with the lift, drag, and moment polars has given the opportunity to assess different numerical settings such as the Spalart–Allmaras and shear stress transport turbulence models with the quadratic constitutive relation approach (QCR-2000) and to discuss the comparison between computational fluid dynamics results and wind-tunnel data. Concerning the Reynolds number increase, it has appeared that the main part of drag reduction comes from the friction () and viscous pressure drag () components. The prediction of pitching moment increments due to Reynolds number variations still needs to be significantly improved. Finally, for an angle of attack above 4.00 deg, by the use of unsteady Reynolds-averaged Navier–Stokes computations, an unsteady buffet phenomenon has been observed and analyzed.

Assessment of advanced RANS turbulence models for prediction of complex flows in compressors

Reynolds-Averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) has been widely used in compressor design and analysis. However, reasonable prediction of compressor flow and its impact on compressor performance remains challenging. In this study, Menter’s Shear Stress Transport (SST) model and its variants, as well as the ω-based Reynolds stress model (Stress-BSL) are assessed. For a single rotor (Rotor 67), under the peak efficiency operating condition, all studied turbulence models predict its performance with reasonable accuracy; under the off-design conditions, SST with Helicity correction (SST-Helicity) shows superiority in predicting the effect of flow on the spanwise distribution of aerodynamic parameters. For Darmstadt’s 1.5-stage transonic axial compressor, SST-Helicity outperforms SST, SST with the Quadratic Constitutive Relation (SST-QCR) and Stress-BSL in predicting the performance as well as the spanwise distribution of aerodynamic parameters. At the design rotating speed, the stall margin given by SST-Helicity (20.90%) is the closest to the experimental measurement (24.81%), which is more than twice that by SST (8.71%) and 1.5 times that by SST-QCR (14.14%). This paper demonstrates that SST-Helicity model, together with a good quality and sufficiently refined grid, can capture the compressor flow features with reasonable accuracy, which results in a credible prediction of compressor performance and stage matching.

A Turbo-Oriented Data-Driven Modification to the Spalart–Allmaras Turbulence Model

Abstract The Spalart–Allmaras (SA) turbulence model is one of the most popular models applied to compressors, but it often over-predicts blockage size and hence under-predicts the stall margin. In this paper, a novel modification to the SA model is proposed to improve the prediction of compressor near-stall flows. The modification is based on the dimensionless vortical pressure gradient, which identifies blockage cells featured by 3D swirling, adverse pressure gradient, and low-momentum flows. It unblocks the compressor passage by enhancing the eddy viscosity in the identified blockage cells; whereas in canonical 2D flows the modification is automatically switched off. The model coefficients are calibrated via Bayesian inference, which considers the uncertainties involved in experiments and computational fluid dynamics (CFD) simulations of turbomachinery. The rotor exit radial profile data of NASA Rotor 67 at peak-efficiency and near-stall points are used for calibration. The calibrated model is tested extensively in four compressors covering both tip blockage and corner separation as well as both industrial and laboratory Reynolds number and Mach number. For the NASA Rotor 67 and the TUDa-GLR-OpenStage, the proposed model predicts more accurate stall margins at all operating speeds due to the tip unblocking effect. For the BUAA Stage B rotor, the proposed model predicts the tip blockage size and thus the stall margin more accurately. For the LMFA NACA65 cascade, the proposed model with the quadratic constitutive relation (QCR) achieves significant improvement in predicting the exit profiles due to the unblocking effect on the corner separation. The proposed model, termed as SA-PGω in this work, is a promising engineering tool for future Reynolds-averaged Navier–Stokes (RANS) simulations of compressor near-stall flows.

Numerical investigation of corner separation flow using Spalart-Allmaras model with various modifications

Three-dimensional corner separation is an inherent flow feature in axial compressors which blocks flow path and worsens the performance. To find an effective turbulence model for such flow, numerical simulations of the corner separation flow are conducted in the prescribed velocity distribution (PVD) cascade using the Spalart-Allmaras (SA) model with various modifications. Turbulence models studied here include the SA model, the SA-neg model, the SA-noft2 model, the SA-R model, the SA-RC model, the SA-Helicity model, the SA-KL model, the SA-S model and their combinations with the quadratic constitutive relation (QCR) corrections. Compared with the experiment, the SA-Helicity model predicts more accurately than SA model with other commonly used modifications which indicates the importance of reasonably considering turbulent non-equilibrium transport characteristics in large-scale vortices. With the QCR corrections for considering turbulence anisotropy, the blending models perform better than the original SA model but most of them still overpredict the separation region. Among them, the SA-Helicity-QCR2013 model which provides a similar result as the SA-Helicity model performs the best. In further validations based on the transonic compressor NASA Rotor 67, the SA-Helicity and SA-Helicity-QCR2013 models can still give more accurate predictions, which is reflected in the stall margin as well as the spanwise distributions of total pressure and total temperature at both the near peak efficiency and the near stall conditions. The universality of the helicity correction is proved to a certain extent. In future turbulence modeling studies of turbomachinery internal flows, it is necessary to reasonably consider turbulence non-equilibrium and turbulence anisotropy in large-scale vortices.

Investigations on Turbulence Model Uncertainty for Hypersonic Shock-Wave/Boundary-Layer Interaction Flows

Shock-wave/boundary-layer interaction (SWBLI) is a fundamental scientific problem that restricts the breakthrough of high-speed flight technology. Owing to the complex shock structure and boundary-layer separation in the interaction region, the widely used eddy-viscosity turbulence models have large uncertainties and model errors in the simulation of such flows. To make more reasonable aerodynamic/aerothermal predictions and identify the critical parameters affecting the prediction results, Bayesian uncertainty quantification and sensitivity analysis are conducted for the three turbulence models [Spalart–Allmaras (SA), SA with quadratic constitutive relation (SA-QCR), and shear-stress transport (SST) models] and the hypersonic SWBLI cases. First, the prior variances and Sobol indices are calculated to obtain a preliminary understanding of the parameter variability for turbulence models. Then, the posterior distributions of the model closure coefficients and posterior uncertainties of the wall pressure and thermal flux are systematically analyzed and compared. The results indicate that the prediction performances of the three models can be effectively enhanced by Bayesian estimation. The peak of the posterior uncertainty of the SST model is the largest among the three models, whereas that of the SA-QCR model is the smallest. The results of the Bayesian model evaluation demonstrate that the SA-QCR model has the highest reliability.

NASA Juncture Flow Computational Fluid Dynamics Validation Experiment

The purpose of the NASA Juncture Flow experiment was to acquire high-quality flowfield details deep in the corner of a wing–body junction specifically for the purpose of computational fluid dynamics (CFD) validation. A truncated DLR F6 wing was used along with a flat-sided fuselage with windows that allowed both laser Doppler velocimetry and particle image velocimetry measurements to be made very close to the corner. This paper describes the experiment and its results. It also makes comparisons between the wind-tunnel data and Reynolds-averaged Navier–Stokes CFD results using a new version of a quadratic constitutive relation that was recently developed to improve separated corner-flow predictions. This validation experiment provides useful flowfield data that can be used to test the ability of CFD methods and models to accurately represent separated juncture flow physics.

Improved Supersonic Turbulent Flow Characteristics Using Non-Linear Eddy Viscosity Relation in RANS and HPC-Enabled LES

A majority of the eddy viscosity models for supersonic turbulent flow are based on linear relationship between Reynolds stresses and mean strain rate. The validity of these models can be improved by introducing non-linearity in relation as RANS models offer advantages in terms of reduced turnaround times typical of industry applications. With these benefits, the present work utilizes quadratic constitutive relation (QCR) with Menter’s k omega SST model to characterize the flowfield of rectangular jets. The sensitivity of this model with QCR, weighted towards diffusion, dissipation, and a combination of both, is addressed. Viscous large eddy simulations (LES) with WALE subgrid scale models are employed for qualitative comparisons using a commercial solver. Massively parallel LES are enabled by the new in-house 1088-core computing cluster at the University of Cincinnati and are also used for benchmarking. The nearfield results are validated with available experimental data and show good agreement in both fidelities. Flow characteristics, including the shear layer profiles, Reynolds stresses, and turbulence kinetic energy (TKE) and its production are compared. LES reveal higher TKE production in the regions with highest Reynolds stresses. It is comparatively lower in QCR RANS. As a special case of TKE analysis in jets, a preliminary investigation of retropropulsion is outlined for rectangular nozzles for the first time. Improved flow behavior by implementation of a non-linear relationship between Reynolds stresses and mean strain rate is demonstrated.

Analysis and extension of the quadratic constitutive relation for RANS methods

AbstractThe quadratic constitutive relation was proposed as an extension of minimal complexity to linear eddy-viscosity models in order to improve mean flow predictions by better estimating turbulent stress distributions. However, the successes of this modification have been relatively modest and are limited to improved calculations of flow along streamwise corners, which are influenced by weak secondary vortices. This paper revisits the quadratic constitutive relation in an attempt to explain its capabilities and deficiencies. The success in streamwise corner flows cannot be entirely explained by significant improvements in turbulent stress estimates in general, but is instead due to better prediction of the particular turbulent stress combinations which appear in the mean streamwise vorticity equation. As a consequence of this investigation, a new formulation of turbulent stress modification is proposed, which appears to better predict the turbulent stress distributions for a variety of flows: channel flow, equilibrium boundary layers, pipe flow, separated boundary layers and square duct flow.

Turbulence modelling analysis in a corner separation flow

This paper studies RANS turbulence modelling for a linear compressor cascade corner separation flow, using large-eddy simulation results as reference. The Boussinesq and the quadratic constitutive relations are investigated with two versions of Wilcox’s k−ω turbulence model through a priori and a posteriori analyses. In the a priori analysis the quadratic constitutive relation shows improvement on the alignment between the rate-of-strain tensor and the Reynolds stress tensor for the inlet region, compared to the Boussinesq constitutive relation. But the improvement is less significant in the highly vortical region. Using the turbulent kinetic energy and the specific dissipation rate provides a fairly good estimate of the turbulent viscosity. The turbulent kinetic energy budget is also investigated. Large-eddy simulation results present non-equilibrium turbulence, i.e. the production and dissipation are not balanced, whereas the RANS models predicts equilibrium turbulence.

Improvements to the Quadratic Constitutive Relation Based on NASA Juncture Flow Data

Modifications to the quadratic constitutive relations (QCRs) of Spalart (“Strategies for Turbulence Modelling and Simulation,” International Journal of Heat and Fluid Flow, Vol. 21, 2000, pp. 252–263) and Mani et al. (“Predictions of a Supersonic Turbulent Flow in a Square Duct,” AIAA Paper 2013-0860, Jan. 2013) are proposed. These modifications result from computational fluid dynamics (CFD) studies in the FUN3D and OVERFLOW solvers of the flowfield in the body boundary layer and near-corner region of the NASA Juncture Flow model. When used in conjunction with the Spalart–Allmaras turbulence model with rotation/curvature correction, the new relations improve the CFD Reynolds normal stress predictions, as compared with the experiment. This improvement leads to better predictions of separated corner flow behavior across a wide range of angles of attack that were measured, somewhat beyond the original QCR effect. The model’s performance in a square duct, driven by turbulence-induced vortices, is slightly improved. The QCR remains fairly simple to program and compatible with other eddy-viscosity models.

High-Lift Simulations of the Japan Aerospace Exploration Agency Standard Model

The Japan Aerospace Exploration Agency standard model high-lift configuration was analyzed computationally using both the OVERFLOW and LAVA codes for the third AIAA High-Lift Prediction Workshop. Simulations were performed both with and without the nacelle/pylon feature as prescribed by the workshop test cases. Without the nacelle/pylon, evidence of multiple solutions was observed when a quadratic constitutive relation was used in the turbulence modeling; however, time-accurate simulations seemed to favor one solution over the other. With the nacelle/pylon, no evidence of multiple solutions was observed. Laminar–turbulent transition modeling was also applied to the configuration, and it had an overall favorable impact on the lift predictions.

Sixth Drag Prediction Workshop Results Using FUN3D with k-kL-MEAH2015 Turbulence Model

The Common Research Model wing/body configuration is investigated with the k-kL-MEAH2015 turbulence model implemented in FUN3D. This includes results presented at the Sixth Drag Prediction Workshop and additional results generated after the workshop with a nonlinear quadratic constitutive relation variant of the same turbulence model. The workshop-provided grids are used, and a uniform grid refinement study is performed at the design condition. A large variation between results with and without a reconstruction limiter is exhibited on “medium” grid sizes, indicating that the medium grid size is too coarse for drawing conclusions in comparison with experiment. This variation is reduced with grid refinement. At a fixed angle of attack near design conditions, the quadratic constitutive relation variant yielded decreased lift and drag compared with the linear eddy-viscosity model by an amount that was approximately constant with grid refinement. The k-kL-MEAH2015 turbulence model produced wing–root junction flow behavior consistent with wind-tunnel observations.

Experimental Investigations on Common Research Model at ONERA-S1MA–Drag Prediction Workshop Numerical Results

This paper aims to present some of the experimental and numerical results obtained with the NASA–Boeing Common Research Model at ONERA. The wind tunnel model used in the present study is the ONERA Large Reference Model (1/16.835), which has the same geometry as the Common Resarch Model considered in the latest AIAA Drag Prediction Workshops. Experimental data have been collected from the ONERA-S1MA wind tunnel at Mach numbers between 0.30 and 0.95 and a mean aerodynamic chord Reynolds number of 5 million for four different configurations: wing/body only and wing/body with horizontal or vertical tails or both. Force and moment and pressure and surface flow measurements have been performed. Concerning the numerical study, all the Reynolds-averaged Navier–Stokes computations have been completed with the structured solver elsA, and the Spalart–Allmaras and with shear stress transport turbulence models have been used in addition to the quadratic constitutive relation. In this paper, configuration effects (increments due to horizontal and/or vertical tails) are assessed both numerically and experimentally for several Mach numbers and angles of attack. The delicate issue of flow separation at the wing/body junction is also addressed with the support of oil flow visualizations. elsA and S1MA skin pressure distributions are presented; the agreement is satisfactory except for some outboard wing sections at high lift levels. Finally, comparisons of drag and moment values including computational fluid dynamics and test data from different wind tunnels are proposed (S1MA, NASA Ames Research Center and National Transonic Facility, and European Transonic Wind Tunnel).

Numerical Investigations of the NASA Common Research Model with Aeroelastic Twist

Numerical investigations of the NASA Common Research Model from the 6th AIAA CFD Drag Prediction Workshop are performed with in-house solver MFlow. The solver is based on a cell-centered finite volume method and is capable of handling various element types. Hybrid grids supplied by the committee and generated in-house are used in the simulations. Grid convergence properties, polar lines, and pressure distributions are analyzed and compared with experimental data. A nearly linear convergence property is achieved with grid refinement for own-generated hybrid grids. The drag increment due to nacelle and pylon is well predicted. As angle of attack increases, the simulations with the original Spalart–Allmaras or shear-stress transport model exhibit a large side-of-body separation, whereas the results obtained with quadratic constitutive relation present a negligible separation bubble, which is more reasonable judging from experimental data. The solver shows good agreement with experiment. The predictions are much improved with the static aeroelastic effect considered in simulations.

Improved compressor corner separation prediction using the quadratic constitutive relation

This work presents the implementation and study of the quadratic constitutive relation nonlinear eddy-viscosity model with representative compressor application, for which the corner separation has been poorly predicted with the widely used linear Boussinesq eddy-viscosity model. With the introduction of the Reynolds stress anisotropy, the secondary flow of the second kind and its effect on the corner flow can be well captured and this results in greatly improved prediction of pressure coefficient, total pressure loss coefficient and the corner separation size. Without the quadratic constitutive relation model, the separation size and loss are generally over-estimated. The mechanism of the improvement is studied using both the vortex dynamics and the momentum equation. It is proved that quadratic constitutive relation model consumes low CPU time and provides much improved compressor corner separation prediction without worsening the convergence property.

Supersonic Corner Flow Predictions Using the Quadratic Constitutive Relation

A series of Reynolds-averaged Navier–Stokes simulations is performed for a supersonic wall-bounded turbulent corner flow. These simulations are compared to a high-order implicit large-eddy simulation for the same geometry and flow conditions. Inclusion of the quadratic constitutive relation in the Reynolds-averaged Navier–Stokes simulation results in significant improvement in qualitative agreement with the large-eddy simulation: specifically, the presence of secondary flow (a counter-rotating vortex pair). The range of valid values for the constant in the quadratic constitutive relation formulation is explored. Additionally, the effects on this range from different turbulence models and momentum thickness Reynolds numbers for the corner are also examined. These effects of the quadratic constitutive relation are explored using both the finite difference fluid solver OVERFLOW and the finite volume fluid solver US3D. The results indicate that the quadratic constitutive relation term directly affects the strength of the vortex pair in the secondary flow and that its influence appears directly dependent on all the aforementioned parameters.

Analysis on phase properties of a transmission line model with non-linear elements

Abstract The theme of this paper is to analyze phase response of a conventional transmission line (Tx line) unit cell with the presence of a non-linear element. In this analysis, a non-linear inductor and non-linear capacitor are used in two Tx line models, firstly, forward wave supporting structure (low pass in nature) and secondly composite right hand left hand meta-material (CRLH-MTM) structure (bandpass in nature) which supports both forward and backward waves. For the non-linearity, a quadratic constitutive relation is considered between charge and voltage for a capacitor. For an inductor it is between the flux and the current. This analysis is to investigate the possibility of producing the negative group delay in the unit cell of Tx model in the presence of a non-linear element. This implies the opposite phase velocity and group velocities in the dispersion relationship for the Tx line model.

Investigation of a Nonlinear Reynolds-Averaged Navier–Stokes Closure for Corner Flows

This study gives an insight into the corner flow around a wing mounted on a flat plate. Reynolds-averaged Navier–Stokes simulations and experimental data are compared for a Reynolds number based on the wing chord () equal to and an angle of attack of 12 deg. The flat plate and wing boundary layers are fully turbulent, and the Reynolds number based on the momentum thickness of the incoming boundary layer is equal to . It is shown that the Reynolds-averaged Navier–Stokes simulation using the Spalart–Allmaras model with the Boussinesq closure predicts a massive corner flow separation, whereas on the experimental side, the separation is confined to a small area. Better predictions are obtained when resorting to the Spalart–Allmaras model with the quadratic constitutive relation closure. A comprehensive investigation of the numerical results using the experimental data is performed to get a better understanding of the behavior of the Spalart–Allmaras quadratic constitutive relation model for the present corner flow.

Japan Aerospace Exploration Agency Studies for the Second High-Lift Prediction Workshop

This paper summarizes Japan Aerospace Exploration Agency studies using a structured grid solver UPACS and an unstructured grid solver TAS-code for the second AIAA CFD High-Lift Prediction Workshop test cases and follow-on problems. Computations are performed for the grid convergence study using the configuration without any slat tracks and flap track fairings on a series of multi-block structured grids and a series of unstructured grids provided by the High Lift Prediction Workshop committee. In this paper, the following are mainly discussed: 1) grid convergence verification study of the turbulence models using a two-dimensional bump case, 2) grid convergence studies for Spalart–Allmaras and shear stress transport turbulence models using UPACS, 3) change of grid convergence and predicted flow fields with and without a nonlinear quadratic constitutive relation model for Spalart–Allmaras and shear stress transport turbulence models using UPACS including evaluation at higher angles of attack, and 4) comparison studies using different solvers UPACS and TAS on different grid systems with and without the quadratic constitutive relation model for the Spalart–Allmaras model.