Reynolds-averaged Turbulence Model Research Articles

Evaluation of Reynolds-averaged Navier-Stokes turbulence models in open channel flow over salmon redds

Evaluation of Reynolds-averaged Navier-Stokes turbulence models in open channel flow over salmon redds

Non-unique machine learning mapping in data-driven Reynolds-averaged turbulence models

Recent growing interest in using machine learning for turbulence modeling has led to many proposed data-driven turbulence models in the literature. However, most of these models have not been developed with overcoming non-unique mapping (NUM) in mind, which is a significant source of training and prediction error. Only NUM caused by one-dimensional channel flow data has been well studied in the literature, despite most data-driven models having been trained on two-dimensional flow data. The present work aims to be the first detailed investigation on NUM caused by two-dimensional flows. A method for quantifying NUM is proposed and demonstrated on data from a flow over periodic hills and an impinging jet. The former is a wall-bounded separated flow, and the latter is a shear flow containing stagnation and recirculation. This work confirms that data from two-dimensional flows can cause NUM in data-driven turbulence models with the commonly used invariant inputs. This finding was verified with both cases, which contain different flow phenomena, hence showing that NUM is not limited to specific flow physics. Furthermore, the proposed method revealed that regions containing low strain and rotation or near pure shear cause the majority of NUM in both cases—approximately 76% and 89% in the flow over periodic hills and impinging jet, respectively. These results led to viscosity ratio being selected as a supplementary input variable (SIV), demonstrating that SIVs can reduce NUM caused by data from two-dimensional flows and subsequently improve the accuracy of tensor-basis machine learning models for turbulence modeling.

Numerical Analysis into the Improvement Performance of Ducted Propeller by using Fins: Case Studies on Types B4-70 and Ka4-70

Numerical analysis was conducted to assess the impact of fins on the B4-70 and Ka4-70 propeller performance. The study explored different fin variations, specifically bare fins, Propeller Boss Cap Fins (PBCF), and propeller nozzles, using computational fluid dynamics (CFD) simulations. To obtain the best results, the researchers utilized the explicit algebraic stress model (EASM) based on Reynolds-Averaged Navier-Stokes (RANS) equations and turbulence modelling. The primary goal of this study was to improve the energy efficiency of ships by examining various propeller configurations, both open and ducted. The overall conclusions indicated that the B4-70 PBCF convergent and Ka4-70 PBCF divergent with the addition of nozzle 19A exhibited the highest efficiency based on the EASM analysis. The CFD simulation results for both B4-70 and Ka4-70 propellers, utilizing a nozzle 19A with added boss cap fins, revealed several noteworthy phenomena. Firstly, for the B4-70 propeller, efficiency (η0) at J = 0.6 to J = 0.8 showed an increase of 1% to 2%. Secondly, concerning the Ka4-70 propeller, efficiency (η0) at J = 0.6 to J = 0.8 increased by 2% to 10%. These findings clearly demonstrate that the use of an ESD, such as the nozzle 19A with added boss cap fins, enhances the propulsion performance of the ship. It is evident that the CFD approach remains suitable and reliable for overall simulations.

Errors and uncertainties in CFD validation for non-equilibrium turbulent boundary layer flows at high Reynolds numbers

NATO AVT-RTG-349 was dedicated to the validation of computational fluid dynamics (CFD) methods based on the Reynolds-Averaged Navier-Stokes (RANS) equations and statistical turbulence models for non-equilibrium turbulent-boundary-layer flows at high Reynolds numbers. This paper describes and discusses the errors and uncertainties arising in the comparison of RANS simulation results with experimental data from wind-tunnel experiments. These errors and uncertainties are associated with the CFD grid and the discretization, the physical modelling, the measurement accuracy, and the differences in the flow conditions between the experimental facility and the computational set-up. The results show the need for a grid-convergence study using systematically-refined families of CFD grids. The two major sources of errors are the RANS turbulence model and the uncertainty originating from the differences between the computational set-up and the wind-tunnel. Then two possible paths for future research are described: future CFD mesh generation, and future validation experiments at high Reynolds numbers.

Study on the Evolution Pattern of Complex Topographic Flow Field in Typical Tableland and Hogback Ridge

The flow field in complex mountainous wind farms is highly intricate, affecting the accuracy of wind resource assessments and the reliability of wind turbine operation. In order to study the evolving characteristics of the flow field in typical complex terrains under different inflow wind directions and wind speeds, this paper, based on the open-source software OpenFOAM, adopts a neutral atmospheric boundary layer model combined with Reynolds-averaged turbulence model to construct a high-precision flow field model for complex terrains. The research reveals that this model requires fewer computational resources and can effectively simulate the flow characteristics around complex terrains. Wind resource distribution in tableland terrains and hogback ridge terrains is more influenced by the height of the mountains, and the recovery speed of the wake behind the mountains decreases with a reduction in slope. In hogback ridge terrains, under a wind direction of 0 °, the wind acceleration effect at the top of the hogback ridge is stronger than under a wind direction of 270°, as it experiences more intense compression from the mountains. This paper can provide reference for micro-siting in complex terrain.

Development of a Benchmark for Thermal Striping in Gas-Cooled Fast Reactors

This paper presents the development of a benchmark for predicting thermal striping through simulation. This work utilized large eddy simulation and will be used to benchmark future models. The testing domain was created using both the STRUCT and the Reynolds-averaged Navier-Stokes turbulence models and is based on an earlier design of the General Atomics Fast Modular Reactor upper plenum. The plenum features two adjacent, identical hexagonal bundles each with a center-placed axial rod drive, with a hot left coolant stream and a cold right coolant stream. The simulation solves the nondimensional Navier-Stokes equations, with temperature accounted as a passive scalar. First- and second-order flow statistics were obtained after 600 convective time units of averaging. The first-order statistics reveal that the hot jet is damped by a recirculatory flow from the near wall. At the same location, the second-order statistics show strong oscillations both in velocity and temperature. The power spectral density was utilized to determine that a low-frequency oscillation occurs here that is within the range of interest for thermal striping. Furthermore, proper orthogonal decomposition was used to identify coherent structures that confirm the oscillatory behavior, indicative of thermal striping. Overall, this benchmark can aid in the development of future models for predicting thermal striping in nuclear reactors, potentially leading to improved reactor safety and performance.

Characteristics Investigations of Ducted Ka4-70 Series Propeller with Boss Cap Fins Using Numerical and Experimental Method

Unconventional system that are generally adopted for ship propulsion are Ducted Propellers. These devices have recently been studied with medium-fidelity computational fluid dynamics code (based on the potential flow hypothesis) with promising results. Numerical and experimental comparison of ducted propeller with PBCF, case studies with Propeller Ka4-70 used combination ducted and PBCF Divergent. The study was done numerically using computational fluid dynamics (CFD) approach. The solver is based on the Reynolds-Averaged Navier-Stokes (RANS) solutions and turbulence modelling explicit algebraic stress model (EASM). The test data was obtained from CFD simulations consisting of the open propeller and combination Nozzle plus PBCF, but the experiment was done to Nozzle and PBCF only. All measurements were carried out from J = 0 to J = 1.0 with speeds from 0 m/s to 2.445 m/s. The results of the comparative investigation cases between numerical and experiment analysis from Ka4-70 propellers with Nozzle 19A and PBCF Divergent appears that between CFD and experiments, several phenomena are seen. (i) the Ka4-70 propeller without Nozzle 19A and PBCF divergent experienced large pressure at low-speed J = 0.1 to high-speed J = 0.7, but Ka4-70 propeller with Nozzle and PBCF divergent reach highest pressure at J = 0.1 to J = 0.5; (ii) the Ka4-70 propeller without 19A nozzle and PBCF divergent increases the flow velocity at the boss cap fins but does not increase the axial induce velocity, while Ka4-70 propeller using nozzle and PBCF divergent increases the axial induce velocity of the blade, but does not increase the flow velocity of the boss cap fins; (iii) Ka4-70 propeller without Nozzle and PBCF value increase of propeller η0 to 12% when ESD added in the form of Nozzle and PBCF when J is high, from J = 0.7 to J = 1.0. ; (iv) Ka4-70 propellers with Nozzle 19A and PBCF Divergent has very similar η0 from J=0 to J=1.0. CFD approach are still appropriate to be relied upon for the overall simulation.

CFD modelling of initial stages of dam-break flow

Cases of dam failures are seen almost every year globally, and the propagation of rapidly varying unsteady flows resulting from those dam failures has significant environmental and economic consequences. A three-dimensional Computational Fluid Dynamics model was created to solve unsteady Reynolds-Averaged equations to investigate the generation and propagation of dam-break flows and reflected flood waves in the presence of a trapezoidal bottom obstacle. The dam-breaks were modelled using OpenFOAM based on the Volume of Fluid method employing the Finite Volume Method, and the performance of various Reynolds-Averaged Navier-Stokes turbulence models, including the realizable k − ε, SST k − ω (Shear Stress Transport k − ω), and v2 − f models, has been evaluated. To assess the capabilities of the different turbulence models, quantitative comparisons of numerical simulations with laboratory experiments were completed, and it was found that the SST k − ω model performed better in predicting the free surface profiles and negative bore propagation speeds than the realizable k − ε and v2 − f models.

CFD SIMULATION AND VALIDATION FOR MIXING VENTILATION SCALED-DOWN EMPTY AIRCRAFT CABIN USING OPENFOAM

An investigation into the spread of the COVID-19 virus within a confined space including an aircraft cabin is essential in order to find out the mechanism. However, this is time-consuming and limited in scope, so a computational fluid dynamics (CFD) simulation is used instead. Therefore, a prior study and an appropriate choice of turbulence model are required before the simulation. The main objective of this study is to validate and evaluate the results predicted by the Open Source Field Operation and Manipulation (OpenFOAM) software through comparison with the experimental data from the literature which was conducted using particle image velocimetry (PIV) measurement. Three different Reynolds-averaged Navier-Stokes turbulence models were selected; Re-normalisation Group k - ɛ (RNG), Realizable k - ɛ (RLZ) and Low Reynold Number (LRN) to simulate a mixing ventilation system of a scaled-down model of empty aircraft cabin. In the RNG and LRN model cases, a fairly large circulation flows were observed on the right and left sides of the model representing the passenger area. The results were also evaluated quantitatively using the factor of two of observations (FAC2) for the velocity components and turbulent kinetic energy (TKE) with root mean square error (RMSE) for the former and normalised mean square errors (NMSE) for the latter. The simulation results showed that RNG and LRN are capable of predicting the flow field well. However, for TKE prediction LRN performed better than RNG which concluded that LRN is the suitable turbulence model in simulating flow fields in investigated case.

Data-driven Reynolds-averaged turbulence modeling with generalizable non-linear correction and uncertainty quantification using Bayesian deep learning

The past few years have witnessed a renewed blossoming of data-driven turbulence models. Quantification of the concomitant modeling uncertainty, however, has mostly been omitted, and the generalization performance of the data-driven models is still facing great challenges when predicting complex flows with different flow physics not seen during training. A robust data-driven Reynolds-averaged turbulence model with uncertainty quantification and non-linear correction is proposed in this work with the Bayesian deep neural network. In this model, the Reynolds stress tensor is decomposed into linear and non-linear parts. The linear part is taken as the usual linear eddy viscosity model while the non-linear counterpart is learned by a Bayesian deep neural network. Independent tensor bases of invariants and tensors constituted by mean strain rate tensor and rotation rate tensor are embedded into the neural network to effectively consider key turbulence features in different flows. The proposed model is well validated through numerical simulations of four canonical flows that significantly deviate in geometrical configurations and/or Reynolds numbers from those in the training data. With the non-linear corrections of embedded invariants and tensors representing key features of turbulence, the proposed model not only improves the predictive capabilities of Reynolds-averaged turbulence models on the same mesh but also has better generalization performance when simulating complex turbulent flows with large scale separation. In addition, this model allows us to quantitatively demonstrate the confidence interval of the predicted flow quantities that are originated from the model itself.

Numerical investigation of the aerodynamic and wake characteristics of a floating twin-rotor wind turbine under surge motion

To elevate the power capacity of floating wind turbines and cut costs, development towards large-scale systems is underway. However, this requires solving the problem of fatigue loading on large, flexible blades, as well as corresponding improvements to production and installation technology. As an alternative to larger turbines, twin-rotor floating wind turbines have been proposed. The aerodynamic and wake interference characteristics of this twin-rotor are not yet well understood. In this study, we have investigated the power and wake characteristics of a twin-rotor floating wind turbine based on unsteady Reynolds-averaged Navier-Stokes turbulence model for a comprehensive analysis. Specifically, the power and wake recovery of a twin-rotor floating horizontal-axis wind turbine (FHAWT) under surge motion are compared in detail with those of a single floating wind turbine and bottom-fixed case, while both the effects of the three wind directions (β = 0°, 45°, 90°) and spacing ratios (S/D = 1.2, 1.375, 1.5) are considered. The results suggest that the mean power of the twin-rotor is enhanced by a maximum of 13% under surge motion compared to the single rotor, the small spacing ratio promotes the power gain. Surge motion plays a significant role in wake interaction between the twin-rotor and facilitates wake recovery. As compared to the baseline case, both power and wake recovery performance of the twin-rotor are boosted at β = 45°, with an increase in power of 12.7% as opposed to the single rotor. The modified Jensen wake model is proposed for the far-wake region and a modified Gaussian wake model is also suggested for different cases. This study can provide affiliated guidance for the application of the twin-rotor floating wind turbine.

CFD analysis on the performance of a coaxial rotor with lift offset at high advance ratios

The aerodynamic performance of an isolated coaxial rotor in forward flight is analyzed by a high-fidelity computational fluid dynamics (CFD) approach. The analysis focuses on the high-speed forward flight with an advance ratio of 0.5 or higher, which is the ratio of the forward speed to the rotor tip speed. The effect of the degree of the rolling moment on the rotor thrust, called lift offset, is studied in detail. The coaxial rotor model is a pair of contrarotating rotors, each rotor consisting of two untwisted blades with a radius of 1.016 m. The pitch angle of the blades is controlled by both collective and cyclic as in a conventional single main-rotor helicopter. CFD analysis is performed using a flow solver based on the compressible Navier-Stokes equations with a Reynolds-averaged turbulence model. Laminar/turbulent transition in the boundary layer is taken into account in the calculation. The rotor trim for target forces and moments is achieved using a gradient-based delta-form blade pitch angle adjusting technique in conjunction with CFD analysis. The reliability of the calculations is confirmed by comparison with published wind tunnel experiments and two comprehensive analyses. Applying the lift offset improves the lift-to-effective drag ratio (lift-drag ratio) and reduces thrust fluctuations. However, in the case where the advance ratio exceeds 0.6, the lift-drag ratio drops significantly even if the lift offset is 0.3. The thrust fluctuation also increases with such a high advance ratio. Detailed analysis reveals that the degradation of aerodynamic performance and vibratory aerodynamic loads is closely related to the pitch angle control to compensate for the reduction in thrust on the retreating side due to the increased reverse flow region. It is effective to reduce the collective and longitudinal cyclic pitch angles for the improvement of the aerodynamic performance of coaxial rotors with an appropriate lift offset.

Turbulence model form errors in separated flows

Physics-based uncertainty quantification is applied to a boundary layer over a flat plate with a statistically stationary separation bubble to assess sources and dynamics of model error (ME) in two-equation Reynolds-Averaged Navier-Stokes turbulence models in separated flows. Two distinct ME modes are found that correspond to the qualitative behavior of ME in a turbulent wall-bounded flow (upstream of the separation bubble) and turbulent free-shear flow (within the separation bubble), with a superposition of these ME modes in intermediate regions. A complex understanding of ME results as it changes throughout the flow, but is ultimately comprised of the elements of canonical flows.

Internal energy balance and aerodynamic heating predictions for hypersonic turbulent boundary layers

The internal energy equation for a compressible fluid can be cast in a form that relates the wall heat flux for a turbulent boundary layer to various terms integrated across the layer. We derive and utilize such a relation to investigate the ability of several Reynolds-averaged Navier-Stokes turbulence models to predict wall heat flux in a Mach 11 turbulent boundary layer. Data from a Direct Numerical Simulation (DNS) provide a means for assessing detailed contributions of the integrated model terms to the wall heat flux, helping to determine whether each model is able to obtain the ``right answer for the right reasons.''

A numerical study on natural ventilation promotion and control in experimental house with fluid diode window

The objective of this study is to investigate the effects of installing a fluid diode window (FDW) in buildings to promote natural ventilation by controlling ventilation paths. An FDW is a porous plate with an internal structure that provides varying resistance to airflow depending on the flow direction. An FDW was installed in the bathroom of an experimental house to prevent polluted air from flowing into the adjacent multipurpose room during ventilation. Computational fluid dynamics simulations with a steady Reynoldsaveraged turbulence model and tracer gas method were performed to evaluate the performance of the FDW. The results show that an open window or FDW in the bathroom reduced the concentration and increased the ventilation rate. In the same wind direction at which the airflow exits through the window or FDW, it flows out at almost the same volumetric flow rate. In the cases where the airflow enters the bathroom through the window or FDW, the effect of preventing backflow from the bathroom window to the multipurpose room can be observed when the FDW was utilised, resulting in a reduction or even prevention of the gas entering the multipurpose room.

Comparison of ethylene combustion mechanisms for the simulation of a supersonic combustion experiment

Hydrocarbon fuels have certain advantages for high speed flight over liquid hydrogen, but are difficult to simulate accurately due to the expense and complexity of the available reaction mechanisms. This paper selects 5 reduced-order chemical reaction schemes for hydrocarbon combustion and applies them to the numerical simulation of an ethylene fuelled supersonic combustion experiment. The experiment consists of an axisymmetric diffuser-cavity-combustor model flowpath, tested at the University of Queensland's T4 reflected shock tunnel at an equivalent Mach 8 flight condition and dynamic pressure of 100 kPa. Data obtained from pressure sensors on the walls is compared to a set of 3D Computational Fluid Dynamics simulations of the flowpath, which employ Reynolds-Averaged Navier-Stokes turbulence modelling and quasi-laminar chemical reactions. Simulations without fuel injection are first performed to assess the underlying validity of the numerical modelling, and the subsequent wall pressure predictions match the results of a no-fuel experiment to within the ≈ 18 kPa margins of uncertainty. In the fuel-on cases however, two shockwaves are observed to have discrepancies in their positions of up to 60 mm, and some of the pressure predictions are outside of the ≈ 25 kPa mean uncertainty of the fuel-on measurements. Of the five tested mechanisms, three correctly predicted ignition of the fuel and two did not, the latter providing a poor comparison to experimental data. The three successful mechanisms produced very similar results to each other, which broadly replicate the pressure rise seen in the experiments but not the precise location of the combustion induced shockwaves. The results imply that any of these three schemes could be used, with caution, to roughly predict ignition behaviour and combustion dynamics as long as conditions are similar enough to those considered in this paper. Future work should be directed at relaxing this caveat by additional testing at different conditions and applying the same mechanisms to other experimental geometries.

Endwall Heat Transfer and Cooling Performance of a Transonic Turbine Vane With Upstream Injection Flow

Abstract Flow fields near the turbine vane endwall region are very complicated due to the presence of high three-dimensional passage vortices and endwall secondary flows. This makes it challenging for the endwall to be effectively cooled by using traditional endwall cooling methods, such as impingement cooling combined with local film cooling inside the vane passage. One effective endwall cooling scheme: coolant injection flow through discrete holes upstream of the vane leading edge on the endwall, has been considered by many gas turbine companies. The present paper focuses on endwall film cooling effectiveness evaluation with upstream coolant injection through discrete holes. Detailed experimental and numerical studies on endwall heat transfer and cooling performance with coolant injection flow through upstream discrete holes are presented in this paper. High-resolution heat transfer coefficient (HTC) and adiabatic film cooling effectiveness values were measured using a transient infrared thermography technique on an axisymmetric contoured endwall. The endwall tested was a scaled up inner endwall of an industrial transonic turbine vane with double-row discrete cylindrical film cooling holes located 0.39Cx upstream of the vane leading edge. The tests were performed in a transonic linear cascade blowdown wind tunnel facility. Conditions were representative of a land-based power generation turbine with exit Mach number of 0.85 corresponding to exit Reynolds number of 1.5 × 106, based on exit condition and axial chord length. A high turbulence level of 16% with an integral length scale of 3.6%P was generated using inlet turbulence grid to reproduce the typical turbulence conditions in real turbine. Low temperature air was used to simulate the typical coolant-to-mainstream condition by controlling two parameters of the upstream coolant injection flow: mass flowrate to determine the coolant-to-mainstream blowing ratio (BR = 2.5, 3.5), and gas temperature to determine the density ratio (DR = 1.2). To highlight the interactions between the upstream coolant flow and the passage secondary flow combined with the influence on the endwall heat transfer and cooling performance, a comparison of CFD predictions with experimental results was performed by solving steady-state Reynolds-averaged Navier–Stokes (RANS) using the commercial CFD solver ansys fluent V.15. A detailed numerical method validation was performed for four different Reynolds-averaged turbulence models. The realizable κ−ε model was validated to be suitable to obtain reliable numerical solution. The influences of a wide range of coolant-to-mainstream blowing ratios (BR = 1.0, 1.5, 1.9, 2.5, 3.0, 3.5) were numerically studied. Complex interactions between coolant injections and secondary flows in vane passage were presented and discussed. An optimal value of the blowing ratio for the present upstream discrete film hole is also suggested based on the current study. Results indicate that for lower values of BR, the endwall coolant coverage from the upstream double-row discrete holes is strongly controlled by the passage secondary flow, thus the cooling effectiveness is very poor. As the BR increases, the strong secondary flow in vane passage can be suppressed by the coolant injections and begin to be almost eliminated when BR increases to a critical value (BR = 2.5–3.0). Beyond the critical BR, most of the injected coolant begins to lift off from the endwall and penetrate significantly into the mainstream flow, yielding inefficient endwall cooling performance.

Comprehensive validation of experimental and numerical natural ventilation predictions based on field measurement with experimental house

In this study, field measurements of the natural ventilation rate of an experimental house with a symmetrical plan and two openings in the east-west direction were conducted under various external wind direction conditions. A constant injection tracer gas flow technique was used to evaluate the ventilation rate. The 10-min average indoor concentrations and velocities at the centres of the openings were measured and analysed according to each wind direction. Following this measurement, wind tunnel experiments (WTEs) and computational fluid dynamics (CFD) simulations with a steady Reynolds-averaged turbulence model were also performed to compare their prediction accuracies against that of the field measurement data. The results show that the mean values for the velocity and concentration obtained from the WTE and CFD generally agreed with the field measurements for many wind directions. However, the WTEs and CFD simulations overestimate the concentrations (underestimate the ventilation rates), especially in the south and north wind directions perpendicular to the opening direction, and the extent of this overestimation is particularly pronounced for the CFD simulations. The prediction accuracy of the CFD simulations may be improved by using a turbulence model that considers the large-scale fluctuations of airflow around the building. However, the main reason for the overestimations by both CFD and WTEs in these wind directions was that the unsteadiness of the external wind direction during the averaging time was not well considered in these methods. This might be the next issue to be solved for more accurately reproducing the actual ventilation rate.

Density-based turbulence damping at large-scale interface for Reynolds-averaged two-fluid models

The overestimation of turbulence near large-scale interfaces in Reynolds-averaged per-phase turbulence models is a well-known problem. Herein, this is addressed by analyzing the turbulence transport equations and identifying the intrinsic limitation in the formulation of per-phase turbulence models for stratified flow. A novel density-based turbulence damping approach is proposed and compared to a conventional phenomenological damping model and a mixture model for pressurized co- and counter-current as well as gravity-driven stratified air-water flows. The novel approach performs well for all test cases. It is shown that the turbulence damping in the less dense phase is more important than that towards the denser phase. The density-based damping approach leads to more realistic turbulent kinetic energy profiles across the interface region compared to typical existing phenomenological damping approaches. Further, the novel approach does not rely on calibration parameters nor interface indicator functions and is thus well suited for further development of per-phase generalized models.

WITHDRAWN: Density-based turbulence damping at large-scale interface for Reynolds-averaged two-fluid models

Abstract The overestimation of turbulence near large-scale interfaces in Reynolds-averaged per-phase turbulence models is a well-known problem. Herein, this is addressed by analyzing the turbulence transport equations and identifying the intrinsic limitation in the formulation of per-phase turbulence models for stratified flow. A novel density-based turbulence damping approach is proposed and compared to a conventional phenomenological damping model and a mixture model for pressurized co- and counter-current as well as gravity-driven stratified air-water flows. The novel approach performs well for all test cases. It is shown that the turbulence damping in the less dense phase is more important than that towards the denser phase. The density-based damping approach leads to more realistic turbulent kinetic energy profiles across the interface region compared to typical existing phenomenological damping approaches. Further, the novel approach does not rely on calibration parameters nor interface indicator functions and is thus well suited for further development of per-phase generalized models.