RNG Turbulence Model Research Articles

Numerical simulation of the effect of downstream material on scouring-sediment profile of combined spillway-gate

ABSTRACT Overflow structures are among the most important hydraulic structures used for measuring flow, controlling floods in reservoirs, and regulating water levels in open channels. Alternative options, such as combined structures like spillway-gates, are preferred due to their compatibility with natural and ecological needs. This study investigates the impact of different soil gradations downstream on the scouring profile of combined spillway-gate structures. Scouring and sedimentation downstream of the spillway-gate were examined under various particle sizes of 0.0008, 0.001, and 0.0014 m, with a constant density in both free and submerged flow conditions using the FLOW-3D software. In this study, the k-ε, k-ω, LES, and RNG turbulence models were evaluated, and the RNG turbulence model was selected among that group. In free flow conditions, the highest sediment deposition occurred with the smallest particle diameter. For larger particle diameters, the void spaces between the particles reduce friction and increase the movement threshold, leading to increased scouring and decreased sediment height. In submerged flow conditions, the changes in scouring for different particle sizes were minor, with results being closely aligned. In submerged flow conditions, increasing the particle diameter resulted in a decrease in sediment deposition in the post-scouring area.

A novel ventilation method: Experimental measurement and numerical simulation of vertical single-row inclined slit ventilation

Existing ventilation technologies often struggle to balance the limited airflow supply with the demands of indoor ventilation and air circulation in winter. This paper introduces a novel ventilation method, vertical single-row inclined slit ventilation (VSISV), which channels airflow through multiple inclined slits spaced at specific intervals along a vertical supply air duct. By ensuring continuity of the upstream and downstream slit jets, a continuous downward airflow is formed, enhancing both winter ventilation performance and indoor air circulation. The method was experimentally measured, with a focus on comparison to the IJV system. At the same airflow rate, it not only efficiently transports the supply air heat from the upper indoor area to the floor but also exhibits strong horizontal diffusion capability. The RNG turbulence model was selected for numerical simulation, focusing on indoor thermal comfort, air quality, and energy-saving potential. The results indicate that, at the same airflow rate, this method effectively maintains high indoor thermal comfort, directs airflow downward from the ceiling, reduces thermal stratification in the upper zone, achieves a more uniform and elevated indoor temperature, enhances indoor air quality, and demonstrates substantial energy-saving potential. This provides valuable insights for energy conservation, emission reduction, and the optimization in winter building ventilation systems.

Twisted tapes in solar chimney-earth air heat exchangers for equipment downsizing and passive cooling

Harnessing solar and geothermal energy in hot and arid climates is an effective strategy to reduce building energy consumption. This research investigates integrating Twisted Tapes (TT) into Earth Air Heat Exchanger (EAHE) pipes within a Solar Chimney (SC)-EAHE system, aiming to reduce EAHE pipe length while meeting ventilation requirements and decreasing cooling demand. Two key innovations are introduced: first, this study pioneers the integration of TT into a SC-EAHE system; second, it comprehensively explores TT geometry optimization, analyzing the effects of width ratio and rotation rate across 77 configurations. Numerical simulations are conducted in ANSYS Fluent using the k-ε RNG turbulence model to investigate thermal performance and airflow characteristics. Findings reveal that TT integration results in a 153 % increase in Nusselt number. Additionally, it is shown that a 1 m SC diameter achieves ventilation of 4.8 Air Changes per Hour (ACH) using TT with a width ratio and rotation rate of 0.5, while reducing EAHE pipe length by 2.6 %. Increasing the SC diameter to 1.25 m, with a TT width ratio of 0.6 and rotation rate of 1.25, leads to a 13 % (35 m) reduction in EAHE pipe length while meeting a minimum outlet temperature of 290.15 K and ACH requirements. However, the benefit diminishes beyond a 1.25 m SC diameter. This study contributes to understanding of integrating TT into SC-EAHE systems, addressing the critical need for sustainable energy solutions and offering a novel and energy-efficient approach for passive cooling in extreme climates.

EXPERIMENTAL INVESTIGATION AND CFD SIMULATION OF THE ORIFICE FLOW METER TO MEASURE TWO-PHASE FLOW

In this study, an investigation is conducted to examine various methods for measuring the two-phase flow of water and air. The two-phase flow loop and its equipment, including the orifice flow meter, were designed and manufactured in accordance with relevant standards. By employing the orifice flow meter in the two-phase flow loop and determining the total mass flow rate of the two-phase flow passing through it, the pressure drop of the orifice flow meter was determined for different values of the flow rate and volume fractions of the water and air phases in the two-phase flow. The Reynolds number of the two-phase flow ranged from 700 to 11000, while the air volume fraction ranged from 10% to 45%. We observed that, for the orifice flow meter, the slope of the discharge coefficient variation in relation to the two-phase flow Reynolds number was higher at low Reynolds numbers, where a laminar flow pattern was established. As the two-phase flow Reynolds number increased, the flow pattern transitioned from laminar to plug flow, resulting in a decrease in the slope. The orifice flow meter under two-phase flow conditions was simulated using computational fluid dynamics (CFD) with different turbulence models. The results indicated that the k-epsilon RNG turbulence model yielded more accurate results compared to other turbulence models. This study provides a foundation for the measurement of two-phase flows in the oil and gas industries.

Study on the interaction characteristics of hull-propeller-appendages of autonomous underwater vehicle based on unsteady conditions

In order to improve the hydrodynamic performance of autonomous underwater vehicle (AUV), a new method is proposed to study the AUV hull-propeller-appendages interaction in coupled mode, considering the coupling effect between the AUV hull, propeller and appendages, and to predict the self-thrust point. Based on the Reynolds mean Navier-Stokes (RANS) solver and the RNG k−ε turbulence model, the hydrodynamic performance of the propeller and AUV is predicted. The error between the predicted results and the experimental results is within 5%, which verified the correctness of the modeling method based on CFD. Finally, the thrust and torque generated by the open-water propeller and the propeller after the boat, the hydrodynamic characteristics of the naked AUV and the AUV with appendages, and the interaction between hull-propeller-appendages under different working conditions are studied. Experiments are carried out in Qiandao Lake, Zhejiang Province, China. The numerical simulation results of AUV at 1.76 m/s are compared with the experimental results of 1.5 m/s, and the correctness of the proposed interaction method is verified. The research results not only provide theoretical guidance for the design and operation of AUV, but also have important significance for improving the performance and reliability of AUV in complex underwater environments.

Numerical Investigation of Flow Distribution in PEMFC and Pemwe Stacks Considering Two-Phase Flow in the Stack Headers

PEM fuel cell (PEMFC) and PEM water electrolyzer (PEMWE) stacks are assembled by connecting a series of unit cells. The stack headers are conduits formed by the ports of these unit cells; see Fig. 1(a). The flow field in the headers affects the flow uniformity of the stack, which significantly impacts the stack’s performance and lifespan. This study aims to investigate the fluid flows in the stack headers of PEMFC and PEMWE. A specific objective of this study is to understand how two-phase flow (liquid water and gases) in the headers would impact the flow distribution inside the stacks.The present study investigates the flow inside a stack by two-dimensional (2D) and three-dimensional (3D) models, as shown in Figure 1(b) and (c) respectively. The 3D model was constructed with an inlet header and an outlet header with flow resistors in between to represent the unit cells.1,2 CFD simulations using ANSYS Fluent were performed over a 100-cell PEMFC stack based on the k-ε RNG turbulence model. The two-dimensional 100-cell PEMFC stack model was built with COMSOL Multiphysics and considers the diffusion and transport, proton and electron conduction, electrochemical reactions on the catalytic layer, and heat transfer in the unit cells. The flow distribution obtained from the 3D model was used as the boundary conditions for the 2D model to calculate the effects of flow distribution on the overall performance of the stack. It should be pointed out that in the 3D models of PEMFC and PEMWE headers, simulations using the volume of fluid (VOF) model to track the liquid water droplets in a PEMFC and bubbles in a PEMWE were carried out to gain insights into the impact of two-phase flow on stack header flow sharing.The simulation results show the existence of unstable and highly transient turbulence within the PEMFC outlet header, as seen in Figs. 1(d) and 1(e). Jet flows from individual cells cause vortices inside the header, reducing the effective flow area. Liquid water accumulates on the inner wall of the header and is swept out by the cross-flow. In the dead-end region, liquid water undergoes helical motion under the influence of vortices, resulting in a longer residence time in the header and making it less prone to be expelled. This study parametrically investigated the effects of header size and air stoichiometry. As the cross-sectional area of the inlet header decreases from A=1,225 mm2 to A=857.5 mm2, the coefficient of mass flow variation within the stack increases, indicating a decrease in the uniformity of flow distribution inside the stack, as shown in Fig. 1(f). The air stoichiometry also affects the flow distribution inside the stack; as the air stoichiometry λ decreases from 2.2 to 1.4, the uniformity of gas distribution in the stack deteriorates significantly.The methodology used in the present study provides a reference for the parameter design and operating conditions of PEMFC and PEMWE stacks. It helps to gain insights into the optimization of stack header design and energy management.

Numerical analysis on hydrodynamic performance and wake recovery of helical-bladed hydrokinetic turbine: Impact of section inclination angle

In the context of exploiting renewable sources, cross-flow helical hydrokinetic turbine is conceded as one of the promising choices to unlock the potential in free-flowing water bodies. This study examines the impact of blade section inclination angle (−10° to +8°) and operational parameters, namely, tip speed ratio (0.6–1.4) and inflow velocity (1.0–2.5 m/s) on the performance as well as wake recovery of a 4-bladed rotor. The coefficients of power and section-averaged mean streamwise velocity deficit values are computed by solving the unsteady Reynolds-Averaged Navier-Stokes equations coupled with k-ε RNG turbulence model using numerical simulations to assess the performance and velocity recovery, respectively. The power coefficients are seen to be improved with an increase in negative section inclination angle and are found optimum in the range from −6° to −10° at a tip speed ratio of 1.0. The simulation results also reveal that the section inclination angle and tip speed ratio parameters significantly influence the velocity recovery. Furthermore, the performance decreases as the inflow velocity increases; however, the wake recovery is unaffected, with an identical wake length of 13 times the rotor diameter along the streamwise direction at a tip speed ratio of 1.0.

Numerical simulation of flow over short crested weirs - case study: Quarter-circular crested weir

In this study, the flow characteristics over a quarter-circular crested spillway (QCCS) were investigated using computational fluid dynamics (CFD) simulations. The simulations included an in-depth analysis of parameters such as flow velocity, pressure distribution, and streamline patterns. The RNG turbulence model was used to simulate the flow field and hydraulic characteristics. In particular, the simulations were performed at the design discharge. The numerical results were carefully compared with laboratory observations, showing that there is good agreement between the simulated results and the observed data. The results of the numerical simulation showed that the streamlines are completely in line with the curvature of the crest and that there is no separation zone (separation of the streamlines from the surface of the crest). furthermore, there is no negative pressure zone along the crest, which means that the risk of cavitation is eliminated. The highest value of velocity and the lowest pressure of flow occur at the outlet end of the crest.

CFD analysis of artificially roughened solar air heater: a comparative study of C-Shape, reverse C-Shape, and reverse R-Shape roughness element

Solar Air Heater (SAH) serves as a pivotal device for converting solar radiation into heat applications. This study focuses on enhancing the performance of SAH through modifications to the absorber surface by introducing ribs as artificial roughness. A two-dimensional CFD analysis investigates the impact of C-shape, reverse C-shape, and reverse R-shape ribs on thermal and hydraulic performance within the SAH channel. RNG (k-ϵ) turbulence model is employed for numerical simulation and model accuracy is validated against established correlations and literature results. The numerical analysis involves the designing and modelling of the SAH using a constant pitch ratio (P/e) of 14.285 and a height ratio (e/D) of 0.021. The Reynolds number (Re) from 4000 to 18000 was considered for the present study. The result reveals that the present rib geometries significantly enhance heat transfer and fluid mixing, leading to substantial improvements in Nusselt number (Nu) and friction factor (f) by 2.5 to 3.3 times and 2.8 to 5.05 times respectively over smooth duct SAH. The C-shape rib configuration attains the highest Thermohydraulic Performance Factor (THPF) at approximately 2.0078 for a Re of 4000 over reverse C-shape, and reverse R-shape ribs.

Numerical analysis of turbulent flow and heat transfer enhancement using V-shaped grooves mounted on the rotary kiln’s outer walls

Rotary kilns have been widely employed in various industrial uses, especially the cement production. This article deals with enhancing the thermal performance of a rotary kiln duct with V-shaped grooves mounted on the outer wall. Four V-shaped grooves with different depths h/D ranging from 0.1 to 0.4 were designed. The Reynolds Averaged Navier–Stokes equations (RANS) of two-dimensional steady-state flow are used to model the governing flow equations by using the finite volume approach (FVM) in FLUENT. k-ε standard, k-ε Realizable, k-ω SST and k-ε RNG turbulence models of the RANS approach and the k -ω SST model has been adopted to validate CFD results. In this study, the numerical results have revealed that the increase in groove depth decrease the temperature of the rotary kiln’s outer wall than the smooth walls and gives the largest Nu number, especially for the groove with h/D =0.3 and 0.4 depths.

The influence of blade curvature radius on the internal unsteady characteristics of a centrifugal pump

AbstractTo study the influence of the blade curvature radius on the flow‐induced vibration and noise of the centrifugal pump, the internal sound field of the IS 80‐65‐160 centrifugal pump was numerically calculated by adjusting the blade wrap angles to change the blade curvature radius. The RNG (Reynolds Time Average Simulation) turbulence model was selected for the numerical calculation of the centrifugal pump. The numerical simulation of the whole flow channel of the centrifugal pump was carried out using Ansys Fluent software, and the external characteristic curve of the centrifugal pump was obtained. The influence of different blade wrap angles on the internal pressure, velocity and Turbulent Kinetic Energy of the centrifugal pump was obtained. On the basis of the sound–structure coupling equation, the unsteady sound field of the centrifugal pump was calculated, and the time domain and frequency domain characteristics of the pressure pulsation at each monitoring point were obtained. The calculation results showed that the dynamic and static interference between the impeller and the volute tongue mainly caused the vibration of the centrifugal pump. By appropriately increasing the blade wrap angles, the pressure pulsation in the volute fluid domain could be effectively reduced. The vibration and noise test bench of the centrifugal pump was set up to verify the test. The experimental results showed that at the double‐blade frequency, the variation trend of the experimental value and the simulated value of the outflow flow‐induced noise of the centrifugal pump were consistent, which verified the accuracy of the sound field simulation calculation. When the blade wrap angle , it could effectively reduce the internal sound pressure level and the flow‐induced vibration and noise of the centrifugal pump. By adopting measures such as grid encryption at the position of the tongue, nonstationary calculation and comparison of different turbulence models, the accuracy of the calculation results of the internal flow field and sound field of the centrifugal pump is effectively improved, and the accuracy and reliability of the experimental results are ensured. The above research results had certain reference significance and application value for improving the working efficiency of centrifugal pumps and reducing flow‐induced vibration and noise.

Effect of Blade Twist on The Flow Characteristics of Gas-Liquid Two-Phase Flow in a Spiral Axial Flow Pump

The mixing of oil and gas forms the foundation of deep-sea oil and gas extraction and transportation. However, traditional conveying equipment has low efficiency and high failure rates. In this study, a spiral axial flow gas and liquid multiphase pump was used as the base model. The Eulerian multiphase flow model and RNG turbulence model were used for numerical simulations to analyze the internal flow field of the multiphase pump. A modification scheme was proposed to twist the airfoil shape and create a twisted vane. The twisted blade with the center of the hub-side flange chord length as the twisting center was twisted in the counterclockwise direction to help reduce the relative volume of gas in the flow channel. When the twisted vane with the hub-side airfoil type trailing edge point as the twisting center was twisted in the suction side direction, it helped to accelerate the movement of the gas-liquid mixture at the trailing edge of the back of the vane and further reduced the low velocity zone at the back of the vane. When the twist center is located at the hub side wing type trailing edge point of the twist vane, the twist degree is 0.214. This results in the maximum head and efficiency of the pump, improves gas phase aggregation phenomenon, and enhances the performance of the multiphase pump.

Experimental and numerical investigation of the thermal and dynamic behavior of a heated vortex multijet system impacting a flat plate

This study concerns the experimental and numerical study of the thermal and dynamic behavior of a configuration of a system of vortex jets impacting a flat plate, The objective of this study is to study the behavior of the thermal and dynamic field of vortex blowing of hot air from a multi-jet system impacting a flat plate. The experimental test bench comprising a support of three diffusers of diameter D, impacting the perpendicular plate. A uniform inlet temperature (T, T, T) is imposed such that the impact height H = 4D. The vortex is obtained by a vortex generator made up of 12 fins arranged at 60° from the vertical, placed just at the outlet of the diffuser. A thermo-anemometer device, to measure the blowing temperature at the point in question. The system was numerically simulated by the fluent code using a k-ε RNG turbulence model. It should be noted that the multi-jet system first appears as a free jet: going from the injection orifice to the impact zone, the axial velocity weakens, the jet undergoes considerable deflection, this is the deflection zone the velocities become mainly radial and the thickness of the boundary layer increases radially: this is the parietal flow zone, the structure of the velocity field has two zones of intense deflection with a wall jet on both sides other, favoring a good development of the resulting jet. The results show that this configuration (T, T, T) gave a better optimized distribution of temperature and velocity on the surface of the plate. This homogenization of the temperatures results from a better thermal transfer of the plate.The k-ε RNG model gave acceptable results, which coincide with those of the experimental results.

CFD simulation of a vortex-controlled diffuser for a jet engine burner

A computational flow analysis of an ideal vortex-controlled diffuser (VCD) was carried out. The simulation model used is the compressible Reynolds averaged Navier-Stokes equations(RANS), with the application of the RNG based k-ε turbulence model. The effects of important parameters like static pressure recovery, bleed fraction, position of bleed slot, have been studied and comparisons were made with respect to VCD without the bleed configuration and the following features were revealed: radial profiles of velocity at the inlet, mid-planes and exit planes, including diffuser effectiveness (i.e. static pressure recovery), diffuser efficiency, reattachment length and diffuser total pressure loss. Results obtained by applying the RNG turbulence model show an instantaneous improvement in the diffuser efficiency that happens at reasonably minimal suction rates. From the calculations, it has been verified and shown in the analysis that the effect of the bleed positioning offers advantages in relation to where it is located.

Entropía y análisis de Hurst para la determinación de la complejidad de series temporales de flujo bifásico

In this work is presented a hydrodynamic and mass transfer study using the computational fluid dynamic (CFD) for typical experimental measurements and theoretical calculations for pyrite (FeS2) dissolution in acid media using ozone as an oxidizing agent, which occurs in a laboratory glass reactor. The hydrodynamics for the reactor employed, was evaluated with the Navier-Stokes equations (κ-ɛ) RNG turbulence model and the VOF multi-phase model. The results of the hydrodynamic simulation study show that the reactor can operate at a speed range of 1.2 to 4.4 m/s at a total pressure of 12.5 Pa; the Reynolds numerical profile indicates that the flow has a turbulent behavior inside the reactor. For the mass transfer simulation, the model of finite velocity-Eddy dissipation was used for solving the kinetic of the reaction, and was performed for 3 different particle fractions: -106 +75 µm, -38 +25 µm and -25 µm. The simulation results present satisfactorily differences of 17%, 13% and 2% respectively, in comparison with experimental data for pyrite dissolved.

Design and optimization of a runner for a gravitational vortex turbine using the response surface methodology and experimental tests

In this work, four runners for a gravitational vortex turbine (GVT) were initially analyzed numerically with the aim of selecting the one with the best efficiency (η). The selected rotor was optimized using the response surface methodology (RSM). Four design factors were considered: the number of blades (M), the twist angle (λ), the relationships between the runner upper (D) and lower (d) diameters, and the upper chamber diameter (Dcd); i.e., D/Dcd and d/Dcd, respectively. For the numerical analysis, a three-dimensional (3D) computational domain in ANSYS Fluent software with the K−ϵ RNG turbulence model and the six degrees of freedom (6-DoF) user defined function (UDF) method was utilized for the unsteady flow simulations. The η versus (vs.) the angular velocity (ω) curve was monitored during the CCD for all the treatments tested. The highest η was 0.522 under optimal design conditions; i.e., for M, λ, D/Dcd and d/Dcd equal to 6, 55°, 0.5 and 0.23, respectively. The optimal runner was built using a 3D printing and was experimentally tested utilizing a hydraulic bench. The experimental and numerical η vs. ω curves were compared. A difference of 5.1% between the maximum values of η was found.

Numerical investigation of air entrainment and wall temperature of a real-scale infrared suppression device with slots of gradient height

The high-temperature funnel walls of the infrared suppression (IRS) device are significant sources of infrared radiation on warships, so that the cooling of the funnel walls becomes indispensable for achieving infrared stealth. In this study, the air entrainment and wall temperature distribution of a real-scale IRS device are investigated in detail. Numerical computations are performed to solve the relevant governing equations in a 2D computational domain. The RNG turbulence model and the enhanced wall treatment are used in the simulations. The present work innovatively proposes to optimize the performance of the IRS device by varying the height ratio of slots between funnels. The results show that the device has the optimal air entrainment rate and temperature distribution on the funnel walls for a slot height ratio of 1.6. In addition, the impacts of nozzle Reynolds number, funnel and nozzle overlap on the air suction and funnel wall temperature are discussed as well. When the nozzle Reynolds number gets higher, an increment in air entrainment and a decrement in funnel wall temperature are observed. The negative funnel overlap and negative nozzle overlap increase the air entrainment, but adversely affect the cooling of the funnel walls.

Effect of geometric shapes of chimney weir on discharge coefficient

This study numerically investigated the effects of geometric parameters of chimney weirs on the discharge coefficient (Cd). Here, chimney weirs with apex angles 74°, 84°, 94.4°, 106° and 116° at two crest heights were examined. The RNG turbulence model was selected due to its greater accuracy compared to k-ε, k-ω and LES methods for these types of flows. With increasing discharge and crest height, the value of Cd decreases. The effect of constant Cd showed an increase in the percentage of discharge deviation from the discharge values obtained from numerical solution. Based on dimensionless variables such as ratio of water head to the crest height, the ratio of crest length to the crest height, the ratio of the vertical distance between weir crest and the first point where the weir slope changes to the crest height, and the vertex angle of the inverted triangle, non-linear polynomial regression relations were obtained. Highlights A numerical investigation of the discharge coefficient of Chimney weir with different geometry has been done using FLOW-3D software. Chimney weirs have high reliability in measuring discharge due to less sensitivity to changes in upstream water depth. This investigation improves the design of hydraulic control structures. New equation was developed for calculating the discharge coefficient of Chimney weir.

Unmanned aerial vehicle fuel tank structure design optimization based on Flow-EFD fluid simulation

Fuel quantity indication and fuel low level warning design is an important technology for unmanned aerial vehicle design with long distance flight capability. Since the size and shape of overflow holes opened on fuel tank rib structure impact the fuel leakage rate and in turn impact whether fuel low level warning will be triggered at expected time, this article use RNG-turbulence model based on Flow-EFD fluid simulation to compare three different overflow hole design schemes and get the conclusion that several rectangular holes design has negative impact on fuel level descending and will not cause nuisance fuel low level alert under large pitch angle.

Entropy generation and CO2 emission in presence of pulsating oscillations in a bifurcating thermoacoustic combustor with a Helmholtz resonator at off-design conditions

In this work, we develop a 2D numerical model of a Y-shaped bifurcating combustor with a Helmholtz resonator attached. Propane (C3H8) is fueled and burnt with air by applying single-step eddy dissipation combustion model and k−ϵ−RNG turbulence model for simplicity. To validate the numerical findings, experimental measurements are conducted on a bifurcating Y-shaped thermoacoustic combustor with an off-design Helmholtz resonator implemented. It is found that the frequency and amplitude of the dominant mode as experimentally measured agree well with the numerical results. Further agreement is obtained between numerically and theoretically predicted mode-shapes. With the model validated, it is applied to gain insights on the entropy generation and nonlinearity of the pulsating oscillations and the flow fluctuations across the resonator neck. It is found that the nonlinearities are originated in the unsteady heat release rate of the premixed propane flame, and the mass flow rate across the resonator neck. In addition, the rate of the entropy production depends strongly on the temperature fluctuations. Approximately 99% of the entropy production rate involves with the temperature oscillations. Furthermore, it is non-uniformly distributed along the combustor. In addition, the energy conversion rate between total heat release rate and the acoustical energy production rate is less than 0.0001%. Finally, the production of CO2 is increased exponentially, and then reduced gradually in the axial flow direction. In general, the present work provides a low-cost numerical tool of a bifurcating thermoaocustic system. It could be applied to predict the acoustic signature of the combustor and to examine and evaluate the performance of the Helmholtz resonator on attenuating self-sustained thermoacoustic oscillations.