Reynolds-averaged Equations Research Articles

Fast Computation of Hydrodynamic Pressure in Lubricated Contacts: Which Lp Loss is Most Suitable for Physics-Informed Neural Networks Solving the Reynolds Equation?

The frictional behavior of pneumatic seals significantly impacts the functionality of fluid power systems, particularly in fast-switching applications where precision and responsiveness are critical. However, the complex relationship between component properties and friction often makes experimental characterization infeasible or prohibitively expensive. To address this challenge, the Institute for Fluid Power Drives and Systems (ifas) developed the ifas Dynamic Seal Simulation (DDS), an iterative elastohydrodynamic lubrication (EHL) simulation capable of accurately solving the relevant partial differential equations (PDEs) [3]. Since iterative solvers are computationally intensive, neural networks have been explored as a more efficient alternative. While traditional neural networks offer computational advantages, they often lack the ability to understand the physical context of the systems they model, potentially limiting their accuracy and reliability. Physics-Informed Neural Networks (PINNs) have been introduced to overcome these limitations. PINNs integrate the governing physical laws directly into the training process, allowing them to grasp the system’s physical context. This approach opens up new possibilities, including more robust training and the ability to extrapolate beyond the training domain, thereby providing a more reliable and efficient tool for modeling the friction of seals in fluid power systems. In this paper, a previously validated hydrodynamic PINN framework [8, 9] is utilized to solve a variant of the averaged Reynolds equation across three scenarios: divergent, convergent, and curved gaps. The investigation focuses on four p-norm training metrics: L1, L2, L22, and L∞. The results indicate that the commonly used L22 metric is the most suitable for the scenarios examined.

Physics-Informed Deep Learning for Lubricated Contacts with Surface Roughness as Parameter

Physics-informed neural networks (PINNs) are developed to solve variants of averaged Reynolds equations for accurately and time-efficiently modeling pressure build-up and cavitation in sealing contacts and journal bearings with rough surfaces. We use microscale coefficients provided through Patir and Cheng’s average flow model or homogenization to integrate roughness or texture height into these macroscale equations. Based on these equations we implement parameter-dependent PINNs to solve multi-case scenarios with varying roughness or texture heights, thus investigating the adaptability and generalizability of PINNs for modeling rough lubricated interfaces. The results demonstrate the promising potential of PINNs to accelerate tribological system computations.

Surface texture transfer in skin-pass rolling under mixed lubrication

This paper presents a novel integrated approach to modeling surface texture transfer in skin-pass rolling under mixed elasto-plasto-hydrodynamic lubrication (EPHL). The innovation lies in combining discrete fast Fourier transform (DC-FFT) for precise characterisation of elastically deformed asperities on the roll surface, dynamic explicit finite element analysis (FEA) for capturing cross-scale deformations, and a transient average Reynolds equation for governing the lubrication flow. By integrating these methods, the model addresses the complex interplay between elastic roll deformation, microscale asperity-lubricant interactions, and elastoplastic strip deformation, providing a more comprehensive understanding of texture transfer mechanisms. In addition, the model predictions are validated by experimental results. Furthermore, this study investigates the effects of rolling speed and surface pattern orientation, revealing that higher speeds reduce texture transfer while surface patterns aligned with the rolling direction enhance it. These insights demonstrate the potential of this integrated modeling approach for advancing the field of skin-pass rolling.

Modeling and Validation of the Sealing Performance of High-Pressure Vane Rotary Actuator

The EHRA (Electro-Hydraulic Rotary Actuator), using a vane rotary actuator, has the advantages of a high torque density and integration and is expected to become a joint actuator for robots. This research focuses on the sealing characteristics of various parts of a vane rotary actuator. The average Reynolds equation was used to analyze the leakage characteristics at the gap. A detailed theoretical analysis was conducted on the internal leakage mechanism of a vane rotary actuator using an X-ring as the dynamic seal for the rotor vane. According to the path of internal leakage, different sealing forms are considered as a series or parallel, and the Newton iteration method is used to obtain the total internal leakage characteristics of a vane rotary actuator. It was also considered that the deformation of the vane rotary actuator caused a thicker gap, leading to an increase in internal leakage. The calculation results are consistent with the experimental data. The analysis results indicate that when estimating the internal leakage of a vane rotary actuator, it is necessary to take the pressure of the high-pressure chamber and output shaft position as inputs. This research provides a reference for an analysis of the method of internal leakage for vane rotary actuators. It provides theoretical support for designing a vane rotary actuator with more minor internal leakage and a higher volumetric efficiency.

Research on turbulence model parameters for the prediction of waterjet propulsion pump hydrodynamic performance

Abstract The hydrodynamic performance of waterjet propulsion pumps has an important impact on improving the speed of waterjet propulsion ships. High-precision prediction of the hydrodynamic performance of water-jet propulsion pumps based on CFD is of great significance to improving the pump design quality. Based on the Reynolds-averaged Naiver-Stokes equation, this paper conducts a grid-independence verification on a water jet propulsion pump based on hydrodynamic test data, uses the Latin hypercube sampling method to generate SST k-ω turbulence model parameter space calculation samples, performs steady-state calculation on all samples, analyzes the sensitivity of changes in the parameters of the turbulence model to the prediction of the external characteristics of the waterjet propulsion pump, and finally obtains the influence of changes in the parameters of the turbulence model on the prediction of the external characteristics of the pump. Among them, the sensitivities of the three parameters β *, β1 and a1 are higher. The results show that the prediction accuracy of the external characteristics of the water jet propulsion pump can be effectively improved by correcting these three highly sensitive parameters, and the correction values of the three key parameters are obtained. The prediction error of the external characteristics of the water jet propulsion pump is reduced from 4.3% to less than 2.1%. A feedforward neural network is used to establish a mapping model between the three turbulence model parameters and the head coefficient and power coefficient of the waterjet propulsion pump in order to achieve rapid prediction. The genetic algorithm is applied to optimize the weights, thresholds and other coefficients in the neural network, thereby improving the prediction accuracy. Based on the optimized neural network, the genetic algorithm is used to optimize the three turbulence model parameters, and the optimized coefficients are obtained. The optimized turbulence model parameters are put into the hydrodynamic model for RANS simulation to verify the degree of optimization of the model coefficients by the genetic algorithm.

Nonlinear friction dynamics of water-lubricated bearings under transient shocks considering cavitation and turbulence effects

This study proposes a transient elastohydrodynamic mixed lubrication (EHML) model for water-lubricated bearings (WLBs) that incorporates cavitation and turbulence effects to evaluate the nonlinear friction dynamic performance under transient shock loads. The generalized average Reynolds equation is discretized using the control volume technique (CVM), and the constrained system is solved using the Fischer-Burmeister-Newton-Schur (FBNS) method. The validity of the model is verified by a comparison of experimental data from published literature. On this basis, the effects of cavitation, turbulence, shock load amplitude, direction, time, and rotational speed on WLB nonlinear friction dynamics characteristics are investigated. The results show that cavitation induces hydrodynamic loss under transient shock conditions, significantly increases contact time and load, and exacerbates the hydrodynamic instability phenomenon. In contrast, the turbulence effect effectively reduces frictional contact. The stability of WLBs under transient shock conditions is closely related to the load offset angle. Reducing the load offset angle improves the anti-shock stability of the bearings. Nevertheless, an increase in the amplitude and duration of the shock load may result in a deterioration of the frictional contact behavior. Increasing the rotational speed appropriately favors accelerating the lubrication regime transition and improving WLB’s anti-shock stability. This study provides a reference for enhancing WLB anti-shock performance and optimizing structural design.

LOAD-CARRYING CAPACITY OF PIVOTED CURVED SLIDER BEARINGS SUBJECT TO SURFACE ROUGHNESS PATTERN AND NON-NEWTONIAN FLUID FLOW BEHAVIOR

The present paper analyzes the Rabinowitsch fluid (RF) behavior on a pivoted curved slider with surface roughness (SR) with the aid of the perturbation technique. In order to account for the random roughness of the bearing surfaces, a stochastic random variable with a nonzero mean, variance, and skewness was taken into consideration. The averaged Reynolds equation (RE) was developed using a stochastic random variable. The load carrying capacity (LCC) and centre of pressure (COP) were obtained by solving the relevant stochastically averaged RE with appropriate boundary conditions. Bearing performance was observed to suffer as a result of transverse SR; nevertheless, the bearing system's efficiency can be increased in the case of negatively-skewed SR. It was demonstrated that, unlike a traditional lubricant, the bearing can withstand a LCC even in the absence of flow. The outcomes of the current paper can be helpful in various modern mathematical physics engineering applications.

Impact of ventilation efficiency on restraining hazardous toilet plume post-flushing in indoor environments

This study analyzed the hybrid ventilation efficiency of indoor toilet plumes generated post-flushing with various designs. Single-sided natural ventilation and hybrid ventilation were numerically investigated to determine the impact of vent locations commonly found in multistorey buildings. An in-situ experiment was performed to identify the size distribution of the plume with time series. A computational fluid dynamics model based on the Reynolds-averaged naiver-Stokes equations and baseline k-ω turbulence equations was used to predict aerosol pollutant transmission. The aerosol-laden particles receded gradually impacted by the convection of the ambient source. Besides flush volume, temperature and ventilation scenarios were also related to the generation and droplet transmission. The results indicated that the ventilation rates were generally increased with decreased residence time if the turbulence was not suppressed. This study suggested that appropriate ventilation design could significantly reduce the average particle residence time, in which case it could also reduce the rate of virus infection.

Numerical analysis on lubrication failure of main bearings with lemon-shaped bearing shells in high-power diesel engine

The lubrication status of crankshaft main bearings significantly affects the stability and reliability of diesel engine operation. This paper primarily investigates the severe failure of main bearings during developing an industrial high-power 6 V diesel engine and analyzes the causes of bearing failure. Main bearings using lemon-shaped bearing shells exhibited failure after a short period of operation. Based on the multi-body dynamics model, average Reynolds equation, and contact model, a lubrication model for the main bearings was established, and the lubrication characteristics of each main bearing were simulated. The research shows that the abrupt peaks of the lemon-shaped bearing shells on the first and second main bearings are located within the area where the bearings suffer from burst pressure, resulting in additional hydrodynamic pressure and asperity contact pressure on the bearing surfaces. This makes the bearing surface prone to wear and high temperature, which are the primary causes of bearing failure. Under the operating conditions, surface damage of the bearings leads to particle abrasion and high temperature. Subsequently, the lubricating oil gradually deteriorates with the combined action of high temperature and wear particles, causing the main bearings to blacken and adhesive wear.

Space Weather with Quantified Uncertainties: Improving Space Weather Predictions with Data-Driven Models of the Solar Atmosphere and Inner Heliosphere

To address Objective II of the National Space Weather Strategy and Action Plan “Develop and Disseminate Accurate and Timely Space Weather Characterization and Forecasts” and US Congress PROSWIFT Act 116–181, our team is developing a new set of open-source software that would ensure substantial improvements of Space Weather (SWx) predictions. On the one hand, our focus is on the development of data-driven solar wind models. On the other hand, each individual component of our software is designed to have accuracy higher than any existing SWx prediction tools with a dramatically improved performance. This is done by the application of new computational technologies and enhanced data sources. The development of such software paves way for improved SWx predictions accompanied with an appropriate uncertainty quantification. This makes it possible to forecast hazardous SWx effects on the space-borne and ground-based technological systems, and on human health. Our models include (1) a new, open-source solar magnetic flux model (OFT), which evolves information to the back side of the Sun and its poles, and updates the model flux with new observations using data assimilation methods; (2) a new potential field solver (POT3D) associated with the Wang–Sheeley–Arge coronal model, and (3) a new adaptive, 4-th order of accuracy solver (HelioCubed) for the Reynolds-averaged MHD equations implemented on mapped multiblock grids (cubed spheres). We describe the software and results obtained with it, including the application of machine learning to modeling coronal mass ejections, which makes it possible to improve SWx predictions by decreasing the time-of-arrival mismatch. The tests show that our software is formally more accurate and performs much faster than its predecessors used for SWx predictions.

Diurnally Varying Ekman Layer in a Rossby Wave

Abstract Weak but persistent synoptic-scale ascent may play a role in the initiation or maintenance of nocturnal convection over the central United States. An analytical model is used to explore the nocturnal low-level jets (NLLJ) and ascent that develop in an idealized diurnally varying frictional (Ekman) boundary layer in a neutrally stratified barotropic environment when the flow aloft is a zonally propagating Rossby wave. Steady-periodic solutions are obtained of the linearized Reynolds-averaged Boussinesq-approximated equations of motion on a beta plane with an eddy viscosity that is specified to increase abruptly at sunrise and decrease abruptly at sunset. Rayleigh damping terms are used to parameterize momentum loss due to radiation of inertia–gravity waves. The model-predicted vertical velocity is (approximately) proportional to the wavenumber and wave amplitude. There are two main modes of ascent in midlatitudes, an afternoon mode and a nocturnal mode. The latter arises as a gentle but persistent surge induced by the decrease of turbulence at sunset, the same mechanism that triggers inertial oscillations in the Blackadar theory of NLLJs. If the Rayleigh damping terms are omitted, the boundary layer depth becomes infinite at three critical latitudes, and the vertical velocity becomes infinite far above the ground at two of those latitudes. With the damping terms retained, the solution is well behaved. Peak daytime ascent in the model occurs progressively later in the afternoon at more southern locations (in the Northern Hemisphere) until the first (most northern) critical latitude is reached; south of that latitude the nocturnal mode is dominant.

Effect of surface roughness on lubrication performance of water-lubricated bearing for energy recovery turbocharger

In this study, the lubrication performance of the water-lubricated bearing of the energy recovery turbocharger was studied considering the surface roughness. By simulating the rough surface of the bearing with the Weierstrass-Mandelbrot function and combining the water film thickness equation, Reynolds average equation, Greenwood-Tripp contact model and motion equation, the theoretical model of water-lubricated bearing is established and its solution is developed. The variational rules for the hydrodynamic pressure and force, asperity contact pressure and force, and other performance parameters of the water lubricated bearings during start-up and stable operation were obtained by solving this model, and the effect of different surface roughness on the start-up performance and steady operation lubrication performance is studied. The results show that the hydrodynamic force increases rapidly and the asperity contact force sharp decreases during the early start-up stage, and the smaller the surface roughness, the faster the hydrodynamic increase and asperity contact force decrease. The thickness distribution and pressure distribution of water film with different surface roughness have slight differences in stable operation. Therefore, the surface roughness has an important effect on the lubrication performance of the water lubricated bearing in the start-up stage, but has little effect on the stable operation.

Numerical investigation of mixed-phase turbulence induced by a plunging jet

In nature and engineering applications, water jet plunging acts as a key process causing interface breaking and generating mixed-phase turbulence. In this paper, high-resolution numerical simulations of the plunging of a water jet into a quiescent pool were performed to investigate the statistical properties of mixed-phase turbulence, with a special focus on the closure problem of the Reynolds-averaged equation. We conducted phase-resolved simulations, with the air–water interface captured using a coupled level-set and volume-of-fluid method. Various cases were performed to analyse the effects of the Froude number and Reynolds number. The simulation results showed that the turbulence statistics are insensitive to the Reynolds number under investigation, while the Froude number influences the flow properties significantly. To investigate the closure problem of the mean momentum equation, the turbulent kinetic energy (TKE) and turbulent mass flux (TMF) and their transport equations were analysed further. It was discovered that the balance relationship of the TKE budget terms remained similar to many single-phase turbulent flows. The TMF is an additional unclosed term in mixed-phase turbulence over the single-phase turbulence. Our simulation results showed that the production term in its transport equation was highly correlated to TKE. Based on this finding, a closure model for the production term of TMF was further proposed.

Co-adjustable guide vane and diffuser vane to improve the energy generation potential of an axial-flow pump as turbine

Operating an axial-flow pump as turbine (APAT) with a high head can be a cost-effective solution for energy recovery applications. However, axial-flow pumps are characterized by a low head, which makes it difficult for an APAT to achieve high efficiency. In this study, computational fluid dynamics–based numerical simulations were performed using steady and unsteady Reynolds-averaged Navier-Stocks equations and a shear stress transport reattachment modification turbulence model to investigate how the energy performance and operating range of an APAT can be improved under low-head conditions. The experiment was conducted to evaluate the accuracy and reliability of the numerical results. Adjusting the diffuser vane (DV) and guide vane (GV) were identified as candidate solutions. The adjusting mechanism was designed and presented in both 2D drawings and 3D model. Further simulations were performed to evaluate the effect of the clearances at the hub and shroud of the GV and DV. While the clearances at the GV had no adverse effect on the energy performance, clearances at the DV greatly reduced the energy performance because of the formation and growth of leakage vortex flows. Adjusting the DV improved the energy performance more than adjusting the GV. A co-adjustable GV and DV is a comprehensive solution that was found to increase the efficiency of the APAT by 40.15 % under low-head conditions and weaken vibration and noise by suppressing the separation flow in the GV domain.

Evaluation of Various Flow Control Methods in Reducing Drag and Aerodynamic Heating on the Nose of Hypersonic Flying Objects

Effective deduction of air heating load and drag is a critical issue in hypersonic vehicle engineering applications. In this research, seven various geometrical models have been proposed to study and compare the effect of each configuration on the flow field, drag, and aerodynamic heating deduction under the same flow conditions. The presented configurations in this study: (a) blunt-body geometry as a reference of comparison, (b) blunt-body geometry with a spike, (c) blunt-body geometry with an counter flow jet, (d) blunt-body geometry with a spike and counter flow jet, (e) blunt-body geometry with a spike and aerodisk, (f) blunt-body geometry with a spike, aerodisk, and root counter flow jet, (g) blunt-body geometry with a spike, four aerodisks and root counter flow jet. The Reynolds-Averaged equations have been solved using the Finite Volume Method (FVM) along with the shear stress turbulence model (k-ω SST). The flow is assumed compressible, steady-state, and axisymmetric with a free stream Mach number of 6. According to the study of each configuration’s performance related to the parameters of drag, maximum pressure, and maximum heat flux factors on the blunt-body walls, (g) configuration with a drag factor of 0.2699, maximum pressure factor of 209.8, and maximum heat flux factor of 25.1, has the most deduction on the blunt-body walls among the seven configurations. The deduction percentage of drag, maximum pressure, and maximum heat flux factors of (g) configuration to (a) configuration are %72.1, %94.5, and %79.9, respectively, which significantly diminished drag and heat flux. Also, the best configuration scenarios for drag and aerodynamic heating deduction are geometrical models of g, f, d, e, c, b, and a, respectively.

The vortex dynamics characteristics in a pump-turbine: A rigid vorticity analysis while varying guide vane openings in turbine mode

In off-design conditions, pump-turbines generate vortex structures of varying scales and intensities, jeopardizing energy conversion due to unstable flow phenomena. While off-design pump-turbine internal flow dynamics have been analyzed, understanding the spatiotemporal evolution of complex vortex structures within runner blade passages, which are prone to inducing hydraulic instabilities, has to be still addressed properly. This paper applies the rigid vorticity theory, also known as Liutex, to explore vortex characteristics in the pump-turbine under different guide vane openings during turbine mode operation. The investigation focuses on three main aspects: i) spatial distribution characteristics, ii) temporal evolution characteristics, and iii) vortex morphological features along with deformation mechanisms. Results showed that the suction surface side of the blade's trailing edge commonly experiences significant shear effects, while vortices tend to emerge on both pressure and suction surface sides. Under small Guide Vane Opening (GVO) condition, vortices in blade passages show periodic evolution, and Enstrophy of the Rigid Vorticity Dilatation Term (ERDT) predominantly contributes to the helical deformation of these vortices. Conversely, under large GVO condition, vortices maintain high helicity and remain stable. This study provides insights to reduce hydraulic instability in pump-turbines and optimize energy conversion efficiency. The proposed Reynolds-averaged transport equation for enstrophy of the rigid vorticity and the relative streamline coordinate system provide advanced tools for analyzing unstable flow structures.

Hybrid lubrication model study of slip ring combination seal under the influence of frictional heat

This paper presents an experimental and theoretical investigation into the factors influencing the sealing performance of combined slip rings. Factoring in the frictional heat from micro convex bodies and convective heat exchange between the slip ring, oil film, and air, we formulate a thermoelastic flow hybrid lubrication model for the combined slip ring seal. This model calculates the distribution of oil film thickness, pressure, temperature, velocity, and viscosity in the sealing zone, drawing on the generalized average Reynolds equation and the transmembrane average energy equation, in conjunction with the heat conduction equation of the slip ring. Utilizing the Archard wear model, we also examine the wear characteristics of the combined slip ring seal, providing insights into seal wear under these conditions. The model enables an analysis of the interplay between parameters and their impact on seal performance. The method proposed accurately predicts friction and leakage in line with experimental data, thereby providing a theoretical foundation for further numerical investigation of seal characteristics.

Thermal Characteristics of Dry Gas Seal in Startup Process Considering Microscale Effects

The heat generated during the startup process of dry gas seals, such as the friction heat of the asperities, viscous shear heat, and the expansion heat of the gas film, easily leads to seal failure and endangers the stable operation of turbomachinery. This study combines the statistically modified contact model and the average Reynolds equation by considering the adiabatic index and the microscale effects of dry gas seals to explore the heat generation characteristics of dry gas seals formed in series at multiple times. In particular, the change laws and influencing factors among the components, including the friction heat of asperities and the shear heat and expansion heat of the gas film and total heat, are investigated. A comparative analysis indicates that the slip flow effect increases the total heat during the startup process. The friction heat is much greater than expansion heat and shear heat. Despite constant acceleration, exponential acceleration, or Harrison acceleration, the heat changes are uneven during the startup process. In particular, the heat changes sharply in the early stage and slowly in the later stage. The Harrison acceleration mode is the most conducive to sealing stability.

A Multi-Parameter-Coupled Mixed Lubrication Model for Rolling Interface Analysis: An Investigation on Interfacial Oil Pressure and Oil Film Thickness

Utilizing the average Reynolds equation, lubrication friction theory, metal rolling deformation theory, and temperature energy theory, a multi-parameter-coupling mixed lubrication model considering the influence of surface roughness with Gaussian distribution characteristics under rolling conditions was established. The results show that the stress distribution of the lubricating oil at the rolling interface conforms to the typical characteristics of line contact stress distribution. The oil film pressure does not increase significantly with the increase in the reduction ratio. A second peak pressure appears before the outlet. The second peak pressure reaches its maximum when ε = 0.2. The rolling speed has little effect on oil film pressure, but greatly affects oil film thickness. The influence of rolling speed mainly occurs in the outlet zone and at the second peak pressure location. The oil film thickness increases with rolling speed. During rolling, the temperature rise in the rolls exhibits two peak values. The first peak appears near the neutral point, and the second peak appears near the outlet, which is also where the second peak pressure is located. The influence of temperature will reduce the peak stress on the rolls. The overall stress on the rolls will also decrease.

Study on lubrication performance of engine piston skirt with bionic design

As the main friction pair of engine, the friction loss of piston-cylinder liner assembly in the working process has become the main reason of engine friction power consumption. Many species in the biological world have formed non-smooth forms of resistance reduction after thousands of years of survival and evolution. Many people have applied this non-smooth shape of drag reduction on organisms to the surface of friction pairs in construction machinery. Based on LX108 engine, the wear and erosion resistance of Scapharca subcrenata surface is applied to the piston skirt, which is the main friction pair in the engine. Nine test schemes were designed according to orthogonal test. This design selects three factors, namely, groove distribution type, groove depth and width, groove spacing, and each factor includes three levels. The macroscopic fluid lubrication state of the whole piston skirt with bionic shape is studied. Through the change of oil film thickness caused by thermal-mechanical coupling deformation of skirt, combined with the average Reynolds equation, the hydrodynamic pressure of lubricating oil, shear stress, and skirt friction force are obtained to verify the contribution of bionic shape to piston drag reduction, wear reduction, wetting increase, and friction power consumption. The simulation and test results show that the lubrication of all parts of the bionic groove piston skirt is better than that of the standard piston; When the depth and width of groove is 0.8 mm and the spacing is 10°, the thicker the oil film thickness of skirt is; The bionic piston whose oil film bearing area is 74%–87% of the standard piston has the smallest normal pressure and friction; When the upper end of the skirt is arranged with groove shape and the lower end is arranged with narrow groove shape between wide grooves, the lubrication effect is better.