Статті в журналах: "Reynolds's equation"

1

Wang, Guo Ping, Hua Ling Chen, She Miao Qi, and Lie Yu. "Key Problem of Solving Nonlinear Reynolds Equation." Applied Mechanics and Materials 241-244 (December 2012): 2751–57. http://dx.doi.org/10.4028/www.scientific.net/amm.241-244.2751.

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Finite element equation of the nonlinear dimensionless Reynolds equation, based on the Galerkin finite element method, was derived. Three key points of solving the equation was studied in detail, i.e. Boolean matrix was calculated under the nonlinear conditions, and a method of integrating discrete element equations was provided; Nonlinear algebraic equations set, resulted from integrated finite element equations, was obtained and a method how to substitute boundary conditions into the algebraic equations was presented; A method of calculating the Jacobi matrix of the equations set were described in this paper. All of them are crucial to solve the nonlinear Reynolds equation and helpful for promoting the further research on compliant foil gas bearing.

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2

Pereira, Bruno M. M., Gonçalo A. S. Dias, Filipe S. Cal, Kumbakonam R. Rajagopal, and Juha H. Videman. "Lubrication Approximation for Fluids with Shear-Dependent Viscosity." Fluids 4, no. 2 (May 28, 2019): 98. http://dx.doi.org/10.3390/fluids4020098.

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We present dimensionally reduced Reynolds type equations for steady lubricating flows of incompressible non-Newtonian fluids with shear-dependent viscosity by employing a rigorous perturbation analysis on the governing equations of motion. Our analysis shows that, depending on the strength of the power-law character of the fluid, the novel equation can either present itself as a higher-order correction to the classical Reynolds equation or as a completely new nonlinear Reynolds type equation. Both equations are applied to two classic problems: the flow between a rolling rigid cylinder and a rigid plane and the flow in an eccentric journal bearing.

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3

Bair, S., and M. M. Khonsari. "Reynolds Equations for Common Generalized Newtonian Models and an Approximate Reynolds-Carreau Equation." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 220, no. 4 (April 2006): 365–74. http://dx.doi.org/10.1243/13506501jet79.

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4

Lai, Y. G., and R. M. C. So. "On near-wall turbulent flow modelling." Journal of Fluid Mechanics 221 (December 1990): 641–73. http://dx.doi.org/10.1017/s0022112090003718.

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The characteristics of near-wall turbulence are examined and the result is used to assess the behaviour of the various terms in the Reynolds-stress transport equations. It is found that all components of the velocity-pressure-gradient correlation vanish at the wall. Conventional splitting of this second-order tensor into a pressure diffusion part and a pressure redistribution part and subsequent neglect of the pressure diffusion term in the modelled Reynolds-stress equations leads to finite near-wall values for two components of the redistribution tensor. This, therefore, suggests that, in near-wall turbulent flow modelling, the velocity-pressure-gradient correlation rather than pressure redistribution should be modelled. Based on this understanding, a methodology to derive an asymptotically correct model for the velocity-pressure-gradient correlation is proposed. A model that has the property of approaching the high-Reynolds-number model for pressure redistribution far away from the wall is derived. A similar analysis is carried out on the viscous dissipation term and asymptotically correct near-wall modifications are proposed. The near-wall closure based on the Reynolds-stress equations and a conventional low-Reynolds-number dissipation-rate equation is used to calculate fully-developed turbulent channel and pipe flows at different Reynolds numbers. A careful parametric study of the model constants introduced by the near-wall closure reveals that one constant in the dissipation-rate equation is Reynolds-number dependent, and a preliminary expression is proposed for this constant. With this modification, excellent agreement with near-wall turbulence statistics, measured and simulated, is obtained, especially the anisotropic behaviour of the normal stresses. On the other hand, it is found that the dissipation-rate equation has a significant effect on the calculated Reynolds-stress budgets. Possible improvements could be obtained by using available direct simulation data to help formulate a more realistic dissipation-rate equation. When such an equation is available, the present approach can again be used to derive a near-wall closure for the Reynolds-stress equations. The resultant closure could give improved predictions of the turbulence statistics and the Reynolds-stress budgets.

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5

Lee, Seungsoo, and Dong Whan Choi. "On coupling the Reynolds-averaged Navier-Stokes equations with two-equation turbulence model equations." International Journal for Numerical Methods in Fluids 50, no. 2 (2005): 165–97. http://dx.doi.org/10.1002/fld.1049.

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6

Zigrang, D. J., and N. D. Sylvester. "A Review of Explicit Friction Factor Equations." Journal of Energy Resources Technology 107, no. 2 (June 1, 1985): 280–83. http://dx.doi.org/10.1115/1.3231190.

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A review of the explicit friction factor equations developed to replace the Colebrook equation is presented. Explicit friction factor equations are developed which yield a very high degree of precision compared to the Colebrook equation. A new explicit equation, which offers a reasonable compromise between complexity and accuracy, is presented and recommended for the calculation of all turbulent pipe flow friction factors for all roughness ratios and Reynold’s numbers.

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7

Wen, Chengwei, Xianghui Meng, and Wenxiang Li. "Numerical analysis of textured piston compression ring conjunction using two-dimensional-computational fluid dynamics and Reynolds methods." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 232, no. 11 (January 31, 2018): 1467–85. http://dx.doi.org/10.1177/1350650118755248.

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The Reynolds equation, in which some items have been omitted, is a simplified form of the Navier–Stokes equations. When surface texturing exists, it may unreasonably reveal the tribological effects in some cases. In this paper, both the two-dimensional computational fluid dynamics method, which is based on the Navier–Stokes equations, and the corresponding one-dimensional Reynolds method are adopted to analyze the performance of the textured piston compression ring conjunction. To conduct a comparison between these two methods, the modified Elrod algorithm for Jakobsson–Floberg–Olsson cavitation model is chosen to solve the Reynolds equation. The results show that the Reynolds method is somewhat different from the computational fluid dynamics method in the minimum oil film thickness, pressure distribution, and cavitation at given operating conditions. Moreover, for a low ratio of texture depth to length, the Reynolds equation is still suitable to predict the overall effects of the designed groove textures. The simulation results also reveal that it is not always beneficial for the tribological performance and sometimes may increase the total friction force when the ring is textured.

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8

Wang, Fei Han, Guo Xin Yan, and Shi Jiang Zhu. "Applying Upwind Difference and Central Difference to Discrete N-S Equation Described by Stream Function." Advanced Materials Research 950 (June 2014): 205–8. http://dx.doi.org/10.4028/www.scientific.net/amr.950.205.

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To study sudden expansion flow, it took N-S equation described by stream function as governing equation and got discrete equations with upwind difference and central difference. The discrete equations are applied to calculate sudden expansion flow. The results showed that when the Reynolds number is lower, stream function distribution calculated with central difference is similar to with upwind difference and that when the Reynolds number is higher, the calculation with central difference becomes unstable, even not converged. It showed that for sudden expansion flow, the upwind difference can simulate well and the results are satisfied

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9

Liu, Jing Yuan, and Chun Hian Lee. "Development of A Two-Equation Turbulence Model for Hypersonic Shock Wave and Turbulent Boundary Layer Interaction." Applied Mechanics and Materials 66-68 (July 2011): 1868–73. http://dx.doi.org/10.4028/www.scientific.net/amm.66-68.1868.

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For hypersonic compressible turbulence, the correlations with respect to the density fluctuation must not be neglected. A Reynolds averaged K-ε model is proposed in the present paper to include these correlations, together with the Reynolds averaged Navier-Stokes equations to describe the mean flowfield. The K-equation is obtained from Reynolds averaged single-point second moment equations which are deduced from the instantaneous compressible Navier-Stokes equations. Under certain hypotheses and scales estimation of the compressible terms, the K-equation is simplified. The correlation terms of the fluctuation field appearing in the resulting K-equation, together with a conventional form of the ε-equation, are thus correlated with the variables in the average field. The new modeling coefficients of closure terms are optimized by computing the hypersonic turbulent flat-plate measured by Coleman and Stollery [J. Fliud Mech., Vol. 56 (1972), p. 741]. The proposed model is then applied to simulate hypersonic turbulent flows over a wedge compression corner angle of 34 degree. The predicting results compare favorably with the experimental results. Also, comparisons are made with other turbulence models. Additionally, an entropy modification function of Harten-Yee’s TVD scheme is introduced to reduce artificial diffusion near boundary layers and provide the required artificial diffusion to capture the shockwaves simultaneously.

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10

Yin, Zegao, Zhenlu Wang, Bingchen Liang, and Li Zhang. "Initial Velocity Effect on Acceleration Fall of a Spherical Particle through Still Fluid." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/9795286.

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A spherical particle’s acceleration fall through still fluid was investigated analytically and experimentally using the Basset-Boussinesq-Oseen equation. The relationship between drag coefficient and Reynolds number was studied, and various parameters in the drag coefficient equation were obtained with respect to the small, medium, and large Reynolds number zones. Next, some equations were used to derive the finite fall time and distance equations in terms of certain assumptions. A simple experiment was conducted to measure the fall time and distance for a spherical particle falling through still water. Sets of experimental data were used to validate the relationship between fall velocity, time, and distance. Finally, the initial velocity effect on the total fall time and distance was discussed with different terminal Reynolds numbers, and it was determined that the initial velocity plays a more important role in the falling motion for small terminal Reynolds numbers than for large terminal Reynolds number scenarios.

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11

Johansson and, Lars, and Ha˚kan Wettergren. "Computation of the Pressure Distribution in Hydrodynamic Bearings Using Newton’s Method." Journal of Tribology 126, no. 2 (April 1, 2004): 404–7. http://dx.doi.org/10.1115/1.1631009.

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In this paper an algorithm is developed where Reynolds’ equation, equilibrium equations and non-negativity of pressure are formulated as a system of equations, which are not differentiable in the usual sense. This system is then solved using Pang’s Newton method for B-differentiable equations.

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12

Elsharkawy, A. A., and B. J. Hamrock. "EHL of Coated Surfaces: Part II—Non-Newtonian Results." Journal of Tribology 116, no. 4 (October 1, 1994): 786–93. http://dx.doi.org/10.1115/1.2927333.

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A complete non-Newtonian elastohydrodynamic lubrication solution for multilayered elastic solids is introduced in this paper. A modified form for the Reynolds equation was derived by incorporating the circular non-Newtonian fluid model associated with a limiting shear strength directly into the momentum equations that govern the instantaneous equilibrium of a fluid element inside the lubricated conjunction. The modified Reynolds equation, the elasticity equations of multilayered elastic half-space, the lubricant pressure-viscosity equation, the lubricant pressure-density equation, and the load equilibrium equation were solved simultaneously by using the system approach. The effects of the surface coating on pressure profiles, film shapes, and surface shear stress profiles are shown. Furthermore, the effects of coating thickness on the minimum film thickness and on the coefficient of friction are presented for different coating materials. The results show that for hard coatings non-Newtonian fluid effects on the pressure profiles and film shapes are significant because of the increase in the contact pressure.

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13

Mitsuya, Y. "Modified Reynolds Equation for Ultra-Thin Film Gas Lubrication Using 1.5-Order Slip-Flow Model and Considering Surface Accommodation Coefficient." Journal of Tribology 115, no. 2 (April 1, 1993): 289–94. http://dx.doi.org/10.1115/1.2921004.

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A 1.5-order modified Reynolds equation for solving the ultra-thin film gas lubrication problem is derived by using an accurate higher-order slip-flow model. This model features two key differences from the current second-order slip-flow model. One is the involvement of an accommodation coefficient for momentum. The other is that the coefficient of the second-order slip-flow term is 4/9 times smaller than that for the current model. From the physical consideration of momentum transfer, the accommodation coefficient is found to have no affect on the second-order slip-flow term. Numerical calculations using the 1.5-order modified Reynolds equation are performed. The results are compared with those obtained using three kinds of currently employed modified Reynolds equations: those employing the first- and second-order slip-flow models and those utilizing the Boltzmann equation. These comparisons confirm that the present modified Reynolds equation provides intermediate characteristics between those derived from the first- and second-order slip-flow models, and produces an approximation closer to the exact solution resulting from the Boltzmann-Reynolds equation.

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14

Almqvist, T., and R. Larsson. "Some Remarks on the Validity of Reynolds Equation in the Modeling of Lubricant Film Flows on the Surface Roughness Scale." Journal of Tribology 126, no. 4 (October 1, 2004): 703–10. http://dx.doi.org/10.1115/1.1760554.

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The objective of this paper is to investigate the flow in a lubricant film on the surface roughness scale and to compare the numerical solutions obtained by two different solution approaches. This is accomplished firstly by the CFD-approach (computational fluid dynamic approach) where the momentum and continuity equations are solved separately, and secondly the Reynolds equation approach, which is a combination and a simplification of the above equations. The rheology is assumed to be both Newtonian and non-Newtonian. An Eyring model is used in the non-Newtonian case. The result shows that discrepancies between the two approaches may occur, primarily due to a singularity which appears in the momentum equations when the stresses in the lubricant attain magnitudes that are common in EHL. This singularity is not represented by the Reynolds equation. If, however, the rheology is shifted to a non-Newtonian Eyring model the deviations between the two solution approaches is removed or reduced. The second source of discrepancies between the two approaches is the film thickness to wavelength scale ω. It will be shown that the Reynolds equation is valid until this ratio is approximately O10−2.

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15

Dai, F., and M. M. Khonsari. "Generalized Reynolds Equation for Solid-Liquid Lubricated Bearings." Journal of Applied Mechanics 61, no. 2 (June 1, 1994): 460–66. http://dx.doi.org/10.1115/1.2901467.

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The continuum theory of mixture was employed to derive a generalized form of the Reynolds equation for the lubrication problems involving lubricants that contain solid particles. The derivation of the governing equations and the boundary conditions are presented. The governing equations are two coupled partial differential equations that must be solved simultaneously for the solid volume fraction and the pressure distribution in a hydrodynamic bearing. The boundary conditions allow the particles to slip at the boundary surfaces. The formulation takes the interaction between the solid and the fluid constituents into consideration. Extensive numerical results are presented for the performance parameters in finite journal bearings lubricated with granular powder mixed with Newtonian oil.

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16

Et. al., Alaa Waleed Salih. "Influence Of Rotation, Variable Viscosity And Temperature On Peristaltic Transport In An Asymmetric Channel." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 6 (April 11, 2021): 1047–59. http://dx.doi.org/10.17762/turcomat.v12i6.2417.

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In this paper we investigates the effects of each rotation, variable viscosity and temperature on the peristaltic phenomena in an asymmetric channel. The motion and heat equations are obtained in Cartesian coordinates, the dimensionless form of the governing equation are controlled by many dimensionless number e.g. Reynolds, Hartmann, Grashof , Prandle …These equations are nonlinear and to simplify ,the long wave length and low Reynolds number is used. The resulting dimensionless equation are then solved analytically by using perturbation expansion about Reynold model viscosity number. The effects of different parameter on axial velocity, stream function, pressure rise and heat distribution are analysis graphically by using the mathematica package.

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17

Kalita, Tapash Jyoti, and Punit Kumar. "Effect of Load Variation on Elastohydrodynamic Lubrication Film Collapse." Applied Mechanics and Materials 592-594 (July 2014): 1366–70. http://dx.doi.org/10.4028/www.scientific.net/amm.592-594.1366.

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Elastohydrodynamic line contact simulations have been carried out in the present study. A practical situation of transient EHL film collapse has been analyzed. The aim is to observe the effect of variation of maximum Hertzian pressure (PH) on transient behavior of EHL film thickness (H).The analysis is based upon classical Reynolds equation considering time variation. The simulation results pertaining to EHL film thickness calculated using linear pressure-viscosity relationship have been compared for different values of load. It has been observed that film thickness reduces with increase in load. Similar results are obtained using exponential pressure-viscosity relationship and compared with those for linear pressure-viscosity. The EHL equations are solved by discretizing Reynolds equation and load equilibrium equation along with other equations using Newton-Raphson technique with the help of a computer code.

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18

Le, Anh Dung, and Thi Thanh Hai Tran. "Numerical Modelization of the Oil Film Pressure for a Hydrodynamic Tilting-Pad Thrust Bearing." Journal of Science and Technology - Technical Universities 30.7, no. 146 (November 2020): 25–30. http://dx.doi.org/10.51316/30.7.5.

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This study analyses the hydrodynamic characteristic of the tilting pad thrust bearing. Research content is simultaneously solving the Reynolds equation, force equilibrium equation, and momentum equilibrium equations. Reynolds equation is solved by utilizing the finite element method with Galerkin weighted residual, thereby determines the pressure at each discrete node of the film. Force and momentums are integrated from pressure nodes by Gaussian integral. Finally, force and momentum equilibrium equations are solved using Newton-Raphson iterative to achieve film thickness and inclination angles of the pad at the equilibrium position. The results yielded the film thickness, the pressure distribution on the whole pad and different sections of the bearing respected to the radial direction. The high-pressure zone is located at the low film thickness zone and near the pivot location.

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19

Stahlberg, J. "An Improved Reynolds Technique for Approximate Solution of Linear Stochastic Differential Equations." Symposium - International Astronomical Union 157 (1993): 251–52. http://dx.doi.org/10.1017/s0074180900174224.

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Our starting point is a formal linear stochastic differential equation of first order (higher order equations can be transformed to systems of these) where I, a, W are stochastic functions with and analogously for a and W. I, a, and W are allowed to depend on the element ω of a set Ω in which a probability measure is defined in the usual way (see e.g. Doob, 1953; de Witt-Morette, 1981). To get a solution of eq.(1) for the mean intensity we treat the problem according the Reynolds averaging technique in the usual manner : The stochastic equation is changed into an infinte hierarchical system of equations for the correlations.

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20

Casey, M. V., and C. J. Robinson. "A unified correction method for Reynolds number, size, and roughness effects on the performance of compressors." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 225, no. 7 (August 5, 2011): 864–76. http://dx.doi.org/10.1177/0957650911410161.

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An equation is derived that relates the changes in turbomachinery efficiency with Reynolds number to the changes in the friction factor of an equivalent flat plate. This equation takes into account the different Reynolds number and roughness dependencies of the individual components, and can be used for whole stages and multistage machines. The new method is sufficiently general to correct for changes in Reynolds number due to changes in fluid properties or speed, changes in machine size, or changes in the surface roughness of components for all types of turbomachinery, but is calibrated here for use on axial and radial compressors. The method uses friction factor equations for a flat plate which include fully rough behaviour above an upper critical Reynolds number, a transition region depending on roughness and a region with laminar flow below the lower critical Reynolds number. The correction equation for efficiency includes a single empirical factor. Based on a simple loss analysis and a calibration with over 30 sets of experimental test data covering a wide range of machine types, a suggestion for the variation of this factor with specific speed has been made. Additional correction equations are derived for the shift in flow and the change in pressure rise with Reynolds number and these are also calibrated against the same data.

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21

Shirazi, S. A., and C. R. Truman. "Prediction of Turbulent Source Flow Between Corotating Disks With an Anisotropic Two-Equation Turbulence Model." Journal of Turbomachinery 110, no. 2 (April 1, 1988): 187–94. http://dx.doi.org/10.1115/1.3262179.

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An anisotropic form of a low-Reynolds-number two-equation turbulence model has been implemented in a numerical solution for incompressible turbulent flow between corotating parallel disks. Transport equations for turbulent kinetic energy and dissipation rate were solved simultaneously with the governing equations for the mean-flow variables. Comparisons with earlier mixing-length predictions and with measurements are presented. Good agreement between the present predictions and the measurements of velocity components and turbulent kinetic energy was obtained. The low-Reynolds-number two-equation model was found to model adequately the near-wall region as well as the effects of rotation and streamline divergence, which required ad hoc assumptions in the mixing-length model.

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22

Fan, Yi Qiang, M. Miyatake, S. Kawada, Bin Wei, and S. Yoshimoto. "Inertial effect on gas squeeze film for large radius disc excited by standing waves with complex modal shapes." International Journal of Modern Physics B 33, no. 24 (September 30, 2019): 1950282. http://dx.doi.org/10.1142/s0217979219502825.

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In order to investigate the gas inertial effect on bearing capacity of acoustic levitation on condition of complex exciting shapes, a new kind of numerical model including inertial effect in cylindrical coordinates was proposed. The inertial terms in Navier–Stokes equations are packaged to derive modified Reynolds equations. The amplitudes of standing waves were tested by distance probe in experiment and film thickness equation were reconstructed by sum of the sinusoidal functions. The theoretical and experimental results implied that the inertial effect is strongly related to the exciting modal shapes. It is concluded that the proposal of modified Reynolds equation can provide more optimized numerical solutions to solve the problems about the deviation between theoretical and experimental data.

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23

Hall, Philip. "A phase-equation approach to boundary–layer instability theory: Tollmien-Schlichting waves." Journal of Fluid Mechanics 304 (December 10, 1995): 185–212. http://dx.doi.org/10.1017/s0022112095004393.

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Our concern is with the evolution of large-amplitude Tollmien-Schlichting waves in boundary-layer flows. In fact, the disturbances we consider are of a comparable size to the unperturbed state. We shall describe two-dimensional disturbances which are locally periodic in time and space. This is achieved using a phase equation approach of the type discussed by Howard & Kopell (1977) in the context of reaction-diffusion equations. We shall consider both large and O(1) Reynolds number flows though, in order to keep our asymptotics respectable, our finite-Reynolds-number calculation will be carried out for the asymptotic suction flow. Our large-Reynolds-number analysis, though carried out for Blasius flow, is valid for any steady two-dimensional boundary layer. In both cases the phase-equation approach shows that the wavenumber and frequency will develop shocks or other discontinuities as the disturbance evolves. As a special case we consider the evolution of constant frequency/wavenumber disturbances and show that their modulational instability is controlled by Burgers equation at finite-Reynolds-number and by a new integro-differential evolution equation at large-Reynolds-numbers. For the large Reynolds number case the evolution equation points to the development of a spatially localized singularity at a finite time.

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24

Sadeghi, Farshid, and Thomas A. Dow. "Thermal Effects in Rolling/Sliding Contacts: Part 2—Analysis of Thermal Effects in Fluid Film." Journal of Tribology 109, no. 3 (July 1, 1987): 512–17. http://dx.doi.org/10.1115/1.3261489.

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A two dimensional numerical solution to the problem of thermal elastohydrodynamic lubrication of rolling/sliding contacts was obtained using a finite difference formulation. The technique involves the simultaneous solution of the thermal Reynolds’ equation, the elasticity equation, and the two dimensional energy equation. A pressure and temperature dependent viscosity for a synthetic paraffinic hydrocarbon lubricant (XRM-109F) was considered in the solution of the Reynolds’ and energy equations. The experimental pressure and surface temperature measurements obtained by Dow and Kannel [1] were used in evaluating the results of the numerical analysis for the cases of pure rolling and slip conditions.

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25

KIM, INCHUL, SAID ELGHOBASHI, and WILLIAM A. SIRIGNANO. "On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers." Journal of Fluid Mechanics 367 (July 25, 1998): 221–53. http://dx.doi.org/10.1017/s0022112098001657.

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The existing model equations governing the accelerated motion of a spherical particle are examined and their predictions compared with the results of the numerical solution of the full Navier–Stokes equations for unsteady, axisymmetric flow around a freely moving sphere injected into an initially stationary or oscillating fluid. The comparison for the particle Reynolds number in the range of 2 to 150 and the particle to fluid density ratio in the range of 5 to 200 indicates that the existing equations deviate considerably from the Navier–Stokes equations. As a result, we propose a new equation for the particle motion and demonstrate its superiority to the existing equations over a range of Reynolds numbers (from 2 to 150) and particle to fluid density ratios (from 5 to 200). The history terms in the new equation account for the effects of large relative acceleration or deceleration of the particle and the initial relative velocity between the fluid and the particle. We also examine the temporal structure of the near wake of the unsteady, axisymmetric flow around a freely moving sphere injected into an initially stagnant fluid. As the sphere decelerates, the recirculation eddy size grows monotonically even though the instantaneous Reynolds number of the sphere decreases.

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26

Gans, R. F. "Lubrication Theory at Arbitrary Knudsen Number." Journal of Tribology 107, no. 3 (July 1, 1985): 431–33. http://dx.doi.org/10.1115/1.3261103.

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It is demonstrated that the slip flow Reynolds equations for ultra low clearance gas bearings can be derived from kinetic theory by an approximation scheme appropriate for arbitrary Knudsen numbers. Thus the usefulness of the slip flow Reynolds equation is extended to cases where it would not be expected to hold.

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27

Sotiropoulos, F., and V. C. Patel. "Application of Reynolds-Stress Transport Models to Stern and Wake Flows." Journal of Ship Research 39, no. 04 (December 1, 1995): 263–83. http://dx.doi.org/10.5957/jsr.1995.39.4.263.

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ABSTRACT The Reynolds-averaged Navier-Stokes equations are solved to assess the importance of the turbulence model in the prediction of ship stern and wake flows. Solutions are obtained with a two-equation scalar turbulence model and a seven-equation Reynolds-stress tensor model, both of which resolve the flow up to the wall, holding invariant all aspects of the numerical method, including solution domain, initial and boundary conditions, and grid topology and density. Calculations are carried out for two tanker forms used as test cases at recent workshops, and solutions are compared with each other and with experimental data. The comparisons reveal that the Reynolds-stress model accurately predicts most of the experimentally observed flow features in the stern and near-wake regions whereas the two-equation model predicts only the overall qualitative trends. In particular, solutions with the Reynolds-stress model clarify the origin of the stern vortex.

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28

Hsiao, Hsing-Sen S., Bernard J. Hamrock, and John H. Tripp. "Finite Element System Approach to EHL of Elliptical Contacts: Part I—Isothermal Circular Non-Newtonian Formulation." Journal of Tribology 120, no. 4 (October 1, 1998): 695–704. http://dx.doi.org/10.1115/1.2833767.

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The column continuity equation is used in formulating a modified Reynolds equation for elastohydrodynamic lubrication of elliptical contacts. A finite element method (FEM), here the Galerkin weighting method with isoparametric Q9 elements, is used to discretize the weak form of the Reynolds equation. In addition to the nodal pressures and the offset film thickness, the locations of the two-dimensional irregular free boundary are explicitly solved for by simultaneously forcing the essential and the natural Reynolds boundary conditions. Newton-Raphson’s iterations with a user-friendly yet efficient meshless scheme (i.e., automatic meshing-remeshing) are finally applied to solve these equations. A decoupled circular non-Newtonian fluid model is adapted in a way to illustrate the implementation of this new solution method. Extensive results will be given in Part II.

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29

Hashimoto, H. "Surface Roughness Effects in High-Speed Hydrodynamic Journal Bearings." Journal of Tribology 119, no. 4 (October 1, 1997): 776–80. http://dx.doi.org/10.1115/1.2833884.

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This paper describes an applicability of modified Reynolds equation considering the combined effects of turbulence and surface roughness, which was derived by Hashimoto and Wada (1989), to high-speed journal bearing analysis by comparing the theoretical results with experimental ones. In the numerical analysis of modified Reynolds equation, the nonlinear simultaneous equations for the turbulent correction coefficients are greatly simplified to save computation time with a satisfactory accuracy under the assumption that the shear flow is superior to the pressure flow in the lubricant films. The numerical results of Sommerfeld number and attitude angle are compared with the experimental results to confirm the applicability of the modified Reynolds equation in the case of two types of bearings with different relative roughness heights. Good agreement was obtained between theoretical and experimental results.

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30

Kumar, M. Phani, Sudipta De, Pranab Samanta, and Naresh Chandra Murmu. "A comprehensive numerical model for double-layered porous air journal bearing at higher bearing numbers." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 232, no. 5 (August 2, 2017): 592–606. http://dx.doi.org/10.1177/1350650117724054.

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Modeling air/gas lubricated double-layered porous journal bearing requires the solution of compressible Reynolds equation. It is observed that at higher bearing numbers, treatment of Reynolds equation with finite difference method using second-order central difference exhibits instabilities due to convective term dominance. To address such instabilities, Reynolds equation is discretized using finite volume and a third-order interpolation scheme to obtain variable values at grid point centers. Multistage Runge–Kutta method and biconjugate gradient stabilized method are used for solving governing equations at film and porous regions, respectively. Steady state and stability characteristics of finite double-layered porous air journal bearings considering Beavers–Joseph velocity slip at porous-film interface are obtained. Numerically stable results are obtained for bearing numbers up to 150 and feeding parameter values ranging from 0.01 to 10.

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31

Shah, Rehan Ali, Aamir Khan, and Amjad Ali. "Parametric analysis of magnetic field-dependent viscosity and advection–diffusion between rotating discs." Advanced Composites Letters 29 (January 1, 2020): 2633366X1989637. http://dx.doi.org/10.1177/2633366x19896373.

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The constitutive expressions of unsteady Newtonian fluid are employed in the mathematical formulation to model the flow between the circular space of porous and contracting discs. The flow behavior is investigated for magnetic field-dependent (MFD) viscosity and heat/mass transfers under the influence of a variable magnetic field. The equation for conservation of mass, modified Navier–Stokes, Maxwell, advection diffusion and transport equations are coupled as a system of ordinary differential equations. The expressions for torques and magnetohydrodynamic pressure gradient equation are derived. The MFD viscosity [Formula: see text], magnetic Reynolds number [Formula: see text], squeezing Reynolds number [Formula: see text], rotational Reynolds number [Formula: see text], magnetic field components [Formula: see text], [Formula: see text], pressure [Formula: see text] and the torques [Formula: see text], [Formula: see text] which the fluid exerts on discs are discussed through numerical results and graphical aids. It is concluded that magnetic Reynolds number causes an increase in magnetic field distributions and decrease in tangential velocity of flow field, also the fluid temperature is decreasing with increase in magnetic Reynolds number. The azimuthal and axial components of magnetic field have opposite behavior with increase in MFD viscosity.

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32

Canuto, V. M. "Theoretical modeling of convection II. Reynolds Stress Model." Proceedings of the International Astronomical Union 2, S239 (August 2006): 19–34. http://dx.doi.org/10.1017/s1743921307000063.

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AbstractThe Reynolds Stress Model (RSM) yields the dynamic equations for the second-order moments (e.g., heat fluxes) needed in the equations for the mean variables (e.g., mean temperature). The RSM equations are in general time dependent and non-local. We first discuss the “buoyancy only” case and the tests of the non-local model against a variety of data. We also “plumenize” the model in order to exhibit the up-down flows that characterize convection so as to show that a non-local RSM is fully equipped to account for the “plume aspect” of buoyant flows. Next, we extend the RSM to account for stable and/or unstable stratification and shear, a formalism that is needed to describe the overshooting region contributed by differentail rotation. We conclude by discussing the equation for the dissipation of turbulent kinetic energy which plays a key role in any RSM.

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33

Walicka, A., E. Walicki, P. Jurczak, and J. Falicki. "Thrust Bearing with Rough Surfaces Lubricated by an Ellis Fluid." International Journal of Applied Mechanics and Engineering 19, no. 4 (November 1, 2014): 809–22. http://dx.doi.org/10.2478/ijame-2014-0056.

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Abstract In the paper the influence of bearing surfaces roughness on the pressure distribution and load-carrying capacity of a thrust bearing is discussed. The equations of motion of an Ellis pseudo-plastic fluid are used to derive the Reynolds equation. After general considerations on the flow in a bearing clearance and using the Christensen theory of hydrodynamic rough lubrication the modified Reynolds equation is obtained. The analytical solutions of this equation for the cases of a squeeze film bearing and an externally pressurized bearing are presented. As a result one obtains the formulae expressing pressure distribution and load-carrying capacity. A thrust radial bearing is considered as a numerical example.

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34

Shi, Jianghai, Hongrui Cao, and Xuefeng Chen. "Effect of angular misalignment on the dynamic characteristics of externally pressurized air journal bearing." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 234, no. 2 (June 25, 2019): 205–28. http://dx.doi.org/10.1177/1350650119858243.

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This paper studies the effect of angular misalignment on the dynamic characteristics of externally pressurized air journal bearing with four degrees of freedom. The linear perturbation method is applied to the Reynolds equation to obtain the steady-state equation and perturbation equations, and then the finite difference method and an iterative method are used to solve the Reynolds equation. Various eccentricity ratios and rotating speeds are taken into considerations for the comparisons of dynamic stiffness coefficients and damping coefficients of the air journal bearing under different tilt angles. In addition, the influences of perturbation frequency ratio on the dynamic performances of the air journal bearing are also estimated.

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35

ZHOU, MING-DE, HONG-YU ZHU, KAI CHEN, and ZHEN-SU SHE. "STREAMWISE VORTICES AND REYNOLDS STRESSES OF WALL-BOUNDED TURBULENT SHEAR FLOWS." Modern Physics Letters B 24, no. 13 (May 30, 2010): 1425–28. http://dx.doi.org/10.1142/s0217984910023785.

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Reynolds Equation is the most commonly used governing equation in turbulence. However, its application in wall-bounded turbulent shear flows may involve a defect. In general, Reynolds averaging should be ensemble averaging and Reynolds stresses are supposed to express all the actions of turbulence on the mean field. In statistically steady three-dimensional flows, Reynolds stresses are usually defined as correlations of temporal velocity fluctuations so that they cannot contain the influences of steady components of streamwise vortices. This is believed to be one of the reasons why many closure models in RANS meet problems in flows where streamwise vortices play significant roles. In this paper, Spatial-Temporal (S-T) averaged Reynolds stresses were defined, which separates the turbulence actions caused by temporal or spatial velocity fluctuations. DNS data for a fully developed channel flow were then used to check balancing of equations. Comparison showed that the balancing errors in the S-T averaged Reynolds equations were obviously smaller than those in the temporal averaged one, in particular, in the near wall region where the streamwise vortices located. Thus, a combination of traditional model with a supplemental model expressing influences of streamwise vortices might be a way out to improve the turbulence modeling.

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36

Sumbatyan, M. A., Ya A. Berdnik, and A. A. Bondarchuk. "AN ITERATIVE METHOD FOR THE NAVIER-STOKES EQUATIONS IN THE PROBLEM OF A VISCOUS INCOMPRESSIBLE FLUID FLOW AROUND A THIN PLATE." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 66 (2020): 132–42. http://dx.doi.org/10.17223/19988621/66/11.

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In this paper, the problem on a viscous fluid flow around a thin plate is considered using the exact Navier–Stokes equations. An iterative method is proposed for small velocity perturbations with respect to main flow velocities. At each iterative step, an integral equation is solved for a function of the viscous friction over the plate. The collocation method is used at each iteration step to reduce an integral equation to a system of linear algebraic equations, and the shooting method based on the classical fourth-order Runge-Kutta technique is applied. The solution obtained at each iteration step is compared with the Harrison–Filon solution at low Reynolds numbers, with the classical Blasius solution, and with the results computed using the direct numerical finite-volume method in the ANSYS CFX software for moderate and high Reynolds numbers. The proposed iterative method converges in a few steps. Its accuracy is rather high for small and large Reynolds number, while the error can reach 15% for moderate values.

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37

Esmaeili, L., and B. Schweizer. "Coupling of the Reynolds Fluid-Film Equation with the 2D Navier-Stokes Equations." PAMM 11, no. 1 (December 2011): 567–68. http://dx.doi.org/10.1002/pamm.201110273.

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38

Bayada, Guy, and Mich�le Chambat. "The transition between the Stokes equations and the Reynolds equation: A mathematical proof." Applied Mathematics & Optimization 14, no. 1 (April 1986): 73–93. http://dx.doi.org/10.1007/bf01442229.

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39

Sarangi, M., B. C. Majumdar, and A. S. Sekhar. "On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part I: Formulation of Stiffness and Damping Coefficients." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 219, no. 6 (June 1, 2005): 411–21. http://dx.doi.org/10.1243/135065005x34071.

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The problems of stiffness and damping characteristics of isothermal elastohydrodynamic mixed lubricated point contact are evaluated numerically considering surface roughness effect including asperity contact load. A set of equations under steady-state and dynamic conditions is derived from the classical Reynolds equation, using linear perturbation method. The elasticity equation and steady-state Reynolds equation are solved simultaneously for finding the steady-state pressure distribution, using finite difference method. Then, the set of perturbed equations is solved for the dynamic pressure distribution in the contact. A Gaussian surface roughness is adopted to model both surface roughness and mixed lubrication. Total load capacity of the contact is calculated from the lubricant film pressure and contact pressure distribution. Results are compared with those of smooth isothermal cases. The stiffness and damping coefficients of the contact are determined using the dynamic pressures. The asperity contact stiffness is calculated separately. Effect of various design parameters on stiffness and damping characteristics of a ball bearing is investigated.

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40

van Odyck, D. E. A., and C. H. Venner. "Stokes Flow in Thin Films." Journal of Tribology 125, no. 1 (December 31, 2002): 121–34. http://dx.doi.org/10.1115/1.1506317.

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Present understanding of the mechanisms of lubrication and the load carrying capacity of lubricant films mainly relies on models in which the Reynolds equation is used to describe the flow. The narrow gap assumption is a key element in its derivation from the Navier Stokes equations. However, the tendency in applications is that lubricated contacts have to operate at smaller film thickness levels, and because engineering surfaces are never perfectly smooth, locally in the film this narrow gap assumption may violated. In addition to this geometric limitation of the validity of the Reynolds equation may come a piezoviscous and compressibility related limitation. In this paper the accuracy of the predictions of the Reynolds model in relation to the local geometry of the gap is investigated. A numerical solution algorithm for the flow in a narrow gap has been developed based on the Stokes equations. For a model problem the differences between the pressure and velocity fields according to the Stokes model and the Reynolds equation have been investigated. The configuration entails a lower flat surface together with an upper surface (flat or parabolic) in which a local defect (single asperity) of known geometry has been embedded. It is investigated how the magnitude of the differences develops as a function of the geometric parameters of the film and the feature. Finally, it is discussed to what extend for these problems a perturbation approach can provide accurate corrections to be applied to the Reynolds solution.

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41

Xu, H., and E. H. Smith. "A New Approach to the Solution of Elastohydrodynamic Lubrication of Crankshaft Bearings." Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 204, no. 3 (May 1990): 187–97. http://dx.doi.org/10.1243/pime_proc_1990_204_094_02.

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A study of the elastohydrodynamic lubrication of crankshaft bearings in internal combustion engines is presented in this paper. A new method for simulating the performance of a finite width bearing and two-dimensional flexible structure is developed and introduced. The method is based on a finite element model of the structure and finite difference approximation of the Reynolds equation. In order to reduce the number of unknowns in the derived simultaneous equations, a modified form of the Reynolds equation is employed. The non-linear system of integro-differential equations is solved by a Newton—Raphson method. Analysis of a dynamically loaded Ruston—Hornsby bearing is performed and the method is found to be both rapid and robust. The influence of mesh and time step sizes upon the solution accuracy is studied.

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42

Yu, Liyuan, Richeng Liu, and Yujing Jiang. "A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures." Geofluids 2017 (July 6, 2017): 1–17. http://dx.doi.org/10.1155/2017/2176932.

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Selecting appropriate governing equations for fluid flow in fractured rock masses is of special importance for estimating the permeability of rock fracture networks. When the flow velocity is small, the flow is in the linear regime and obeys the cubic law, whereas when the flow velocity is large, the flow is in the nonlinear regime and should be simulated by solving the complex Navier-Stokes equations. The critical conditions such as critical Reynolds number and critical hydraulic gradient are commonly defined in the previous works to quantify the onset of nonlinear fluid flow. This study reviews the simplifications of governing equations from the Navier-Stokes equations, Stokes equation, and Reynold equation to the cubic law and reviews the evolutions of critical Reynolds number and critical hydraulic gradient for fluid flow in rock fractures and fracture networks, considering the influences of shear displacement, normal stress and/or confining pressure, fracture surface roughness, aperture, and number of intersections. This review provides a reference for the engineers and hydrogeologists especially the beginners to thoroughly understand the nonlinear flow regimes/mechanisms within complex fractured rock masses.

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43

Hong, Sung-Ho, Sang-Ik Son, and Kyung-Woong Kim. "A Comparative Study of the Navier-Stokes Equation & the Reynolds Equation in Spool Valve Analysis." Journal of the Korean Society of Tribologists and Lubrication Engineers 28, no. 5 (October 31, 2012): 218–32. http://dx.doi.org/10.9725/kstle-2012.28.5.218.

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44

Moore, J., and J. G. Moore. "Osborne Reynolds: Energy Methods in Transition and Loss Production: A Centennial Perspective." Journal of Turbomachinery 117, no. 1 (January 1, 1995): 142–53. http://dx.doi.org/10.1115/1.2835632.

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Osborne Reynolds’ developments of the concepts of Reynolds averaging, turbulence stresses, and equations for mean kinetic energy and turbulence energy are viewed in the light of 100 years of subsequent flow research. Attempts to use the Reynolds energy-balance method to calculate the lower critical Reynolds number for pipe and channel flows are reviewed. The modern use of turbulence-energy methods for boundary layer transition modeling is discussed, and a current European Working Group effort to evaluate and develop such methods is described. The possibility of applying these methods to calculate transition in pipe, channel, and sink flows is demonstrated using a one-equation, q-L, turbulence model. Recent work using the equation for the kinetic energy of mean motion to gain understanding of loss production mechanisms in three-dimensional turbulent flows is also discussed.

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45

Walicka, A., and E. Walicki. "Mechanical Parameters of the Squeeze Film Curvilinear Bearing Lubricated with a Prandtl Fluid." International Journal of Applied Mechanics and Engineering 21, no. 4 (December 1, 2016): 967–77. http://dx.doi.org/10.1515/ijame-2016-0058.

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Abstract Based upon a Prandtl fluid flow model, a curvilinear squeeze film bearing is considered. The equations of motion are given in a specific coordinate system. After general considerations on the Prandtl fluid flow these equations are used to derive the Reynolds equation. The solution of this equation is obtained by a method of successive approximation. As a result one obtains formulae expressing the pressure distribution and load-carrying capacity. The numerical examples of the Prandtl fluid flow in gaps of two simple bearings are presented.

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46

Walicka, A., E. Walicki, P. Jurczak, and J. Falicki. "Curvilinear Squeeze Film Bearing with Porous Wall Lubricated by a Rabinowitsch Fluid." International Journal of Applied Mechanics and Engineering 22, no. 2 (May 24, 2017): 427–41. http://dx.doi.org/10.1515/ijame-2017-0026.

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AbstractThe present theoretical analysis is to investigate the effect of non-Newtonian lubricant modelled by a Rabinowitsch fluid on the performance of a curvilinear squeeze film bearing with one porous wall. The equations of motion of a Rabinowitsch fluid are used to derive the Reynolds equation. After general considerations on the flow in a bearing clearance and in a porous layer using the Morgan-Cameron approximation the modified Reynolds equation is obtained. The analytical solution of this equation for the case of a squeeze film bearing is presented. As a result one obtains the formulae expressing pressure distribution and load-carrying capacity. Thrust radial bearing and spherical bearing with a squeeze film are considered as numerical examples.

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47

Vakhrushev, Aleksandr, and Eugene Molchanov. "Hydrodynamic Modeling of Electrocodeposition on a Rotating Cylinder Electrode." Key Engineering Materials 654 (July 2015): 29–33. http://dx.doi.org/10.4028/www.scientific.net/kem.654.29.

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The mathematical model of hydrodynamic mathematical modeling of copper electrodeposition on rotating cylinder electrode are presented. Mass transfer of electrolyte ions is described by diffusion-convection equation. Reynolds-averaged Navier–Stokes equations with Low Reynolds k-e model are used to describe turbulent flow of electrolyte. The results of mathematical modeling are in good agreement with the published experimental data

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48

Shen, Feng, Cheng-Jin Yan, Jian-Feng Dai, and Zhao-Miao Liu. "Recirculation Flow and Pressure Distributions in a Rayleigh Step Bearing." Advances in Tribology 2018 (June 21, 2018): 1–8. http://dx.doi.org/10.1155/2018/9480636.

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Flow characteristics in the Rayleigh step slider bearing with infinite width have been studied using both analytical and numerical methods. The conservation equations of mass and momentum were solved utilizing a finite volume approach and the whole flow field was simulated. More detailed information about the flow patterns and pressure distributions neglected by the Reynolds lubrication equation has been obtained, such as jumping phenomenon around a Rayleigh step, vortex structure, and shear stress distribution. The pressure distribution of the Rayleigh step bearing with optimum geometry has been numerically simulated and the results obtained agreed with the analytical solution of the classical Reynolds lubrication equation. The simulation results show that the maximum pressure of the flow field is at the step tip not on the lower surface and the increment of the strain rate from Navier-Stokes equation is approximately 49 percent greater than that from Reynolds theory at the step tip. It is also shown that the position of the maximum pressure of the lower surface is a little less than the length of the first region. These results neglected by the Reynolds lubrication equation are important for designing a bearing.

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49

Walicka, A., and E. Walicki. "Mechanical Parameters of the Curvilinear Squeeze Film Bearing Lubricated by a Gecim-Winer Fluid." International Journal of Applied Mechanics and Engineering 22, no. 2 (May 24, 2017): 465–73. http://dx.doi.org/10.1515/ijame-2017-0030.

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AbstractBased upon a Gecim-Winer fluid flow model, a curvilinear squeeze film bearing is considered. The equations of motion are given in a specific coordinates system. After general considerations on the Gecim-Winer fluid flow these equations are used to derive the Reynolds equation. The solution of this equation is obtained by a method of successive approximation. As a result one obtains formulae expressing the pressure distribution and load-carrying capacity. The numerical examples of the Gecim-Winer fluid flow in gaps of two simple bearings: radial and spherical are presented.

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50

Yano, Hideo, Katsuya Hirata, and Masanori Komori. "An Approximate Method for the Drags of Two-Dimensional Obstacles at Low Reynolds Numbers." Journal of Applied Mechanics 63, no. 4 (December 1, 1996): 990–96. http://dx.doi.org/10.1115/1.2787257.

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We propose a new simple method of computing the drag coefficients of two-dimensional obstacles symmetrical to the main-flow axis at Reynolds numbers less than 100. The governing equations employed in this method are the modified Oseen’s linearized equation of motion and continuity equation, and the computation is based on a discrete singularity method. As examples, simple obstacles such as circular cylinders, rectangular prisms, and symmetrical Zhukovskii aerofoils are considered. And it was confirmed that the computed drags agree well with experimental values. Besides optimum shapes of these geometries, which minimize the drag coefficients, are also determined at each Reynolds number.

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