Equivalent Clearance Model for Solving Thermohydrodynamic Lubrication of Slider Bearings With Steps

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Hideki Ogata ◽  
Joichi Sugimura
Keyword(s):  
Reynolds Equation ◽  
Energy Equation ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Differential Method ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Fluid Film Bearings ◽  
Clearance Model

This study focuses on the thermohydrodynamic lubrication (THD) analysis of fluid film bearings with steps on the bearing surface, such as Rayleigh step. In general, the Reynolds equation does not satisfy the continuity of fluid velocity components at steps. This discontinuity results in the difficulty to solve the energy equation for the lubricants by finite differential method (FDM), because the energy equation needs the velocity components explicitly. The authors have solved this issue by introducing the equivalent clearance height and the equivalent gradient of the clearance height at steps. These parameters remove the discontinuity of velocity components, and the Reynolds equations can be solved for any bearing surfaces with step regions by FDM. Moreover, this method results in pseudocontinuous velocity components, which enables the energy equation to be solved as well. This paper describes this method with one-dimensional and equal grids model. The numerical results of pressure and temperature distributions by the proposed method for an infinite width Rayleigh step bearing agree well with the results obtained by solving full Navier–Stokes equations with semi-implicit method for pressure-linked equations revised (SIMPLER) method.

Hydrodynamic Behaviors of the Gas-Lubricated Film in Wedge-Shaped Microchannel

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Xueqing Zhang ◽  
Qinghua Chen ◽  
Juanfang Liu
Keyword(s):  
Wall Temperature ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Stokes Equations ◽  
Load Capacity ◽  
Velocity Slip ◽  
Navier Stokes ◽  
Hydrodynamic Properties ◽  
Obvious Effect ◽  
Navier Stokes Equations

As for the micro gas bearing operating at a high temperature and speed, one wedge-shaped microchannel is established, and the hydrodynamic properties of the wedge-shaped gas film are comprehensively investigated. The Reynolds equation, modified Reynolds equation, energy equation, and Navier–Stokes equations are employed to describe and analyze the hydrodynamics of the gas film. Furthermore, the comparisons among the hydrodynamic properties predicted by various models were performed for the different wedge factors and the different wall temperatures. The results show that coupling the simplified energy equation with the Reynolds or modified Reynolds equations has an obvious effect on the change of the friction force acting on the horizontal plate and the load capacity of the gas film at the higher wedge factor and the lower wall temperature. The velocity slip weakens the squeeze of the gas film and strengths the gas backflow. A larger wedge factor or a higher wall temperature leads to a higher gas film temperature and thus enhances the rarefaction effect. As the wall temperature is elevated, the load capacity obtained by the Reynolds equation increases, while the results by the Navier–Stokes equations coupled with the full energy equation rapidly decrease. Additionally, the vertical flow across the gas film in the Navier–Stokes equations weakens the squeeze between the gas film and the tilt plate and the gas backflow.

Thermohydrodynamic Lubrication Analysis of Slider Bearings With Steps on Bearing Surface

2009 ◽  
Author(s):  
Hideki Ogata
Keyword(s):  
Reynolds Equation ◽  
Energy Equation ◽  
Fluid Velocity ◽  
Differential Method ◽  
Fluid Film ◽  
Bearing Surface ◽  
Fluid Film Bearings ◽  
Temperature Distributions ◽  
Finite Differential Method ◽  
Good Agreement

This study focuses on the thermohydrodynamic lubrication analysis of fluid film bearings with step on the surface such as a Rayleigh step bearing. In general, the Reynolds equation does not satisfy the continuity of the fluid velocity components at steps. This discontinuity results in the difficulty to solve the energy equation for the lubricants, because the energy equation needs the velocity components explicitly. The author has solved this problem by introducing the equivalent clearance height and the equivalent gradient of clearance height at steps. These parameters remove the discontinuity of velocity components and the energy equation as well, so that one can solve these equations on all of the bearing surfaces including the step region by finite differential method (FDM). The numerical results of pressure and temperature distributions by the proposed method for a Rayleigh step bearing were compared with the results obtained by a commercial CFD package. These results showed good agreement with each other. This method is extended to 2D unequal grid problems.

scholarly journals A rigorous derivation of the stationary compressible Reynolds equation via the Navier–Stokes equations

2018 ◽  
Vol 28 (04) ◽  
pp. 697-732 ◽  
Author(s):  
I. S. Ciuperca ◽  
E. Feireisl ◽  
M. Jai ◽  
A. Petrov
Keyword(s):  
Reynolds Equation ◽  
Stokes System ◽  
Stokes Equations ◽  
A Priori ◽  
Limit Problem ◽  
Navier Stokes ◽  
Rigorous Derivation ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier–Stokes system on a thin domain. In particular, the existence of solutions to the Navier–Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the one-dimensional case.

Modeling of the Vortex-Structure in a Particle-Laden Mixing-Layer

2005 ◽  
Author(s):  
Jens A. Melheim ◽  
Stefan Horender ◽  
Martin Sommerfeld
Keyword(s):  
Vortex Structure ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Mixing Layer ◽  
Equation Model ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Calculated Concentration ◽  
Langevin Model ◽  
Particle Velocities

Numerical calculations of a particle-laden turbulent horizontal mixing-layer based on the Eulerian-Lagrangian approach are presented. Emphasis is given to the determination of the stochastic fluctuating fluid velocity seen by the particles in anisotropic turbulence. The stochastic process for the fluctuating velocity is a “Particle Langevin equation Model”, based on the Simplified Langevin Model. The Reynolds averaged Navier-Stokes equations are closed by the standard k-epsilon turbulence model. The calculated concentration profile and the mean, the root-mean-square (rms) and the cross-correlation terms of the particle velocities are compared with particle image velocimetry (PIV) measurements. The numerical results agree reasonably well with the PIV data for all of the mentioned quantities. The importance of the modeled vortex structure “seen” by the particles is discussed.

Pulsatile Waterhammer Subject to Laminar Friction

1972 ◽  
Vol 94 (2) ◽  
pp. 467-472 ◽  
Author(s):  
D. A. P. Jayasinghe ◽  
H. J. Leutheusser
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Present Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Arbitrary Velocity ◽  
Velocity Input ◽  
Special Case ◽  
Signalling Problem

This paper deals with elastic waves which may be generated in a fluid by the sudden movement of a flow boundary. In particular, an analysis of the classical piston, or signalling problem is presented for the special case of arbitrary velocity input into a stationary fluid contained in a circular, semi-infinite waveguide. The decay of the pulse, as well as the resulting flow development in the inlet region of the pipe are analyzed by means of an asymptotic expansion of the suitably nondimensionalized Navier-Stokes equations for a compressible, nonheat-conducting Newtonian fluid. The results differ significantly from those of the more conventional one-dimensional approach based on the so-called telegrapher’s equation of mathematical physics. The present theory realistically predicts the growth of a boundary layer both in time and position and, hence, it appears to represent the transient fluid motion in a manner which is physically more appealing.

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