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.

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.

Relationship between accuracy and number of velocity particles of the finite-difference lattice Boltzmann method in velocity slip simulations

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Minoru Watari
Keyword(s):  
Relaxation Process ◽  
Stokes Equations ◽  
Velocity Slip ◽  
Simulation Models ◽  
Knudsen Layer ◽  
Navier Stokes ◽  
Flow Area ◽  
Navier Stokes Equations ◽  
Conservation Of Momentum ◽  
Boltzmann Method

Relationship between accuracy and number of velocity particles in velocity slip phenomena was investigated by numerical simulations and theoretical considerations. Two types of 2D models were used: the octagon family and the D2Q9 model. Models have to possess the following four prerequisites to accurately simulate the velocity slip phenomena: (a) equivalency to the Navier–Stokes equations in the N-S flow area, (b) conservation of momentum flow Pxy in the whole area, (c) appropriate relaxation process in the Knudsen layer, and (d) capability to properly express the mass and momentum flows on the wall. Both the octagon family and the D2Q9 model satisfy conditions (a) and (b). However, models with fewer velocity particles do not sufficiently satisfy conditions (c) and (d). The D2Q9 model fails to represent a relaxation process in the Knudsen layer and shows a considerable fluctuation in the velocity slip due to the model’s angle to the wall. To perform an accurate velocity slip simulation, models with sufficient velocity particles, such as the triple octagon model with moving particles of 24 directions, are desirable.

Influence of Inertia Terms on High Pressure Gap Flow Applications in Hydraulics

2019 ◽  
Author(s):  
Felix Fischer ◽  
Andreas Rhein ◽  
Katharina Schmitz
Keyword(s):  
High Pressure ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Fluid Phase ◽  
Simulation Models ◽  
Rigid Bodies ◽  
Navier Stokes ◽  
Dependent Manner ◽  
Fluid Inertia ◽  
Navier Stokes Equations

Abstract Hydraulic pumps, which reach pressures up to 3000 bar, are often realized as plunger-piston type pumps. In the case of a common-rail pump for diesel injection systems, the plunger is driven by a cam-tappet construction and the contact during suction stroke is maintained by a helical spring. Many hydraulic piston-based high pressure pumps include gap seals, which are formed by small clearances between the two surfaces of the piston and the bushing. Usually the gap height is in the magnitude of several micrometers. Typical radial gaps are between 0.5 and 1 per mil of the nominal diameter. These gap seals are used to allow and maintain pressure build up in the piston chamber. When the gap is pressurized, a special flow regime is reached. For the description of this particular flow the Reynolds equation, which is a simplification of the Navier-Stokes equations, can be used as done in the state of the art. Furthermore, if the pressure in the gap is high enough — 500 bar and above — fluid-structure interactions must be taken into account. Pressure levels above 1500 or 2000 bar indicate the necessity for solving the energy equation of the fluid phase and the rigid bodies surrounding it. In any case, the fluid properties such as density and viscosity, have to be modelled in a pressure dependent manner. This means, a compressible flow is described in the sealing gap. Viscosity changes in magnitudes while density remains in the same magnitude, but nevertheless changes about 30 %. These facts must be taken into account when solving the Reynolds equation. In this paper the authors work out that the Reynolds equation is not suitable for every piston-bushing gap seal in hydraulic applications. It will be shown that remarkable errors are made, when the inertia terms in the Navier-Stokes equations are neglected, especially in high pressure applications. To work out the influence of the inertia terms in these flows, two simulation models are built up and calculated for the physical problem. One calculates the compressible Reynolds equation neglecting the fluid inertia. The other model, taking the fluid inertia into account, calculates the coupled Navier-Stokes equations on the same geometrical boundaries. Here, the so called SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm is used. The discretization is realized with the Finite Volume Method. Afterwards, the solutions of both models are compared to investigate the influence of the inertia terms on the flow in these specific high pressure applications.

Analysis of Three-Lobe Bearings Having Non-Newtonian Lubricants by a Finite Element Method

1983 ◽  
Vol 7 (1) ◽  
pp. 7-11 ◽  
Author(s):  
D.V. Singh ◽  
R. Sinhasan ◽  
S.P. Tayal
Keyword(s):  
Shear Stress ◽  
Stokes Equations ◽  
Load Capacity ◽  
Navier Stokes ◽  
Shear Strain Rate ◽  
Navier Stokes Equations ◽  
Viscosity Term ◽  
Static Performance ◽  
Iteration Technique ◽  
Element Method

Additives are extensively used in the commercial lubricants to improve their specific qualities. These lubricants are therefore non-Newtonian and their nonlinear relations between shear stress and shear strain rate are generally represented by cubic shear stress laws. The Navier-Stokes equations and the continuity equation in clindrical coordinates, representing the flow-field in the clearance space of each lobe of the three-lobe hydrodynamic journal bearings having Newtonian fluids, are solved by the finie element method using Galerkin’s technique. The solution for non-Newtonian lubricants is obtained by an iteration technique modifying the viscosity term in each iteration. The static performance characteristics have been obtained for both Newtonian and the non-Newtonian lubricants. The load capacity and friction of the bearing decrease with increase in the nonlinearity of the lubricant whereas the end flow is relatively unaffected.

Inertial effect on gas squeeze film for large radius disc excited by standing waves with complex modal shapes

2019 ◽  
Vol 33 (24) ◽  
pp. 1950282 ◽  
Author(s):  
Yi Qiang Fan ◽  
M. Miyatake ◽  
S. Kawada ◽  
Bin Wei ◽  
S. Yoshimoto
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Standing Waves ◽  
Inertial Effect ◽  
Squeeze Film ◽  
Navier Stokes ◽  
Large Radius ◽  
Navier Stokes Equations ◽  
Sinusoidal Functions

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.

Heat Transfer From a Circular Cylinder at Low Reynolds Numbers

1998 ◽  
Vol 120 (1) ◽  
pp. 72-75 ◽  
Author(s):  
V. N. Kurdyumov ◽  
E. Ferna´ndez
Keyword(s):  
Circular Cylinder ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Order Unity ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Prandtl Numbers ◽  
High Degree

A correlation formula, Nu = W0(Re)Pr1/3 + W1(Re), that is valid in a wide range of Reynolds and Prandtl numbers has been developed based on the asymptotic expansion for Pr → ∞ for the forced heat convection from a circular cylinder. For large Prandtl numbers, the boundary layer theory for the energy equation is applied and compared with the numerical solutions of the full Navier Stokes equations for the flow field and energy equation. It is shown that the two-terms asymptotic approximation can be used to calculate the Nusselt number even for Prandtl numbers of order unity to a high degree of accuracy. The formulas for coefficients W0 and W1, are provided.

Laminar Pipe Flow With Mass Transfer Cooling

1965 ◽  
Vol 87 (2) ◽  
pp. 252-258 ◽  
Author(s):  
Y. Peng ◽  
S. W. Yuan
Keyword(s):  
Temperature Distribution ◽  
Pipe Flow ◽  
Energy Equation ◽  
Stokes Equations ◽  
Porous Wall ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Coolant Injection ◽  
Laminar Pipe Flow ◽  
The Heat Transfer Coefficient

The effect of foreign coolant injection at the wall on the temperature distribution of a laminar flow of a fluid with variable transport and thermodynamic properties in a porous-wall pipe has been investigated. The velocity components, mass concentration, and temperature distribution were obtained by the solution of the Navier-Stokes equations, the diffusion equation, and the energy equation. A perturbation method was used to solve the first equations for small flows through the porous wall, and the eigenvalues in the latter two equations were calculated with the aid of the CDC 1604 computer. The results from this investigation depict the significant differences in both velocity distribution and temperature distribution between the present case of hydrogen coolant and the case of air coolant [1]. The results also show that the heat transfer coefficient at the wall in the present case is considerably smaller than the case of air-coolant injection.

DYNAMIC BEHAVIOR OF A STEADY FLOW IN AN ANNULAR TUBE WITH POROUS WALLS AT DIFFERENT TEMPERATURES

2009 ◽  
Vol 19 (09) ◽  
pp. 2939-2951 ◽  
Author(s):  
JACQUES HONA ◽  
ELISABETH NGO NYOBE ◽  
ELKANA PEMHA
Keyword(s):  
Steady Flow ◽  
Shooting Method ◽  
Energy Equation ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Navier Stokes ◽  
Specific Mass ◽  
Navier Stokes Equations ◽  
Different Temperatures ◽  
Annular Tube

In this paper, the axisymmetric flow of a viscous fluid through a porous annular tube with walls kept at different temperatures is studied theoretically. The physical properties of the fluid remain constant, notably its specific mass, its dynamic viscosity and its thermal diffusivity. The nondimensional parameters which the solutions of the problem depend on are defined. A numerical integration using the shooting method is applied for solving the Navier–Stokes equations and the energy equation. Bifurcation diagrams are presented and enable to highlight significant properties of the flow. Some thermal behaviors corresponding to specific values of the parameters are performed. Asymmetric solutions of the steady flow are described and some results about velocity components are also analyzed.

Determination of Dynamic Coefficients in a Hydrodynamic Journal Bearing Based on the 3-D Navier-Stokes Equations and Considering Cavitation Effects

2012 ◽  
Author(s):  
Changhu Xing ◽  
Minel J. Braun
Keyword(s):  
Pressure Distribution ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Maximum Magnitude ◽  
Navier Stokes Equations ◽  
Dynamic Coefficients ◽  
Finite Perturbation ◽  
Hydrodynamic Journal Bearing

Dynamic coefficients are very important for the stability of a hydrodynamic journal bearing and therefore for its design. In order to determine the stiffness, damping and added mass coefficients of the hydrodynamic bearing, the finite perturbation method around its stabilization position was employed. Based on the Reynolds equation with Gumbel cavitation algorithm, the maximum magnitude of the perturbation was judged by comparing results from finite perturbation (numerical way) to those from infinitesimal perturbation (additional analytical equations need to be derived based on order analysis), as well as theoretical analysis. Using the determined perturbation amplitude, the full three-dimensional Navier-Stokes equations in CFD-ACE+ were used to evaluate coefficients from an actual lubricant and compare to those obtained with Reynolds equation. Finally, a homogeneous gaseous cavitation algorithm is coupled with the Navier-Stokes equation to establish the pressure distribution in the bearing. When gas concentration was varied, the pressure distribution as well as the dynamic coefficients changed significantly.

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