Theoretical and Numerical Investigation of Gas Lubrication Processes on the Basis of Aerodynamic Equations

The dimensionless parameters of the complete system of Navier-Stokes equations of a compressible gas are estimated with reference to a typical gas bearing. It is found that the three-dimensional compressible boundary layer equations should be used as the determining equations for describing gas lubrication processes. After introducing certain assumptions with respect to the dimensionless parameters in the determining equations, an equation for the pressure, the generalized Reynolds equation, is obtained.Use of the spectral method of analysis makes it possible to transform the generalized Reynolds equation into a system of ordinary differential equations. An analytic solution of the entire boundary value problem is obtained for a journal bearing with fairly small eccentricity. By comparing the numerical results obtained using both the solution of the generalized Reynolds equation and the traditional theory it is possible to estimate the effect of the inertia forces, dissipation processes, and heat transfer.     

考虑磁头表面高度不连续性气膜润滑的数值模拟与有效算法

随着磁头滑块的飞行高度不断降低,给气体润滑方程的数值求解带来了诸如计算时间过长、甚至计算发散等方面的问题。为了获得1Tbit/in2的存储密度,磁头滑块尾部的最小飞行高度接近1.5nm。本文基于作者提出的修正气膜润滑方程的线性流率(LFR)模型,考虑磁头滑块表面高度的不连续性,建立了基于有限体积法的气膜润滑方程离散格式,并把网格自适应技术与多重网格法应用到离散方程的迭代算法中,发展了可模拟最小飞行高度为0.5nm时磁头滑块压力分布的数值模拟方法与有效算法。文中以一个具有复杂表面形状的磁头滑块为例,检验了计算方法与算法的有效性。数值结果表明:在磁头滑块最小飞行高度较低时,必须要考虑滑块表面高度的不连续性,否则就得不到收敛的数值计算结果;与FK-Boltzmann模型相比,LFR模型具有较高的计算效率,采用网格自适应技术与多重网格法能有效地提高求解气膜润滑方程的计算效率。     

离心惯性力效应对超临界二氧化碳干气密封流场与密封特性影响分析

为揭示离心惯性力效应对S-CO₂干气密封流场与密封特性的影响规律,以螺旋槽干气密封为研究对象,引用考虑离心惯性力效应的Reynolds方程,在考虑气膜真实气体效应、黏度随压力与温度双重变化的同时,基于N-S方程与能量守恒定律,建立了绝热状态下考虑离心惯性力效应作用的能量控制方程. 然后,采用有限差分法对压力控制方程与能量控制方程进行耦合求解,并对考虑离心惯性力效应与没有考虑离心惯性力效应下的压力分布、温度分布以及密封特性进行了分析讨论. 研究表明:离心惯性力效应具有削弱流场内压力与温度的作用;从避免凝结流动角度考虑,离心惯性力效应引起的温降将不利于S-CO₂干气密封;考虑离心惯性力效应作用时,气膜开启力在不同槽深与转速下存在最佳工况点,泄漏率随着转速的增加显著减小,而离心惯性力效应与膜厚之间没有强交互作用;考虑离心惯性力效应作用的气膜开启力、泄漏率、出口温度均比不考虑离心惯性力效应作用的小,且这种差异随着转速的增大而增加,而随着膜厚的变化没有改变. 这些结果为进一步研究S-CO₂干气密封奠定了一定的理论基础.      

A method for determining valve coefficient and resistance coefficient for predicting gas flowrate

A method is introduced to determine the valve flow coefficient and resistance coefficient with the experiment of air discharging from a reservoir, and with the least squares method to fit the cumulative molar quantities discharged. The test valve is an angle-seat valve (Type 2632, Bürkert) with different apertures. At pressure difference of about 6 bar, the choked flow occurs when the valve aperture over 60%. Both the valve coefficient and resistance coefficient model can exactly predict the flowrate for the non-choked flow, while there are larger deviations for the choked flow. The modified equation for the choked flow can improve the prediction. In the resistance coefficient model, the value of resistance coefficient and the discharged cumulative molar quantities obtained with both the compressible and incompressible assumption are very close. The compressibility of air is negligible within the experimental pressure difference of about 6 bar. The additivity of the resistance coefficient makes the model more convenient to use.     

Inertia effects in hydromagnetic lubrication

Summary A theoretical investigation of inertia effects in a squeeze film bearing with an electrically conducting lubricant in the presence of a uniform transverse magnetic field is presented. The two cases of infinitely long rectangular plates and circular plates as bearing surfaces are considered. It is shown that the load supporting capacity of the bearing increases and the squeeze decreases if the lubricant inertia effects are taken into account. However, the inertia effect becomes smaller when the strength of the magnetic field increases.     

Effect of Volume Energy Supply on the Stability of a Subsonic Vortex Flow

Calculations of the stability of an axisymmetric vortex flow of viscous heat-conducting gas with volume energy supply are presented. The unperturbed axisymmetric vortex flow was found numerically using a quasi-cylindrical approximation of the Navier-Stokes equations under the assumption of constant peripheral-velocity circulation in the ambient co-current flow. The volume energy supply in the viscous vortex core was modeled by an additional source term in the energy equation. The stability characteristics of the viscous vortex flow in a longitudinal vortex with respect to both axisymmetric and non-axisymmetric three-dimensional waves traveling along the vortex axis and corresponding to both positive and negative values of the azimuthal wave number were found using the time-dependent formulation of the linear stability theory for compressible three-dimensional plane-parallel flows.     

基于连续模型和离散模型的微尺度三维凸台轴承的气体润滑分析

为了考察应用图案化盘片实现超高密度磁存储时磁头/磁盘界面气体润滑设计理论的有效性,本文设计了1种三维凸台轴承构型,分别使用直接模拟蒙特卡罗方法（DSMC）和气体分子薄膜润滑理论（MGL）方法进行了模拟计算,考察了相对滑动速度、轴承最小间隙、凸台高度和入射速度方向对气体轴承压力分布的影响.结果表明：对于三维凸台微尺度气体轴承,MGL方法的计算结果依然与DSMC方法的结果相差不大.存在凸台接触,轴承间隙为零的情况下,气体轴承仍然具有一定的承载能力,且压力分布形状随凸台高度的变化表现出与常规非接触式气体轴承不同的规律.     

纳米尺度气体滑动轴承的DSMC模拟

依据硬盘驱动系统中读写头与盘面的构造,结合现有的楔型和阶跃型气体滑动轴承的特点,提出了一种气体滑动轴承结构,并采用直接模拟Monte Carlo(DSMC)法数值研究了滑动轴承的最小间距、移动速度和系统温度对轴承承载能力的影响.计算结果表明,所设计轴承的承载能力随移动速度的增大而增加,随最小间距的增加和系统温度的升高而减小;轴承的承载能力对轴承最小间距的变化更敏感.     

A two-phase boundary-layer model for laminar mixed-convection condensation with a noncondensable gas on inclined plates

A finite-volume-based numerical model for mixed-convection laminar film condensation from a flowing mixture of a vapor and a heavier noncondensable gas on inclined isothermal flat plates is presented. The full boundary layer equations for the liquid film and the vapor-gas mixtures (including liquid inertia and energy convection terms) are solved implicitly with appropriate liquid-mixture interface conditions. Results were obtained for three mixtures, covering wide ranges of liquid Prandtl number and free-stream gas concentration in the forced-convection, mixed-convection and free-convection flow regimes. The effects of liquid inertia were found to be significant only for low-Prandtl-number fluids and lower gas concentrations. The effects of liquid energy convection were found to be significant only for high-Prandtl-number fluids and to be most significant for mixed-convection condensation. Received on 3 March 1998     

Perturbation solutions of weakly compressible Newtonian Poiseuille flows with Navier slip at the wall

We consider both the planar and axisymmetric steady, laminar Poiseuille flows of a weakly compressible Newtonian fluid assuming that slip occurs along the wall following Navier’s slip equation and that the density obeys a linear equation of state. A perturbation analysis is performed in terms of the primary flow variables using the dimensionless isothermal compressibility as the perturbation parameter. Solutions up to the second order are derived and compared with available analytical results. The combined effects of slip, compressibility, and inertia are discussed with emphasis on the required pressure drop and the average Darcy friction factor.     

MODELLING OF RELEASE OF GAS FROM HIGH-PRESSURE PIPELINES

The problem investigated is the break of a high-pressure pipeline carrying natural single-phase gas which may condensate (retrograde) when the pressure drops. Single-phase non-ideal gas is assumed using a general- ized equation of state. Taking advantage of the choked massflow condition, the break is split into a pipe flow problem and a dispersion flow problem, both solved using a finite difference control volume scheme. The transient flow field from the pipeline break location is expanded analytically, using an approximation of the governing equations, until ambient pressure is reached and matched to the corresponding gas dispersion flow field using as subgrid model a jet box with a time-varying equivalent nozzle area as an internal boundary of the dispersion domain. The turbulence models used for the pipe and dispersion flow fields are an empirical model of Reichard and the k–ϵ model for buoyant flow respectively. The pipe flow simulations indicate that the flow from the pipeline might include dispersed condensate which will affect quantitatively the mass flow rate from the pipeline and qualitatively the gas dispersion if the condensate rains out. The transient dispersion simulation shows that an entrainment flow field develops and mixes supersaturated gas with ambient warmer air to an unsaturated mixture. Because of the inertia of the ambient air, it takes time to develop the entrainment flow field. As a consequence of this and the decay of the mass flow with time, the lower flammability limit of the gas–air mixture reaches its most remote downstream position relatively early in the simulation (about 15 s) and withdraws closer to the break location.     

The Effect of Bubbly Oil on the THD Lubrication of Tilting-pad Journal Bearings Subjected to Rotating Unbalance Load

Little work has been published on the contribution of entrained air and/or gas bubbles on the lubricant viscosity and hence on the hydrodynamic performance of bearings. In this work, a thermo-hydrodynamic analysis of the performance of dynamically loaded tilting-pad journal bearing lubricated by bubbly oil is carried out. The non-steady Reynolds equation for compressible fluid and the energy equation are solved iteratively using finite difference method to study the effect of air bubbles on the bearing performance characteristics. The effect of bubbles content on the lubricant viscosity is considered.The results showed that the value of maximum pressure slightly increases with increasing the bubble ratio up to 0.2, while the film thickness deteriorates at very high bubble content. The pressure peak moves in the downstream direction with the increase of bubble ratio ₁, which would be of great importance in deciding the pivot location in tilting-pad bearings.     

Unified gas-kinetic simulation of slider air bearing

The unified gas-kinetic scheme (UGKS) is presented and used in this letter to study the slider air bearing problem. The UGKS solutions are first validated by comparison with direct simulation Monte Carlo results. After validation, the UGKS is used to study the air-bearing problem under different non-equilibrium conditions. On the surface of the slider, the dependency of the gas pressure and normal force on the Mach and Knudsen numbers are fully evaluated. The non-equilibrium effect on the force loading in the whole transition regime up to the free molecular limit is also studied.     

Aerodynamic thermal environment around transonic tube train in choked/unchoked flow

The aerodynamic thermal environment in an evacuated tube transport (ETT) system is an important factor in ensuring the operational safety of tube trains, where choking can further worsen air flow and aerodynamic heating. A compressible flow solver based on total variation diminishing (TVD) schemes was used to calculate the transonic aerodynamic behaviour of a capsule train in a confined space under low pressure conditions, and the flow fields and aerodynamic heating effect on the train were obtained. The results showed that the flow state around the train could be classified into choked and unchoked flow according to the blockage ratio (BR) and train speed based on the Kantrowitz limit. The wall viscosity caused a difference in the boundary-layer flow and potential flow in the annular space between the train and the tube with an increase in the BR. The choked flow was driven forward by the train, passing through its throat at the speed of sound. Owing to the complicated compressible flow in the tube, the thermal environment around the train gave rise to extreme temperature changes on its surface. In transonic choked flow, the temperature rise of the train head reached a maximum of 525 K, whereas local cooling could occur in the afterbody, causing the surface temperature to fall below the ambient temperature under certain conditions. The findings can be used to guide the design of ETT systems.     

Compressible gas-liquid mixture flow at abrupt pipe enlargements

A detailed investigation was made of the flow of compressible gas-liquid mixtures through sudden enlargements in diameter of circular pipes. One-dimensional analysis shows that the dimensionless pressure rise varies with mixture void fraction and mixture momentum, while the establishment of choking conditions at the enlargement is controlled by the length of pipe downstream in which frictional pipe flow occurs. The flows were found to exhibit two characteristic modes, jet flow and submerged flow, with intermediate flows displaying unsteady oscillation between these modes. The distance to the downstream position of maximum pressure increased steadily with mixture void fraction when the upstream pipe outlet was choked, varying from 5 to 50 times the downstream pipe diameter. If the flow was not choked, this distance was much smaller and showed discrete fixed values associated with the mode of flow.

One-dimensional analysis accurately predicted maximum pressure, but when flow was choked at the enlargement the calculation was sensitive to the pressure in the region of separated flow surrounding the central jet in the enlargement. Although analysis of maximum pressure in terms of flow expansion and normal shock gave a general indication of the maximum pressure (which was thus concluded to depend on the general flow processes expected in the enlargement), accurate prediction of maximum pressures will depend on empirical knowledge of the separated flow region pressures. The maximum pressure rise was found to be in the range extending down to 0.3 of the upstream pipe outlet pressure and reduced with void fraction; it was also influenced by the enlargement area ratio. Flows in the approach and outlet pipes were found to be compressible, frictional pipe flows of the Fanno type, with somewhat reduced friction factors occurring in the outlet pipe.     

螺旋槽端面微间隙高速气流润滑密封特性

考虑入口气流压力损失和出口阻塞效应,建立了微间隙端面高速气体润滑密封分析数学模型,对螺旋槽端面微间隙高速气流润滑密封特性进行研究.重点分析了不同密封间隙、密封压力和转速等工况条件下,入口压力损失和出口阻塞效应对开启力、泄漏率及气膜刚度等密封特性参数的影响规律.结果表明:高速气体阻塞效应使出口压力高于环境压力,压力损失使入口气膜压力下降,导致泄漏率和气膜刚度明显下降,并使开启力增加.随着密封压力和密封间隙的增加,阻塞效应增强,导致泄漏率和气膜刚度显著降低.密封压力10 MPa时,泄漏率降低可达20%,气膜刚度的下降可达30%以上.     

An investigation into non-linear vibrations of thin plates

The variational and modified forms of the von Kármán-type non-linear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods. An approximate analysis of the non-linear vibrations of thin elastic plates including inplane inertia is presented. The quantitative study confirms that the inplane inertia effects are negligible for thin plates provided the non-linearity is not too large. It is observed that the non-linear inertia terms in the transverse equation of motion should be retained in any such study. The analysis is simplified by neglecting the inplane inertia and applied to constrained and unconstrained plates. A different type of inplane boundary condition termed ‘the partially constrained’ is studied, and the inadequacy of replacing the unconstrained condition by means of an average-zero stress condition is clearly demonstrated. It is observed that in most of the cases considered the Galerkin method yields lower bounds for the non-linear coefficient of the modal equation. In all cases the Galerkin results yield less stiff models than the Rayleigh-Ritz method. The general significance of the convergence of the two methods beyond the scope of the title problem is highlighted.     

One-Dimensional Matrix-Fracture Transfer in Dual Porosity Systems with Variable Block Size Distribution

Most of the developed models for fractured reservoirs assume ideal matrix block size distribution. This assumption may not be valid in reality for naturally fractured reservoirs and possibly lead to errors in prediction of production from the naturally fractured reservoirs especially during a transient period or early time production from the matrix blocks. In this study, we investigate the effect of variable block size distribution on one- dimensional flow of compressible fluids in fractured reservoirs. The effect of different matrix block size distributions on the single phase matrix-fracture transfer is studied using a recently developed semi-analytical approach. The proposed model is able to simulate fluid exchange between matrix and fracture for continuous or discrete block size distributions using probability density functions or structural information of a fractured formation. The presented semi-analytical model demonstrates a good accuracy compared to the numerical results. There have been recent attempts to consider the effect of variable block size distribution in naturally fractured reservoir modeling for slightly compressible fluids with a constant viscosity and compressibility. The main objective of this study is to consider the effect of variable block size distribution on a one-dimensional matrix-fracture transfer function for single-phase flow of a compressible fluid in fractured porous media. In the proposed semi-analytical model, the pressure variability of viscosity and isothermal compressibility is considered by solving the nonlinear partial differential equation of compressible fluid flow in the fractured media. The closed form solution provided can be applied to flow of compressible fluids with variable matrix block size distribution in naturally fractured gas reservoirs.     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

The upward flow of air betweem two waterfalls: An unusual choked flow problem

An experimental and theoretical investigation of the effect of air rising between two opposed waterfalls is presented. It is found experimentally that as the air flow is increased, the waterfalls are drawn more closely together until a critical air flow rate is reached at which the waterfalls collapse together. A theoretical analysis of this phenomenon is presented and the collapse condition is shown to be analogous to the choked flow of air through a nozzle the cross-sectional area of which is strongly pressure dependent. This dependency results in a very low effective “sonic” velocity and “choked” flow and the theoretical predictions are in reasonable agreement with the experiment results. The relevance of this work to the PWR refill problem is discussed.