Surface Roughness Effects in the Region Between High Wave Number and High Bearing Number Limited Lubricant Flows

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Roughness Length ◽  
Wave Number ◽  
Roughness Effects ◽  
High Wave Number ◽  
High Wave ◽  
Lubrication Equation ◽  
Order Of Magnitude ◽  
Bearing Number ◽  
Gas Bearing

The ability to predict surface roughness effects is now well established for gas bearings that satisfy the requirements for either high wave number–limited or high bearing number–limited conditions. However, depending on the parameters involved, a given bearing configuration may not satisfy either of these limited requirements for analysis of roughness effects. Well-established methods for the analysis of surface roughness effects on gas lubrication are not yet available outside of these two limited regions. With that as motivation, this paper then reports an analytical investigation of rough surface gas-bearing effects for the region bounded on one side by high wave number–limited conditions and on the other by high bearing number–limited effects. It emphasizes the gas-bearing region, where shear-driven flow rate and pressure-driven flow rate due to surface roughness are of the same order of magnitude. This paper makes use of the compressible continuum form of the Reynolds equation of lubrication together with multiple-scale analysis to formulate a governing lubrication equation appropriate for the analysis of striated roughness effects collectively subject to high bearing number (Λ→∞), high inverse roughness length scale (β→∞), and unity order of magnitude-modified bearing number based on roughness length scale (Λ2=Λ/β=O(1)). The resulting lubrication equation is applicable for both moving and stationary roughness and can be applied in either averaged or un-averaged form. Several numerical examples and comparisons are presented. Among them are results that illustrate an increased sensitivity of bearing force to modified bearing number for Λ2=O(1). With Λ2 in this range, bearings with either moving or stationary roughness exhibit increased force sensitivities, but the effects act in opposite ways. That is, while an increase in modified bearing number causes a decrease in force for stationary roughness, the same increase in modified bearing number causes an increase in force for moving roughness.

Combined Effects of Surface Roughness and Rarefaction in the Region Between High Wave Number-Limited and High Bearing Number-Limited Lubricant Flows

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Knudsen Number ◽  
Analytical Methods ◽  
Wave Number ◽  
Solution Technique ◽  
High Wave Number ◽  
High Wave ◽  
Lubrication Equation ◽  
Wide Range ◽  
Bearing Number

Analytical methods and techniques are required for design and analysis of low clearance gas-bearings that account for the combined influence of surface roughness and Knudsen number. Analytical methods for the lubrication equation are currently available for bearings that are either high wave number-limited or high bearing number-limited. There are few useful analytical methods in the range between these limiting extremes that account for the combined effect of roughness and rarefaction. That is the focus of this paper as it extends the work reported by White (2013, “Surface Roughness Effects in the Region Between High Wave Number and High Bearing Number-Limited Lubricant Flows,” ASME J. Tribol., 135(4), p. 041706) to include rarefaction effects. Results of an analytical study will be reported that investigates a wedge bearing geometry using perturbation methods and multiple-scale analysis over a wide range of Knudsen numbers for roughness on moving and stationary surfaces. The solution technique developed allows nonlinear aspects of the lubrication equation to be retained in the analysis. Solutions will be presented graphically and discussed. Results indicate that most of the bearing sensitivity to Knudsen number can be accounted for by a modified form of the bearing number.

The Effect of Two Sided Surface Roughness on Ultra-Thin Gas Films

1983 ◽  
Vol 105 (1) ◽  
pp. 131-137 ◽  
Author(s):  
J. W. White
Keyword(s):  
Surface Roughness ◽  
Current Method ◽  
Solution Procedure ◽  
Lubrication Equation ◽  
Write Heads ◽  
Stationary Surface ◽  
Simple Geometry ◽  
Load Carrying ◽  
Bearing Number ◽  
Gas Bearing

The influence of two sided striated surface roughness on bearing load carrying capacity is analyzed for very low clearance gas films. As was done for the case of stationary surface roughness [1], a model lubrication equation appropriate for extremely high gas bearing number films is solved analytically for several simple geometry bearings. The analytic solution provides information on the exact relationship between pressure and roughness which makes it possible to ensemble average the lubrication equation before solution, greatly simplifying the solution procedure. It is found that the translating surface roughness has an influence on load similar to that caused by the stationary surface. Exact solutions with the current method are compared with those of the theory attributed to Christensen and To̸nder. The results are strikingly different and serve to bring attention to the fact that for high bearing number compressible lubrication, the Christensen-To̸nder theory is inappropriate. The results reported here should find application in the computer peripherals area where read/write heads now routinely hover over a spinning disk at clearances of 0.25 micron.

Comparison of Moving and Stationary Surface Roughness Effects on Bearing Performance, With Emphasis on High Knudsen Number Flow

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Knudsen Number ◽  
Multiple Scale ◽  
Air Bearing ◽  
Roughness Effects ◽  
Lubrication Equation ◽  
Wide Range ◽  
Transverse Roughness ◽  
Number Flow ◽  
Gas Bearing

Low clearance gas bearing applications require an understanding of surface roughness effects at increased levels of Knudsen number. Because very little information has been reported on the relative air-bearing influence of roughness location, this paper is focused on a comparison of the effects of moving and stationary striated surface roughness under high Knudsen number conditions. First, an appropriate lubrication equation will be derived based on multiple-scale analysis that extends the work of White (2010, “A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces,” ASME J. Tribol., 132, p. 021701). The resulting roughness averaged equation, applicable for both moving and stationary roughness over a wide range of Knudsen numbers, allows an arbitrary striated roughness orientation with regard to both (1) the direction of surface translation and (2) the bearing coordinates. Next, the derived lubrication equation is used to analyze and compare the influences produced by a stepped transverse roughness pattern located on the moving and the stationary bearing surface of a wedge bearing geometry of variable inclination. Computed results are obtained for both incompressible and compressible lubricants, but with an emphasis on high Knudsen number flow. Significant differences in air-bearing performance are found to occur for moving versus stationary roughness.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

An Averaging Technique for the Analysis of Rough Surface High Bearing Number Gas Flows

1999 ◽  
Vol 121 (2) ◽  
pp. 333-340 ◽  
Author(s):  
James W. White
Keyword(s):  
Surface Roughness ◽  
Rough Surface ◽  
Numerical Solutions ◽  
Scale Effects ◽  
Wavy Surface ◽  
Universal Property ◽  
Flying Height ◽  
Lubrication Equation ◽  
Product Term ◽  
Bearing Number

Earlier analytical solutions by White (1980, 1983, 1992, 1993) included Couette effects, transverse diffusion, and mass storage in a model lubrication equation for narrow width wavy surface high bearing number gas films. The model lubrication equation did not include longitudinal diffusion effects due to the high bearing number restriction. Crone et al. (1991), however, reported numerical solutions of the full Reynolds equation for a gimbal mounted slider subject to wavy surface roughness. The first objective of this work is to reconcile the differences observed between the reported results of White and those of Crone et al. for moving and stationary roughness. The second objective is to describe how to best apply what appears to be a universal property of a high bearing number gas film subjected to a rough surface. Each solution of the model lubrication equation by White (1980, 1983, 1992, 1993) produced a product term based on local gas pressure and clearance (Z = Ph) that is independent of roughness details but which is dependent on the statistical properties of the roughness. In the present work, this characteristic is treated as a universal property of all high bearing number rough surface gas films. The product variable Z = Ph is introduced into the generalized full lubrication equation, and the resulting lubrication equation is ensemble averaged before a solution is attempted. This removes the short length and time scale effects due to the surface roughness. Solution of the ensemble averaged equation for Z(x, y, t) then follows by standard analytical or numerical methods. The unaveraged pressure is then given by P(x, y, t) = Z(x, y, t)/h(x, y, t) and the ensemble averaged or mean pressure at a point is computed from Pm(x, y, t) = Z(x, y, t)E(1/h(x, y, t)), where E(1/h) represents the ensemble average of 1/h. Using this technique, numerical solutions of the full generalized lubrication equation based on kinetic theory were obtained for a low flying gimbal mounted slider. Results indicate that the nominal flying height increases and the minimum flying height decreases as surface roughness increases.

scholarly journals Stable and Accurate Filtering Procedures

2020 ◽  
Vol 82 (1) ◽  
Author(s):  
Tomas Lundquist ◽  
Jan Nordström
Keyword(s):  
Boundary Conditions ◽  
High Frequency ◽  
Wave Number ◽  
Main Concern ◽  
High Wave Number ◽  
High Wave ◽  
Artificial Dissipation ◽  
High Frequency Oscillations ◽  
Energy Bound ◽  
Energy Stable

AbstractHigh frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the $$\pi $$π-mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summation-by-parts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.

scholarly journals A robust domain decomposition method for the Helmholtz equation with high wave number

2016 ◽  
Vol 50 (3) ◽  
pp. 921-944 ◽  
Author(s):  
Wenbin Chen ◽  
Yongxiang Liu ◽  
Xuejun Xu
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Mesh Size ◽  
Relaxation Parameter ◽  
Point Of View ◽  
Wave Number ◽  
Theoretical Point ◽  
High Wave Number ◽  
High Wave

In this paper we present a robust Robin−Robin domain decomposition (DD) method for the Helmholtz equation with high wave number. Through choosing suitable Robin parameters on different subdomains and introducing a new relaxation parameter, we prove that the new DD method is robust, which means the convergence rate is independent of the wave number k for kh = constant and the mesh size h for fixed k. To the best of our knowledge, from the theoretical point of view, this is a first attempt to design a robust DD method for the Helmholtz equation with high wave number in the literature. Numerical results which confirm our theory are given.

Theoretical Approach to Turbulent Lubrication Problems Including Surface Roughness Effects

1989 ◽  
Vol 111 (1) ◽  
pp. 17-22 ◽  
Author(s):  
H. Hashimoto ◽  
S. Wada
Keyword(s):  
Surface Roughness ◽  
Velocity Distribution ◽  
Boundary Layers ◽  
Theoretical Approach ◽  
Turbulent Boundary Layers ◽  
Distribution Law ◽  
Roughness Effects ◽  
Relative Roughness ◽  
Lubrication Equation ◽  
Lubrication Characteristics

A new theoretical approach to turbulent lubrication problems including the surface roughness effects is described. On the basis of a logarithmic velocity distribution law in the turbulent boundary layers, the resistance laws for pressure and shear flows in the lubricant film are formulated separately in both cases of smooth and homogeneous rough surfaces. Moreover, combining the bulk flow concept proposed by Hirs with the formulated resistance laws, the generalized turbulent lubrication equation including the surface roughness effects is derived. Some numerical results for the modified turbulence coefficients are presented in the graphic form for different values of relative roughness, and the effects of surface roughness on the turbulent lubrication characteristics are generally discussed.

Sign in / Sign up

Export Citation Format

Share Document