Application of the Pseudospectral Method to Thermohydrodynamic Lubrication

Schumack, Mark

doi:10.1002/(sici)1097-0363(19961215)23:11<1145::aid-fld459>3.0.co;2-v

Cited by 18 publications

(9 citation statements)

References 15 publications

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“…The governing equations are transformed to nondimensional forms by using the terms with a þ superscript similar to the convention used in the literature [3][4][5]. The steady-state Reynolds equation becomes…”

Section: Governing Equationsmentioning

confidence: 99%

“…Recent examples include the work of Rodkiewicz [2], Schumack [3] (who used a spectral element scheme for thermohydrodynamic lubrication (THDL) analysis), and Kumar et al [4,5]. They analyzed numerically the load-carrying mechanism of a slider bearing under thermal influence by solving the coupled nonlinear partial differential equations governing the fluid film with either isothermal or adiabatic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Three-Dimensional Modeling of Thermohydrodynamic Lubrication in Slider Bearings Using Streamline Upwind Petrov-Galerkin Method

Zengeya

Gadala

Segal

2006

Numerical Heat Transfer, Part A: Applications

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A three-dimensional thermohydrodynamic lubrication model which couples the Reynolds and energy equations is developed in the finite-element program Sepran. The model uses the streamline upwind Petrov-Galerkin method. Model results indicate the peak temperature is not on the mid-plane surface. This position shifts toward the mid-plane as the width-to-length ratio is reduced from 10 to 1 as well as when pressure boundary conditions are altered in such a way that the inlet/outlet pressure is higher than the side pressure. The adiabatic temperature profiles of an infinite slider and a square slider are compared. The wider slider shows a higher peak temperature. The side flow plays a major role in determining the value and position of the peak temperature. Model results also indicate peak side flow at a width-to-length ratio of 2.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Modeling of Thermohydrodynamic Lubrication in Slider Bearings Using Streamline Upwind Petrov-Galerkin Method

Zengeya

Gadala

Segal

2006

Numerical Heat Transfer, Part A: Applications

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“…The presence of geometrical nonlinearities, coupled with material nonlinearities such as inertia (high-speed lubrication, and die casting) and non-Newtonian effects (injection molding), makes the problem difficult to investigate. The method was also applied to the flow of complex fluids [25,26]. Although the lubrication assumption remains the basis for the simulation of flow of thin films, it is mostly used with inertial effects neglected [2].…”

Section: Introductionmentioning

confidence: 99%

A spectral approach to inertial confined thin‐film flow

Laure

Khayat

2012

Numerical Methods in Fluids

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International audienceA spectral approach is proposed to determine the flow field of a thin film inside narrow channels of arbitrary shape. Although the method is easily extended to transient flow, only steady flow is considered here. The flow field is represented spectrally in the depthwise direction in terms of orthonormal shape functions, which together with the Galerkin projection lead to a system of ordinary differential equations that can be solved using standard methods. The method is particularly effective for nonlinear flow, including nonlinearities of geometrical or material origins. The validity of the proposed method is demonstrated for a flow with inertia, and, unlike the depth-averaging method, is not limited to a flow at small Reynolds number. The problem is closely related to high-speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing it with the fully two-dimensional finite-element solution, and the widely used depth-averaging method from shallow-water theory. It is found that a low number of modes are usually sufficient to secure convergence and accuracy. The influence of inertia is examined on the velocity and pressure fields. The pressure distributions reflect excellent agreement between the low-order spectral method and the finite-element solution, even at moderately high Reynolds number. The depth-averaging solution is unable to predict accurately (qualitatively and quantitatively) the high-inertia flow. Comparison of the velocity field reflects the expected discrepancy in a boundary layer formulatio

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“…At first, Elrod (1991) used finite difference methods to solve the Reynolds equation, and Legendre collocation method in the cross-film direction to solve the energy equation. Later, Schumack (1996) solved Reynolds equation and energy equation by applying Legendre collocation method in axial, lateral, and cross-film directions. Talmage and Carpino (1997) developed a pseudospectral-finite-difference method based on Elrod's method and solved two-dimensional Navier-Stokes equations and energy equation.…”

Section: Introductionmentioning

confidence: 99%

An Efficient General Fluid-Film Lubrication Model for Plane Slider Bearings

Shyu

Jeng

2002

Tribology Transactions

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ment is made in this study by combining the original model with a proposed in the former study. Pressure is obtainedfr-om the bulk-B = bearing width (m) B* = slenderness ratio = B l L C,, = specific heat at constant pressure (J / kg K) h(x,zj = film thickness (m) h, = film thickness at inlet (m) h2 = film thickness at exit (m) h* = convergence ratio = hi / h2 i , , = total number of grid points in s-direction ke = effective thermal conductivity = km+ k, (W / m K) km = molecular thermal conductivity (W / m K) k,nar = total number of grid points in z-direction k, = eddy thermal conductivity (W / m K) L = bearing length (m) N = highest degree of Legendre polynomial p = pressure (Pa) po = inlet pressure (Pa) 2 p* -= normalized pressure = (p-p,)h2 /(pp,L) P,,(c) = Legendre polynomial of degree n Pe = Peclet number = ~~~~u , , h~/ k~,~ Re* = modified Reynolds number = Po~,h,2/P,~ Rem = mean Reynolds number = pou,(h,+h2)/2p, T = temperature (K) To = inlet temperature (K) T,, = Legendre coefficient of temperature (K) 2 T* = normalized temperature = Re*C,(T-T,)/U, u = axial velocity component (m / s) u = film-averaged axial velocity component (m / s) u* = normalized axial velocity component = ulu, u, = axial velocity component at the runner surface (m / s) un = Legendre coefficient of u (m / s) I = velocity component through the film thickness (m / s) 3 = v -0.5 ( 5 + 1) [gu+ g w ] ( m / s ) fin = Legendre coefficient of 6 (m / s 2 L .+a/l W* = normalized load = h2& J-B,2 @-pJdz dx/p,u,B L' w = lateral velocity component (m / s) z3 = film-averaged lateral velocity component (m / s) MI,, = Legendre coefficient of w (m / s) x = axial coordinate (m) x* = normalized axial coordinate = x / L y = coordinate through the film thickness (rn) z = lateral coordinate (m) pe = effective viscosity = pm + p, (Pa s) pm = molecular viscosity (Pa. s) po = molecular viscosity at inlet = pn, ( po, To ) (Pa . S) p, = eddy viscosity (Pa. s) p = density (kg / m3) = film-averaged density (kg / m3) 3 po = density at inlet = p ( p,, To ) (kg / m ) 5 = normalized coordinate through film thickness, 5 = (2y -h)/h -5 = coordinate through film thickness with stretching, ( = l n (~) / l n (~) b = runner surface, y = 0 m = molecular quantity n = degree of Legendre polynomial; Legendre coefficient o = inlet quantity (reference quantity)Downloaded by [UNAM Ciudad Universitaria] at 14:51 20 December 2014 S. SHYU AND Y. JENG j7ow model. Comparisons of this model with the original model arc made for plane slider bearings in four flow regimes: laminar isothermal, lanlinar thermal, turbulent isothermal, nnd turbulent thern~al. Res~lts show good agreement between the two models, arid the model developed here is one order of magnitude faster thun the original model for turbulent thernzal flow.

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Application of the Pseudospectral Method to Thermohydrodynamic Lubrication

Cited by 18 publications

References 15 publications

Three-Dimensional Modeling of Thermohydrodynamic Lubrication in Slider Bearings Using Streamline Upwind Petrov-Galerkin Method

Three-Dimensional Modeling of Thermohydrodynamic Lubrication in Slider Bearings Using Streamline Upwind Petrov-Galerkin Method

A spectral approach to inertial confined thin‐film flow

An Efficient General Fluid-Film Lubrication Model for Plane Slider Bearings

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