lubricants Article
Accuracy and Grid Convergence of the Numerical Solution of the Energy Equation in Fluid Film Lubrication: Application to the 1D Slider Silun Zhang 1,2 , Mohamed-Amine Hassini 1 1 2
*
and Mihai Arghir 2, *
ERMES-IMSIA, EDF Lab Paris-Saclay, 91120 Palaiseau, France;
[email protected] (S.Z.);
[email protected] (M.-A.H.) Institut Pprime, UPR CNRS 3346, Université de Poitiers, ISAE ENSMA, 86962 Chasseneuil Futuroscope, France Correspondence:
[email protected]; Tel.: +33-(0)549496540
Received: 13 September 2018; Accepted: 22 October 2018; Published: 26 October 2018
Abstract: The present work is focused on the numerical solution of the complete energy equation used in fluid film lubrication. The work was motivated by the fact that the complete energy equation has no analytical solution that can be used for validations. Its accuracy and computation time are related to the employed numerical method and to the grid resolution. The natural discretization method (NDM) applied on different grids is systematically compared with the spectral method (the Lobatto Point Colocation Method or LPCM) with different polynomial degrees. A one dimensional inclined slider is used for the numerical tests, and the energy equation is artificially decoupled from the Reynolds equation. This approach enables us to focus all the attention on the numerical solution of the energy equation. The results show that the LPCM is one or two orders of magnitude more efficient than the NDM in terms of computation time. The energy equation is then coupled with the Reynolds equation in a thermo-hydrodynamic analysis of the same 1D slider; the numerical results confirm again the efficiency of the LPCM. A thermo-hydrodynamic analysis of a two-lobe journal bearing is then presented as a practical application. Keywords: energy equation; numerical solution; lobatto point quadrature; legendre polynomials
1. Introduction The flow of thin lubricant films in journal and thrust bearings is most often described by the Reynolds equation of lubrication coupled with the energy transport equation. The numerical solution of these two coupled equations is a problem that has been solved for many decades [1]. However, it requires a computational effort that can render transient analyses very time consuming. Therefore, efficient solvers and adequate coupling strategies are of major importance to perform complex analyses in a reasonable amount of time. If film rupture and reformation (traditionally designated as “cavitation”) are absent, and if the flow regime is laminar and isothermal, then the Reynolds equation is an elliptic linear differential equation. A direct solver can be used after its discretization in the thin film plane. The energy equation contains convective transport terms, conductive transport terms across the thin film, and dissipative source terms arising from its coupling with the Reynolds equation. Its character is therefore not entirely elliptic, and its solver is different from the one used for the Reynolds equation. Moreover, the energy equation must also be discretized across the thin fluid film, and the number of discretization points in this direction must be large enough to capture temperature gradients near the walls. Therefore, the energy equation requires a substantially higher computational
Lubricants 2018, 6, 95; doi:10.3390/lubricants6040095
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effort compared to the Reynolds equation. Developing an accurate and efficient solver for the energy equation is then an important step toward solving non-isothermal lubrication problems. The task is not simple because there is no analytical solution of the complete energy equation to be used for validations. For example, if the analytical solution of the laminar and isothermal thin film flow in a one dimensional (1D) slider can be used to validate the solver of the Reynolds equation, no similar solution exists for the energy equation. For validation, a numerical solver for the energy equation will have to be tested with different wall boundary conditions, and the numerical results must be checked for various grid densities. Such results are absent from the literature. As mentioned above, the energy equation must also be discretized over the film thickness. The normal practice is to increase the number of discretization cells across the thin film until a grid independent numerical solution is reached, and this renders its solution time consuming. In the natural discretization methods (NDMs), such as the finite difference or finite volumes methods, the variation of the temperature between two cells is linear. This leads to a significant number of discretization cells across the thin film. An efficient approach for solving the energy equation was introduced by Elrod and uses the Legendre polynomials to describe the temperature variation across the thin film [2]. The coefficients of the Legendre polynomials expansion were first calculated by using a Galerkin approach. Later, Elrod used the collocation method at the Lobatto points; that is, the roots of the derivative of the highest order of the used Legendre polynomial. This method, known as the “Lobatto Points Collocation Method (LPCM)”, proved to be more efficient [3]. Despite its efficiency, the method is barely used, likely because of its complexity. Thus, Moraru [4] extended the LPCM to compressible films by describing the variation of the density across the thin film with Legendre polynomials. Feng and Kaneko [5] used the LPCM to solve the energy equation in aerodynamic foil bearings. Lehne et al. [6] presented a comprehensive review of the numerical solution strategies of the coupled Reynolds and energy equations, including the LPCM. Except for the cited references, the literature is scarce in references dealing with the LPCM. This work presents a systematic comparison between the natural discretization method (NDM) of the energy equation and its LPCM approximation. In order to decouple the Reynolds and energy equation, the viscosity is supposed to be constant and the flow laminar. A one dimensional (1D) inclined slider is used for the numerical tests. The Reynolds equation then has an analytical solution, and the analysis is entirely focused on the energy equation. The results for the 1D slider compare both the number of points and the computational time required by the NDM and by the LPCM to obtain grid independent solutions. The results show how the NDM is excessively time consuming when high accuracy is sought, and the net economy of computational time brought by the LPCM approach. A thermo-hydrodynamic analysis of the 1D slider with the coupled Reynolds and energy equation is then performed. The analysis highlights the same conclusions, namely the large superiority of the LPCM compared to NDM in terms of computational effort. Lastly, a two-lobe journal bearing with axial supply grooves is analyzed to demonstrate the accuracy that can be expected with simple, non-triggered boundary conditions for the energy equation. 2. The Numerical Solution of the Energy Equation Based on the Natural Discretization Method The 1D inclined slider shown in Figure 1 is used in References [2,3] for the resolution of the Reynolds equation coupled with the energy equation. The current work is focused on the resolution of the energy equation decoupled from the Reynolds equation, for which an analytical solution is used. This gives an accurate numerical solution of the energy equation without the influence of the coupling with pressure and velocities affected by numerical uncertainties. The numerical data used is detailed in Table 1.
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Figure 1. The 1D inclined slider.
Figure 1. The inclined slider. Table 1. The data1D used for the 1D slider. Physical Characteristics
Values
Physical Characteristics
Table 1. The data used for the 1D slider. 800 kg/m3 2000 [J/kgK] Values 0.14 [ W/mK] 800 kg/m 0.081 [Pa·s] 2000 31.946J/kgK [m/s]
Density ρ0 Specific Heat Capacity C p Physical Characteristics Thermal Conductivity λ Density 𝜌 Dynamic Viscosity η0 Specific HeatOperating Capacity 𝐶 U Lower Wall Speed
Values
Inlet thickness h1 1.8288e−4 [m] 4 [m] Outlet thickness h 0.9144e− Physical Characteristics Values 2 Slider length L 0.18288 [m] 1.8288𝑒 Inlet thickness ℎ Ambient temperature Ta 20 [◦ C] Outlet thickness 0.9144𝑒 Ambient pressure Paℎ 1 [bar]
m m 0.14 W/mK 0.18288 m Thermal Conductivity 𝜆 Slider length 𝐿 0.081 Pa s 20 of °C an Ambient temperature 𝑇𝑎 Dynamic Viscosity 𝜂 The conservative form of the energy equation for the lubrication thin film flow 31.946 m/s 1 bar Lower Wall Operating Speed Ambient pressure 𝑃𝑎 incompressible fluid writes [1]: 𝑈
2
2
∂(energy uT ) ∂(vT ∂ T ∂ulubrication thin film flow of an ) The conservative form ofρ0 Cthe (1) + equation = λ for + ηthe p 0 ∂x ∂y ∂y ∂y2 incompressible fluid writes [1]: It contains the convection transport terms on its left hand side, and the temperature diffusion and 𝜕(𝑢𝑇) The𝜕(terms 𝑣𝑇) on the 𝜕 𝜕𝑢 are simplified according to 𝑇 hand side dissipation terms on its right (1) = 𝜆 right + 𝜂 𝜌 𝐶hand side. + 𝜕𝑥that is, only 𝜕𝑦 derivatives𝜕𝑦across the thin 𝜕𝑦 film are taken into account. the lubrication thin film assumption; The dimensionless coordinate y is introduced to take into account that the film thickness h is It contains the convection transport terms on its left hand side, and the temperature diffusion not constant. and dissipation terms on its right hand side. The terms according y= yh( x )on the right hand side are simplified(2)
to the lubrication thin film assumption; that is, only derivatives across the thin film are taken into Following this coordinate transformation, the energy Equation (1) for the 1D slider becomes: account. 2 ∂(uT ) y𝑦 dh ∂T account 1 ∂u (uT ) 1 ∂(vTto ) take 1into is∂introduced that the film thickness ℎ is The dimensionless coordinate (3) ρ0 C p − + =λ 2 + η0 2 ∂x h dx ∂y h ∂y ∂y ∂y h h not constant. The slider is discretized with 2D computational cells as depicted in Figure 2. Following the (2) 𝑦 = 𝑦ℎ(𝑥) variable change described by Equation (2), the computational domain is rectangular and orthogonal. The computational cells have four plane faces,the denoted by lower-case corresponding to their Following this coordinate transformation, energy Equationletters (1) for the 1D slider becomes: direction (e, w, n, and s) with respect to the central node P.
𝜌 𝐶
𝜕(𝑢𝑇) 𝑦 𝑑ℎ 𝜕(𝑢𝑇) 1 𝜕(𝑣𝑇) 1 𝜕²𝑇 1 𝜕𝑢 − + =𝜆 + 𝜂 𝜕𝑥 ℎ 𝑑𝑥 𝜕𝑦 ℎ 𝜕𝑦 ℎ 𝜕𝑦² ℎ 𝜕𝑦
(3)
The slider is discretized with 2D computational cells as depicted in Figure 2. Following the variable change described by Equation (2), the computational domain is rectangular and orthogonal. The computational cells have four plane faces, denoted by lower-case letters corresponding to their direction (e, w, n, and s) with respect to the central node P.
𝜕(𝑢𝑇) 𝑦 𝑑ℎ 𝜕(𝑢𝑇) 1 𝜕(𝑣𝑇) 1 𝜕²𝑇 1 𝜕𝑢 − + =𝜆 + 𝜂 𝜕𝑥 ℎ 𝑑𝑥 𝜕𝑦 ℎ 𝜕𝑦 ℎ 𝜕𝑦² ℎ 𝜕𝑦
𝜌 𝐶
(3)
The slider is discretized with 2D computational cells as depicted in Figure 2. Following the variable change described by Equation (2), the computational domain is rectangular and orthogonal. Lubricants 2018, 6, 95 of 23 The computational cells have four plane faces, denoted by lower-case letters corresponding to4their direction (e, w, n, and s) with respect to the central node P.
Figure 2. 2. A cell and and the the notation notation used used for for aa 2D 2D grid. grid. Figure A typical typical 2D 2D computational computational cell
In the natural discretization numerical approach, the energy equation is solved using the finite volume method [7]. Equation (3) is therefore integrated over the control volumes corresponding to the 2D computational cells: ρ0 C p
R xe R yn xw
ys
h
=
i
y dh ∂(uT ) 1 ∂(vT ) dxdy h dx ∂y + h ∂y R R 2 y x 2 λ h12 xwe y n ∂ T2 dxdy + η0 h12 ∂u ∂y P ∆x∆y s ∂y P P
∂(uT ) ∂x
−
(4)
For example, the convection transport terms in the x direction can be expressed as: Z xe Z y n ∂ ( uT ) xw
ys
∂x
dxdy = [(uT )e − (uT )w ]∆y
(5)
An upwind discretization technique is used for the convective transport terms to avoid numerical instabilities [7]. For example, at the east face of the control volume, the temperature Te is up-winded based on the fluid flow direction. Mathematically, this can be expressed as: 1 − sgn(ue ) 1 − sgn(ue ) + T + T uT = u − T ( )e e P E P 2 2
(6)
This yields the discretization form of Equation (3):
( ae + aw + an +
as ) TP − ae TE − aw TW − an TN − as TS − a P ( TN − TS ) 2 η0 ∂u + ∆x∆y = ρ Cλ h2 ( TN − 2TP + TS ) ∆x ∆y ∂y ρ C h2 0 p P
where:
0 p P
(7)
P
ue [sgn(ue ) − 1] uw [sgn(uw ) − 1] ∆y; aw = ∆y; 2 2 vn [sgn(vn ) − 1] ∆x vs [sgn(vs ) − 1] ∆x an = ; as = ; 2 hP 2 hP u P y P dh ∆x aP = 2 dx P h P
ae =
(8)
As stated in the introduction, the goal of the present work is to investigate the accuracy of the numerical solution of the energy equation given by the NDM and LPCM methods. Therefore, in order to avoid the uncertainties introduced by the numerical solution of the Reynolds equation, its analytic solution for the 1D slider is used.
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Several other simplifying assumptions are needed for decoupling the Reynolds and the energy equations. Thus, the viscosity is considered constant and the flow regime laminar. Following the analytical solution, the pressure and the velocity in the 1D slider are [1]: p( x ) =
6η0 UL h h 2 −2 1 h −1 − 1 h − h1 − h2 h1 + h2 h1 + h2
u( x, y) = h2
y(y − 1) dp + ( 1 − y )U 2η0 dx
(9)
(10)
The velocity v component across the thin film is deduced by integrating the continuity Equation (11) over the film thickness: ∂u 1 ∂v y dh ∂u + − =0 ∂x h ∂y h dx ∂y
(11)
With the boundary conditions v( x, y = 0) = 0 and v( x, y = 1) = 0, this yields: v( x, y) = −h
Z y ∂u 0
∂x
dy +
dh dx
Z y ∂u 0
y
∂y
dy
(12)
It is to be underlined that in the following approach only the boundary condition at y = 0 is used, while the second boundary condition at y = 1 is used to check the accuracy of the numerical integration. The discretized system given by Equation (7) is solved using any numerical procedure for a linear system with a positive definite matrix. Grids with 80 equidistant cells in the x direction and different numbers of cells across the film thickness were used. The number of equidistant grid cells across the film thickness are Ny = 10, 20, 40, 50, 80, 100, and 160. In a first test, the ambient temperature of 20 ◦ C was imposed on the lower and upper walls and at the inlet. The dimensionless wall temperature gradient was monitored (This is equivalent to monitoring the wall heat fluxes). This dimensionless wall temperature gradient is defined by: 1 ∂T ( x, y) ∂T ( x, y) = ∂y Ta ∂y
(13)
where Ta = 20 ◦ C is a reference temperature. The results are depicted in Figures 3 and 4, and show that the curves become superposed starting with Ny = 40 cells. An estimation of the accuracy is obtained by using the relative difference of the wall temperature gradients which is defined in Equation (14): s 1 n
ε=
∑in=1
∂T Ny ( xi ,y) ∂y
s 1 n
∑in=1
−
∂T re f ( xi ,y) ∂y
∂T re f ( xi ,y) ∂y
2
2 f or y = 0 or y = 1
(14)
where T re f is the solution obtained with the finest grid Ny = 160. The variation of these errors with the number of grid cells across the film thickness is depicted in Figure 5a, and the computational time is depicted in Figure 5b. A rapid increase in the computational time with the grid cells must be underlined. For example, the calculation case with Ny = 80 grid cells required 1.844 s, while the computational time for the case with Ny = 160 cells was one order of magnitude higher. However, for the purpose of accuracy, the mesh with Ny = 160 was chosen as the reference solution in the following and its numerical values of the wall temperature gradients are given in Table A1 in Appendix B.
computational time with the grid cells must be underlined. For example, the calculation case with Ny = 80 grid cells required 1.844 s, while the computational time for the case with Ny = 160 cells was one order of magnitude higher. However, for the purpose of accuracy, the mesh with Ny = 160 was chosen as the reference solution in the following and its numerical values of the wall temperature gradients are given in 6, Table Lubricants 2018, 95 A1 in Appendix B. 6 of 23
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Lubricants 2018,3.6,Dimensionless x FOR PEER REVIEW Figure temperature gradient at the lower wall for the natural discretization method6 of 22
Figure solution 3. Dimensionless temperature gradient at(hthe/hlower wall for the natural discretization method (NDM) and imposed wall temperatures 1 2 = 2, Nx = 80 cells). (NDM) solution and imposed wall temperatures (h1/h2 = 2, Nx = 80 cells).
Figure 4. Dimensionless temperature gradient at the upper wall for the NDM solution and imposed Figure 4. Dimensionless temperature gradient at the upper wall for the NDM solution and imposed wall temperatures (h1/h2 = 2, Nx = 80 cells). Figure 4. Dimensionless wall temperatures (h1 /h2temperature = 2, Nx = 80 gradient cells). at the upper wall for the NDM solution and imposed wall temperatures (h1/h2 = 2, Nx = 80 cells).
(a)
(b)
(b) for y direction grid Figure 5. (a) Relative(a) difference and (b) computational time of the NDM solution /h2 2==2,2,Nx Nx= =8080cells). cells). refinements (h11/h Figure 5. (a) Relative difference and (b) computational time of the NDM solution for y direction grid refinements (h1/h2 = 2, Nx = 80 cells).
3. The Numerical Solution of the Energy Equation Based on the Lobatto Points Collocation Method 3. The Numerical Solution of the Energy Equation Based on the Lobatto Points Collocation
Method The Lobatto Point Collocation Method (LPCM) is based on the approximation of the temperature withThe Legendre polynomials across Method the film(LPCM) thickness. Because polynomials are defined on Lobatto Point Collocation is based on Legendre the approximation of the temperature the interval −1, 1 , the following coordinate transformation is used: with Legendre polynomials across the film thickness. Because Legendre polynomials are defined on
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3. The Numerical Solution of the Energy Equation Based on the Lobatto Points Collocation Method The Lobatto Point Collocation Method (LPCM) is based on the approximation of the temperature with Legendre polynomials across the film thickness. Because Legendre polynomials are defined on the interval [−1, 1], the following coordinate transformation is used: y=
( ζ + 1) h 2
(15)
where ζ is the new dimensionless coordinate across the film thickness. Following the coordinate transformation, the energy Equation (1) becomes: ρC p
∂(uT ) (ζ + 1) dh ∂(uT ) 2 ∂(vT ) 4 ∂T 4 ∂u 2 − + =λ 2 + η0 2 ∂x h dx ∂ζ h ∂ζ h ∂ζ h ∂ζ
(16)
For an incompressible lubricant with constant viscosity, only the variable temperature T is approximated across the film thickness: N
T ( x, ζ ) =
∑ Tˆ j (x) Pj (ζ )
(17)
j =0
where N is the highest order of the Legendre polynomials, Pj (ζ ) is the Legendre polynomial of degree j, and Tˆ j ( x ) is the polynomial coefficients of temperature of degree j. This description holds for every point in the 2D computational domain. Compared to the NDM, which computes directly the temperature by solving the energy equation discretized over the film thickness, the LPCM calculates the polynomial coefficients of temperature Tˆ j . Based on the work of Mahner et al. [6], the coefficients of the Legendre polynomials expansion can be obtained using different methods, but it is accepted that the most reliable is the collocation method using the Lobatto points. The Lobatto points are the roots of the derivative of the Legendre polynomial of the highest degree (i.e., the roots of dPN /dζ). For a given position in the x direction, the temperature is replaced by its approximation from Equation (17) across the fluid film, and the energy Equation (16) is enforced to hold true for each Lobatto point, ζ j , j ∈ [1, . . . , N − 1]. This leads to N − 1 partial differential equations with the unknown Tˆ j . The boundary conditions are applied at the two walls, ζ = −1 and ζ = 1, which leads to the other two equations. In total, a system of N + 1 equations for the N + 1 unknown Tˆ j is obtained. Figures 6 and 7 depict the dimensionless wall temperature gradient obtained with an increasing degree of the Legendre polynomials. The difference between the NDM solution and the LPCM solution at the lower wall (Figure 6) in the inlet section is reduced by increasing the degree of the Legendre polynomial.
to the other two equations. In total, a system of N + 1 equations for the N + 1 unknown 𝑇 is obtained. Figures 6 and 7 depict the dimensionless wall temperature gradient obtained with an increasing degree of the Legendre polynomials. The difference between the NDM solution and the LPCM solution at the lower wall (Figure 6) in the inlet section is reduced by increasing the degree of the Legendre polynomial. Lubricants 2018, 6, 95 8 of 23
Figure 6. 6, Dimensionless temperature gradient at the lower wall for the Lobatto Points Collocation 8 of 22 Lubricants 2018, x FOR PEER REVIEW
Figure 6. Dimensionless temperature gradient at the lower wall for the Lobatto Points Collocation Method (LPCM) solution with imposed wall temperatures (h1 /h2 = 2, Nx = 80 cells). Method (LPCM) solution with imposed wall temperatures (h1/h2 = 2, Nx = 80 cells).
Figure 7. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed Figure 7. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h1 /h2 = 2, Nx = 80 cells). wall temperatures (h1/h2 = 2, Nx = 80 cells).
The reference results used for comparison are given by the NDM with Ny = 160 grid cells. The reference results used for comparison are given by the NDM with Ny = 160 grid cells. The The relative difference between the reference results and the wall temperature gradient obtained with relative difference between the reference results and the wall temperature gradient obtained with the the LPCM is defined as: LPCM is defined as: s 2∂T N
ε=
𝜀=
1 n 1n ∑i=12𝜕𝑇 ∑ 𝑛 s 1
( xi ,ζ ) Lobbato (𝑥 , 𝜁)
𝜕𝜁 n
∂T re f
−𝜕𝑇
−
∂ζ
dT re f
1n ∑i=1 𝑑𝑇 ∑ 𝑛
( x ,y) NDM i (𝑥 ) dy , 𝑦
( xi ,y) NDM (𝑥 , 𝑦)
2
dy
𝑑𝑦
2
(18) (18)
𝑑𝑦
where ζ= −1−1 forfor LPCM, and y =𝑦 1=or1 yor= 𝑦0 = for0NDM. whereζ 𝜁==11oror 𝜁= LPCM, and for NDM. The relative difference is depicted in Figure 8a. This relative difference decreases rapidly withwith the The relative difference is depicted in Figure 8a. This relative difference decreases rapidly increase of the degree of the Legendre polynomial. However, the lower and upper wall results show the increase of the degree of the Legendre polynomial. However, the lower and upper wall results different trends. A higherAdegree the Legendre is required to a grid show different trends. higherofdegree of the polynomial Legendre polynomial is reach required to independent reach a grid solution at the lower wall. This is due to the more predominant thermal effect at the driving wall;atthat independent solution at the lower wall. This is due to the more predominant thermal effect the is, the lower wall. driving wall; that is, the lower wall.
The computational effort is depicted in Figure 8b. Compared with the reference solution, the computational time of the LPCM solution is divided by ten, since only a limited expansion of the Legendre polynomials is necessary to obtain grid independent results. For comparison, the temperature fields obtained with the LPCM (N = 12) and the NDM (Ny = 160) are depicted in Figures A1 and A2 in Appendix A, respectively. The obtained polynomial
where 𝜁 = 1 or 𝜁 = −1 for LPCM, and 𝑦 = 1 or 𝑦 = 0 for NDM. The relative difference is depicted in Figure 8a. This relative difference decreases rapidly with the increase of the degree of the Legendre polynomial. However, the lower and upper wall results show different trends. A higher degree of the Legendre polynomial is required to reach a grid Lubricants 2018, 6, 95 9 of 23 independent solution at the lower wall. This is due to the more predominant thermal effect at the driving wall; that is, the lower wall. The Thecomputational computationaleffort effortisisdepicted depictedininFigure Figure8b. 8b.Compared Comparedwith withthe thereference referencesolution, solution,the the computational computationaltime timeofofthe theLPCM LPCMsolution solutionisisdivided dividedby byten, ten,since sinceonly onlyaalimited limitedexpansion expansionofofthe the Legendre Legendrepolynomials polynomialsisisnecessary necessarytotoobtain obtaingrid gridindependent independentresults. results. For fieldsobtained obtainedwith withthe theLPCM LPCM = 12) and NDM = Forcomparison, comparison, the the temperature temperature fields (N(N = 12) and thethe NDM (Ny(Ny = 160) 160) are depicted in Figures A1 A2 in Appendix A, respectively. The polynomial obtained polynomial are depicted in Figures A1 and A2and in Appendix A, respectively. The obtained coefficients coefficients of temperature degree j, 𝑇 (𝑋in),Table are given Table A2 Appendix B. The indicates present of temperature of degree j, Tˆof , are given A2 in in Appendix B. in The present analysis j X that theindicates LPCM results areLPCM more accurate formore the same grid for density. analysis that the results are accurate the same grid density.
(a)
(b)
Figure 8. (a) Relative difference and (b) computational time of the LPCM solution for Y direction grid refinements (h1 /h2 = 2, Nx = 80 cells)
4. Further Comparison of Numerical Results Obtained by NDM and LPCM The previous results were obtained for a linear converging 1D slider with an inlet/outlet film thickness ratio h1 /h2 = 2, imposed wall temperatures (Dirichlet boundary condition), and completely decoupled from the Reynolds equation. Other simple cases of the converging 1D slider were also investigated and bring interesting conclusions. 4.1. Different Geometrical Configurations of the 1D Slider Two other inlet/outlet film thickness ratios, h1 /h2 = 4 and h1 /h2 = 8, were investigated while keeping the same wall temperature boundary conditions and the decoupled Reynolds equation. The increased inlet/outlet film thickness ratio leads to a slower convergence of the NDM results with the y grid refinements. The best NDM solution obtained with Ny = 160 was considered as the reference results, and the number of computational cells in the x direction was kept the same (Nx = 80 cells). The results obtained for h1 /h2 = 4 are depicted in Figures 9–11. Again, the lower and upper wall results show different trends. The upper wall temperature gradient reaches grid convergence starting with the Legendre polynomials of 11 degrees, while the resolution of the lower wall gradient needs N = 16 Legendre polynomials. However, Figure 11b shows that even when using a high number of Legendre polynomials, the LPCM requires one order of magnitude less computational time than the NDM for the same accuracy.
results show different trends. The upper wall temperature gradient reaches grid convergence starting with the Legendre polynomials of 11 degrees, while the resolution of the lower wall gradient needs N = 16 Legendre polynomials. However, Figure 11b shows that even when using a high number of Legendre polynomials, the LPCM requires one order of magnitude less computational time than the Lubricants 2018,the 6, 95 10 of 23 NDM for same accuracy.
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Figure 9. 6, Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed10 of 22 Lubricants 2018, x FOR PEER REVIEW
Figure 9. Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed wall temperatures (h1 /h2 = 4, Nx = 80 cells). wall temperatures (h1/h2 = 4, Nx = 80 cells).
Figure 10. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed Figure 10. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed Figure 10. Dimensionless gradient at the upper wall for the LPCM solution with imposed wall temperatures (h1/h2 = temperature 4, Nx = 80 cells). wall temperatures (h1 /h2 = 4, Nx = 80 cells). wall temperatures (h1/h2 = 4, Nx = 80 cells).
(a) (a)
(b) (b)
Figure 11. 11. (a) (a) Relative Relativedifference differenceand and(b)(b) computational time of the LPCM solution Y direction Figure computational time of the LPCM solution for Yfor direction grid Figure 11. (a) Relative difference and (b) computational time of the LPCM solution for Y direction grid refinements (h 1 /h 2 = 4, Nx = 80 cells). refinements (h1 /h2 = 4, Nx = 80 cells). grid refinements (h1/h2 = 4, Nx = 80 cells).
The results obtained for h1/h2 = 8 are depicted in Figures 12–14. The same remarks as for h1/h2 = 4 The resultsexcept obtained 1/h2 = 8 are depicted Figures 12–14. remarks as for h1/hwas 2= 4 can be drawn, for for thehfact that this time, ainhigher order of The the same Legendre polynomials can be drawn, for the fact that this time, a higher order to of the Legendre was needed for the except grid convergence solution, and this is related increased polynomials inlet/outlet film needed for theFigure grid convergence solution, and this is time related to the increased film thickness ratio. 14b shows that the computational of the NDM with Nyinlet/outlet = 160 increases thickness ratio. Figure shows thatfrom the computational time thethe NDM with Ny = 160 increases for this calculation case.14b It can be seen Figures 11b and 14bofthat computational effort of the
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The results obtained for h1 /h2 = 8 are depicted in Figures 12–14. The same remarks as for h1 /h2 = 4 can be drawn, except for the fact that this time, a higher order of the Legendre polynomials was needed for the grid convergence solution, and this is related to the increased inlet/outlet film thickness ratio. Figure 14b shows that the computational time of the NDM with Ny = 160 increases for this calculation case. It can be seen from Figures 11b and 14b that the computational effort of the NDM increases one order of magnitude with increasing the ratio h1 /h2 from 4 to 8. This was not the case for the LPCM, which required the same computational time for these two cases, one or two orders of magnitude lower than the NDM. Therefore, LPCM remains largely superior to NDM in terms of the computational time. The 2018, temperature fields obtained with the LPCM (N = 16) and the NDM (Ny = 160) are depicted Lubricants 6, x FOR PEER REVIEW 11 of in 22 Figures A3–A6 in Appendix A and its numerical values of the wall temperature gradients are given in Lubricants 2018, 6, x FOR PEER REVIEW 11 of 22 Table A1 in Appendix B.
Figure 12. Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed Figure 12. Dimensionless gradient at the lower wall for the LPCM solution with imposed wall temperatures (h1/h2 =temperature 8, Nx = 80 cells). Figure 12. Dimensionless temperature gradient at the lower wall for the LPCM solution with imposed wall temperatures (h1 /h2 = 8, Nx = 80 cells). wall temperatures (h1/h2 = 8, Nx = 80 cells).
Figure 13. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed Figure 13. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h1 /h2 = 8, Nx = 80 cells). wall temperatures (h1/h2 = 8, Nx = 80 cells). Figure 13. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed wall temperatures (h1/h2 = 8, Nx = 80 cells).
Figure 13. Dimensionless temperature gradient at the upper wall for the LPCM solution with imposed Lubricants 6, 95 wall 2018, temperatures (h1/h2 = 8, Nx = 80 cells).
(a)
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(b)
Lubricants 2018, 6, (a) x FOR PEER REVIEW 12 of 22 Figure and(b) (b)computational computationaltime timeofof LPCM solution Y direction Figure14. 14. (a) Relative Relative difference difference and thethe LPCM solution for for Y direction grid
grid refinements (h21/h 8, Nx = 80 cells). refinements (h1 /h = 28,= Nx = 80 cells).
4.2. Different Thermal Boundary Conditions Applied to 1D Slider 4.2. Different Thermal Boundary Conditions Applied to 1D Slider The previous cases dealt with imposed wall temperatures, while the wall temperature gradient previousresult. cases dealt imposed wall wallwas temperature was aThe calculation In thewith following test, thetemperatures, lower wall of while the 1Dthe slider adiabatic;gradient that is, was⁄∂y a calculation following test, theupper lowerwall wallisofTthe 1D = slider was adiabatic; that is, (∂𝑇 ) = 0result. while In thethe temperature of the 30 °C. The film thickness ◦ C. The film thickness ∂T/∂y = 0 while the temperature of the upper wall is T = 30 ( ) 1/h2 = 4 and the inlet temperature T = 20 °C. For theUpperW LPCM, 10 and 14 Lobatto points ratio is hLowerW ◦ C. For the LPCM, 10 and 14 Lobatto points were ratio isused h1 /hand and results the inletwere temperature Tinlet 20 NDM 2 = 4 the were compared to=the (Nx = 160 cells and Ny = 160 cells, all used and theFigures results 15 were to the NDM (Nx = 160 cells and Ny =wall 160temperature cells, all equidistant). equidistant). andcompared 16 show the consistent resolution of the upper gradient Figures 15 and 16 show the consistent resolution of the upper wall temperature gradient and of the and of the lower wall temperature. lower wall temperature.
Figure 15. Dimensionless temperature gradient at the upper wall. Figure 15. Dimensionless temperature gradient at the upper wall.
A different test case consists of imposing different wall temperatures, TUpperW = 30 ◦ C, TLowerW = 20 ◦ C, and Tinlet = 20 ◦ C. The LPCM was performed with 10 and 14 Lobatto points, and the previous resolution was used for the NDM. The results are depicted in Figures 17 and 18, and show that discrepancies exist between the NDM solution and the LPCM solution at the lower wall (Figure 18) in the inlet section. Again, this difference is reduced with the increase of the degree of the Legendre polynomials. It is necessary to highlight that the LPCM method approximates the temperature gradient at the walls based on the polynomial coefficients of temperature over the film thickness, while the NDM method calculates the temperature gradient with finite differences over a limited number of boundary cells. Thus, the two methods have different approximation orders. In some extreme configurations, it is necessary to increase the degree of the Legendre polynomial in order to reach the grid independent solution.
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Figure 15. Dimensionless temperature gradient at the upper wall.
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thickness, while the NDM method calculates the temperature gradient with finite differences over a limited number of boundary cells. Thus, the two methods have different approximation orders. In thickness, while the NDM method calculates the temperature gradient with finite differences over a some extreme configurations, it is necessary to increase the degree of the Legendre polynomial in limited number of boundary cells. Thus, the two methods have different approximation orders. In order to reach the grid independent solution. some extreme configurations, it is necessary to increase the degree Figure 16. Temperature at the lower wall.of the Legendre polynomial in 16. Temperature at the lower wall. The temperature fields of Figure these two last cases are depicted in Figures A7–A10 of Appendix A order to reach the grid independent solution. andThe its numerical values of the wall temperatures and temperature gradients are given inAppendix Table A1 A temperature fields cases are are depicted depictedininFigures FiguresA7–A10 A7–A10 Thedifferent temperature fields of of these these two two last cases ofof Appendix A = 30 °CA, in Appendix B. test case consists of imposing different wall temperatures, T and andtemperature temperaturegradients gradients are given Table anditsitsnumerical numericalvalues valuesof of the the wall wall temperatures temperatures and are given in in Table A1A1 = 20 °C, and T = 20 °C. The LPCM was performed with 10 and 14 Lobatto points, and T AppendixB.B. ininAppendix the previous resolution was used for the NDM. The results are depicted in Figures 17 and 18, and show that discrepancies exist between the NDM solution and the LPCM solution at the lower wall (Figure 18) in the inlet section. Again, this difference is reduced with the increase of the degree of the Legendre polynomials. It is necessary to highlight that the LPCM method approximates the temperature gradient at the walls based on the polynomial coefficients of temperature over the film
Figure 17. Dimensionless temperature gradient at the upper wall. Figure 17. Dimensionless temperature gradient at the upper wall. Figure 17. Dimensionless temperature gradient at the upper wall.
Figure 18. Dimensionless temperature gradient at the lower wall. Figure 18. Dimensionless temperature gradient at the lower wall. Figure 18. Dimensionless temperature gradient at the lower wall.
5. Results for the Energy Equation Coupled with the Reynolds Equation
5. Results for thepresented Energy Equation Coupledthe withgrid the refinements Reynolds Equation The above results showed required for obtaining accurate solutions of the energy equation when decoupled from the Reynolds equation. In reality, the viscosity The above presented results showed the grid refinements required for obtaining accurate
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5. Results for the Energy Equation Coupled with the Reynolds Equation The above presented results showed the grid refinements required for obtaining accurate solutions of the energy equation when decoupled from the Reynolds equation. In reality, the viscosity of a liquid lubricant depends strongly on the temperature, and therefore decoupling is very difficult. For this reason, the data found in the literature always considered the complete thermo-hydrodynamic analysis. Lubricants 6, x FOR PEER REVIEW (THD) coupled analysis was subsequently performed for the 14 of 22 This2018, thermo-hydrodynamic 1D slider case. The results of the LPCM were compared with the data from Ref. [3]. Their grid convergence convergence and the computational time were analyzed bywith comparisons withThis NDM results. This step and the computational time were analyzed by comparisons NDM results. step completes the completesof the validation of energy the decoupled equation the previous sections. validation the decoupled equationenergy described in thedescribed previous in sections. A temperature temperature dependent dependent viscosity viscosity following following ananexponentially exponentiallydecaying decayinglaw lawη ( T𝜂(𝑇) A ) == . ( T(− T ) − 0.045 ) 0.13885𝑒 was used replacethe theconstant constantviscosity viscosityη0𝜂. .The Therest restofofthe thegeometrical geometrical ambient was 0.13885e used toto replace and physical parameters were the same as in the previous case. The inlet and wall temperatures were and physical parameters were the same as in the previous case. The inlet and wall temperatures were equal to the ambient reference temperature. As for the variations of the temperature across the thin equal to the ambient reference temperature. As for the variations of the temperature across the thin film,the theterms termsin inthe thegeneralized generalizedReynolds Reynoldsequation equationwith with variable variable viscosity viscosity were were also also discretized discretizedby by film, Legendre polynomials, and the coefficients were obtained by collocation at the Lobatto points. Legendre polynomials, and the coefficients were obtained by collocation at the Lobatto points. Asfor forRef. Ref. computational domain 1Dwas slider was discretized Nx = 30 As [3],[3], the the computational domain of the of 1D the slider discretized using Nx =using 30 equidistant equidistant cellsflow in the main flow direction 10across Lobatto across the thickness. cells in the main direction and 10 Lobatto and points thepoints film thickness. Thefilm Reynolds and The the Reynolds and the energy equations were numerically solved following a segregated approach. energy equations were numerically solved following a segregated approach. Figure19a 19acompares comparesthe thepressure pressurevariation variationin inthe the1D 1Dslider sliderobtained obtainedwith withthe theLPCM LPCMand andthe theone one Figure given in Ref. [3]. Figure 19b presents the variation of the outlet temperature difference across the film given in Ref. [3]. Figure 19b presents the variation of the outlet temperature difference across the film thickness.Both Boththethe pressure the temperature variations obtained are in thickness. pressure and and the temperature variations obtained with the with LPCMthe areLPCM in agreement agreement with Ref. [3].with Ref. [3].
(a)
(b)
Figure19. 19.(a) (a)Pressure Pressurevariation variationininthe the1D 1Dslider slider(h(h 1/h2 = 2) coupled coupled thermo-hydrodynamic thermo-hydrodynamic (THD) (THD) Figure 1 /h 2 = 2) 1/h2 = 2) solution, and and (b) (b)outlet outlettemperature temperaturevariation variationacross acrossthe thefilm filmthickness thicknessininthe the1D 1Dslider slider(h(h solution, 1 /h 2 = 2) coupledTHD THDsolution. solution. coupled
This slider was also used to compare the efficiency of theof LPCM with the NDM. This1D 1Dnon-isothermal non-isothermal slider was also used to compare the efficiency the LPCM with the Several were performed with the NDM method in order to check the to grid convergence NDM. calculations Several calculations were performed with the NDM method in order check the grid and for obtaining that could serve ascould a reference. These tests used seven grid refined in grid the convergence and results for obtaining results that serve as a reference. These tests used seven yrefined direction (Nyy =direction 10, 20, 40, 80, 20, 100, equally control volumes), a constant in the (Ny60, = 10, 40,and 60, 120 80, 100, and spaced 120 equally spaced controlwhile volumes), while number of 30 controlofvolumes was used in was the xused direction. relative difference ε K difference in terms of 𝜀wall a constant number 30 control volumes in the The 𝑥 direction. The relative in temperature between two successive grids is definedgrids as: is defined as: terms of wallgradients temperature gradients between two successive r ∂TK +1(𝑦) (y) 2 𝜕𝑇 1 1 ∑n 𝜕𝑇∂T(𝑦) K (y) − ∑ i =1 − n ∂y 𝜕𝑦∂y 𝜕𝑦 ε = 𝑛 r 𝜀K = ∂TK +1 (y) 2 1 n ∑ i =1 n ∂y 𝜕𝑇 (𝑦) 1 ∑ 𝑛 𝜕𝑦
(19) (19)
where the subscript K indicates the grid refinement level in the y direction. Figure 20a shows that a minimum number of 40 cells in the y direction is necessary to reach a satisfactory grid-independent solution. The computational time is depicted in Figure 20b. The solution obtained by the NDM with 120 equally spaced cells over the film thickness is considered as the reference for the 1D thermo-hydrodynamic slider test case.
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where the subscript K indicates the grid refinement level in the y direction. Figure 20a shows that a minimum number of 40 cells in the y direction is necessary to reach a satisfactory grid-independent solution. The computational time is depicted in Figure 20b. The solution Lubricants 2018, 6, x FOR PEER REVIEW 15 of 22 obtained by the NDM with 120 equally spaced cells over the film thickness is considered as the Lubricants 2018, x FOR PEER REVIEW 15 of 22 reference for 6, the 1D thermo-hydrodynamic slider test case.
(a)
(b)
(a) difference 𝜀 of the NDM for different grid(b)refinements, and (b) Figure 20. (a) Relative computational time of the NDM for different numbers of cells, over the film thickness for the 1D THD Figure 20. Relative difference ε K of 𝜀the NDM different refinements, (b) computational 20.(a)(a) Relative difference of theforNDM forgrid different grid and refinements, and (b) slider. time of the NDM forofdifferent numbers of cells, over the 1D THDfor slider. computational time the NDM for different numbers offilm cells,thickness over the for filmthe thickness the 1D THD slider.
The LPCM LPCM results results obtained obtained with with different different numbers numbers of of Lobatto Lobatto points points were were compared compared with with the the The reference NDM NDM solution. In In Figure Figure 21a, 21a, the the relative relative difference difference drops drops rapidly rapidly and and remains remains below below 1%, 1%, reference The LPCM solution. results obtained with different numbers of Lobatto points were compared with the starting with 13 Lobatto points. Figure 21b shows that the computational time for the LPCM does not starting 13 Lobatto Figure that the computational timeand for the LPCM does 1%, not referencewith NDM solution.points. In Figure 21a,21b theshows relative difference drops rapidly remains below exceed 2 s, while the reference method takes about 54 s. exceed s, while the reference takes aboutthat 54 s. starting2with 13 Lobatto points.method Figure 21b shows the computational time for the LPCM does not exceed 2 s, while the reference method takes about 54 s.
(a)
(b)
(a)difference (b) Figure 21. 21. (a) NDM NyNy = 120, andand (b) Figure (a) Relative Relative difference𝜀ε Nbetween betweenthe theLPCM LPCMand andthe thereference reference NDM = 120, (b) computational of the LPCM compared to reference the reference NDM 120 results, for1D theTHD 1D computational timetime of the LPCM compared to the NDM Ny =Ny 120=results, for the Figure 21. (a) Relative difference 𝜀 between the LPCM and the reference NDM Ny = 120, and (b) THD slider.slider. computational time of the LPCM compared to the reference NDM Ny = 120 results, for the 1D THD slider. of a Two-Lobe Journal Bearing 6. Example
6. Example of a Two-Lobe Journal Bearing The coupled ReynoldsJournal and energy equations represent a solver for thermo-hydrodynamic 6. Example of a Two-Lobe Bearing The coupled Reynolds and energy equations represent a solver for thermo-hydrodynamic problems in lubricated journal and thrust bearings. However, pure validations are more difficult problems in lubricated journal and thrust bearings. However, pure validations are more difficult in coupled Reynoldsbearings and energy equations represent effects, a solversuch for as thermo-hydrodynamic in theThe context of lubricated because of the additional the film rupture and the context of lubricated bearings because of the additional effects, such as the film rupture and problems in (cavitation), lubricated journal bearings. pure are more difficult in reformation which and mustthrust be modeled andHowever, dealt with Ref.validations [8,9]. reformation (cavitation), which must be modeled and dealt with Ref. [8,9]. the context of lubricated bearings because of the additional effects, such as the film rupture and Not the least, several user-defined heat transfer parameters must be specified, and this is Not the least, several user-defined heat transfer parameters must be specified, and this is mainly reformation (cavitation), which must be modeled and dealt with Ref. [8,9]. mainly done on a trial and error basis. The uncertainties in this kind of problem may therefore done on a trial and error basis. The uncertainties in this kind of problem may therefore be quite Not the least, several user-defined heat parameters be specified, and this is mainly be quite important, and comparisons withtransfer experimental datamust are not exactly validations of the important, and comparisons with experimental data are not exactly validations of the numerical done on a trial and error basis. The uncertainties in this kind of problem may therefore be quite numerical approaches. approaches. important, and comparisons with experimental data are not exactly validations of the numerical For the completeness of the present work, the numerical analysis of a real journal bearing is approaches. presented in the following. The recent experimental results published by Giraudeau et al. [10] in 2016 For the completeness of the present work, the numerical analysis of a real journal bearing is for a two-lobe journal bearing with axial supply grooves were used for comparing numerical presented in the following. The recent experimental results published by Giraudeau et al. [10] in 2016 predictions with experimental results. The length of the tested bearing was 68.4 mm and its diameter for a two-lobe journal bearing with axial supply grooves were used for comparing numerical
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For the completeness of the present work, the numerical analysis of a real journal bearing is presented in the following. The recent experimental results published by Giraudeau et al. [10] in 2016 for a2018, two-lobe bearing with axial supply grooves were used for comparing numerical Lubricants 6, x FORjournal PEER REVIEW 16 of 22 predictions with experimental results. The length of the tested bearing was 68.4 mm and its diameter μm.100 Themm. bearing was lubricated an ISOwas VG 46 oil supplied a constant pressure of 0.17 and was The radial assembly by clearance 68 µm, while theatradial bearing clearance was MPa 143 µm. a temperature 43 °C. The oiloilcharacteristics used for theofcalculations: The bearing was of lubricated by anfollowing ISO VG 46 supplied at a were constant pressure 0.17 MPa and𝜌a= 850 kg/m ,of𝐶43=◦ C. 2000 and = 0.13 W/(mK). viscosity the oil was ρ0.0416 Pa ∙ s 3at temperature TheJ/(kgK), following oil𝜆characteristics wereThe used for the of calculations: = 850 kg/m , ◦ and Pa ∙ sλ at The variation is described decaying The C p40 = °C 2000 J/(0.0191 kgK), and = 60 0.13°C. W/ of by thean oilexponentially was 0.0416 Pa ·s at 40 law. C and (mK ). The viscosity ◦ C. 3500 angular rpm and is the static load 6 kN. Only the results for the lower loaded lobe 0.0191 Pa·speed s at 60was The variation described bywas an exponentially decaying law. The angular speed are 3500 shown Cavitation was dealt with bythe using thefor algorithm [11].here. This was rpmhere. and the static load was 6 kN. Only results the lowerintroduced loaded lobeinareRef. shown algorithm was usesdealt a freewith boundary formulation of the incompressible problem Cavitation by using the algorithm introduced in Ref.Reynolds [11]. Thisequation. algorithmThe uses a free was thenformulation solved withofan efficient solver based on the Fischer–Burmeister Newton algorithm, boundary the incompressible Reynolds equation. The problemform, was then solved with an and Shur’s complement. efficient solver based on the Fischer–Burmeister form, Newton algorithm, and Shur’s complement. Theshaft shaftwas wasconsidered consideredtotohave havea aconstant constanttemperature temperatureestimated estimatedfrom fromexperiments, experiments,while while The adiabaticwall wallconditions conditionswere wereimposed imposedon onthe thebushing. bushing.The Thecomputational computationaldomain domainwas wasdiscretized discretized adiabatic using3232× 1616cells cellsinincircumferential circumferentialand andaxial axialdirections, directions,while while1111Lobatto Lobattopoints pointswere wereused usedtoto using describethe thetemperature temperaturevariation variationacross acrossthe thefluid fluidfilm. film. describe Figure22a,b 22a,bdepicts depictsthe thepressure pressureand andthe thetemperature temperaturevariations variationsininthe thecircumferential circumferentialdirection direction Figure thebearing bearingmid midplane. plane.The Thepredicted predictedpressures pressuresshow showgood goodagreement agreementwith withthe themeasurements, measurements, ofofthe whilethe thepredicted predicted temperatures show a reasonable agreement. quality of the prediction could while temperatures show a reasonable agreement. TheThe quality of the prediction could be be improved by including the thermal deformation of the bushing and refining the boundary improved by including the thermal deformation of the bushing and refining the boundary conditions conditions forequation. the energy equation. for the energy
(a)
(b)
Figure Figure22. 22.(a) (a)Comparison Comparisonofofmeasured measuredpressures pressuresand andcurrent currentnumerical numericalresults resultsininthe themid midplane planeofof the loaded lobe (3500 rpm, 6 kN load), and (b) comparisons of measured temperatures and current the loaded lobe (3500 rpm, 6 kN load), and (b) comparisons of measured temperatures and current numerical numericalresults resultsininthe themid midplane planeofofthe theloaded loadedlobe. lobe.
7.7.Conclusions Conclusions The Thepresent presentwork workfocused focusedon onthe thenumerical numericalsolution solutionofofthe theenergy energyequation equationininvery verysimple simpletest test cases as part of thin fluid films lubrication. The work was motivated by the fact that the cases as part of thin fluid films lubrication. The work was motivated by the fact that thecomplete complete energy energyequation equationhas hasno noanalytical analyticalsolution solutionthat thatcan canbe beused usedfor forcomparisons comparisonsininvalidations. validations.Therefore, Therefore, an energy equation decoupled from the Reynolds equation in the case of a 1D an energy equation decoupled from the Reynolds equation in the case of a 1Dslider sliderwas wasimagined, imagined, for which the analytical solution of the velocity field was used. The reference values used in for which the analytical solution of the velocity field was used. The reference values used inall allcases cases presented presentedare aredetailed detailedininAppendix AppendixB.B.The Thesecond secondconcern concernwas wasan anefficient efficientsolver solverfor forthe theenergy energy equation. equation.Two Twonumerical numericalmethods methodswere werecompared: compared:The TheNDM NDMbased basedon onaafinite finitevolume volumediscretization, discretization, and the LPCM based on a Legendre polynomial approximation of the temperature across and the LPCM based on a Legendre polynomial approximation of the temperature acrossthe thefilm film thickness. one or or two two orders orders of of magnitude magnitudemore moreefficient efficientthan thanthe theNDM NDMin thickness.The The LPCM LPCM proved proved to to be be one interms termsofofcomputation computationtime. time.This This negligible if one takes into account that when coupled, is is notnot negligible if one takes into account that when coupled, the
Reynolds and the energy equations are numerically solved in a segregated and iterative manner. The thermo-hydrodynamic calculation of a 1D slider (i.e., for coupled Reynolds and energy equations) confirms this conclusion. Comparisons of the LPCM thermo-hydrodynamic analyses with the experimental results obtained in a two lobe journal bearing were also presented. However, they are not considered as real
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the Reynolds and the energy equations are numerically solved in a segregated and iterative manner. The thermo-hydrodynamic calculation of a 1D slider (i.e., for coupled Reynolds and energy equations) Lubricants 2018, 6, x FOR PEER REVIEW 17 of 22 confirms this conclusion. Lubricants 2018, 6, x FOR REVIEW 17 of 22 Comparisons ofPEER theused. LPCM thermo-hydrodynamic with experimental results energy equation were Alternatively, the resultsanalyses presented forthe the 1D slider may be obtained used for in a two lobe journal bearing were also presented. However, they are not considered as real validations validating the first development steps of any solver based on the energy equation for thin film flows. energy equation were used. Alternatively, the results presented for the 1D slider may be used for of the numerical model, since very simple and intuitive boundary conditions of the energy equation validating the first development steps ofM.A.; any solver based on energy equation for film flows. Author Contributions: Conceptualization, Methodology, S.Z.the and M.A.; S.Z.;thin Validation, S.Z. were used. Alternatively, the results presented for the 1D slider may be Software, used for validating the first and M.-A.H.; Formal Analysis, M.A.; Data Curation, S.Z.; Writing-Original Draft Preparation, S.Z. and M.A.; development steps of any solver based on the energy equation for thin film flows. Author Contributions: Conceptualization, M.A.; Software, S.Z.; M.-A.H; Validation, S.Z. Writing-Review & Editing, M.-A.H. and M.A.; M.A.; Methodology, Visualization, S.Z. S.Z; and Supervision, M.A. and Project and M.-A.H.; Formal Analysis, M.A.; Data Curation, S.Z.; Writing-Original Draft Preparation, S.Z. and M.A.; Administration, M.A.; Funding Acquisition, M.-A.H. Author Contributions: Conceptualization, M.A.; Methodology, S.Z. and M.A.; Software, S.Z.; Validation, Writing-Review & Formal Editing,Analysis, M.-A.H. M.A.; and M.A.; Visualization, Supervision, Draft M.A. Preparation, and M.-A.H;S.Z. Project S.Z. and M.-A.H.; Data Curation, S.Z.; S.Z; Writing-Original and Funding: This research no external funding. Administration, M.A.; Funding Acquisition, M.-A.H. M.A.; Writing-Review &received Editing, M.-A.H. and M.A.; Visualization, S.Z; Supervision, M.A. and M.-A.H; Project Administration, M.A.; Funding Acquisition, M.-A.H. Acknowledgments: Thereceived authorsno thank Électricité de France (EDF) R&D for their financial support. Funding: This research external funding. Funding: This research received no external funding. Conflicts of Interest:The Theauthors authorsthank declare no conflict of interest. Acknowledgments: Électricité de France (EDF) R&D for their financial support. Acknowledgments: The authors thank Électricité de France (EDF) R&D for their financial support. Conflicts of Interest: The authors declare no conflict of interest. Appendix Temperature Fields Conflicts of A. Interest: The authors declare no conflict of interest.
Case 1: ℎTemperature /ℎ = 2, 𝑇 = 20 °C, 𝑇 Appendix Appendix A. A. Temperature Fields Fields
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
Case /ℎ = 2, 𝑇 = 20 ◦°C, 𝑇 = 20◦ °C, 𝑇 = 20 = 20 ◦ °C, 𝑇 ◦ °C. Case 1: 1: hℎ1 /h 2 = 2, TU pperW = 20 C, TLowerW = 20 C, Tinlet = 20 C, Toutlet = 20 C.
Figure A1. LPCM, N = 12. Figure A1. LPCM, N = 12. Figure A1. LPCM, N = 12.
Figure A2. NDM, Ny = 80.
Figure A2. NDM, Ny = 80.
Case 2: ℎ /ℎ = 4, 𝑇
= 80. = 20 Figure °C, 𝑇 A2. NDM, = 20Ny°C, 𝑇
= 20 °C, 𝑇
= 20 °C.
Case 2: ℎ /ℎ = 4, 𝑇
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
= 20 °C, 𝑇
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2
U pperW
20 ◦ C, TLowerW = 20 ◦ C, Tinlet = 20 ◦ C, Toutlet = 20 ◦ C.
Figure A3. LPCM, N = 16. Figure A3. LPCM, N = 16. Figure A3. LPCM, N = 16.
Figure A4. NDM, Ny = 160. Figure A4. NDM, Ny = 160.
Figure A4. NDM, Ny = 160.
Case 3: ℎ /ℎ = 8, 𝑇
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
Case 3: ℎ /ℎ = 8, 𝑇
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
Figure A5. LPCM, N = 16. Figure A5. LPCM, N = 16.
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Figure A4. NDM, Ny = 160.
Case /ℎ = = 8, = 20 20 ◦ C, °C,T𝑇 = 20◦ °C, 𝑇 = 20 = 20 °C. ◦ °C, 𝑇 ◦ Case 3: 3: hℎ1 /h 8, T𝑇U pperW = 2 LowerW = 20 C, Tinlet = 20 C, Toutlet = 20 C.
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Figure A5. LPCM, N = 16. Figure A5. LPCM, N = 16.
Figure A6. NDM, Ny = 160. Figure A6. NDM, Ny = 160.
Case 4: ℎ /ℎ = 4, 𝑇
= 30 °C, (𝜕𝑇⁄𝜕𝑦)
= 0, 𝑇
Figure A7. LPCM, N = 16.
= 20 °C, 𝑇
= 20 °C.
Figure A6. NDM, Ny = 160. Lubricants 2018, 6, 95
Case 4: ℎ /ℎ = 4, 𝑇
A6. NDM, Ny = 160. ⁄𝜕𝑦 ) = 30 Figure °C, (𝜕𝑇 = 0, 𝑇
= 20 °C, 𝑇
= 20 °C.
(𝜕𝑇⁄𝜕𝑦) Case /ℎ = = 4, = 30 30 ◦ C, °C,(∂T/∂y 𝑇 == 𝑇 = 20 = ◦20 ◦ C,°C, Case 4: 4: hℎ1 /h 4, T𝑇U pperW = 2020 Toutlet C. °C. ) LowerW ==0,0,Tinlet 2
Figure A7. LPCM, N = 16. Figure A7. LPCM, N = 16. Figure A7. LPCM, N = 16.
Figure A8. NDM, Ny = 160. Figure A8. NDM, Ny = 160.
Case 5: ℎ /ℎ = 4, 𝑇
Ny = 160. = 30 Figure °C, 𝑇 A8. NDM, = 20 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
Case 5: ℎ /ℎ = 4, 𝑇
= 30 °C, 𝑇
= 20 °C, 𝑇
= 20 °C.
= 20 °C, 𝑇
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Case 5: h1 /h2 = 4, TU pperW = 30 ◦ C, TLowerW = 20 ◦ C, Tinlet = 20 ◦ C, Toutlet = 20 ◦ C.
Figure A9. LPCM, N = 16. Figure A9. LPCM, N = 16. Figure A9. LPCM, N = 16.
Figure A10. NDM, Ny = 160. Figure A10. NDM, Ny = 160.
Figure A10. NDM, Ny = 160.
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Appendix B. Reference Values Table A1. Result of the all cases obtained by the best NDM mesh.
Case 1 Case 2 Case 3 Case 4 Case 5
X [-]
0.00625
0.11900
0.24400
0.36900
0.49400
0.61900
0.74400
0.86900
0.99400
dT/dy_LowerW dT/dy_UpperW dT/dy_LowerW dT/dy_UpperW dT/dy_LowerW dT/dy_UpperW T_LowerW [◦ C] dT/dy_UpperW dT/dy_LowerW dT/dy_UpperW
1.55718 −0.13495 1.43830 −2.46419 0.830646 −5.17655 20.276 −0.41141 0.938225 −0.42745
5.21441 −0.96439 5.35156 −2.34816 3.894458 −4.86225 22.293 −0.33314 5.39037 −0.34021
6.77197 −1.92252 7.30481 −2.24448 5.609075 −4.47603 24.839 −0.28279 7.32993 −0.28590
7.44975 −3.09524 8.42416 −2.22769 6.852488 −4.02043 27.649 −0.32754 8.39752 −0.32154
7.53430 −4.61156 8.87026 −2.42486 7.746559 −3.49895 30.669 −0.59665 8.86711 −0.59344
7.14248 −6.64528 8.57660 −3.13238 8.195229 −2.97106 33.738 −1.37867 8.55326 −1.36122
6.36616 −9.46179 7.37678 −5.03152 7.845554 −2.75667 36.470 −3.38145 7.32938 −3.32850
5.34304 −13.49330 5.21935 −10.04817 5.840858 −4.76198 38.132 −8.58824 5.15078 −8.46418
4.35192 −19.48457 3.57358 −24.05486 3.052226 −25.4916 38.967 −23.9895 3.62624 −23.39503
Table A2. Coefficients of Legendre polynomials Tˆ j X describing temperature variations for Case 1, N = 12. X [-] jth Degree 1 2 3 4 5 6 7 8 9 10 11 12 13
0.00625
0.11900
0.24400
0.36900
0.49400
0.61900
0.74400
0.86900
0.99400
20.39 −0.09 −0.16 −0.07 −0.10 0.03 −0.04 0.06 −0.04 0.04 −0.04 0.03 −0.02
23.98 −1.09 −2.41 −0.07 −0.99 0.78 −0.37 0.32 −0.19 0.07 −0.02 0.00 0.00
27.32 −1.70 −5.04 0.28 −1.76 1.18 −0.43 0.24 −0.10 0.01 0.01 −0.01 0.00
30.51 −1.73 −7.85 0.43 −2.35 1.20 −0.30 0.12 −0.02 −0.01 0.01 0.00 0.00
33.61 −1.04 −10.74 0.09 −2.77 0.93 −0.11 0.03 0.02 −0.01 0.01 0.00 0.00
36.64 0.54 −13.60 −1.00 −3.14 0.47 0.06 0.00 0.03 −0.01 0.00 0.00 0.00
39.70 3.24 −16.28 −3.10 −3.62 −0.14 0.17 0.00 0.03 0.00 0.00 0.00 0.00
42.97 7.36 −18.67 −6.52 −4.55 −0.90 0.20 0.05 0.04 0.00 0.00 0.00 0.00
46.43 12.66 −20.42 −11.08 −6.18 −1.72 0.12 0.12 0.05 0.01 0.00 0.00 0.00
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