1 Introduction Commonly manufactured by sintering, porous journal bearings (PJB) have non-rigid seats consisting of a network of microscopic pores. Their main advantage is the ability to store the lubricant within their porous seat and release it in the lubrication gap during operation. This oil circulation insures the desired separation of the moving surfaces (shaft and journal) such that PJBs are able to operate for a lifetime without the need of external supply of lubricant. The flow through the porous seat is therefore crucial to the lubrication process. However, the highly interconnected network of pores have a random behaviour and cannot be characterized by exact methods. Two quantities are used to describe the porous material, namely the scalar porosity φ and permeability tensor ϕ. While the first is a straightforward ratio between the void volume and total volume, permeability is not only related to φ, but it is also influenced by the shape/ size of the pores and fluid viscosity. In studying PJBs, one approach is to apply the Darcy’s law [1] to obtain the pressure through the porous material. The flow in this region is then coupled with the flow in the lubrication gap, commonly given by the Reynolds equation, and the resulting system solved numerically or analytically (see eg. [2, 3]). Let ϕ r denote the characteristic value of the radial component of ϕ. Varying the design variable defined as Ψ = ϕ r λ/ c 3 gives insight into the system response in limiting cases, like, say very high permeabilities. This situation is in our focus here. However, its values can be increased up to only a certain value until numerical so- Aus Wissenschaft und Forschung 11 Tribologie + Schmierungstechnik · 65. Jahrgang · 4/ 2018 Numerical investigation of highly permeable thin-walled porous bearings including cavitation * I. Neacsu, B. Scheichl** Eingereicht: 25.02.2018 Nach Begutachtung angenommen: 15.03.2018 Diese Studie betrachtet eine Erweiterung des bestehenden theoretischen Schmierströmung-Modells, welches in vorangegangenen Arbeiten (mit Ko-autoren) bereits diskutiert wurde, in Zusammenhang mit selbstschmierenden Gleitlagern, die sich durch einen sehr dünnen und entsprechend hoch permeablen Lagersitz auszeichnen. Die Strömung des Schmiermittels wird durch eine entsprechend modifizierte Reynolds-Gleichung beschrieben. Die vorteilshafte Reduktion der physikalischen Einflußgrößen auf nur vier dimensionslose Paramter erleichtert die numerische Behandlung des Problems. Seine Lösung untermauert vorhandene semi-empirische Ergebnisse mit hinreichender Genauigkeit, was weitere Modellieraktivitäten in diese Richtung vielversprechend erscheinen lässt. Schlüsselwörter Sintergleitlager, Modifizierte Reynolds-Gleichung, Schmiertheorie This work is an extension of the lubrication model presented in preceding papers (jointly with co-authors) and considers the case of a self-lubricating journal bearing with a very thin but simultaneously highly permeable porous seat. A general modified Reynolds equation is employed in order to capture this situation. The particular configuration is studied numerically with ease as the system is advantageously governed by only four leading non-dimensional quantities. The results are promising regarding further research efforts since they resemble the behaviour put forward on a semi-empirical basis in the literature available with satisfactory accuracy. Keywords Porous journal bearings, modified Reynolds equation, lubrication theory Kurzfassung Abstract * Based on the contribution presented at OeTG Symposium, TFZ Wiener Neustadt (Austria), 22 November 2017 ** Dr. techn. Ioana-Adina Neacsu AC2T research GmbH, Wiener Neustadt, Austria Priv.-Doz. Dipl.-Ing. Dr. techn. Bernhard Scheichl Österreichische Tribologische Gesellschaft, 2700 Wiener Neustadt, Austria Technische Universität Wien, Institute of Fluid Mechanics and Heat Transfer, 1060 Vienna, Austria T+S_4_2018.qxp_T+S_2018 05.06.18 11: 15 Seite 11 Ê 8 89 : ; < 8,- 8= > ? @ 8 8= : ; A 8,- 8= > " 87BCD 89 E ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ 71D Ê Ê ÊÊÊÊÊ Ê ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ ; <FA " B ' ? ! )G <FA 79F =DF B " 1 ? ÊHÊ cos 9I ÊÊÊÊÊ 72D (1), (2). Its numerical solution is obtained in the same manner as for the full bearing. A conservative numerical scheme is used to discretize the system and an artificial pressure-saturation relation helps to linearize the system. In this way, the lowest pressure that the system can achieve is given by the cavitation pressure, denoted in a nondimensional form P C . The numerical results for the reduced problem are presented in the following section. 3 Numerical results The key parameters that were varied in the present numerical study are ε and Kˆ, while Γ and P C are kept constant (Γ = 0.9, P C = -0.01). The eccentricity ratio takes the usual values between 0 and 1, while Kˆ is varied between 10 -1 and 10 2 . The results of this study are given in Figures 1-5, by evaluating the normalized friction coefficient μ n = μ r/ c, the Sommerfeld number defined as So = w/ p ref r l, where w is the applied load and the attitude angle. Looking at the representation of the normalized friction coefficient μ n versus the inverse of the Sommerfeld number (Figure 1) we can remark the same behavior obtained by Cameron et. al. [7]. It appears that for each value of Kˆ there exists a critical Sommerfeld number which marks the maximum loading capacity that can be achieved. Beyond the critical value, the friction coefficient increases abruptly, and hydrodynamic lubrication cannot be achieved. As Kˆ increases, the value of the critical Sommerfeld varies only slightly as compared to lower values of Kˆ . Also, for very small values of 1/ So all curves collide on the same line, given by Petroff’s friction for lightly loaded bearings. Depicting the normalized friction coefficient versus the eccentricity ratio (Figure 2) shows the asymptotic behaviour of μ n around both limits of ε. We are able to obtain a solution for ε →1 due to the modified pressure coefficients in the leading equation. However, this implies that the friction coefficient achieves a minimum, which in the present configuration appears to occur around ε = 0.65, for any value of Kˆ . For the evaluation of the cavitation behaviour we resort to plotting the minimum gap pressure versus the varied key parameters. We consider the lubricant to have cavitated when Pˆ attains the value of P C . This appears to be the case of the majority of the analysed combinations of parameters, as one can see that P C covers exclusively the region defined by ε > 0.5, denoting medium to high loadings. Cavitation-free region is encountered for the case of low eccentricities and larger coupling parameter, and specifically, as ε →0, for any value of Kˆ the lubricant remains in a coherently fully liquid state. Finally, we look at the bearing attitude angle, defined as the angle between the line of centres and the line of the applied load. Its variation with the Sommerfeld number and eccentricity ratio is depicted in Figure 4 and Figu- Aus Wissenschaft und Forschung 12 Tribologie + Schmierungstechnik · 65. Jahrgang · 4/ 2018 lutions cannot be obtained. This motivates the present work, where our interest is to study a reduced system, governed by a single equation, and which is able to tackle the extreme case of very high permeability. 2. Governing equations The model adopted in this work is based upon a rigorous reduction process applied to an appropriate modification of the Reynolds equation within the well-known limitations of lubrication theory. This has originally been developed for the specific case of a finite-width porous bearing with a circular bush and seat where the effects of vaporous cavitation are included [4]. Here we consider the hypothetical limiting case where the porous seat is very thin (its thickness λ is very small), but characterised by a very high permeability. If we keep the notation for the design parameters K = 12ϕ r r/ c 3 , Λ = λ/ r, where r and c are the bearing radius and gap radial clearance, respectively, the configuration we are interested in can be studied by considering Kˆ =KΛ as of O(1) when K >> 1 and Λ << 1. These assumptions lead to a reduced equation in Pˆ (denoting the pressure in the gap, made non-dimensional with a reference pressure p ref = 6ωη(r/ c) 2 , where ω is the journal rotation speed and η is the lubricant viscosity) written in the form (see [5, 6]): (1) (2) Herein θ and z denote respectively the circumferential and axial coordinates, the latter made non-dimensional with the bearing length l. H is the non-dimensional film thickness defined as a function of the bearing eccentricity ratio ε, Γ is a non-dimensional parameter relating the bearing diameter to the bearing length Γ = (2r) 2 / l, and Φ θ,z is the non-dimensional permeability, made nondimensional with ϕ r , in the azimuthal and axial directions. In the term on the right hand side, S denotes the lubricant saturation, i. e. the density ratio between the twophase mixture in the cavitation zone and the one in the fully liquid film. The typical associated boundary conditions stay fully intact in this reduced setting: periodicity in the circumferential coordinate θ, symmetry with respect to the axial coordinate z, and prescribing the ambient pressure level at the edge of the bearing. In the full simulations the bearing is assumed to be encapsulated in a solid casing such that the oil remains contained in the matrix. This condition and the ambient pressure requirement at the edge of the bearing cannot be fulfilled simultaneously as this would imply a separate treatment of that particular region. It is interesting that both the seepage flow through the seat and the lubricant flow through the gap are described by the single equation (1) of the rigorously reduced problem T+S_4_2018.qxp_T+S_2018 05.06.18 11: 15 Seite 12 re 5, respectively, where the curves are smoothed out by using spline interpolation. The overall values lie close to each other, indicating only a small displacement of the shaft inside the bearing for the analysed configuration. The particular behaviour observed for all the combinations ε and Kˆ , namely a local maximum at a given So is related and due to the occurrence of cavitation. The deviations from the typical 270°attitude angle for relatively light loads and a fully submerged bearing are relatively small and, as one might expect, promoted by higher loads but, on the other hand, diminished by an increased permeability of the seat, as acting as an additional lubricant reservoir. 4 Conclusions An advantageous single equation was obtained in order to describe the case of a very thin and highly permeable porous journal bearing in a most rational manner. The numerical study accompanying this reduced problem confirms results found in literature, and covers a large spectrum of variables. As indicated, this approach allows for a self-consistent and appealing explanation of the minimum friction value under variation of the Sommerfeld number. We finally point to the quantitative asymptotic behaviour for very lightly and heavily loaded bearings resorting to (1) as put forward in [5, 6]. Current research concerns the inclusion of microscopic effects such as by distributed asperities of the gap surfa- Aus Wissenschaft und Forschung 13 Tribologie + Schmierungstechnik · 65. Jahrgang · 4/ 2018 Figure 1: Normalized friction coefficient vs. the inverse of the Sommerfeld number for various coupling parameters Figure 2: Normalized friction coefficient vs. the eccentricity ratio for various coupling parameters Figure 4: Bearing attitude angle vs. Sommerfeld number for distinct values of Kˆ Figure 5: Polar plot of the bearing attitude angle vs. the eccentricity ratio for distinct values of Kˆ Figure 3: Influence of ε and Kˆ on the minimum lubrication pressure T+S_4_2018.qxp_T+S_2018 05.06.18 11: 15 Seite 13 [2] Meurisse M. H., Giudicelli B.: A 3D conservative model for self-lubricated porous journal bearings in a hydrodynamic steady state. Journal of Tribology 121, 529 - 537 (1999) [3] D’Agostino V., Senatore A.: Analytical solution for twodimensional Reynolds equation for porous journal bearings. Industrial Lubrication and Tribology 58, 110 - 117 (2006) [4] Scheichl B., Neacsu I.A., Kluwick A.: A novel view on lubricant flow undergoing cavitation in sintered journal bearings. Tribology International 88, 189 - 208 (2015) [5] Neacsu I.A., Scheichl B., Vorlaufer G., Eder S., Franek F., Ramonat L.: Experimental validation of the simulated steady-state behaviour of porous journal bearings. Journal of tribology 138(3), 031703 (2016) [6] Neacsu I.A., Scheichl B.: Numerical simulation of cavitating flow through porous journal bearings: the case of highly permeable seats. Proceedings of the OeTG Symposium (2017) [7] Cameron A., Morgan V. T., Stainsby A. E.: Critical conditions for hydrodynamic lubrication of porous metal bearings. Proceedings of the Institution of Mechanical Engineers 176, 761 - 770 (1962) Aus Wissenschaft und Forschung 14 Tribologie + Schmierungstechnik · 65. Jahrgang · 4/ 2018 ces and associated micro-cavitation. This challenge can, in principle, be mastered by a systematic extension of the current asymptotic approach by multiple scaling. In the same spirit, this applies to inertial effects and other ones definitely beyond lubrication theory, which, as we feel, have not appreciated correct attention in computational schemes so far. Nevertheless, these are of interest in view of a most reliable, state-of-the-art prediction of the bearing behaviour. 5 Acknowledgement This work was funded by the Austrian COMET Program (Project K2 XTribology, Grant No. 849109) and was carried out at the Excellence Centre of Tribology (AC2T research GmbH). References [1] Whitaker S.: Flow in porous media: A theoretical derivation of Darcy’s law. Transport in porous media 1, 3 - 25 (1986) Bestellcoupon Tribologie und Schmierungstechnik „Richtungsweisende Informationen aus Forschung und Entwicklung“ Getriebeschmierung - Motorenschmierung - Schmierfette und Schmierstoffe - Kühlschmierstoffe - Schmierung in der Umformtechnik - Tribologisches Verhalten von Werkstoffen - Minimalmengenschmierung - Gebrauchtölanalyse - Mikro- und Nanotribologie - Ökologische Aspekte der Schmierstoffe - Tribologische Prüfverfahren Bestellcoupon Ich möchte Tribologie und Schmierungstechnik näher kennen lernen. Bitte liefern Sie mir ein Probeabonnement (2 Ausgaben), zum Vorzugspreis von € 39,-. So kann ich die Zeitschrift in Ruhe prüfen. Wenn Sie dann nichts von mir hören, möchte ich Tribologie und Schmierungstechnik weiter beziehen. Zum jährlichen Abo-Preis von € 189,- Inland bzw. € 198,- Ausland. Die Rechnungsstellung erfolgt dann jährlich. Das Jahresabonnement ist für ein Jahr gültig; die Kündigungsfrist beträgt sechs Wochen zum Jahresende. 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