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Application of CFD to model oil–air flow in a grooved two-disc system
International Journal of Heat and Mass Transfer 91 (2015) 293–301
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Application of CFD to model oil–air flow in a grooved two-disc system Wei Wu ⇑, Zhao Xiong, Jibin Hu, Shihua Yuan National Key Laboratory of Vehicular Transmission, Beijing Institute of Technology, Beijing 100081, PR China
a r t i c l e
i n f o
Article history: Received 20 May 2014 Received in revised form 16 July 2015 Accepted 23 July 2015
Keywords: Rotating disc Rotor-stator discs Two-phase flow Drag torque Wet clutch Tesla pump
a b s t r a c t The flow field of an open grooved two-disc system was studied. The system includes a rotating finite disc and a stationary finite disc. The rotating disc has radial grooves. The numerical results for the air–oil two-phase flow inside the open grooved two-disc system calculated by the CFD code FLUENT were proposed. The results are discussed and compared with the published test results. The results indicate that the groove affects the transitional characteristics from a single-phase flow to an air–oil two-phase flow of the flow field. In the same flow area, the radial oil flow coefficient is enhanced with a larger groove number. The oil mainly discharges through the groove and the air flows into the non-grooved area from the upstream side of the groove. The effects of the angular velocity on the oil volume fraction and the drag torque become greater when the disc is grooved. It is weakened in the high-velocity operations. The drag torque can be reduced by increasing the groove number in the same flow area. The axial force applied on the disc is gradually decreased with the increase of the angular velocity and becomes close to zero finally. The increase of the groove number reduces the maximum axial force. The results can be used for the optimisation of the Tesla pump and wet clutch. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The investigation of viscous flow between rotating discs has received considerable attention, e.g., exact solutions of the Navier–Stokes equations in certain geometric limiting cases, and practical, e.g., computer storage devices, clutches and turbomachinery applications. 1.1. Literature review of the rotor–stator disc flow Von Karman studied the flow over a rotating disc when the fluid at infinity approaches a non-rotating condition [1]. The azimuthal fluid velocity gradually decreases from the velocity of the disc to zero when moving further away from the disc. The region in which these velocity changes occur will be termed here as the ‘‘Von Karman boundary layer’’ or ‘‘Ekman layer’’ [2]. Bödewadt [3] studied the flow over an infinite stationary plane, where fluid rotated with a uniform angular velocity to an infinite distance above the plane. In this situation, the azimuthal velocity component gradually decreases to zero with a decreasing distance from the disc. Here, the region of fluid in which the velocity changes take place is generally termed as the ‘‘Bödewadt boundary layer’’. ⇑ Corresponding author at: Room 412, Building 9, Beijing Institute of Technology, Beijing 100081, PR China. Tel.: +86 10 68914786; fax: +86 10 68944487. E-mail address: [email protected] (W. Wu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.092 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
Batchelor [4] and Stewartson [5] described the flow structure of both the infinite rotor–stator and the infinite counter-rotating discs theoretically. Batchelor [4] suggested that when one disc is rotating at a constant angular velocity, while the other remains stationary, for moderate Reynolds numbers, three distinct layers would form between the discs. Near the rotating disc a Von Karman boundary layer would form, whereas near the stationary disc a Bödewadt boundary layer would develop. According to Batchelor, these two boundary layers are separated by a core in which viscous effects are negligible. The fluid in this non-viscous core would have a constant angular velocity, a constant axial velocity, and zero radial velocity. Stewartson [5] opposed this view by predicting that indeed a Von Karman boundary layer would develop near the rotating disc, but that the main body of the fluid would be at rest. In this situation, both the Bodewadt boundary layer and the non-viscous core would be absent. In the real case of discs of finite radius, both the enclosed disc configuration and the open disc configuration were investigated. Daily and Nece studied experimentally an enclosed rotor–stator flow and identified four flow regimes: two laminar and two turbulent, each having either merged or separated boundary layers [6]. It was found by Brady and Durlofsky that the boundary conditions at the edge of the disc greatly determine the flow pattern actually emerging between the two discs. In an open disc configuration, the flow field would tend towards a Stewartson-type flow [7]. Gan and Macgregor [8] measured the velocity field in rotor–stator
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Nomenclature C 1e C 2e Cl F Fz g Gk h H j k minlet moutlet n nw Ng p Q r r1 r2 ReH Rer t tw
turbulence model constant turbulence model constant turbulence model constant external force axial force on the disc gravity acceleration production term of the turbulent kinetic energy depth of the groove disc spacing iteration number turbulent kinetic energy inlet oil mass flow outlet oil mass flow surface normal vector unit normal vector groove number pressure oil flow rate radial coordinates internal radius external radius reynolds number based on the height rotational Reynolds number time tangent vector
and counter-rotating disc systems without superimposed through flow and concluded that the Batchelor-type flow in rotor–stator. Schouveiler et al. [9] studied the nature of flow between a rotor and a stator disc and found coexistence of annular and spiral rolls. In a later study by Lopez [10], it was shown that the motion of the cylindrical shroud does have a profound impact on the fluid motion inside the cavity. When the full velocity field was calculated inside the cavity together with the no-slip conditions at the cylindrical end wall, it was found that a co-rotating shroud, on the one hand, resulted in Batchelor types of flow. For open disc systems, Rotor–stator flow presents annular and/or spiral rolls on rotor disc plane and Ekman layer on rotor disc in gap flow of single-cell structure [11]. Moreover, recent work by Lopez et al. [12] has revealed that the self-similar solution is indeed able to describe the flow of a fluid near a confined rotating disc up to a radial position of 80% of the disc radius. The Reynolds numbers are up to the order 1e3.
T ui xi z
turbulent drag velocity components coordinate directions axial coordinate
Greek symbols ak turbulence model constant ae turbulence model constant e turbulent kinetic energy dissipation rate em tolerance value eT tolerance value h azimuthal coordinate hw static contact angle r surface tension coefficient j surface curvature l dynamic viscosity leff effective dynamic viscosity q density of the oil or air phase qair air density qoil oil density t velocity vector uair air volume fraction uoil oil volume fraction x angular velocity
between a grooved rotating disc and a smooth stationary disc has also been examined analytically and experimentally [19]. A transition regime from laminar to turbulent flow has been tentatively identified. Kitabayashi et al. [20] presented a theoretical model based on single phase laminar flow in the gap between the discs of the wet clutch. The model assumes a full oil film at all the speeds. Multiphase flows over the disc have been investigated and it was found that the formation of multiphase flows between the two discs has an enormous effect on mass and heat transfer. The two-phase flow in the grooved disc system is the simplified theoretical model for wet clutches and viscous pumps [21,22], as
1.2. Single-phase and two-phase grooved disc flow Today the study of rotating disc flow is still very meaningful. A proper description of the flow pattern between two discs is also crucially important. It can be seen that most of the studies on two-disc flow were associated with the smooth disc and single-phase flow for their wide applications in rotating machinery. The discs were clean and no surface texturing was considered. However, more complicated discs are used for real applications. The grooved disc system is an important pattern to achieve better performance. The single phase flow in the grooved disc system is a simplified theoretical model for many applications e.g. mechanical seals and thrust bearings [13–17]. Muijderman concerns radial single phase inflow with a smooth rotor and grooves on the stator used in spiral groove bearings [18]. The analysis evaluates separately the flow in the grooves and the portion of the gap directly above the grooves and the flow in the portion of the gap above a non-grooved segment of the stator. The radial single phase outflow
Fig. 1. Simplified model.
W. Wu et al. / International Journal of Heat and Mass Transfer 91 (2015) 293–301
shown in Fig. 1. Kato et al. [23] were the first to propose a model that considers the rupture of the full oil film due to cavitation between the discs. The turbulent flow is considered. Yuan et al. [24] presented a mathematical model and verified it by using the commercial computational fluid dynamics code FLUENT and experiments. They formulated the model by assuming non-grooved discs, but eventually approximated the model for the grooved discs by multiplying the torque with the fraction of area which is not grooved. A mathematical model based on continuity and Navier–Stokes equations, considering laminar flow in the gap between the discs was presented and the model is capable of predicting the drag [21]. Another model is also proposed to predict the drag of a disengaged wet clutch at different rotation speeds, clearances, disc sizes and oil temperatures [25]. There is a good degree of agreement between the DT trends derived from the proposed model and the test results for the same condition. Some tested results about the two-phase flow inside the grooved discs of the wet clutch have been presented [26,27]. However, most studies for application focused on the drag forecast and the flow field is not investigated in detail, especially at the air–oil two-phase flow operation. For the gas–liquid two-phase flow simulation, two methods to dynamic flow simulations have been discussed. The first is a method where both the liquid motion and the gas motion are considered in a homogeneous way, which is referred to as Euler-Euler method [28]. The second method treats only the liquid-phase motion in an Eulerian representation and computes the motion of the dispersed gas-phase fluid elements in a Lagrangian way, which is referred to as Euler–Lagrange method [29]. Both the Euler/Euler method and the Euler/Lagrange method rely on physical models for the interfacial transfer of momentum, heat and mass. For this reason, simulation methods are usually used for such problems which do not resolve details of the interface [30]. The Volume of Fluid method employs a piecewise constant scalar field instead of marker particles to locate the position of both fluid phases [31] and has been used widely. Then, the Simple Line Interface Calculation (SLIC) [32] technique and the Piecewise Linear Interface Calculation (PLIC) [33] method are developed to obtain a reconstruction with sufficient accuracy and smoothness. The level-set technique was introduced by Osher and Sethian [34] for gas–liquid two-phase flow simulation. Similar to the VOF method, the level-set technique greatly reduces the complexity of a description of the interface, especially when topological changes such as pinching and merging occur. The objective of this analysis is to characterise the air–oil flow between an open grooved two-disc system. There are two finite discs, one rotating at various angular speeds and the other at rest. The rotating disc has radial grooves. The discs are separated by a small gap. The geometric configuration and the governing parameters are described first. After the basic equations are established considering air–oil two-phase flows, the numerical results of the flow field between the discs are obtained using the CFD code FLUENT 6.1. The results are discussed and compared with published experiments.
2. Problem description Based on the simplified model shown in Fig. 1, the physical situation is schematically shown in Fig. 2, including a rotating finite disc and a stationary finite disc with a disc spacing H. The upper rotating disc has radial grooves and the depth of the groove is h. It rotates with an angular velocity x. The angular width of the radial groove is varied according to the number of grooves since the total area of the grooves is set to be constant in one disc. The flow field is described by a cylindrical coordinate system (r, h
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Fig. 2. Schematic drawing of the open grooved disc system.
and z), being the radial, azimuthal and axial coordinates, respectively. The position r = 0 is identified with the rotation axis and the plane z = 0 is identified with the stationary disc. r1 and r2 are the internal and external radius. The inlet is completely filled with the oil of a flow rate Q, as a mass flow boundary. The outlet is set to be the pressure boundary conditions. In the real case, the single-two phase transition of the flow field is affected by operation parameters such as the angular velocity and two-disc system structure parameters such as the disc spacing. The flow field has an important effect on the operational characteristics of the disc fluid machine. The drag torque of the wet clutch changes with the single-two phase transition of the flow field [21]. The viscous pump efficiency is also affected by the flow field parameters [22,35]. 3. Mathematical modelling 3.1. Governing equations Considering the physical situation of two discs, one stationary and one rotating, the equation of continuity representing the mass conservation is written as following [36]. The modelling is based on the isothermal operation.
@q þ r ðq~ tÞ ¼ 0 @t
ð1Þ
where q denotes the fluid density, t is the time and ~ t is the velocity vector. The momentum equations are written as:
@ðq~ tÞ F þ r ðq~ t~ tÞ ¼ r ðlðr~ t þ ðr~ tÞT ÞÞ r p þ q~ g þ~ @t
ð2Þ
where l is the dynamic viscosity, p is the pressure, ~ g is the gravity F is the external force due to the surface tension acceleration and ~ and wall adhesion at the interface [37]. Surface tension and wall adhesion represent the interaction of the air, oil, and wall. This will affect the external force in the momentum equation. The additional volume force is expressed as follows [38]:
2qjr~ n ~ F¼ qoil þ qair
ð3Þ
where ~ n is the surface normal vector, j is the surface curvature, r is the surface tension coefficient, qoil is the oil density and qair is the air density. The curvature of the surface near the wall, where the air–oil interface meets the solid surface, is adjusted. The local curvature of this interface is determined by the static contact
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angle, hw, which is the angle between the wall and the tangent of the interface at the wall. The surface normal at the wall is given by [39],
~ t w sin hw n ¼~ nw cos hw þ ~
ð4Þ
tw are the unit normal vector and tangent vector nw and ~ where ~ of the wall surface, respectively. The analysis of the whole flow passage involves the air and oil flows. The volume of fluid (VOF) method developed by Hirt and Nichols [33] is used. A volume fraction uoil is used to mark the volume fraction of the oil phase in the VOF method. Then, uoil = 0 represents a cell that is empty of the oil and uoil = 1 represents a cell that is full of the oil. If 0 < uoil < 1, it represents the interface between the oil phase and the air phase. Thus, the uoil is defined by:
8 in the oil > < uoil ¼ 1 uoil ¼ 0 in the air > : 0 < uoil < 1 at the interface
ð5Þ
The subscript oil represents the oil phase properties. The interface between the oil and air phases is tracked by solving a continuity equation for the volume fraction of the oil phase. From Eq. (1), the continuity equation for the volume fraction of the oil phase is given by [37]:
@ ðu q Þ þ r ðuoil qoil~ tÞ ¼ 0 @t oil oil
ð6Þ
where qoil is the oil density. Because there are two phases, Eq. (2) for the volume fraction is solved only for the secondary phase that is defined as the oil phase. The volume fraction for the primary phase that is defined as the air phase is obtained by the following constraint:
uoil þ uair ¼ 1
ð7Þ
where uair is the air volume fraction. The subscript air represents the air phase properties. The properties that appear in Eq. (2) are volume-fractionaveraged properties. The density and the dynamic viscosity for the air–oil two-phase flow are given by
q ¼ uoil qoil þ uair qair
ð8Þ
l ¼ uoil loil þ uair lair
ð9Þ
where qair is the air density, lair is the air dynamic viscosity and loil is the oil dynamic viscosity. 3.2. Numerical method
turbulent has been explained as approximately 2.0e4 for a representative length of disc radius [42,43]. A disc angular velocity of 150 rad/s, giving a rotational Reynolds number of 5.4e6, which exceeds the critical Rer value of 2.0e4 for rotating disc. Thus, the flow field of the open grooved disc system cannot be treated as a laminar flow. In the calculation, the renormalization group (RNG) k–e turbulence model is used. The RNG k–e turbulence model is widely adopted for the rotating or swirling flow. For weakly to moderately strained flows, the RNG model tends to give results comparable to the standard k–e model. In rapidly strained flows, the RNG model yields a lower turbulent viscosity than the standard k–e model. Thus, the RNG model is more responsive to the effects of rapid strain and streamline curvature than the standard k–e model, which explains the superior performance of the RNG model for certain classes of flows including rotational flow [44,45]. Further, Comparison with available limited experimental data shows that CFD results display reasonable agreement. Predicted results also show that the RNG k–e model performs better to describe the behaviour of the performance of a spray dryer fitted with a rotary disc atomizer in a cylinder-on-cone chamber geometry. It is similar in form to the standard k–e model, but the effect of swirl on turbulence is included in the RNG mode enhancing accuracy for swirling flows [46]. The RNG k–e turbulence model is derived from the instantaneous Navier–Stokes equations, using the renormalization group method. The turbulence momentum equation is
@ðqkÞ @ ðqkui Þ @ @k þ Gk þ qe þ ¼ ak leff @t @xi @xi @xi The turbulence dissipation rate equation is
@ðqeÞ @ðqeui Þ @ @e C 1e e e2 þ þ ¼ ae leff Gk C 2e q @t @xi @xi @xi k k
the height of the cavity, ReH = xH2 =l [41]. In the following, we will use the rotational Reynolds number Rer as the general physical parameter. The transition Reynolds number from laminar to
Table 1 Technical data of the open grooved disc system. Value
r1 r2 h Q H
47.5 mm 59.5 mm 0.37 mm 200 ml/min 0.20 mm
ð11Þ
where, k is the turbulent kinetic energy, e is the turbulent kinetic energy dissipation rate, leff the is effective dynamic viscosity, Gk the is production term of the turbulent kinetic energy caused by the average velocity gradient. xi and ui represent the coordinate directions and the velocity components, respectively. C 1e , C 2e , C l , ak and ae are the turbulence model constants. The mesh images of the flow field are shown in Fig. 3. The flow field was divided by a structured mesh. To ensure the accuracy and validity of numerical results, a careful check for the grid independence of the numerical solutions has been made. When the groove number is 12, three sets of mesh were generated with different grid densities, as presented in Table 2. The abscissa is the element quality on a scale from 0 (worst) to 1 (best). All values of the element quality are higher than 0.9, indicating that the all meshes
The technical data of the open grooved disc system is presented in Table 1. The Reynolds number Re is the major physical parameter of the flow. In the literature, the Reynolds number is based either on the external radius of the cavity, Rer = xr22 =l [40], or on
Parameter
ð10Þ
Fig. 3. Simulated zone and view of mesh of the flow field for Ng = 12.
W. Wu et al. / International Journal of Heat and Mass Transfer 91 (2015) 293–301 Table 2 Specifications for three different meshes.
Total number of nodes Minimum volume (m3) Minimum face area (m2)
Mesh 1
Mesh 2
Mesh 3
358,800 1.567941e14 8.999956e11
472,752 1.511754e14 8.999902e11
533,255 1.461819e14 8.999863e11
meet a high-quality criterion. The calculated results of the average oil volume fraction with four different angular velocities are also given, as presented in Table 3. The value differences between the mesh density of 472,752 elements and the other two are smaller than 5% (0.93% 3.33%) in the same velocity. The mesh density presented in Table 3 does not change the average oil volume fraction in any appreciable way. Due to the different groove numbers, the refinements of meshes of 78,234 472,752 elements are chosen for analysis. In the calculation, the air phase was set as the incompressible main phase. The oil phase was set as the incompressible second phase. The volume fraction of the oil phase in the inlet was set as 1.0. The mass flow conservation and the turbulent torque were used as the calculation convergence conditions as follows.
jminlet moutlet j < em jminlet j
ð12Þ
jT jþ1 T j j < eT T jþ1
ð13Þ
where, minlet is the inlet oil mass flow, moutlet is the outlet oil mass flow, em and eT are the tolerance values. The subscript j is the iteration number. 4. Results and discussion 4.1. Flow visualisation Fig. 4 shows the simulated results of the flow field under different operation conditions. The measured visualisations of the flow field are published by Takagi et al. [26]. The air phase in the figure is a little transparent. Fig. 4(a) shows the variation of the flow field under different groove width and the angular velocity is 104.7 rad/s. In the same angular velocity, the air flow becomes more obvious near the outlet with more grooves in the simulation. The measured results have similar trends. The radial position of the phase interface between the pure oil phase and the air–oil mixture is a little closer to the inner radius in the groove due to the larger flow area. It is conducive to the heat diffusion of the disc. Further, it seems that the stability of the phase interface is affected less with a larger groove number. A larger groove number also makes the pure oil phase along the radial direction be gradually reduced at the same angular velocity. Fig. 4(b) presents the flow field under different angular velocities and the groove number is 12. It seems that the air phase in the flow field increases gradually along the radial direction with a higher angular velocity. Although the oil flow rate is constant, the oil volume fraction in the flow field decreases. The phase interface between the pure oil and the air–oil mixture is obvious. When the angular velocity becomes higher, the oil phase inside the groove is more than the outside region of the groove. Table 3 Calculated oil volume fraction under different mesh specifications and angular velocities. Angular velocity
104.7 rad/s
157.0 rad/s
209.3 rad/s
261.7 rad/s
Mesh 1 Mesh 2 Mesh 3
0.72486 0.73549 0.74235
0.24522 0.25011 0.25320
0.2101 0.2147 0.2204
0.17985 0.18605 0.18984
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Compared with the published numerical results [21,26,27], the change of the flow field has the same trend in the present study. The effect of the groove and the single-two phase transition of the flow field are described in detail. The radial groove is easy for machining and widely used. The detailed analysis on the flow field of the grooved disc is useful for the optimisation. Fig. 5 is an enlarged flow field around the groove. The groove number is 12 and the angular velocity is 104.7 rad/s. Fig. 5(a) presents the air–oil distribution. The oil flows from the inner end face of the flow field. As the angular velocity increases, the oil mainly discharges through the groove. The non-grooved space in the radial direction beside the outer end face is filled with the air. The state of the oil flow affects the temperature distribution of the disc. Further, a smaller groove number will make the temperature distribution of the grooved disc become more nonuniform. It is not helpful for the thermal stability of the grooved disc system. Fig. 5(b) is the streamlines of the flow field around the groove. The air–oil distribution of the figure is a little transparent. It seems that the oil phase gradually moves outwards with the enhanced centrifugal force. The formation of the continuous oil phase is still circular. However, the radius of the continuous oil phase in the gap is gradually reduced with the increase of the angular velocity. A large space filled with the air phase is formed in the downstream side of the groove. 4.2. Volume fraction With the increasing of the angular velocity, the flow field inside the grooved disc system changes from a single-phase flow to an air–oil two-phase flow. For a constant oil flow rate of the inlet, the oil volume fraction of the flow field decreases with a higher angular velocity. The transitional characteristics of the flow field are important to get a good performance for such as the wet clutch [47]. Fig. 6 presents the average oil volume fraction with the angular velocity for different groove numbers. Fig. 6(a) is the simulated and measured results of the average oil volume fraction inside the non-grooved disc system. It is seen that the simulated results have the same trend with the experimental data [26]. When the velocity is higher than 100 rad/s in the calculation, the oil volume fraction begins to drop. The value of the oil volume fraction reaches the minimum and becomes essentially unchanged with a higher angular velocity than 200 rad/s. It seems that the effects of the angular velocity on the oil volume fraction variation in the high-velocity operations become smaller. With the increasing of the angular velocity, the relation of the air phase between the inside of flow field and the outside become balanced. The effects of the angular velocity on the oil volume fraction variation become greater when the disc is grooved, as shown in Fig. 6(b). The oil volume fraction decreases more rapidly with the grooved disc. The variations of the oil volume fraction with the angular velocity are similar for the same flow area. Although the grooves under different operation conditions have the same flow area, the flow coefficient in the groove is enhanced with a larger groove number in the same flow area. It results in smaller oil volume fractions at the same angular velocity with different groove numbers, as shown in Fig. 6(b). Further, the effects of the angular velocity on the oil volume fraction with the grooved disc in the high-velocity operations are also gradually weakened the same as with the non-grooved disc. Fig. 7 presents the distribution of the oil volume fraction alone different coordinate directions with different angular velocities. The groove number is 12. Fig. 7(a) and (b) propose the oil volume fraction distributions inside and outside the groove, respectively. It is seen that the oil volume fractions decrease alone the radial direction both inside and outside the groove. Further, the oil volume fraction decreases more rapidly outside the groove than
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(a) Variation of flow field with the groove number for ω = 104.7 rad/s.
(b) Variation of flow field with the angular velocity for the same value of Ng. Fig. 4. Simulated results of the flow field under different operation conditions.
inside. The oil volume fraction decreases with the increase of the angular velocity until the oil volume fraction is zero. It is seen that the groove is the main path of the oil flow in the high-velocity operations. The oil volume fraction inside the groove is higher than the non-grooved space, especially in the high-velocity operations, as shown in Fig. 7(c). Fig. 7(c) presents the oil volume fraction distribution in the circumferential direction. The groove number is 12. It seems that the oil volume fraction distribution is not uniform along the circumferential direction. The peak values of the oil volume fraction appear periodically due to the existence of the groove. With a larger angular velocity, the oil volume fraction
shows a downward trend and the non-uniformity of the oil volume fraction distribution is enhanced. As the angular velocity increases, the centrifugal force increases. More oil phase cannot sustain the negative pressure and also cannot satisfy the flow continuity. As a result, the negative pressure space of the flow field becomes large [21]. The air flows into the non-grooved space more. 4.3. Turbulent drag The turbulent drag is important for the energy transfer of the grooved disc system. In some automobile power train equipping
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(a) Distribution of the oil phase and air phase.
299
Fig. 9 presents the variation of the axial force on the disc with the angular velocity. It seems that the axial force decreases gradually with the increase of the angular velocity. The maximum axial force is also reduced by the groove. The value of the axial force becomes close to zero finally. It is because that the axial force is mainly provided by the oil phase. The increase of the angular velocity results in the reduction of the oil volume fraction inside the flow field. It makes the axial force provided by the air phase, which is much smaller, become the main part of the axial force. Thus, the axial force has an overall downward trend with the increase of the angular velocity. The existence of the groove reduces the external axial force. The discs are unrestrained axially at the disengaged mode of the wet clutch. The external axial force is useful for separating the discs. Too small axial force may cause disc unexpectable axial moving, especial in the high-velocity operation. It causes higher drag due to the mechanical contact of the discs. Thus, the smaller axial force decreases the axial stability of the discs. It may result in the damage of the clutch due to the mechanical contact of the discs in the high-velocity operation. An enough oil volume fraction of the flow field should be guaranteed in the high-velocity wet clutch design.
(b) Streamlines of the flow field beside the groove. Fig. 5. Enlarged the flow field beside the groove.
several wet clutches, the turbulent drag in the disengagement process of the wet clutch gives rise to energy loss and the efficiency of is reduced. For a Tesla pump, the viscous drag force, produced due to the relative velocity between the rotor and the working fluid, causes power consumption [47]. A higher turbulent drag is not useful to increase the efficiency of the Tesla pump. Fig. 8 presents the variation of the drag torque with the angular velocity. The simulated and measured [26] results of the drag torque with the non-grooved disc and the grooved disc is given in Fig. 8(a). The groove number is 12. It seems that the numerical and tested results have the same overall trend. As the angular velocity increases, the torque increases and then decreases faster at first. The effects of the angular velocity on the torque become smaller with a higher angular velocity. And then the decreasing amplitude of the torque becomes unapparent. Fig. 8(b) presents the variation of the torque for different groove numbers. The maximum torque decreases significantly with the grooved disc. The existence of the groove makes the oil film rupture more easily, which further suppresses the increase of the torque. It is helpful for reducing the torque by increasing the groove number and the un-loaded energy loss of the wet clutch can be reduced. For the Tesla pump, more grooves are helpful for achieving higher efficiency. However, the increase of the groove number also causes an increased processing cost.
(a) Simulated and measured results of the non-grooved disk.
(b) Simulated results of the grooved disk for different groove numbers. Fig. 6. Variation of average oil volume fraction with the angular velocity for different values of Ng.
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(a) Oil volume fraction distribution with the radial coordinate inside the groove.
(b) Oil volume fraction distribution with the radial coordinate between grooves.
(a) Simulated and measured drag torques at Ng= 0 and Ng=12.
(b) Simulated drag torques for different groove numbers. Fig. 8. Variation of drag torque with the angular velocity for different values of Ng.
(c) Oil volume fraction with the circumferential coordinate. Fig. 7. Variation of average oil volume fraction with the radial and circumferential coordinates for different values of x.
Fig. 9. Variation of axial force with the angular velocity for different values of Ng.
W. Wu et al. / International Journal of Heat and Mass Transfer 91 (2015) 293–301
5. Conclusions and future work The air–oil two-phase flow inside the open grooved two-disc system was studied. The detailed discussions on the simulated and measured results are presented. The results suggest the following. (1) The radial flow coefficient of the oil is enhanced with a larger groove number in the same flow area. With a higher angular velocity, the oil mainly discharges through the groove. The air flows into the non-grooved area from the upstream side of the groove. (2) The effects of the angular velocity on the oil volume fraction variation become greater when the disc is grooved. The oil volume fraction changes less in the high-velocity operations. The differences in the oil volume fraction values are small in the same flow area with different groove numbers. (3) The drag torque applied on the disc can be reduced by increasing the groove number in the same flow area. The effects of the angular velocity on the torque become smaller with a higher angular velocity. (4) The axial force applied on the disc is mainly provided by the oil phase. It is gradually decreased with the increase of the angular velocity. The maximum axial force is reduced by the groove. The results can be used for the optimised design of the Tesla pump and wet clutch. The energy transfer characteristics of the air–oil two-phase flow needs more effort to study. Work on these topics is currently underway in the National Key Laboratory of Vehicular Transmission at the Beijing Institute of Technology. Acknowledgment This work is supported the National Natural Science Foundation of China (Grant No. 51305032). The authors thank the works of Prof. Y. Okano. References [1] T. Von Kármán, Über laminare und turbulente reibung, ZAMM-J. Appl. Math. Mech. 1 (4) (1921) 233–252. [2] W.G. Cochran, The flow due to a rotating disc, Math. Proc. Cambridge Philos. Soc. 30 (3) (1934) 365–375. [3] K.M.P. Van Eeten, J. Van der Schaaf, J.C. Schouten, G.J.F. Van Heijst, Boundary layer development in the flow field between a rotating and a stationary disk, Phys. Fluids 24 (2012) 033601. [4] G.K. Batchelor, Note on a class of solutions of the Navier–Stokes equations representing steady rotationally symmetric flow, Q. J. Mech. Appl. Math. 4 (1) (1951) 29–41. [5] K. Stewartson, On the flow between two rotating coaxial disks, Math. Proc. Cambridge Philos. Soc. 49 (2) (1953) 333–341. [6] J.W. Daily, R.E. Nece, Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks, J. Fluids Eng. 82 (1) (1960) 217–232. [7] J.F. Brady, L. Durlofsky, On rotating disk flow, J. Fluid Mech. 175 (1987) 363– 394. [8] X.P. Gan, S.A. Macgregor, Experimental study of the flow in the cavity between rotating disks, Exp. Thermal Fluid Sci. 10 (3) (1995) 379–387. [9] L. Schouveiler, P. Le Gal, M.P. Chauve, Stability of a traveling roll system in a rotating disk flow, Phys. Fluids 10 (11) (1998) 2695–2697. [10] J.M. Lopez, Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall, J. Fluid Mech. 359 (1998) 49–79. [11] C.Y. Soong, C.C. Wu, T.P. Liu, T.P. Liu, Flow structure between two co-axial disks rotating independently, Exp. Thermal Fluid Sci. 27 (3) (2003) 295–311. [12] J.M. Lopez, F. Marques, A.M. Rubio, M. Avila, Crossflow instability of finite Bodewadt flows: Transients and spiral waves, Phys. Fluids 21 (11) (2009) 114107. [13] I. Etsion, Y. Kligerman, G. Halperin, Analytical and experimental investigation of laser-textured mechanical seal faces, Tribol. Trans. 42 (3) (1999) 511–516. [14] I. Etsion, State of the art in laser surface texturing, J. Tribol. 127 (1) (2005) 248– 253.
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Desmet, Model for predicting drag torque in open multi-disks wet clutches, J. Fluids Eng. 136 (2) (2014) 021103. [22] D. Blanchard, P. Ligrani, B. Gale, Single-disk and double disk viscous micropumps, Sens. Actuators, A 122 (1) (2005) 149–158. [23] Y. Kato, T. Murasugi, H. Hirano, T. Shibayama, Fuel Economy Improvement Through Tribological Analysis of the Wet Clutches and Brakes of an Automatic Transmission, JSAE Technical Paper No. 9305436. [24] Y. Yuan, A.L. Eysion, H. James, Z. Qian, An Improved hydrodynamic model for open wet transmission clutches, J. Fluids Eng. 129 (3) (2007) 333–337. [25] S. Pahlovy, S. Mahmud, M. Kubota, M. Ogawa, N. Takakura, Multiphase drag modeling for prediction of the drag torque characteristics in disengaged wet clutches, SAE Int. J. Com. Veh. 7 (2) (2014) 441–447. [26] Y. Takagi, H. Nakata, Y. Okano, M. Miyagawa, N. Katayama, Effect of two-phase flow on drag torque in a wet clutch, J. Adv. Res. Phys. 2 (2) (2011) 021108. [27] C.R. Aphale, J.H. Cho, W.W. Schultz, S.L. Ceccio, T.Y. Oshioka, H. Hiraki, Modeling and parametric study of torque in open clutch plate, J. Tribol. 128 (2) (2006) 422–430. [28] A. Sokolichin, G. Eigenberger, Gas liquid flow in bubble columns and loop reactors. Part I. Detailed modelling and numerical simulation, Chem. Eng. Sci. 49 (1994) 5735–5746. [29] A. Lapin, A. Ltibbert, Numerical simulation of the dynamics of two-phase gasliquid flow in bubble columns, Chem. Eng. Sci. 49 (1994) 3661–3674. [30] W. Martin, Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications, Microfluid. Nanofluid. 12 (6) (2012) 84–886. [31] R. Crocce, M. Griebel, M.A. Schweitzer, A parallel level-set approach for two-phase flow problems with surface tension in three space dimensions, Technical Report 157, Universitat Bonn, Sonderforschungsbereich 611, 2004. [32] W.F. Noh, P. Woodward, SLIC (simple line interface calculation), in: A.I. Van Der Vooren, P.J. 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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Application of CFD to model oil–air flow in a grooved two-disc system Wei Wu ⇑, Zhao Xiong, Jibin Hu, Shihua Yuan National Key Laboratory of Vehicular Transmission, Beijing Institute of Technology, Beijing 100081, PR China
a r t i c l e
i n f o
Article history: Received 20 May 2014 Received in revised form 16 July 2015 Accepted 23 July 2015
Keywords: Rotating disc Rotor-stator discs Two-phase flow Drag torque Wet clutch Tesla pump
a b s t r a c t The flow field of an open grooved two-disc system was studied. The system includes a rotating finite disc and a stationary finite disc. The rotating disc has radial grooves. The numerical results for the air–oil two-phase flow inside the open grooved two-disc system calculated by the CFD code FLUENT were proposed. The results are discussed and compared with the published test results. The results indicate that the groove affects the transitional characteristics from a single-phase flow to an air–oil two-phase flow of the flow field. In the same flow area, the radial oil flow coefficient is enhanced with a larger groove number. The oil mainly discharges through the groove and the air flows into the non-grooved area from the upstream side of the groove. The effects of the angular velocity on the oil volume fraction and the drag torque become greater when the disc is grooved. It is weakened in the high-velocity operations. The drag torque can be reduced by increasing the groove number in the same flow area. The axial force applied on the disc is gradually decreased with the increase of the angular velocity and becomes close to zero finally. The increase of the groove number reduces the maximum axial force. The results can be used for the optimisation of the Tesla pump and wet clutch. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The investigation of viscous flow between rotating discs has received considerable attention, e.g., exact solutions of the Navier–Stokes equations in certain geometric limiting cases, and practical, e.g., computer storage devices, clutches and turbomachinery applications. 1.1. Literature review of the rotor–stator disc flow Von Karman studied the flow over a rotating disc when the fluid at infinity approaches a non-rotating condition [1]. The azimuthal fluid velocity gradually decreases from the velocity of the disc to zero when moving further away from the disc. The region in which these velocity changes occur will be termed here as the ‘‘Von Karman boundary layer’’ or ‘‘Ekman layer’’ [2]. Bödewadt [3] studied the flow over an infinite stationary plane, where fluid rotated with a uniform angular velocity to an infinite distance above the plane. In this situation, the azimuthal velocity component gradually decreases to zero with a decreasing distance from the disc. Here, the region of fluid in which the velocity changes take place is generally termed as the ‘‘Bödewadt boundary layer’’. ⇑ Corresponding author at: Room 412, Building 9, Beijing Institute of Technology, Beijing 100081, PR China. Tel.: +86 10 68914786; fax: +86 10 68944487. E-mail address: [email protected] (W. Wu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.092 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
Batchelor [4] and Stewartson [5] described the flow structure of both the infinite rotor–stator and the infinite counter-rotating discs theoretically. Batchelor [4] suggested that when one disc is rotating at a constant angular velocity, while the other remains stationary, for moderate Reynolds numbers, three distinct layers would form between the discs. Near the rotating disc a Von Karman boundary layer would form, whereas near the stationary disc a Bödewadt boundary layer would develop. According to Batchelor, these two boundary layers are separated by a core in which viscous effects are negligible. The fluid in this non-viscous core would have a constant angular velocity, a constant axial velocity, and zero radial velocity. Stewartson [5] opposed this view by predicting that indeed a Von Karman boundary layer would develop near the rotating disc, but that the main body of the fluid would be at rest. In this situation, both the Bodewadt boundary layer and the non-viscous core would be absent. In the real case of discs of finite radius, both the enclosed disc configuration and the open disc configuration were investigated. Daily and Nece studied experimentally an enclosed rotor–stator flow and identified four flow regimes: two laminar and two turbulent, each having either merged or separated boundary layers [6]. It was found by Brady and Durlofsky that the boundary conditions at the edge of the disc greatly determine the flow pattern actually emerging between the two discs. In an open disc configuration, the flow field would tend towards a Stewartson-type flow [7]. Gan and Macgregor [8] measured the velocity field in rotor–stator
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Nomenclature C 1e C 2e Cl F Fz g Gk h H j k minlet moutlet n nw Ng p Q r r1 r2 ReH Rer t tw
turbulence model constant turbulence model constant turbulence model constant external force axial force on the disc gravity acceleration production term of the turbulent kinetic energy depth of the groove disc spacing iteration number turbulent kinetic energy inlet oil mass flow outlet oil mass flow surface normal vector unit normal vector groove number pressure oil flow rate radial coordinates internal radius external radius reynolds number based on the height rotational Reynolds number time tangent vector
and counter-rotating disc systems without superimposed through flow and concluded that the Batchelor-type flow in rotor–stator. Schouveiler et al. [9] studied the nature of flow between a rotor and a stator disc and found coexistence of annular and spiral rolls. In a later study by Lopez [10], it was shown that the motion of the cylindrical shroud does have a profound impact on the fluid motion inside the cavity. When the full velocity field was calculated inside the cavity together with the no-slip conditions at the cylindrical end wall, it was found that a co-rotating shroud, on the one hand, resulted in Batchelor types of flow. For open disc systems, Rotor–stator flow presents annular and/or spiral rolls on rotor disc plane and Ekman layer on rotor disc in gap flow of single-cell structure [11]. Moreover, recent work by Lopez et al. [12] has revealed that the self-similar solution is indeed able to describe the flow of a fluid near a confined rotating disc up to a radial position of 80% of the disc radius. The Reynolds numbers are up to the order 1e3.
T ui xi z
turbulent drag velocity components coordinate directions axial coordinate
Greek symbols ak turbulence model constant ae turbulence model constant e turbulent kinetic energy dissipation rate em tolerance value eT tolerance value h azimuthal coordinate hw static contact angle r surface tension coefficient j surface curvature l dynamic viscosity leff effective dynamic viscosity q density of the oil or air phase qair air density qoil oil density t velocity vector uair air volume fraction uoil oil volume fraction x angular velocity
between a grooved rotating disc and a smooth stationary disc has also been examined analytically and experimentally [19]. A transition regime from laminar to turbulent flow has been tentatively identified. Kitabayashi et al. [20] presented a theoretical model based on single phase laminar flow in the gap between the discs of the wet clutch. The model assumes a full oil film at all the speeds. Multiphase flows over the disc have been investigated and it was found that the formation of multiphase flows between the two discs has an enormous effect on mass and heat transfer. The two-phase flow in the grooved disc system is the simplified theoretical model for wet clutches and viscous pumps [21,22], as
1.2. Single-phase and two-phase grooved disc flow Today the study of rotating disc flow is still very meaningful. A proper description of the flow pattern between two discs is also crucially important. It can be seen that most of the studies on two-disc flow were associated with the smooth disc and single-phase flow for their wide applications in rotating machinery. The discs were clean and no surface texturing was considered. However, more complicated discs are used for real applications. The grooved disc system is an important pattern to achieve better performance. The single phase flow in the grooved disc system is a simplified theoretical model for many applications e.g. mechanical seals and thrust bearings [13–17]. Muijderman concerns radial single phase inflow with a smooth rotor and grooves on the stator used in spiral groove bearings [18]. The analysis evaluates separately the flow in the grooves and the portion of the gap directly above the grooves and the flow in the portion of the gap above a non-grooved segment of the stator. The radial single phase outflow
Fig. 1. Simplified model.
W. Wu et al. / International Journal of Heat and Mass Transfer 91 (2015) 293–301
shown in Fig. 1. Kato et al. [23] were the first to propose a model that considers the rupture of the full oil film due to cavitation between the discs. The turbulent flow is considered. Yuan et al. [24] presented a mathematical model and verified it by using the commercial computational fluid dynamics code FLUENT and experiments. They formulated the model by assuming non-grooved discs, but eventually approximated the model for the grooved discs by multiplying the torque with the fraction of area which is not grooved. A mathematical model based on continuity and Navier–Stokes equations, considering laminar flow in the gap between the discs was presented and the model is capable of predicting the drag [21]. Another model is also proposed to predict the drag of a disengaged wet clutch at different rotation speeds, clearances, disc sizes and oil temperatures [25]. There is a good degree of agreement between the DT trends derived from the proposed model and the test results for the same condition. Some tested results about the two-phase flow inside the grooved discs of the wet clutch have been presented [26,27]. However, most studies for application focused on the drag forecast and the flow field is not investigated in detail, especially at the air–oil two-phase flow operation. For the gas–liquid two-phase flow simulation, two methods to dynamic flow simulations have been discussed. The first is a method where both the liquid motion and the gas motion are considered in a homogeneous way, which is referred to as Euler-Euler method [28]. The second method treats only the liquid-phase motion in an Eulerian representation and computes the motion of the dispersed gas-phase fluid elements in a Lagrangian way, which is referred to as Euler–Lagrange method [29]. Both the Euler/Euler method and the Euler/Lagrange method rely on physical models for the interfacial transfer of momentum, heat and mass. For this reason, simulation methods are usually used for such problems which do not resolve details of the interface [30]. The Volume of Fluid method employs a piecewise constant scalar field instead of marker particles to locate the position of both fluid phases [31] and has been used widely. Then, the Simple Line Interface Calculation (SLIC) [32] technique and the Piecewise Linear Interface Calculation (PLIC) [33] method are developed to obtain a reconstruction with sufficient accuracy and smoothness. The level-set technique was introduced by Osher and Sethian [34] for gas–liquid two-phase flow simulation. Similar to the VOF method, the level-set technique greatly reduces the complexity of a description of the interface, especially when topological changes such as pinching and merging occur. The objective of this analysis is to characterise the air–oil flow between an open grooved two-disc system. There are two finite discs, one rotating at various angular speeds and the other at rest. The rotating disc has radial grooves. The discs are separated by a small gap. The geometric configuration and the governing parameters are described first. After the basic equations are established considering air–oil two-phase flows, the numerical results of the flow field between the discs are obtained using the CFD code FLUENT 6.1. The results are discussed and compared with published experiments.
2. Problem description Based on the simplified model shown in Fig. 1, the physical situation is schematically shown in Fig. 2, including a rotating finite disc and a stationary finite disc with a disc spacing H. The upper rotating disc has radial grooves and the depth of the groove is h. It rotates with an angular velocity x. The angular width of the radial groove is varied according to the number of grooves since the total area of the grooves is set to be constant in one disc. The flow field is described by a cylindrical coordinate system (r, h
295
Fig. 2. Schematic drawing of the open grooved disc system.
and z), being the radial, azimuthal and axial coordinates, respectively. The position r = 0 is identified with the rotation axis and the plane z = 0 is identified with the stationary disc. r1 and r2 are the internal and external radius. The inlet is completely filled with the oil of a flow rate Q, as a mass flow boundary. The outlet is set to be the pressure boundary conditions. In the real case, the single-two phase transition of the flow field is affected by operation parameters such as the angular velocity and two-disc system structure parameters such as the disc spacing. The flow field has an important effect on the operational characteristics of the disc fluid machine. The drag torque of the wet clutch changes with the single-two phase transition of the flow field [21]. The viscous pump efficiency is also affected by the flow field parameters [22,35]. 3. Mathematical modelling 3.1. Governing equations Considering the physical situation of two discs, one stationary and one rotating, the equation of continuity representing the mass conservation is written as following [36]. The modelling is based on the isothermal operation.
@q þ r ðq~ tÞ ¼ 0 @t
ð1Þ
where q denotes the fluid density, t is the time and ~ t is the velocity vector. The momentum equations are written as:
@ðq~ tÞ F þ r ðq~ t~ tÞ ¼ r ðlðr~ t þ ðr~ tÞT ÞÞ r p þ q~ g þ~ @t
ð2Þ
where l is the dynamic viscosity, p is the pressure, ~ g is the gravity F is the external force due to the surface tension acceleration and ~ and wall adhesion at the interface [37]. Surface tension and wall adhesion represent the interaction of the air, oil, and wall. This will affect the external force in the momentum equation. The additional volume force is expressed as follows [38]:
2qjr~ n ~ F¼ qoil þ qair
ð3Þ
where ~ n is the surface normal vector, j is the surface curvature, r is the surface tension coefficient, qoil is the oil density and qair is the air density. The curvature of the surface near the wall, where the air–oil interface meets the solid surface, is adjusted. The local curvature of this interface is determined by the static contact
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angle, hw, which is the angle between the wall and the tangent of the interface at the wall. The surface normal at the wall is given by [39],
~ t w sin hw n ¼~ nw cos hw þ ~
ð4Þ
tw are the unit normal vector and tangent vector nw and ~ where ~ of the wall surface, respectively. The analysis of the whole flow passage involves the air and oil flows. The volume of fluid (VOF) method developed by Hirt and Nichols [33] is used. A volume fraction uoil is used to mark the volume fraction of the oil phase in the VOF method. Then, uoil = 0 represents a cell that is empty of the oil and uoil = 1 represents a cell that is full of the oil. If 0 < uoil < 1, it represents the interface between the oil phase and the air phase. Thus, the uoil is defined by:
8 in the oil > < uoil ¼ 1 uoil ¼ 0 in the air > : 0 < uoil < 1 at the interface
ð5Þ
The subscript oil represents the oil phase properties. The interface between the oil and air phases is tracked by solving a continuity equation for the volume fraction of the oil phase. From Eq. (1), the continuity equation for the volume fraction of the oil phase is given by [37]:
@ ðu q Þ þ r ðuoil qoil~ tÞ ¼ 0 @t oil oil
ð6Þ
where qoil is the oil density. Because there are two phases, Eq. (2) for the volume fraction is solved only for the secondary phase that is defined as the oil phase. The volume fraction for the primary phase that is defined as the air phase is obtained by the following constraint:
uoil þ uair ¼ 1
ð7Þ
where uair is the air volume fraction. The subscript air represents the air phase properties. The properties that appear in Eq. (2) are volume-fractionaveraged properties. The density and the dynamic viscosity for the air–oil two-phase flow are given by
q ¼ uoil qoil þ uair qair
ð8Þ
l ¼ uoil loil þ uair lair
ð9Þ
where qair is the air density, lair is the air dynamic viscosity and loil is the oil dynamic viscosity. 3.2. Numerical method
turbulent has been explained as approximately 2.0e4 for a representative length of disc radius [42,43]. A disc angular velocity of 150 rad/s, giving a rotational Reynolds number of 5.4e6, which exceeds the critical Rer value of 2.0e4 for rotating disc. Thus, the flow field of the open grooved disc system cannot be treated as a laminar flow. In the calculation, the renormalization group (RNG) k–e turbulence model is used. The RNG k–e turbulence model is widely adopted for the rotating or swirling flow. For weakly to moderately strained flows, the RNG model tends to give results comparable to the standard k–e model. In rapidly strained flows, the RNG model yields a lower turbulent viscosity than the standard k–e model. Thus, the RNG model is more responsive to the effects of rapid strain and streamline curvature than the standard k–e model, which explains the superior performance of the RNG model for certain classes of flows including rotational flow [44,45]. Further, Comparison with available limited experimental data shows that CFD results display reasonable agreement. Predicted results also show that the RNG k–e model performs better to describe the behaviour of the performance of a spray dryer fitted with a rotary disc atomizer in a cylinder-on-cone chamber geometry. It is similar in form to the standard k–e model, but the effect of swirl on turbulence is included in the RNG mode enhancing accuracy for swirling flows [46]. The RNG k–e turbulence model is derived from the instantaneous Navier–Stokes equations, using the renormalization group method. The turbulence momentum equation is
@ðqkÞ @ ðqkui Þ @ @k þ Gk þ qe þ ¼ ak leff @t @xi @xi @xi The turbulence dissipation rate equation is
@ðqeÞ @ðqeui Þ @ @e C 1e e e2 þ þ ¼ ae leff Gk C 2e q @t @xi @xi @xi k k
the height of the cavity, ReH = xH2 =l [41]. In the following, we will use the rotational Reynolds number Rer as the general physical parameter. The transition Reynolds number from laminar to
Table 1 Technical data of the open grooved disc system. Value
r1 r2 h Q H
47.5 mm 59.5 mm 0.37 mm 200 ml/min 0.20 mm
ð11Þ
where, k is the turbulent kinetic energy, e is the turbulent kinetic energy dissipation rate, leff the is effective dynamic viscosity, Gk the is production term of the turbulent kinetic energy caused by the average velocity gradient. xi and ui represent the coordinate directions and the velocity components, respectively. C 1e , C 2e , C l , ak and ae are the turbulence model constants. The mesh images of the flow field are shown in Fig. 3. The flow field was divided by a structured mesh. To ensure the accuracy and validity of numerical results, a careful check for the grid independence of the numerical solutions has been made. When the groove number is 12, three sets of mesh were generated with different grid densities, as presented in Table 2. The abscissa is the element quality on a scale from 0 (worst) to 1 (best). All values of the element quality are higher than 0.9, indicating that the all meshes
The technical data of the open grooved disc system is presented in Table 1. The Reynolds number Re is the major physical parameter of the flow. In the literature, the Reynolds number is based either on the external radius of the cavity, Rer = xr22 =l [40], or on
Parameter
ð10Þ
Fig. 3. Simulated zone and view of mesh of the flow field for Ng = 12.
W. Wu et al. / International Journal of Heat and Mass Transfer 91 (2015) 293–301 Table 2 Specifications for three different meshes.
Total number of nodes Minimum volume (m3) Minimum face area (m2)
Mesh 1
Mesh 2
Mesh 3
358,800 1.567941e14 8.999956e11
472,752 1.511754e14 8.999902e11
533,255 1.461819e14 8.999863e11
meet a high-quality criterion. The calculated results of the average oil volume fraction with four different angular velocities are also given, as presented in Table 3. The value differences between the mesh density of 472,752 elements and the other two are smaller than 5% (0.93% 3.33%) in the same velocity. The mesh density presented in Table 3 does not change the average oil volume fraction in any appreciable way. Due to the different groove numbers, the refinements of meshes of 78,234 472,752 elements are chosen for analysis. In the calculation, the air phase was set as the incompressible main phase. The oil phase was set as the incompressible second phase. The volume fraction of the oil phase in the inlet was set as 1.0. The mass flow conservation and the turbulent torque were used as the calculation convergence conditions as follows.
jminlet moutlet j < em jminlet j
ð12Þ
jT jþ1 T j j < eT T jþ1
ð13Þ
where, minlet is the inlet oil mass flow, moutlet is the outlet oil mass flow, em and eT are the tolerance values. The subscript j is the iteration number. 4. Results and discussion 4.1. Flow visualisation Fig. 4 shows the simulated results of the flow field under different operation conditions. The measured visualisations of the flow field are published by Takagi et al. [26]. The air phase in the figure is a little transparent. Fig. 4(a) shows the variation of the flow field under different groove width and the angular velocity is 104.7 rad/s. In the same angular velocity, the air flow becomes more obvious near the outlet with more grooves in the simulation. The measured results have similar trends. The radial position of the phase interface between the pure oil phase and the air–oil mixture is a little closer to the inner radius in the groove due to the larger flow area. It is conducive to the heat diffusion of the disc. Further, it seems that the stability of the phase interface is affected less with a larger groove number. A larger groove number also makes the pure oil phase along the radial direction be gradually reduced at the same angular velocity. Fig. 4(b) presents the flow field under different angular velocities and the groove number is 12. It seems that the air phase in the flow field increases gradually along the radial direction with a higher angular velocity. Although the oil flow rate is constant, the oil volume fraction in the flow field decreases. The phase interface between the pure oil and the air–oil mixture is obvious. When the angular velocity becomes higher, the oil phase inside the groove is more than the outside region of the groove. Table 3 Calculated oil volume fraction under different mesh specifications and angular velocities. Angular velocity
104.7 rad/s
157.0 rad/s
209.3 rad/s
261.7 rad/s
Mesh 1 Mesh 2 Mesh 3
0.72486 0.73549 0.74235
0.24522 0.25011 0.25320
0.2101 0.2147 0.2204
0.17985 0.18605 0.18984
297
Compared with the published numerical results [21,26,27], the change of the flow field has the same trend in the present study. The effect of the groove and the single-two phase transition of the flow field are described in detail. The radial groove is easy for machining and widely used. The detailed analysis on the flow field of the grooved disc is useful for the optimisation. Fig. 5 is an enlarged flow field around the groove. The groove number is 12 and the angular velocity is 104.7 rad/s. Fig. 5(a) presents the air–oil distribution. The oil flows from the inner end face of the flow field. As the angular velocity increases, the oil mainly discharges through the groove. The non-grooved space in the radial direction beside the outer end face is filled with the air. The state of the oil flow affects the temperature distribution of the disc. Further, a smaller groove number will make the temperature distribution of the grooved disc become more nonuniform. It is not helpful for the thermal stability of the grooved disc system. Fig. 5(b) is the streamlines of the flow field around the groove. The air–oil distribution of the figure is a little transparent. It seems that the oil phase gradually moves outwards with the enhanced centrifugal force. The formation of the continuous oil phase is still circular. However, the radius of the continuous oil phase in the gap is gradually reduced with the increase of the angular velocity. A large space filled with the air phase is formed in the downstream side of the groove. 4.2. Volume fraction With the increasing of the angular velocity, the flow field inside the grooved disc system changes from a single-phase flow to an air–oil two-phase flow. For a constant oil flow rate of the inlet, the oil volume fraction of the flow field decreases with a higher angular velocity. The transitional characteristics of the flow field are important to get a good performance for such as the wet clutch [47]. Fig. 6 presents the average oil volume fraction with the angular velocity for different groove numbers. Fig. 6(a) is the simulated and measured results of the average oil volume fraction inside the non-grooved disc system. It is seen that the simulated results have the same trend with the experimental data [26]. When the velocity is higher than 100 rad/s in the calculation, the oil volume fraction begins to drop. The value of the oil volume fraction reaches the minimum and becomes essentially unchanged with a higher angular velocity than 200 rad/s. It seems that the effects of the angular velocity on the oil volume fraction variation in the high-velocity operations become smaller. With the increasing of the angular velocity, the relation of the air phase between the inside of flow field and the outside become balanced. The effects of the angular velocity on the oil volume fraction variation become greater when the disc is grooved, as shown in Fig. 6(b). The oil volume fraction decreases more rapidly with the grooved disc. The variations of the oil volume fraction with the angular velocity are similar for the same flow area. Although the grooves under different operation conditions have the same flow area, the flow coefficient in the groove is enhanced with a larger groove number in the same flow area. It results in smaller oil volume fractions at the same angular velocity with different groove numbers, as shown in Fig. 6(b). Further, the effects of the angular velocity on the oil volume fraction with the grooved disc in the high-velocity operations are also gradually weakened the same as with the non-grooved disc. Fig. 7 presents the distribution of the oil volume fraction alone different coordinate directions with different angular velocities. The groove number is 12. Fig. 7(a) and (b) propose the oil volume fraction distributions inside and outside the groove, respectively. It is seen that the oil volume fractions decrease alone the radial direction both inside and outside the groove. Further, the oil volume fraction decreases more rapidly outside the groove than
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(a) Variation of flow field with the groove number for ω = 104.7 rad/s.
(b) Variation of flow field with the angular velocity for the same value of Ng. Fig. 4. Simulated results of the flow field under different operation conditions.
inside. The oil volume fraction decreases with the increase of the angular velocity until the oil volume fraction is zero. It is seen that the groove is the main path of the oil flow in the high-velocity operations. The oil volume fraction inside the groove is higher than the non-grooved space, especially in the high-velocity operations, as shown in Fig. 7(c). Fig. 7(c) presents the oil volume fraction distribution in the circumferential direction. The groove number is 12. It seems that the oil volume fraction distribution is not uniform along the circumferential direction. The peak values of the oil volume fraction appear periodically due to the existence of the groove. With a larger angular velocity, the oil volume fraction
shows a downward trend and the non-uniformity of the oil volume fraction distribution is enhanced. As the angular velocity increases, the centrifugal force increases. More oil phase cannot sustain the negative pressure and also cannot satisfy the flow continuity. As a result, the negative pressure space of the flow field becomes large [21]. The air flows into the non-grooved space more. 4.3. Turbulent drag The turbulent drag is important for the energy transfer of the grooved disc system. In some automobile power train equipping
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(a) Distribution of the oil phase and air phase.
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Fig. 9 presents the variation of the axial force on the disc with the angular velocity. It seems that the axial force decreases gradually with the increase of the angular velocity. The maximum axial force is also reduced by the groove. The value of the axial force becomes close to zero finally. It is because that the axial force is mainly provided by the oil phase. The increase of the angular velocity results in the reduction of the oil volume fraction inside the flow field. It makes the axial force provided by the air phase, which is much smaller, become the main part of the axial force. Thus, the axial force has an overall downward trend with the increase of the angular velocity. The existence of the groove reduces the external axial force. The discs are unrestrained axially at the disengaged mode of the wet clutch. The external axial force is useful for separating the discs. Too small axial force may cause disc unexpectable axial moving, especial in the high-velocity operation. It causes higher drag due to the mechanical contact of the discs. Thus, the smaller axial force decreases the axial stability of the discs. It may result in the damage of the clutch due to the mechanical contact of the discs in the high-velocity operation. An enough oil volume fraction of the flow field should be guaranteed in the high-velocity wet clutch design.
(b) Streamlines of the flow field beside the groove. Fig. 5. Enlarged the flow field beside the groove.
several wet clutches, the turbulent drag in the disengagement process of the wet clutch gives rise to energy loss and the efficiency of is reduced. For a Tesla pump, the viscous drag force, produced due to the relative velocity between the rotor and the working fluid, causes power consumption [47]. A higher turbulent drag is not useful to increase the efficiency of the Tesla pump. Fig. 8 presents the variation of the drag torque with the angular velocity. The simulated and measured [26] results of the drag torque with the non-grooved disc and the grooved disc is given in Fig. 8(a). The groove number is 12. It seems that the numerical and tested results have the same overall trend. As the angular velocity increases, the torque increases and then decreases faster at first. The effects of the angular velocity on the torque become smaller with a higher angular velocity. And then the decreasing amplitude of the torque becomes unapparent. Fig. 8(b) presents the variation of the torque for different groove numbers. The maximum torque decreases significantly with the grooved disc. The existence of the groove makes the oil film rupture more easily, which further suppresses the increase of the torque. It is helpful for reducing the torque by increasing the groove number and the un-loaded energy loss of the wet clutch can be reduced. For the Tesla pump, more grooves are helpful for achieving higher efficiency. However, the increase of the groove number also causes an increased processing cost.
(a) Simulated and measured results of the non-grooved disk.
(b) Simulated results of the grooved disk for different groove numbers. Fig. 6. Variation of average oil volume fraction with the angular velocity for different values of Ng.
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(a) Oil volume fraction distribution with the radial coordinate inside the groove.
(b) Oil volume fraction distribution with the radial coordinate between grooves.
(a) Simulated and measured drag torques at Ng= 0 and Ng=12.
(b) Simulated drag torques for different groove numbers. Fig. 8. Variation of drag torque with the angular velocity for different values of Ng.
(c) Oil volume fraction with the circumferential coordinate. Fig. 7. Variation of average oil volume fraction with the radial and circumferential coordinates for different values of x.
Fig. 9. Variation of axial force with the angular velocity for different values of Ng.
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5. Conclusions and future work The air–oil two-phase flow inside the open grooved two-disc system was studied. The detailed discussions on the simulated and measured results are presented. The results suggest the following. (1) The radial flow coefficient of the oil is enhanced with a larger groove number in the same flow area. With a higher angular velocity, the oil mainly discharges through the groove. The air flows into the non-grooved area from the upstream side of the groove. (2) The effects of the angular velocity on the oil volume fraction variation become greater when the disc is grooved. The oil volume fraction changes less in the high-velocity operations. The differences in the oil volume fraction values are small in the same flow area with different groove numbers. (3) The drag torque applied on the disc can be reduced by increasing the groove number in the same flow area. The effects of the angular velocity on the torque become smaller with a higher angular velocity. (4) The axial force applied on the disc is mainly provided by the oil phase. It is gradually decreased with the increase of the angular velocity. The maximum axial force is reduced by the groove. The results can be used for the optimised design of the Tesla pump and wet clutch. The energy transfer characteristics of the air–oil two-phase flow needs more effort to study. Work on these topics is currently underway in the National Key Laboratory of Vehicular Transmission at the Beijing Institute of Technology. Acknowledgment This work is supported the National Natural Science Foundation of China (Grant No. 51305032). The authors thank the works of Prof. Y. Okano. References [1] T. Von Kármán, Über laminare und turbulente reibung, ZAMM-J. Appl. Math. Mech. 1 (4) (1921) 233–252. [2] W.G. Cochran, The flow due to a rotating disc, Math. Proc. Cambridge Philos. Soc. 30 (3) (1934) 365–375. [3] K.M.P. Van Eeten, J. Van der Schaaf, J.C. Schouten, G.J.F. Van Heijst, Boundary layer development in the flow field between a rotating and a stationary disk, Phys. Fluids 24 (2012) 033601. [4] G.K. Batchelor, Note on a class of solutions of the Navier–Stokes equations representing steady rotationally symmetric flow, Q. J. Mech. Appl. Math. 4 (1) (1951) 29–41. [5] K. Stewartson, On the flow between two rotating coaxial disks, Math. Proc. Cambridge Philos. Soc. 49 (2) (1953) 333–341. [6] J.W. Daily, R.E. Nece, Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks, J. Fluids Eng. 82 (1) (1960) 217–232. [7] J.F. Brady, L. Durlofsky, On rotating disk flow, J. Fluid Mech. 175 (1987) 363– 394. [8] X.P. Gan, S.A. Macgregor, Experimental study of the flow in the cavity between rotating disks, Exp. Thermal Fluid Sci. 10 (3) (1995) 379–387. [9] L. Schouveiler, P. Le Gal, M.P. Chauve, Stability of a traveling roll system in a rotating disk flow, Phys. Fluids 10 (11) (1998) 2695–2697. [10] J.M. Lopez, Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall, J. Fluid Mech. 359 (1998) 49–79. [11] C.Y. Soong, C.C. Wu, T.P. Liu, T.P. Liu, Flow structure between two co-axial disks rotating independently, Exp. Thermal Fluid Sci. 27 (3) (2003) 295–311. [12] J.M. Lopez, F. Marques, A.M. Rubio, M. Avila, Crossflow instability of finite Bodewadt flows: Transients and spiral waves, Phys. Fluids 21 (11) (2009) 114107. [13] I. Etsion, Y. Kligerman, G. Halperin, Analytical and experimental investigation of laser-textured mechanical seal faces, Tribol. Trans. 42 (3) (1999) 511–516. [14] I. Etsion, State of the art in laser surface texturing, J. Tribol. 127 (1) (2005) 248– 253.
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