Plain hydrodynamic bearings in the turbulent regime — A critical review

Wear, 72 (1981) 13 - 28 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

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PLAIN HYDRODYNAMIC BEARINGS IN THE TURBULENT REGIME - A CRITICAL REVIEW

VINAY KUMAR

Mechanical Engineering Department, Regional Engineering College, Kurukshetra (Havana), Pin 132119 (India)

(Received March 21, 1980; in revised form July 31, 1980)

Summary

The state of the art of non-laminar behaviour in bearings is presented with the emphasis on those aspects which are concerned with the performance of journal bearings under turbulent film operation. Analytical and relevant experimental studies are described critically and their advantages and limitations are compared.

1. Introduction

There has recently been much interest from both an analytical and an experimental point of view in the operation of bearings beyond the laminar regime. The state of the art in this area is reviewed in this paper with emphasis on those studies (analytical and relevant supporting experimental work) which can lead to the determination of the characteristics of plain journal bearings [l] . The reference list provides a bibliography of the subject. Addi- tional reviews and literature on various aspects of the subject may be found in the references quoted, particularly in refs. 1 - 16.

2. Non-laminar behaviour (superlaminar flow) in bearings

Non-laminar flow occurs in bearings owing to their high speed of operation and the use of unconventional lubricants (such as water or liquid metals) of low kinematic viscosity. These process fluids are used to simplify equipment design, to overcome shaft sealing problems or to reduce contam-

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ination. A high velocity and/or a low kinematic viscosity lead to a large Reynolds number and to a departure from laminar flow which cannot be adequately described by conventional Reynolds lubrication theory. Some such situations arise in (i) bearings lubricated with liquid sodium in nuclear reactor applications, (ii) oil-lubricated journal and thrust bearings in large power generation equipment, (iii) water-lubricated journal bearings for large boiler feed pumps, (iv) air cushion support for a tracked hovercraft vehicle, (v) bearings and seals in advanced design gas turbine engines, (vi) bearings in cryogenic equipment employing liquified gases and (vii) mercury-lubricated bearings in power generation systems utilizing mercury as the working fluid.

An appreciation in terms of the relative dominance of inertia affects of the significance of a lubricant with a low kinematic viscosity can be gained [ 21 by reviewing the mathematical foundation of Reynolds’ theory.

The significance of an inertia effect is more than the mere prevalence of transient and/or convective forces. Two types of breakdown during flow can occur if the kinematic viscosity is low. One type of breakdown [ 171 is the generation of toroidal secondary vortices (Taylor vortices); this is caused by the centrifugal force of a curved layer of flow (called transitional flow) which occurs when (UC/u)(C/R) “’ = 41 1 This results in the problem of . . bearing instability; the instability can be remedied by the use of lobed or tilting-pad bearings. As soon as this critical speed is exceeded the torque, which at lower speeds was proportional to the speed, is no longer linearly proportional and increases at a much more rapid rate. The second type of flow instability is caused by the tendency of a shear flow to dissipate its kinetic energy in random fluctuations known as turbulence; in this condition the mean shear stress is approximately proportional to the mean kinetic energy or to the square of mean flow rate. The criterion for turbulence to occur is UC/2 > 1000. Turbulence is the type of flow instability which is found most frequently (Fig. 1) in thin film bearings using process fluids with C/R between 10d3 and 3 X 1O--3 ; this range of C/R values is commonly found in practice. Figure 1 shows that this statement is correct: the flow regime with secondary vortices in the absence of turbulence occupies a relatively small portion in the graph. Because of the relative importance of turbulence, only work which relates to the analysis of hydrodynamic turbulent film bearings (more particularly to plain journal bearings) is reviewed. A critical review of porous metal bearings with laminar and turbulent films was presented by Vinay Kumar [ 181.

The basic analysis of fluid film bearings is of necessity a complex mathematical procedure involving, in its simplest form, a numerical solution of the governing Reynolds equation. For laminar conditions, the Reynolds equation is reasonably well defined and poses no theoretical difficulties although the usual simplifying assumptions of isothermal conditions, newtonian lubricants, rigid (circular) surfaces etc. are still involved. However, the situation is different with turbulent flow because it is also required that the Reynolds equation takes into account the local eddying motions due to turbulence of the lubricant flow within the clearance spaces.

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Fig. 1. Reynolds number us. clearance ratio in turbulent flow bearings.

The distinguishing effects of a turbulent film in plain journal bearings were first reported by Wilcock [ 31 as follows: (i) a major increase in the drag loss, the temperature and the fluid film pressure and hence in the load capacity; (ii) an increase in the attitude angle at each value of the eccentric- ity ratio, at least for purely hydrodynamic full journal bearings.

This was the starting point of efforts to understand the phenomena and to establish rules of behaviour useful for predicting turbulent bearing performance. Smith and Fuller [ 191 published data on power losses in full journal bearings operating in the laminar, transition and turbulent regimes; these data are still used as the experimental reference basis for checking current engineering theories on turbulent bearings. Vohr [20] later reported a detailed experimental study of superlaminar flow between non-concentric cylinders. Low viscosity silicones containing suspended metal particles were used with large clearances and a transparent outer cylinder to permit flow observation. Vortex patterns were observed at predicted transition Reynolds numbers and also random turbulence was observed at higher Reynolds numbers of up to 32 000.

These experimental observations stimulated an interest in analytical prediction; this essentially involves solving the two basic equations, i.e. the momentum (Navier-Stokes) and continuity equations. For fully developed turbulent flow (which has been tacitly assumed when certain parameters such as the Reynolds number are high enough although it is not clear from either an analytical or an experimental viewpoint when or whether such a flow exists) it is generally assumed that turbulent stresses dominate. Constantinescu [ 211 states that the mean fluid inertia stresses are of the same order of mag- nitude as the viscous terms and that the inertia stresses contribute little to

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the mode of flow. However, the viscous terms are retained in the motion equations in order to avoid having to make allowances for supplementary approximations in the laminar sublayer at the solid boundaries. Because of this, the mean fluid inertia terms are neglected so that the governing equations [21,22] become

aP a26 -=p_ + &Ju’u’ ) ax ay2 ay

aP a26 --=P-- + q-&q a.2 ay2 ay

aii a& +-=o

G a.2

Although these three equations constitute the starting point for the mathematical treatment of turbulent bearing films, they are unsolvable as such because the number of variables exceeds the number of equations. To overcome this difficulty, the relation between the fluctuating components a, u’ w’ and the time-averaged velocities u, w must be established either experimentally or analytically. Extensive experimental work can be related to purely fundamental theory. The analytical approach is possible only with the use of turbulence models; this approach involves empirical rela- tions and a judicious choice of constants to confirm experimental observations. There are five distinct turbulence models and therefore five correspond- ing lubrication theories. These theories are based on (a) the mixing length, (b) the eddy viscosity, (c) the midchannel velocity, (d) the bulk fluid theory and (e) the kinetic energy of turbulence. Thus there are altogether six ways of dealing with turbulent lubricant films.

3. The purely fundamental theory

It might be possible to measure averages of products of flow velocity fluctuations in lubricant films for a large number of cases and to use the experimental data for substituting into and thus solving the Navier-Stokes equations. Although sufficient experimental data are not yet available, this method has been shown to be promising by Carper et al. [23], Laufer [24] and Reichardt [ 251 with experiments on flow under the influence of a pressure gradient in a channel of rectangular section and by Burton and Hsu [26] with experiments on the flow induced in the annulus between cylindrical surfaces by sliding one of the surfaces; this in some cases was com- bined with a flow component under the influence of a pressure gradient. However, the measuring procedure (e.g. the hot wire method) is tedious and it is difficult to obtain reliable results. Solving the Navier-Stokes equations with the aid of such experimental data is the most fundamental approach available for predicting the pressure build-up and flow in a turbulent lubri-

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cant film but considerable computational effort is involved. Thus this method is slow and complicated when the loadcarrying capacity and leakage of turbulent bearings are calculated. However, Burton and Hsu [26] main- tain that such investigations would permit a critical judgment of less fundamental theories and of the bearing calculations made on the basis of these theories.

4. The mixing length theory

This theory is also known as Constantinescu’s theory. Constantinescu treated the problem of turbulent shearing stresses in the x coordinate by introducing Prandtl’s mixing length theory. According to this theory

au au --pu’v’ = pl, - -

I I ay ay

where the mixing length I, is related to the independent variable y by means of a mixing length constant K, . An analogy to pipe flow is made by assuming a linear variation in the mixing length:

I, = K,,,Y O< Y< h/2

and

kn =Kn(h-_y) h/2< Y< h

The total shear stress for two-dimensional flow is then

aii aii T =p - +pKm2y2 -

w ay Specific details of Constantinescu’s work are documented in refs. 27 - 32. He considered strong Couette flows and hence application of his results to hybrid and hydrostatic bearings may lead to errors. Also, the buffer zone receives no attention; this leads to a discontinuity in the shearing stress. In addition, non-planar flows found in finite-width journal bearings require more careful consideration. Finally, a difficulty arises in specifying the value of the mixing length constant Km owing to the close proximity of the two walls. Constantinescu recommended [32] that the mixing length constant K, should be treated as a function of the local Reynolds number; the relation is K, = 0.125Re0.“. The numerical variation in K, is contained in ref. 27.

Constantinescu obtained the following constant coefficients for the Reynolds equation by means of a linearization method:

1 - = 12 + 0.53(K,2Re)0.‘25 GX

and

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;- = 12 + 0.296(K,2Re)0.65 L

Constantinescu and Galetuse [ 301 have also shown that

PU- h a~ 7=--~c+ --

h 2 ax

where

Fc = 1 + 0.0525(K,2Re)0.75

Constantinescu’s work is also applicable to turbulent compressible lubrication when the appropriate density terms are included in the Reynolds equation.

Chou and Saibel [ 331 used the mixing length concept by assuming an algebraic variation of the form

L12 Oc Y(2b -Y)

and were able to calculate the general features of the pressure distribution in a long bearing with a turbulent film. However, this assumed relationship does not lead to a fully satisfactory prediction for Couette flow or for simple displacement flow. When the mixing length constant K, was varied numerically by Arwas et al. [22] up to K, = 0.4, they found that K, should be smaller than 0.4 to reproduce the bearing data of Smith and Fuller [ 191. Wheeler [ 341 found that K, = 0.2 provided a good fit. This observation has, unfor- tunately, led to speculation that bearing films are so thin that the scale of turbulence is affected. This is not the case because Couette’s thin film results and large scale flow results lie along the same line when appropriately non- dimensionalized [ 351. The discrepancy is caused by a change in the mixing length near a wall [36] , i.e. by a laminar sublayer where the mixing length appears to vary as the cube of the distance to the wall [37] . When this change is accounted for, the discrepancy between experimental and theoretical results is corrected by 40%. Reynolds [ 381 recognized the sublayer problem and carried out an analysis involving a three-layer description of the flow which allowed for the sublayer.

In his analysis of crossed Couette and displacement flows, Constantinescu initially dealt with these independently, ignoring coupling effects. It has been shown [39] that the suggestion of Prandtl would permit the use of a mixing length in such flows and that a weak pressure flow acting across a strong Couette flow could be computed accurately even if the pressure flow led to first-order changes in the mixing length distribution. These ideas were extended to codirectional Couette and displacement flows [ 361.

5. The eddy viscosity theory

The eddy viscosity model evolved from Prandtl’s logarithmic “law of wall” which derived from fully developed turbulent pipe flow data [40 -

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461. In most cases of turbulent flow near a wall, the major changes in velocity take place in a region of almost constant stress, The universal correlation of the law of wall is based on the assumption that the shear stress is every- where constant and equal to the shear stress at the wall. The assumption is

$=f y; 112

IO i = f(y+)

Ng and Pan [47] represented turbulent Reynolds stress by means of an “eddy viscosity” (in much the same way as the laminar viscous shearing stresses are related by the absolute viscosity to the appropriate velocity gradients) so that the turbulent stress is

d: -pziv’ = pE -

dy

where PE is the eddy kinematic viscosity analogous to the laminar fluid kinematic viscosity. Thus the total constant stress flow condition is

7 = 7,

The functionality for the eddy viscosity is in turn determined to be that which best fits the wall law profile (which is very close to the Couette flow profile between the wall and midchannel). The adopted formula [ 481 is that of Reichardt:

Ng [ 481 found that this formula correlated with all regions of pipe flow data if k = 0.4 and 6,+= 10.7 are adopted. One of the main reasons for the choice of Reichardt’s law of wall is that it is easy to use because only p, Y and T need to be known to obtain the eddy viscosity at any point in the flow. Results either for pressure flow alone or.for a weak pressure flow imposed on a Couette flow are also both very close to predictions of the fixed mixing length relationship. Ng [48] used the wall shear stress in the law of wall to correlate the time mean velocity profile in the region near a boundary. Ng and Pan [47] later based the velocity parameter on local shear and assumption of the isotropy of turbulent momentum transport enabled the application of this model to non-planar flows in finite bearings [14, 491. The empirical expressions for the constant coefficients in the turbulent lubrication equation have been developed by Taylor [ 501 using curve fitting. An alter- native though slightly less accurate representation is contained in refs. 4 and 51.

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Ng and Pan [47] specifically stated that their theory is applicable to incompressible films. The precise reason why compressible lubricants were not investigated was not explained. Elrod and Ng [52] extended Ng and Pan’s work to permit its application to turbulent hydrostatic and hybrid bearings. For this case, l/G, and l/G, depend on the dimensionless pressure gradient (h3/pv) Vp and the Couette Reynolds number. Curves describing this dependence have been given in ref. 52. Attractive simplifications are given by Constantinescu [53] . For dominant Couette flow, the shearing stress acting on the surfaces in the x direction is given by

vu- h 3~ 7=---7c&-_

h 2 ax

and the data of Ng and Pan have been curve fitted by Taylor [ 501 and by Constantinescu [ 531. The eddy viscosity concept has also been applied to bearings by Chou and Vohr [54]. It has now been accepted that the approach of Ng, Pan and Elrod is more attractive than that of Constantinescu and will lead to more accurate predictions. However, it should be noted that there are many similarities between the two approaches. Both use empirical information to specify a turbulent viscosity relating the apparent stress to the mean velocity gradient and both yield the same form of the turbulent lubrication equations. Both approaches rely on experimental measurements. Ng and Pan’s work [47] is more empirical because their method relies on an empirical formula for eddy diffusivity, the constants of which have been chosen to agree with known results. Constantinescu’s work also depends on the determination of the mixing length constant K, [ 311.

The main difference between these two approaches relates to three- dimensional (i.e. non-planar flow) situations. Further calculations of bearings should not be based on these approaches because the empirical constants necessary for predicting the profile must be fitted to measurements of actual flow velocity profiles.

6. The midchannel velocity theory

Burton [26], aware of the shortcomings of the above theories, arrived at a simpler theoretical approach by interrelating all the basic characteristics of a lubricant film, e.g. the pressure gradient, sliding velocity and shear stresses at both surfaces, to the characteristic velocity of flow, i.e. midchannel velocity profiles. This theory is so simple that inertia effects can readily be considered. A somewhat different approach of Burton [ 35, 361, which is related to bulk flow, is based on the observation that velocity profiles are the same for Couette and Poiseuelle flows [7] . From the data of Robertson [55], it appears that this theory may be most accurate [7] for codirectional flows. However, experimental data for crossed flows [ 56, 571 suggest that the mixing length prediction (or eddy viscosity approach) is the most ac-

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curate. Burton [7] thought that without an improved theoretical basis or better experimental data the accuracy of the approximation will not be in- creased. However, the uncertainty is fortunately small compared with the predicted magnitudes of the data. Computations based on this theory are confined to very simple types of self-acting bearings and the extension of the theory to externally pressurized and hybrid bearings is extremely difficult.

7. The bulk flow theory

The three theories discussed here are based on information obtained from experiments regarding (a) fluctuating velocity components owing to turbulence or (b) velocity profiles time averaged so as to eliminate fluctuating velocity components.

This approach is subject to difficulties in measuring flow velocities and in processing the experimental data. Therefore Hirs [ 581 attempted a suffi- ciently accurate description of the build-up and flow of the pressure in a lubricant film; this attempt was based on correlational data of flow relative to each of the two bearing surfaces. Hirs adopted the bulk flow approach so that a physical representation of the turbulent transport mechanism was not required. Thus Hirs’ bulk theory is based on an analogy between turbulent flow under the influence of a pressure gradient and turbulent flow caused by the sliding of a surface. Davies and White [ 591 and Coutte [ 601 found that in both types of turbulent flow the wall shear stress depends on the density, the viscosity, the mean flow velocity with respect to the particular surface for which the shear stress is considered and the fluid film thickness. Combin- ing their findings, it can be shown that in either type of turbulent flow this dependence requires at least two dimensionless groups; these are a friction factor and a Reynolds number. When the curve for the experimental pressure flow in the turbulent regime is plotted graphically it is found to be close to the curve representing drag flow. Burton [26] found an even closer agreement between the two flow types when the midchannel velocity was used instead of mean flow velocity.

Hirs [58] found evidence that these dependences are also insensitive to fairly general combinations of the two flow components such as mutually perpendicular flows where one flow is a pressure flow and the other flow is a drag flow. The insensitivity of the wall shear stress to the type of flow led Hirs to treat the flow in lubricant films by bulk flow theory. Bulk approaches had been used earlier by Smith and Fuller [19] and by Tao [61, 621. Fuller’s [19] experimental results are interpreted on the basis of a bulk flow approach relating the displacement flow to the wall shear stress. Later Tao [62] used a similar approach and extended it to crossed as well as codirectional Couette and pressure flows. It is clear that these early approaches did not account adequately for the coupling between turbulent Couette and pressure flows. The starting point of the approach is that the simple formula

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relating the friction factor to the Reynolds number, i.e. f = C, Re” as used in other areas of fluid mechanics, is utilized to represent the relationship between the wall shear stress and the mean flow velocity relative to the wall for both Couette and Poiseuille flow and combinations of the two in the empirical form

7 pu,h m -=~l -

gpkl12 i 1 P

The constants C, and m are empirically chosen or derived from assumed velocity profiles. (Blasins’ law conforms to the friction factor representation given.) Such a theory is simpler to apply but more dependent on empirical knowledge.

A particular advantage of this approach is that empirical constants used in the theory can be derived from bulk flow measurements without the determination of velocity profiles. Consequently, there is no recourse to the schemes adopted by others to represent the turbulent stresses in terms of the mean velocity gradient.

There is a relationship between the shear stress exerted on a boundary by a Couette flow and by a Poiseuille flow which has the same mean flow velocity, other parameters being equal. This relationship enables an approach whereby a “total” pressure gradient can be used to represent both Couette flow and Poiseuille flow. This concept is useful in extending the work to finite-width situations. Hirs presented a wide range of results to verify his bulk flow theory experimentally. The constants involved in his basic relationship were determined from published experimental work for a range of conditions including various Couette and Poiseuille flow situations, different lubricant film profiles, various surface finish conditions and different groov- ing arrangements. Thus Him theory can accommodate rough and grooved surfaces by making the appropriate experimental measurements to deter- mine the constant coefficients and the exponent. This is not true for other turbulent lubrication theories. Him stated that little experimental work has been carried out on turbulent flow with fluid inertia effects present; the analysis of such conditions is still difficult. However, it appears that this approach can readily accommodate such effects.

Because of the nature of his bulk flow theory Hirs did not arrive directly at the turbulent Reynolds equation predicted by the other two approaches. This is a consequence of his method, which formulates the equation in such a way that an advantage is obtained in comparing experimental and theoretical work. However, this has the consequent disadvantage that the average flow velocities are not known a priori and hence substitution into the continuity equation to yield the Reynolds differential equation for the pressure is not possible.. However, accepting the general turbulent Reynolds equation as derived by the phenomenological approaches of Constantinescu and of Ng, Pan and Ehod, it is possible to cast the results of Him work into the form already considered [58] . It is concluded that for

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Reynolds’ numbers greater than lo4 the law of wall approach of Ng, Pan and Elrod is in excellent agreement with the predictions of the bulk flow theory. A discrepancy between the two approaches exists in the so-called transition region, up to a Reynolds’ number of 104. Hirs claimed that in this region the law of wall is not in good agreement with experimental evidence and cites the statement of Elrod and Ng to this effect. Since the bulk flow theory is based directly on experimental evidence, this theory is said to be more applicable in this region and it is generally the most reliable turbulent lubrication theory.

The turbulent lubrication theories of Constantinescu and of Ng, Pan and Elrod can now be summarized. Both theories predict the same form of the turbulent Reynolds equation. The work of Ng, Pan and Ehod allows the consideration of Couette and Poiseuille flows and general combinations of the two and it is generally accepted that it is a more precise theory than that of Constantinescu. Hence the numerical predictions of the second theory are to be preferred. The approach utilized by Him is completely different and based solely on experimental observations. It does not render a turbulent Reynolds equation directly but the results can be fitted to the form used by the other workers. Agreement with the predictions of Ng, Pan and Elrod for Reynolds’ numbers greater than lo4 is good but below this value a divergence exists. It is claimed that the bulk flow theory is marginally better in this regime of relatively low Reynolds’ numbers. It is therefore pertinent to discuss critically the overall limitations of the theories when applied to the design of fluid film bearings.

(i) It would not be expected that the results of Constantinescu, or of Ng, Pan and Elrod, would be precise in the transition region between laminar and fully developed turbulent flow because the empirical information used does not apply directly to such conditions and the balance of forces in this region is not well understood. Hence omissions may be present in the theoretical approach. The work of Him should be better in this respect if it is assumed that his basic premise is applicable over the whole range of Reynolds’ numbers. An extension of the work of Ng and coworkers to cover the transition region has been reported [ 531.

(ii) At very high Reynolds’ numbers (e.g. 50 000) it would be expected that mean fluid inertia forces would become significant. As no account is taken of these in the work of Constantinescu or of Ng, Pan and Elrod, significant errors could occur. Also, since little experimental evidence is available under such conditions, the approach of Him is not applicable.

(iii) In the case of thrust bearing design it is possible in certain cases that consideration should be given to the effect of the inlet development length of flow. For laminar Poiseuille flow in a channel of width 2a the inlet development length L, of flow is given by L,/2a = O.O$Rer, where Rep is the Poiseuille Reynolds number based on the channel width and the mean velocity. At very high Reynolds’ numbers the development length of flow could become a significant proportion of the length of the bearing. The flatter velocity profiles encountered in turbulent lubrication will relieve this problem.

NC AND PAN

1 1 c,=-) cz=-

kx “2

,04 10' IO‘ 105

REYNOLDS No. -

Fig. 2. Gx and G, vs. the Reynolds number in bearings with dominant Couette flow.

Comparison of the predictions based on the approaches of Constantinescu and of Ng and Pan have been presented [ 501 where percentage errors are quoted with respect to the theoretical values of Ng and Pan. Over the range of film thickness ratios, ratios of width to length and mean Reynolds’ numbers considered, the predictions of the two theories for the normalized centre of pressure and for volume flows differ by a maximum of 3% and in many cases by much less. The error in the prediction of the normalized frictional force on the moving surface is generally of the order of 6% but the prediction of the normalized load capacity can be subject to a somewhat greater error. For example, the maximum errors in the prediction of the normalized load capacity by the two theories for mean Reynolds’ numbers of 5000,lO 000 and 50 000 are about 17%, 25% and 42% respectively. The largest errors occur at the smallest value of the ratio of width to length considered. However, it has already been stated that the approach of Ng and Pan is to be preferred to that of Constantinescu. Hence Ng and Pan’s predictions for the plane inclined slider will now be compared with those of Hirs [50] . At extremely high Reynolds’ numbers the agreement between the theories is excellent. At a mean Reynolds number of 50 000 the maximum errors for all conditions considered for the normalized values of the load capacity, the centre of pressure, the frictional force and the volume flows are less than 2%, 0.25%, 3.5% and 0.25% respectively. With a decrease in the mean Reynolds number the agreement becomes less satisfactory, particularly in the case of the load capacity. The centre of pressure and lubricant flow predictions are always very close. In the low Reynolds number region, near transition, divergence between the predictions of Ng and Pan and of Him would be expected (Fig. 2). The maximum error in the normalized load capacity for mean Reynolds’ numbers of 1000 and 2000 is approximately 30%. While the maximum error in the normalized frictional forces

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lb3

103 681i+ 6Blos

REYNOLDS NUMBER -

Fig. 3. Friction factor Cf us. the Reynolds number.

is about 6% for a mean Reynolds number of 2000, it rises to 20% for a mean Reynolds number of 1000. This large percentage is caused primarily by the relative variations in ?, (Fig. 3).

It is possible to conclude that for purposes of design it would make little difference whether the approaches of Ng and Pan or of Hirs were used for bearing design if the mean Reynolds number is high, e.g. 10 000 or more. This is particularly true when it is recalled that the design of fluid film thrust bearings is still a difficult exercise. It is not easy to account accurately for the effect of viscosity variations, elastic and thermal distortions, hot oil carry-over and load sharing between pads. For mean Reynolds’ numbers less than 10 000 a divergence begins to appear in the predicted value of the load capacity. However, even the maximum error of 30% in the load capacity is equivalent to a discrepancy of only 16% in the prediction of the minimum film thickness. These comments on design predictions are made regardless of the appropriateness or correctness of the theoretical and empirical approaches used in the turbulent lubrication theories.

8. The kinetic energy model of turbulence

Recently a different turbulence model has been applied by Ho and Vohr [5] ; this is essentially an improvement of earlier theories. Since all phenomenological models of turbulent flow are based on empiricism, there always exists a degree of uncertainty as to the accuracy of such models, particularly when they are applied to flow situations for which experimental data are not available. Consequently, the more evidence that can be gathered in support of such theories, the more confidence can be had in their applica- bility. The kinetic energy theory presents such evidence in the form of calculations of lubricant film velocity profiles which are based on a model of

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turbulent flow different from either of the earlier theories; these calculations are in good agreement with these previous theories. Hence this provides some corroboration of these theories. The turbulent model used in this energy theory is that proposed by Kolmogorov [63] and later independently by F’randtl [ 641. Their hypothesis is similar to the eddy viscosity hypothesis in that they propose that the turbulent shear stress - pu)uI can be expressed in terms of an effective lubricant viscosity multiplying the gradient du/dy of the mean velocity. The novel aspect in their approach is the suggestion that this effective viscosity could be taken to be locally proportional to the square root of the time-averaged kinetic energy of turbulence multiplied by a suitable scale of length i,, i.e. uT = CrpZ,E”’ where E = (uf2 + Y’~ + w12)/2 and C, is a constant. A differential equation for the turbulent energy is then solved in conjunction with the Reynolds equation. Such an approach has appeal because it does not, in principle, depend on any empirical model. The kinetic energy approach has other advantages over the mixing length or the eddy viscosity models in that it can be readily applied to non-planar flows without additional generalization and also to situations where mean-flow convective effects are significant. However, it greatly increases the computational difficulty. The results obtained also agree with the simpler theories of turbulent flow. However, the application of the energy model to the analysis of turbulent journal bearings or thrust bearings is still lacking.

References

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2 C. H. T. Pan and J. H. Vohr, Superlaminar flow in bearings and seals. In R. A. Burton (ed.), Bearing and Seal Design in Nuclear Power Machinery, American Society of Mechanical Engineers, New York, 1967, pp. 219 - 250.

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13 Z. S. Safar, Inertia and thermal effects in turbulent flow journal bearings, Wear, 53 (1979) 325 - 335.

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22 E. B. Arwas, B. Sternlicht and R. J. Wernick, Analysis of plain cylindrical journal bearings in turbulent regime, J. Basic Eng., 86 (June 1964) 387 - 395.

23 H. J. Carper, J. U. Hellhecker and E. Logan, The effect of a change in wall roughness on turbulence intensity and shear stress distribution in a two-dimensional channel flow, 8th Midwestern Mechanics Conf., Case Institute of Technology, 1963.

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Plain hydrodynamic bearings in the turbulent regime — A critical review

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