land joints

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International Vol. 29. No. 1, pp. 69-16, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights resewed 0301-679X196iS15.OfJ +O.oO

0301-679X(95)00036-4

Pressure and flow characteristics in a shallow hydrostatic pocket with rdunded pocket/land joints M. Dzodzo*.

M. J. Braun*

and I?. C. Hendricks’

The paper studies the development of the flow and pressure maps in a shallow hydrostatic bearing pocket and, on a comparative basis, discusses the effects of the pocket-to-land exit geometry when it takes different shapes (sharp 90” angle, and rounded with different radii of curvature). The numerical simulation uses a dimensionless formulation of the Navier-Stokes equations written in primitive variables for a body fitted coordinate system, and applied through a collocated grid. The model includes on one hand the coupling between the pocket flow and a finite length feedline flow, and the pocket and the adjacent lands on the other hand. Geometrically, all pockets have the same footprint, same land length, and same capillary feedline. The numerical simulation uses the Reynolds number (Re) based on the runner velocity, and the inlet jet strength (F) as dynamic similarity parameters, while the radius of curvature of the pocket/land joint is used as a geometric parameter. The study treats the laminar ranges of the Re number.

Introduction In recent years there has been sustained activity towards improved modelling of the flow in a hydrostatic pocket. Numerical and experimental studies have generally aimed at a better understanding of the physics of jet and upstream flow mixing, improvement of the prediction of the loss coefficients, a better prediction of the inertia pressure drops, of the downstream cavitation inception, and the overall effect of all these factors on the dynamics of the bearing. From the standpoint of a comprehensive bearing analysis most of the recent work has concentrated towards the implicit lumped treatment of the pocket performance in the numerical environment of the total solution of * Departmenr of Mechanical Engineering, Akron, USA ‘NASA Lewis Research Centre, Cleveland,

University USA

of

Akron,

the Reynolds equation for the entire hydrostatic journal bearing (HJB). A rather complete review of this literature is presented by Braun et al. 1 in their study of the steady state flow structure and pressure patterns in a hydrostatic cavity of variable crosssection. Numerous authors have analysed the overall thermofluid and dynamic performance of the hydrostatic bearing. Thus, Artiles et ~l.~, Castelli and Shapiro3, Redecliffe and Voh?, Belousov and Yus, Braun and Wheelefl, San Andres’, Goryunov and Belousov8, and Davies and Leonard9 have presented rather complete analytical and numerical analyses of the pressures and/or dynamic coefficients development in multi-pocketed HJB. Overall experimental studies concerning the flow characteristics, frictional characteristics, pressures, and dynamics of hydrostatic bearings were performed by Shinkle and Hornung’O, Leonard

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Notation A

dimensionless cell surface area A, = AX& A,, = hx, dimensionless parameter of the jet strength characteristic length h = R&000 Jacobian of the coordinate transformation; y’ = y’(xj) number of control volume cells in the domain dimensionless pressure P = pIp(vI

F

h J n

P

h)*

dimensionless Cartesian component velocity vector (uihlv) = r/& + U2@ dimensionless covariant velocity components (normal to the cell faces) Reynolds number, Re = U, for Y* = 310 + c radius of the shaft dimensionless average control cell mass imbalance curvilinear coordinate dimensionless general curvilinear coordinate xi = xjlh Cartesian coordinate dimensionless Cartesian coordinate yi = y’lh cofactor (ith row and jth column) of the Jacobian matrix kinematic viscosity density

ui

Ui

Re

y’ yi

Pi vi P

observed hydrostatic jet penetration, the formation of the secondary eddies, both upstream and downstream of the main jet, and the fluid turn around zone (TAZ) in the upstream portion of the pocket land. The TAZ becomes particularly dominant as the velocity of the shaft increases. In the limit, no jet-borne fluid exits at the upstream end of the pocket, and only a thin layer of fluid attached to the rotating shaft is carried through the control surface of the cavity. Using a 2-D Navier-Stokes model, Braun and Dzodzo3* obtained good qualitative numerical agreement with the experimental workWs31. In a comprehensive review paper, Hendricks et ~1.~~ present results that also shed light on the physics of pressure build-up in shallow pockets. San Andres and Velthuis34 for shallow pockets, and Braun et al.’ for deep pockets, developed steady state codes for the simulation of the laminar flow in the pockets of an HJB. Both authors confirmed that inertia induced pressure drops occur at the pocket edges when the fluid flows outward. Braun1,35 has also shown that the pressure in the recess is not uniform, and, as the shaft angular velocity increases, the pressure in the upstream portion of the deep pocket decreases, starting thereafter a process of recovery in the central region, followed by a sharp increase in the downstream (exit) of the pocket. This sequence of phenomena is assigned to a combination of backstep and Rayleigh step type flows. Braun and Dzodzo3* have further investigated the effects of the shape of the pocket (square, ramped Rayleigh step, arc of circle) on both the mixing flow patterns and the development of the pressure immediately under the runner.

Scope of work and Daviesll, Ho and Chenl*-14, Aoyama et af.i5, Chaomleffel and Nicholas16, Scharrer et al. 17,18, Spica et al.19 and Franchek and Childs*O. Finally, for the thermofluid and dynamic behaviour of an HJB, comparisons between analytical/numerical models and their own experiments were presented by Heller*l, Betts and Roberts**, Bou-Said and Chaomleffe123, and Hunt and Ahmed24. Thus, while there is a reasonable body of literature concerning the overall HJB design and performance, relatively few researchers have concentrated on. the mechanics particular to the flow in HJB cavities, and even less attention has been given to the idea of designing the HJB starting from the characteristics of the individual pockets, and using that information as a fundamental building block towards the construction of an entire HJB. Ettles25 gave an insightful analysis of the flow in a bearing groove without hydrostatic effects, while Heckelman and Ettles26 studied the viscous and inertial pressure effects at the inlet of a bearing cavity film. Pan*‘, Constantinescu and Galetuse28, and Tichy and Bourgin29 also looked at pressure entrance effects at the inlet of a bearing film. Recent experimental work by Braun30 and Batur31 presented flow visualization detailing the fundamentals of the flow and patterns both in the pocket of an HJB contiguous lands. Using particle tracers, 70

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and Braun experiments the pressure and on its the authors 29 Number

The work presented here specifically continues the work of Braun et al. 1,32,35,and discusses on a comparative basis the effects of change in the radius of curvature at the pocket/land joints (PLJ) of a shallow pocket (Fig 1) with an aspect ratio D/L = 0.021. The capillary restrictor feedline, the pocket and the adjacent lands are modelled in a synthesis configuration that has direct pertinence to the overall hydrostatic journal bearing performance. Other parameters used in this study are the Reynolds number based on the runner velocity, and the jet strength. The jet penetration, the interaction of the shear layer with the cavity flow, the interaction of the lands with the pocket domain through square or rounded joints, all affect the flow patterns, as well as the magnitude, profile, and the trends of the pressures, inertia pressure drops and velocities in the pocket.

Formulation

of the mathematical

model

Geometry

The geometry of the shallow hydrostatic pocket is shown in Fig 1. The physical dimensions are presented in Table 1. The pocket is fitted at the bottom with a capillary feedline of length L2, and lands of length M. The main physical parameter that differentiates the various geometries of the pocket presented in this 1 1996

Pressure

x”f x,

/“I

M

and flow

,5

.*-c.c. \JL

It\

/ DOMAIN

),C


bOMAlN

D

characteristics:

M

M. Dzodzo

et al.

3 A

2

Fig 1 Geometry of the pocket

Table 1 The dimensions

L M : B L2 RS h = R,/lOOO

of the considered

pockets

Dimensionless

in

m

464.758 267.621 10 5 100 300 1000 1

0.697 1 0.4149 0.015 0.0075 0.15 0.45 1.5 0.0015

0.01770728 0.01053926 0.000381 0.0001905 0.00381 0.01143 0.0381 3.81*10-5

study is the radius of curvature R of the PLJ (Fig 1, details Bl, B2, and B3). All geometries introduce the assumption that the cavity width is much larger than its length or depth, thus reducing the problem to a two-dimensional case, in coordinates ( Y1, Y,). The process is steady-state, and the fluid is Newtonian and incompressible. The properties of the fluid (p,v) are kept constant. Dimensionless

equations

for the fluid

model

To obtain a certain degree of generality in the discussion of the results, the governing equations of motion are cast in a dimensionless form. The characteristic length has been chosen to be h = R,/lOOO, where R, is the radius of the shaft. The equations are written in a general form for a nonorthogonal body fitted coordinate system in accordance with Peric’s36 formulation. The collocated control volume cells and the corresponding local arbitrary coordinates X, X are presented in Fig 1 (details A, Bl, B2, and B3). Using these notations, and the dimensionless variables defined in the nomenclature, the continuity equation takes the form:

while the momentum equations for Cartesian velocity components Ui (i = 1,2) can be written as:

(2)

where pi, represents the cofactor of the ith row and the jth column in the Jacobian coordinate transformation matrix .I y' = y’(x’) (3) where yi represents the Y,,Y, reference Cartesian system of coordinates. For two-dimensional applications the cofactors are:

The advantage of this formulation is its applicability to the irregular geometry of the pocket as described by Braun and Dzodzo32. The computational domain was split into three blocks (Fig l), each representing a different functional portion of the hydrostatic pocket: i) the restrictor (Domain 1); ii) the pocket (Domain 2), and iii) the land region and the region under the

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runner (Domain 3). For each block the local nonorthogonal coordinates X, X were used. Boundary

conditions

The boundary conditions ends of the cavity are: +);

z=O; 1

1

on the lands at the open

Y1=O;

Y,=M+L+M=lOOO;

310I Y2 5 310+ c;c=s

(5)

To justify the application of the boundary conditions of Eq (5), the authors have run preliminary tests to ensure that the lands were indeed long enough to guarantee a fully developed flow condition. The pressure on the boundaries was assigned to a reference environment pressure, P = 0, on the land’s inlet and exit: P=O;

Y,=O;Y,=M+L+M=lOOO; 310 I Y2 I310 + c; c = 5

(6)

All other pressures inside the domain were calculated taking the boundary pressures as the reference pressure. As a mean of further verification of the algorithm, a symmetrical mass outflow at both ends of the pocket was obtained for the case when the runner is stationary, and the prescribed pressure boundary conditions are the same. For this case, there was no need to prescribe the two exit lands mass outflow ratio37. In the entrance region where the fully developed, perpendicular hydrostatic jet enters the capillary feedline, one can write:

u2=-F[Y1-(s-~)][Y,-(s+~)]; Y*=O;

B s--sy,ss+-

B

2

where S = n/r + L/2 = 500. The strength was varied by changing the dimensionless F from F = 1*10-4 to F = 4*10p4. The boundary conditions U,=V=Re;

(7)

2

of the jet parameter

at the driving lid are:

U,=O;

Y,=31O+C;

OsY,sM+L+M

(8)

During the parametric study, the horizontal velocity of the runner is set to U, = Re = 1 and 8 respectively. The rest of the boundary conditions set the magnitudes of the velocities to U1 = 0 and 17, = 0 along the walls. In order to match the three computational domains (Fig 1) with each other, the pressure and velocity values calculated at the boundaries of the contiguous domains were used as their respective boundary conditions. This procedure is similar to the ‘slab by slab’ method of calculation used by Pratap and Spalding3* for staggered grids. For the work presented here the procedure has been adapted to collocated cells in the same manner as described by Dzodzo39. Numerical

procedure

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Figure 1 presents in details A, Bl, B2, and B3 parts of the discrete grid superimposed on the pocket cavity. The computational domains (1, 2, and 3) contain 20 x 20, 108 x 10 and 208 x 5 control cells, respectively. The convergence criterion was set through the calculation of the overall average dimensionless mass imbalance, S, , for each of the component domains. This criterion can be expressed as: S, = (XjUI,A,

- UI,A, + UzgAg - UznA,[); < 1O-4

(9) Using S, < 10e4 translates in a difference, between all dimensionless mass inflow and all outflow over the summed domains, of less than 1.5%. Results

and discussion

The parametric numerical experiments were divided in two groups: a) the first involved flow and pressure studies at high Reynolds number and low F jet strength (U = Re = 8, F = 1 *10-4), while b) the second group used a high F and a low Reynolds number (U = Re = 1, F = 4*10p4). For both a) and b) the clearance was kept constant at C = 5, while the radius of curvature at the PLJ was varied from a sharp 90” corner to rounded corners of radii R = 5 and 10 respectively. An appreciation of the physical magnitudes of these variables can be obtained from an inspection of Table 2. Figure 2 presents in sequential order the right hand side (RHS) and left hand side (LHS) of the pocket for the change in roundness of PLJ from 90” square (Fig 2A) to R = 5 (Fig 2B), and R = 10 (Fig 2C) respectively. The central portion of the pocket containing the region where the restrictor exit merges into the bottom of the pocket is shown in Figs 3A1, 3A2 and 3A3. One can see, generally, in the downstream RHS regions of the pocket, Figs 2A, B, and C, a flattened recirculating pillow which represents an excrescence of the modified vertical cell (MOVC) Table 2 Runner velocities and mean feeding jet velocities as functions of Re and F respectively Re = U linear speed (m/s) rot. speed (rpm)

F average jet (m/s) inlet velocity

A finite volume method based on the finite difference method was applied for the numerical implementation of Eqs (1) through (8). The method used non72

orthogonal coordinates36 and a collocated gridN. The SIMPLE procedure4* was used for solving the resulting set of equations. To calculate the convection flux through the control volume cell face a second-order central differencing scheme is implemented using the ‘deferred correction’ approach similar to the one presented by Khosla and Rubin43.

For p 100*10-6 38.1*1O-6

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=

9UOkg/m3, m2/s) and m

1

8

2.625 658 l”lO-4 0.4374

21.0 5263.4

,cL = 9*7O-2 kgims RS = 1.5 in = 38.1 mm,

4'10-4

1.7498 (V = w/p h = Ff.JlOOO

= =

Pressure

LBS

DETAIL

AU

A ’ AU -

! T!u

DETAIL

DU

DETAIL

AD

DhTAIL

DD

and flow

U=8

characteristics: Cl2 ,ARANCE

M. Dzodzo

et a/.

U=l

U=8

U=8

R=S TAZ DETAIL

U=8

CU

DLTAIL

CD

U=8 NIS I

F = h-

-

Fig 3 Flow patterns in the central portion of the pocket and the restrictor exit for PLJ square, and R = 5 and 10. Re = U = (8, and 1); F = (l*lO-“, and 4*10p4); c=s

+A2 0

I”‘

Fig 2 Flow patterns in the pocket with PLJ square and with R = 5 and 10 (Re = U = 8, F = 1*10-4 and c = 5)

that can be seen in its entirety in Figs 3A1, A2, and A3. Further inspection of the RHS in all three Figs 2 reveals that the increase in roundness of PLJ causes the bottom recirculating pillow to decrease slightly in height, due to an extra amount of fluid which exits the downstream land. The ram-jet effect mentioned by Ettles25 and Braun and Dzodzo32 can be observed in the Ram-Jet Impingement Regions (RJIR). The ram-jet position and structure are presented in Details AD, BD, and CD. The LHS of the pocket presents a flow Turn Around Zone (TAZ) that seems to move upward with the increase in the radius of the PLJ. The most remarkable part, however, is the total lack of fluid outflow in this zone. We shall see later that this phenomenon, due to the domination of the Couette flow, contributes decisively to the pattern of the pressure profiles under the runner. Details AU, BU, and CU show clearly the structure of the TAZ zone. The MOVCs in Fig 3 represent the remnants of the Central Vertical Cell zones described by Braun et al. 1,32,34, and have shifted completely into the exit

R=lO

region of the capillary restrictor feedline, Figs 3A1, 3A2 and 3A3. As the PLJ radius is increased, the dominating Couette effect (Re = U = 8, F = 1*10p4) causes both the flat pillow region, and the MOVC to diminish in size. In turn, the smaller MOVC causes an increase in the effective flow area for the restrictor inlet jet. The increase in PLJ radius also contributes to a decrease in the resistance to flow in its region. This situation is also likely to influence considerably the discharge coefficient at the exit of the restrictor. The authors believe that both the MOVC and the excrescent (pillow) recirculation zone are part of the mechanism of pressure build-up. Thus with a larger MOVC, a higher pressure needs to build between the lid and MOVC, in order to move the fluid out of the pocket and onto the lands. Flow patterns in the pocket with dominating jet (Poiseuille) effects (Re = 1, F = 4*10e4, C = 5) Figure 3 presents a comparison of the flow in the central region of the pocket when the flow is a) Couette shear layer dominated (Figs 3A1, 3A2, 3A3,

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with Re = 8, and F = 1*10p4),

and b) jet strength dominated (Figs 3B 1, 3B2, 3B3, with Re = 1, F = 4* 10e4). The PLJ is varied from square to rounded with R = 5, and R = 10, respectively. The major qualitative difference between the two cases is the total suppression of the MOVC in the case of the jet dominated flow (Figs 3B), and a pronounced change in the geometry of the TAZ zones, as can be seen from the inspection of Figs 4A, 4B, and 4C. One can notice that in all three cases the pocket’s downstream RHS does not contain any flattened recirculation pillow for any PLJ shape. With the increase in PLJ roundness, the RJIR zones are pushed towards the bottom of the pocket. The LHS portion of the pocket contains very sharp and narrow TAZ zones that are located in the vicinity of the runner (Detail AU, BU, CU), and not in the cavity of the pocket, as was the case of the Couette dominated flow (Fig 2, details AU, BU, and CU). Analysis of the data revealed that the change in PLJ does not affect significantly the TAZ apex location.

DFPAIL

AU

DETAIL

AD

DETAIL

BD

TAZ

DETAIL

DU

T&Z

Comparison of pressure profiles between the Couette and Poiseuille dominated flows under PW change

Figure 5 presents the pressure profiles under the runner for the Couette dominated flow when Re = U = 8 and F = 1*10p4. The three curves in the figure represent pressure levels (starting from the top) when the PLJs are square, and at R = 5 and R = 10, respectively. The dominating Couette effect is responsible for the continuous ramp type increase of the pressure in the pocket. While the Rayleigh step of the pocket is responsible for the RHS slope of the curve, the TAZ zones seen in details AU, BU, and CU of Fig 2, are responsible for the positive slope of the pressure curve on the LHS of the curve. The fact that the TAZ completely shuts off the fluid exit in the upstream portion of the pocket is also responsible for the lack of any inertia pressure drop in that zone (Fig 5, detail A). In the downstream PLJ zone one can see inertia pressure drops (IPD) generated by the exiting flow (detail B). However, the IPDs decrease as the PLJ radius grows. This trend can be attributed to the decrease in the resistance to flow due to the change in PLJ. Figure 6 presents the pressure profiles for the same pocket with the same variations in PLJ, but when the flow is jet dominated (Re = U = 1, F = 4*10p4). The pressure profiles are almost symmetric with respect to the centre of the pocket, and the maximum pressure is generated in the region where the jet impacts the runner. This pressure pattern is readily explained if one looks at the flow patterns of Fig 4, where the flow is exiting both at the upstream and downstream sides of the pocket. The pressure gradient shows a drop in the direction of the lands, and one can see in details A and B the inertia effects at work. The IPDs are much larger (almost 4.5 times) than in the Couette dominated flow, and are not symmetric. Thus the downstream IPD (detail B), is higher by almost a factor of 1.3 than the upstream IPD (detail A). This situation can be attributed to the fact that the downstream IPD is due to the added

POCKET

LAND!

T,AZ LUS

@P DETAIL

CU

TAZ

DETAIL

CD

racy

II

200

300

400

500

Fm 240

Fig 4 Flow patterns in the pocket with PLJ square and with R = 5 and 10 (Re = U = 1, F = 4*10p4 and c = 5) Tribology

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600

700

600

;zm$ 260

260

Yl

74

! LAND

y

(Yllh)

720

740

760

Y1 (Yllh)

Fig 5 Pressure distribution under the runner for the Couette dominated flow. (Re = U = 8, F = 1*10p4 and C = 5). PLJ is varied from square to R = 5 and 10

Volume 29 Number 1 1996

Pressure

LAND!

100

POCKET

300

400

’ LAND

500

600

800

700

cir

Yl

(Yllh)

@L$iJqz$F$; 240

260

280

y1 (Yl/h)

720

740

760

Y1 (q/h)

Fig 6 Pressure distribution under the runner for the jet dominatedflow (Re = U = 1, F = 4*10e4and C = 5). PW is varied from square to R = 5 and 10

effects of the Couette and Poiseuille flows, while for the upstream IPD these effects subtract.

and flow

characteristics:

M. Dzodzo

et al.

Couette dominated flow one observes no fluid outflow at the pocket upstream, and consequently the shape of PLJ is of little consequence to the inertia pressure drops, which are generally negligible. For the Poiseuille dominated flow the IPDs are comparable in order of magnitude for both the upstream and downstream exits of the pocket, with PLJ playing an equally important role on both sides. However, the IPDs are different in magnitude. Thus, at the downstream exit, the additive effects of Poiseuille and Couette generate a larger pressure drop than at the upstream exit, where the two effects work against each other. As for the pressure profiles, the Couette dominated flow generates a continuously ramped pressure across the pocket with a mild IPD at the downstream exit. This is in stark contrast with the Poiseuille generated pressure profile which is relatively even across the pocket, but presents large IPDs at the upstream and downstream pocket exits. The ramped pressure profile of the Couette dominated flow requires, according to Shinkle and Hornung”, that the jet inlet on the floor of the pocket be positioned in the centre of the pocket. The more even profile of the Poisseuille dominated flow renders the position of the jet less consequential.

Conclusions Figure 7 is introduced as a corollary to the numerical experiments presented heretofore. The resistance network is used to rationalize and classify the results presented above. One has to keep in mind that the experiments assumed a constant mass inflow (Mi”) at the entrance of the pocket restrictor, a constant runner velocity and a constant reference pressure (P& at the lands’ exit (Eqs (6), (7) and (8)). The resistance of the restrictor (R,) is in series with those of the pocket (R,), PLJ (RPLJ), and the land (R,). The numerical experiments, whether Couette dominated or Poiseuille dominated, had in common the systematic variation of RpU by rounding the pocket exits to the lands. When the resistance is largest (PLJ square exit) it induces a higher pressure in the pocket, necessary to move out the constant mass of fluid. Conversely when PLJ is rounded the necessary pressure drop to move the same amount of fluid goes down, and thus the pressure in the pocket itself goes down. In the

Acknowledgements The authors wish to thank Mr Jim Walker for his continuous support during the execution of this work. The work was funded and performed under the auspices of NASA Lewis Research Center, Cleveland, Ohio, USA. References Braun, M. J., Choy, F. K. and Zhou, Y. M. The effects of a hydrostatic pocket aspect ratio and its supply orifice position and attack angle on steady state flow patterns, pressure and shear characteristics. ASME J. Tribal. 1993, 115, 4, 678-685 Artiles, A., Walowit, J. and Shapiro, W. Analysis of fluid film journal bearings with turbulence and inertia effects. Advances in Computer Aided Bearing Design. ASME, 1982 Castelli, V. and Shapiro, W. Improved methods for numerical solutions of the general fluid film lubrication problem. J. Lubr. Technol.

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89, 71-79

Redecliffe, J. and Vohr, J. Hydrostatic bearings for cryogenic rocket engine turbopumps. ASME J. Lub. Technol. July 1969, 557-575 Belousov, A. I. and Yu, A. R. Dynamic characteristics of liquid film in hybrid hydrostatic bearing. Sow. Aeronaut. 1976, 21, 3. 16-19 Braun, M. J. and Wheeler, R. L. A fully coupled variable properties thermohydraulic model for a cryogenic hydrostatic journal bearing. J. Tribal. 1987, 109. 405-417 San Andres, L. A. Analysis of turbulent hybrid bearings with fluid inertia effects. J. Tribal. 1990. 112, 699-707 Goryunov, L. and Belousov, A. Certain design features of radial hydrostatic bearings for high-speed machines. Izvsnya VUZ. Aviatsionnya

Fig 7 Comparative corollary forpressure curve development under Couette and Poiseuille dominated flow Tribology

Tekhnika,

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Davies, P. B. and Leonard, R. The dynamic behavior of multirecess hydrostatic journal bearings. IMechE Tribology Group Convection, May 1970, Vol. 184. Pt3L, 139-137 Shinkle, J. and Hornung, K. Frictional characteristics of liquid hydrostatic journal bearings. ASME J. Basic Eng, Ser. D 196.5, 87, 163-169

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et al.

11. Leonard, R. and Davies, P. B. An experimental investigation of the dynamic behavior of a four recess hydrostatic journal bearing. IMechE Externally Pressurized Bearings, Paper C29, 1971, 245-261

28.

Constantinescu, V. N. and Galetuse, S. Pressure drop due to inertia forces in step bearings. J. Lubric Technol. 1976, 98,

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Tichy, J. A. and Bourgin, P. The effect of inertia in lubrication flow including entrance and initial conditions. J. Appl. Mech.,

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12. Ho, Y. S. and Chen, N. N. S. Performance characteristics of a capillary-compensated hydrostatic journal bearing. Wear, 1979, 52, 285-295 13.

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Ho, Y. S. and Chen, N. N. S. Dynamic characteristic of a hydrostatic journal bearing. Wear, 1980, 63, 13-24

14. Ho, Y. S. and Chen, N. N. S. Pressure distribution in a six pocket hydrostatic bearing. Wear, 1984, 98, 89-100 15. Aoyama, T., Inasaki, I. and Yonetsu, S. Limiting conditions of hydrostatic bearings. Bull. Jap. Sot. Prec. Eng. 1982, 16, 2,

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16. Chaomleffel, J. P. and Nicholas, D. Experimental investigations of hybrid journal bearings. Tribol. Intern. 1986, 19, 5, 253-259 17. Scharrer, J. K., Tellier, J., Hibbs, R. A study of the transient performance of the hydrostatic journal bearings: Part I-Test apparatus and facility. STLE preprint No. 91-TC-3B-1, 1991 18. Scharrer, J. K., Tellier, J. and Hibbs, R. A study of the transient performance of the hydrostatic journal bearings: Part II-Experimental results. STLE preprint No. 91-TC-3B-2, 1991 19. Spica, P. W., Hannum, N. P., and Meyer, S. D. Evaluation of hybrid hydrostatic bearing for cryogenic turbopump application. NASA TM872254, 1986 20.

Francket, N. M. and Chills, D. W. Experimental test results for four high-speed, high-pressure, orifice-compensated hybrid bearings. J. Tribol. 1994, 116, 147-153

21.

Heller, S. Static and dynamic performance of externally pressurized fluid film journal bearings in the turbulent regime. ASME

22.

Betts, C. and Roberts, W. H. A theoretical and experimental study of a liquid-lubricated hydrostatic journal bearing. Proc. ZMechE 1969, 183, Pt. 1, 32, 647-657

J. Lubr.

23.

52, 759-765

Braun, M. J. Discussion. Tribol. Trans. 1990, 31. Braun, M. J. and Batur, C. Non-intrusive field quantitative flow measurements aided processing, Part 2: The case of hydrostatic 30.

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33, 4, 547-550

laser based full by digital image bearing. Tribal.

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Braun, M. J. and Dzodzo, M. B. Effects of hydrostatic pocket shape on the flow patterns and pressure distribution. 5th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-5), May 8-12 1994, Maui,

Hawaii

Hendricks, R. C., Steinetz, B. M., Atbavale, M. M., Prekwas, A. J., Braun, M. B., Dzodzo, M. B., Choy, F. K., Kudriavtsev, V. V., Mullen, R. L. and von Pragenau, G. L. Interactive development of seals, bearings, and secondary flow systems with the power stream. 5th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-5), May 8-12 1994, Maui, Hawaii 34. San Andres, L. A. and Velthuis, J. F. M. Laminar flow in a recess of a hydrostatic bearing. STLE preprint 91-TC-3B3, STLEiASME Tribology Conference, St. Louis, Missouri, 33.

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Braun, M. J., Zhou, Y. M. and Choy, F. K. Transient flow patterns and pressures. Characteristics in a hydrostatic pocket. ASME J. Tribol. 1994, 116, 1, 139-146 36. Peric, M. A finite volume method for the prediction of threedimensional fluid flow in compled ducts. PhD Thesis, University of London, 1985 37. Sbyy, W. Numerical outflow boundary condition for NavierStokes flow calculations by a line iterative method. AZAA J.

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