Pressure Drop In A Hydrostatic Pocket. Experimental and Theoretical Results

The Third Body Concept / D. Dowson et al. (Editors) 0 1996 Elsevier Science B.V. All rights reserved.

423

Pressure Drop In A Hydrostatic Pocket. Experimental and Theoretical Results M. Arghir, S.E. Attar, D. Nicolas Laboratoire de Mtcanique des Solides URA CNRS,Universitt de Poitiers 40, Avenue du Recteur Pincau, 86022 POITIERS CEDEX

-

This paper presents experimental 'and numerical results of the pressure drop in a hydrostatic pocket. The pure hydrostatic flow was investigated for laminar and turbulent regimes. The good agreement between experiment and theory obtained in the turbulent regime enables to propose a relation for the pressure drop coefficient. 1. INTRODUCTION

Geometric discontinuities cannot be handled by the Reynolds equations or by the film thickness averaged (bulk flow) equations. The hypothesis involved in deducing this equations (neglecting the transversal velocity, constant pressure across the filni thickness) exclude the existence of a flow pattern with recirculation zones. The effect of geometric discontinuities is introduced in the equations of the lubrication theory using concentrated pressure droplrise coefficients. This coefficients are usually deduced from experimental information. During the last years the force of the numerical methods offered an alternative tool for predicting unconventional flows in bearings. Because no approximations are made, the integration of the complete system of the flow governing equations may be considered as a riumerical experiment. This is true for the laminar flow regime, when no modelling assumptions are made and the only difficulty is the numerical integration of a non-linear system of differential equations. For the turbulent flow regime the nccessity to describe the turbulent behavioiir introduces modelling assumptions. Concentrated inertia effects act different if the flow is a shear driven or a pressure driven one. The pure shear driven flow correspoiids to the limiting case of a hydrostatic bearing working in centred position. As first shown by Constantinescu and Galetuse [4] the pressure may rise or may drop, depending on the surface Reynolds number and the ratio of the film heights. The pressure driven flow is the limiting case of a flow through an orifice. The fluid is accelerated in the land region and a pressure drop occiirs. For an inviscid flow or if the protruding corner would be

rounded the pressure drop is given by Bernoulli's formula. In real flows, the streamlines are curved after the inlet. A small recirculation zone may be located on the step wall after the contraction. This curvature reduces the flow section and so enhances the pressure drop. This effect may be quantified by a pressure drop coefficient.

5 = - AP2

pu, 2

Experimental investigation of this effect were done by Chaomleffel and Nicolas [3] and by Constantinescu [S] for the laminar and the turbulent flow regimes. Numerical studies were done by SanAndres and Velthuis 191 and by Braun and Dzodzo 121. Their results concerned only the laminar flows. In this paper we present experimental and numerical results obtained for a pressure driven flow in a step. The results are obtained for laminar and turbulent flow regimes up to Rep=12000. 1.1. Notations A, coefficient in (A3) CI , y, coefficients in (14) h, thickness k, k', turbulent kinetic energy, non-dimensional k,, k,, coefficients in (2) I,, mixing length p, pressure P, film thickness averaged pressure Q, flow rate Rep, Reynolds number of the Poiseuille flow R, Rp, local Reynolds numbers s, parameter defined by (22)

424

- pocket length

- pocket depth - pocket pressure level - ineasured temperature

Figure 1 Experimental device To, reference temperature u, v, velocity components u,, friction velocity lit, yt. non-dimensional velocity and distance ii,,,, film thickness averaged vclocity s,y, Cartesian co-ordinates x, E, coeflicients in (22) h,reference dynamic viscosity p, coefficient in (4) E, E'. dissipation, non-dimensional @, general variable p, dynamic viscosity p,,u~l> pLsl~, turbulent and effective viscosity p, dcnsity T. slicar strcss 5. pressure drop coefliciciit

!,=o.o751n

h,=2mm+hf or 5mm+hr Po=0.5.10SPa to 8.105Pa T=18"C to 30°C

A close circuit pump with a 1.2 m 3 reservoir is used to supply the fluid. The temperature of the fluid is not controlled so heating occurs during a measuring set. The average exit tempcrature is continuously measured by a thermocouple. The pressure is measured by a classical manometer. The flow rate Q is determined using a volumetric counter. 2.2. Bulk model of the flow

Three flow regions can be distinguished on figure 2a: I. the pocket region 11 the step region 111. the thin film region The theoretical pressure variation is presented on figure 2b. The flow in regions I and 111 is governed by the Reynolds equation presented by Frine et al. [6]

Re, I 1 0 0 0 1000
(3)

2. EXPERIMENTAL DEVICE AND RESULTS 2.1. Description o f the ex~ierimcntaldevice

The experimental device presented on tlie figure

1 is made of two plates separated by lateral sheets

which enable to eliminate the lateral flow and to control tlie height of tlie thin film The lower plate has ;I pocket and is provided at one edge with ten supply orifices in order to have a uniforni flow along the width L. The upper plate is flat and is provided with pressure measuring locations in its median plane. The characteristics are:

- axial width, - thin film length, - thin film thickness

L=O. 1In e =o. 1nl h,=O. lmin to 0.541niii f

The viscosity is considered to vary with the teinpcrature.

(4) In the region I the pressure PSI is calculated by linear extrapolation of the measured results.

425 bulk velocity in the thin film zone.

The pressure gradient in the region I is very small compared to the pressure gradient in the region I11 because the height ha is always much larger than the height hr. 2.3. Measurement accuracy An accurate evaluation of the pressure drop coefficient imposes that the surfaces should be parallel, the film thickness should be well known and the pressure and the flow rate measurements should be of known tolerances. A finite element calculation of the deformations produced by the film pressure shows that the plates should be thick enough. A 40 mm thickness of the aluminium plates was appropriate. The tightening of the screw-bolts is made using a torque wrench in order to ensure an uniform distribution of the stresses. The film thickness in the land region is not a priori known because the lateral sheets are squeezed ‘after the tightening of the screw-bolts. However we can estimate this thickness using the measured flow rate Q and pressure P4.

Figure 2a Flow geometry

Figure 2b Theoretical pressure distribution

and the flow rate is given by Q

11:

I -

P3 -Po ll,

$.I

In the region 111 the pressure Pn2and the flow rate are: 1, P,, = P4 -

1,

C’,. = 1, -5.10-3m

(7)

@a> In the region 11, the pressure variation is modelled by the pressure drop coefficient 5 that relies the pressure variation due to the step to the

The accuracy of the flow rate measurements is estimated to 1%. The accuracy of the pressure measurement is in the order of 2% and it can attain 4% for small values. The accuracy of the temperature measurement is in the order of 2%. Taking into account this accuracy, the experimental uncertainty of the calculated film height is between 2% and 3%. That is experimentally validated because for the same test conditions, calculated film heights vary less than 2%. 2.4. Experimental results The pressure drop coefficient is given by

426

3

&* *

i

........... ...

ha/hf : 4C

*

*

I2

I0 A

65

0

4.8 4.5

0 0

0

15.3 9.7 9.3

2000

4000

6000

8000

10000 Rep

12000

Figure 3 Experimental pressure drop coefficient The experimental results are prcscnted on the figure 3 as function of the Reynolds number for different film thickncss ratio h./hr. Each symbol represents a different test, i.e. many supply prcssures are considered for the same geometry The dispersion of the results is rnoderatc. Because the measured pressure accuracy may attain 4% for small values, the dispersion is more important for low Reynolds numbers. So, for Reynolds numbers less than 2000, the accuracy of 5 may attain lo%, but it is around 6% for higher Reynolds numbers. At Reynolds numbers less than 1500, the pressure drop coefficient is very depcndcnt of this; it decreases rapidly with increasing Reynolds number. If the Reynolds number becomes high, the pressure drop coeflicient is around 1.5. Apparently the pressure drop coeflicient docs not depend on the film thickness ratio, its variation being less than the measurements accuracy.

3. THEORETICAL MODEL

3.1. Flow governing equations

The mathematical model consists of the time averaged Navier-Stokes equations. The equations are numerically integrated on a two-dimensional domain.

The turbulence model is the classical k-E turbulence model introduced by Launder and

427 Spalding IS]. For the coefficient C, intervening in the relation (13) an expression introduced by Rodi II3 I was used.

This expression is better adapted for situations when the equilibrium between the kinetic turbulence energy generation, Gk, and the dissipation E is not satisfied. The form of the source terms So was presciited by many authors, e.g. in [ I ] . The equations are discretized using the finite volume method. The coupling between the pressure field and the velocity field is solved using the SIMPLE algorithm proposed by Patankar [ 111. The “power law” was used for discretizing the convective terms. The control volumes for the components of the velocity are staggercd from the grid points to avoid an unfavourable decoupling between the pressure and the velocity fields. The grid consists of variable spaced rectangles. Very line grid spacing in both directions were used in the vicinity of the protruding corner of the domain. The grid spacing varies like a geometric progression towards the entrance and the esit section and towards the upper wall. The computational domain do not covers all the length of the experimental device. The entrance and the exit sections are considered far away from the step zone in order lo have fully developed flow. The numerical algorithm has two steps: an iniplicit predictor step and an explicit corrector. At the beginning of the predictor step an estimation of the pressure field is used in the momentum equations which are implicitly solved. The new estimations of the velocity ficld are used to calculate the source term in the discretized pressure correction equation. This equation is also implicitly solved in the predictor step. The resolution of the linear systems is made using a line-by-line relaxation algorithm combined with a red-black columns numbering. In the corrector step. velocities and pressures are adjusted considering that the pressure perturbation in a point determines a

proportional velocity perturbation only in the same point. The convergence of the iterative process is controlled by the source terms of the pressure correction equation:

The same accuracy is used for the transport equations residues. 3.2. Boundary conditions In the inlet section we suppose that the flow is parallel to the walls:

laminar flow regime u(y) = 6 u mg 1 - i ) turbulent flow regime I

-

~ ( y ) = ~ u ~ [ 2 - m i n ( i , l - i ), ]n ~2 3 (18) The inlet average velocity, u,,,, is calculated from the Reynolds number given by (3a). A constant kinetic energy profile, corresponding to 15% turbulence degree level is considered in the inlet section. Even if the entrance boundary conditions are not very accurate, the distance from the inlet section to the step zone is large enough for having a fully developed flow before the contraction. The additive character of the pressure in an incompressible flow enables the selection of the exit section of the computational domain. The position of the exit section is selected analysing the pressure distribution in the land zone. In the vicinity of the exit section the effects due to the contraction are attenuated and the average pressure distribution must regain a linear variation. The boundary conditions for a fully developed flow are imposed in the exit section.

428

am

-=0

, @={u,v,~,E}

ax

The atmospheric pressure is considered as a reference value and is imposed in one of the control volumes near the exit section. The wall boundary conditions for the laminar flow regime are the typical no-slip conditions.

In turbulent flows numerical boundary conditions usually replace the no-slip conditions. These are obtained using the logarithmic law at the wall in the first grid point near the boundary, Schiestel [lo]. They are imposed as gradient type boundary conditions for the velocity component parallel to the walls. For the turbulent quantities k and F. the logarithmic law provides an estimation in the first grid point near tlie walls. 2

~ = 0 . 4 , E = 7.8 /

(224

\2

dr

= w.ll u, = -

U + =-

'

U

u,

y + = -f y u r

P

(22c)

=-ursign(up f 2 -uwRI,)

The logarithmic law is solved as a non-linear equation in the first grid point near the boundary. The solution sp and so u, is used to calculate the gradient of the velocity component parallel to the wall. The estimations of k and E in the first grid point near the wall are used as boundary conditions on a computational domain with shifted boundaries. The point P must always lie in the zone wlicre logarithmic law is valid, 11.5
Figure 4 Detail of the structured grid of the wall boundary conditions neglects the intermediate sublayer (1 1.5
429

0.25

5

0.20

4

g 0.15 c

5

z.

0.10

0.05

1

0.00

0

0

I0

20 x [inm]

30

40

Figure 5a. Pressure variation for laminar flow on figure 5 for laminar and turbulent flow regimes. Streamlines in the pocket are presented on figure 6. It can be seen that the pressure level in the pocket is constant. As expectcd, the pressure has a linear v'ariation in the land region. In turbulent flow the effect of the contraction is more pronounced and the development zone which follows after the step is longer. In estimating the pressure drop due to the contraction this development length is ignored and the linear pressure variation is extrapolated to the step wall. The difference is calculated between the pressure level in the pocket and the extrapolated land pressure.

The experimental and theoretical results are presented on figure 7. The laminar flow results are detailed on figure 8. The results were curve-fitted separately for the laminar regime and for the turbulent regime. The pressure drop coefficient is approached by the following relations: 7.78 , 300 < Re,, < 2000 Re:*

<=-

40 x Imml

60

80

Figure 5b. Pressure variation for turbulent flow 0.0051

E

t

0.004

0.003

Y

0.002 0.001

0'0000.021

0.023

0.025

0.027

0.029

Figure 6. Streamlines in the pocket 3.45

4. COMPARISONS AND DISCUSSIONS

20

0

Re, > 2000

A good agreement between theory and experiment is obtained for the turbulent regime. The results obtained for Rp>5000 are greater as the values measured by Chaomleffel [3]. His values are very close to 1. The pressure drop coefficient obtained from our numerical results can be approached by the relation: c n ic

5 = 1.96 + J U J J

Re b34

430 do not allow to appreciate the disagreement between the measured results and the calculated one. It seems that for very small Reynolds numbers (
5

r; 4

3

2 1 Rep O . O E + O 2.OE+3 4.OE+3 6.OE+3 8.OE+3 1.OE+4 1 . 2 E + 4

Figure 7 Pressure.drop coefficient - theoretical and experimental results

Experiment

r 5

4

ACKNOWLEDGEMENTS This work was performed with the partial financial support of Rtgion Poitou-Charentes, Contract no. 94/RPC-B-6. The authors also express their gratitude to S. Brochet for his contribution.

REFERENCES 1. Argliir,

3

2.

2 1 0

400

800

1200

1600 Rep2000

Figure 8 Pressure drop coefficient for small Reynolds numbers As seen on figure 8, for Reynolds number between 300 and 1200 the theoretical results give a quasi-constant value of the pressure drop coefficient. Our theoretical results show the same behaviour as San Andres' and Velthuis 191 results but the values are somewhat greater. This differences could be due to different grid arrangements. The experimenkd variation of E, are less rapid than the theoretical ones. 5. CONCLUSIONS

For the pure hydrostatic regime an abrupt ch,ange of the flow section entrains important pressure drops due to coupling between inertia and

viscous effects. For laminar flow, the experimenkil uncertainties

3. 4.

5.

6. 7.

8.

9.

M., Frene, J., "Determination des caractkristiques statiques et dynainiques des joints rainurks fonctionnmt en position centree", Rapport Final de Contrat, juin 1995. Braun, M. J., Dzodzo, M., "Effects of the Feedline and the Hydrostatic Pocket Depth on the Flow Pattern and Pressure Distribution", Paper No. 94-Trib-27, ASMEISTLE Tribology Conference, Maui, Hawaii, October 16-19, 1994. Chaomleffel, J.P., Nicolas, D., "Experimental Investigation of Hybrid Journal Bearings", Tribologv International, 19, 5, pp 253-259, 1985. Constantinescu, V., N., Galetuse, S., "Pressure Drop Due to Inertia Forces in Step Bearings", ASME Journal of Lubrication Technology, 98, pp 167-174, 1976. Constantinescu, V.N., "On conditions at the inlet edge of a lubricating film operating at large Reynolds numbers", (in Romanian) The 541 Conference on Friction, Lubrication and Wear, Bucharest, Sept. 1987. Frhe, J., Nicolas, D., Degueurce, R., Berthe, D., Godet, M.,"Lubrfication hydrodynamique. Paliers et Butbes",Editions Eyrolles, Paris, 1990. Hwang, Y.H., Liou, T.M., "Expressions €or k and E Near Walls", AIAA Journal, pp. 477479, March 1991. Launder, B.E., Spalding, D.B., "The Numerical Computation of Turbulent Flows", ComputerMethods in Applied Mechanics and Engineering, Vol. 3, pp. 269-289, 1974. San Andres, L.A., Velthuis, J.F.M., "Laminar Flow in a Recess of a Hydrostatic Bearing", Tribology Transactions,35,4, pp 738-744, 1992.

43 1 10. Schiestel, R.,

"Modc'lisation et sinrulation des c'coulemerrts turbulents",Hennes, Paris, 1993. I 1. Patankar, S., 1980, "Nunrerical Heat Trotisfir And Fluid Flow",Hemisphere Pub. Corp, 1980. 12.Patankar, S.V., Spalding, B., "Heatarid mass transfer in boundary layers. a general calculation procedure", Intertext Books, London, 1970. 13. Rodi, W., "A New Algebraic Relation For Calculating The Reynolds Stresses",U h M , 56, T2 19-22 I , 1976.

APPENDIX The approach used to reformulate the turbulent wall boundary conditions was suggested by Pak2nkar and Spalding [ 121. Practical relations for the velocity profiles can be obtained using van Driest's mixing length relation. Very close to the wall. the transport equations can be drastically simplified.

calculated.

Using relation (A5) which replaces the logarithmic law, the non-dimensional velocity up' is calculated and so the friction velocity ur and the wall shear stress are determined. Relation (A5) has the advantages of being continuous from the wall to the validity domain of the logarithmic law and do not necessitates the resolution of a non-linear equation. In Schiestel [lo] relation (A5) is approximated by a polynomial expression:

s=

(5)

= R-'

- 0.1561.R4.4J- 0.08723-R".' +

0.03713*R4"*, O < R < l o J (A7)

au

T I = jl-

aY

(A3) Introducing the definitions of the shear stresses in equation (Al) and after integration one obtains the non-dimensional velocity profile: u+=

11'

+'C

Jo

2dq' 1+/1+4[1:(q')]2

The result of the numerical integration of (A4). = u+(y+), is putted in a form appropriate to be

used for calculations in the first grid point near the boundary. I = f ( y + u + )= f(R) 11'

This relation enables tlie direct calculation of uz without solving the non-linear relation of the logarithmic law in the first grid point P near the wall. The distance yp and the velocity up are known. and the local Reynolds number RP can then be

Thc approxiiliation given by this relation is not correct because it calculates negative values for s. We deduced a different approximation of the relation (A5)

g , (r), - 0.1704 < r < 3.2 g2(r), 3.2 < r < 5.5

(A81

g3(r), 5.5 < r < 11.7947

r = ln(R)

(A84

g,(r)=0.00126646-0.50471.r +0.00242796.r2

(ASb)

g2(r)=-0.0131839-0.425012.r -0.0444457.8+0.0073206 1-r2

(A8c)

g3(r)=5.756S3-4.12612-r+0.838655.? -0.0881695.r3+0.00464523.r4 -9.75491.10-5.rS

(A84

The boundary conditions for k and E must take into account that the first grid point close to the wall might lie under the domain of the logarithmic law. Hwang and Liou [7] proposed the following relations.

432

k * = 3 [ (Ay+y'* - I 1 - 2 ,

E*=-

(Ay')" + 1

and the profiles of k and E are:

k *'

a)y+ 2-7

XYi

k * = 3 [ 0.771S(y+ ) I 0.77 1S(y

6 ' 9

-l]1-2

+1

k" 0 . 4 +~ b)O
where

=-

k' = 0.0135(y')2 E+

and &=0.45 was determined from experimental data. The other two constants, A and a, are calculated imposing the continuity of k and E at y+=y:. For y=:7 one obtains: A=0.8748,

a,=O.OO 1353

(A 12)

= 0.09 +0.001353(y')2

Using the relations (A5), (A13) and (A14) do not offers to the presented k-E model the capability to handle very low Reynolds number flows. On the other hand, it is possible to treat flows with large separation or reattachment zones when the point P lies bellow the domain of the logarithmic law without increasing the computational effort.