Turbulent Effect on the Dynamic Response ... - ScienceDirect

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ScienceDirect Procedia Technology 23 (2016) 193 – 200

3rd International Conference on Innovations in Automation and Mechatronics Engineering, ICIAME 2016

Turbulent effect on the dynamic response coefficients of finite journal bearings lubricated with micropolar fluid Subrata Dasa*, Sisir Kumar Guhaa a

Indian Institute of Engineering Science and Technology, Shibpur, Howrah - 711103, West Bengal, India

Abstract The aim of the present work is to analyse theoretically the dynamic response coefficients of fluid film of finite hydrodynamic journal bearing with micropolar lubrication in turbulent regime. The governing non-dimensional Reynolds equations for steady state and dynamic film pressures have been solved numerically to obtain the film pressure distributions which are used to determine the dynamic response coefficients in terms of stiffness and damping coefficients of fluid film. These response coefficients are used to obtain the linear stability characteristics in terms of critical mass parameter and whirl ratio. It is found that the stiffness and damping coefficients both increases with increase in Reynolds number. It is also observed that the turbulence of the micropolar lubrication has an adverse effect on stability and the whirl ratio. © Ltd. This is an openby access article under the CC BY-NC-ND license © 2016 2016Elsevier The Authors. Published Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of the Organizing Committee of ICIAME 2016. Peer-review under responsibility of the organizing committee of ICIAME 2016 Keywords: Micropolar hydrodynamic lubrication; Journal Bearings; Dynamic response coefficients; Stability; Turbulence.

1. Introduction It is observed that in case of many modern machinery such as turbomachinery operating with large diameters at relatively high speed and in machines using low viscosity fluids as lubricant, the non-laminar flow frequently occurs. In the field of turbulent lubrication, Constantinescue [1-3] used the concept of Prandtl’s mixing length to derive the modified Reynolds equation applicable to turbulent hydrodynamic lubrication. Ng et al [4] and Elrod et al

* Corresponding author. Tel.: +91-9163902661; fax: +91-33-2668-2916. E-mail address: [email protected]

2212-0173 © 2016 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIAME 2016 doi:10.1016/j.protcy.2016.03.017

194

Subrata Das and Sisir Kumar Guha / Procedia Technology 23 (2016) 193 – 200

[5] used the concept of Reichardt’s eddy diffusivity for the analysis of bearings operating in turbulent regime. Taylor et al [6] suggested the application of the existing lubrication theories developed by Ng, Pan and Elrod [4-5]. In the literatures mentioned so far, the lubricants were considered as Newtonian fluid. But in practice, most of the lubricants exhibit non-Newtonian characteristics due to the presence of long-chained additives and suspended particles or dirt. The micropolar fluid model is better suited for the design of journal bearings using such lubricants. Therefore, to analyze the lubrication problem dealing with such fluids, the theory of micropolar fluids [7] which are characterized by the presence of suspended rigid microstructure particles has been applied. Several research works [8-11] had been carried out using the micropolar theory of lubrication to find out the effect of non-Newtonian lubricants on the bearing performances. Faralli et al [13] first utilised the turbulent lubrication theory proposed by Constantinescue while analyzing the steady-state characteristics of worn spherical bearing lubricated with non-Newtonian fluid in turbulent regime. Gautam et al [12] and Das et al [14] carried out the theoretical investigation on the static performance characteristics of journal bearings under micropolar lubrication with turbulent effect. Recently, Rana et al [15] conducted approximate dynamic analysis of short journal bearings operating with micropolar lubrication in turbulent regime. In the present work, an attempt has been made to extend the micropolar lubrication theory to find out the effect of turbulence on the response coefficients of the fluid film. The stability parameters are also estimated using the values of response coefficients at various Reynolds number. Nomenclature AT

BT C Cz D Dij D ij Dz Fi Fi h h kz kT lm L M M N p p

pi pi

R Re Reh Sij S ij

Constant parameter of turbulent shear coefficient for circumferential flow Exponential constant parameter of turbulent shear coefficient for circumferential flow Radial clearance, m Constant parameter of turbulent shear coefficient for axial flow Journal diameter, m Damping coefficients of micropolar fluid film, i r ,I and j r ,I , N s/m Dimensionless damping coefficients of micropolar fluid film, D ij 2 Dij C 3 / P:R 3 L , i r ,I and j r ,I Exponential constant parameter of turbulent shear coefficient for axial flow Force components along r - and I - directions, i r and I , N Non-dimensional force components along r - and I - directions, F i Fi C 2 P: 2 R 3 L , i r and I Local film thickness, m Non-dimensional film thickness, h h C Turbulent shear coefficient z direction Turbulent shear coefficient along circumferential direction Non-dimensional characteristics length of micropolar fluid, lm C / Bearing Length, m Mass parameter, kg Non-dimensional mass parameter, M MC 3 PR 3 L Coupling number Micropolar film pressure in the film region, Pa Non-dimensional film pressure in the film region, p pC 2 P:R 2 Local micropolar film pressure in the film region, i 0, 1 and 2 for the steady-state and first order perturbed film pressures along r- and I - directions Local micropolar film pressure in the film region, p i pi C 2 P:R 2 , i 0, 1 and 2 for the steady-state and first order perturbed film pressures along r- and I - directions Radius of the journal, m Mean or average Reynolds number defined by radial clearance, C , Re U:RC P Local Reynolds number defined by the local film thickness, h , Re h h. Re U:Rh P Stiffness coefficients of micropolar fluid film, i r ,I and j r ,I , N/m Stiffness damping coefficients of micropolar fluid film, S ij 2S ij C 3 P:R 3 L , i r ,I and j r ,I


t U W W0 W0 z z

195

Time, s Velocity of journal, U :R , m/s Load in bearing, N Steady state load in bearing, N Non-dimensional steady state load in bearing, W 0 W0 C 2 P: 2 R 3 L Cartesian coordinate axis along the bearing axis, m Non-dimensional Cartesian coordinate axis along the bearing axis, z 2 z L Eccentricity ratio Steady-state eccentricity ratio Perturbed eccentricity ratio Whirl ratio, O Z p : Attitude angle, rad Steady state attitude angle, rad Micropolar fluid functions along circumferential and axial directions Non-dimensional micropolar fluid functions along circumferential and axial directions Characteristics length of the micropolar fluid Newtonian viscosity coefficient, Pa s Angular velocity of the orbital motion of the journal centre, rad/s Angular velocity of journal, rad/s Circumferential coordinate, rad, T x R ; Circumferential coordinate where the film cavitates, rad Non-dimensional time, W Z.t

H H0 H1 O

I

I0

) x, z )T ,z /

P Zp : T Tc W

2. Analysis: 2.1. Modified Reynolds equation A schematic diagram of a hydrodynamic journal bearing with the circumferential coordinate system used in the analysis is shown in Fig. 1. The modified Reynolds equation in non-dimensional form [12-13] applicable for twodimensional flow of micro-polar lubricant with the turbulent effect under the dynamic condition is represented as follows:

w wT

2 ª wpº § D · w ª wpº , , ) h l N ¨ ¸ « T » «) z h, lm , N » m wT ¼ © L ¹ w z ¬ wz ¼ ¬

1 1 2I c w h w h 2 wT wW

(1)

where,

)T , z h, lm , N

§ hl N · h h 1h N ¸; coth¨ m 2 ¨ ¸ 2 lm 2 kT , z lm © ¹ 3

2

wI wW The values of the turbulent coefficients, AT , BT , C z and D z are obtained from references [6, 13]. kT

12 AT Re h

BT

, kz

12 C z Re h

Dz

and I c

It is assumed that the journal undergoes a whirling motion in an elliptical orbit about its mean steady-state position ( H 0 ,I0 ) with amplitudes Re al H1eiOW and Re al H 0I1eiOW along the line of centres and perpendicular to the line of centres respectively. For the first order perturbation, the non-dimensional pressure and film thickness can be expressed as

196


T I O

CH 0

O

b

b

h

I1

R

O

j

:

I0 O

CH1

r

W

j

Whirling orbit of Journal Centre

CH 0I1

Fig. 1. Configuration of the journal bearing showing the whirling orbit of the journal centre

p

p0 p1H 1e iOW p2H 0I1e iOW ; h

kT

kT 0 kT 1H 1e iOW

where, h0 1 H 0 cosT ; H

kT 0 12 aT Re h0

h0 H 1e iOW cos T H 0I1e iOW sin T ½ ° ¾ kT 2H 0I1e iOW ; k z k z 0 k z1H 1e iOW k z 2H 0I1e iOW ° ¿

bT

(2)

H 0 H1eiOW ; I I0 I1eiOW and O Z :.

;

k z 0 12 c z Re h0

(3a)

dz

(3b)

bT ½ a b cos T Re h bT ½ h ; (3c) kT 2 ® T T ¾ 0 ®aT bT sin T Re h0 ¾ h0 0 ¯ ¿ ¯ ¿ dz ½ dz ½ (3d) k z1 k z2 ®c z d z cos T Re h0 ¾ h0 ; ®c z d z sin T Re h0 ¾ h0 . ¯ ¿ ¯ ¿ Substituting equation (2) into equation (1) and collecting the zeroth and first order terms of H1 and H 0I1 gives the following set of equations in p 0 , p1 and p 2 :

kT 1

w wT

2 ª w p0 º § D · w ª w p0 º ) h , l , N ¨ ¸ « T0 0 m » «) z 0 h0 , l m , N » wT »¼ © L ¹ w z «¬ w z »¼ ¬«

w wT

2 ª w p0 º w p1 º § D · w 2 p1 w ª c » «)T 0 h0 , l m , N cosT «)T 0 h0 , l m , N » ¨ ¸ ) z 0 h0 , l m , N 2 T L T w wT »¼ w »¼ © ¹ «¬ wz ¬«

2

w 2 p0 §D· ¨ ¸ ) z 0c h0 , l m , N cosT 2 ©L¹ wz w wT

1 w h0 2 wT

(4)

2 ª w p0 º w p2 º § D · w 2 p2 w ª c » «)T 0 h0 , lm , N sin T «)T 0 h0 , lm , N » ¨ ¸ ) z 0 h0 , lm , N 2 T T L w w wT »¼ »¼ © ¹ «¬ wz ¬« 2

w 2 p0 §D· ¨ ¸ ) z 0c h0 , lm , N sin T 2 ©L¹ wz

)T 0 h0 , l m , N

3

(5)

1 sin T iO cosT 2

§ 1 w h0 ·¸ 1 cosT iO ¨¨ sin T ¸ H 2 0 wT ¹ ©

(6)

2

§ h l N · h0 h 1 h0 N ¸ 0 coth¨ 0 m ¨ ¸ 2 kT 0 l m 2 2 l m © ¹

(7a)

197


) z 0 h0 , l m , N

) T 0c h0 , l m , N

2

3

§ h l N · h0 h 1 h0 N ¸ 02 coth¨ 0 m ¨ ¸ 2 lm 2 k z 0 lm © ¹

2

h0 kT 0

aT bT Re h0 ° ®3 kT 0 °¯

bT

(7b)

2 ½ 1 § h l N · N 2 h0 § h l N · h0 N ° ¸ ¸ coth¨ 0 m cosech 2 ¨ 0 m ¾ 2 ¨ ¸ ¨ ¸ lm 2 4 2 °¿ l m © ¹ © ¹

(7c)

dz ½ 2 2 § h l N · N 2 h0 § h l N · h0 ° c z d z Re h0 ° 1 h0 N ¸ ¸ (7d) coth¨ 0 m cosech 2 ¨ 0 m ®3 ¾ 2 ¨ ¸ ¨ ¸ 2 4 2 k z0 ° k z0 lm lm ° © ¹ © ¹ ¯ ¿ Equations (4) to (6) are discretised using central finite difference method and solved with Gauss-Seidel iterative technique using successive over-relaxation scheme satisfying the following boundary conditions to obtain the dynamic film pressure distribution.

) z 0c h0 , l m , N

p i (T , z )

0, i

0, 1, 2 at z

1 ;

w pi w z

0, i

0, 1, 2 at z

0;

p 0 (T c , z ) w p 0 (T c , z ) wT

0.

2.2. Stiffness and Damping coefficients With the dynamic pressure fields known, the non-dimensional components of stiffness and damping coefficients are obtained as follows: § 1 Tc § 1 Tc · · S rr Re al¨¨ p1 cosT dT d z ¸¸ ; S Ir Re al¨¨ p1 sin T dT d z ¸¸ (8a) ©0 0 ©0 0 ¹ ¹

³³

D rr

³³

§ 1 Tc · Im aginary ¨¨ p1 cosT dT d z ¸¸ O ; ©0 0 ¹

³³

DIr

§ 1 Tc · Im aginary ¨¨ p1 sin T dT d z ¸¸ O ©0 0 ¹

³³

(8b)

where, S ij

2S ij C 3 P:R 3 L and D ij

2 Dij C 3 P:R 3 L

The other four stiffness and damping coefficients S II , S rI , DII and D rI can be represented by analogy. 2.3. Stability characteristics The stability of the journal is analysed by combining the equations of motion and the resultant film forces Fr and FI in r and I directions respectively. Referring to Fig. 1, the equation of motion of the rigid journal, assuming the rotor to be rigid, can be written as 2 ª d 2H ª d 2I dH dI º § dI · º FI W0 sin I MC «H 2 2. . » 0 (9) Fr W0 cos I MC « 2 H ¨ ¸ » 0 ; dt dt ¼» dt © ¹ »¼ «¬ dt ¬« dt For steady-state condition the equation of motion is written as FI 0 W0 sin I0 0 Fr 0 W0 cosI0 0 ;

(10)

Using the equations (2), (9) and (10), the following equations result in ª º 1 W0 MW 0 D rr cosI0 DIr sin I0 » « D rr S II S rr DII D rI S Ir S rI DIr 2 H O D rr DII ¬« 0 ¼»

(11)

198


M W O 0

2 4

ª º § W 0 cos I0 · «M W 0 ¨ S rr S II ¸ D rr DII D rI DIr » O2 ¨ ¸ H0 »¼ © ¹ ¬«

S rr S II S rI S Ir

W0

H0

S

rr

cos I0 S Ir sin I0

(12)

0

From the known values of stiffness and damping coefficients, the values of critical mass parameter and whirl ratio can be obtained by solving the above equations. 3. Results and discussions: A parametric study has been carried out to theoretically analyse the dynamic response coefficients of the fluid film and the results have been exhibited in Figs 2 and 3. In the present analysis, the inclusion of two non-dimensional parameters, viz. lm and N 2 imposes the condition of micropolar lubrication. When N 2 → 0 or lm→ ∞ the micropolar effect becomes insignificant and the lubricant behaves as Newtonian fluid. Fig 2 depicts the variation of stiffness coefficients with l m for different values of Reynolds number. It is observed that the magnitude of both direct and cross stiffness coefficients increase with increase in Reynolds number. This effect is more pronounced at lower values of l m i.e., when the lubricant behaves as micropolar fluid. The micropolar effect enhances the effective viscosity of the lubricant by either increasing the material characteristic length or by decreasing the radial clearance. In case of journal bearings the radial clearance is very small. Hence film pressures increase with decrease in l m which results in higher values of the stiffness coefficients, for particular values of Re and N.

a

15

36

Laminar Re=4000 Re=8000

30

Re=2000 Re=6000 Re=12000

b

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4


12

Re=2000 Re=6000 Re=12000

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

24 9

Srr

Sφφ

18

6 12 3

6

0

0 10

c

20

30

lm

40

50

35


30

Re=2000 Re=6000 Re=12000

60

10

70

d

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

20

lm

40

50

60

70

40


35 30

25

30

Re=2000 Re=6000 Re=12000

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

-Sφr

Srφ

25 20 15 10

20 15 10

5

5

0

0 10

20

30

40

lm

50

60

70

10

20

30

40

lm

Fig. 2. Variation of Stiffness coefficients with l m for different values of Re. (a) S rr Vs. l m ; (b) S II Vs. l m ; (c) S rI Vs. l m ; (d) S Ir Vs. l m

50

60

70

199


a

100


80

Re=2000 Re=6000 Re=12000

b

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

140


120

Re=2000 Re=6000 Re=12000

100 60

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

Dφφ

Drr

80

40

60 40

20 20 0

0 10

c

20

lm

40

50

60


Re=2000 Re=6000 Re=12000

d

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

20

10

70

60 50

30

40

lm

50

60

70

30


25

40

20

30

15

Re=2000 Re=6000 Re=12000

L/D = 1.0 N 2 = 0.3 ε 0 = 0.4

-Dφr

-Drφ

30

20

10

10

5

0

0 10

20

30

40

50

60

70

20

10

lm

30

40

50

60

70

lm

Fig. 3. Variation of Damping coefficients with l m for different values of Re. (a) D rr Vs. l m ; (b) DII Vs. l m ; (c) D rI Vs. l m ; (d) D Ir Vs. l m

Since the values of turbulent shear coefficients are greater than 12, turbulence is effectively equivalent to operating with a lubricant of higher viscosity. As a result the dynamic film pressures and consequently the stiffness coefficients increase with Reynolds number for all l m values. The variation of damping coefficients with l m for different values of Reynolds number is presented in Fig 3. It is noted that the magnitude of both direct and cross damping coefficients increase with increase in Reynolds number. This effect is found to be more prominent for micropolar fluids (lm→ 0). The reason for such enhanced components of damping coefficients at lower values of l m and at higher values of Re is same as that mentioned in case of stiffness coefficients. The significance of the present analysis is that the values of the response coefficients are used to compute the stability parameters in terms of critical mass parameter and whirl ratio, which in turn are used to predict the stability of the journal bearing system operating in turbulent regime. Table 1 shows the values of stability parameters, i.e., Critical mass parameter and whirl ratio. The table suggests that the effect of turbulence is to reduce the stability of journal bearing system as the values of Critical mass parameter decreases and whirl ratio increases with increase in Reynolds number. Table 1. Values of Stability Parameters at L/D = 1.0, lm = 20.0, N2 = 0.3 and ε0 = 0.4 Flow Conditions

Laminar

Re = 2000

Re = 4000

Re = 6000

Re = 8000

Re = 12000

M

7.52298

6.40429

5.925011

5.80416

5.40849

5.05960

O

0.53786

0.54457

0.54420

0.55610

0.56984

0.58191

Since, the results on the stability parameters for finite journal bearing lubricated with micropolar fluid under turbulent regime are not available in the literature so far, the validation of results of the present analysis is exhibited

200


in Table 2 by comparing the values of the stability parameters using laminar flow conditions with those obtained by Das et al [11]. The comparison shows that the values of the stability parameters follow similar trends in both works. Table 2. Values of Stability Parameters at L/D = 1.0, lm = 20.0 and ε0 = 0.5

N2

0.1

0.3

0.5

0.7

M

7.51947+

7.64593+

7.93225+

8.11943+

7.93#

8.17#

8.39#

8.62#

O

+

#

+

0.54102

0.54099

0.54081

#

#

#

0.5265 +

+

0.5251

0.5244

0.54062+ 0.5235#

Results obtained in the present analysis Results obtained by Das et al [11]

4. Conclusions: From the present analysis the following conclusions can be drawn: x The effect of the Reynolds number is to increase the magnitudes of the dynamic response coefficients. x The critical mass parameter decrease with increase in Reynolds number. Such trend of variation is not observed in case of whirl ratio. x The effect of turbulence is more pronounced in case of micropolar fluid compared to that in Newtonian fluid. References [1] Constantinescu, VN. Analysis of Bearings Operating in Turbulent Regime. J Basic Eng Trans ASME 1962; 84(1): 139-151. [2] Constantinescu, VN. Theory of turbulent lubrication. Int Symp Lubr Wear., Houston; 1964: 159. [3] Constantinescu, VN. Lubrication in the turbulent regime. AEC-tr-6959, U. S. Atomic Energy Commission, Division of Technical Information; 1968. [4] Ng, CW and Pan, CHT. A linearized turbulent lubrication theory. J Basic Eng Trans ASME 1965; 675-688. [5] Elrod, HG and Ng, CW. A theory of turbulent fluid films and its application to bearings. J Lubr Technol Trans ASME 1967; 89(3): 356-363. [6] Taylor, CM and Dawson, D. Turbulent lubrication theory- Application to design. J Lubr Technol Trans ASME 1974; 96: 36-46. [7] Eringen, A. Theory of micropolar fluids. J Math Mech 1966; 16: 1-18. [8] Allen, S and Kline, K.. Lubrication theory of micropolar fluids. J Appl Mech 1971; 38( 3): 646-650. [9] Prakash, J and Sinha, P. Lubrication theory of micropolar fluids and its application to a journal bearing. Int J Eng Sci 1975; 13: 217-232. [10] Tipei, N. Lubrication with micropolar fluids and its application to short bearings. J Lubr Technol Trans ASME 1979; 101: 356-363. [11] Das, S, Guha, SK and Chattopadhyay, AK. Linear stability analysis of hydrodynamic journal bearings under micropolar lubrication. Tribol Int 2005; 38: 500-507. [12] Gautam, SS and Samanta, S. Analysis of short bearing in turbulent regime considering micropolar lubrication. WASET 2012; 68: 14001405. [13] Faralli, M and Belfiore, NP. Steady-state analysis of worn spherical bearing operating in turbulent regime with non-Newtonian lubricants Int Conf Tribol. AITC – AIT. Parma. Italy 2006. [14] Das, S and Guha, SK. On the Steady-state Performance Characteristics of Finite Hydrodynamic Journal Bearing under Micro-Polar Lubrication with Turbulent Effect. WASET 2013; 7(4): 18-24. [15] Rana, NK, Gautam, SS and Samanta, S. Approximate Analysis of Dynamic Characteristics of Short Journal Bearings in Turbulent Micropolar Lubrication. J Inst Eng India Ser C 2014; 95(4): 383-388.