From a compressible fluyid model to new mass conserving cavitation algorithms

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Submitted on 6 Jun 2013

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From a compressible fluyid model to new mass

conserving cavitation algorithms

Guy Bayada

To cite this version:

June 6 2013

From a compressible fluid model to

New mass conserving cavitation algorithms

G. BAYADA

Institut Camille Jordan ,UMR CNRS 5208, Université de Lyon Mathématiques -INSA de Lyon

F 69621Villeurbanne Cedex Tel 33472438312 Fax 33472438529

Jakobson.Floberg.Olsson /Elrod.Adams mass conserving cavitation model is very popular ,

however numerically difficult to manage.

The main idea of this paper is that the J.F.O./E.A. model can be viewed as a compressible

Reynolds equation with a specific densitypressure law. Approximation of this pressure

-density law originating from a 3-dimensional model of vaporous cavitation has been

proposed. This physically justified approximation covers the wide range of the pressure

including the liquid-vapor transition. New mass conserving algorithms are proposed . One of

them is implicit. Main advantage of this implicit algorithm is that it can be implemented very

easily: exactly as the well known Christopherson algorithm. Moreover it can be extended to

unsteady cavitation problems without difficulties.

Various numerical examples are given to

show the stability of this kind of approximation as well as the great diminution of time

computation by using some over relaxation parameter.

Abstract:

relation is obtained from a barotropic-isentropic assumption. It can be viewed as an

approximation of the Jakobson-Floberg-Olsson /Elrod Adams cavitation model. Two

algorithms are proposed to solve it. The first one is explicit and needs a important number of

nodes. The second is implicit and can be used for steady-state and unsteady problems. Its

implementation is easy and needs only minor modifications for a computer code in which

cavitation is ignored. It can also be used to compute the solution of the usual J.F.O./E.A.

model. Faster convergence is obtain using a relaxation parameter.

123456789AB376573CDEFA6FDF5CA5E6288F2A23C578A2DF5CADFDF5CAD56FBE

1

1

Highlights :

A new cavitation model as an approximation of usual

Jakobsson-Floberg-Olsson/Elrod-Adams model

Algorithm easily implemented for finite difference and finite element discretization

Algorithm available for both steady state and unsteady problems

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≥

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cav

,

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≤

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≤

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,

(

p

−

p

cav

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−

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)

=

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The Reynolds equation is written for a model device with a small gap h(x

1

,x

2

) in which the

upper part is fixed and the flat lower part has a constant velocity u along the x

1

-main

direction. Neglecting variation of the temperature, viscosity

µ

and density

ρ

are assumed to be

only function of the pressure.

t

h

p

(

+

x

h

p

(

u

=

)

x

p

p

1

2

h

(

x

)

x

p

p

1

2

h

(

x

∂

∂

∂

∂

∂

∂

∂

∂

+

∂

∂

∂

∂

(

)

)

(

)

)

2

/

)

(

)

(

12

)

(

)

(

12

2 1 3 2 1 3 1

ρ

ρ

ρ

ρ

(2)

Although this equation is written for transient situations, it will firstly be considered for a

steady state situation, so neglecting the time derivative term. Dynamic aspects will be treated

in section 5 .

one for pure liquid and one of mixture. The input data are: the velocities of the sound, c

v

and

c

l

, the density

ρ

v

and

ρ

l

and the viscosities

µ

v

and

µ

l

in each of the pure regimes. From these

data, it is possible to compute the transition pressure between these 3 regimes: p

vm

between

vapour and mixture and p

sat

between mixture and liquid:

) 3 ( with ) 3 ( log ) 3 ( 2 2 2 2 2 2 2 2 2 2 2 2 c c 1 c 1 ) 1 (1 c 1 c 1 = N b ) c 1 c 1 ( N c 1 = p a c 1 = p l l v v l v l l v v l l v v v v sat v v vm − − −

It is convenient to introduce the void fraction

α

defined by:

α

=(

ρ

−

ρ

_l)/(

ρ

_v−

ρ

_l)

Pressure-density relations are:

(4c)

1

log

)

(

(4b)

)

(

)

(4a)

)

2 2 2 l v l l v v l v v sat l l 2 l sat v 2 v

if

)

3

)

c

1

+

3

)

(

c

(1

1

1

c

1

(

N

P

p

if

1

1

c

+

P

=

p(3

if

1

c

=

p(3

ρ

ρ

ρ

α

ρ

ρ

ρ

ρ

<

<

−

+

=

>

−

<

In the mixture region, viscosity-density relation can be chosen as:

M132=3 1v

113 2

(7)

A comparison of the present “full compressible” law with the one issued from the E.A./JFO

model is given in figure (1). It is clear that the graph of the present law can be very close to

the one of J.F.O./E.A. and so are also the pressures. This is especially true if the value p

cav

used in the J.F.O./E.A model is chosen close to p

sat

. In other words; the new model can be

viewed as an approximation of the J.F.O./E.A model. This assertion will be numerically

studied in the following section 4.

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7 Conclusion: The fully compressible model recently proposed to deal with vaporous

cavitation is very close to the usual mass flow conservation Jakobson-Floberg-Olsson/Elrod-

Adams model. However it is easier to solve due to the fact that positivity of the pressure is

not required. Two algorithms are proposed for this new model. A first one which is based on a

sequence of resolutions of linear systems requires a relatively great number of nodes to

easy to implement. It is very similar to the well-known Christopherson algorithm with a

Gauss-Seidel over relaxation parameter. The most important feature is that the non linear

discrete equation can be exactly solved at each iteration. Introduced first for steady state

problem, it is shown how it can be extended to unsteady problems. Moreover it can be also

used to solve the J.F.O./E.A. cavitation model by approximating the J.F.O./E.A

density-pressure curve by a linear approximation. The time computation can be greatly reduced by

using a successive- over-relaxation procedure.

2 2 2 2 2 2

[1] Dowson D , Taylor CM. Fundamental aspects of cavitation in bearings,” in Cavitation and

related phenomena in lubrication, Mechanical Engineering Publications for the Institute of

Tribology, The University of Leeds,1975; pp. 15-25.

[2] Christopherson DG. A new mathematical method for the solution of film lubrication

problem. Inst. Mech. Eng. Proc. 1941;146:126-135.

[3] Jakobsson B, Floberg L. The finite journal bearing considering vaporization. Göteborg:

Transactions of the Chalmers University Technology 190; 1957.

[4] Elrod HG, Adams L. A Computer program for Cavitation and Starvation. In: Dowson,

Engineering Publications for the Institute of Tribology, The University of Leeds; 1975, p.

37-41.

[5] Etsion L, Ludwig LP. Observation of Pressure variation in the Cavitation region of

submerged Journal bearings. Journal of Lubrication Technology 1982; 104: 157-163.

[6] Braun MJ, Hendricks RC. An experimental Investigation of the Vaporous/Gaseous Cavity

Characteristics of an Eccentric Journal Bearing, ASLE Transactions 1983; 27(1):1-14.

[7] Bayada G, Chambat M. Analysis of a free boundary problem in partial lubrication.

Quartely Applied Math. 1983; 40(4):369-375.

[8] Vijayaraghavan D, Keith Jr. An efficient, robust and time accurate Numerical Scheme

applied to a cavitation algorithm. Journal of Tribology 1990; 112: 42-51.

[9]Kumar A, Booker JF. A finite Element cavitation Algorithm: Application and Validation.

Journal of Tribology 1991;113: 255-261.2

[10] Vincent B, Maspeyrot P, Frêne J. Starvation and cavitation effects in finite journal

bearings. In : D. Dowson et al editors. Leeds Lyon Symposium on Lubricants and

Lubrification, Elsevier Science Tribology series, 30;1995, p. 455-464.

[11] Bayada G, Chambat M, Vázquez C. Characteristics method for the formulation and

computation of a free boundary cavitation problem. Journal Computing and Applied Math.

1988; 98 (2):191–212.

[13]2197B9C92252)BDD25!2G9C92G222AB$$9557957257.6BC89C#26BDCA79A2

)BED7A9BC2AB22AE2;EF97CA2)9D29C2A2@C2B26789A7A9BC950BEC7D22B2!9FBDB#2

3>1>M21432?>3.?15

[14] Chupin L, Sart R. Compressible flows: New existence results and justification of the

Reynolds asymptotic in thin film. Asymptotic Analysis 2012;76(3-4):193-231.

[15

] Bayada G, Chupin L. Compressible fluid model for hydrodynamic lubrication cavitation.

Journal of Tribology 2013;To appear. J.F.O./E.A

[16] Sahlin F, Almquist A, Larsson R, Glavatskih S. A cavitation algorithm for arbitrary

lubricant compressibility. Tribology International 2007;40:1294-1300.

[17] Ausas R, Jai M, Buscaglia GC. A mass-conserving algorithm for dynamical lubrication

problems with cavitation. Journal of Tribology 2009;131:031702-1.

[18] Ausas R, Ragot P, Leiva J, Jai M, Bayada G, Buscaglia GC. The impact of the cavitation

model in the analysis of a microtextured lubricated journal bearing. Journal of Tribology

2007;129(3):868-875.

[19] Optasanu V, Bonneau D. Finite Element mass-conserving cavitation algorithm in pure

squeeze motion: Validation/application to a connecting-rod small end bearing. Journal of

Tribology 1986;122: 162-169.

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