Franc J-P. Fundamentals of Cavitation

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1 - INTRODUCTION – THE MAIN FEATURES OF CAVITATING FLOWS 11

where cavitation inception is expected. If T is the operating temperature of the liquid

and Dp a pressure difference that characterizes the system, the cavitation number

(also called cavitation parameter, or T

HOMA cavitation number) is defined by:

ppT

- ()

(1.2)

For example:

— in the case of a gate:

downstream v

upstream downstream

ppT

()

(1.3)

— for a foil placed at a submersion depth h in a horizontal free surface channel

where the pressure on the surface is p

and the flow velocity U:

pghpT

()

(1.4)

— for a pump:

inlet v

ppT

- ()

(1.5)

where V

is the velocity at the periphery of the runner.

It should be noted that the cavitation number is defined using dynamical parameters

and not geometrical ones.

In a non-cavitating flow, this non-dimensional parameter cannot be considered as a

scaling parameter. The difference

has no physical significance for a single

phase flow as it cannot be obtained by integration of the pressure gradient along

a real path. The cavitation parameter becomes a similarity parameter only at

cavitation inception.

1.4.2. CAVITATION NUMBER AT INCEPTION,

The number s

is the value of the parameter s

corresponding to cavitation

inception at any point of the flow system. Cavitation appears because of either

a decrease in pressure at the reference point (i.e., the ambient pressure) or an

increase in the Dp-value. For experimental convenience (in particular for improved

repeatability), the number s

vd,

corresponding to cavitation disappearance from an

initial regime of developed cavitation, may also be used.

Operation in non-cavitating conditions requires that :

vvi

(1.6)

FUNDAMENTALS OF CAVITATION12

The threshold s

depends on all the usual factors considered in fluid mechanics

such as flow geometry, viscosity, gravity, surface tension, turbulence levels, thermal

parameters, wall roughness and the gas content of the liquid in terms of dissolved

and free gases (i.e., gas nuclei, see chap. 2).

In general, the smaller the value of s

for a given system, the better behaved is the

flow. For example, for the flow around a 10 mm diameter circular cylinder, the

cavitation inception number is about 1.5, whereas for elliptical cylinders at zero

incidence, with a chord length of 80 mm and axis ratios of 1/4 and 1/8, the

-values are 0.45 and 0.20, respectively.

When s

becomes smaller than s

, cavitation usually becomes increasingly

developed. Very exceptionally, it may happen that after initial development,

cavitation finally disappears as the consequence of a further lowering of s

(see

chap. 6 and 8).

In many circumstances, particularly for the numerical modeling of cavitating flows,

the following estimate is taken for s

Cp=-

min

(1.7)

where Cp

min

is the minimum pressure coefficient, which is normally negative. The

pressure coefficient Cp at a point M is defined by the relation:

(1.8)

In this expression, p

is the absolute pressure at the reference point, as in

relation (1.2). Two assumptions lie behind equation (1.7). First, cavitation occurs

at the point of minimum pressure and second, the pressure threshold value is that

of the vapor pressure. Clearly, these assumptions may be over-restrictive. Thus,

the estimate (1.7) must be considered cautiously.

1.4.3. RELATIVE UNDERPRESSURE OF A CAVITY,

If a developed cavity is attached to the low-pressure surface of a blade, or if a large

number of bubbles are present, the pressure in the region covered by the cavity is

uniform. Referring to this pressure as p

, a non-dimensional parameter known as

the relative underpressure of the cavity is defined as:

(1.9)

It is a true scaling parameter, as the numerator expresses an actual pressure

difference inside the flow domain. This number is extensively used in the numerical

modeling of flows with developed cavities and plays an important role in cavity

dynamics. It is often, but improperly, referred to as the cavitation number.

1 - INTRODUCTION – THE MAIN FEATURES OF CAVITATING FLOWS 13

Usually, the pressure p

in the cavity is the sum of two components: the vapor

pressure p

and a partial pressure p

due to the presence of non-condensable gas

inside the cavity. If this last term is negligible, the relative underpressure of the

cavity s

becomes equal to the cavitation parameter s

, which probably explains

the confusion referred to above.

1.5. SOME HISTORICAL ASPECTS

The word "cavitation" appeared in English scientific literature at the end of the

nineteenth century. It seems that the problem of cavitation in rotating machinery

handling liquids was identified by T

ORRICELLI, and later by EULER and NEWTON. In

the middle of the nineteenth century, D

ONNY and BERTHELOT measured the

cohesion of liquids. The negative effect of cavitation on the performance of a ship

propeller was first noted by P

ARSONS (1893), who built the first cavitation tunnel.

The cavitation number was introduced by T

HOMA and LEROUX around the years

1923-1925.

Subsequently, many experiments were carried out to study the physical aspects of

the phenomenon and to examine its effects on industrial systems. Theoretical and

numerical approaches were widely used. There were two main fields of research.

The first focused on bubble dynamics [R

AYLEIGH 1917, LAMB 1923, COLE 1948,

LAKE 1949, PLESSET 1949]. The simplicity of the spherical shape made their studies

(either theoretical or experimental) relatively easy. A large amount of work has

been published on bubble dynamics.

The second field was related to developed cavities or supercavities and was based

on the old wake theory [H

ELMHOLTZ 1868, KIRCHHOFF 1869, LEVI-CIVITA 1907,

ILLAT 1913, RIABOUCHINSKI 1920

]. This theory considers wakes as regions of

uniform pressure, limited by surfaces on which the tangential velocity is not

continuous. It is more suited to cavitating wakes than to single phase wakes. Later,

ULIN (1953) and WU (1956) made use of linearization procedures to adapt the

theory to the case of slender bodies such as wings and blades.

Vortical cavitation was only considered more recently, in particular by G

ENOUX

and CHAHINE (1983) and by LIGNEUL (1989), who studied the cavitating torus and

tip vortices, respectively.

3. Those references can be found in JACOB's book: Introduction mathématique à la mécanique

des fluides.

FUNDAMENTALS OF CAVITATION14

REFERENCES

BIRKHOFF G. & ZARANTONELLO E.H. –1957– Jets, wakes and cavities.

Academic Press Inc.

LAKE F.G. -1949- The tensile strength of liquids: a review of the literature.

Harvard Acoustics Res. Lab. TM 9, June.

OLE R.H. -1948- Underwater explosions. Princeton University Press.

ENOUX P. & CHAHINE G.L. -1983- Équilibre statique et dynamique d'un tore de

vapeur tourbillonnaire. J. Méc. Théor. Appl. 2(5), 829-857.

ACOB C. -1959- Introduction mathématique à la mécanique des fluides.

Gauthier-Villars Ed.

KNAPP R.T., DAILY J.W. & HAMMITT F.G. –1970– Cavitation. McGraw-Hill Book

Company Ed., 578 p.

AMB H. -1923- The early stages of a submarine explosion. Phil. Mag. 45, 257 sq.

ESIEUR M. -1998- Vorticity and pressure distributions in numerical simulations

of turbulent shear flows. Proc. 3

Int. Symp. on Cavitation, vol. 1,

Grenoble (France), April 7-9, 9-18.

EVKOVSKY Y.L. –1978– Structure of cavitating flows. Sudostroenie Publishing House,

Leningrad (Russia), 222 p.

IGNEUL P. -1989- Theoretical tip vortex cavitation inception threshold.

Eur. J. Mech. B/Fluids 8, 495-521.

ERNIK A.D. –1966– Problems of cavitation. Sudostroenie Publishing House,

Leningrad (Russia), 439 p.

LESSET M.S. -1949- The dynamics of cavitation bubbles. J. Appl. Mech. 16, 277 sq.

AYLEIGH (Lord) -1917- The pressure developed in a liquid during the collapse of

a spherical cavity. Phil. Mag. 34, 94 sq.

OZHDESTVENSKY V.V. –1977– Cavitation. Sudostroenie Publishing House,

Leningrad (Russia), 247 p.

TULIN M.P. -1953- Steady two-dimensional cavity flows about slender bodies.

DTMB, Rpt 834.

U T.Y.T. -1956- A free streamline theory for two-dimensional fully cavitated

hydrofoils. J. Math. Phys. 35, 236-265.

2. NUCLEI AND CAVITATION

2.1. INTRODUCTION

2.1.1. LIQUID TENSION

In chapter 1, possible differences between the actual value of the cavitation

threshold and the vapor pressure were discussed. In real flows as in laboratory

flows, liquids can actually sustain absolute pressures lower than the vapor

pressure at the operating temperature and even negative pressures, i.e. tensions.

To explain these discrepancies, one must first refer to the classical data relating to

liquid breakdown. In the nineteenth century [D

ONNY 1846, BERTHELOT 1850,

EYNOLDS 1882], experiments demonstrated that a liquid at rest could sustain

negative pressures without vaporization occurring. For water, the values were of

the order of several tens of bars. More recent experiments [T

EMPERLEY 1946, BRIGGS

1950, REES & TREVENA 1966] have shown that the experimental values are rather

scattered (for example, B

RIGGS obtained 277 bars). They depend on the experimental

procedure, the preliminary treatment of the liquid (for example, degassing or

pressurization over a long period) and the degree of cleanliness of the container

wall. It is often not clear whether the limit corresponds to a loss of cohesion in the

bulk liquid or a loss of adhesion of the liquid to the walls.

These experimental values are lower than the estimates calculated from theoretical

models. For example, considering a microscopic bubble with a diameter of the order

of the intermolecular distance d

, the expression

Sd/

(S is the liquid surface

tension) gives a value of 7,000 bars (with

01ª .nm

, S = 0.072 N/m for water).

This estimate is obviously open to criticism, as such a very small size is not

compatible with the assumption of a continuum. If the molecular nature of the

liquid is then taken into account, together with oscillations due to temperature

fluctuations, the limiting tension is reduced by about one order of magnitude.

Another estimate can be derived from the V

AN DER WAALS equation (see § 1.1.2,

fig. 1.2). The ordinate of the minimum M on the isotherm can be considered a

limiting value of the tension sustained by a liquid. In the case of water, it is about

500 bars at room temperature.

2.1.2. CAVITATION NUCLEI

The differences observed with respect to the vapor pressure p

(T) in typical

experiments on cavitation are much smaller than the experimental results and

FUNDAMENTALS OF CAVITATION16

theoretical estimates mentioned above. They don’t usually exceed a few bars at

most for tap water. Thus, for liquids currently used in industry, the existence of

points of weakness in the liquid continuum is to be expected. Those points are

formed by small gas and vapor inclusions and operate as starting points for the

liquid breakdown. They are known as cavitation nuclei. Numerous experiments

show that those nuclei actually exist. Their size is between a few micrometers and

some hundreds of micrometers. They remain spherical at this scale due to surface

tension. They can be referred to as microbubbles.

The assumption of heterogeneities inside a homogeneous medium in order to

explain phase changes is common in thermophysics, for example in boiling,

condensation, and solidification. Nuclei also proved to be the origin of great

differences in cavitation inception found in the past when tests on similar bodies

were made in different facilities.

Various questions arise concerning nuclei: How do they appear? Are they stable?

What is their effect on liquid cohesion and then on conditions of cavitation

inception? How can they be measured? How can the nucleus content of a liquid be

characterized?

Nuclei are present either on walls or in the liquid bulk. Surface nuclei consist of gas

trapped in small wall crevices that are not filled with the liquid (see § 2.3.2). The

wetting capacity of the liquid is therefore of great importance. It is possible that

bulk nuclei are produced by cosmic rays, i.e., by a mechanism of energy deposition

similar to the one used in bubble chambers for the experimental study of atomic

particles. Another example of micronic bubble production by energy deposition is

found in the breakdown of insulating liquids subjected to high voltage [A

ITKEN et al.

1996, J

OMNI et al. 1999]. However, the most efficient way to produce nuclei is the

reduction in pressure of a saturated liquid. Regions downstream of developed

cavities may also be an abundant source of nuclei, as it will be seen in chapter 7.

Once present, nuclei evolve under two main influences. First, free nuclei (i.e., not

attached to a wall) rise due to gravity. Second, all nuclei exchange gas via diffusion

with the dissolved gas present in the surrounding liquid. In general, as the mass

diffusion coefficients are very small, the diffusion process is slow and typical

diffusion times are long, of the order of a second (see § 2.3.2). This is a large value

in comparison with the time necessary for bubble collapse, which typically takes

milliseconds (see chap. 3). Thus, in the following section, mechanical equilibrium

of a spherical nucleus is assumed and the mass of enclosed gas is supposed

constant. The problem of gas diffusion will be considered at the end of the chapter

as it concerns the stability of gas nuclei over a long period.

The void fraction resulting from the presence of free nuclei is extremely small. For

example, for a concentration of one hundred nuclei per cubic centimeter (this value

is actually rather high) with a diameter of 0.1 mm, the void fraction is 0.52

¥ 10

–4

Thus, the liquid density remains practically unchanged. The same conclusion holds

for the velocity of sound in the liquid.

2 - NUCLEI AND CAVITATION 17

2.2. EQUILIBRIUM OF A NUCLEUS

2.2.1. EQUILIBRIUM CONDITION [BLAKE 1949]

Consider a spherical microbubble, containing gas and vapor, in equilibrium within

a liquid at rest. The liquid is supposed capable of withstanding pressures below

the vapor pressure p

and even tensions. Its metastable state can be represented by

a point on the branch AM of the V

AN DER WAALS curve in figure 1.2. The microbubble

is the point of weakness from which the breakdown of the liquid medium can begin.

2.1

Microbubble in a liquid

The bubble radius R is considered sufficiently

small for the difference in hydrostatic pressure

2r gR to be negligible compared to the pressure

difference corresponding to the surface tension

2S R/

. This condition requires that R be smaller

than the limiting value

(S/ g)r

, namely 2.7 mm

for water. It is fulfilled in this case since micro-

microbubbles of diameter smaller than about 0.5 mm only are considered. The

pressure can then be considered uniform in the bulk of the surrounding liquid

(expressed as p

•

) and the microbubble is actually spherical.

The equilibrium of the interface requires the following condition to be satisfied:

ppp

gv•

=+-

(2.1)

in which p

is the partial pressure of the gas inside the bubble, S the surface tension

and R the radius.

It is assumed that the pressure changes slowly so that mechanical equilibrium is

still satisfied and that heat transfer between gas and liquid is possible. However,

the change in pressure must be rapid enough to ensure that gas diffusion at the

interface is negligible. In other words, the transformation is assumed isothermal

and the mass of gas constant.

For the initial state, denoted by subscript 0, equation (2.1) is written:

ppp

gv•

=+-

(2.2)

∞

FUNDAMENTALS OF CAVITATION18

As the gas pressure is inversely proportional to the volume in an isothermal

transformation, then from equation (2.1) one obtains:

gv•

(2.3)

The curve p

•

(R) is shown in figure 2.2. The two mechanisms considered in

equation (2.1), i.e.,

— the internal pressure effect, which tends to increase the bubble size, and

— the surface tension effect, which acts in the opposite direction, result in the

existence of a minimum given by:

(2.4)

The corresponding locus is also represented in figure 2.2.

For a given liquid and a given gas at a fixed temperature, a nucleus is characterized

by the mass of gas it contains, which is proportional to the quantity p

. To define

a nucleus with a constant mass of gas, one can use either one of the doublets

•

), (R

), etc. or preferably one of the quantities R

or p

2.2.2. STABILITY AND CRITICAL PRESSURE OF A NUCLEUS

Consider another nucleus containing more gas and having a radius

RR'

under

pressure p

•0

. The equilibrium curve of this nucleus is shown in figure 2.2. The

corresponding gas pressure p'

is smaller than p

, as seen in equation (2.2).

2.2

Equilibrium of a spherical nucleus

Suppose a virtual transformation is

carried out resulting in the radius

of the first nucleus becoming R'

that the surface tension terms are

identical. In its new state, the first

nucleus has a gas pressure given

pRR

g0 0 0

(/')

. This is lower

than p'

as can be seen from a

comparison of the mass of gas

enclosed in each nucleus. This is

locus of minima

∞

∞0

2 - NUCLEI AND CAVITATION 19

proportional to p

. Thus, the balance of forces tends to return the first nucleus

to its initial state. Hence, the mechanical equilibrium of the spherical nucleus is

stable on the branch of the curve that has a negative slope. It is straightforward to

check that the equilibrium is unstable on the other branch.

The minimum pressure, p

, that the nucleus can withstand under stable conditions

is a limiting value referred to as the critical pressure. The difference

is the

static delay to cavitation. Expression (2.4) suggests that the smaller the nucleus, the

greater the delay.

Figure 2.3 gives another representation of the equilibrium curves, limited to the

stable domain, in the case of air nuclei in water. It is based on the following

equation:

v•

(/)

()

obtained by combination of equations (2.3) and (2.4). The evolution of a given

nucleus, characterized by its critical pressure, under varying external pressures, can

be followed on a vertical line. On this diagram, the larger nuclei, which are more

likely to provoke a breakdown of the liquid medium, have the smaller static delay.

1000

100

0.001 0.01 0.1 1

Static delay p

– p

[bars]

Nuclei radius [micrometers]

∞

= p

∞

= p

∞

= 0.1 bar

∞

= 1 bar

∞

= 10 bars

= – ———

– p

3√3

– p

R = —— ———

2.3 - Radius of equilibrium of air nuclei in water

for various external pressures p

••

FUNDAMENTALS OF CAVITATION20

2.2.3. NUCLEUS EVOLUTION IN A LOW PRESSURE REGION

Via BLAKE's model the evolution of an isolated nucleus passing through a low-

pressure region of limited extent is more easily understood. Consider, for example,

the evolution of a nucleus in a Venturi (fig. 2.4). The local pressure at a distance s

in this approximately one-dimensional flow is denoted p(s). For the nucleus, this

local pressure p(s) plays the role of p

•,

as introduced in section 2.2.1.

Two cases must be considered according to the value of the minimum pressure p

min

with regard to the critical pressure p

of the nucleus.

® If

cmin

, the nucleus grows slightly and then returns to its initial state as it

passes through the throat.

® If

cmin

, it becomes unstable and much larger than it was initially. Figure 2.4

shows its evolution in the (R,p

•

) diagram, as deduced from dynamic calculations

(see chap. 3). Due to inertial forces, its maximum size is reached at point D, a

little downstream from throat C. At D, it is far from equilibrium, so that the

increasing external pressure compels it to collapse violently.

Throat

AB DE

∞

min

2.4 - Typical nucleus evolution in a Venturi. Unstable case: p

min

This type of bubble evolution can be applied to many practical situations in which

the time spent in the low-pressure region is long compared to other characteristic

times, particularly the collapse time (see chap. 3 for an estimation of this time).

Note that, on the stable branch of the equilibrium curve (fig. 2.2), small oscillations