Franc J-P. Fundamentals of Cavitation

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2 - NUCLEI AND CAVITATION 21

around equilibrium can be expected. Thus, if the time required for the pressure

drop is of the same order –or even shorter– than the natural period of oscillation of

the nucleus, the bubble size will not grow monotonically but will undergo

oscillations.

2.3. HEAT AND MASS DIFFUSION

In section 2.2, it was assumed that the nucleus undergoes an isothermal

transformation with a constant mass of gas. The conditions of validity of both

assumptions and the consequences of possible transfers of gas with the

surrounding liquid are now examined.

2.3.1. THE THERMAL BEHAVIOR OF THE GAS CONTENT

If, contrary to the assumption made in section 2.2.1, the change in pressure around

the nucleus occurs in such a short time that heat transfer between the gas and the

liquid cannot be achieved, the transformation of the gas is effectively adiabatic. In

this case, equations (2.4) for the critical radius and critical pressure have to be

replaced by the following:

/( )

=--

13 1

(2.5)

where g is the ratio of the gas heat capacities. This model is not entirely consistent

since the vapor pressure is still taken at a constant temperature.

To decide on the correct gas behavior in each particular case, the heat transfer must

be analysed. A rather simple way is to proceed as follows [P

LESSET & HSIEH 1960].

Consider a nucleus which undergoes a pressure drop during a time Dt, such that

its radius increases from R to

RR+D

. For simplicity, the bubble is supposed to

contain only non-condensable gas, and no phase change is considered. During this

transformation, the gas temperature inside the bubble is assumed to change from T

TT+D

. The temperature variation DT can be determined from the energy

balance of the gas:

pr pRcTQpRR

gvg

=-DD D[]

(2.6)

1. The question of resonant excitation of nuclei by turbulent fluctuations in the liquid flow

was considered by C

RIGHTON and FFOWCS WILLIAMS (1969). They showed that such a

mechanism is not plausible because the high resonant frequencies of nuclei are

associated with sizes of turbulent fluctuations that are very small in comparison with

nuclei sizes (see § 3.4.1).

FUNDAMENTALS OF CAVITATION22

The left-hand side of this equation is the variation of the internal energy of the gas

and c

stand for the density and specific heat at constant volume), whereas the

two terms on the right-hand side are the heat received by the bubble and the work

of the surface pressure forces respectively.

Because of the temperature variation DT, a thermal boundary layer develops inside

the liquid (see fig. 2.5). The temperature gradient induces a heat flux by conduction

which determines the heat DQ transferred from the liquid to the bubble. If a

is the

thermal diffusivity of the liquid, the boundary layer thickness is of the order of

, so that FOURIER’s law leads to the following estimate:

DDDQ

Rt c t RT

ª- =-lp lr p

ll l

(2.7)

where l

and c

are the conductivity and heat capacity of the liquid respectively.

2.5

Thermal boundary layer

and schematic temperature distribution

in the liquid around the bubble

The energy balance (2.6) allows an estimation

of the temperature variation of the gas in

the bubble:

(2.8)

where DT

ad.

is the temperature variation in the adiabatic case

DQ = 0

gvg

(2.9)

and Dt

is a characteristic time defined by:

gvg

()r

(2.10)

The transformation can be considered adiabatic if the exchanged heat is much

lower than the internal energy variation or, in terms of the characteristic times, if

the transit time Dt is much smaller than the characteristic time Dt

, which can be

interpreted as the time required for heat transfer.

T+∆T

√a

∆t

∆Q

Thermal

boundary layer

2 - NUCLEI AND CAVITATION 23

Inversely, if

DDtt

, equation (2.8) gives

DT = 0

and the transformation can be

considered as isothermal. In other words, the time available for heat transfer is so

large that heat transfer can proceed until thermal equilibrium is reached.

In the case of the water/air system at ambient temperature (

= 06.//WmK

= 1 000

,/kg m

cJkgK

=4 180,//

cJkgK

=715 / /

DtR

= 0 023

.()r

. For

example, with

kg m= 3

and

Rmm= 05.

(2)

the required time is

Dts

= 0 051. m

a very small value.

For the transit time, we take the case of a nucleus that passes through a low-

pressure region 1 cm in length at a velocity of 50 m/s.

(3)

The corresponding transit

time

Dtms= 02.

is about four thousand times greater than the time Dt

required for

heat transfer. Thus, it can be inferred that, in many practical water-air systems, the

transformation can be considered isothermal.

As the characteristic time Dt

is proportional to the radius squared, the question

must be re-examined if the nucleus explodes and becomes a macroscopic bubble of

very large radius (see chap. 3).

2.3.2. GAS DIFFUSION AND NUCLEUS STABILITY

As stated in the previous sections, gas can be present in a liquid in two forms,

dissolved, or trapped in free nuclei. The question of any exchange between the

forms is examined here on the basis of the physical laws concerning the saturation

of the liquid and the transfer of gas in non-equilibrium conditions.

Dissolved gas and nuclei in a static liquid

2.6

Diffusive equilibrium between a liquid

and the atmosphere above it

The diffusion equilibrium between a liquid and the

atmosphere above it is described by H

ENRY's law:

CHTp

si i i

= ()

(2.11)

in which p

is the partial pressure of gas i in the atmosphere (fig. 2.6), C

is its

concentration at saturation in the liquid and H

the corresponding HENRY's constant.

is expressed in kg/m

, and H

in (m/s)

–2

2. These values are chosen large enough to produce adiabatic behavior.

3. These values are rather extreme and again favorable to adiabatic transformation.

GAS

LIQUID

FUNDAMENTALS OF CAVITATION24

If diffusion equilibrium is not achieved, a concentration gradient C exists, resulting

in a mass flux given by F

ICK's law:

q D grad C

ii i

(2.12)

In equation (2.12), D

is the diffusivity coefficient of element i. The balance of mass

transfer for a small domain gives the classic diffusion equation:

∂

(2.13)

In the case of air dissolved in water, concentrations are often expressed in parts per

million, a concentration of 1 ppm being equivalent to 10

–3

kg/m

. The concentrations

at saturation in water are 19 ppm for pure nitrogen and 43 ppm for pure oxygen,

under an external pressure of 1 bar. Taking into account the partial pressure of each

element in the atmosphere, 0.79 bar and 0.21 bar respectively, one obtains 15 ppm

for nitrogen and 9 ppm for oxygen, i.e., 24 ppm in total under atmospheric pressure.

Then, the H

ENRY constant is 0.24 ¥ 10

–6

(s/m)

while the diffusivity coefficient is

Dms=¥

210

for air in water.

Let us now consider the equilibrium of a spherical nucleus in a static liquid

(fig. 2.7). Mechanical equilibrium requires that relation (2.1) be satisfied:

•

=+-

ppp

(2.14)

Its diffusive equilibrium is expressed, from (2.11), by:

CHp

(2.15)

2.7

Nucleus in a static liquid

Two cases must be considered:

® If the concentration far from the nucleus, C

•

is smaller than C

, the gas tends to migrate

from the nucleus to the liquid. The radius

decreases, the surface tension term increases.

With a constant pressure p

•

far from the

nucleus, the term p

grows and increases the diffusive imbalance and the nucleus

tends to be resorbed.

GAS

LIQUID

∞

2 - NUCLEI AND CAVITATION 25

® If C

•

is larger than C

, the opposite phenomenon occurs: the nucleus volume

increases and the diffusive imbalance continues to increase, as in the previous

case.

In conclusion, the double mechanical and diffusive equilibrium of the nucleus is

always unstable. The conclusion is the same if surface tension is neglected, but the

difference between the concentrations remains constant and the instability is

weaker.

PSTEIN and PLESSET (1950) give estimates of the characteristic time for air diffusion

in the absence of surface tension. The integration of equation (2.13) leads to the

following equation for the evolution of the radius R of the nucleus:

•

(2.16)

where r

is the gas density. If the first term is ignored, i.e. if the nucleus radius is

assumed small with respect to the thickness of the concentration boundary layer,

the total resorption time is:

res

DC C

•

(2.17)

Equation (2.17) gives 10,420 seconds, for the air/water system at atmospheric

pressure, with the following values:

Rmm

Ckgm

= 0 024

•

= 0

kg m=1

. The table below gives the values of t for a nucleus of initial

radius 0.01 mm and

/.r=002

. The parameter f is the ratio

s•

. The

calculated time is the total time required for nucleus resorption if

f <1

. If

f >1

, it is

the time needed by the nucleus to grow to a radius ten times larger than its initial

value.

0 0.25

0.50 0.75

1 1.25

1.50 1.75

2.0 5.0

(s)

1.05 1.44

2.21 4.58

• 466

228 149 110 46

If surface tension is taken into account, a resorption time of about 6 seconds is found

for

f =1

. In most practical situations, the characteristic diffusion time appears to be

at least of the order of one second.

Nucleus stability with respect to gas diffusion

According to the previous analysis, nuclei cannot exist for long periods of time in

static liquids. Other mechanisms must be invoked to explain their commonly

observed long lifetimes.

FUNDAMENTALS OF CAVITATION26

It was previously assumed that the nucleus is surrounded by an infinite liquid

medium that constitutes a source (or sink) capable of giving (or receiving) an

unlimited amount of gas. Actually, the gas transfer involves a limited domain of

liquid close to the nucleus [M

ORI 1977, CHA 1981, TSAI 1990, ACHARD & CANOT

1992]. The equilibrium can thus be stable since each nucleus has a limited capacity

of gas transfer. However, such a mechanism requires all nuclei to be the same size

–which is not the case in reality– since the gas concentration and the pressure are

uniformly distributed inside the domain. This is a weakness of this model.

The stability of nuclei can also be explained by assuming that they are enveloped

by a thin organic film that prevents gas transfer [F

OX & HERZFELD 1954]. Such

microbubbles actually exist in the sea, but their response to cavitation has not been

systematically studied.

2.8

Model of a crevice as a nucleus

[from K

NAPP et al., 1970]

Finally, small crevices in solid walls or in solid

particles carried by the liquid flow can shelter

stable nuclei [H

ARVEY et al. 1950]. Consider, for

example, a conical crevice (fig. 2.8) with an angle 2a,

which is partly filled by a liquid

. If the contact angle

between the liquid and the wall q

is larger than

pa/2 +

, which is the case in figure 2.8, the interface

curvature is reversed and the surface tension term

in equation (2.14) changes sign. If the inequality

s•

holds, the gas migrates

towards the liquid and the volume of the trapped gas decreases. Consequently,

both the radius R and the gas pressure p

decrease while the external pressure

remains constant. The difference

s•

diminishes and the situation tends to a

stable equilibrium.

Liquid convection and rectified diffusion

In the previous paragraphs, it was assumed that the nucleus did not move with

respect to the surrounding liquid. If there is a relative motion as is the case for

liquids flowing past wall crevices, the liquid convection continuously renews the

gas source at a small distance from the nucleus and gas transfer is considerably

increased [P

ARKIN & KERMEEN 1962].

LIQUID

GAS

2α

2 - NUCLEI AND CAVITATION 27

Another situation deserves attention. If a nucleus undergoes pressure oscillations

due, for example, to an ultrasonic oscillation field, and if the excitation frequency is

properly adjusted, the size of the nucleus will vary with the imposed pressure. If

the diffusive equilibrium corresponding to the mean pressure is reached, then,

during half a period of growth phase, the internal pressure of the nucleus becomes

lower than the mean pressure and the gas migrates towards it. The reverse gas flux

tends to take place during the half period when the size of the nucleus becomes

smaller. On the whole, as the exchange area is larger in the first half period, there is

a net flux of gas to the advantage of the nucleus. This phenomenon, called rectified

diffusion, was studied by P

LESSET and HSIEH (1960) who give the following

expression for the time required for doubling the nucleus radius:

•

(2.18)

where eis the relative amplitude of the pressure oscillations. For

Rmm

01= .

kg m=1

e=05.

(a rather large value), in the case of air-water one finds

t=2 300,s

. Thus, for most industrial situations where pressure fluctuations are due

to turbulence, this mechanism of nucleus growth by rectified diffusion is not very

efficient.

2.4. NUCLEUS POPULATION

2.4.1. MEASUREMENT METHODS

In the case of real liquids, nuclei of various sizes are present. From the previous

static analysis, it is expected that the response of the liquid to the pressure variations

imposed by the flow will strongly depend upon the concentration of cavitation

nuclei. The measurement of the nucleus population is therefore essential. Two

main kinds of measuring techniques are used:

— optical methods, which give the equilibrium radius R

of the nuclei under an

external pressure p

•0

;

— dynamic methods, which give the critical pressure p

of the nuclei [OLDENZIEL

1982].

In addition, it is necessary to determine the volume of the liquid sample analyzed.

For optical methods such as holography and DOPPLER phase anemometry, the

nucleus population is characterized by a histogram which gives, per unit volume,

the number of nuclei of radius R

in a bandwidth DR

. It is expressed in terms of

nuclei/cm

/DR

(fig. 2.9-a).

FUNDAMENTALS OF CAVITATION28

water with a very

small nuclei content

∆R

n [nuclei/cm

/∆R

]

N [nuclei/cm

]

2.9 - Characterization of the nucleus population by

(a) a size-based histogram obtained by optical methods or

(b) a cumulative histogram of critical pressure obtained by dynamic methods

Histogram 2 corresponds to a much smaller nuclei content than histogram 1.

In the case of dynamic methods, the nuclei go through a low-pressure region, such

as a Venturi. All nuclei with a critical pressure higher than the minimum are

destabilized. They explode and are counted subsequently by an ultrasonic

transducer sensitive to the noise they emit during collapse. A cumulative

histogram

N(p

) is obtained (fig. 2.9-a and 2.10-a), with N given in nuclei/cm

Comparison between optical and dynamic methods requires a conversion between

the critical pressure and the equilibrium size R

by means of the equilibrium

relation (2.2).

Dynamic methods enable us to count nuclei with a diameter of less than 10 mm,

which is the usual limit of the holographic technique. Generally, optical methods

give higher concentrations (by up to more than one order of magnitude) than

4. Note that a population characterized by a variable x can be described by either a

distribution function n(x) or a cumulative histogram N(x) where N is the number

of elements whose size is larger than x. N and n are connected by the equation:

Nx nudu

() ()=

•

2 - NUCLEI AND CAVITATION 29

0.8

0.6

0.4

0.2

– 2000 – 1500 – 1000 – 500 0

0.1

0.01

0.001

1 10 100 1000

Nuclei/cm

Dynamic method

Doppler phase anemometr

Holography

– p

[mbar]

[mm]

2.10 - (a) Typical histogram of nuclei population

as a function of critical pressure (dynamic method of measurement)

and (b) comparison of cumulative histograms in size

for three different measuring techniques

[from Bassin d'Essais des Carènes, 1992]

dynamic methods. This could be due either to the difficulty in estimating the

volume of the liquid sample in holography, or to the fact that all objects counted

are not necessarily cavitation nuclei, but may be solid particles or microbubbles

with an organic envelope. The difference between dynamic and holographic

methods is lower for carefully filtered water.

FUNDAMENTALS OF CAVITATION30

In recent years, the use of Venturi devices has become common in test facilities

such as cavitation tunnels. They offer several advantages in comparison with

optical techniques.

® Nuclei are characterized by their critical pressure, which is the relevant quantity

in cavitation. In particular, it has been shown that concentration of nuclei

measured with a Venturi closely agrees with the density of bubbles that

explode on the upper surface of a 2-D foil [B

RIANÇON-MARJOLLET et al. 1990].

® Very significant volumes of liquid, typically 0.1 m

, are used for counting.

® The measurement can be carried out in a short time, approximately ten minutes

for a complete histogram.

® The relative uncertainties regarding the size of the nuclei are fewer, at least for

small nuclei.

Details of the design of Venturi devices can be found in P

HAM et al. (1997).

It must be noted that detection of collapse by piezoelectric transducers can lead to

an overestimate of the bubble concentration for two main reasons:

— big bubbles may be broken into smaller ones during their collapse;

— in some circumstances, the same bubble can collapse, then rebound and collapse

again.

In both cases, multipeaking is observed in the noise signal, leading to possible

errors in the measurement of the bubble population.

Modern cavitation tunnels are also equipped with systems of nucleus seeding. The

concentration of nuclei which can explode in the vicinity of a body tested in a

laboratory water flow in which nuclei are injected is typically of the order of one

nucleus per cubic centimeter. Very large cavitation effects can be obtained with

such a concentration.

2.4.2. CONDITIONS FOR INCEPTION OF BUBBLE CAVITATION

The largest critical pressure p

in a nucleus population, which is the critical pressure

of the biggest nucleus, is called the susceptibility pressure of the liquid (see fig. 2.9-b).

It is an important element in the onset of bubble cavitation.

If thermal and dynamic delays in cavitation inception are assumed negligible,

cavitation inception occurs as soon as the biggest nucleus is destabilized, i.e., as

soon as the minimum pressure is lower than the susceptibility pressure (fig. 2.11).

Hence, the condition for bubble cavitation inception is:

min

(2.19)

In figure 2.11, the pressure distribution 1 corresponds to cavitation inception, while

a larger number of nuclei are destabilized in the case of pressure distribution 2.