Franc J-P. Fundamentals of Cavitation

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7 - PARTIAL CAVITIES 163

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APPENDIX: SONIC VELOCITY IN A LIQUID/VAPOR MIXTURE

WITH PHASE CHANGE

We consider a bubbly flow in thermodynamic equilibrium in which the void

fraction is a. The bubbles are assumed to be filled with saturated vapor only.

Surface tension is neglected and thus the pressure in the liquid is equal to the vapor

pressure p

. In addition, the mixture is treated as homogeneous, which requires

that the wave length be much larger than the size of the dispersed phase, i.e. the

bubble size.

A unit volume of mixture contains a volume a of vapor and

()1 -a

of liquid. The

corresponding masses of vapor and liquid are ar

and

()1 -a r

where r

and r

are the vapor and liquid densities respectively. The density of the mixture is

rar ar=+-

()1

Assuming this volume undergoes a pressure variation from p

, then in

order to calculate the velocity of sound

, we must evaluate dr.

The pressure variation induces a phase change. If dm is the mass of liquid changed

into vapor, the mass of vapor is now

ar d

and the vapor density becomes

∂r

∂

+=+

. Thus the vapor volume becomes:

ar d

@+ -

(7.26)

FUNDAMENTALS OF CAVITATION166

Calculations are similar for the liquid, with dm changing sign and a changed to

1 -a

. The recalculated liquid volume is then:

() ()11

-- --a

(7.27)

with density:

ar a r

()

+-- -

(7.28)

@- -

11 1

(7.29)

Thus the sound velocity c is given by:

1111 1

222 22

drr

(7.30)

The phase change dm depends on the nature of the thermodynamic transformation.

If the transformation is adiabatic and the latent heat of vaporization is taken from

the liquid whose temperature varies by dT, the heat balance gives:

dardmL c T

=- -()1

(7.31)

where L is the latent heat of vaporization and c

the liquid heat capacity. The

contribution of the vapor to the previous heat balance is negligible, except for void

fractions close to unity. In the case of vaporization (

dm > 0

), the liquid temperature

obviously decreases (

dT < 0

Equation (7.31) gives:

-()1

(7.32)

The term

in the last equation is the inverse of the slope of the vaporization-

condensation curve. It is calculated on the basis of the C

LAPEYRON relation (5.45)

and approximated by

according to equation (5.46).

7 - PARTIAL CAVITIES 167

Returning to equation (7.30) we obtain finally:

222 2

()

(7.33)

In the case of water at room temperature, we have the following orders of magnitude

(see § 5.3.1):

cms

@ 380 /

and

ckgms

2 500@ ,//

cms

@ 1 500,/

and

ckgms

22510@ .//

()

.//

kg m s

Except in very special cases and particularly for void fractions close to unity, the

last term in equation (7.33) is predominant because of the phase change. It reduces

the speed of sound dramatically compared to the case without phase change (for

which it is well known that the velocity of sound is smaller than the values for each

phase separately). For example, for a void fraction of 50%, equation (7.33) gives the

speed of sound as 0.08 m/s whereas, ignoring the last term, it is about 3 m/s (i.e.

without phase change).

8. BUBBLES AND CAVITIES

TWO-DIMENSIONAL FOILS

This chapter is mainly concerned with developed cavitation on two-dimensional

hydrofoils as it occurs on blades of rotating machinery or lifting hydrofoils. The

mechanisms which govern inception and development of attached cavitation and

traveling bubble cavitation are presented in sections 8.1 and 8.2 respectively. It is

shown that attached cavities are strongly connected to the boundary layer, while

bubble cavitation chiefly depends on the pressure distribution and the liquid

nuclei content. The possible competition and interaction between both modes of

cavitation is examined in section 8.3 while roughness effects are examined in

section 8.4. From an experimental viewpoint, the control of cavitation patterns

requires specific control of the water quality by means of equipment devoted to de-

aeration, nuclei seeding and nuclei measurement.

8.1. ATTACHED CAVITATION

In chapters 6 and 7, the main features of attached cavities, partial cavities as well

as supercavities, were reviewed. Two critical regions were pointed out: cavity

detachment and cavity closure. The latter was treated in detail in chapter 7 for

partial cavitation. Here the focus is on cavity detachment, whose location is, a

priori, unknown in the case of a wall of continuous slope.

8.1.1. CAVITATION INCEPTION ON A CIRCULAR CYLINDER

The case of a circular cylinder or a sphere is particularly representative of the

connection between cavitation inception and the boundary layer.

In non-cavitating conditions, the drag coefficient of a cylinder presents a sudden

decrease for a critical value of the R

EYNOLDS number. This value depends to a

small extent upon the operating conditions (such as wall roughness or turbulence

level) but usually lies around 3.10

. It is well-known that this drop corresponds to

the transition to turbulence of the boundary layer flow.

Figure 8.1 shows the corresponding states of the boundary layer, as observed

from visualizations, together with the variation of the drag coefficient versus the

EYNOLDS number.

FUNDAMENTALS OF CAVITATION170

Laminar

boundary layer

Laminar

separation bubble

Turbulent

boundary layer

SUBCRITICAL SUPERCRITICAL

CRITICAL

TRANSCRITICAL

1.0

0.5

0.0

EYNOLDS

number

Drag coefficient

Separate transient

bubbles

Band-type

cavitation

Shear

cavitation

Laminar

separation

Laminar

separation

Turbulent

separation

Turbulent

separation

Transition

Reattachment

8.1 - Drag coefficient of a circular cylinder versus the REYNOLDS number

and associated cavitation inception patterns and boundary layer state

Below the critical value of the R

EYNOLDS number, the boundary layer is laminar

and separates from the body at an angle of between 70 and 80 degrees with

respect to the stagnation point. The wake is thick and the drag is large. Transition

to turbulence occurs away from the wall, in the free shear layer. It progressively

approaches the separation point as the R

EYNOLDS number is increased.

In the transcritical region, the boundary layer is dramatically modified. Transition

to turbulence is so close to the cylinder that the boundary layer reattaches on the

wall. Laminar separation, transition to turbulence and turbulent reattachment are

8 - BUBBLES AND CAVITIES ON TWO-DIMENSIONAL FOILS 171

concentrated within a small region and form the so-called "separation bubble". The

boundary layer, which has become turbulent, can withstand a stronger adverse

pressure gradient than if it were laminar and separates later, at about 130 degrees.

The wake becomes narrower and the drag smaller.

Photographs on figure 8.2 show the corresponding cavitation patterns at inception.

Although the difference in R

EYNOLDS numbers is very small, the cavitation looks

very different.

Re = 270 000,

, cavitation is not attached to the body, but appears in the wake, in

the form of vapor filaments which are characteristic of shear cavitation (see chap. 11).

8.2

Cavitation patterns

at inception in the vicinity

of the critical R

EYNOLDS number

for a circular cylinder

(a)

Re ,= 270 000

(b)

Re ,= 290 000

Conversely, for a slightly higher

EYNOLDS number of 290,000,

cavitation appears as a narrow

sheet of vapor attached to the wall

at an angle of about 115 degrees

(visible on figure 8.2-b on the

upper surface of the cylinder). It

is generally referred to as band-

type cavitation. It is initially made

up of small bubbles, originating in nuclei trapped in the laminar separation bubble,

where they grow slowly by diffusion of dissolved air. A further decrease in the

ambient pressure results in the transformation of the band into a continuous

attached vapor cavity. In this transcritical regime, the cavitation band, in which the

pressure is close to the vapor pressure, develops downstream of the minimum

pressure location. Hence, the liquid particles are in a metastable state while

traveling through a region where the pressure is lower than the vapor pressure.

For large values of the R

EYNOLDS number, the flow regime is supercritical.

Transition to turbulence moves upstream of the laminar separation and suppresses

the laminar separation bubble. Cavitation appears as separate transient bubbles

–on the condition that water contains nuclei which can actually be activated by the

minimum pressure– or as three-dimensional vapor cones attached to isolated

roughness elements.

FUNDAMENTALS OF CAVITATION172

8.1.2. CAVITY PATTERNS ON A TWO-DIMENSIONAL FOIL

In the case of hydrofoils, various cavity flow patterns can be observed according to

the angle of attack and the cavitation number. For a proper observation of attached

cavities in a hydrodynamic tunnel, it is essential to eliminate as much as possible

traveling bubble cavitation in favor of attached cavities, and hence to use strongly

deaerated water, so that almost no nucleus is activated.

Figure 8.3 gives a mapping of the various cavity flow patterns which have been

observed on a NACA 16012 hydrofoil, for a fixed R

EYNOLDS number, when the

incidence and the cavitation number (or simply the ambient pressure) are modified

RANC &MICHEL 1985].

For small values of the cavitation number, supercavitation is observed for any

angle of attack. The supercavity detaches itself from the rear part of the foil for an

angle of attack of around zero (region 1), and the detachment point progressively

moves upstream, towards the leading edge, as incidence increases. The detachment

line is almost straight in the spanwise direction in regions 1 and 3, whereas it

becomes strongly three-dimensional in the intermediate zone 2.

In region 1, the detachment point is far downstream of the point of minimum

pressure, which is close to the point of maximum foil thickness. Thus, as for trans-

critical flow around a cylinder, the liquid particles are in a metastable state in front

of the cavity. This is why deaerated water is needed to observe this cavity flow

regime.

For high values of the cavitation number, the pattern progressively evolves from a

partial pure vapor cavity (region 3'), to a two-phase cavity (region 4) with a smaller

void fraction and, finally, to cavitation in the shear layer bordering the recirculating

zone due to stall at high angle of attack (region 5).

Within a very narrow domain of attack angles (around 4 degrees), the behavior of

the cavity flow is rather unexpected. When the cavitation number is decreased from

non-cavitating conditions, a leading edge cavity appears first. As the cavitation

number is further decreased, the leading edge cavity completely disappears

before a new cavity develops, at sufficiently low values of the cavitation number.

The disappearance of the leading edge cavity is associated with an unexpected

displacement of the detachment point, which moves downstream as s

decreases

until the cavity disappears. This behavior, connected to the S-shaped limiting

curve, results from the strong interaction between the cavity and the boundary

layer. Similar behavior of the detachment point was also observed by M

ICHEL

(1988) on a different foil, for velocities of up to 30 m/s, and approximately the

same s

-values.

8 - BUBBLES AND CAVITIES ON TWO-DIMENSIONAL FOILS 173

PARTIAL CAVITATION

SHEAR CAVITATION

TWO-PHASE

CAVITY

SUPERCAVITATION

1.2

0.81

0.49

0.33

0.21

0.13

0.07

0.045

0481216

Angle of attack [degree]

Cavitation number σ

NON CAVITATING

8.3 - Cavity patterns on a NACA 16012 hydrofoil at

Re 10

in strongly deaerated water

The foil is set at mid-height in the free surface channel of a hydrodynamic tunnel.

The foil chord is 0.10 m and the channel height 0.40 m [from F

RANC & MICHEL, 1985].