HYDRODYNAMICS OF PUMPS

by Christopher Earls Brennen     © Concepts NREC 1994

CHAPTER 6.
BUBBLE DYNAMICS, DAMAGE AND NOISE

 

6.1 INTRODUCTION

We now turn to the characteristics of cavitation for σ <σi. To place the material in context, we begin with a discussion of bubble dynamics, so that reference can be made to some of the classic results of that analysis. This leads into a discussion of two of the deleterious effects that occur as soon as there is any cavitation, namely cavitation damage and cavitation noise. In the next chapter, we address another deleterious consequence of cavitation, namely its effect upon hydraulic performance.

 

6.2 CAVITATION BUBBLE DYNAMICS

Two fundamental models for cavitation have been extensively used in the literature. One of these is the spherical bubble model which is most relevant to those forms of bubble cavitation in which nuclei grow to visible, macroscopic size when they encounter a region of low pressure, and collapse when they are convected into a region of higher pressure. For present purposes, we give only the briefest outline of these methods, while referring the reader to the extensive literature for more detail (see, for example, Knapp, Daily and Hammitt 1970, Plesset and Prosperetti 1977, Brennen 1994). The second fundamental methodology is that of free streamline theory, which is most pertinent to flows consisting of attached cavities or vapor-filled wakes; a brief review of this methodology is given in chapter 7.

Virtually all of the spherical bubble models are based on some version of the Rayleigh-Plesset equation (Plesset and Prosperetti 1977) that defines the relation between the radius of a spherical bubble, R(t), and the pressure, p(t), far from the bubble. In an otherwise quiescent incompressible Newtonian liquid, this equation takes the form
......(6.1)
where ν, , and ρL are respectively the kinematic viscosity, surface tension, and density of the liquid. This equation (without the viscous and surface tension terms) was first derived by Rayleigh (1917) and was first applied to the problem of a traveling cavitation bubble by Plesset (1949).

The pressure far from the bubble, p(t), is an input function that could be obtained from a determination of the pressure history that a nucleus would experience as it travels along a streamline. The pressure, pB(t), is the pressure inside the bubble. It is often assumed that the bubble contains both vapor and noncondensable gas, so that
......(6.2)
where TB is the temperature inside the bubble, pV(TB) is the vapor pressure, mG is the mass of gas in the bubble, and KG is the gas constant. However, it is convenient to use the ambient liquid temperature far from the bubble, T∞, to evaluate pV. When this is done, it is necessary to introduce the term, Θ, into equation 6.1 in order to correct for the difference between pV(TB) and pV(T∞). It is this term, Θ, that is the origin of the thermal effect in cavitation. Using the Clausius-Clapeyron relation,
......(6.3)
where ρV is the vapor density and is the latent heat.

Figure 6.1 Typical solution, R(t), of the Rayleigh-Plesset equation for a spherical bubble originating from a nucleus of radius, R0. The nucleus enters a low pressure region at a dimensionless time of 0 and is convected back to the original pressure at a dimensionless time of 500. The low pressure region is sinusoidal and symmetric about a dimensionless time of 250.

Note that in using equation 6.2 for pB(t), we have introduced the additional unknown function, TB(t), into the Rayleigh-Plesset equation 6.1. In order to determine this function, it is necessary to construct and solve a heat diffusion equation, and an equation for the balance of heat in the bubble. Approximate solutions to these equations can be written in the following simple form. If the heat conducted into the bubble is equated to the rate of use of latent heat at the interface, then
......(6.4)
where (∂T/∂r)r=R is the temperature gradient in the liquid at the interface and kL is the thermal conductivity of the liquid. Moreover, an approximate solution to the thermal diffusion equation in the liquid is
......(6.5)
where αL is the thermal diffusivity of the liquid (αL=kL/ρL cPL where cPL is the specific heat of the liquid) and t is the time from the beginning of bubble growth or collapse. Using equations 6.4 and 6.5 in equation 6.3, the thermal term can be approximated as
......(6.6)
where
......(6.7)
In section 7.7, we shall utilize these relations to evaluate the thermal suppression effects in cavitating pumps.

For present purposes, it is useful to illustrate some of the characteristic features of solutions to the Rayleigh-Plesset equation in the absence of thermal effects (Θ =0 and TB(t)=T∞). A typical solution of R(t) for a nucleus convected through a low pressure region is shown in figure 6.1. Note that the response of the bubble is quite nonlinear; the growth phase is entirely different in character from the collapse phase. The growth is steady and controlled; it rapidly reaches an asymptotic growth rate in which the dominant terms of the Rayleigh-Plesset equation are the pressure difference, pV-p, and the second term on the right-hand side so that
......(6.8)
Note that this requires the local pressure to be less than the vapor pressure. For traveling bubble cavitation, the typical tension (pV-p) will be given nondimensionally by (-Cpmin-σ) (see equations 5.2 and 5.4) so the typical growth rate is given by
......(6.9)
While this growth rate may appear, superficially, to represent a relatively gentle process, it should be recognized that it corresponds to a volume that is increasing like t3. Cavitation growth is therefore an explosive process to be contrasted with the kind of boiling growth that occurs in a kettle on the stove in which dR/dt typically behaves like t-½. The latter is an example of the kind of thermally inhibited growth discussed in section 7.7.

Figure 6.2 The maximum size to which a cavitation bubble grows (according to the Rayleigh-Plesset equation), RM, as function of the original nuclei size, R0, and the cavitation number, σ, in the flow around an axisymmetric headform of radius, RH, with /ρL RH U2=0.000036 (from Ceccio and Brennen 1991).

It follows that we can estimate the typical maximum size of a cavitation bubble, RM, given the above growth rate and the time available for growth. Numerical calculations using the full Rayleigh-Plesset equation show that the appropriate time for growth is the time for which the bubble experiences a pressure below the vapor pressure. In traveling bubble cavitation we may estimate this by knowing the shape of the pressure distribution near the minimum pressure point. We shall represent this shape by
......(6.10)
where s is a coordinate measured along the surface, D is the typical dimension of the body or flow, and Cp* is some known constant of order one. Then, the time available for growth, tG, is given approximately by
......(6.11)
and therefore
......(6.12)
Note that this is independent of the size of the original nucleus.

One other feature of the growth process is important to mention. It transpires that because of the stabilizing influence of the surface tension term, a particular tension, (pV-p), will cause only bubbles larger than a certain critical size to grow explosively (Blake 1949). This means that, for a given cavitation number, only nuclei larger than a certain critical size will achieve the growth rate necessary to become macroscopic cavitation bubbles. A decrease in the cavitation number will activate smaller nuclei, thus increasing the volume of cavitation. This phenomenon is illustrated in figure 6.2 which shows the maximum size of a cavitation bubble, RM, as a function of the size of the original nucleus and the cavitation number for a typical flow around an axisymmetric headform. The vertical parts of the curves on the left of the figure represent the values of the critical nuclei size, RC, that are, incidentally, given simply by the expression
......(6.13)
where the factor κ is roughly unity (Ceccio and Brennen 1991). Note also from figure 6.2 that all the unstable nuclei grow to roughly the same size as anticipated earlier.

Turning now to the collapse, it is readily seen from figure 6.1 that cavitation bubble collapse is a catastrophic phenomenon in which the bubble, still assumed spherical, reaches a size very much smaller than the original nucleus. Very high accelerations and pressures are generated when the bubble becomes very small. However, if the bubble contains any noncondensable gas at all, this will cause a rebound as shown in figure 6.1. Theoretically, the spherical bubble will undergo many cycles of collapse and rebound. In practice, a collapsing bubble becomes unstable to nonspherical disturbances, and essentially shatters into many smaller bubbles in the first collapse and rebound. The resulting cloud of smaller bubbles rapidly disperses. Whatever the deviations from the spherical shape, the fact remains that the collapse is a violent process that produces noise and the potential for material damage to nearby surfaces. We proceed to examine both of these consequences in the two sections which follow.

 

6.3 CAVITATION DAMAGE

Figure 6.3 Photograph of localized cavitation damage on the blade of a mixed flow pump impeller made from an aluminium-based alloy.

Figure 6.4 Cavitation damage on the blades at the discharge from a Francis turbine.

Figure 6.5 Cavitation damage to the concrete wall of the 15.2m diameter Arizona spillway at the Hoover Dam. The hole is 35m long, 9m wide and 13.7m deep. Reproduced from Warnock (1945).

Perhaps the most ubiquitous problem caused by cavitation is the material damage that cavitation bubbles can cause when they collapse in the vicinity of a solid surface. Consequently, this aspect of cavitation has been intensively studied for many years (see, for example, ASTM 1967, Knapp, Daily, and Hammitt 1970, Thiruvengadam 1967, 1974). The problem is complex because it involves the details of a complicated unsteady flow combined with the reaction of the particular material of which the solid surface is made.

As we have seen in the previous section, cavitation bubble collapse is a violent process that generates highly localized, large amplitude disturbances and shocks in the fluid at the point of collapse. When this collapse occurs close to a solid surface, these intense disturbances generate highly localized and transient surface stresses. Repetition of this loading due a multitude of bubble collapses can cause local surface fatigue failure, and the detachment of pieces of material. This is the generally accepted explanation for cavitation damage. It is consistent with the appearance of cavitation damage in most circumstances. Unlike the erosion due to solid particles in the flow, for which the surface appears to be smoothly worn with scratches due to larger particles, cavitation damage has the crystalline and jagged appearance of fatigue failure. To illustrate this, a photograph of localized cavitation damage on the blade of a mixed flow pump, fabricated from an aluminium-based alloy, is included as 6.3. More extensive damage is illustrated in figure 6.4 which shows the blades at discharge from a Francis turbine; here the cavitation damage has penetrated the blades. Cavitation damage can also occur in much larger scale flows. As an example, figure 6.5 shows cavitation damage suffered by a spillway at the Hoover dam (Warnock 1945, Falvey 1990).

Figure 6.6 Axial views from the inlet of the cavitation and cavitation damage on the hub or base plate of a centrifugal pump impeller. The two photographs are of the same area, the left one showing the typical cavitation pattern during flow and the right one the typical cavitation damage. Parts of the blades can be seen in the upper left and lower right corners; relative to these blades the flow proceeds from the lower left to the upper right. The leading edge of the blade is just outside the field of view on the upper left. Reproduced from Soyama, Kato and Oba (1992) with permission of the authors.

In hydraulic devices such as pump impellers or propellers, cavitation damage is often observed to occur in quite localized areas of the surface. This is frequently the result of the periodic and coherent collapse of a cloud of cavitation bubbles. Such is the case in magnetostrictive cavitation testing equipment (Knapp, Daily, and Hammitt 1970). In many pumps, the periodicity may occur naturally as a result of regular shedding of cavitating vortices, or it may be a response to a periodic disturbance imposed on the flow. Examples of the kinds of imposed fluctuations are the interaction between a row of rotor vanes and a row of stator vanes, or the interaction between a ship's propeller and the nonuniform wake behind the ship. In almost all such cases, the coherent collapse of the cloud can cause much more intense noise and more potential for damage than in a similar nonfluctuating flow. Consequently, the damage is most severe on the solid surface close to the location of cloud collapse. An example of this phenomenon is included in figure 6.6 taken from Soyama, Kato and Oba (1992). In this instance, clouds of cavitation are being shed from the leading edge of a centrifugal pump blade, and are collapsing in a specific location as suggested by the pattern of cavitation in the left-hand photograph. This leads to the localized damage shown in the right-hand photograph.

Currently, several research efforts are focussed on the dynamics of cavitation clouds. These studies suggest that the coherent collapse can be more violent than that of individual bubbles, but the basic explanation for the increase in the noise and damage potential is not clear.

 

6.4 MECHANISM OF CAVITATION DAMAGE

The intense disturbances that are caused by cavitation bubble collapse can have two separate origins. The first is related to the fact that a collapsing bubble may be unstable in terms of its shape. When the collapse occurs near a solid surface, Naude and Ellis (1961) and Benjamin and Ellis (1966) observed that the developing spherical asymmetry takes the form of a rapidly accelerating jet of fluid, entering the bubble from the side furthest from the wall (see figure 6.7). Plesset and Chapman (1971) carried out numerical calculations of this ``reentrant jet'', and found good agreement with the experimental observations of Lauterborn and Bolle (1975). Since then, other analytical methods have explored the parametric variations in the flow. These methods are reviewed by Blake and Gibson (1987). The ``microjet'' achieves very high speeds, so that its impact on the other side of the bubble generates a shock wave, and a highly localized shock loading of the surface of the nearby wall.

Figure 6.7 The collapse of a cavitation bubble close to a solid boundary. The theoretical shapes of Plesset and Chapman (1971) (solid lines) are compared with the experimental observations of Lauterborn and Bolle (1975) (points) (adapted from Plesset and Prosperetti 1977).

Parenthetically, we might remark that this is also the principle on which the depth charge works. The initial explosion creates little damage, but does produce a very large bubble which, when it collapses, generates a reentrant jet directed toward any nearby solid surface. When this surface is a submarine, the collapse of the bubble can cause great damage to that vessel. It may also be of interest to note that a bubble, collapsing close to a very flexible or free surface, develops a jet on the side closest to this boundary, and, therefore, traveling in the opposite direction. Some researchers have explored the possibility of minimizing cavitation damage by using surface coatings with a flexibility designed to minimize the microjet formation.

The second intense disturbance occurs when the remnant cloud of bubbles, that remains after the microjet disruption, collapses to its minimum gas/vapor volume, and generates a second shock wave that impinges on the nearby solid surface. The generation of a shock wave during the rebound phase of bubble motion was first demonstrated by the calculations of Hickling and Plesset (1964). More recently, Shima et al. (1981) have made interesting observations of the spherical shock wave using Schlieren photography, and Fujikawa and Akamatsu (1980) have used photoelastic solids to examine the stress waves developed in the solid. Though they only observed stress waves resulting from the remnant cloud collapse and not from the microjet, Kimoto (1987) has subsequently shown that both the microjet and the remnant cloud create stress waves in the solid. His measurements indicate that the surface loading resulting from the remnant cloud is about two or three times that due to the microjet.

Until very recently, virtually all of these detailed observations of collapsing cavitation bubbles had been made in a quiescent fluid. However, several recent observations have raised doubts regarding the relevance of these results for most flowing systems. Ceccio and Brennen (1991) have made detailed observations of the collapse of cavitating bubbles in flows around bodies, and have observed that typical cavitation bubbles are distorted and often broken up by the shear in the boundary layer or by the turbulence before the collapse takes place. Furthermore, Chahine (personal communication) has performed calculations similar to those of Plesset and Chapman, but with the addition of rotation due to shear, and has found that the microjet is substantially modified and reduced by the flow.

Figure 6.8 Examples of cavitation damage weight loss as a function of time. Data from vibratory tests with different materials (Hobbs, Laird and Brunton 1967).

Figure 6.9 Cavitation erosion rates in a centrifugal pump as a function of the flow rate relative to the design flow rate (Pearsall 1978 from Grist 1974).

The other important facet of the cavitation damage phenomenon is the reaction of the material of the solid boundary to the repetitive shock (or ``water hammer'') loading. Various measures of the resistance of particular materials to cavitation damage have been proposed (see, for example, Thiruvengadam 1967). These are largely heuristic and empirical, and will not be reviewed here. The reader is referred to Knapp, Daily, and Hammitt (1970) for a detailed account of the relative resistance of different materials to cavitation damage. Most of these comparisons are based, not on tests in flowing systems, but on results obtained when material samples are vibrated at high frequency (about 20kHz) in a bath of quiescent liquid. The samples are weighed at regular intervals to determine the loss of material, and the results are presented in the form typified by figure 6.8. Note that the relative erosion rates, according to this data, can be approximately correlated with the structural strength of the material. Furthermore, the erosion rate is not necessarily constant with time. This may be due to the differences in the response of a collapsing bubble to a smooth surface as opposed to a surface already roughened by damage. Finally, note that the weight loss in many materials only begins after a certain incubation time.

The data on erosion rates in pumps is very limited because of the length of time necessary to make such measurements. The data that does exist (Mansell 1974) demonstrates that the rate of erosion is a strong function of the operating point as given by the cavitation number and the flow coefficient. The influence of the latter is illustrated in figure 6.9. This curve essentially mirrors those of figures 5.16 and 5.17. At off-design conditions, the increased angle of incidence leads to increased cavitation and, therefore, increased weight loss.

 

6.5 CAVITATION NOISE

Figure 6.10 Bubble natural frequency, ωP, in Hz as a function of the bubble radius and the difference between the equilibrium pressure and the vapor pressure (in kg/m sec2) for water at 300°K.

Figure 6.11 Typical acoustic signal from a single collapsing bubble (from Ceccio and Brennen 1991).

The violence of cavitation bubble collapse also produces noise. In many practical circumstances, the noise is important not only because of the vibration that it may cause, but also because it advertizes the presence of cavitation and, therefore, the likelihood of cavitation damage. Indeed, the magnitude of cavitation noise is often used as a crude measure of the rate of cavitation erosion. For example, Lush and Angell (1984) have shown that, in a given flow at a given cavitation number, the rate of weight loss due to cavitation damage is correlates with the noise as the velocity of the flow is changed.

Prior to any discussion of cavitation noise, it is useful to identify the natural frequency with which individual bubbles will oscillate a quiescent liquid. This natural frequency can be obtained from the Rayleigh-Plesset equation 6.1 by substituting an expression for R(t) that consists of a constant, RE, plus a small sinusoidal perturbation of amplitude, , at a general frequency, ω. Steady state oscillations like this would only be maintained by an applied pressure, p(t), consisting of a constant, , plus a sinusoidal perturbation of amplitude, , and frequency, ω. Obtaining the relation between the linear perturbations, and , from the Rayleigh-Plesset equation, it is found that the ratio, /, has a maximum at a resonant frequency, ωP, given by
......(6.14)
The results of this calculation for bubbles in water at 300°K are presented in figure 6.10 for various mean pressure levels, . Note that the bubbles below about 0.02μm are supercritically damped, and have no resonant frequency. Typical cavitation nuclei of size 10→ 100μm have resonant frequencies in the range 10→ 100kHz. Even though the nuclei are excited in a highly nonlinear way by the cavitation, one might expect that the spectrum of the noise that this process produces would have a broad maximum at the peak frequency corresponding to the size of the most numerous nuclei participating in the cavitation. Typically, this would correspond to the radius of the critical nucleus given by the expression 6.13. For example, if the critical nuclei size were of the order of 10-100μm, then, according to figure 6.10, one might expect to see cavitation noise frequencies of the order of 10-100kHz. This is, indeed, the typical range of frequencies produced by cavitation.

Figure 6.12 The acoustic impulse, I, produced by the collapse of a single cavitation bubble. Data is shown for two axisymmetric bodies (the ITTC and Schiebe headforms) as a function of the maximum volume prior to collapse. Also shown are the equivalent results from solutions of the Rayleigh-Plesset equation (from Ceccio and Brennen 1991).

Figure 6.13 Typical spectra of noise from bubble cavitation for various cavitation numbers as indicated (Ceccio and Brennen 1991).

Figure 6.14 Typical spectra showing the increase in noise with increasing cavitation in an axial flow pump (Lee 1966).

Fitzpatrick and Strasberg (1956) were the first to make extensive use of the Rayleigh-Plesset equation to predict the noise from individual collapsing bubbles and the spectra that such a process would produce. More recently, Ceccio and Brennen (1991) have recorded the noise from individual cavitation bubbles in a flow. A typical acoustic signal is reproduced in 6.11. The large positive pulse at about 450μs corresponds to the first collapse of the bubble. Since the radiated acoustic pressure, pA, in this context is related to the second derivative of the volume of the bubble, V(t), by
......(6.15)
(where ℓ is the distance of the measurement from the center of the bubble), the pulse corresponds to the very large and positive values of d2V/dt2 that occur when the bubble is close to its minimum size in the middle of the collapse. The first pulse is followed in figure 6.11 by some facility-dependent oscillations, and by a second pulse at about 1100μs. This corresponds to the second collapse; no further collapses were observed in these particular experiments.

A good measure of the magnitude of the collapse pulse in figure 6.11 is the acoustic impulse, I, defined as the area under the curve or
......(6.16)
where t1,t2 are the times before and after the pulse when pA=0. The acoustic impulses for cavitation on two axisymmetric headforms (ITTC and Schiebe headforms) are compared in figure 6.12 with impulses predicted from integration of the Rayleigh-Plesset equation. Since these theoretical calculations assume that the bubble remains spherical, the discrepancy between the theory and the experiments is not too surprising. Indeed, the optimistic interpretation of figure 6.12 is that the theory can provide an order of magnitude estimate of the noise produced by a single bubble. This could then be combined with the nuclei number density distribution to obtain a measure of the amplitude of the noise (Brennen 1994).

Figure 6.15 The relation between the cavitation performance, the noise and vibration produced at three frequency levels in a centrifugal pump, namely the shaft frequency (squares), the blade passage frequency (upsidedown triangles) and 40kHz (circles) (Pearsall 1966-67).

The typical single bubble noise shown in figure 6.11 leads to the spectrum shown in figure 6.13. If the cavitation events are randomly distributed in time, this would also correspond to the overall cavitation noise spectrum. It displays a characteristic frequency content in the range of 1→ 50kHz (the rapid decline at about 80kHz represents the limit of the hydrophone used to make these measurements). Typical measurements of the noise produced by cavitation in an axial flow pump are illustrated in figure 6.14, and exhibit the same features demonstrated in figure 6.13. The signal in figure 6.14 also clearly contains some shaft or blade passage frequencies that occur in the absence of cavitation, but may be amplified or attenuated by cavitation. Figure 6.15 contains data obtained for cavitation noise in a centrifugal pump. Note that the noise at a frequency of 40kHz shows a sharp increase with the onset of cavitation; on the other hand, the noise at the shaft and blade passage frequencies show only minor changes with cavitation number. The decrease in the 40kHz cavitation noise as breakdown is approached is also a common feature in cavitation noise measurements.

The level of the sound produced by a cavitating flow is the result of two factors, namely the impulse, I, produced by each event (equation 6.16) and the event rate or number of events per second, . Therefore, the sound pressure level, pS, will be
......(6.17)
Here, we will briefly discuss the scaling of the two components, I, and , and thus the scaling of the cavitation noise, pS. We emphasize that the following equations omit some factors of proportionality necessary for quantitative calculations.

Both the experimental observations and the calculations based on the Rayleigh-Plesset equation, show that the nondimensional impulse from a single cavitation event, defined by
......(6.18)
(where U and D are the reference velocity and length in the flow), is strongly correlated with the maximum volume of the cavitation bubble (maximum equivalent volumetric radius = RM), and appears virtually independent of the other flow parameters. In dimensionless terms,
......(6.19)
It follows that
......(6.20)
The evaluation of the impulse from a single event is then completed by some estimate of the maximum bubble size, RM. For example, we earlier estimated RM for traveling bubble cavitation (equation 6.12), and found it to be independent of U for a given cavitation number. In that case I is linear in U.

Modeling the event rate, , can be considerably more complicated than might, at first sight, be visualized. If all the nuclei flowing through a certain known streamtube (say with a cross-sectional area, AN, in the upstream reference flow), were to cavitate similarly then, clearly, the result would be
......(6.21)
where N is the nuclei concentration (number/unit volume). Then the sound pressure level resulting from substituting the expressions 6.21, 6.20, and 6.12 into equation 6.17, is
......(6.22)
where we have omitted some of the constants of order unity. For the simple circumstances outlined, equation 6.22 yields a sound pressure level that scales with U2 and with D4 (because AN is proportional to D2). This scaling with velocity does correspond to that often observed (for example, Blake, Wolpert, and Geib 1977, Arakeri and Shangumanathan 1985) in simple traveling bubble flows. There are, however, a number of complicating factors. First, as we have discussed earlier in section 6.2, only those nuclei larger than a certain critical size, RC, will actually grow to become cavitation bubbles, and, since RC is a function of both σ and the velocity U, this means that N will be a function of RC and U. Since RC decreases as U increases, the power law dependence of pS on velocity will then be Um where m is greater than 2.

Different scaling laws will apply when the cavitation is generated by turbulent fluctuations, such as in a turbulent jet (see, for example, Ooi 1985, Franklin and McMillan 1984). Then the typical tension and the typical duration of the tension experienced by a nucleus, as it moves along an approximately Lagrangian path in the turbulent flow, are very much more difficult to estimate. Consequently, estimates of the sound pressure due to cavitation in turbulent flows, and the scaling of that sound with velocity, are more poorly understood.

REFERENCES

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Last updated 12/1/00.
Christopher E. Brennen