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Chapter 3
3.
CAVITATION
Cavitation is a hydrodynamic phenomenon that can occur in propellers, pumps for rockets or ships as well as artificial hearts and it can be generated in any kind of fluid.
As already seen in the introduction chapter, cavitation is an unwanted phenomenon where nucleation, growth and collapse of bubbles can develop structural damages and failures. Hence, the inception of cavitation may lead to unlike behavior of flow, degrading the performance of turbomachines and therefore it is fundamental to understand the conditions in which it presents.
3.1.
FUNDAMENTAL PARAMETERS
In liquid flows, hydrodynamic effects lead to local low pressure regions where the pressure becomes lower than the vapor pressure pV. An important parameter which characterizes the system is
the cavitation number
( )
r V p p T p σ = − ∆ (3.1)where pr is the reference pressure in a region where cavitation is expected and ∆p is a pressure
difference that characterizes the system. In the specific case of a pump it becomes
( )
( )
(
)
1 1 2 2 1 1 2 2 V V L p L T p p T p p T V r σ ρ ρ ∞ ∞ − − = = Ω (3.2)where p1 is the inlet pressure, ρL is the density of the liquid, Vp is the peripheral velocity of the
impeller. It is important to note that the cavitation number is an overall parameter.
For high values of cavitation number it is quite clear that a single phase flow will be present, otherwise, reducing it, nucleation will start at some specific values σi called incipient cavitation
number. For values below σi the extent of vapor bubbles will increase and different type of cavitation
will manifest.
Moreover the static pressure p(x), where x is the distance along a streamline, is nondimensionalized by means of the pressure coefficient Cp defined in the following equation (3.3), which is a function of
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the geometry and Reynolds number which in this case is given by 2 1
2ΩrT /ν. In the idealized case of an inviscid, frictionless liquid, Cp(x) becomes independent from Re.
(
)
1 1 1 2 2 ( ) ( ) ( ) ( ) 1 1 2 L 2 L T p x p p x p p x p Cp x p ρ U ρ r − − − = = = ∆ Ω (3.3)In the absence of cavitation, fluid velocities and Cp are independent with respect to p and therefore a variation in the inlet pressure will lead to a change in the other pressures in such a way that Cp remains the same. Note also that Cp is a negative number.
Furthermore, there will be a specific position in which Cp will be a minimum
min 1 min 2 1 2 p p Cp U ρ − = (3.4)
In the hypothetical case in which vapor bubbles start to manifest as the vapor pressure is reached, the cavitation inception occurs at the value of -Cpmin
min
i Cp
σ = − (3.5)
Hence in this case the liquid can’t withstand any tension (pmin = pV) and even if it is a crude
approximation, it is usually useful to adopt it as a basic guideline.
In Figure 3.1 the trend of Cp along a streamline is shown and it is clear that if σ is above -Cpmin, the
pressure over the path is always greater than pV whereas for values of σ below -Cpmin there is a region
in which the nucleus experiences p<pV for a finite time.
Another nondimensional parameter is the so called Thoma’s cavitation factor defined as:
1 2 1 ( ) ( ) T V TH T T p p p p σ = − −
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3.2.
NUCLEI AND CAVITATION INCEPTION
Two principal reasons can cause a deviation from the hypothesis stated in the equation 3.5:
• Nucleation does not occur at vapor pressure. In a pure liquid a positive tension ∆pC is
needed to disrupt the flow so that nucleation may need a lower value
min 2 1 2 C i L p Cp U σ ρ ∞ ∆ = − − (3.6)
In this case the process is termed homogeneous cavitation. If the contaminant gas is widely present, then ∆pC can be negative, leading to a higher value of σi with respect to -Cpmin.
• A finite time is required for nucleus to reach observable size where p<pV-∆pC and therefore
a minor sign should be present in equation 3.6.
If the flow is not assumed to be inviscid, Cpmin will depends on Reynolds number.
The basic characteristcs of cavitation inception parameter are the following: • σi can be lower in presence of a tensile strength,
• σi can be lower as a consequence of the residence time effect,
• σi can be higher if a contaminant gas is present,
• Steady viscous effects can cause a dependence of σi from Re,
• σi can be higher if the flow is turbulent instead of laminar due to the transient vortices in
which cavitation can occur at the core.
Due to these effects the equation 3.5 is not an applicable criterion from the engineering point of view.
In real liquids homogeneous cavitation does not occur since the liquid always contains contaminant gas and solid particles that are points of weakness (nuclei of cavitation). The presence of these contaminant can’t be avoided and due to them, the cavitation inception may be different between facilities performing the same tests or even in the same facility with different processed water.
Cavitation itself can be the origin of nuclei since it creates bubbles which need a long residence time to be absorbed by the liquid. Because of this behavior, the water tunnel should be long enough to dissolve the bubbles, avoiding the situation in which the incoming flow at the inlet contains bubbles.
Two kinds of nuclei can be present, suspended and surface nuclei. Whereas the latter is originated at walls and consists in gas trapped in wall crevices, the former has more importance in leading to cavitation inception. It is also possible that bulk nuclei can be produced by the action of cosmic rays that transfer localized energy to the liquid. As previously stated, suspended nuclei can be composed by microbubbles or solid particles. In the former case the bubble is in equilibrium if
2 /
g V N
p p= +p − S R (3.7)
where RN is the radius of the microbubble, S is the surface tension and pg is the partial pressure of the
gas within the bubble. Hence the pressure of the liquid has to fall below pg+pV-2SRN to let the bubble
grow to a visible size.
Note that if the bubble is considered as filled only with vapor, pg is neglected in equation 3.7.
If an isothermal transformation with constant gas mass is assumed the initial state is given by
0 g0 V 2 / 0
p =p + p − S R (3.8)
28
[
]
30 0/ 2 /
g V N
p p R R= +p − S R (3.9)
In equation 3.9 two contributions are present: the internal pressure and the surface tension. Whereas the former tends to increase the size of the bubble, the latter tends to decrease it and as result of the two contributions a minimum is reached:
0 0 0 3 2 / g C p R R S R = (3.10) 4 3 C V C S p p R = − (3.11)
The two equations 3.10 and 3.11 define a locus of minima as shown in Figure 3.2.
Figure 3.2 Liquid equilibrium pressure in a isothermal and gas mass constant process. Franc [4]
In the branch of the curve with negative slope the mechanical equilibrium is stable due to the fact that if a bubble with larger radius '
0
R with a corresponding gas pressure of p'g0< pg0 and the same
surface tension is considered, the resulting forces tend to decrease the radius until R0.
Hence pC is the minimum pressure at which nuclei can remain in stable conditions.
3.3.
BUBBLE DYNAMICS
This chapter will deal with the main dynamic aspects of bubble growth in order to give essential indications on the cavitation collapse phenomena which is of primary concern due to its deleterious effects (cavitation damage and cavitation noise).
The model used extensively is the one in which Rayleigh-Plesset equation is exploited. Spherical bubble model defines a relation between the pressure p∞ far outside the bubble and its radius R. If the
liquid is considered incompressible, Newtonian and inviscid, and the spherical bubble is considered saturated with vapor with constant air mass as well as adiabatic situation (absence of thermal effects), it is possible to obtain the fundamental relation:
29 2 3 (t) ( ) 2 4 2 L RR R pB p t RS RR ρ + = − ∞ − − µ 3 2 0 B 0 3 (T ) ( ) 2 4 2 V g R S R RR R p p t p R R R γ ρ + = − ∞ + − − µ (3.12)
where γ is the ratio between cpg and cvg (heat gas capacities) whereas µ, S and ρ are respectively the
dynamic viscosity, the surface tension and the density of the liquid. If the liquid is assumed without viscosity, the last term vanishes.
The Rayleigh-Plesset equation can also be seen as an energy balance noting that it can be written as follows: 3 2 3 0 2 2 0 2 V ( ) g R 4 8 16 d R R p p t p R R SRR RR dt R γ πρ ∞ π π πµ = − + − − (3.13)
where the term on the left-hand is the kinetic energy variation whereas the three terms on the right are respectively the pressure forces, the surface tension forces and the dissipation rate as a consequence of the viscosity of the liquid.
R(t) can be obtained by means of a numerical integration of equation 3.12. If the viscosity effects, noncondensable gas and surface tension are neglected, a solution for the first collapse can be obtained. An additional hypothesis is that the bubble is in equilibrium before the initial time at which it is considered to be at pressure pV. After the initial time a higher constant pressure is applied, resulting in
the collapse of the bubble in a characteristic Rayleigh time τ. In these conditions the Rayleigh-Plesset equation lead to the following:
2 3 3 3 V 0 2 (p p )(R R ) 3 R R ρ = − ∞− − (3.14)
From which it is possible to obtain the evolution of R and R as represented in Figure 3.3.
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As can be seen from the previous figure, the velocity of the radius tends to infinity reaching speeds of about 720 m/s for R/R0 = 1/20 and therefore at the final stages the compressibility should be taken
into account. Furthermore a singularity is present for 𝑅̇ when R tends to zero.
The general case is shown in Figure 3.4 in which the microbubble is in equilibrium at the initial time when its radius is R0 and the pressure of the fluid is changed from p∞0 to a lower pressure p∞(t)
and then to p∞0 again. The first bubble collapse is followed by rebounds and collapses that are
attenuated by dissipation.
Figure 3.4 Solution of Rayleigh-Plesset equation. Brennen [1]
The liquid pressure around a bubble, p(r,t), can be derived from
3 0 4 V p p R R R ρ ∞− = − (3.15)
obtaining the following expression for the nondimensional pressure Π
3 4 3 0 0 3 4 3 ( , ) p (r,t) 4 1 3 3 V R R p r t R R p∞ p ∞ r R r R − Π = = − − − − (3.16)
This equation can be represented as in Figure 3.5 and as can be seen from it there is a maximum for bubbles smaller than 0.63R0. When R/R0 is small enough the maximum of the nondimensional
pressure and the distance at which this maximum is reached are
3 0 max 0.157 R R Π ≅ max 1.59 r R ≅ (3.17)
The catastrophic collapse of the bubble is a consequence of two facts:
• The bubble does not resist to the motion of the liquid as the inside pressure is constant, • Liquid volume conservation, due to spherical symmetry, concentrates the liquid motion to
a smaller region as the collapse proceeds.
It has to be noted that during the collapse, the incompressibility is not the only assumption that is violated but also the spherical symmetry of the bubble is lost.
31
Figure 3.5 Pressure field around bubbles during collapse phase. Franc [4]
Consider the case in which a nucleus is in equilibrium condition under a pressure p∞0 at t < t0, with
a step variation to a value p∞ < p∞0 applied at t = 0. If viscous effects are ignored whereas surface
tension is considered, an expression for 𝑅̇2= 𝑓(𝑅) can be obtained from equation 3.12. For the
equilibrium of a bubble with R = R0, a double root of this expression is obtained thus 𝑅̇0= 0 and
𝑅̈0= 0. To be stable, the equilibrium needs an additional condition which consists in f (R0) < 0 and therefore the equilibrium is stable only on the descending branch of p∞(R).
If a root R1 > R0 exists, the radius R oscillates between two different values with a certain period. Furthermore the ratio R1/R0 characterizes the motion and for high ratios it becomes strongly nonlinear as the effects of the partial gas pressure is dominant for R close to R0 but it is negligible for large R and close to R1 (see Figure 3.6 case b). In this case the oscillation is composed by collapses and
explosions.
On the other hand when the root R1 approaches the root R0, the motion tends to be harmonic (case
a).
If there is not any other root different from R0, the sign of R does not change and the bubble grows
indefinitely as shown in case d.
In the case of double root in R1 the situation is the one depicted in case c of Figure 3.6 and it is a
limited case between b and d.
It can be seen that in the previous considerations the damping effect has been neglected but what happens in reality is that explosions and collapses are slowed by viscosity. It has been demonstrated (Poritsky 1952 and Shu 1952) that a viscosity of about 1200 times of the one of water corresponds to a collapse that lasts an infinite time. Hence in water case the collapse phase is not affected significantly from damping effects.
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Figure 3.6 (a) linear oscillations - (b) non linear oscillations - (c) limiting case between bubble oscillation and unlimited growth - (d) unlimited growth. Franc [4]
Another assumption which actually in real situations does not often occur is the step change of the pressure. If the variation of the pressure of the liquid far from the bubble is considered as slightly oscillating, then the radius of the bubble will oscillates too with
( )
0(
1 cos( ))
p t∞ = p∞ +ε ωt (3.18)[
]
0 1 (t) R R= +κ (3.19) where κ is small.In many circumstances also the assumption of small oscillation of the pressure is not accurate and the response of the bubble is not linear at all, exhibiting nonperiodic oscillations or subharmonics, as shown respectively in Figure 3.7 a and b where
a. ε = 1.5; ω/ω0 = 0.0154
33
Figure 3.7 Two examples of gas bubble under oscillating pressure. In case (a) strong nonlinearities are visible whereas in case (b) subharmonics appears. Franc [4]
The presence of a nonuniform pressure outside the cavity produces a nonspherical configuration of the bubble in such a way that in the higher pressure region a re-entrant jet appears into the bubble, where the gas and vapor pressures are uniform. These situations may arise in moving bubbles within a static liquid or in the collapse/explosion of a bubble when it is close enough to a solid wall or a free surface.
Indeed a solid wall would alter the sphericity of the interface and from the Laplace equation ∆φ = 0, along with the appropriate boundary conditions, it is possible to obtain the velocity potential φ at any instant t. Moreover, the velocities at the interface can be obtained from 𝑉�⃗ = 𝛻𝜑 whereas the generalized Bernoulli equation gives the lagrangian derivative of the velocity potential:
2 interface . 2 V p p d V V dt t ϕ ϕ ϕ ρ ∞− ∂ = + ∇ = + ∂ (3.20)
where the bubble is considered as filled only with vapor (pB = pV) and the surface tension is neglected.
Hence, assuming that the bubble’s interface nodes move with fluid velocity 𝑉�⃗, the position of the bubble interface at t+dt can be found and from the equation 3.20 it is possible to find the new velocity potential at the interface. The new boundary conditions can then be used in the Laplace equation and the cycle proceed as many boundary value problems for each time step t+dt.
The computational results of this model (Plesset-Chapman) for a bubble with R0 = 1 mm, which
center is at a distance of h = 1mm from the wall, can be seen in Figure 3.8. It is possible to observe that the opposite face, with respect to the wall, develops a hollow, and a re-entrant jet occurs, striking the solid wall and dividing the bubble in more parts. In the figure the time is nondimensionalized through the reference time 𝑅0�ρ⁄(𝑝∞− 𝑝𝑉).
The paths of the particles, during growth and collapse, are shown in Figure 3.9 where the ratio between the distance from the wall and the maximum radius of the bubble R is taken as unity.
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Figure 3.8 Plesset-Chapman’s numerical results of a collapsing bubble close to a solid wall. Franc [4]
Figure 3.9 Paths of the particles during growth and collapse of a vapor bubble. Franc [4]
3.4.
CAVITATING PUMP PERFORMANCE
Turbomachines such as inducers are intended to operate under moderate or heavy cavitation which, as previously seen, affects their performance and therefore it is important to obtain also the cavitating pump performance. This kind of characteristic can be represented by means of a family of curves Ψ(σ) obtained for each value of flow coefficient. A typical trend of these curves for a specific flow coefficient is represented in Figure 3.10.
It is possible to find three important cavitation numbers as σ is reduced. The first special cavitation number is the one at which cavitation occurs and it is termed cavitation inception number, σi, as
defined in chapter 3.2. At lower values the phenomenon of cavitation increases until it starts to affect the performance of the turbomachine. This is usually defined by a certain percentage loss in the head coefficient and the value at which occurs is termed critical cavitation number. Typical head loss value
35
for the definition of this special number can be 2, 3 or 5%. The third number that characterizes the cavitating performance curve is the breakdown cavitation number that is the level at which there will be a major deterioration of the performance and the head rise will tend to zero.
Figure 3.10 Typical cavitating performance curve Ψ(Φ,σ) with the three special cavitation numbers. Brennen [2]
A family of curves can also be represented in a single diagram as in Figure 3.11 where different flow coefficients are used. In the figure it is also possible to observe that cavitation inception occurs at low values of σ at the design flow coefficient but it increases as Φ is decreased. At low flow coefficients, the breakdown will occurs at higher values of cavitation number and it is also more abrupt than situations at higher flow coefficients.
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3.5.
THERMAL EFFECTS ON CAVITATION
An increase in the liquid temperature leads to a higher tendency to cavitate, as the vapor pressure increases and smaller pressure variations are needed.
Vaporization requires transfer heat between liquid bulk and bubble interface and therefore a thermal equilibrium equation is needed to find the unknown temperature inside the cavity.
If the change in pressure of the liquid occurs in a short time, the gas undergoes an adiabatic transformation since there can’t be a heat transfer between gas and liquid. On the other hand if a bubble with noncondensable gas only which undergoes a pressure variation during ∆t time, with an increasing radius from R to R+∆R, is considered, a change in the gas temperature will occur from T to T+∆T. This temperature variation will set a thermal boundary layer at the liquid around the bubble and therefore a heat flux will be present. The transformation can thus be taken as adiabatic if the internal energy variation is much more than the transferred heat. Conversely, ∆T is neglected and transformation becomes isothermal. In the latter case the transit time is so long compared with the time needed for heat transfer that a thermal equilibrium can be reached.
Hence in many practical situations the transformation can be considered as isothermal whereas, for bubbles with very large radius (like after a nucleus explosion that leads to a macroscopic bubble), the situation has to be re-examined.
In the vaporization process the latent heat comes from the liquid at the interface and if the temperature inside the bubble is smaller than the liquid temperature, then also the vapor pressure inside the bubble is smaller than the vapor pressure in the liquid at infinity. Hence the pressure difference between bubble and liquid will decrease and the bubble’s growth will reduce.
If the bubble contains both contaminant gas and vapor, the bubble pressure is given by the following equation G G B B G B V B 3 3m R T ( ) (T ) p (T ) p (T ) 4 B V p t p R π = + = + (3.21)
where RG is the gas constant, mG is the mass of the gas and TB is the temperature of the bubble.
However, in the evaluation of the vapor pressure it is convenient to use the temperature of the liquid bulk T∞ instead of TB. Hence from the Clausius-Clapeyron relation
(T ) (T ) 1 (T T ) (T )L(T )(T T ) T V V B V V B B L L L p p dp dT ρ ρ ρ ρ ∞ ∞ ∞ ∞ ∞ ∞ − Θ = = − = − (3.22)
where L is the latent heat, a new form of equation 3.21 can be found:
G G B V L 3 3m R T ( ) p (T ) 4 B p t R ρ π ∞ = − Θ + (3.23)
Clearly in the evaluation of the parameter Θ, the temperature TB(t) is unknown and therefore
further relations are needed. From a first thermal diffusion equation (T T )B r R L T r α t ∞ = − ∂ = ∂ (3.24)
where 𝛼𝐿 = 𝑘𝐿/𝜌𝐿𝑐𝑃𝐿 is the liquid thermal diffusivity and t is the time from bubble creation.
Furthermore equating the rate of use of latent heat with the heat conducted it is possible to obtain another important relation as follows:
37 V r R L L T dR r k dt ρ = ∂ = ∂ (3.25)
where kL is the liquid thermal conductivity.
Substituting these two equations in (3.23) it is possible to obtain the new expression of thermal term: 2 2 2 (T ) V L PL L L tdR tdR dt dt c T ρ ρ α ∞ ∞ Θ = = ∑ (3.26)
Considering water as the working fluid, at two different temperatures, 20 °C and 70 °C, the parameter Σ changes by several orders of magnitude and therefore the thermodynamic effects can be considered negligible only at 20 °C.
The evolution of the parameter Σ with temperature can be represented for any fluid as in Figure 3.12. From this figure it is possible to observe that at low temperatures, liquid oxygen and water are not affected by thermal effects whereas the liquid hydrogen behaves as a hot liquid and the effects of the temperature are not negligible.
Figure 3.12 Thermodynamic function Σ for different working fluids. Franc [4]
Although the effect of the temperature variation in vapor pressure, and therefore in cavitation number, has been already took into account in the performance by means of the difference between
38
inlet pressure and vapor pressure, another different effect can be observed in cavitation performance, as shown in Figure 3.13. It can be seen that for higher temperatures the performance will increase, delaying the cavitation breakdown that will therefore occur at a lower cavitation number. This effect can be recognized in different type of pumps operating with different liquids.
To explain this kind of behavior, the travel of a single bubble can be considered. As it passes through a low pressure region, its size will increase by vaporization of the liquid at the boundary. At this level two different situations may arise:
• At low temperatures the density of the vapor is low and therefore a small amount of liquid will vaporize (low mass rate). For this reason the latent heat needed is low and the vapor pressure inside the bubble will slightly fall below the vapor pressure of the liquid bulk. Hence the difference between the internal pressure and the liquid pressure is not significantly influenced by thermal effects.
• At high temperatures the vapor density inside the cavity is much more bigger of some order of magnitude and the mass rate of vaporization is high. The heat transferred to the bubble is high and a thick thermal boundary layer will be set at the interface and therefore the temperature inside the cavity will be much lower than the temperature of the liquid bulk and so the vapor pressure. Hence the force for bubble growth will be less than expected and this reduction in bubble growth will lead to less disruption of the flow and consequently the performance will improve.
Figure 3.13 Cavitation performance curves for a centrifugal pump operating at different water temperature. Brennen [2]
In the Rayleigh-Plesset equation the thermal effects are taken into account by means of the factor Θ. It is possible to define a critical time tC at which the thermal effect starts becoming significant.
39
important and the radius of the bubble will increases with √𝑡. Brennen showed that the relation obtained for tC is as follows:
2 3 2 3 1 1 min 2 2 2 2 1 1 ( ) 2 B T T c p L T p p R R t C R φ φ φ σ ρ − Ω Ω Ω = − − Ω Σ Σ (3.27)
Defining 𝛽 = 𝑡𝐶ϕ𝛺 as the value that separate the two different behavior influenced or not by
thermal effects, the equation 3.27 will define a critical breakdown cavitation number
2 pmin 2 3 1 2 b T C R σ β φ ∑ = − − Ω (3.28)
If thermal effects can be neglected Σ will be equal to zero and therefore σb will be -Cpmin.
Figure 3.14 Critical cavitation number ratio for different liquids and pumps. Brennen [2]
3.6.
CAVITATION EROSION
Collapsing bubbles near solid walls lead to material damage due to the violent process that produces shocks and highly localized and transient surface stresses which can exceed the material’s resistance (ultimate strength, fatigue limit, yield strength). The continuous collapse of a high number of bubbles at the same point can cause surface fatigue failures with a consequent release of material and therefore the surface will appear as crystalline and jagged (see Figure 1.4 and Figure 3.15). In propellers and pump impellers it can be a localized phenomenon that involves only certain areas due to periodical cloud cavitation.
40
Figure 3.15 Superficial fatigue failure due to cavitation damage on a mixed flow pump impeller. Brennen [1]
Indeed cavitation erosion is caused by shock waves and/or impulsive pressure as a consequence of microjets as previously said in chapter 3.3. However after the collapse a remnant cloud is present, composed by small bubbles that will collapse collectively again reproducing the phenomena of shock waves and microjets.
The velocity of the liquid relative to the wall influences the mass loss rate 𝑚̇ as follows:
0
( )n
m k V V= − (3.29)
where n is between 4 and 9 and V0 is a threshold velocity at which erosion occurs. A high value of n
means that the erosion damage is severe.
This mass loss rate has a time evolution. At the beginning of the exposure to cavitation (incubation period) there is no mass loss due to the plastic deformation and pitting of the material surface that occurs at this time. After this first period the erosion rate increases in time (acceleration period) reaching a steady state. A further attenuation period can be observed for long tests (see Figure 3.16).
Cavitating flows can become more aggressive as the flow turns unsteady, amplifying the violence of collapsing bubbles. Collective collapses, as a result of self-oscillating partial cavities, can also lead to a more violent environment from the erosion point of view with respect to a single collapsing bubble.
Microjets can occur in a nonsymmetrical collapsing bubble and, in the presence of a solid wall, the microjet is directed towards the wall if they are close to each other. These microjets can reach velocities, Vj, of the order of 100 m/s and the pressure rise can be estimated by means of the
Joukowski-Allievi formula:
j
p ρcV
∆ = (3.30)
where c is the sound’s velocity. In water the pressure pulse can reach values of 150 Mpa and its duration (of the order of microseconds) is fixed by the diameter of the jet.
41
Figure 3.16 Typical trend of mass loss rate. Franc [4]
In conclusion, shock waves and microjets leads to pressure pulses that are close to the yield strength of common metals.
Moreover the erosion rate has a strong dependence on the flow coefficient and cavitation number. A lower cavitation number will increase cavitation and therefore the effects seen before will be more significant, leading to higher weight loss.
3.7.
REFERENCES
[1] C.E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press, 1995. [2] C.E. Brennen, Hydrodynamics of Pumps, Oxford University Press, 1994.
[3] B.Lakshminarayana, Fluid Dynamics of Heat Transfer of turbomachinery, John wiley & sons Inc., 1996.
[4] J.P. Franc, J.M. Michel, Fundamentals of cavitation, Kluwer Academic Publishers, 2004. [5] Poritsky H., The collapse or growth of a spherical bubble or cavity in a viscous fluid.
Proc. 1st US Nat. Congress of Appl. Mech, 813 sq, 1952.
[6] Plesset M.S. & Mitchell T.P., On the stability of the spherical shape of a vapor cavity in a liquid. Quart. Appl. Math. 13, 419-430, 1955.
[7] Plesset M.S. & Prosperetti A., Bubble dynamics and cavitation. Ann. Rev. Fluid Mech. 9, 145-164, 1977.
[8] Plesset M.S. & Chapman R.B., Collapse of an initially spherical cavity in the neighbourhood of a solid boundary. J. Fluid Mech. 47-2, 283-290, 1971.
[9] Blake J.R. & Gibson D.C., Cavitation bubbles near boundaries. Ann. Rev. Fluid Mech. 19, 99-128, 1987.
