**2. Liquid-solid impact**

Liquid-solid impact is important in many engineering and industrial applications like the erosion of turbine blades due to high-speed impacts of condensed droplets in the expanding steam, cavitation damage in hydraulic components, erosion of aircraft wings due to the impact of rain droplets, erosion of soil due to rain droplets and impacts of water waves on river banks and erosion of embankments on the seashore.

The impingement of a liquid on a solid in the form of a jet or high-speed droplet was analysed by Joukowski in 1898, who first described the importance of compression waves in the liquid, mentioning the formation of high pressures arising on the liquid-solid impact. In 1928, Cook recognised the same concept in the form of his water hammer equation. In his theory, he explained the high pressure on liquid-solid impact by the formation of compression waves in the liquid taking into account the compressibility of the liquid. He proved that the water hammer pressure is many times higher than the steady pressure of a jet at the same velocity. He related it to the pressure Pimpact arising from the compressible nature of impacting liquid also known as the water hammer equation:

$$P\_{input} = \rho\_l \mathbf{C}\_l \mathbf{V}\_{input} \tag{1}$$

over which the compressed shock envelope occurred is given by radius Rcontact, which is of the

The liquid-solid interaction is further complicated when the solid surface becomes deformed from erosion, usually exhibiting peaks and craters. For example, a drop falling on a peak or slope may not develop the full impact pressure, and on falling in a crater, it may produce

The formation of the shock envelope was explained by Lesser in 1981 using the Huygens principle. As shown in **Figure 3**, in the initial regime, when the contact edge velocity is greater than that of a shock wave, at each instant the expanding liquid edge will emit an expanding

zones, one with expanding wavelets and another outside the wavelets where liquid is still not affected by impact. In the initial regime, the droplet edge will coincide with these wavelets and form the shock envelope. In the second regime (**Figure 4**), when the edge velocity is lower than the shock speed, the wavelet travels up to the free edge of the droplet, and the

The geometrical acoustic model from Lesser gives the detailed pressure distribution field inside the impacting drop [5, 6]. According to the model, the pressure at the centre of the

expanding droplet contact edge. This high pressure at the contact edge attains a maximum

**Figure 2.** Initial regime of droplet impact on surface with contact edge velocity higher than shock wave speed. The liquid is compressed in the shock envelope giving the maximum pressure. The shock envelope is composed by many wavelets.

Cl

. As release waves take an extra time of rVimpact/Cl2

**Figure 1.** Idealised diagram of early stages of liquid drop impact. From Heymann [4].

compressed liquid trapped in the shock envelope flows away laterally [5].

the flow will occur for the total time of 3rVimpact/2Cl2

increased pressure due to shock wave collisions [4].

wavelet moving with acoustic velocity Cl

impact is the water hammer pressure of ρ<sup>l</sup>

(r is the radius of droplet), whereas the time taken to complete this radius

An Overview of Droplet Impact Erosion, Related Theory and Protection Measures in Steam…

[1].

, so the compressed nature of

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93

. So at each instant, the liquid will consist of two

Vimpact; there are even higher pressures at the

order of rVimpact/Cl

is rVimpact/2Cl2

From Field [1].

where ρ<sup>l</sup> is the mass density of the liquid; Cl is the acoustic speed in the liquid, which, with some limitations, represents the speed of shock wave propagation in it; and Vimpact is the impact velocity [3]. This equation plays an important role in this work as it provides a means of scaling the impact pressure with impact velocity.

As illustrated by **Figures 1** and **2**, when a liquid droplet with a curved surface approaches a solid surface, then at the first instant of impact, the contact area increases with a velocity greater than the shock wave speed in the liquid. The exact value of this velocity depends upon the radius of the droplet and the velocity with which the droplet approaches the surface. So in the initial regime, the contact edge spreads out with a velocity greater than that of the shock wave, and the liquid is compressed within the shock envelope giving maximum pressure at the impact surface. This maximum pressure reduces to static pressure when the shock envelope overtakes the contact edge and release waves can enter the liquid. The contact region An Overview of Droplet Impact Erosion, Related Theory and Protection Measures in Steam… http://dx.doi.org/10.5772/intechopen.80768 93

**Figure 1.** Idealised diagram of early stages of liquid drop impact. From Heymann [4].

different possible phenomena including chemical attack, oxidation and solid particles carried by the steam was tried to be explained (Coles 1904) [1]. However by the 1920s, experimental studies focusing on the erosion of steam turbine blades by droplet impact had been started [2]. In 1928, Cook presented his water hammer equation in which he estimated the pressure generated when a liquid column impacts on a solid surface. In his theory, he proved that the pressure generated at the liquid-solid impact is sufficient to exceed the yield strength of many steel alloys typically used for steam turbine blades [1]. The following section aims to highlight the phenomenon of liquid-solid impact and to provide a brief review of the scientific findings

Liquid-solid impact is important in many engineering and industrial applications like the erosion of turbine blades due to high-speed impacts of condensed droplets in the expanding steam, cavitation damage in hydraulic components, erosion of aircraft wings due to the impact of rain droplets, erosion of soil due to rain droplets and impacts of water waves on

The impingement of a liquid on a solid in the form of a jet or high-speed droplet was analysed by Joukowski in 1898, who first described the importance of compression waves in the liquid, mentioning the formation of high pressures arising on the liquid-solid impact. In 1928, Cook recognised the same concept in the form of his water hammer equation. In his theory, he explained the high pressure on liquid-solid impact by the formation of compression waves in the liquid taking into account the compressibility of the liquid. He proved that the water hammer pressure is many times higher than the steady pressure of a jet at the same velocity. He related it to the pressure Pimpact arising from the compressible nature of impacting liquid

*Pimpact* = *ρ<sup>l</sup> Cl Vimpact* (1)

some limitations, represents the speed of shock wave propagation in it; and Vimpact is the impact velocity [3]. This equation plays an important role in this work as it provides a means

As illustrated by **Figures 1** and **2**, when a liquid droplet with a curved surface approaches a solid surface, then at the first instant of impact, the contact area increases with a velocity greater than the shock wave speed in the liquid. The exact value of this velocity depends upon the radius of the droplet and the velocity with which the droplet approaches the surface. So in the initial regime, the contact edge spreads out with a velocity greater than that of the shock wave, and the liquid is compressed within the shock envelope giving maximum pressure at the impact surface. This maximum pressure reduces to static pressure when the shock envelope overtakes the contact edge and release waves can enter the liquid. The contact region

is the acoustic speed in the liquid, which, with

and developments in this field.

92 Cavitation - Selected Issues

**2. Liquid-solid impact**

river banks and erosion of embankments on the seashore.

also known as the water hammer equation:

is the mass density of the liquid; Cl

of scaling the impact pressure with impact velocity.

where ρ<sup>l</sup>

over which the compressed shock envelope occurred is given by radius Rcontact, which is of the order of rVimpact/Cl (r is the radius of droplet), whereas the time taken to complete this radius is rVimpact/2Cl2 . As release waves take an extra time of rVimpact/Cl2 , so the compressed nature of the flow will occur for the total time of 3rVimpact/2Cl2 [1].

The liquid-solid interaction is further complicated when the solid surface becomes deformed from erosion, usually exhibiting peaks and craters. For example, a drop falling on a peak or slope may not develop the full impact pressure, and on falling in a crater, it may produce increased pressure due to shock wave collisions [4].

The formation of the shock envelope was explained by Lesser in 1981 using the Huygens principle. As shown in **Figure 3**, in the initial regime, when the contact edge velocity is greater than that of a shock wave, at each instant the expanding liquid edge will emit an expanding wavelet moving with acoustic velocity Cl . So at each instant, the liquid will consist of two zones, one with expanding wavelets and another outside the wavelets where liquid is still not affected by impact. In the initial regime, the droplet edge will coincide with these wavelets and form the shock envelope. In the second regime (**Figure 4**), when the edge velocity is lower than the shock speed, the wavelet travels up to the free edge of the droplet, and the compressed liquid trapped in the shock envelope flows away laterally [5].

The geometrical acoustic model from Lesser gives the detailed pressure distribution field inside the impacting drop [5, 6]. According to the model, the pressure at the centre of the impact is the water hammer pressure of ρ<sup>l</sup> Cl Vimpact; there are even higher pressures at the expanding droplet contact edge. This high pressure at the contact edge attains a maximum

**Figure 2.** Initial regime of droplet impact on surface with contact edge velocity higher than shock wave speed. The liquid is compressed in the shock envelope giving the maximum pressure. The shock envelope is composed by many wavelets. From Field [1].

**Figure 3.** Impact of droplet on a surface and subsequent Huygens wavelet construction forming shock envelope separating disturbed and undisturbed liquid. From Lesser [5].

value of 3ρ<sup>l</sup> Cl Vimpact just before the shock envelope overtakes the contact edge. The reason for this high pressure is the bunching of the wavelets at the contact edge as the contact edge velocity deceases. This high pressure of 3ρ<sup>l</sup> Cl Vimpact at the contact edge lasts only for a short duration of time and has a very small effect on the surface damage as compared to the sustained damage caused by the ρ<sup>l</sup> Cl Vimpact pressure.

The pressure field under the liquid impact was investigated experimentally by Rochester and Brunton (1974) and Rochester (1979) who recorded the pressure distribution using piezoelectric ceramic gauges embedded in the impact surface. The ratio of edge-to-central pressures was about 2.8, which is quite close to the theoretical estimations [1].

#### **2.1. Jetting angle and time**

The angle at which jetting starts was given by Bowden and Field in 1964 by the relations:

$$\beta\_c = \operatorname{Sin}^{-1} \{ M\_i \} \tag{2}$$

where r is the radius of the droplet. However, experimental studies found even greater angles and hence greater jetting times than those predicted by theory. Hancox and Brunton explained this by the effects of viscosity, which delays the onset of jetting [7]. However, this explanation was not logical due to the high pressures and velocities involved in this phenomenon. Lesser [5] suggested that the deformability of the target has a major effect

and PMMA as the target surfaces, confirmed the Lesser theory that target admittance has a major effect on the critical angle at which jetting commences [8]. However, this delay in jetting is too small to explain the completely different jetting times obtained by theory and experiments. Lesser and Field [6] tackled the problem in a different way. According to their theory, as the shock wave travels upwards, the liquid particles would be ejected by the release wave in a direction perpendicular to the local droplet surface (**Figure 5**). In this way, paths of ejected liquid particles would cross each other. They argued that during the initial stage of droplet impact, the edge angle β is very small, and the gap between the droplet surface and impact surface would practically be closed by the jet of ejected liquid particles and hence cannot be detected [8]. Using this theory to explain the delay

which the shock wave overtakes the contact edge and starts to spall liquid into the air gap and βj is the angle at which this spalled liquid moves ahead of the contact edge and can

Impact pressure on a solid surface upon the collision of a droplet is given by the water ham-

wave propagation in it and Vimpact is the impact velocity [3]. In 1933, de Haller pointed out that this pressure is valid only when the impacting surface is rigid [9]. In the case of a compressible

is the mass density of the liquid, Cl

<sup>1</sup> <sup>+</sup> *<sup>ρ</sup><sup>l</sup> <sup>C</sup>* \_\_\_\_*<sup>l</sup> ρ<sup>s</sup> Cs*

. Experiments, conducted by Field et al. in 1985 using steel

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An Overview of Droplet Impact Erosion, Related Theory and Protection Measures in Steam…

and β<sup>j</sup>

, where βc is the angle at

is the speed of shock

(4)

on increasing the jetting angle β<sup>c</sup>

be observed as a jet [6].

**2.2. Impact pressure**

mer pressure as ρ<sup>l</sup>

Cl

solid, the resulting pressure would be

in jetting, Field et al. (1988) suggested two values of β; β<sup>c</sup>

Vimpact. Here ρ<sup>l</sup>

*Pimpact* <sup>=</sup> *<sup>ρ</sup><sup>l</sup> Cl <sup>V</sup>* \_\_\_\_\_\_\_\_ *impact*

**Figure 5.** Trajectories of liquid particles upon the impact of a droplet. From Lesser and Field [6].

where Mi is the impact Mach number based on the liquid speed of sound and velocity for impact with a rigid target. The time at which jetting starts and at which the pressure reaches its maximum value is given by the relation:

**Figure 4.** The second regime in the liquid droplet impact when the edge velocity is lower than the shock wave velocity. The expanding wavelet travels up to the free edge of the liquid, and the trapped liquid is released away in the form of lateral jetting. From Lesser [5].

where r is the radius of the droplet. However, experimental studies found even greater angles and hence greater jetting times than those predicted by theory. Hancox and Brunton explained this by the effects of viscosity, which delays the onset of jetting [7]. However, this explanation was not logical due to the high pressures and velocities involved in this phenomenon. Lesser [5] suggested that the deformability of the target has a major effect on increasing the jetting angle β<sup>c</sup> . Experiments, conducted by Field et al. in 1985 using steel and PMMA as the target surfaces, confirmed the Lesser theory that target admittance has a major effect on the critical angle at which jetting commences [8]. However, this delay in jetting is too small to explain the completely different jetting times obtained by theory and experiments. Lesser and Field [6] tackled the problem in a different way. According to their theory, as the shock wave travels upwards, the liquid particles would be ejected by the release wave in a direction perpendicular to the local droplet surface (**Figure 5**). In this way, paths of ejected liquid particles would cross each other. They argued that during the initial stage of droplet impact, the edge angle β is very small, and the gap between the droplet surface and impact surface would practically be closed by the jet of ejected liquid particles and hence cannot be detected [8]. Using this theory to explain the delay in jetting, Field et al. (1988) suggested two values of β; β<sup>c</sup> and β<sup>j</sup> , where βc is the angle at which the shock wave overtakes the contact edge and starts to spall liquid into the air gap and βj is the angle at which this spalled liquid moves ahead of the contact edge and can be observed as a jet [6].

#### **2.2. Impact pressure**

value of 3ρ<sup>l</sup>

94 Cavitation - Selected Issues

where Mi

Cl

tained damage caused by the ρ<sup>l</sup>

**2.1. Jetting angle and time**

lateral jetting. From Lesser [5].

velocity deceases. This high pressure of 3ρ<sup>l</sup>

separating disturbed and undisturbed liquid. From Lesser [5].

*β<sup>c</sup>* = *Sin*<sup>−</sup><sup>1</sup>

*Tj* <sup>=</sup> *rVimpact* \_\_\_\_\_\_

its maximum value is given by the relation:

Cl

Vimpact just before the shock envelope overtakes the contact edge. The reason

Vimpact at the contact edge lasts only for a short

) (2)

<sup>2</sup> (3)

for this high pressure is the bunching of the wavelets at the contact edge as the contact edge

**Figure 3.** Impact of droplet on a surface and subsequent Huygens wavelet construction forming shock envelope

duration of time and has a very small effect on the surface damage as compared to the sus-

The pressure field under the liquid impact was investigated experimentally by Rochester and Brunton (1974) and Rochester (1979) who recorded the pressure distribution using piezoelectric ceramic gauges embedded in the impact surface. The ratio of edge-to-central

The angle at which jetting starts was given by Bowden and Field in 1964 by the relations:

impact with a rigid target. The time at which jetting starts and at which the pressure reaches

2 *Cl*

**Figure 4.** The second regime in the liquid droplet impact when the edge velocity is lower than the shock wave velocity. The expanding wavelet travels up to the free edge of the liquid, and the trapped liquid is released away in the form of

(*Mi*

is the impact Mach number based on the liquid speed of sound and velocity for

Cl

Vimpact pressure.

pressures was about 2.8, which is quite close to the theoretical estimations [1].

Impact pressure on a solid surface upon the collision of a droplet is given by the water hammer pressure as ρ<sup>l</sup> Cl Vimpact. Here ρ<sup>l</sup> is the mass density of the liquid, Cl is the speed of shock wave propagation in it and Vimpact is the impact velocity [3]. In 1933, de Haller pointed out that this pressure is valid only when the impacting surface is rigid [9]. In the case of a compressible solid, the resulting pressure would be

**Figure 5.** Trajectories of liquid particles upon the impact of a droplet. From Lesser and Field [6].

where subscripts l and s denote liquid and solid, respectively. In the water hammer equation, its descendants and the Lesser acoustic model 1981, the shock speed has been approximated by the acoustic speed of the liquid itself. However, Heymann in 1968 [10] explained that this approximation is valid only for the lower-impact speed regimes and if impact speed exceeds certain limits, then this simple relation does not hold anymore. In his model, he argued that the impact pressure is not uniform along the impact line. While the pressure at the centre of impact is water hammer pressure, there are even higher pressures found at the contact edge. The pressure at the contact edge increases gradually as the contact perimeter grows with time, while the pressure at the impact centre deceases. The maximum pressure found at the contact edge is up to three times the water hammer pressure at the moment of shock wave lifting up the droplet free surface. For higher-impact velocities, the impact pressure can be approximated by seeking the dependence of shock wave speed on particle speed change across the shock front as;

$$P\_{input} = \rho\_l \mathbf{C}\_l V\_{input} \left( 1 + \frac{k V\_{input}}{\mathbf{C}\_l} \right) \tag{5}$$

a shock speed in the range of 2600–3000 m/s, which is substantially higher than the ambient speed of sound in water [12]. By using the same conditions as used by Haller, the shock speed

Jetting occurs when the critical angle is reached, and the shock travels up the free surface of the drop. Bowden and Brunton (1961) suggested a relationship between the jetting angle and

β > βc, the jetting velocity is greater for smaller values of β. The particles that form jetting first travel normally to the drop surface and towards the target surface. They also cross each other's path on rebound, and the particles which travel closest to the target surface are those which are ejected later. In certain impact speed ranges, the jetting velocity is found to be up to 10 times the impact velocity [8]. This is further verified by Field et al. (1989) by high-speed photography. Haller (2002) numerically found that the jetting velocity of up to 6000 m/s can be obtained for a 100-μm droplet impacting on a surface with impact speed of 500 m/s. By using the same conditions as used by Haller and using Eqs. (2) and (7), the jetting velocity

Field et al. [8] observed that when a droplet impacts on a solid target, then after the initial regime with a high-pressure zone in the centre of impact, expansion waves come from the free surface and jetting commences (**Figure 6**). These expansion waves have the same magnitude as the compression waves, and the liquid is brought back to the initial ambient conditions. These expansion waves cross each other and bring the liquid into negative pressure and cause cavitation. These cavities collapse near the solid surface, produce both shocks and microjets, add pressure near the solid surface and contribute to the damage of the target surface [8]. Haller (2002) numerically studied the formation of cavitation during the impact of a 100-μm droplet on a solid surface with an impact speed of 500 m/s (**Figure 7**). His picture of droplet impact shows that after lifting up the droplet free surface, the shock wave reflects normally to the droplet free surface as expansion waves. These expansion waves create cavitation in the middle of the drop. Contrary to the cavitation picture given by Field in 1985, expansion waves in Haller's simulations are focused only in the middle of a drop and have no significant effect on the damage of the surface. However, Rein reported that upon the droplet impact, cavitation fields can be observed above the interface between the target surface and the liquid as well as below the apex of droplet. However, only the cavitation formed at the interface is well known

β\_\_

An Overview of Droplet Impact Erosion, Related Theory and Protection Measures in Steam…

is the jetting velocity and β is the jetting angle. Field et al. found that provided

<sup>2</sup>) (7)

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97

calculated by Heymann's Eq. (6) comes out to be 2500 m/s.

Vj = V*impact* Cot (

**2.4. Jetting velocity**

jetting speed as follows:

comes out to be 3000 m/s.

for severe erosion [13].

**2.5. Cavitation**

where Vj

The value of *K* is found to be 2 for water. So if a water drop impacts on a solid surface with an impact speed of 500 m/s, the impact pressure would be 1250 MPa, considerably above the yield strength of many alloys.

#### **2.3. Shock wave speed**

The water hammer equation ρ<sup>l</sup> Cl Vimpact has been derived from momentum considerations using an idealised case where the parameters are assumed to be invariant. This approximation is valid for relatively lower-impact velocities where the shock wave speed C can be reasonably approximated by the acoustic velocity of the liquid C<sup>l</sup> . However in the case of high-speed liquid impact, the compressibility can be taken into account in the variation of density ρ<sup>l</sup> and/ or shock wave speed C, and the water hammer pressure is needed to correct for the mass transport across the shock front due to compressibility.

Heymann [10] proposed an approximate relationship for water for the shock wave velocity C as a function of particle velocity change ΔU as follows:

$$\mathbf{C} = \mathbf{C}\_{l} + \mathbf{K}\Lambda\mathbf{U} \tag{6}$$

where Cl is ambient speed of sound and ΔU is the liquid particle velocity change across the shock front. K is some constant, and with the help of experimental data, he found K = 2 for water. This equation is limited for Mach number Mi < 1.2. Actually K is not a constant, and for very large Mach numbers, K approaches unity as k = ρ/(ρ − ρο ), where ρ is the density in the compressed state. The value of particle velocity change across the shock front during the initial regime of the impact is found to be equal to the impact velocity [11]. Haller also observed the same effect when he numerically calculated the shock wave speed by considering a 100 μm droplet impacting on a solid surface with an impact speed of 500 m/s. Within the first stage of impact where the shock wave is still in contact with the contact edge, he found a shock speed in the range of 2600–3000 m/s, which is substantially higher than the ambient speed of sound in water [12]. By using the same conditions as used by Haller, the shock speed calculated by Heymann's Eq. (6) comes out to be 2500 m/s.

#### **2.4. Jetting velocity**

where subscripts l and s denote liquid and solid, respectively. In the water hammer equation, its descendants and the Lesser acoustic model 1981, the shock speed has been approximated by the acoustic speed of the liquid itself. However, Heymann in 1968 [10] explained that this approximation is valid only for the lower-impact speed regimes and if impact speed exceeds certain limits, then this simple relation does not hold anymore. In his model, he argued that the impact pressure is not uniform along the impact line. While the pressure at the centre of impact is water hammer pressure, there are even higher pressures found at the contact edge. The pressure at the contact edge increases gradually as the contact perimeter grows with time, while the pressure at the impact centre deceases. The maximum pressure found at the contact edge is up to three times the water hammer pressure at the moment of shock wave lifting up the droplet free surface. For higher-impact velocities, the impact pressure can be approximated by seeking the dependence of shock wave speed on particle speed change across the shock front as;

The value of *K* is found to be 2 for water. So if a water drop impacts on a solid surface with an impact speed of 500 m/s, the impact pressure would be 1250 MPa, considerably above the

using an idealised case where the parameters are assumed to be invariant. This approximation is valid for relatively lower-impact velocities where the shock wave speed C can be reasonably

or shock wave speed C, and the water hammer pressure is needed to correct for the mass

Heymann [10] proposed an approximate relationship for water for the shock wave velocity C

C = C*<sup>l</sup>* + KΔ*U* (6)

shock front. K is some constant, and with the help of experimental data, he found K = 2 for water. This equation is limited for Mach number Mi < 1.2. Actually K is not a constant, and

the compressed state. The value of particle velocity change across the shock front during the initial regime of the impact is found to be equal to the impact velocity [11]. Haller also observed the same effect when he numerically calculated the shock wave speed by considering a 100 μm droplet impacting on a solid surface with an impact speed of 500 m/s. Within the first stage of impact where the shock wave is still in contact with the contact edge, he found

is ambient speed of sound and ΔU is the liquid particle velocity change across the

liquid impact, the compressibility can be taken into account in the variation of density ρ<sup>l</sup>

*kV* \_\_\_\_\_\_ *impact*

Vimpact has been derived from momentum considerations

*Cl* ) (5)

. However in the case of high-speed

), where ρ is the density in

and/

*Pimpact* = *ρ<sup>l</sup> Cl Vimpact*. (1 +

Cl

for very large Mach numbers, K approaches unity as k = ρ/(ρ − ρο

approximated by the acoustic velocity of the liquid C<sup>l</sup>

transport across the shock front due to compressibility.

as a function of particle velocity change ΔU as follows:

yield strength of many alloys.

The water hammer equation ρ<sup>l</sup>

**2.3. Shock wave speed**

96 Cavitation - Selected Issues

where Cl

Jetting occurs when the critical angle is reached, and the shock travels up the free surface of the drop. Bowden and Brunton (1961) suggested a relationship between the jetting angle and jetting speed as follows:

$$\mathbf{Vj} = \mathbf{V}\_{input} \mathbf{Cost} \left(\frac{\beta}{2}\right) \tag{7}$$

where Vj is the jetting velocity and β is the jetting angle. Field et al. found that provided β > βc, the jetting velocity is greater for smaller values of β. The particles that form jetting first travel normally to the drop surface and towards the target surface. They also cross each other's path on rebound, and the particles which travel closest to the target surface are those which are ejected later. In certain impact speed ranges, the jetting velocity is found to be up to 10 times the impact velocity [8]. This is further verified by Field et al. (1989) by high-speed photography. Haller (2002) numerically found that the jetting velocity of up to 6000 m/s can be obtained for a 100-μm droplet impacting on a surface with impact speed of 500 m/s. By using the same conditions as used by Haller and using Eqs. (2) and (7), the jetting velocity comes out to be 3000 m/s.

#### **2.5. Cavitation**

Field et al. [8] observed that when a droplet impacts on a solid target, then after the initial regime with a high-pressure zone in the centre of impact, expansion waves come from the free surface and jetting commences (**Figure 6**). These expansion waves have the same magnitude as the compression waves, and the liquid is brought back to the initial ambient conditions. These expansion waves cross each other and bring the liquid into negative pressure and cause cavitation. These cavities collapse near the solid surface, produce both shocks and microjets, add pressure near the solid surface and contribute to the damage of the target surface [8]. Haller (2002) numerically studied the formation of cavitation during the impact of a 100-μm droplet on a solid surface with an impact speed of 500 m/s (**Figure 7**). His picture of droplet impact shows that after lifting up the droplet free surface, the shock wave reflects normally to the droplet free surface as expansion waves. These expansion waves create cavitation in the middle of the drop. Contrary to the cavitation picture given by Field in 1985, expansion waves in Haller's simulations are focused only in the middle of a drop and have no significant effect on the damage of the surface. However, Rein reported that upon the droplet impact, cavitation fields can be observed above the interface between the target surface and the liquid as well as below the apex of droplet. However, only the cavitation formed at the interface is well known for severe erosion [13].

**3. Droplet impact erosion**

erosion damage with the fatigue mechanism.

**3.1. Time dependence of erosion rate**

as follows:

[14].

Thomas and Brunton [14] investigated repeated liquid impacts for several materials. Erosion curves are drawn for each material and then generalised into one curve, which shows the presence of three stages (**Figure 8**). The first stage is the incubation period during which no weight loss occurred, but some plastic or brittle deformation was noted. In stage 2, pits formed and grew by the removal of material. In stage 3, the erosion rate fell down to a lower value. The growth of small depressions (stage 1) into pits was explained by the stress concentrations. Even though the average stress is low, local soft points of materials may account for yielding. At the start, these local disturbances are very rare, so the first depression would appear with some delay. Later on, with the formation of many depressions, the erosion rate would arise. In stage 2, the tangential flow over the roughened surface also greatly influences the erosion phenomenon; work hardening and eventual fracture of the material occur. In stage 3, the rate of erosion declines again since the drop is broken up by the roughened surface; also the impact is no longer normal to the surface. They tried to compare the constant

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With less intense but repeated impacts, there is no immediate material loss, but randomly disposed dimples gradually develop, and the surface undergoes gradual deformation and work hardening. The material loss may occur through the propagation of fatigue-like cracks that intersect to release erosion fragments. In materials with non-uniform structure, damage will initiate at weak spots. In brittle materials, circumferential cracks may form around the impact site, which are caused by the tensile stress waves propagating outwards along the surface [4]. Heymann (1969) [4] characterised the repetitive impact erosion in five different stages (**Figure 9**)

**Figure 8.** The development of erosion in a number of materials eroded at an impact velocity of 125 m/s with a water jet diameter of 1.5 mm. (a) Experiment results and (b) three-stage model for erosion process. From Thomas and Brunton

**Figure 6.** Formation of cavitation at the impact of a jet on a solid surface. From Field et al. [8].

**Figure 7.** Formation of shock wave in a liquid droplet upon impact on a solid surface. From Haller et al [12].
