coupling bubble and material dynamics to model …€¦ · coupling bubble and material dynamics...

Paper No.xx-xxxx [DOI: 10.1299/xxx.2014xxx000x] © 2016 The Japan Society of Mechanical Engineers

1

Bulletin of the JSME

(DRAFT)

Coupling bubble and material dynamics to

model cavitation peening and pitting

Georges L. Chahine1, Anil Kapahi1, Chao-Tsung Hsiao1, and Jin-Keun Choi1 1DYNAFLOW, INC.

10621-J Iron Bridge Rd, Jessup, MD, 20794, USA

E-mail: [email protected]

Received 25 February 2016

Abstract

The effects of cavitation bubble dynamics on material peening and pitting is investigated numerically using a

coupled fluid and material dynamics approach. The model is applied here to the study of peening and pitting of

metallic materials resulting from non-spherical cavitation bubble collapse near the material. Bubble reentrant jet

impact and shock wave emission from the jet impact and from the collapse of the remaining bubble ring can

induce permanent micro-deformation, pitting, and residual stresses, which modify the roughness of the material

and harden it through pre-stressing. These effects are investigated through a parametric study for different bubble

material standoff distances. Quantities such as bubble collapse peak pressure, pit depth, and residual stresses

depend strongly on bubble standoff distance, which is an important factor in whether hardening or erosion of the

material occurs.

Key words : peening, pitting, erosion, fluid-structure interaction

1. Introduction

Cavitation is known for its deleterious effects such as loss of performance, noise, and material erosion (Kim et al.).

It initiates when the local pressure in the liquid drops below a critical pressure (e.g. the liquid vapor pressure) due to local

high velocities or accelerations of the liquid and this drives nuclei (microbubbles always present in liquids) to grow

explosively. When the pressure along the path of these bubbles returns to a high value, volume implosions occur resulting

in high pressure pulses and shock waves. Many pioneering studies (Plesset and Chapman)(Naude and Ellis) have shown,

experimentally as well as analytically, that the collapse of these bubbles near boundaries result in high-speed reentrant

liquid jets, which penetrate the highly deformed bubbles and strike the nearby rigid boundary generating water hammer

like impact pressures followed by remaining bubble volume collapse and emission of shock waves. Both shock waves

and high-speed reentrant liquid jets produce high local stresses in the adjacent material and are responsible for material

deformation and damage.

Useful applications of cavitation are expanding and include disinfection (Loraine et al.), organic components

oxidation (Kalumuck et al.), material cleaning (Georges L. Chahine et al.), and material strengthening through peening

(Nalla et al.). When the intensity of the stresses generated by bubble collapse exceeds the yield strength of the material,

permanent deformation and residual stresses remain in the material, which can increase its yield strength and harden it.

This loading mechanism can be done in a controlled manner to increase the life of the material against fatigue through

cavitation peening (Odhiambo). This phase of accumulation of residual stress is also related to the incubation period of

material erosion and pitting. Therefore, cavitation can result in desired residual stresses improving the fatigue life of

material or to damaging erosion and mass loss depending upon its intensity and duration of exposure.

Some of the critical material problems involve local fracture and initiation of corrosion at the surface. Therefore, the

life of equipment and its resistance to corrosion depends on its surface properties. Various methods, both chemical and

mechanical, have been used to improve the surface properties of metallic materials (Bhushan and Gupta). One of the

mechanical surface treatments is peening (Kanou et al.), which involves purposely creating local plastic deformations

Chahine, Kapahi, Hsiao and Choi, Mechanical Engineering Journal, Vol.00, No.00 (2016)

[DOI: 10.1299/xxx.2016xxx000x] © 2016 The Japan Society of Mechanical Engineers

leading to work hardening of the metal and imparting of residual compressive stresses (Wang et al.). This results in

improved fatigue life as the imparted compressive stresses prohibit the propagation of micro-cracks from the surface to

the bulk of the material.

Several methods have been developed for peening each using a different mechanism to impact the worked material

surface and induce plastic micro-deformation. These include solid collision, liquid collision, and laser treatment. One of

the first methods based on solid collisions involves usage of metallic shots in the form of impacting solid particles and is

named “shot peening” (e.g. Kobayashi et al. 1998). It is a cold working process with the projectiles being accelerated by

compressed air jets. This has been used for treating turbine discs (Taricco), engine parts (Zhuang and Wicks), landing

gears (Jayaraman et al.), airframe components (Sharp et al.), nuclear reactor vessel (Hur et al.), .. etc. Other methods

involve wet shot peening (“Microstructure and Fatigue Properties of Ti-6Al-4V Titanium Alloy Treated by Wet Shot

Peening of Ceramic Beads--《The Chinese Journal of Nonferrous Metals》2013 年 01 期”) ultrasonic shot peening

(Abramov et al.) and laser peening (Peyre et al.; Mannava et al.). In the ultrasonic and laser peening, cavitating bubbles

are formed from the deposited energy, which then collapse sending shock waves inside the material to be peened. Also,

liquid collision peening methods are applied (Grinspan and Gnanamoorthy), which use droplet impact from a high speed

liquid jet and are used in boiling and pressurized water reactors (Morinaka et al.). For submerged liquid jet peening, or

cavitating jets peening (Chahine, Choi, and Frederick), the load on the material is also due to bubble collapse (Odhiambo).

Cavitating jets are used for cleaning, cutting, drilling, and for controlled evaluation of materials’ resistance to cavitation

erosion (Chahine, Franc, and Karimi). The cavitation intensity can be varied in a wide range through adjustment of the

jet velocity, diameter, angle, standoff distance relative to the worked surface, and the ambient pressure in which they are

discharged.

The work presented here considers the fundamental mechanisms of the interaction between a collapsing bubble and

a nearby material. It describes detailed simulations of single bubble collapse near a material and models both the bubble

dynamics and the material response in order to demonstrate both peening leading to beneficial residual stresses and pitting

which is a precursor to erosion and damage.

2. Methodology

The numerical approach to model the coupled bubble/material dynamics is part of a general hybrid approach, which

we developed to simulate fluid structure interaction (FSI) problems involving shock waves and highly deforming bubbles

(Hsiao et al.). The bubble growth phase and the reentrant jet development are handled with a highly accurate Boundary

Element Method (BEM) (Chahine, Kalumuck, and Duraiswami). Reentrant jet impact on the other side of the bubble

and/or on the material and the following bubble ring collapse can result in shock wave formation and are handled using

a compressible solver (Wardlaw and Luton) (Kapahi, Hsiao, and Chahine), which seamlessly continues the BEM solution

using a link procedure. This hybrid procedure, illustrated in Fig 1, is an efficient way of handling bubble dynamics near

boundaries without compromising the physics of the problem.

Flow Stage Fluid Codes

Co

up

ler

Inte

rfa

ce

STRUCTURE

CODE

Tim

e

SHOCK PHASE COMPRESSIBLE

Compressible-

Incompressible

Link

BUBBLE PHASE INCOMPRESSIBLE BEM

Incompressible

Compressible

Link

REBOUND PHASE COMPRESSIBLE



Fig 1. Schematic diagram of the numerical approach used to simulate the interaction between a highly inertial bubble or a

cavitation bubble and a structure.

The dynamics of the material is modeled using the finite element model DYNA3D (McKeown et al.). In the results

shown in this work, metallic alloys Aluminum 7075 and Stainless Steel A2205 are used to illustrate cavitation peening

and pitting. These were selected because this study was part of an extensive series of simulations and tests reported in

(Kim et al.). These metallic alloys are modeled using an elastic-plastic model with linear slopes, one for the initial elastic

regime and the second, a tangent modulus, for the plastic regime (Dill). The details of various numerical algorithms used

for this work are given below.

2.1 Boundary Element Model

The Boundary Element Method (BEM) is used to model the incompressible liquid phase of the bubble dynamics

(Chahine and Kalumuck). The Laplace equation, 2 0, is solved for the velocity potential, , defined through

u , where u is the velocity vector. Green’s theorem is used to solve the Laplace equation

2 2G G d G G d

nS

S. (1)

In this expression is the domain of integration having elementary volume d andS includes all boundary surfaces of

such as the surface of the modeled bubble and the nearby surface to be cleaned. n is the local normal unit vector and

1/G x y is Green's function, where x corresponds to a fixed point in and y is a field point on the boundary surface

S. Eq. (1) reduces to Green’s formula

( ) ( ) ( , ) ( , ) ( ) ,S

Ga G dS

n n

x y x y x y y (2)

where a is the solid angle under which x sees the domain, . Eq. (2) provides a relationship between and /n at

the boundary surfaceS.Thus, if either of these two variables (e.g. ) is known everywhere on the surface, the other

variable (e.g. /n) can be obtained.

To solve Eq. (2) numerically, the surfaces of all objects in the computational domain are discretized into triangular

panels. To advance the solution in time, the coordinates of all surface nodes, y, are advanced according to /d dt y .

The velocity potential on the bubble surface nodes is obtained through the time integration of the material derivative of

, i.e. d/dt, which can be written as

. ,d

dt t

(3)

where /t can be determined from the Bernoulli equation:

1.

2lgz p p

t

. (4)

p is the hydrostatic pressure at infinity at z=0 where z is the vertical coordinate. lp is the liquid pressure at the bubble

surface, which balances the internal pressure and the surface tension,

l v gp p p C , (5)

where is the surface tension, and C is the local bubble wall curvature. The bubble internal pressure is composed of vp

the vapor pressure and gp the gas pressure. gp is assumed to follow a polytropic law, which relates the gas pressure to

the gas volume, ,V and reference value, 0gp , and 0V .

0 ,

k

g gp p

0

V

V (6)

where k is a gas compression constant; 1k corresponds to isothermal compression, while /p vk c c corresponds to

adiabatic compression. In this model, as in the conventional Rayleigh-Plesset bubble dynamics model (Plesset and

Prosperetti), phase change (evaporation and condensation) at the bubble wall is implicitly taken into account. Evaporation

and condensation are very fast compared to the bubble dynamics time scale and as a result vapor comes in and out of the

bubble quasi-instantaneously such that the vapor pressure remains constant. However, the gas inside of the bubble does



not have time to enter or exit and is subjected to compression and expansion determined by Eq. (6).

2.2 Compressible flow model

The compressible liquid phase of the bubble dynamics is addressed in this study using a multi-material compressible

Euler equation solver based on a finite difference method (Kapahi, Hsiao, and Chahine). Continuity and momentum

equations for the compressible liquid can be written as follows in Cartesian coordinates:

Q E F G

St x y z

, (7)

where

2

2

2

0

0

, , , , 0 .

t t t t

u v w

vuu u p wu

Q v E uv F v p G wv S

w uw vw gw p

e gwe p u e p v e p w

(8)

In Eq. (8) p is the pressure, u, v, and w are the velocity components in the x, y, z directions respectively, e is the

specific internal energy, and et = e+0.5(u2+v2+w2) is the specific total energy. The system is closed by using an equation

of state for each material, which provides the pressure as a function of the material specific internal energy and the

density. Here, a -law (with =1.4) is used for the gas-vapor mixture.

=( -1)p e, (9)

and the Tillotson equation is used for water (Anderson):

2 30 0

0

( ) , 1p p e e A B C

. (10)

, A, B, C are constants and 0 ,p 0 ,e and 0 are the reference pressure, specific internal energy, and density respectively. 92.20 10 Pa,A 99.54 10 Pa,B 101.48 10 Pa, C 5

0 1.0 10 Pa ,p 30 1000 kg/m , 5 2 2

0 3.54 10 m / s ,e

and 0.28.

The compressible flow solver 3DYNAFS-COMP uses a high order Godunov scheme (Colella). It can solve the

Riemann problem and construct a local flow solution that abruptly changes between adjacent cells. The numerical method

is based on a higher order MUSCL scheme and tracks each material. To improve efficiency, an approximate Riemann

problem solution replaces the full problem. The MUSCL scheme is augmented with a mixed cell approach to handle

shock wave interactions with fluid or material interfaces. This approach uses a Lagrangian treatment for the cells

including an interface and an Eulerian treatment for cells away from interfaces. A re-map procedure is employed to map

the Lagrangian solution back to the Eulerian grid (Wardlaw and Luton)(Kapahi, Hsiao, and Chahine). The CFL number

for fluid solver is calculated here as follows:

( )CFL maxu c t

x

, (11)

where maxu is the maximum value of all absolute local material speeds, c is the sound speed, t and x are the

time step and the grid spacing respectively. The selected value of the CFL is used to compute the adaptive time step

during the computation. The sound speed, c, in Eq. (11) is obtained as follows:

/ .c p (12)

2.3 Compressible-incompressible link procedure

Both incompressible and compressible flow solvers are able to model the full bubble dynamics on their own.

However, each method has its shortcomings when it comes to specific parts of the bubble history. The BEM based



incompressible flow solver is efficient, reduces the dimension of the problem by one (line integrals for an axisymmetric

problem, and surface integrals for a 3D problem) and thus allows very fine gridding and increased accuracy with

reasonable computation times. It has been shown to provide reentrant jet parameters and speed accurately. However, it

has difficulty pursuing the computations beyond surface impacts (liquid-liquid and liquid solid).

On the other hand, the compressible flow solver is most adequate to model shock wave emission and propagation,

liquid-liquid, and liquid-solid impacts. The method requires, however, very fine grids and very small time steps to resolve

shock wave fronts. This makes it appropriate to model time portions of the bubble dynamics. Concerning the bubble-

liquid interface and the reentrant jet dynamics, the procedure is diffusive since the interface is not directly modeled and

the computed reentrant jet characteristics are usually less accurate than obtained with the BEM approach.

Hence our novel approach combines the advantages of both methods and consists in executing the following steps:

1. Setup the initial flow field using the Eulerian compressible flow solver, 3DYNAFS-COMP, and run the simulation

until the initial shock fronts exit the domain and the remnant flow field can be assumed to be incompressible.

2. Transfer at that instant to the Lagrangian BEM potential flow solver, 3DYNAFS-BEM, all the flow field variables

needed by the solver: geometry, bubble pressure, boundary velocities to specify the moving boundary’s normal

velocities, / n .

3. Solve for bubble growth and collapse using fine BEM grids to obtain a good description of the reentrant jet until

the point where the jet is very close to the opposite side of the bubble.

4. Transfer the solution back to the compressible flow solver with the required flow variables. To do so, compute

using the Green’s equation all flow field quantities on the Eulerian grid.

5. Continue solution progress with the compressible code to obtain pressures due to jet impact and remnant bubble

ring collapse.

The transfer mechanism between the two methods (compressible and incompressible) is fully conservative and we

observed no instabilities following the link procedure.

2.4 Finite Element Model:

The materials’ dynamics response to loadings, which are characteristic of cavitation, is studied using the finite

element model, DYNA3D, developed by the Lawrence Livermore National Laboratory. DYNA3D (McKeown et al.) is a

non-linear explicit solid and structural dynamics solver. It is based on a lumped mass formulation and solves the structure

momentum equation:

MU+CU+KU = F , (13)

where M is the mass matrix, U is the displacement vector, C is the damping matrix, K is the stiffness matrix, and F is

the matrix of the applied loads on the structure. DYNA3D is appropriate for problems where high strain rate dynamics or

stress wave propagation effects are important. Many material models are available to represent a wide range of material

behavior, including elasticity, plasticity, thermal effects, rate dependence, damage, and failure of elements.

Material Models Used

In this study, two metal alloys (Aluminum 7075 and Stainless Steel A2205), were examined. These were part of an

extensive experimental and numerical study reported in (Kim et al.). The alloys were modeled as elastic-plastic with two

linear slopes, one for the initial elastic regime and the other for the plastic regime. The parameters of the model used in

this study are shown in Table 1. The stress-strain curves corresponding to the two materials are shown in Fig 2.

Metallic

Alloys

Yield

Stress

(MPa)

Young’s

Modulus

(GPa)

Tangent Modulus

(MPa)

Elongation at

Break

Density

(g/cm3)

Al 7075 503 71.7 670 0.11 2.81

A2205 515 190 705 0.35 7.88

Table 1. Material properties of the metal alloys simulated in this study.



Fig 2. Stress-strain curves for aluminum Al 7075 and stainless steel A 2205.

3. Problem Setup

To illustrate the bubble / material dynamics, we consider a 50 µm bubble in equilibrium at 105 Pa with the surrounding

liquid and subject it to a pressure drop to 103 Pa for 2.415 ms followed by a pressure rise to 107 Pa. The time dependence

of this pressure can also be written as follows:

5

3

7

10 Pa; 0,

( ) 10 Pa; 0 2.415ms,

10 Pa; 2.415ms.

t

p t t

t

(14)

The bubble dynamics near the wall up to the point of reentrant jet impact can be simulated using the 3D BEM solver

(Chahine and Kalumuck). Fig 3 compares the bubble radius versus time between the Rayleigh-Plesset solution and the

3D solution. The three-dimensional dynamics results in a reduction of the bubble maximum volume relative to the free

field Rayleigh-Plesset solution due to the material wall confinement effects.

Fig 3. Comparison of the equivalent radius versus time of the deforming bubble with the Rayleigh-Plesset solution. Initially

the bubble is spherical of radius 50 μm and is located at a distance of X = 1.5 mm above a flat material surface. It is then

subjected to the pressure field described by Equation (14).



As illustrated in Fig 4 from the BEM solution, the applied pressure field results in the bubble expansion to an

equivalent radius of 2 mm followed by a strong collapse with reentrant jet formation and pressure loading of the nearby

metallic plate located at a standoff of 1.5 mm. It is more appropriate to present the results in terms of the non-dimensional

standoff X , which is defined as the ratio of the distance of the bubble center to the wall, X, to the bubble maximum

radius, Rmax.

max/ .X X R (15)

Fig 4. Bubble shape during (a) growth phase and (b) collapse phase. 3DYNAFS-BEM results: R0=50 μm, 0.75.X

The BEM solution at the moment of touchdown is then mapped to provide initial conditions for 3DYNAFS-COMP,

which then computes the dynamics of the reentrant jet impact and the ensuing multi-connected bubble and couples this

dynamics with the materials dynamics to provide stress distributions as shown in Fig 5. A high pressure wave is emitted

when the jet impacts the other side of the bubble (Fig 5a) and then when it impacts the wall (Fig 5c). Another high pressure

or shock wave is emitted with the remaining bubble ring collapses (Fig 5d). These instantaneous high loads induce high

stresses to substantial depths in the material as illustrated in Fig 5e-h.

The deformation of the central node of the plate along with the pressures recorded at that location is shown in Fig 6.

Note that there is a sharp rise in the pressure due to the reentrant jet impact. The subsequent bubble ring collapse results

in an even higher pressure, 1.5 µs later, as seen in the figure.

The contours of the residual stresses in the peened material are shown in Fig 7a. The figure also indicates the increase

in roughness due to the formation of a microscopic pit under the bubble. The resulting depth of the deformation of the

surface is negligible compared to that generated by solid body collisions as done in shot peening (Bagherifard et al.).

Finally, the temporal variation of the residual stress developed in the material along the depth is shown in Fig 7b. Note

that the residual compressive stress saturates with time and increases the yield strength of the material.

Fig 5. (a)-(d) Bubble contours and liquid pressures versus time, (e)-(h) corresponding material stress contours during

bubble collapse after reentrant jet impact. R0 = 50 µm, Rmax = 2 mm, Pd = 10 MPa and 0.75.X

1.5 mm

Time Time

(a) (b)

(a) (b) (c) (d)

(e) (f) (g) (h)



Fig 6. Central node displacement and pressure versus time for R0 = 50 µm, Rmax = 2 mm, Pd = 10 MPa and 0.75.X

Fig 7. (a) Residual equivalent stress in the material and pit shape. (b) Residual stress with depth at different times.

This loading mechanism for a single bubble collapse shows that cavitation generated impacts done in a controlled

manner can increase the resistance of the material against fatigue. The tiny pit formed during the bubble collapse process

also increases the roughness of the material making it more conducive to bonding. Therefore, by controlling the intensity

of the cavitation and the duration of exposure, the residual stresses are increased and the fatigue life of the material

improved.

4. Effect of standoff

This section describes the effects of bubble standoff on the formation of pits and the resultant peening effects in the

material. Besides standoff, the same conditions used above are used in this section. Fig 8 shows the time history of the

bubble equivalent radius for different initial non-dimensional standoff distances: X 0.5, 0.75, 1.0, and 1.5, while

Fig 9 shows the shape of the bubble in each case at a time close to the reentrant jet touching the other side of the bubble.

Compressible flow computations were conducted after these points in time. In all cases shown, the bubble starts at

0 50 ,R m grows to a bubble maximum Rmax = 2 mm, and is then subjected to a pressure driving the collapse

10 MPa.dP The duration of the pressure drop, t, was adjusted such that the sudden pressure rise, Pd, was imposed at

the time when the bubble radius reached 2 mm.

(a) (b)



Fig 8. Evolution of the equivalent bubble radius for different normalized standoff values at bubble maximum volume.

R0 = 50 µm, Rmax = 2 mm, Pd = 10 MPa.

In Fig. 8, the bubble shapes close to the time of reentrant jet impact show that the collapse occurs earlier (larger

bubble volume at time of impact) and the reentrant jet becomes more pronounced when the bubble is closer to the wall.

At the larger standoffs, the bubble volume shrinks significantly before the jet develops, while closer to the wall the

reentrant jet develops much earlier while the bubble still has a large volume when the jet reaches the opposite side of the

bubble. From these contours one can expect very different pressure loadings on the material surface for different values

of X .

Fig 10a compares the momentum average jet velocity,momV , at the touchdown moment (time the jet reaches the other

side of the bubble) for different standoff distances. momV is defined as:

1

,mom J d V V VV

(16)

where JV is the velocity of the liquid at a field point inside the jet, and V is the jet volume. It is seen that the jet velocity

increases as the standoff distance is increased. This is because the largest bubble wall speed is achieved when the bubble

is spherical (G L Chahine et al.) since the bubble wall has then enough time to accelerate. A better illustration of the

energy in the jet could be the total momentum of the jet,momV V , at the moment it touches the opposite side of the bubble.

This is shown in Fig 10b.

Fig 9. Bubble contours at the time of compressible-incompressible link for different normalized standoff distances between

the bubble and the wall for R0 = 50 m, Rmax = 2 mm, and Pd = 10 MPa.

X =0.5

X =0.75

X =1.0

X =1.5

X =0.5

X =0.75

X =1.0

X =1.5



Fig 10. Variations with the bubble wall normalized standoff distance of a) the momentum averaged jet velocity, Vmom, and

b) the jet momentum at touchdown for R0 = 50 m, Rmax = 2 mm, and Pd = 10 MPa.

Here we can see that there is an optimum distance ( ~ 0.75)X at which the energy in the jet is maximal. Similar

observations have been reported by both numerical studies (Chahine) (Jayaprakash, Chahine, and Hsiao) and

experimental studies (Brujan et al.) (Harris et al.). Actually, a higher jet velocity does not necessarily result in a higher

impact pressure on the wall because the distance between the jet front and the wall at the touchdown moment is also very

important. This is illustrated in the material deformation results below.

Fig 11. Pressure versus time at the center of the Al 7075 plate for different bubble plate normalized standoff distances

for R0 = 50 m, Rmax = 2 mm, and Pd = 10 MPa.

Fig 11 shows the pressure versus time monitored at the plate center for different standoff distances. All standoff

distances induce pressure waves containing both a jet impact, which results in a shock wave, and a shock wave emitted

by the remaining bubble ring collapse. However, the relative intensity of the two types of shock depends on the standoff.

It is seen that the pressure loading due to the jet impact is much higher for smaller standoff, especially for 0.5X , since

in this case the reentrant jet directly impacts on the material surface when it penetrates the other side of the bubble. As

the standoff increases, the magnitude of the pressure due to the jet impact is reduced because the high speed liquid has

to travel a longer distance while submerged before reaching the material surface. For 1.5X , only one significant

pressure peak with a typical exponential decay is observed because the jet touchdown occurs almost at the same time as

when the bubble reaches the minimum size and no significant jet pressure reaches the wall. Instead, a shock wave type

pressure profile is observed. Fig 11 also illustrates the time period during which the impact pressures exceed the yield

stress of the material. One has to be careful however to not confuse impact pressures at the surface and actual stresses

X =0.5

X =0.75

X =1.0

X =1.5

Yield stress

(a) (b)

Standoff, X Standoff, X



imparted inside the material.

Fig 12. Variations of (a) pit depth, (b) pit radius, and (c) pit volume with the normalized standoff distance for Al 7075 and

A 2205 for R0 = 50 m, Rmax = 2 mm and Pd = 10 MPa.

The influence of the normalized standoff distance on pit characteristics is very important and relevant for damage

assessment. Fig 12 shows three pit characteristics for the two metallic alloys studied: depth, radius, and volume

respectively as functions of the normalized standoff distance. Fig 12 shows that pit depth and volume continually decrease

when the standoff distance increases. However, as for the jet momentum, pit radius goes through a maximum when the

standoff distance is close to 0.75X . This is expected since the jet momentum correctly reflects the impact energy

contained in the jet and should correlate better the jet speed with the pit formed. The volume, not provided directly by

the software, was approximated by the volume of a cone with the same base diameter and height. Actually, the shape of

the pit varies with standoff as shown in Fig 13. At the smallest standoffs, the pit radius is smaller with 0.5X than with

0.75X , while the pit depth is larger with 0.5X than with 0.75X . These shapes reflect the impact of the jet, which

produces a thin and deep deformation and the bubble ring collapse, which generates a wider and shallower deformation.

Fig 13. Comparison of pit shape between normalized distances 0.5X and 0.75,X for aluminum Al7075 for R0 = 50 m,

Rmax = 2 mm, and Pd = 10 MPa.

The effect of standoff on the residual stresses left in the material can be seen in Fig 14. The stress magnitude is higher

for smaller standoffs. This is due to the proximity of the bubble to the metallic surface leading to higher impact pressures.

On the other hand, the propagation of the residual stress to deeper areas is higher for larger standoff value. This can be

due to the duration of the pressure peaks, which gets longer as the standoff increases as shown in Fig 10.

Standoff X

Pit

Vo

lum

e,

m3

0.4 0.6 0.8 1 1.2 1.4 1.60

1000

2000

3000

4000

5000

6000

A2205

AL7075

(b) (a) (c)

X =0.5

X =0.75

Standoff, X Standoff, X Standoff, X



Fig 14. Comparison of induced residual stresses for Al7075 (a) 0.5,X (b) 0.75,X (c) 1.0,X and for A2205

(d) 0.5,X (e) 0.75,X (f) 1.0.X

The change in residual stress with depth for different standoffs for both materials is shown in Fig 14. The figure

clearly shows that the residual stresses are higher for lower standoff values and the stress is induced much deeper for the

higher standoff values.

Fig 15. Comparison of variation of induced residual stress with depth for different standoffs (a) Al7075 (b) A2205.

5. Conclusion

Material pitting and peening due to cavitation bubble collapse was studied by modeling the dynamics of growing

and collapsing cavitation bubbles near deforming materials with an initial flat surface. The bubble nucleus, initially at

equilibrium with the surrounding liquid near wall, was subjected to a time dependent pressure field. The pressure first

(a) (b)

X =0.5

X =0.75

X =1.0

X =0.5

X =0.75

X =1.0

(a) (b) (c)

(d) (e) (f)



dropped to a value below the bubble critical pressure, stayed at this pressure for a prescribed time, and then rose to a high

pressure value. As a result the nucleus first grew explosively then collapsed violently near the wall forming a fast reentrant

jet, which hit the wall and deformed it permanently when the collapse intensity was high enough to result in stresses

exceeding the material elastic limit. The pressure loading on the material surface during the bubble collapse is found to

be due to both the reentrant jet impact and to the collapse of the remaining bubble ring. The high pressure loading results

in high stress waves, which propagate radially from the loading location into the material and cause the deformation. A

pit (permanent deformation) is formed when the local equivalent stresses exceed the material yield stress. These

compressive stresses stay in the material as residual stresses and have benefits related to improved fatigue life of the

material. The tiny pits then formed can help in improving the bonding strength of the material.

A parametric study involving the initial normalized standoff distance between the bubble and the material showed

that the normalized standoff distance affects the jet characteristics in a non-monotonic fashion. Higher jet velocities occur

at the larger normalized standoff distances. However, the energy in the jet is maximum at a normalized standoff distance

close to 0.75.X A higher jet velocity does not necessarily result in a higher impact pressure, since the impact pressure

also depends on the distance between the wall and the jet front at the touchdown moment. A more concentrated pressure

loading on the material surface is obtained for smaller standoffs where the jet touches down and the bubble ring collapses

very close to the wall. Such concentrated pressure loadings result in deeper but narrower pits. As a result, the shape of

the pit, i.e. the ratio of pit radius to depth, does not vary monotonically with standoff. This also affects the induced

residual stresses which are higher and shallower for small standoffs and smaller and deeper for larger standoffs.

Acknowledgements

This work was conducted under partial support from DYNAFLOW, INC. internal IR&D and from the Office of Naval

Research under Contract N00014-15-C-0052, monitored by Dr. Ki-Han Kim. We appreciate this support.

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