*Corr. Author’s Address:  University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, Maribor, Slovenia, ignacijo.bilus@um.si 71
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81 Received for review: 2021-08-10
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2021-12-17
DOI:10.5545/sv-jme.2021.7364 Original Scientific Paper Accepted for publication: 2022-01-13
Cavitation Erosion Modelling on a Radial Divergent  
Test Section Using RANS
Kevorkijan, L. – Lešnik, L. – Biluš, I.
Luka Kevorkijan – Luka Lešnik – Ignacijo Biluš
*
University of Maribor, Faculty of Mechanical Engineering, Slovenia
Cavitation is the phenomenon of fluid evaporation in hydraulic systems, which occurs due to a pressure drop below the value of the vapor 
pressure. For numerical modelling of this generally undesirable phenomenon, which is often associated with material damage (erosion), there 
are various mathematical vapor transfer models that have been validated in the past. There are different approaches to predicting cavitation 
erosion, which have  mostly been experimental in the past. Recently various numerical models have been developed with the development of 
numerical simulations. They describe the phenomenon of cavitation erosion based on different theoretical considerations, such as Pressure 
wave hypothesis, Microjet hypothesis, or a combination of both. 
In the present paper, an analysis of the Schnerr-Sauer transport cavitation model was used, upgraded with an erosive potential energy 
model based on pressure wave hypothesis for cavitation erosion prediction. The extended numerical model has been applied to the case 
of a radial divergent test section in three different mathematical formulations. The results of simulation were compared and validated 
to experimental work performed by other authors. The study shows that the distribution of surface accumulated energy agrees with the 
experimental results, although certain differences exist between formulations. The applied method appears to be appropriate for further use, 
and to be extended to materials response modelling in the future. 
Keywords: cavitation, erosion, erosive potential energy, numerical simulation
Highlights
• 	 The cavitation erosion process can be predicted by cavitation potential energy modelling. 
• 	 Numerical definition of erosive power is possible with different formulations of cavitation potential. 
• 	 Cavitation energy focusing accomplishes accurate prediction of cavitation erosion damage.
0  INTRODUCTION
Cavitation is the phenomenon that consists of the 
formation, activity and collapse of vapor structures 
inside a liquid medium. Usually, the phenomenon 
is undesired, since it causes many negative effects. 
Among those negative effects the cavitation erosion is 
the most complex one, since it combines complicated 
hydrodynamics with solid material response at the 
mechanical and metallurgical perspectives. The 
research of cavitation phenomenon dates to the late 
19
th
 and early 20
th
 centuries. In 1917, Lord Rayleigh 
[1] derived the equation for calculation of the pressure 
in a liquid during the collapse of a spherical bubble. 
In 1949, Plesset [2] upgraded this work by writing the 
differential equation of the dynamics of a spherical 
bubble, today known as the Rayleigh-Plesset 
equation. Mostly experimental research of cavitation 
damage was conducted during the theoretical research 
of the cavitation phenomenon.. Hammit [3] studied 
cavitation damage on the case of fluid flow through 
a Venturi channel, and introduced the erosion model 
based on the energy spectrum of a vapor cavity. Vogel 
et al. [4] observed the dynamics of the collapse of a 
bubble along a solid wall, and derived the equation 
for calculation of the potential energy of the collapse. 
By the end of the 20
th
 century, two hypotheses were 
known about the origin of cavitation damage. The first 
hypothesis followed the high value of the pressure 
in the liquid around the bubble as it was calculated 
by Lord Rayleigh [1]. The existence of a pressure 
wave was confirmed by many experimental studies, 
for example, by Harrison [5]. A second, “microjet” 
hypothesis was proposed by Kornfeld and Suvorov 
[6], while Plesset and Chapman [7] predicted the 
microjet velocity numerically, and claimed that a 
microjet can damage the material.
Based on the pressure wave hypothesis, Fortes 
Patella et al. [8] presented an approach to erosion 
assessment with an energy cascade that transfers the 
initial potential energy of large vapor structures to the 
smallest structures, and eventually to the surface walls. 
A key contribution was the definition of the potential 
power of cavitation structures based on the total 
derivative of potential energy. The model upgrade is 
described in a paper of Leclercq et al. [9], where all of 
the vapor structures, even those not in direct contact 
with the wall, were considered. In the paper by Carrat 
et al. [10] the results of numerical analysis, including 
the amplitude and frequency of the pressure waves, 
were presented and validated experimentally. Based 
on their findings, Schenke and Terwisga [11] proposed 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
72 Kevorkijan, L. – Lešnik, L. – Biluš, I.
a model of forecasting the erosion threat where energy 
generated by the vapor structures is conserved and 
transferred to the wall surface via the pressure wave 
released after the collapse. They also introduced 
a continuous formulation of energy transfer from 
wall-distant structures to the wall surface. Li [12] 
introduced the erosion intensity function, which is 
based on pressure only.
On the other hand, Dular et al. presented the 
semi-empirical cavitation erosion model based on the 
microjet hypothesis [13]. The same research group 
have contributed a lot mainly to the experimental field 
of cavitation erosion research, [14] and [15]. Based on 
their experimental findings and semi-empirical model, 
Peters et al. [16] introduced the multilevel modelling 
of two-phase flow, where the  Euler-Euler and Euler-
Lagrange approach were combined.
Currently, the research contributions are focused 
on the use of existing models on practical problem 
geometries [17], which are, in fact, more complex 
than in the past, due mainly to the rapid progress of 
computational capabilities [18]. Brunhart et al. [19] 
introduced the use of erosion indicators on the case 
of gaps within a diesel fuel pump. Gomez Santos et 
al. [20], similarly, presented the comparison of erosion 
models in engine injectors operating with ethanol. 
Despite intensive research work and numerous 
publications in the field of cavitation erosion, the 
derived models have still not been perfected, and offer 
many opportunities for upgrading and improvement. 
The reason for this is mainly the lack of consensus 
on the predominant mechanism of cavitation erosion. 
Following this, Arabnejad et al. [21] joined both 
erosion mechanisms. In their work, the collapse 
induced kinetic energy is divided into the kinetic 
energy of the pressure wave and kinetic energy of the 
microjet.
In the present work, a model by Schenke and 
Terwisga [11] has been applied to a radial divergent 
test section, which is a common test case for 
cavitation studies found in the literature. The mesh 
was generated with a meshing module within ANSYS 
Fluent 2020 R2, cavitating flow and cavitation 
erosion were calculated using the computational 
fluid dynamics (CFD) software ANSYS Fluent 2020 
R2, where the model to predict cavitation erosion by 
Schenke and Terwisga [11] was implemented via a 
user defined function (UDF) which was written in the 
C programming language, compiled and linked to the 
solver. 
Different authors analyzed cavitation on this flow 
case numerically, but they applied different cavitation 
erosion models [16]. In this work, we applied an 
erosive potential energy model based on the pressure 
wave hypothesis with three different mathematical 
formulations. The transfer of energy from the 
cavitation structures to the surface was accomplished 
with and without the energy focusing method. The 
numerical results show good trends and agreement 
with the experimental results, with certain limitations, 
which could be a subject of further studies and model 
optimizations. 
1  CAVITATION EROSION 
1.1  Homogeneous Mixture Method  
In the following, we focus on the general formulation, 
and treat the fluid as a continuous mixture consisting 
of the volume fraction of the vapor phase  and the 
volume fraction of the liquid phase (1 – α). The flow 
is considered as isothermal, pure phases are treated 
as incompressible. The mixing rule applies to the 
properties of the mixture, the mixture density ( ρ) and 
the mixture viscosity ( μ) 
      
vl
1, (1)
      
vl
1, (2)
where the liquid and vapor densities are ρ
l
 =  
998.85 kg/m
3
 and ρ
v
 = 0.01389 kg/m
3
. The liquid and 
vapor viscosities are μ
l
 = 0.0011 Pa·s and μ
v
 = 9.63· 
10
–6
 Pa·s.
1.2  Conservation of Vapor Phase Mass
The vapor mass is defined as m
v
 = ρ
v
 V
v
 = ρ
v
 α V , where  
V
v
 is the vapor volume and V is the mixture volume. 
According to this, the mass flow of vapor is
  m
Dm
Dt
dV
DV
Dt
dV
v
V
v
V
v



, (3)
and the source of the vapor fraction is 
 
V
v
v
V
vv
t
ud VS dV












 



, (4)
where 

u is mixture velocity. If we write Eq. (4) in 
differential form and consider pure phases 
incompressible, we get
 


 



t
uS
v


, (5)
where S
αv
 presents the source of the vapor phase.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
73 Cavitation Erosion Modelling on a Radial Divergent Test Section Using RANS 
1.3  Conservation of the Mixture Mass
Similarly to Eq. (5), the mass conservation for the 
mixture is written as
 


 


t
u


0. (6)
From      
vl
1� and 





l
vl
 we 
can write 
 


   
 




 t
uu u
l
vl





, (7)
and after rearrangement 
 


  





 t
uu
vl



, (8)
which presents the conservation equation of the 
mixture mass written by the vapor volume fraction α.
1.4  Conservation of the Mixture Momentum
Within a homogeneous mixture, the approach for the 
momentum conservation is written for the mixture as
 


   






u
t
uu p , (9)
where τ is the viscous stress tensor and is evaluated as
   





uu
T
.  (10)
From Eq. (9) it is noticeable that body forces 
(gravity) are omitted in this work. The flow was 
considered as incompressible, while compressible 
behavior of the mixture is exhibited during the phase 
transition, as in the work of [21] and [22].
1.5 Turbulence Modelling
To model turbulent flow, the RANS approach was 
adopted, with the SST k  –  ω model. In order to 
address the transient cavitation effects that arise due 
to the compressibility of a liquid-vapor mixture, the 
turbulent viscosity μ
t
 is modified according to Reboud 
et al. [23], and Coutier-Delgosha et al. [24] as follows
 



t
fk
SF
a








1
1
2
1
max
*
,
, (11)
where k is the turbulence kinetic energy, ω is the 
specific rate of dissipation of the turbulence kinetic 
energy, α* is a damping coefficient, S is the strain rate 
magnitude, F
2
 is the second blending function and 
α
1
 is the model constant equaling 0.31. Correction 
of turbulent viscosity is achieved by modifying 
the mixture density function f( ρ) such that a lower 
turbulent viscosity is achieved in the presence of both 
the liquid and vapor phases
 f
v
v
n
lv
n



 
 
 
1
, (12)
where n is an arbitrary exponent with a recommended 
value of 10.
This modification of turbulent viscosity was 
implemented in ANSYS Fluent via a UDF.
1.6  Cavitation Modelling
For the cavitation modelling, the additional 
conservation equation of vapor mass is solved.
 


 



v
iv
t
u
v


S. (13)
The Schnerr-Sauer cavitation model was used, 
which states that mass transfer source term S
α
v
 is
S
if








v
vl
vv
b
v
l
v
vl
vv
b
v
R
pp
pp
R
pp

 
 

 

1
32
3
1
32
3
 









l
v
pp if
,
 (14)
where the bubble radius is
 R
n
b
v
v

 














0
13
4
3
1
/
. (15)
The bubble number density was specified with a 
value 1·10
11
. The vapor pressure was set to p
v
 = 1854 
Pa.
1.7  Cavitation Erosion Potential
Hammit [3] and V ogel et al. [4] described the potential 
energy of a cavitation bubble in terms of work done as 
a consequence of the pressure difference between the 
vapor inside the bubble and the surrounding liquid. 
If we analyze the spherical cavitation bubble with 
volume V
V
, then the potential energy of the cavitating 
bubble E
pot
 equals 
 Ep pV
pot dv V
  , (16)
where p
d
 is the pressure in the surrounding liquid and 
p
v
 is the vapor pressure inside the bubble. Authors 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
74 Kevorkijan, L. – Lešnik, L. – Biluš, I.
[11] highlighted the problem of defining p
d
 for the 
purposes of numerical simulations, where the pressure 
in the surrounding liquid  is evaluated locally in each 
computational grid cell. In that case p
d
 equals p
v
 and 
the calculated potential energy equals zero. In this 
work p
d
 is taken as a time-averaged value in each 
computational grid cell as proposed in [11]
 p
t
ptdt
d
t
 

1
0
*
*
, (17)
where p(t) is an instantaneous pressure that is then 
time-averaged over the current simulated time t
*
.
The cavitation potential power P
pot
 is introduced 
as the total derivative of the potential energy E
pot
 P
Dp p
Dt
V
DV
Dt
pp
pot
dv
V
V
dv

 
  . (18)
Since p
v
 = const. (isothermal flow) and 
pf xt
d
  

,. const the potential power equals
 P
Dp
Dt
V
DV
Dt
pp
pot
d
V
V
dv
   . (19)
The first term in Eq. (19) is neglected, as it has 
been found that it is at least an order of magnitude 
lower than the second term [22]. Finally, the cavitation 
potential power is written as
 P
DV
Dt
pp
pot
V
dv
  . (20)
To calculate the cavitation potential power 
numerically, it is appropriate to calculate the 
instantaneous change of the volume specific potential 
energy [11] or cavitation erosion potential 
  e
P
V
pot
pot
cell
= , (21)
where V
cell
 presents the volume of the computational 
grid cell. A positive value of 
 e
pot
 is then a consequence 
of an increase of the vapor cavity volume, and a 
negative value is a consequence of the reduction in 
cavity volume. It is possible to rearrange Eq. (20) and 
rewrite it in modified forms, dependent on different 
variables. In the present study we used three equation 
forms, namely, a form dependent on the vapor volume 
fraction e
pot,α
, a form dependent on the velocity 
divergence e
pot u ,div
 and a form dependent on the vapor 
phase source term e
pot S
v
,
α
.
1.7.1  Cavitation Erosion Potential  e
pot, α
If we calculate the total derivative in  Eq. (16), 
consider the relations V  →  V
cell
 and V
cell
 = const. and 
combine it with Eq. (21) we get 
 


e
t
up p
pot dv ,
.


 


 






  (22)
Eq. (22) defines the cavitation erosion potential 
with vapor volume fraction α. It describes the change 
of potential due to the bubble volume decrease during 
collapse.
1.7.2  Cavitation Erosion Potential 
 
e
pot divu ,
Combining the mass conservation Eq. (8) and 






l
vl
 with Eq. (21) we get the cavitation 
erosion potential 
 
e
pot divu ,
 
 



ep pu
pot divu dv
lv
,
.  




 (23)
Eq. (23) presents the second form of cavitation 
erosion potential, which is dependent on velocity 
divergence. 
1.7.3 Cavitation Erosion Potential  e
pot S
v
,
α
Combining the conservation Eqs. (5) and (8) we get
 


 





t
uS
l
v


, (24)
and a third form of equation for erosion potential
  ep pS
pot Sd v
l
v
v
,
.




  (25)
In Eq. (25) the volume change is now expressed 
via the mass transfer source term S
α
v
.
1.8  Collapsing Bubble Energy Balance
If it is assumed that, as the vapor cavity is collapsing, 
reduction of it’s potential energy is converted 
instantaneously into the radiated power, then 
evaluation of Eqs. (22), (23) and (25) gives this 
instantaneous radiated power
  ee
rad pot
 , (26)
where a negative sign is needed to obtain a positive 
value for instantaneous radiated power, as only a 
reduction in cavitation erosion potential 
 e
pot
< 0  

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
75 Cavitation Erosion Modelling on a Radial Divergent Test Section Using RANS 
represents vapor cavity collapse, and only the collapse 
stage is considered in the model.
To predict the cavitation erosion on a solid 
surface better, the potential energy conservation 
hypothesis was  introduced by Schenke et al. [25] and 
Mellisaris et al. [22].  It states that during the vapor 
cavity collapse potential energy is first converted into 
the kinetic energy of the vapor cavity interface, and is 
accumulating until the last moment of collapse, when  
it is finally released. This delayed release of energy 
from the collapsing cavity is named energy focusing, 
as the release of energy is more focused towards the 
center of the collapse compared to the case where 
energy is released instantaneously as described by Eq. 
(26). This accumulation and delayed release of energy 
is described by the additional conservation equation 
for collapse induced kinetic energy ε
 


  


t
ue t
ir ad


 , (27)
where 

u
i
 presents the collapse induced velocity, and  
 et
rad
 presents the radiated power of the pressure 
wave released in the bubble center at the final stage 
of collapse.
The discretized explicit model for transport Eq. 
(27) could be written as:
 



||
tt t
t
t
pot u
t
t
t
Ke tP







   






 11 1 
. (28)
P
u
 in Eq. (28) presents the projection operator, 
which ensures that the specific kinetic energy of the 
collapse accumulates at the inner side of the bubble 
interface, K presents the conservation parameter 
that ensures global energy conservation and β is the 
criterion which determines whether the induced 
energy is released at a given moment. 
 P
u
u
u











max, ,






0 (29)
 K
t
PdV
ed V
V
u
V
pot






, (30)
 


 



10
0
,
.
if and
,e lse.
pp
v
 (31)
p
∞
 in Eq. (29) presents  the external pressure 
infinitely far from the bubble. This pressure represents 
the threshold for the propagation of the pressure wave 
outwards from the point of collapse, and the pressure 
at the exit from the domain was used for its value. The 
criterion α
v
 = 0 represents the end of the collapse when 
there is no more vapor phase in the computational cell.
The energy released after the collapse (at the end 
of time step Δt) is expressed as the specific power of 
the pressure wave 
 
 e
t
rad
tt
t





 |
. (32)
An infinite wave propagation speed is assumed in 
this model. The energy received by a surface element 
S is then expressed in terms of the instantaneous 
surface specific impact power
 
 

ee
xx n
xx
dV
S
V
rad
cellS
cellS

  










1
4
3

, (33)
where 

x
cell
 is the position vector of the computational 
cell center, 

x
S
 is the position vector of the wall surface 
element center, and 

n is the surface element normal. 
By integrating Eq. (33) over the simulated time (t), we 
obtain instantaneous surface specific impact energy 
with units [J/m
2
]
 ee dt
S
t
S


0
 . (34)
The described model to predict cavitation was 
implemented inside ANSYS Fluent via a UDF, 
which was written in the C programming language, 
compiled, and linked to the solver. Eqs. (22), (23), 
(25) and (28) to (34) were evaluated at the end of 
every time step. Energy accumulated on a given face 
of the surface is then stored as a sum of the previously 
stored value and newly computed value by Eq. (33).
2  NUMERICAL TEST CASE AND SETUP
2.1  Geometry
The conventional radial divergent test section was 
chosen for the test and validation of the cavitation 
erosion models [26]. The radial divergent test section 
geometry is shown in Fig. 1.
2.2  Computational Grid and Boundary Conditions
A computational grid of a 45° sector was generated 
in ANSYS Fluent Meshing using ANSYS Mosaic 
meshing technology, which resulted in Poly-Hexcore 
grid topology. Additionally, a prismatic inflation layer 
was generated on the walls. The final mesh consisted of 
1,396,703 cells with a minimum orthogonal quality of 
0.28 and maximum aspect ratio of 65. The maximum 
wall y
+
 value was 51.9 and the minimum wall y
+
 value 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
76 Kevorkijan, L. – Lešnik, L. – Biluš, I.
was 0.174. Special care was taken to achieve y
+
 < 5  
around the radius before the radial divergent test 
section, and to achieve 30 < y
+
 < 300 on the walls of the 
extended inlet and outlet parts of the computational 
domain. The computational domain was extended 
downstream with the outlet boundary positioned 100 
mm in the radial direction from the radial divergent 
test section axis to improve numerical convergence. 
The mesh was refined around the radial divergent 
test section radius for improved resolution of 
cavitation dynamics, and, therefore, cavitation erosion 
prediction. 
Fig. 1.  Radial divergent  test section 
The results of the grid independence study are 
presented in Table 1. A medium grid was chosen 
as it provided us with an acceptable computation 
time and numerical uncertainty, as given by the grid 
convergence index (GCI). All simulations were 
performed on a personal computer using 10 CPU 
cores of the Intel Core i9-10940X stock CPU with 64 
GB of RAM. The time needed to obtain convergence 
on the medium mesh was approximately 10 days. 
For GCI calculations we chose the force exerted on 
the bottom plate, the fluid velocity at the outlet and 
turbulence kinetic energy at the outlet. In all three 
cases the estimated numerical error was less than 5 % 
for both the medium and the fine grids [27].
Table 1.  Computational grid independence study
Grid
Number of 
cells
Force on 
bottom 
wall [N]
Outlet 
velocity 
[m/s]
Outlet turbulence 
kinetic energy 
[J/kg]
Coarse 475271 3802 3.64 0.196
Medium 1396703 3798 3.57 0.195
Fine 5175737 3799 3.51 0.187
GCImedium [%] - 0.02 4.42 0.51
GCIfine [%] - 0.001 2.59 1.99
Fig. 2.  Computational grid; a) isometric view of the whole grid 
with shown boundary conditions, b) detailed isometric view of the 
refined grid, and c) detailed front view of the refined grid
2.3  Numerical Setup
Solution of the Navier-Stokes equations was 
achieved using the SIMPLE algorithm, gradients 
were evaluated using the least squares cell-based 
method, interpolation of pressure cell face values was 
achieved by the PRESTO! interpolation scheme and 
the QUICK spatial discretization scheme was adopted 
for all equations. Temporal discretization of equations 
was achieved by the Bounded Second-Order Implicit 
time integration scheme.
A time step of 1·10
–6
 s was chosen, which 
resulted in the maximum Courant number C = 1.59. 
The iterative solution of equations within each 
time step was limited to 100 iterations, however, a 
scaled residuals convergence criterion of 1·10
–6
 was 
achieved before this limitation. Overall, 0.012249 s of 
physical time were simulated.
3  RESULTS
The extent of cavitation predicted by the Schnerr-
Sauer cavitation transport model at different times 
of numerical simulation is shown in Fig. 3. The 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
77 Cavitation Erosion Modelling on a Radial Divergent Test Section Using RANS 
black dotted line shows the cavitation closure line 
positioned 32 mm in a radial direction from the radial 
divergent test section axis, which was determined 
experimentally [26]. From the cavitation extent 
comparison, it is visible that the predicted cavitation 
sheet closure coincides with the experimental 
position, as cavitation number σ = 0.9 was achieved. 
According to this, it can be concluded that the used 
cavitation model, boundary conditions and physical 
properties were determined properly to predict 
the cavitation dynamics correctly. Qualitative 
assessment of cavitation erosion by the numerical 
model is presented in Fig. 4, with three approaches 
for calculating cavitation erosion potential compared 
to the picture of the sample from the experiment by 
[26]. Both energy focusing, where radiated power was 
obtained from Eq. (32), and non-focusing approaches, 
where radiated power was obtained from Eq. (26), are 
Fig. 3.  Transient sheet cavitation displayed as an iso-surface of α = 0.9 at different times; a) top view; b) three- dimensional display of 
cavitation structures viewed from the side
Fig. 4.  Comparison between the cavitation erosion models and experiment [26]; a) accumulated energy on the surface without energy 
focusing; b) accumulated energy on the surface with energy focusing

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
78 Kevorkijan, L. – Lešnik, L. – Biluš, I.
The total vapor volume time evolution in the 
numerical domain is shown in  Fig. 5 for the time 
interval 0.002 s to 0.012 s. It is apparent that the total 
volume of vapor in the domain changes periodically, 
which confirms the transient nature of the extent 
of cavitation in the radial divergent test section. 
Additionally, Fig. 6 displays the local transient 
behavior of the vapor cloud shedding, with minimums 
in pressure between the peaks, indicating the presence 
of cavitation. It was observed that vapor cloud 
shedding occurs with higher frequency than the global 
(total) change in vapor volume in the domain. In a half 
of a total vapor volume change period (the dashed 
line frame in Figs. 5 and 6), at least three transitions 
occur between the liquid and vapor phases. Within 
the simulated time of 0.012249 s multiple vapor 
cloud shedding events occurred, which allowed for a 
consistent analysis of cavitation erosion.
Fig 7. shows the time evolution of the total 
cavitation potential power in the domain within this 
time period, calculated with a formulation based on 
the partial derivative of vapor volume fraction . 
Fig. 7.  Total cavitation potential power in the domain
Within the observed time window (the dash lined 
frame in Figs. 5 and 6) release of accumulated kinetic 
energy for the energy focusing case is presented in 
Fig. 8. Comparison between Figs. 7 and 8a further 
demonstrates and explains the difference in extent 
and magnitude of the predicted accumulation of 
energy between the energy focusing case and the non-
focusing case. In the non-focusing case cavitation 
potential power was released instantaneously, so Fig. 
7. also represents the total radiated power of pressure 
waves in the domain, for the non-focusing case. It is 
again apparent that energy focusing results in fewer 
events with higher magnitude of pressure wave power 
radiation (at the end of vapor collapse). Because of 
presented. Energy focusing resulted in a sharper, more 
pitted image compared to a more blended picture 
without energy focusing. Since in the case of energy 
focusing all previously accumulated collapse induced 
kinetic energy is released only at the final stage of 
collapse, it is reasonable to expect accumulation of 
radiated energy on the surface in a smaller area, and 
higher magnitude compared to the non-focusing 
case. In both cases, with energy focusing and without 
energy focusing, it is apparent that modelling 
cavitation erosion potential based on partial derivative 
of vapor volume fraction (∂ α / ∂t) and source term 
in vapor transport equation (S
αv
) produced a similar 
erosion pattern and location of accumulated energy 
on the bottom plate surface, which are comparable 
with the experiment (shown in the background of 
Fig. 4). However, calculation based on the divergence 
of velocity 




u did not predict a similar erosion 
pattern and location of accumulation of energy on the 
bottom plate. 
The comparison of the eroded zone position was 
similar for both (focusing and non-focusing) cases. 
The deviation of the position of the damaged area 
between the results of the numerical simulation and 
the experiment visible in Fig. 4 indicates a strong 
sensitivity of the numerical erosion prediction model 
on correct prediction of the cavitation dynamics.
Fig. 5.  Total vapor volume time evolution in the domain
Fig. 6.  Pressure time evolution, 20 mm from the radial divergent 
test section axis in a radial direction 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
79 Cavitation Erosion Modelling on a Radial Divergent Test Section Using RANS 
the energy conservation by the model, kinetic energy 
reduces at the end of vapor collapse when wave energy 
is released. Two pronounced events of release of 
energy are visible in Fig. 8. at around 0.0115 s, first a 
smaller peak occurred, followed by a larger peak (Fig. 
8a). Hence, there are two pronounced corresponding 
drops in kinetic energy (Fig. 8b).
4  DISCUSSION
The presented study results show that the used 
mathematical-physical model with different 
formulations for prediction of vapor phase potential 
power, and, thus, cavitation erosion, predicts the 
position of damaged surfaces on the geometry 
of the radial divergent test section satisfactorily, 
which confirms the findings of other researchers 
on different flow cases [22] and [25] and different 
cavitation erosion models [16]. A more detailed 
comparison of the positions of damaged surfaces 
indicates discrepancies between the individual 
formulations. Given the fact that a verified cavitation 
transfer model was used, it can be concluded that the 
largest deviation of the damage positions from the 
experimentally observed ones occurred in the case 
of using a formulation with velocity divergence, both 
in the case of focusing and without focusing. The 
phenomenon was a consequence of numerical error, 
and is consistent with the observations of researchers 
who analyzed the flow over an isolated wing and 
marine propeller [22] and [25]. The formulations 
with the source term and the derivative of the vapor 
phase fraction confirm the importance of a correct 
description of the dynamics of cavitation structures. 
There was practically no difference in the calculated 
positions of damage between the applied formulations 
with energy focusing and without energy focusing. 
Conserving the potential energy of cavitation by 
focusing indicates an appropriate step towards a better 
agreement of the shape of the damaged surface with 
the experimental observations, where the damage 
was more concentrated and the eroded surface was 
characteristically pitted, while, in the case without 
focusing, the damage was more continuous and did 
not reflect the experimentally observed damage of 
eroded surfaces [26]. It has to be noted however, that 
to directly link the material pitting prediction to the 
cavitating flow simulation a change of scales would 
be necessary. Furthermore,  material work hardening 
would have to be taken into account via proposed 
material strengthening models [26] and [28].
5  CONCLUSIONS
The performed analysis confirms that the formulation 
of cavitation erosion potential with the source term 
is comparable to the formulation with the derivative 
of vapor phase volume fraction in the case under 
consideration. An appropriate description of cavitation 
dynamics is crucial. The Schnerr-Sauer cavitation 
model predicts the extent and dynamics of cavitation 
correctly. In relation to that, it would be sensible to 
use cavitation transfer models of other authors, such 
as Singhal et al. [29] and Kunz et al. [30]. However, 
from the thermodynamic point of view  this approach 
to model cavitation is not as appropriate, as discussed 
in [31]. The applied model transfers potential energy 
from cavitation structures to the surface, is energy 
conservative, which makes it a good basis for 
predicting the response of various materials exposed 
to the phenomenon of cavitation. In modelling 
their response to cavitation, the model used in this 
a)    b) 
Fig. 8.  a) Total radiated power of pressure waves generated by cavitation implosions in the domain;  
b) total vapor collapse induced kinetic energy of vapor in the domain

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 71-81
80 Kevorkijan, L. – Lešnik, L. – Biluš, I.
work could be upgraded by taking into account the 
mechanical properties of the material, which would 
allow the calculation of the actual depth of damage 
on different surfaces. This would allow a quantitative 
evaluation of the model and it’s different formulations 
used in this work.
6  ACKNOWLEDGEMENTS
The authors wish to thank the Slovenian Research 
Agency (ARRS) for the financial support in the 
framework of the Research Programme P2-0196 
Research in Power, Process and Environmental 
Engineering.
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