284 DOI: 10.5545/sv-jme.2025.1351
Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 9-10   ▪  Y 2025 © The Authors. CC BY 4.0 Int. Licencee: SV-JME
Numerical Investigation of Erosion Due to Particles  
in a Cavitating Flow in Pelton Turbine
Luka Kevorkijan
1
 − Matjaž Hriberšek
1
 − Luka Lešnik
1
 − Aljaž Škerlavaj
2
 − Ignacijo Biluš
1 
1 University of Maribor, Faculty of Mechanical Engineering, Slovenia 
2 Scotta Turboinštitut, Slovenia 
 ignacijo.bilus@um.si
Abstract Erosion of Pelton turbine components due to cavitation and particle-laden flow is a major challenge in hydropower applications, particularly in 
sediment-rich river environments. This study presents a numerical investigation on how solid particles contribute to the erosion of a Pelton runner. Computational 
fluid dynamics (CFD) simulations were conducted using ANSYS CFX 2023 R2, incorporating a Lagrangian particle tracking approach and the Finnie abrasion 
model to predict erosion patterns under varying sediment concentrations. The results indicate that, under normal sediment conditions, particle erosion does not 
significantly contribute to blade tip damage. However, under extreme sediment loading, the predicted erosion patterns closely match real-world observations, 
particularly at the blade tip. 
Keywords Pelton turbine, solid particle erosion, cavitation, CFD simulation
Highlights
 ▪ Numerical CFD analyzis was used to evaluate erosion in Pelton turbine runners.
 ▪ Particle erosion is negligible under normal sediment conditions but significant under extreme loading.
 ▪ Erosion patterns predicted by simulations align with real-world turbine wear.
1 INTRODUCTION
Multiphase flows occur in wide range of devices in process and 
energy engineering. In some cases, their occurrence is intentional 
due to the process taking place between the phases in the flow, such 
as in spray coating (spray towers), preparation of suspensions in the 
pharmaceutical and food industries (fluidized bed devices, mixing 
reactors), distillation, drying, air conditioning (air conditioning and 
ventilation systems), combustion in thermal machines (thermal power 
plants, internal combustion engines), coating removal (sandblasting), 
and many other processes [1].
However, multiphase flow can also arise unintentionally as a 
consequence of natural phenomena or engineering process, for 
example, solid particles in emissions during combustion (combustion 
in internal combustion engines, thermal power plants, fires), 
sediments in river flows (flow through hydraulic turbine machines) 
and sand particles in wind flow (wind erosion in deserts).
In general, multiphase flows are classified into stratified and 
dispersed types, with flows containing solid dispersed phases, such 
as particles in liquid flow, being a special case of the latter. Due to the 
frequent occurrence of such flows in process and energy devices, the 
interaction of particles with the walls of the devices is of significant 
engineering interest. These interactions can result in material 
loss from the wall surface, commonly referred to as abrasion. A 
particularly relevant issue is the damage to the flow components of 
turbine machines operating in rivers polluted with sediments [2].
In 2023, approximately 750 million people worldwide still 
lacked access to electricity [3]. In the preceding year, 4300 TWh 
of electricity worldwide was produced in hydropower plants,  
accounting for about 15 % of total global electricity production. 
Annual growth of production was 2 %, with projections suggesting 
an increase of 4 % by 2030 [4]. 
As further adoption of hydraulic turbines to produce electricity is 
pursued, several engineering challenges have emerged [5]. Hydraulic 
turbines are conventionally designed for operation within normal 
operating limits. When operating outside these limits, i.e. within 
temporary operating limits, the operating time is limited according 
to IEC 60609 standard [6]. If a turbine is operating in temporary 
operating limits in such a regime that pressure drops below vapor 
pressure, cavitation can occur. Cavitation can cause erosion of turbine 
components, independent of other erosion causes. Another challenge 
arises in rivers with suspended sediments, where the flowing water 
carries solid particles, which can cause additional erosion, termed 
abrasion.
To improve turbine design with respect to erosion phenomena, 
or to predict erosion in existing turbine designs, computational fluid 
dynamics (CFD) simulations coupled with erosion models are often 
utilized. In the past, various modelling approaches for cavitating 
flow have been adopted, and several cavitation erosion models 
have been developed. Research in this field remains ongoing, with 
recent efforts focused on phenomenological models applied to a 
range of engineering applications. Leclercq et al. [7] developed a 
cavitation erosion model based on earlier work by Fortes-Patella et 
al. [8] in which all cavitation collapses are considered by projecting 
cavitation erosion potential from interior cells to the wall using a 
discrete formulation. Implemented in Code_Saturne, this model was 
successfully applied to predict cavitation erosion on a NACA65012 
hydrofoil. Schenke and van Terwisga [9] proposed a continuous 
formulation for the projection of cavitation potential to walls, while 
Melissaris et al. [10] later improved the model by considering energy 
focusing during cavity collapse. Using this improvement, they were 
able to predict more spatially focused cavitation erosion patterns in 
the case of a KCD-193 model propeller. Arabnejad et al. [11] further 
advanced modeling by considering two different mechanisms, both 
pressure waves and microjets, depending on the distance from the 
wall at which cavity collapse occurs. These complex models have 
been successfully used to predict erosion for various cases; however, 

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open questions  remain regarding modeling assumptions, particularly 
the cavitation collapse driving pressure [12]. 
In addition to above presented complex models, simpler models 
have been developed and successfully applied to predict cavitation 
erosion in different hydraulic systems. These simpler models are 
often applied to complex geometries or cases in which change in 
geometry needs to be considered. Such was the study by Brunhart 
et al. [13], where it has been determined that for the cavitation 
erosion prediction within a diesel fuel pump, where dynamic mesh 
was adopted, good agreement with experiment was obtained using 
an erosion indicator based on the recorded maximum of the squared 
total time derivative of pressure. Santos et al. [14] used three erosion 
indicators to predict cavitation erosion in gasoline direct injection 
(GDi) injector, where again dynamic mesh was adopted in the 
simulation. For the prediction of cavitation erosion in Pelton turbine, 
Jošt et al. [15] adopted criteria previously proposed by Rossetti et 
al. [16], which relate material damage to presence of water vapor in 
contact with the wall, rapid reduction in volume of this water vapor 
and the volume fraction of air mixed with the water (and water 
vapor) in the observed region. Based on this approach, Jošt et al. [15] 
concluded that damage of the observed Pelton rotor blades is not the 
result of cavitation erosion as the collapses are too slow. 
Erosion of material due to particles has long been a topic of 
investigation of a distinct branch of engineering research – tribology. 
It stems from early research of contact forces due to friction by 
physicists such as Coulomb and later Hertz. Due to previously 
mentioned industry applications in which multiphase flows and 
subsequent abrasion occur, research into particle abrasion focused 
on experimental studies to obtain empirical predictive models of 
abrasion. 
Such early study was conducted by Finnie [17], in which basic 
parameters that influence abrasion were determined, notably the 
influence of the particle impact angle on abrasion. Similarly, Bitter 
[18,19] identified different influencing parameters and proposed 
an empirical abrasion model, following by Grant and Tabakoff 
conducted studies [20,21] of abrasion in helicopter turbines and 
proposed a particle wall rebound model alongside their own abrasion 
model. Ahlert [22] proposed another abrasion model based on 
experimental investigation of particles impacting AISI 1018 steel 
sample. For AISI 4130 steel Forder et al. [23] conducted similar 
study and applied the abrasion model to predict abrasion in control 
valve using CFD simulations. Later multiple models have been 
proposed, with increasing complexity with respect to number of 
parameters considered, such as studies by Oka et al. [24] and Oka 
and Yoshida [25] and by Det Norske Veritas society (DNV) [26]. Over 
time adoption of empirical abrasion models in CFD has increased, 
Gnanavelu et al. [27] used this approach to reduce the number of 
experiments needed in their study of abrasion. 
These empirical abrasion models have been applied to predict 
particle abrasion in different hydraulic systems by using CFD 
simulations of flows containing solid particles, where particles are 
considered as points and are tracked in Lagrangian frame. Peng 
and Cao [28] studied abrasion of pipe bends in piping found in oil 
industry, by comparing multiple abrasion models used in numerical 
simulations with experimental results, they concluded that McLaury 
model [29] in conjunction with the Grant and Tabakoff particle-wall 
rebound model [20,21] was the most accurate in predicting abrasion 
due to particles in liquid flow. Messa et al. [30] conducted a numerical 
study of abrasion in Pelton turbine injectors, where they applied the 
model by Oka et al. [24] and Oka and Yoshida [25] to predict abrasion 
of the nozzle seat and needle for different needle openings and needle 
vertex angles. They found enhanced abrasion for low openings 
and lower needle vertex angles. Many similar analyzes exist in the 
literature, indicating the need to better understand abrasion of Pelton 
injectors, particularly for full scale Pelton turbine injectors, which 
has recently been analyzed by Liu et al. [31]. 
In general, abrasion of turbine components is prevalent also on 
turbine runners, specifically rotor blades. Kumar and Bhingole 
[32] conducted a CFD study of a combined effect of cavitation 
and particle erosion on Kaplan turbine, with varying particle size 
and concentration and determined that larger particles and larger 
concentrations of particles produced more abrasion on the runner. 
Similarly, effects of particle concentration and diameter on abrasion 
characteristics obtained by CFD simulations on a Pelton turbine 
runner have been studied by Li et al. [33], where they concluded 
that the diameter of the particles mainly effected the distribution of 
predicted abrasion regions and concentration mainly influenced the 
intensity of abrasion. Han et al. [34] also considered cavitation to 
have an influence on particle abrasion of Pelton runner, which they 
predicted using the Finnie [17] model. They concluded that cavitation 
has a clear influence on particle abrasion development, especially due 
to its effect on motion of smaller particles at the jet interface (air-
liquid interface) [34]. Table 1 summarizes the mechanisms of erosion 
and highlights the studies where erosion was modelled specifically in 
the case of Pelton runner.
Table 1. Summary of studies of erosion modelling highlighting which erosion mechanism was 
modelled
Reference Mechanism of erosion 
Leclercq et al. [7] Cavitation collapse
Melissaris et al. [10] Cavitation collapse
Arabnejad et al. [11] Cavitation collapse
Brunhart et al. [13] Cavitation collapse
Santos et al. [14] Cavitation collapse
Rossetti et al. [16] Cavitation collapse
Finnie [17] Particle impact
Bitter [18,19] Particle impact
Ahlert [22] Particle impact
Oka, Okamura and Yoshida [24,25] Particle impact
Det Norske Veritas society [26] Particle impact
Peng and Cao [28] Particle impact
Messa, Mandelli and Malavasi [30] Particle impact
Liu et al. [31] Particle impact.
Kumar and Bhingole [32] Particle impact and cavitation collapse
Li et al. [33] Particle impact
Han et al. [34] Particle impact 
In the present study, we investigated numerically, whether the 
cause of erosion of Pelton runner, found in the previous study by Jošt 
et al. [15], could be due to solid particles present in the water flow. 
For this purpose, we extended the modelling of flow through Pelton 
turbine by including Lagrangian particle tracking and applying Finnie 
abrasion model within ANSYS CFX 2023 R2 [17]. Two particle 
concentrations were considered, one for regular river conditions and 
one for the case of heightened presence of particles, for example due 
to the extreme weather phenomena. Unlike previous studies, such as 
the one by Han et al. [34], we considered fully transient behavior of 
the flow including particle motion and Pelton rotor rotation. For this 
purpose, a sliding mesh approach was adopted, specifically a rotating 
mesh was used for the rotor region. With this approach we managed 
to avoid using a simplification to steady-state.

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2 METHODS AND MATERIALS
2.1 Case Description
A Pelton runner from previous study is considered [15], where a 
numerical simulation of the existing prototype-scale Pelton turbine 
was conducted for the case of cavitating flow. The purpose of that 
study was to determine if cavitation could be the cause of erosion on 
the blade tip observed after prolonged operation as shown in Fig. 1.
The authors [15] concluded that cavitation alone could not explain 
the erosion in the region of the blade tip. The main question then 
arose, whether that damage could be the result of turbine operating in 
a river laden with sediments (solid particles). 
The presence of sediment particles was confirmed by electron 
microscope imaging of the river water sample and both size and 
chemical composition of sediment particles were determined. 
Sediment particles were found to be in range between 30 µm and 80 
µm and are visible on an electron microscope image shown in Fig. 2.
It was then found that sand particles, which are agglomerated 
to form a sediment particle, are silica particles (SiO2). Chemical 
composition of sediment for a wider sampling region Spectrum 1 
and a sampling point Spectrum 2 (elongated particle visible in Fig. 
2) is shown in Table 2, where Oxygen (O) and Silicone (Si) have the 
highest fraction of all elements present in sediment sample for both 
sampling regions Spectrum 1 and Spectrum 2. 
Table 2. Chemical composition of sediment
Element Spectrum 1 Spectrum 2 Element Spectrum 1 Spectrum 2
O 45.43 48.21 K 2.42 0.58
C 13.68 16.17 Mg 0.87 -
Si 17.12 28.55 Na 0.54 0.44
Al 6.32 0.83 Cl - 0.11
Ca 9.29 3.42 Ti 0.33 0.18
Fe 3.67 1.51 Mn 0.32 -
Total 100 100
2.2 Mathematical Model
An incompressible, turbulent, multiphase flow of water jet with 
cavitation and solid particles is considered. Multiphase flow of liquid 
water and due to cavitation water vapor contained within a jet, which 
forms an interface with respect to surrounding gas (air), is modelled 
using a homogeneous mixture approach. Mixture density ( ρ) and 
mixture viscosity ( μ) are determined by mixing rule as:
    
ll vv gg
, (1)
    
ll vv gg
, (2)
where φ
l
 is liquid volume fraction, φ
v
 is the vapor volume fraction 
and φ
g
 is gas volume fraction. Similarly, ρ
l
 and ρ
v
 are liquid and 
a)    b) 
Fig. 1. Damage of a Pelton rotor blade tip (in red bracket) after prolonged operation: a) view of the back side of the blade, view of the front side of the blade
Fig. 2. Electron microscope image of the river sediment with two sampling positions  
to determine chemical composition indicated as Spectrum 1 and Spectrum 2
vapor density respectively and ρ
g
 is gas density. Finally, μ
l
 and μ
v
 
are liquid and vapor dynamic viscosity respectively, while μ
g
 is gas 
dynamic viscosity. The governing equations of cavitating flow can 
then be written, the continuity equation as:


 


t
u 0, (3)
and momentum equation as:


  


u
uu S
M
t
p   , (4)
where Eqs. (3) and (4) are written for the mixture of liquid and vapor 
phases which share the same velocity u and pressure p. In Eq. (4) τ 
is mixture shear stress tensor and S
M
 is the momentum source term 
accounting for the presence of particles. Since the multiphase flow 
is considered as a mixture of two phases, an additional equation for 
transport of vapor volume is needed:


 



v
v
l
t
m
u

, (5)

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where  m is the interface mass transfer rate due to cavitation, for 
which a cavitation model by Zwart et al. [35] was adopted in this 
study:
 m
F
r
R
pp
pp
F
R
pp
p
e
nuc vv
B
v
l
v
c
vv
B
v
l


  



31 2
3
32
3




,
,
if
if p p
v







, (6)
where F
e
 is the evaporation coefficient with a recommended value 
of 50 [35], F
c
 is the condensation coefficient with a recommended 
value of 0.01, r
nuc
 is the nucleation site volume fraction with a default 
value of 5×10
−4
 and R
B
 is the bubble radius upon which the model is 
derived, with a recommended value of 10
−6 
m [35]. From Eq. (6) it is 
evident that the mass transfer rate was considered negative in the case 
of evaporation, when pressure p is bellow vapor pressure p
v
, which 
was 1300 Pa for water. Likewise, in Eq. (6) the mass transfer rate is 
positive in the case of condensation, when the pressure p is above 
vapor pressure p
v
.
Although more advanced turbulence models (hybrid Reynolds-
averaged Navier-Stokes and large Eddy simulation (RANS/LES), 
and large Eddy simulation (LES)) have recently been used in the 
studies of turbulent, cavitating flows [36], even for some engineering 
applications [37,38], RANS two-equation models still represent a good 
balance between accuracy and calculation times for most engineering 
applications, like a Pelton turbine. Therefore, in this study turbulence 
was modelled using a RANS approach, specifically the k – ω SST 
two-equation turbulence model was used. Two additional transport 
equation are introduced, one for the turbulent kinetic energy k:


  


















 





k
t
kk GYS
t
k
kkk
u ,
 (7)
and one for the specific turbulence dissipation rate ω:


  


















 

 



	


 
t
GYDS
t
u , (8)
where G
k
 and G
ω
 are production terms for the turbulent kinetic 
energy and the specific turbulence dissipation rate respectively, Y
k
 
and Y
ω
 are the dissipation terms for the turbulent kinetic energy and 
for the specific turbulence dissipation rate respectively, σ
k
 and σ
ω
 are 
the turbulent Prandtl numbers for k and for ω respectively, S
k
 and S
ω
 
are the source terms. Since the k – ω scale-adaptive simulation model 
(SST) is a blended turbulence model, consisting of standard k – ε and 
standard k – ω model, an additional term cross-diffusion term D
ω
 is 
introduced in Eq. (8) due to reformulation of k – ε model for blending 
with k – ω model. RANS approach results in additional turbulent 
viscosity μ
l
, which is in the case of k – ω SST model written as:




t
k
SF
a







1
1
2
1
max,
,
*
 (9)
where α
*
 is the turbulent viscosity damping coefficient, a
1
 is a model 
constant with value of 0.31 [39], S is the strain rate magnitude and 
F
2
 is the second blending function. A detailed description of the 
turbulent model used is available in [39].
Particles represent a discrete phase for which Lagrangian tracking 
is adopted within previously described continuous phase (mixture 
of two continuous phases) which was considered in Eulerian frame. 
In our study drag force and virtual mass force were considered, 
gravity effects (buoyancy) were neglected as the inertia of the flow, 
due to high fluid velocity, had a dominant influence on the motion 
of particles. Then, an additional motion equation for particles due 
to drag force and virtual mass force can be written in the form of 
particle acceleration as:
d
dt d
CR e
D
Dt
d
dt
l
pp
Dp
l
p
v
uv
uv
  






18
24
1
2
2




, (10)
where ν is the particle velocity, ρ
v
 is the particle density, d
p
 is the 
particle diameter and the particle Reynolds number Re
p
 is defined as:
Re
d
p
pp
l



u-v
, (11)
The drag coefficient C
D
 is calculated using the Schiller Naumann 
correlation [40]:
C
Re
Re
Re
Re
D
p
p
p
p












24
10 15
1000
04 4 1000
0 687
.
,
.,
.
.
if
if
 (12)
Since the volume fraction of the particles in particle-laden flow 
is low, interactions between particles are neglected. However, 
interaction of particles with the wall must be considered as it is 
one of the boundary conditions. For this, Hard Sphere Model is 
adopted, where particles are considered as nondeformable during 
their collision with the wall. Rebound of particles from the wall is 
then described with two coefficients of restitution, one in wall normal 
direction:
e
v
v
n
n
n
=
,
,
,
2
1
 (13)
and one in tangential direction:
e
v
v
t
t
t
=
,
,
,
2
1
 (14)
where v
n,1
 and v
n,2
 are particle velocities in the wall normal direction 
before and after rebound respectively, and v
t,1
 and v
t,2
 are particle 
velocities in the wall tangential direction before and after rebound 
respectively. Particle velocity in both directions after rebound (v
n,2
 
and v
t,2
) are calculated for the known coefficients of restitution (e
n
 
and e
t
), which were determined by using Grant and Tabakoff model 
[19], where coefficients of restitution are given as functions of 
particle impact angle γ as:
e
n
  0 993 17 61 56 04 9
23
.- .. .,   (15)
e
t
 0 988 17 61 56 04 9
23
.... .   (16)
To predict erosion of the wall due to impacting particles, we used 
an empirical model by Finnie [17], which gives the erosion rate as:
E
v
f
R
n










v
0
 , (17)
where ν
0
 is the empirical reference velocity with a value of 3321 m/s 
for steel, n is the velocity exponent with a value of 2.4 and f ( γ) is an 
impact angle function, given as:
f 

 
 

 







1
3
1
3
23
1
3
2
2
cos; tan
sins in ,t an
.
if
if
 (18)
2.3 Boundary Conditions and Physical Properties
Based on the previously described sediment analyzis, solid silica 
particles with density ρ
p
 = 2650 kg/m
3
 of sizes between 30 µm and 
80 µm were considered in numerical simulation. To account for the 
varying size of particles we used the Rosin-Rammler particle size 
distribution, where the mass fraction of particles above a certain 
particle diameter d
p
 is defined as:

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Re
d
d
p
e



















, (19)
where d
e
 is the size constant and χ is the size distribution 
parameter. We considered a particle distribution with d
e
= = = 50 μm and 
χ = 1.1. The resulting cumulative mass fraction distribution is shown 
in Fig. 3. 
Fig. 3.  Rosin-Rammler particle size distribution used in numerical simulation,  
represented with a cumulative mass fraction over particle diameters
Varying shape of particles was not considered in numerical 
simulations; therefore, particles were assumed to be spherical. Two 
different cases of particle loading of the flow were considered, 
one for the regular river flow where volume fraction of particles 
is 0.006 % and one for heightened particle loading scenario where 
volume fraction of particles is 10 times higher, resulting in volume 
fraction of particles of 0.06 %. Properties of liquid water, water vapor 
and air are presented in Table 3.
Table 3. Properties of continuous phases (liquid water, water vapor and air) used in numerical 
simulation
Material Density [kg/m
3
] Dynamic viscosity [Pa s]
Liquid water 999.18 0.00114029
Water vapor 0.02308 9.86e-6
Air 1.185 1.83e-5
Boundary conditions were defined as shown in Fig. 4. For the 
liquid jet flow, velocity components were prescribed as a function 
of coordinates over the nozzle area, which were determined with 
previous numerical simulation of flow through the injector [15] 
for the mean velocity magnitude of 105.233 m/s and 5 % turbulent 
intensity at the inlet. For the particles entering the domain with the 
jet, zero-slip velocity condition was used while injection of particles 
was realized by prescribing their number rate and mass flow rate. 
This is presented in Table 4 for both particle loading scenarios.
Fig. 4. Boundary conditions marked by colors: green – inlet, blue – outlet, red – no-slip wall, 
purple – symmetry and yellow – opening
Table 4. Particle injection definition at the inlet
Particle loading 
scenario
Volume fraction 
[%]
Number rate  
[s
–1
]
Mass flow rate 
[kg/s]
Normal 0.006 2.81794e+7 0.048875
Heavy 0.06 2.81794e+8 0.48875
At the outlet, static pressure was prescribed with a value of 
101,300 Pa for continuous phase, and particles are given the escape 
boundary condition by default. Same conditions were prescribed for 
the opening boundary condition. 
To reduce computational demands, symmetry boundary condition 
was used at symmetry plane as shown in Fig. 4. For particles, 
however, this symmetry plane represented a wall, the fact that only 
half of the full volume (domain) was considered was accounted 
for when calculating particle inlet number rate and mass flow rate 
presented in Table 4. 
Remaining surfaces (rotor blades and hub) were treated as no-
slip walls for the continuous phase and solid walls with a rebound 
boundary condition for particles.
2.4 Mesh and Numerical Setup
To further reduce the computational demands, we considered only 5 
rotor blades in the geometrical model for the meshing, since mesh 
around the blades require refinement resulting in higher mesh cell 
density. We used the mesh from the previous study [15], where a 
mixed hexahedral and tetrahedral mesh with 10.75 million cells 
was used. The mesh is presented in Fig. 5. The mesh consisted of 
two main regions, stationary and rotating. Rotating region of mesh 
   
Fig. 5. Mesh showing: a) full mesh, b) detailed view of the blade region on the symmetry plane, and c) detailed view around a single blade at cross-section

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was prescribed mesh rotation, such that 0.1° of rotor rotation was 
achieved per time step, and in total 27° of rotor rotation was achieved 
during the simulation.
Since the simulation was fully transient, a second order backward 
Euler transient scheme was adopted and for the advection terms 
high resolution scheme was used. Within each time step maximum 
10 iterations were performed, however a residual target of 1e-4 was 
achieved before this limit.
3  RESULTS AND DISCUSSION
First, we present the results for the flow of sand particles in water 
jet impacting Pelton rotor blades, which is shown in Fig. 6 for the 
normal particle loading scenario and in Fig. 7 for the heavy particle 
loading scenario. In Figs. 6 and 7 we observe jet (green) impacting 
Pelton rotor blades shown at different times, with visible cavitation 
(magenta) shown as iso-surface with 20 % volume fraction of vapor 
and sand particles (black) for a normal particle loading scenario and 
heavy particle loading scenario, respectively. Rotor blade on which 
subsequent erosion is studied is highlighted with orange color. At the 
beginning in a) the blade is yet to come in contact with the jet, in 
following moments b) through e) it passes through the jet, particles in 
the jet impact the blade and in f) finally jet is cut-off by next passing 
blade
The difference in particle loading is clearly shown and discernible 
when comparing Figs. 6 and 7. Due to the modelling approach taken 
in this work, jet development and cavitation development are not 
influenced by particles, therefore they are identical for both particle 
loadings, as seen when comparing Figs. 6 and 7. 
        
      
Fig. 6. Impact of jet with particles on Pelton rotor blade at different times for normal particle loading scenario
      
      
Fig. 7. Impact of jet with particles on Pelton rotor blade at different times for heavy particle loading scenario

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Particle motion follows the motion of the water jet. Attached 
cavitation pocket behind the second blade, visible at time t = 
0.0114 s on Figs. 6 and 7, redirects the particles around it. Within 
the simulated time, the second blade of five blades considered in 
this simulation passes through the impacting water jet carrying sand 
particles completely. As the second blade passes through the jet, 
particles entrained in the jet impact the back side of the blade as well.
This is why erosion prediction results will be presented only for 
the second blade, since particle erosion is expected to occur on both 
the front and the back side of the blade. Particle erosion on the front 
side of the blade is shown in Fig. 8, where contour of time integral 
of Eq. (17) is shown for both particle loading scenarios. Two distinct 
erosion zones are marked as A – blade tip region, and B – blade 
bucket. Difference in the pattern of zone B is observed, for heavier 
particle loading case (Fig. 8c) a more spread-out pattern emerges. 
Presented simulation results show that the extent of abrasion is 
higher in the heavy particle loading case, while in both normal 
and heavy loading scenarios two distinct erosion zones are formed 
on the front side of the blade. One is at the tip of the blade and is 
more pronounced in the heavy particle loading scenario. Second one 
is in the middle of the blade bucket and is again more pronounced 
in the heavy particle loading scenario. The difference between the 
erosion zone B pattern in Fig. 8 can be explained by larger number 
of particles in the flow in the heavy particle loading scenario. Due 
to larger number of particles, which take up more configurations 
in space (in the water jet), they produce a more spread-out pattern. 
These results of erosion patterns for the front side of the blade are 
also in general agreement with reference lab-scale experimental 
investigation of Pelton bucket by Umar et al. [41], where they also 
observed two distinct erosion zones (one around the blade splitter 
and one in the middle of the bucket). Direct comparison is however 
limited, by different Pelton geometry, operating parameters and 
sediment concentration. 
Fig. 8. Erosion on the front side of the Pelton rotor blade: a) real operating blade, b) normal particle loading scenario simulation, and c) heavy particle loading scenario simulation 
Fig. 9. Erosion on the back side of the Pelton rotor blade: a) real operating blade, b) normal particle loading scenario simulation , c) heavy particle loading scenario simulation

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Power Engineering
Similarly, we present erosion on the back side of the blade in Fig. 
9. Three distinct erosion zones are marked as A – blade mid-bucket 
region, B – blade bucket edge, and C – blade tip region. On the blade 
of real operating Pelton rotor, three erosion zones can be identified, 
(Fig. 9a), one in the middle of the bucket, one at the edge, splitting the 
blade in two halves, and one around the blade tip. All three regions 
are observed in the case of the simulation with the heavy particle 
loading, however, for the normal particle loading condition only the 
region in the middle of the bucket (Fig. 9b) region marked with A is 
predicted. In general, it is observed that the extent of erosion is less 
on the back of the blade than on the front side.
Since from the previous study a research question was whether 
particles could be the cause of blade tip damage, a detailed view of 
the blade tip is shown in Fig. 10. Results of simulation with heavy 
particle loading only are shown in Fig. 10a), as in the normal particle 
loading no erosion of the tip was predicted (Fig. 9). Under normal 
particle loading of the river, tip damage observed after operation of 
a real Pelton rotor as seen in Fig. 10b), can’t be attributed to particle 
erosion. However, under heavy loading conditions, particle erosion 
could cause damage to the tip, simulation results for these conditions 
show good agreement. Erosion rate is highest at the tip and the 
spreads out with lower intensity, and black lines in Fig. 10 indicate 
the extent of erosion spread. 
Finally, we give brief discussion of model sensitivity to different 
parameters or modelling scenarios. Particles were assumed 
to be spherical, however real river sediment particles come in 
different shapes. An example of idealized non-spherical shape is a 
superellipsoid, for which Wedel et al. [42] found that Lagrangian 
tracking gives better particle motion than by using simpler shape 
factors, indicating the complexity of this problem. This difference 
in particle motion could then be reflected in the erosion pattern at 
the wall. The influence of particles on erosion could also be due to 
irregular shape of particles, for example angular particles are known 
to be more erosive. Yasser, Zhou and El- Emam [43] conducted 
detailed computational fluid dynamics - discrete element method 
(CFD-DEM) simulations of different angular particles and spherical 
particles in pipe elbow and found that particles with fewer corners 
(but therefore sharper edges) produced more erosion, while pattern 
of erosion was more localized. In addition to that, even for spherical 
particles different drag models exist. Likewise, there are several 
approaches to model cavitation and within the vapor transport 
equation modelling approach adopted in this study, several algebraic 
cavitation models exist. For an overview of different cavitation 
models, we refer the reader to Folden and Aschmoneit [44].
4 CONCLUSIONS
This study presents a numerical investigation into the erosion of 
Pelton turbine rotor due to solid particles in a cavitating flow. By 
extending previous research that focused on cavitation-induced 
erosion, we incorporated Lagrangian particle tracking and employed 
the Finnie abrasion model [17] to assess the effects of sediment-laden 
water on turbine blades. Our findings indicate that, under normal river 
conditions, particle-induced erosion is not a significant contributor 
to the observed blade tip damage but can cause erosion of the blade 
bucket. However, under heavy sediment loading scenario, as a result 
of extreme weather phenomena, erosion predictions closely align 
with real-world observation on an actual operating Pelton rotor, 
suggesting that heavy particle concentrations can lead to substantial 
material loss, particularly at the blade tip.
The results highlight the necessity of considering both cavitation 
and particle erosion when evaluating turbine durability in sediment-
rich environments. Future work could focus on refining erosion 
models by incorporating particle shape effects, varying material 
properties, and exploring mitigation strategies such as optimized 
blade coatings or operational adjustments to minimize erosive wear.
Nomenclature
α
*
 turbulent viscosity damping coefficient, [-]
γ impact angle, [°]
χ size distribution parameter, [-]
φ
l
 liquid volume fraction, [-]
φ
v
 vapor volume fraction, [-] 
φ
g
 gas volume fraction, [-]
μ mixture viscosity, [Pa s] 
μ
l
 liquid dynamic viscosity, [Pa s] 
μ
v
 vapor dynamic viscosity, [Pa s]
μ
g
 gas dynamic viscosity, [Pa s]
ω specific turbulence dissipation rate, [s
−1
] 
ρ mixture density, [kg/m
3
] 
ρ
l
 liquid density, [kg/m
3
]
ρ
g
 gas density, [kg/m
3
]
Fig. 10. Erosion of the tip of the blade: a) simulation for heavy particle loading condition, b) actual Pelton rotor blade tip 

Power Engineering
292   ▪   SV-JME   ▪   VOL 71   ▪   NO 9-10 ▪   Y 2025
ρ
p
 particle density, [kg/m
3
]
ρ
v
 vapor density, [kg/m
3
]
σ
k
 turbulent Prandtl number for the turbulent kinetic energy, [-]
σ
ω
 turbulent Prandtl number for the specific turbulence dissipation 
rate, [-]
τ mixture shear stress tensor, [Pa] 
a
1
 model constant, [-]
C
D
 drag coefficient, [-]
d
p
 particle diameter, [m]
d
e
 size constant, [-] 
e
n
 coefficient of restitution in wall normal direction, [-]
e
t
 coefficient of restitution in wall tangential direction, [-]
F
e
 evaporation coefficient, [-] 
F
d
 condensation coefficient, [-]
F
2
 second blending function, [-]
f( γ) is an impact angle function, [-]
G
k
 production term for the turbulent kinetic energy, [m
2
/s
2
]
G
ω
 production term for the specific turbulence dissipation rate,  
[m
2
/s
3
]
k turbulent kinetic energy, [m
2
/s
2
] 
 m interface mass transfer rate, [kg/s]
n velocity exponent, [-] 
p pressure, [Pa]
p
v
 vapor pressure, [Pa]
R mass fraction of particles, [-]
R
B
 bubble radius, [m]
Re
p
 particle Reynolds number, [-]
S strain rate magnitude, [s
−1
] 
S
M
 momentum source term, [N
S
k
 source term for the turbulent kinetic energy, [m
2
/s
2
]
S
ω
 source term for the specific turbulence dissipation rate, [m
2
/s
3
]
t time, [s]
u velocity, [m/s]
ν particle velocity, [m/s]
v
n
 particle velocity in the wall normal direction, [m/s]
v
t
 particle velocity in the wall tangential direction, [m/s]
v
0
 empirical reference velocity, [m/s]
Y
k
 dissipation term for the turbulent kinetic energy, [m
2
/s
2
]
Y
ω
 dissipation term for the specific turbulence dissipation rate,  
[m
2
/s
3
] 
References
[1] Vovk, N., Ravnik, J. Numerical modeling of two-phase flow inside a wet flue gas 
absorber sump. Energies 16 8123 (2023) DOI:10.3390/en16248123.
[2] Gautam S., Acharya, N., Lama, R., Chitrakar, S., Neopane, H.P. Zhu, B., Dahlhaug, 
O.G. Numerical and experimental investigation of erosive wear in Francis runner 
blade optimized for sediment laden hydropower projects in Nepal Sust. Ener Tech 
Assess 51 101954 (2022) DOI:10.1016/j.seta.2022.101954.
[3] IEA. SDG7: Data and Projections (2024) https://www.iea.org/reports/sdg7-data-
and-projections, accessed 2025-09-22.
[4] Kent, R. Renewables. Plast Eng 74 56-57 (2018) DOI:10.1002/peng.20026.
[5] Giljen, Z., Nedeljković, M., Cheng, Y. The influence of pump-turbine specific 
speed on hydraulic transient processes. Stroj vest-J Mech E 70 231-246 (2024) 
DOI:10.5545/sv-jme.2023.776.
[6] CEI/IEC 60609-1:2004 Hydraulic Turbines, Storage Pumps and Pump-Turbines 
-Cavitation Pitting Evaluation -Part 1: Evaluation in Reaction Turbines, Storage 
Pumps and Pump-Turbines. International Organization for Standardization, 
Geneva
[7] Leclercq, C., Archer, A., Fortes-Patella, R., Cerru, F. Numerical cavitation intensity 
on a hydrofoil for 3D homogeneous unsteady viscous flows. Int J Fluid Mach Syst 
10 254-263 (2017) DOI:10.5293/IJFMS.2017.10.3.254.
[8] Fortes Patella, R., Reboud, J.-L., Briancon Marjollet, L. A phenomenological and 
numerical model for scaling the flow agressiveness in cavitation erosion. EROCAV 
Workshop (2004) https://www.researchgate.net/publication/281921326, 
accessed 2025-09-22.
[9] Schenke, S., van Terwisga, T.J.C. An energy conservative method to predict 
the erosive aggressiveness of collapsing cavitating structures and cavitating 
flows from numerical simulations. Int J Multiph Flow 111 200-218 (2019) 
DOI:10.1016/j.ijmultiphaseflow.2018.11.016.
[10] Melissaris, T., Schenke, S., Bulten, N., van Terwisga, T.J.C. On the accuracy of 
predicting cavitation impact loads on marine propellers. Wear 456-457 203393 
(2020) DOI:10.1016/j.wear.2020.203393.
[11] Arabnejad, M.H., Svennberg, U., Bensow, R.E. (2021). Numerical assessment of 
cavitation erosion risk using incompressible simulation of cavitating flows. Wear 
464-465 203529 (2021) DOI:10.1016/j.wear.2020.203529.
[12] Kevorkijan, L., Pezdevšek, M., Biluš, I., Hren, G. Cavitation erosion modelling - 
comparison of different driving pressure approaches. In. J Simul Model 22 233-
244 (2023) DOI:10.2507/IJSIMM22-2-639.
[13] Brunhart, M., Soteriou, C., Daveau, C., Gavaises, M., Koukouvinis, P., Winterbourn, 
M. Cavitation erosion risk indicators for a thin gap within a diesel fuel pump. Wear 
442-443 203024 (2020) DOI:10.1016/j.wear.2019.203024.
[14] Gomez Santos, E., Shi, J., Venkatasubramanian, R., Hoffmann, G., Gavaises, M., 
Bauer, W. Modelling and prediction of cavitation erosion in GDi injectors operated 
with E100 fuel. Fuel 289 119923 (2021) DOI:10.1016/j.fuel.2020.119923.
[15] Jošt, D., Škerlavaj, A., Pirnat, V., Morgut, M. Nobile, E. Detailed analyzis of flow in 
two pelton turbines with efficiency and cavitation prediction. Int J Fluid Mach Syst 
12 388-399 (2019) DOI:10.5293/IJFMS.2019.12.4.388.
[16] Rossetti, A., Pavesi, G., Ardizzon, G., Santolin, A. Numerical analyzes of 
cavitating flow in a Pelton turbine. J Fluid Eng-T ASME 136 081304 (2014) 
DOI:10.1115/1.4027139.
[17] Finnie, I. Erosion of surfaces. Wear 3 87-103 (1960) DOI:10.1016/0043-
1648(60)90055-7.
[18] Bitter, J.G.A. A study of erosion phenomena: Part I. Wear 6 5-21 (1963) 
DOI:10.1016/0043-1648(63)90003-6.
[19] Bitter, J.G.A. A study of erosion phenomena: Part II. Wear 6 169-190 (1963) 
DOI:10.1016/0043-1648(63)90073-5.
[20] Grant, G., Tabakoff, W. Erosion prediction in turbomachinery resulting from 
environmental solid particles J Aircraft 12 471-478 (1975) DOI:10.2514/3.59826.
[21] Grant, G., Tabakoff, W. An Experimental Investigation of the Erosive Characteristics 
of 2024 Aluminum Alloy. NTIS, Springfield (1973).
[22] Ahlert, K.R. Effects of Particle Impingement Angle and Surface Waiting on Solid 
particle Erosion of AISI 1018 Steel. MScThesis (1994) University of Tulsa, Tulsa.
[23] Forder, A., Thew, M., Harrison, D. A numerical investigation of solid particle 
erosion experienced within oilfield control valves. Wear 216 184-193 (1998) 
DOI:10.1016/S0043-1648(97)00217-2.
[24] Oka, Y.I., Okamura, K. Yoshida, T. Practical estimation of erosion damage caused 
by solid particle impact: Part 1: Effects of impact parameters on a predictive 
equation. Wear 259 95-101 (2005) DOI:10.1016/j.wear.2005.01.039.
[25] Oka, Y.I., Yoshida, T. Practical estimation of erosion damage caused by solid 
particle impact: Part 2: Mechanical properties of materials directly associated with 
erosion damage. Wear 259 102-109 (2005) DOI:10.1016/j.wear.2005.01.040.
[26] Det Norske Veritas. Erosive Wear in Piping Systems. https://rules.dnv.com/docs/
pdf/dnvpm/codes/docs/2015-08/RP-O501.pdf, accessed on 2025-09-22
[27] Gnanavelu, A., Kapur, N., Neville, A., Flores, J.F., Ghorbani, N. A numerical 
investigation of a geometry independent integrated method to predict 
erosion rates in slurry erosion. Wear 271 5-6 712-719 (2011) DOI:10.1016/j.
wear.2010.12.040.
[28] Peng, W., Cao, X. Numerical simulation of solid particle erosion in pipe bends 
for liquid-solid flow. Powder Technol 294 266-279 (2016) DOI:10.1016/j.
powtec.2016.02.030.
[29] McLaury, B.S., Shirazi, S.A., Shadley, J.R., Rybicki, E.F. Modeling erosion in chokes. 
Proceedings of ASME Fluids Eng. (1996) 773-782.
[30] Messa, G.V., Mandelli, S., Malavasi, S. Hydro-abrasive erosion in Pelton turbine 
injectors: A numerical study. Renew Energ 130 474-488 (2019) DOI:10.1016/j.
renene.2018.06.064.
[31] Liu, J., Pang, J. Liu, X., Huang, Y., Deng, H. Analysis of sediment and water flow 
and erosion characteristics of large Pelton turbine injector. Processes 11 1011 
(2023) DOI:10.3390/pr11041011.
[32] Kumar, D., Bhingole, P.P. CFD based analysis of combined effect of cavitation 
and silt erosion on Kaplan turbine. Mater Today-Proc 2 2314-2322 (2015) 
DOI:10.1016/j.matpr.2015.07.276.
[33] Li, L., Lu, J., Gong, Y., Zhao, H., Liu, X., Zhu, B. Sediment erosion characteristics 
of Pelton turbine runner: Effects of sediment concentration and diameter. Renew 
Energ 220 119679 (2024) DOI:10.1016/j.renene.2023.119679.
[34] Han, L., Guo, C., Yuan, Y., Li, D., Liu, Y., Iranzo, A., Qin, D. Investigation on sediment 
erosion in bucket region of Pelton turbine considering cavitation. Phys Fluids 36 
023305 (2024) DOI:10.1063/5.0179387.
[35] Zwart, P.J., Gerber, A.G. Belamri, T. A two-phase flow model for predicting 
cavitation dynamics. Proceedings of International Conference on Multiphase Flow 
152 (2004).

SV-JME   ▪   VOL 71   ▪   NO 9-10 ▪   Y 2025   ▪   293
Power Engineering
[36] Decaix, J., Goncalvès, E. Investigation of three-dimensional effects on a 
cavitating Venturi flow. Int J Heat Fluid Fl 44 576-595 (2013) DOI:10.1016/j.
ijheatfluidflow.2013.08.013.
[37] Drešar, P. Development of Biofluid Pump. PhD Thesis University of Ljubljana, 
Ljubljana (2020). (in Slovene)
[38] Zhang, X. Transient flow characteristics of a pressure differential valve with 
different valve spool damping orifice structures. Stroj vestn-J Eng E 70 141-158 
(2024) DOI:10.5545/sv-jme.2023.691.
[39] Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering 
applications. AIAA J 32 1598-1605 (1994) DOI:10.2514/3.12149.
[40] Schiller, L., Naumann, A. A Drag Coefficient Correlation. Z Ver Dtsch Ing 77 318-
320 (1935).
[41] Umar, B.M., Wang, Z., Chitrakar, S., Thapa, B., Huang, X., Poudel, R., Karna, A. 
Experimental erosion flow pattern study of Pelton runner buckets using a non-
recirculating test rig. Energies 17 4006 (2024) DOI:10.3390/en17164006.
[42] Wedel, J., Steinmann, P., Štrakl, M., Hriberšek, M., Ravnik, J. Shape matters: 
Lagrangian tracking of complex nonspherical microparticles in superellipsoidal 
approximation. Int J Multiphase Flow 158 104283 (2023) DOI:10.1016/j.
ijmultiphaseflow.2022.104283.
[43] Yasser, E., Zhou, L., El-Emam, M.A. Numerical analysis of particle shape influence 
on erosion and flow behavior in a 90-degree elbow pipe under Solid-Liquid flow. 
Adv Powder Technol 36 104928 (2025) DOI:10.1016/j.apt.2025.104928.
[44] Folden, T.S., Aschmoneit, F.J. A classification and review of cavitation models with 
an emphasis on physical aspects of cavitation. Phys Fluids 35 081301 (2023) 
DOI:10.1063/5.0157926.
Acknowledgements The authors wish to thank dr. Dragica Jošt for her 
assistance in the transfer and explanation of the previous case, as well as her 
kind openness for discussion. Additionally, we thank the Slovenian Research 
Agency (ARRS) for its financial support in the framework of Research Program 
P2-0196 in Power, Process, and Environmental Engineering.
Received: 2025-04-16, revised: 2025-07-13, accepted: 2025-07-24  
as Original Scientific Paper 1.01.
Data Availability The data that support the findings of this study are not 
available due to project confidentiality.
Author Contribution Luka Kevorkijan: Formal Analysis, Investigation, Writing 
– original draft, Software, Visualization; Matjaž Hriberšek: Conceptualization, 
Methodology, Project Administration, Supervision; Luka Lešnik: Investigation, 
Writing – review & editing, Visualization; Aljaž Škerlavaj: Conceptualization, 
Data curation, Methodology, Validation; Ignacijo Biluš: Investigation, 
Supervision, Writing- original draft, Writing – review & editing.
Numerična raziskava erozije zaradi delcev v kavitirajočem toku 
skozi Peltonovo turbino
Pozetek Erozija komponent Peltonove turbine zaradi kavitacije in toka 
z delci predstavlja velik izziv pri hidroenergetskih sistemih, zlasti v rekah, 
bogatih s sedimenti. V tej študiji je predstavljena numerična raziskava vpliva 
trdnih delcev na erozijo rotorja Peltonove turbine. Simulacije računalniške 
dinamike tekočin (CFD) so bile izvedene z uporabo programa ANSYS CFX 
2023 R2, pri čemer sta bila vključena Lagrangev pristop sledenja delcev in 
Finniejev model abrazije za napovedovanje erozijskih vzorcev pri različnih 
koncentracijah sedimentov. Rezultati kažejo, da pri običajnih pogojih 
sedimentacije erozija zaradi delcev ne prispeva bistveno k poškodbam konic 
lopatic. Vendar pa pri ekstremni obremenitvi s sedimenti napovedani erozijski 
vzorci tesno ustrezajo dejanskim opazovanjem, zlasti na konici lopatice.
Ključne besede Peltonova turbina, erozija zaradi trdnih delcev, kavitacija, 
CFD računalniška dinamika tekočin