Andreas Petersa,∗, Hemant Sagara , Udo Lantermanna , Ould el Moctara of Ship Technology, Ocean Engineering and Transport Systems, University of Duisburg-Essen, Bismarckstr. 69, 47057 Duisburg
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a Institute
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Numerical Modelling and Prediction of Cavitation Erosion
Abstract
This paper addresses the prediction of cavitation erosion using a numerical flow
solver together with a new erosion model. Numerical flow simulations were conducted with an implicit, pressure-based Euler-Euler multiphase flow solver
in combination with the developed erosion model. The erosion model refers to the microjet hypothesis and uses information from the flow solution to assess
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the occurrence of microjets in specific areas. The ability of the numerical code to simulate cavitating flows was shown by comparison with experimental tests
of sheet cavitation over a NACA 0009 hydrofoil. The numerical prediction of
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cavitation erosion was compared to measured erosion in experimental tests of an axisymmetric nozzle and shows good agreement regarding the erosive areas in general and the areas of highest erosion. Aim of this work is the assessment of erosion sensitive areas, as well as the erosion potential of cavitational flow during the incubation period.
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Keywords: cavitation, erosion, multiphase flow, computational fluid
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dynamics, erosion model, Reboud’s correction
∗ Corresponding
author Email addresses:
[email protected] (Andreas Peters),
[email protected] (Hemant Sagar),
[email protected] (Udo Lantermann),
[email protected] (Ould el Moctar)
Preprint submitted to Wear
June 7, 2016
1. Introduction Cavitation occurs, when small gas filled cavitation nuclei reach low pressure
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regions. When the local field pressure approaches the saturation pressure of
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the fluid, an evaporation process is started. This causes the cavitation nuclei to grow to vapour filled “cavitation bubbles”. As soon as the pressure inside
the bubbles exceeds the surrounding field pressure, the bubbles will suddenly collapse and condensate. In addition to cavitation bubbles, different types of cavitation may be identified, containing accumulations of these single cavitation
bubbles. They can be distinguished depending on their shape and dynamical behaviour. “Sheet cavitation” is often formed at the leading edge, on the suction
side of a hydrofoil. This type of cavitation can be more or less stationary, where the change of its form is only marginal. Depending on the flow conditions,
though, the sheet cavitation may also show a strong transient behaviour, where
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its growth and shrinkage are harmonic. The so called “cloud cavitation” may be shed from an unsteady sheet cavitation by a “re-entrant jet” as described by Callenaere et al. [1] and Decaix and Goncalv`es [2]. This re-entrant jet is a
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result of the cavitation due to a blocking effect on the liquid flow. The flow is going to be reversed at the end of the sheet cavitation. Analog to the dynamics of single bubbles, the volume of a vapour cloud oscillates. The collapse of the cloud cavitation occurs further downstream in higher pressure regions. The instantaneous implosion of these cavitation clouds leads to the genera-
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tion of pressure waves of high amplitudes. These pressure waves are regarded as a main mechanism giving rise to erosion. Fortes-Patella et al. [3] suggested a cavitation erosion model, where the potential energy of the macroscopic cav-
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itation structures is regarded as the main factor that generates erosion. The potential energy of a cloud cavitation is supposed to be converted into acoustic energy of pressure waves, which travel through the fluid and are able to damage a surface directly. The mechanism of cloud collapse is also regarded as the main damaging mechanism by Wang and Brennen [4]. In their model, the fully non-linear
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Rayleigh-Plesset equation is used to simulate the interactions of a spherical bubble cloud, consisting single cavitation bubbles, with the liquid. The acoustic
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pressure radiated by the spherical cloud is calculated and employed to quantify the damage potential of the cloud.
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Further erosion models concentrate on different flow properties, which may be calculated by using numerical methods. Li [5] stated a numerical erosion model, where the absolute pressure needs to exceed a threshold pressure for
erosion to be predicted. Nohmi et al. [6] proposed multiple indices based on pressure, vapour volume fraction and their derivatives. These indices may then be used to assess the local erosion from the flow simulation.
Another hypothesis implies that the radiated pressure waves are not able to damage a surface directly since their pressure amplitudes are strongly attenuated when moving through the fluid. It is believed that these pressure waves
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initiate the oscillation and collapse of other cavitation bubbles. When bubbles
collapse close enough to a material surface, this process is always asymmetrical as the flow through the bubble is disturbed by the surface itself [7]. This, in turn, produces a liquid waterjet, also called “microjet”, which flows through the
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bubble and breaks it up into partial cavities. The collapse process was investigated experimentally and theoretically by Brujan et al. [8] as well as numerically by Lauer et al. [9]. Because of the asymmetrical collapse process, the microjet is almost always pointing towards the material surface. As shown by Field [10] and Haller Kneˇzevi´c [11], the impact of the jet onto the wall leads to the
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generation of a shock wave radiated perpendicularly away from the wall. This phenomenon induces a very high pressure near the wall, the so called “water
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hammer pressure”, which may be higher than the yield strength of common steel materials and able to damage a solid surface [12]. Dular and Coutier-Delgosha [13] and Dular et al. [14] stated an erosion model, where the velocity of a microjet needs to exceed a certain velocity threshold to be erosive for a certain material. Both the amount of pits on a surface, as well as the totally damaged surface is calculated further on. In the present approach, a numerical erosion model has been developed fol3
lowing the work of Dular and Coutier-Delgosha [13] to predict the most threatened areas of erosion – the erosion sensitive areas – as well as the intensity of
2. Numerical Method
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erosive impacts during the incubation period – the erosion potential.
In the present work, the open source CFD software package OpenFOAM [15] was used to simulate cavitating flows and develop models to predict cavitation erosion. 2.1. Euler-Euler Two Phase Flow
Cavitating flows are multiphase flows involving phase changes. In the present work, the Euler-Euler approach is adopted, which deals with both liquid and
vapour phase on a fixed Eulerian grid, where the flow is treated as a homoge-
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neous mixture of the two incompressible, isothermal phases. A Volume of Fluid
(VoF) method is utilized to track interfaces between the phases. For this approach, the equations for conservation of mass and momentum of the mixture are defined as:
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∂ρ ∂(ρui ) + = 0, ∂t ∂xi
∂(ρui ) ∂(ρui uj ) ∂p ∂ + + =µ ∂t ∂xj ∂xj ∂xj
∂ui ∂uj + ∂xj ∂xi
(1)
+ Svf .
(2)
ui is the velocity in the coordinate direction xi and p is the pressure. t is
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the time and ρ and µ are the density and dynamic viscosity of the homogeneous mixture. Svf are source terms due to volume forces like gravitation.
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Characteristic for the Euler-Euler approach with a VoF method is the in-
troduction of a volume fraction α, which defines the volume of vapour or liquid occupied in the current cell. The vapour volume fraction is given by: α=
Vv , Vv + Vl
(3)
with the volumes of liquid Vl and vapour Vv of the cell. The densities and viscosities of the pure phases are constant. The mixture properties of density 4
and dynamic viscosity can then be calculated with the volume fraction: µ = αµv + (1 − α)µl .
(4)
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ρ = αρv + (1 − α)ρl ,
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ρl and ρv are the densities µl and µv are the viscosities of pure liquid and pure
vapour, respectively. α is obtained from an additional convective transport equation:
∂α ∂αui + =S, ∂t ∂xi
(5)
where S is the source term of the net phase change, defined as: S = Se − Sc .
(6)
The terms on the right hand side are the phase change rates of evaporation Se
and condensation Sc . These source terms are obtained from a cavitation model.
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2.2. Schnerr-Sauer Cavitation Model
In the present work, the cavitation model by Sauer and Schnerr [16] is applied. The model is based on the fact that the vapour phase can be defined by
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a finite number of single bubbles per volume of liquid. The vapour volume is then a function of the number of bubbles per liquid and of their size. Therefore, a definition of the vapour volume fraction is stated: α=
nb 4/3 πRb3 , 1 + nb 4/3 πRb3
(7)
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with nb as the number of bubbles per volume of liquid and Rb the bubble radius. Furthermore, the dynamics of bubbles are embedded by using a simplified form of the Rayleigh-Plesset equation. In the Rayleigh-Plesset equation the pressure
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difference pb − p is the dominant term, with pb being the pressure inside the bubble. Therefore, effects due to inertia, surface tension, viscosity and relative velocities can be neglected, which leads to the following simplified form: r 2 pb − p dRb = . dt 3 ρl
5
(8)
(9)
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continuity equation by using expressions 7 and 8: r ρv ρl 3α(1 − α) 2 pb − p Se = , for p < pv , ρ Rb 3 ρl r ρv ρl 3α(1 − α) 2 p − pb Sc = , for p ≥ pv . ρ Rb 3 ρl
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The source terms for evaporation and condensation are then deduced from the
Depending on the local field pressure, a process of evaporation or condensation is started, causing the vapour volume to either grow or shrink. 2.3. Turbulence Effects on Cavitation
The turbulence in a flow may have an essential influence on the dynamics of cavitation. The standard turbulence models are not able to model unsteady cav-
itating flows with incompressible flow solvers. They do not standardly account for turbulence effects on the vapour pressure. Additionally, they do often fail to
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enable the re-entrant jet, which causes the harmonic cloud shedding. This leads
to a steady sheet cavitation most of the time, as first noticed by Reboud et al. [17]. Therefore, some additional effects of turbulence have to be considered to
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calculate cavitating flows. According to Singhal et al. [18] it could be shown in experimental investi-
gations that turbulent pressure variations have an effect on the local vapour pressure. This can be accounted for by calculating the turbulent pressure fluctuations as follows:
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p0turb = 0.39ρm k
(10)
with the turbulent kinetic energy k. This induces a higher vapour pressure,
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which is defined as: pv = psat +
p0turb , 2
(11)
where psat is the saturation pressure. Besides the increase of the vapour pressure in cavitating flows, compress-
ibility effects on turbulence play an important role. The presence of vapour in a liquid leads to a significant decrease of the speed of sound. It may change considerably and be even lower than the speed of sound for either pure liquid 6
or pure vapour, as remarked by Kiefer [19]. For that reason, it has to be dealt with supersonic regions in cavitating flows.
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Using Direct Numerical Simulation (DNS), Heinz [20] could show that one result of growing compressibility is the reduction of turbulent kinetic energy k.
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These compressibility effects are not considered by the common two equation turbulence models for incompressible RANS methods. As a consequence, the
turbulent viscosity is generally overestimated. Reboud et al. [17] therefore proposed a correction of the turbulent viscosity, which accounts for these effects
in interface regions between the two phases. The turbulent viscosity can be calculated as: µt = f (ρ) Cµ k 2 /ω ,
with f (ρ) = ρ
(standard)
(12)
where Cµ is a constant and ω is the specific turbulent dissipation rate. f (ρ)
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is a function of the density. In the standard form this function is equal to the density itself, whereas Reboud et al. stated the following formula: n
f (ρ) = ρv +
(ρ − ρv )
n−1
(ρl − ρv )
,
with n = 10
(Reboud) .
(13)
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n is an exponential coefficient. A comparison between the density functions is shown in Figure 1. It can be seen that the density function by Reboud et al. is much lower in the mixture region than the linear function. This, in turn, leads to a reduction of the turbulent viscosity in that region.
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3. Erosion Modelling
3.1. New Approach for an Erosion Model
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The present erosion model is based on the hypotheses stated by Dular and
Coutier-Delgosha [13] that the damaging of a surface is caused by single bubbles collapsing and forming microjets. The collapse of a cloud cavitation is emitting a pressure wave, whose amplitude is however attenuated towards the material surface. Nevertheless, this pressure wave is able to induce the oscillation and collapse of bubbles in other flow regions. If these bubbles are collapsing close enough to a material surface, they are going to form microjets. 7
1000 standard Reboud’s correction
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600 400 200 0 0
200
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f (ρ) [kg/m3]
800
400
ρ
600
800
1000
[kg/m3]
Figure 1.
According to Dular and Coutier-Delgosha, the local velocity of the microjet
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is one of the critical quantities for the prediction of cavitation erosion. The
most probable case for a microjet is chosen with the dimensionless stand-off distance γ = H/Rb (H: distance from bubble center to surface) of 1.1 based
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on former investigations by Dular and Coutier-Delgosha [13] and Dular et al. [14]. This represents a typical collapse scenario of a bubble approaching the surface. Once the bubble is in the vicinity of the surface, it will start to deform quickly. Additionally, a stand-off distance of γ = 1.1 may lead to microjets of high damaging potential. Lohrberg [21] derived a formula for the approximation of jet velocity, which was compared to velocities from different theoretical and
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experimental investigations, which is shown in Figure 2. The formula for the jet velocity by Lohrberg [21] is derived based on the
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Kelvin impulse, as described by Blake [22] and assumes the collapse of a spherical bubble as well as the conversion of the potential bubble energy into the energy of the jet impulse. These assumptions, as well as the overall validity of the formula are questionable. In Figure 2 the formula shows an opposing tendency for higher γ values compared to the experimental results from Philipp and Lauterborn [23]. However, it does show a fair agreement with the velocities from the other
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Figure 2.
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investigations for a dimensionless wall distance of γ ≈ 1. In the present erosion model, only γ = 1.1 is considered. The following formula is derived from the equation of Lohrberg [21] to approximate the jet velocities for the specific case of γ = 1.1:
r
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vjet ≈ 10.8
p − pv , ρl
(14)
If water at 20 ◦ C as well as a liquid pressure of p = 100 kPa are consid-
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ered, a jet velocity of 107 m/s is calculated using Equation 14. For the present erosion prediction this approximation shows acceptable agreement with the experimental data in Figure 2 and Figure 3 since the used erosion model predicts √ qualitative values only. The dependency of vjet ∝ p − pv given by Equation
14 was also found and validated by Chahine [24] in numerical investigations. To find a dependency of the local jet velocity vjet on γ, the measurements by Philipp and Lauterborn [23] are the best choice to the knowledge of the present 9
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Figure 3.
authors. The velocities are shown in Figure 3. It becomes clear that for γ > 0.5, a higher γ leads to a lower impact velocity of the microjet.
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Based on the local jet velocity, the water hammer stress or water hammer pressure can be approximated. This pressure is supposed to be the main damaging mechanism. It is expressed as: phammer ≈ vjet ρl cl ,
(15)
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with cl being the speed of sound of pure liquid and ρl being its density. When the water hammer stress reaches a given stress, plastic flow is caused on the material surface. This is supposed to be the case, when the local jet velocity
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exceeds a critical velocity. An expression for the critical velocity was stated by Lush [25], where a one-
dimensional impact of a liquid mass on a solid surface was observed. When an accelerated liquid mass, a microjet, hits a surface, a shock wave is generated, travelling perpendicularly away from the surface. The microjet causes a decrease of velocity and an increase of pressure at the wall. The governing equations for
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conservation of mass and momentum can then be solved, using the Tait equation of state to relate pressure and density. When the critical velocity is exceeded
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expression for the critical velocity vcrit reads as follows: s py py −1/n vcrit = 1− 1+ , ρl B
t
by the jet velocity, a solid material of yield strength py is being deformed. The
(16)
with B = 300 MPa and n = 7 being standard coefficients for liquid water in the Tait equation.
A part of the energy resulting from the water hammer stress is spent to reach
the limit for plastic flow. This pressure can be approximated using the critical velocity as:
pplastic ≈ vcrit ρl cl .
(17)
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The remaining part of phammer is exerted to deform the surface: pdef ≈ vdef ρl cl = (vjet − vcrit )ρl cl .
(18)
pdef is the deformation pressure and vdef the deformation velocity.
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Information about traces, position and amount of single bubbles is not in-
cluded in the Euler-Euler approach. Therefore, in the present work, the areas of potential erosion are identified by looking at the local flow conditions. Afterwards, the potential of erosion is determined by using dimensionless coefficients. To enable the occurrence of microjets, conditions have to be fulfilled, which
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control the presence of cavitation bubbles in the vicinity of the considered surfaces. Whether there are bubbles near to a face on a boundary, depends on the
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flow properties in the cells in its vicinity. It is therefore examined, if vapour is present within a given distance to the surface, which enables the movement of single bubbles towards the surface. For each face of a surface, geometrical shapes are used to select cells around all faces of the considered boundary and collect them in volumetric cell zones. In the present approach, these zones are generated by selecting all the cells located in spheres around the face centres of the regarded boundary. This method is sketched in Figure 4, where the face is 11
marked red, the cell zone is marked as a green sphere and a white cell is shown
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inside of that zone.
Figure 4.
The generation of cell zones has to be conducted before the simulation is run
and exactly one zone is assigned to every face of the considered wall boundary. Figure 5 shows the control volumes, which are collected for a face by using the
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sphere.
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R
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: collected cells
Figure 5.
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The erosion calculation is conducted following the flow simulation in each
time step. It is started by checking conditions depending on the surrounding flow field for all the faces on the solid surface and the cells in their associated cell zones. For each face, the following conditions are checked (see Fig. 6): • Vapour Condition: Is at least a small amount of vapour present in any cell of the current cell zone? This condition is examined, by checking if 12
the vapour volume fraction α is greater than a threshold value. In this work, a threshold value of 0.01 is chosen based on investigations of former
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are able to move towards the solid surface.
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simulations. When this condition is fulfilled, it is predicted that bubbles
• Jet Velocity Condition: The implosion of bubbles near a solid wall
will only have an impact on the material, if the local pressure, caused by a microjet, exceeds a given threshold. It is therefore calculated, whether the
local jet velocity is greater than the critical velocity, which is constant for a given flow problem and solid material. The local jet velocity is calculated
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based on the flow properties on the regarded face.
R
α ≥ 0.01
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vjet > vcrit
solid surface
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Figure 6.
Figure 6 shows the described investigation for a cell–face relationship. In
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this case, a cell can be found within the assigned cell zone, where α ≥ 0.01 and where vjet ≥ vcrit at the considered face. It is then enabled that a bubble can possibly move to the vicinity of the surface and form a microjet. The velocity of this microjet is then high enough to induce an impact on the surface, with pressure magnitudes exceeding the yield strength of the contemplated material. This leads to a deformation of the surface during the incubation period.
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To determine the erosion intensity of such an impact, the dimensionless intensity coefficient cintensity is introduced, which is calculated for every impact
vjet phammer ≈ . vcrit pplastic
(19)
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cintensity =
t
on a face of the considered surface in particular:
The coefficient relates the local jet velocity vjet to the critical velocity vcrit
as well as the water hammer pressure phammer to the pressure needed to reach plastic flow pplastic . It is a measure of the intensity of erosion of a single microjet
impact on a face. For faces which are not impacted in the current time step,
cintensity is equal to 0. Impacted faces have a value greather than 1, since the jet velocity is always higher than the critical velocity in this case.
The damage inflicted on a surface due to cavitation erosion, depends on the amount of impacts on that surface and the aggressiveness of each single im-
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pact. The erosion potential is therefore calculated based on the accumulation of
cintensity on each face. To obtain a qualitative erosion prediction, all the erosion impacts of the whole temporal history as well as of the whole spatial domain are taken into account and used for normalization. The cintensity values are summa-
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rized for each face explicitly over the temporal history (see Fig. 7). Afterwards, the erosion of every face as the accumulation of cintensity is normalized by the total sum of cintensity in the domain. The deformation coefficient is introduced
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as:
cdef = P
n
P
T
cintensity,t
t
N
(
P
t
T
cintensity,t )n
.
(20)
t is the time step index and T the total erosion calculation time. n is the
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face index and N the total number of eroded faces. An example of this calculation is sketched in Figure 7. A surface of 9 faces
is investigated for erosion potential and shown from top view. In the first time step of the calculation of erosion prediction 3 faces are impacted. As described before, the intensities of these impacts are calculated in terms of the intensity coefficient cintensity for every face. Faces that have not been impacted have
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values of 0 and four faces is impacted in the second time step. This procedure is repeated for every time step and the values of cintensity are accumulated for
t
every face explicitly. To yield a dimensionless coefficient cdef for the erosion potential of a face, the sum of the accumulated cintensity values from all faces is
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used. Therefore, in the final step the deformation coefficient cdef is calculated
according to Equation 20 for every face, which is the fraction of erosion on this face compared to the total predicted erosion. For example, on the right
of Figure 7 one can see the calculated cdef field, which shows that 29 % of the whole predicted erosion takes place on the face in the centre of the grid. cintensity
cintensity
0.0 0.0 0.0
0.0 1.2 0.0
1.1 1.3 0.0
+
1.2 1.4 1.1 0.0 0.0 0.0
time step 1
time step 2
+ ... =
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0.0 1.2 0.0
P
cintensity
cdef
6.4 44.9 18.4
0.02 0.16 0.07
24.1 82.2 32.4
0.09 0.29 0.11
5.2 45.3 23.7
0.02 0.16 0.08
after n time steps
Figure 7.
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The final deformation coefficient cdef does though combine both the intensity of single impacts (from Equation (19)) as well as the amount of impacts on a face (sums of impacts). It is limited between 0 and 1 by normalization. The sum of cdef of all faces is then equal to 1. A value of 0 means that no erosion is predicted on this face and a high number corresponds to a high erosion potential
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for the considered face.
The erosion calculation is a post-processing procedure, which is executed
in every time step of the simulation after the flow solution. The coupling in
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this case is just in one direction, so that the calculated erosion potential has no influence on the flow solution. This is a reasonable hypothesis, since the predictions are going to be made for the incubation period only, where the material deformation is linear, but there is almost no material loss. A change of geometry during this period can therefore be neglected since the flow conditions are hardly influenced. This approach enables the comparison of erosion prediction
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with erosion during the incubation period, where a linear deformation rate can
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be assumed.
4. Validation of the Numerical Approach to Simulate Cavitation on
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NACA 0009
Since there is a lack of data to validate both the cavitating flow, as well as the prediction of erosion for one case, the validation of the present approach to predict cavitation erosion is divided into two steps: 1. Validation of the cavitating flow for a hydrofoil. 2. Validation of the new erosion model for an internal nozzle flow.
The method used to simulate cavitating flows is validated for the 3-D tunnel
flow around a modified NACA 0009 hydrofoil. This case is chosen to validate cavitation, because sufficient information is available from [26]. Therefore, pres-
obtained and compared.
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sure distributions, force coefficients and frequencies of the cavitation cycles are
The pressure-based, cavitating flow solver interPhaseChangeFoam out of the
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OpenFOAM framework was used. The second-order, implicit Crank-Nicolson method is applied for temporal discretization. Central differencing schemes are utilized for spatial discretization of all variables despite turbulence variables, which are discretized by a first order upwind scheme. All necessary modifications, as described in Section 2, are applied. Experimental data from [26] is adduced for comparison with the numerical results. The inlet velocity is set
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to 20 m/s at an incidence angle of 2.5◦ . The cavitation number for this case is
σ = 0.81. A time step size of 1 · 10−5 is defined. To model turbulence, the
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k-ω-SST turbulence model is applied together with the corrections described in Section 2.3. Logarithmic wall functions are used at all wall boundaries for friction modelling. The cavitation model by Sauer and Schnerr is chosen to model the processes of evaporation and condensation. The parameters of bubble number per volume of the liquid and bubble radius are set to nb = 5 · 108 and Rb = 1 · 10−5 , respectively.
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The numerical setup is generated analog to the measurement section of the experiments. Wall boundaries are chosen for the sides, as well as for the hydro-
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foil and the top and bottom boundaries of the domain. An inlet boundary is defined upstream and an outlet boundary downstream from the NACA profile.
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A hexahedric block is generated around the foil to apply additional grid refinement. Another block is placed at the trailing edge for further mesh resolution.
Thin prismatic cells are used near wall boundaries to adopt logarithmic wall functions. The grid topology of the solution domain is shown in Figure 8. Wall
Outlet
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Inlet
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Wall
Figure 8.
The numerical simulations show harmonic variations in form and length of
the sheet cavitation. However, a shedding process of the whole sheet cavitation
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detaching from the hydrofoil to form a macroscopic cloud cavitation can not be observed. Sheet cavitation is formed downstream from the leading edge, on the
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suction side of the NACA profile. Figure 9 shows the hydrofoil in perspective view with images being separated by ∆timage = 10−3 s. The velocity magnitude is shown in a slice parallel to the chord of the hydrofoil and the iso-surface of 10 % vapour volume fraction is depicted in light blue. The figure shows that sheet cavitation is generated in the high velocity regions on the suction side of the foil, where the lowest pressure values are located. Figure 9 (a) shows the fully developed sheet cavitation on the hydrofoil’s suction side. From (b) to (e) small 17
re-entrant jets develop towards the leading edge of the foil, which cut through the sheet cavitation in different positions. This leads to small vapour clouds
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being shed from the sheet cavitation, which collapse closely behind it (see (f) to (g)). While small cavitation structures are shed, the sheet cavitation already
(a)
(b)
(d)
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(c)
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recovers its initial length until it is almost fully developed again in (h).
(e)
(f)
(g)
(h)
Figure 9.
The behaviour of sheet cavitation in the numerical simulation is harmonic,
as it is in the experimental investigations. The vapour volume fraction shows a mostly harmonic behaviour with varying amplitudes (see Fig. 10). A dominant frequency of this process can clearly be identified by looking at the fast Fourier transform (FFT) of the temporal history of the vapour volume fraction. The 18
frequency can be traced back to the variations in form and length of the sheet
0.18
0.14 0.12 0.1 0
0.02
0.04
0.06
0.08
0.1
Time [s]
0.2 0.15
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0.16
0.25
t
Amplitude of Volume Fraction [-]
Vapour Volume Fraction [10−3 ]
cavitation. This frequency is analyzed for different mesh sizes.
0.1
0.05 0
0
100
200
300
400
500
Frequency of Cavitation [Hz]
Figure 10. 4.1. Grid Independence Study
For the described case, a grid independence study was conducted to find
the grid size, which is the best compromise between efficiency and precision.
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Simulations with three different grids were investigated. The edge length of the cells between the generated medium grid and the other two grids differs by the factor 23/4 . As previously described, the main frequencies of the harmonic variations are obtained by conducting fast Fourier transforms of the temporal
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progress of the vapour volume fraction. A comparison of these frequencies, as well as of the lift and drag coefficients is listed in Table 1. Here, f is the main frequency of the harmonic variations of the sheet cavi-
tation. The force coefficients of drag cD and lift cL can be calculated by: F 1/2ρv 2 A
ce
cF =
,
(21)
ref
where F is the force of drag D or lift L. v is the velocity in the far field and
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Aref is the wetted area of the hydrofoil. The simulation with the coarse grid is not able to show a harmonic behaviour
of the vapour volume fraction. For this case, more than one frequency is having a strong influence on the transient behaviour of cavitation. A characteristical frequency can not be identified, though. For the simulations with medium and fine grid, such a frequency can clearly be found. The same frequencies can also be found by looking at the temporal progress of the lift force. The frequency of 19
the medium grid simulation is deviating by 5.1% from the fine grid simulation,
4.2. Validation of the Simulation of Sheet Cavitation
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whereas the force coefficients are deviating by less than 4%.
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The results of the numerical simulations are compared to results from experimental measurements (see Table 2). v is the velocity in the far field and σ
is the cavitation number of the specific case. The results of experimental and numerical tests are in good agreement. The forces acting on the hydrofoil are
generally depending on length, form and frequency of the cavitation. Here, the deviation for the lift coefficient of the numerical simulation is less than 2 %. The
calculated characteristic frequency is deviating by 11.3 % from the measured one from experimental results.
In the measurements the characteristic frequency of cavitation was derived
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from pressure sensors. These sensors did show an unsteady pressure behaviour, which is much less harmonic than the progression of the calculated vapour volume in the simulations. Additionally, frequencies of rotating components in front of the measurement section can not be hidden in the results of the mea-
pt ed
surements. Besides these discrepancies in obtaining the cavitation frequencies, deviations between the measured and calculated values may be caused by modelling of cavitation and turbulence. The influence of turbulence modeling on
Ac
ce
cavitating flow was explained in chapter 2.3.
20
1 numerical experimental
0.6 0.4
t
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
0.2
an us cr ip
Pressure Coefficient [-]
0.8
0.4
0.6
0.8
1
x/c [-]
Figure 11.
Instantaneous distributions of the pressure coefficient around the foil are
M
depicted in Figure 11. It becomes clear, that the length of the sheet cavitation
and the general pressure distributions of experiment and simulation are similar.
pt ed
5. Prediction of Cavitation Erosion in an Axisymmetric Nozzle The phenomenon of cavitation erosion in an axisymmetric nozzle was exper-
imentally investigated by Franc and Riondet [27]. The test section of this case is shown in Figure 12. Liquid water is flowing into an inlet cylinder of 16 mm diameter from top to bottom. A radial divergent outlet part is connected to
ce
the inflow cylinder via a 1.0 mm radius. The flow through the nozzle generates sheet and cloud cavitation at this radius, which is highly unsteady. The mean length of the sheet cavitation is stated to be of approximately 25 mm length
Ac
from the radius. The forming and collapsing of the vapour structures causes the bottom target plate to be eroded in a radial distance of 19 mm to 32 mm from the central axis. The velocity at the nozzle inlet is 31 m/s and a cavitation number of 0.9 is configured.
21
Uin
Uout
cavitation
an us cr ip
2.5 mm
t
16 mm
1 mm
erosion target plate
Figure 12.
The described case is used for the validation of the numerical prediction
of erosion. Simulations were conducted for a quarter of the nozzle geometry. Physical conditions are realized by setting the inlet velocity in the simulation to
31 m/s and the outlet pressure to 10.1 bar to obtain the same cavitation number as in the experiment. The inlet boundary is located 50 mm upstream of the
test section. A diffuser like pressure reservoir is attached at the outlet of the
M
solution domain in a radial distance of 100 mm from the central axis. 5.1. Simulation of the Quarter Nozzle
An unstructured hexahedral mesh is generated for a quarter of the nozzle.
pt ed
Refinements are positioned at the test section in the radius region and in different radial distances outward from the central axis. The meshed test section of the grid with about 2.14 million cells is depicted in Figure 13. A high mesh resolution of the radius is important, since the geometry of the radius has a significant influence on the characteristics of the cavitation. Prism layers are
ce
generated at all wall boundaries and logarithmic wall functions are applied. The same properties of the numerical flow solver are adopted as for the simulations
Ac
of the NACA 0009 hydrofoil. To obtain a maximum CFL number of less than 5, time step sizes are varied between 2.5 · 10−7 s and 5.0 · 10−7 s depending on the
spatial discretization. The model by Schnerr and Sauer is again used to model the processes of evaporation and condensation involved in the cavitating flow.
22
t an us cr ip
Figure 13.
The simulation of the flow through the nozzle shows that vapour is generated at the connecting radius which is also observed for the experimental results. It
can be seen from Figure 14 that the vapour structures spread further downstream. The cavitation structures look unsteady and arbitrary in both radial
M
and circumferential direction. Though, multiple harmonics can be identified for
Figure 14.
ce
pt ed
this process, as is shown in more detail below.
Figure 15 shows the temporal progress of the vapour volume fraction α (left)
Ac
and the fast Fourier transform (FFT) of this progress (right). From looking at the temporal progress of the generation and collapsing of vapour, the unsteady behaviour is obvious. This makes it difficult to identify one single characteristic frequency of cavitation, which can also be seen from the frequency analysis in Figure 15 (right). Nevertheless, multiple dominant frequencies can be found. Within a range of 300 to 400 Hz dominant frequencies are related to the macroscopic generation and condensation of vapour. Further dominant, but smaller 23
amplitudes, can be found at frequencies of 1200 to 1300 Hz and around 2500 Hz, which both correspond to small local collapses of vapour. The smaller collapses
t
were investigated in more detail, as shown below. The present case was also numerically investigated by Mihatsch et al. [28] using a density-based approach.
an us cr ip
There, the authors stated a characteristic frequency of 408 Hz and two higher
frequencies of 1139 Hz and 1182 Hz, that are in good agreement with the ones identified by the pressure-based approach used in the present study. This shows
Vapour Volume Fraction [10−3]
ulate cavitating flows. 1.5 1.25 1 0.75 0.5 0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
M
0
Amplitude of Volume Fraction [-]
the ability of the numerical method, which is used in the present work, to sim-
9
7.5 6
4.5 3
1.5 0
0
1000
2000
3000
4000
5000
Frequency of Cavitation [Hz]
Figure 15.
pt ed
Figure 16 shows the temporal progress (left) and the FFT (right) of the
vertical pressure force on the bottom boundary FZ . The temporal progress shows many negative peaks of high magnitudes, which indicate forces directed onto the bottom target plate. These impacts occur especially with frequencies of 1250 to 1300 Hz and further subharmonics, which can be seen from looking
ce
at the FFT of the vertical force (Fig. 16 (right)). Frequencies between 300 and 400 Hz are less dominant for the vertical force than for the cavitation itself, but
Ac
a correlation is still apparent. Comparing both the FFT of the vertical force on the bottom boundary (Fig. 16 (right)) to the one of the cavitation (Fig. 15 (right)) it is obvious that both the range of 300 to 400 Hz as well as the range of 1200 to 1300 Hz show local maximum peaks. It can therefore be assumed that the forces are directly related to the process of cavitation.
24
3e+07 2.5e+07
-6000 -10000 -14000
1e+07 5e+06 0
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [s]
an us cr ip
-18000
2e+07 1.5e+07
t
Amplitude of FZ [N]
FZ [N]
2000 -2000
0
2000
4000
6000
8000
10000
Frequency of FZ [Hz]
Figure 16. 5.2. Local Cloud Collapses
The frequency analyses of the vapour volume fraction and the vertical force
on the bottom boundary show the same characteristics. Although, higher fre-
quencies are more dominant for the vertical force. Therefore, the correlation between cavitation and the vertical force on the bottom boundary was investi-
M
gated in more detail. Local collapses of small vapour structures are observed. Since an incompressible solver is used, the propagation of pressure waves can not be resolved, but just the local pressure maxima are identified. Figure 17 (left) shows the temporal history of both the vapour volume fraction and the
pt ed
vertical pressure force on the bottom boundary from 0.0515 s to 0.0545 s. When looking at the variation of the vapour volume fraction, it can be seen that local minima are responsible for the high magnitudes of the vertical force. The variations of the volume fraction are rather small compared to the global variations shown in Figure 15 (left). High magnitude peaks of the force appear alongside
ce
local collapses of vapour volume. A detailed view of an impact of high pressure is illustrated in Figure 17 (right). The minimum of the force is reached just
-2000
1.05
-6000
1
-10000
0.95
-14000
0.9 -18000 0.0515 0.052 0.0525 0.053 0.0535 0.054 0.0545
0.908
2000
0.906
-2000
0.904
-6000
0.902
-10000
0.9
-14000
0.898 0.053900
0.053905
Time [s] α
0.053910 Time [s]
α
FZ
25
FZ
0.053915
-18000 0.053920
FZ [N]
2000
1.1
Vapour Volume Fraction [10−3]
1.15
FZ [N]
Vapour Volume Fraction [10−3]
Ac
before the vapour volume has reached its local minimum.
Figure 17. The high magnitude force peaks on the bottom boundary are a consequence
t
of multiple collapses of small vapour clouds. Figure 18 shows the sequence
an us cr ip
of collapses in the simulation domain from top view. The pressure peak that occurs after each collapse may be amplified by the preceding collapses. The
high pressures appearing during a cloud collapse are able to induce an even more rapid collapse of another vapour structure in its vicinity. The highest
pressure peak is generated during a local cloud collapse at around 0.05391375 s (t0 + 10 · 10−6 ). These collapses of small vapour structures show, that a high
peak in the temporal history of the force (Fig. 16 (left)) is caused. Though, they remain almost unnoticed in the progress of the vapour volume fraction (Fig. 15 (left)), since the collapsing vapour volume is small.
Although, these collapses are rapid and cause high pressures at the surface,
M
these collapses do not always automatically lead to a predicted erosion impact
for the microjet erosion model. Since the collapses very often take place too far away from the surface, they would not introduce the collapse of single bubbles
pt ed
near that surface, if no vapour is found in its vicinity. This hypothesis is especially valid for the present approach to predict cavitation erosion. Although, the highest pressures created by local cloud collapses in the conducted simulations are in the range of 10 MPa to 20 MPa, these pressures are not high enough to damage common considered materials, whose yield strengths are usually one
ce
order of magnitude higher. In contrast to these pressures, the water hammer pressures of the predicted microjet impacts can be estimated by Equation 15. A jet velocity of 240 m/s, for example, which is predicted to be common for the
Ac
given case, would lead to a water hammer pressure of over 350 MPa, being in the range of the yield strength of the material.
26
Pressure [Pa]
Pressure [Pa] 1 · 107
5 · 106
5 · 106
0
0
∆timage = 2 · 10
t0 = 0.05390375 s Pressure [Pa] 1 · 107
Pressure [Pa]
0
∆timage = 7 · 10
s
Pressure [Pa]
−6
1 · 107
pt ed
5 · 106
5 · 106
0
0
∆timage = 9 · 10−6 s
∆timage = 10 · 10−6 s
ce
Pressure [Pa]
Ac
s
Pressure [Pa]
M
1 · 107
∆timage = 11 · 10
s
5 · 106
0
∆timage = 4 · 10
−6
1 · 107
5 · 106
−6
an us cr ip
t
1 · 107
Pressure [Pa]
1 · 107
1 · 107
5 · 106
5 · 106
0 −6
0 −6
∆timage = 14 · 10
s Figure 18.
27
s
5.3. Grid Independence Study A grid independence study was conducted to find the spatial resolution,
t
which is needed to resolve the phenomena of cavitating flow. Therefore, cases
an us cr ip
with 0.417·106 , 0.78·106 , 1.66·106 and 2.14·106 control volumes were simulated. The average vapour volume fraction is calculated from its temporal progress for
four simulations of different mesh sizes (Fig. 19 (left)). The results from the two cases with coarse meshes deviate by over 10 % from the volume fraction of
the case with the finest mesh. The average volume fraction, of the case with 1.66 · 106 control volumes, shows a deviation of just 0.55 %, though. Therefore,
the FFT of the temporal progress of volume fraction is compared for the two
cases with finer grids in Figure 19 (right). Although, the temporal progress is highly unsteady in both cases, the FFT shows the same frequency ranges where
M
Amplitude of Volume Fraction [-]
0.001 0.00095 0.0009 0.00085 0.0008 0.00075 0.0007
pt ed
Average Volume Fraction [-]
dominant frequencies appear, which have already been stated.
0.00065
0
0.5
1
1.5
Number of Control Volumes
2
2.5
[106]
10
1.66 · 106 CVs 2.14 · 106 CVs
8 6 4 2 0 0
500
1000
1500
2000
Frequency of Cavitation [Hz]
Figure 19.
5.4. Erosion Prediction
ce
As shown above, a mesh of 1.66 · 106 control volumes delivers sufficient
accuracy, while maintaining acceptable computational efficiency and is therefore
Ac
selected for further investigations. The developed erosion model is used to predict the erosion sensitive areas and the erosion potential in these areas for the flow through the nozzle. A material yield strength of 400 MPa is chosen for the calculation of the critical jet velocity vcrit according to the yield strength of stainless steel 316 L, which has been employed by Franc and Riondet [27]. The value of cdef for every face is the fraction of erosion in that face compared to
28
the whole computational domain. High values of cdef mark areas of high erosion potential. Values of 0 mark areas, where no erosion is predicted. cdef
0
an us cr ip
t
0.0005
cdef
0.0005
0
Figure 20.
The correlations between cavitation and predicted erosion are depicted in Figure 20. The unsteady and non-uniform shedding of cavitation structures
has already been addressed. The numerically predicted erosion takes place in regions where the majority of macroscopic vapour structures collapse. Erosion
M
is mainly predicted within an area of a certain radial distance from the central
Ac
ce
pt ed
axis. This leads to a circular area of predicted erosion.
simulation
experiment
Figure 21.
The comparison of the eroded target sample and the numerical erosion prediction on the bottom boundary is depicted in Figure 21 from top view. The left 29
half shows the numerical prediction and the right half shows the eroded sample from the experimental investigations of [27, 29]. The red circle marks the region,
t
where the highest erosion was measured in the experiments, as stated by Franc and Riondet [27]. It is apparent that the numerical method is very well able to
an us cr ip
predict the area of erosion.
To make a statistical erosion prediction the erosion potential is summarized for radial intervals along the circumferential direction. The generation of cell
zones is important to investigate the vicinity of the faces of a certain surface (see 3.1). The size of these zones affects the probability of an impact being predicted
by the erosion model. A larger cell zone would result in a higher probability of an
impact, because a higher number of candidate cells is selected. The size of these cell zones is varied and compared with the experimental damage distribution from [27].
R = 0.5 mm R = 1.0 mm
M
0.175 0.15 0.125 0.1
pt ed
Erosion Potential cdef [-]
0.2
0.075
0.05
0.025
0
10
15
20
25
30
35
Figure 22.
Ac
ce
Radial Distance from Central Axis [mm]
The distribution of the numerically predicted erosion for simulations of dif-
ferent cell zone sizes is shown in Figure 22. The size of the cell zones is varied through the radius R of the spheres, which are used for cell selection. The
deformation coefficient cdef is summarized for intervals of size 1 mm. For the erosion calculation with the bigger cell zone erosion is predicted to take place
30
within a radial distance of approximately 10 mm to 30 mm. The distribution for the prediction with smaller cell zones is thinner and more concentrated in
t
the area around the highest erosion potential. The highest erosion potential is calculated within a range of 20 mm to 23 mm for both calculations. The graph
an us cr ip
clarifies that a smaller cell zone results in a smaller area predicted to be eroded. Further on, the maximum of the erosion potential is higher for the smaller cell zone, because there are less impacts apart from the peak region. t = 1.08 · 10−3 t = 2.05 · 10−3 t = 4.38 · 10−3 t = 5.77 · 10−3
0.175 0.15 0.125 0.1 0.075 0.05 0.025 0 10
M
Erosion Potential cdef [-]
0.2
15
20
25
s s s s
30
35
pt ed
Radial Distance from Central Axis [mm]
Figure 23.
The statistical erosion prediction for the case with the small cell zones (R =
0.5 mm) is depicted for different calculation times in Figure 23. The basic erosion sensitive regions are already predicted after 1.08 · 10−3 s of calculation
ce
time. After 4.38 · 10−3 s the statistical prediction has almost converged as it does not change significantly any more. The duration of a cycle of the low
Ac
frequency (350 Hz) of the vapour volume fraction is 2.86 · 10−3 s. The statistical
erosion prediction is therefore converged after approximately 2 low frequency cycles of cavitation. This is due to the comparably high number of impacts over the circumference of the quarter nozzle geometry, which was considered in the conducted simulations. This enables for an early statistical prediction for the present case.
31
70
0.15
60
0.125
50
0.1
40
0.075
30
0.05
20
0.025
10
0
0
10
15
20
t
0.175
Measured Depth of Erosion [mm]
80 simulation measurement
an us cr ip
Erosion Potential cdef [-]
0.2
25
30
35
Radial Distance from Central Axis [mm]
Figure 24.
The distribution of cdef from the numerical erosion prediction is compared
M
to the depth of penetration from surface measurements by Franc and Riondet
[27]. Figure 24 shows the comparison of the numerically predicted erosion from the simulation, using the smaller cell zones (R = 0.5 mm), and the erosion depth
pt ed
from the measured target plate. The prediction of the area of highest erosion shows good agreement with the measurements, as the radial position of the maxima differs by just about 1 mm. The total distribution is more concentrated in the measurement, where the main eroded area is located within a radial position of 19 mm to 24 mm. The simulation predicts erosion within a distance
ce
of about 14 mm to 30 mm.
Ac
6. Conclusions In the present approach, a new cavitation erosion model for 3-D Euler-Euler
flow simulations has been developed to numerically predict the areas of erosion and assess the erosion potential in these areas. Following the approach of Dular and Coutier-Delgosha [13], the new model predicts the formation of microjets near surfaces, based on additional conditions regarding the surrounding flow field. Further on, dimensionless coefficients were introduced to enable a quali32
tative prediction of erosion, based on the amount of impacts and their intensity in a certain area. The ability to simulate cavitating flows around hydrofoils
t
was shown, which enables for the prediction of cavitation erosion for simple hydrodynamic flow problems. By comparing the numerical results to experi-
an us cr ip
mental results from [27, 29], it could be shown that the erosion model is well able to find erosion sensitive areas and differentiate the intensity of erosion in those. The developed erosion model did clearly show correlations between the
cycles of cavitation and the erosion impacts. The identified eroded areas are mostly related to unsteady phenomena like the closure of sheet cavitation or the generation and collapse of cloud cavitation, which agrees well with former investigations from literature [30, 31].
In further investigations the new erosion model will be validated for simple as
well as complex 3-D flow problems in maritime environments such as hydrofoils
M
and ship propellers. The conditions responsible for the erosion prediction are going to be validated in detail and enhanced. Additionally, a quantitative pre-
diction of erosion may be established by correlating the numerically predicted erosion to measured erosion from experimental tests. Predictions of water ham-
pt ed
mer pressures caused by microjets can be calculated and used for simulations of structure analysis or fluid-structure-interaction in the future.
7. Acknowledgement
ce
The present study was funded by the German Federal Ministry of Economic
Affairs and Energy.
Ac
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[11] K. Haller Kneˇzevi´c. High-Velocity Impact of a Liquid Droplet on a Rigid Surface: The Effect of Liquid Compressibility. PhD Thesis, Swiss Federal
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Institute of Technology Zurich, Zurich, Switzerland, October 2002.
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[12] M. S. Plesset and R. B. Chapman. Collapse of an Initially Spherical Vapor Cavity in the Neighborhood of a Solid Boundary. Journal of Fluid Mechanics, 47(2):283–290, 1971.
[13] M. Dular and O. Coutier-Delgosha. Numerical Modelling of Cavitation
Erosion. International Journal for Numerical Methods in Fluids, 61:1388– 1410, February 2009.
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[15] OpenFOAM Foundation. OpenFOAM. [http://www.openfoam.org/], 2014.
[16] J. Sauer and G. H. Schnerr. Unsteady Cavitating Flow - a New Cavitation Model Based on Modified Front Capturing Method and Bubble Dynamics.
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[18] A. K. Singhal, M. M. Athavale, H. Li, and Y. Jiang. Mathematical Basis and Validation of the Full Cavitation Model. Journal of Fluids Engineering,
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[22] J. R. Blake. The Kelvin Impulse: Application to Cavitation Bubble Dy-
namics. The Journal of the Australian Mathematical Society - Series B: Applied Mathematics, 30(2):127–146, October 1988. [23] A. Philipp and W. Lauterborn.
Cavitation Erosion by Single Laser-
Produced Bubbles. Journal of Fluid Mechanics, 361:75–116, 1998.
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of Fluid Mechanics, 135:373–387, 1983.
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[28] M. Mihatsch, S. J. Schmidt, M. Thalhamer, N. A. Adams, R. Skoda, and U. Iben. Collapse Detection in Compressible 3-D Cavitating Flows and As-
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sessment of Erosion Criteria. WIMRC 3rd International Cavitation Forum, 2011.
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[30] C. E. Brennen, G. Reismann, and Y.-C. Wang. Shock Waves in Cloud Cavitation. Twenty-First Symposium on Naval Hydrodynamics, 1997.
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tures and Cavitation Erosion. Wear, 300:55–64, 2013.
t
[31] M. Petkovˇsek and M. Dular. Simultaneous Observation of Cavitation Struc-
37
Figure Captions Figure 1. Reboud’s Correction: comparison of density functions for cal-
t
culation of turbulent viscosity
an us cr ip
Figure 2. Comparison of Microjet Velocities: comparison of microjet
velocities from theoretical and experimental investigations taken from Lohrberg [21]
Figure 3. Measured Microjet Velocities: microjet velocities at impact onto a solid surface taken from Philipp and Lauterborn [23]
Figure 4. Collecting Cells Around a Hydrofoil: using a sphere to collect cells around a face (red) of a hydrofoil in a cell zone
Figure 5. Selecting Cells Around a Face: cells whose cell centres are located within the sphere (green) are collected in a cell zone for the face (red)
assigned cell zone
M
Figure 6. Microjet Conditions: examination of a face and a cell in its
Figure 7. Calculation of Erosion Potential: calculation of the deformation coefficient cdef for a simple surface of 9 faces
pt ed
Figure 8. Mesh of Hydrofoil: 3-D view of solution domain and mesh Figure 9. NACA 0009 from Perspective View: velocity field around
NACA 0009 hydrofoil with a cycle of forming and detaching sheet cavitation Figure 10. Volume Fraction of NACA 0009: temporal progress (left)
and FFT (right) of the vapour volume fraction for a simulation with 3.73 · 106
ce
CVs
Figure 11. Pressure Distribution on NACA 0009: comparison of cal-
culated and measured [26] pressure coefficients
Ac
Figure 12. Sketch of Nozzle: experimental test section of nozzle geometry
based on [27] Figure 13. Mesh of Nozzle: mesh and refinement regions in the test section Figure 14. Cavitation at Nozzle: isosurface of αv = 0.1 from top view
(left) and perspective view (right)
38
Figure 15. Volume Fraction of Nozzle: temporal progress (left) and FFT (right) of the vapour volume fraction for a simulation with 2.14 · 106 CVs
t
Figure 16. Vertical Force on Bottom Boundary: temporal progress
with 2.14 · 106 CVs
an us cr ip
(left) and FFT (right) of the vertical force on the bottom plate for a simulation
Figure 17. Correlations between Cavitation and Vertical Force: cor-
relations between the temporal progress of vapour volume (red line) and the vertical force on the bottom plate (blue line)
Figure 18. Local Cloud Collapses: top view of local collapses of small vapour clouds leading to a high pressure peak on the bottom boundary
Figure 19. Grid Independence Study: comparison of the average vapour
volume fraction of four different mesh sizes (left) and FFT of the volume fraction for the two fine meshes (right)
M
Figure 20. Correlations between Cavitation and Erosion: correlations
between cavitation and predicted erosion from perspective view (left) and top view (right)
Figure 21. Comparison of Numerical Prediction to Experimental
pt ed
Erosion: comparison of predicted erosion from numerical simulation (left) and the eroded target sample from experiments [29] (right) Figure 22. Erosion Prediction for Different Cell Zone Sizes: distribu-
tion of the erosion potential cdef over the radial distance from the central axis for different cell zone sizes
ce
Figure 23. Convergence of Erosion Prediction: temporal evolution of
the distribution of the erosion potential cdef for the case with R = 0.5 mm
Ac
Figure 24. Comparison of Damage Distributions: erosion prediction
from simulation and measured erosion depth by [27]
39
cD [-]
cL [-]
f [Hz]
coarse grid
0.29
0.0184
0.328
230.00
medium grid
1.03
fine grid
3.73
an us cr ip
cells [106 ]
t
Tables
0.0171
0.336
160.55
0.0178
0.342
169.16
Table 1: numerical results of the flow around a NACA 0009 hydrofoil for different mesh sizes
v [m/s] 20.00
experimental
20.07
numerical (medium)
20.00
numerical (fine)
cD [-]
cL [-]
f [Hz]
0.82
0.021
0.331
-
0.81
-
-
0.81
0.0171
0.336
160.55
0.0178
0.342
169.16
M
experimental
σ [-]
20.00
0.81
179
Ac
ce
pt ed
Table 2: experimental and numerical results of the flow around a NACA 0009 hydrofoil
40
