Assessment of cavitation erosion risk indicated by pressure impact exceeding material strength threshold

Qiu, Ning; Zhu, Han; Xu, Pei; Che, Bangxiang; Wu, Jie; Zhou, Wenjie; Wang, Chuan

doi:10.1063/5.0167100

Cavitation erosion can seriously affect the performance of hydraulic machinery. To assess its risk, the cavitation erosion that covered a NACA0015 hydrofoil was investigated using experiment and simulation. In particular, the material strength threshold was considered in the experiment, and soft specimens were employed instead of common painting surfaces. High-speed cameras were utilized to capture cavity morphology, while pressure sensors simultaneously measured the impact intensity of cavitation collapse. Additionally, an erosion assessment method based on pressure impact was used in the numerical simulation. The results indicate that the cavitation erosion risk can be indicated by the intensity of pressure impact, which is caused by the collapse of cavity. The high erosion risk appears near 60% of chord length where shedding cloud collapses. Due to the influence of middle-entrant jet and side-entrant jets, the erosion risks are different at various span-wise locations. Additionally, the response differs for different materials exposed to cavitation. When the material strength threshold is low, the chord-wise difference of cavitation erosion is not significant, because the slight impact can also generate erosion pits. When the threshold is high, the chord-wise difference of erosion becomes larger, and erosion pits are more centralized.

I. INTRODUCTION

Cavitation erosion is defined as the material damage due to the collapse of cavities near a surface. The collapses may induce high-intensity shock loads exceeding the material threshold, resulting in irreversible material loss. Deformed and incomplete components in fluid machinery would lead to undesirable flow states, resulting in severe vibration, loud noise, power loss, etc. Consequently, numerous studies have been devoted to investigate the mechanism of cavitation erosion. These investigations can be broadly categorized into two perspectives: macroscopic and microscopic approaches.

The initial focus of cavitation erosion research was on the collapse of micro single bubbles. In an unbounded flow domain, even a small disturbance at any position on the bubble surface could lead to an unstable bubble boundary, resulting in a non-spherical collapse.¹ Plesset and Chapman² proposed that if the collapse occurs near a neutral wall, the presence of the solid wall promotes asymmetry of the flow field around the bubble along the wall-normal direction. Then, the kinetic energy would be released in the form of microjets and impact the solid wall during the collapse process. Sun et al.³ studied the oscillation process of the laser-induced bubble on solid walls. They indicated that the bubble shape during collapse is only related to the dimensionless parameter $γ$ (the distance from the bubble center to the wall divided by the maximum bubble radius). Wang et al.⁴ studied the collapse process of a bubble based on a lattice Boltzmann method. They indicated that compared to an unbounded flow domain, the neutral wall could affect the collapse velocity, resulting in a broader distribution of microjets and a shorter collapse time. Additionally, the velocity of these micro-jets can reach 130–170 m/s and lead to significant water hammer pressure.⁵ Zhang⁶ measured the water hammer pressure caused by microjets on a rigid boundary, which could reach 1–2 GPa and have the potential to cause material damage. In a recent study by Xie,⁷ the relationship between erosion intensity and impact angle was established, showing that the maximum stress occurs when water hammer impacts the solid wall at an angle of 90°. However, the radiation pressure caused by microjets resulting from the collapse of bubbles in the far field is relatively small. The material damage of cavitation erosion is also related to the propagation of shock waves.

Shock waves are another mechanism of energy released by collapsed bubbles, and their influence on the material should not be ignored. In 1978, Singeo⁸ studied the instantaneous shock loads generated by a collapsed bubble. They found that pressure could reach approximately 10³–10⁴ MPa. Cui et al.⁹ experimentally demonstrated that shock waves produced by bubble collapse can even break ice, confirming the potentially irreversible damage to solid surfaces caused by shock waves. Supponen et al.¹⁰ studied the collapse process of a single bubble and observed the presence of multiple shock waves, which are closely related to the collapsed stage. Recently, Luo et al.¹¹ studied the mechanism of interaction between a cavity bubble and an air bubble using experiment. They indicated that the air bubble has the potential to suppress the shock waves. Compared to the direct observation of microjets, capturing and distinguishing shock waves are more challenging. Fortunately, advances in equipment have increased the feasibility of difficult experiments. Zhai et al.¹² compared cavitation erosion characteristics caused by microjets and shock waves using a high-speed dynamic acquisition system and an electric pulse-induced bubble system, indicating that microjets are the primary factor for cavitation erosion on elastic materials, while damage caused by shock waves is minimal. This difference may be related to the dimensionless distance parameter $γ$ ⁠: the distance from the bubble center to the wall.Microjets dominate cavitation erosion near the wall, while shock waves dominate cavitation erosion in the far field.^13,14 In industrial applications, the material loss of cavitation erosion is usually caused by the collapse of macro-cavity formed by the cluster of bubbles.

Therefore, it is necessary to investigate the macroscale mechanisms of cavitation erosion. Pitting experiments^15,16 offer direct observations of the recorded erosive collapse of large-scale cavities. Franc et al.¹⁷ compared cavitation damage of aluminum, bronze, and stainless steel using pitting experiments, highlighting the varying material responsiveness to collapsed cavities. Zhuang et al.¹⁸ measured the mass loss rate under cavitation conditions. They indicated that the mass rate does not follow a linear trend with time; instead, it exhibits two stages: the slow accumulation stage and a subsequent rapid loss stage. Lu et al.^19,20 found that a rough surface can effectively inhibit the growth of the cavitation region. Due to the time-consuming nature of pitting experiments, researchers often employ paint coatings as substitutes. Li et al.²¹ applied paint coatings on a NACA0015 hydrofoil, capturing the cavitation erosion caused by attached cavities and cloud cavitation. Nahon et al.²² compared cavitation erosion on different blades and validated the advantages of reverse design optimization within a short time, utilizing pressure sensors and high-speed photography as auxiliary equipment. However, the painting surfaces not only respond to cavity collapse but also to turbulent pulsations and inflow impact, which may affect the correct assessment of cavitation erosion.

The employ of soft metallic foils or small test specimens may be more reasonable. Dular et al.²³ attached copper foils to the hydrofoil surface, comparing the cavitied structure and cavitation erosion captured by high-speed cameras.²⁴ A pit may be formed by the collapse of multiple cavities,²⁵ and damage only occurs during the collapse of cloud cavitation. The size of the cloud and its distance to the wall at the moment of collapse do not significantly affect the extent of the damage; however, irregular or “fragmented” cavitation clouds could cause the most significant damage.²⁶ Qiu et al.^27,28 tested the cavitation erosion distribution on a hydrofoil surface using circular specimens and metallic foils. They established the relationship between cavitation stages, pressure pulsation, and cavitation erosion, noting that the severity of cavitation is greatly influenced by pressure pulsation near the wall. The shedding and deformation of cavitation clouds could lead to pressure pulsations and drive the collapse of small bubbles near solids. Indeed, due to the long time and expensive price of experiments to assess the material damage, widespread industrial application remains challenging.

Benefiting from the rapid advancements in computer performance²² and the progressive refinement of numerical methods over the past decade,²³ the currently simulated techniques have significantly reduced the time required to assess cavitation damage. This allows for early prediction of cavitation erosion during the initial stages of design, leading to improved economic efficiency. Based on the concept of microjets, Peters et al.⁵ proposed a cavitation model based on impact quantity and intensity in specific erosion areas. Additionally, inspired by the concept of shock waves, Pereira²⁹ and Patella et al.^30–32 treated the internal energy of the cavity as potential energy, developing a function of pressure and cavity volume to describe the energy released by collapsed cavities. Considering both microjets and shock waves, Arabnejad et al.^33–35 proposed a cavitation erosion prediction method based on kinetic energy. Their analysis of cavitation on hydrofoils and water jet pumps emphasized that non-uniform inflow is one of the reasons leading to severe cavitation erosion. Schenke et al.^36,37 proposed a prediction method considering cavity potential energy and radiation angles. This approach successfully predicted the distribution of cavitation erosion on water jet pumps and cargo ships. Wang et al.³⁸ utilized an Eulerian–Lagrangian method to precisely capture the cavitating flow and then presented a new asymmetric bubble collapse model. They highlighted the high risk of cavitation erosion originating from bubble disturbance and cavity collapse near the cavitation closure line.

Based on the literature review, few experiments and simulations have taken into account the effect of material thresholds on cavitation erosion so far. This paper is intended to eliminate this drawback since this issue is of particular interest for industrial applications.

A. Outline of the present work

This present work specifically focuses on the influence of material thresholds on predicting cavitation erosion and favors a pressure pulsation-based cavitation erosion risk prediction method proposed by Li et al.,²¹ which has proven beneficial for engineering applications as it does not require excessive flow details.³⁹ In the present study, pressure fluctuations are captured to quantitatively analyze the cavitation impact at different locations. Specimens of damaged copper and aluminum are used as qualitative evidence. The predicted cavitation erosion based on artificially set thresholds is assessed.

The following is a brief outline of the organizational structure of this study. Section II provides an introduction to the experimental equipment, techniques employed in this work, and the key parameters of a NACA0015 hydrofoil. Section III provides a detailed description of the numerical methods and mesh generation scheme. Section IV validates the accuracy of the simulations based on experimental results. In Sec. V, experimental and simulation results are discussed. The analysis focuses on the pressure pulsations caused by cavitation, the propagation process of pressure impact loads, the cavitation erosion caused by different cavitation structures, and the influence of material thresholds on cavitation erosion. Finally, a summary and conclusions are presented.

II. EXPERIMENTAL METHODOLOGY

The experiment was conducted in the cavitation tunnel at the School of Aeronautics and Astronautics at Zhejiang University. For this study, a NACA0015 hydrofoil was employed, with a chord length of 100 mm (1 C) and a span length of 200 mm (2 C). The angles of attack can be adjusted by rotating the central axis. During the experiment, the cavitating flow can be observed through the transparent windows of the flow channel. High-speed cameras were utilized to capture cavity morphology, while pressure sensors simultaneously measured the impact intensity of cavitation collapse. The sampling frequency was set at 10 kHz. The equipment parameters, including the device version, the measured range, and relative error, are summarized in Table I, meeting the requirements for experiments and analysis. The synchronous acquisition system and the locations of the pressure sensors are illustrated in Fig. 1.

TABLE I.

The equipment parameters.

	Device version	Measured range	Error
High speed camera	Fastcam-SA4	0–50000 F/s (sampling rate)	$\pm$ 0.1 ms
Pressure sensor	Keller-4LC	0–5 bar (absolute pressure)	$\pm$ 0.25 % (maximum accuracy) $\pm$ 0.3 % (maximum stability)

	Device version	Measured range	Error
High speed camera	Fastcam-SA4	0–50000 F/s (sampling rate)	$\pm$ 0.1 ms
Pressure sensor	Keller-4LC	0–5 bar (absolute pressure)	$\pm$ 0.25 % (maximum accuracy) $\pm$ 0.3 % (maximum stability)

FIG. 1.

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Schematic diagram of NACA0015 hydrofoil in the cavitation test: (a) chord-wise position of monitors, (b) span-wise position of monitors, and (c) synchronous acquisition system.

III. NUMERICAL METHODOLOGY

A. Cavitation model

The cavitation process is controlled by the mass transfer equation defined as

\frac{\partial (ρ_{v} α_{v})}{\partial t} + \nabla (ρ_{v} α_{v} u) = \dot{m} = {\dot{m}}^{+} - {\dot{m}}^{-},

(1)

where

\dot{m}

is the cavitation source term, including the evaporation source term

{\dot{m}}^{+}

and condensation source term

{\dot{m}}^{-}

⁠; the subscripts

ρ_{v}

⁠,

α_{v}

⁠,

u

⁠, and

t

represent the vapor density, vapor volume fraction, velocity of liquid, and flow time, respectively.

Here, the Sauer–Schnerr cavitation model,⁴⁰ simplifying from the Rayleigh–Plesset (R–P) equation, is used for the cavitation source term. The R–P equation describing the growth and collapse of a single bubble in liquid is defined as

R_{B} \frac{d^{2} R_{B}}{d t^{2}} + \frac{3}{2} {(\frac{d R_{B}}{d t})}^{2} + \frac{2 σ_{s t}}{R_{B}} = \frac{p_{v} - p_{ref}}{ρ_{l}},

(2)

where

R_{B}

represents the bubble radius,

σ_{s t}

represents the surface tension,

p_{v}

represents the saturated pressure at the corresponding temperature,

ρ_{l}

is the liquid density, and

p_{ref}

represents the reference pressure, respectively.

Based on Eqs. and , the source term could be rewritten as follows:

\dot{m} = \frac{ρ_{l} ρ_{v}}{ρ_{m}} \frac{d α_{v}}{d t},

(3)

where

ρ_{m}

is the mixed-phase density.

The following equation is used to define the vapor volume fraction

α_{v}

and avoid defining too many empirical coefficients, which affect calculation accuracy,

α_{v} = \frac{n_{b} \frac{4}{3} π R_{B}^{3}}{1 + n_{b} \frac{4}{3} π R_{B}^{3}},

(4)

where

n_{b}

represents the number of bubbles per unit volume of liquid.

Equations and are combined and derived as follows:

\dot{m} = \frac{ρ_{l} ρ_{v}}{ρ_{m}} \frac{3 α_{v} (1 - α_{v})}{R_{B}} \frac{d R_{B}}{d t} .

(5)

Ignoring the second-order term in the Rayleigh–Plesset equation, the change rate of bubble radius can be obtained from Eq. ,

\frac{d R_{B}}{d t} = sign (p_{v} - p) \sqrt{\frac{3}{2} \frac{abs (p_{v} - p)}{ρ_{l}}} .

(6)

Equations and are combined and derived as follows:

{\dot{m}}^{+} = \frac{3 α_{v} (1 - α_{v})}{R_{B}} \cdot \frac{ρ_{v} ρ_{l}}{ρ_{m}} {(\frac{2}{3} \frac{p_{v} - p}{ρ_{l}})}^{1 / 2}, p < p_{v},

(7)

m^{-} = \frac{3 α_{v} (1 - α_{v})}{R_{B}} \cdot \frac{ρ_{v} ρ_{l}}{ρ_{m}} {(\frac{2}{3} \frac{p - p_{v}}{ρ_{l}})}^{1 / 2}, p \geq p_{v},

(8)

where

p

is the surrounding pressure near the macro-cavity. The bubble radius can be derived from Eq. as

R_{B} = {(\frac{α_{v}}{{1 - α}_{v}} \frac{3}{4 π} \frac{1}{n_{b}})}^{1 / 3},

(9)

where

n_{b}

is selected to be 1 × 10¹¹.⁴¹

B. Turbulence model

The shear stress transfer $k - ω$ (SST) model⁴² with the mixture density function is used to simulate turbulent flow. The modified model, considering the transfer of turbulent shear stress, is more accurate and reliable. It has advantages in calculating reverse pressure gradients and flow characteristic around hydrofoils.

The governing equations of k and

ω

are described as follows:

\frac{\partial (ρ k)}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ U_{j} k) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} - β^{*} ρ k ω,

(10)

\frac{\partial (ρ ω)}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ U_{j} ω) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ω}}) \frac{\partial ω}{\partial x_{j}}] + α \frac{ω}{k} P_{k} - β ρ ω^{2} + 2 (1 - F_{1}) ρ \frac{1}{σ_{ω 2} ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}},

(11)

where

P_{k}

is the turbulence generation rate, k is the kinetic energy, and

ω

is the specific rate of dissipation. The other parameters are selected as

β^{*} = 0.09

⁠,

α = 5 / 9

⁠,

σ_{k} = 2

⁠,

σ_{ω} = 2

⁠,

β = 0.075

⁠, and

σ_{ω 2} = 1 / 0.856

⁠.

F_{1}

and

F_{2}

are blending functions, expressed as follows:

F_{1} = tanh (Φ_{1}^{4}),

(12)

F_{2} = tanh (Φ_{2}^{2}),

(13)

Φ_{1} = min (max (\frac{\sqrt{k}}{β ω y}, \frac{500 μ}{ρ y^{2} ω}), \frac{4 ρ k}{σ_{ω, 2} D_{w}^{+} y^{2}}),

(14)

D_{W}^{+} = max (2 ρ \frac{1}{σ_{ω, 2} ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}}, 1.0 \times 10^{- 10}),

(15)

Φ_{2} = max (\frac{2 \sqrt{k}}{β ω y}, \frac{500 μ_{m}}{ρ y^{2} ω}),

(16)

where y is the nearest distance from the wall, and

μ_{m}

is the dynamic viscosity.

The turbulent viscosity

μ_{t}

can be described as

μ_{t} = f (ρ_{m}) \frac{C_{μ} k}{ε} .

(17)

The mixture density function can be defined as

f (ρ_{m}) = ρ_{v} + {(\frac{ρ_{v} - ρ_{m}}{ρ_{v} - ρ_{l}})}^{n} (ρ_{l} - ρ_{v}) = ρ_{v} + \frac{{(ρ_{m} - ρ_{v})}^{n}}{{(ρ_{l} - ρ_{v})}^{n - 1}},

(18)

where n is the correction index. With the increase in n, the turbulent viscosity in the two-phase mixing region can be more accurately corrected. Regarding the research object and cavitation structure characteristics in this condition, the coefficients are selected as n = 10 (Ref. 43) and

C_{μ}

= 0.09.

C. Erosion assessment method

According to the macroscopic cavitation theory proposed by Pereira,²⁹ the potential energy of macroscopic cavitation structure is released during collapse and can be defined as⁴⁴

E_{pot} = \int (p_{d} - p_{v}) d V_{v},

(19)

where

p_{d}

represents the surrounding pressure driving cavitation collapse, and

V_{v}

represents the cavitation volume.

The surrounding pressure

p_{d}

could be substituted by the surface averaged pressure

\bar{p_{o}}

⁠. The potential power

P_{pot}

for the cluster of bubbles can then be defined as follows:

P_{pot} = \frac{d E_{pot}}{d t} = \sum_{o = 1}^{N} (\bar{p_{o}} - p_{o}) \frac{\partial V_{v_{o}}}{\partial t} + \sum_{i = 1}^{N} \frac{\partial \bar{p_{o}}}{\partial t} V_{v_{o}},

(20)

where

\bar{p_{o}}

and

V_{v_{o}}

represent the surface averaged pressure and vapor volume for the oth cell, and N is the total number of the discretized cells in the macro-cavity. The vapor volume for the oth cell can be replaced by the product of its vapor volume fraction

α_{o}

and the volume of the microcavity

V_{c o}

⁠. Equation can be written as follows:

P_{pot} = \sum_{o = 1}^{N} (\bar{p_{o}} - p_{v}) \frac{\partial α_{o}}{\partial t} V_{c o} + \sum_{o = 1}^{N} \frac{\partial \bar{p_{o}}}{\partial t} α_{o} V_{c o} .

(21)

Li²¹ conducted a sensitivity analysis on the two terms on the right-hand side of the equation. The results indicate that the time derivative of local pressure

\partial p / \partial t

has the best correlation with experimental erosion. Adding the time derivative of the local vapor volume fraction

\partial α / \partial t

did not improve this correction. It is worth noting that cavitation erosion is the cumulative result of repeated damage on the material surface. Then, the accumulated erosion function can be defined as

e_{pot} = \sum_{h = 1}^{M} I_{h} and I_{h} = {\begin{matrix} \frac{\partial p}{\partial t} \begin{matrix} , & \frac{\partial p}{\partial t} \geq ε_{c} \end{matrix} \\ 0 \begin{matrix} , & \frac{\partial p}{\partial t} < ε_{c}, \end{matrix} \end{matrix}

(22)

where the subscript h refers to each individual time instant.

ε_{c}

is the threshold.⁴⁵ When

e_{pot}^{I_{h}} \geq ε_{c}

⁠, it is considered that the hydrofoil will suffer a high risk of cavitation erosion, while there is always no cavitation risk when

e_{pot}^{I_{h}} < ε_{c}

⁠. M is the number of the events with a value of

\partial p / \partial t

higher than

ε_{c}

⁠.

I_{h}

is an intensity function used to qualitatively assess the risk of cavitation erosion, where ∂p/∂t can be seen as the pressure impact within a certain time instant.

D. Numerical setup

The physical parameters used in the simulation are shown in Table II.

TABLE II.

Physical parameters.

Parameters	Value
Temperature T	32.8 $\pm$ 0.5 °C
Angle of attack AoA	8 $\pm$ 0.05 deg
Liquid phase density $ρ_{l}$	994.727 kg/m³
Liquid dynamic viscosity $μ_{l}$	7.52054 × 10⁻⁴ $Pa s$
Gas phase density $ρ_{v}$	3.534 × 10⁻² $kg / m^{3}$
Gaseous dynamic viscosity $μ_{v}$	1.00923 × 10⁻⁵ $Pa s$
Saturated vapor pressure $p_{v}$	4979.18 Pa

Parameters	Value
Temperature T	32.8 $\pm$ 0.5 °C
Angle of attack AoA	8 $\pm$ 0.05 deg
Liquid phase density $ρ_{l}$	994.727 kg/m³
Liquid dynamic viscosity $μ_{l}$	7.52054 × 10⁻⁴ $Pa s$
Gas phase density $ρ_{v}$	3.534 × 10⁻² $kg / m^{3}$
Gaseous dynamic viscosity $μ_{v}$	1.00923 × 10⁻⁵ $Pa s$
Saturated vapor pressure $p_{v}$	4979.18 Pa

Figure 2 shows the calculation domain and boundary conditions. In order to avoid the calculation divergence caused by turbulent disturbance downstream of the hydrofoil, the hydrofoil center is placed at a distance of 2.5 C from the inlet of the calculation domain and 7.5 C from the outlet. The cross section of the domain is $2 C \times 2 C$ ⁠.

FIG. 2.

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Calculation domain and boundary conditions.

The boundary parameters are shown in Table III. The inlet velocity is 10 m/s, which is consistent with the experiment. Because the water tunnel is a closed cycle and the disturbance is relatively large, the inlet turbulence intensity is set as 5%. The outlet is set as a pressure outlet with a value of 74 610 Pa. The cavitation number can be calculated as

σ = \frac{P_{\infty} - P_{v}}{0.5 ρ_{l} U_{\infty}^{2}},

(23)

where

P_{\infty}

is the far field pressure without disturbance, and the average pressure at the outlet surface of the calculation domain is taken in the simulation. Reynolds number

R e

can be calculated according to the inlet flow velocity

U_{\infty}

and characteristic length L. In this work, the characteristic length L is equal to the hydrofoil chord length C,

R e = \frac{ρ_{l} \times U_{\infty} \times L}{μ_{l}} = 1.32 \times 10^{6} .

(24)

TABLE III.

Boundary parameters.

Parameters	Value
Reynolds number $R e$	1.32 $\times$ 10⁶
Cavitation number $σ$	1.4
Inlet velocity $U_{\infty}$	10 m/s
Inlet turbulence intensity	5%
Outlet pressure	74 610 Pa
Outlet pressure fluctuation	5 %

Parameters	Value
Reynolds number $R e$	1.32 $\times$ 10⁶
Cavitation number $σ$	1.4
Inlet velocity $U_{\infty}$	10 m/s
Inlet turbulence intensity	5%
Outlet pressure	74 610 Pa
Outlet pressure fluctuation	5 %

Considering the influence of sidewalls on the evolution of cloud cavitation, the walls are set as non-slip walls, the same as the hydrofoil surface. In the unsteady cavitation calculation, the time step

Δ t

is estimated based on the empirical equations,⁴⁶^,

Δ t = \frac{T_{ref}}{200},

(25)

T_{ref} = \frac{C}{U_{\infty}},

(26)

where

T_{ref}

is the characteristic timescale. The time step was set as

Δ t = 5.0 \times 10^{- 5} s

⁠. The maximum Courant number is less than 70, and its RMS value is less than 3, which meets the requirements of implicit algorithm.^47–49 The total unsteady calculation duration is set as 0.5 s. In order to ensure the precision of simulation, the RMS residual convergence threshold of each time step is set as 1.0 × 10⁻⁵. The calculation iterated 3–20 times in each loop to ensure the accuracy of results. Segregated solver is used to solve the velocity and pressure distribution of vapor phase and liquid phase. A high-resolution scheme is used for convection term calculation. A second-order backward Euler scheme is used for transient term discretization. The Volume of Fluid (VOF) method is used to capture the interface between vapor phase and liquid phase.

E. Grid-independent check

In order to ensure the accuracy of numerical calculation, hexahedral structured mesh is conducted. O-block topology is adopted around the hydrofoil, and the mesh is refined near the hydrofoil surface to capture the cavitation structure in the boundary layer. The grid number at each direction in the computational domain is systematically increased to ensure a constant scale change rate and an unchanged topological structure. The grid number successively increases by four times. The mesh information is shown in Table IV. The lift and drag coefficients are compared to determine the final mesh number. The lift coefficient

C_{l}

and drag coefficient

C_{d}

are defined as follows:

C_{l} = \frac{2 F_{l}}{ρ_{l} U_{\infty}^{2} A},

(27)

C_{d} = \frac{2 F_{d}}{ρ_{l} U_{\infty}^{2} A},

(28)

A = C \times S,

(29)

where

F_{l}

represents the hydrofoil lift force,

F_{d}

represents the drag force,

A

is the equivalent area, and S is the span length of hydrofoil.

TABLE IV.

Mesh independence verification.

Scheme	Grid number	Lift coefficient, $C_{l}$	Relative error, %	Drag coefficient, $C_{d}$	Relative error, %
1	300 000	0.79665	⋯	0.03449	⋯
2	830 000	0.90024	11.50	0.02469	39.69
3	2 350 000	0.94006	4.20	0.02159	14.36
4	8 140 000	0.94678	0.70	0.01982	8.93
5	14 000 000	0.94721	0.04	0.0195	1.6

Scheme	Grid number	Lift coefficient, $C_{l}$	Relative error, %	Drag coefficient, $C_{d}$	Relative error, %
1	300 000	0.79665	⋯	0.03449	⋯
2	830 000	0.90024	11.50	0.02469	39.69
3	2 350 000	0.94006	4.20	0.02159	14.36
4	8 140 000	0.94678	0.70	0.01982	8.93
5	14 000 000	0.94721	0.04	0.0195	1.6

With the increase in the grid number, the relative error of lift and drag coefficients decreases gradually. Considering the accuracy and efficiency, the mesh with 8.14 × 10⁶ cells is selected for subsequent calculation. To meet the mesh requirements of SST model, the scale change ratio of mesh along the flow direction and perpendicular to the wall is less than 1.3, and the maximum $y^{+}$ near the foil is maintained at less than 60.^37,42,50 The mesh details are shown in Fig. 3.

FIG. 3.

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Mesh details.

IV. NUMERICAL VALIDATION

To verify the reliability of numerical simulation, the temporal–spatial evolution process of cavitation is captured and compared with the experimental results, as illustrated in Fig. 4. As can be seen from the figure, the simulation captured the transient shedding process well.

FIG. 4.

$Temporal–spatial evolution of cavitation volume. (The red curve is the experimental result captured by the high-speed camera, while the cavitation morphology of simulation is represented by iso-surface with a vapor volume fraction of 10%.)$

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Temporal–spatial evolution of cavitation volume. (The red curve is the experimental result captured by the high-speed camera, while the cavitation morphology of simulation is represented by iso-surface with a vapor volume fraction of 10%.)

The typical frequency of pressure pulsation at monitor P2 is adopted to calculate the Strouhal number. Figure 5 shows the power spectral density characteristics. The Strouhal number captured by experiment is 0.57, and the simulation result is 0.60. The relative error is 5.3%, which is within a reasonable range.^46,51,52

FIG. 5.

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Strouhal number and power spectral density (PSD).

The cavitation evolution in a typical cycle is shown in Fig. 6. The cavitation structure at each instant is described by the top view of experiment, and the cavitation structure of numerical simulation is represented by iso-surface with a vapor volume fraction of 10%. The high pressure caused by cloud cavitation collapse is well captured, and the local collapse caused by side-entrant jets and middle-entrant jet is also well captured.

FIG. 6.

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Cavitation evolution in a single cycle.

V. RESULTS AND DISCUSSION

A. Pressure pulsation characteristic

The captured pressure pulsation is non-dimensionalized. The pressure coefficient can be defined as

C_{P} = \frac{p - p_{v}}{\frac{1}{2} ρ_{l} U_{\infty}^{2}} .

(30)

The pressure fluctuations at seven monitors are shown in Fig. 7.

FIG. 7.

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Pressure pulsation. (a) Time domain signals and (b) power spectral density.

The fluctuation amplitude captured by monitors at the downstream of the hydrofoil is higher than others. At the same chord length, the pressure coefficients of monitors vary with different span-wise locations. The main frequency of pressure pulsation at each monitor is 57 Hz, as shown in Fig. 7(b). The amplitude of PSD corresponding to the main frequency is high downstream, and the amplitudes corresponding to the main frequency are different at various span-wise locations.

In order to analyze the cause of this difference, three span-wise positions were selected that are 50 mm (z = 0.5 C), 100 mm (z = C), and 150 mm (z = 1.5 C) away from the front observation window, as shown in Fig. 1(b). Figure 8 shows the pressure fluctuations on them. The vapor content in the attached cavity area is higher, and the pressure is lower. There is a region of pressure increase downstream of the cavity closure line. The low-pressure area near the trailing edge of hydrofoil is related to the motion of cloud cavitation. The cyclical evolution of cavitation is the cause of periodic fluctuation of pressure.

FIG. 8.

$Pressure fluctuations at different locations: (a) z = 0.5 C, (b) z = 1.0 C, and (c) z = 1.5 C (the write curves are the contour line with a vapor volume fraction of 10%).$

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Pressure fluctuations at different locations: (a) z = 0.5 C, (b) z = 1.0 C, and (c) z = 1.5 C (the write curves are the contour line with a vapor volume fraction of 10%).

B. Cavitation erosion characteristics considering material threshold

The collapse of macro-cavity causes the local high pressure. Pressure impact $\partial p / \partial t$ can be considered as the main indicator of cavitation erosion. To explore the response and erosion of different materials to impact loads, round polished specimens were installed flush with the hydrofoil surface and were easy to disassemble. The specimen must be polished into the mirror surface before carrying out the cavitation erosion experiment, in order that the erosion pits can be distinguished. Soft materials copper (Cu) and aluminum (Al) were used for experiment due to the very low erosion rate of the water tunnel. The placement of the specimens is shown in Fig. 9, and the diameters of the specimens were 12 mm. The operation lasted for 15 h under the steady cavitation condition that the angle of attack is 8°, cavitation number is 1.4, and water temperature is 32.8 °C.

FIG. 9.

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Specimens for cavitation erosion experiment.

Cavitation erosion is not only related to cavitation dynamic, but also to material characteristic. The response also differs for different materials exposed to cavitation. Therefore, it is important to set a reasonable strength threshold when predicting cavitation erosion.

The time percentage of pressure impact exceeding different strength thresholds, obtained from simulation, was analyzed, as shown in Figs. 10 and 11. High pressure impact mainly concentrates near the chord length of x = 0.6 C. There is a difference in impact intensity along the span-wise direction, and the erosion risk at M4 and M6 is higher than that of M2 and M8. When the threshold was set as 0, high erosion risk spreads over the whole chord-wise direction, and the chord-wise difference of impact intensity is not significant. In the experimental result illustrated in Fig. 12, this phenomenon was also captured for softer material aluminum with a lower strength threshold. With the increase in the threshold, the accumulation of pressure impact at M1 position is lower than that at downstream monitoring points, which is consistent with the experiment.

FIG. 10.

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Pressure impact fluctuation of monitor points in 0.2 s.

FIG. 11.

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Time percentage of pressure impact exceeding given thresholds at different locations: (a) $ε_{c} = 0$ ⁠; (b) $ε_{c} = 1 e + 06$ ⁠; and (c) $ε_{c} = 1 e + 07$ ⁠.

FIG. 12.

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Cavitation erosion of materials with different strength thresholds.

The cavitation erosion extent of the specimens was analyzed under the metallographic microscope as shown in Fig. 13. There is obvious cavitation erosion and oxidation corrosion on the aluminum material surface. There are span-wise differences in erosion intensity for M3 and M5.

FIG. 13.

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Cavitation erosion of aluminum.

The cavitation erosion distribution on the surface of copper foil was captured after running for 1 h, as shown in Fig. 14. The quantitative result was extracted by binarization of erosion images and quantified in the form of cavitation erosion coverage radio, as shown in Fig. 15. The cavitation erosion coverage radio on the copper surface has an increased trend along the chord length.

FIG. 14.

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Cavitation erosion of copper.

FIG. 15.

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Cavitation erosion coverage ratio of copper.

C. Propagation of cavitation impact

Figure 16 shows the relationship between the near-wall cavity volume change rate $\partial α / \partial t$ ⁠, vapor volume fraction $α$ ⁠, and pressure impact $\partial p / \partial t$ (i.e., pressure partial derivative) at monitor M1.

FIG. 16.

$Vapor volume fraction, cavitation volume change rate, and pressure change rate in a single cycle.$

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Vapor volume fraction, cavitation volume change rate, and pressure change rate in a single cycle.

It can be seen that the change of pressure impact near the wall in a single cycle can be divided into four stages. In stage A, attached cavity grows and the cavity volume change rate is higher than 0. A slight pulsation indicated by the red arrow first appears in the cycle since the low pressure remains stable after a decrease. In stage B, the cavity volume rate gradually decreased and the pressure impact $\partial p / \partial t$ stabilized around 0. In stage C, when the re-entrant jet moves to this position, the local vapor volume fraction decreases, and the cavity begins to break away from the wall. Pressure begins to rise at the same time and reaches the first peak in a short time after the vapor volume deformation rate drops to the lowest when collapse. In stage D, the attached cavity shed off completely, and the unsteady deformation of the shedding cloud leads to the pulsation of local pressure. Therefore, the collapse of attached cavity can be reasonably indicated by the variation of pressure impact and vapor volume fraction. The change rate is related to the propagation of cavitation impact.

The impact load propagation of cloud cavitation structure can be shown by side-view distribution of the pressure impact, the cavitation change rate, and vapor volume fraction, as illustrated in Fig. 17.

FIG. 17.

$Propagation of cavitation impact load caused by cloud cavitation. (The black curve is the contour line with a vapor volume fraction of 10%.)$

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Propagation of cavitation impact load caused by cloud cavitation. (The black curve is the contour line with a vapor volume fraction of 10%.)

The trend of growing attached cavity and shedding collapsing clouds can be captured, accompanied by the propagation of impact load. With the passage of time, the shedding cloud collapses and the vapor volume fraction in the core decreases. For the growing attached cavity, the vapor volume fraction increases near the cavity closure line. The high raise rate of pressure mainly appears near 60% of chord length where shedding cloud collapses. The pressure raise in region RA is caused by the collapse of small broken bubble cluster moving downstream. Between the attached cavity and the shedding cloud, the pressure increases sharply when the shedding cloud collapses, as shown in RB region in Fig. 17. The pressure raise area propagates quickly when the shedding cloud volume sharply decreases.

The variation of pressure impact and the simultaneous cavitation structure at typical instants in a single cycle were captured, as shown in Fig. 18. The distribution of pressure impact corresponds to the cavitation structure. The impact loads at instant I, instant II, and instant VI are mainly caused by the volumetric deformation of shed cloud cavity. Large area of impact exists at instant III, and the volumetric deformation is the largest. At instant IV, accompanying the re-entrant jet proceeds upstream, local collapse of the cavity causes the upstream propagation of impact load. At instant V, when the attached cavity begins to shed, the separation of the attached cavity from the wall causes large area of the hydrofoil suffer impact.

FIG. 18.

View large Download slide

Impact intensity at typical instants.

D. Statistical analysis of predicted cavitation erosion

In Sec. IV C, the propagation of cavitation impact throughout space and time has been clarified. In this subsection, the time–frequency characteristics of cavitation volume deformation and impact intensity are further displayed to predict cavitation erosion, as shown in Fig. 19. Figures (a)–(c) signify the different span-wise positions on the hydrofoil surface, the interval is 25% of span length, and those are 50 mm (z = 0.5 C), 100 mm (z = C), and 150 mm (z = 1.5 C) away from the front observation window, as shown in Fig. 1(b).

FIG. 19.

View large Download slide

The time–frequency relationship between cavitation volume deformation and pressure impact: (a)–(c) cavitation evolution process; (a′)–(c′) time domain characteristic of pressure impact; and (a′′)–(c′′) frequency domain characteristic of pressure impact.

Figures 19(a)–19(c) show the time-space characteristic of cavitation structure. The yellow arrow shows the re-entrant jet movement trend, and the dotted line shows the time when the jet cuts off the attached cavity. Cavitation structure is affected by the sidewall, causing different features between side-entrant jet and middle-entrant jet. The middle-entrant jet moves from the closure line of the attached cavity to the leading edge, while the side-entrant jet moves from the sidewall to the midstream and leading edge of the hydrofoil. The middle-entrant jet reaches the leading edge of hydrofoil earlier, which causes local cavitation break away from the wall and collapse earlier.

Figures 19(a′)–19(c′) show the time-space characteristics of pressure impact. The cavitation structure boundary was marked by the black solid line. Attached cavity grows in region QA, and the pressure impact is slight in this area. The movement of re-entrant jet leads to local cavitation collapse and causes the pressure impact to increase in region QB. After the attached cavity broke away from the wall, cloud cavitation gathered and rotated in region QC, resulting in a certain strength of pressure impact. The extremely unstable cloud cavitation causes local collapse in region QD. The impact intensity is of large magnitude and lasts for a longer time in this area.

Figures 19(a′′)–19(c′′) show the frequency domain characteristics of pressure impact. The frequency of impact, impact location, and the power spectral density (PSD) of impact were analyzed. High impact intensities are mainly concentrated in the middle and downstream of hydrofoil, which last for a long time and have high risk of cavitation erosion. High erosion risk of cavitation in the upstream area is marked by the black dotted line. There are significant differences in different span-wise locations. Due to the influence of middle-entrant jet, the amplitude of power spectral density at the span-wise location of z = C is the highest, and the erosion risk is higher than on other span-wise locations.

The accumulation of pressure impact is an important indicator of cavitation erosion. The different erosion risks at various locations on hydrofoil surface are shown in Fig. 20. The impact intensities are accumulated in a single cavitation evolution cycle. The accumulated erosion function $e_{pot}$ is nondimensionalized compared with the maximum value. The time-average value during a cavitation evolution cycle is illustrated in Fig. 20(a), and the root mean square value is illustrated in Fig. 20(b). Figure 20(c) shows the cavitation erosion distribution demonstrated in the literature.²¹

FIG. 20.

Cavitation erosion risks at different locations: (a) time-averaged value; (b) RMS result; and (c) erosion result illustrated by Li.21

View large Download slide

Cavitation erosion risks at different locations: (a) time-averaged value; (b) RMS result; and (c) erosion result illustrated by Li.²¹

The time-averaged result in Fig. 20(a) shows that around 60% of chord length, high erosion risk with high-intensity pressure impact lasts for a longer time. The root mean square result in Fig. 20(b) highlights the intensity difference. The cavitation erosion risk caused by cloud cavitation evolution consists of different factors. The erosion region EA is influenced by the middle-entrant jet when moving upstream. The erosion region EB corresponds to the collapse caused by side-entrant jets. The erosion area EC is the largest area suffered by impact released by cloud cavitation when deforming and collapse. The pressure impact caused by cloud cavitation collapse is the highest, and the cavitation erosion risk is higher than the upstream area where attached cavity alleviates the impact for a long duration.

Figure 21 shows the cavitation risk distribution accumulated in a single evolution cycle after statistically analyzed. The blue circle corresponds to the nine specimens in the cavitation erosion experiment. The highest erosion risk is mainly located near 0.575 C in the chord-wise direction. The duration of impact acting on the specimens was compared with the experimental results. The distribution is different in the span-wise direction, and the erosion of M3 and M5 follows the erosion experiment well, as illustrated in Figs. 13 and 14.

FIG. 21.

View large Download slide

Distribution of cavitation erosion risk accumulated in an evolution cycle.

VI. CONCLUSION

In this present work, the erosion risk by cavitating flow over a NACA0015 hydrofoil was investigated using simulation and experiment. In particular, the material strength thresholds were considered, and soft specimens were employed in the experiment instead of painting surfaces. The conclusions may be summarized as follows:

The collapse of cavities can be reasonably indicated by the variation of pressure and vapor volume fraction. The pressure partial derivative $\partial p / \partial t$ (i.e., pressure impact) is related to the propagation of cavitation impact. For the attached cavity, the re-entrant jet can lead to the shedding and local collapse, resulting in pressure impact. The high pressure impact mainly appears near 60% of chord length where shedding cloud collapses.
The pressure pulsation is dominated by the cavitation evolution. Its fluctuation amplitude captured by monitors at the downstream of the hydrofoil is higher than others, but the main frequency of pressure at seven monitors is consistent. The local collapse of cavities has an influence on the amplitude and impact intensity of pressure, which are different at various span-wise locations.
The effect of the material thresholds was well captured by the experiment and simulation. The response differs for different materials exposed to cavitation. When the material strength threshold is low, the chord-wise difference of cavitation erosion is not significant, because the slight impact can also generate erosion pits. When the threshold is high, the chord-wise difference of erosion becomes larger, and erosion pits are more centralized.
High impact intensities are mainly concentrated in the middle and downstream of hydrofoil, which last for a long time and have high risks of cavitation erosion. At the upstream covered by the attached cavity, the middle-entrant jet easier leads to cavitation erosion than the side-entrant jets, because the power spectral density amplitude of pressure impact at the span length of 100 mm exceeds that at 50 and 150 mm, and its frequency is lower.
The local collapse of cavitation causes the difference in impact intensity and erosion risk. The cavitation erosion risks on the foil include the risk induced by the upstream motion of the middle-entrant jet, the risk influenced by the side-entrant jet, and the most significant erosion risk caused by the shedding and collapse of the cavitation cloud.

ACKNOWLEDGMENTS

This study was supported by the National Natural Science Foundation of China (No. 51806082), China Postdoctoral Science Foundation (Nos. 2020M671363 and 2021T140282), Postdoctoral Science Foundation of Jiangsu Province (No. 2020Z298), Entrepreneurial Doctor Program of Jiangsu Province (No. 18SCBS016), and the National Natural Science Foundation of China (No. 51979240 and 52009013).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Ning Qiu: Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Han Zhu: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal). Pei Xu: Investigation (equal); Software (equal); Writing – original draft (equal). Bangxiang Che: Resources (equal); Supervision (equal); Visualization (equal). Jie Wu: Methodology (equal); Software (equal); Writing – review & editing (equal). Wenjie Zhou: Data curation (equal); Formal analysis (equal); Supervision (equal). Chuan Wang: Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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Author(s)

Assessment of cavitation erosion risk indicated by pressure impact exceeding material strength threshold

I. INTRODUCTION

A. Outline of the present work

II. EXPERIMENTAL METHODOLOGY

III. NUMERICAL METHODOLOGY

A. Cavitation model

B. Turbulence model

C. Erosion assessment method

D. Numerical setup

E. Grid-independent check

IV. NUMERICAL VALIDATION

V. RESULTS AND DISCUSSION

A. Pressure pulsation characteristic

B. Cavitation erosion characteristics considering material threshold

C. Propagation of cavitation impact

D. Statistical analysis of predicted cavitation erosion

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Assessment of cavitation erosion risk indicated by pressure impact exceeding material strength threshold

I. INTRODUCTION

A. Outline of the present work

II. EXPERIMENTAL METHODOLOGY

III. NUMERICAL METHODOLOGY

A. Cavitation model

B. Turbulence model

C. Erosion assessment method

D. Numerical setup

E. Grid-independent check

IV. NUMERICAL VALIDATION

V. RESULTS AND DISCUSSION

A. Pressure pulsation characteristic

B. Cavitation erosion characteristics considering material threshold

C. Propagation of cavitation impact

D. Statistical analysis of predicted cavitation erosion

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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