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Cavitation is a complex flow phenomenon including unsteady characteristics, turbulence, gas-liquid two-phase flow. This paper provides a numerical investigation on comparing the simulation performance of three different models in OpenFOAM-Merkle model, Kunz model and Schnerr-Sauer model, which is helpful for understanding the cavitation flow. Considering the influence of vapor-liquid mixing density on turbulent viscous coefficient, the modified SST
*k-ω* model is adopted in this paper to increase the computing reliability. The InterPhaseChangeFoam solver is utilized to simulate the two-dimensional cavitation flow of the Clark-Y hydrofoil with three cavitation models. The hydrodynamic performance including lift coefficient, drag coefficient and cavitation flow shape of the hydrofoil is analyzed. Through the comparison of the numerical results and experimental data, it is found that the Schnerr-Sauer model can get the most accurate results among the three models. And from the simulation point of water and water vapor mixing, the Merkle model has the best water and water vapor mixing simulation.

When the local pressure of flow field decreases below the saturated vapor pressure, explosive vaporization of liquid medium will occur, which is called cavitation. Cavitation is a complex flow phenomenon including unsteady characteristics, turbulence, gas-liquid two-phase flow. Due to the unsteady characteristics of the cavitation bubble, the pressure fluctuation during the collapse stage of the cavitation bubble will cause noise and vibration. According to the form of cavitation, the cavitation can be divided into sheet cavitation, vortex cavitation, cloud cavitation and bubble cavitation. Except for sheet cavitation, other types of cavitation have strong unsteady characteristics. Cavitation is inevitable on the mechanical surfaces of high-speed fluids such as hydrofoils and propellers which lead to blade erosion damage, accompanied by noise, vibration and other adverse effects. Therefore, more and more attention has been paid to the study of cavitation phenomena and unsteady characteristics of cavitation flows, both experimentally and numerically.

Early researchers [

Model experiment is of great importance for cavitation research. However, the high cost, severe scale effect and long period of the above model tests restrict our further research on cavitation problems. The development of numerical methods and computer processors has prompted researchers to further explore and study cavitation problems through numerical simulation. Potential flow simulation was used in the early stage [

At present, the single-phase homogeneous model is mainly used to deal with the problem of cavitation flow. According to different definitions of the density field of y, the cavitation model can be classified into two types: one is based on the state equation and the other is based on the transport equation. Based on the equation of state, the density of a single medium is assumed to be a single-valued function of pressure, and the conversion between vapor and liquid is controlled by a function of density and pressure. Isentropic vaporization model, isothermal model and Rayleigh-Preset model represent three major branches. Shin [

This paper mainly simulates the hydrodynamic characteristics of the two-dimensional Clark-Y hydrofoils by using the interPhaseChangeFoam solver under the OpenFOAM platform. The lift coefficient, drag coefficient and cavitation flow of the hydrofoil are analyzed. The SST k-omega turbulence model is utilized, and the eddy viscosity coefficient of the turbulence model is modified to restrict the viscosity of the water vapor mixing zone. The simulation performance in two-dimensional cavitation flow of the Merkle model, Kunz model and Schnerr-Sauer model are also carried out. The structure of this paper is organized as follows: Section 1 presents the specific numerical simulation methods adopted in this study; in Section 2, the computational model for analysis is described and the results are discussed in Section 3; finally, several concluding remarks are made in Section 4.

Cavitation flow belongs to two-phase flow and there is continuous mass transformation between vapor phase and liquid phase in the two-phase flow. In order to solve a cavitation flow problem, the following equations need to be solved: continuity Equation (7), phase Equations (2) and (3) and momentum Equation (8). Before solving the problem, the volume fraction is define

α = lim δ → 0 δ V l δ V l + δ V V (1)

In the cavitation flow, the continuity equation of each phase is given as:

∂ α l ∂ t + ∇ ⋅ ( α l U l ) = m ˙ / ρ l (2)

∂ α v ∂ t + ∇ ⋅ ( α v U v ) = − m ˙ / ρ v (3)

In the above equations, l denotes liquid phase, V denotes vapor phase, α is their volume fraction, ρ is their density, U is their velocity vector, m ˙ is the mass exchange between two phases. In the process of solution, only one phase equation needs to be solved because of the relationship between gas phase and liquid phase.

α v = 1 − α l (4)

For the mixed medium consisting of vapor and liquid phases, it is satisfied that:

∂ ρ ∂ t + ∇ ( ρ u → ) = 0 (5)

ρ = α l ρ l + ( 1 − α l ) ρ v (6)

Among them, u is the velocity of mixed medium and ρ is the density of mixed medium.

The continuity equations for the simulation of cavitation flow are obtained:

∇ u → = m ˙ ( 1 / ρ l − 1 / ρ v ) (7)

According to the assumption of single-phase homogeneity, the momentum equation of vapor-liquid two-phase matter can be obtained:

∂ ( ρ m u j ) ∂ t + ∂ ( ρ m u i u j ) ∂ x j = − ∂ p ∂ x j + ∂ ∂ x j [ μ ( ∂ u i ∂ x j + ∂ u j ∂ x i ) ] − ∂ τ i j ∂ x j (8)

The density and viscosity coefficients of mixed media are defined as follows:

ρ m = ρ l α l + ρ V ( 1 − α l ) (9)

μ m = μ l α l + μ V ( 1 − α l ) (10)

In this paper, three cavitation models based on transport equation are used.

The Merkle cavitation model is based on a vapor-liquid two-phase flow model, which derives the interphase mass transfer rate from the mixed density. The formula is:

m ˙ e = C e ρ v α l min ( 0 , P − P v ) 0.5 U ∞ 2 t ∞ ρ l (11)

m ˙ c = C c ρ v α l ( 1 − α l ) max ( 0 , P − P v ) 0.5 U ∞ 2 t ∞ ρ l (12)

In the Merkle model, U ∞ is the reference speed; t ∞ is the reference time. C e and C c are empirical coefficients, which are C e = 1 and C c = 80 , respectively.

m ˙ e and m ˙ c are expressions of mass exchange during vaporization and condensation, respectively.

Kunz modified the Merkle model and simplified the Ginzburg-Landau potential for the condensation rate. It can be seen that the final condensation rate is proportional to the volume fraction of vapor, while the evaporation rate is proportional to the volume fraction of liquid phase and the difference of vapor-hydraulic pressure.

R e = C d e s t ρ v α l min [ 0 , p − p s a t ] ρ l U ∞ 2 2 t ∞ (13)

R c = C p r o d ρ v α l 2 ( 1 − α l ) t ∞ (14)

In the formula of the Kunz model, U ∞ is the free-flow velocity; t ∞ = L / U ∞ is the characteristic time scale, where L is the feature length; C d e s t and C p r o d are empirical constants. Generally, C d e s t is the evaporation coefficient, and C p r o d is the condensation coefficient. The appropriate value should be selected according to the specific model. The general value range is 0.2 ≤ C d e s t = C p r o d ≤ 105 .

For the two empirical coefficients in the above formula, there is no uniform reference value, so different empirical coefficients need to be taken in different situations. Kunz model is considered to have the following characteristics: for sheet cavitation, the force at the vapor-liquid interface is more balanced, the pressure and velocity change range is small; it can deal with the high density ratio of cavitation flow, and the dependence on empirical constants is relatively high.

The Schnerr-Sauer model is derived from Rayleigh-Plesset equation based on bubble dynamics theory. In Schnerr-Sauer model, the vapor volume fraction is expressed as follows:

α v = n o 4 3 π R 3 n o 4 3 π R 3 + 1 (15)

In practical applications, considering the phase change is divided into two types, liquefaction and evaporation. The Schnerr-Sauer model equation is deduced:

m ˙ c = C c 3 ρ v ρ l α v ( 1 − α v ) ρ R sgn ( P V − P ) 2 | P V − P | 3 ρ l (16)

m ˙ v = − C v 3 ρ v ρ l α v ( 1 − α v ) ρ R sgn ( P V − P ) 2 | P V − P | 3 ρ l (17)

The bubble radius R satisfies the following Formula (18), and can be obtained by the deformation of the Formula (15). C c and C v are the condensation and vaporization coefficients respectively. The Schnerr-Sauer model has proved to be a relatively reliable model after a large number of practical applications and verification. The researchers roughly summarize the following characteristics: It can be applied to mixed flows with a large number of spheres. In the case of cavitation; the expression of condensation rate and vaporization rate is symmetrical; it is used more frequently in cavitation flow around an approximate cylinder or sphere, which can better simulate the retroreflection flow.

R = ( α v 1 − α v ⋅ 3 4 π n 0 ) 1 / 3 (18)

In SST k-omega model, k-omega model is used near the wall and k-epsilon model is used in the far field. At present, it is one of the widely used turbulence model, which can effectively avoid the problem that the k-omega model is too sensitive to the size of the entrance turbulence. Since the RANS turbulence model is developed on the basis of a single-phase and completely incompressible medium, the turbulent viscosity at the end of the cavitation will be over-predicted when the RANS model is used to simulate the cavitation flow.

Reboud et al. [

With the help of Reboud’s idea, this paper modifies the SST k-Omega model:

μ t = f ( ρ ) C μ k 2 ε (19)

f ( ρ ) = ρ v + ( 1 − α ) n ( ρ l − ρ v ) (20)

where k is turbulent energy, ω is turbulent dissipation rate, C ω are constant, and ρ is fluid density. When the water vapor content is the same, after introducing the density function, especially in the vapor-liquid mixing region with high water vapor content, the influence of the over-predicted viscosity on the cavitation flow can be reduced, which can limit the excessive turbulence of the water vapor mixing zone in the cavitation tail, to simulate the Re-entrant and cavitation shedding more precisely. The parameter n in the modified turbulence model should be greater than 1, and some scholars recommend a value of not less than 10. After research [

The 2D computational domain is shown in

A region-wide background grid region is created firstly, then encrypted in the OpenFOAM via the snappy HexMesh method. This process is for better prediction of the forces on the Clark-Y two-dimensional hydrofoil and the cavitation on the suction surface of the blade. A sketch of the grid in the computational domain is shown in

part is the outer net, which is the initial mesh of the entire flow field; the length is twice the length of the chord, and the height is 2D.

The variables are initialized as follows:

Velocity U: The entrance is a fixed value U = 10 m/s, and the exit is a zero gradient. Pressure p: Reference pressure is obtained by the cavitation number through Equation (21). The cavitation number in the present work is 0.8.

σ = P − P V 1 2 ρ U 2 (21)

The surface of the hydrofoil can form a layer of cavitation. In this paper, three models are used to simulate the cavitation flow of two-dimensional hydrofoil Clark-Y. The cavitation clouds of Clark-Y hydrofoil computed by three different models are shown in the following figures, from Figures 3-5. As can be seen from the figures, a thin layer of cavitation generally appears on the surface of hydrofoil and grows from the front of hydrofoil. This layer of cavitation is close to the surface of hydrofoil, also known as sheet cavitation. Sheet cavitation is one of the main cavitation forms of hydrofoil.

The cyclic variation of the cavitation flow in a period of the numerical simulation can be summarized according to the simulation results:

1) A sheet-like cavitation zone is formed from the front of the hydrofoil to the surface like the cavitation zone in

2) The sheet cavitation zone grows correspondingly. When the upstream flow flows to the rear, the low-pressure zone becomes longer and longer. The sheet cavitation zone can be seen from

3) Cavitation zone begins to break and then shed. Because the sheet cannot continue to grow after the cavitation grows to a certain extent. Then the cavitation zone will shed, which is caused by the re-entrant jet. The shedding phenomena in the cavitation zone can be seen from

4) Formation of cloud cavitation. The falling cavitation forms cloud-like cavitation under the shear action of upstream inflow and return jet, and moves downstream. At the same time, the next period of sheet cavitation begins at the front of hydrofoil. The formation of cloud cavitation can be found from

From the simulation point of water and water vapor mixing, the Merkle model has the best water and water vapor mixing simulation and is more accurate than the Schnerr-Sauer model. The effect of the Kunz model in simulating water vapor mixing is very general. In the cavitation cloud diagram of

Compared with the cavitation flow diagrams of model tests (

cavitation flow in Merkle model shows a complete cavitation period. From

The time history curve of lift and drag coefficients of two-dimensional hydrofoils calculated by three models are shown in

Different Simulation | Hydrodynamic Characteristics | |
---|---|---|

Drag Coefficient | Lift Coefficient | |

Merkle Model | 0.1460 | 0.7529 |

Kunz Model | 0.1478 | 0.7800 |

Schnerr-Sauer Model | 0.1390 | 0.7564 |

Experiment | 0.118 | 0.8 |

Data [ | 0.125 | 0.66 |

result of Schnerr-Sauer model is more accurate than the results of Kunz model and Merkle model.

The figures for cavitation simulation in Schnerr-Sauer model is close to the experimental cavitation image. The phenomenon of cavitation in Schnerr-Sauer model fracture and shedding with time is more obvious which is in good agreement with the experimental results. And Schnerr-Sauer model can capture abundant small vortices accurately. The velocity vector diagram during cavitation shedding is shown in

In this paper, a numerical investigation on comparing the simulation performance of three different models in OpenFOAM-Merkle model, Kunz model and Schnerr-Sauer model is provided. The SST k-omega model is modified and the modified SST k-omega model can better reflect the initial, development and collapse of the vacuole, so as to better study the cavitation problem. The latter SST k-omega model is a more accurate turbulence model for predicting cavitation. And three different cavitation models Kunz, Merkle and Schnerr-Sauer are used to simulate the cavitation around Clark-Y hydrofoil by using interPhaseChangeFoam solver in OpenFOAM. The Merkle model has the best water and water vapor mixing simulation and is more accurate than the Schnerr-Sauer

model. The hydrodynamic characteristics are demonstrated and compared. Merkle model and Schnerr-Sauer model are better than Kunz model in simulating unsteady cavitation flow, which are in good agreement with the experimental results, and the unsteady characteristics of cavitation change periodically. Compared with model Merkle, model Schnerr-Sauer can capture abundant small vortices and simulate unsteady cavitation flow around two-dimensional hydrofoils more accurately. The hydrodynamic performance including lift coefficient, drag coefficient and cavitation flow shape of the hydrofoil is analyzed. With Schnerr-Sauer model, a more accurate simulation result is achieved. Further research on re-entry jet shows that the vortex structure at the tail of hydrofoil is the main cause of cavitation shedding. In the follow-up study, the Schnerr-Sauer model can be used to further study the hydrodynamic characteristics of E779A propeller under the wake flow field.

This work is supported by the National Natural Science Foundation of China (51879159, 51490675, 11432009, 51579145), Chang Jiang Scholars Program (T2014099), Shanghai Excellent Academic Leaders Program (17XD1402300), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (2013022), Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23/09) and Lloyd’s Register Foundation for doctoral student, to which the authors are most grateful.

The authors declare no conflicts of interest regarding the publication of this paper.

Liu, Y.J., Wang, J.H. and Wan, D.C. (2019) Numerical Simulations of Cavitation Flows around Clark-Y Hydrofoil. Journal of Applied Mathematics and Physics, 7, 1660-1676. https://doi.org/10.4236/jamp.2019.78113