# Prediction of the Cavitation over a Twisted Hydrofoil Considering the Nuclei Fraction Sensitivity at 4000 m Altitude Level

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{cav}increases with the decreasing of cavitation coefficient C

_{σ}. At a specific C

_{σ}, high altitude and few nuclei will cause smaller size of cavitation. The smaller C

_{σ}is, the higher the sensitivity Δf

_{cav}is. The larger C

_{σ}is, the higher the relative sensitivity Δf

_{cav}

^{*}is. On the twisted foil, flow incidence angle increases from the sidewall to mid-span with the decreasing of the local minimum pressure. When C

_{σ}is continually decreasing, the size of cavitation extends in spanwise, streamwise and thickness directions. The cavity is broken by the backward-jet flow when C

_{σ}becomes small. A tail generates and the cavity becomes relatively unstable. This study will provide reference for evaluating the cavitation status of the water pumps and hydroturbines installed on a plateau with high altitude level.

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Numerical Method for Turbulent Flow

_{t}is eddy viscosity, σ is model constant, C

_{ω}is the turbulence dissipation term, F

_{1}is the zonal blending function, l

_{k−ω}is called the turbulence scale that l

_{k-ω}= k

^{1/2}β

_{k}ω, where β

_{k}is the model constant. The DES method uses a zonal treatment term that min(l

_{k−ω}, C

_{DES}·L

_{mesh}), where L

_{mesh}is the maximum mesh element dimension and C

_{DES}is a constant. When l

_{k−ω}is larger than C

_{DES}·L

_{mesh}, Large Eddy Simulation will be activated. Otherwise, SST model is used in the Reynolds-averaged mode.

#### 2.2. Cavitation Model

_{v}is the saturation pressure, ρ

_{v}is the vapor density, f

_{vnuc}is the nuclei volume fraction, R

_{B}is the nuclei average radius, F

_{e}and F

_{c}are coefficients of evaporization and condensation which are commonly F

_{e}= 50 and F

_{c}= 0.01.

#### 2.3. Vapor Volume Fraction

_{atm}and altitude H

_{alt}have the approximate relationship as:

_{d}are constants that C

_{d}

_{1}= 1.013 × 10

^{5}, C

_{d}

_{2}= 1.259 × 10

^{1}, C

_{d}

_{3}= 6.476 × 10

^{−4}. Based on Henry’s law, the nuclei volume fraction f

_{vnuc}will be different if the altitude H

_{alt}is different. The original f

_{vnuc}at H

_{alt}= 0 m is 5 × 10

^{−4}. In this case, the altitude conditions that H

_{alt}= 0 m, 1000 m, 2000 m, 3000 m and 4000 m are comparatively studied. Therefore, the values of f

_{vnuc}are listed in Table 1.

## 3. Case and Setup

#### 3.1. Important Dimensionless Parameters

_{σ}:

_{v}is the saturation pressure, p

_{ref}and v

_{ref}are reference pressure and velocity, respectively, which are usually measured at the upstream of hydrofoil, ρ is density. Therefore, C

_{σ}can be adjusted by changing the value of p

_{ref}. Secondly, the pressure coefficient C

_{p}can be defined as:

_{σ}values can be compared for different altitudes with considering that the altitude caused a difference on nuclei volume fraction f

_{vnuc}.

#### 3.2. Flow Domain of Hydrofoil

_{m}is the maximum thickness of foil. Parameter C

_{a}and C

_{b}are constants that C

_{a}= 0.2, C

_{b}

_{1}= 0.2969, C

_{b}

_{2}= 0.126, C

_{b}

_{3}= 0.3516, C

_{b}

_{4}= 0.2843 and C

_{b}

_{5}= 0.1015. This twisted hydrofoil has a different installation angle α at a different span. It has the law of α as:

_{m}is the maximum installation angle at mid-span, which is 11 degrees. α

_{w}is the installation angle on the wall side, which is −2 degrees in this case.

_{ref}, where L

_{ref}is 100 mm. The domain size is L

_{1}= 10.5 L

_{ref}, L

_{2}= 3.0 L

_{ref}and L

_{3}= 1.5 L

_{ref}. The foil center locates 3.0 L

_{ref}downstream to the inlet. In this case, the domain is simplified as a half of the entire flow region that s = z/c is within 0~1.

#### 3.3. CFD Setup

^{+}was from 0.47 to 23.75. As introduced above, the DES method and Zwart cavitation model were used in this numerical study. The fluid is water at 20 °C. As indicated, boundary conditions are set on the domain including a velocity inlet, a pressure outlet, a symmetry boundary at mid-span, a no-slip wall on foil surface, slip wall boundaries on upper wall, lower wall and side wall. The inlet velocity v

_{in}is 6.97 m/s, which means that the Reynolds number Re is 1.05×10

^{6}. The reference location for C

_{p}and C

_{σ}is the inlet boundary. To have a better study of cavitation, the same C

_{σ}situations are compared for different altitude levels. For a specific value of v

_{in}, the inlet–outlet pressure difference is almost unchanged. Thus, p

_{ref}at inlet can be adjusted to an expectable value by setting a specific pressure value at outlet. Steady-state simulation is firstly conducted. It will converge after the RMS residual of momentum and continuity equation is less than 1 × 10

^{−4}or finish after 1000 iterations. Transient simulation is conducted based on steady-state simulation. The total time is 1s and the time step is 1 × 10

^{−5}s. The maximum iteration number for each time step is 10. The convergence criterion is also RMS residual less than 1 × 10

^{−4}.

## 4. Numerical-Experimental Verification

_{p}between CFD prediction and experimental data [12,36]. Three different spanwise positions in which s = 0.6, 0.8 and 1.0 are compared, especially focusing on the low-pressure side where cavitation usually occurs. The CFD predicted C

_{p}curves are accurate on the three spanwise surfaces. The CFD simulation can be used for further analyses of the cavitating flow in the following sections.

## 5. Cavitation Vapor Proportion at Different Altitudes

#### 5.1. Variation Law

_{alt}, the cavitation vapor proportion f

_{cav}in the fluid domain is defined as:

_{cav}is the cavitation vapor volume and V

_{fluid}is the fluid domain volume.

_{cav}among different H

_{alt}at different C

_{σ}. With the decreasing of C

_{σ}from 2.713 to 1.071, f

_{cav}continually increases to a high level. The smaller C

_{σ}is, the quicker f

_{cav}increases. It represents the increasing of cavitation vapor in the entire fluid domain. However, there are differences among different H

_{alt}situations. Figure 5 shows the cavitation vapor proportion among different H

_{alt}at specific C

_{σ}values. The tendency is similar in all the 9 situations that f

_{cav}decreases with the increasing of the H

_{alt}level.

#### 5.2. Sensitivity Analysis

_{alt}, it is necessary to analyze the sensitivity of f

_{cav}on H

_{alt}and f

_{vnuc}. The difference between maximum and minimum f

_{cav}among different H

_{alt}at a specific C

_{σ}is defined as the sensitivity Δf

_{cav}. For a better comparison, the relative sensitivity Δf

_{cav}

^{*}is defined as:

_{cav}

^{a}is the average vapor proportion of all the 5 H

_{alt}situations.

_{cav}at different C

_{σ}. The H

_{alt}-average vapor proportion f

_{cav}

^{a}shows the same variation tendency as in Figure 4. f

_{cav}

^{a}increases with the decreasing of C

_{σ}. The sensitivity Δf

_{cav}has almost the similar tendency of f

_{cav}

^{a}. The smaller C

_{σ}is, the greater the difference is among different altitudes. However, there is a special local peak region when Δf

_{cav}drops to a low level around C

_{σ}= 1.8. It means that the difference between maximum and minimum f

_{cav}is locally higher.

_{σ}, the absolute difference of f

_{cav}among different H

_{alt}is also small. Therefore, it is necessary to compare the variation of relative sensitivity Δf

_{cav}

^{*}. As shown, the tendency is completely different. It is in a W-shape with two slowly variating regions and two rapidly rising regions, as indicated. The first rapidly rising region is about C

_{σ}= 1.5~1.9. The secondly rapidly rising region is about C

_{σ}= 2.5~2.7. Generally, when the size of cavitation is small, the relative sensitivity Δf

_{cav}

^{*}is higher. In the high altitude plateau area, it is necessary to consider the influence of altitude level on cavitation inception.

## 6. Flow Behaviors Considering Altitude Level

#### 6.1. Pressure Distribution Law on Foil Surface

_{p}on different spanwise positions (0 ≤ s ≤ 1) of foil without considering cavitation. The maximum pressure coefficient C

_{pmax}is similar (about 1.0) for different s. This high pressure is because of the local flow striking on the foil lower surface, as shown in Figure 8. The minimum pressure coefficient C

_{pmin}varies with s. The larger s is, the smaller C

_{pmin}is. This is because of the local flow separation on the foil upper surface, as is also shown in Figure 8.

_{pmin}and installation angle α at different spanwise s positions. There is a significant inverse relationship between C

_{pmin}and α. The larger the installation angle is, the larger the flow incidence angle is. It indicates the stronger and stronger flow separation and pressure drop when incidence angle is increasing.

#### 6.2. Turbulent Flow around Foil

_{v}at different spanwise s positions with indication of vectors. The uniformed velocity C

_{v}is defined as:

_{in}is the velocity at inlet.

_{v}regions. Firstly, it is the local flow striking region on the leading-edge on the foil lower surface. Secondly, it is the wake region downstream to the foil trailing-edge. At s = 0.8, installation angle α increases to about 7.86 degrees. An obvious flow separation region occurs on the foil upper surface with low C

_{v}. The leading-edge striking region and the trailing-edge wake region are wider. At s = 1.0 (mid-span), installation angle α increases to 9 degrees, which is relatively large. It is obvious that the flow separation region on the foil upper surface is much wider. The leading-edge striking region and the trailing-edge wake region are also wider.

_{v}is used [37]. It can be non-dimensionalized to the velocity helicity coefficient C

_{vhe}by:

_{vhe}on the mid-span plane and foil upper surface. The vortex-shedding phenomenon can be seen with the indication of the vortex-shedding route (VSR). On the mid-span plane, obvious VSR can be found from leading-edge (LE), along the upper surface, to trailing-edge (TE) and towards downstream. On the foil upper surface, VSR is complex and mainly along the diagonal line from the LE-mid-span corner to the TE-wall corner. Generally speaking, local vortical flow moves along the slope of the twisted foil surface.

#### 6.3. Development of Cavitation at H_{alt}- = 4000 m

_{alt}= 4000 m and f

_{vnuc}= 3.01×10

^{−4}, the cavitating flow is simulated and analyzed in Figure 11. The development of cavitation is comparatively studied from C

_{σ}= 2.713 to C

_{σ}= 1.071. The scale of cavitation continually increases in different directions. Figure 11 mainly shows the region of cavitation covering on the foil upper surface. Leading-edge (LE) is on the left side. Two parameters are defined to have a quantitative comparison. One is the length of cavity-covered area l

_{cav}and another is the width of cavity-covered area w

_{cav}. Figure 11 also includes the mid-span view of cavitation. Two parameters are defined in this view. One is the maximum thickness of cavity on mid-span t

_{cav}. Another is the total length of the attached part of the cavity on mid-span, which is denoted as l

_{cav}

^{*}. In general, l

_{cav}, w

_{cav}, t

_{cav}and l

_{cav}

^{*}increase with the decreasing of cavitation coefficient C

_{σ}.

_{cav}, w

_{cav}, t

_{cav}and l

_{cav}

^{*}are compared in Figure 12. These four parameters are normalized against the foil chord length c. The growth rate dφ/dC

_{σ}is also analyzed between each two conditions.

_{cav}, the growth rate is relatively low, within C

_{σ}= 2.173~1.784. The value of d(l

_{cav}/c)/dC

_{σ}is lower than 0.05. The value of l

_{cav}/c increases from about 0.026 to about 0.051. When C

_{σ}is smaller than 1.784, the growth rate of l

_{cav}becomes much higher. The value of d(l

_{cav}/c)/dC

_{σ}is about 0.09~0.40. From C

_{σ}= 1.784 to C

_{σ}= 1.071, the value of l

_{cav}/c strongly increases from about 0.051 to 0.202. Cavity covers about 1/5 of the foil surface along x direction at C

_{σ}= 1.071.

_{cav}, the growth rate d(w

_{cav}/c)/dC

_{σ}is stable around 0.3. The increasing of w

_{cav}/c against C

_{σ}is almost linear. From C

_{σ}= 2.173 to C

_{σ}= 1.071, w

_{cav}/c increases from about 0.158 to about 0.636. At C

_{σ}= 1.071, cavity covers more than 3/5 of the half foil surface along the y direction.

_{cav}, the growth rate is relatively stable. The value of d(t

_{cav}/c)/dC

_{σ}is lower than 0.03. The increasing of t

_{cav}/c against C

_{σ}is almost linear, except in a small range between 1.784 and 1.480. In this range, a special phenomenon occurs in which a tail can be seen on the profile of cavity. This is because of the backward-jet flow (indicated in Figure 13 as an example), and the cavity becomes much thicker. From C

_{σ}= 2.173 to C

_{σ}= 1.071, t

_{cav}/c increases from about 0.001 to about 0.023. This thickness (t

_{cav}/c = 0.023 at C

_{σ}= 1.071) is about 1/4 of the maximum foil thickness.

_{cav}

^{*}, the growth rate is similar to l

_{cav}. From C

_{σ}= 2.713 to C

_{σ}= 1.480, the value of d(l

_{cav}

^{*}/c)/dC

_{σ}is lower than 0.2. The value of l

_{cav}

^{*}increases from about 0.019 to 0.162. From C

_{σ}= 1.480 to C

_{σ}= 1.071, the value of d(l

_{cav}

^{*}/c)/dC

_{σ}is 0.36~0.44. The value of l

_{cav}

^{*}/c obviously increases from about 0.162 to 0.326. Comparing with l

_{cav}, both the value and the growth rate of l

_{cav}

^{*}is higher. This is also because of the tail on cavity. The total length of cavity is bigger than the cavity length covering on foil surface.

_{σ}. When C

_{σ}is at a higher level, the small-scale cavity is attached on foil surface. When C

_{σ}becomes lower, cavity on large-incidence-angle spans is broken by the backward-jet from small-incidence-angle spans. A tail is generated on the cavity and the cavity becomes relatively unstable. The growth of cavity becomes quicker, especially in streamwise (length) direction.

## 7. Conclusions

- (1)
- With the decreasing of cavitation coefficient C
_{σ}, the scale of cavitation continually increases and the increasing is quicker and quicker. The nuclei volume fraction f_{vnuc}has obvious influence on cavitation. The size of cavitation is different at different altitude levels. If the altitude is higher within 0~4000 m, the f_{vnuc}is lower and the size of cavitation is smaller. The difference of the size of cavitation among altitude levels is bigger when C_{σ}is small. That is, the sensitivity Δf_{cav}is high. On the contrary, the relative sensitivity Δf_{cav}^{*}, which is the ratio between Δf_{cav}and the absolute cavitation fraction f_{cav}, is high when C_{σ}is large. When C_{σ}is 1.071, the Δf_{cav}^{*}between 0 m and 4000 m altitudes is about 4.6%. When C_{σ}increases to 2.713, the Δf_{cav}^{*}can be up to about 22.8%. It means that the cavitation volume fraction sensitivity should be considered in judging the inception cavitation of water pumps and hydro-turbines in the plateau environment. - (2)
- For this twisted hydrofoil, the installation angle and flow incidence angle are different at different spans. The incoming flow will cause local high pressure on the lower surface of hydrofoil. There will be a local low pressure site on the foil upper surface due to flow separation. This low pressure will cause cavitation. From sidewall to mid-span, the installation angle increases and the minimum pressure decreases. With the decreasing of C
_{σ}, the size of cavitation extends along the spanwise direction, streamwise direction and thickness direction. The growth rate is high in the spanwise (cavity width) and streamwise (cavity length) directions and low in thickness direction. When the size of cavitation is large enough, it will be broken by backflow-jet flow. A tail generates and the cavity becomes relatively unstable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Comparison of cavitation vapor proportion f

_{cav}among different H

_{alt}at different C

_{σ}.

**Figure 8.**Minimum pressure coefficient C

_{pmin}and installation angle α at different spanwise s positions with local contour of C

_{p}on the mid-span plane.

**Figure 9.**Velocity coefficient C

_{v}at different spanwise s positions with indication of velocity vectors.

**Figure 10.**Velocity helicity coefficient C

_{vhe}on the mid-span plane and foil upper surface. VSR: vortex-shedding route.

**Figure 11.**Variation of the cavitation vapor volume fraction f

_{v}on the mid-span plane and foil upper surface with the decreasing of cavitation coefficient C

_{σ}. LE: leading-edge.

**Figure 12.**Variation of the length, width and thickness of cavitation bubble on mid-span plane and foil upper surface.

**Figure 13.**Indication of the backward-jet flow breaking the attached cavity at low C

_{σ}. LE: leading-edge.

Altitude H_{alt} | Nuclei Volume Fraction f_{vnuc} |
---|---|

0 m | 5 × 10^{−4} |

1000 m | 4.38 × 10^{−4} |

2000 m | 3.88 × 10^{−4} |

3000 m | 3.48 × 10^{−4} |

4000 m | 3.01 × 10^{−4} |

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**MDPI and ACS Style**

Luo, H.; Tao, R.
Prediction of the Cavitation over a Twisted Hydrofoil Considering the Nuclei Fraction Sensitivity at 4000 m Altitude Level. *Water* **2021**, *13*, 1938.
https://doi.org/10.3390/w13141938

**AMA Style**

Luo H, Tao R.
Prediction of the Cavitation over a Twisted Hydrofoil Considering the Nuclei Fraction Sensitivity at 4000 m Altitude Level. *Water*. 2021; 13(14):1938.
https://doi.org/10.3390/w13141938

**Chicago/Turabian Style**

Luo, Hongying, and Ran Tao.
2021. "Prediction of the Cavitation over a Twisted Hydrofoil Considering the Nuclei Fraction Sensitivity at 4000 m Altitude Level" *Water* 13, no. 14: 1938.
https://doi.org/10.3390/w13141938