# Large Eddy Simulation of Periodic Transient Pressure Fluctuation in a Centrifugal Pump Impeller at Low Flow Rate

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^{2}

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## Abstract

**:**

_{d}(Q

_{d}indicates design load) and 0.75Q

_{d}when the rotating speed is maintained. Research shows that in one period, the inlet flow rate will twice reach 0.5Q

_{d}, and among the impeller of one moment is a stall state, but the other is a non-stall state. In the process of flow development, the evolution of low-frequency pressure fluctuation shows an obviously sinusoidal form, whose frequency is insensitive to the monitoring position and equals to that of the flow rate. However, inside the impeller, the phase and amplitude in the stall passages lag behind more and are stronger than that in the non-stall passages. Meanwhile, the strongest region of the high-frequency pressure fluctuation appears in the stall passages at the transient rising stage. The second dominant frequency in stall passages is 2.5 times to that in non-stall passages. In addition, similar to the pressure fluctuation, the evolution of the low-frequency head shows a sinusoidal form, whose phase is lagging behind that by one-third of a period in the inlet flow rate.

## 1. Introduction

## 2. Model Descriptions

#### 2.1. Geometric Model

_{d}= 3.06 L/s. The outer radius of the impeller is 95 mm, and the inlet height and outlet height are 13.8 and 5.8 mm, respectively. Several other parameters of the centrifugal pump impeller are listed in Table 1.

_{d}and 0.75Q

_{d}when the rotating speed is maintained, as shown in Figure 2, which can be expressed as follows:

_{d}and 0.25Q

_{d}are the equilibrium and amplitude flow rates, respectively; n/6 is the frequency. The initial phase of the function is zero. Five instantaneous points, including two maximum (1 and 5), one minimum (3), and two equilibrium (2 and 4) flow rates, are selected to investigate the behavior of pressure fluctuation. At this time, the evolution of flow rate includes a dropping stage (instantaneous points from 1 to 3) and a rising stage (instantaneous points from 3 to 5). For comparison, three flow rates (0.25Q

_{d}, 0.5Q

_{d}, 0.75Q

_{d}) during the quasi-steady conditions are also performed.

#### 2.2. Mesh Generation

_{d}was studied. As shown in Figure 4, with the increase of the number of grids, the head of the centrifugal pump impeller gradually increases. After 10 million grids, the head of the centrifugal pump impeller becomes stable. Compared with 14 million grids, the relative error of the head calculated with 12 million grids was less than 0.1%. Considering the efficiency of the calculation process and the minimum requirements of the LES computing grid, the 12 million grids number model was finally selected for follow-up research.

## 3. Numerical Considerations

## 4. Simulation Verification

_{d}and 0.5 Q

_{d}conditions, the head of the impeller under a 0.75 Q

_{d}condition is quite different from the fitted head curve. There are two reasons for this phenomenon. First, Pedersen [5] only conducted experiments on this centrifugal pump impeller under 0.25 Q

_{d}and 1.0 Q

_{d}conditions. The head curve in Figure 7 is the result depicted by Pedersen [5], which is not absolutely accurate. Second, according to the previous numerical calculation results, this impeller has a hump near the 0.7 Q

_{d}condition, so the head of the impeller will be raised in the 0.75 Q

_{d}condition. The distributions of the relative velocity for both the two methods are similar, and present obviously “two-channel” (one of the two adjacent channels is in a stall state and the other is in a non-stall state) phenomenon under a quarter load, as shown in Figure 8left,right. The relationship between stall passage and non-stall passage is central symmetry about the impeller rotating shaft.

_{d}by LDV (laser doppler velocimetry) and PIV (particle image velocimetry) experiments. As shown in Figure 9, at r/R

_{2}= 0.5, in the passages A, the experiment results show that the value of mean radial velocity from the pressure surface to the suction surface gradually increases. Compared with the experiment, while the LES results of this paper are similar to the experimental results, the amplitude of the pressure surface is higher. This remarkable difference is mainly attributed to the influence of pre-rotation. This problem is explained in detail in Byskov’s research results [5]. In passages B, the stall phenomenon is obvious in the experimental velocity curve, which shows that mean radial velocity changes sharply in the passage span at r/R

_{2}= 0.5. The mean radial velocity indicates the reverse flow along the suction side, and the position and size of LES and PIV peaks are similar. Because the peak value of mean radial velocity predicted by LES deviates slightly to the suction side, there is a small reverse vortex on the pressure side.

_{2}= 0.98 is shown in Figure 9. The LES simulation results are consistent with the experimental results. In passages A, LES results show that the radial velocity amplitude of the impeller suction surface is smaller than that observed by Pedersen’s experiment [36]. This may be due to the stronger jet structure on the pressure surface in LES, which limits the exit velocity of the suction surface. Meanwhile, due to the leakage in the experiment, the jet flow of the impeller is reduced. In passages B, the experimental results are consistent with those of LES in this paper, but LES captures stronger impeller outlet reflow. By comparing the results of the experiment and the LES, it can be found that the LES can effectively capture the internal flow of the impeller, but there are differences in the details.

## 5. Results and Discussion

#### 5.1. Pressure Distribution

_{p}) is defined as [37]

_{2}= 0.5). It is indicated that, at all instantaneous flow rates, the pressure coefficient increases gradually with the axial position increase, and there are several low pressure regions at the impeller inlet. However, the difference of the low-pressure regions is significant at different instantaneous flow rates. A low-pressure region is observed at the suction front edge at instantaneous 0.75Q

_{d}, whose phenomena are similar in the six passages. The “two-channel” pattern characteristics appear in the low-pressure region at instantaneous 0.25Q

_{d}. It means that the pressure of one passage is obviously lower than that of another passage, which is one of the stall characteristics. Unlike instantaneous 0.75Q

_{d}and 0.25Q

_{d}, the other instantaneous flow rate includes rising and dropping stages, and those low-pressure characteristics are different. For example, considering the instantaneous 0.5Q

_{d}, the low-pressure region in rising and dropping stages is similar to that at 0.25Q

_{d}and 0.75Q

_{d}, respectively.

_{2}= 0.5). The results show that the maximum value of the relative velocity is mainly concentrated in the impeller inlet and outlet regions. The area of those regions decreases with the instantaneous flow rate decreases. Although the value of the relative velocity is opposite of that for static pressure, similar channel pattern characteristics can be obtained in the region of the impeller inlet. Moreover, these channel pattern characteristics will be developed and evolved in the impeller passage. According to the above analysis, it can be concluded that non-stall is observed on instantaneous 0.75Q

_{d}, whereas the stall will appear at instantaneous 0.5Q

_{d}and 0.25Q

_{d}.

_{2}= 0.5), the distributions of pressure and relative velocity during a quasi-steady condition, including three constant flow rates (0.75Q

_{d}, 0.5Q

_{d}, and 0.25Q

_{d}), are also shown in Figure 10 and Figure 11, respectively. Note that non-stall is observed on 0.75Q

_{d}, whereas the stall will appear at 0.5Q

_{d}and 0.25Q

_{d}. It is indicated that the stall characteristics are not varied by a periodic transient condition under the three flow rates. However, there are still several differences between the transient and quasi-steady conditions, especially at 0.5Q

_{d}. For example, due to the internal flow field being expanded during a transient condition at the dropping stage, the two-channel phenomenon is weaker than that for the quasi-steady condition. An opposite effect is at the rising stage. Therefore, the transient flow has a significant effect on the internal flow in the centrifugal pump impeller, even the stall.

#### 5.2. Pressure Fluctuation in the Impeller Mid-Height

_{d}are shown in Figure 12right. It is shown that, similar to the transient condition under a corresponding instantaneous flow rate, the high-frequency pressure fluctuation at Passage B is larger than that at Passage A. However, there are no obvious periodic characteristics during the quasi-steady condition, especially in the low-frequency pressure fluctuation.

_{Z}, which is the same as the frequency of the flow rate. The second dominant frequency of stall and non-stall passages are 0.084f

_{BPF}(blade passing frequency) and 0.070f

_{BPF}, respectively. However, the amplitude of the frequency of pressure fluctuations for stall passages is significantly larger than that for non-stall passages. For example, the relationship between the amplitude of the second dominant frequency of stall and non-stall passages is 2.5 times. However, there is no frequency with an amplitude greater than 140 during the quasi-steady condition at 0.5Q

_{d}, as shown in Figure 14right. In addition, the amplitude of the first dominant frequency during the transient condition is obviously greater than that during the quasi-steady condition. Moreover, the second dominant frequency and its higher frequencies will be enlarged because of the coupling of the different frequencies. By analyzing the frequency spectrum of the quasi-steady state and transient flow, it can be obtained that the disturbance at the inlet of the centrifugal pump impeller enhances the complexity and intensity of the pressure pulsation in the impeller.

#### 5.3. Pressure Fluctuation in the Impeller Axial Plane

## 6. Discussion

_{d}and 0.25Q

_{d}. The reliability of the LES data was verified by comparing with existing experimental data.

_{d}and 0.25Q

_{d}, respectively. Moreover, the stall phenomenon at instantaneous 0.5Q

_{d}in the rising stage was more significant than that in the dropping stage. The stall passages will not shift with the time development. That is, the symmetry of the stall passage and non-stall passage remains unchanged.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | channel |

B | channel |

C_{p} | pressure coefficient |

C_{r} | radial velocity |

d | blade thickness |

h | height |

H | head |

n | rotational speed |

p | monitor point |

P | static pressure |

P_{i} | monitor point static pressure |

$\overline{P}$ | average static pressure |

Q | flow rate |

Q_{d} | design flow rate |

R_{1} | inlet radius |

R_{2} | outlet radius |

R_{b} | blade curvature radius |

t | time |

T | the time of one cycle of inlet flow change |

U_{2} | circumferential velocity at impeller outlet |

V | relative velocity |

x | coordinate components |

y+ | dimensionless wall distances |

z | section height |

Z | number of blades |

λ_{1} | inlet blade angle |

λ_{2} | outlet blade angle |

## Acronyms

FFT | Fast Fourier Transform |

LES | Large Eddy Simulation |

RANS | Reynolds-Averaged Navier-Stokes |

SST | Shear Stress Transfer |

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**Figure 3.**Grids of centrifugal pump impeller: (

**a**) view of centrifugal pump impeller, (

**b**) blade front view, (

**c**) blade middle view.

**Figure 6.**y+ distribution on the centrifugal pump impeller during a quasi-steady condition. 0.25Q

_{d}(

**left**); 0.50Q

_{d}(

**middle**); 0.75Q

_{d}(

**right**).

**Figure 7.**Performance curve [5] of the centrifugal pump impeller under different quasi-steady conditions with low flow.

**Figure 8.**Relative velocity vectors during a quasi-steady condition at 0.25 Q

_{d}in the impeller mid-height, z/b

_{2}= 0.5, given at radial position of r/R

_{2}= {0.5,0.65,0.75,0.9}. Compared using large eddy simulation (LES) (

**left**) and measured using laser doppler velocimetry (LDV) (

**right**) [5].

**Figure 9.**Blade-to-blade distributions of the mean radial velocity <C

_{r}>/U

_{2}measured with an experiment (

**right**) [36] and LES simulation (

**left**) at 0.25 Q

_{d}. r/R

_{2}= 0.50 (

**up**); r/R

_{2}= 0.98 (

**down**).

**Figure 10.**Distributions of the pressure coefficient under a transient condition at the impeller mid-height, z/b

_{2}= 0.5.

**Figure 12.**Distributions of pressure fluctuation at the impeller inlet in the impeller mid-height, z/b

_{2}= 0.5: (

**left**) transient condition; (

**right**) quasi-steady condition at two-quarter load.

**Figure 13.**Distributions of pressure fluctuation during the transient condition in the impeller mid-height, z/b

_{2}= 0.5, given at radial positions of r/R

_{2}= {0.50, 0.58, 0.66, 0.74, 0.82, 0.90}: (

**left**) suction surface, (

**right**) pressure surface; (

**top**) non-stall passages; and (

**bottom**) stall passages.

**Figure 14.**Distributions of pressure frequency spectra at the impeller inlet in the impeller mid-height, z/b

_{2}= 0.5: (

**left**) transient condition; (

**right**) quasi-steady condition at two-quarter load.

**Figure 15.**Monitoring points in the impeller axial plane, x = 0.0, given at radial positions of r/R = {0.25, 0.50, 0.7, 0.9}.

**Figure 16.**Distributions of pressure fluctuation in the impeller axial plane, x = 0.0, given at radial positions of r/R

_{2}= {0.25, 0.50, 0.7, 0.9}: (

**left**) transient condition; (

**right**) quasi-steady condition for two quarter-load.

Geometry | Symbol | Size |
---|---|---|

Number of blades | Z | 6 |

Inlet radius | R_{1} | 35.5 mm |

Outlet radius | R_{2} | 95 mm |

Inlet height | h_{1} | 13.8 mm |

Outlet height | h_{2} | 5.8 mm |

Blade thickness | d | 3 mm |

Blade curvature radius | R_{b} | 70 mm |

Inlet blade angle | λ_{1} | 19.7° |

Outlet blade angle | λ_{2} | 18.4° |

Boundary Conditions | |

Inflow | Inlet-velocity |

Outflow | Outlet-pressure |

Wall | No-slip and smooth wall |

Numerical Setup | |

Absolute criteria of residual | 10^{-6} |

Number of time steps | 8640 |

Time step | 0.00023(s) |

Max iterations/Time step | 10 |

Turbulence intensity (inlet) | 5% |

Turbulence viscosity rate (inlet) | 10% |

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**MDPI and ACS Style**

Kuang, R.; Chen, X.; Zhang, Z.; Zhu, Z.; Li, Y.
Large Eddy Simulation of Periodic Transient Pressure Fluctuation in a Centrifugal Pump Impeller at Low Flow Rate. *Symmetry* **2021**, *13*, 311.
https://doi.org/10.3390/sym13020311

**AMA Style**

Kuang R, Chen X, Zhang Z, Zhu Z, Li Y.
Large Eddy Simulation of Periodic Transient Pressure Fluctuation in a Centrifugal Pump Impeller at Low Flow Rate. *Symmetry*. 2021; 13(2):311.
https://doi.org/10.3390/sym13020311

**Chicago/Turabian Style**

Kuang, Renfei, Xiaoping Chen, Zhiming Zhang, Zuchao Zhu, and Yu Li.
2021. "Large Eddy Simulation of Periodic Transient Pressure Fluctuation in a Centrifugal Pump Impeller at Low Flow Rate" *Symmetry* 13, no. 2: 311.
https://doi.org/10.3390/sym13020311