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To investigate the unsteady flow characteristics and their influence mechanism in the volute of centrifugal pump, the Reynolds time-averaged N-S equation, RNG

The internal flow of the centrifugal pump is a complex three-dimensional unsteady turbulent flow, often accompanied by flow separation, cavitation, hydraulic vibration, etc., which affect the stability of flow in the centrifugal pump. Among them, the rotor-stator interaction will produce periodic pressure fluctuation, which will lead to the intensification of pump vibration, noise enhancement, and performance decay, seriously affecting the safe and stable operation of the unit. Especially in the tongue area, since the fluid flow from the impeller outlet impacts the volute tongue, the rotor-stator interaction at this place should be the strongest, which is one of the key areas for the study of unsteady flow in centrifugal pumps. For instance, Kelder [

Dynamic mode decomposition (DMD) is a data-driven algorithm for extracting dynamic information from unsteady experimental measurements or numerical simulations. For such simulations and experiments provide large-scale data, it is necessary to understand essential phenomena from the data provided. So it can be used to analyze the main characteristics of complex unsteady flows or to establish low-order flow-field dynamics models. The DMD method was first proposed by Schmid [

The object of this paper is a conventional single-stage single-suction centrifugal pump. The main parameters of the pump are listed in Table

Main design parameters of the centrifugal pump.

Design parameter | Design value |

| |

Nominal flow rate | 750m^{3}/h |

Rated head | 36m |

Rated rotational speed | 1710 r/min |

Number of blades | 6 |

Diameter of impeller inlet _{1} | 0.245m |

Diameter of impeller outlet _{2} | 0.325m |

Blade wrap angle | 120° |

Blade outlet angle _{2} | 27° |

Base circle diameter of volute _{3} | 0.34m |

For the entire computational domain (composed of suction pipe, impeller, volute, and outlet pipe), structured hexahedral mesh is generated using Ansys-ICEM mesh generation tool. The overall calculation domain is shown in Figure ^{−5}.

Structured grid of the centrifugal pump.

As shown in Figure

Test stand.

External characteristic curve of the centrifugal pump.

Dynamic mode decomposition is a method for extracting the mode in a flow based on flow-field snapshots, so that the flow structure can be accurately described. For the linear flow, the DMD method can extract the modes that can characterize the global flow stability. For the nonlinear flow, the DMD method can describe the flow structure in which the observations (such as velocity and pressure) dominate. The mathematical derivation process of it is introduced below.

Suppose there is a set of observation data matrices that vary with time:

where N is the total number of snapshots of the flow field, and the column vector_{i} is the data of the

As the acquired snapshot data increases, we can further assume that the vector formed by the snapshot data eventually tends to be linearized. Thus, the last flow-field snapshot can be represented as a linear combination of all previous snapshots:

From formula (

From formula (

From the previous analysis, we can see that the eigenvalues of matrix

Finally, the DMD modes can be obtained as follows:

It should be pointed out that the eigenvalue_{i} contains the information of the mode Φ_{i}. When the eigenvalues are expressed in the complex plane, the modes on the unit circle of the complex plane are relatively steady, while the modes whose eigenvalues are not on the unit circle are unsteady.

The corresponding frequency_{i} and growth rate _{i} can be mapped to a subspace_{i}.

By simple transformation there are

So, from the previous derivation, a snapshot at any time instant

From the DMD mode definition formula (_{i} denotes the amplitude of the_{1}.

Substituting formula (

Then its snapshot sequences can be expressed as

In the unsteady calculation, the spectrum analysis is performed to obtain the characteristic frequency of the unsteady flow inside the centrifugal pump. The dimensionless pressure coefficient_{p} is introduced to describe the pressure fluctuation characteristics of each monitoring point. The expression is as follows:

where_{i} is the static pressure value of the monitoring point at a certain time, _{ave} is the average value of static pressure in one cycle, _{2} is the circumferential velocity at the impeller outlet, m/s. The monitoring points_{1},_{2},_{3}, and_{4} are arranged inside the volute as shown in Figure _{1} and_{3} are set to monitor pressure fluctuation characteristics of fluid flow in volute tongue and baffle entrance region;_{2} is set to monitor the pressure fluctuation characteristics of fluid flow at the volute inlet except for the tongue and the baffle entrance region; and P4 is set to monitor the pressure fluctuation characteristics at the volute outlet. By comparing the pressure fluctuation characteristics at these four points, the main factors causing unsteady flow in volute are explored.

Locations of monitoring points.

Figure _{1}=171Hz), and the pressure amplitude at 2_{1}=342HZ) is also prominent. Figure _{1}=169.59Hz and the 2_{1}=339.17HZ. Comparing the results of spectrum analysis for the two operating conditions, it can be seen that the dominant pressure fluctuation main frequency and the high amplitude pressure fluctuation frequency are basically consistent. However, compared with the nominal flow-rate condition, it produces a certain low frequency fluctuation at less than the blade passing frequency in the low flow-rate condition.

Spectrum characteristic analysis for the low flow-rate condition.

Spectrum characteristic analysis for the nominal flow-rate condition.

Based on the previous CFD unsteady calculation results, DMD analysis was performed on 120 flow-field speed snapshots with time interval_{j} is defined as the correlation coefficient._{j} is the modal amplitude and_{j} is the eigenvalue and

Distribution of DMD eigenvalues for the low flow-rate condition.

Distribution of correlation coefficients for the low flow-rate condition.

Figure

DMD velocity mode contours for the low flow-rate condition.

First-order mode

Second-order mode

Third-order mode

Fourth-order mode

Fifth-order mode

Sixth-order mode

Figures

Figures

Distribution of DMD eigenvalues for the nominal flow-rate condition.

Distribution of correlation coefficients for the nominal flow-rate condition.

Figure

DMD velocity mode contours for the nominal flow-rate condition.

First-order mode

Second-order mode

Third-order mode

Fourth order mode

Fifth order mode

Sixth order mode

Figures

Figures

In order to further observe the effect of dynamic mode decomposition on the extraction of flow-field characteristics in volute of centrifugal pump, a reduced order model of unsteady flow field in volute was established based on formula (

Transient velocity contour inside the volute for the low flow-rate condition.

Original flow field at T/2

Reconstructed flow field at T/2

Transient velocity contour inside the volute for the nominal flow-rate condition.

Original flow field at T/2

Reconstructed flow field at T/2

Since the first-order mode is an average flow mode, it does not change during the entire rotating period of the centrifugal pump. In order to further study the unsteady flow structure in a certain mode, the mode can be superimposed on the first-order average flow mode to observe the oscillation law of the mode. The second-order mode caused by rotor-stator interaction is superimposed on the average flow mode to reconstruct the flow field of a single mode, so as to observe the variation of the unsteady structure with time in the mode.

Figure

Reconstructed velocity contour with second-order mode for the nominal flow-rate condition.

In this paper, the internal flow of the centrifugal pump is numerically simulated, and the velocity field of the mid-span section of the centrifugal pump volute is analyzed by dynamic mode decomposition for the low flow-rate condition and the nominal flow-rate condition. The characteristic frequency of the flow field and the flow mode of different frequencies are extracted, and the reduced order analysis of the original flow field is realized. In the extracted flow modes of different frequencies, the first-order mode is the flow structures occupying the dominant position of the original flow field for the low flow-rate condition and the nominal flow-rate condition. The second- and third-order modes are the main oscillation modes of the original flow field, and the two modes’ characteristic frequency is consistent with the blade passing frequency and 2x blade passing frequency obtained by FFT, which proves that the rotor-stator interaction between the impeller and the volute is the main reason for the unsteady flow in the volute. Next, comparing the mode velocity contours of each order for the condition of low flow rate and nominal flow rate, when the flow rate of the centrifugal pump is reduced, the existence of the tongue and the diaphragm will have a certain inhibitory effect on the unstable flow structure inside the volute, but it is more prone to occur in low frequency unsteady flow. Finally, the flow field of a single mode is reconstructed by superimposing the second-order mode to the first-order mode under the rated condition. And the results show that six high-speed fluid clusters dominated at the volute inlet rotate periodically with the blade. Generally speaking, the combination of DMD method and CFD unsteady calculation method is beneficial to understand the flow mechanism in the volute of centrifugal pump by extracting the coherent structure of unsteady flow.

Nominal rate flow (m^{3}/h)

Pump head for nominal rate flow (m)

Nominal rotational speed (r/min)

Time step (s)

Number of blades

_{1}:

Diameter of impeller inlet (m)

_{2}:

Diameter of impeller outlet (m)

Blade wrap angle

_{2}:

Blade outlet angle

_{3}:

Base circle diameter of volute (m)

Pressure coefficient

Static pressure value of the monitoring point at a certain time (Pa)

Average value of static pressure in one cycle (Pa)

Circumferential velocity at the impeller outlet (m/s)

Observation data matrix

Total number of snapshots

Data of the i th snapshot

Linear transformation matrix

Residual vector

Companion matrix

Unitary matrix

Diagonal matrix

Eigenvalues of matrix S

Eigenvectors of matrix S

Corresponding frequency

Growth rate

_{i}:

DMD modes

A subspace

Matrix with

Matrix with

Amplitude of the i th mode

Correlation coefficient

Mode Frobenius norm.

Dynamic mode decomposition

Fast Fourier transformation

Computational fluid dynamics.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors have approved the final version of the manuscript submitted. There is no financial or personal interest.

The authors are grateful to the financial support from the National Natural Science Foundation of China (Research Project No. 51866009).