The harmonic balance method was used for the flow simulation in a centrifugal pump. Independence studies have been done to choose proper number of harmonic modes and inlet eddy viscosity ratio value. The results from harmonic balance method show good agreements with PIV experiments and unsteady calculation results (which is based on the dual time stepping method) for the predicted head and the phaseaveraged velocity. A detailed analysis of the flow fields at different flow rates shows that the flow rate has an evident influence on the flow fields. At 0.6
Flow in centrifugal pumps produces a complex threedimensional phenomenon involving turbulence, secondary flows, separations, and so forth (Brennen [
A number of researchers have contributed to the understanding of complex unsteady phenomena by using CFD [
Complex geometry: structured mesh is not easy to generate and difficult to converge; a great number of cells are needed to get more details on the curved blades.
The interaction between impeller and volute requires an unsteady solution process to calculate the time dependent flow field. In addition, the blade position with respect to the volute tongue must be taken into account. This can be partially accomplished in a quasisteady way: calculating the steady solution with different grid positions. Nonetheless, it is always much better if the code is able to perform an unsteady flow calculation at the same time that slides the impeller grid at each time step.
As computer power increases, increasingly sophisticated computer fluid dynamic models of unsteady flows about turbo machines have been developed. Some researchers have performed timeaccurate Large Eddy Simulations (LES) of unsteady flows about pumps [
To analyze the nonlinear hydrodynamics of pump impeller, in the turbo machinery community, the need for efficient unsteady flow solvers led investigators to develop efficient mixed time and frequency domain techniques. Hall et al. [
A number of investigators have used this technique to solve interesting fluid dynamic problems. Such problems include the vortex shedding of a circular cylinder [
Consider the threedimensional unsteady NavierStokes equations. In Cartesian coordinates, these equations are given by
The Fourier series is truncated to a small number of harmonics
We denote this vector of flow variables by
The Fourier series is then substituted back into the NavierStokes equations and rearranged into the individual harmonics. Consider
This gives a system of
Using this transformation (discrete inverse Fourier transformation) and substituting this into the equations in the frequency domain give
This gives a system of
The method has some similarities to the dual time step method, used by Davis et al. [
The harmonic balance method is implemented into the inhouse code SPARC [
In order to solve for these numerically very stiff equations we are using the preconditioning method proposed by Shouqi et al. [
The shrouded impeller having backward curved blades with a vaneless single tongue volute obtained from the optimization method described in [
Main characteristics of the pump.
Characteristics  Value 

Rated flow, 
6.3 m^{3}/h 
Rated head, 
8 m 
Rotating speed, 
1450 rpm 
Impeller inlet diameter, 
50 mm 
Impeller outlet diameter, 
160 mm 
Volute inlet diameter, 
172 mm 
Blade outlet width, 
6 mm 
Blade outlet angle, 
27° 
Blades number, 
6 
Flow coefficient 
0.133 
Head coefficient 
0.0028 
Specific speed, 
47 
Re at the impeller inlet 

Re in the impeller  ~10^{6} 
Due to the asymmetric volute, the flow in one blade flow passage is not periodic, so the computational domain includes a rotational zone (the whole impeller and the flow of inlet tube, while the wall of the inlet tube should be set as fixed wall) and stationary zones (volute and outlet tube). For numerical stability reasons, and to minimize boundary condition effects, this computational domain is extended upstream and downstream. The inlet tube with the length of five times the diameter is also transparent, and the outlet tube is extended to the length about three times the diameter of the outlet tube, which are corresponding with the position of pressure sensors, respectively. The CAD of this configuration, as well as the boundary conditions, is shown in Figure
Structured H–O–H grid topology cells are generated to define the impeller, the inlet tube, and the volute (together with the outlet tube). The geometry has been simplified in the front wearing ring; that means that the leakage flow through it is not considered. A 3level full multigrid technique was used, the mesh refinement zones are defined near the blades inlet and the volute tongue, and the grids numbers are shown in Table
Mass flow rate condition at the inlet and static pressure outlet boundary conditions has been used, according to the values measured in the experiment (Figure
The calculations were done using our inhouse developed code called SPARC. The NS equations were solved, respectively, with the HB method and the dual time stepping method. We have been using 2 harmonic modes giving
Pump model.
Sketch of the open test rig.
Flow and boundary condition domains.
The 2nd level grid: 982784 points.
The turbulence was modeled with the SpalartAllmaras oneequation model. The harmonic balance calculations have been done with a threestage RungeKutta scheme (CFL number of about 2.2) using full multigrid, implicit residual smoothing, and local time stepping to accelerate the convergence rate considerably. For the unsteady calculation, a secondorder accurate dual time stepping approach was used. All convergence acceleration techniques used for the HB method were also applied as well. The physical time step used here was
To select an economic harmonics value is considered next. 1, 2, and 3 harmonic modes were investigated based on the 2nd level mesh. The calculated mean head with different harmonic modes and their CPU time is shown in Table
The predicted head for different harmonics.
Calculation methods  HB_1 mode  HB_2 modes  HB_3 modes  Dual time stepping  Experiment 

Time consuming  8 h  12 h  16 h  ~14 d  — 
Average head (m)  7.517  7.570  7.694  7.748  7.75 
Relative error  −3.01%  −2.32%  −0.73%  0  — 
Mesh information.
Multigrid levels  Grid number (3rd level) 



1st level  2nd level  3rd level  Impeller  Inlet  Volute  Total  Moving wall  wall  
Original  122,848  982,784  7,862,272  3,084,288  1,945,600  2,832,384  7,862,272  15–100  >100 
Refined  231,888  1,855,104  14,840,832  8,552,448  1,704,960  4,583,424  14,840,832  1–10  <20 
To verify the results from HB method, we compared the absolute velocity inside the volute with the results from unsteady calculation and PIV. The absolute velocity is compared at the same radius (
Absolute velocity results from different HM modes (for the 2nd level mesh) and PIV.
The arc line
The mean absolute velocity results
The dimensionless velocity is approximately 0.5 in a large part of the channel. Close to the volute tongue (angle about 20°), there is a zone with low absolute velocity. The results from the PIV test are uniform compared with those from CFD. With just one mode we can get similar results with PIV, while with the increasing of the number of modes, the results are converging toward that of the unsteady calculation.
The difference between the unsteady calculation and the harmonic balance method is largest in the range of the angle between 80° and 120°. There the unsteady calculation shows a larger dimensionless velocity than the harmonic balance one. Since the mesh, the numerical scheme, and the turbulence model used are the same, only the number of modes used for the comparison makes the difference. In the case of a pump with a spiral volute, we must calculate the whole impeller and volute. That means that the time slices used by the harmonic balance have to be equally distributed around 360°. When averaging these results for the harmonic balance method we have only a small number of solutions available compared with a large number in the case of dual time stepping method. Therefore, we believe that increasing the number of harmonics will give a better prediction for the velocity distribution.
From the calculations, we got the
The number of grid points of the refined mesh is shown in Table
The number of the cells inside the boundary layer is sufficient according to our experience, it is confirmed by the predicted results for the pump performance, and so it allows the analysis of the main flow phenomena involved.
One mode was chosen to verify the mesh independence, as shown in Figure
Head value under different mesh number.
The ratio
Figure
The predicted head compared with the test results.
In Figures
Dimensionless absolute velocity
The contours of absolute velocities at different flow rates for the first position are shown in Figure
Absolute velocity at different flow rates.
Figure
Contour of relative velocity and their visualization at different flow rates and position of tongue.
Eddy viscosity ratio is shown in Figure
Contour of eddy viscosity ratio at different flow rates.
Harmonic balance method provides flow information at
Absolute velocity distribution at 0.6
Figure
Relative velocity and its visualization at different positions with position of tongue.
0.6
0.4
In this work, harmonic balance method has been applied to a low specific speed radial volute pump to predict the unsteady flow fields. The following conclusions can be drawn.
Some works were done investigating mode independence of the harmonic balance method. The second finest mesh and suitable turbulence boundary conditions have been chosen for the investigation of the performance curve. It resamples a good compromise between computational effort and accuracy.
The CFD results show good agreements with PIV experiments for the predicted head.
A detailed analysis of the flow fields at different flow rates shows that the flow rate has a very evident influence on the flow fields. At 0.6
The flow fields at different positions at 0.6
The harmonic balance method predicts very similar flow fields like an unsteady simulation but being more than ten times faster.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research has been done while the second author visited the ISTM of KIT, Germany. She has been supported by the National Natural Science Foundation of China (no. 51009072) and the Foundation for senior person with ability in Jiangsu University of China (no. 08JDG040), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.