Modulation effect on rotor-stator interaction subjected to fluctuating rotation speed in a centrifugal pump

Figures & data

Figure 1. Pump model.

Table 1. Geometry and performance parameters.

Item	Value
Blade numbers of impeller	6
Pump design head Hr (m)	33.6
Design flow rate Qr (m^3/h)	16
Outlet width of impeller b1 (mm)	7
Diameter of inlet D1 (mm)	50
Diameter of outlet D2 (mm)	32
Outside diameter of impeller D3 (mm)	162
Base circle diameter of volute D4 (mm)	170
Design rotational speed ω0 (rad/s)	100π
Rotational frequency fn (Hz)	50
Type number Kr	0.271

Figure 2. Mesh of fluid domain: (a) Boundary layers; (b) Impeller.

Figure 3. Independence analysis: (a) Grid number; (b) Time-step.

Figure 4. y plus contour.

Figure 5. Layout of the closed test system. 1-water reservoir, 2-flowmeter, 3-piezoresistive pressure sensors, 4-pump, 5-piezoelectric presser sensor, 6-valve.

Figure 6. Comparison of experimental and simulated results: (a) pump performance; (b) pressure fluctuation.

Table

Algorithm 1: Implementation process of VMD
Initialize $\left\{{\hat{u}}_k^{\text{1}}\right\}$, $\left\{{\omega }_k^{\text{1}}\right\}$, ${\hat{\lambda }}^{\text{1}}$, $n\leftarrow \text{0}$
repeat
$n\leftarrow n+\text{1}$
for  k = 1: K  do
Update ${\hat{u}}_k$ for all $\omega \ge \text{0}$:
${\displaystyle {\hat{u}}_k^{n+\text{1}}\left(\omega \right)\leftarrow \frac{\hat{f}\left(\omega \right)-{\sum }_{i<k}{\hat{u}}_i^{n+\text{1}}\left(\omega \right)-{\sum }_{i>k}{\hat{u}}_i^n\left(\omega \right)+\frac{{\hat{\lambda }}^n\left(\omega \right)}{\text{2}}}{\text{1}+\text{2}\alpha {\left(\omega -{\omega }_k^n\right)}^{\text{2}}}}$	(10)
Update ${\omega }_k$:
${\displaystyle {\omega }_k^{n+\text{1}}\leftarrow \frac{{\int }_{\text{0}}^{\mathrm{\infty }}\omega {\left|{\hat{u}}_k^{n+\text{1}}\left(\omega \right)\right|}^{\text{2}}d\omega }{{\int }_{\text{0}}^{\mathrm{\infty }}{\left|{\hat{u}}_k^{n+\text{1}}\left(\omega \right)\right|}^{\text{2}}d\omega }}$	(11)
end for
Dual ascent for all
${\displaystyle \omega \ge \text{0}}$: ${\displaystyle {\hat{\lambda }}^{n+\text{1}}\left(\omega \right)\leftarrow {\hat{\lambda }}^n\left(\omega \right)+\tau \left(\hat{f}\left(\omega \right)-\sum _k{\hat{u}}_k^{n+\text{1}}\left(\omega \right)\right)}$	(12)
until convergence: ${\sum }_k{\Vert {\hat{\mathrm{u}}}_k^{n+\text{1}}-{\hat{\mathrm{u}}}_k^n\Vert }_{\text{2}}^{\text{2}}/\Vert {\hat{\mathrm{u}}}_k^n{\Vert }_{\text{2}}^{\text{2}}<ϵ$.

Figure 7. Flowchart of quasi–bivariate VMD.

Figure 8. Effect of fluctuating rotation speed on radial force: (a) orbit diagram; (b) frequency spectra.

Figure 9. Effect of fluctuating rotation speed on pressure pulsation: (a) time domain; (b) frequency domain.

Figure 10. Effect of fluctuating rotation speed on pressure pulsation and radial force intensity.

Figure 11. Time history of the pressure pulsation around the impeller outlet: (a) stationary rotation speed; (b) fluctuating rotation speed.

Figure 12. Time history of the velocity pulsation around the impeller outlet: (a) stationary rotation speed; (b) fluctuating rotation speed.

Figure 13. Absolute velocity contour on pump mid-plane.

Figure 14. Vortex structure comparisons in different moments.

Figure 15. Decomposition results of pressure pulsation using VMD: (a) stationary rotation speed; (b) fluctuating rotation speed.

Figure 16. Decomposition results of pressure pulsation in the frequency domain: (a) stationary rotation speed; (b) fluctuating rotation speed.

Figure 17. Hilbert spectrum at fluctuating rotation speed.

Figure 18. Effect of fluctuating rotation speed on instantaneous amplitude and frequency.

Figure 19. Marginal spectrum at fluctuating rotation speeds.

Figure 20. Time history of pressure pulsation under BPF: (a) stationary rotation speed; (b) fluctuating rotation speed.

Figure 21. Time history of pressure pulsation under FF.

Table

b1	=	outlet width of impeller, mm
Cp	=	pressure coefficient
CF	=	force coefficient
Cv	=	velocity coefficient
$C_{\text{p}}^{\ast }$	=	pressure pulsation intensity
$C_{\text{F}}^{\ast }$	=	force pulsation intensity
${\overline{C}}_{\text{p}}$	=	average pressure coefficient
${\overline{C}}_{\text{F}}$	=	average force coefficient
$C_{\text{P0}}^{\ast }$	=	pressure pulsation intensity at stationary rotation speed
$C_{\text{F0}}^{\ast }$	=	force pulsation intensity at stationary rotation speed
D1	=	diameter of inlet, mm
D2	=	diameter of outlet, mm
D3	=	outside diameter of impeller, mm
D4	=	base circle diameter of volute, mm
F	=	force, N
fn	=	rotational frequency, Hz
f1	=	fluctuation frequency of the rotating speed, Hz
g	=	gravity, m^2/s
H	=	head, m
Hr	=	pump design head, m
Kr	=	type number
K	=	mode number
N	=	signal length
P	=	power, W
p	=	pressure, Pa
Q	=	flow rate, m^3/s
Qr	=	design flow rate, m^3/h
T	=	impeller rotation period
t	=	time, s
v	=	velocity, m/s
y	=	vertical height of the first boundary layer to the wall, m

Data availability statement

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