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TECHNICAL PAPERS

# Response of Backflow to Flow Rate Fluctuations

[+] Author and Article Information
Xiangyu Qiao, Yoshinobu Tsujimoto

Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

Hironori Horiguchi

Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japanhoriguti@me.es.osaka-u.ac.jp

1

Author to whom correspondence should be addressed.

J. Fluids Eng 129(3), 350-358 (Aug 31, 2006) (9 pages) doi:10.1115/1.2427081 History: Received November 07, 2005; Revised August 31, 2006

## Abstract

The response of backflow at the inlet of an inducer to the flow rate fluctuation is studied by using three-dimensional numerical calculations based on the $k-ϵ$ turbulence model for the discussion of its effect on cavitation instabilities. It is first shown that the size of the backflow region can be correlated with the angular momentum in the upstream and the phase of the backflow significantly delays behind the quasi-steady response even at a very low frequency. It is then shown that the conservation relation of angular momentum is satisfied with minor effects of the shear stress on the boundary. The supply of the angular momentum by the negative flow is shown to be quasi-steady due to the fact that the pressure difference across the blade causing the backflow is quasi-steady at those frequencies examined. A response function of the angular momentum in the upstream to flow rate fluctuation is derived from the balance of the angular momentum and the results of the numerical calculations. This clearly shows that the backflow responds to the flow rate fluctuation as a first-order lag element. The effects of the backflow cavitation on cavitation instabilities are discussed assuming that the delay of cavity development is much smaller than the delay of the backflow. It was found that the backflow cavitation would destabilize low frequency disturbances due to the effects of the positive mass flow gain factor but stabilize high frequency disturbances due to the effect of the cavitation compliance.

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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 1

Inducer geometry

Figure 2

Computational grid

Figure 3

Comparison of the results by steady calculation and experimental results: (a) Noncavitating performance (uncertainty in ϕ=±0.005 and in ψ=±0.02 in the experimental results) and (b) location of the upstream edge of the backflow region (Uncertainty in z∕Dt=±0.05 in the experimental results)

Figure 4

Flow field at the inlet of the inducer, ϕ=0.078: (a) Initial flow field and (b) after one cycle of oscillation with f∕fn=0.125

Figure 5

Inlet flow rate fluctuation with f∕fn=0.125

Figure 6

Backflow region and velocity vector at the instants shown in Fig. 5, with f∕fn=0.125: (a)ϕ=0.078 (increasing), (b)ϕ=0.088 (maximum), (c)ϕ=0.078 (decreasing), (d)ϕ=0.068 (minimum), and (e)ϕ=0.078 (increasing)

Figure 7

Angular momentum (AM: Angular momentum in the upstream, AMB: Angular momentum supply by the backflow, AMN: Angular momentum removal by the normal flow): (a)f∕fn=0.0625, (b)f∕fn=0.125, and (c)f∕fn=0.25

Figure 8

Location of upstream edge of the backflow region in the flow rate fluctuation (uncertainty in z∕Dt=±0.05 in the experimental results): (a)f∕fn=0.0625, (b)f∕fn=0.125, and (c)f∕fn=0.25

Figure 9

Angular momentum conservation relation: (a)f∕fn=0.0625, (b)f∕fn=0.125, and (c)f∕fn=0.25

Figure 10

The region with negative axial flow, ϕ=0.078: (a) Negative axial velocity and (b) velocity field in the meridional plane, θ=−10deg

Figure 11

The instantaneous pressure distribution along the blade for ϕ=0.068, 0.078, and 0.088 at r∕R=0.99, with f∕fn=0.25

Figure 12

Unsteady pressure performance of the impeller (uncertainty in ϕ=±0.005 and in ψ=±0.02 in the experimental results): (a)f∕fn=0.0625, (b)f∕fn=0.125, and (c)f∕fn=0.25

Figure 13

Response function of the backflow

Figure 14

Phase lag of the backflow (11)

Figure 15

Resonant frequency f0 and the critical frequency f90(11)

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