There always appear unsteady characteristics during start-up periods of pumps, which can lead to instability of the entire system. However, the lack of a method for quickly and accurately predicting pump start-up performance makes it difficult to analyze the performance of the overall system during the period. To this end, a theoretical model to predict pump transient performance under fast start-up conditions is established in the present study. The theoretical prediction model for pump steady performance is built based on loss modeling first. Then, the balance between pump transient head and pipeline system is considered to determine transient pump performance. A time stepping algorithm is proposed to solve transient pump performance during start-up periods. The established theoretical model and the corresponding time stepping algorithm are applied to a mixed flow pump under fast start-up conditions with various acceleration time. The predicted evolution of pump head shows good agreement with experimental measurements, and average relative errors are within 10% for both steady and transient conditions. In addition, the theoretical model is applied to analyze the transient performance of oscillating curves and impact head. The mechanism for predicted oscillating results and the relation between peak impact head and the acceleration of rotation speed are revealed.

Bubbles-in-chain is a two-phase fluid flow phenomenon in natural environments and engineering practices with unique bubble–bubble interactions. The interactions among bubbles lead to a periodically varying local drag coefficient, which has been rarely studied. Here, we report a numerical investigation of rising bubbles in a chain using large eddy simulation and coupled volume-of-fluid and level-set methods to examine the local drag coefficient and its predicting models. We first validated the bubble diameter and rise velocity with an experimental dataset. The local drag coefficient was then calculated and compared with the traditional Weber number based model. We propose a modified Weber number model with a trigonometric relation, defining the upper and lower limits of the drag coefficient as a result of bubble–bubble interactions. With the proposed model, the predicted drag coefficient shows an acceptable accuracy with errors less than 7%. The resolved flow field in the bubble wake confirms the correlation among the drag coefficient, wake velocity, and rising velocity. This mechanism provides a stable and periodical variation of the drag coefficient in the bubbles-in-chain system.