The study of bubble swarm distributions in a direct-contact heat exchanger was addressed both theoretically and experimentally. But recently developed approach based on uniform design for measuring bubbles uniformity and mixing efficiency within rectangular or square region cannot be used to characterize the mixture homogeneity and mixing time within circular region. This paper discussed the difference between rectangular or square region and circular region. A quantitative metric of mixing uniformity is used to characterize the bubble patterns within circular region. Simulation studies are also used to illustrate the proposed methodologies while the experimental case considered demonstrate that the proposed approach evaluated the experimental data rather well. The space-time features of the mixing transient have been successfully derived when the bubble swarms or other objects distribute in a circular region. This work brings new insights to evaluate mixing quality of different systems practically, which has a high degree of accuracy (the percentage of enhancing achieved is about 40%). (C) 2017 Elsevier Ltd. All rights reserved.