Many contemporary data sources in a variety of domains can naturally be represented as fully-dynamic streaming graphs. How to design an efficient online streaming clustering algorithm on such graphs is of great concern. However, existing clustering approaches are inappropriate for this specific task because: (1) static clustering approaches require expensive computational cost to cluster the graph for each update and (2) the existing streaming clustering neither could fully support insertion/deletion of edges nor take temporal information into account. To tackle these issues, in this work, firstly we propose an appropriate streaming clustering model and design two new core components: streaming reservoir and cluster manager. Then we present an evolution-aware bounded-size clustering algorithm to handle the edge additions/deletions. It requires the clusters to satisfy the maximum cluster-size constraint, and maintains the recency of edges in the temporal sequence and gives high priority to the recent edges in each cluster. The experimental results show that the proposed BSC algorithm outperforms current online algorithms and is capable to keep track of the evolution of graphs. Furthermore, it obtains almost one order of magnitude higher throughput than the state-of-the-art algorithms.