Outsourcing storage has emerged as an effective solution to manage the increasing volume of data. With the popularity of pay-as-you-go payment models in outsourcing storage, data auditing schemes that prioritize timeliness can be valuable evidence for elastic bill settlement. Unfortunately, existing data auditing schemes do not sufficiently consider timeliness during auditing. Furthermore, practical data auditing schemes should have the capability to check the integrity of scalable data. In this paper, we propose a blockchain-based dynamic data auditing scheme with strong timeliness to ensure that data stored in outsourcing storage systems remain intact. Our scheme encapsulates timestamps into homomorphic verifiable tags to simultaneously check data integrity and timestamp validity. To achieve dynamicity, we utilize the Merkle hash tree to store the tags, allowing for block-level dynamic operations. Additionally, by leveraging the transparency, non-repudiation, and tamper resistance of blockchain technology, we design a blockchain-based data auditing framework to prevent malicious behavior from all entities. We then formally prove the soundness and privacy of our scheme. Finally, we conduct theoretical analysis and experimental evaluation to demonstrate that the performance of our scheme is of acceptable efficiency to existing works in terms of computation cost, communication overhead, and storage overhead.