With the rapid growth of manuscript submissions, finding eligible reviewers for every submission has become a heavy task. Recommender systems are powerful tools developed in computer science and information science to deal with this problem. However, most existing approaches resort to text mining techniques to match manuscripts with potential reviewers, which require high-quality textual information to perform well. In this paper, we propose a reviewer recommendation algorithm based on a network diffusion process on a scholar-paper multilayer network, with no requirement for textual information. The network incorporates the relationship of scholar-paper pairs, the collaboration among scholars, and the bibliographic coupling among papers. Experimental results show that our proposed algorithm outperforms other state-of-the-art recommendation methods that use graph random walk and matrix factorization and methods that use machine learning and natural language processing, with improvements of over 7.62% in Recall, 5.66% in Hit Rate, and 47.53% in Ranking Score. Our work sheds light on the effectiveness of multilayer network diffusion-based methods in the reviewer recommendation problem, which will help to facilitate the peer-review process and promote information retrieval research in other practical scenes.