We have created few document sets in a document library in my site. Within the same site we have a enterprise search center subsite. I tried to search a document set in search center but nothing is shown. But a document in the document set is displaying when searched.
Is there any limitation for document set for not to come under search results?
First, make sure that Search Service App and related accounts have "full read" permission with proper configuration on the site & document library content (as described here).
Document sets share common metadata amongst the documents in the container and there are no limitations (for crawling, document types, etc). After a search crawl, in the Search Results page you can refine the filters on the results to get the document type, fields, managed types (shared metadata), etc.
Another approach would be to search directly within the document sets - as described in this blog post: SharePoint 2010 + Search within document sets.
