I was waiting for google to crawl my website automatically without out 
having to submit a site map. It never happened so. I was wondering how 
long does it take for google to index a website once site map is submitted. 


Here are a few answers i found on the web. 

New website with no or few external links and is not updated frequently: 
6-12 months. 

Popular site with lots of links and has been around for years and is updated on 
a regular bases: Indexed Everyday or several times a day (news sites).

Tip: Try getting as many links as possible. Even start dropping a link to your site 
sites like Yahoo answers will help

If you can, link your new website from another site already in the google index.


creating a google site map, it may or may not help you get listed faster but it won't hurt.
https://www.google.com/webmasters/sitemaps/docs/en/protocol.html

You could also try google webmaster Central
http://www.google.com/webmasters