No, not by default, but this can be set up using project rules. From the Projects screen, select a project and view its details. Under General > Assignments > Rules, you can assign the rule 'Automatically email invoice when processed.' After processing an invoice, CORE prompts you with a message whether you want to email the invoice. Select Yes and the invoice is sent to the Client Contact automatically.
Note: This rule does not apply to retainer invoices and they are not sent automatically. You can email them manually from the Retainers screen. Check CORE Help Center for details.