by default drupal commerce does not have an email sent when an order status has been completed. this is a rule just import it and straightaway you can use it.

send an email notification after an order status has been set to completed

{ "rules_send_email_after_order_status_is_completed" : {
    "LABEL" : "send email after order status is completed",
    "PLUGIN" : "reaction rule",
    "REQUIRES" : [ "rules", "entity" ],
    "ON" : [ "commerce_order_update" ],
    "IF" : [
      { "data_is" : { "data" : [ "commerce-order:status" ], "value" : "completed" } }
    ],
    "DO" : [
      { "mail" : {
          "to" : "[commerce-order:mail]",
          "subject" : "Order [commerce-order:order-number] at [site:name]",
          "message" : "Thank you. We have completed and processed now your order. Your Order Number is: [commerce-order:order-number] Order History. You can view your order history with us at any time by logging into our website at [site:login-url] Order Status You can also follow the status of your current order at: [commerce-order:customer-url] Thank you for being our customer!",
          "language" : [ "" ]
        }
      }
    ]
  }
}