I have a php-script (in a normal LAMP environment) that runs a couple of housekeeping-tasks at the end of script.
I use flush() to push all the data to the client, which works fine (the page is fully loaded), but the browser still waits for data (indicated by the "loading"-animation) which is confusing for the user but of course clear because Apache cannot know whether PHP will generate more output after flush() - in my case it never does, however.
Is there a way to tell the client that the output is finished and the http-connection should be closed immediately even though the script keeps running?
It sounds like you have a long running script performing varioous tasks. Especially it appears to script goes on doing things after it has sent the reply to the client. This is a design that opens a whole lot of potential problems. You should re-think your architecture.
Keep house keeping tasks and client communication strictly separate. For example you could have a client request processed and trigger internal sub requests (which you can detach from) or deligate tasks to a cron like system. Then offer a second view to the client which visualized the progress and result of those tasks. This approach is much safer, more flexible and easier to extend when required. And your problem at hand is solved, too :-)
you can use this function fastcgi_finish_request() special function to finish request and flush all data while continuing to do something time-consuming (video converting, stats processing etc.); http://php.net/manual/en/install.fpm.php but you need to install FPM for it like
<?php
echo "You can see this from the browser immediately.<br>";
fastcgi_finish_request();
sleep(10);
echo "You can't see this form the browser.";
?>