I've been doing a bit of local research on the water that once flowed through this.
Historically it was known as the Tumble or the Clockmillburn, and drained the somewhat wet land of Arthur's Seat towards the sea, where it discharged nearby Fillyside.
It also had another important purpose; it was the route by which the effluent of the Old Town's chamber pots and the Nor Loch were disposed of. In reality it was an open sewer.
But why waste all that rich, brown water? After all, this was the time before chemical fertilisers and animal and human waste were important for agriculture. To capture the waste, there was a system of settling ponds around what is now Meadowbank and Restalrig. The solids were extracted for sale as fertiliser and the water used to irrigate a system of meadows.
This is how the area of Meadowbank got it's name; it was on the bank of the river and it was one of these irrigated Meadows. Fillyside and Craigentinny Meadows also.
Over time the burn was formalised into a system of man-made ditches and drains and had 2 routes under what is now London Road, one through what is now this tunnel and the other parallel to Clockmill Lane. When the railway was being constructed in the late 1830s, the burn was still referred to as the "Foul Burn" as although the system of human waste recover and open sewers were largely gone, they were still in living memory. The industry of Edinburgh now made use of the river to dump its waste, the New Street Gasworks was said to add its own particular smell and a certain effervescence to the water. So, they couldn't just stop the burn up and it was run in a culvert big enough that it was later converted to a footpath when the burn was diverted in a smaller pipe.
I'm trying to write a more complete story of the burn for a blog. Might get round to finishing it, but there's lots of detail.