It wasn't such an easy journey in the early years of the city. Goose Creek, renamed as "Tiber Creek" after the River Rome is built by, seperated the hill on which the Capitol stands and the ridge upon which the White House is built. James Sterling Young wrote that "Pennsylvania Avenue remained a desolate country road, where, 'every turn of your wagon wheel...is attended with danger.' Flash floods and spring tides made the avenue impassable altogether, necessitating either a long detour via highland roads or a trip by packet boat running between Georgetown, Greenleaf's Point (near the Navy Yard), and Alexandria to effect communication between the marooned communities at either end of the avenue." Although a bridge had been planned by L'Enfant - and preparatory work done on clearing the route, no bridge was built until 1832.
As far as I can make out, The route of the Tiber (see map at http://www.capitolhillhistory.org/library/04/Jenkins%20Hill.html) is as follows - a stream known as the "Reedy Branch" met the shorter "Goose Creek" near what is today the junction of New York Avenue and North Capitol Street, and then flows southwards in the vicincity of the rail tracks which run into Union Station. Just to the south-west of Union Station it met with another triutary, the Oden. It widened as it approached the site of the Capitol then turned westwards towards the Potomac. It was later canalised at this later stretch, into a canal which ran approximately along the route of Constitution Avenue. (Although an 1894 newspaper article suggests this was the course of the ancient James Creek, to the north of the Tiber - http://www.sewerhistory.org/articles/compon/1894_aen03/index.htm). The Tiber emptied into the Potomac close to the current position of the Washington Monument. Another tributary ran from F Street, between 9th and 10th Streets (under what is now the FBI Building and the Justice Department) joining the Tiber 20 yards south of Pennsylvania Avenue.
The drawing is described as "Tiber Creek. North East of the Capitol. Washington, D.C." Pen and black ink drawing by August Kollner, 1839. A train can be seen - so presumably this is that stretch of the river where it runs southwards.
Please let me know if you know of any sources which can throw further light on this forgotten river.