Medium Transport; Low Wing; 2 Turbojet Eng
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 4
Situational Awareness; Distraction
Conflict Ground Conflict; Critical; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Issued New Clearance; Flight Crew Executed Go Around / Missed Approach
Human Factors; Weather
I was working local control in a south operation. I had Aircraft Y holding in position on RWY19 with Aircraft X on final for RWY19. There was a small layer of clouds in between the buildings in Rosslyn and the Kennedy Center (our 2 mile final visual landmark). This small area of clouds did not prevent the aircraft from seeing the runway; and none of the previous aircraft on final disappeared from my view. During this operation; shortly after putting Aircraft Y in position; Aircraft X disappeared into the cloud just as they were passing the buildings over Rosslyn. I cleared Aircraft Y knowing that Aircraft X was somewhere between a two and three mile final. I could not tell exactly where Aircraft X was because they were behind a cloud that blocked my sight; but they were still showing outside of two miles on the radar. When I cleared Aircraft Y for takeoff; I said 'traffic is inside of a three mile final'. Shortly after takeoff roll; Aircraft X came out of the clouds and I saw they were inside of a two mile final. At that point I knew this may be an operation that I may not have runway separation minimums. I asked Aircraft X to square his last turn if able; to try to achieve the 6;000 ft and airborne. A few seconds after that I could tell this would not work and issued go-around instructions to Aircraft X on what I would estimate to be just inside of a half mile final. His instructions were to turn right to a heading of 250 and maintain 3;000 ft. He read back correctly; but did not turn and flew over top the runway while climbing. I then issued an instruction to turn right to a 270 heading to ensure separation from Aircraft Y departing.  Even after this; I observed Aircraft X continuing on runway heading. I then issued traffic and told him to turn immediately. After this; he finally started to turn. I would estimate that Aircraft X flew at least 2 miles after I issued the initial go-around heading before they started their turn.  I recommend that I could have issued a traffic alert and said to turn immediately earlier than I did. I also think that I could have held on to the departing aircraft longer and turned them eastbound to ensure diverging headings from the trailing go-around aircraft that was issued a westbound turn. In my experience; it is not the norm in a go-around for an aircraft to fly for so long after being issued a turn; to not turn. I am not a pilot myself; so I'm not sure what is considered a reasonable amount of time to 'clean up' the airplane while going around. Again; I from what I've experienced; this is not the norm. I would even consider this to potentially be a deviation.
DCA Tower Controller realized a plan was not going to work and sent an arrival around which came close to a departure.