This fabric insignia was cut from a German aircraft reportedly shot down by Lieutenant Frank Luke, Jr., an ace with the 27th Aero Squadron of the United States Army Air Service (USAAS). Luke's total of 18 enemy kills included four aircraft. Therefore the source of this fabric insignia was a Fokker D.VII, a Halberstadt C, or a Hannover CL aircraft.
The straight black cross, often called the "Balkenkreuz," was a World War I insignia adopted by the German Air Service in mid-April 1918. Variations of the straight cross design replaced the earlier "cross pattée" insignia on all German aircraft for the duration of the conflict and beyond.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.