One of the most regular questions about the 146/Avro type. The flap howl is is an aerodynamic harmonic between the inner flap edge and the fuselage side at certain settings, it makes the ascending tone as the flaps come in on departure, and the opposite descending tone as they are deployed on approach. It was apparently a surprise on first flight, the wind tunnel tests hadn't identified it. BAe did indeed develop a modification to overcome it but it weighed 100 kg, two pax, and I always thought nobody took it until reading above that the RAF VIP ones put it on.
I've heard cabin crew say the descending tone is the cue to them to give cabin secure. On departure from London City on 28 and turning right, the flaps normally come in overhead Stratford and the Olympic stadium at 3,000 feet. The howl from this is quite audible when standing out on the train station platform at Stratford as the aircraft passes overhead. I was wondering if the commentators were going to pick up on it in the Olympics !