Calling it sound design & then stating no field recording seems somewhat at odds but anyway...
First, disregarding movement for a minute, a real jet engine doesn't sound like white noise. If you looked at the spectrum of of a jet engine it would have very pronounced harmonics, so recreating those would be a starting point, whether its by comb filtering white noise, or by adding very narrow band whines etc. to white noise..noise. Also if you band pass it you may lose the bottom end, and jet engines have a lot of low frequency energy, so I'd be more inclined to use a low pass filter it with lots of resonance
Second, the reason it goes whoosh is based on physics so I would then mess with doppler and possibly flanging or phasing.... but that depends on the perspective: are we onboard? is Is it passing by? or Or has it already gone by & its the away sound you are after? or Or is it approaching? each Each of these are totally different scenarios..scenarios.