Triumph Tiger 800 - Tuning - Airbox
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In my opinion the standard air filter should flow enough air to keep up with the
flow through the throttle bodies so I am not planning to upgrade the air filter
anytime soon.
This is the view of the standard Air Intake on my 2015 Tiger 800,
circled in blue is the location of the one and only mounting
screw.  The Air Intake is easy to remove and a bit fiddly to replace.
View with the Air Intake removed, circled in Red is the large
diameter intake pipe, circled in yellow is part of a thin rubber
skirt that stretches across the top of the frame possibly to direct
air, keep water out or to maybe to keep noise in.
Photo of the Air Intake from an early Tiger 800, note the
narrow moulding and the small cross section
Photo of the Air Intake from my Tiger 800 XCx 2015.  No I have not
been cutting away the moulding, this is stock Triumph, clearly  
effort has been made in improve air flow into the airbox.  The
inside diameter of the air intake is 47mm which is larger than the
single intake on my stock Tiger 1050.  However the 800 intake
faces rearward, on the 1050 the intake faced forward.
As I write this in October 2015 no one seems to have attempted an airbox mod on a Tiger 800 2015.   It was easy to tweak my
Tiger 955i and Tiger 1050 with airbox mods to gain modest but tangible  improvements in performance. ( I did not make changes
to my Explorer, it  had more than enough power as standard).  I do not want to spoil the road manners of the standard 800 motor.  
On investigation It seems that Triumph have got there before me because the air intake on the 2015 model is less restrictive.  
Like all manufacturers Triumph are forced to restrict airflow into the airbox in order to eliminate induction noise.  So I am thinking
will my Tiger 800 run better with the Air Intake moulding removed?  Also see Tuning Summary page.
Airbox - My Conclusion
When I put my Tiger 800 XCx on the Dyno we proved that the Arrow Tune + the
Leo Vince Evo2 silencer improved power delivery, especially at the bottom end.  
On my final run removing the plastic Air Intake improved low end power slightly
and gave the best fuelling.

I therefore decided to make a number of runs on the road with and without the
Air Intake in place and measure the results using the Torque App on my Galaxy
Note 4 phone.  I repeatedly tried Top Gear Roll-Ons from 2,500 revs to
measure 40-60, 60-80 and 80-100 times.  Without the air intake there was
more induction noise, this additional noise was obvious in the Dyno room but
less so on the road.

On the road the best results occurred when the Air Intake was in place.  This
surprised me, there is little difference but the initial pickup 3 - 4,000 revs and 4 -
5000 revs was better with the plastic Air Intake in place.   I assume that
Triumph have spent a lot of time in a wind tunnel and the Air Intake is designed
to trap and redirect air as it flows through the Tiger's bodywork.  I thought
removing the Intake would help, it does in static air on the Dyno but not when
riding on the road.  Overall this is not a bad thing because I can leave it in place
and I do not have to tell my insurance company or risk invalidating my warranty.
In my youth I spent many hours gas flowing all sorts of engines.  The Air Intake
moulding has many rough edges and the area below the intake hole is
particularly bad so I spent time smoothing these out with a rotary file to improve
air flow.  
Shame really, I was hoping that removing the Air Intake would result in a
marked improvement. I am sure that fitting a larger, shorter bellmouth close to
the airbox would really help top end but I am not going to bother, too much
hassle to get it set up correctly.  If I want to go really fast I can ride my MV.