“Don’t leave your roof bars on because they create wind resistance and cause your car to use more fuel through the ‘drag’ effect.
Roof bars tend to weigh between 3kg to 5kg but the aerodynamic factor is greater. An empty roof rack affects fuel consumption by about 10%.”
https://www.rac.co.uk/drive/advice/how-to/fuel-saving-tips/
See tip 10.
Chowdhury et al. (2012) measured drag forces of passenger car add-ons through a series of wind tunnel tests. At high driving speeds (> 80 km/h), the unloaded and loaded roof rack (carrying a ladder) resulted in 10%~22% and 13%~28% increases in aerodynamic drag depending on cross-wind effects and speeds. The drag forces can be translated into 7.5%~17.6% and 10%~22.4% increases in vehicle motion resistance and similar Fuel Consumption penalties.
Chowdhury, H., Alam, F., Khan, I., Djamovski, V., Watkins, S., 2012. Impact of Vehicle
Add-ons on Energy Consumption and Greenhouse Gas Emissions. Procedia Engineering,
Vol. 49, pp. 294-302.
There is quite a few studies out there. Please note this was a US study and their high driving speeds are not so high. Aerodynamic drag increases as a Square (x2 speed =x4 drag)