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In smaller aircraft, a high wing is less likely to naturally enter a spiral dive; the low CG (center of gravity) relative to the CL (center of lift) means the plane tends to return to a wings-level attitude instead of staying in a bank if a thermal kicks a wing up. This isn't as much a concern with something as big as your pictured airliner concept.
Larger high-wings mean there isn't a big wing spar and fuel systems running across or just underneath your cargo deck. This is a major reason military transports are high-wing, so the deck can be flat without taking up usable cargo volume raising the entire deck above the wing spar (the high wing also keeps the engines safe from the hazards of "unimproved" landing strips in forward bases). However, in a commercial airliner, it means that spar is now running through your passenger cabin, which is why you don't see it much on jet airliners (the Dash 8 and other turboprops use it by necessity, and typically require a "camel hump" to get the spar up out of the cabin). This concept seems to split the difference, using a thin brace underneath the fuselage to reinforce a thinner high-wing spar.
High wings, especially swept wings on high-mass planes, have an increased tendency to Dutch roll. This is a "stable instability" where one wing gains lift due to a slight crosswind or yaw/sideslip, lifting that wing and inducing a bank, which creates a sideslip toward the other side which lifts that wing and induces a sideslip in the other direction. This ultimately results in a circular or figure-8 rocking motion of the fuselage not unlike the rolling of a ship, which can make passengers queasy.
High-wing aircraft tend to be more susceptible to this because their center of gravity is below their center of lift, and so their natural tendency is to return to wings-level, but on large heavy aircraft this settling takes time, during which the plane develops the sideslip that lifts the opposing wing, causing the overcorrection that continues the rolling motion. To counteract this, large high-wing aircraft are usually designed with some anhedral (downward) angle in the wing. This first lowers the center of lift, as some of the wing is now closer to the center of mass. More importantly, the lift vector of each wing now points outward from the fuselage instead of directly up in opposition to gravity. These two changes combine to relax the plane's natural tendency to self-level, and thus the tendency to Dutch roll, at the cost of keeping the pilot a little busier making small inputs to keep the plane level.
Low-wing aircraft tend to avoid Dutch roll in the opposite way; their center of gravity is above the wings' center of lift. Because of this, the plane will tend to remain in a banked angle, similar to a high-wing with anhedral but for different reasons. Because this can lead to instabilities like spiral departures including the dreaded spiral dive, low aircraft wings are typically given a dihedral (upward) angle, which gives the aircraft some self-stabilizing characteristics at low bank angles while still avoiding Dutch roll.
Lastly, as illustrated by aeroalias's answer, high wings have high engine mounts. Most modern airliner designs have under-wing engines which can be accessed from very near the ground (either standing or on a simple stepladder), and removed from the aircraft while supported with fairly simple cradles on casters. Engines mounted high on the airframe (such as on high-wing aircraft but also the fuselage-mounted engines of T-tails and the centerline engine of tri-engines like the DC-10 and 727) can require a cherrypicker to get to, and powerful accordion lifts to support the engine as it's being removed. This increases maintenance costs and reduces the opportunity for meaningful inspections at the gate.
The SUGAR Volt and SUGAR Freeze concepts use a high wing purely to make effective bracing possible. It is a design for a world where oil costs $200 per barrel and above. Mainly, two things make its aerodynamics much more efficient than current designs:
- A very high aspect ratio of its wing, enabled by the bracing, and
- A lower flight Mach number, which in turn makes the wing more efficient by requiring less sweep. Boeing doesn't give a Mach number in their web page, but I would bet it is somewhere around 0.75. See here for a discussion of a very similar design.
Todays airliners use a low wing to enable them to stow away their long landing gears and to put the wing spar below the passenger deck. Long gears make it possible to stretch the fuselage and still be able to rotate during take-off. Large high-wing aircraft, with their low fuselage position, are easier to load and unload, at the price that the fuselage taper has to start shortly aft of the landing gear, so no stretching is possible.
Both concepts use rather speculative engine designs, in case of the SUGAR Volt even one with hybrid natural gas / electric propulsion with onboard batteries. The SUGAR Freeze uses a less speculative engine burning liquefied natural gas. In the SUGAR study, Boeing set the goal to reduce fuel consumption by 70%.
The position of the wing is determined by the required stability, the necessity to land in unprepared runways, etc.
In general, the high wing aircraft have more lateral stability compared to the low wing design. In case of low wing design, the desired stability is achieved by dihedral.
Another advantage of high wing is that the aircrafts can be landed on unprepared and semi-prepared runways as the high wing (and engine position) offers some protection against debris.
Also, in case of high wing aircraft, the ground effect is lower. This prevents the smaller aircrafts from 'floating' and improves the landing characteristics.
Form structural point of view, the high wing aircrafts can have a flat floor for easy loading/unloading and using standard pallets. In case of low wing, the spar will run through that section of the fuselage. On the other hand, a high wing design limits the cabin height for the same reason.
In case of civil aircraft, the ease of maintenance is one the main reasons for using low wing.