Von Kármán–Gabrielli diagram

The von Kármán–Gabrielli diagram (also Gabrielli–von Kármán diagram, GvK diagram) is a diagram which compares the efficiency of transportation methods by plotting specific tractive force, or specific resistance (ε = P/mgv = E/mgL) against velocity (v). It first appeared in Theodore von Kármán's ASME Thurston Lecture, and in the 1950 paper on this subject by Giuseppe Gabrielli and von Kármán.

Details

In GvK diagram, the x-axis is the vehicle velocity, and the y-axis is the dimensionless specific resistance. The same kind of vehicle may have multiple different operating velocities, with different corresponding specific resistances, consequently each kind of vehicle generally correspond to a whole region on the GvK diagram, but only the lower-edge of the region is plotted, since specific resistance can always be artificially increased by wasting energy, but not decreased beyond a lower limit depending on the physical and engineering constraints.

The specific resistance is a dimensionless quantity defined by

ϵ

:=

P

/

m

g

v

{\displaystyle \epsilon :=P/mgv}

where

P

{\displaystyle P}

is the total power consumption by the vehicle,

m

{\displaystyle m}

is the total mass of the vehicle,

g

{\displaystyle g}

is the gravitational constant, and

v

{\displaystyle v}

is the velocity of the vehicle. The inverse of specific resistance is defined as transport efficiency.

Alternative forms of specific resistance

For example, if we have a freight vehicle in which most of the total mass is in its freight, and we have freight to carry over a distance of

L

{\displaystyle L}

, but we have no requirement for how long it should take, then the total energy consumption of the vehicle over the journey is

E

=

P

t

=

P

(

L

/

v

)

=

ϵ

(

m

g

L

)

{\displaystyle E=Pt=P(L/v)=\epsilon (mgL)}

so we can interpret the specific resistance as a ratio for how much resistance the vehicle faces. A low resistance means it needs to consume less energy to carry the same load through the same distance.

The power is equal to the drag force times velocity. For aircraft in cruise flight the lift is equal to the weight (L=mg) and the engine thrust is equal to the drag (T=D).