Appendix A: How X-Plane Works
From X-Plane Wiki
X-Plane’s Blade Element Simulation Explained
X-Plane assimilates the geometric shape of any aircraft and then figures out how that aircraft will fly. It does this by an engineering process called "blade element theory,” which involves breaking the aircraft down into many small elements and then finding the forces acting on each little element many times per second. These forces are then converted into accelerations which are then integrated to velocities and positions. Of course, all of this technical theory is completely transparent to the end users—they just fly!
X-Plane goes through the following steps to propagate the flight:
Done only once (during initialization), X-Plane breaks the wing(s), horizontal stabilizer(s), vertical stabilizer(s), and propeller(s) (if equipped) down into a finite number of elements. The number of elements is decided by the user in Plane-Maker. Ten elements per side per wing or stabilizer is the maximum, and studies have shown that this provides roll rates and accelerations that are very close to the values that would be found with a much larger number of elements.
This is done twice per cycle. The aircraft’s linear and angular velocities, along with the longitudinal, lateral, and vertical arms of each element are considered to find the velocity vector of each element. Downwash, propwash, and induced angle of attack from lift augmentation devices are all considered when finding the velocity vector of each element. Propwash is found by looking at the area of each propeller disk, and the thrust of each propeller. Using local air density, X-Plane determines the propwash required for momentum to be conserved. Downwash is found by looking at the aspect ratio, taper ratio, and sweep of the wing, as well as the horizontal and vertical distance of the "washed surface" (normally the horizontal stabilizer) from the "washing surface" (normally the wing), and then going to an empirical look-up table to get the degrees of downwash generated per coefficient of lift.
The airfoil data entered in Part-Maker is two dimensional, so X-Plane applies finite wing lift-slope reduction, finite-wing CLmax reduction, finite-wing induced drag, and finite-wing moment reduction appropriate to the aspect ratio, taper ratio, and sweep of the wing, horizontal stabilizer, vertical stabilizer, or propeller blade in question. Compressible flow effects are considered using Prandtl-Glauert, but transonic effects are not simulated other than an empirical mach-divergent drag increase. In supersonic flight, the airfoil is considered to be a diamond shape with the appropriate thickness ratio; pressures behind the shock waves are found on each of the plates in the diamond-shaped airfoil and summed to give the total pressures on the foil element.
Using the coefficients just determined in step three, as well as the areas determined during the first step and dynamic pressures (determined separately for each element based on aircraft speed, altitude, temperature, propwash and wing sweep), the forces are found and summed for the entire aircraft. Forces are then divided by the aircraft mass for linear accelerations, and moments of inertia for angular accelerations.
Back to Work
The process is repeated from step two, and the whole thing is run over again at least 15 times per second. Aren't computers great?
Advantages of Blade Element Simulation
This method of computing the forces on the airplane is much more detailed, flexible, and advanced than the flight model that is used by most other flight simulators. Most other simulators use something called "stability derivatives" to compute how an airplane flies. This technique involves simply forcing the nose to return to a centered position along the flight path with a certain acceleration for each degree of offset from straight-ahead flight of the airplane—for every degree of angle of attack the nose is raised, the nose should return to center with a certain acceleration. This is a perfectly nice rule of thumb, but is far too simplistic to use across the flight envelope of the airplane.
Stability derivatives will not normally take into proper account the asymmetric affects of engine failures, the chaotic effects of turbulence, stalls, and spins, and the myriad of dynamic effects that are generated by the props of planes and the rotors of helicopters, such as spiraling slipstream, P-factor, and translational lift. As well, these simplifications can not easily consider such effects as transonic drag rise and compressibility which effect different parts of the airplane in different ways at different speeds, angles of attack, sideslips, and rotation rates.
Stability derivatives will typically say, "Okay, we are flying at Mach 0.8, so we add 5% to our drag due to compressibility,” in a situation where blade element theory will say, "Okay, we are flying at Mach 0.8, but the wings are swept at 45 degrees, and the plane is in a 5 degree right side-slip, so the effective sweep on the left wing is only 40 degrees, but the effective sweep on the right wing is 50 degrees, and the plane is rotating at 10 degrees per second to the right, so the advancing wing has an extra 10 knots of speed at the wingtip due to this rotation, but the retreating wingtip has 10 knots less speed due to this rotation, and the roll rate is 30 degrees per second to the right, which increases the angle of attack from nothing at the center of the plane to 2 degrees at the right wingtip and negative 2 degrees at the left wingtip, and the plane is pitching up at 10 degrees per second, which adds 1.5 degrees of angle of attack to the tail and takes away 0.1 degrees angle of attack on the main wing because it is in front of the center of gravity, and the changes in angle of attack cause increase in induced drag on the horizontal stab reduction in induced drag on the forward wing.”
Furthermore, the above is only a gross approximation—the simulator does this for each piece of the wing, horizontal stabilizer, vertical stabilizer, and propeller blade to really build a model of what the airplane is doing.
In other words, the commonly-used "stability derivatives" are gross over-simplifications of how an airplane flies, and blade element theory figures out the forces on each little bit of the airplane. Blade element theory is much more robust, and it can give greater accuracy in a much wider variety of flight conditions. As well, stability derivatives cannot predict how an airplane will fly. The aircraft model’s creator has to figure out how the plane will fly and then use the stability derivative to mindlessly spit that performance back out. Only blade element theory can accurately predict what an airplane of a given geometry will do. Microsoft Flight Simulator cannot predict how an airplane will fly for the user. Whoever designed the airplane has to tell the simulator how the airplane should fly, and the simulator then spits that information back to the user—nobody actually learns anything. With blade element theory, though, used in X-Plane, a user can enter the shape of an airplane and then fly that plane in the simulator. X-Plane will figure out how a plane of that shape and weight and power should fly!