F10 – Lecture Notes: Aircraft Performance Analysis

F10 – Lecture Notes 1. Aircraft Performance Analysis Aircraft Performance Analysis Drag breakdown The drag on a subsonic aircraft can be broken down as follows. D = Do + Dp + Di where Do = “parasite” drag of fuselage + tail + landing gear + . . . Dp = wing profile drag Di = induced drag We now use the wing airfoil drag polar cd (c� ; Re) to give the wing profile drag, and use lifting line to give the induced drag. The nondimensional total drag coefficient is then D CL2 So � CD = CDo + cd (CL ; Re) + 1 �V2S S � eAR 2 (1) where the “�” subscript on the flight speed V has been dropped. The parasite drag coefficient CDo is based on its own reference area So , which may or may not be the same as the wing’s reference area S. The factor So /S allows any convenient So to be used. Flight power The mechanical power P needed for constant-velocity flight is given by πp P = V (D + W sin �) (2) where W is the weight, � is the climb angle, and πp is the propulsive efficiency. Typically P is defined as the motor shaft power, in which case πp is the propeller efficiency. L L V V � D D � W W sin � W In level flight, � = 0, and the power is πp P = V D = 1 1 � V 3 S CD 2 (3) The flight speed V is given by the Lift = Weight condition, together with the definition of the lift coefficient CL . L = 1 � V 2 S CL = W 2 V = � 1/2 2 W/S � CL The ratio W/S is called the wing loading , and has the units of force/area, or pressure. The level-flight power equation (3) then takes the following form. 1 P = πp � 1/2 2 W/S � W CD (4) 3/2 CL Power dependencies Equation (4) indicates how the level flight power depends on the quantities of interest. The following dependencies are particularly worthy to note: 1 πp P � 1 P � S 1/2 P � W 3/2 CD P � 3/2 CL Note the very strong effect of overall weight W . This indicates that to reduce flight power, considerable effort should be directed towards weight reduction if that’s possible. An important consideration is that the proportionality with each parameter assumes that all the other parameters are held fixed, which is difficult if not impossible to do in practice. For example, increasing the wing area S is likely to also increase the weight W , so the net influence on the flight power requires a closer analysis of the area-weight relation. 3/2 The flight power is seen to vary as the inverse of the ratio CL /CD , must be maximized to obtain minimum-power flight, or maximum-duration flight. But again, increasing the 3/2 maximum achievable CL /CD will typically require changes in the other aircraft parameters. 3/2 For example, increasing the aspect ratio AR will typically increase the maximum CL /CD , but it will also increase the weight W which offsets the benefit. Drag and power polars One convenient way to examine the aircraft’s power-requirement characteristics is in a power polar . This is a variation on the more common drag polar, with the vertical CL axis being 3/2 replaced by CL . An example power polar is shown below, showing three curves: Only CDp , which assumes CDo = 0 and AR = �. Total CD , assuming CDo So /S = 0.01, and e AR = 10. Total CD , assuming CDo So /S = 0.01, and e AR = 5. 2 3/2 The slope of the line tangent to each polar curve indicates the maximum CL /CD value, and the point of contact gives the CL at which this condition occurs. For the two example polars, we have: For AR = 10: For AR = 5 : 3/2 (CL /CD )max = 16.0 at CL ≡ 0.90 3/2 (CL /CD )max = 10.0 at CL ≡ 0.70 The AR = 10 aircraft can therefore be expected to have a power requirement which is lower by a factor of 10.0/16.0 = 0.625. The expected duration is longer by the reciprocal factor 16.0/10.0 = 1.6. But this assumes that the larger AR has no other adverse effect on the other important parameters affecting P , which is very unlikely. It’s useful to compare the effects of increasing the parasite drag CDo versus increasing the induced drag CDi via aspect ratio. The optimum CL (i.e. optimum speed) varies in opposite direction. In general, larger CDo favors lower speed, while smaller AR favors higher speed. Increasing CDo Decreasing AR 3/2 3/2 CL CL CD CD 3