Dash time from Mach 0.9 to Mach 1.2:
Since drag coefficient is based on lift coefficient, and lift coefficient changes with velocity, there was no simple mathematical expression that would express the dash time. Therefore the dash time was calculated numerically.
It was assumed that the dash is executed at constant altitude; therefore lift remains at the constant value of the weight of the aircraft,
L = Wfull = 98,000 kg.
Since
L = Cl * S * 0.5 ρ V2
The lift coefficient can be found as
Cl = L/(S * 0.5 ρ V)
Lift coefficient must decrease as velocity increases for lift to remain the same. The drag coefficient is a function of lift coefficient as shown in figure 1.
Figure 1: Cd vs Cl for VAT model
green = Mach 0.6
blue = Mach 0.9
red = Mach 1.2
The differential equation which describes the motion of the aircraft is then:
M(dV/dt) = T – Cd*S*0.5*0.5 ρ V2
For 0.025 second increments of time, the drag coefficient was calculated from the Cd vs Cl curve. Then the drag was calculated and the velocity was calculated at each time increment as:
ΔV = (T – Cd*S*0.5*0.5 ρ V2)/M
It was then found that Mach 1.2 was reached after 31.1 seconds at full weight.
Figure 2: Velocity vs. Time plot of F-16 at 3km altitude
dash time = 31.1 seconds (full weight)
Sensitivity of dash time to drag coefficient:
The allowable variation in dash time is 4 seconds. The drag coefficient was increased until the dash time turned out to be 33.1 seconds, and then decreased until the dash time resulted in 29.1 seconds. It was found that the drag coefficient can vary by ± 0.0015, or ± 4% of an average overall Cd of 0.0253.
The weight of the aircraft also significantly affects dash time. Using a similar method, it was found that a weight variation of 6% of the full weight would yield a + 2 second variation in dash time. Dash time increases with weight for two reasons: First, more lift is generated, resulting in an increase in induced drag. Secondly, the larger mass takes more time to accelerate. The opposite is true when weight decreases.
Calculating the Radius of Action:
Range is given by the Breguet Range equation:
Range = (h/g)*(L/D)* η*ln(Wfull/W0),
where h is the heating value of the fuel, g is the acceleration of gravity, (L/D) is the ratio of lift to drag, η is the overall aircraft efficiency, Wfull and W0 are the full and empty weights of the aircraft.
h= 42 MJ/kg
g= 9.8 m/s
η= 0.23 (typical value for low Bypass Ratio engine aircraft)
Wfull= 98,000 N
W0= 66,800 N
Figure 3 shows L/D as a function of lift. As the fuel is consumed during flight, the weight of the aircraft decreases. The two vertical lines in figure 3 show the full and empty weights of the aircraft. Assuming the aircraft will be traveling at Mach 0.9, the average L/D is 8.8.
Figure 3: L/D as a function of Lift
green = mach 1.2
blue = mach 0.9
red = mach 0.6
average L/D for mach 9 = 8.8
The Breguet range equation then yields:
Range = 3,320 km
Radius of action = Range/2 = 1,660 km.
Sensitivity of range to L/D:
The variation in range with respect to the lift/drag ratio is equal to the partial derivative of range with respect to (L/D) multiplied by the variation in (L/D):
ΔRange = δ(Range)/ δ(L/D) * Δ(L/D);
δ(Range)/ δ(L/D) = (h/g)* η*ln(Wfull/W0) = 3.77e+005
allowable variation in range = ΔRange = 50 nautical miles
ΔRange = 92.6 km
92,600 = 3.77e+005* Δ(L/D)
Therefore L/D can vary by 0.245 if the range can vary by 50 nautical miles (92.6 km).