1. Cruise condition. You do not yet know about high-speed aerodynamics. We will cover that later in the semester. For now, note the following:

At supersonic speeds, the correct expression for dynamic pressure is : 0.5gPinfMinf^2

where

g is gamma, the ratio of specific heats of air at constant pressure and constant volume (you will learn this in thermodynamics). Here all we have to know is the value of gamma for air, which is 1.4.

At very high Mach numbers, we may have to use a different value for gamma, but that is not of concern to the supersonic airliner or business jet.

Pinf and Minf are the freestream static pressure and Mach number, respectively.
 
 

Example: Flight Mach number of 3.0 at an altitude where the pressure is 10% of sea-level static pressure:

Pinf = 10132.5 Pascals (that's another name for Newtons per meter^2)

Minf = 3.0

Freestream Dynamic pressure qinf = 0.5*10132.5*9 = 45596.25 N/m^2

Suppose we had used 0.5rUinf^2, we would have got the same value (check for yourself).

The expression for lift of a wing of area S, at angle of attack a radians , at a supersonic flight Mach number Minf is:

CL = 4a / sqrt(Minf^2 -1)

You don't have to worry about lift-induced drag at supersonic speeds: the wing cannot feel the wake being left behind. The drag at supersonic speeds is large, being mostly due to "shocks", "wave drag", and "skin friction draq". No easy way to estimate these for a new airplane, so we'll just take values of drag coefficients for comparable aircraft and use those with some slight projected improvement because ours will be a future airplane.
 
 

The following data on supersonic-cruise aircraft are from Ref 1:

Roskam, J., "Airplane Design, Part1: Preliminary Sizing of Airplanes". Roskam Aviation and Engineering Corporation, 1989.
 

Aircraft	TOW (lbs)	Empty Weight	Max Fuel Wt	Source
Concorde	389,000	172,000	202,809	Roskam, J. (1)
Tupolev TU144	396,830	187,400	209,440	Roskam, J. (1)
Boeing 969-512BA	340,194	162,510	155,501	Roskam, J. (1)
Boeing 969-512BB	750,000	358,270	342,824	Roskam, J. (1)
SM-SST	56,200	25,200	29,800	Roskam, J. (1)
GD-F111A	91,500	47,500	NA	Roskam, J. (1)
GD-B58A	160,000	58,000	98,250	Roskam, J. (1)
NAA-B70A				Roskam, J. (1)
NASA Supersonic Cruise Fighter	47,900	19,620	NA	Roskam, J. (1)
Rockwell B-1B	477,000	NA		Roskam, J. (1)

CLmax ranges from 1.6 to 1.8 on takeoff, and 1.8 to 2.2 on landing, for supersonic cruise airplanes.

For fighters and business jets, the landing CLmax may be as high as 2.6.
 
 

How to determine the Wing Area needed, based on landing speed.

We will assume that the maximum landing weight is only 80% of takeoff weight, and that the lift coefficient is in the middle of the specified range for CLmax, above. The landing is to be accomplished at an airport which is at 5000 feet altitude on a standard day (see pressure and density from the standard atmosphere). The maximum landing speeds (stall speed at the landing weight) are taken as:

150 knots (subsonic jets)

180 knots (supersonic cruise airplanes).

This gives you enough information to find the minimum wing area needed. Add a little bit to it to be on the safer side because the wing effectiveness may go down due to installation effects.

See if the selected value of wing area gives a reasonable value of wing loading for your type of aircraft.
 
 

Typical Mission for a Fighter to intercept high-speed missiles

Weapons load: 6 missiles. (see AMRAAM for example of weight class and size, though these will be rocket+ airbreathing supersonic-combustion ramjet (scramjet) missiles.

Takeoff with full weapons and fuel load; climb to 35,000 feet. Cruise at Mach 0.6 for 1 hour.

Climb at max. cruise thrust to 100,000 feet, accelerating to Mach 3. Loiter for 15 minutes.

Dash at full thrust (afterburners) at Mach 5 for 2 minutes. Launch two missiles. Combat turn at 7 Gs. Dash at Mach 5, 1 minute. Launch 2 missiles. Combat turn at 7Gs,dash at Mach 5, 1 minute, 2 missiles. Glide to landing 100 miles away. Landing speed no greater than 200 knots, CLmax = 1.8. Fuel reserve at landing for 10 minutes of low-altitude flight at Mach 0.5.
 
 

Alternative Design /Mission Strategy for the Missile Interceptor discussed above

Fighters are set up in pairs, to be carried to 40,000 feet by a modified C-5-based tanker, one under each wing. On takeoff, the fighters' engines are operated at some moderate setting to augment the thrust of the C-5 and to provide for emergency escape. The C-5 takes the fighters up to altitude and tops off their fuel tanks at 40,000 feet, flying at Mach 0.8, so that they leave their engines running, but with minimal power. On receiving the signal that the missiles are inbound, the fighters are dropped from the C-5, and accelerate to Mach 5, 100,000 feet, and continue the mission from there.

The loiter time at 40,000 feet is now increased to 2 hours, but has no impact on the fighters' fuel requirement because it is assumed that the fuel tanks are kept topped off by the C-5 until it is time to drop the fighters.

This way the fighters' takeoff weight can be reduced, because they do not need fuel to climb to 40,000 feet, reach Mach 0.8, or loiter at 40,000 feet.