Why increase chord length for supersonic aircraft (swept vs delta wings)?

Question

TL/DR

For supersonic transport and a given sweep angle, why are delta wings used instead of swept wings? If you write the drag equations, a sweep of $55^{\circ}$ seems sufficient with no modification to chord length. Why then do almost all supersonic passenger aircraft have delta wings?

Calculation

Let's start w/ the weight of a Boeing 737, then hypothetically set the cruise altitude to ~24000 m, where air density is 10% of 12000 m (the 737's normal cruise altitude). I'll use MTOW here as the weight of the aircraft at cruise (which is high, but will provide an upper bound on the calculations). To generate enough lift, we have: $$ F_L = \frac{\rho v^2 s C_L}{2}$$ where $s$ = surface area. If we're a high altitude where $\rho$ is 10% of what it normally is, then to generate the same lift $F_L$, you need to either change $v, C_L, $ or $s$. I wrote a Python script to iterate through the following:

• Increasing surface area from $s$ to $2s$
• Increasing lift coefficient $C_L$ from $0.5$ to $1.2$

The resulting graph looks like this: $$ $$ The x-axis is $C_L$. From this graph, we can calculate new values for $s$ and $C_L$, and the resulting $F_L$. $F_L$ is much lower than the lift $F_{L737}$ required to lift a 737 , hence we will need to increase $v$ significantly. Using our new values for $s$ and $C_L$, the velocity required to generate enough lift is $$ v = \sqrt{\frac{2F_{L737}}{\rho s C_L}} $$

The velocity required is about Mach 1.5. Now we calculate the drag force, which will determine the force required to keep this plane in the air. $$ F_D = \frac{\rho v^2 s}{2} \left( c_{Di} + c_{Dp} + c_{Dw} \right) $$ where $c_{Di}, c_{Dp}$, and $c_{Dw}$ are the coefficients of induced, parasitic, and wave drag, respectively. $$ c_{Di} = \frac{C_L^2}{2 \pi A_R e} $$ $$ c_{Dp} = constant $$ $$ c_{Dw} = 20(M - M_{cr})^4 $$ $A_R$ is the aspect ratio, $M$ is simply $v$ expressed as Mach, and $M_{cr}$ is the critical Mach number. This is where I don't get delta wings vs swept wings. $$ $$ Using the Korn equation (p.18 here), $c_{Dw}$ is: $$c_{Dw} = 20\left( M +\sqrt[3]{\frac{0.1}{80}}+\frac{t/l}{\cos^2\lambda}+\frac{c_i}{10 \cos^3 \lambda} - \frac{k}{\cos\lambda} \right)^4 $$ $l$ is the chord length and $\lambda$ is sweep angle. $c_{Di}$, when $A_R = \frac{b}{l}$ is: $$ c_{Di} = \frac{C_L^2l}{2 \pi b e} $$ It seems that just sweeping the wings back (increasing $\lambda$ to $55^{\circ}$) would be sufficient to reduce wave drag without reducing $l$. Why then do supersonic aircraft have delta wings where $l$ is quite large? It seems like chord length $l$ does not matter too much in supersonic, but the resulting $A_R$ changes would cause pretty inefficient subsonic flight. Is there a component of drag here for the wing that I'm missing?

Open to all answers, but mathematically justified would be awesome.

Answer 1

As sweep reduces the effective velocity over the wing, it reduces drag. However, the same is true for lift, see the image below:

Source

With increasing sweep the effective airfoil becomes thinner, and lift reduces. As a consequence of this, the surface area needs to increase to make landing performance acceptable. You can do this by increasing the chord of a swept wing, or more structurally efficient, by creating a delta wing.

Answer 2

Open to all answers, but mathematically justified would be awesome.

Aerodynamics is what keeps aeroplanes in the air, and is ruled by physics. The equations used in OP are simplified physics equations, and the main thing to observe is if the simplified equation is applicable to the case under consideration.

1. The first equation used in OP, usually written as $F_L = C_L \cdot ½ \rho V^2 \cdot S$, is valid in low sub-sonic conditions, when the compressibility of air can be neglected - for airspeeds lower than Mach 0.4 - 0.5. In these conditions $C_L$ is reasonably independent of dynamic pressure, at higher airspeeds this is not the case anymore and $C_L$ will change with M. As can be seen in the graph underneath, from TU Delft course Introduction Supersonic Aerodynamics D 25-A by Prof. dr. -Ing Erdmann. The equation is of very limited value in transsonic and supersonic cases where compressibility of air plays a role, and should not be used without considering the validity conditions.

2. The Korn equation is a generalised equation that can be used for design calculation of expected drag of the wing. In transsonic circumstances, as pointed out by @PeterKämpf in a comment, so with M between 0.8 and 1.2. Not to be used at M > 1.2, when the shockwaves are the main determinant of supersonic drag.

Pic earlier used in this answer

What is the use of delta wings for supersonic aeroplanes? It keeps the wing inside the shock cone - would it be sticking out of the shock cone, additional compressibility drag would be created at wing bit that sticks out.

Why delta instead of swept? Image above from Torenbeek, first used in this answer, shows four bombers with planform shapes ranging from pure sweep to pure delta. Consider a strongly swept wing @ 55°: there is a triangular gap between the aft wing edge and the fuselage. Fill this space up with wing structure to obtain a delta wing, and:

• Wing area increases, reducing wing loading.
• Structural stiffness increases, reducing flutter & wing twist problems.

Note that the depicted aeroplanes are high subsonic ones - the delta wing was considered advantageous in this cruise speed region!

Pic source

One supersonic fighter jet with straight swept wings instead of delta wings, The English Electric Lightning, looks like a delta with the inner aft triangle of the wing shifted further aft, to become the swept horizontal tail.

Answer 3

Using a delta wing instead of a highly swept wing for supersonic aircraft has positive effects both from an aerodynamic and a structural point of view.

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From an aerodynamic point of view, the biggest source of drag in supersonic flight is the “wave drag” i.e. the drag generated by the shocks (waves). It can be derived (mathematically from the small-disturbance potential equation) that it exists an optimal distribution of the cross-sectional area of the aircraft which minimises the wave drag. The cross-sectional area is defined as the area obtained slicing the aircraft from nose to tail, as depicted in this picture:

Definition of cross-sectional area. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Area_rule_unifilar_drawing.svg/1092px-Area_rule_unifilar_drawing.svg.png

This ideal distribution is called "Whitcomb area rule": moving from nose to tail, the cross-sectional area smoothly increases from zero at the nose till a maximum at the center and afterward gently decreases toward zero at the tail. How the cross-sectional area is distributed among fuselage, wing and other components doesn't matter. This picture shows how the area rule was used on the fuselage of the F-102 to regularise its cross-section in order it to be as smooth as possible:

How the area rule works in practice. Source: http://www.aerospaceweb.org/question/aerodynamics/q0104.shtml

So why a delta wing instead of a highly swept wing? A delta wing (eventually combined with a small reduction of the fuselage’s cross-section toward the center, like in the F-102 of the previous picture or in the most recent iteration of Boom Overture) helps in achieving this smooth change of the cross-section. Note that this result could possibly be achieved also with a swept wing if the swept is forward or adding so called anti-shock bodies.

A second important aerodynamic aspect is to have the entire wing inside the Mach cone. When the wing lies inside the Mach cone, the component of the speed perpendicular to the leading edge is always subsonic and that not only avoids the generation of wave drag but mitigates the Mach tuck as well. The following slide depicts this aspect:

The right wing lies completely inside the Mach cone giving less drag at supersonic speeds. In the slides also some other nice examples are presented. Source: https://www.slideserve.com/grace/mae-3241-aerodynamics-and-flight-mechanics

For an aircraft flying at Mach 2 (like the Concorde), the line connecting the nose with the far end of the wing should have a sweep bigger than some 60° since the Mach cone has an angle $\mu$ of 30°.

So again, why a delta wing instead of a highly swept wing? The lifting area of a delta wing is concentrated toward the fuselage and therefore the wingspan can be reduced in respect of a highly swept wing, the area being the same. This gives a more compact wing which is more easily fit inside the boundary of the Mach cone.

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From a structural point of view the delta wing is more efficient than a simply swept wing. This cutaway of a F-111 helps in visualising this:

F-111 cutaway. Source: https://www.pinterest.de/pin/391672498829049341/

The F-111 was a supersonic aircraft with variable swept wing. On the cutaway I highlighted in red the load-path, from wingtip till fuselage, when the wing was swept backward of some 70°. What would have been the same load-path if the wing were a delta instead? The much shorter blue line gives the answer.

So again, why a delta wing instead of a highly swept wing? Because the spars can leave the fuselage perpendicularly to it giving a much shorter and more structurally efficient (i.e. lighter) “tip-to-tip” load path. Just as an another example, this cutaway shows how the spars of the Concorde looked like:

Concorde cutaway. Source: https://www.flightglobalimages.com/cutaways/civil-aviation-1949-present-cutaways/bae-concorde-cutaway-drawing-1571101.html?prodid=693

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Obviously a delta wing has also an impact on other aspects: in general it provides more room for fuel, engines, avionics and payload. Furthermore the weight of the wing is closer to the roll-axis, giving a more agile (combat) aircraft.

As usual in the aerospace world, it is always a matter of compromises :-)