Delta Arrow Wings: Advantages For Civil Supersonic Flight

There is more than one way to skin a cat; this cat happens to be supersonic drag. There are many theories on how to achieve efficient civil supersonic flight, each with distinct advantages. One method that I have always favored is the simple and reliable delta wing. No complex construction, boundary layer control systems, or reliance on laminar flow. Just a big triangle without the need for high lift devices. There are tradeoffs but if your goal is to go supersonic without complexity, a good place to start is with the delta.

It is a fact that sweeping a wing delays supersonic drag rise and raises the critical mach number. It is also a fact that sweeping a wing with no special treatments will cause all kinds of hellacious stability problems at low speeds. While I could go into at least 4 pages of descriptions, case studies and NACA test data, I’m trying to write less like a mad scientist this year. So I’ll limit the following formulas to the basics in trying to get the point across.

An object moving faster than the speed of sound in air will produce a shockwave. The angle created by this shock cone (it’s a three dimensional wave) is dependent on the speed of the object. The faster the object, the smaller the angle created by the shock cone. With low supersonic Mach numbers, it is entirely possible to sweep a wing enough to contain it within the subsonic wake of the cone. The formula for determining the half-angle is:

1 / Mach#  = sin * cone angle

As stated, this is the half angle formula. To get what the entire cone would look like if drawn whole and not bisected, simply multiply the result by 2. For example, say that my aircraft is going Mach 1.3 and the wing is swept 54 degrees. With a cone half-angle of  50.3 degrees, my wing is definitely within the confines of the wake. This has a significant effect on reducing wave drag.

In addition to reducing wave drag, critical Mach number can also be reduced from sweep. Simply put, air is tricked into thinking that the wing has a longer chord and accelerates over the top of the wing at a slower rate. The formula for this effect is:

Vmach (cos * sweep angle) = Effective Vmach

This formula determines the Mach velocity over the wing when sweep is introduced. This may or may not be lower than the critical Mach number for that airfoil section. To determine what the new critical Mach number is when sweep is accounted for, the following formula is appropriate:

Mcrit / cos * sweep angle = Effective Mcrit

In effect, a highly swept wing can delay the critical Mach number to supersonic velocities. A straight wing with a relatively low Mcrit of Mach 0.7 would have an effective Mcrit of Mach 1.19 when swept to 54 degrees. This holds a lot of promise for reducing drag in the low supersonic speed range.

It’s not all free soda and candy for the swept wing. As mentioned before, there are very serious effects to consider aerodynamically. Swept wings stall at the tips first, creating unstable rolling and pitching moments during the stall. In other situations, pitch-up may occur when the horizontal stabilizer gets caught in the wing’s flowfield. Slow speed handling is degraded by spanwise flow, the same phenomenon that helps to reduce drag at higher speeds. Maximum lift coefficient for a given angle of attack is also reduced, leading to sometimes extreme pitch attitudes at slow speeds. There are other issues but these are the most critical to control and stability.

A straight wing that is swept may be troublesome, but delta wings have distinct advantages  that make them attractive for our purposes. The double delta design is a derivative that is configurable to nearly any range of speeds. The two main variants of the double delta are the “shovel”, with the low sweep segment in front, and the “arrow” with the high sweep segment in front. In either configuration, the forward segment produces low-pressure vortices that drift over the aft segment and delay the stall to a much higher angle of attack (non-linear lift). If properly balanced, a double delta will display highly favorable stability characteristics as well.  For this discussion, we will deal with the delta arrow variant since it is customized for supersonic speeds.

LoFlyte test vehicle at USAF Museum in Dayton, OH. Designed for hypersonic waveriding flight, this basic design is applicable to low supersonic flight as well.

A delta arrow consists of a highly swept leading edge section and a less severely swept aft wing section. This ensures that the leading edge of the wing remains behind the shockwave at moderate supersonic speeds while retaining adequate lift reserves for slow speed operations. As mentioned earlier, swept wings have a reduced lift curve slope with the degree of sweep directly correlated to the reduction (provided airfoil section and thickness remain constant). This disadvantage becomes an advantage in supersonic flight. All aircraft experience a rearward shift in the center of lift which reduces maneuverability and increases drag. Rather than rely on large control surface deflections to correct this situation, the delta arrow’s forward wing segment provides the lift required with minimal drag.

Airfoil thickness is a strong modifier of total drag at high subsonic speeds. A thin airfoil has far less drag than a thick one, but in trade, it has a lower lift curve slope, and less space for structure and fuel. A way around this is to sweep a moderately thick airfoil so that the effective thickness is reduced while retaining actual space inside for the structure and fuel. This phenomenon is more pronounced with large amounts of sweep, so the volume inside the forward segment of a delta arrow is quite extensive.

Maneuvrability is closely tied to the wing loading and center of gravity location. A heavily loaded wing will have a larger angle of attack in 1G flight, reducing the amount of lift available for aggressive maneuvering, regardless of aspect ratio. A reduced angle of attack can be achieved by moving the CG aft but within certain limits. Locating the CG too far aft would render the aircraft uncontrollable, even with fly-by-wire. A forward CG will reduce the control response and increase static angle of attack, but enhance stability. Balance between the two extremes will be determined by the aircraft’s purpose and desired handling capabilities.

The reduction in lift coefficient for a given angle of attack is subject to the same cosine formula that was applied to the critical Mach number earlier. Therefore the formula is as follows:

Cl ( cos * sweep angle) = Effective Cl

Assume an aircraft requires a Cl of 0.2 to sustain level flight at a given speed. If the aft wing is swept at 54 degrees, the effective Cl would be 0.12, demanding the aircraft increase its angle of attack to create 0.2 Cl. An alternative is to reduce wing loading, reducing the required Cl and by association, angle of attack. This reduction in lift is beneficial for flying in turbulence as the reaction of the aircraft to disturbances will be markedly reduced. While some people may not consider turbulence reaction to be a reason to reduce Cl slope, those who fly low-level, high-speed profiles, especially over warm areas or near mountains may have differing viewpoints. Not all civilian designed turbine aircraft have to be business jets.

So what does this mean in plain English? It means that economical and safe civil supersonic flight is possible. The reason we have not been able to achieve this so far is that industry has been focused on improving efficiency of existing designs. To integrate these advantages, a radical departure in construction has to be undertaken. The wing must be blended with the fuselage to keep drag and weight to a minimum without sacrificing strength. Thrust to weight has to be increased to ensure adequate acceleration is available at high altitudes. Low wing loading will not only improve induced drag numbers while subsonic, but reduce the impact of sonic booms at higher speeds.

The aviation industry has been hearing for at least 15 years about proposed civil supersonic aircraft. In each case the designs were business jets. With the tumultuous world economy of the early 21st century, no builder or prospective buyer has seen fit to invest in such a jet for understandable reasons. To date, no one has proposed building a manned research aircraft of much smaller size to investigate actual performance, handling, environmental effects and integration with the current ATC environment. The cost of a purpose-built test aircraft would be far less than attempting to build a full size business jet requiring full Part 25 certification. Something to think about for the frugal mavericks among us.

Light Jet Thrust Formula (#1)

For determining minimum static thrust required for non flat-rated turbofan and turbojet engines.

As temperature and or altitude increases, the amount of thrust produced by an engine will subsequently decrease. On larger turbines, flat rating is available to supply rated thrust above sea level and ISA. For smaller engines that may not have the thermodynamic capacity to flat rate, the designer must take into account the degradation of thrust. Failure to do so will result in unsatisfactory performance, inability to meet expected book values and possible payload restrictions.

The simple fact is as engines get larger, the weight of passengers becomes less of a fraction of the static thrust. Those who wish to build cabin class transportation aircraft are probably going to find their optimum engine (or engines) in the 2000lb thrust class or larger. Use of small jet engines under 500lbs thrust is therefore likely to be associated with very light multi engine craft, experimental/test vehicles and unmanned aircraft. Development of small jet engines however, should not be deterred by market analysis that may not fully understand the different potential uses beyond business and high-end general aviation. Too often advances are cut short by the opposite nature of engineering and profit, which work on completely different time scales.

The urge to use as small an engine as possible should be avoided when building light jet aircraft for the simple fact that passengers cannot be reduced in weight. Smaller engines can operate closer to their optimal TSFC at cruise altitude, but like all of aviation, there has to be a tradeoff for other flight envelopes. Therefore if one wishes to have a 4 place jet with a 200 lb allowance for each passenger, using twin 400 lb thrust engines for a total of 800lbs will provide a 1:1 thrust to payload ratio. This number will gradually change as airframe, fuel, avionics and interior is added. By the time a weight of say 3200lbs is reached (1600lbs empty weight, 800lbs payload and 800lbs fuel…very minimal levels), the total thrust to weight has declined to 4:1. This figure is at sea level on a standard day. Take the same aircraft to Colorado Springs at 65 degrees and the thrust to weight balloons to 5:1. Each pound of thrust has to push over 5lbs of aircraft, thus increasing takeoff distance, time to climb, and fuel consumed over a given stage. Humidity and air pressure may also conspire to rob the engines of performance.

The following formula is very simple as it is meant to be a rough estimate to warn a designer of low thrust levels at a nominal altitude of 6,000 ft MSL and 65 degrees. This altitude also equates to approximately 0.785 with respect to air density at SL ISA. But we’re conservative and/or lazy so the easy to remember 0.75 ratio is the rule of thumb to remember. The thrust lapse rate is based on a low bypass turbofan engine (less than 3:1) and will vary with fan size, nozzle design and turbine temperature. Your engine will vary so consult the appropriate manufacturer tables when advancing beyond the initial planning stage of your design. By using a reduced thrust computation when setting gross weight, unpleasant surprises can be avoided while in the flight test stage.

Tr₁/.75=Tr₂

Thrust to weight ratio 1 = A/C weight divided by thrust at sea level, ISA.

Thrust to weight ratio 2 = A/C weight divided by thrust at roughly 6000 feet MSL, ISA.

If you really want to scare yourself, multiply 0.75 by the static thrust of your selected engines (assuming they are not flat rated) and see what number you get!