# Moving from 2D wing to 3D wing design

In terms of the design perspective of an Aircraft, once we decide on the airfoil that would be used for the wing, how would an Aerodynamics Engineer (or you) proceed to construct the wing? The platform of my wing would be a rectangular + tapered wing. So, I am only looking to find the optimal Taper ratio and dihedral angle.

The main idea would be to get as close to the elliptical lift distribution as possible. But, how would we approach it? It is not as if we would try all kinds of combinations of taper ratio and dihedral and compare all of them and see which is closest to elliptic dist. It would be time-consuming and tedious as there would be a large number of combinations (possibilities).

Why wouldn't you try all kinds of combinations? This is one of the fundamental ways design is performed.

Design is inherently an inverse process -- we start with desired performance and work to find the system that achieves it. In rare situations, we can analytically solve inverse problems directly, but most of the time, we end up having to use some sort of guess and check or repeated evaluation.

In some cases, people have run many solutions before us -- we can use their results, charts, curves, or experience. But you have to find that set of data or knowledge and make sure it applies to your situation. In this case, you're really relying on someone else running lots of combinations.

If I were to assume you're at the start of your Senior Design project -- let me assure you, running a lot of cases and doing a lot of iterative guess and check is going to become a very familiar activity.

You say you already want a planform with poly-taper -- rectangular inboard and simple taper outboard (Like a Cessna 172). The dihedral will not substantially effect the load distribution (how close to elliptical). So why not set up a tool and compare wings with taper of 1.0, 0.8, 0.6, 0.4, 0.2, 0.0.

If you use zero twist and an un-cambered airfoil, you can isolate the effects of the planform. Run all the cases at the same angle of attack. Record the lift and drag coefficients. Both will be different, so you will want to process them into equivalent induced drag factors -- either Oswald e or the K in CD0+KCL^2. You know that the zero angle of attack case should result in zero CL and zero CDi.

You can also plot the six lift distributions on top of one another. You might also want to plot the lift coefficient distributions and compare them.

For this, you'll want to use AVL, Tornado, XFLR5, VSPAERO, or another similar tool.

An aerodynamic engineer would maybe strive for an elliptical circulation. But not an airplane engineer.

In 1933 Ludwig Prandtl published "Über Tragflügel kleinsten induzierten Widerstandes" in the Zeitschrift für Flugtechnik und Motorluftschiffahrt. 17 years later R.T. Jones came to the same conclusions in NACA-TN-2249. The 2013 article "Lift Distributions for Minimum Induced Drag with Generalized Bending Moment Constraints" by David J. Pate and Brian J. German goes one step further and optimises drag at a lower and structural mass at a higher lift coefficient.

The optimum wing for lowest drag has less than elliptical lift towards the tips and more taper, but more span than a wing with elliptical circulation. This reduces the root bending moment and, consequently, wing weight. If less lift has to be produced to carry the wing, drag is also lower. Scaling laws mean that structure becomes more important as aircraft grow in size and mass. You know that elephants have much more massive legs relative to their body size than antelopes (or even ants, for an even more drastic comparison), since body mass scales with the cube of linear dimension while structural strength scales only with the square of linear dimension. This means that wing spar weight will be proportionally higher for larger aircraft.

As a consequence, insects have more elliptic wings than albatrosses, and model aircraft have optimum wings which are much more elliptic than the optimum wing of an airliner. The optimum shifts from an elliptic load distribution at very small scales to an almost triangular distribution at large scales.

Once you know the size of your airplane, you vary taper and span such that a given excess lift (needed to carry the fuselage, tail, systems, crew and payload) can be produced with the lowest drag. This requires to consider aerodynamics and structure together. Chord is the result of the required wing volume.

Sometimes the process may seem direct, but other elements come into play. As an initial part of the process, the designer may have an idea regarding the plane, its weight, and required power. If the design proceeds, scheduling may begin where estimates are made of the materials required to assemble the plane. During this process, the weight of the aircraft is more closely estimated. Afterwards, comes refinements such as finishing the design of the wing and wing spar. But even after a plane is built, the design process may not be complete. Charles O'Neill presents an analysis of the wing of three successive versions of the Piper Cherokee, each new wing a modification and improvement of its predecessor. Here is a link to Dr. O'Neill's analysis. Information in his analysis includes the wing section, a perspective on why that airfoil was chosen, how the airfoil compliments construction of the wing, and how aerodynamics of the wing were improved by successive modifications on an already successful design. Dr. O'Neill also comments on the methods used and resulting limitations in his analysis. In my estimation, a worthwhile read given the perspective of the question being asked.

how would we approach it? It is not as if we would try all kinds of combinations of taper ratio and dihedral and compare all of them.

You're right.

Unless you are designing a radically new aircraft then there's no need to reinvent the wheel.

What you would do is just using a similar airplane as a basis and build upon it your new project. You want to design a new long-range, wide-body twin-engine jet airliner? You use the A330 as a basis and develop what you will call A350. You want to design the best-selling among four-seat, single-engine aircraft? You use the Cessna 170 as a basis and develop what you will call Cessna 172. And so on.

The basic aerodynamics characteristics of most of the wing layouts out there are well known so there's really no need to run thousands of simulations to discover that, for example, the wing for a jetliner should have a taper ratio around 0.4, aspect ratio around 9.5, laminar airfoils, thick airfoil at the root (around 16%) and thinner toward the tip (around 8%), sweep angle around 35°, root airfoil with prolonged chord in order to accommodate the main landing gears and some 5° of dihedral.

For other wing layouts, any book about preliminary airplane design contains the relevant general guidelines.