To offer an interpretive clarification: By definition 2D and 3D imply two-dimensional and three-dimensional fluid flow. A 2D profile is, by definition, a wing section or airfoil. There are many applications for 3D flow analysis and proprietary/patented design of wings and engine turbines. For instance, understanding 3D flow would be necessary in computational fluid dynamics for innovative winglet design to lower the drag of a wing and thereby improve fuel economy, or in the specialized modification of fairings on a wing or along the fuselage to limit flow interference and thereby reduce aerodynamic drag. By definition, flaps usually means a segment along the trailing edge of a wing that can be extended, and deflected downward, to increase lift. For heavy aircraft, such flaps are slotted. The design for improved lift typically involves, in addition to the use of slotted flaps, leading-edge slats (a drooping, forward extension of the leading edge of a wing) that opens a slot to improve airflow over the upper surface of the wing and augment the improved lift from slotted flaps.
And now onward to airfoils. There are airfoils that have been patented. Many of those airfoils are designed for specialized or specific applications be they wings, propellers, or turbine blades. Patented design usually means the designer developed and applied new and original special processes, innovative procedures, or applications in the design and development of their airfoil. By originality of process and design, these are considered unique and patentable. Examples have been offered elsewhere in the answer to this question.
However, airfoil design in general may not involve this type of innovative originality, even though the designer's resulting airfoil is, in itself, highly original. The typical development of airfoils for low-speed applications uses mathematical constructs involving conformal mapping and applications in fluid mechanics. The mathematical basis for these designs originated in the early 20th century. Developments in the theory were advanced during the Second World War through a careful review of mathematical theory and verification by wind-tunnel testing. This work was done by the National Advisory Committee for Aeronautics, the predecessor of NASA. A considerable amount of this work has been published and is electronically available. Among special advances in pre-1950 airfoil design were the use of boundary-layer control to lower airfoil drag, development of advanced flap systems to improve lift, and the development of more efficient propellers.
After 1950, the specialized design of airfoils diverged. High-speed flight and the development of heavier, jet-powered aircraft, had different requirements and resulted in different advances in airfoil design. Many of these advances are considered proprietary and by their very nature, are obscure. In many cases, these proprietary features have been patented. This same element also persisted in the design of low-speed airfoils. Although the features of a design were easily recognized, understood, and attributed to known theory, backing out or reversing the process to obtain the procedure followed in developing the design would be protracted and considered largely impractical. Unless, of course, one had critical and key insights to the applied theory. Consequently, such designs have been considered the intellectual property of the designer. For all practical intents and purposes, this has been considered as good as a patent.
So if the point of a patent is to protect one's intellectual property and hopefully make money in licensing the use of that property, why would a designer simply put all that hard work out to be used for free? Actually, a lot of the design work that is not patented was not developed for free. Airfoils were specifically designed for specialized applications in aeronautics by companies providing research grants for the development of the airfoil. A general requirement is the publication of the results of the research, even though the specific procedures used in the design are not disclosed. Features that are disclosed are related to the profile itself, and its aerodynamic tests in a wind tunnel or on an actual aircraft. The resulting payment for the research, and goodwill brought to the designer through recognition of the work, is usually satisfactory.