面积律

概观

跨音速面积律

二次世界大战以后，许多飞机开始改成使用喷气发动机作为动力来源，但是它们的速度都在马赫数1以下，因为它们遇到同样的问题，就是音障。

即使到了1950年代，美国空军的YF-102使用三角翼，依然无法突破音障。1951年至1952年，经过多次的实验以后，NACA兰利航空实验室的航空工程师、美国著名空气动力学家理查德·惠特科姆（Richard T.Whitcomb）发现，如果以从机头至机尾的飞机中心轴（x）来看飞机的截面积（A），在截面积变化曲线越平滑的话（dA/dx越小），产生的跨音速阻力就会越小。惠特科姆将此知识用在YF-102上，使后继机YF-102A能真正突破马赫数1，它也是第一架采用跨音速面积律的飞机。早期利用跨音速面积律的飞机有个特色，就是它的机身会像可乐瓶，因为要满足面积律，增加机翼截面积的部分，就必须在机身减少截面积。因此，F-102有着可乐瓶状的机身。

超音速面积律

由于跨音速面积律的理论基础有限，1953年美国航空工程师Robert T. Jones提出超音速面积律。这使得面积律的应用范围从跨音速能够延伸到超音速。与跨音速面积律不同，超音速面积律的截面是以与马赫锥面平行的截面积为准。亦可以想成，当马赫数趋近于1时，马赫角会趋近90度，而马赫锥会趋近于平面，刚好与跨音速面积律的截面相同。

The Whitcomb area rule, named after a US National Advisory Committee for Aeronautics (NACA) engineer Richard Whitcomb and also called the transonic area rule, is a design procedure used to reduce an aircraft's drag at transonic speeds which occur between about Mach 0.75 and 1.2. For supersonic speeds a different procedure called the supersonic area rule, developed by NACA aerodynamicist Robert Jones, is used.

Transonic is one of the most important speed ranges for commercial and military fixed-wing aircraft today, with transonic acceleration an important performance requirement for combat aircraft and which is improved by reductions in transonic drag.

Description

At high-subsonic flight speeds, the local speed of the airflow can reach the speed of sound where the flow accelerates around the aircraft body and wings. The speed at which this development occurs varies from aircraft to aircraft and is known as the critical Mach number. The resulting shock waves formed at these zones of sonic flow cause a sudden increase in drag, called wave drag. To reduce the number and strength of these shock waves, an aerodynamic shape should change in cross sectional area as smoothly as possible from front to rear.

Transonic area rule

The area rule says that two airplanes with the same longitudinal cross-sectional area distribution have the same wave drag, independent of how the area is distributed laterally (i.e. in the fuselage or in the wing). Furthermore, to avoid the formation of strong shock waves the external shape of the aircraft has to be carefully arranged so that the cross-sectional area changes as smoothly as possible going from nose to tail. At the location of the wing, the fuselage is narrowed or "waisted".

Sears–Haack body

A superficially related concept is the Sears–Haack body, the shape of which allows minimum wave drag for a given length and a given volume. However, the Sears–Haack body shape is derived starting with the Prandtl–Glauert equation which approximately describes small-disturbance subsonic flows, as well as Ackeret Theory, which closely describes supersonic flow. Both methods lose validity for transonic flows where the area rule applies, due to assumptions made in their derivations. So although the Sears–Haack body shape, being smooth, will have favorable wave drag properties according to the area rule, it is not theoretically optimum.

History

Germany

The area rule was discovered by Otto Frenzl when comparing a swept wing with a w-wing with extreme high wave drag while working on a transonic wind tunnel at Junkers works in Germany between 1943 and 1945. He wrote a description on 17 December 1943, with the title Anordnung von Verdrängungskörpern beim Hochgeschwindigkeitsflug ("Arrangement of Displacement Bodies in High-Speed Flight"); this was used in a patent filed in 1944. The results of this research were presented to a wide circle in March 1944 by Theodor Zobel at the Deutsche Akademie der Luftfahrtforschung (German Academy of Aeronautics Research) in the lecture "Fundamentally new ways to increase performance of high speed aircraft."

Subsequent German wartime aircraft design took account of the discovery, evident in slim mid-fuselage of aircraft including the Messerschmitt P.1112, P.1106 and Focke-Wulf 1000x1000x1000 type A long-range bomber, but also apparent in delta wing designs including the Henschel Hs 135.

Applications

The first aircraft where the area rule was consequently implemented was the German bomber testbed Junkers Ju-287 (1944). Other corresponding German designs were not completed due to the end of the war or even remained in the planning stage.

When the area rule was re-discovered by Whitcomb, it was made available to the US aircraft industry on a secret basis for military programs from 1952 and it was reported in 1957 for civilian programs.

Convair and Grumman, with Whitcomb's help, used it concurrently to design the Grumman F-11 Tiger and to redesign the Convair F-102. The Grumman F-11 Tiger was the first of the two aircraft to fly and had been designed using the area rule from the outset. The Convair F-102 Delta Dagger had to be redesigned as it had been unable to reach Mach 1 although its design speed was Mach 1.2. The expectation that it would reach design speed had been based on optimistic wind-tunnel drag predictions.