Lift - Lift

Japanese: 揚力 - ようりょく（英語表記）lift

Of the forces acting on an object moving through a stationary fluid (a general term for liquids and gases), this is the component perpendicular to the direction of travel. An object moving through a stationary fluid generally receives a force F from the fluid. This force is the resultant of resistance D , which acts in the opposite direction to the direction of travel, and lift L. Heavier-than-air aircraft can fly steadily and level because the lift acting on the wings balances the weight of the aircraft.

An object moving at a uniform speed in a stationary fluid is the same as a stationary object in a uniform flow when considered relatively speaking ( Figure A shows this case). Below, we will explain the cause of lift acting on a wing from two different perspectives: (1) from the perspective of the law of action and reaction between the wing and the fluid, and (2) from the perspective of integrating the fluid force acting on the wing surface.

[1] When a wing is tilted in a uniform, horizontal flow, the flow changes direction to a downward direction. If there was no wing, the flow would be horizontal, so this means that the fluid receives downward momentum from the wing. This is because the wing exerts a downward force on the fluid. Therefore, as a reaction to this (the third law of motion, or the law of action and reaction), the wing receives an upward force, or lift, from the fluid.

[2] Figure B shows the results of a computer-aided numerical analysis of the flow around a certain airfoil. There is a uniform flow from left to right at about 70 meters per second (Mach number 0.2), and the inclination of the airfoil relative to the direction of the uniform flow (angle of attack α ) is 12°. The grayscale in Figure B represents the pressure distribution, with darker colors indicating higher pressure and lighter colors indicating lower pressure. Point A in the figure is called a stagnation point. The darker colors around the stagnation point indicate that the pressure is higher than the pressure P _∞ of the uniform flow in the absence of the airfoil. The airflow flowing above the flow passing through the stagnation point goes around the leading edge of the rounded (large curvature) airfoil. At this time, the pressure that was high at the stagnation point drops rapidly, and the pressure becomes smaller than P _∞ . After that, as it flows along the upper side of the airfoil, the pressure on the upper side that has dropped below P _∞ gradually returns to P _∞ as it flows toward the trailing edge. On the other hand, the airflow below the stagnation point flows along the underside of the wing. Generally, the underside has a straight (small curvature) shape, and the pressure that is high at the stagnation point gradually returns to P _∞ as it flows toward the trailing edge. Therefore, compared to P _∞ , the pressure on the upper side of the wing is generally lower, and the pressure on the lower side of the wing is generally higher. In other words, as a result of the suction force acting on the upper side of the wing and the push force acting on the lower side of the wing, an upward lift force is generated on the wing as a result of these forces.

If the density of a fluid is ρ and the flow velocity is U , the momentum is proportional to ρU , so the change in momentum of the fluid per unit time, and ^therefore the lift acting on the wing, is proportional to ρU2 . In general, the lift force L acting on an airplane with a wing area S is

The proportionality coefficient C _L is called the lift coefficient. It is known that the lift coefficient changes with the angle of attack (the angle between the uniform flow direction and the wing) as shown in Figure C. When the angle of attack is small (generally in the range of about ±10 degrees), the lift coefficient changes in proportion to the angle of attack. In the case of an ideal two-dimensional wing (a wing with an infinitely long width), the theory of fluid mechanics states that the slope (proportionality coefficient) is 2π/radian (a slope of about 0.11 per degree. Radian is a unit of angle), and in the case of a typical three-dimensional wing (a wing with a finite width such as the main wing of an airplane), it is smaller than this value. When the angle of attack increases, a vortex is formed behind the wing, and lift actually decreases ( Figure C ). This phenomenon is called a stall. This is because the boundary layer on the wing surface cannot develop along the wing surface, making it difficult for the flow to change downward in direction, and therefore the momentum to change. In this case, the pressure drops in the vortex region behind the wing, so the wing is pulled backward. In other words, a large resistance acts on the wing. The angle of attack at which this stall occurs is called the stall angle. The value of the lift coefficient that is maximum just before the stall is called the maximum lift coefficient ( _CL ) _max . The maximum lift coefficient is determined by the shape of the wing and the uniform flow conditions (flow speed and fluid viscosity), and is one of the important values that determine the performance of an airplane.

[Isao Imai and Taro Imamura, June 18, 2019]

"Rheology" by Ichiro Tani (1967, Iwanami Shoten)" ▽ "Mechanics of Fluids" by Yasuyoshi Nakayama (Revised Edition, 1998, Yokendou)" ▽ "Triton Fluid Mechanics, Vol. 1, 2nd Edition, by D.J. Triton, translated by Tetsuya Kawamura (2002, Index Publishing)"

[Reference] | Flow | Airplane | Wing | Fluid

Object in a uniform flow (Figure A)
©Shogakukan ">

Object in a uniform flow (Figure A)

Air flow and pressure distribution around the airfoil (Figure B)
©Shogakukan ">

Air flow and pressure distribution around the airfoil (Figure B)

Change in lift coefficient with angle of attack and stall (Figure C)
©Shogakukan ">

Change in lift coefficient versus angle of attack and stall (Figure...

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

静止している流体（液体と気体の総称）中を運動する物体に働く力のうち、進行方向に垂直な成分。静止流体中を運動する物体は、一般に流体から力Fを受ける。その力は進行方向に逆向きの抵抗Dと揚力Lの合力である。空気より重い飛行機が定常的に水平飛行をすることができるのは、翼(よく)に働く揚力が飛行機の重量と釣り合うからである。

　静止している流体中を物体が等速運動するのは、相対的に考えると一様な流れの中に静止している物体が置かれているのと同じである（図Aはこの場合を示す）。以下では、翼に働く揚力の原因について、〔1〕翼と流体との間の作用・反作用の法則の観点、〔2〕翼表面に働く流体力を積分する観点、の二通りの見方を示す。

〔1〕一様かつ水平な流れの中に翼を傾けて置くと、流れは下向きに方向を変える。翼がなければ流れが水平方向であったことから、流体は下向きの運動量を翼からもらったことを意味する。これは翼が流体に下向きの力を及ぼしたためである。したがって、その反作用（運動の第三法則すなわち作用・反作用の法則）として、翼は流体から上向きの力、揚力を受ける。

〔2〕図Bはある翼型周りの流れについて、コンピュータを用いて数値解析した結果である。左から右方向に約70メートル毎秒（マッハ数0.2）の一様な流れがあり、一様な流れの方向に対する翼の傾き（迎角α）は12°である。図Bのグレースケールは圧力分布を表しており、濃い色ほど圧力が高く逆に薄ければ圧力が低い。図中の点Aはよどみ点とよばれる。よどみ点周囲の色が濃くなっていることから、翼が存在しない場合の一様な流れの圧力P_∞と比較して高い圧力となっている。よどみ点を通る流れより上側を流れる気流は丸みを帯びた（曲率の大きい）翼の前縁をまわりこむ。この時、よどみ点で高くなった圧力は急激に低下し、圧力はP_∞より小さくなる。その後、翼の上面側に沿って流れながら、P_∞より低下した上面側の圧力は後縁に向かって流れるにつれて緩やかにP_∞に戻る。一方よどみ点を通る流れより下側を流れる気流は、翼の下面側に沿って流れる。一般的に下面側は直線的な（曲率の小さい）形状をしており、よどみ点で高くなった圧力は、後縁に向かって流れるにつれて緩やかにP_∞に戻る。したがって、P_∞と比較すると翼上面側の圧力は全般に低く、翼下面側の圧力は全般に高くなっている。つまり翼上面側には吸い上げる力、翼下面側には押し上げる力が働く結果、その合力として上向きの揚力が翼型に発生する。

　流体の密度をρ、流速をUとすると、運動量はρUに比例するから、単位時間当りの流体の運動量の変化、したがって翼に働く揚力はρU²に比例する。一般に翼面積Sを有する飛行機に働く揚力Lは

の形に表される。比例係数C_Lは揚力係数（lift coefficient）とよばれる。揚力係数は迎角（一様な流れの方向と翼のなす角）に対して、図Cのような変化をすることが知られている。迎角が小さい場合（一般的には±10度程度の範囲）には、迎角に比例して揚力係数は変化する。理想的な二次元翼型（無限に長い幅を有する翼）の場合、流体力学の理論によればその傾き（比例係数）は2π/radian（1°あたり約0.11の傾き。radianは角度の単位）であり、一般的な三次元翼（飛行機の主翼のように有限の幅を有する翼）の場合はこの値より小さくなる。迎角が大きくなると翼の背後に渦ができて、揚力はかえって減少する（図C）。この現象を失速（stall）という。これは翼表面の境界層が翼表面に沿って発達できなくなり、流れの下向きの方向変化、したがって運動量変化がおこりにくくなるためである。なおこの場合、翼の背後の渦領域では圧力が低下するために、翼は後方に引かれる。つまり、翼には大きい抵抗が働くことになる。この失速が発生する迎角を失速角（stall angle）という。また失速直前において最大となる揚力係数の値を最大揚力係数(C_L)_max（maximum lift coefficient）という。最大揚力係数は翼の形状や一様な流れの条件（流れの速さや流体の粘性）によって決まり、飛行機の性能を決める重要な値の一つである。