# Drag suffered by moving bodies

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Why does a swimmer have so much difficulty moving through the water? Why do large ships consume so much energy to overcome water resistance? And in such light air, how can birds, cyclists, cars and airplanes overcome this resistance to advancing called drag. It is also this resistance due to the viscosity of the surrounding air that prevents droplets of mist and other small particles from falling (this is called viscous drag). For vehicles or other large objects, the drag is called turbulent: it is related to the loss of energy due to the movement of the displaced fluid. As for surface ships, they also lose energy through another mechanism, this time related to gravity, since waves are generated by the bow lifting the water from which the ship takes its place.

## 1. Viscous drag: examples of fog and rain

### 1.1. The viscous friction

Imagine a flat object, as thin as a razor blade, moving in its plane within a viscous fluid such as honey or oil (Figure 1). Even at low speeds, an effort must be made to advance this plate against resistance due to fluid friction. Locally, each surface unit of this plate is subjected to a tangential force that opposes movement. This friction force is proportional to the viscosity of the fluid (see article Fluids and solids) and the velocity gradient present in the immediate vicinity of the wall.

At the scale of the entire plate, it is the result of these local forces that constitutes the drag resistance force that can be described as viscous as long as other contributions to friction do not mask this effect. This is the case of a spoon falling into the honey, or a smooth, tapered boat hull in slow motion. The flow is then called laminar, which means that each fluid particle follows its own current line, without encroaching on those of its neighbours. This behaviour is different from the turbulent regime discussed below.

The slow flow around a sphere (Figure 2) is one of the best known because of the simplicity of this geometry and because this shape is the one acquired by small drops or bubbles under the effect of surface tension (capillarity). According to a common practice, the observation is made in a reference point linked to the sphere: rather than an object moving in a fluid at rest, we consider the flow around a fixed object.

The viscous drag of a small sphere is given by Stokes’ formula [1] according to which the drag F is proportional to the diameter d of the sphere and its velocity U with respect to the surrounding fluid, F = 3πµdU The same law applies for honey, water, air, or any other ordinary fluid (called Newtonian) taking into account its dynamic viscosity µ. The same type of formula actually applies to any small object in slow motion, the coefficient 3π being the only element specific to the spherical shape. As expected for friction, drag is always a force aligned with the speed of the object, and in the opposite direction.

### 1.2. The speed limit of a small object in free fall

In a vacuum, under the effect of gravity, any object, lead or feather, falls with the same uniformly accelerated movement (see the article The laws of dynamics). In a fluid such as water or air, it is also subjected to Archimedes’ thrust, which is none other than the result of hydrostatic pressure forces (see the article Archimedes’ thrust and lift). This force is equal to and opposite to the weight of the displaced fluid, so that the object will fall with reduced acceleration, or rise, depending on whether it is heavier or lighter than the surrounding fluid.

But the object does not accelerate indefinitely because the drag increases with its speed, until it exactly compensates for the driving force resulting from its weight and Archimedes’ thrust. The total force undergone by this object is then zero, as well as its acceleration in accordance with the law of dynamics. The speed that has become constant is called the falling limit speed. It is very quickly reached for a small and light object that we are considering here. Note that it can go upwards for an object lighter than the ambient fluid, such as a bubble or a drop of oil in water.

The maximum falling speed can easily be calculated using the Stokes formula given above. Weight is the product of the acceleration of gravity g ≃ 9.81 m.s-2 by the volume of the sphere, (1/6)πd3, and by the density ρc of the body. The Archimedes’ thrust is given by the same formula, using the density of the object ρf of the fluid instead of the body. By cancelling the sum of the three forces (see Figure 3), we can deduce that the maximum falling speed is equal to (1/18)g (ρc-ρf)d2/µ. It is all the lower the higher the viscosity µ is, and it increases as the square of the diameter d. This is why small objects stay in suspension for a long time (see Focus The world of small suspended objects).

### 1.3. No fog without drag

For example, consider a droplet of mist, which has a diameter of about 20 µm (0.02 mm). Taking into account the density of the water (1000 kg.m-3) and the viscosity of the air= 2 10-5 kg.s-1.m-1), the maximum falling speed of this droplet is about 1 cm.s-1. However, the slightest breath of the surrounding air has a much higher speed. Under these conditions, the droplet remains suspended and moves with the wind. The same applies to pollen grains and all particles smaller than about 20 microns. This explains the long duration of pollution peaks in the calm air of a beautiful high.

The mechanism that causes the fog to disappear, when it occurs, is therefore not its fall to the ground, but its evaporation following sunlight, which may be accompanied by condensation on the ground that remains cold, in the form of dew or frost.

### 1.4. From fog to rain

In a cloud sufficiently charged with water, the coalescence of the droplets produces larger and larger drops. When their diameter becomes about a millimeter, they fall as rain. If the air flow remained laminar as a drop of diameter d = 2 mm, 100 times larger than a fine droplet of fog, would have a falling speed 10,000 times greater and could reach 100 m.s-1 (360 km/h). In reality, as we will see later, the drag then becomes turbulent, which increases friction and decreases the free fall speed.

In addition, at millimetric scales, the surface tension is no longer strong enough to maintain the spherical shape of the drop. As shown in Figure 4, the drop deforms under the influence of the pressure forces induced by the flow. Beyond a size of about 5 mm (right column in Figure 4), instabilities develop and are able to break the drop by reforming droplets of about 2 mm, which allows the B, C, D, E process to renew itself, until the drop falls.

## 2. Turbulent drag

### 2.1. The Reynolds number

When a large enough body moves in a fluid as low in viscosity as air or water, the flow becomes unstable. This instability is at the origin of the erratic movement of a dead leaf. At higher speeds, a wake of disordered eddies forms downstream of the object, as can be seen at the back of a boat: this is the turbulent wake (see Focus The turbulent wake).

The first scientific characterization of the concept of turbulence was carried out by the Irish physicist and engineer Osborne Reynolds [2]. Its name is attributed to one of the most important ratios in fluid mechanics, called the Reynolds number. This number, noted Re, is equal to the ratio of the time of action of the viscosity (which can be estimated at d2/ν, where ν is the kinematic viscosity [3]) to the transit time of the fluid near the obstacle (near d/U). Its expression is therefore Re = Ud/ν. Turbulence occurs when this Reynolds number exceeds a critical value, always much higher than the unit, which depends on the flow considered.

Although turbulence is a very complex phenomenon, the modelling of which remains problematic, the Reynolds number has the great merit of providing rules of similarity between flows around objects of identical geometric shapes but of different sizes, and with fluids of different viscosities. For example, we can predict that in the same fluid, the wake of an object enlarged by a factor of 10 will become turbulent at a speed 10 times slower. In practice, the wakes of fairly large objects, such as balloons, cars or boats, are always turbulent.

### 2.2. The drag coefficient

This Reynolds similarity makes it possible to express the turbulent drag according to the expression F = (1/2)CX S ρf U2, where CX is called drag coefficient. This force is proportional to the area S of the maximum cross section swept by the body, also called master torque [4], as well as to the quantities already encountered ρf and U. The CX drag coefficient depends on the shape of the object, but also on the Reynolds number.

For a sphere in a laminar flow the Stokes formula presented above leads to CX = 24/Re. On the other hand, at large Reynolds numbers, CX tends towards a constant that depends only on the shape of the object, which gives great interest to this quantity. Figure 5 shows the variations of CX as a function of Re for a transverse disc and a sphere. The transition between laminar and turbulent regimes occurs between Re’s values 100 and 1000 and, when Re becomes very large, CX tends towards a constant, different for the two forms. The sphere being the best profiled of the two: its CX is lower.

This behaviour can be understood by estimating the energy to be expended to set in motion the column of fluid that the body sweeps in its movement [5]. A movement of the entire swept section thus corresponds to Cx ≃ 1, as is the case with the transverse disc. A more streamlined object spreads the fluid less violently, reducing the cross-section of the turbulent wake. The most efficient production cars reach Cx ≃ 0.25. It is in fact the CXS product that controls the drag, which is then easily calculated by multiplying this effective area by the density of the fluid and the square of the velocity.

### 2.3. A power dissipation proportional to the cube of the speed

The presence of a drag imposes a loss of energy, equal to the work of this force. The corresponding power, or energy lost per unit of time, is obtained by multiplying the drag by the travel speed. Since the force is proportional to U2, the power consumed is proportional to U3. It therefore increases considerably with speed.

This power is converted into the kinetic energy of the fluid (that of all turbulent fluctuations) and eventually dissipates into heat by the action of viscosity. The resulting temperature rise is generally imperceptible because the heat is diluted in a large mass of fluid. However, for a meteorite or spacecraft entering the atmosphere at very high speed, the heating becomes such that it can raise the object’s external temperature to thousands of degrees and lead to its destruction.

### 2.4. The role of dynamic pressure: drag and lift

While the drag of a slow object is controlled by viscosity, that of a fast object is mainly due to the pressure force induced by the flow, called dynamic pressure. As we have seen, this pressure allows the surrounding fluid to be evacuated laterally from upstream to downstream, so that the moving body can take its place. This difference in pressure causes an overall thrust on the body, which opposes its movement.

At high speed the effect of viscosity becomes negligible, and this overpressure at the stopping point can be estimated at ρfU2/2 using the Bernouilli relationship [6]. Multiplying this overpressure by the transverse area S gives the turbulent drag law stated above. In reality, it is the transverse area of the turbulent wake, in the order of CxS rather than the area S of the object itself, that controls the drag.

In the case of an asymmetric body, such as an aircraft wing or sail, another dynamic pressure force appears, perpendicular to the speed, called lift (see the article Archimedes’ thrust and lift). It is added to the drag, which is aligned with the speed and in the opposite direction. These two forces are proportional to the density of the fluid and the square of the velocity, and their ratio is therefore constant. As we will see later on, this one characterizes the ability of an aircraft to glide: it is called its fineness.

### 2.5. Some examples of free fall speed limits

In the case of fog and rain, it is easy to estimate the Reynolds number, knowing the kinematic viscosity of the air, ν = 1.5 10-5 m2s-1. For the mist droplet, with d = 20×10-6 m (or 0.02 mm) and U = 10-2 m.s-1, Re ≃ 10-2 is found very clearly in the laminar regime. On the other hand, for the drop with a diameter of d = 2 mm with U = 100 m.s-1, Re = 13 000 is obtained in the turbulent regime. But this falling speed limit is overestimated since it does not take into account turbulent friction. The weight of the drop is (1/6)πgρfd3 = 4 10-5 N, equalizing it with the turbulent drag based on CX = 0.5, one reaches a speed of 6 m.s-1, which gives Re ≃ 800, at the beginning of the turbulent regime.

In the case of a man in free fall, with a mass M = 80 kg, i.e. a weight of 800 N, the drag can be estimated with a section CXS = 1 m2, which gives a speed of 50 m.s-1, i.e. 180 km/h. It takes about 5 s to reach this speed limit, which represents a head (1/2)gt2 ≃ 125 m. The speed limit is thus quickly reached. The jumper then presses his weight on the air and no longer feels any acceleration. Once the parachute is deployed, the speed limit is greatly reduced due to its large transverse area CX S which considerably increases drag by a factor of about 100. The same drag is thus obtained for a speed 10 times lower (because proportional to U2), leading to a limit speed 10 times lower, in the order of 5 m.s-1.

### 2.6. The importance of fluid density

We have seen that the turbulent drag is proportional to the density of the fluid. At an altitude of 34,000 m, the density of the air is 100 times lower than near the ground; the same drag is then obtained at a speed 10 times higher. The maximum falling speed, for which the drag balances the weight, is therefore 10 times greater, i.e. 500 m.s-1. During his 2012 freefall jump in 2012, from an altitude of 39,000 m, parachutist Felix Baumgartner approached this speed [7], reaching 372 m.s-1 (1,340 km/h), after 45 s of fall, at an altitude of about 30,000 m.

On the other hand, the drag is much larger in water, 800 times denser than air at low altitude, which explains why a solid object falls much more slowly than in air (Archimedes’ thrust also slows the fall, and even cancels it out for an object that floats, but it plays a more limited role than the drag for a dense object such as a stone).

This density effect also explains why a cyclist can reach a speed of 40 km/h, whereas good swimmers rarely exceed 4 km/h. This factor of 10 in speed translates into a factor of 1,000 in power spent (we have seen that it increases as U3). This factor is more or less compensated by the ratio of densities between air and water, so that the power expended is more or less the same in both cases.

## 3. The gravitational drag on the surface of the water

Another effect that slows down the swimming or walking of ships is the wave drag, or gravity drag. The pressure at the tip of the bow exerts a double action. As we have seen, it spreads the upstream water on the sides, from which the ship will take its place. But much of this water is also lifted above the open surface by the bow and falls back on either side of the ship. The potential energy acquired during the uplift (Figure 6) is transformed into kinetic energy during its fall, which extends below the water equilibrium surface. Then, like a pendulum, this water oscillates, rising and falling, while moving away from the ship. This generates the waves seen on the sides of ships as well as those of a swan moving slowly on the calm water of a lake (Figure 7). This gravitational drag transfers part of the energy consumed by the ship to the waves. It is controlled by gravity.

We have seen that viscous and turbulent trails depend on the Reynolds number (Figure 5). The gravitational drag is controlled by another parameter, called Froude number [8], which is expressed as Fr = U /(gl)1/2 where l is the length of the moving object. Experience shows that this drag increases considerably as the number of Froude approaches the unit. The wavelength of the waves produced (distance between two successive ridges) is then close to the length of the boat [9]. The latter spends a considerable amount of energy to ride his own wave, instead of simply lifting the water from the neighbourhood.

Figure 8 illustrates this situation clearly different from that in Figure 7. Exceeding this limit Fr = 1 becomes almost impossible for a classic design boat. Only a longer boat can be faster. It is only by using lift, by using a suitable hull (such as a windsurfing board) or lateral fins (foils), that this limit can be exceeded (see the article Archimedes’ thrust and lift).

## 4. How to fight the drag?

As small as it is, the trail slows down movement. It can therefore only be maintained if a driving force compensates it. If there is a difference between the driving force and the drag, the body accelerates or slows down, depending on whether the difference is positive or negative. Thus in the case of the free fall discussed above, gravity accelerates the body until the drag accurately compensates for the weight.

Any horizontal movement is regularly slowed down by drag in the absence of power. The horizontal speed of the balls and balloons thus decreases along their trajectory. Their maximum value is imposed by the initial impact (about 260 km/h for a tennis ball or football) and the length of the trajectory then depends on the drag. For example, for a golf ball, the speed record set by world champion Jason Zuback is 320 km/h; the total length of the trajectory was then 400 m. It could have been twice as high in the absence of drag, for a purely ballistic movement subjected to gravity alone.

### 4.1. The energy cost of speed

A cyclist, even on a horizontal road and without headwinds, must pedal to overcome the resistance to advancing. The transverse area of its CXS wake is between 0.2 and 0.4 m2 depending on its position on the bicycle, more or less upright [10]. Thus for a speed of 15 m/s (54 km/h), the drag will be between 27 and 54 N, and the power to be provided by the cyclist to fight it (obtained by multiplying this force by the speed) is 400 to 800 W. To this must be added the power expended to fight against the mechanical friction of the machine, which is almost independent of speed. Only exceptional champions can maintain power [11] of 400 W beyond a few minutes (in comparison, the average power supplied by a horse is estimated at 735 W, the value of the horsepower). This explains why the cycling hour record is precisely 54 km (Bradley Wiggins in 2015). Limiting your speed to 12 m/s (43 km/h) requires half the power, which is more accessible.

For a well-profiled production car, the CXS product reaches a value of about 0.6 m2. At a speed of 28 m.s-1 (100 km/h), this leads to a drag of 280 N, or 8 kW of power used to overcome the aerodynamic drag alone. This remains modest for an average car, with a typical power of 50 kW. On the other hand, reaching 200 km/h requires fighting a trail 4 times stronger. The energy to be expended for a given path, equal to the product of force and displacement, is therefore 4 times greater. Since the displacement is then twice as fast, the power required (i.e. the energy expended per unit of time), is 8 times greater, i.e. 64 kW instead of 8 kW to compensate for the aerodynamic drag alone. This is only accessible to high-powered cars.

### 4.2. Different propulsion modes

The rower of the cover image, while pushing the water back with his paddles, releases eddies whose trace on the free surface is clearly visible. Here, it is a drag force that is applied to the oars and pushes the boat forward. The same applies to pedal boats and old wheel boats.

In the case of an aircraft, it is the propellers or engines that provide the propulsion force. Each propeller blade is profiled like an aircraft wing and is therefore subjected to a lift directed perpendicular to its rotational speed. It is designed with additional torsion to effectively direct the propeller thrust forward. In the case of reactors, it is a large number of propellers housed in a fairing that optimizes their operation.

In the case of birds in beaten flight, it is the effort of their muscles that produces a flapping of their wings and provides both the lift and propulsion required, thanks to the release of a vortex at each flapping (Read the article Archimedes’ thrust and lift). Swimmers’ beats, with or without fins, also release eddies in their wake; it is this mechanism that explains their propulsion. It is also a form of lift perpendicular to the movement of the flapping fins or wings.

### 4.3. The gliding flight

The glider is more original: the experienced pilot looks for updrafts, such as thermal winds, towards which he guides his aircraft to get it up in the air. In this case, the drag, directed in the direction of the wind, can have an upward component, which adds to the lift and thus participates in the glider’s ascent. Gliding birds, such as seabirds and raptors (Figure 9), widely use these methods to minimize their efforts.

As we have seen, lift and drag are proportional to the square of the speed, and their ratio, called fineness, also represents the inverse of the descent slope in resting air, as shown in Figure 10. It is then the component of the weight Mg projected along the trajectory that pushes the glider, while its transverse component is compensated by the lift, as if the glider was descending on an inclined plane. The most efficient glider bird, the albatross, has a glide ratio of 20, which is similar to that of airliners such as the Airbus A320, with a glide ratio of 17. Modern gliders do better, with a fineness of about 50. In resting air, such a glider can cover a horizontal distance of 100 km for a loss of altitude of only 2,000 m.

The speed of a flying object is thus imposed by the constraint that the lift must balance the weight, with however a possible adjustment range by the angle of incidence with respect to the trajectory. A light and large glider can fly slowly and take advantage of rising winds to rise. On the other hand, to go faster, an aircraft will have to be designed with a smaller wingspan, at the cost of higher energy consumption. Another strategy is to fly higher: at an altitude of 10,000 m, the density of the air is three times lower than on the ground, so that the same drag and lift forces, proportional to the square of the speed, are achieved for a speed multiplied by √3 ≃ 1.73.

## 5. Towards greater efficiency

### 5.1. Inspired by animals

Observation of the animal world shows remarkable examples of adaptation to flight or swimming, to minimize drag. Some dolphins are thus able to maintain a speed of 30 km/h, so that we have long sought mysterious mechanisms to suppress turbulence through the flexibility of their skin. But more recent research shows that it is essentially the optimal shape of their bodies, and their exceptional muscle power [12] that gives them these impressive abilities.

As for birds, they know how to reduce their drag by retracting their legs into their plumage and lying as far as possible in the direction of their trajectory. The majestic gliding flight of the osprey (Figure 9) illustrates well how these birds have developed their know-how and anatomy to minimize their efforts.

### 5.2. Controlling turbulence

While imitation of nature remains an excellent source of inspiration for our means of transport, scientific and technological progress allows us to do better. To minimize drag and therefore the amount of fuel consumed, car and aircraft manufacturers use significant resources, both high-performance numerical simulations and wind tunnel experiments. These allow the aerodynamic profile of the vehicle to be optimized and the influence of each detail to be tested, visualizing the flow using markers such as the smoke used in Figure 11.

Since turbulence is an essential factor controlling drag, extensive research is being conducted to reduce its effects. This can be achieved by using passive elements such as roughness or scratches, or active elements to prevent the growth of turbulent disturbances. However, there are practical and economic constraints to the development of these systems. In the case of aircraft, a known solution to limit drag is to reattach the air threads to the rear of the wings and fuselage by sucking them through a multitude of small holes; however, this is not implemented due to maintenance problems related to plugging the holes. In liquids, it is possible to reduce turbulent drag remarkably by adding polymers, even in minute concentrations. But this is of course limited to industrial applications in closed environments, for example for pipelines.

In the case of racing boats, the use of foil lift has allowed spectacular progress by freeing itself from the gravity drag. The application to transport ships is much more problematic. The search for speed always represents a significant cost in energy, which is difficult to offset by efficiency gains.

#### References and notes

Cover image. The drag suffered by the canoe must be compensated by the effort of the rower. The balance of these forces determines the speed of the canoe [Source: pixabay].

[1] George Gabriel Stokes, an Irish physicist who was responsible for the modern formulation of viscosity effects (1819-1903).

[2] Osborne Reynolds, (1842-1912) was an Irish engineer and physicist to whom important contributions to fluid dynamics are due, including the identification of hydrodynamic instabilities using an ink net (Proc. Roy. Soc. London, 35, 84-99, Jan. 1883), in an experiment is still cited today.

[3] The dynamic viscosity of a fluid, often noted at μ, is relevant for evaluating the viscous frictional force. It is counted in kg.m-1.s-1, a unit sometimes called poiseuille, in homage to the 19th century French physician and physicist Jean-Louis Marie Poiseuille. In the study of the effects of viscosity on flow, it is often preferable to use the kinematic viscosity of the fluid ν=µ/ρf, the ratio between its dynamic viscosity and its density (see the article Diffusion and propagation in air at rest). This is counted in m2s-1.

[4] Strictly speaking, the word “torque” refers to the straight section of a hull, the “master torque” being the largest section of all. It is therefore a word from marine vocabulary that we use here to designate the equivalent size about any body or vehicle moving in a fluid.

[5] In a time dt, it scans a cylinder of length Udt and section S, thus a mass ρfSUdt. To communicate the velocity U to this mass requires the contribution of kinetic energy ½ ρfSU3dt, which must be the work of the friction force FUdt. This leads to the formula of the drag force with Cx=1.

[6] Daniel Bernoulli (1700-1782), a Swiss doctor, physicist, mathematician and philosopher, professor at the University of Basel, was particularly interested in the study of fluid flow. Bernoulli’s theorem, published in 1738 in the book Hydrodynamica, plays a fundamental role in fluid mechanics. It requires that p+ρfu2/2= p0 along a power line, where p is the local pressure and p0 its value at the upstream stop point where the velocity cancels out (we are placed in the reference linked to the body).

[7] It then exceeded the speed of sound, which provides additional braking force: the energy of the fall is no longer used only to spread the air around the body, but to compress it and emit a shock wave, a high amplitude sound wave, the famous sound wall. This wave drag is also well known for boats, but this time by the emission of waves, governed by gravity.

[8] This number is named after William Froude (1810 -1879), a British engineer and naval architect who was at the origin of the first scientific studies of gravity drag.

[9] The Froude number physically represents the ratio between the boat’s U speed and the speed of wave propagation in deep water, which depends on their wavelength l per c=(gl)1/2 . The wake waves follow the boat as it moves, and for this to happen, they must propagate at the same speed as the boat. The relationship c≃U then leads to a first estimate of the wavelength of the wake, and this reaches the length of the boat when the Froude number is equal to 1.

[10] See http://sportech.online.fr/sptc_idx.php?pge=spfr_xfd.html

[11] This mechanical power supplied corresponds to a total power consumed by the body of about 1.9 kW, the efficiency being about 21%, see http://www.agoravox.fr/culture-loisirs/sports/article/puissance-et-performance-en-159520. This power consumed is related to the quantity of oxygen captured by the breath.

[12] F.E. Fisch and G.V. Lauder, 2006, Ann. Rev. Fluid Mech. https://www.yumpu.com/en/document/view/46389631/passive-and-active-flow-control-by-swimming-fishes-and-mammals.

The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: MOREAU René, SOMMERIA Joël (February 5, 2019), Drag suffered by moving bodies, Encyclopedia of the Environment, Accessed September 14, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/en/physics/drag-suffered-moving-bodies/.

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# 运动物体受到的阻力

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为什么人在水中移动如此困难？为什么大型船只需要消耗那么多的能量来克服水的阻力？而在如此轻盈的空气中，鸟类、骑自行车的人、汽车和飞机又是如何克服这份阻力前进的呢？也正是这种因空气的黏粘性而产生的阻力（黏粘性阻力），阻止了雾滴和其他微小颗粒的下落。对于车辆或其他大型物体，它们的阻力还来源于湍流（紊乱的流动）阻力，它与流体位移造成的能量损失有关。而水面上的船只会通过另一种机制损失能量，这一机制与重力有关——船体与水的相互作用使得船的周围产生波浪，从而产生了兴波阻力。

## 1. 黏粘性阻力：以雾和雨为例

### 1.1. 黏粘性摩擦

想象一个薄如刀片的平板在蜂蜜或油这样黏粘稠的液体（图1）中运动，即使是低速运动，平板也必须克服流体摩擦产生的阻力才能向前运动。从局部看，该板的每个表面单元都受到与运动方向相反的切向力。切向力的大小与流体的黏粘度（见文章《流体和固体》）以及壁面附着流体的法向速度梯度成正比。

从整体看，这些由局部力构成的阻力产生的结果就是粘性摩擦效应。掉进蜂蜜里的勺子下落和流线型锥形船体缓慢运动都属于这种情况。物体周围的流动称为层流，这意味着每个流体颗粒的运动都比较规则，相邻的流线也比较规则。这种行为与下面讨论的湍流状态不同。

小球在流体中的缓慢运动（图2）是一个非常著名的实验，因为球体的几何形状简单，并且小水滴或气泡受表面张力作用也会形成类似形状。通常，我们并非在静止的流体中观察物体的运动，而是在与球体固结的参考系中观察流动。

小球所受的黏粘性阻力可由斯托克斯公式[1]得到，根据该公式，阻力F大小与球体的直径d及其相对于周围流体的速度U成正比，即F=3πµdU。该定律在考虑动力黏粘度μ的情况下可适用于蜂蜜、水、空气或任何其它普通流体（统称为牛顿流体）。实际上此类公式适用于缓慢运动的任何小物体，特指球形物体阻力公式的系数。和摩擦力一样，阻力总是与速度方向相反，大小成正比。

### 1.2. 小物体自由落体的速度限制

在真空中，受重力的作用，任何物体——无论是铅块还是羽毛，都会以同样的加速度匀加速落下（见《动力学定律》。在水或空气等流体中，物体也要受到阿基米德浮力，这正是静水压力的结果（见《阿基米德的浮力和升力》。这种力与被物体所排挤出流体的重力大小相等，方向相反，因此，物体将以较小的加速度下降或上升——这取决于它与周围流体密度的相对大小。

但物体不会无限加速下落，因为阻力随着速度的增加而增加，直到完全平衡了由它的重力和阿基米德浮力产生的驱动力。这时，物体所受的合力为零，根据动力学定律，它的加速度就为零。此时速度所达到的恒定值被称为下降速度极限。对于我们这里考虑的轻小物体来说，很快就能达到这个速度极限。请注意，比周围流体轻的物体，如水中的一个气泡或一滴油，会向上运动。

利用上面给出的斯托克斯公式，可以很容易计算出来最大下降速度。重力是重力加速度g9.81m·s²乘以球体的体积 (1/6)p，再乘以球体的密度ρc。阿基米德的浮力由同样的公式给出，用流体的密度ρf代替物体密度ρc。通过三个力的平衡（见图3），我们可以推导出最大下落速度等于 (1/18)g(ρcf)d²/µ。最大下落速度不仅随着粘度μ增大而降低，而且随着直径d平方的增大而增大，这就是为什么小物体能长时间保持悬浮状态的原因（见焦点：《小物体悬浮的世界》）。

### 1.3. 雾随风动

假设一个直径约为20微米（0.02毫米）的雾滴，考虑到水的密度（1000 kg·m-3）和空气的黏粘度（μ= 2*10-5 kg·s-1·m-1），该液滴的最大下降速度约为1 cm·s-1。然而，周围空气的轻微运动都比这个速度大得多。在这种条件下，液滴保持悬浮状态，随风而动。这同样适用于花粉和所有直径小于20微米的颗粒。这也就解释了为什么在美丽的高空中，大气平静时污染峰值持续时间很长。

因此，当雾产生时，导致雾消失的机理并不是雾落到地面上，而是雾在阳光照射后蒸发，或者是随着温度降低，在寒冷的地面上以露水的形式凝结

### 1.4. 从雾到雨

在充满水汽的云中，水滴会凝聚产生越来越大的水滴。当其直径达到约一毫米时，就会像一样落下来。如果气流仍然是层流，此时直径d=2毫米的水滴，比细小的雾滴大100倍，其下降速度将是雾滴的10000倍，可以达到100m/s（360km/h）。实际上，此时空气阻力为湍流情况的阻力，摩擦力增加，自由落体速度降低（我们后面也会讲到）。

此外，在毫米尺度上，表面张力已经不足以维持液滴的球形。如图4所示，液滴在空气流动引起的压差阻力作用下发生变形。如果液滴的尺寸超过5mm（见图4. E），就会变得不稳定，从而发生液滴的破碎，重新形成2mm左右的液滴，因此液滴下落的整个过程就是B、C、D、E的不断循环的过程。

## 2. 湍流阻力

### 2.1. 雷诺数

当一个足够大的物体在空气或水这样低黏粘度的流体中运动时，流动就会变得不稳定。这种不稳定性也是落叶不规则运动的根源。在较高的速度下，就像船只驶过后留下的湍流尾迹那样，在物体的下游会形成一个无序涡流尾迹（见焦点《湍流尾迹》）。

爱尔兰物理学家和工程师奥斯本·雷诺（Osborne Reynolds）首次对湍流概念进行了科学定性[2]。为了纪念他，流体力学中最重要的无量纲量用他的名字命名，称为雷诺数。这个数，记为Re，等于流动的黏粘性扩散时间尺度（可估计为d²/ν，其中ν为运动黏粘度[3]）与流体绕过障碍物的时间尺度(约为d/U)之比。所以，雷诺数的表达式为Re=Ud/ν。当雷诺数超过一个临界值时，就会发生湍流，这个临界值总是比1大得多，具体取决于所考虑的流动情况。

尽管湍流这一现象非常复杂，且建模仍然是个问题，但雷诺数体现了流体（甚至不同黏粘度的流体）围绕几何形状相同但大小不同的物体流动的相似性规则。例如，我们可以预测，在相同的流体中，如果一个物体尺度扩大了10倍，那么它的速度需要减小到原来的1/10才能形成相似的湍流流动。在实践中，如气球、汽车或船等这些相当大的物体的尾部总是存在湍流。

### 2.2. 阻力系数

通过这种雷诺相似性可以得到湍流阻力的表达式为F=(1/2)Cx S ρf U2，其中Cx称为阻力系数。该力大小与物体的最大迎风面积S（也称为主迎风面积[4]）成正比，同时也与前面提到的流体密度ρf和相对运动速度的平方成正比，阻力系数Cx取决于物体的形状，也取决于雷诺数。

对于一个处于层流中的球体，通过上面提出的斯托克斯公式可以得到Cx=24/Re。另一方面，当雷诺数足够大时，Cx则会趋于定值，而这个定值只取决于物体的形状，这使得物体外形对减阻具有重要意义。图5显示了圆盘和球体的CxRe之间函数关系。Re在100到1000之间为层流到湍流的转捩区，当Re变得非常大时即湍流时，Cx趋于一个常数，但圆盘和球体的Cx最终趋于的常数不同。由于球体的外形优于圆盘，所以其Cx值较低。

要理解以上现象，可以估算物体在运动过程中排开流体所消耗的能量[5]。像圆盘这样整个截面的横扫运动对应的阻力系数Cx≃1，而流线型的物体推开流体的所需的力就小得多，它减少了湍流尾迹的横截面大小。减阻性能最好的汽车阻力系数可以达到Cx≃0.25。事实上，控制阻力的是CxS乘积，可以看作是有效面积，用它和流体密度、速度的平方的乘积就很容易计算出阻力了。

### 2.3. 消耗功率与速度的立方成正比

阻力造成了能量损失，即阻力做了负功。相应的功率，或单位时间内的能量损失，是将阻力乘以运动速度得到的。由于阻力与成正比，消耗的功率就与成正比。因此，消耗功率随速度增加而急剧增长

这种消耗的能量会转化为流体的动能（包含所有湍流脉动的动能），并在流体黏粘性的作用下最终耗散为热量。由于热量被稀释在大量的流体中，因此产生的温升一般难以察觉。但是，对于以极高速度进入大气层的陨石或航天器来说，产生的热量会使它们表面的温度上升到数千度并导致其毁灭。

### 2.4. 动压的作用：阻力和升力

物体缓慢运动所受的阻力是由黏粘度控制的，而快速运动的物体所受的阻力主要是由流动引起的压力差，称为动压。正如我们所看到的，这种压差阻力使其周围的流体在侧向绕开物体时从上游向下游流动，以至于运动的物体可以替换原来流体所在的位置。这种压力差会对运动物体整体产生阻力，从而阻碍物体运动。

物体在高速运动时，黏粘性效应可以忽略，利用伯努利方程[6]可以估算出流体的动压ρfU²。将这个动压乘以横截面积S，就可以得到上面所说的湍流阻力。实际上，控制阻力的是湍流尾迹的横截面积，大小为CxS，而不是物体本身的面积S

对于不对称的物体，如飞机的机翼或风帆，通过表面压力积分还会得到与速度垂直的力，称为升力（见《阿基米德的浮力和升力》）。它和阻力都是力的分解，而阻力的方向与速度方向平行且相反。这两个力的大小均与流体的密度和速度的平方成正比，因此它们的比值是不变的。后面我们会知道，这个比值可反映飞机滑行能力，叫做升阻比

### 2.5. 关于自由落体速度极值的例子

对于雾和雨来说，只要知道空气的运动黏粘度ν=1.5*10-5 m2/s，就很容易估算出雷诺数。对于雾滴，d = 20×10-6 m (或0.02 mm)，U =10-2 m/s，雷诺数Re≃10-2，流动处于层流状态。而对于雨滴，直径d=2 mm，以速度为U=100 m/s下降的物体，雷诺数为Re=13000，处于湍流状态。但是，由于没有考虑湍流摩擦阻力，下落速度被高估了。若液滴的重量为(1/6)πgρfd3=4*10-5 N，用阻力系数Cx=0.5计算的湍流阻力进行平衡，下落速度达到6 m/s，对应的雷诺数Re≃800，刚好在湍流区的边缘。

假如一个人从高处自由下落，其质量M=80 kg，即重量为800 N，阻力可以用有效截面积CxS=1m2来估计，得出最快下落速度为50m/s，即180km/h。达到这一极限速度大约需要5s，落体高度为 (1/2)gt2 ≃125 m。因此，很快就会达到速度极限，随后，跳伞者在空中下落时，不再感觉到任何加速度。一旦降落伞展开，其横截面积CxS增加了100倍，意味着阻力增大100倍，速度极值就会大大降低。因此，要获得同样大小的阻力，速度就要下降到（因为与U²成正比），即5m/s。

### 2.6. 流体密度的重要性

如前文所述，湍流阻力大小与流体的密度成正比。在34000米的高度，空气的密度比地面附近低100倍，需要10倍的速度才能产生相同的阻力。因此，阻力与重力平衡时的最快下降速度值就要增大10倍，即500 m/s。2012年，跳伞运动员费利克斯·鲍姆加特纳（Felix Baumgartner）从39000米高空自由下跳，45秒后，在约30000米的高度，他的下落速度达到了372m/s（1340km/h），接近预估速度[7]

在水里，阻力则要大得多，水的密度是低空时空气密度的800倍，这也解释了为什么固体在水中下落的速度要比在空气中慢得多（阿基米德浮力也会减慢下落的速度，甚至可以抵消重力，让物体漂浮起来。但对于石头等密度大的物体来说，它的作用要比阻力小很多）。

这种密度效应也解释了为什么一个人骑自行车速度可以达到40km/h，而优秀的游泳运动员游泳时的速度很少超过4km/h。速度增加10倍，则消耗功率增加1000倍（如前文所述，消耗功率与U3成正比）。但由于空气和水的密度不同，所以骑自行车的人和游泳运动员所消耗的功率大致相同。

## 3. 水面上的重力阻力（兴波阻力）

另一个减缓船舶行进的力是兴波阻力（wave drag），或叫重力阻力（gravity drag）。船艏的压力具有双重作用，正如我们所看到的，行进中的船艏把上游的水推向两侧，然后占据原来水的位置。但是拨开的水高度被提升高于原来的水平面，随后向船的两侧回落。水在上升过程中获得的势能（图6）在其下降过程中转化为动能，下降到水平面以下。然后，水体上下振荡，像钟摆一样，形成远离船体的波动，这就产生了在船舷上看到的波浪。天鹅在平静的湖面上缓慢移动也会产生类似的波浪（图7）。这种重力阻力将船舶消耗的部分能量转移到波浪上，该阻力受重力影响。

我们已经知道，黏粘性和湍流尾迹取决于雷诺数（图5）。重力阻力由另一个参数控制，称为弗劳德数（Froude number）[8]，它表示为Fr = U /(gl)1/2，其中l是运动物体的长度。经验表明，当弗劳德数接近1时，重力阻力会大大增加。此时所产生波的波长（两个连续的波峰之间的距离）就接近于船的长度[9]。船体将消耗相当大的能量来驾驭自己产生的波浪，而不是简单地提升附近水面。

图8与图7的情况明显不同。对于经典的船型来说，超过Fr=1这个极限几乎不可能。船必须更长才能更快。只有通过产生额外的升力，比如使用合适的船体（如帆板）或侧翼（水翼），船只才能超过这个极限（见《阿基米德的浮力和升力》

## 4. 如何对抗阻力

尽管阻力再小，仍会降低运动速度。因此，只有在有驱动力与之平衡时才能维持运动状态不变。只要驱动力和阻力之差不为零，物体就会加速或减速，这取决于差值是正还是负。因此，在上面讨论的自由落体情况中，重力使物体加速，直到阻力准确地平衡了重力。

在没有动力的情况下，任何水平运动都会因阻力而减慢。因此，球和气球的水平速度沿其轨迹下降。速度的最大值来自最初的冲击力（网球或足球约为260km/h），而轨迹的长度则取决于阻力。例如，对于高尔夫球，世界冠军杰森·祖巴克（Jason Zuback）创下的速度记录是320km/h，轨迹总长度是400m。在竖直上抛运动中，不受阻力的情况下，物体达到的高度大概是受阻力情况的两倍。

### 4.1. 速度的能源成本

一款外形优良的量产汽车，其CxS约为0.6m2。在速度为28 m/s （100 km/h）时，会产生280 N的阻力，或者说为了克服空气阻力需要输出8 kW功率。这对于一辆额定功率为50kW的普通汽车来说仍然是适度的。当速度达到200 km/h时，所产生的阻力也会增加到原来的4倍。因此，给定路程所要消耗的能量（等于力和位移的乘积）增大到原来的4倍。由于此时速度是原来的2倍，因此所需的功率（即单位时间内所消耗的能量）是原来的8倍，即需要64千瓦而不是8千瓦才能平衡空气阻力。这只有大马力汽车才能做得到。

### 4.2. 不同的推进模式

封面图中的划船者用桨向后划水时，可在水面上见到一个个清晰的旋涡。在这里，作用在桨上的是一种阻力，它推动着船前进。脚踏船和老式轮船也是同样的原理。

为飞机提供推进力的是螺旋桨或发动机。每个螺旋桨叶片的外形像飞机机翼，因此受到垂直于其转动平面的“升力”。在设计中需要加入扭矩，才能有效地产生向前的推力。在发动机的整流罩中，安装有大量桨片以优化运行。

鸟类扑翼飞行时，肌肉做功带动翅膀的扇动，每次扇动都会释放出旋涡（见《阿基米德的浮力和升力》），从而产生了升力和推力。游泳者的拍打（不管有没有游泳鳍）也会在其尾迹中释放涡流；它们的推进力正是来自这种机制。这也是一种垂直于鳍或翼挥拍运动的升力型推进模式。

### 4.3. 滑翔飞行

滑翔机的例子更加经典，经验丰富的飞行员会寻找上升气流，如热风，来引导飞机在空中上升。这时，风的阻力有一个向上的分量，增加了抬升作用，从而助力滑翔机上升。滑翔的鸟类，如海鸟和猛禽（图9），就广泛使用这些方法以减少它们的飞行能耗。

我们已经知道，升力和阻力与速度的平方成正比，它们的比值称为升阻比，也代表静止空气中物体下降斜率的倒数，如图10所示。那么，滑翔机在倾斜的平面上下降时，推动滑翔机的是重力Mg沿轨迹投影的分量，而其横向分量则由升力来平衡。效率最高的滑翔鸟——信天翁的滑翔比（升阻比）为20，与空客A320等客机相当（后者的滑翔比为17）。现代滑翔机做得更好，其升阻比约为50。在静止空气中，这样的滑翔机仅需要下降2000米，水平距离上就可以滑行100公里。

因此，飞行物体的速度受到如下约束：升力必须平衡重力，调整范围取决于其相对于轨迹的倾斜角。一架轻而大的滑翔机可以缓慢飞行，并利用上升的气流而上升。另一方面，为了飞得更快，飞机必须设计更小的翼展，代价是能量消耗更高。另一种策略是飞得更高：在1万米的高空，空气的密度比地面上低3倍，由于阻力和升力与速度的平方成正比，为了获得同样大小的力，速度需要增大到原来的3≃1.73倍。

## 5. 提高效率

### 5.1. 灵感来自于动物

观察动物世界，我们会发现很多例子，在飞行和游泳时最大限度减少运动的阻力。一些海豚能够在水中保持30km/h的游速，我们一直在寻找这种保持高速的神秘机制，比如它们通过皮肤弹性来抑制湍流等。但最近的研究表明，本质上，赋予它们惊人运动能力的是优良的身体外形和特殊的肌肉力量[12]

至于鸟类，它们知道如何通过将腿缩进羽毛中，并尽可能贴合自己的轨迹方向来减少阻力。鱼鹰美妙的滑翔飞行（图9）很好地诠释了鸟类是如何进化身体结构，提高飞行技能以减少它们的飞行能耗的。

### 5.2. 控制湍流

在从自然界获得灵感的同时，科学和技术的进步也在为我们助力。汽车和飞机制造商使用了大量研究手段来最大限度降低阻力，从而减少燃料消耗，包括高性能数值模拟和风洞实验。使用烟雾令流动可视化（如图11），可帮助优化车辆的空气动力学外形，并测试每个细节的影响。

湍流是控制阻力的一个重要因素，人们对其展开了深入研究，以降低其影响，例如可以通过主动方法，或增加粗糙度或划痕等被动方法避免湍流扰动的增长，从而实现减阻。然而，这些方法存在着可行性和经济上的限制。以飞机为例，已知的限制阻力的解决方案是依靠机翼后半段的多个小孔吸收气流，从而使绕机翼的气流贴体，然而，由于小孔会堵塞，这并没有实施下去。在液体中，通过添加聚合物，即使是微量浓度的聚合物，也可以显著降低湍流阻力。当然，这只限于封闭环境下的工业应用，例如管道。

在赛艇中，人们使用翼型产生升力来减小兴波阻力取得了巨大进步。但要应用于运输船则会有许多问题。追求速度总是意味着巨大的能源消耗，这是提高效率所难以弥补的。

#### 参考资料及说明

[1] 斯托克斯（George Gabriel Stokes，1819-1903）, 爱尔兰物理学家，在流动粘性效应的数学表达上做出了卓越贡献。

[2] 雷诺（Osborne Reynolds，1842-1912），爱尔兰工程师、物理学家，在流体力学中贡献卓越，特别是关于流动稳定性的开创性实验（1883），至今仍被引述。

[3] 流体的动力粘度，通常用μ表示，与流动的粘性摩擦力大小有关。单位是kg/(m∙s)，有时叫做泊肃叶(poiseuille)，以纪念19世纪法过物理学家泊肃叶（Jean-Louis Mare Poiseuille）。在研究中，流动的粘性也可用运动学粘度来表示，ν=µ/ρf，即流体的动态粘度与密度之比（参见文章Diffusion and propagation in air at rest），单位是m2/s。

[4] 严格地说，“迎风面”一词应指船体的竖直截面，“主迎风面”是最大的截面。因此这个海洋专业词汇被我们用来表示物体或航行器在流体中运动的等价面积。

[5] 在一个时间步长dt内，截面积为S的物体扫过的长度为Udt，则对应的流体质量为ρf SUdt。估算的动能为½ρfSU3dt，与阻力做功大小FUdt相当，这就得到阻力系数Cx=1。

[6] 伯努利(Daniel Bernoulli, 1700-1782)，瑞士医生、物理学家、数学家、哲学家，巴塞尔大学的教授，他的主要兴趣是对流动的研究。伯努利定理，于1738年在出版的水动力学著作中发表，是流体力学的基础理论之一。它表述为沿着流线p+ρfU2/2=p0，其中p是当地压强，p0是来流制止的压强。

[7] 超过声速时存在额外的阻力：下落的能量不仅用于推开周围的空气，还使得空气压缩，形成激波，即一种高振幅声波，产生声障。船舶的兴波阻力则与此不同，它是与重力有关的。

[8] 这个无量纲数称为弗劳德数，弗劳德（William Froude, 1810-1879），英国工程师、海军建筑师，他是第一位研究重力阻力（兴波阻力）的科学家。

[9] 弗劳德数表达了船体速度与重力波传播速度之比，与波长l有关。尾迹波传播速度与船体的速度相当。速度关系即cU可用来估算尾迹波长，当Fr=1时，它与船体长度相当。

[10] 参考 http://sportech.online.fr/sptc_idx.php?pge=spfr_xfd.html

[11] 相对于总能耗，机械能的消耗大约为1.9kW，效率约21%，参考http://www.agoravox.fr/culture-loisirs/sports/article/puissance-et-performance-en-159520，这部分能耗与呼吸的氧吸收有关。

[12] F.E. Fisch and G.V. Lauder, 2006, Ann. Rev. Fluid Mech. https://www.yumpu.com/en/document/view/46389631/passive-and-active-flow-control-by-swimming-fishes-and-mammals.

The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: MOREAU René, SOMMERIA Joël (April 12, 2024), 运动物体受到的阻力, Encyclopedia of the Environment, Accessed September 14, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/zh/physique-zh/drag-suffered-moving-bodies/.

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