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Paul G . Hewitt

PrincipleBernoulli’s Understanding

Bernoulli’s principleas it applies toaerodynamic lift

everal physics teachers I’ve recently spoken with saytheir understanding of Bernoulli’s principle is unclear,particularly when the principle is applied to aerody-

namic lift. Some teachers favor using Newton’s laws insteadof Bernoulli’s principle to explain the physics behind lift.Some also consider Bernoulli’s principle too difficult to ex-plain to students and avoid teaching it altogether. The fol-lowing simplified treatment of the principle ignores most ofthe complexities of aerodynamics and hopefully will encour-age teachers to bring Bernoulli back into the classroom.

S

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Daniel Bernoulli (1700-1782)

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Understanding BernoulliBernoulli’s principle states that when an incompress-ible, smoothly flowing fluid gains speed, internal pres-sure in the fluid decreases, and vice versa. Ignoringchanges in temperature and density, and energy dissi-pated by friction, Bernoulli’s principle can be expressedin units of pressure, 1/2 ρv2 + ρgy + p = constant,where ρ is the density of the fluid, v its speed, y itselevation, and p its internal pressure. If y doesn’tchange, then an increase in v means a decrease in p, andvice versa. When v = 0 (the motionless condition re-ferred to as hydrostatics), Bernoulli’s equation reduces tothe familiar ∆p = ρg∆y, which gives the increase inpressure with increasing depth in a motionless fluid.

Bernoulli’s equation is derived for the smooth steadyflow (laminar flow) of constant-density fluid. In practiceit works well even when density undergoes smallchanges. For flow speeds too great, the flow may be-come turbulent and follow changing, curling paths (ed-dies). This exerts friction on the fluid and transformssome of its energy to thermal energy. Then, Bernoulli’sequation doesn’t hold (although in practice it oftenworks surprisingly well!).

Fluid flow is best analyzed via streamlines—thesmooth paths, or trajectories, of fluid particles. Stream-lines are visible when dyes or smoke are passed throughevenly spaced openings, as in a wind tunnel. Becausethe flow is continuous, flow speed is greater where theflow is constricted, such as in a pipe when the cross-sectional area is narrowed. Faster fluid flow in narrowregions is a consequence of the principle of continuity.Squeezing the end of a garden hose, for example, in-creases the speed of ejected water.

Bernoulli’s equation applies to points on specificstreamlines. Changes in velocity and pressure shouldproperly be referenced along the same streamline (justas potential difference and gravitational potential en-ergy need reference points). The velocity and pressureat a point on one streamline should not be comparedwith the velocity and pressure on another streamline,especially for unparallel streamlines.

Demonstrating pressurePressure changes can be illustrated with a Venturi tube.Pressure changes are also nicely evident in water thatcontains air bubbles because bubbles provide a visualpressure reference (Figure 1). The volume of a bubbledepends on the surrounding water pressure. When wa-ter gains speed, pressure is lowered and the bubbles arebigger. In slower, higher-pressure water, bubbles aresqueezed smaller.

The decrease of fluid pressure with increasingspeed may at first seem surprising to students, particu-larly if they fail to distinguish between the pressurewithin the moving fluid (inappropriately called static

pressure) and the pressure by the fluid on somethingthat interferes with its flow (dynamic pressure).

Many people miss this distinction. My science men-tor 50 years ago, Jacque Fresco, a designer of novelaircraft, is one of a large group of accomplishedpeople who overlook the distinction between staticand dynamic pressure. He said that when he puts avane in the path of swiftly flowing water, pressure onthe vane increases with increasing water speed. Howcan pressure decrease when the experiment shows oth-erwise? When I was a physics student, I was similarlybothered. I had seen movie footage of people beingknocked off their feet by high-speed water from firehoses. How can we say pressure goes down as speedincreases? Again, the pressure within the water mustbe distinguished from the pressure exerted by the wa-ter on whatever interrupts its flow. Jets of high-speedwater are used to cut steel and have very little internalpressure. However, the stream exerts enormous pres-sure on the steel that halts it. When the momentum ofmoving water or anything else is suddenly reduced,the impulse exerted can be relatively huge.

The pitot tube—which sticks out in front of anairplane’s nose or wing into the undisturbed airflowand is used for determining airplane speed—illustratesthese different pressures (Figure 2). Air coming to restat the back of the tube’s central duct produces dynamicpressure. Small holes along the side of the tube, perpen-dicular to the flowing air, measure atmospheric pres-sure. Even though the air may be rushing by these holesat hundreds of kilometers per hour, the pressure of theair is atmospheric (at the airplane’s altitude). To seethis, consider a streamline at the holes and the samestreamline a long way in front of the plane. In bothlocations the speed of air relative to the plane is thesame (when you measure speed and pressure elsewherealong the same streamline, then you’re doing Bernoulliproperly). If soap bubbles were in the air to indicateambient air pressure, the bubbles would have the samesize at both locations—whether seen at rest or whizzingby at high speed. Therefore, atmospheric pressure infront of the plane and at the holes is the same, whichexplains why a pitot tube is often called a static pressuretube. Quite importantly, high-speed air doesn’t neces-sarily mean low-pressure air. Pressure differences arecaused by changes in the speed of air or any fluid; ifthere are no changes in speed along a streamline, thenthere will be no differences in pressure.

Applying the principleClassroom demonstrations of Bernoulli’s principle in-clude hovering a beach ball in an air stream, blowingbetween a pair of air-filled balloons, and blowing acrossa sheet of paper. The teacher can ask students if anyonehas noticed anything unusual about the soft top of a

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Wind blowing over a peaked roof is a good ex-ample to start with (Figure 3). Air pressure insidethe house may be the same as in the wind well re-moved from the house. Again, fast moving air doesnot necessarily mean low-pressure air. But above theroof, the flow of air is constrained in a way similar toair passing through a narrowed pipe. This is indi-cated by the crowded streamlines above the peakedroof. Curvature of the streamlines and their closenessindicate that pressure at that point is reduced. Thepressure difference on the roof due to inside and out-side air does not need to be very much because asmall average pressure difference over a large roofarea can produce a formidable upward force. Inshort, the roof experiences lift.

The blown-off roof can be viewed as similar to anairplane wing. In both cases a greater pressure belowpushes the roof and wing into a region of lesser pres-sure above. Wings come in a variety of designs. Whatthey all have in common is that air is made to flowfaster over the top surface than the bottom surface. Acommon misconception is that air flows faster overthe top because of a longer path along the top thanunderneath (air molecules on top move faster to keepup with molecules flowing below). But molecularmotion is random—even molecules in air at restdon’t stay close to one another, and there’s no reasonfor molecules to keep up with one another whileflowing over and under a wing. Air does flow fasterover the top of a wing experiencing lift, but not be-cause it resists shearing from the air below. The airflows faster because of continuity (for much the samereason air flows faster in a narrowed pipe or anyother constricted region).

Most often, but not always, different speeds of air-flow over and beneath a wing are accomplished by adifference in the curvatures (camber) of the upper andlower surfaces of the wing and an upward tilt (theangle of attack). The result is more crowded stream-lines along the top wing surface (Figure 4, p. 54).When the average pressure difference over the wing ismultiplied by the surface area of the wing, a net up-ward force is attained—the vertical component beinglift (or more correctly, the component of force perpen-dicular to the direction of flight). Lift is greater whenthe plane has a large wing area and is traveling fast. Aglider does not have to travel fast due to a very largewing area relative to its weight. At the other extreme,a fighter plane designed for high speed has a smallwing area relative to its weight. Consequently, it musttake off and land at high speeds.

Interestingly, wings without camber provide lift.Balsa-wood model planes, for example, have flatwings. The slot that holds the wing is cut at a smallangle to give it an angle of attack. Airflow above and

convertible car in motion. The top puffs up—anotherdemonstration of Bernoulli’s principle.

A puzzler for many teachers, however, is aerody-namic lift. Some controversy exists as to whether lift isbest explained by Bernoulli’s principle or by Newton’slaws (Smith 1972). I submit that in most cases Ber-noulli is the key. Just as we ignore forces when apply-ing momentum conservation to collisions, we can dothe same with Bernoulli’s principle—knowledge ofpressure differences is all we need.

F I G U R E 1

Pressure changes are nicely evident in water thatcontains air bubbles.

F I G U R E 2

The pitot tube illustrates different pressures.

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Mercury

F I G U R E 3

Wind blowing over a peaked roof illustratesBernoulli’s principle.

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below the wing is similar to that of the camberedwing—greater pressure below and less on top.

Years ago I did a lot of painting, mainly on Masoniteboard rather than on canvas. Masonite board is sold in 4× 8 ft. slabs. I remember buying a slab and putting it onthe roof of my car. The salesperson was concerned that Irefused his offer to tie it down. Nope, I said, we’re lessthan a mile away so save the rope—my wife and I willhold it down by hand. What a surprise to find that thedrooped board made an airfoil, which allowed a top

speed of 25 kph. Any greater speed would rip the boardfrom our grip. This experience added to my understand-ing that airliners weighing many tons can fly. Theirwings have more surface area and they go considerablyfaster than 25 kph!

Early in my teaching career I put a lot of effort intotrying to explain aerodynamic lift in terms of billiard-ball physics, via differences in molecular impacts on thetop and bottom surfaces of wings. All I achieved was asense of futility, especially when experiments showedthat molecules flowing past the wing surface don’tmake direct impact on the surface anyway. Impact isinstead on thin boundary layers of air that are carriedwith the wing—layers with circulation patterns of theirown that can greatly alter the effective airfoil shape. Iended my quest for a molecular-collision model when afriend showed me dust on the surface of whirling fanblades. A small wonder that aerodynamicists investi-gate parcels of air rather than molecules. I’ve since beenresigned to hydrodynamics being too complex for a bil-liard-ball physics explanation.

Some physicists attribute aerodynamic lift to onlyNewton’s laws, maintaining that lift occurs because air is

Demonstrating Bernoulli’s principlein the classroom.Soft drink cans.Begin by suspending a pair of soft drink cans fromstrings so that approximately a centimeter of spaceexists between the cans. Students then blow be-tween the cans. Their breath increases in speed as itmoves between the cans, reducing pressure there.The cans will then click together as Bernoulli’s prin-ciple predicts. The cans can also be placed on a fewparallel straws on a tabletop. When students blowbetween the cans, the cans roll together and click,once again demonstrating Bernoulli’s principle.

Table tennis.Students begin this activity by lining a cardboardtube with sandpaper. Next, students make a pre-diction about what will happen when they sling atable tennis ball from the tube sidearm, and whetherthey can reproduce their results or improve theirlauncher. The sandpaper will produce friction thatwill make the ball skid down the tube and emergespinning. Students will observe that the ball breaksfrom a straight-line path (streamlines around a spin-ning ball are asymmetric and produce curving ofthe ball). This is the reason that table tennis paddleshave a rough surface—so that the ball spins whenit is properly struck.

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Closer streamlines above the wing's top surfaceindicate reduced pressure. Differences in pressurebetween the bottom and top wing surfaces producean upward force—with components of lift and drag.

Lift

Drag

F I G U R E 5

Very little downward deflection of streamlines isshown in this typical wind-tunnel photo.

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constantly being pushed downward. Bernoulli is seen assuperfluous, and lift is seen as the result of downwarddeflected air (Waltham 1998; Swartz 1999).

By whatever means, if air produces an upwardforce on a wing, the wing produces a downward forceon the air—no exceptions! But if other forces act onthe air, it may not acquire downward momentum.Where, for example, is the downward momentum ofair inside the blown-off peaked roof in a hurricane orinside the convertible car with the puffed-up top? In-terior air is unaccelerated as the roof pushes down onit and the floor pushes up. Downward reaction forceon the interior air contributes to its atmospheric pres-sure—not to any downward momentum. The interiorair goes nowhere. Advocates of non-Bernoullian liftmight say that air behind the peaked roof or convert-ible top is swept downward. But that air isn’t what’spushing up on the roof or top. The air in the interiordoes that. Neither Newton nor the impulse-momen-tum theorem explains lift for the roof or car top. Pres-sure difference does, however, and that’s whatBernoulli’s principle is all about.

When wind blows against a wall and bounces fromthe wall, the force on the wall isn’t affected by whathappens to the wind after it bounces. A golf player mayexert more force on a ball when he or she followsthrough on their swing. However, after the ball leavesthe club, subsequent swing motion doesn’t affect the ball.Similarly, when air interacts with a wing, what happensto the air after the interaction doesn’t affect lift. Whetherthe air is downwashed or rides on the cushion of airbelow, after interaction, lift is no longer affected.

When one blows horizontally across the curved topof a sheet of paper, air pressure there is reduced. Thepaper is pushed upward by the atmospheric pressurebelow. Whereas the bottom of an aircraft wing canpush oncoming air downward, the sheet of paper can-

not. Any downward push on low-pressure air crossingover the top of the paper is by the weight of some 30km of air above. Likewise for blown-off roofs, convert-ible cars, and airplane wings.

Very little downward deflection of streamlines isshown in the typical wind-tunnel photo shown in Fig-ure 5. The more crowded streamlines above the airfoiland wider-spaced ones below indicate differences inpressure that produce lift. Even with no downwash atall, if the average pressure difference on the wing mul-tiplied by the wing area equals the weight of the craft,lift is sufficient for flight. If air careens off the bottomproducing downwash, then even greater pressure is ex-erted on the lower surface of the wing, producing agreater upward force.

There are situations, however, where lift is better ex-plained by Newton. Consider, for example, the upwardmotion of a piece of plywood held at an angle when astream of water from a fire hose is deflected off it. Thelift force should be very close to the rate of change ofmomentum of the deflected stream, with no other sig-nificant force deflecting the stream other than that pro-vided by the slab of wood. Invoking Bernoulli’s principlewould be comparing water pressure on the bottom of theslab to nonexistent water pressure on the top.

Is the little girl with the broken umbrella in Figure6 correct? Or should she credit Isaac Newton for hermisfortune? And when students are asked if birdswere able to fly before the time of Daniel Bernoulli,should teachers further ask if birds could fly beforethe time of Isaac Newton?

The next time you’re in an airplane marveling at thewonder of such an auditorium in flight, make a toast toDaniel Bernoulli. Go ahead and make a secondary toastto Newton if you wish. Unless you’re in a helicopter,which is when you’d want to make your primary toastto Isaac Newton. Nevertheless, don’t be hesitant toteach Bernoulli’s principle in the classroom…it’s defi-nitely worth exploring. n

Paul G. Hewitt is Professor Emeritus at the City Collegeof San Francisco, San Francisco, CA 94112; e-mail:pghewitt@aol.com.

Acknowledgments

This article is the outgrowth of wonderful discussions with friendsPaul Doherty, Marshall Ellenstein, Ken Ford, and Walter Steiger.

References

Smith, N.F. 1972. Bernoulli and Newton in fluid dynamics. ThePhysics Teacher 10:451–455.

Swartz, C. 1999. Quibbles, misunderstandings, and egregious mis-takes. The Physics Teacher 37:297–305.

Waltham, C. 1998. Flight without Bernoulli. The Physics Teacher36:457–462.

F I G U R E 6

Is the little girl in this picture correct? Or should shecredit Isaac Newton for her misfortune?

COURTESY OF THE AUTHOR