Clearing certain misconception in the common explanations of the aerodynamic lift
CClearing certain misconception in the common explanations of the aerodynamic lift
Clearing certain misconception in thecommon explanations of the aerodynamiclift N avinder S ingh ∗ , K. S asikumar R aja , P. J anardhan Physical Research Laboratory, Ahmedabad, [email protected]
October 30, 2018
Abstract
Air travel has become one of the most common means of transportation. The most common questionwhich is generally asked is: How does an airplane gain lift? And the most common answer is via theBernoulli principle. It turns out that it is wrongly applied in common explanations, and there are certainmisconceptions. In an alternative explanation the push of air from below the wing is argued to be the liftgenerating force via Newton’s law. There are problems with this explanation too. In this paper we tryto clear these misconceptions, and the correct explanation, using the Lancaster-Prandtl circulation theory,is discussed. We argue that even the Lancaster-Prandtl theory at the zero angle of attack needs furtherinsights. To this end, we put forward a theory which is applicable at zero angle of attack. A new length scaleperpendicular to the lower surface of the wing is introduced and it turns out that the ratio of this lengthscale to the cord length of a wing is roughly ± for typical NACA airfoils that we analyzed.This invariance points to something fundamental. The idea of our theory is simple. The "squeezing" effectof the flow above the wing due to camber leads to an effective Venturi tube formation and leads to highervelocity over the upper surface of the wing and thereby reducing pressure according the Bernoulli theoremand generating lift. Thus at zero angle of attack there is no need to invoke vortex and anti-vortex pairgeneration. In fact vortex and anti-vortex pair generation cannot be justified. We come up with the equationfor the lift coefficient at zero angle of attack:C l = c (cid:90) c dx (cid:18) h c ( h c − f ( x )) − (cid:19) . Here C l is the lift coefficient, h c is our new length scale perpendicular to the lower surface of the wing andf ( x ) is the functional profile of the upper surface of the wing. a r X i v : . [ phy s i c s . pop - ph ] O c t . I ntroduction The issue of the mechanism of the aerody-namical lift is one of the most vexed one[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The reason is the com-plexity of a real fluid flow over an airfoil whichrenders inappropriate the direct or oversimpli-fied applicability of the standard argumentsrelated to Bernoulli’s theorem and Newton’sdynamical laws. In the next section, we con-sider them one by one.
II. W rong explanation no . 1: thestandard explanation and " distance " argument The standard explanation of aerodynamic lift isbased on an application of the Bernoulli theo-rem [7]. In this explanation of the aerodynamiclift over an airfoil, it is generally argued thatair flow is faster over the wing as compared tothat underneath it, and this is due to the pecu-liar shape of an airfoil. Faster air flow leads tolower pressure over the top of a wing due toBernoulli’s theorem as compared to its lowersurface, and thus aerodynamic lift is generateddue to pressure difference above and belowthe wing. In other words it is a suction lift.This is all fine, but the reason given for fasterairflow over the top of a wing is not correct.It is generally argued that air has to travel alonger distance on the upper surface due to ∗ Cell Phone: +919662680605; Landline:00917926314457. more curvature of upper surface as comparedto lower surface as air parcels which departat the leading edge has to meet at the trailingedge.This explanation based on "distance" argu-ment is fundamentally flawed. Air parcelsare not "living beings" that they have to meetagain! In fact two air parcels which depart atthe leading edge never meet again at the trail-ing edge, as the air parcel which flows overthe top of the wing has much more speed thanthat which flows underneath. The upper onealways reaches before the lower one. So, theexplanation based on the above “distance argu-ment” is not correct.
III. W rong explanation no . 2:N ewton ’ s action - reaction In the literature, an alternative explanation isalso found[1, 5]. In this explanation it is ar-gued that it is not the Bernoulli theorem thatleads to lesser pressure at the top and generateaerodynamic lift, rather it is the deflection ofthe air stream in the downward direction dueto an angle of attack of the wing. The down-ward deflected air reacts back on the wing viaaction-reaction law of Newton and imparts up-wards momentum to the wing thus lift. Inother words the lift is generated due to thepush of the air from below the wing.This explanation is also problamatic on manyaccounts:1. If we use Newton’s theory, the calcu-2learing certain misconception in the common explanations of the aerodynamic liftlated lift is proportional to angle of attacksquared, not linearly proportional to theangle of attack as observed experimentally[8].2. In reality air flow is not like bombardingbullets on the lower surface when the wingis at a finite angle of attack and deflectingbullets impart impulse in the opposite di-rection. Fluid flow is much more complex.The upper air stream which flows on thetop of the wing is also deflected in thedownward direction due the characteris-tic fluid motion (see Figure 1) that is theCoanda effect [11].
UpperLower
Figure 1:
Air is not only deflected by the lower surfacebut also by the upper surface due to Coandaeffect.
3. The calculated lift slowly due to this mech-anism (deflecting from the lower surface)is too less to account for the actual value oflift. In actual practice the suction lift dueto Bernoulli’s effect is an order of magni-tude more than lift due to deflection of airfrom the lower surface.4. Also, this explanation cannot explain thefinite value of lift at zero angle of attack, because then there is no deflecting surface.Another common problem is the wrong ap-plication of Newton’s action-reaction. It is notthe third law of Newton that is applicable here(third law is about the applied forces and reac-tion forces on a solid body, and their equal andopposite magnitudes). Rather it is the secondlaw that needs to be applied consistently. Therate of change of total momentum transferredin the downward direction (of the air mass de-flected by the lower surface of the wing plus the air mass deflected downwards by the up-per surface due to the characteristic motion offluid sometimes called the Coanda effect givesthe upward induced lift force. And as in thecase of spinning tennis ball, the calculationsdone by total momentum transfer method orby correct application of the Bernoulli theoremmust give consistent answers.But the question in the Bernoulli explanationremains unanswered: why do higher speedforms on the upper surface of the wing? Whatis the mechanism? In the next section an an-swer to this question is given.
IV. T he correct explanation : thecirculation theory of lift The correct explanation of the aerodynamic liftwas given by Lancaster and Prandtl and theircoworkers [8, 9]. It is based on the circulationtheory of lift. In the circulation theory of lift,in addition to the laminar flow around theairfoil there is also a circulatory flow around3learing certain misconception in the common explanations of the aerodynamic liftthe wing. This circulatory flow is generateddue to the formation of vortex and anti-vortexpair. The vortex formation around the wingleads to higher air speed at the top of the wing.Then Bernoulli’s theorem can be consistentlyapplied, and mechanism of the aerodynamiclift becomes evident. The circulation theoryis in good agreement with observations [8].To illustrate the Lancaster-Prandtl circulationtheory of lift we discuss the following pointsone by one:1. The mechanism of starting Vortex genera-tion.2. The Helmholtz theorem and Vortex - An-tivortex pair generation.3. Induced circulation around the wing pro-file, and the Kutta condition.4. Induced circulation and increased speedon the top surface of the wing.(1) Starting Vortex generation: The most im-portant ingredient in the circulation theory oflift is the generation of the starting vortex. Inthe standard literature it is only presented thatwith the generation of a vortex an anti-vortexmust be generated to satisfy the Helmholtz the-orem [8, 9]. But the question remain why thereshould be a vortex created when a wing at afinite angle of attack is moved from rest. Herewe present the actual reason behind it.When a wing is started from rest at a finiteangle of attack, the air flow is little faster onthe top of the leading edge due to effective
Starting vortex(a)(b)(c)Point BPoint A
Figure 2:
Starting vortex generation
Venturi formation. Similar effective venturi isformed below the trailing edge and leads tohigher speed of the air stream that is leavingfrom below the wing (point B in Figure 2a).The air stream that comes from the top of thewing slows down as the effective area of crosssection widens above the trailing edge (Figure2). Thus when these two air streams meet atthe trailing edge, the air stream which comesfrom below the wing will have higher speed(this whole scenario is true only at the startof the motion of the wing from its initial restposition). Due to finite viscosity of the air, theair streams from below the wing, which aremoving at higher speed tends to curl up as thestream from the upper surface has lower speed.
This curling up provides the seed for the formationof the starting vortex (Figure 2) which departs asthe wing move forward . With the generation ofstarting vortex we next consider the formationof vortex and anti-vortex pair.(2) The Helmholtz theorem and Vortex - An-4learing certain misconception in the common explanations of the aerodynamic lifttivortex pair generation: By Helmholtz the-orem, this starting vortex cannot exist alone.The net or total circulation in fluid must be con-served with time[8, 9]. If there is no circulationinitially in a fluid flow then a vortex genera-tion has to be accompanied with an anti-vortexgeneration so that net effect is the nullificationof the vortex antivortex pair. Actually, startingvortex generation due to speed difference ofthe two streams is the result of a finite viscos-ity i . Thus by Helmholtz theorem there is ananti-vortex generated around the wing sectionin response to the starting vortex (Figure 3). Circulation Starting vortex
Figure 3:
Vortex anti-vortex pair generation due toHelmholtz theorem. (3) Induced circulation around the wing pro-file, and the Kutta condition: With the abovepicture in mind it is easy to understand theKutta condition. In the literature Kutta con-dition is presented as smooth leaving of theflow from the trailing edge. But this statementhides the whole mechanism of the setting upof smooth flow at the trailing edge. And in theliterature it is generally applied in the calcu-lations of the lift coefficient without knowing i It is important to note that application of Helmholtztheorem in the present case is only an approximation, asfriction is unavoidable in a real flow. why this condition has to come. At the start ofa wing this condition is not justified. This con-dition is justified when steady state flow pat-tern sets up. Thus in authors’ opinion it shouldbe stated with this background made clear be-fore hand [8, 10]. Initially, the flow speed ofthe stream coming from the lower side of thewing is higher. Then with the generation ofthe starting vortex, an anti-vortex is generatedover the wing section. This anti vortex leads toincreased speed of the stream that leaves theupper surface. It will continue to increase untilthe speeds of both the streams match, and flowleaves the trailing edge smoothly: the Kuttacondition becomes satisfied.(4) Induced circulation and increased speedon the top surface of the wing: With this justi-fication of the Kutta condition one obtains anincreased speed of the air stream over the topof the wing span. One can resolve the totalflow over the wing surface into two compo-nents: (1) stream line flow due to the motionof the wing, and (2) circulatory flow due to thegeneration of anti-vortex (Figure 4).Now the stage is set for the consistent ap-plication of the Bernoulli theorem. The anti-vortex leads to higher speed over the top of thewing and lower speed comparably at the lowersurface of the wing (much like in the case ofspinning tennis ball). The higher speed at thetop of the wing leads to lower pressure andleading to the aerodynamic lift.This aerodynamic lift linearly increases withthe angle of attack as stronger circulation sets5learing certain misconception in the common explanations of the aerodynamic lift (a)(b)(c) In fi nite velocity about sharp trailing edgeFlow leaves trailingedge smoothly Figure 4:
The total flow over the top of a wing canthought of a superposition of circulatory flowand linear flow. around the wing. The quantitative calculationsagree very well with the observations [8].However, at larger angle of attack, flow sep-aration from the upper surface of the winghappens, and turbulence sets in. This leads tostall and loss of lift. The circulation theory isnot competent enough to account for it. It isnot theoretically possible to calculate the ex-act value of the angle of attack at which stallhappens within the Lancaster-Prandtl theory[8, 9]. Another problem with the circulationtheory of lift is that it is not possible to ex-plain the origin of a “seed” for the formationof the starting vortex at zero angle of attack. Atzero angle of attack there is no flow squeezingbelow the trailing edge of the wing for a flatbottom wing, thus the faster air stream frombelow the wing is not there; in other words, the‘seed’ for the starting vortex formation cannotbe justified. Due to this reason authors havedeveloped an alternative explanation for the observed lift seen at zero angle of attack forasymmetrical airfoils. The theory is as follows.
V. A different perspective at zeroangle of attack
In view of the above discussion, a theory ofaerodynamical lift at zero angle of attack froma cambered airfoil is developed. Our theorywhich we call "effective Venturi tube formation"theory is different from the standard circula-tion theory of Lancaster and Prandtl. The ideaof our theory is simple. The "squeezing" ef-fect of the flow above the wing due to camberleads to an effective Venturi tube formation andleads to higher velocity over the upper surfaceof the wing and thereby reducing pressure ac-cording the Bernoulli theorem and generatinglift. However, at finite angle of attack vortexgeneration has to be taken into account andthe total lift then is given by Lancaster-Prandtltheory. Our theory is only valid at zero angleof attack.
VI. F ormulation of the theory
Consider that we have a very long wing of cordlength c . Let f ( x ) be defines the curvatureof the upper surface of the wing (Figure 5).Wing’s leading edge is at the origin and itslower flat surface is along x-axis. Suppose thatwing is moved through air with velocity V ∞ or a laminar flow is set up over a stationerywing from left with the same velocity. Imaginea height h c along the y-axis (Figure 5).6learing certain misconception in the common explanations of the aerodynamic lift xy h y = f(x)o Flow direction c Figure 5:
Side view of the wing profile with various pa-rameters defined.
Consider two rectangular cross-sectional ar-eas of width L . The first one (situated far inthe upwind from the leading edge) has height h c (again measured from the x-axis). The othercross-sectional area is on the top of the wingwith same width L but with height h c − f ( x ) where f ( x ) is height of the wing due to cam-ber at position x . Then by continuity equation( A V = A V ) we have: h c LV ∞ = L ( h c − f ( x )) V ( x ) (1)where V ( x ) is the local flow velocity over thewing at position x . From the above equation v ( x ) = h c V ∞ h c − f ( x ) . (2)Clearly, flow speed is increased as compared to V ∞ and this leads to lower pressure accordingto the Bernoulli’s theorem:12 ρ V ∞ + P ∞ = ρ V ( x ) + P ( x ) . (3)Here ρ is the density of air assuming it to be in-compressible and P ( x ) is the local pressure atpoint x ( P ∞ is the free stream pressure). Fromthe above equation local pressure can be writ-ten as P ( x ) = P ∞ − ρ ( V ( x ) − V ∞ ) . (4)On substituting for V ( x ) in the above equa-tion and writing the reduction in pressure as ∆ P ( x ) = P ∞ − P ( x ) we have ∆ P ( x ) = ρ V ∞ (cid:18) h c ( h c − f ( x )) − (cid:19) . (5)And the lift generated is given by L f = L (cid:82) c dx ∆ P ( x ) with lift coefficent C l defined as C l = L f ρ V ∞ ( Lc ) . (6)We have C l = c (cid:90) c dx (cid:18) h c ( h c − f ( x )) − (cid:19) . (7)This is the other main result of the presentcontribution. VII. R esults
We apply our theory to some NACA air foilswith known experimental data, and find thatthe h c is more or less a constant number. Forexample, with cord length 1 cm, the func-tional profile of the upper surface of the airfoilmodel 2306 can be approximated as f ( x ) = − + x − x + x − x + x . It has experimentally de-termined lift coefficient C l = h c for this given value of C l and profile f ( x ) . Wefind that h c = C l h c No. Model f(x) (cm)(1) (2) (3) (4) (5)1 2306 − + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x − + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x + x − x + x − x + x Table 1:
The Table describes the various airfoil models, their profiles, experimentally measured lift coefficients and thecalculated h c values using the new theory for zero angle of attack. wing cord, above which flow can be treatedundisturbed. This is reasonable intuitively. Weinspected 10 airfoil models shown in column(2) of the Table 1 and their corresponding up-per surface profiles are listed in column (3) ofthe Table 1. The experimentally measured liftcoefficient ( C l ) [12], ii , iii , iv and theoretically cal-culated h c were listed in column (4) and (5)respectively. In our study of the 10 airfoils themean h c = h c = C l for the ii http://airfoiltools.com/airfoil/details?airfoil=n22-il iii http://airfoiltools.com/airfoil/details?airfoil=rhodesg32-il iv http://airfoiltools.com/airfoil/details?airfoil=s7055-il wing section of an airfoil model ‘6309’. Fromequation 7, we calculated C l and it is approxi-mately 0.3891 which is close to experimentallymeasured value 0.4043 for this wing section. VIII. C onclusion
In this article we have resolved various mis-conceptions and refuted various wrong expla-nations in the issue of the mechanism of theaerodynamic lift. Actual reason for higher airspeed on the top of the wing due to circu-lation mechanism is explained in the simplelanguage. To that end, the physical principlesof the Lancaster-Prandtl theory are explained.We also point out that at zero angle of attackthe generation of starting vortex and antivortex8learing certain misconception in the common explanations of the aerodynamic liftpair is questionable. The physical reason forthe lack of “seed” formation for vortex - an-tivortex pair is given. To this end we have de-veloped a theory at zero angle of attack whichseems to be in good agreement with the exper-imental data. R eferences [1] Jearl Walker. The Flying Circus of Physics .[2] D. L. Webster. What Shall We Say aboutAirplanes?
American Journal of Physics ,15:228–237, May 1947.[3] Klaus Weltner. A comparison of expla-nations of the aerodynamic lifting force.
Americal Journal of Physics , 1987.[4] N. H. Fletcher. Mechanics of flight.
PhysicsEducation , 10:385–389, July 1975.[5] Paul A Tipler and Gene Mosca.
Physics forscientists and engineers . Macmillan, 2007.[6] Eberhardt Scott Anderson, David W.
Un-derstanding flight . 2009.[7] JD Cutnell and KW Johnson. Physics (p.466.), 1998.[8] Theodore Von Kármán.
Aerodynamics .Number BOOK. Cornell University press:Mc Graw-Hill company, 1963.[9] Oskar Karl Gustav Tietjens and LudwigPrandtl.
Fundamentals of hydro-and aerome-chanics , volume 1. Courier Corporation,1957. [10] John David Anderson Jr.
Fundamentals ofaerodynamics . Tata McGraw-Hill Educa-tion, 2010.[11] I. Reba. Applications of the Coanda Effect.