the choice between bernoulli's or newton's model in ... choice between bernoulli's or...

PROFESSIONAL PERSPECTIVES

INTERNATIONAL JOURNAL OF SPORT BIOMECHANICS, 1990, 6, 235-245

The Choice Between Bernoulli's or Newton's Model in Predicting

Dynamic Lift

Eric J. Sprigings and James A. Koehler

This paper questions the appropriateness of using a model based on Bernoulli's theorem to explain dynamic lift in sport. The authors discuss the relative merits of an alternative model based on Newton's second and third principles.

The forces acting on a body as a result of its motion through a fluid are of fundamental interest to sport biomechanists. This is especially true for aquatic sports but it also holds true for general motion of a body through air. Only under special circumstances where the relative magnitudes of the fluid forces are small compared to the other external forces acting on the body can these fluid forces be ignored. It is important from a sport performance point of view not only to be aware that such forces exist but also to try and develop strategies that can best take advantage of these forces. In order for the biomechanist to predict the magnitude and direction of the fluid forces that will act on the body under a given set of conditions, he or she first needs a model to work from. Generally biomechanists have drawn upon a number of established models in fluid mechanics to explain such phenomena as lift, drag, buoyancy, and hydrostatic pressure (Brancazio, 1984). However, without a doubt it is the phenomenon of lift that has attracted the most attention in sport in recent years (Allman, 1987; Held, 1983; Mason, Sweetham, & Pursley, 1985; Rernizov, 1984; Soong, 1982; Watanabe, 1983). It is with regard to the theoretical models used to explain this lift phenomenon that this paper will focus upon.

Lift is defined as the component of force that acts on a body in a direction perpendicular to the path of the fluid flow. It is this component of force that is responsible for the observed curve in a baseball pitch, the sliced path of a golf ball, or the extra support, for example, that one obtains in the water from sculling the hands. It has become increasingly evident in swimming (Brown & Counsilman, 1971; Counsilman, 1971; de Groot & Ingen Schenau, 1988; Maglischo, 1982; Schleihauf, 1979, 1983) that the best competitive swimmers use predominantly

Eric J. Sprigings is with the College of Physical Education, University of Saskatchewan, Saskatoon, Sask. S7N OWO, Canada. J.A. Koehler is with the Institute of Space and Atmospheric Physics, University of Saskatchewan.

236 Sprigings and Koehler

lift in breaststroke to propel themselves through the water and that the other three competitive strokes also rely to some extent on lift to provide forward propulsion. In order to take advantage of this lift phenomenon, a theoretical model must be employed that is conceptually simple yet capable of allowing the user to draw the proper connection between cause and effect.

From the current sport literature (Brancazio, 1984; Counsilman, 1977; de Groot & Ingen Schenau, 1988; Hay, 1985; Kreighbaum & Barthels, 1985; Maglischo, 1982; Schleihauf, 1977; Townend, 1984) it is evident that the two most common theoretical models used to explain the lift phenomenon are Bernoulli's principle for nonspinning objects and the Magnus effect for spherical spinning objects. Generally, in most discussions on lift for nonspinning objects, the explanation is given in terms of the asymmetrical shape of an airplane wing. It is generally stated that aircraft fly because airfoils have a longer upper surface than lower one and thus the air flowing over the top has a longer distance to travel compared to air flowing below the wing and hence must travel faster. The author next invokes Bernoulli's theorem to explain that the faster flow of air over the top of the wing as compared to the bottom of the wing will result in a relative decrease in pressure for the wing's upper surface. This pressure differential between one side of the wing and the other is then credited with producing the so-called lift effect.

For spinning spherical objects, the term Magnus effect (Hay, 1985) is employed to describe the resulting curved path taken by such an object. However, the explanation as to why a spinning ball will curve is still given in terms of Bernoulli's principle. The gist of the argument is based on the relative velocity of air flow moving past one side of the ball in comparison to the other. The idea is that the spinning ball carries around with it a thin boundary layer of air that will slow down the oncoming air on one side and speed up the flow on the other, thus creating a difference in pressure (Bernoulli's theorem) between the two sides of the ball. This difference in pressure is then credited with creating an unbalanced force on the ball which then presumably is responsible for the ball's curved flight.

Both of these models have a long history of use in most introductory physics textbooks (Beyer & Williams, 1957; Gettys, Keller, & Skove, 1989; Ostdiek & Bord, 1987; Resnick & Halliday, 1966) for explaining the principle of dynamic lift, and it is for this reason that authors in sport literature have tried to adhere to these models in explaining the lift phenomenon in athletics. However, it is most evident that these same sport authors frequently find themselves struggling in their attempt to extract meaningful information from the Bernoulli model as to how best to improve the lift component. The reason for this is that the Bernoulli model conjures up in our mind the picture of an asymmetrical wing, which tends to focus our attention on shape as the primary variable that governs lift. For example, Maglischo (1982) begins his explanation of the lift phenomenon by invok- ing Bernoulli's theorem to explain the flow dynamics around a curved upper surface of an airplane wing. From this often cited example, he concludes, "The amount of lift force is proportional to the difference in pressure between the two wing surfaces which is, in turn, dependent upon the shape of the wing surfaces and the forward speed of the airplane" (p. 12).

Having made this seemingly obligatory reference to Bernoulli's theorem in explaining lift, Maglischo next introduces the key variable that governs lift, which is the angle of attack. However, it is clearly evident that his explanation

Predicting Dynamic Lift 237

of lift, in terms of this variable, is not based upon Bernoulli's principle but instead upon Newton's third law: "the wing is pitched upward at an angle of 25 degrees, causing an increase in lift force. This occurs because the air passing under the wing is deflected downward. In accordance with Newton's third law of motion, for every action there is an equal and opposite reaction, the deflection of air downward causes a counterforce to be exerted on the wing in an upward direction" @. 12). Thus, even though tradition calls for lift to be explained in terms of a model based on Bernoulli's theorem, it is clear that Maglischo found the model lacking in the conceptual ease with which it could be used to explain the importance of the angle of attack in developing lift.

Counsilman (1971), in his classic piece of work, states, "A wing provides aerodynamic lift through the camber (curvature) of its surfaces. Because the upper surface is more highly cambered than the lower surface, the air moving over the top surface is forced to move more quickly. This results in a lower pressure on the upper surface as compared with the lower surface and results in aerodynamic lift (Bernoulli's Principle)" (p. 61). Again it is evident that this commonly used example of a cambered airplane wing has focused attention on shape rather than on the angle of attack. In truth the statement is very misleading. It would be a simple matter to orientate the leading edge of the wing at an attack angle that would produce zero lift, or even negative lift, even though the shape of the wing was not altered. In fairness, Counsilman does arrive at the correct conclusion that the pitch of the hand (i.e., angle of attack) is more responsible for lift than camber, but it is doubtful whether such a conclusion was reached by strictly adher- ing to the model explained in his statement above.

The use of the Bernoulli model to explain lift tends to suggest that small changes in shape should cause large changes in lift. In reality, however, this is not the case. As any aeronautical engineer will tell you, aircraft performance is not very sensitive to airfoil shape. Yes, shape is a factor in that it can be used to provide a smooth directional change for the fluid flow past the wing, but it is only one of a number of factors, the primary one being the angle of attack. This is certainly in agreement with our everyday experiences. For example, if a person were driving along in a car and stuck her hand out the window, it would become immediately obvious to her that it is the angle of attack of her hand into the wind that plays the major role in lift, not the shape of the hand. Curving the back of the hand so as to make a nonsymmetrical airfoil with a longer upper surface than a lower one may cause some change in lift, but it is not as important as the angle of attack. Further, a superficial application of Bernoulli's model would suggest that one could not obtain lift from the oncoming air if the top and bottom of the hand assumed a symmetrical shape. This is certainly not the case, as will quickly be attested to the upward deflection of a flat plate (i.e., symmetrical top and bottom) that is held at an angle to the oncoming air.

This paper does not dispute the correctness of Bernoulli's theorem. How- ever, it does question the appropriateness of using Bernoulli's theorem to explain the lift phenomenon. As we will see, there is an alternative model that is more comprehensive and certainly easier to use.

To be useful from a qualitative point of view, the Bernoulli model must be able to provide us with a clear visual understanding of the factors responsible for altering the flow rate over the top and bottom portions of the wing. This is certainly not the case. As was mentioned previously, the model tends to focus


our attention more on the shape of the object than on its angle of attack. To use Bernoulli's theorem in a quantitative manner to calculate lift, one must have a detailed knowledge of what the air speeds are everywhere in the vicinity of the lifting surface. This kind of information is difficult to obtain even by experiment (e.g., using models of airfoils in wind tunnels), and it would be nearly impossible to obtain for such irregular and flexible surfaces as the hands of swimmers in water.

Fortunately, there is an alternative way of explaining lift that is just as valid and lends itself more readily to both qualitative and quantitative analyses. That method is to use Newton's principles, specifically Newton's second and third principles. Consider the case of an airfoil moving through air. As the air mass comes in contact with the airfoil, two things occur; the air mass is slowed down and its direction is changed. In other words, the momentum of the air mass has been changed. From Newton's second principle we know that the airfoil must have applied a force to the air mass for this to have occurred. We also know from Newton's third principle that the airfoil will receive an equal and opposite force back on itself. It is this reaction force back on the airfoil that we are primarily interested in. From it, the components of lift and drag can easily be determined. The fact that both lift and drag are predicted is in itself an advantage over the Bernoulli model. The Bernoulli model only attempts to explain the lift component; no attempt is made to predict drag. In practical applications both of these components must be known. For example, in a swimming stroke it is usually the swimmer's intent to maximize the component of the reaction force that acts in the forward direction. The lift component is just a portion of the total reaction force and, as would be expected, does not always point in the intended direction of travel. Thus, to dwell exclusively on the lift component without any attention to the drag component would be an analytical blunder.

The Newtonian approach lends itself nicely to both qualitative and quantitative analyses. To be used quantitatively, one must be able to measure both the magnitude and the direction of change in momentum of the fluid. To some extent the criticism leveled at the Bernoulli theorem explanation, namely that it requires additional information in order to be used quantitatively, is also true for the Newtonian approach. To make a precise estimate of lift requires precise knowledge of the change in motion of all parts of the working fluid (air in the case of wings or water in the case of a swimmer's hands). In general, this sort of information can be measured empirically with the help of streamline photographs, and from this information, accurate estimates of lift and drag can be determined for the airfoil itself.

In sport, a real strength of the Newtonian approach is that many useful ap- proximations can be made in order to provide us with useful qualitative information that can be used in the decision making process as to what constitutes good technique. We can estimate the forces that are involved as well as determine what variables affect the lift (and the drag) generated. By making a number of simplifying assumptions, we can derive an approximate relationship between things such as velocity and lift, which gives us insight into the nature of the processes. The following is an example of the information that can be extracted using this approach.

Consider a swimmer's hand sculling through the water, as shown in Figure la. We shall initially approximate the hand by an infinitely thin sheet of length, 1, and of width, w, which has the same projected area "A" as the swimmer's


Figure 1 - (a) Motion of a swimmer's hand in water; (b) Equivalent surface area of swimmer's hand.

Figure 2 - Artist's impression of streamline flow around flat plate.

hand. Consider this area moving through the water with some angle of attack, a, as shown in Figure lb.

Figure 2 shows the approximate fluid flow in cross-section. As stated before, without streamline photographs or a more complex mathematical model we cannot know precisely how the fluid behaves in the vicinity of the plate, thus


we shall approximate the fluid flow by making the following simple assumption- that all the fluid in the area of the plate projected along the original direction of flow is deflected downward through an angle a. For the time being we will also assume that the magnitude of the fluid velocity does not change during this deflection. As well, in the subsequent analysis it is convenient to assume that the plate is stationary and that the fluid flows past it; this is a simple change of reference system and has no effect on the results.

Let the initial fluid flow velocity, vi, be as shown in Figure 3; then after passing the leading edge of the plate, the fluid flow direction will be v,. The resulting change in velocity will be dv. Since the mass of the fluid has had its momentum changed, there must have been a force exerted on it in the direction of dv. This force is a consequence of Newton's second principle:

F = dpldt (1)

Figure 3 - A fluid particle's interaction with the hand assuming no friction forces.

To calculate the change in momentum per unit time, we need to consider what mass of fluid is affected per second (the mass of fluid that would flow through the projected area of the plate per second) multiplied by the change in velocity. The projected sea is A sin(a), and letting g be the density of the fluid and v be the magnitude of the velocity, vi, we get

v A g &(a)

for the mass of fluid affected per unit time so that Equation 2 below gives the change in momentum per unit of time;

(V A g sin(a)) dv (2)

where dv is a vector quantity as shown in Figure 3 and has a magnitude of

2v sin(a12). (3)


The force on the plate is then equal in magnitude to this force and opposite in direction. From Equations 2 and 3, the magnitude is

The direction of this force is not quite perpendicular to the direction of initial flow, so the resultant force given by Equation 4 will have a component of force along the direction of fluid flow and a component perpendicular to the direction of flow (Figure 4). These two components are called the induced drag @) and the lift (L,), respectively. The induced drag gets its name from its direction; it is in the direction of the fluid relative to the object. In the case of a swimmer's hand, it would be the direction opposite to its motion relative to the water. In either case its existence is a result only of the directional change that the fluid has undergone. Thus, induced drag can be considered a consequence of the creation of lift and is quite separate from either profile or surface drag, which will be discussed later. Using Figure 4, it is easy to show that,

and

It is worthwhile now to examine some of the characteristics of Equation 4. Suppose the angle of attack is very small, less than about lo0, for example. We can then make the usual estimate that the sine of the angle is approximately equal to the angle expressed in radians. Under these conditions, lift is proportional to the square of the velocity for any given angle of attack and is highly dependent on the angle of attack; in this simple approximate model, it is propor-

Force on Hand

- + 2 Y-

Direction of -I

Motion of Hand JA

Drag, D

Figure 4 - Breakdown of the resultant dynamic fluid force into components of induced lift and drag.


tional to the square of the angle of attack. In real life, airfoils may be cambered (curved in cross-section) rather than just flat and have generally complex shapes. For this reason, the exact way in which the lift depends on the angle of attack may differ from this admittedly simple model. In aeronautical textbooks, the lift is usually described by the following equation:

where CL is called the coefficient of lift and is a quantity whose dependence on the angle of attack is measured for any particular airfoil and is normally displayed graphically. It is this dependence that is shown, for example, in the article by Schleihauf (1983).

In the derivation of the above equations, the assumption was made that the magnitude of the fluid velocity did not change during deflection by the hand. This was a simplifying assumption that was useful in showing the nature of induced drag but, as we will see, certainly not a necessary one. Let us now look at a more realistic case in which the fluid is not only deflected by the airfoil but is also slowed down. The slowing down of the deflected fluid can easily be in- corporated into the model, as illustrated in Figure 5.

Figure 5 - A fluid particle's interaction with the hand assuming friction forces to be present.

As can be seen, the velocity vector v, is given the same angle of deflection as in Figure 3, but its length is shortened to depict the decrease in the fluid's speed after contacting the leading edge of the hand. Using simple vector addition, where vi + dv = v,, we can determine the vector dv which represents the fluid's change in velocity. Knowledge of the direction of dv allows one to predict, using Newton's third principle, the direction of the reaction force that acts back on the hand. As before, this reaction force can be broken down into its components of lift and drag. However, this time the component of drag represents not only what we earlier referred to as induced drag but also to two other forms of drag, which a r e generally called profile and surface drag. (Wave drag also-provides a source of resistance against a swimmer's progress through the water, but its


effects will not be considered in this paper since we are primarily interested in predicting the forces that act on airfoils, or hydrofoils such as a swimmer's hand, which operate predominantly in a single medium.)

Surface drag, as the name would suggest, occurs as a result of the retarding frictional forces between the surface of the airfoil and the fluid. As the fluid passes over the surface of the airfoil, frictional forces are created that slow both the fluid and the airfoil down. Surface drag is a function of the roughness of the surface, the density of the fluid, the surface area, and the relative fluid velocity.

Profile drag results from the fact that any real body moving through a fluid will have to displace fluid as it moves. Even a spherical, nonspinning ball moving through air-which does not show any lift and hence no induced drag-will have to displace air as it moves and thus impart some momentum to the air. This will result in a net retarding force on the ball called profile drag, sometimes referred to as form drag. The profile drag is proportional to the density of the fluid, to the square of the velocity, to the projected area along the direction of flow, and to some coefficient that depends on the shape of the moving body.

In aeronautical textbooks, the total component of drag is usually expressed as

where CD represents the coefficient of drag. Using the Newtonian model, it is evident that this coefficient of drag for a lifting surface is normally composed of three parts. The first part is the induced drag, which varies with the square of the velocity and is highly dependent on the angle of attack. The other two parts, surface drag and profile drag, are also dependent on the square of the velocity and to some extent on the angle of attack. In practice, CD is a measured quantity usually displayed, like the coefficient of lift, as a graphical function of the angle of attack.

From the foregoing discussion, it is clear that the Newtonian model can predict factors that determine both the lift and drag of any body acting as an airfoil (or hydrofoil). However, as was alluded to earlier, the use of the model does not end here but can just as easily be used to predict the curved flight of a rotating spherical object. Most physics textbooks generally draw upon the Magnus effect to explain this curved flight, but in truth the Magnus effect predicts nothing that could not be obtained from the Newtonian model. If one looks at wind tunnel photographs of air flowing past a rotating sphere (Zafiratos, 1976), it is readily apparent that the air flow in the vicinity of the sphere has been disturbed. More air mass is deflected to one side of the sphere than the other. It is this asymmetrical deflection pattern of the air mass that can be used to predict (Newton's 2nd and 3rd principles) the resulting force that acts on the rotating sphere. From this resultant force arises the component of lift which has traditionally been accounted for by the so-called Magnus effect.

The strength of the Newtonian model in explaining curved motion of a rotating sphere is that it considers other factors besides rotation in accounting for the resultant lift component of force. This becomes important if we look at the example of a knuckleball pitch. A model based on the Magnus effect has great difficulty trying to explain the erratic flight behavior of a knuckleball. That is because a well thrown knuckleball only rotates about a quarter revolution on its way to home plate (Allman, 1987). As would be expected, this fact poses a serious problem to a model such as the Magnus effect, which relies strictly on the interaction of a rotating boundary layer of air with the oncoming airflow to explain


the erratic curvature of the ball's flight path. It is obvious that some variable other than angular velocity must be responsible for the ball's erratic motion.

Watts and Sawyer (1975), who conducted wind tunnel tests on the knuckleball, concluded that it is the changing profile presented by the seams that is primarily responsible for the erratic behavior of the knuckleball. Apparently, the subtle changes in the seam profile disturb the oncoming airflow asymmetrically. That is, the air mass is deflected one way, then the other, depending on the particular profile that the slowly rotating ball presents to the oncoming air. While no one would suggest that the erratic motion of the knuckleball can be predicted easily with any model, it is at least possible to show through wind tunnel tests that the results are consistent with the Newtonian model. A model based on the Magnus effect is unable to make such a claim.

Summary

The intent of this paper is not to question whether Bernoulli's principle can be used to explain lift; it most certainly can. However, that model suffers from the complexity of the information that one needs for applying it successfully. Our point is that a model based on Newton's laws provides a simple, direct, yet more comprehensive explanation of lift. We feel that from a practical point of view in sports, the predictive potential of a model based on Newton's laws is superior to that of one based on Bernoulli's theorem.

The main features of lift and drag can be derived using an approximate method that relies on the application of Newton's second and third principles rather than Bernoulli's equation. The advantages of the Newtonian model, in comparison with the Bernoulli model, are that (a) it can be used to provide both qualitative and quantitative results; (b) it is intuitively easier to understand; (c) it allows one to derive relatively simple equations that show, to a reasonable first approximation, the level of dependency that the components of lift and drag have on certain variables that are controllable by the athlete; (d) it provides us with a model that is not only capable of predicting lift but also one that is capable of accounting for the different types of drag; and finally, (e) the Newtonian model allows us to predict the total reaction force that results from the dynamic interaction with the fluid. From a practical point of view, this is usually a more meaningful result since good sport technique normally calls for the maximizing of the total forward force component and not just the maximizing of lift.

References

Allman, W. (1987). Pitching rainbows: The untold physics of the curve ball. In E. Schrier & W. Allman (Eds.), Newton at the bat: The science in sports (pp. 3-14). New York: Charles Scribner's Sons.

Allman, W. (1987). What makes a knuckleball dance? In E. Schrier & W. Allman (Eds.), Newton at the bat: The science in sports (pp. 15-19). New York: Charles Scribner's Sons.

Beyer, R., & Williams, A. (1957). College physics (pp. 208-210). Englewood Cliffs, NJ: Prentice-Hall.

Brancazio, P. (1984). Sport science (pp. 359-379). New York: Simon & Schuster.


Brown, R., & Counsilman, J. (1971). The role of lift in propelling the swimmer. In J. Cooper (Ed.), Selected topics on biomechanics: Proceedings of the C.Z. C. Symposium on Biomechanics @p. 179-188). Chicago: The Athletic Institute.

Counsilman, J. (1971). The application of Bernoulli's Principle to human propulsion in water. In L. Lewillie & J. Clarys (Eds.), First Zntemtional Symposium on Bio- mechanics of Swimming @p. 59-71). Brussels, Belgium: Universit6 Libre de Bruxelles.

Counsilman, J. (1977). Competitive swimming manual (pp. 224-231). Bioomington, IN: Counsilman Co., Inc.

de Groot, G., & Ingen Schenau, G. (1988). Fundamental mechanics applied to swimming: Technique and propelliig efficiency. In B. Ungerechts, K. Wilke, & K. Reischle (Eds.), Swimming science V @p. 17-29). Champaign, IL: Human Kinetics.

Gettys, W., Keller, F., & Skove, M. (1989). Classical and modem physics @p. 336-342). Toronto: McGraw-Hill.

Hay, J. (1985). The biomechanics of sports techniques (3rd ed., pp. 185-187). Englewood Cliffs, NJ: Prentice-Hall.

Held, D. (1983). Javelin: Aerodynamics and selection. Track Technique, 84, 2671. Kreighbaum, E., & Barthels, K. (1985). Biomechanics: A qualitative approach for studying

human movement (2nd ed., pp. 401-416). New York: Macmillan. Maglischo, E. (1982). Swimming faster (pp. 11-46). Palo Alto, CA: Mayfield. Mason, B., Sweetham, B., & Pursley, D. (1985). Angling in on getting a lift. Swimming

Technique, 21, 20-26. Ostdiek, V., & Bord, D. (1987). Inquiry intophysics (pp. 160-163). New York: West. Remizov, L. (1984). Biomechanics of optimal flight in ski-jumping. Jouml of Bio-

mechanics, 17, 167-171. Resnick, R., & Halliday, D. (1966). Physics @p. 449-451). Toronto: Wiley & Sons. Schleihauf, R. (1977). A biomechanical analysis of freestyle aquatic skill. In J. Counsilrnan,

Competitive swimming manual @p. 232-247). Bloomington, IN: Counsilman Co., Inc.

Schleihauf, R. (1979). A hydrodynamic analysis of swimming propulsion. In J. Terauds & W. Bedingfield (Eds.), Swimming ZZZ (pp. 70-109). Baltimore: University Park Press.

Schleihauf, R. (1983). Swimming propulsion: A hydrodynamic approach. In G. Wood (Ed.), Collected papers on sports biomechanics (pp. 165- 175). Nedlands: The University of Western Australia.

Soong, T. (1982). Biomechanical analysis and applications of shot put and discus and javelin throws. In D. Ghista (Ed.), Human body dynamics: Impact, occupational, and athletic aspects @p. 462497). London: Clarendon Press.

Townend, S. (1984). Mathematics in sport (pp. 108-123). Toronto: Wiley & Sons. Watanabe, K. (1983). Aerodynamic investigation of arm position during the flight phase

in ski jumping. In H. Matsui & K. Kobayaski (Eds.), Biomechanics VZZZ-B: Proceedings of the Eighth Zntemtional Congress of Biomechanics @p. 856-860). Champaign, IL: Human Kinetics.

Watts, R., & Sawyer, E. (1975). Aerodynamics of a knuckleball. American Jouml of Physics. 43, 960-963.

Zafiratos, C. (1976). Physics @. 381). Toronto: Wiley & Sons.

the choice between bernoulli's or newton's model in ... choice between bernoulli's or...

Documents