On Wings of Minimum Induced Drag

Contents

1 Introduction

In 1933, Ludwig Prandtl published a paper titled “Über Tragflügel kleinsten induzierten Widerstandes” (“On Wings of Minimum Induced Drag”) (Prandtl, 1933). In this paper, he postulated that the elliptic lift districution, commonly considered to be optimum, no longer results in minimum induced drag when structural considerations are taken into account and the wing span constraint is removed.

Unfortunately, his paper was never published in the English language. Copyright expired on 15 August 2023 (70 years after Ludwig Prandtl’s death), and since then anyone has been free to provide an open-access translation. An excellent translation of the German publication was published by Hunsaker & Phillips (2020) prior to this date, as a result of which it is unfortunately not openly accessible.

In this article I include my own, openly accessible translation of Prandtl’s paper, alongside some commentary and ancillary information. The text in the main column of Sec. 3 is a translation of Prandtl’s paper, and the text in the margin is my commentary. The translation was carried out without knowledge of existing translations of Prandtl’s paper.

2 List of symbols

3 “On Wings of Minimum Induced Drag” – a translation of Prandtl’s paper with commentary

3.1 Problem formulation

A wing with given lift and prescribed wingspan has the smallest induced drag if its lift is distributed semi-elliptically. However, the constraint of the prescribed wingspan is rather significant, and so the claim that the elliptic lift distribution is best in general would be most incorrect. The larger the wing span, the smaller the induced drag. If in a special case the wing span of an aeroplane is limited by the requirement that it must fit through a specific hangar gate, then it would be appropriate to distribute the lift elliptically over the prescribed wingspan. If, however, such a constraint is not present, other aspects need to be considered. The wingspan cannot be made arbitrarily large since spar weight would then increase too much. A problem formulation that would do justice to appropriate aeronautical considerations might be that the weight of non-structural parts is given, and that a wing form is to be found which minimises total drag (induced plus zero-lift drag), which is also a function of the spar weight. It would be very difficult to formulate this as a variational problem.

In the present work a simpler question shall be asked, one which can be answered precisely. In particular, a reasonable limit on the wing span can also be imposed by prescribing the moment of inertia⁠[4][4]: “Moment of inertia” is a direct translation, however second moment of lift might be a more appropriate term. This is not to be confused with the wing spar’s contribution to the aeroplane’s moment of intertia about its roll axis. That moment of inertia, assuming Prandtl’s approximation of wing spar weight in Eqn. 2 is valid, would be proportional to ${\int }_{-b/2}^{b/2}M(x)x^{2}dx$. $L{r}^2$ of the lift distribution, alongside the lift $L$. $r$ is then the “radius of gyration” of the lift distribution. One gets to the moment of inertia of the lift distribution by assuming that the spar weight at each span location is proportional to the wing bending moment $M$ at that location. This assumption would only be exactly true if the spar had the same section depth everywhere and if the weight of the web was negligible compared to the weight of the flanges.⁠[5][5]: The stress at the top of the wing spar is $\sigma =Md/(2I)$, where $I$ is the second moment of area. Neglecting the web’s contribution to $I$, we have $I=t_{f}w{d}^2/2$ (provided $t_f\ll d$), and thus $$t_{f}w=\frac{M}{d\sigma }.$$ (1)Assumig constant section depth $d$ and constant $\sigma$ (constant safety factor), the flange’s area $t_{f}w$ is then proportional to the bending moment $M$, and so is the flanges’ mass per unit length. The shear stress in the web is $\tau =S/(d{t}_w)$ (neglecting the contribution of stresses in the web to the shear flow), where $S$ is the shear force at a given span location. At the wing root, $S=L/2$ and so the mass per unit length of the web is constant there and has no bearing on the resulting lift distributions, however the shear forces in the rest of the wing and corresponding web weights are a function of the chosen lift distribution, and so to simplify our problem we must neglect these weights. But even so, one has already gotten closer to what one really wants to express, if one prescribes that for a wing $${\int }_{-b/2}^{+b/2}M(x)dx[6]$$ (2)[6]: In Prantl’s paper, Eqn. 2 was typeset as ${\int }_{-b/2}^{b/2}Mdx$. should not exceed a given value. Doubling this integral and integrating by parts twice readily yields the above mentioned moment of inertia of the lift. If one now expresses the lift using Kutta-Joukowski’s theorem $$L=\rho V\int \Gamma dx,$$ (3)then for $x>0$ $$M(x)=\rho V{\int }_x^{b/2}\Gamma (x^{\prime })(x^{\prime }-x)d{x}^{\prime }[7]$$ (4)[7]: In Prandtl’s paper, Eqn. 4 was typeset as $M(x)=\rho V{\int }_x^{b/2}\Gamma (x^{\prime }-x)d{x}^{\prime }$. This could easily be misinterpreted, since $\Gamma$ is a function of x. The notation used in Eqn. 4 dispels this ambiguity. and so ∫0b/2​Mdx=[xM]0+b/2​−∫0b/2​xdxdM​dx[8], (5)[8]: In Prandtl’s paper, the final $dx$ on the r.h.s. of Eqn. 5 was typeset instead as $d{x}^{\prime }$. This is incorrect, as the integration is over $x$, not $x^{\prime }$.where dxdM​=−ρV∫xb/2​Γ(x′)dx′[9][10]. (6)[9]: This result may be obtained by differentiation of Eqn. 4 using the Leibniz Integral Rule.[10]: In Prandtl’s paper, Eqn. 6 was typeset as $$\frac{dM}{dx}=-\rho V{\int }_x^{b/2}\Gamma dx.$$This is incorrect, as the integration is over $x^{\prime }$, not $x$.The first expression vanishes.⁠[11][11]: The first term on the r.h.s. of Eqn. 4 that is, since the bending moment at the wingtip is zero. The second partial integration now straightforwardly⁠[12][12]: Starting with $${\int }_0^{b/2}Mdx=\rho V{\int }_0^{b/2}x{\int }_x^{b/2}\Gamma (x^{\prime })d{x}^{\prime }dx,$$integrating by parts a second time gives $$\begin{array}{rl}{\int }_0^{b/2}Mdx=& \rho V{\left[\frac{x^2}{2}{\int }_x^{b/2}\Gamma (x^{\prime })d{x}^{\prime }\right]}_0^{b/2}\\ & -\rho V{\int }_0^{b/2}\frac{x^2}{2}\left[-\Gamma (x)\right]dx,\end{array}$$ (7)where the differentiation of Eqn. 6 was again done using the Leibniz Integral Rule. The first term on the r.h.s of Eqn. 7 vanishes, and we are left with Eqn. 8. yields $${\int }_0^{b/2}Mdx=\rho V{\int }_0^{b/2}\frac{x^2}{2}\Gamma dx,$$ (8)which agrees with our above claim.

Our mathematical problem shall thus be: Minimise $$D_i=\rho {\int }_0^{b/2}\Gamma wdx$$ (9) with the side conditions $$L=\rho V{\int }_{-b/2}^{b/2}\Gamma dx=\text{given}$$ (10) and Lr2=ρV∫0b/2​Γx2dx=given[13]. (11)[13]: In Prandtl’s paper, Eqns. 9, 10, and 11 were all typeset as indefinite integrals.Here $w$ is the downwash velocity for a given circulation distribution, according to lifting line theory. Apparently prescribing the radius of gyration $r$ also imposes a limit on the wing size, by defining the median dimension of the wing, rather than a wing span limit. In fact, this problem statement results in completely different lift distributions than the elliptical one.

3.2 Solution

The task is made up of two parts, firstly the variational problem is to be solved, which tells us, according to what principle the lift is to be distributed within the chosen wingspan, and secondly a minimisation problem, which for the case where the wing span is left unconstrained, selects, according to the variational problem, among the allowable solutions the one which results in the lowest drag.

The variational problem is comparatively simple to solve, if we make use of one of A. Betz’ ideas, namely that it is valid to apply the variations $\delta \Gamma$ instead of on the wing itself on a far rearwardly translated auxiliary wing, whose influence on the flow in front can be neglected, and which is subject to the downwash $2w$.⁠[14][14]: This idea was presented by Betz (1919), where Betz showed that propellers with constant downwash across their blades are most efficient. This argument, however, is not necessary to justify the present assumption, namely that the changes in downwash $\delta w$ caused by the additional circulations $\delta \Gamma$ can be neglected. Starting with Eqn. 14 and writing $w+\delta w$ instead of $w$, we have $$\delta D_i=\rho {\int }_0^{b/2}\delta \Gamma (w+\delta w)dx=0.$$This simplifies to Eqn. 14 when neglecting second order effects. If the circulation $\Gamma (x)$ is a solution, then a variation of $D_i$ due to the presence of such additional circulations $\delta \Gamma$, which in turn satisfy the side conditions, must vanish. Evidently, these side conditions are $$\delta L=\rho V{\int }_{-b/2}^{b/2}\delta \Gamma dx=0$$ (12) and $$\delta (L{r}^2)=\rho V{\int }_0^{b/2}\delta \Gamma x^{2}dx=0.$$ (13)It now becomes evident that δDi​=ρ∫0b/2​δΓwdx=0[15], (14)[15]: In Prandtl’s paper, Eqns. 12, 13, and 14 were all typeset as indefinite integrals. However, it is important to keep in mind that the integration is from the wing root to the tip, and that the wing span $b$ must be specified. Recall that in our problem formulation we only constrained the wing spar weight, not the span. Therefore, our work is not complete by having found the shape of the circulation associated with the lowest induced drag via Eqn. 15 for a given wingspan and spar weight. We have yet to determine the wingspan of the wing with circulation according to Eqn. 15 and a given spar weight which has the lowest induced drag overall.if $$w=C_1+C_2{x}^2,$$ (15)since then, by means of the side conditions Eqn. 12 and Eqn. 13, Eqn. 14 is satisfied as well.

The circulation belonging to the value of $w$ specified in Eqn. 15, however, is known from earlier developments in wing theory (Prandtl (1918), reprinted by Prandtl & Betz (1927, p. 32)).

If one puts $$\Gamma =({\Gamma }_0+{\Gamma }_2{\xi }^2)\sqrt{1-{\xi }^2},$$ (16)whereby $\xi$ is an abbreviation for $x/(b/2)$, then one obtains a formula for the downwash that is exactly as per Eqn. 15, and in fact $$C_1=\frac{1}{2b}\left({\Gamma }_0-\frac{1}{2}{\Gamma }_2\right)$$ (17) and $$C_2=\frac{6{\Gamma }_2}{b^3}.$$ (18)

The variational problem is thus solved, and it is further a matter of finding the values of ${\Gamma }_0$ and ${\Gamma }_2$ that are compatible with the conditions of the task. Considering that ${\Gamma }_2$ will turn out to be negative, we introduce $$-{\Gamma }_2/{\Gamma }_0=\mu$$ (19) and thus get $$\Gamma ={\Gamma }_0(1-\mu {\xi }^2)\sqrt{1-{\xi }^2}$$ (20) and $$w=\frac{{\Gamma }_0}{2b}\left(1+\frac{\mu }{2}-3\mu {\xi }^2\right).$$ (21)Our side conditions are now $$L=\frac{\pi }{4}\rho bV{\Gamma }_0\left(1-\frac{\mu }{4}\right)$$ (22) and Lr2=64π​ρb3VΓ0​(1−2μ​)[16]. (23)[16]: Substituting Eqn. 20 into Eqn. 10 and Eqn. 11 yields $$L=\rho bV{\Gamma }_0{\int }_0^1(1-\mu {\xi }^2)\sqrt{1-{\xi }^2}d\xi$$ and $$L{r}^2=\frac{\rho b^{2}V{\Gamma }_0}{4}{\int }_0^1{\xi }^2(1-\mu {\xi }^2)\sqrt{1-{\xi }^2}d\xi$$ respectively. Substituting $\sin \theta =\xi$, we have $$L=\rho bV{\Gamma }_0{\int }_0^{\pi /2}(1-\mu {\sin }^2\theta ){\cos }^2\theta d\theta$$ and $$L{r}^2=\frac{\rho b^{2}V{\Gamma }_0}{4}{\int }_0^{\pi /2}{\sin }^2\theta (1-\mu {\sin }^2\theta ){\cos }^2\theta d\theta$$ respectively, which may be integrated to yield Eqn. 22 and Eqn. 23.By division, this yields at once $$r^2=\frac{1}{16}{b}^2\frac{1-\frac{\mu }{2}}{1-\frac{\mu }{4}}$$ (24) or $$b=4r\sqrt{\frac{1-\frac{\mu }{4}}{1-\frac{\mu }{2}}}.$$ (25)By substitution into Eqn. 22 one finds herefrom $${\Gamma }_0=\frac{L}{\pi \rho Vr}\sqrt{\frac{1-\frac{\mu }{2}}{{\left(1-\frac{\mu }{4}\right)}^3}}.$$ (26)According to Prandtl & Betz (1927, p. 32), the induced drag amounts to $$D_i=\frac{\pi \rho {\Gamma }_0^2}{8}\left(1-\frac{\mu }{2}+\frac{{\mu }^2}{4}\right).$$ (27)Considering Eqn. 26, this yields $$D_i=\frac{L}{8\pi \rho V^2{r}^2}\frac{\left(1-\frac{\mu }{2}\right)\left(1-\frac{\mu }{2}+\frac{{\mu }^2}{4}\right)}{{\left(1-\frac{\mu }{4}\right)}^3}.$$ (28)The behaviour of the function of $\mu$, which shall be referred to here as $f(\mu )$, can be seen from the table below, in which the values of $b/(4r)$ and ${\Gamma }_0$ are stated as well. It can easily be seen that the minimum of $f(\mu )$ is at $\mu =1$, which was also be proven by investigation of the derivative of $f(\mu )$, which vanishes exactly at $\mu =1$. However, at that point the function has no ordinary minimum, but an inflection point, and decreases further for the values $\mu >1$. However, here our task loses its reasonable sense as negative lift occurs at the wing tips and thus negative bending moments, which of course do not correspond to negative spar weights, but again positive spar weights. One would thus in the case of a change of sign of $M$ need to integrate the absolute value of $M$ instead of $M$ and then the bases of our calculation would become invalid. Therefore, the largest reasonable value of $\mu$ is simultaneously the best value. Incidentally, according to the table, the values of $\mu$ which are significantly below $1$ are not much worse. However, in the context of our task the elliptic lift distribution is significantly worse.

In the figure the circulation distributions according to Eqn. 20 and the corresponding downwash velocities according to Eqn. 21 are shown, for a given value of the radius of gyration $r$ and a given ${\Gamma }_0$.⁠[18][18]: A graph more in keeping with the task’s conditions might have shown circulation distributions of constant total lift. In such a graph, the bell-shaped circulation distribution would have had a wing-root circulation ${\Gamma }_0$ about 9% larger than that of the elliptical distribution. Here the curves $a$ represent the elliptic lift distribution ($\mu =0$), the curves $b$ and $c$ belong to $\mu =1/2$ and $\mu =1$. The figure shows that from our point of view the in recent times favoured pointed wings deserve preference over those with approximately rectangular planform, but that the degree of tapering doesn’t matter too much in detail.⁠[19][19]: According to Tbl. 1 the difference in induced drag of circulations $a$ and $b$ in Fig. 2 is only about 2%. However, it can also be seen from Fig. 2 that the degree of upwash at the wing tip for circulation $c$ is significantly larger than for circulation $b$. This has implications for proverse yaw (see Sec. 4.2). Therefore, it may be that the degree of tapering doesn’t matter too much with regards to the induced drag, but circulation $c$ may still be a much more desirable distribution than $b$ when considering aircraft handling qualities.

3.3 Summary

The solution to the problem of the lift distribution, which for given wing lift and given moment of inertia of the wing lift results in the smallest induced drag, is found. This lift distribution is not elliptical, and more closely resembles that of a pointed wing.

4 Further commentary

4.1 Implications for aircraft design

In his paper, Prandtl showed that the elliptic lift distribution is no longer optimal when structural considerations are taken into account as well, and a smaller induced drag can be achieved by choosing the bell-shaped lift distribution instead. Since the idea of the elliptic lift distribution being most efficient is so deeply engrained in the minds of many aeronautical engineers, it is worth stating explicitly the implications of Prandtl’s findings:

In deriving the bell-shaped lift distribution, the wingspan was left unconstrained, and the structural weight was prescribed. In deriving the elliptic lift distribution, the wingspan is prescribed, and the structural weight left unconstrained. In practical aircraft design, however, there are usually limits on both the structural weight as well as the wing span. How is an aircraft designer to choose one lift distribution over another? Prandtl’s findings offer an answer to this question. Our initial baseline design might have a certain weight budget allocated to the wing spar. Neglecting the web’s weight, the wing spar’s weight per unit length is $2\rho g{t}_{f}w$. Using Eqn. 1, and assuming constant section depth for our wing spar⁠[20][20]: This is a rather limiting assumption, as spar section depth usually varies quite substantially along the length of the wing. Consequently, the present calculation should merely be used to guide initial conceptual design, and needs to be followed up with numerical optimisation. For the purposes of this calculation, we can consider an average section depth along the length of the wing., we have for the total wing spar weight $$W_{spar}=\frac{2\rho g}{{\sigma }_{max}d}{\int }_{-b/2}^{b/2}M(x)dx,$$ (29)where ${\sigma }_{max}$ is the maximum allowable stress resulting from the design load. In combination with Eqn. 11 and Eqn. 24, this yields $$\frac{8{\sigma }_{max}d}{\rho g}\frac{1}{b^2}\frac{W_{spar}}{W}=\sqrt{\frac{1-\frac{\mu }{4}}{1-\frac{\mu }{2}}},$$ (30)recognising that $L=W$, where $W$ is the total aircraft weight. Note that the r.h.s. of Eqn. 30 decreases with increasing $\mu$. $\mu$ can be thought of as a wing structural efficiency factor, with greater $\mu$ corresponding to a more efficient wing (but maintaining $0\le \mu \le 1$). To determine the wing span associated with the minimum induced drag, we may substitute $\mu =1$, and solve for $b$. If the result is less than the maximum allowable wing span, we are not span-limited, and may choose the optimum lift distribution. If the calculated wing span is more than the maximum allowable wing span, we are span limited. The task is then to determine the optimum lift distribution given the wing span limit. Substituting the maximum allowable value of $b$, we can solve for $\mu$. If the l.h.s. of Eqn. 30 evaluates to more than $1$ (which would result in a negative value of $\mu$), we have oversized our relative wing spar weight budget $W_{spar}/W$. In this case, the elliptic lift distribution needs to be chosen ($\mu =1$), and the wing spar weight budget reduced accordingly.

4.2 Impact on handling qualities

From Fig. 2, it can be seen that for $\mu =1$ (the bell-shaped distribution $c$), there is considerable upwash towards the wing tip. This is the region where the ailerons are positioned. If an aileron is now deflected downwards, as would be the case on the right wing of an aeroplane subjected to a left-roll command, then the additional lift and corresponding circulation (which may be considered infinitesimal, so as not to impact the distribution of downwash across the wing) at the aileron location results in a proverse yawing moment, yawing the aeroplane into the turn. This is contrary to the experience of any fixed-wing pilot, who would apply left rudder in this situation, to counteract the aircraft’s adverse yawing moment. It therefore seems that an aircraft’s handling qualities can be improved by choosing the bell-shaped lift distribution. This observation also explains how birds are able to fly without a vertical stabiliser. While it seems sensible that birds’ wings would have lift distributions which have minimum induced drag for a given structural weight rather than a given wingspan, we can also deduce the approximate shape of a birds’ lift distribution from the shape of its wings. Albatrosses, for example, have pointed rather than elliptically shaped wings.

References