Aerotecnica M&S 100 Years Ago: A Study on the Possibility of a Dynamic Gliding Flight

Dear Reader,

The preface of this issue of Aerotecnica Missili & Spazio (ATMS) is dedicated to the paper “ON THE POSSIBILITY OF DYNAMIC GLIDING” (in Italian “Sulla possibilità del volo a vela dinamico”), published by Prof. Enrico Pistolesi in December 1922.

This paper is an interesting read to understand how, without the means of computing such as the present ones, scientists used the tools of mathematical analysis to bring developments to aeronautical sciences.

1 ON THE POSSIBILITY OF DYNAMIC GLIDING (translation of the manuscript “Sulla possibilità del volo a vela dinamico” by E. Pistolesi, 1922)

It is known that gliding and flight without engines, in its simplest form, does not differ from ordinary hovering flight and that the possibility of maintaining or increasing the altitude is conditioned on the existence of ascending streams.

In fact, if the vertical component of the stream exceeds the minimum descent speed of the aircraft, we can have an ascent speed equal to the difference of the two. In this case, gliding is called static sail flight [1].

Different from this would be the dynamic sail flight in which the wind direction, or at least the average direction of it, is horizontal but the wind itself is pulsating or gusty, such as to give the aircraft appropriate accelerations. These accelerations, if conveniently exploited, would allow the aircraft to maintain altitude and rise above the starting point. The various ways in which dynamic flight could be implemented have been studied by various authors such as Betz, Knoller, Ahlborn, v. Karman, Prandtl and Blume (\* MERGEFORMAT [2,3,4,5,6,7]), etc. and their studies have recently been summarized in a brilliant paper by W. Hoff ( \* MERGEFORMAT [8]). These studies show that dynamic flight is possible, but of course, certain conditions must be met.

This brief critical study aims at establishing the limits within which the dynamic flight is possible; in this study, we present the above research activities of the authors mentioned before, either in its original form or in another that more easily allows a brief quantitative analysis. Thus, it is easy to verify whether the current construction technologies make it possible not to cross the aforementioned limits and, therefore, if the dynamic flight, at the current state of the today constructions, is an achievable reality or a chimera.

There is a controversy about this point among the experts on the subject: even more so, since in the recent races in Germany, France and England, where surprising results have been obtained, what has actually been done is only to exploit ascending currents. It is not the case, for now, to speak on Dynamic Flight.

The brief notes that follow, without pretending to solve the dispute, would seem to support this opinion, at least for some of the various ways conceived for dynamic flight. More likely, it would seem to present the dynamic spiralling flight, which is observed to be implemented by some sailing birds.

However, this study is intended to be only a modest contribution to the study of flight without an engine, a topic that fascinates both experts and laymen of Aerodynamics today.

1.1 The Karman theory

A way in which the dynamic flight could be implemented is that envisaged by Karman ( \* MERGEFORMAT [9]), based on the pulsating structure of the wind. For this, Karman refers to a mechanical model (this model was considered for the first time by A. Bazin in 1890 and, some years later, by Lancester).

A way (achievable with a wooden surface) of sinuous shape (Fig. 1) moves with a motion time-dependent, e.g., with a sinusoidal law.

Fig. 1

(from the original paper)

Let U be the translation speed of this shape; we can put:

$$U={u}_{0}+{u}_{1}sen(\lambda t)$$

where $\lambda =\frac{2\pi }{T}$, and T is the period.

A sphere, placed on this moving way is subjected, for the accelerations of this, to impulses that, under appropriate conditions, are able to lift it to heights higher than the starting one. Let W be the horizontal component of the speed of the sphere with respect to the guide, where W varies with a sinusoidal law with the same period of the motion of the guide:

$$W={w}_{0}+{w}_{1}sen\left(\lambda t+\psi \right).$$

We can study the motion of the sphere with respect to the guide as if it is fixed, provided to add the accelerations due to the motion of the guide itself. Let us also assume that there is no friction.

That said, if the sphere passes from the height y₀ to y, it is subjected to the increase $mg\left( {y - y_{0} } \right)$ of the potential energy, which equals the decrease of the living force and the work of the accelerations impressed by the motion of the guide.

Said $\varphi$ the angle of the guide with the horizontal at a given point, the speed of the sphere will be:

$$\frac{W}{cos\varphi }=W\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}$$

and, therefore, his living force will be

$$\frac{1}{2}m{W}^{2}\left(1+{\left(\frac{dy}{dx}\right)}^{2}\right).$$

Finally, the acceleration due to the motion of the guide will be:

$${u}_{1}\lambda \mathit{cos}(\lambda t)$$

to which corresponds a horizontal force:

$${-m u}_{1}\lambda \mathit{cos}(\lambda t)$$

and then the work:

$$-m {u}_{1}\lambda {\int }_{0}^{t}\mathit{Wcos}\left(\lambda t\right)dt.$$

It follows, therefore:

$$mgy+\frac{1}{2}m{W}^{2}\left[1+{\left(\frac{dy}{dx}\right)}^{2}\right]=cost-m {u}_{1}\lambda {\int }_{0}^{t}\mathit{Wcos}(\lambda t)dt$$

(1)

That is:

$$y=cost-\frac{1}{2g}{\left\{{w}_{0}+{w}_{1}sen\left(\lambda t+\psi \right)\right\}}^{2} \left[1+{\left(\frac{dy}{dx}\right)}^{2}\right]-\frac{1}{g}{u}_{1}{w}_{0}sen\left(\lambda t\right)+\frac{{u}_{1}{w}_{1}}{4g}\mathit{cos}\left(2\lambda t\right)cos\psi +-\frac{{u}_{1}{w}_{1}}{4g}\mathit{sen}\left(2\lambda t\right)sen\psi -\frac{{u}_{1}{w}_{1}}{2g}\lambda t sen\psi .$$

Thus, the expression of y consists of a periodic part and a last term

$$-\frac{{u}_{1}{w}_{1}}{2g}\lambda t sen\psi$$

which, after ony period, undergoes an increase:

$$\Delta y= -\frac{{\pi u}_{1}{w}_{1}}{g} sen\psi .$$

Of course, if $sen\psi$ is negative, the increase is positive, that is, the sphere really rises above the starting point.

In first approximation, to calculate the trajectory of the sphere, that is the shape on which the sphere rolls without friction, KARMAN assumes that w₁ and u₁, are small w.r.t w₀ and that ${\left(\frac{dy}{dx}\right)}^{2}$ is small w.r.t the unity.

With such assumptions, the expression of y is simplified:

$$y=cost-\frac{1}{2g}{{w}_{0} }^{2}-\frac{1}{g}{w}_{0}{w}_{1}sen\left(\lambda t+\psi \right)-\frac{1}{g}{w}_{0}{u}_{1}sen\left(\lambda t\right)$$

that is:

$$y = y_{0} - \frac{{w_{0} }}{g}\left( {u_{1} + w_{1} \cos \psi } \right)sen\left( {\lambda t} \right) - \frac{1}{g}w_{0} w_{1} sen\left( \psi \right)\cos \left( {\lambda t} \right),$$

where y₀ includes all the constants (therefore y₀ is not y(0)); or, also:

$$y={y}_{0}-hsen\left(\lambda t+\delta \right),$$

where:

$h=\frac{{v}_{0}}{g}\sqrt{{u}_{1}^{2}+2{u}_{1}{w}_{1}cos\psi +{w}_{1}^{2}}$

and:

$$tg\delta =\frac{{w}_{1}sen\psi }{{u}_{1}+{w}_{1}cos\psi } .$$

So far, the theory of Karman. But we must take the drag into account, which, in the case of the aircraft, consists of friction and cannot be neglected.

1.2 The effect of drag

Assuming, for simplicity’s sake, that a constant drag coefficient ${\chi }_{\infty }$ is constant (it is an average value), it will be necessary to add the following term to the right-hand side of (1):

$${-aS\chi }_{\infty }{\int }_{o}^{t}{\left(\frac{W}{\mathit{cos}\varphi }\right)}^{3}dt.$$

Assuming again, for simplicity, $\mathrm{cos}\varphi =1,$ we have:

$${-aS\chi }_{\infty }{\int }_{o}^{t}{\left({w}_{0}+{w}_{1}sen(\lambda t+\psi )\right)}^{3}dt.$$

To the expression of y, it is necessary to add the following term, in which we put mg = Q (Q is the weight of the aircraft):

$${-\frac{aS}{Q}\chi }_{\infty }\left\{{w}_{0}^{3}t+\frac{3}{2}{w}_{0}{w}_{1}^{2}t-\frac{3{w}_{0}^{2}{w}_{1}}{\lambda }\mathit{cos}\left(\lambda t+\psi \right)-\frac{3}{4}{w}_{0}{w}_{1}^{2}\mathit{sen}2\left(\lambda t+\psi \right)-\frac{{w}_{1}^{3}}{\lambda }\mathit{cos}\left(\lambda t+\psi \right){\mathit{sen}}^{2}\left(\lambda t+\psi \right)\right\} .$$

Therefore, the increase $\Delta y$ after each period becomes:

$$\Delta y= -\frac{\pi {u}_{1}{w}_{1}}{g}sen\psi -\frac{2\pi }{\lambda }\frac{{a\chi }_{\infty }}{{q}_{s}}\left({w}_{0}^{3}+\frac{3}{2}{w}_{0}{w}_{1}^{2}\right),$$

(2)

where q_s indicates the load per square meter.

Finally, the approximate expression of y will be modified as follows:

$$y={y}_{0}-\frac{{w}_{0}}{g}\left({u}_{1}+{w}_{1}cos\psi +3\frac{ag{\chi }_{\infty }}{{q}_{s}\lambda }{w}_{0}{w}_{1}sen\psi \right)sen\left(\lambda t\right)-\frac{{w}_{0}}{g}\left({w}_{1}sin\psi -3\frac{ag{\chi }_{\infty }}{{q}_{s}\lambda }{w}_{0}{w}_{1}cos\psi \right)cos\left(\lambda t\right).$$

(3)

Putting

$$y= {y}_{0}-h\mathrm{^{\prime}}sen(\lambda t+\delta )$$

(4)

we will obtain the following expression of the amplitude of a half-oscillation:

$${h}^{\mathrm{^{\prime}}2}= {h}^{2}+\frac{{w}_{0}^{2}}{{g}^{2}}\left\{\frac{{9{a}^{2}{g}^{2}{\chi }_{\infty }^{2}}}{{q}_{s}^{2}{\lambda }^{2}}{w}_{0}^{2}{w}_{1}^{2}+6\frac{ag{\chi }_{\infty }}{{q}_{s}\lambda }{w}_{0}{w}_{1}{u}_{1}sen\psi \right\}$$

$${h}^{\mathrm{^{\prime}}2}= {h}^{2}-\frac{6{w}_{0}^{3}a{\chi }_{\infty }}{{q}_{s}\lambda }\left\{-\frac{{u}_{1}{w}_{1}}{g}sen\psi +6\frac{a{\chi }_{\infty }}{{q}_{s}\lambda }\frac{3}{2}{w}_{0}{w}_{1}^{2}\right\}$$

(5)

From a comparison with (2), it can be deduced that the quantity subtracted from h² is certainly positive, that is, definitely:

$${h}^{\mathrm{^{\prime}}}<h.$$

In the expression (2) of $\mathrm{\Delta y},$ we can omit the term containing ${\mathrm{w}}_{1}^{2}$ and write:

$${{q}_{s}=a{\chi }_{\infty }w}_{0}^{2},$$

(6)

where ${\upchi }_{\infty }$ represents the value of ${\upchi }_{\mathrm{y}}$ corresponding to the average velocity ${\mathrm{w}}_{0}$, that is, approximately, the mean value of ${\upchi }_{\mathrm{y}}$.

It follows:

$${w}_{0}^{3}= \frac{{q}_{s}^\frac{3}{2}}{{a}^\frac{3}{2}{\chi }_{yo}^\frac{3}{2}}$$

and therefore:

$$\Delta y= -\frac{\pi {u}_{1}{w}_{1}}{g}sen\psi -\frac{2\pi }{\lambda }{\chi }_{\infty }{a}^{-\frac{1}{2}}{{\chi }_{y0}}^{-\frac{3}{2}}{q}_{s}^\frac{1}{2} .$$

(7)

The following main consequence can be deduced from (7): for the possibility of a dynamic flight of the type studied so far, low values of the wing load, small drag and high average lift are required and, also, that at high altitude dynamic gliding flight is more difficult than at low altitude. Formula (7) would confirm what the intuition of the practitioners had foreseen: low wing loads and highly lifting profiles are needed.

Within the limits of the first approximation, the wing lift is the sum of the weight, mg, and the force of inertia due to the vertical acceleration, $m\frac{{d}^{2}y }{d{t}^{2}}.$ We obtain:

$$a S{ \chi }_{y}{W}^{2}=mg+m{h}^{\mathrm{^{\prime}}{\lambda }^{2}}sen\left(\lambda t+{\delta }^{\mathrm{^{\prime}}}\right).$$

(8)

Of course, to calculate ${\chi }_{y}$ we should put for W its actual value ${w}_{0}+{w}_{1}(sen(\lambda t+\psi )$ but, in a first approximation, we can replace the variable speed W with its average value ${w}_{0}$; thus:

$${\chi }_{y}={\chi }_{y0}+{\chi }_{y0}\frac{h\mathrm{^{\prime}}{\lambda }^{2}}{g}sen\left(\lambda t+{\delta }^{\mathrm{^{\prime}}}\right) ,$$

(9)

where:

$$X_{Y} = {\raise0.7ex\hbox{$Q$} \!\mathord{\left/ {\vphantom {Q {aSwo^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${aSwo^{2} }$}}$$

The amplitude of the half-oscillation ${\chi }_{y}$ is:

$${\chi }_{y0}\frac{h\mathrm{^{\prime}}{\lambda }^{2}}{g}.$$

In general, it will be convenient that ${\chi }_{y}$ does not exceed a certain fraction of ${\chi }_{y0},$ determined by the position that ${\chi }_{y0}$ can assume between its minimum and the maximum values. If we assume zero as a practical minimum, it will be, at most:

$$\frac{{h\mathrm{^{\prime}}}^{2}{\lambda }^{2}}{g}\le 1 .$$

Since h' < h, the above condition will certainly be verified when:

$$\frac{h{\lambda }^{2}}{g}\le 1,$$

That is, when:

$$\frac{{w}_{0}{\lambda }^{2}}{{g}^{2}}\sqrt{{u}_{1}^{2}+2{u}_{1}{w}_{1}cos\psi +{w}_{1}^{2}}\le 1 .$$

(10)

Given the value of ${w}_{0}$ (e.g. by fixing ${q}_{s} e {\chi }_{y0}$), $\Delta y$ results from the condition: $sen\psi =-1, {u}_{1}={w}_{1}=\frac{{g}^{2}}{\sqrt{2}{w}_{0}{\lambda }^{2}}$ and then:

$$\Delta y= \frac{\pi {g}^{3}}{2{w}_{0}^{2}{\lambda }^{4}}-\frac{a{\chi }_{\infty }}{{q}_{s}}\frac{2}{{\lambda }^{3}}{w}_{0}^{3}.$$

Putting, e.g.: ${\chi }_{yo}=0.25,$ ${q}_{s}=10 \mathrm{kg}/{\mathrm{m}}^{2}$ (from which: ${v}_{0}=18 \mathrm{m}/\mathrm{sec}$), the maximum value of ${w}_{1}$ will be ${w}_{1}=\frac{96}{25.5 {\lambda }^{2}}$. With a period of 6.28 s, i.e. $\lambda =1$, we could obtain:

$${u}_{1}={w}_{1}\le 3.8\frac{m}{sec}.$$

It is easy to remark that, given these orders of magnitude, h' differs only slightly from h, so that, in a rough calculation, it will be possible to identify h' with h.

1.3 The limits of dynamic flight

We will use the previous conclusions to establish the limits within which the dynamic flight is possible.

Assuming again ${u}_{1}={w}_{1}$ and $cos\psi =0\mathrm{ for simplicity},$ it results:

$${h}^{^{\prime}}=\approx h={w}_{0}{w}_{1}\frac{\sqrt{2}}{g}$$

or:

$${h}^{^{\prime}}=\approx \frac{\sqrt{2}}{g} {q}_{s}^\frac{1}{2}{a}^{-\frac{1}{2}}{\chi }_{y0}^{-\frac{1}{2}}{w}_{1} ,$$

so that:

$${\chi }_{y}={\chi }_{y0}\left[1+\frac{{w}_{1}{\lambda }^{2}}{{g}^{2}}\sqrt{2}{q}_{s}^\frac{1}{2}{a}^{-\frac{1}{2}}{\chi }_{y0}^{-\frac{1}{2}} sen(\lambda t+\delta \mathrm{^{\prime}})\right] .$$

It follows that ${\chi }_{y}$ is between a maximum of:

$${\chi }_{y0}\left\{1+\frac{{w}_{1}{\lambda }^{2}}{{g}^{2}}\sqrt{2}{q}_{s}^\frac{1}{2}{a}^{-\frac{1}{2}}{\chi }_{y0}^{-\frac{1}{2}}\right\}$$

and a minimum of:

$${\chi }_{y0}\left\{1-\frac{{w}_{1}{\lambda }^{2}}{{g}^{2}}\sqrt{2}{q}_{s}^\frac{1}{2}{a}^{-\frac{1}{2}}{\chi }_{y0}^{-\frac{1}{2}}\right\}.$$

If we want that ${\Delta }_{y}$ is not negative, it must also be, for (7):

$${{w}_{1}}^{2}\ge \frac{2g}{\lambda }{\chi }_{\infty }{q}_{s}^\frac{1}{2}{a}^{-\frac{1}{2}}{\chi }_{y0}^{-\frac{1}{2}}$$

(11)

Therefore, the maximum of ${\chi }_{y}$ will not be less than:

$${\chi }_{y0}\left\{1+2{\lambda }^\frac{3}{2}{q}_{s}^\frac{3}{4}{a}^{-\frac{3}{4}}{g}^{-\frac{3}{2}}{\chi }_{\infty }^\frac{1}{2}{\chi }_{y0}^{-\frac{5}{4}}\right\}$$

(12)

and the minimum not higher than:

$${\chi }_{y0}\left\{1-2{\lambda }^\frac{3}{2}{q}_{s}^\frac{3}{4}{a}^{-\frac{3}{4}}{g}^{-\frac{3}{2}}{\chi }_{\infty }^\frac{1}{2}{\chi }_{y0}^{-\frac{5}{4}}\right\} .$$

(13)

If we want, for example, that this minimum is not less than zero, it must be:

$$2{({\lambda }^{2}{q}_{s})}^\frac{3}{4}{g}^{-\frac{3}{2}}{K}_{\infty }^\frac{1}{2}{K}_{y0}^{-\frac{5}{4}}<1$$

(14)

That is:

$${\lambda }^{2}{q}_{s}<\frac{{g}^{2}}{{2}^{4/3}}{K}_{\infty }^{-\frac{2}{3}}{K}_{y0}^\frac{5}{3}$$

(15)

or, also:

$${K}_{y0}\ge \sqrt[5]{\frac{16{K}_{\infty }^{2}}{{g}^{6}}}{\left({\lambda }^{2}{q}_{s}\right)}^\frac{3}{5} .$$

(16)

Putting, to fix ideas, ${K}_{\infty }$ = 0.003, it follows:

$${K}_{y0}\ge 0.011{\left({\lambda }^{2}{q}_{s}\right)}^\frac{3}{5} .$$

(17)

Since, when (14) is satisfied the expression (12) grows with ${K}_{y0}$ (how can be verified easily) we can substitute the expression of ${K}_{y0}$ in (15) into (12) to determine the lower limit of the maximum value of ${K}_{y}$. This lower limit, which is of course equal to 2 ${K}_{y0}$, must not exceed a certain value depending on the airfoil of the aircraft; let it be ${K}_{y*}.$

Let's put, to fix the ideas, ${K}_{y*}=a$ ${\chi }_{y*}=0.075$; from (14a) it follows:

$${\lambda }^{2}{q}_{s}<\frac{{g}^{2}}{8}{K}_{\infty }^{-\frac{2}{3}}{K}_{y0}^\frac{5}{3}$$

or:

$${\lambda }^{2}{q}_{s}<0.161{K}_{\infty }^{-\frac{2}{3}}.$$

(18)

If, for example, ${K}_{\infty }$ = 0.003 it results:

$${\lambda }^{2}{q}_{s}<7.7.$$

(19)

We can find another limitation based on the following considerations. In fact, we have already found:

$${w}_{1}\le \frac{{g}^{2}}{\sqrt{2}{w}_{0}{\lambda }^{2}}$$

(20)

or:

$$\frac{{w}_{1}}{{w}_{0}}\le \frac{{g}^{2}}{\sqrt{2}{{w}_{0}}^{2}{\lambda }^{2}} .$$

Because ${w}_{0}^{2}={q}_{s}$ ${K}_{y0}^{-1}$, it follows:

$$\frac{{w}_{1}}{{w}_{0}}\le \frac{{g}^{2}{K}_{y0}}{\sqrt{2}{q}_{s}{\lambda }^{2}}$$

$${q}_{s}{\lambda }^{2}\le \frac{{g}^{2}{K}_{y*}}{2\sqrt{2}\frac{{w}_{1}}{{w}_{0}}} ,$$

(21)

and, assuming ${K}_{y*}=0.075$:

$${q}_{s}{\lambda }^{2}\le \frac{2.58}{2\sqrt{2}} .$$

(22)

It is not easy to establish within what limits ${\frac{{w}_{1}}{{w}_{0}} }$ may vary in practice, even in the absence of precise data on wind fluctuations. Admitted, for example, as a practical lower limit $\frac{{w}_{1}}{{w}_{0}}$= ¼, we would get:

$${q}_{s}{\lambda }^{2}\le 10.3.$$

expression (22) may be written as:

$$\frac{{w}_{1}}{{w}_{0}}\le \frac{2.58}{{q}_{s}{\lambda }^{2}} .$$

(23)

From (2), we also have:

$${w}_{1}^{2}\ge 2g\frac{{K}_{\infty }}{{q}_{s}}\frac{{w}_{0}^{3}}{\lambda } ,$$

from which:

$$\frac{{w}_{1}^{2}}{{w}_{0}^{2}}\ge 2g\frac{{K}_{\infty }}{{q}_{s}}\frac{{w}_{0}}{\lambda }$$

$${\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{2}\ge 2g\frac{{K}_{\infty }}{{K}_{y0}^\frac{1}{2}}\frac{1}{\lambda {q}_{s}^\frac{1}{2}}$$

and:

$${q}_{s}{\lambda }^{2}\ge 4{g}^{2}\frac{{{K}_{\infty }}^{2}}{{K}_{y0}}\frac{1}{{\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{4}}$$

(24)

because ${K}_{\infty }$ cannot be higher than 0.0375:

$${q}_{s}{\lambda }^{2}\ge \frac{10300}{{\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{4}}{{K}_{\infty }}^{2} .$$

(25)

It is easy to verify that (23) and (18) are not independent.

From the comparison of (25) and (18), the following condition arises:

$$\frac{10300}{{\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{4}}{{K}_{\infty }}^{2}<0.161{{K}_{\infty }}^{-\frac{2}{3}}$$

$${{K}_{\infty }}^\frac{8}{3} <\frac{0.161}{10300}{\left( \frac{{w}_{1}}{{w}_{0}}\right)}^{4}$$

$$\frac{{w}_{1}}{{w}_{0}}>1.59{{K}_{\infty }}^\frac{2}{3} .$$

(26)

Thus, if we fix ${K}_{\infty }=0.003,$ it follows:

$$\frac{{w}_{1}}{{w}_{0}}>0.33,$$

(27)

that is, the half-amplitude of the fluctuation of w must be not less than 1/3 of the average value.

From the foregoing, albeit with a criterion of large approximation, it is clear that it is very difficult to carry out in practice a dynamic gliding of the type studied. Assumed for ${K}_{y*}$ and ${K}_{\infty }$ the values 0.075 and 0.003 used previously, the following inequalities must occur simultaneously:

$${\lambda }^{2}{q}_{s}\le 7.72$$

$$\frac{{w}_{1}}{{w}_{0}}\ge 0.33$$

$$\frac{{w}_{1}}{{w}_{0}}\le \frac{2.58}{{\lambda }^{2}{q}_{s}} .$$

The last two inequalities limit the range of choice of the ratio $\frac{{w}_{1}}{{w}_{0}}$ for which just the only value $\frac{{w}_{1}}{{w}_{0}}=\frac{1}{3}$ is possible when ${\lambda }^{2}{q}_{s}=7.72.$

The first inequality severely limits the field of the wing loads in relation to the frequency of fluctuations. It appears that high frequencies must necessarily correspond to low wing loads.

Thus, if ${q}_{s}$ varies between 5 and 10, the upper limit varies from 1.24 to 0.88, corresponding to periods of 5" and 7" × 2, respectively.

Now, let us illustrate the previous results with a practical example:

let be: ${q}_{s}=7\frac{kg}{{m}^{2}} {\lambda }^{2} {q}_{s}=7 ;$ then$\lambda =1$.

From (17), we have:

$${K}_{y0}\ge 0.0337.$$

Now, we assume:

$${K}_{y0}=0.035$$

and it follows:

$${w}_{0}= \sqrt{\frac{{q}_{s}}{{K}_{y0}}}=\sqrt{\frac{7}{0.035}}=14.1\frac{\mathrm{m}}{\mathrm{sec}}. \left(51\frac{\mathrm{km}}{\mathrm{h}}\right).$$

Assuming ${K}_{\infty }=0.003,$ we will have:

$${\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{2}>\frac{2g}{{\left({\lambda }^{2}{q}_{s}\right)}^\frac{1}{2}}\frac{{K}_{\infty }}{{K}_{y0}^\frac{1}{2}}$$

$${\left(\frac{{w}_{1}}{{w}_{0}}\right)}^{2}gt;0.119\,\frac{{w}_{1}}{{w}_{0}}gt;0.345 .$$

We put:

$$\frac{{w}_{1}}{{w}_{0}}=0.39\,{w}_{1}=5.5\frac{m}{sec},$$

and, finally:

$${u}_{1}=5.5\mathrm{ mm}$$

or, also:

$$sen\psi =-1 cos\psi =0 \left(\psi =-\frac{\pi }{2}\right) .$$

Thus, from (2) we obtain:

$$\Delta y=3.09-2.97=0.12\mathrm{ m},$$

and, because the period is 6.28 s, the average $\Delta y$ per second is only 19 mm.

We have, also:

$$y={y}_{0}-6.52 sen\left(\lambda t\right)+7.92 cos(\lambda t)$$

or:

$$y={y}_{0}+10.26 sen\left(\lambda t+129^\circ 20\mathrm{^{\prime}}\right)$$

$$y={y}_{0}-10.26 sen\left(\lambda t-50^\circ {40}^{\mathrm{^{\prime}}}\right).$$

Therefore, the value of the half-oscillation is 10.26 m. to be compared with the space travelled in one period. In general:

$$w={w}_{0}+{w}_{1} sen(\lambda t+\psi )$$

$$x={w}_{0}t-\frac{1}{\lambda }{w}_{1}\mathit{cos}\left(\lambda t+\psi \right)+c$$

(28)

And, in the case, at hand:

$$w={w}_{0}-{w}_{1}cos t$$

$$x={w}_{0}t+{w}_{1}sen t+c$$

The space travelled with respect to the wind in a period $\left(T=\frac{2\pi }{\lambda }\right)$ is, therefore:

$${w}_{0}\frac{2\pi }{\lambda } .$$

In our case:

$${w}_{0}\frac{2\pi }{\lambda }= 88.5m$$

and finally, we have:

$${K}_{y}=0.035\left[\left(1-1.046 sen\left(\lambda t-50^\circ {40}^{\mathrm{^{\prime}}}\right)\right)\right].$$

Therefore, the extreme values of ${K}_{y}$ are − 0.0011 and 0.0717; they are acceptable, even though the minimum is negative. This result is not surprising since inequalities (which even in this case are verified), are extreme and necessary, but not sufficient, conditions.

In addition, the research has been conducted with general and not strict criteria.

Figure 2 shows of the various quantities introduced before and Fig. 3 shows the trajectory of the aircraft, in the case of zero wind speed and, therefore, it results:

Fig. 2

(from the original paper)

Fig. 3

(from the original paper)

$$u=5.5\left(1+sen t\right).$$

For the tracking of the trajectory, it is necessary to take as abscissa:

$$s=\left({w}_{0}-{u}_{0}\right)t+{w}_{1}sen t+{u}_{1}\mathit{cos}t+c$$

(29)

that is, in our case:

$$s+5.5=8.6t+{7.78}_{1}sen (t+45^\circ )$$

Figure 3 shows the poor likelihood of such a flight (observers of the flight of sailing birds are almost unanimous in ruling out that a type of “Russian mountings” flight, such as that represented in Fig. 3, could be actually practiced by the birds).

1.4 A different procedure

Results, similar in all respects to the previous ones, can be achieved by a rather different procedure, which can serve to control the one followed so far. Therefore, for example, it will serve to show that to exploit the pulsations of the wind in a dynamic gliding flight, it is necessary to be able to vary harmoniously the coefficient of thrust, according to the variation of the wind speed; this can be obtained both by manoeuvring the entire aircraft, if this is a rigid body, and (what would certainly be more ready and effective) by acting on the wing structure or by varying the trim with respect to the body, or by varying the curvature and shape of the lifting surfaces.

We write the equations of motion in a vertical plane. Let U be the horizontal component of wind speed, u its vertical component, V is the horizontal component of the speed of the aircraft, v its vertical component. Given again m the mass of the aircraft and S its surface, we have:

$$\left\{\begin{array}{l}a S {\chi }_{y}\left(V-U\right)\sqrt{{\left(V-U\right)}^{2}+{\left(v-u\right)}^{2}}-\\ -a S {\chi }_{\infty }\left(v-u\right)\sqrt{{\left(V-U\right)}^{2}+{\left(v-u\right)}^{2}}-mg=m\frac{dv}{dt}\\ a S {\chi }_{y}\left(v-u\right)\sqrt{{\left(V-U\right)}^{2}+{\left(v-u\right)}^{2}}-\\ -a S {\chi }_{\infty }\left(V-U\right)\sqrt{{\left(V-U\right)}^{2}+{\left(v-u\right)}^{2}}-mg=m\frac{dV}{dt}\end{array}\right.$$

(30)

Assuming that the wind is horizontal (and therefore u = 0), that v² can be neglected with respect to (U − V)² (in the first approximation) and neglecting the term in ${\chi }_{\infty }$ in the first equation, we get:and placing, at first,

$$\left\{\begin{array}{l}a S {\chi }_{y}{\left(V-U\right)}^{2}-mg=m\frac{dv}{dt}\\ -a S {\chi }_{y}v\left(V-U\right)- aS {\chi }_{\infty }{\left(V-U\right)}^{2}=m\frac{dV}{dt}\end{array}\right.$$

(31)

Putting V-U = W = w₀ + w₁ sen(λt), from the first of (25), it follows:

$$mv=-mgt+aS\int {\chi }_{y}{\left[{w}_{0}+{w}_{1}sen\left(\lambda t\right)\right]}^{2}dt+C .$$

If we admit that ${\chi }_{y}$ is constant, and we place ${K}_{y}$ in place of ${a\chi }_{y}$, we get:

$$mv=-mgt+{K}_{y}S\int {\left[{{w}_{0}}^{2}+{2{w}_{o}w}_{1}sen\left(\lambda t\right)+{{w}_{1}}^{2}{sen\left(\lambda t\right)}^{2}\right]}dt+C ,$$

that is:

$$mv=\left({K}_{y}S{{w}_{0}}^{2}+{K}_{y}S\frac{{{w}_{1}}^{2}}{2}- mg\right)t-2\frac{{K}_{y}S}{\lambda }{w}_{0}{w}_{1}\mathit{cos}\left(\lambda t\right)-$$

$$-\frac{{K}_{y}S}{4\lambda }{{w}_{1}}^{2}sen\left(2\lambda t\right)+C .$$

For v to be periodic, the term in t must disappear, that is, we have:

$${K}_{y}S\left({{w}_{0}}^{2}+\frac{{{w}_{1}}^{2}}{2}\right)=mg ,$$

that is:

$${q}_{s}={K}_{y}\left({{w}_{0}}^{2}+\frac{{{w}_{1}}^{2}}{2}\right) .$$

Because $\frac{{{w}_{1}}^{2}}{2}$ can be disregarded with respect to ${{w}_{0}}^{2},$ it would result simply ${q}_{s}={K}_{y}{{w}_{0}}^{2}$.

Now, we substitute the value of v into the second equation, and, after simple calculations, we find:

$$mV=-St\left({K}_{\infty }{{w}_{0}}^{2}+{K}_{\infty }\frac{{{w}_{1}}^{2}}{2}+C{K}_{y}{w}_{0}\right)+\mathfrak{I},$$

where con $\mathfrak{I}$ indicates a group of periodic terms.

It follows that V, after any period, decreases of the quantity:

$$\frac{gT}{{q}_{s}}\left(\frac{{K}_{\infty }}{{K}_{y}}{q}_{s}+C{K}_{y}{w}_{0}\right)=gT\left(\frac{{K}_{\infty }}{{K}_{y}}+\frac{c}{{q}_{s}}{K}_{y}{w}_{0}\right)$$

which cannot be zero if C is not negative.

This shows what we had said: that, to have a gliding flight without loss of altitude, it is necessary that ${K}_{y}$ depend periodically on time.

But we can also show that this is not sufficient and that the relative speed, W = (U-V), between the wind and the aircraft must vary periodically as well.

In fact, let us put:

$${K}_{y}={b}_{0}+{b}_{1}sen(\lambda t)$$

and neglect the periodic variation of U-V, or, simply, put ${w}_{1}=0$.

Then, from the first of (25), we derive:

$$mv=-mgt+{v}_{0}^{2}S\int {\left[{b}_{0}+{b}_{1}sen(\lambda t)\right]}dt$$

$$mv=\left({w}_{0}^{2}S{b}_{0}-mg\right)t-\frac{1}{\lambda }{w}_{0}^{2}S{b}_{1}\mathit{cos}\left(\lambda t\right)+C .$$

Arguing as before, we find:

$${w}_{0}^{2}S{b}_{0}=mg$$

$${w}_{0}^{2}{b}_{0}={q}_{s} ,$$

and, from the second of (25) it results:

$$mV=-\left({K}_{\infty }{{w}_{0}}^{2}+S{w}_{0}{b}_{0}c\right)t+\mathfrak{I}\mathfrak{^{\prime}}$$

being ${\mathfrak{I}}^{\mathrm{^{\prime}}}$ a periodic function. In order not to have a decrease of V after any period, C must be negative, which proves the assertion.

The research could continue in the event that both (V-U) and Ky are harmonically variable and would lead, of course, to results similar to those previously found. But we omit it for the sake of brevity.

1.5 Dynamic spiral flight

A dynamic gliding flight can take place when the aircraft flies spiralling, rather than straight.

This way was also made clear by KÀRMÀN.

The mechanical model, designed by KLEMPERER, consists of a spiral guide, with a double motion: of rotation around its own axis and of revolution around an external axis parallel to the first one (Fig. 4). A sphere moving along the guide is subjected to the tangential acceleration due to the motion of revolution, b, and the acceleration of gravity, g; things can be adjusted so that the resultant, r, is normal to the guide. If we apply a certain speed to the sphere on the guide and a motion of rotation to the guide with peripheral speed opposite to that of the sphere, provided that there is no friction between the sphere and the guide, the sphere will rise. This scheme is difficult to achieve in practice with an aircraft; but it is possible to replace, to the constant centrifugal acceleration, the variable acceleration due to a pulsating wind.

Fig. 4

(from the original paper)

Let be:

$$\lambda {u}_{1}sen(\lambda t)$$

the acceleration due to the periodic wind variations. If the aircraft flies along a circular trajectory so that the direction of motion makes the angle $\lambda t$ with the direction of the wind (Fig. 5), the tangential acceleration will result:

Fig. 5

(from the original paper)

$$b=\lambda {u}_{1}{sen}^{2}\left(\lambda t\right)$$

That is, always positive. The average acceleration in a complete turn will be:

$${b}_{m}=\frac{2\lambda {u}_{1}}{\lambda T}{\int }_{0}^{T/2 }{sen}^{2}\left(\lambda t\right)dt=\frac{\lambda {u}_{1}}{2} .$$

This acceleration is composed with that of gravity to have the average direction of the resultant r (and then the average slope of the trajectory); we will obtain:

$$\left(tg\varphi \right)=\frac{{b}_{m}}{g}=\frac{\lambda {u}_{1}}{2g}$$

Note that the component of the speed of the aircraft in the direction of the wind is, in general:

$${w}_{0}+{w}_{1}cos\lambda t ,$$

if ${w}_{1}$ indicates the speed with which the aircraft travels the spiral, and with ${w}_{0}$ ia a flight speed added to that.

In addition, the amount of height increase, $\Delta y,$ in the period $T=\frac{2\pi }{\lambda }$ becomes: ${v}_{1}{\left[tg\varphi \right]}_{m}T$, that is:

$$\Delta y=\frac{\pi {u}_{1}{v}_{1}}{g} ;$$

this formula is identical to that already obtained in the paragraph 1, when we put $\psi =-\frac{\pi }{2}$. This result was easily predictable, according to KÀRMÀN.

If we introduce the aerodynamic drag, by way of first approximation, we must subtract the quantity $\frac{{W}^{2}{K}_{\infty }S}{m}=\frac{{W}^{2}{K}_{\infty }g}{{q}_{s}}$ from b, where W indicates speed of the aircraft relevant to the wind. Instead of assuming ${w}_{1}$ small compared to ${w}_{0}$ (as in the previous paragraphs), now make the opposite case, in which ${w}_{0}=0$ and neglect, for simplicity, the fluctuations of ${w}_{1}$. We obtain:

$${\left[tg\varphi \right]}_{m}=\frac{1}{g}\left(\frac{\lambda {u}_{1}}{2}-\frac{{w}_{1}^{2}{K}_{\infty }g}{{q}_{s}}\right) ,$$

so that:

$$\Delta y=\frac{\pi {u}_{1}{w}_{1}}{g}-\frac{2\pi }{\lambda }{w}_{1}^{3}\frac{{K}_{\infty }}{{q}_{s}} .$$

We deduce immediately:

$$\frac{{u}_{1}}{g}-\frac{2}{\lambda }{w}_{1}^{2}\frac{{K}_{\infty }}{{q}_{s}}>0$$

or also, indicating with ${K}_{y0}$ an average value of ${K}_{y}$:

$$\frac{{u}_{1}}{g}>\frac{2}{\lambda }\frac{{K}_{\infty }}{{K}_{y0}} ,$$

$$\lambda {u}_{1}>19.62\frac{{K}_{\infty }}{{K}_{y0}} .$$

Taken 1/10 as a practical minimum of $\frac{{K}_{\infty }}{{K}_{y0}}$, we get (approximately):

$$\lambda {u}_{1}>2 .$$

This limit can be easily exceeded in practice. Perhaps, other limitations could arise from a more in-depth analysis of the problem: but, at first glance, this kind of gliding seems more easily achievable than the previous one. It is understood that the frequency of wind fluctuations cannot be too strong, since the aircraft must make, in a period, a whole turn of its spiral.

1.6 The dynamic flight of Betz and its limits

Finally, there is another possible way to exploit the energy of the wind for dynamic flight, and it occurs when the wind direction varies with a sinusoidal law.¹

In this case, we will need (24), which we will simplify with the usual limitations, also neglecting the fluctuations of V-U and placing, at first, v = 0. Thus, we obtain:

$$\left\{\begin{array}{l}S{K}_{y}{(V-U)}^{2}-mg=m\frac{dv}{dt}\\ S{K}_{y}u\left(V-U\right)-S{K}_{\infty }{(V-U)}^{2}=m\frac{dV}{dt}\end{array}\right.$$

(32)

and, putting W = V − U = w₀, we get

$$\left\{\begin{array}{l}S{K}_{y}{w}_{0}^{2}-mg=m\frac{dv}{dt}\\ S{K}_{y}u{w}_{0}-S{K}_{\infty }{w}_{0}^{2}=m\frac{dV}{dt}\end{array}\right.$$

(33)

From the second of (33), putting:

$$u={u}_{1}sen(\lambda t)$$

We get:

$${SK}_{y}{u}_{1}{ w}_{0}sen\left(\lambda t\right)-S{K}_{\infty }{w}_{0}^{2}=m\frac{dV}{dt} .$$

We also put:

$${K}_{y}={b}_{0}+{b}_{1}sen(\lambda t+\beta )$$

to obtain:

$$S{u}_{1}{ w}_{0}{b}_{0}sen\left(\lambda t\right)+S{u}_{1}{ w}_{0}{b}_{1}sen\left(\lambda t\right)sen\left(\lambda t+\beta \right)-S{K}_{\infty }{w}_{0}^{2}=m\frac{dV}{dt}.$$

Integrating for one period $\left(\frac{2\pi }{\lambda }\right),$ you get:

$$0=-S{K}_{\infty }{w}_{0}^{2}\frac{2\pi }{\lambda }+S{u}_{1}{w}_{0}{b}_{1}cos\beta \frac{\pi }{2}$$

and it follows:

$$2{K}_{\infty }{w}_{0}={u}_{1}{b}_{1}cos\beta .$$

(34)

From the first of (33), integrating for an entire period, we also obtain:

$$S{b}_{0}{w}_{0}^{2}-mg=0 ,$$

from which:

$${b}_{0}{w}_{0}^{2}={q}_{s} .$$

(35)

If, instead of a horizontal flight, we have an uphill trajectory with ascending speed v, from the second of (32) we get:

$${SK}_{y}\left(u-v\right){w}_{0}-S{K}_{\infty }{w}_{0}^{2}=m\frac{dV }{dt} ,$$

(36)

so that:

$$2\left({K}_{\infty }{w}_{0}+{b}_{0}v\right)={u}_{1}{b}_{1}cos\beta .$$

(37)

The most favourable condition is obviously represented by $cos\beta =1$, for which:

$${K}_{\infty }{w}_{0}=\frac{{u}_{1}{b}_{1}}{2}-{b}_{0}v .$$

(38)

It follows:

$${K}_{\infty }{w}_{0}\le \frac{{u}_{1}{b}_{1}}{2} ,$$

(39)

and, for (28):

$${{K}_{\infty }}^{2}{q}_{s}\le \frac{{{u}_{1}}^{2}{{b}_{1}}^{2}{b}_{0}}{4} .$$

(40)

Now, since ${b}_{0}+{b}_{1}$ (maximum value of ${K}_{y}$) can not exceed a certain value, which we can still establish, to fix the ideas, in 0.075, it follows that we have the maximum of ${b}_{0}{{b}_{1}}^{2}$ when ${b}_{1}=2{b}_{0}$ and then ${b}_{0}+{b}_{1}=3{b}_{0}$.

It follows ${b}_{0}=0.025$ and:

$${{K}_{\infty }}^{2}{q}_{s}\le \frac{15.6}{{16}^{6}}{{u}_{1}}^{2} .$$

In this case, the lower limit of the ${K}_{y}$ is: − 0.025.

If assume as zero the lower limit of ${K}_{y}$, we obtain instead:

$${{K}_{\infty }}^{2}{q}_{s}\le \frac{13.2}{{10}^{6}}{{u}_{1}}^{2} .$$

Putting ${K}_{\infty }=0.003,$ it follows:

$${u}_{1}\ge \left\{\begin{array}{c}0.76\\ 0.83\end{array}{q}_{s}\right.$$

We can remark how the variations of the vertical speed of the wind should be very strong, at least for values ${q}_{s}$ between 6 and 12, as it is occurs in general.

The research can be deepened taking the fluctuations in the vertical speed, v, of the aircraft into account.

For this reason, the first of (32) is integrated:

$$mv=-mgt+S{{w}_{0}}^{2}{b}_{0}t-\frac{S{{w}_{0}}^{2}{b}_{1}}{\lambda }\mathit{cos}\left(\lambda t+\beta \right)+c .$$

Place, as usual:

$$mg=S{{w}_{0}}^{2}{b}_{0} ,$$

that is:

$${{{b}_{0}w}_{0}}^{2}={q}_{s} ,$$

it follows:

$$mv=c-\frac{mg}{{b}_{0}\lambda }{b}_{1}cos(\lambda t+\beta ) ,$$

or even:

$$v={c}_{1}-\frac{g{b}_{1}}{\lambda {b}_{0}}\mathit{cos}\left(\lambda t+\beta \right).$$

Substituting in (26b) gives:

$${Sw}_{0}\left({b}_{0}+{b}_{1}sen\left(\lambda t+\beta \right)\right)\left({u}_{1}sen\lambda t-{c}_{1}+\frac{g{b}_{1}}{\lambda {b}_{0}}\mathit{cos}\left(\lambda t+\beta \right)\right)-$$

$$-{SK}_{\infty }{{w}_{0}}^{2}=m\frac{dV}{dt} .$$

By integrating for one period, you get:

$$2\left({K}_{\infty }{w}_{0}+{b}_{0}{c}_{1}\right)={u}_{1}{b}_{1}cos\beta .$$

(41)

That is, a relationship perfectly similar to (41).

Therefore, the above considerations are also valid in this case.

We conclude that also this type of dynamic flight is difficult to be achieved in practice, requiring very low wing loads and exceptionally favourable wind conditions, perhaps never achievable practically.

It is, therefore, very likely, as mentioned in the beginning, that gliding without decreasing, or even with an increase in altitude, is always a static flight, that is, obtained by exploiting the ascending streams or even a wise combination of static flight and dynamic flight.