4.1 Inferences from performance claims

In horizontal flight, the lift must balance the weight of the aircraft. Therefore, we can estimate the lift coefficient for the aircraft described in Lanier’s patents by considering the weight.

The Lanier Vacuplane of 1935 had a gross weight (including the pilot) of 574 pounds ( $M=260$ kg), a take-off/landing speed as low as 30 mph ( $\approx 48$ km/h or $13.3$ m/s), a cruising speed for level flight of 80 mph ( $U=128$ km/hr $\approx 35.6$ m/s) and a wing area of 73 square feet ( $A\approx 7$ m $^2$ ) [Reference Bowers4]. Using the density of air $\rho =1.23$ kg/m $^3$ at 15 $^\circ $ C and atmospheric pressure, the lift equation (2.1) gives a lift coefficient $C_L\approx 2Mg/(\rho U^2A) \approx 0.47$ , where $g=9.8$ m/s $^2$ is the acceleration due to gravity. This is comparable with lift coefficients of conventional aircraft.

The Lanier Paraplane Commuter 110 (see [Reference Bowers4]) was built by Lanier aircraft corporation around 1949, 16 years after the original patents were submitted, and is of unknown design. This aeroplane had similar take-off and cruise speeds but a greater mass (640 kg), and roughly 30% greater wing area, giving a lift coefficient of $C_L\approx 0.88$ , again within conventional values.

Further to this, we can estimate the drag coefficient by considering the maximum speed. The 1935 Vacuplane [Reference Bowers4, 19] had a 36 horsepower engine ( $P_E\approx 27$ kW), and an estimated top speed of $96$ mph ( $\approx 154$ km/h or $43$ m/s). Drag $D=C_D \rho U^2 A/2$ (2.2) and the power required to overcome this drag is $P=DU \approx 3.4\times 10^5 C_D$ Watts, so $P\approx 340 C_D$ kW. By comparison, we see that the drag coefficient is $C_D \approx 0.079$ .

The Lanier Paraplane Commuter 110 had a maximum speed of $165$ mph ( $\approx 74$ m/s), a $150\,\mbox {Hp} \approx 112$ kW engine and a slightly greater wing area. Using the same approach, $P\approx 2300 C_D$ kW, and therefore the drag coefficient $C_D \approx 0.049$ .

Subsequent calculations use an airfoil shape approximating the Clark-Y wing [Reference Jones12]. For comparison, the lift and drag coefficients of this wing at $0^\circ \ (6^\circ )$ angle of attack are $C_L=0.36\ (0.80)$ and $C_D=0.0217\ (0.045)$ .

These basic calculations are built on less than ideal data extracted from popular literature. Nevertheless, they indicate that there is nothing extraordinary in the behaviour of the Lanier aircraft.

A further implication in some of the popular literature and Lanier’s patents (see Section 3) is that there is a buoyancy effect of air being sucked out of the wing cavity. However, it is easy to show that the effect of reducing air density within the wings would have an almost negligible effect, perhaps lightening the aeroplane by a few hundred grams. For example, the total weight of air in a wing cavity with a volume of two cubic metres (estimated for the Lanier XL-4) is approximately 2.4 kilograms or approximately $1\%$ of the total weight. However, since not all of the air could be evacuated, this is a very generous upper bound. In heavier, larger craft, this proportion would be greatly reduced.

In a stall situation, the pressure would equalise between the inside and outside of the wing, causing the air to rush back in, negating any buoyancy effect in free flight. It may be that the effect of drag on the lighter and slatted (and hence rougher) wings is greater than that on the engine and cabin, causing the plane to right itself as it falls, but this will depend on the plane’s attitude at stall.

4.2 Inviscid theory for an elliptic airfoil

In this section, we assume a large wing span relative to its chord length so that we may solve for the two-dimensional flow about the cross-section of the wing. We also assume an elliptic cross-section and, hence, compute the lift on an elliptic airfoil, the centre of which is at height H above the ground, using an integral equation method to determine the effects of wing thickness, angle of attack and proximity to the ground. The lift on an airfoil can be determined by inviscid flow theory with the ground effect included using the method of images, that is, we consider the flow around the ellipse and its image such that the ground is a line of symmetry between the two. Assuming an inviscid, incompressible fluid, the flow is irrotational so that the problem reduces to that of solving for the velocity potential $\Phi $ , where the velocity field $\mathbf {q}=\nabla \Phi $ , and $\Phi $ must satisfy Laplace’s equation $\nabla ^2 \Phi = 0$ .

One way to do this is to compute the complex potential $w(z)= \Phi +i\Psi $ , where $z=x+iy$ , y being vertical distance above the ground and $\Psi $ is the streamfunction. For our problem, the complex potential is given by

(4.1) $$ \begin{align} w(z) = Uz+\frac {i\Gamma}{2\pi}\log(z^2+H^2) +\chi(z), \end{align} $$

where the first term represents the free stream flow with velocity U, the second the circulation around the airfoil and the third, $\chi (z)$ , is to be determined to satisfy the boundary conditions for the flow. From complex function theory, we have that the velocity potential $\Phi $ satisfies Laplace’s equation provided $w(z)$ is an analytic function. The streamfunction $\Psi $ must be constant on the surface of the airfoil so that there is no flow through the surface of the airfoil, that is, $\mathbf {q} \cdot \mathbf {n}=0,$ where $\mathbf {n}$ is the normal to the boundary of the airfoil which has upper and lower surfaces $y=f^{\pm }(x)$ . Similarly, $\Psi $ is constant on the ground $z=x+i0$ .

Using Cauchy’s integral formula to integrate the function $\chi (z)=\xi +i\eta $ around a contour including the airfoil, its image on reflection in the line of symmetry of the ground and a circle of infinite limiting radius leaves

$$ \begin{align*}\chi(z_0)=\frac {1}{2\pi i} \int_\gamma \ \frac {\chi(z)}{z-z_0} \ dz,\end{align*} $$

where $\gamma $ consists of that part of the contour around the surface of the airfoil and its image. Defining arclength s, where $ds=\sqrt {dx^2+dy^2}$ , and using the chain rule together with the symmetry of the airfoil and its image, it can be shown that the real part of $\chi $ , that is, $\xi $ , is given by the integral equation

(4.2) $$ \begin{align} \xi(s) & =\displaystyle{\frac{1}{\pi}\int_0^{s_L} \frac{\xi(t)[{y'(t)\Delta x - x'(t)\Delta y}] - \eta(t) [{x'(t)\Delta x+y'(t)\Delta y}]}{(\Delta x^2 + \Delta y^2)}}\nonumber \\[.1cm] & \quad {+ \frac{\xi(t)[{y'(t)\Delta x - x'(t)\Delta y_+}] - \eta(t) [{x'(t)\Delta x+y'(t)\Delta y_+}]}{(\Delta x^2 + \Delta y^2_+)} \,dt,} \end{align} $$

where $s_L$ is the arclength from the trailing edge of the body to the leading edge then back, $\Delta x = x(t)-x(s)$ , $\Delta y = y(t)-y(s)$ and $\Delta y_+ = y(t)+y(s)$ .

Thus, the method is to write the surface of the airfoil in parametric form $(x(s),y(s))$ , and then take a discrete form of the integral using steps in arclength, $s_k,\, k=1,2,\ldots ,N$ . Replacing the integral by a sum, the unknown $\xi _k=\xi (s_k)$ can be obtained by solving N equations in N unknowns. Further details of the method can be found in [Reference Scott20].

We also know from (4.1) that the function $\chi (z)=\xi +i\eta $ is made up of the following components:

$$ \begin{align*} \xi(s)&=\Phi(s)-Ux(s)+\frac{\Gamma}{2\pi}(\beta_1(s)-\beta_2(s)),\\ \eta(s)& = \Psi_0 - Uy(s) - \frac{\Gamma}{2\pi}\ln\bigg[\frac{\rho_1(s)}{\rho_2(s)}\bigg], \end{align*} $$

where $\Psi _0$ is the (constant) value of the streamfunction on the airfoil surface,

$$ \begin{align*} &\rho_1=(x^2+(y-H)^2)^{1/2}, \quad \rho_2=(x^2+(y+H)^2)^{1/2} \end{align*} $$

are the distances between $z=\pm iH$ and points $z=x(s)+iy(s)$ on the surface, and

$$ \begin{align*} &\beta_1=\arctan\bigg(\frac{y-H}{x}\bigg), \quad \beta_2=\arctan\bigg(\frac{y+H}{x}\bigg) \end{align*} $$

are the angles of lines between $z=\pm iH$ and points on the surface. Thus, $\eta $ is known everywhere on the surface, and the integral equation (4.2) can be used to find $\xi $ and hence the velocity potential.

The crucial factor in determining the lift is the Kutta condition, which requires that the flow detaches smoothly from the end of the airfoil. The circulation $\Gamma $ must be chosen to ensure this condition is satisfied. This was achieved by allowing $\Gamma $ to be one of the unknowns and including an extra equation to enforce this condition. In the case of an ellipse having a blunt end, it was required that the stagnation point forms at the trailing edge (i.e. rear-most point) of the ellipse.

A Fortran program was written to implement this solution method and simulations were performed using this code for various values of wing chord, thickness, angle of attack and height above the ground. Figure 1 shows an increase in the theoretical lift $C_L$ with increasing angle of attack $\alpha $ for elliptic airfoils at large height ( $H\rightarrow \infty $ ) above the ground (no ground effect) with maximum dimensionless thickness $\tau =0.1,\,0.2$ (scaled with chord-length $\mathcal C$ ). These data are compared with the lift on a flat plate (or thin symmetric airfoil), computed using the analytic formula $C_L=2\pi \sin \alpha $ , and with the lift coefficient for the Clark-Y wing which has $\tau \approx 0.12$ . Clearly, the nonsymmetric Clark-Y airfoil performs much better at small angle of attack than those used for numerical experiments. Moreover, its lift coefficient continues to increase through to $\alpha \approx 18.5^\circ $ beyond which it will stall (lose lift). In contrast, the other airfoils will undergo stall at much lower angles of attack, which is not apparent from the figure because separation and stall were not computed. The effect of wing thickness is seen for the elliptic airfoil; doubling $\tau $ from 0.1 to 0.2 increases the lift by approximately $10\%$ at each angle of attack.

Figure 1 Lift coefficient $C_L$ versus angle of attack $\alpha $ for elliptic airfoils of (dimensionless) thickness (a) $\tau =0.1$ and (b) $\tau =0.2$ . Also shown for comparison is $C_L$ versus $\alpha $ for (c) a flat plate as given by the analytic formula, and (d) a Clark-Y wing having $\tau \approx 0.12$ for which stall occurs at $\alpha \approx 18.5^\circ $ .

Figure 2 shows the effect of proximity to the ground $h=H/\mathcal C$ on the lift coefficient for airfoils of several different values of (dimensionless) thickness $\tau $ . It is clear that ground effect increases lift when the ground is within one or two chord-lengths $\mathcal C$ of the airfoil, suggesting that ground effect can be neglected from our deliberations.

Figure 2 Lift coefficient $C_L$ versus (dimensionless) height $h=H/{\mathcal C}$ above ground for elliptic airfoils of (dimensionless) thickness (a) $\tau =0.05$ , (b) $\tau =0.1$ and (c) $\tau =0.2$ , all at angle of attack $\alpha =10^\circ $ .

In general, wing profile designs must balance lift with drag. Our results confirm that thicker wings tend to give greater lift for a given speed. However, they also tend to have increased drag making it more difficult to attain speed. In addition, thicker wings at higher speed are more likely to induce separation of the flow and, hence, stall.

At the time of Lanier’s patents, wings were generally narrow in profile. However, one of his patents includes an illustration of a conventional wing together with the slatted wing of the patent design [Reference Lanier18]. The slatted wing is drawn much thicker than the conventional wing and if this was the case in practice, then this might explain an increase in lift for the Lanier aeroplane. Note, however, that the simple calculations in the previous section suggest that the lift of the Lanier craft was not exceptional compared to conventional airfoils such as the Clark-Y wing.

5.1 Two-dimensional viscous flow over an airfoil

The form drag of an airfoil is due to the viscosity of the fluid and to determine this, as well as its lift, we cannot use inviscid theory. Here, we consider two-dimensional viscous flow around thick and thin airfoils, with and without Lanier-type cavities, at different angles of attack. We necessarily neglect induced drag, which is a three-dimensional effect as described earlier, although this can be significant, especially for short wings. We have also made no attempt to compute form drag from the aircraft fuselage, focusing rather on the trade-off between lift and drag for a “slat-wing” airfoil compared to a conventional airfoil.

The geometries of the thick airfoils used in the simulations are shown in Figure 3. The basic airfoil shape superficially resembles the Clark-Y wing [Reference Jones12]. The chord length ${\cal C}$ of the airfoil is eleven times the nose radius R. For comparison with the conventional shape, the flow around a similar shape with a cut away cavity and slats, to resemble the Lanier “slat-wing” design, is also considered. The thin airfoil geometries are obtained by halving lengths in the vertical direction, giving a less blunt elliptical nose. Again, a conventional closed airfoil and a “slat-wing” geometry are considered.

Figure 3 Typical “thick” airfoil geometry. Conventional airfoil shown solid; cavity and vertical slats shown dashed. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

The flow is assumed to be two-dimensional, incompressible, steady and laminar, with a Reynolds number of 10. Although, Reynolds numbers of order $10^5$ are to be expected, this was about the maximum that Fastflo could reliably handle. Further, the flow would almost certainly be turbulent, but only laminar flow was considered. Despite these drawbacks, the simulations still allow a comparison of the fundamental behaviour of a conventional airfoil and a Lanier “slat-wing” airfoil. Separation of the flow from the airfoil is expected to occur at lower angles of attack for the blunt-nosed thick airfoil than for the thin airfoil.

The continuity and steady Navier–Stokes equations must be solved for the flow around four different airfoils (thick/thin $\times $ conventional/“slat-wing”), subject to no-slip on the airfoil boundary. A reference frame is adopted that moves with the airfoil at speed U. The horizontal and vertical axes are x and y, respectively, with the origin at the tip of the nose of the airfoil (see Figure 3). Let $u,v$ be the $x,y$ components of velocity scaled with U, and let p denote pressure scaled with $\rho U^2$ . Lengths are scaled with the nose radius R. Then the dimensionless continuity equation is

(5.1) $$ \begin{align} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \end{align} $$

and the Navier–Stokes equations are

(5.2) $$ \begin{align} u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}& = -\frac{\partial p}{\partial x} + \frac{1}{Re}\bigg(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\bigg), \end{align} $$

(5.3) $$ \begin{align} u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}& = -\frac{\partial p}{\partial y} + \frac{1}{Re}\bigg(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\bigg), \end{align} $$

where $Re=UR/\nu $ is the Reynolds number, $\nu $ being the kinematic viscosity of air ( $\approx 1.5\times 10^{-5}$ m $^2$ /s). On the boundary of the airfoil, we have $u=v=0$ . Far upstream of the airfoil, the flow is taken to be a uniform stream of magnitude U at angle of attack $\alpha $ . Sufficiently far above and below the airfoil, we expect the flow to be that of a uniform stream also. As the solution must be done over a finite computational domain, a square far-field boundary is defined with sides of length $20R$ around the airfoil, aligned with the far-upstream flow and with centre at the tip of the nose of the airfoil $(x,y)=(0,0)$ . Thus, on the inlet (left), top and bottom boundaries, the velocity is specified as $u=\cos \alpha ,\ v=\sin \alpha $ . The outlet (right) boundary is assumed to be a stress-free boundary. The far-field boundary is sufficiently far from the airfoil that the prescribed boundary conditions do not have significant impact on the solution.

The finite-element PDE solver Fastflo was used to solve for the flow. Fastflo’s automatic mesh generator was used to generate an unstructured mesh of approximately 1900 6-node triangles over the computational domain, with elements clustered more densely near the airfoil. The “augmented Lagrangian method” [7, Section 13.3] and quadratic basis functions were used to solve for pressure and velocity.

Having solved for velocity and pressure, lift and drag forces per unit wing-span were found by integrating the pressure around the surface of the airfoil $d\Omega $ , that is,

$$ \begin{align*} \frac{\mathbf{F}}{\rho U^2R}=\oint_{d\Omega} p\,d\mathbf{r}. \end{align*} $$

Resolving the force per unit span obtained into two components $\mathbf {F}=(F_x,F_y)$ , the drag $D=F_x\cos \alpha -F_y\sin \alpha $ in the direction of the uniform stream, and the lift $L=F_x\sin \alpha +F_y\cos \alpha $ normal to it. From these, the lift and drag coefficients were computed as

$$ \begin{align*} C_L=\frac{2L}{11\rho U^2 R},\quad C_D\approx\frac{2D}{11\rho U^2 R}. \end{align*} $$

Table 1 shows these coefficients for different angles of attack $\alpha $ for each of the four airfoils considered, while they are plotted against $\alpha $ in Figures 4 and 5, respectively. Figure 6 shows the ratio of lift to drag, i.e. $C_L/C_D$ , again versus angle of attack. In Figures 7–10, we show stream lines around the airfoils and velocity vectors near the upper surface behind the nose of the airfoils.

Table 1 Lift ( $C_L$ ) and drag ( $C_D$ ) coefficients at angle of attack $\alpha $ for thick and thin conventional and “slat-wing” airfoils. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 4 Lift coefficient $C_L$ versus angle of attack $\alpha $ for the (a) thick conventional, (b) thick “slat-wing”, (c) thin conventional and (d) thin “slat-wing” airfoils. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 5 Drag coefficient $C_D$ versus angle of attack $\alpha $ for the (a) thick conventional, (b) thick “slat-wing”, (c) thin conventional and (d) thin “slat-wing” airfoils. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 6 Ratio of lift to drag $C_L/C_D$ versus angle of attack $\alpha $ for the (a) thick conventional, (b) thick “slat-wing”, (c) thin conventional and (d) thin “slat-wing” airfoils. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 7 Thick, conventional airfoil. Streamlines and velocity vectors. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 8 Thick, “slat-wing” airfoil. Streamlines and velocity vectors. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 9 Thin, conventional airfoil. Streamlines and velocity vectors. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 10 Thin, “slat-wing” airfoil. Streamlines and velocity vectors. (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

A comparison of the curve for the Clark-Y wing in Figure 1 with those for the thin wings in Figure 4 shows the lift coefficients to be of a similar order of magnitude at the same angle of attack and gives some assurance that the general behaviour of the wings under investigation is captured by the low Reynolds number simulations. It is expected that at higher Reynolds number, the boundary layers will be thinner and the lift coefficients a little larger. In keeping with known aerodynamic behaviour, the lift coefficient for the thick wings is larger than for the thin wings at small angle of attack, however, the slope of the curve $C_L$ versus angle of attack $\alpha $ is greater for thin wings than for thick wings so that this situation reverses at larger angle of attack. (For thin symmetric wings, $C_L/\!\sin \alpha \sim 2\pi $ , while $C_L/\!\sin \alpha \sim 4$ for the thin asymmetric airfoils considered here.) For the thin wings, there is a sudden decrease in slope beyond $\alpha =15^\circ $ signalling imminent stall (loss of lift). This is due to flow separation which occurs at approximately $\alpha =15^\circ $ , as seen in the plots of streamlines and velocity vectors given in Figures 9 and 10. The thicker wings experience flow separation at a lower angle of attack $\alpha \sim 10^\circ $ , as seen in Figures 7 and 8, but do not exhibit such a sudden reduction in lift. It is readily seen that the conventional airfoil and corresponding “slat-wing” airfoil, whether thick or thin, differ little from one another in terms of lift coefficient.

As is to be expected, the drag coefficient increases with angle of attack, as seen for all airfoils in Figure 5. For the thin airfoils, we have $C_D\approx 0.1\ (0.15)$ at $\alpha =0^\circ \ (5^\circ )$ compared with $C_D\approx 0.022\ (0.045)$ for the Clark-Y wing. It is expected that with thinner boundary layers at higher Reynolds numbers, drag coefficients will be lower than indicated by our simulations. There is a significant increase in drag for the thin airfoils from $\alpha =10^\circ $ to $\alpha =15^\circ $ , which may be attributable to a relatively large increase in the projection of the surface area normal to the flow, an effect which would be smaller for thicker airfoils. There appears to be a slight increase in drag for the thick “slat-wing” airfoil compared to the thick conventional airfoil, but the curves of $C_D$ versus $\alpha $ differ little between the two thin airfoils.

The curves of lift to drag ratio $C_L/C_D$ versus angle of attack (Figure 6) indicate that both the thick and the thin airfoils are most efficient over the range $\alpha =5\text {--}10^\circ $ . For the thick airfoils, it is evident that the conventional shape is superior to the “slat-wing” in giving slightly more lift, less drag and, consequently, a higher lift to drag ratio at a fixed angle of attack, over the full range of angles considered. For the thinner airfoils, the conventional and “slat-wing” profiles are very similar except at $\alpha =10^\circ $ where the conventional wing again appears to be superior.

Although we are mindful of the fact that the computations are not very accurate, the clear message emerging from our work is that our simulations certainly provide no evidence to support the “slat-wing” over conventional airfoils, but, if anything, the reverse.

5.2 Viscous flow over a slot

Further numerical simulations were done to illustrate the general features of flow over a slot cavity. The typical dimensionless geometry used for these is shown in Figure 11. The continuity and Navier–Stokes equations (5.1)–(5.3) were solved, again using Fastflo and the augmented Lagrangian method with quadratic basis functions. At the inlet (left boundary), we specified the flow to be that of a unit uniform stream ( $U=1$ ), while the outlet (right) was defined to be a stress-free boundary. The lower boundary containing the cavity is, of course, a no-slip boundary ( $u=v=0$ ); at the upper boundary, we specified no normal flow ( $v=0$ ) and no tangential stress, that is, this is a slip boundary. A mesh of approximately 3000 6-node triangles was used over the computational domain.

Figure 11 Geometry used for simulations of flow over a slot cavity. The slot aspect ratio is defined as $\varrho =w/d$ . (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

Figure 12 Flow in the vicinity of a slot of aspect ratio $\varrho =3/2$ ( $d=1.5,\ w=1$ ) at various angles of inclination $\beta $ . The colour of contours from blue to red indicates the change in value from lowest to highest. (Colour available online.) (Reproduced from [Reference Hocking, Stokes, Sweatman and Wake10].)

The effects of slot aspect ratio ( $\varrho =w/d$ ) and slot angle $\beta $ were considered to a limited extent. Pressure contours and streamlines are shown in Figure 12 for a cavity of depth $d=1.5$ and width $w=1$ at angles of inclination $\beta =60^\circ , 90^\circ , 120^\circ $ . These were computed at a Reynolds number of $Re=1000$ ; at higher Reynolds numbers, convergence difficulties were experienced. The results shown are typical of cavities of both larger and smaller aspect ratio, although the width and depth of the slot does vary the vortex flow and pressure. As can be seen, a vortex develops in the slot. The pressure at the centre of this vortex is lower than the average pressure in the surrounding fluid, but the overall pressure in the slot is very similar to that in the fluid immediately above the slot. This confirms the earlier findings that slots in the upper surface of a wing make little difference to its lifting capacity.