Generalized Weissinger’s L-method for prediction of curved wings operating above a free surface in subsonic flow

Abstract

The classical Weissinger’s L-method is generalized to the lifting problem for steadily advancing curved wings subject to the wing-in-ground (WIG) effect above a large body of water in subsonic flow, and the free surface defines the boundary between the air and water. Unlike the traditional analysis of the lifting problem, the essential techniques focus on finding the three-dimensional free surface Green’s function generated by the isolated horseshoe vortex in the upper layer of the stratified fluid where the air is regarded as weakly compressible and the water is incompressible. The numerical calculation is implemented using Weissinger’s L-method. Finally, the effects of the curved geometry on WIG effect in the vicinity of a free surface in subsonic flow are discussed. Extensive numerical examples are carried out to show the lift properties for three-dimensional swept and dihedral wings operating in the vicinity of a free surface as a function of the sweep or dihedral angle for different clearance-to-chord ratios and Mach numbers. Interestingly, for high Froude numbers, the free surface effectively becomes rigid, and it can safely be treated as a solid surface.

Appendices

Appendix 1

In this part, we aim to verify that the elevation of a free surface is small. Based on the one-dimensional continuity equation, we have [32]

$$\begin{aligned} Uh=\left( {U+\frac{\partial \varphi _a }{\partial x}} \right) \left[ {h-t\left( {x,y} \right) -\zeta \left( {x,y} \right) } \right] . \end{aligned}$$

(72)

Thus, the free surface elevation can be obtained from Eq.(72):

$$\begin{aligned} \zeta \left( {x,y} \right) =h-t\left( {x,y} \right) -\frac{Uh}{U+\frac{\partial \varphi _a }{\partial x}}. \end{aligned}$$

(73)

Based on Bernoulli equation in (7), the free surface elevation can be expressed in the form

$$\begin{aligned} \zeta \left( {x,y} \right)&= -\frac{\rho _w }{g\left( {\rho _w -\rho _a } \right) }\left[ {U\frac{\partial \varphi _w }{\partial x}+\frac{1}{2}\left( {\frac{\partial \varphi _w }{\partial x}} \right) ^{2}+\frac{1}{2}\left( {\frac{\partial \varphi _w }{\partial y}} \right) ^{2}+\frac{1}{2}\left( {\frac{\partial \varphi _w }{\partial z}} \right) ^{2}} \right] \nonumber \\&+\frac{\rho _a }{g\left( {\rho _w -\rho _a } \right) }\left[ {U\frac{\partial \varphi _a }{\partial x}+\frac{1}{2}\left( {\frac{\partial \varphi _a }{\partial x}} \right) ^{2}+\frac{1}{2}\left( {\frac{\partial \varphi _a }{\partial y}} \right) ^{2}+\frac{1}{2}\left( {\frac{\partial \varphi _a }{\partial z}} \right) ^{2}} \right] . \end{aligned}$$

(74)

Due to the small density ratio of air to water, the density ratio can be expressed as

$$\begin{aligned} \frac{\rho _a }{\rho _w }=\delta _1 \rho , \end{aligned}$$

(75)

where $\rho$ is an $O(1)$ quantity. By substituting asymptotic expansions of the velocity potentials in air and water in Eqs. (17) and (18) into Eq. (74) and preserving the first-order expression, we can obtain

$$\begin{aligned} \zeta (x)=-\frac{\delta _1 U\frac{\partial \varphi _w^{\left( 1 \right) } }{\partial x}}{g\left( {1-\delta _1 \rho } \right) }. \end{aligned}$$

(76)

From Eq. (76), it can be argued that the free surface deformation induced by the foil has the same order as $\delta _{1}$.

Appendix 2

In this part, we aim to derive the perturbed velocity components induced by the vortex filament with an arbitrary contour in the subsonic regime using the Biot–Savart law, which reads

$$\begin{aligned} \mathbf{u}=-\frac{\Gamma }{4\pi }\int \limits _C {\frac{\mathbf{R}\times \text{ d }\mathbf{l}}{R^{3}}}. \end{aligned}$$

(77)

Due to the scale stretching of the $x$ coordinate in the subsonic flow, the vectors R and dl can be expressed as

$$\begin{aligned} \mathbf{R}=\left( {\frac{x-\xi }{\beta },\;y-\eta ,\;z-\zeta } \right) \end{aligned}$$

(78)

and

$$\begin{aligned} \text{ d }\mathbf{l}=\left( {\frac{1}{\beta }\text{ d }\xi ,\;\text{ d }\eta ,\;\text{ d }\zeta } \right) . \end{aligned}$$

(79)

Thus, the perturbed velocity components in the Ox, Oy, and Oz directions are

$$\begin{aligned} u&= -\frac{\Gamma }{4\pi }\int \limits _C {\frac{\left( {y-\eta } \right) \text{ d }\zeta -\left( {z-\zeta } \right) \text{ d }\eta }{\left[ {\left( {\frac{x-\xi }{\beta }} \right) ^{2}+\left( {y-\eta } \right) ^{2}+\left( {z-\zeta } \right) ^{2}} \right] ^{3/2}}} =\frac{\Gamma }{4\pi }\int \limits _C {\frac{\partial }{\partial y}\left( {\frac{1}{R}} \right) \text{ d }\zeta -\frac{\partial }{\partial z}\left( {\frac{1}{R}} \right) \text{ d }\eta } ,\end{aligned}$$

(80)

$$\begin{aligned} v&= -\frac{\Gamma }{4\pi }\int \limits _C {\frac{\left( {z-\zeta } \right) \frac{\text{ d }\xi }{\beta }-\left( {\frac{x-\xi }{\beta }} \right) \text{ d }\zeta }{\left[ {\left( {\frac{x-\xi }{\beta }} \right) ^{2}+\left( {y-\eta } \right) ^{2}+\left( {z-\zeta } \right) ^{2}} \right] ^{3/2}}} =\frac{\Gamma }{4\pi }\int \limits _C {\frac{1}{\beta }\frac{\partial }{\partial z}\left( {\frac{1}{R}} \right) \text{ d }\xi -\beta \frac{\partial }{\partial x}\left( {\frac{1}{R}} \right) \text{ d }\zeta }, \end{aligned}$$

(81)

and

$$\begin{aligned} w=-\frac{\Gamma }{4\pi }\int \limits _C {\frac{\left( {\frac{x-\xi }{\beta }} \right) \text{ d }\eta -\left( {y-\eta } \right) \frac{\text{ d }\xi }{\beta }}{\left[ {\left( {\frac{x-\xi }{\beta }} \right) ^{2}+\left( {y-\eta } \right) ^{2}+\left( {z-\zeta } \right) ^{2}} \right] ^{3/2}}} =\frac{\Gamma }{4\pi }\int \limits _C {\beta \frac{\partial }{\partial x}\left( {\frac{1}{R}} \right) \text{ d }\eta -\frac{1}{\beta }\frac{\partial }{\partial y}\left( {\frac{1}{R}} \right) \text{ d }\xi }. \end{aligned}$$

(82)