An analytical investigation of two-dimensional and three-dimensional biplanes operating in the vicinity of a free surface

Abstract

In the present article, the classical two- and three-dimensional lifting theories are generalized to the biplane operating in proximity to a free surface. The singularity distribution method is employed to calculate the lifting force for a two-dimensional biplane subjected to wing-in-ground effect in the vicinity of a free surface, and the three-dimensional correction is carried out by the aid of the Prandtl lifting line theory. The essential techniques lie in finding the three-dimensional Green’s function for the system of horseshoe vortices operating above a free surface and ensuring numerical implementation. Extensive numerical examples are carried out to show the lift coefficient for the two- and three-dimensional biplanes in the vicinity of a free surface with the variation of the clearance-to-chord ratio and the height-to-chord ratio. Incidentally, the induced (inviscid) drag coefficients as well as the lift-to-drag ratio for a three-dimensional biplane are also computed. Good agreement can be found among results obtained from this study and the experiment.

Appendices

Appendix 1

Equations 17 and 18 are integral equations. For the integral equation with one integrand, an analytical solution may be derived. However, obtaining the analytical solution to an integral equation with more than two integrands is almost impossible. Thus, we should seek the solution to Eqs. 17 or 18 from a numerical position. Here, we only exhibit the numerical solution to Eq. 14.

The purpose of solving Eq. 17 is obtaining the numerical solution to the vortex strength distribution. By separating the real and imaginary parts of Eq. 17, we can obtain the vertical velocity component

$$ \begin{aligned} v_{\text{lv}} & = \frac{1}{2\pi }\int\limits_{0}^{c} {\frac{{\gamma_{\text{l}} \left( \xi \right){\text{d}}\xi }}{x - \xi }} - \frac{1}{2\pi }\int\limits_{0}^{c} {\frac{{\gamma_{\text{l}} \left( \xi \right)\left( {x - \xi } \right){\text{d}}\xi }}{{\left( {x - \xi } \right)^{2} + 4h^{2} }}} \\ & \quad - Re\int\limits_{0}^{c} {\frac{{{\text{i}}ak\gamma_{\text{l}} \left( \xi \right)}}{\pi }{\text{d}}\xi \int\limits_{0}^{\infty } {\frac{{{\text{e}}^{{ik\left( {x - \xi + 2{\text{i}}h} \right)}} }}{{k - \mu + a\left( {k + \mu } \right)}}{\text{d}}k} } \\ & \quad - \int\limits_{0}^{c} {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} \gamma_{1} \left( \xi \right){\text{e}}^{{ - 2k_{0} h}} \cos k_{0} \left( {x - \xi } \right){\text{d}}\xi } \\ & \quad + \frac{1}{2\pi }\int\limits_{0}^{c} {\frac{{\gamma_{\text{u}} \left( \xi \right)\left( {x - \xi } \right){\text{d}}\xi }}{{\left( {x - \xi } \right)^{2} + d^{2} }}} - \frac{1}{2\pi }\int\limits_{0}^{c} {\frac{{\gamma_{\text{u}} \left( \xi \right)\left( {x - \xi } \right){\text{d}}\xi }}{{\left( {x - \xi } \right)^{2} + \left( {2h + d} \right)^{2} }}} \\ & \quad - Re\int\limits_{0}^{c} {\frac{{{\text{i}}ak\gamma_{\text{u}} \left( \xi \right)}}{\pi }{\text{d}}\xi \int\limits_{0}^{\infty } {\frac{{{\text{e}}^{{ik\left( {x - \xi + 2{\text{i}}h + {\text{i}}d} \right)}} }}{{k - \mu + a\left( {k + \mu } \right)}}{\text{d}}k} } \\ & \quad - \int\limits_{0}^{c} {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} \gamma_{\text{u}} \left( \xi \right){\text{e}}^{{ - k_{0} \left( {2h + d} \right)}} \cos k_{0} \left( {x - \xi } \right){\text{d}}\xi }. \end{aligned} $$

(55)

In Eq. 55, the Kutta condition stating γ _u = γ _l = 0 at the trailing edge should be imposed. For the purpose of solving Eq. 55 numerically, the transformation into the trigonometric variables is introduced. The length scale can be expressed as

$$ x = \frac{c}{2}\left( {1 - \cos \theta } \right),\quad \xi = \frac{c}{2}\left( {1 - \cos \theta^{\prime}} \right) , $$

(56)

and its derivative is

$$ {\text{d}}\xi = \frac{c}{2}\sin \theta^{\prime}{\text{d}}\theta^{\prime} . $$

(57)

It should be noted from Eq. 56 that the trailing edge is at θ = 0, and the leading edge is at θ = π. By taking the Kutta condition into account, the vortex distribution can be expanded in the form of trigonometric series [28, 29]

$$ \gamma_{\text{l}} = 2U\left[ {A_{0} \cot \left( {\frac{1}{2}\theta^{\prime}} \right) + \sum\limits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}} } \right] , $$

(58)

and

$$ \gamma_{\text{u}} = 2U\left[ {B_{0} \cot \left( {\frac{1}{2}\theta^{\prime}} \right) + \sum\limits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}} } \right] . $$

(59)

Equations 58 and 59 are equal to zero at θ = π. Substituting Eqs. 56–59 into Eq. 55, we can obtain

$$ \begin{aligned} v_{\text{lv}} = & - UA_{0} + U\sum\limits_{n = 1}^{\infty } {A_{n} \cos n\theta } \\ & \quad - \frac{1}{\pi }\int\limits_{0}^{\pi } {\frac{{U\left[ {A_{0} \left( {1 + \cos \theta^{\prime}} \right) + \sum\nolimits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} } \right]\left( {\cos \theta - \cos \theta^{\prime}} \right){\text{d}}\theta^{\prime}}}{{\left( {\cos \theta - \cos \theta^{\prime}} \right)^{2} + 16\left( {{h \mathord{\left/ {\vphantom {h c}} \right. \kern-\nulldelimiterspace} c}} \right)^{2} }}} \\ & \quad - Re\int\limits_{0}^{\pi } {\frac{{{\text{i}}akcU}}{\pi }\left[ {A_{0} \left( {1 + \cos \theta^{\prime}} \right) + \sum\limits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} } \right]{\text{e}}^{{\chi_{\text{l}} }} E_{1} \left( {\chi_{\text{l}} } \right){\text{d}}\theta^{\prime}} \\ & \quad - \int\limits_{0}^{\pi } {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} cU{\text{e}}^{{ - 2k_{0} h}} A_{0} \left( {1 + \cos \theta^{\prime}} \right)\cos \left[ {\frac{{ck_{0} }}{2}\left( {\cos \theta - \cos \theta^{\prime}} \right)} \right]{\text{d}}\theta^{\prime}} \\ & \quad - \int\limits_{0}^{\pi } {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} cU{\text{e}}^{{ - 2k_{0} h}} \sum\limits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} \cos \left[ {\frac{{ck_{0} }}{2}\left( {\cos \theta - \cos \theta^{\prime}} \right)} \right]{\text{d}}\theta^{\prime}} \\ & \quad + \frac{1}{\pi }\int\limits_{0}^{\pi } {\frac{{U\left[ {B_{0} \left( {1 + \cos \theta^{\prime}} \right) + \sum\nolimits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} } \right]\left( {\cos \theta - \cos \theta^{\prime}} \right){\text{d}}\theta^{\prime}}}{{\left( {\cos \theta - \cos \theta^{\prime}} \right)^{2} + 4\left( {{d \mathord{\left/ {\vphantom {d c}} \right. \kern-\nulldelimiterspace} c}} \right)^{2} }}} \\ & \quad - \frac{1}{\pi }\int\limits_{0}^{\pi } {\frac{{U\left[ {B_{0} \left( {1 + \cos \theta^{\prime}} \right) + \sum\nolimits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} } \right]\left( {\cos \theta - \cos \theta^{\prime}} \right){\text{d}}\theta^{\prime}}}{{\left( {\cos \theta - \cos \theta^{\prime}} \right)^{2} + 4\left[ {{{\left( {2h + d} \right)} \mathord{\left/ {\vphantom {{\left( {2h + d} \right)} c}} \right. \kern-\nulldelimiterspace} c}} \right]^{2} }}} \\ & \quad - Re\int\limits_{0}^{\pi } {\frac{{{\text{i}}akcU}}{\pi }\left[ {B_{0} \left( {1 + \cos \theta^{\prime}} \right) + \sum\limits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} } \right]{\text{e}}^{{\chi_{u} }} E_{1} \left( {\chi_{\text{u}} } \right){\text{d}}\theta^{\prime}} \\ & \quad - \int\limits_{0}^{\pi } {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} cU{\text{e}}^{{ - k_{0} \left( {2h + d} \right)}} B_{0} \left( {1 + \cos \theta^{\prime}} \right)\cos \left[ {\frac{{ck_{0} }}{2}\left( {\cos \theta - \cos \theta^{\prime}} \right)} \right]{\text{d}}\theta^{\prime}} \\ & \quad - \int\limits_{0}^{\pi } {\frac{{a\left( {1 - a} \right)}}{a + 1}k_{0} cU{\text{e}}^{{ - k_{0} \left( {2h + d} \right)}} \sum\limits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}\sin \theta^{\prime}} \cos \left[ {\frac{{ck_{0} }}{2}\left( {\cos \theta - \cos \theta^{\prime}} \right)} \right]{\text{d}}\theta^{\prime}} \\ \end{aligned} $$

(60)

where the trigonometric relationship $\cos \left( {n - 1} \right)\theta - \cos \left( {n + 1} \right)\theta = 2\sin \theta \sin n\theta$ and the complex exponential integral E ₁ are used. In addition, the expressions of χ _l and χ _u are

$$ \chi_{\text{l}} = - \frac{1 - a}{1 + a}2\mu h + {\text{i}}\mu \frac{c}{2}\frac{1 - a}{1 + a}\left( {\cos \theta^{\prime} - \cos \theta } \right) $$

(61)

and

$$ \chi_{\text{u}} = - \frac{1 - a}{1 + a}\mu \left( {2h + d} \right) + {\text{i}}\mu \frac{c}{2}\frac{1 - a}{1 + a}\left( {\cos \theta^{\prime} - \cos \theta } \right) $$

(62)

Equation (56) becomes a linear equation system with N unknown numbers. In the same way, Eq. 18 can also be rewritten in the same form by the aid of the trigonometric series. Then, unknown numbers of the linear equation system becomes 2N. Finally, we can obtain the vortex strength distribution along the lower and upper foils by solving the 2N-dimensional linear system of equation.

Appendix 2

In this appendix, we shall give a general approach to solve the integro-differential equation. Calculating the derivative of Eq. 27 with respect to z at x = 0 and z = h yields the velocity of downwash:

$$ w_{\text{W}} = \left. {\frac{{\partial \Upphi_{\text{W}} }}{\partial z}} \right|_{\begin{subarray}{l} x = 0 \\ z = h \end{subarray} } = \frac{1}{4\pi }\int\limits_{{ - {\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{\Upgamma \left( \eta \right){\text{d}}\eta }}{{\left( {y - \eta } \right)^{2} }}} $$

(63)

Equation 63 is an integral with singularity of second order. It is easy to divergent when we use the finite integral concept. Thus, reducing the singularity order can benefit from computation time. Integration by parts is performed by Eq. 63

$$ \begin{aligned} w_{\text{W}} = & \frac{1}{4\pi }\int\limits_{{ - {\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}} {\Upgamma \left( \eta \right){\text{d}}\frac{1}{y - \eta }} \\ & = \left. {\frac{1}{4\pi }\frac{\Upgamma \left( \eta \right)}{y - \eta }} \right|_{{ - {\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}} - \frac{1}{4\pi }\int\limits_{{ - {\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{{\text{d}}\Upgamma \left( \eta \right)}}{y - \eta }} \\ \end{aligned} $$

(64)

The boundary condition that the vortices vanish at the tips should be imposed. Therefore, Eq. 64 becomes:

$$ w_{\text{W}} = - \frac{1}{4\pi }\int\limits_{{ - {\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{\sigma \mathord{\left/ {\vphantom {\sigma 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{{{{\text{d}}\Upgamma \left( \eta \right)} \mathord{\left/ {\vphantom {{{\text{d}}\Upgamma \left( \eta \right)} {{\text{d}}\eta }}} \right. \kern-\nulldelimiterspace} {{\text{d}}\eta }}}}{y - \eta }{\text{d}}\eta } $$

(65)

The formula Eq. 65 can also be found in [37]. We make the following substitutions, which are standard for solving the lifting line equation:

$$ y = - \frac{\sigma }{2}\cos \theta ,\;\eta = - \frac{\sigma }{2}\cos \theta^{\prime} $$

(66)

and

$$ \Upgamma \left( {\theta^{\prime}} \right) = \sigma U\sum\limits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}} $$

(67)

where the representations for $\Upgamma \left( {\theta^{\prime}} \right)$ ensure that the vorticity vanishes at the end of the lifting line. Using the above coordinate change, we can calculate the derivative of Eq. (58) used in the integrals:

$$ {\text{d}}\eta = \frac{\sigma }{2}\sin \theta^{\prime}{\text{d}}\theta^{\prime} $$

(68)

Substituting the above substitutions, Eq. (65) becomes the integrals with respect to $\theta^{\prime}$ with singularity of order two. After some calculations, Eq. (65) can be rewritten as

$$ w_{\text{W}} = \frac{1}{2\pi }\int\limits_{0}^{\pi } {\frac{{\sum\nolimits_{n = 1}^{\infty } {nA_{n} \cos n\theta^{\prime}} }}{{\cos \theta - \cos \theta^{\prime}}}{\text{d}}\theta^{\prime}} $$

(69)

Equation 69 is known as Glauert integral, which equals:

$$ w_{\text{W}} = \frac{1}{2}\sum\limits_{n = 1}^{\infty } {nA_{n} \frac{\sin n\theta }{\sin \theta }} $$

(70)

However, the velocity of downwash on the upper wing induced by the wake of the lower wing is not integrable. And there are two different circulation expressions:

$$ \Upgamma_{\text{l}} \left( {\theta^{\prime}} \right) = \sigma U\sum\limits_{n = 1}^{N} {A_{n} \sin n\theta^{\prime}} ,\;\Upgamma_{\text{u}} \left( {\theta^{\prime}} \right) = \sigma U\sum\limits_{n = 1}^{N} {B_{n} \sin n\theta^{\prime}} $$

(71)

Because of the symmetry characteristics of circulation, the coefficients of even terms in Eq. (71) are equal to zero. We suppose that there are N terms in each circulation expression. Thus, there are 2N unknown numbers. We can assign 2N values to θ, and then an equation system with 2N unknowns will be obtained. Thus, we can get the expressions of Γ_l and Γ_u.