Aerodynamic Characteristics of Low Speed Wing Flow

Abstract

This chapter introduces the basic concept and principle of flow around the low-speed wing. It includes wing geometric characteristics and parameters, aerodynamic coefficients of the wing, low-speed flow characteristics of high aspect ratio straight wing, vortex model of the low-speed wing, Prandtl’s lifting-line theory, stall characteristics of high aspect ratio straight wing, low-speed aerodynamic characteristics of the swept-back wing, lifting-surface theory of wing, low-speed aerodynamic characteristics of low aspect ratio wing, engineering estimation of low-speed aerodynamic characteristics of wing, and aerodynamic characteristics of the control surface.

Exercises

• 1. There is a straight trapezoidal wing, S = 35 m², taper ratio $\eta = 4$, and wingtip chord length b₁ = 1.5 m, find the aspect ratio λ of the wing.

• 2. Given the sweep angle χ₀ of the delta wing and the aspect ratio λ, try to prove from the geometric relationship that λ tanχ₀ = 4.

• 3. Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8. The airfoil profile is thin and symmetrical. Calculate the lift and induced drag coefficient of the wing at an angle of attack of 5°. Assume that δ = τ.

• 4. Given a trapezoidal swept-back wing, the wingtip chord length b₁, the wing root chord length b₀, the taper ratio η = b₀/b₁, and the spanwise length l, try to derive the calculation formulas of area, aspect ratio, the tangent of middle line sweep angle, and mean aerodynamic chord length of the wing from the geometric relationship.

  

$$ \begin{aligned} S & = b_{1} \frac{1 + \eta }{2}l \\ \lambda & = \frac{{l^{2} }}{S},\frac{{b_{1} }}{l} = \frac{2}{\lambda (1 + \eta )},b_{0} = \frac{2l\eta }{{\lambda (1 + \eta )}},tg\chi_{0} - tg\chi_{1} = \frac{4(\eta - 1)}{{\lambda (\eta + 1)}} \\ b_{A} & = \frac{2}{S}\int\limits_{0}^{l/2} {b(z)^{2} {\rm{d}}z} = \frac{4}{3}\frac{l}{\lambda }\left[ {1 - \frac{\eta }{{(1 + \eta )^{2} }}} \right] \\ \end{aligned} $$

• 5. Consider a rectangular wing with an aspect ratio of 6, an induced drag coefficient δ = 0.055, and a zero-lift angle of attack of −2°. When the angle of attack is 3.4°, the induced drag coefficient of the wing is 0.01. Calculate the induced drag coefficient of a similar wing (a rectangular wing with the same airfoil profile) with the same angle of attack and an aspect ratio of 10. Assume that the inducing factors δ and τ of drag and lift slope are, respectively, equal (i.e., δ = τ). Similarly, for λ = 10, δ = 0.105.

• 6. Assume that the spanwise circulation distribution of a straight wing with a high aspect ratio is parabolic, $\Gamma (z) = \Gamma_{0} \left[ {1 - \left( \frac{2z}{l} \right)^{2} } \right]$, as shown in the figure. If the total lift is equal to the elliptical circulation distribution wing, try to find the corresponding relationship of $\Gamma_{0}$ and $w_{i}$ on the symmetry plane of the two circulation distributions.

Problem 6

• 7. The measured lift slope of NACA 23,012 airfoil is 0.1080 per degree, and α_L=0 = −1.3. Consider the finite wing using this airfoil, λ = 8, taper ratio = 0.8. Assume that δ = τ. Calculate the lift and induced drag coefficient of the wing with the geometric angle of attack 7°.

• 8. Given that the spanwise circulation distribution of a high aspect ratio wing is $\Gamma \left( z \right) = \Gamma_{0} \left[ {1 - \left( \frac{2z}{l} \right)^{2} } \right]^{3/2}$, try to use the lifting-line theory to solve

  • (1) Downwash speed $w_{i}$ at z = l/4

  • (2) Downwash speed $w_{i}$ at z = l/2

Tip: use integral

$$ \int\limits_{0}^{\pi } {\frac{{{\text{cos}}\left( {n\theta } \right)}}{{{\text{cos }}\theta - {\text{cos}}\theta_{1} }}} {\text{d}}\theta = \pi \frac{{{\text{sin}}\left( {{\text{n}}\theta_{1} } \right)}}{{{\text{sin}}\theta_{1} }} $$

• 9. If the wing is replaced by a II-shaped horseshoe vortex line, the span length of the attached vortex is L, as shown in Fig. 6.6 of the exercises, try to prove

  • (1) The downwash angle at position a after the attached vortex at the middle of the wing is

    $$ \alpha_{{\text{i}}} = \frac{{C_{L} }}{2\pi \lambda }\left[ {1 + \frac{{\sqrt {a^{2} + (l/2)^{2} } }}{a}} \right] $$

    where $C_{L}$ is the lift coefficient and λ is the aspect ratio.

  • (2) If the airfoil has $C_{l\infty }^{a} = 2\pi$, assuming that the aspect ratio correction adopts the correction of the elliptical wing, the rate of change of the downwash angle at a position the back of the middle wing to the angle of attack is

$$ \frac{{{\rm{d}}\alpha_{i} }}{{{\rm{d}}\alpha }} = \frac{1}{\lambda + 2}\left[ {\frac{{\sqrt {a^{2} + (l/2)^{2} } }}{a} + 1} \right] $$

And calculate the value of $\frac{{{{\rm{d}}}\alpha }}{{{{\rm{d}}}\alpha }}$; when $\lambda = {8,}\;a = 0.4l$.

Problem 9

• 10. A light single-engine general aircraft has a wing area of 15 square meters, a wingspan of 9.7 m, and a maximum gross weight of 1111 kg. The wing adopts NACA 65–415 airfoil, the lift coefficient is 0.1033 per degree, and α_L=0 = −3°. Assume τ = 0.12. If the aircraft cruises at 120 mph at standard sea level at its maximum gross weight. Calculate the geometric angle of attack of the wing in straight and level flight.

• 11. An airplane with a weight of G = 14700 N, Cruising (Y = G) at h = 3000 m with $V\infty = 300$ km/h, wing area $S = 17\, \rm{m}^{2}$, λ = 6.2, NACA23012 airfoil ($a_{0\infty } = - 4^{ \circ } ,C_{l}^{\alpha } = 0.108/\left( {^{ \circ } } \right)$ , non-twisted elliptical plane shape. Try to calculate $C_{L}$, $\alpha$, and $C_{Dv}$.

• 12. There is a monoplane with a weight of G = 7.38 × 10⁴ N, the wing is an elliptical plane shape, the wingspan length is L = 15.23 m, and it is flying straight at the sea level with a speed of 90 m/s, try to calculate the induced drag D_v and the value of $\Gamma_{0}$ at the wing root profile.

• 13. Try to prove that if a horseshoe vortex line with a span length ofland a strength of the circulation $\Gamma_{0}$ of the wing root profile of the original wing is used to simulate the total lift of the elliptical wing with a span length of $\mathrm{l}$, it can be obtained

  $$ \frac{{l^{\prime} }}{l} = \frac{\pi }{4} $$

• 14. A curved airfoil, $a_{0\infty } = - 4^{ \circ } ,C_{l}^{\alpha } = 2\pi /rad$. If this airfoil is placed on an elliptical wing with λ = 5 and no twist, try to find the $C_{L}$ at $\alpha = 8^{ \circ }$.

• 15. For a flat delta wing with λ = 3, known $C_{L\infty }^{\alpha } = 2\pi$, try to use the engineering calculation method to find the values of $C_{L}^{\alpha }$ and $\frac{{x_{F} }}{{b_{A} }}$ at small $\alpha$.

• 16. For a rectangular wing, $\lambda = {6,}l = 12$ m, wing load G/S = 900 N/m². Try to calculate the induced drag and the ratio of induced drag to total lift when the aircraft is flying at sea level with $V\infty =150\mathrm{ km}/\mathrm{h}$.

• 17. The C_L-$\alpha$ curve of a non-twisted straight wing with $\lambda = {9,}\;\;\eta = {2}{\text{.5}}$ under a certain Reynolds number is shown in the figure. From the figure, it can be obtained that $a_{0\infty } = - 1.5^{ \circ } ,C_{l}^{\alpha } = \frac{0.084}{{\left( \circ \right)}},$ C_lmx = 1.22. If other parameters remain the same, but λ is reduced to 5, find the $a_{0}$ and $C_{{\text{l}}}^{\alpha }$ at this time and draw the C_L-$\alpha$ curve of the wing when λ = 5.

Problem 17

• 18. Consider a finite wing with an aspect ratio of 6. Suppose the lift is distributed in an elliptical shape. The lift slope of the airfoil profile is 0.1/degree. Calculate and compare the lift coefficients of (a) straight wing and (b) swept-back wing, the half-chord sweep angle is 45°.

• 19. A civil aircraft uses a trapezoidal swept-back wing with $\lambda = {8,}\;\;\eta = {2,}\;\;\chi_{{0}} = {45}^{{0}}$. Use the engineering calculation method to calculate the $C_{L}^{\alpha }$ and $\frac{{x_{F} }}{{b_{A} }}$ of the wing.

• 20. When the angle of attack is small, the polar curve of the wing can be expressed as a parabola ${\text{C}}_{D} = C_{D0} + \lambda C_{L}^{2}$, try to prove

  $$ K_{\max } = \left( {\frac{{C_{L} }}{{C_{D} }}} \right)_{\max } = \frac{1}{{2\sqrt {\lambda C_{D0} } }},\left( {C_{L} } \right)_{{K_{\max } }} = \sqrt {\frac{{C_{D0} }}{\lambda }} $$