Finite Wing Theory

One of the most vital uses of potential flow theory was the analysis of lifting surfaces such as the wings of an aircraft, since the boundary conditions on a complicated geometry can substantially muddle the attempt to solve the problem by analytical means which, in turn, necessitates some simplifying assumptions to obtain the solution. In this chapter, these assumptions will be linked to the definition of the three-dimensional thin wing problems.

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Notes

In the expression of Biot and Savart law for calculating the induced velocity, if we curl our fingers from the line segment \(\left( \overrightarrow<\mathrm

>\right) \) toward \(\hat<\mathrm>\) , the thumb shows the direction of induced velocity.

In Cartesian space, suppose a general ellipse is defined by the equation \(\frac>>+\frac>>=1\) , where a and b are the lengths of semi-major and semi-minor axes, respectively. The area enclosed by this ellipse will be \(\pi ab\) .

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Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, India Mrinal Kaushik
  1. Mrinal Kaushik
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Appendices

Summary

The properties associated with an airfoil section which are indeed the same as the properties of a wing of infinite span. These properties are different for the wings of finite span attached to a real aircraft. This is because, unlike an airfoil which is a two-dimensional object, a wing is essentially a three-dimensional body, that is, there will be a component of flow in the spanwise direction. That is, the flow over the wings is three-dimensional in nature and hence their aerodynamic properties are quite different from those of its airfoil sections.

The downwash produced by shedding trailing edge vortices from the wing tips and its effect on the inclination of the local relative wind has two major consequences on the local airfoil section. The actual angle of attack as seen by the airfoil locally is, in fact, lower than the geometric angle of attack \(\mathrm <\left( \alpha \right) >\) . This angle of attack is referred to as effective angle of attack \(\mathrm <\left( \alpha _\right) >\) for a three-dimensional wing. That is,

$$ \mathrm =\alpha -\alpha _> $$

Also, the effective freestream velocity \(\mathrm <\left( U_\right) >\) will now become

However, for small downwash \(\mathrm <\left( w\approx 0\right) >\)

$$\begin \mathrm >&\mathrm \end$$

Besides, the downwash induced by these trailing edge vortices from the wing tips leads to an additional component of drag known as induced drag.

The concepts of vortex sheets and vortex filaments are advantageous in evaluating the aerodynamic characteristics of wings of finite span. From a directed line segment \(\mathrm <\left( dl\right) >\) of a vortex filament, the induced velocity can be calculated by using the below mentioned Biot–Savart law.

The velocity induced by a straight vortex filament of the finite length is given as

$$\begin \mathrm >\,&\mathrm <4\pi d>\left( \cos \theta _-\cos \theta _\right) > \end$$

In the lifting line model, developed by Ludwig Prandtl, a wing is numerically described by an infinite number of horseshoe vortices and these bound vortices pass through the aerodynamic centers of the airfoils, which in turn creates the lifting line. Besides, the trailing edge vortices starting at the lifting line and shed downstream toward the infinity are basically responsible for inducing the downwash at the lifting line, and consequently, modify the local angles of attack. The circulation distribution \(\mathrm \) is calculated from the accompanying relation

For the symmetric aerodynamic load distribution, defined as \(\mathrm \left[ 1-\left( \frac\right) ^\right] ^>>\) , a summary of important relations is described below.

For a symmetric elliptical lift distribution over the wingspan, both induced downwash and induced angle are constant along the span.

$$ \mathrm > $$> $$

The total lift acting on the complete wingspan for a symmetric elliptic lift distribution is

$$\begin \mathrm \,&\mathrm <=\frac<\pi b>\rho U_\Gamma _> \end$$

and the expression for \(\mathrm >\) is

$$\begin \mathrm >&\mathrm <=\frac<\pi >\frac\frac>> \end$$

The overall induced drag for a symmetric elliptical loading is given by

$$\begin \mathrm >\,&\mathrm <=\frac<\pi >\rho \Gamma _^> \end$$

In addition, the coefficient of induced drag is

$$\begin \mathrm >>\,&\mathrm <=\frac<\pi >\frac^>=\frac<\pi >\fracC_^> \end$$

This relation can also be written as

$$\begin \mathrm >>\,&\mathrm ^><\pi AR>> \end$$

where \(\mathrm >>\) is the aspect ratio of a finite wing.

For the symmetric general aerodynamic load distribution, given by , a summary of important relations is described as follows.

The generalized expression for the induced angle \(\mathrm <\left( \alpha _\right) >\) is

For a wing of finite span, the lift coefficient is given by

$$\begin \mathrm >\,&\mathrm <=\pi A_<1>AR> \end$$

and the coefficient of induced drag is

$$\begin \mathrm >>\,&\mathrm <=\pi AR\sum _^<\infty >nA_^=\frac^><\pi AR>\left( 1+\delta \right) =\frac^><\pi eAR>> \end$$

Exercises

1.1 Descriptive Type Questions

  1. 1. Show that the integral on the right-hand side of Kelvin’s circulation theorem vanishes, if the fluid is barotropic.
  2. 2. Prove that the elliptical spanwise lift distribution leads to a constant downwash along the span.
  3. 3. Consider a vortex sheet, where the velocities above and below the sheet are 6 and 4 \(\mathrm >\) , respectively. The element of this vortex sheet is 0.4 \(\mathrm \) wide, which rolled up into a line vortex after some time. Calculate the strength of the vortex.
  4. 4. Determine the wing loading of an aircraft weighing 2000 \(\mathrm \) , if the wing planform area is 18 \(\mathrm >\) .
  5. 5. Plot the curve between the induced drag coefficient and the lift coefficient for an elliptical load distribution over the wing of aspect ratio 7.5.
  6. 6. If the aspect ratio of a glider, having elliptical planform wing, is 6.5. Calculate the change in minimum angle of glide, if the aspect ratio is doubled. Assume \(\mathrm =\text +\text C_^>\) .
  7. 7. An airplane is flying at 150 \(\mathrm >\) in a steady level flight. If the aircraft weighs 80 \(\mathrm \) and have elliptical wing of span 16 \(\mathrm \) , determine the induced drag.
  8. 8. Describe the motion of a vortex pair (a) when their circulations are equal and in the same direction, and (b) when their circulations are equal but in opposite directions.
  9. 9. Determine the flow field due to (a) a vortex filament, which is in the form of a circular ring (vortex ring), (b) a plane vortex sheet, which is formed by a distribution of horseshoe vortex filament, and (c) an infinite row of point vortices of equal strength distributed along a straight line at equal intervals.
  10. 10. Find the path of a vortex bounded by two walls perpendicular to each other.

1.2 Multiple Choice Questions

  1. 1. According to Prandtl’s lifting line theory, which of the following shape of the wing has minimum induced drag?
    1. (a) elliptical
    2. (b) straight rectangular
    3. (c) straight tapered
    4. (d) tapered sweptback
    1. (a) will remain same
    2. (b) will be increased two times
    3. (c) will be increased three times
    4. (d) will be increased four times
    1. (a) 10
    2. (b) 20
    3. (c) 0.1
    4. (d) 30
    1. (a) longitudinal stability
    2. (b) lateral stability
    3. (c) directional stability
    4. (d) both (b) and (c)
    1. (a) decrease
    2. (b) increase
    3. (c) remain same
    4. (d) cannot say
    1. (a) increase
    2. (b) decrease
    3. (c) remain same
    4. (d) increase or decrease depending upon the shape of the flap.
    1. (a) less
    2. (b) more
    3. (c) same
    4. (d) cannot say
    1. (a) One wing stalls slightly before the other.
    2. (b) The nose drops gently and the wings remain level throughout.
    3. (c) The nose rises, pushing the wing deeper into the stalled state.
    4. (d) All the above are FALSE.
    1. (a) increased by three times
    2. (b) increased by nine times
    3. (c) \(\frac<3<\mathrm >>\) of the original
    4. (d) \(\frac<9<\mathrm >>\) of the original
    1. (a) doubled
    2. (b) four times the original
    3. (c) eight times the original
    4. (d) \(\frac<8<\mathrm >>\) of the original

    1.2.1 Keys

    1. 1. (a)
    2. 2. (c)
    3. 3. (b)
    4. 4. (d)
    5. 5. (b)
    6. 6. (c)
    7. 7. (a)
    8. 8. (b)
    9. 9. (d)
    10. 10. (c)

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    Kaushik, M. (2019). Finite Wing Theory. In: Theoretical and Experimental Aerodynamics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1678-4_6

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    • DOI : https://doi.org/10.1007/978-981-13-1678-4_6
    • Published : 16 December 2018
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