**2. Drag polar of aircraft**

Taking as a reference any paper on aerodynamics, we can find the formulas for lift *L* and drag *D* forces [2]:

*Flight Vehicle Performance DOI: http://dx.doi.org/10.5772/intechopen.92105*

$$\begin{cases} \quad L = c\_L \frac{\rho v^2}{2} \mathcal{S}\_{r\!\!f}, \\\\ D = c\_D \frac{\rho v^2}{2} \mathcal{S}\_{r\!\!f}, \end{cases} \tag{1}$$

where *cL* is the dimensionless lift coefficient; *ρ* is the freestream density; *v* is the magnitude of freestream speed, which is taken equal to aircraft speed; *Sref* is the reference area, and *cD* is the dimensionless drag coefficient.

From the system of Eq. (1)

$$\frac{L}{D} = \frac{c\_L}{c\_D}.$$

The ratio *k* ¼ *L=D* is usually called lift-to-drag ratio and in eastern literature it is defined as aerodynamic quality of aircraft. This ratio has interesting properties and it changes with changes of angle of attack *α*.

Let us consider now the lift coefficient *cL* and its relations with angle of attack *α* (**Figure 3**), which is similar to the analogous relation for 2-D airfoils. The function *cL* ¼ *fL*ð Þ *α* can be found using experiments with aircraft model in wind tunnels or using methods of computational fluid dynamics (CFD).

As we can see from the graph of function *cL* ¼ *fL*ð Þ *α* (**Figure 3**) there is an angle of attack *αcr* at which the lift coefficient is maximal, that is, *cL* max . The flight at critical angle of attack *αcr* will lead to stall, resulting aircraft crash. At angle of attack *αts*, the tip stall processes are started and the rate of increase of lift coefficient is decelerated, allowing it to get its maximum value *cL* max and decrease sufficiently. At the point *αts* ð Þ ,*cts* of starting tip stall, the effects of shaking of aircraft are started [3]. The range between angles of attack *α*0, which is called zero lift angle of attack,

**Figure 3.** *Lift coefficient of aircraft versus angle of attack.*

Based on the model of force balance, we will study flight vehicle performance at several steady flight regimes, but before getting there let us consider the aerodynamic forces—lift and drag, and the effect of their relations on the aerodynamic

Taking as a reference any paper on aerodynamics, we can find the formulas for

quality of aircraft.

*Simple model of force balance.*

**Figure 2.**

**154**

**Figure 1.**

*Aerodynamics*

*Forces acting on aircraft.*

**2. Drag polar of aircraft**

lift *L* and drag *D* forces [2]:

and *αts* is the range of regular flights of aircraft and as it can be observed at this range the function *cL* ¼ *fL*ð Þ *α* is linear.

Before getting to the drag coefficient study, let us consider the existing types of the drag force. The drag force *D* can be represented as a sum of parasitic drag *Dp*, which consists of form drag *Df* and skin friction drag *Dsf* [4], lift-induced *Di* drag, wave drag *Dw*, and interference drag *Dif* :

$$D = D\_f + D\_{\mathcal{f}} + D\_i + D\_w + D\_{\mathcal{f}},$$

or, which is the same as

$$\mathcal{L}\_D \frac{\rho v^2}{2} \mathcal{S}\_{r\notin} = \left(\mathcal{c}\_{Dp} + \mathcal{c}\_{Di} + \mathcal{c}\_{Dw} + \mathcal{c}\_{D\hat{f}}\right) \frac{\rho v^2}{2} \mathcal{S}\_{r\notin},$$

where *cDp* ¼ *cDf* þ *cDsf* is the parasitic drag coefficient, *cDf* is the form drag coefficient, *cDsf* is the skin friction drag coefficient, *cDi* is the lift-induced drag coefficient, *cDw* is the wave drag coefficient, and *cDif* is the inference drag coefficient.

where *<sup>λ</sup>* <sup>¼</sup> *<sup>b</sup>*<sup>2</sup>

*Flight Vehicle Performance*

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

*On lift-induced drag.*

**Figure 5.**

lift generation:

(**Figure 6**).

**157**

area, and *e* is the span efficiency.

when there are shock waves near the aircraft.

aircraft model in wind tunnels and using CFD tools.

*cD*<sup>0</sup> at zero lift angle of attack *α*0.

attack, it is possible to draw drag polar of aircraft (**Figure 7**).

tioned ratio, also known as aerodynamic quality of aircraft:

*=Sref* is the aspect ratio, *b* is the wing span, *Sref* is the wing reference

*cD* ¼ *cD*<sup>0</sup> þ *cDi*, (3)

The wave drag *Dw* is a consequence of the compressibility of the air and occurs

The mutual influence of the parts of the aircraft is called interference. It occurs due to a change in the velocity field, as a result of which the nature of the flow around the aircraft changes leading to generation of interference drag *Dif* .

Based on the review of drag components, we can divide them into components related to the lift generation or lift-induced drag, and components not related to the

where *cD*<sup>0</sup> is the component of drag coefficient not related with the lift generation and is called zero lift drag coefficient. Usually *cD*<sup>0</sup> is taken as constant and not related to the angle of attack, while *cDi* is proportional to the square of lift coefficient *cL*, which linearly depends on angle of attack *α* in range of the regular flight regimes. Thus the function *cD* ¼ *fD*ð Þ *α* should have the graph of parabolic form

Similar to *cL* ¼ *fL*ð Þ *α* , function *cD* ¼ *fD*ð Þ *α* can be found using experiments with

Using estimated or experimental results for *fL*ð Þ *α* and *fD*ð Þ *α* at set of angles of

Based on the drag polar conditions for the best lift-to-drag ratio, zero lift drag and maximal lift can be found. By drawing tangent 1 to the curve of drag polar from the origin of coordinate frame *cDcL*, the best lift-to-drag ratio can be found, which corresponds to the angle of attack *αbldr*; the tangent 2 to the drag polar parallel to axis *cD* defines maximal value of lift coefficient *cL* max at critical angle of attack *αcr*, and tangent 3 to the drag polar parallel to axis *cL* defines zero lift drag coefficient

The angle of attack *αbldr* of best lift-to-drag ratio has sufficient role in the flight of aircraft, as the flight with this angle provides the maximum value of the men-

*<sup>k</sup>*max <sup>¼</sup> *<sup>f</sup> <sup>L</sup>*ð Þ *<sup>α</sup>bldr*

*f <sup>D</sup>*ð Þ *αbldr*

*:*

Parasitic drag is the pressure difference in front of and behind the wing. The pressure difference depends on the shape of the wing airfoil, its relative thickness *c* and curvature. The larger the relative thickness of the wing airfoil, the greater the form drag (also known as pressure drag); on the other hand, the lower the relative thickness of the wing airfoil, the greater the effect of skin friction drag [5] (**Figure 4**).

The lift-induced drag is the result of the flow tilt (**Figure 5**). Due to the pressure difference above and under the wing on its tips, vortices are generated, leading to the downwash of air from upper surface with velocity *u*. Thus the effective flow speed *veff* becomes the vector sum of the freestream air speed *v* and downwash speed *u*. The direction of the effective flow speed differs from the freestream velocity's direction by angle *δα*, so the effective angle of attack *αeff* is defined as:

$$a\_{\mathcal{G}} = a + \delta a.$$

With an increase in the angle of attack or lift coefficient, the pressure difference under and above the wing increases quickly, and the coefficient of lift-induced drag increases according to the quadratic law [2]:

**Figure 4.** *Parasitic drag components.*

**Figure 5.** *On lift-induced drag.*

and *αts* is the range of regular flights of aircraft and as it can be observed at this

Before getting to the drag coefficient study, let us consider the existing types of the drag force. The drag force *D* can be represented as a sum of parasitic drag *Dp*, which consists of form drag *Df* and skin friction drag *Dsf* [4], lift-induced *Di* drag,

*D* ¼ *Df* þ *Dsf* þ *Di* þ *Dw* þ *Dif* ,

*Sref* ¼ *cDp* þ *cDi* þ *cDw* þ *cDif*

where *cDp* ¼ *cDf* þ *cDsf* is the parasitic drag coefficient, *cDf* is the form drag coefficient, *cDsf* is the skin friction drag coefficient, *cDi* is the lift-induced drag

coefficient, *cDw* is the wave drag coefficient, and *cDif* is the inference drag coefficient. Parasitic drag is the pressure difference in front of and behind the wing. The pressure difference depends on the shape of the wing airfoil, its relative thickness *c* and curvature. The larger the relative thickness of the wing airfoil, the greater the form drag (also known as pressure drag); on the other hand, the lower the relative thickness of the wing airfoil, the greater the effect of skin friction drag [5] (**Figure 4**). The lift-induced drag is the result of the flow tilt (**Figure 5**). Due to the pressure difference above and under the wing on its tips, vortices are generated, leading to the downwash of air from upper surface with velocity *u*. Thus the effective flow speed *veff* becomes the vector sum of the freestream air speed *v* and downwash speed *u*. The direction of the effective flow speed differs from the freestream velocity's direction by angle *δα*, so the effective angle of attack *αeff* is defined as:

*αeff* ¼ *α* þ *δα:*

*cDi* <sup>¼</sup> *<sup>c</sup>*<sup>2</sup> *L πλe*

With an increase in the angle of attack or lift coefficient, the pressure difference under and above the wing increases quickly, and the coefficient of lift-induced drag

*<sup>ρ</sup>v*<sup>2</sup>

2 *Sref* ,

, (2)

range the function *cL* ¼ *fL*ð Þ *α* is linear.

*Aerodynamics*

wave drag *Dw*, and interference drag *Dif* :

*cD ρv*<sup>2</sup> 2

increases according to the quadratic law [2]:

**Figure 4.**

**156**

*Parasitic drag components.*

or, which is the same as

where *<sup>λ</sup>* <sup>¼</sup> *<sup>b</sup>*<sup>2</sup> *=Sref* is the aspect ratio, *b* is the wing span, *Sref* is the wing reference area, and *e* is the span efficiency.

The wave drag *Dw* is a consequence of the compressibility of the air and occurs when there are shock waves near the aircraft.

The mutual influence of the parts of the aircraft is called interference. It occurs due to a change in the velocity field, as a result of which the nature of the flow around the aircraft changes leading to generation of interference drag *Dif* .

Based on the review of drag components, we can divide them into components related to the lift generation or lift-induced drag, and components not related to the lift generation:

$$
\mathfrak{c}\_{\rm D} = \mathfrak{c}\_{\rm D0} + \mathfrak{c}\_{\rm Di}, \tag{3}
$$

where *cD*<sup>0</sup> is the component of drag coefficient not related with the lift generation and is called zero lift drag coefficient. Usually *cD*<sup>0</sup> is taken as constant and not related to the angle of attack, while *cDi* is proportional to the square of lift coefficient *cL*, which linearly depends on angle of attack *α* in range of the regular flight regimes. Thus the function *cD* ¼ *fD*ð Þ *α* should have the graph of parabolic form (**Figure 6**).

Similar to *cL* ¼ *fL*ð Þ *α* , function *cD* ¼ *fD*ð Þ *α* can be found using experiments with aircraft model in wind tunnels and using CFD tools.

Using estimated or experimental results for *fL*ð Þ *α* and *fD*ð Þ *α* at set of angles of attack, it is possible to draw drag polar of aircraft (**Figure 7**).

Based on the drag polar conditions for the best lift-to-drag ratio, zero lift drag and maximal lift can be found. By drawing tangent 1 to the curve of drag polar from the origin of coordinate frame *cDcL*, the best lift-to-drag ratio can be found, which corresponds to the angle of attack *αbldr*; the tangent 2 to the drag polar parallel to axis *cD* defines maximal value of lift coefficient *cL* max at critical angle of attack *αcr*, and tangent 3 to the drag polar parallel to axis *cL* defines zero lift drag coefficient *cD*<sup>0</sup> at zero lift angle of attack *α*0.

The angle of attack *αbldr* of best lift-to-drag ratio has sufficient role in the flight of aircraft, as the flight with this angle provides the maximum value of the mentioned ratio, also known as aerodynamic quality of aircraft:

$$k\_{\max} = \frac{f\_L(a\_{bldr})}{f\_D(a\_{bldr})}.$$

*cL* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *πλe c*ð Þ *<sup>D</sup>* � *cD*<sup>0</sup> p *:*

At point *cD*ð Þ *αbldr* , this derivative is the same as the slope *k*max of tangent 1 from

*cD*ð Þ *αbldr*

Based on the Eq. (4) we can find the maximal value of lift-to-drag ratio:

*<sup>k</sup>*max <sup>¼</sup> <sup>1</sup> 2

graph of the function *k*ð Þ¼ *α fL*ð Þ *α =fD*ð Þ *α* can be plotted (**Figure 8**) and the range of angles near the *αbldr* studied to find the effective flight regimes and

To examine the dependency of the lift-to-drag ratio on angles of attack, the

¼

ffiffiffiffiffiffiffi *πλe cD*<sup>0</sup>

*:*

r

<sup>¼</sup> *πλ<sup>e</sup>*

p

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *πλe c*ð Þ *<sup>D</sup>* � *cD*<sup>0</sup> <sup>p</sup> *:*

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *πλe c*ð Þ *<sup>D</sup>*ð Þ� *αbldr cD*<sup>0</sup>

,

*cD*ð Þ *αbldr*

*cD*ð Þ¼ *αbldr* 2*cD*0*:* (4)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *πλe c*ð Þ *<sup>D</sup>* � *cD*<sup>0</sup>

*∂cD*

Taking *cD* derivative of *cL* we get:

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

*Flight Vehicle Performance*

*∂cL ∂cD* ¼ *∂*

*πλe* 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *πλe c*ð Þ *<sup>D</sup>*ð Þ� *αbldr cD*<sup>0</sup>

which can be easily transformed to

patterns [3, 6].

**Figure 8.**

**159**

*Lift-to-drag ratio versus angle of attack.*

p

**Figure 7**, so we can write down the following expression:

<sup>p</sup> <sup>¼</sup> *cL*ð Þ *<sup>α</sup>bldr*

**Figure 6.** *Drag coefficient of aircraft versus angle of attack.*

Let us consider Eq. (3) taking into account Eq. (2):

$$c\_D = c\_{D0} + \frac{c\_L^2}{\pi \lambda \epsilon}.$$

The above expression is also called drag polar equation, with the use of which we can represent the non-negative values of lift coefficient as

*Flight Vehicle Performance DOI: http://dx.doi.org/10.5772/intechopen.92105*

$$c\_L = \sqrt{\pi \lambda e (c\_D - c\_{D0})}.$$

Taking *cD* derivative of *cL* we get:

$$\frac{\partial c\_L}{\partial c\_D} = \partial \frac{\sqrt{\pi \lambda e (c\_D - c\_{D0})}}{\partial c\_D} = \frac{\pi \lambda e}{2 \sqrt{\pi \lambda e (c\_D - c\_{D0})}} \cdot \frac{1}{2}$$

At point *cD*ð Þ *αbldr* , this derivative is the same as the slope *k*max of tangent 1 from **Figure 7**, so we can write down the following expression:

$$\frac{\pi \lambda e}{2\sqrt{\pi \lambda e (c\_D(a\_{bldr}) - c\_{D0})}} = \frac{c\_L(a\_{bldr})}{c\_D(a\_{bldr})} = \frac{\sqrt{\pi \lambda e (c\_D(a\_{bldr}) - c\_{D0})}}{c\_D(a\_{bldr})},$$

which can be easily transformed to

$$\mathcal{L}\_D(a\_{bldr}) = \mathcal{L}\_{D0}.\tag{4}$$

Based on the Eq. (4) we can find the maximal value of lift-to-drag ratio:

$$k\_{\text{max}} = \frac{1}{2} \sqrt{\frac{\pi \lambda \mathcal{e}}{c\_{D0}}}.$$

To examine the dependency of the lift-to-drag ratio on angles of attack, the graph of the function *k*ð Þ¼ *α fL*ð Þ *α =fD*ð Þ *α* can be plotted (**Figure 8**) and the range of angles near the *αbldr* studied to find the effective flight regimes and patterns [3, 6].

**Figure 8.** *Lift-to-drag ratio versus angle of attack.*

Let us consider Eq. (3) taking into account Eq. (2):

**Figure 6.**

*Aerodynamics*

**Figure 7.**

**158**

*Drag polar of aircraft.*

*Drag coefficient of aircraft versus angle of attack.*

can represent the non-negative values of lift coefficient as

*cD* ¼ *cD*<sup>0</sup> þ

*c*2 *L πλe :*

The above expression is also called drag polar equation, with the use of which we

Examined material is one of core bases of aircraft performance, and the results obtained through the above analysis are used in studies of different flight paths and patterns and will be referred in next subsection dedicated to the Zhukovsky curves.
