## Abstract

In this chapter, the problem of flight vehicle performance is described. Performance parameters, such as lift-to-drag ratio, maximum and minimum level flight speed, speeds for the best rate of climb, steepest climb, maximum range and endurance, and most economical climb are described using graphical methods, such as drag polar, Zhukovsky curves with combination of analytical derivations. The approach of graphical description of flight vehicle performance allows to understand the physical basics of the aerodynamic properties of flight vehicles easier and to develop deeper connectivity between their interpretations. In addition, flight envelope and operational limits are discussed using both analytical and graphical methods for better understanding.

### Keywords

- aerodynamics
- flight vehicle
- drag polar
- flight performance
- Zhukovsky curves

## 1. Introduction

During the flight, aircraft is usually loaded by four forces (Figure 1)—gravity force

As we can see from the Figure 1, the vector of aircraft speed

In problems of flight vehicle performance analysis, usually a simpler model of the balance of forces is used: the side forces

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 quality of aircraft.

## 2. Drag polar of aircraft

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

where

From the system of Eq. (1)

The ratio

Let us consider now the lift coefficient

As we can see from the graph of function

Before getting to the drag coefficient study, let us consider the existing types of the drag force. The drag force

or, which is the same as

where

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

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

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]:

where

The wave drag

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

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:

where

Similar to

Using estimated or experimental results for

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

The angle of attack

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

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

Taking

At point

which can be easily transformed to

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

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

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.

## 3. Zhukovsky curves

Let us now consider steady horizontal flight. The scheme on Figure 2 will be transformed to the following form (Figure 9):

In steady horizontal flight, we have the following equation of the force balance:

which is same as:

From the first equation of the above system, we can find that

and by substituting the value

Based on the above result, we can state that the required thrust

As we can see, zero lift drag

Let us now define the conditions of minimal drag or, which is the same as, minimal required thrust at steady horizontal flight:

The above expression can be rewritten as:

It is obvious that the left-hand side of the above is zero lift drag

or

Thus, the drag coefficient is equal:

As we remember from the drag polar, the same condition is true for maximal lift-to-drag ratio, so the conditions for minimal required thrust and maximal lift-to-drag ratio are the same. To complete the calculations of all parameters for minimal required thrust, let us derive the expressions for lift coefficient and air speed:

so,

and from Eq. (6)

We can also find the characteristics of available thrust

Based on the available information, Nikolay Zhukovsky developed a graphical method for the analysis of the range of the horizontal flight speeds at different altitudes. His method is based on the plotting curves of zero lift drag and lift-induced drag versus air speed at different altitudes, graphically calculating their sum, and plotting the dependencies of available thrust on air speed at corresponding altitudes for graphoanalytic estimation of ranges of available air speeds for horizontal flight at different altitudes. The set of curves obtained through the above-described procedure in memory of him are called Zhukovsky curves. The graph with curves of required and available thrusts or powers at certain altitude is also called performance diagram.

Let us implement the method proposed Zhukovsky for any altitude, for example, do a plot for the altitude of zero meters above sea level (Figure 11) and define the speed characteristics of the horizontal flight.

From Zhukovsky curves at sea level altitude, we can find the maximum speed

or

For flight at higher altitudes, we can get conditions when the required thrust at the above minimum speed is much higher than the available thrust at that altitude (Figure 12). In such cases, minimum speed defined as minimum thrust speed

The idea of plotting Zhukovsky curves at sea level flight allows us to have the same graph at any altitude

where

Based on the Zhukovsky curves based on IAS, we can define the theoretical or static ceiling of horizontal flight (Figure 12), which is an altitude where the horizontal flight is possible only with IAS equal to

There are also defined such concepts as speed of maximum endurance

Let us now consider descending flight or glide (Figure 14a) and ascending (Figure 14b) flight of aircraft.

In descending flight, the throttle is usually set to minimum, so the thrust can be neglected and consider the gliding flight (Figure 14a), for which we can write down the following equations of force balance:

where

Based on the above we can get:

The flattest glide corresponds to the minimum magnitude of flight path angle

The above means that the flattest glide, resulting the longest gliding distance, is also related to maximum lift-to-drag ratio

For the ascending flight the force balance is presented as follows:

As we know, the rate of climb is the projection of airspeed to the vertical plane:

From second equation of Eq. (7) we can get:

or which is same as:

where

From the above we can find the maximum of ROC, that is,

The angle

which corresponds to

In ideal jet aircraft case, the angle of steepest climb is defined at the speed of maximum lift-to-drag ratio. The most economical ascending flight is defined by the operational regime of aircraft engine at which the minimum fuel will be consumed for aircraft to climb to the required altitude.

## 4. Flight envelope and operational limits of aircraft

For any aircraft, performance and operational limits are defined. Performance limits (Figure 16) are mostly defined by the aerodynamic configuration of aircraft. On the other hand, operational limits are based on the type of aircraft, structural and engine limits, wind resistance parameters, and maximum Mach number. All these limits are presented in diagrams of altitude

All the speeds presented on Figure 16 were described in the previous subsection. Operational limits related to the type of aircraft are described from the point of view of aircraft application: if an aircraft is a passenger jet, it should apply to the requirements of comfort for passengers and not exceed load factors of comfortable flight; on the other hand, if an aircraft is a jet fighter its operational limits from the point of view of load factors should be derived from compromise between the prevention of health issues of pilot that may occur and maneuverability for the combat use.

Structural limits are mostly related to aircraft strength, while the engine limits can be the result of its design and performance at higher altitudes. Wind resistance limits can be derived from the requirements of operational use or comfortability of flight for passengers.

Maximum Mach number can be defined from the conditions of aeroelastic effects and vibrations, effects of shifting aerodynamic center, causing severe pitching moments, which can lead to the crashes, or loosing effectiveness of aerodynamic surfaces. For example, for jet aircraft L-39 the critical Mach number is

Finally, an example of flight envelope for analysis is presented in Figure 17 where the intersection of all limiting conditions is described.

## 5. Conclusion

The material described in this chapter involves flight vehicle analysis using graphical and analytical tools for better understanding of the physical aspects of flight core parameters and development of strong and meaningful connections between them. The material from this chapter can be useful in the preliminary design and prototyping of flight vehicles and for finding the paths for further developments and improvements in the design.

## Acknowledgments

In this work, the name of Prof. Nikolay Zhukovsky (January 17, 1847–March 17, 1921) is mentioned many times, while his name may be unfamiliar to many readers. Nikolay Zhukovsky was one of the first scientists who established the mathematical base of aerodynamics; he was the initiator and the first head of the Central Aerohydrodynamic Institute (TsAGI), and on his initiative was created Zhukovsky Air Force Engineering Academy—alma mater of three famous Soviet aircraft designers Sergei Ilyushin, Artem Mikoyan, and Alexander Yakovlev.

Science lives in the research schools, and great scientists are those who both do great discoveries and develop the next generations of discoverers. Zhukovsky Air Force Engineering Academy is my alma mater as well, and with these few words I would like to express my appreciation to all my teachers who helped me on my way of professional and personal development.

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