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Chapter 21

# Performance – Descent

## Abstract

The chapter is split into two parts. The first one develops fundamental relationships required to evaluate the characteristics of gliding flight. The second one develops an assortment of methods to predict a number of descent characteristics of an airplane.

### Keywords

Equations of motion; free-body; descent; glide; planar; angle-of-descent; rate-of-descent; equilibrium glide speed; sink rate; minimum sink rate; minimum angle-of-descent; best glide; glide distance

Outline

## 21.1 Introduction

For powered aircraft, the analysis of gliding flight is essential from the standpoint of safety. For unpowered flight, glide performance is what sets one sailplane apart from another. Analysis of descent provides a very important insight into how efficient an airplane is. It can even expose possible handling problems. For instance, a powered airplane with a high glide ratio (L/D ratio) will find this feature very favorable if it experiences engine failure and must rely on this property to get to the nearest emergency airport. However, if the L/D ratio is very high, this will actually make it harder to land as it would have to fly at a very shallow approach path angle to keep the airspeed low. The shallow angle would not only make it more difficult to assess where it will touch-down, but would also tend to make the airplane float once it enters ground effect, compounding the difficulty. Figure 21-1 shows an organizational map displaying the descent among other members of the performance theory. FIGURE 21-1 An organizational map placing performance theory among the disciplines of dynamics of flight, and highlighting the focus of this section: descent performance analysis.

This section will present the formulation of and the solution of the equation of motion for the descent, and present practical, as well as numerical solution methodologies, that can be used both for propeller and jet-powered aircraft. When appropriate, each method will be accompanied by an illustrative example using the sample aircraft. The primary information we want to extract from this analysis is characteristics like minimum rate-of-descent (ROD), best (lowest) angle-of-descent (AOD), the corresponding airspeeds, the AOD for a given power setting, and unpowered glide distance.

In general, the methods presented here are the “industry standard” and mirror those presented by a variety of authors, e.g. Perkins and Hage , Torenbeek , Nicolai , Roskam , Hale , Anderson  and many, many others. Also note that sailplane design and glide analyses methods are detailed in Appendix C4, Design of Sailplanes.

### 21.1.1 The Content of this Chapter

• Section 21.2 develops fundamental relationships necessary to evaluate the characteristics of gliding flight, most importantly the equations of motion for descending flight.

• Section 21.3 presents an assortment of methods to predict the various descent characteristics of an airplane.

## 21.2 Fundamental Relations for the Descent Maneuver

In this section, we will derive the equation of motion for the descent maneuver, as well as all fundamental relationships used to evaluate its most important characteristics. First, a general two-dimensional free-body diagram will be presented to allow the formulation to be developed. Only the two-dimensional version of the equation will be determined as this is sufficient for all aspects of aircraft design.

### 21.2.1 General Two-dimensional Free-body Diagram for an Aircraft

The free body for the descending flight is developed in a similar manner to that for the climbing flight. Figure 21-2 shows a free-body diagram of an airplane moving along a flight path. The x- and z-axes are attached to the center of gravity (CG) of the airplane, just as for the climb and have identical orientation with respect to the datum of the airplane. The angle between the datum and the tangent to the flight path is the angle-of-attack and the force (or thrust) generated by the airplane’s power plant, T, may be at some angle ε with respect to the x-axis. The figure shows that this coordinate system can change its orientation with respect to the horizon depending on the maneuver being performed. Then, the angle between the horizon and the x-axis, the descent angle, is denoted by θ, where these three simple rules apply: If θ > 0, then the aircraft is said to be climbing. If θ = 0 then the aircraft is said to be flying straight and level (cruising). If θ < 0, then the aircraft is said to be descending. This chapter only considers the final rule. FIGURE 21-2 A two-dimensional free-body of the airplane in a powered gliding flight.

The free-body diagram of Figure 21-2 is balanced in terms of inertia, mechanical, and aerodynamic forces. The lift is the component of the resultant aerodynamic force generated by the aircraft that is perpendicular to the flight path (along its z-axis). The drag is the component of the aerodynamic force that is parallel (along its x-axis). These are balanced by the weight, W, and the corresponding components of T. The presentation of Figure 21-2 can now be used to derive the planar equations of motion for the descent maneuver, which are sufficient to accurately predict the vast majority of descent maneuvers.

### 21.2.2 Planar Equations of Motion (Assumes No Rotation about Y-axis)

The equations of motion for gliding flight can be derived using the free-body diagram in Figure 21-2: (21-1) (21-2)

The equations of motion can be adapted for descending flight by making the following assumptions:

(1) Steady motion implies dV/dt = 0.

(2) The descent angle, θ, is a non-zero quantity.

(3) The angle-of-attack, α, is small.

(4) The thrust angle, ε, is 0°.

Equations of motion for a steady unpowered (T = 0) descent: (21-3) (21-4)

Equations of motion for a steady powered (T > 0) descent: (21-5) (21-6)

Vertical airspeed: (21-7)

AOD is also known as angle-of-glide (AOG) or glide angle. FIGURE 21-3 Airspeed components during climb.

## 21.3 General Descent Analysis Methods

### 21.3.1 General Angle-of-descent

The angle-of-descent is the flight path angle to the horizontal and is computed from:

Unpowered descent: (21-8)

Powered descent: (21-9)

The right approximations (≈) are valid for low descent angles, θ, and when the CG is not too far forward, as this can put a high load on the stabilizing surface and invalidate the approximation LW. Many airplanes, in particular sailplanes, have such high glide ratios that landing becomes difficult. For this reason, they are equipped with speed brakes or spoilers, which are panels that deflect from the wing surface and cause flow separation, increasing drag and reducing lift. The same holds for high-speed jets.

#### Derivation of Equation (21-9)

We get Equation (21-8) by dividing Equation (21-4) by (21-3): We get Equation (21-9) from Equation (21-6): QED

### 21.3.2 General Rate-of-descent

The rate at which an aircraft reduces altitude is given below: (21-10)

The above expression has units of ft/s or m/s. Generally, the units preferred by pilots are in terms of feet per minute or fpm for general aviation, commercial aviation, and military, but m/s for sailplanes and some nations that use the metric system. To convert Equation (21-10) into units of fpm multiply by 60.

#### Derivation of Equation (21-10)

Begin by multiplying Equation (21-4) by V, and then rewrite V sin θ using Equation (21-7): QED

EXAMPLE 21-1

Plot the rate-of-descent for the Learjet 45XR at S-L, 15,000 ft, and 30,000 ft at a weight of 20,000 lbf (assuming no thrust). Plot the descent rate as a function of true airspeed in knots (KTAS).

Solution

Sample calculation for the sample aircraft gliding at 175 KCAS (LDmax) at S-L and at 20,000 lbf (no thrust). Note that LDmax is calculated in EXAMPLE 19-5. At S-L the density is 0.002378 slugs/ft3. Therefore: This amounts to 1147 fpm. The rate-of-descent for other airspeeds is plotted in Figure 21-4. Figure 21-5 shows the corresponding glide angle and L/D for the aircraft at S-L. Figure 21-6 shows how important performance characteristics, such as the airspeed for minimum power required and best glide ratio, can be extracted from the rate-of-descent plot. FIGURE 21-4 A flight polar, also known as ROD vs airspeed graph. FIGURE 21-5 ROD, L/D and glide angle superimposed on the same graph (at S-L). FIGURE 21-6 Important characteristics extracted from the flight polar (at S-L).

### 21.3.3 Equilibrium Glide Speed

Equilibrium glide speed is the airspeed that must be maintained to achieve a specific glide angle for a specific AOA. One common use is to determine the airspeed required to maintain a specific flight path angle, θ. (21-11)

The lift coefficient can be determined based on the AOA required for the airspeed using CL = CLo + C·α

#### Derivation of Equation (21-11)

From Equation (21-3) we get: QED

EXAMPLE 21-2

During flight testing, the pilot of an SR22 wants to maintain a 3° glide path angle at an AOA of 5°. What airspeed must be maintained? Assume a test weight of 3250 lbf, ISA at S-L conditions, CLo = 0.4 and C = 5.5 /rad.

Solution

Lift coefficient: Knowing that the wing area is 144.9 ft², we can now compute the airspeed necessary to maintain the said glide path angle: ### 21.3.4 Sink Rate

Sink rate is the rate at which an aircraft loses altitude. This is most commonly expressed in terms of feet per minute or meters per second. If the lift and drag coefficients can be determined for a specific glide condition (e.g. from knowing the AOA), the sink rate can be computed from:

Straight and level sink: (21-12)

The above expression will return the sink rate in terms of ft/s or m/s. To convert to fpm multiply by 60. If the airplane is turning and the bank angle is given by the bank angle φ, then the sink rate increases and amounts to:

Sink rate while banking at φ: (21-13)

#### Derivation of Equations (21-12) and (21-13)

Substitute Equation (21-11) into (21-7) to get: (i)

Divide Equation (21-4) by (21-3) to get: (ii)

Substitute Equation (ii) into (i) to get: If we assume cos θ ∼ 1 we get Equation (21-12).

To get Equation (21-13) we refer to Figure 19-26 and see that W = L·cos ϕ = qS·CL·cos ϕ. When level (ϕ = 0°) the same relationship is W = L = qS·CL. This shows that the lift really depends on the product CL·cos ϕ. Therefore, it is more accurate to replace the lift coefficient in the above formulation with the product.

QED

### 21.3.5 Airspeed of Minimum Sink Rate, VBA

Just like the rate-of-climb, the magnitude of the sink rate varies with airspeed. This implies it has a minimum value that would be of interest to the operator of the vehicle as the kinetic energy of the vertical speed is then also at a minimum and, thus, may have an impact on survivability in an unpowered glide (as impact energy is a function of the square of the speed). It turns out that if the simplified drag model applies, the minimum sink speed can be calculated directly as follows. Note that this expression only holds in air mass that is neither rising nor sinking (see Appendix C4, Design of Sailplanes). (21-14)

#### Derivation of Equation (21-14)

Inspection of Equation (21-12) reveals that when CL1.5/CD is maximum, VV is minimum. The airspeed at which this takes place has already been derived as Equation (19-14).

QED

### 21.3.6 Minimum Angle-of-descent

This angle results in a maximum glide distance from a given altitude and is of great importance to both glider pilots and pilots of powered aircraft. (21-15)

Note that the drag model yields a θmin which is independent of altitude.

#### Derivation of Equation (21-15)

Equation (19-18) gives the maximum lift-to-drag ratio for the simplified drag model (repeated below for convenience) and is inserted into Equation (21-8): QED

EXAMPLE 21-3

Determine the minimum angle-of-descent for the sample aircraft flying at S-L at a weight of 20,000 lbf.

Solution This amounts to 3.7°.

### 21.3.7 Best Glide Speed, VBG

The best glide speed is the airspeed at which the airplane will achieve maximum range in glide. It is a matter of life and death for the occupants of an aircraft, as is evident from its inclusion in 14 CFR 23.1587(c)(6), Performance Information. A part of pilot training requires this airspeed to be remembered in case of an engine failure. It can be calculated using Equation (21-16) below. Note that this expression only holds in air mass that is neither rising nor sinking (see Appendix C4, Design of Sailplanes). (21-16)

#### Derivation of Equation (21-16)

Using Equation (21-11) and the assumption that at the best glide angle cos θ ≈ 1, we get: It was demonstrated in the derivation for Equation (19-18) that at LDmax the lift coefficient . Inserting this into the above expression and manipulating leads to: QED

EXAMPLE 21-4

Determine the airspeed of minimum angle-of-descent for the sample aircraft flying at 30,000 ft and S-L at a weight of 20,000 lbf.

Solution

At 30,000 ft the density is 0.0008897 slugs/ft3. Therefore: At S-L the density is 0.002378 slugs/ft3. Therefore: This amounts to 175 KCAS in both cases.

### 21.3.8 Glide Distance

For a powered airplane, knowing how far one can glide in case of an emergency is not just a matter of safety, but of survivability. Such information is required information for the operation of GA aircraft by 14 CFR Part 23, §23.1587(d)(10), Performance Information, and must be determined per §23.71, Glide: Single-engine Airplanes (see below) and presented to the operator of the aircraft. This is typically done in the form of a glide chart, which shows clearly how far the airplane will glide for every 1000 ft lost in altitude.

§ 23.71 GLIDE: SINGLE-ENGINE AIRPLANES

The maximum horizontal distance traveled in still air, in nautical miles, per 1000 feet of altitude lost in a glide, and the speed necessary to achieve this must be determined with the engine inoperative, its propeller in the minimum drag position, and landing gear and wing flaps in the most favorable available position.

During the design phase, this distance can be calculated from the following expression (which is shown schematically in Figure 21-7). Note that this expression only holds in air mass that is neither rising nor sinking (see Appendix C4, Design of Sailplanes). (21-17) FIGURE 21-7 Distance covered during glide can be estimated using the L/D ratio.

#### Derivation of Equation (21-17)

First we note the following relation between the speed and distance: Assuming that θ is small, we can say that . Therefore, using Equation (21-5) we get: Using Equation (21-3) and the assumption that for small angles cos θ ≈ 1, we get: QED

EXAMPLE 21-5

Determine the maximum glide distance for the sample aircraft flying at 30,000 ft.

Solution

Using the maximum LD calculated in EXAMPLE 19-7 we get (LDmax = 15.45): ## Variables    ## References

1. Perkins CD, Hage RE. Airplane Performance, Stability, and Control. John Wiley & Sons 1949.

2. Torenbeek E. Synthesis of Subsonic Aircraft Design. 3rd ed. Delft University Press 1986.

3. Nicolai L. Fundamentals of Aircraft Design. 2nd ed. 1984.

4. Roskam J, Lan Chuan-Tau Edward. Airplane Aerodynamics and Performance. DARcorporation 1997.

5. Hale FJ. Aircraft Performance, Selection, and Design. John Wiley & Sons 1984; pp. 137–138.

6. Anderson Jr JD. Aircraft Performance & Design. 1st ed. McGraw-Hill 1998.

7. Raymer D. Aircraft Design: A Conceptual Approach. AIAA Education Series 1996.

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