Climbing and Gliding Caveat

Since this course comprises both aircraft performance and flight mechanics, the nomenclature can be a little confusing when trying to keep things consistent across the disciplines.

Aircraft performance uses simplified models of aircraft mechanics to determine basic relationships for flight. Using these to predict climbing and gliding flight is possible, and will be completed in the following, but some explanation of the nomenclature will be described first.

Flight Mechanics Angles

To describe the aircraft attitude a set of angles are used called Euler angles - these will be fully described in Module 3. The aircraft pitch is defined as the angle between the horizontal and the earth axis, and is given the symbol \(\theta\).

To describe the aircraft flight path a set of angles describing the orientation of the aircraft velocity vector with respect to the earth are used. These are the flight path angle \(\gamma\), and the track angle, \(\tau\).

To describe the aerodynamic incidence, the aerodynamic angles \(\alpha\) and \(\beta\) are used. \(\alpha\) is angle of attack, and \(\beta\) is sideslip.

The angles above will be fully explored and utilised in the later modules, but an appreciation of the longitudinal angles is required here:

Fig. 20 Relationship between longitudinal angles

Aircraft Performance

In aircraft performance, the climb or glide angle is denoted by \(\theta\) in numerous texts - for example :cite:`Anderson:1999AP` - and this angle represents the angle between the aircraft flight direction and the horizontal plane.

Looking at Fig. 20, you will see that this should really be denoted by \(\gamma\). In the following, the glide or climb angle will be denoted by \(\gamma\), but you should be aware that in most texts you will see it denoted by \(\theta\).

In reality, \(\gamma=\theta\) if and only if (iff) \(\alpha=0\).

The distinction is made because this course later requires us to use \(\theta\), and I don’t want you getting confused about what the angles mean.

Since the flight path angle, \(\gamma\), is defined as positive in a glide and negative in a climb (positive \(\gamma\) gives positive \(W_e\) which is velocity towards the earth), it leads to requiring a negative angle continually in the climb expressions. :cite:`Yechout:2014vg` uses a good approach and introduces

\[\bar{\gamma}=\left|\gamma\right|\]

to avoid the negative expression. For the following, \(\bar{\gamma}\) will be used.

Gliding Flight (Unpowered Descent)

The parameters determine above, the lift and hence speeds for minimum thrust and minimum power are important for determination of an aircraft’s glide characteristics.

Why gliding?

If you’ve never experienced flight in a glider, I’d thoroughly recommend it (I’m a recent convert to gliding from powered flight).

It will make you a more rounded aeronautical engineer, and enable you to fully understand the significance of the parameters we’re talking about here.

I’m trying to get a flying club set up for our school, so hopefully it’ll happen soon…

With no thrust, the aircraft has \(T=0\) but \(D>0\), so there is no longer force equilibrium. The aircraft will descend at the glide angle \(\bar{\gamma}\). In a steady glide, the horizontal component of weight is equal to the drag.

Fig. 21 Forces on aircraft in a glide

What’s actually happening in a glide?

In the preceding section, the aircraft was in equilibrium - the lift was equal to the weight, and the thrust was equal to the drag.

Drag is the aerodynamic force opposing motion, and has an associated power - which is the rate that the aircraft does work on the flow in the direction of motion. During cruise, the thrust power (\(T\cdot V\)) is equal to the power associated with drag.

With no input power (\(T=0\)), in order to maintain a constant forward speed, some power has to be input to the aircraft to maintain horizontal equilibrium. As stated, part of the weight vector (\(W\cdot\sin\bar{\gamma}\)) opposes the drag.

The associated power the rate of change of gravitational potential energy. In a glide, potential energy is being exchanged for kinetic energy to maintain forward speed.

Glide Angle

In Fig. 21, the forces are parallel and perpendicular to the flight path as before, but you should note that this is no longer parallel with the ground. In the horizontal direction, equilibrium gives \(W\cdot\sin\bar{\gamma}=D\) and in the vertical direction, equilibrium gives \(L=W\cdot\cos\bar{\gamma}\). Hence the glide angle can be determined from simple trigonometry:

\[\begin{split}\begin{aligned}\tan\bar{\gamma}&=\frac{D}{L}\\ &=\frac{C_D}{C_L}\end{aligned}\end{split}\]

So it can be seen that the best/shallowest glide (the smallest \(\bar{\gamma}\)) occurs at the minimum drag to lift ratio - i.e., at \(C_{L,md}\).

The glide angle dictates the glide distance, which is very important to be aware of in the cockpit in a glider as this dictates the furthest distance that can be covered along the ground.

The aircraft speed equation can be used with \(L=W\cdot\cos\bar{\gamma}\) in place of \(L=W\) - which shows that the speed that the shallowest glide occurs is not \(V=V_{md}=\left[\frac{B}{A}\right]^\frac{1}{4}\) as \(V_{md}\) as calculated previously is only valid for cruise.

\[\begin{split}\begin{aligned} V&=\sqrt{\frac{L}{\tfrac{1}{2}\rho\cdot S\cdot C_{L,md}}}\\ &=\sqrt{\frac{W\cdot\cos\bar{\gamma}}{\tfrac{1}{2}\rho\cdot S\cdot C_{L,md}}}\\ &=\sqrt{\frac{W}{\tfrac{1}{2}\rho\cdot S\cdot C_{L,md}}}\cdot\sqrt{\cos\bar{\gamma}}\\ &=V_{md}\cdot\sqrt{\cos\bar{\gamma}}\end{aligned}\end{split}\]

Small angle assumption?

The speed for minimum slide is therefore slower than the minimum drag speed in cruise. Compare the ratio of the two speeds for a range of glide angles:

import numpy as np
import plotly.graph_objects as go

# Make a vector of reasonable glide ratios - say 30 - 80
glide_ratios = np.linspace(2, 80, 1000)
glide_angles = np.arctan(1/glide_ratios)

Vbestglide_on_Vmd = np.sqrt(np.cos(glide_angles))


fig = go.Figure()
fig.add_trace(go.Scatter(x=glide_ratios, y=glide_angles*180/np.pi))

fig.update_layout(
    title="Glide Angle vs. Glide Ratio",
    xaxis_title="Glide Ratio",
    yaxis_title="Glide Angle / deg",
)

fig.show()

fig = go.Figure()
fig.add_trace(go.Scatter(x=glide_ratios, y=Vbestglide_on_Vmd))

fig.update_layout(
    title="$\\text{Ratio of best glide speed and }V_{md}$",
    xaxis_title="Glide Ratio",
    yaxis_title="$\\frac{V_{bestglide}}{V_{md}}$",
)

fig.show()