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→Lift and Drag Coefficients
More information is available under [[Pressure, Atmosphere and Instrumentation]].
The true air density and the true airspeed '''always ''' appear together as a compound quantity \( \frac{1}{2} \rho V^2 \) which is referred to as the '''dynamic pressure''' or '''dynamic head'''.
The density of air is not a constant: it depends on pressure (which most notably depends on altitude) and temperature. This causes a major inconvenience as we have to assert a value to it in order to arrive at any numerical results directly useful for flying: e.g. you do not check any non-dimensional quantities in the cockpit, you read the instruments instead which tells you the airspeed in knots or the altitude in feet.
In practise, the instrument that tells you \( V_e \) is the '''air speed indicator (ASI)''', and \( V_e \) is known as '''indicated airspeed'''. Based on the discussions above, you should realise that:
# Indicated airspeed is directly related to the dynamic head.
# The dynamic head is the only way true airspeed affects glider aerodynamics(before it disintegrates by overspeeding).
# Therefore, the glider's aerodynamics is affected only by indicated airspeed, not true airspeed (apart from the never-exceed speed).
# We should tabulate performance figures and draw polar graphs using indicated airspeed.
The drag is more complex: the drag on an aeroplane has three components:
# Friction drag, this is the drag caused by the air sticking onto the glider and trying to slow it down. Imagine flying a glider in honey which is quite rather sticky. The friction drag coefficient \( C_{DF} \) is approximately a constant for a given glider.# Pressure drag, this is the drag associated with the glider trailing a wake. This is also known as the form drag because it is related to the form of the glider being not fully aerodynamic. You would intuitively think that a Land Rover Discovery has more drag than a Jaguar fastback: because the Discovery is not streamlined while the fastback is, and this is what pressure drag is about. The pressure drag coefficient \( C_{DP} \) is '''approximately ''' a constant for a given glider, because its form does not change in flight. Were this approximation not to be made, the following derivation can remain unaltered by pretending this variation is a part of the induced drag.# Induced drag, this is the drag caused by having lift. There is no free lunch in aerodynamics and wherever you have lift you must have drag, no matter how well good your design is. The induced drag coefficient \( C_{DI} \) takes the following form:
\[ C_{DI} = \frac{k}{\pi A} C_L^2 \]
Where \( A \) is the aspect ratio of the wings (how slender the wings are), and \( k \) is a factor that depends on the wing design. This drag component increases quadratically with \( C_L \).
== Relationship Between Lift and Drag: the Parabolic Polar ==
The following relationship between \( C_D \) and \( C_L \) is fundamental to the discussions that follow:
\[ C_D = C_{D0} + \frac{k}{\pi A} C_L^2 \]
This is a parabolic function. It is this function that is referred to when talking about a 'parabolic polar': the actual (and more useful) polar curve that we shall derive is '''not''' a parabola.
=== A statement of the task that follows ===
From the relationship presented above, and making use of the following facts or assumptions:
# Mass of the glider remains constant.
# Energy is conserved.
# The air is still.
# The density of air is uniform and known, or rather and better, we work in the corrected ('''indicated''') airspeed system.
We will be deriving a one-to-one relationship between indicated airspeed and sink rate.
== A General Method of Solution ==
Before making further assumptions and simplifications, a general method of solution is worth presenting. The algebraic difficulties, as we shall see, is formidable, but it lends itself nicely to numerical methods.
Re-arranging the definitions of \( C_L \) and \( C_D \):
\[ W \cos(\theta) = C_L \times \frac{1}{2} \rho V^2 S \]
\[ W \sin(\theta) = C_D \times \frac{1}{2} \rho V^2 S \]
The two expressions can both be squared and added together. Notice that \( \cos^2(\theta) + \sin^2(\theta) =1 \), the following is arrived at:
\[ W^2 = (C_L^2 + C_D^2) \times (\frac{1}{2} \rho V^2 S) \]
Or rather, in the more insightful form:
\[ C_L^2 + C_D^2 = \frac{W^2}{\frac{1}{2} \rho V^2 S} \]
The right hand side of the expression above is a function of indicated airspeed only, because air density is a constant for it to be compatible with the indicated airspeed.
Substitute the parabolic relationship between \( C_D \) and \( C_L \) into the expression above, we have:
\[ f(C_L) = g(V) \]
Where \( f(x) \) and \( g(x) \) are functions that are too cumbersome to typeset. Keep in mind that \( C_{D0} \) is embedded in \( g(x) \).
Solving the above which the author does not believe is analytically possible:
\[ C_L = \frac{W \cos(\theta)}{\frac{1}{2} \rho V^2 S} = h(V) \]
In words, a relationship between the lift coefficient and the indicated airspeed can be arrived at.
The above can be further re-arranged, such that:
\[ \cos(\theta) = \frac{h(V) \rho V^2 S}{2W} \]
This is a relationship between the glide slope and the airspeed. From here on, determining the sink rate from the glide slope and airspeed is a trivial geometrical task, so the required relationship between airspeed and sink rate is essentially derived.
