What is missing from these diagrams of the forces in slips and skids?

Question

See these images that are widely reproduced in many different on-line ground school materials.

A) Is the magnitude of the wing's lift vector illustrated correctly in each of the three cases? Should it really be the same size in each case?

B) What real aerodynamic force generated by the aircraft has been completely omitted from the second two cases, but should be included to make the diagrams more comprehensible? Specifically, to explain why the vector labelled "load" is not the same in all the figures?

Assume that the aircraft is maintaining a constant altitude and airspeed regardless of whether the turn is coordinated, slipping, or skidding.

(The same diagrams could apply to gliding flight as well, in which case we would assume that the aircraft was maintaining a constant airspeed and a constant descent rate in relation to the surrounding airmass.)

The intent of the question is to address a major flaw in the diagrams, not to nitpick small errors on the part of the artists. The second diagram is the best one to focus our attention on, because the bank angle is clearly drawn to be identical in every case, and the horizontal and vertical components of the lift vector are clearly drawn to be identical in every case.

A word about the images included here-- both images are widely reproduced in many different on-line ground school materials. For example, the first image appears as Figure 3-21 from this "Aerodynamics in Flight" section from an on-line ground school. For another example, see page 12 of this document. It may have originally been published in the FAA's "Pilot's Handbook of Aeronautical Knowledge". The second image appears as figure 5-35 on page 5-24 of the FAA's "Pilot's Handbook of Aeronautical Knowledge" (2016 edition). It also may be found in various on-line ground school materials-- see for example this one.

Answer 1

A) Is the magnitude of the wing's lift vector illustrated correctly in each of the three cases? Should it really be the same size in each case?

No (on both counts).

B) What real aerodynamic force generated by the aircraft has been completely omitted from the second two cases, but should be included to make the diagrams more comprehensible? Specifically, to explain why the vector labelled "load" is not the same in all the figures?

Side-slip.

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This is my attempt at better describing the forces in a slipping, skidding and coordinated turn. These thoughts represent my best guess and should be considered the basis for further discussion rather than fact.

The biggest and somewhat major limitation of my diagram (that I'm aware of), is that it doesn't show the critical role of yaw (due to my limited ability to draw, rather than by design).

Ideally it would show a yaw away from the turn for a slip and a yaw toward the turn for a skid.

This would also serve to clarify that the aircraft depicted is viewed from behind (only relevant to the slip ball displacement shown).

Another limitation of the diagram, due to it's 2D nature, is that it doesn't illustrate conventional drag (along the longitudinal axis) and how it varies with yaw during a slip or skid. ie: the force acting perpendicular to and out of the screen. This is important as the vector diagram doesn't facilitate conceptualising total drag during uncoordinated flight.

The key to understanding slipping, skidding and coordinated turns (as I understand it), is recognising the effect of side-slip on the total aerodynamic force (TAF) which is defined as the net aerodynamic force experienced by the aircraft. As per the previous paragraph, this diagram only considers the TAF along the lateral axis and so doesn't provide the complete picture, though it does serve to fully explain the different turn rates and resulting turning radii.

To be clear, I'm using the term side-slip to refer to the aerodynamic condition where an aircraft has an asymmetrical airflow about it's longitudinal axis resulting in a degree of lateral motion with respect to the relative airflow.

The air impinging on the exposed side of the aircraft effectively creates a lateral force, which is labelled side-slip in the diagrams.

The diagram depicts forces on the basis that, regardless of whether the turn is coordinated, slipping, or skidding - the aircraft is maintaining a constant;

• angle of bank; and
• altitude (thus lift [$TAF_{vertical}$] always equals weight); and
• airspeed

This means the main variables are;

• AoA (power) - indirectly represented by the vector labelled TAF

Quick recap on the aerodynamics of a turn:  

In any turn (assuming the constants above), the power requirements increase  
proportional to the rate of turn.  

In vector speak, a turn is achieved by tilting the TAF away from the vertical  
in the direction of the turn. The resulting horizontal component provides the  
centripetal acceleration required, thus the greater the tilt, the  
greater the rate of turn.

The consequence of tilting the TAF away from the vertical is a reduction in  
lift. In order to maintain altitude during a turn, the TAF must therefore be  
increased in order to keep the vertical component equal to what it was in  
level flight. This represents an increased power requirement.

In the diagram, the length of the TAF vector varies in each scenario, therefore each has a different power requirement.

For any given side-slip angle (representing a given amount of drag along the longitudinal axis), a skidding turn requires more power than a slipping turn due to the higher value of $a_c$. I am making the assumption that the lateral drag is the same in both cases (but I don't know if this is the case).

I posit that at some angle of side-slip, the power required for a slipping turn will exactly equal that required for a coordinated turn. In this scenario, some of the power used to create the turning force ($a_c$) during a coordinated turn is instead used to combat the additional frontal drag associated with a slipping turn, which results in a reduced rate of turn.

It seems a skidding turn does a better job at turning than a coordinated turn. This is true, the rate of turn is higher, so why not use a skidding turn to turn faster than a coordinated turn? Aerodynamic hazards aside, it's simply less efficient.

Key to realise is that the rate of turn achieved per unit of power (ie: efficiency) is less in an uncoordinated turn (slip or skid) as power is wasted in both dragging the aircraft sideways and forwards (additional drag is created in the direction of travel as the aircraft presents a greater cross-section to the relative airflow).

Thus, in order to achieve a given turn radius, it is more efficient (requires less power) to perform a coordinated turn at a higher angle of bank than a skidding turn at a lesser angle of bank or a coordinated turn at a lower angle of bank than a slipping turn at a higher angle of bank.

• turn radius

  The turn radius is determined by nothing other than the horizontal component of the TAF, identified as $a_c$ in the diagrams.

  A higher centripetal acceleration results in a smaller turn radius and a lower centripetal acceleration results in a larger turn radius.

Insights:

• the TAF varies according to how coordinated flight is, in order to keep the value of lift ($TAF_{vertical}$) constant (ie: to maintain altitude in the turn)
• apparent gravity / load and how it changes is explained simply as the reaction force equal and opposite to the varying TAF (rather than a result of centripetal force and it's fictitious sibling centrifugal force)
• in a level slipping turn, side-slip opposes the horizontal component of lift ($a_c$) increasing the radius of the turn
• in a level skidding turn, side-slip reinforces the horizontal component of lift ($a_c$) reducing the radius of the turn

Comments & criticisms welcome.

Answer 2

By including the vector labeled "centrifugal force", the illustrators have signalled that they are basing their reference frame on the aircraft itself, not the earth or the airmass.

The reference frame based on the aircraft is not a valid inertial reference frame.

The aircraft cannot accelerate with respect to itself. The net force in the aircraft's own reference frame must be zero. Yet that's not what we see illustrated in the diagrams for the "slipping" and "skidding" cases.

It would better to omit the "lift" vector entirely and just show the load vector, weight vector, and "centrifugal force" vector, than to give the false impression that the actual aerodynamic forces generated by the aircraft are identical in all three cases.

What is missing from the "slipping" and "skidding" diagrams is the aerodynamic sideforce generated by the fuselage as it flies sideways through the air. When this vector is added to the lift vector, we end up with a net aerodynamic force vector that is equal in magnitude and opposite in direction to the vector labelled "load".

To make this work, the lift vector must be reduced in size in the slipping case, and must be increased in size in the skidding case.

Drawn this way, the diagrams would help the reader to understand the real reason why the inclinometer ball is displaced to the side in a slip or a skid. Fundamentally, it is due to the aerodynamic force created by the airflow striking the side of the fuselage. As a result, the net aerodynamic force vector is no longer "straight up" (i.e. parallel to the vertical fin) in the aircraft's own reference frame. So the ball, and the pilot's body, and the other aircraft contents, tend to be displaced toward the low wingtip in a slip, and toward the high wingtip in a skid.

Drawn correctly, the diagrams would teach the reader this concept, regardless of whether or not the choice is made to include the "centrifugal force" vector or not.

Drawn correctly, the diagrams would also teach the reader that the load "felt" by the airplane, slip-skid ball, pilot's body, etc, is the direct result of the aerodynamic forces generated by the aircraft. The "load" vector must always be the mirror image of the net aerodynamic force vector.

Drawn correctly, the diagrams would help the reader understand why increasing or decreasing the lift force by moving the stick or yoke forward or aft during a normal coordinated turn doesn't make the slip-skid ball deflect toward one side or the other, even though the turn rate is altered. As long as the net aerodynamic force vector acts straight "up" in the aircraft's reference frame, the apparent "load" vector must act straight "down" in the aircraft's reference frame, regardless of whether the turn rate is "right" for the bank angle and airspeed, or has been temporarily boosted or decreased via a pitch control input. (Naturally, such variations in the lift force will also cause the flight path to curve skywards or earthwards-- for a given bank angle, there's only one value for the lift vector that permits a stabilized turn at constant airspeed.)

Starting from the "load" vectors that we've been given here, what would the corrected diagrams look like? They would look like the top row of diagrams described in the remainder of this answer.

Diagrams to be added-- for now we'll have to use our imagination.

I'm referencing the second diagram specifically, the one from the 2016 edition of the "Pilot's Handbook of Aeronautical Knowledge", where the aircraft are clearly all drawn at exactly the same bank angle.

Imagine four rows of diagrams, each based on the diagram referenced above, but modified as follows--

First row -- forces in aircraft reference frame (not a valid inertial reference frame)

Weight and "centrifugal force" and "load" as illustrated in original. Note that "load" is the vector sum of weight and "centrifugal force".

The figures will include a net aerodynamic force vector (not shown in original). The net aerodynamic force vector must be exactly equal and opposite to "load" in all three figures.

Lift vector is correct in fig 1 of original (coordinated), and same as net aerodynamic force. So fig 1 of first row of answer is essentially same as original.

Lift vector needs to be shorter in fig 2 (slip), and there needs to be an aerodynamic sideforce vector acting at right angles to the lift vector, pointing to the right side of the page and upwards. That is the force that is missing from the diagram. It is generated by the air hitting the side of the fuselage. The vector sum of lift plus sideforce is the net aerodynamic force vector, and it must be exactly equal and opposite to the "load" vector.

Lift vector needs to be longer in fig 3 (skid), and there needs to be an aerodynamic sideforce vector acting at right angles to the lift vector, pointing to the left side of the page and downwards. That is the force that is missing from the diagram. It is generated by the air hitting the side of the fuselage. The vector sum of lift plus sideforce is the net aerodynamic force vector, and it must be exactly equal and opposite to the "load" vector.

With the figures drawn this way, we are able to see that we actually "load up" the wing in a skid, in the sense that we force it to create more lift than we'd normally need for a given bank angle, assuming that we are not letting the flight path curve earthward. The turn rate is also increased, and the turn radius is decreased.

(On the other hand, if we hold the turn radius fixed and leave the bank angle unconstrained, then a skid actually "unloads" the wing, because it involves a shallower bank angle.)

Second row-- forces acting on aircraft in earth reference frame (or in the reference frame of airmass moving at constant speed)--(these are valid inertial reference frames, at least from an elementary viewpoint that views gravity as a real force.)

Same as above but "centrifugal" force and "load" are omitted. A net force vector may be added which is the vector sum of net aerodynamic force and weight. It is horizontal, and exactly equal and opposite to the (omitted) "centrifugal force" vector. It is the centripetal force vector that causes the turn. For a given bank angle, it is smaller in the slip, and larger in the skid, than in coordinated flight.

Third row-- aerodynamic forces only-- same as above but now weight is also omitted. Now there is less clutter to take our attention away from the net aerodynamic force vector. Note that the net aerodynamic vector is aligned with the aircraft's sense of "up" (i.e. the direction of the vertical fin is pointing) in coordinated flight, but not in the slip or skid. This is arguably the most important row of diagrams. It shows us what the pilot really "feels".

Or if it makes more sense to us-- we can have a fourth row-- "apparent inertial force" "felt" by the pilot-- exactly equal and opposite to the net aerodynamic force vector on row 3. Just this one force vector for each figure, with all aerodynamic forces omitted. It is valid to say that this apparent inertial force is caused purely by the net aerodynamic force vector. It is also valid to note that it is exactly equal to the vector sum of gravity and "centrifugal force", though those won't be included on the fourth row of diagrams. It is also valid to observe that the "apparent inertial force" vector is exactly the same thing as the vector labelled "load" in the top row of diagrams.

Unlike the diagrams in the question, the diagrams in the answer won't show the wing's lift vector decomposed into horizontal and vertical components. No real insight is provided by doing that, especially when we completely omit the aerodynamic sideforce vector generated by the airflow hitting the side of the fuselage.

Unlike the diagrams in the question, the diagrams in the answer won't give the illusion that somehow the net aerodynamic force vector is "balanced with" (equal and opposite) the load vector, or the vector sum of weight and centrifugal force, in the coordinated turn, but not in the slip or the skid. That is simply not true. The load vector is equal and opposite to the net aerodynamic force vector in all three cases. From one point of view, the net aerodynamic force vector is what causes the load vector.

Unlike the diagrams in the question, the diagrams in the answer won't give the illusion that the "centrifugal force" vector is exactly "balanced with" (equal and opposite) the horizontal component of the net aerodynamic force vector in the coordinated turn, but not in the slip or the skid. Again, that is simply not true. The "centrifugal force" vector is equal and opposite to the horizontal component of the net aerodynamic force vector in all three cases. Because fundamentally, in this specific case where the vertical component of acceleration is constrained to be zero, we can observe that the centrifugal force vector is entirely caused by the horizontal component of the net aerodynamic force vector.

Unlike the diagrams in the question, the diagrams in the answer won't give the false impression that some mysterious thing, presumably somehow related to turn rate, yet apparently somehow unrelated to any actual aerodynamic force, is magically affecting the amount of "centrifugal force" the aircraft is generating in the slip or the skid.

If the diagrams in the answer actually include the airplane figure (I'm not much of an artist), it should be drawn yawed to the high side of the turn in the slip and yawed toward the low side of the turn in the skid. The flight path will be drawn coming straight out of the page toward the viewer, so the little arrows suggesting that the plane is sliding down and left in the slip, and up and right in the skid, will be omitted.

One interesting question is whether the aerodynamic sideforce vectors referenced above should be considered to include the sideways thrust vector cause by yawing the thrust line sideways relative to the aircraft. As described above, it seems they should, but interestingly, the sideways component of thrust vector has no tendency to displace the slip-skid ball, because it has no sideways component relative to the slip-skid ball, or relative to the pilot's seat for that matter. If we include the sideways thrust vector as part of the aerodynamic sideforce vector, then we are not really using a reference frame fully aligned with the aircraft itself, but rather a reference frame aligned with the direction of the flight path through the airmass at any given instant. It's interesting to think of other analogs-- for example, perhaps a flat-bottomed sled on an icy lake performing a turn by yawing sideways and then firing a thruster that was aimed purely in the aftwards direction in relation to the driver's seat-- that would be equivalent to a flat skidding turn that was somehow performed using only the sideforce generated by yawing the thrust line to one side, with the aerodynamic sideforce from the air hitting the side of the fuselage somehow playing only a negligible role. One airborne analog would be a perfectly spherical airship with a fixed motor in the back, plus thrusters that could establish any desired yaw angle between the flight path and the heading. Skidding turns in such an aircraft would not disturb the slip-skid ball.