Reply to Brian Abraham
G'day Brian! I'm sure you are aware that two different lift coefficients are defined. There is the aircraft lift coefficient C_L (capital C) based on the lift on the aircraft and the wing planform area; and there is also the section lift coefficient c_L (lower case c) based on two-dimensional flow around an airfoil section.
On 9 December on this thread you correctly stated that induced drag is proportional to the square of the coefficient of lift. That is a correct reference to the aircraft lift coefficient C_L. In your latest post you comment about the lift coefficient on the flapped part of the wing, and a different lift coefficient on the unflapped part of the wing. That is correct, but in your latest post you are referring to section lift coefficients.
In the well-known mathematical equation for induced drag coefficient, the reference to the square of the coefficient of lift is a reference to the aircraft lift coefficient.
Aircraft lift coefficient is defined by a mathematical equation which says C_L is equal to the lift divided by dynamic pressure and wing planform area. In my previous post I addressed this mathematical equation explicitly and in detail. You must admit your latest post comments about lift coefficient, but makes no attempt to address a mathematical equation. It is all subjective. We don't tackle mathematical equations by subjective prose.
(Section lift coefficient c_L is also defined by a mathematical equation.)
We are straying from the matter in question. My point is that the change in induced drag when trailing-edge flaps are extended can't be explained by a change in lift coefficient because changing flap setting (but changing nothing else) doesn't alter the aircraft lift coefficient (unless the flaps are area-changing Fowler flaps). What changes is the Oswald Efficiency Number (or Span Efficiency Factor) and this changes in such a way that extending partial-span flaps increases induced drag.
In your 9 December post you gave a web link to a NASA site that contains the formula for induced drag coefficient. There, in the denominator, is e, called the efficiency factor. That is the one I am calling Oswald Efficiency Number.
I am very happy to discuss mathematical formulae for induced drag, and lift coefficients, but we need to remember that mathematical formulae can't be solved by intuition or subjective assessments. Best regards.