F 104 V-n diagram

(This question has been edited to reflect a change in perspective: based on answers to the related question What is causing these "corners" on this F-104 V-n diagram?, I now believe that the curves on the upper left edge of the envelope are intended to represent the stall speed throughout their entirety, both above and below the distinct "corners". If the aircraft is being limited by something other than the actual stall along some portions of these curves, then it would be helpful to clarify and explain that in an answer either to the present question or the one linked above.)

In this V-n diagram for an F-104 (source: this Wikipedia link), why does the stall speed at any given G-loading (in terms of IAS) decrease with altitude-- at least in the part of the envelope that is below the sharp "corners"?

And why is the opposite is true for data points above the "corners"?

For example-- (numbers are approximate)--

Below the "corners"--

20,000', 5G, 415 knots IAS, versus 30,000', 5G, 375 knots IAS

30,000', 3G, 330 knots IAS, versus 40,000', 3G, 295 knots IAS

60,000', 1G, 175 knots IAS, versus 70,000', 1G, 140 knots IAS

Above the "corners"--

30,000', 5G, 375 knots IAS, versus 40,000', 5G, 475 knots IAS

And why do the curves cross each other?

(For example, at about 350 KIAS and 3.5G, the 40,000' curve crosses the 30,000' curve.)

Related ASE questions and answers-- but some of which seem to suggest the opposite relationship should be the case (at least for the "below the corners" cases)--

Why does indicated stall speed change? (see question and all answers)

What causes a slight increases in indicated stall speed with altitude? (see question and all answers)

How does the IAS stall speed vary with increasing altitude? (see question and all answers)

  • $\begingroup$ @ymb1-- lots of tags deleted there (edit #4), was that all really necessary? Among other things, stall speed is arguably not unrelated to aircraft performance, and physics is a pretty broad brush that would seem to include this question. $\endgroup$ – quiet flyer Apr 17 at 18:24
  • $\begingroup$ chat.stackexchange.com/transcript/message/57682815#57682815 $\endgroup$ – ymb1 Apr 17 at 18:46
  • $\begingroup$ After much magnification, it seems each altitude has only one step. Notice S.L. and 20,000 feet do not have a step. The get progressively lower (lower IAS) with altitude. $\endgroup$ – Robert DiGiovanni Apr 18 at 16:55
  • $\begingroup$ @RobertDiGiovanni -- yes, agreed $\endgroup$ – quiet flyer Apr 18 at 16:58

As an airplane approaches Mach 1, all pressure changes grow with the Prandtl-Glauert factor of $\frac{1}{\sqrt{1-Ma^2}}$. Therefore, the lift curve slope increases so the wing produces more lift at the same angle of attack and dynamic pressure the closer its Mach number is to 1. On wings with thicker airfoils and higher aspect ratio the maximum lift coefficient drops in the transsonic region; however, thin wings of low aspect ratios are unaffected by this.

NACA TN 3469 by Edward Polhamus compares wings with varying sweep angle, thickness and aspect ratio. The plot below shows that the Prandtl-Glauert effect is strongest for unswept wings and the transsonic lift bucket disappears for thin wings with an aspect ratio of 4 and lower. The F-104 had a 3.36% airfoil and an aspect ratio of 2.45.

NACA TN 3469 figure 2

This effect also shows up in the maximum lift; the plot below is from Sighard Hoerner's Fluid Dynamic Lift, chapter 4. It shows how the Mach number affects maximum lift: Small leading edge radii show a loss of maximum lift at moderate Mach numbers but perform well at transsonic speeds.

Hoerner FDL figure 4-27

The indicated stall speed decreases with altitude because the lower temperature lowers the local speed of sound and brings the airplane closer to Mach 1 and thus to higher maximum lift. This effect is limited to subsonic speed close to Mach 1 which explains why those corners only show up when the indicated airspeed correlates with Mach close to 1 at that particular altitude.


Plugging in some values using a True Airspeed calculator seems to confirm Peter Kampf's conclusion that the "steps" are indeed coinciding with transonic flight at various IAS and altitudes.

There for, there is no "over lapping" of the curves, though they appear to be on the graph.

One might suggest the flattening of the curve once the aircraft goes supersonic may be related to "Mach tuck", and some sensible engineering related to airframe stress.

By design, and because of the rearward shift of the center of lift, after trimming there simply is not enough "up" stabilizer left to re-approach stall limits in supersonic flight.


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