## Wind-up turns: TLF or MAX thrust?

The main technique to determine the lifting boundary of an airplane is the Wind-up turn (WUT) with the Split-S (SS) being another very good option for fighter aircraft. This is a mandatory item in producing the instantaneous turn rate (ITR or $\omega_{inst}$) performance of a fighter airplane commonly in the form of a doghouse plot. The discussion which follows is on WUT, but same results apply to SS.

During an over-a-beer discussion of a recent meeting of Test Pilots and Flight Test Engineers, some disagreement was encountered on the thrust setting required during the execution of a WUT test maneuver for the purpose of lift boundary generation. The two arguments were the following:

Argument 1: The WUT should be performed with the TLF set for establishing the desired trim shot ($M$). The throttle setting should be kept constant during the maneuver not to affect the measured maneuvering performance and to maintain stable engine parameters.

Argument 2: After the stabilized trim shot at the desired target speed ($M$) with thrust for level flight (TLF), and while establishing the turn at constant targeted $M$, throttle can be advanced to MAX in order to minimize altitude loss, with no effect on the test derived results, considering that the lifting boundary is by definition related to wing’s maximum lift capability only and not affected by engine setting.

It should be reminded at this point, that ITR is related to is the maximum lifting capability which is related to maximum achieved load factor normal to the flight path ($n_{z_{max}}$) encountered during a WUT:

$\omega_{inst_{max}} \sim n_{z_{max}}$

and for a stabilized level turn:

$\omega=\frac{g\sqrt{n_z^2-1}}{V}$

In order to decide on the above arguments, we will examine in detail the effect of thrust on $n_{z_{max}}$ achieved during a WUT.

In the analysis the following considerations are made:

1. The ITR refers to instantaneous level turn performance and not the instantaneous turn rate achieved during the WUT.
2. Both angle of attack and thrust axis angle are accounted for, thus no approximations are made. This allows to clearly see the effects of these angles in the final result and also accounts for high AoA and thrust vectoring.

Let’s start the analysis by considering a turn in the vertical plane only. As the figure below shows, in a pull-up, the thrust component on the vertical stability axis ($z_{s}$), relates to the net thrust magnitude ($F_{net}$), the thrust axis angle ($\delta$) and the angle of attack ($\alpha$).

The load factor normal to the flight path ($n_{z}$) is related to the sum of the “lifting” forces measured on the $z_s$:

$L+F_{net}\sin(\alpha+\delta)=n_zW \Rightarrow$

$\frac{q C_{L}}{W/S}+\frac{F_{net}}{W}\sin(\alpha+\delta)=n_{z}$

And for the case of turning on the lift boundary, maximum achievable load factor ($n_{z_{max}}$) is:

$n_{z_{max}}=\frac{q C_{L_{max}}}{W/S}+\frac{F_{net}}{W}\sin(\alpha_{max}+\delta)$

The above equation indicates that the load factor in a vertical turn at specific weight, and dynamic pressure ($q$), besides $C_{L_{max}}$ it also depends on the thrust magnitude, the angle of attack and the thrust axis angle. A similar dependency is true for the “cockpit g” ($n_{z_c})$ which is read directly by the pilot through the g-meter, or recorded from a body fixed accelerometer (aligned to the $z_b$ axis).

$n_{z_c}=n_{z}\cos(\alpha)$

(some references consider $\cos(\alpha) \approx 1$ and thus “cockpit g” $n_{z_c}=n_{z}$).

For the case of a WUT which considers maneuvering in the oblique plane, the $n_z$ expressions are the same to the pull-up case derived above, as they are weight component independent.

The above analysis indicates that applying MAX thrust during the execution of a WUT does affect the achieved $n_{z_{max}}$ and thus the derived ITR for a level turn.

So does this mean that TLF and not MAX thrust should be applied? No!

The curves in a standard doghouse plot need to be created for a specific thrust setting, this means the the lifting boundary corresponding to the ITR and created through maneuvers like WUT should also be performed with the specific thrust setting.

So technically neither of the aforementioned arguments are fully right as it is common to have doghouse plots generated for throttle settings different than MAX.

The  accurate answer to what throttle setting should be applied during the execution of a WUT for the purpose of ITR curve generation, is the thrust setting desired for the specific curves. This is not to reduce the altitude loss, but because thrust does have an effect in the lifting boundary.

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