#### Investigating the Lift Boundary

Theoretical Basis

Let us begin with introduction of a Pseudo Non-dimensional Parameter ${nW \over δ}$, the utility of which shall be evident as we progress

Lift, $L= {1 \over 2} ρ V^2 S C_L$

also, L=nW, Where W is the Weight of the aircraft

Hence, $nW={1 \over 2} ρ V^2 S C_L$

This can be further treated to give us the following identity (replacing V by Mach number, and substituting known constants)

${nW \over δ}=k P_0 M^2 S C_L$, $\space \rightarrow \{1\}$

Where $k$ is a constant, $δ={P \over P_0}$ the pressure ratio.

Clearly, ${nW \over δ} ∞ f(M^2 ,C_L)$

It can be seen that all other parameters remaining same, greater the weight (W), lesser the ‘g’ that can be pulled. So is the effect of increase in altitude (lesser δ).

Therefore it is convenient to combine the weight and pressure altitude effects with ‘n‘ into a single parameter,   which will then depend only on Mach number and $C_L$ . By using this combined Pseudo non dimensional parameter we reduce the number of independent variables for the analysis.

The maximum useful value of $n W \over δ$ will depend on speed and the highest usable value of CL . The highest usable CL  may be limited by a number of factors. Stalling puts an absolute limit on the value of n and the resulting ‘g’ envelope is the  boundary, but the aircraft may become operationally ineffective before the stall is reached, due to buffet, pitch up, wing rock etc. and the ‘g’ envelopes imposed by these limitations are called buffet boundaries, pitch up boundaries, wing rock boundaries etc. As we know, they are all dependent on α and CL.

At low speeds, before compressibility effects become applicable, CLmax remains reasonably constant so that $n W \over δ$ is proportional to $M^2$ i.e, it increases parabolically with speed. At speeds above about M = 0.3, CLmax decreases, which serves to reduce the rate of increase in  $n W \over δ$ with speed. CLmax  may decrease to such a low value in the transonic speed range that $n W \over δ$  may even decrease with speed. But generally it starts to pick up again at supersonic speeds.

Flight Test Technique

The purpose of flight test is to determine exactly how $({n W \over δ})_{max}$ varies with Mach number so that the graph shown in Figure 1 may be plotted. Once this information is available, the maximum ‘g’ is known at any altitude, speed and weight.

The test consists of first flying straight 1g stalls at various altitudes to enable the various boundaries to be identified and separated at low speeds. The aircraft is then flown into ‘Wind Up Turns (WUT)’ at the various test speeds and the ‘g’ is slowly increased (wound up) until the aircraft reaches the ‘aimed for’ boundary at the test altitude. When the lift boundary lies above the thrust boundary (i.e thrust boundary is reached before lift boundary as n is increased) then the test points must be reached in a diving WUT. The maximum ‘g’ obtained, weight altitude and the test speed will give us the $({n W \over δ})_{max}$  for that speed and thus the plot as shown in Figure 1 can be obtained.

The main technique to determine the lift boundary of an airplane is the Wind-up Turn (WUT). This manoeuvre involves a combined plane turn with a continuously changing bank angle. The wind up turn is accomplished by gradually rolling from a 1-g level condition to obtain maximum desired n, or α . Bank angle and ‘g’ are slowly blended to maintain constant Mach values (±0.02). The onset rate is limited, so nose low pitch attitude must be closely monitored to prevent the Mach from running away.

In lift boundary investigation we mainly consider the maximum normal acceleration n, the radius of turn R and the rate of turn ω. All these are required to be presented in a form to depict their variation with weight, altitude and speed in an easily interpretable manner.

#### The Thrust Boundary

Unlike lift boundary which represents the unsteady instantaneous manoeuvre performance, thrust boundary represents a steady condition. It represents the sustained normal acceleration for a fixed speed, altitude and power setting. There are two main methods to determine an aircraft’s thrust boundary.

Method 1: Sustained/Stabilized Turn Method (or, the Direct Method).  This method is called as direct method, as it comes directly from the definition of thrust boundary. The aircraft is put into a level turn at selected values of height, speed and engine stetting and the ‘g’ is increased till the height and speed can no longer be maintained. This value of ‘g’ is then a point on the thrust boundary of the aircraft, corresponding to the test conditions. We have elaborately discussed this method earlier in PART-3.

Method 2: Level Acceleration-Deceleration Method (or, the Indirect Method). The stabilized turn method is very expensive and time consuming. The basis of Level Acceleration-Deceleration method is that the excess thrust can be used in various ways, and that having measured it in one way, say an accelerated level run, then it is possible to calculate with certain simplifying assumptions, what the turning performance would be. The method involves a level acceleration followed by a decelerated turn at the same altitude. It does not involve flying the aircraft at the actual point on the thrust boundary. The analysis is based on experimental method of performance reduction. The results are not always accurate and needs to be verified by spot checks with stabilized turn method.

Theoretical Background

The direct method needs no additional theoretical analysis than what has been discussed till now. However, the basis of indirect method needs further clarity. Hence we shall go thorough the theoretical aspects of this method.

For a level acceleration flight, a little mathematical deduction will yield

${T \over δ} ={k_1 M^2}+k_2{({({nW \over δ}) \over M})}^2+{WV \over g δ}$, $\space \space \rightarrow \{2\}$

where $k_1$ and $k_2$ are constants

We may simply it further by few assumptions.

During level accelerated run, n is ‘1’ and V is positive. Similarly, during the decelerated turn, n will be greater than the value corresponding to the thrust boundary at the particular speed and V will be negative. For a given engine conditions at a fixed Mach number it is assumed that the thrust is also constant.

For both these conditions, the above equation can take the form

${WV \over g δ}=k_3{ ({nW \over δ})}^2+ k_4$, $\space \space \rightarrow \{3\}$

where $k_3$ and $k_4$ are constants (which are although different for the acceleration and deceleration parts.)

Clearly, for given Mach number, a linear relation exists between ${WV \over g δ}$ and ${ ({nW \over δ})}^2$ .

Till now we had considered that the level acceleration and level declaration are carried out at a constant altitude. Due to inevitable limitations in executing the test point, variations of height will occur and hence would need to be catered for. Fortunately, both the height and speed changes can be provided for by considering $E \over δ V$ in place of $W V \over δ g$, where,

${E \over δ V}={Wh \over δV} +{WV \over δg}$

We can find that with the same assumptions as before, a similar linear relation as in {3} exists between ${E \over δ V}$ and ${ ({nW \over δ})}^2$

Test Execution: Ideally, the test should be performed by initially stabilizing the aircraft at a low airspeed at the test height, and then opening the throttle quickly, and allowing the aircraft to accelerate, maintaining the height constant. When the aircraft has stabilized at its maximum level speed, it is then put into a smooth level turn, maintaining the same height and engine rpm, the ‘g’ being applied gradually so that the airspeed falls slowly and steadily (about 1 knot / sec). To do this the ‘g’ applied must be slowly increased, and then, when the airspeed drops below a certain value, slowly decreased again. In practice the variation of height during the manoeuvre should be minimum possible; allowance is made in the reduction for any height variations.

By using increments method, the values of $\dot V$ and $\dot H$ can be estimated for the mid points of selected intervals. From the recorded values of n, the values of ${ ({nW \over δ})}^2$ can be found.

A convenient number of mach number are selected and values of ${E \over δ V}$ and ${ ({nW \over δ})}^2$ for both accelerated level and decelerated level turn are obtained for these mach numbers. For each mach number a pair of points are plotted on a graph of ${E \over δ V}$ against ${ ({nW \over δ})}^2$  ( one point representing accelerated level and the other for decelerated level  and a straight line connecting them is drawn. The intercept of this line on the ${ ({nW \over δ})}^2$ axis gives the value of ${ ({nW \over δ})}^2$  corresponding to the thrust boundary at this particular mach number. This is found for a wide range of mach numbers. Further, ${ ({nW \over δ})}$  values are plotted against Mach number to give the generalized thrust boundary. A plot  of n for varying mach number can also be obtained at the selected values of weight, pressure, height and power.

This is the last part of a four part tutorial.

PART-3: Flight Testing for Manoeuvre Performance

(Any queries / observations may kindly be posted here as comments. All effort shall be made to revert on the same within a reasonable time-frame )

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