Steady Level-Flight Operations

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n35

37 Steady Level-Flight Operations

Introduction

An airplane’s typical flight operation involves several phases, including takeoff, climb, cruise, turns or maneuvers, descent, and landing. As shown in the figure below, a civil aircraft, such as a commercial airliner, spends much of its flight in the cruise condition flying from one airport to another. This condition is predominantly in straight and level, unaccelerated flight at relatively constant airspeed and high altitudes. A contingency plan for a possible diversion because of bad weather or other issue preventing a landing at the destination must always be accounted for, which will require reserve fuel to be carried.

Typical phases of flight for a civil aircraft such as a commercial airliner. A contingency plan for a possible diversion must be accounted for, which requires reserve fuel.

A military aircraft may spend much more flight time conducting climbs, turns, and maneuvers, and the cruise segment may be much shorter. It may also involve an out-and-return mission profile, i.e., fly to a destination to conduct some operation and then return to base without landing, as shown below. Nevertheless, any airplane’s steady level flight performance is essential in determining its flight range at normal cruising speeds, ceilings, and airspeeds to attain maximum range and/or endurance.

Typical phases of flight for a military aircraft. Military aircraft, more often than not, perform out-and-return or “round-robin” missions.

The analysis pursued to quantify level flight performance depends on the type of airplane, i.e., jet-propelled or propeller-driven airplanes. For this reason, engines have previously been classified into two main groups; one is propeller-driven engines (i.e., piston-prop and turboprop), and the other one is jet engines (i.e., turbojet and turbofan).

The primary consideration in the analysis of airplane flight performance is that the output of a jet engine is quantified in terms of its thrust production. In contrast, the output of an engine driving a propeller is quantified in terms of its power output. However, it must also be recognized that propeller-driven engines convert the power into propulsive thrust by the propeller, which also has its own performance characteristics. For jet engines, power is required to produce the thrust by increasing the momentum of the flow through the engine in the form of a jet velocity.

Therefore, all engines are examined one way or another as power-producing devices, which is one reason why they are often called powerplants. Another reason is that they also generate electrical, hydraulic, and pneumatic power, besides thrust or propulsive power. Also, recognize that fuel is required to produce the power, so for airplane performance, the fuel flow, and hence the fuel burned during flight, is always at the core of the analysis

Learning Objectives

Be aware of an airplane’s general flight performance characteristics in straight and level flight.
Understand the basic differences in the performance analysis of jet-powered versus propeller-driven aircraft.
Review typical thrust required curves for jet-powered aircraft and power required curves for propeller-driven aircraft.
Appreciate the effects of weight and altitude on thrust required and/or power required performance curves and flight characteristics.

Performance Characteristics of an Airplane

The performance characteristics of an airplane depend on its aerodynamic drag, as well as the characteristics of the engines that power it. A large fraction of the total aircraft drag comes from the wing, which depends primarily on its angle of attack and operational Mach number. However, the airframe drag, and everything else other than the wing, is also a significant contributor to the total drag of the airplane.

Once the drag (or an estimate for drag) is determined, the thrust or power requirements for flight can be determined at any given aircraft weight and operational altitude (precisely the density altitude). Using the engine characteristics (thrust developed or power available), then many performance characteristics of the airplane can be calculated, such as its fuel burn, maximum and minimum attainable flight speeds, rates of climb, ceilings, range, endurance, etc.

Drag Model

The simplest drag model for an airplane is to represent its drag as an average (non-lifting) value combined with a value that varies with the square of the lift coefficient. The average value is independent of lift and, in the aggregate, accounts for the profile drag on all airframe surfaces, i.e., the sum of the skin friction and pressure drag over the wings, fuselage, empennage, etc. The other part of the total drag is called induced drag because it is the drag induced on the airplane from the creation of lift; the physics behind this component is tied to the trailing vortex system behind the airplane, as previously discussed.

The drag coefficient for the entire airplane can be expressed as

(1) $\begin{equation*} C_D = C_{D_{0}} + C_{D_{i}} = C_{D_{0}} + k {C_L}^2 \end{equation*}$

where $C_{D_{i}}$ is the induced drag coefficient and $k$ depends on the wing design. Theoretically, the induced drag coefficient can be expressed as

(2) $\begin{equation*} C_{D_{i}} = \frac{{C_L}^2}{\pi A\!R e} = k {C_L}^2 \end{equation*}$

where $A\!R$ is the aspect ratio of the wing and $e$ (the value of $e$ is < 1) is Oswald’s efficiency factor.

The validity of Eq. 1 has been established for many categories and classes of airplanes, an example being shown in the figure below for an airliner. While the fit is imperfect, it allows analytical results to be obtained and also those sufficient for at least preliminary estimates of airplane performance. Using a table look-up process for the values of $C_D$ at a given $C_L$ is an alternative approach, but this requires numerical methods.

The validity of the standard drag polar for an airplane has been well established by reference to measurements.

Other sources of drag can affect the polar, such as wave drag (from the creation of shock waves) and the effects of the stall. However, departures from Eq. 1 from the stall will only affect the airplane’s performance at higher lift coefficients and higher gross weights that may be considered outside of the normal operational flight envelope.

The growth of wave drag as the transonic flight regime is approached is nonlinear in terms of Mach number and angle of attack, so developing a suitable mathematical model for the drag is somewhat more complicated. One approach is to use the Lock model, as previously discussed. This model requires a value of the critical Mach number of the wing, $M_{\rm crit}$ , which is the flight Mach number (defined as $M_{\infty}$ ) at which the onset of supersonic flow first appears.

The wave drag increment can be written as

(3) $\begin{equation*} \Delta C_{D_{w}} = k_w \left( M_{\infty} - M_{\rm crit} \right)^4~\mbox{for $M_{\infty} \ge M_{\rm crit}$} \end{equation*}$

where $k_w$ is a constant. Therefore, including the wave drag increment, the modified drag equation for the airplane can be written as

(4) $\begin{equation*} C_D = C_{D_{0}} + k {C_L}^2 + k_w \left( M_{\infty} - M_{\rm crit} \right)^4 \end{equation*}$

While the Lock equation has been found to give reasonably good predictions of the extra wave, drag on an aircraft, and the consequent effects on the polar, quantitative predictions for a specific aircraft depend on the value of $k_w$ . For example, a value $k_w = 20$ is often used for an airliner designed to cruise in transonic flight where $0.7 \le M_{\rm crit} \le 0.8$ . For airplanes with thin, supersonic airfoils with higher critical Mach numbers, $k_w = 5$ is more appropriate.

Thrust & Power Required for Flight

The total (dimensional) drag on the airplane, $D$ , is then

(5) $\begin{equation*} D = \frac{1}{2} \varrho V_{\infty}^2 S \left( C_{D_{0}} + k {C_L}^2 \right) = T \end{equation*}$

a representative variation being shown in the figure below. The aircraft’s true airspeed through the air is denoted by $V_{\infty}$ . Notice that the drag must be equal to the thrust, $T$ , needed from the propulsive system when lift equals the in-flight weight, i.e., in balance flight or trim, then

(6) $\begin{equation*} L = W \quad \mbox{and}\quad T = D \end{equation*}$

Representative thrust required(to overcome drag) curve for an airplane at a given in-flight weight and altitude.

The corresponding power required for flight is then

(7) $\begin{equation*} P_{\rm req} = T V_{\infty} \end{equation*}$

as shown in the figure below, which again has a characteristic U-shaped curve. Notice that the power required for flight increases rapidly when the airplane flies fast.

Representative power required curve for an airplane at a given in-flight weight and altitude.

Calculation of In-Flight Weight

In the following sections and chapters, there will be frequent references to the in-flight weight of the aircraft. The weight of an aircraft at a given time in its flight is not fixed and it depends on the weight of fuel being burned off, which in turn depends on airspeed, altitude, thrust, throttle setting, and the engine characteristics such as the specific fuel consumption (SFC). For many aircraft, the weight of fuel to be carried for the flight is a significant fraction of the takeoff weight. Therefore, the weight of the fuel burned off must be known to determine accurately the in-flight weight. However, for many performance calculations, the in-flight weight at any point in the flight can be assumed or estimated based on the takeoff weight with a reasonable average estimate of the fuel burn, e.g., in units of weight per hour.

The aircraft’s gross weight on the ramp before taxiing to the runway will be the sum of its empty weight, $W_E$ , plus the useful load, $W_U$ . The aircraft’s empty weight comprises its structure, the engines, internal fixtures, oil(s), hydraulic fluids, etc., and everything else it needs to be ready fly but without loading any payload or fuel. The useful load is the sum of the payload, $W_P$ , and the fuel weight, $W_F$ . Payload is the weight the aircraft carries onboard that pays the bills, such as passengers and their baggage and cargo. But fuel is not payload. The takeoff weight must always be less than or equal to the maximum allowable (certified) gross weight of the aircraft, $W_{\rm \small MGTOW}$ . Therefore, the gross takeoff weight of the aircraft, $W_{\rm GTOW}$ , will be

(8) $\begin{equation*} W_{\rm GTOW} = W_E + W_F + W_P \leW_{\rm \small MGTOW} \end{equation*}$

Another way of expressing this latter sum is in terms of weight fraction, i.e., a component weight as a fraction of the gross weight. Therefore,

(9) $\begin{equation*} \frac{W_E}{W_{\rm GTOW}} + \frac{W_F}{W_{\rm GTOW}} + \frac{W_P}{W_{\rm GTOW}} = \phi_E + \phi_F + \phi_P \le 1 \end{equation*}$

where the $\phi$ values are called weight fractions.

If the average fuel burn rate for the aircraft can be established, i.e., an estimate for $dW_f/dt = \overbigdot{W}_{f}$ , then the in-flight weight, $W$ , after a given time $t$ since the takeoff can be obtained by integration, i.e.,

(10) $\begin{equation*} W =W_{\rm \small GTOW} - \int_{0}^{t} \overbigdot{W}_{f} \, dt \end{equation*}$

the minus sign indicating that weight is reducing as fuel is burned off.

However, for initial estimates of aircraft performance, including flight range and endurance, it is common to use an in-flight weight equal to the gross weight less half the fuel weight at the takeoff condition. A more refined calculation will use an estimate or measurement of the fuel burn during different phases of the flight, including taxi, takeoff, and climb, to better estimate the in-flight weight. Nevertheless, whatever aircraft weight is being used for the performance calculations, its value and how it is being calculated should always be qualified based on the best available information.

Jet-Propelled Airplane

Consider first the performance analysis of a jet-propelled airplane. For the steady level flight condition where $T = D$ , then

(11) $\begin{equation*} T = \frac{1}{2} \varrho V_{\infty}^2 S C_D = \frac{1}{2} \varrho V_{\infty}^2 S \left( C_{D_{0}} + k {C_L}^2 \right) \end{equation*}$

which assumes that there is no extra compressibility drag (i.e., wave drag) at high cruise Mach numbers. The equation to be solved is

(12) $\begin{equation*} T - \frac{1}{2} \varrho V_{\infty}^2 S \left( C_{D_{0}} + k {C_L}^2 \right) = 0 \end{equation*}$

Because in steady flight $L = W$ then

(13) $\begin{equation*} L = \frac{1}{2} \varrho V_{\infty}^2 S C_L = W \end{equation*}$

where $\varrho$ is the air density in which the aircraft is flying, $S$ is the reference wing area, and $C_L$ is the total wing lift coefficient (the assumption here is that the wings generate all of the lift). Notice that $\varrho = \varrho_0 \sigma$ where the value of $\sigma$ comes from the ISA model, i.e.,

(14) $\begin{equation*} L = W = \frac{1}{2} (\varrho_0 \sigma) V_{\infty}^2 S C_L \end{equation*}$

Rearranging this equation, the lift coefficient that needs to be produced on the wing for a given flight speed can be solved for, i.e.,

(15) $\begin{equation*} C_L = \frac{2 W}{\varrho_0 \sigma S V_{\infty}^2} \end{equation*}$

Therefore, after some algebra, the drag becomes

(16) $\begin{equation*} D = \frac{1}{2} \varrho_0 \sigma V_{\infty}^2 S C_{D_{0}} + \frac{2 k W^2}{\varrho_0 \sigma S V_{\infty}^2} \end{equation*}$

Notice that the first term in this latter equation (the profile or zero-lift drag) becomes dominant at higher airspeeds, and the second term (the induced drag) becomes larger at lower airspeeds; the resulting drag curve is U-shaped, as shown previously. Notice also the effects of weight on the induced drag, which is proportional to $W^2$ . Therefore, the equation to be solved is

(17) $\begin{equation*} T - \frac{1}{2} \varrho_0 \sigma V_{\infty}^2 S C_{D_{0}} + \frac{2 k W^2}{\varrho_0 \sigma S V_{\infty}^2} = 0 \end{equation*}$

This problem can be solved graphically (which is easy to visualize, as discussed below) or numerically.

Thrust Available

Now consider the engine thrust. The maximum thrust from a jet engine is available only at sea level. It will decrease with a decrease in air density, i.e., for an increase in altitude, the density decreases, and the thrust available declines for a given throttle setting. Throttle settings are generally specified as takeoff, maximum continuous, cruise, and idle, but military aircraft may have an afterburner thrust selection too.

Clearly, the thrust available (the output of the jet engine) depends on many things, but specifically the airspeed, the density of the air in which the airplane is flying (i.e., the density altitude), and the throttle setting ( $\delta_{T_A}$ ). Therefore, the available thrust from the engine can be written the general form

(18) $\begin{equation*} T_A = f(V_{\infty}, \sigma, \delta_{T_A}) \quad \text{or} \quad T_A = f(M_{\infty}, \sigma, \delta_T) \end{equation*}$

Consequently, for a given density altitude, airplane weight, and engine throttle setting, both the thrust available (from the engine) and the thrust required (equal to the aircraft drag) becomes a function of airspeed and/or Mach number.

Thrust Required

To achieve level flight, then the horizontal equilibrium equation must be satisfied (Eq. 12 or Eq. 17) for a given weight and altitude. To solve the equation, both the thrust available from the engine and also the thrust required (i.e., the drag of the aircraft) must be plotted versus the airspeed (or Mach number) and then determine the precise conditions where the curves coincide, an example being shown in the figure below. At that point, Eq. 17 will be satisfied, and both the airspeed and the thrust required for a given weight and altitude can be determined.

Representative variations in drag and thrust available versus airspeed for a jet-powered airplane.

Notice that the matching thrust available and the required curve can potentially intersect at two points. The intersection at the highest airspeed will correspond to the maximum level flight airspeed for that aircraft at that particular density altitude, aircraft weight, and throttle setting. The possibility of an intersection at the lower airspeeds will be the minimum possible airspeed the aircraft can maintain level flight at that altitude, and is called the thrust-limited minimum airspeed; in some cases this airspeed may be higher than the stall airspeed. In general, an airplane’s safe minimum flyable airspeed at a given altitude will be greater than the thrust-limited minimum airspeed or the stall airspeed. However, in the clean configuration (i.e., no flaps, gear up), the stall airspeed is usually the higher of the two airspeeds.

The thrust available and thrust required curves versus airspeed can be plotted for each altitude and weight of interest. Of course, the exact quantitative relationships between power and airspeed depend on the detailed aerodynamics of the actual airplane as well as the characteristics of the engines, which, as previously mentioned, may not be available other than in numerical form, e.g., tables. As altitude increases and the air density decreases, the maximum thrust available at a given throttle setting decreases, as shown in the figure below. These intersection points then represent the maximum and minimum airspeeds that the aircraft can fly at each altitude for the given weight and throttle setting and so determine the airspeed flight boundary. Notice that at higher altitudes, the achievable lowest airspeed becomes thrust-limited, while at lower altitudes, the achievable speed is generally determined by the onset of stall.

Representative variations in thrust required versus airspeed for a jet-powered airplane at different operating flight altitudes.

There will eventually be some altitude at which the available thrust approaches the minimum drag, where the available thrust curve will touch the required thrust curve at a single point. This condition corresponds (more or less) to the aircraft’s achievable maximum altitude or ceiling. Notice from the figure below that changing the weight of the aircraft significantly affects the thrust curves because the lift coefficient increases at lower airspeeds, then the induced drag increases. As shown in the figure below, increasing flight weight tends to shift the thrust required curve up and slightly to the right.

Representative variations in thrust required versus airspeed for a jet-powered airplane at different operating flight weights.

The effect of configuration also affects the thrust required for flight, as shown in the figure below. The clean configuration is the normal flight condition with flaps, slats (if any), and the landing gear retracted. Lowering the landing gear increases drag, and the deployment of the flaps increases this drag further, which is referred to as the dirty configuration. Notice in the figure that the flight envelope (or corridor) is increasingly constrained between the thrust available from the engines and the stall speed of the aircraft. In the dirty configuration with landing gear down and full flaps, then the thrust-limiting airspeed is reached before the onset of stall. In this case, such narrow allowable airspeed corridors require that the aircraft be flown very precisely, such as during landing.

Representative power required curve for an airplane in the “clean” condition of flight versus the “dirty” configuration with flaps and landing gear down.

While these forgoing graphical results are fairly easy to see, the problem can also be solved analytically under some conditions. Take as a further example a jet-powered airplane where it can be assumed that the propulsive thrust is not substantially dependent on the airspeed for a given altitude, i.e., where it is reasonable to assume that $T =$ constant for a given altitude. The airspeed of the airplane can be found from

(19) $\begin{equation*} T - \frac{1}{2} \varrho S \left( C_{D_{0}} + k {C_L}^2 \right) V_{\infty}^2 = 0 \end{equation*}$

After some algebra and rearrangement of terms, the airspeed for thrust and drag equilibrium can be obtained by solving the quartic equation

(20) $\begin{equation*} c_1 V_{\infty}^4 + c_2 V_{\infty}^2 + c_3 = 0 \end{equation*}$

where

(21) $\begin{equation*} c_1 = -\frac{1}{2} \varrho_0 \sigma \frac{S}{W} C_{D_{0}}, \quad c_2 = \frac{T}{W}, \quad c_3 = -k \frac{2}{\varrho_0 \sigma} \frac{W}{S} \end{equation*}$

Notice that these coefficients contain both the thrust to weight ratio and the wing loading. While there are multiple roots to Eq. 20 they are not all physical, and only two roots will have physical significance. It can be shown that the jet airplane can maintain level flight at a given altitude if

(22) $\begin{equation*} \frac{T}{W} \ge 2 \sqrt{C_{D_{0}} k} \end{equation*}$

which is naturally and intrinsically tied to the aerodynamic characteristics of the aircraft. The limiting condition of this latter result occurs at the ceiling of the aircraft when the minimum and maximum attainable airspeeds coincide, which is

(23) $\begin{equation*} V_{\infty} = -\frac{c_2}{2 c_1} = \frac{T}{\varrho_0 \sigma S C_{D_{0}}} \end{equation*}$

However, this value can only be determined explicitly if the engine thrust is known, which, as previously discussed, is a function of density altitude and throttle setting.

In most cases, the preceding type of problem must be solved numerically or graphically. The procedures described would apply to any drag or thrust variations. The propulsion characteristics of engines are often made available for engineering analysis in graphs or tables to calculate the thrust at each airspeed and altitude. Similar procedures are used to develop a more detailed model of the drag on the aircraft, including the effects of wave drag. For high-performance aircraft, such results are usually determined using a combination of calculations, wind tunnel tests, as well as flight tests and are made available as tables as functions of the angle of attack and Mach number.

Propeller-Driven Airplane

Consider now a propeller-driven airplane. Remember that the output of an engine driving a propeller (e.g., a turboshaft or a piston engine) is quantified in terms of its power. However, it must also be recognized that this power is converted into thrust according to the aerodynamic characteristics and performance of the propeller. There may be some jet thrust from a turboshaft, but usually, this is small enough to be ignored.

The power required for flight can be written as

(24) $\begin{equation*} P_{\rm req} = \frac{T V_{\infty}}{\eta_p} = \frac{1}{\eta_p} \left( \frac{1}{2} \varrho V_{\infty}^3 S \left( C_{D_{0}} + k {C_L}^2 \right) \right) \end{equation*}$

where $\eta_p$ can be considered as the net propulsive efficiency of the propulsive system (engine and propeller combined); notice that this value may not be a constant and will generally vary with airspeed depending on the type of propeller system.

Splitting the foregoing equation (Eq. 24) into its two parts leads to

(25) $\begin{equation*} P_{\rm req} = \frac{1}{\eta_p} \left( \frac{1}{2} \varrho V_{\infty}^3 S C_{D_{0}} \right) + \frac{k}{\eta_p} \left( \frac{1}{2} \varrho V_{\infty}^3 S {C_L}^2 \right) \end{equation*}$

the first term in the foregoing equation being the non-lifting part and the second term being the induced part. Notice that the power associated with the non-lifting part increases with the cube of the airspeed but the second (induced) part depends on the lift coefficient, which as has previously been shown, reduces with increasing airspeed.

Using Eq. 15 gives

(26) $\begin{equation*} {C_L}^2 = \left(\frac{2 W}{\varrho_0 \sigma S V_{\infty}^2}\right)^2 \end{equation*}$

so the power required equation now becomes

(27) $\begin{equation*} P_{\rm req} = \left(\frac{1}{2} \varrho_0 \sigma V_{\infty}^3 S \right) \frac{1}{\eta_p} C_{D_{0}} + \frac{1}{2} \varrho_0 \sigma V_{\infty}^3 S \left( \frac{k}{\eta_p} \right) \left(\frac{2 W}{\varrho_0 \sigma S V_{\infty}^2}\right)^2 \end{equation*}$

which after some simplification leads to

(28) $\begin{equation*} P_{\rm req} = \left( \frac{1}{2} \varrho_0 \sigma V_{\infty}^3 S \right) \frac{1}{\eta_p} C_{D_{0}} + \left( \frac{k}{\eta_p} \right) \frac{2 W^2}{\varrho_0 \sigma S V_{\infty} } \end{equation*}$

This latter equation is simply in the form

(29) $\begin{equation*} P_{\rm req} = A V_{\infty}^3 + \frac{B}{V_{\infty}} \end{equation*}$

where

(30) $\begin{equation*} A = \frac{1}{2} \varrho_0 \sigma \frac{1}{\eta_p} C_{D_{0}} \end{equation*}$

and

(31) $\begin{equation*} B = \frac{k}{\eta_p} \frac{2 W^2}{\varrho_0 \sigma S} \end{equation*}$

Notice that the non-lifting part increases with the cube of the airspeed but the lifting part decreases inversely with airspeed.

Power Available

The power available for flight as a characteristic of the powerplant, i.e., all the power that could be delivered from the engine to drive the propeller and propel the airplane forward. If the shaft power available from the engine is $P_{\rm bp}$ (this is called its brake power), then the power available for flight $P_A$ will be

(32) $\begin{equation*} P_A = \eta_p P_{\rm bp} \end{equation*}$

This result shows that the available power for flight is reduced by the propeller efficiency $\eta_p$ , i.e., not all of the power at the shaft of the engine can be delivered as useful work to the air by the propeller.

The actual power required for flight depends on the drag of the airplane so

(33) $\begin{equation*} P_{\rm req} = D V_{\infty} \end{equation*}$

For steady level flight at a constant airspeed and altitude, the pilot needs to set the throttle so that the power required for flight is equal to the power available, i.e.,

(34) $\begin{equation*} P_{\rm req} = D V_{\infty} = P_A = \eta_p P_{\rm bp} \end{equation*}$

so that the brake power needed from the engine will be

(35) $\begin{equation*} P_{\rm bp} = \frac{P_A}{\eta_p} = \frac{D V_{\infty}}{\eta_p} \end{equation*}$

In performance analyses, the brake (shaft) power matters because it is the power that can be delivered at the shaft that ultimately affects the engine’s fuel consumption. Obviously, the higher the propeller efficiency then, the lower the brake power required, and so the lower the fuel consumption will be.

Naturally, the power available (from the powerplant) may be greater or less than the power required for flight. For example, any excess power available over and above required will allow the airplane to accelerate to a higher airspeed and/or climb to a higher altitude. For this reason, the airplane’s initial takeoff and climb performance is strongly affected by the power available from the powerplant.

Power Required

Representative power required curves for a propeller-driven airplane is shown in the figure below. It can be seen that higher power is required at lower airspeeds, a minimum range of power at some intermediate airspeeds, and then a rapid increase in power is required (with the cube of the airspeed) as higher airspeeds are reached. Aircraft that use propellers will have performance charts given in terms of airspeed as well as flight Mach number, the helical tip Mach number of the propeller being critical. The lowest possible flight airspeed is generally limited by the onset of wing stall and/or buffeting from the onset of flow separation (often causing the aircraft to shake), no matter how much power is available. At higher weights and/or density altitudes, the rapid increase in power required will eventually limit the maximum level flight airspeed of the airplane for a given weight and operational altitude, assuming no other limits or barriers to flight appear.

The power available typically increases with airspeed and levels off over the airspeed range where the airplane would generally fly. Power available will lapse with altitude, as shown in the figure below. Again, analogous to the manner for the jet aircraft, the level flight solution is the intersection of the available power curves, and the power required curves. The highest airspeed solution is the maximum level flight airspeed (at a given altitude, aircraft weight, and throttle setting). However, the minimum speed solution is only valid if that speed is greater than the aircraft’s stall speed. Again, the aircraft’s ceiling can be determined at the airspeed when the minimum power for flight coincides with the power available.

Power required for a propeller-driven airplane at different flight altitudes as a function of its airspeed.

By assuming that $k$ and the propulsive efficiency $\eta_p$ remain constant for all weights and airspeeds then the airspeeds can be solved for in closed form. This is a special case, admittedly, but a reasonable assumption for a constant speed propeller with the engine operating at wide-open throttle. After some algebra, the needed equation to be solved is

(36) $\begin{equation*} c_1 V_{\infty}^3 + \frac{c_2}{V_{\infty}} + c_3 = 0 \end{equation*}$

where

(37) $\begin{equation*} c_1 = -\frac{1}{2} \frac{\varrho_0 \sigma S C_{D_{0}}}{W}, \quad c_2 = -2 k \left( \frac{W}{\varrho_0 \sigma S} \right), \quad c_3 = \frac{\eta_p P_{\rm req}}{W} \end{equation*}$

from which the maximum and minimum speeds can, in principle, be solved for. However, recognize that this is a more difficult problem to solve because the relevant equation is nonlinear.

The effect of configuration, i.e., clean or dirty with the flaps up or down, landing gear up or down, etc., also affects the power required, as shown in the figure below. Just like jet-powered aircraft, notice that the flight envelope is increasingly constrained between the power available and the stall speed. In the dirty configuration with landing gear down and full flaps, such as for landing, then it is possible that a power limiting airspeed can be reached before the onset of stall. In this regard, the pilot must ensure that the airplane carries enough excess airspeed so that in the event of a go-around, there is enough excess power available to climb away from the runway.

Avian Flight

The natural evolution of birds over 100s of millions of years has resulted in sophisticated adaptations to their aerial environment. These adaptations are shaped by natural selection, which favors natural fliers with specific biological characteristics that enhance their flight performance. A question is, therefore, whether the performance avian fliers behave in the same or even a similar manner to airplanes.

Even a casual observation shows that birds exhibit a wide range of wing shapes, each suited to their specific flight requirements. For example, birds that are adapted for long-distance soaring, such as hawks, eagles, pelicans, and albatrosses, have long, high aspect ratio wings that allow them to glide effortlessly. In contrast, birds that are adapted for hovering, such as hummingbirds, have short, low aspect ratio wings with low inertia that enable them to hover in place and maneuver with great agility. These avian wing forms are not random, but have been shaped by evolution and constrained by the physical laws that govern aerodynamics.

The complex nature of the reciprocating flapping wings of birds poses additional challenges in understanding their flight performance characteristics. Unlike fixed-wing aircraft, birds must generate propulsion by flapping their wings, which requires specific adaptations in the muscular and skeletal system to produce aerodynamic forces. The upshot is continuously changing flight parameters during each wingbeat such as wing area, wingspan, aspect ratio, and angle of attack. Wing flapping motion also creates unsteady flows around their wings and potentially makes estimates of flight forces and power requirements more complicated compared to the steady level flight of an airplane.

However, the same fundamental principles of aerodynamics that apply airplanes should also apply to the wings of birds. For example, the shape of a bird’s wing generates lift by creating differences in air pressure between the upper and lower surfaces of the wing, just like an airplane wing generates lift. The effects of the wing tip vortices must cause induced drag, as for an airplane’s wing. To this end, biologists have made measurements in the wind tunnel of the flapping power birds require for flight, which requires measurements of metabolic rates or using other methods. Birds can be trained to fly in the wind tunnel, although it would seem that the measurement approaches used so far have somewhat high experimental uncertainty. Nevertheless, the results in the figure below show that the characteristic U-shaped power curve for an airplane is obtained, the induced drag dominating the power for flight at low flight speeds.

The U-shaped power required for flight curves for avian fliers look remarkably similar to those of airplanes, as might be expected if the same laws of aerodynamics apply.

The unique characteristics of flapping wings also provide birds with notable advantages in flight compared to airplanes. The ability of birds to adapt their wing shapes allows birds to quickly change flight situations. Birds can also adaptively twist their wings to change the angle of attack and lift distribution, enabling them to perform agile maneuvers, the hummingbird being an exquisite example.

In recent years, aeronautical engineers have taken inspiration from these natural fliers to explore new ideas, such as for micro air vehicles that can mimic the wing beat movements. Indeed, the evolution of flight in birds is an excellent example of how biology can inspire other engineering solutions. By understanding the physical laws that govern flight and studying the forms and structures of natural flyers, engineers may be able to gain some insight into designing more efficient and maneuverable aircraft. However, it should also be remembered that avian fliers have developed under evolutionary constraints for which engineers are not so encumbered.

Summary & Closure

The analysis used to quantify level flight performance depends on the type of airplane, i.e., jet-propelled or propeller-driven airplanes. The main difference is that the output of a jet engine is quantified in terms of its thrust production, while the output of an engine driving a propeller is quantified in terms of its power output. A large fraction of the total aircraft drag comes from the wing, which depends on its angle of attack and flight Mach number. However, the airframe drag and everything else other than the wing is also a significant contributor to the total drag of the airplane. Once the drag (or an estimate for drag) is determined, then the thrust or power requirements for flight can be determined at any given aircraft weight and operational density altitude. Using the engine characteristics (thrust available or power available as well as specific fuel consumption), then many performance characteristics of an airplane can now be calculated,

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Consider the addition of a term to the drag polar to account for the onset of wave drag. How will this change the thrust required curve?
If the aerodynamics characteristics of an airplane are available only in table format (i.e., tables of lift and drag), think about how these results can be incorporated into an analysis to find the aircraft’s performance characteristics.
Under what conditions might the minimum level flight airspeed of an airplane be higher than its stall speed?
Why does the thrust available from a turboprop system decrease quickly at higher airspeeds?

Other Useful Online Resources

To learn more about the level flight performance characteristics, follow up with some of these more practical online resources:

An explanation of the power curve and an in-flight example – check out this video.
Another video explaining the power curve of an airplane.
A good ERAU video from the flight school explaining the drag curves of an airplane.

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n35