{"id":485,"date":"2021-09-15T12:24:35","date_gmt":"2021-09-15T12:24:35","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/chapter\/level-flight-performance-envelopes\/"},"modified":"2022-07-18T15:13:00","modified_gmt":"2022-07-18T15:13:00","slug":"level-flight-performance-envelopes","status":"web-only","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/chapter\/level-flight-performance-envelopes\/","title":{"raw":"Level-Flight Performance Envelopes","rendered":"Level-Flight Performance Envelopes"},"content":{"raw":"<div>\n<p style=\"text-align: left\">[latexpage]<\/p>\n\n<h1>Introduction to this Lesson<\/h1>\n<\/div>\nAll aircraft have operational limits in terms of the maximum and minimum airspeeds as\u00a0well as altitudes at which they can fly level in steady, unaccelerated flight, e.g., the\u00a0airspeed versus altitude boundary such as shown in the figure below. Notice\u00a0that, by design, jet fighter aircraft can fly faster and higher over a wider range of flight conditions\u00a0than other airplanes but remember that they also have a broader type of mission. In\u00a0comparison, commercial jet airplanes are very much <em>point designs<\/em> in that they a\u00a0designed primarily to cruise for long periods at a specific airspeed (or Mach number) and altitude. Turboprops are often used for short-haul flights, and while they fly at lower altitudes and airspeeds they are better suited for operating out of shorter runways.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"469\"]<img class=\"wp-image-468 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope.jpg\" alt=\"\" width=\"469\" height=\"408\"> Representative flight envelopes of different types of airplanes in terms of their\u00a0achievable altitudes and airspeeds.[\/caption]\n\n<div>\n\nThe area inside the boundaries that limit normal flight is called the airplane's\u00a0<em>operational flight envelope<\/em>. The limits of the envelope are defined and set based on\u00a0aerodynamics (such as the highest achievable Mach number), the engine power (e.g., a turboprop or piston engine) or\u00a0thrust (i.e., a turbojet or turbofan) that is available, the onset of maximum structural loads, or even something else\u00a0such as the onset of flutter or buffeting. Limits could also be set by excessive\u00a0aerodynamic heating for a supersonic aircraft. The <em>flight corridor<\/em> is often referred\u00a0to as the speed range or band over which the airplane can fly at any given altitude\u00a0without encountering any of the flight limits.\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Objectives of this Lesson<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div>\n<ul>\n \t<li>Understand the meaning of an airplane's flight envelope and a flight corridor.<\/li>\n \t<li><span style=\"font-size: 1rem\">Know about the various factors that may limit the operational flight envelope\u00a0of an airplane, including stall.<\/span><\/li>\n \t<li><span style=\"font-size: 1rem\">Have a general understanding of the phenomenon of wave drag and why it can also limit the flight envelope.<\/span><\/li>\n \t<li><span style=\"font-size: 1rem\">Understand the\u00a0principles associated with drag reduction using supercritical wing design and the area\u00a0rule.<\/span><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<h1>General Comments on the Flight Envelope<\/h1>\n<\/div>\n<div>\n\nThe size and shape of the flight envelope (or flight corridor) will depend on the type of airplane, i.e., whether it is propeller-driven or jet-powered, has an unpressurized or pressurized fuselage, and\/or whether it\u00a0is specifically designed for subsonic or supersonic flight. Naturally, the exact size and\u00a0shape of the envelope for any given airplane also depends on the properties of the\u00a0atmosphere, particularly the density and temperature of the air.\u00a0Generally, the lowest attainable airspeed of an airplane (either jet-powered or propeller-driven) is dictated by the onset of wing stall, which determines the left side boundary on\u00a0the flight envelope. This stalling airspeed will be a function of the airplane's weight and\u00a0altitude, as well as the wing flap settings and sometimes also if the undercarriage is up or down.\n\nThe right side of the boundary will be set by the highest attainable airspeed, which is\u00a0usually limited by the power available (for propeller-driven airplanes) or thrust available\u00a0(for jet engines) to overcome the drag of the airplane, the drag being a function of the\u00a0shape of the airplane as well as its flight Mach number.\u00a0The upper edge of the flight envelope is the maximum attainable altitude, which is\u00a0referred to as the <em>operational ceiling<\/em>. The ceiling is the altitude above which an aircraft cannot\u00a0climb, which is usually defined based on a threshold of a diminishing rate of climb of 100 ft\/min. The\u00a0attainable flight ceiling depends on the excess power available relative to the aircraft's\u00a0aerodynamic and other characteristics, including its weight.\u00a0In some cases, however, such as on most commercial airplanes, the flight ceiling is\u00a0limited by the onset of wave drag or transonic buffet, or by the airplane reaching some\u00a0maximum structural loads associated with the pressurization of the fuselage (which is a\u00a0trade with airframe weight), even though the airplane may have the excess power\u00a0available to achieve higher flight altitudes.\n<h1>Trimmed Lift Coefficient<\/h1>\nAn airplane is said to be in steady level unaccelerated flight\u00a0when the three forces (lift, drag and side force) and the three corresponding moments\u00a0(pitching, rolling, and yawing) on the airplane are perfectly balanced, in which case the\u00a0airplane is said to be in <em>trim<\/em>, as shown in the figure below. The balance of forces in\u00a0steady trim is that vertical equilibrium requires that lift = weight and horizontal\u00a0equilibrium requires that thrust = drag, i.e.,\n\\begin{equation}\nL = W, \\quad \\quad T = D\n\\end{equation}\nthe side force is naturally assumed to be zero in trimmed flight. One other assumption here is that the\u00a0thrust vector's line of action is (primarily) in the flight direction. Of course, full flight trim\u00a0also requires that the airplane have a moment balance in pitch, roll, and yaw about the\u00a0center of gravity.\n\n[caption id=\"attachment_469\" align=\"aligncenter\" width=\"433\"]<img class=\"wp-image-469\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight.jpg\" alt=\"\" width=\"433\" height=\"280\"> In the level flight trim condition the forces and moments on the airplane will be\u00a0in perfect balance.[\/caption]\n\nRemember that the wings generate the lift to overcome the weight and the engines provide the propulsive force to overcome the drag of the airplane, the generation of this thrust requiring a source of power and fuel. In terms of basic aerodynamics, for vertical equilibrium then\n\n<\/div>\n<div>\\begin{equation}\nL = \\frac{1}{2} \\rho V_{\\infty}^2 S C_L = W\n\\end{equation}\nwhere $\\rho$ is the density of the air in which the airplane is flying, $S$ is the reference wing area and $C_L$ is the total wing lift coefficient (the assumption here being that the wings generate all lift). Notice that $\\rho = \\rho_0 \\sigma$ where $\\sigma$ comes from the ISA model, i.e.,\n\\begin{equation}\nL = W = \\frac{1}{2} \\rho_0 \\sigma V_{\\infty}^2 S C_L\n\\end{equation}\nRearranging this equation allows us to solve for the lift coefficient that needs to be\u00a0produced on the wing for a given flight speed, i.e.,\n\\begin{equation}\nC_L = \\frac{2 W}{\\rho_0 \\sigma S V_{\\infty}^2}\n\\label{CL eqn}\n\\end{equation}\nor the flight speed that corresponds to a given lift coefficient, i.e.,\n\\begin{equation}\nV_{\\infty} = \\sqrt{ \\frac{2}{\\rho_0 \\sigma} \\left( \\frac{W}{S} \\right) \\frac{1}{C_L} }\n\\end{equation}\nRecall that the ratio of airplane's weight to its lifting wing area, $W\/S$, is called the\u00a0wing loading. Notice that the lift coefficient is proportional to weight (or to wing loading)\u00a0but decreases with the square of the airspeed. The lift coefficient also increases with\u00a0altitude for a given true airspeed and weight.<\/div>\n<h1>Stalling Airspeeds<\/h1>\n<div>The minimum airspeed that would allow level flight of the airplane is called the <em>stall speed<\/em> or the\u00a0<em>stalling\u00a0speed<\/em>, which is the airspeed corresponding to the angle of attack and lift coefficient at which the wing will stall. This value is called the maximum lift coefficient $C_{L_{\\rm max}}$ and it depends on the nature of the wing used on the airplane, including its planform, its twist, and airfoil section. The actual value of $C_{L_{\\rm max}}$ also depends on whether the flaps and other high lift devices such as slats are retracted or deployed, as shown in the figure below. Notice that flaps are very effective in increasing $C_{L_{\\rm max}$ but the use of a slat can boost the $C_{L_{\\rm max}}$ by a further 50\\%.<\/div>\n<div>\n\n[caption id=\"attachment_470\" align=\"aligncenter\" width=\"420\"]<img class=\"wp-image-470 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax.jpg\" alt=\"\" width=\"420\" height=\"324\"> The maximum attainable lift coefficient depends on whether the flaps or slats are retracted or deployed. If the flaps and slats are retracted, then the wing is said to be in its \"clean\" configuration.[\/caption]\n\nAlthough the value of $C_{L_{\\rm max}}$ may not be exactly known by calculation, it\u00a0can be determined indirectly from flight tests with the airplane from measurements of\u00a0true airspeed and density altitude. After the $C_{L_{\\rm max}}$ for the wing is\u00a0determined, the stall speed in steady level flight can be solved for at any weight and\u00a0density altitude. i.e.,\n\\begin{equation}\nV_{\\rm stall} = \\sqrt{ \\frac{2}{\\rho_0 \\sigma} \\left( \\frac{W}{S} \\right) \\frac{1}{C_{L_{\\rm\u00a0max}} } }\n\\label{stallspeed1}\n\\end{equation}\nusing the value of $\\sigma$ from the ISA model, i.e., based on the prevailing pressure altitude and outside air temperature.\n\nNotice that for a given $C_{L_{\\rm max}}$, the stalling speed depends on the wing loading, i.e., all things being equal an airplane with a higher wing loading will stall at a higher airspeed. If a linear lift curve slope of the wing is assumed, say $C_{L_{\\alpha}}$, then the angle of attack of the wing $\\alpha$ (measured relative to the zero-lift angle) can be calculated using\n\\begin{equation}\n\\alpha = \\frac{2 W}{\\rho_0 \\sigma S C_{L_{\\alpha}} V_{\\infty}^2}\n\\end{equation}\nthe maximum value of $\\alpha$ typically being less than 15$^{\\circ}$ at low Mach numbers and lower than that at higher Mach numbers. However, it is important to appreciate that a wing will stall at any airspeed if the angle of attack is high enough. For this reason, caution must be used when referring to stall speeds.\n\n<\/div>\n<div><\/div>\n<div>In summary, four conclusions can be drawn from the use of Eq.~\\ref{stallspeed1}:<\/div>\n<ol>\n \t<li>Stall speed will increase with increasing weight of the airplane.<\/li>\n \t<li>Stall speed will\u00a0increase with increasing density altitude, i.e., with a lowering of the air density.<\/li>\n \t<li>Stall\u00a0speed will decrease with increasing values of wing $C_{L_{\\rm max}}$, which, as\u00a0previously discussed, can be achieved by the application of wing flaps and\/or leading-edge\u00a0slats.<\/li>\n \t<li><span style=\"text-align: justify;font-size: 1em\">Stall speed will decrease with increasing wing area, an increase in wing area\u00a0also being possible with the use of certain types of flaps, such as Fowler flaps.<\/span><\/li>\n<\/ol>\n<h1>Limiting Cruise Speeds<\/h1>\nThe figure below shows a historical trend as to how the cruise airspeeds for commercial transport airplanes have increased over the\u00a0decades, which naturally is a direct consequence of the rapid advancements and\u00a0maturation of aeronautical technology. Of course, the introduction of the jet engine was\u00a0responsible for the more rapid growth in achievable cruise speeds seen after 1960.\u00a0However, it can also be seen that since the early 1970s, the cruise airspeeds for\u00a0commercial airplanes have all but plateaued, with corresponding achievable cruise\u00a0flight Mach numbers in the range 0.8 to 0.85. There are a couple of exceptions to this\u00a0trend with the British Concorde and the Russian Tu-144, but these airplanes were\u00a0specifically designed for cruising at supersonic Mach numbers. While the use of supercritical wing designs has extended the flight envelope of airliners to higher transonic Mach numbers, the eventual onset of wave drag and buffeting still remains a physical barrier to higher flight conditions.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"538\"]<img class=\"wp-image-471 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear.jpg\" alt=\"\" width=\"538\" height=\"431\"> The cruise airspeeds of commercial transports show that rapid increases\u00a0occurred with the maturation of aeronautical technology, but since 1970 have all but\u00a0reached a plateau.[\/caption]\n<h1>Supercritical Flows &amp; Drag Rise<\/h1>\nOne reason is that cruise speeds for commercial airliners have reached a plateau is\u00a0because of the buildup of high drag on a wing as transonic flow conditions are\u00a0approached, the basic physics of what happens on the wing section being shown in the figure below. The drag buildup from the development of compressibility\u00a0and shock waves takes much thrust and power to overcome, and there are other issues too about\u00a0operating at higher flight Mach numbers such as buffeting, as has been discussed.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"622\"]<img class=\"wp-image-472 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow.jpg\" alt=\"\" width=\"622\" height=\"552\"> As the flight (free-stream) Mach number increases the flow about a wing\u00a0section develops supersonic flow and eventually a shock wave. This shock wave\u00a0strengthens and moves aft over the wing as the Mach number increases, eventually in\u00a0supersonic conditions forming shocks at the wing's leading and trailing edges.[\/caption]\n\nAt some flight (free-stream) Mach number, the local flow at a point on the wing's\u00a0surface reaches sonic conditions, which is called the <em>critical Mach number<\/em>. As the\u00a0free-stream Mach number increases further, a small pocket of supersonic flow\u00a0develops on the section, resulting in a weak shock wave in the flow. As the Mach number\u00a0further increases, the shock strengthens and moves aft over the section and a\u00a0supersonic region is formed. An associated shock wave also develops on the lower\u00a0surface. This is called the well-established transonic flow region, the shock waves\u00a0resulting in an energy loss that manifests as form of drag called <em>wave drag. <\/em>Wave drag causes the total drag on the wing to\u00a0increase rapidly, as shown in the figure below\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"473\"]<img class=\"wp-image-473 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise.jpg\" alt=\"\" width=\"473\" height=\"250\"> There is a large increase in drag and loss of lift on a wing as transonic flight\u00a0conditions develop and before supersonic flight is established.[\/caption]\n\nBecause steep adverse pressure gradients also accompany the shock waves that\u00a0develop during transonic conditions on the wing section, the boundary layer downstream of the shock becomes\u00a0thicker and the profile drag increases. If the shock wave becomes sufficiently strong, flow\u00a0separation may occur, leading to a buffeting aerodynamic phenomenon. Buffeting can\u00a0result in high levels of vibration being transmitted to the airframe, and it is not a viably\u00a0sustained flight condition. The onset of buffeting can also cause aeroelastic concerns, so this must be examined carefully through flight testing. The onset of buffeting is usually a limiting factor in the\u00a0operational flight envelope of most aircraft (unless they are designed for supersonic flight) and is referred to as the <em>buffet boundary<\/em>.\n\nIf and when Mach number approaches unity, the shocks move all the way to the trailing edge of the section. Finally, when the Mach number becomes greater than one, a bow wave appears just ahead of the section, and the shocks at the trailing edge become oblique. For supersonic airplanes, these strong shock waves are responsible for the pressure changes that are heard on the ground that manifest as the impulsive \"boom-boom\" sound known as the <em>sonic boom<\/em> as the airplane passes overhead at supersonic\u00a0speeds. \u00a0The drag rise on the aircraft during the transition from transonic to supersonic flight usually requires excess thrust to be produced using an afterburner. Some aircraft may be subsequently able to cruise supersonically without the use of the afterburner, but it depends on the engine.\n<h1>Reducing Compressibility Drag<\/h1>\nThe minimization of wave drag on the wings as the transonic flight regime is\u00a0approached is obviously key to lowering drag and\/or allowing the airplane to fly faster\u00a0and opening up the flight envelope before significant drag rise is encountered. Lower\u00a0drag means lower thrust and power is required for flight, so less fuel is expended and more flight\u00a0range can be achieved.\n\nWing sweep has a very profound effect on transonic and supersonic drag, as shown in the figure below. The use of swept-back wings reduces the strength of the\u00a0shock waves and prevents the shocks from interfering with the flow over the\u00a0wings and causing flow separation. However, although\u00a0swept wings can help delay this drag rise from compressibility effects, other\u00a0aerodynamic and aeroelastic problems are associated with swept wings, so aircraft designers ten to use as little wing sweep as possible, 20 to 30 degrees being used on may airliners.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"543\"]<img class=\"wp-image-474 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep.jpg\" alt=\"\" width=\"543\" height=\"374\"> The use of sweepback on a wing provides for a very significant reduction in its\u00a0drag.[\/caption]\n\nA visualization of the flow about swept and unswept wings at a low supersonic speed is shown below, which was obtained using the schlieren method. Notice that with the use of sweepback the shockwaves do not interact directly with the wing, which keeps the drag low. With the unswept wings not only are the shock waves stronger but they interact strongly with the wing driving up drag.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"650\"]<img class=\"wp-image-475 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback.jpg\" alt=\"\" width=\"650\" height=\"463\"> Schlieren flow visualization image of the shock wave patterns around two\u00a0airplane models showing the effects of sweptback at Mach 1.2. (NACA image.)[\/caption]\n\nThe figure below shows the difference in the shapes of a conventional airfoil and a supercritical airfoil. The basic principle used in transonic airfoil design is to\u00a0control the flow's expansion to supersonic speed and its subsequent recompression.\u00a0Compared to a conventional wing section, a supercritical wing section is distinctive in\u00a0that it is flatter along the top surface with significant camber at its trailing edge. variations of supercritical airfoil sections are used on all commercial jet airliners.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"565\"]<img class=\"wp-image-476 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical.jpg\" alt=\"\" width=\"565\" height=\"304\"> Supercritical airfoil designs have led to notable reductions in wave drag to\u00a0allow wings to cruise at higher flight Mach numbers.[\/caption]\n\nThe challenges in reaching higher transonic cruise speeds have led to the design of a\u00a0special shape of swept wing called a <em>supercritical wing<\/em>, as shown in the photograph below. The\u00a0supercritical wing evolved from the careful tailoring of the airfoil section(s) with the overall\u00a0wing design to delay the formation and\/or reduce the strength of the shock waves over\u00a0the wing so reducing wave drag. In the early 1970s, NASA modified an airplane to test a supercritical wing in\u00a0place of the conventional wing to reduce the effects of shock waves and wave drag,\u00a0with great success, and aircraft designers have never looked back.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"615\"]<img class=\"wp-image-2010 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalwing-scaled-1.jpg\" alt=\"\" width=\"615\" height=\"304\"> Supercritical airfoils and smoothly varying wing shapes are now\u00a0standard on virtually every modern subsonic commercial transport airplanes.[\/caption]\n<h1>Area Rule<\/h1>\nOther ways of reducing wave drag expanding the flight envelope of the airplane to\u00a0higher cruise speeds include the use of the area rule, which was developed by Richard Whitcomb. To reduce the number and\u00a0intensity of shock waves over an airplane as it approaches transonic and then\u00a0supersonic flight, the basic design principle behind the area rule is that the airplane's\u00a0overall cross-sectional shape should change smoothly with no significant\u00a0discontinuities.\n\nThe principle was proven to work in wind tunnel testing and then applied retroactively to various airplanes, with successful results after flight testing. Early airplanes that were modified to validate the area rule had distinctive if not odd looking \"waisted\" fuselage shapes at the wing root, as shown in the figure below, which were often dubbed as \"flying coke bottles.\" Nevertheless, the notable reductions in drag proved the viability of the area rule concept.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"550\"]<img class=\"wp-image-478 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule.jpg\" alt=\"\" width=\"550\" height=\"379\"> The principle of the \"area rule\" is to maintain a smooth variation in the net cross-sectional area of the airplanes to reduce the compressibility effects and wave drag. Wind tunnel and flight testing have both confirmed the benefits.[\/caption]\n\nLater airplanes were designed with the area rule in mind but were aesthetically more pleasing because of the blending of the wing root area and the careful positioning of engines, the use of large trailing edge anti-shock wing pods or \"canoe\" fairings, and other subtle changes to the shape of the airplane to prevent large changes in effective cross-sectional area. For many commercial airplanes, the wing-mounted \"pod\" engines are placed relatively far forward of the wings to control the change in cross-sectional area of the airplane as the wing is approached.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"607\"]<img class=\"wp-image-479 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization.jpg\" alt=\"\" width=\"607\" height=\"332\"> Surface oil flow visualization showing the signatures of strong shock-induced flow separation on a wing (left) and the weaker shocks and reduced flow separation from the use of \"anti-shock\" extensions at the trailing edge of the wing. (NASA image.)[\/caption]\n\nA careful examination of the most commercial airliners will show some careful contouring of\u00a0the fuselage and wing root to help minimize wave drag according to the principles\u00a0established by the area rule. For the same reason, later versions of the Boeing 747 were also modified with\u00a0a longer upper deck and a shallower transition at its end to keep area changes as progressive as possible. Most airplanes capable of\u00a0transonic or supersonic airspeeds incorporate design features that can be traced back\u00a0to the fundamental principles underlying Whitcomb's area rule.\n<h1>Flight Ceilings<\/h1>\nThe flight ceiling for an airplane is defined based on a demonstrated rate of climb. The\u00a0absolute ceiling is defined when the achievable rate of climb diminishes to zero,\u00a0whereas the service ceiling is defined such that the rate of climb reduces below\u00a0100 ft\/min. The airplane's normal performance ceiling is defined as when the rate of\u00a0climb reduces below 200 ft\/min. The ceiling is reached when the excess power\u00a0available over and above that for level flight at the same airspeed and weight becomes\u00a0diminishingly small.\n\nThe ceiling for most commercial transport airplanes is limited by cabin pressurization\u00a0requirements rather than attainable engine thrust and power, which set a structural\u00a0stress limit on the fuselage; for most airplanes, the cabin pressure is maintained at an\u00a0altitude equivalent to about 6--8,000 ft to allow for good passenger comfort.\u00a0Nevertheless, some passengers may still exhibit some of the symptoms of hypoxia\u00a0(oxygen deprivation) during long flights, which contribute to the malady known as <em>jet\u00a0lag<\/em>. The most modern commercial transport airplanes such as the Boeing 787 maintain the\u00a0cabin pressure at the equivalent of 6,000 ft (i.e., at a higher pressure differential),\u00a0which improves passenger comfort and reduces the effects of jet lag.\n<h1>Representative Flight Envelopes<\/h1>\nThe general idea of a flight envelope has already been introduced, although now having learned about the specifics of airspeed and Mach number, stalling, transonic drag rise, and the thrust\/power required for flight,\u00a0 the characteristics of the flight envelope of an airplane and why it has inherent boundaries can be understood. Stalling speeds always define the low speed end of the envelope, and the onset of transonic drag rise and buffet will define the high speed end of the envelope. The ceiling is defined by the allowable differential pressurization, which is a structural limit not an aerodynamic one as previously explained.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"579\"]<img class=\"wp-image-480 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial.jpg\" alt=\"\" width=\"579\" height=\"486\"> A representative flight envelope for a commercial subsonic transport (jet)\u00a0airplane.[\/caption]\n\nA representative flight envelope for a commercial subsonic transport (jet) airplane is shown in the figure below, with an actual measured flight envelope with test points identified shown in the figure below. In this case, the graphs are defined in terms of the airspeed and the flight Mach number, the significance of the Mach number already being discussed. At lower airspeeds, the envelope is bounded by the stalling speeds, which is in the \"clean\" configuration. The stall region of the flight envelope needs little further elaboration other than it is a complex aerodynamic region involving flight at low airspeeds and high angles of attack, which also depends on how the airplane is configured, e.g., flaps up or down, landing gear up or down, etc. The stall boundary is always defined carefully during flight testing, and usually requires many tests to establish good confidence that the stall boundary and the handling qualities and other characteristics of the airplane at stall have been explored for all combinations of flight (e.g., weights and altitudes).\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"539\"]<img class=\"wp-image-481 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811.jpg\" alt=\"\" width=\"539\" height=\"466\"> An actual measured flight envelope with test points for a commercial airliner,\u00a0which is built up during certification testing over many flights with several different\u00a0airplanes.[\/caption]\n\nAt higher airspeeds, the limits of flight are dictated by the maximum operating Mach\u00a0number, which is called $M_{\\rm mo}$, with the corresponding airspeed being called\u00a0the maximum operating airspeed $V_{\\rm mo}$ or VMO. In operational service, the\u00a0airplane will cruise at an airspeed that is somewhat lower than this recommended\u00a0airspeed (which will appear in the airplane's operating manual and procedures).\n\nWhile fundamental engineering issues are key here, there are non-engineering factors that may limit the actual usable flight envelope. For example, there are\u00a0issues centered around economic requirements, manufacturability, passenger\u00a0ergonomics and safety, airfield requirements, and environmental and noise regulations.\u00a0For example, an airline always wants to maximize its profit because the higher the\u00a0profit per unit weight of payload carried, the higher the profit. In this respect, the empty\u00a0weight of the airplane is critical. The benefit is that not only is the fuel burn lower (i.e.,\u00a0lower costs for a given payload), but the revenue can also be increased by carrying\u00a0more payload. This is one of the reasons why the use of lightweight composite\u00a0materials has become so critically important in modern aircraft design.This not\u00a0because composites are necessarily lighter per se but because they can be better\u00a0tailored to give a better strength to weight ratio.\n\nThe figure below shows the flight envelope of high-performance jet airplanes\u00a0that can reach supersonic flight speeds, at least at higher altitudes. In this case, the\u00a0envelope has again been established with the aid of flight test, which has included\u00a0various types of maneuvers, including accelerations, decelerations, climbs, and\u00a0descents. Notice the relatively broad flight envelope for this airplane in terms of\u00a0attainable altitudes and airspeeds (Mach numbers). However, such high-performance\u00a0airplanes tend to expose the limits of aeronautical technology, which tie closely to the\u00a0limitations imposed by aerodynamics and the strength of the airframe and the engines.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"575\"]<img class=\"wp-image-482 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet.jpg\" alt=\"\" width=\"575\" height=\"496\"> Representative flight envelope of a high-performance jet-powered airplane that\u00a0can reach supersonic flight speeds.[\/caption]\n<h1>Other Limiting Factors on the Flight Envelope<\/h1>\nThe engines themselves may suffer various problems that may limit the airplane's flight boundary, including surge\/stall issues and intake buzz. Intake \"buzz\" is usually associated with supersonic airplanes, which have inlets designed to reduce the flow speeds to subsonic conditions before the flow enters the engine's compressor stage. The buzz phenomenon, if it occurs, involves the interaction between the surface boundary layer flows and the shock waves, which can result in an unsteady flow behavior at the intake to the engine.\n\nEngine surges can occur on all types of jet engines when stall manifests in the\u00a0compressor stage. However, the onset is usually precipitated by a high operating angle\u00a0of the airplane's attack, especially when at lower airspeeds. The phenomenon results in\u00a0a sudden back pressure through the engine, resulting in unstable engine operation. In\u00a0some cases, the combustion process is interrupted to the degree that raw fuel ends up\u00a0burning in the tailpipe, often resulting in a spectacular discharge of smoke and flames.\n\nWhile serious in that surges cause an immediate loss of thrust, they are usually quickly\u00a0self-correcting when the conditions that promoted the problem are removed, i.e., by the\u00a0pilot reducing the angle of attack of the wing and\/or increasing airspeed by pushing\u00a0forward on the control stick. Nevertheless, engine surge conditions have been known\u00a0to manifest more often during the critical takeoff and climb phases of flight, which\u00a0always poses a safety of flight issue. Usually, an engine that suffers from\u00a0surging must be closely inspected for damage before further flight.\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"450\"]<img class=\"wp-image-483 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter.png\" alt=\"\" width=\"450\" height=\"274\"> A mathematical finite-element and aerodynamic model of a Boeing 747 shows\u00a0the potential structural deformations that can occur because of flutter. (NASA\/ZONA\u00a0Technology.)[\/caption]\n\nWing flutter is an aeroelastic phenomenon, a coupling between the aerodynamic loads\u00a0and the elastic deformation of the structure. Wings and tail surfaces are prone to flutter\u00a0at higher airspeeds and Mach numbers, although other parts of the airframe such are\u00a0the engine nacelles and tail surfaces may also be susceptible to such problems. Even on a wing, the onset of flutter is not necessarily catastrophic and\u00a0can manifest as a benign (but often alarming) limit-cycle torsional and\/or bending\u00a0oscillations.\n\nGenerally, however, the avoidance of flutter is a key design requirement. The structural\u00a0dynamics and potential flutter characteristics of the aircraft's structure are carefully\u00a0examined using computer models, the objective being to identify the natural\u00a0frequencies and modes of deformation; as shown below. The parameters\u00a0that may affect the onset of flutter on a wing include the geometry of the wing (its span,\u00a0aspect ratio, thickness, sweep angle, etc.) as well as its structural stiffness, total weight\u00a0and weight distribution, positions and weights of the engines, moments of inertia about\u00a0the bending and torsional axes, etc.\n\n&nbsp;\n\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"470\"]<img class=\"wp-image-484 \" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/fluttermodes.jpg\" alt=\"\" width=\"470\" height=\"272\"> A mathematical finite-element and aerodynamic model of a Boeing 747 shows\u00a0the potential structural deformations that can occur because of flutter. (NASA\/ZONA\u00a0Technology.)[\/caption]\n\nEven with a good understanding of the flexible airframe, however, flutter developments\u00a0can still occur. Flutter usually leads to large structural deformations and even to\u00a0structural failure. Because the onset of flutter conditions on an airplane can be\u00a0potentially catastrophic, wings, in particular, are designed carefully to avoid the problem\u00a0and then verified by flight testing to ensure that flutter will never occur if the airplane is\u00a0flown properly within its normal validated flight envelope.\n<h1>Some Final Comments.<\/h1>\nAll aircraft will have an operational flight envelope that is defined in terms of the airspeeds and altitudes where the aircraft can be flown safely with it aerodynamics and performance limits. In this regard, not all aircraft are created equally. The advent of the supercritical airfoil design and redevelopment of the supercritical wing allowed for a significant increase in the flight envelope of commercial airliners and other aircraft that cruise in transonic flow. The characteristic flat top surfaces of supercritical airfoil is apparent on all modern jetliners. The use of the area rule has allowed not only reductions in wave drag but have allowed aircraft to cruise more efficiently at a higher transonic Mach number.\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">For Further Thought and\/or Discussion<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n \t<li>Think about the nature of the flight envelope for a small, general aviation airplane\u00a0powered by a reciprocating engine and propeller.<\/li>\n \t<li><span style=\"font-size: 1rem\">What factors will limit the lowest and\u00a0highest achievable airspeeds?<\/span><\/li>\n \t<li><span style=\"font-size: 1rem\">What operational flight envelope would a typical helicopter have compared to an\u00a0airplane? A tiltrotor?<\/span><\/li>\n \t<li><span style=\"font-size: 1rem\">Study photos you can find of the Airbus A380. Can you identify any design\u00a0features that tie to the use of the ``area rule'' in its design.<\/span><\/li>\n \t<li>What type of flight envelope would a supersonic transport (SST) aircraft have?\u00a0What factors will limit the highest achievable flight Mach number?<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Other Useful Online Resources<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nTo learn more about the supercritical airfoil check out <a href=\"https:\/\/www.nasa.gov\/pdf\/89232main_TF-2004-13-DFRC.pdf\">this article<\/a> by NASA.\n\n<\/div>\n<\/div>\n&nbsp;","rendered":"<div>\n<p style=\"text-align: left\">[latexpage]<\/p>\n<h1>Introduction to this Lesson<\/h1>\n<\/div>\n<p>All aircraft have operational limits in terms of the maximum and minimum airspeeds as\u00a0well as altitudes at which they can fly level in steady, unaccelerated flight, e.g., the\u00a0airspeed versus altitude boundary such as shown in the figure below. Notice\u00a0that, by design, jet fighter aircraft can fly faster and higher over a wider range of flight conditions\u00a0than other airplanes but remember that they also have a broader type of mission. In\u00a0comparison, commercial jet airplanes are very much <em>point designs<\/em> in that they a\u00a0designed primarily to cruise for long periods at a specific airspeed (or Mach number) and altitude. Turboprops are often used for short-haul flights, and while they fly at lower altitudes and airspeeds they are better suited for operating out of shorter runways.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 469px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-468\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope.jpg\" alt=\"\" width=\"469\" height=\"408\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope.jpg 611w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope-300x261.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope-65x56.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope-225x196.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2021\/09\/flightenvelope-350x304.jpg 350w\" sizes=\"auto, (max-width: 469px) 100vw, 469px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Representative flight envelopes of different types of airplanes in terms of their\u00a0achievable altitudes and airspeeds.<\/figcaption><\/figure>\n<div>\n<p>The area inside the boundaries that limit normal flight is called the airplane&#8217;s\u00a0<em>operational flight envelope<\/em>. The limits of the envelope are defined and set based on\u00a0aerodynamics (such as the highest achievable Mach number), the engine power (e.g., a turboprop or piston engine) or\u00a0thrust (i.e., a turbojet or turbofan) that is available, the onset of maximum structural loads, or even something else\u00a0such as the onset of flutter or buffeting. Limits could also be set by excessive\u00a0aerodynamic heating for a supersonic aircraft. The <em>flight corridor<\/em> is often referred\u00a0to as the speed range or band over which the airplane can fly at any given altitude\u00a0without encountering any of the flight limits.<\/p>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Objectives of this Lesson<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div>\n<ul>\n<li>Understand the meaning of an airplane&#8217;s flight envelope and a flight corridor.<\/li>\n<li><span style=\"font-size: 1rem\">Know about the various factors that may limit the operational flight envelope\u00a0of an airplane, including stall.<\/span><\/li>\n<li><span style=\"font-size: 1rem\">Have a general understanding of the phenomenon of wave drag and why it can also limit the flight envelope.<\/span><\/li>\n<li><span style=\"font-size: 1rem\">Understand the\u00a0principles associated with drag reduction using supercritical wing design and the area\u00a0rule.<\/span><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<h1>General Comments on the Flight Envelope<\/h1>\n<\/div>\n<div>\n<p>The size and shape of the flight envelope (or flight corridor) will depend on the type of airplane, i.e., whether it is propeller-driven or jet-powered, has an unpressurized or pressurized fuselage, and\/or whether it\u00a0is specifically designed for subsonic or supersonic flight. Naturally, the exact size and\u00a0shape of the envelope for any given airplane also depends on the properties of the\u00a0atmosphere, particularly the density and temperature of the air.\u00a0Generally, the lowest attainable airspeed of an airplane (either jet-powered or propeller-driven) is dictated by the onset of wing stall, which determines the left side boundary on\u00a0the flight envelope. This stalling airspeed will be a function of the airplane&#8217;s weight and\u00a0altitude, as well as the wing flap settings and sometimes also if the undercarriage is up or down.<\/p>\n<p>The right side of the boundary will be set by the highest attainable airspeed, which is\u00a0usually limited by the power available (for propeller-driven airplanes) or thrust available\u00a0(for jet engines) to overcome the drag of the airplane, the drag being a function of the\u00a0shape of the airplane as well as its flight Mach number.\u00a0The upper edge of the flight envelope is the maximum attainable altitude, which is\u00a0referred to as the <em>operational ceiling<\/em>. The ceiling is the altitude above which an aircraft cannot\u00a0climb, which is usually defined based on a threshold of a diminishing rate of climb of 100 ft\/min. The\u00a0attainable flight ceiling depends on the excess power available relative to the aircraft&#8217;s\u00a0aerodynamic and other characteristics, including its weight.\u00a0In some cases, however, such as on most commercial airplanes, the flight ceiling is\u00a0limited by the onset of wave drag or transonic buffet, or by the airplane reaching some\u00a0maximum structural loads associated with the pressurization of the fuselage (which is a\u00a0trade with airframe weight), even though the airplane may have the excess power\u00a0available to achieve higher flight altitudes.<\/p>\n<h1>Trimmed Lift Coefficient<\/h1>\n<p>An airplane is said to be in steady level unaccelerated flight\u00a0when the three forces (lift, drag and side force) and the three corresponding moments\u00a0(pitching, rolling, and yawing) on the airplane are perfectly balanced, in which case the\u00a0airplane is said to be in <em>trim<\/em>, as shown in the figure below. The balance of forces in\u00a0steady trim is that vertical equilibrium requires that lift = weight and horizontal\u00a0equilibrium requires that thrust = drag, i.e.,<br \/>\n\\begin{equation}<br \/>\nL = W, \\quad \\quad T = D<br \/>\n\\end{equation}<br \/>\nthe side force is naturally assumed to be zero in trimmed flight. One other assumption here is that the\u00a0thrust vector&#8217;s line of action is (primarily) in the flight direction. Of course, full flight trim\u00a0also requires that the airplane have a moment balance in pitch, roll, and yaw about the\u00a0center of gravity.<\/p>\n<figure id=\"attachment_469\" aria-describedby=\"caption-attachment-469\" style=\"width: 433px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-469\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight.jpg\" alt=\"\" width=\"433\" height=\"280\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight.jpg 320w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight-300x194.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight-65x42.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/trimmedflight-225x146.jpg 225w\" sizes=\"auto, (max-width: 433px) 100vw, 433px\" \/><figcaption id=\"caption-attachment-469\" class=\"wp-caption-text\">In the level flight trim condition the forces and moments on the airplane will be\u00a0in perfect balance.<\/figcaption><\/figure>\n<p>Remember that the wings generate the lift to overcome the weight and the engines provide the propulsive force to overcome the drag of the airplane, the generation of this thrust requiring a source of power and fuel. In terms of basic aerodynamics, for vertical equilibrium then<\/p>\n<\/div>\n<div>\\begin{equation}<br \/>\nL = \\frac{1}{2} \\rho V_{\\infty}^2 S C_L = W<br \/>\n\\end{equation}<br \/>\nwhere $\\rho$ is the density of the air in which the airplane is flying, $S$ is the reference wing area and $C_L$ is the total wing lift coefficient (the assumption here being that the wings generate all lift). Notice that $\\rho = \\rho_0 \\sigma$ where $\\sigma$ comes from the ISA model, i.e.,<br \/>\n\\begin{equation}<br \/>\nL = W = \\frac{1}{2} \\rho_0 \\sigma V_{\\infty}^2 S C_L<br \/>\n\\end{equation}<br \/>\nRearranging this equation allows us to solve for the lift coefficient that needs to be\u00a0produced on the wing for a given flight speed, i.e.,<br \/>\n\\begin{equation}<br \/>\nC_L = \\frac{2 W}{\\rho_0 \\sigma S V_{\\infty}^2}<br \/>\n\\label{CL eqn}<br \/>\n\\end{equation}<br \/>\nor the flight speed that corresponds to a given lift coefficient, i.e.,<br \/>\n\\begin{equation}<br \/>\nV_{\\infty} = \\sqrt{ \\frac{2}{\\rho_0 \\sigma} \\left( \\frac{W}{S} \\right) \\frac{1}{C_L} }<br \/>\n\\end{equation}<br \/>\nRecall that the ratio of airplane&#8217;s weight to its lifting wing area, $W\/S$, is called the\u00a0wing loading. Notice that the lift coefficient is proportional to weight (or to wing loading)\u00a0but decreases with the square of the airspeed. The lift coefficient also increases with\u00a0altitude for a given true airspeed and weight.<\/div>\n<h1>Stalling Airspeeds<\/h1>\n<div>The minimum airspeed that would allow level flight of the airplane is called the <em>stall speed<\/em> or the\u00a0<em>stalling\u00a0speed<\/em>, which is the airspeed corresponding to the angle of attack and lift coefficient at which the wing will stall. This value is called the maximum lift coefficient $C_{L_{\\rm max}}$ and it depends on the nature of the wing used on the airplane, including its planform, its twist, and airfoil section. The actual value of $C_{L_{\\rm max}}$ also depends on whether the flaps and other high lift devices such as slats are retracted or deployed, as shown in the figure below. Notice that flaps are very effective in increasing $C_{L_{\\rm max}$ but the use of a slat can boost the $C_{L_{\\rm max}}$ by a further 50\\%.<\/div>\n<div>\n<figure id=\"attachment_470\" aria-describedby=\"caption-attachment-470\" style=\"width: 420px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-470 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax.jpg\" alt=\"\" width=\"420\" height=\"324\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax.jpg 420w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax-300x231.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax-65x50.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax-225x174.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/CLmax-350x270.jpg 350w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><figcaption id=\"caption-attachment-470\" class=\"wp-caption-text\">The maximum attainable lift coefficient depends on whether the flaps or slats are retracted or deployed. If the flaps and slats are retracted, then the wing is said to be in its &#8220;clean&#8221; configuration.<\/figcaption><\/figure>\n<p>Although the value of $C_{L_{\\rm max}}$ may not be exactly known by calculation, it\u00a0can be determined indirectly from flight tests with the airplane from measurements of\u00a0true airspeed and density altitude. After the $C_{L_{\\rm max}}$ for the wing is\u00a0determined, the stall speed in steady level flight can be solved for at any weight and\u00a0density altitude. i.e.,<br \/>\n\\begin{equation}<br \/>\nV_{\\rm stall} = \\sqrt{ \\frac{2}{\\rho_0 \\sigma} \\left( \\frac{W}{S} \\right) \\frac{1}{C_{L_{\\rm\u00a0max}} } }<br \/>\n\\label{stallspeed1}<br \/>\n\\end{equation}<br \/>\nusing the value of $\\sigma$ from the ISA model, i.e., based on the prevailing pressure altitude and outside air temperature.<\/p>\n<p>Notice that for a given $C_{L_{\\rm max}}$, the stalling speed depends on the wing loading, i.e., all things being equal an airplane with a higher wing loading will stall at a higher airspeed. If a linear lift curve slope of the wing is assumed, say $C_{L_{\\alpha}}$, then the angle of attack of the wing $\\alpha$ (measured relative to the zero-lift angle) can be calculated using<br \/>\n\\begin{equation}<br \/>\n\\alpha = \\frac{2 W}{\\rho_0 \\sigma S C_{L_{\\alpha}} V_{\\infty}^2}<br \/>\n\\end{equation}<br \/>\nthe maximum value of $\\alpha$ typically being less than 15$^{\\circ}$ at low Mach numbers and lower than that at higher Mach numbers. However, it is important to appreciate that a wing will stall at any airspeed if the angle of attack is high enough. For this reason, caution must be used when referring to stall speeds.<\/p>\n<\/div>\n<div><\/div>\n<div>In summary, four conclusions can be drawn from the use of Eq.~\\ref{stallspeed1}:<\/div>\n<ol>\n<li>Stall speed will increase with increasing weight of the airplane.<\/li>\n<li>Stall speed will\u00a0increase with increasing density altitude, i.e., with a lowering of the air density.<\/li>\n<li>Stall\u00a0speed will decrease with increasing values of wing $C_{L_{\\rm max}}$, which, as\u00a0previously discussed, can be achieved by the application of wing flaps and\/or leading-edge\u00a0slats.<\/li>\n<li><span style=\"text-align: justify;font-size: 1em\">Stall speed will decrease with increasing wing area, an increase in wing area\u00a0also being possible with the use of certain types of flaps, such as Fowler flaps.<\/span><\/li>\n<\/ol>\n<h1>Limiting Cruise Speeds<\/h1>\n<p>The figure below shows a historical trend as to how the cruise airspeeds for commercial transport airplanes have increased over the\u00a0decades, which naturally is a direct consequence of the rapid advancements and\u00a0maturation of aeronautical technology. Of course, the introduction of the jet engine was\u00a0responsible for the more rapid growth in achievable cruise speeds seen after 1960.\u00a0However, it can also be seen that since the early 1970s, the cruise airspeeds for\u00a0commercial airplanes have all but plateaued, with corresponding achievable cruise\u00a0flight Mach numbers in the range 0.8 to 0.85. There are a couple of exceptions to this\u00a0trend with the British Concorde and the Russian Tu-144, but these airplanes were\u00a0specifically designed for cruising at supersonic Mach numbers. While the use of supercritical wing designs has extended the flight envelope of airliners to higher transonic Mach numbers, the eventual onset of wave drag and buffeting still remains a physical barrier to higher flight conditions.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 538px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-471 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear.jpg\" alt=\"\" width=\"538\" height=\"431\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear.jpg 538w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear-300x240.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear-65x52.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear-225x180.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/speedsvsyear-350x280.jpg 350w\" sizes=\"auto, (max-width: 538px) 100vw, 538px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">The cruise airspeeds of commercial transports show that rapid increases\u00a0occurred with the maturation of aeronautical technology, but since 1970 have all but\u00a0reached a plateau.<\/figcaption><\/figure>\n<h1>Supercritical Flows &amp; Drag Rise<\/h1>\n<p>One reason is that cruise speeds for commercial airliners have reached a plateau is\u00a0because of the buildup of high drag on a wing as transonic flow conditions are\u00a0approached, the basic physics of what happens on the wing section being shown in the figure below. The drag buildup from the development of compressibility\u00a0and shock waves takes much thrust and power to overcome, and there are other issues too about\u00a0operating at higher flight Mach numbers such as buffeting, as has been discussed.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 622px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-472\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow.jpg\" alt=\"\" width=\"622\" height=\"552\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow.jpg 795w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow-300x266.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow-768x682.jpg 768w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow-65x58.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow-225x200.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalflow-350x311.jpg 350w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">As the flight (free-stream) Mach number increases the flow about a wing\u00a0section develops supersonic flow and eventually a shock wave. This shock wave\u00a0strengthens and moves aft over the wing as the Mach number increases, eventually in\u00a0supersonic conditions forming shocks at the wing&#8217;s leading and trailing edges.<\/figcaption><\/figure>\n<p>At some flight (free-stream) Mach number, the local flow at a point on the wing&#8217;s\u00a0surface reaches sonic conditions, which is called the <em>critical Mach number<\/em>. As the\u00a0free-stream Mach number increases further, a small pocket of supersonic flow\u00a0develops on the section, resulting in a weak shock wave in the flow. As the Mach number\u00a0further increases, the shock strengthens and moves aft over the section and a\u00a0supersonic region is formed. An associated shock wave also develops on the lower\u00a0surface. This is called the well-established transonic flow region, the shock waves\u00a0resulting in an energy loss that manifests as form of drag called <em>wave drag. <\/em>Wave drag causes the total drag on the wing to\u00a0increase rapidly, as shown in the figure below<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 473px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-473\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise.jpg\" alt=\"\" width=\"473\" height=\"250\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise.jpg 392w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise-300x158.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise-65x34.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise-225x119.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/dragrise-350x185.jpg 350w\" sizes=\"auto, (max-width: 473px) 100vw, 473px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">There is a large increase in drag and loss of lift on a wing as transonic flight\u00a0conditions develop and before supersonic flight is established.<\/figcaption><\/figure>\n<p>Because steep adverse pressure gradients also accompany the shock waves that\u00a0develop during transonic conditions on the wing section, the boundary layer downstream of the shock becomes\u00a0thicker and the profile drag increases. If the shock wave becomes sufficiently strong, flow\u00a0separation may occur, leading to a buffeting aerodynamic phenomenon. Buffeting can\u00a0result in high levels of vibration being transmitted to the airframe, and it is not a viably\u00a0sustained flight condition. The onset of buffeting can also cause aeroelastic concerns, so this must be examined carefully through flight testing. The onset of buffeting is usually a limiting factor in the\u00a0operational flight envelope of most aircraft (unless they are designed for supersonic flight) and is referred to as the <em>buffet boundary<\/em>.<\/p>\n<p>If and when Mach number approaches unity, the shocks move all the way to the trailing edge of the section. Finally, when the Mach number becomes greater than one, a bow wave appears just ahead of the section, and the shocks at the trailing edge become oblique. For supersonic airplanes, these strong shock waves are responsible for the pressure changes that are heard on the ground that manifest as the impulsive &#8220;boom-boom&#8221; sound known as the <em>sonic boom<\/em> as the airplane passes overhead at supersonic\u00a0speeds. \u00a0The drag rise on the aircraft during the transition from transonic to supersonic flight usually requires excess thrust to be produced using an afterburner. Some aircraft may be subsequently able to cruise supersonically without the use of the afterburner, but it depends on the engine.<\/p>\n<h1>Reducing Compressibility Drag<\/h1>\n<p>The minimization of wave drag on the wings as the transonic flight regime is\u00a0approached is obviously key to lowering drag and\/or allowing the airplane to fly faster\u00a0and opening up the flight envelope before significant drag rise is encountered. Lower\u00a0drag means lower thrust and power is required for flight, so less fuel is expended and more flight\u00a0range can be achieved.<\/p>\n<p>Wing sweep has a very profound effect on transonic and supersonic drag, as shown in the figure below. The use of swept-back wings reduces the strength of the\u00a0shock waves and prevents the shocks from interfering with the flow over the\u00a0wings and causing flow separation. However, although\u00a0swept wings can help delay this drag rise from compressibility effects, other\u00a0aerodynamic and aeroelastic problems are associated with swept wings, so aircraft designers ten to use as little wing sweep as possible, 20 to 30 degrees being used on may airliners.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 543px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-474 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep.jpg\" alt=\"\" width=\"543\" height=\"374\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep.jpg 543w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep-300x207.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep-65x45.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep-225x155.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Wingdragvssweep-350x241.jpg 350w\" sizes=\"auto, (max-width: 543px) 100vw, 543px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">The use of sweepback on a wing provides for a very significant reduction in its\u00a0drag.<\/figcaption><\/figure>\n<p>A visualization of the flow about swept and unswept wings at a low supersonic speed is shown below, which was obtained using the schlieren method. Notice that with the use of sweepback the shockwaves do not interact directly with the wing, which keeps the drag low. With the unswept wings not only are the shock waves stronger but they interact strongly with the wing driving up drag.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 650px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-475 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback.jpg\" alt=\"\" width=\"650\" height=\"463\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback.jpg 650w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback-300x214.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback-65x46.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback-225x160.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/schlierensweepback-350x249.jpg 350w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Schlieren flow visualization image of the shock wave patterns around two\u00a0airplane models showing the effects of sweptback at Mach 1.2. (NACA image.)<\/figcaption><\/figure>\n<p>The figure below shows the difference in the shapes of a conventional airfoil and a supercritical airfoil. The basic principle used in transonic airfoil design is to\u00a0control the flow&#8217;s expansion to supersonic speed and its subsequent recompression.\u00a0Compared to a conventional wing section, a supercritical wing section is distinctive in\u00a0that it is flatter along the top surface with significant camber at its trailing edge. variations of supercritical airfoil sections are used on all commercial jet airliners.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 565px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-476\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical.jpg\" alt=\"\" width=\"565\" height=\"304\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical.jpg 422w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical-300x161.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical-65x35.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical-225x121.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/conventional-vs-supercritical-350x188.jpg 350w\" sizes=\"auto, (max-width: 565px) 100vw, 565px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Supercritical airfoil designs have led to notable reductions in wave drag to\u00a0allow wings to cruise at higher flight Mach numbers.<\/figcaption><\/figure>\n<p>The challenges in reaching higher transonic cruise speeds have led to the design of a\u00a0special shape of swept wing called a <em>supercritical wing<\/em>, as shown in the photograph below. The\u00a0supercritical wing evolved from the careful tailoring of the airfoil section(s) with the overall\u00a0wing design to delay the formation and\/or reduce the strength of the shock waves over\u00a0the wing so reducing wave drag. In the early 1970s, NASA modified an airplane to test a supercritical wing in\u00a0place of the conventional wing to reduce the effects of shock waves and wave drag,\u00a0with great success, and aircraft designers have never looked back.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 615px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2010\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/supercriticalwing-scaled-1.jpg\" alt=\"\" width=\"615\" height=\"304\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Supercritical airfoils and smoothly varying wing shapes are now\u00a0standard on virtually every modern subsonic commercial transport airplanes.<\/figcaption><\/figure>\n<h1>Area Rule<\/h1>\n<p>Other ways of reducing wave drag expanding the flight envelope of the airplane to\u00a0higher cruise speeds include the use of the area rule, which was developed by Richard Whitcomb. To reduce the number and\u00a0intensity of shock waves over an airplane as it approaches transonic and then\u00a0supersonic flight, the basic design principle behind the area rule is that the airplane&#8217;s\u00a0overall cross-sectional shape should change smoothly with no significant\u00a0discontinuities.<\/p>\n<p>The principle was proven to work in wind tunnel testing and then applied retroactively to various airplanes, with successful results after flight testing. Early airplanes that were modified to validate the area rule had distinctive if not odd looking &#8220;waisted&#8221; fuselage shapes at the wing root, as shown in the figure below, which were often dubbed as &#8220;flying coke bottles.&#8221; Nevertheless, the notable reductions in drag proved the viability of the area rule concept.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 550px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-478 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule.jpg\" alt=\"\" width=\"550\" height=\"379\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule.jpg 550w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule-300x207.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule-65x45.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule-225x155.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/arearule-350x241.jpg 350w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">The principle of the &#8220;area rule&#8221; is to maintain a smooth variation in the net cross-sectional area of the airplanes to reduce the compressibility effects and wave drag. Wind tunnel and flight testing have both confirmed the benefits.<\/figcaption><\/figure>\n<p>Later airplanes were designed with the area rule in mind but were aesthetically more pleasing because of the blending of the wing root area and the careful positioning of engines, the use of large trailing edge anti-shock wing pods or &#8220;canoe&#8221; fairings, and other subtle changes to the shape of the airplane to prevent large changes in effective cross-sectional area. For many commercial airplanes, the wing-mounted &#8220;pod&#8221; engines are placed relatively far forward of the wings to control the change in cross-sectional area of the airplane as the wing is approached.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 607px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-479\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization.jpg\" alt=\"\" width=\"607\" height=\"332\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization.jpg 550w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization-300x164.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization-65x36.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization-225x123.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/Antishock_Bodies_Visualization-350x192.jpg 350w\" sizes=\"auto, (max-width: 607px) 100vw, 607px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Surface oil flow visualization showing the signatures of strong shock-induced flow separation on a wing (left) and the weaker shocks and reduced flow separation from the use of &#8220;anti-shock&#8221; extensions at the trailing edge of the wing. (NASA image.)<\/figcaption><\/figure>\n<p>A careful examination of the most commercial airliners will show some careful contouring of\u00a0the fuselage and wing root to help minimize wave drag according to the principles\u00a0established by the area rule. For the same reason, later versions of the Boeing 747 were also modified with\u00a0a longer upper deck and a shallower transition at its end to keep area changes as progressive as possible. Most airplanes capable of\u00a0transonic or supersonic airspeeds incorporate design features that can be traced back\u00a0to the fundamental principles underlying Whitcomb&#8217;s area rule.<\/p>\n<h1>Flight Ceilings<\/h1>\n<p>The flight ceiling for an airplane is defined based on a demonstrated rate of climb. The\u00a0absolute ceiling is defined when the achievable rate of climb diminishes to zero,\u00a0whereas the service ceiling is defined such that the rate of climb reduces below\u00a0100 ft\/min. The airplane&#8217;s normal performance ceiling is defined as when the rate of\u00a0climb reduces below 200 ft\/min. The ceiling is reached when the excess power\u00a0available over and above that for level flight at the same airspeed and weight becomes\u00a0diminishingly small.<\/p>\n<p>The ceiling for most commercial transport airplanes is limited by cabin pressurization\u00a0requirements rather than attainable engine thrust and power, which set a structural\u00a0stress limit on the fuselage; for most airplanes, the cabin pressure is maintained at an\u00a0altitude equivalent to about 6&#8211;8,000 ft to allow for good passenger comfort.\u00a0Nevertheless, some passengers may still exhibit some of the symptoms of hypoxia\u00a0(oxygen deprivation) during long flights, which contribute to the malady known as <em>jet\u00a0lag<\/em>. The most modern commercial transport airplanes such as the Boeing 787 maintain the\u00a0cabin pressure at the equivalent of 6,000 ft (i.e., at a higher pressure differential),\u00a0which improves passenger comfort and reduces the effects of jet lag.<\/p>\n<h1>Representative Flight Envelopes<\/h1>\n<p>The general idea of a flight envelope has already been introduced, although now having learned about the specifics of airspeed and Mach number, stalling, transonic drag rise, and the thrust\/power required for flight,\u00a0 the characteristics of the flight envelope of an airplane and why it has inherent boundaries can be understood. Stalling speeds always define the low speed end of the envelope, and the onset of transonic drag rise and buffet will define the high speed end of the envelope. The ceiling is defined by the allowable differential pressurization, which is a structural limit not an aerodynamic one as previously explained.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 579px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-480 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial.jpg\" alt=\"\" width=\"579\" height=\"486\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial.jpg 579w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial-300x252.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial-65x55.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial-225x189.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundarycommercial-350x294.jpg 350w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">A representative flight envelope for a commercial subsonic transport (jet)\u00a0airplane.<\/figcaption><\/figure>\n<p>A representative flight envelope for a commercial subsonic transport (jet) airplane is shown in the figure below, with an actual measured flight envelope with test points identified shown in the figure below. In this case, the graphs are defined in terms of the airspeed and the flight Mach number, the significance of the Mach number already being discussed. At lower airspeeds, the envelope is bounded by the stalling speeds, which is in the &#8220;clean&#8221; configuration. The stall region of the flight envelope needs little further elaboration other than it is a complex aerodynamic region involving flight at low airspeeds and high angles of attack, which also depends on how the airplane is configured, e.g., flaps up or down, landing gear up or down, etc. The stall boundary is always defined carefully during flight testing, and usually requires many tests to establish good confidence that the stall boundary and the handling qualities and other characteristics of the airplane at stall have been explored for all combinations of flight (e.g., weights and altitudes).<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 539px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-481\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811.jpg\" alt=\"\" width=\"539\" height=\"466\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811.jpg 627w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811-300x259.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811-65x56.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811-225x194.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/787_Sum_Flight_Envelope_22811-350x303.jpg 350w\" sizes=\"auto, (max-width: 539px) 100vw, 539px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">An actual measured flight envelope with test points for a commercial airliner,\u00a0which is built up during certification testing over many flights with several different\u00a0airplanes.<\/figcaption><\/figure>\n<p>At higher airspeeds, the limits of flight are dictated by the maximum operating Mach\u00a0number, which is called $M_{\\rm mo}$, with the corresponding airspeed being called\u00a0the maximum operating airspeed $V_{\\rm mo}$ or VMO. In operational service, the\u00a0airplane will cruise at an airspeed that is somewhat lower than this recommended\u00a0airspeed (which will appear in the airplane&#8217;s operating manual and procedures).<\/p>\n<p>While fundamental engineering issues are key here, there are non-engineering factors that may limit the actual usable flight envelope. For example, there are\u00a0issues centered around economic requirements, manufacturability, passenger\u00a0ergonomics and safety, airfield requirements, and environmental and noise regulations.\u00a0For example, an airline always wants to maximize its profit because the higher the\u00a0profit per unit weight of payload carried, the higher the profit. In this respect, the empty\u00a0weight of the airplane is critical. The benefit is that not only is the fuel burn lower (i.e.,\u00a0lower costs for a given payload), but the revenue can also be increased by carrying\u00a0more payload. This is one of the reasons why the use of lightweight composite\u00a0materials has become so critically important in modern aircraft design.This not\u00a0because composites are necessarily lighter per se but because they can be better\u00a0tailored to give a better strength to weight ratio.<\/p>\n<p>The figure below shows the flight envelope of high-performance jet airplanes\u00a0that can reach supersonic flight speeds, at least at higher altitudes. In this case, the\u00a0envelope has again been established with the aid of flight test, which has included\u00a0various types of maneuvers, including accelerations, decelerations, climbs, and\u00a0descents. Notice the relatively broad flight envelope for this airplane in terms of\u00a0attainable altitudes and airspeeds (Mach numbers). However, such high-performance\u00a0airplanes tend to expose the limits of aeronautical technology, which tie closely to the\u00a0limitations imposed by aerodynamics and the strength of the airframe and the engines.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 575px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-482 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet.jpg\" alt=\"\" width=\"575\" height=\"496\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet.jpg 575w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet-300x259.jpg 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet-65x56.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet-225x194.jpg 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/boundaryjet-350x302.jpg 350w\" sizes=\"auto, (max-width: 575px) 100vw, 575px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Representative flight envelope of a high-performance jet-powered airplane that\u00a0can reach supersonic flight speeds.<\/figcaption><\/figure>\n<h1>Other Limiting Factors on the Flight Envelope<\/h1>\n<p>The engines themselves may suffer various problems that may limit the airplane&#8217;s flight boundary, including surge\/stall issues and intake buzz. Intake &#8220;buzz&#8221; is usually associated with supersonic airplanes, which have inlets designed to reduce the flow speeds to subsonic conditions before the flow enters the engine&#8217;s compressor stage. The buzz phenomenon, if it occurs, involves the interaction between the surface boundary layer flows and the shock waves, which can result in an unsteady flow behavior at the intake to the engine.<\/p>\n<p>Engine surges can occur on all types of jet engines when stall manifests in the\u00a0compressor stage. However, the onset is usually precipitated by a high operating angle\u00a0of the airplane&#8217;s attack, especially when at lower airspeeds. The phenomenon results in\u00a0a sudden back pressure through the engine, resulting in unstable engine operation. In\u00a0some cases, the combustion process is interrupted to the degree that raw fuel ends up\u00a0burning in the tailpipe, often resulting in a spectacular discharge of smoke and flames.<\/p>\n<p>While serious in that surges cause an immediate loss of thrust, they are usually quickly\u00a0self-correcting when the conditions that promoted the problem are removed, i.e., by the\u00a0pilot reducing the angle of attack of the wing and\/or increasing airspeed by pushing\u00a0forward on the control stick. Nevertheless, engine surge conditions have been known\u00a0to manifest more often during the critical takeoff and climb phases of flight, which\u00a0always poses a safety of flight issue. Usually, an engine that suffers from\u00a0surging must be closely inspected for damage before further flight.<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 450px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-483 size-full\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter.png\" alt=\"\" width=\"450\" height=\"274\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter.png 450w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter-300x183.png 300w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter-65x40.png 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter-225x137.png 225w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/flutter-350x213.png 350w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">A mathematical finite-element and aerodynamic model of a Boeing 747 shows\u00a0the potential structural deformations that can occur because of flutter. (NASA\/ZONA\u00a0Technology.)<\/figcaption><\/figure>\n<p>Wing flutter is an aeroelastic phenomenon, a coupling between the aerodynamic loads\u00a0and the elastic deformation of the structure. Wings and tail surfaces are prone to flutter\u00a0at higher airspeeds and Mach numbers, although other parts of the airframe such are\u00a0the engine nacelles and tail surfaces may also be susceptible to such problems. Even on a wing, the onset of flutter is not necessarily catastrophic and\u00a0can manifest as a benign (but often alarming) limit-cycle torsional and\/or bending\u00a0oscillations.<\/p>\n<p>Generally, however, the avoidance of flutter is a key design requirement. The structural\u00a0dynamics and potential flutter characteristics of the aircraft&#8217;s structure are carefully\u00a0examined using computer models, the objective being to identify the natural\u00a0frequencies and modes of deformation; as shown below. The parameters\u00a0that may affect the onset of flutter on a wing include the geometry of the wing (its span,\u00a0aspect ratio, thickness, sweep angle, etc.) as well as its structural stiffness, total weight\u00a0and weight distribution, positions and weights of the engines, moments of inertia about\u00a0the bending and torsional axes, etc.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 470px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-484\" src=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/fluttermodes.jpg\" alt=\"\" width=\"470\" height=\"272\" srcset=\"https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/fluttermodes.jpg 280w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/fluttermodes-65x38.jpg 65w, https:\/\/integrations.pressbooks.network\/app\/uploads\/sites\/569\/2022\/07\/fluttermodes-225x130.jpg 225w\" sizes=\"auto, (max-width: 470px) 100vw, 470px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">A mathematical finite-element and aerodynamic model of a Boeing 747 shows\u00a0the potential structural deformations that can occur because of flutter. (NASA\/ZONA\u00a0Technology.)<\/figcaption><\/figure>\n<p>Even with a good understanding of the flexible airframe, however, flutter developments\u00a0can still occur. Flutter usually leads to large structural deformations and even to\u00a0structural failure. Because the onset of flutter conditions on an airplane can be\u00a0potentially catastrophic, wings, in particular, are designed carefully to avoid the problem\u00a0and then verified by flight testing to ensure that flutter will never occur if the airplane is\u00a0flown properly within its normal validated flight envelope.<\/p>\n<h1>Some Final Comments.<\/h1>\n<p>All aircraft will have an operational flight envelope that is defined in terms of the airspeeds and altitudes where the aircraft can be flown safely with it aerodynamics and performance limits. In this regard, not all aircraft are created equally. The advent of the supercritical airfoil design and redevelopment of the supercritical wing allowed for a significant increase in the flight envelope of commercial airliners and other aircraft that cruise in transonic flow. The characteristic flat top surfaces of supercritical airfoil is apparent on all modern jetliners. The use of the area rule has allowed not only reductions in wave drag but have allowed aircraft to cruise more efficiently at a higher transonic Mach number.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">For Further Thought and\/or Discussion<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Think about the nature of the flight envelope for a small, general aviation airplane\u00a0powered by a reciprocating engine and propeller.<\/li>\n<li><span style=\"font-size: 1rem\">What factors will limit the lowest and\u00a0highest achievable airspeeds?<\/span><\/li>\n<li><span style=\"font-size: 1rem\">What operational flight envelope would a typical helicopter have compared to an\u00a0airplane? A tiltrotor?<\/span><\/li>\n<li><span style=\"font-size: 1rem\">Study photos you can find of the Airbus A380. Can you identify any design\u00a0features that tie to the use of the &#8220;area rule&#8221; in its design.<\/span><\/li>\n<li>What type of flight envelope would a supersonic transport (SST) aircraft have?\u00a0What factors will limit the highest achievable flight Mach number?<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Other Useful Online Resources<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>To learn more about the supercritical airfoil check out <a href=\"https:\/\/www.nasa.gov\/pdf\/89232main_TF-2004-13-DFRC.pdf\">this article<\/a> by NASA.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":291,"menu_order":38,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-485","chapter","type-chapter","status-web-only","hentry"],"part":456,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapters\/485","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapters\/485\/revisions"}],"predecessor-version":[{"id":486,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapters\/485\/revisions\/486"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/parts\/456"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapters\/485\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/wp\/v2\/media?parent=485"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/pressbooks\/v2\/chapter-type?post=485"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/wp\/v2\/contributor?post=485"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testcloneissuefixthom\/wp-json\/wp\/v2\/license?post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}