AircraftFlightStrategiesDavid Arterburn', Michael Jaye, Joseph Myers, Kip NygrenIntroductionThreeimportantconsiderationsineveryflightoperationarethealtitude(possiblyvariable)at which to travel, the velocity (possiblyvariable)at which to travel, and the amount ofliftthatwe choosetogenerate (attheexpenseoffuelconsumption-againpossiblyvariable)during the flight. It turns out that whenplanningaflightoperation,onecannotjustchooseanydesiredvalueforeachofthesethree quantities; they are dependent upon one another.We can relatethesethree quantities through a setof equations known as the Breguet (pronouncedbre-ga)RangeEquations.Theseequationsarederived inAppendixA.Deriving these equations showsthat oncewedecideto choose constantvaluesforanytwoofaltitude,liftcoefficient,andvelocity,thethirdisautomaticallydetermined.Thustherearethreebasicindependentflightstrategies:constantaltitude/constant liftcoefficient,constantvelocity/constantaltitude,and constantvelocitylconstant lift coefficient.Exercise 1asksyouto analyzehowthethirdquantitymust varyundereach of theseflight strategies.Commercial flight operations aregenerally conducted at constantvelocity/constant lift coefficientin orderto savefuel.Inmilitary operations,however,thereareoftenotherconsiderationsthatoverridecostefficiency,andthus dictatethechoiceofadifferentflightstrategy.Surveillance/reconnaissanceflights generallydictate flying at constant velocity/constant altitude in ordertobest gather required intelligence. Phased air operations are sometimes bettercoordinatedwhenrestrictedtoconstantvelocity.Whenseveralsortiesareintheairat the same time, especially both outbound and inbound, safe airspacemanagementoftendictatesflightsatconstantspecifiedaltitudes.Exercise2asksyoutomorecloselyanalyzewhichflightstrategymaybemostappropriatefor whichmilitary mission.Thus unlikemost commercial operations,themilitaryplannermust bepreparedto operateunderany of several different flightstrategies.The following scenarios demonstrate how different techniques of single variablecalculuscanassistinanalyzingthegoverningequationstoyieldimportantinformation about flight operations.Concepts covered includemodelingwithI Department ofCivil and Mechanical Engineering,USMA31
31 Aircraft Flight Strategies David Arterburn1 , Michael Jaye, Joseph Myers, Kip Nygren1 Introduction Three important considerations in every flight operation are the altitude (possibly variable) at which to travel, the velocity (possibly variable) at which to travel, and the amount of lift that we choose to generate (at the expense of fuel consumption – again possibly variable) during the flight. It turns out that when planning a flight operation, one cannot just choose any desired value for each of these three quantities; they are dependent upon one another. We can relate these three quantities through a set of equations known as the Breguet (pronounced bre-ga¢) Range Equations. These equations are derived in Appendix A. Deriving these equations shows that once we decide to choose constant values for any two of altitude, lift coefficient, and velocity, the third is automatically determined. Thus there are three basic independent flight strategies: constant altitude/constant lift coefficient, constant velocity/constant altitude, and constant velocity/constant lift coefficient. Exercise 1 asks you to analyze how the third quantity must vary under each of these flight strategies. Commercial flight operations are generally conducted at constant velocity/constant lift coefficient in order to save fuel. In military operations, however, there are often other considerations that override cost efficiency, and thus dictate the choice of a different flight strategy. Surveillance/reconnaissance flights generally dictate flying at constant velocity/constant altitude in order to best gather required intelligence. Phased air operations are sometimes better coordinated when restricted to constant velocity. When several sorties are in the air at the same time, especially both outbound and inbound, safe airspace management often dictates flights at constant specified altitudes. Exercise 2 asks you to more closely analyze which flight strategy may be most appropriate for which military mission. Thus unlike most commercial operations, the military planner must be prepared to operate under any of several different flight strategies. The following scenarios demonstrate how different techniques of single variable calculus can assist in analyzing the governing equations to yield important information about flight operations. Concepts covered include modeling with 1 Department of Civil and Mechanical Engineering, USMA
derivatives,numericalintegration,analyticintegration,andgraphical analysis(ofrange strategies).Scenario:A-1oCloseAirSupportYou are thepilot on anA-10 Thunderbolt, Close Air Support (CAS)aircraft.Among the many things for which you are responsible, some of the particularaspects aretodeterminewithin whatradius yourplanecan safely service CAstargets,howlongitcan"loiter"inatargetarea,andwhenitmustreturnforrefueling.Now,an interesting aspect ofyour jobisthat, at times,someofthe instrumentsmalfunction.Thisforcesyoutodouble-checkyourinstruments'accuracythroughothermeans,ortorelyonthese othermeanstoplanyourplane'sflight.Inthisprojectyou aregoingto answer severalquestionsabout theflight of yourcraftbasedprimarilyonyourplane'sfuel consumption.(Yourfuelgaugeisknowntobeworking)Figure1:TheA-10BThunderboltStrategy 1:Flyingat Constant Velocity/Constant Lift CoefficientRangeEquation:You can answer questions regarding howfar the plane can travel by relating thedistancetraveledbytheplanetotheweightoffuelthatitconsumes.Assumethat youfly at constant velocity and with a constant coefficient of lift (thus,youincreasealtitudeovertimeasyourplanegetsprogressivelylighter).Fromourknowledgeoffluiddynamics,wehavethefollowingrelationship(thisandallfollowingrelationshipsarederivedinAppendixA):32
32 derivatives, numerical integration, analytic integration, and graphical analysis (of range strategies). Scenario: A-10 Close Air Support You are the pilot on an A-10 Thunderbolt, Close Air Support (CAS) aircraft. Among the many things for which you are responsible, some of the particular aspects are to determine within what radius your plane can safely service CAS targets, how long it can "loiter" in a target area, and when it must return for refueling. Now, an interesting aspect of your job is that, at times, some of the instruments malfunction. This forces you to double-check your instruments' accuracy through other means, or to rely on these other means to plan your plane's flight. In this project you are going to answer several questions about the flight of your craft based primarily on your plane's fuel consumption. (Your fuel gauge is known to be working). Figure 1: The A-10B Thunderbolt Strategy 1: Flying at Constant Velocity/Constant Lift Coefficient Range Equation: You can answer questions regarding how far the plane can travel by relating the distance traveled by the plane to the weight of fuel that it consumes. Assume that you fly at constant velocity and with a constant coefficient of lift (thus, you increase altitude over time as your plane gets progressively lighter). From our knowledge of fluid dynamics, we have the following relationship (this and all following relationships are derived in Appendix A):
VCdxdwccwwhere x = distance traveled, W = weight, V= velocity, c is the coefficient of fuelCL is 3.839 forconsumption(c=0.3700Ibs.offuel/hr/lbthrust),andtheratioCpconstantlift coefficient.Thus,thedistancetraveled,x,is given by:WfinishVCL1I-dwWcCDWstartExample1:Youtakeoffweighing40,434Ibs(thisweightincludesfuelarmament,andordnance)and youtravelat V=347.5mi/hr.Youarriveatthetargetarea weighing36,434 Ibs.Byuseofa numerical integration techniquewithan increment sizeof 1000 Ibs in yourpartition, estimatethe distanceyouhavetraveled.Doesyouranswerdepend onyour incrementsize?36.434 dWSolution:This requires us to numerically evaluate the integral -3605.840.434W40,434 dWWeusethetrapezoidal rulewhichwerewriteas3605.835,434 Wx=3605.8*(.5*f。*△W+f*△W+.5*f.*△W)=lWand W=1000.Substitutingforfto evaluatethe integral, with f(W)=1/yields:x=3605.8*(.5*35.434*△W+*△W+.5*40,434*△W),whereW,=i/WisthevalueofWinthei'thsubinterval (equaltoW,=35.434+(i-1)*(40.434-35.434)).Thistechniqueis implemented in thefollowingspreadsheet:Initial W:36434Final W:404344Intervals:Delta W:1000Distance:375.6537Wf(W)Partial Sum364342.74469E-050.013723374342.67137E-050.040437384342.60186E-050.066456394340.0918152.53588E-05404342.47317E-050.1041833
33 C W C c V dW dx D L 1 = - , where x = distance traveled, W = weight, V = velocity, c is the coefficient of fuel consumption (c = 0.3700 lbs. of fuel/hr/lb thrust), and the ratio C L C D is 3.839 for constant lift coefficient. Thus, the distance traveled, x, is given by: dW finish W start W W D C L C c V x =- ò 1 . Example 1: You take off weighing 40,434 lbs (this weight includes fuel, armament, and ordnance) and you travel at V = 347.5 mi/hr. You arrive at the target area weighing 36,434 lbs. By use of a numerical integration technique, with an increment size of 1000 lbs in your partition, estimate the distance you have traveled. Does your answer depend on your increment size? Solution: This requires us to numerically evaluate the integral - ò 36,434 40,434 3605.8 W dW , which we rewrite as ò 40,434 35,434 3605.8 W dW . We use the trapezoidal rule 3605.8 * (.5 * * * .5 * * ) 1 1 x f 0 W f W f n W n i = D + å i D + D - = to evaluate the integral, with W f W 1 ( ) = and DW = 1000 . Substituting for f yields: * .5 * 40,434 * ) 1 3605.8 * (.5 * 35,434 * 1 1 W W W x W n i i = D + å D + D - = , where Wi is the value of W in the i'th subinterval (equal to W = 35,434 + (i - 1) * (40,434 - 35,434) i ). This technique is implemented in the following spreadsheet: Initial W: 36434 Final W: 40434 Intervals: 4 Delta W: 1000 Distance: 375.6537 W f(W) Partial Sum 36434 2.74469E-05 0.013723 37434 2.67137E-05 0.040437 38434 2.60186E-05 0.066456 39434 2.53588E-05 0.091815 40434 2.47317E-05 0.10418
Thisyieldsadistancetraveledof375.6miles.Welooktothenextexampletobetteranswerthequestion"is the calculated rangeafunctionof incrementsize?"Example2:Refineyourestimateby increasingthenumberofpartitions.Whatappears tobethe limit as thenumberofpartitions increases without bound?Solution:Repeatingtheaboveprocessfordiffering numbers of subintervalsyields thefollowing sequenceofvaluesforthedistancetraveled:IntervalsDistance375.664375.6537375.6410375.61820375.6129375.6240375.6116375.6100375.6112400375.6112375.584102040100400Number of Intervals (not to scale)The calculatedrangeappearstobeamonotonicallydecreasingfunctionofthenumberofsubintervals(orconversely,amonotonicallyincreasingfunctionofincrement size).This also appearstobea convergent sequence witha limit ofapproximately375.6miles.Notehowfewtermsarerequired(inthiscase)toconverge very closeto the apparent limit ofthenumerical integration scheme.Example3:Nowevaluatethedefiniteintegral tofindthedistancetraveledSolution:Evaluatingthedefinite integral,whichiseasyto doforthis simpledw= 3605.8* In(W) 10;43 = 375.6112 miles. This is inintegrand,yields3605.8W35,434excellentagreementwiththenumericalsolutionabove.Endurance EquationTo determinehow longyou canloiter in thetarget area with agiven amountoffuel,we needto relatethetimettothefuel consumption.Withthehelpof someequations from ourfluid dynamics background, wefind that, ifwe assumethatwe are loitering at a constant velocity, V,and a constant lift coefficient Cr,wehave34
34 This yields a distance traveled of 375.6 miles. We look to the next example to better answer the question “is the calculated range a function of increment size?” Example 2: Refine your estimate by increasing the number of partitions. What appears to be the limit as the number of partitions increases without bound? Solution: Repeating the above process for differing numbers of subintervals yields the following sequence of values for the distance traveled: The calculated range appears to be a monotonically decreasing function of the number of subintervals (or conversely, a monotonically increasing function of increment size). This also appears to be a convergent sequence with a limit of approximately 375.6 miles. Note how few terms are required (in this case) to converge very close to the apparent limit of the numerical integration scheme. Example 3: Now evaluate the definite integral to find the distance traveled. Solution: Evaluating the definite integral, which is easy to do for this simple integrand, yields 3605.8 3605.8 * ln( )| 375.6112 40,434 36,434 40,434 35,434 ò = W = W dW miles. This is in excellent agreement with the numerical solution above. Endurance Equation To determine how long you can loiter in the target area with a given amount of fuel, we need to relate the time t to the fuel consumption. With the help of some equations from our fluid dynamics background, we find that, if we assume that we are loitering at a constant velocity, V, and a constant lift coefficient CL , we have Intervals Distance 4 375.6537 10 375.618 20 375.6129 40 375.6116 100 375.6112 400 375.6112 375.58 375.6 375.62 375.64 375.66 4 10 20 40 100 400 Number of Intervals (not to scale) Distance Traveled
dxdtCL1dwdxdwWCDdtThus, t, the loiter time, is given by:W.Wend1Ci11CLend-dwdwWcCDWcCpWbeginhegiExample4:You arrived at thetarget weighing36,434 Ibs. The S-3 (Air)directs you toreconnoiterthetargetfor15minutes(0.2500hour).Howmuchfuelwill youhaveforyour return trip assuming that the plane weighs 29,784 Ibs with itsarmamentand ordnancebut nofuel?Solution:Substituting into the endurance equation yields36,434dWdw0.2500=-10.3757,whichwerewriteas 0.2500=10.3757WW36,434WeEvaluating yields 0.2500 =10.3757(ln(36,434)-In(Wfmal). Solving for W fimal yieldsWfmal=35,566.6lb.Thismeansthatwewill have35,566.6-29,784=5782.6Ibsoffuel remainingwhenwearereadytoreturn.Strategy2:Flying at Constant VelocityiConstant AltitudeFortactical reasons.youarereguiredtoreturnhomeatconstantvelocityandconstantaltitude.Youmust,therefore,decreaseyourliftasyourplanelightensbydecreasingyourliftcoefficient.Itturnsout,aftersomework,thatwecanderive therelationshipdxVdwcqSCDo+O聘= 0.03700,q = 541.894, S =506.0 ft2 (the surfacewhere a=2.330×10~area ofthewing),andc=0.37ooIbsoffuel/hr/lbthrust.Thus.thedistancetraveled, in miles, is given by:WWarriveZ1arrive1/WcqSCTcqSCDo+aw+0departdepa35
35 dt dW dx dW dx dt c CL CD W = = - 1 1 Thus, t, the loiter time, is given by: dW end W begin W W D C L C c dW end W begin W W D C L C c t = ò - = - ò 1 1 1 1 Example 4: You arrived at the target weighing 36,434 lbs. The S-3 (Air) directs you to reconnoiter the target for 15 minutes (0.2500 hour). How much fuel will you have for your return trip assuming that the plane weighs 29,784 lbs with its armament and ordnance but no fuel? Solution: Substituting into the endurance equation yields = - ò W final W dW 36,434 0.2500 10.3757 , which we rewrite as = ò 36,434 0.2500 10.3757 W final W dW . Evaluating yields 0.2500 10.3757(ln(36,434) ln( ) Wfinal = - . Solving for Wfinal yields Wfinal = 35,566.6 lb. This means that we will have 35,566.6 - 29,784 = 5782.6 lbs of fuel remaining when we are ready to return. Strategy 2: Flying at Constant Velocity/Constant Altitude For tactical reasons, you are required to return home at constant velocity and constant altitude. You must, therefore, decrease your lift as your plane lightens by decreasing your lift coefficient. It turns out, after some work, that we can derive the relationship ( ) 2 1 1 aW Do cqSC V dW dx + = - , where , 11 2.330 10- a = ´ = 0.03700 o D C , q = 541.894, S = 506.0 ft2 (the surface area of the wing), and c = 0.3700 lbs of fuel/hr/lb thrust. Thus, the distance traveled, in miles, is given by: ( ) dW arrive W depart cqSCDo W aW V dW Warrive depart W aW Do cqSC V x ò ÷ ø ö ç è æ + ò = - + = - 2 1 1 2 1 1