High Speed Problems of Aircraft and Experimental Methods 9781400875030

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HIGH SPEED PROBLEMS OF AIRCRAFT AND EXPERIMENTAL METHODS

BOARD OF EDITORS THEODORE VON KARMAN, Chairman HUGH L. DRYDEN HUGH S. TAYLOR COLEMAN DUP. DONALDSON, General Editor, 1956Associate Editor, 1955-1956 JOSEPH V. CHARYK , General Editor, 1952-1956 Associate Editor, 1949-1592 MARTIN SUMMERFIELD, General Editor, 1949-1952 RICHARD S. SNEDEKER, Associate Editor, 1955-

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Thermodynamics and Physics of Matter. Editor: F. D. Rossini Combustion Processes. Editors: B. Lewis, R. N. Pease, H. S. Taylor Fundamentals of Gas Dynamics. Editor: H. W. Emmons Theory of Laminar Flows. Editor: F. K. Moore Turbulent Flows and Heat Transfer. Editor: C. C. Lin General Theory of High Speed Aerodynamics. Editor: W. R. Sears Aerodynamic Components of Aircraft at High Speeds. Editors: A. F. Donovan, H. R. Lawrence High Speed Problems of Aircraft and Experimental Methods. Editors: A. F. Donovan, H. R. Lawrence, F. Goddard, R. R. Gilruth Physical Measurements in Gas Dynamics and Combustion. Editors: R. W. Ladenburg, B. Lewis, R. N. Pease, H. S. Taylor Aerodynamics of Turbines and Compressors. Editor: W. R. Hawthorne Design and Performance of Gas Turbine Power Plants. Editors: W. R. Hawthorne, W. T. Olson Jet Propulsion Engines. Editor: Ο. E. Lancaster

VOLUME VIII HIGH SPEED AERODYNAMICS AND JET PROPULSION mι•

HIGH SPEED PROBLEMS OF AIRCRAFT AND EXPERIMENTAL METHODS

EDITORS

PART 1. A. F. DONOVAN, H. R. LAWRENCE PARTS 2 AND 3. F. E. GODDARD PART 4. R. R. GILRUTH

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1961

COPYRIGHT, 1961, BY PRINCETON UNIVERSITY PRESS

London: OXFORD UNIVERSITY PRESS L. c. CARD 59-11073

Reproduction, translation, publication, use, and dis­ posal by and for the United States Government and its officers, agents, and employees acting within the scope of their official duties, for Government use only, is per­ mitted. At the expiration of ten years from the date of publication, all rights in material contained herein first produced under contract Nonr-03201 shall be in the public domain.

PRINTED IN THE UNITED STATES OF AMERICA

FOREWORD On behalf of the Editorial Board, I would like to make an acknowledgement to those branches of our military establishment whose interest and whose financial sup­ port were instrumental in the initiation of this publi­ cation program. It is noteworthy that this assistance has included all three branches of our Services. The Department of the Air Force through the Air Re­ search and Development Command, the Department of the Army through the Office of the Chief of Ord­ nance, and the Department of the Navy through the Bureau of Aeronautics, Bureau of Ships, Bureau of Ordnance, and the Office of Naval Research made significant contributions. In particular, the Power Branch of the Office of Naval Research has carried the burden of responsibilities of the contractual ad­ ministration and processing of all manuscripts from a security standpoint. The administration, operation, and editorial functions of the program have been cen­ tered at Princeton University. In addition, the Univer­ sity has contributed financially to the support of the undertaking. It is appropriate that special appreciation be expressed to Princeton University for its important over-all role in this effort. The Editorial Board is confident that the present series which this support has made possible will have far-reaching beneficial effects on the further develop­ ment of the aeronautical sciences. Theodore von Kdxmdn

PREFACE Rapid advances made during the past decade on problems associated with high speed flight have brought into ever sharper focus the need for a comprehensive and competent treatment of the fundamental aspects of the aerodynamic and propulsion problems of high speed flight, together with a survey of those aspects of the underlying basic sciences cognate to such problems. The need for a treatment of this type has been long felt in research institutions, universities, and private industry and its poten­ tial reflected importance in the advanced training of nascent aeronautical scientists has also been an important motivation in this undertaking. The entire program is the cumulative work of over one hundred scientists and engineers, representing many different branches of engineer­ ing and fields of science both in this country and abroad. The work consists of twelve volumes treating in sequence elements of the properties of gases, liquids, and solids; combustion processes and chemical kinetics; fundamentals of gas dynamics; viscous phenomena; turbulence; heat transfer; theoretical methods in high speed aerody­ namics; applications to wings, bodies and complete aircraft; nonsteady aerodynamics; principles of physical measurements; experimental methods in high speed aerodynamics and combustion; aerodynamic problems of turbo machines; the combination of aerodynamic and com­ bustion principles in combustor design; and finally, problems of complete power plants. The intent has been to emphasize the fundamental aspects of jet propulsion and high speed aerodynamics, to develop the theoretical tools for attack on these problems, and to seek to highlight the directions in which research may be potentially most fruitful. Preliminary discussions, which ultimately led to the foundation of the present program, were held in 1947 and 1948 and, in large measure, by virtue of the enthusiasm, inspiration, and encouragement of Dr. Theodore von Karman and later the invaluable assistance of Dr. Hugh L. Dryden and Dean Hugh Taylor as members of the Editorial Board, these discussions ultimately saw their fruition in the formal establishment of the Aeronautics Publication Program at Princeton University in the fall of 1949. The contributing authors and, in particular, the volume editors, have sacrificed generously of their spare time under present-day emergency conditions where continuing demands on their energies have been great. The program is also indebted to the work of Dr. Martin Summerfield who guided the planning work as General Editor from 1949-1952. The co­ operation and assistance of the personnel of Princeton University Press and of the staff of this office has been noteworthy. In particular, Mr.

PREFACE

Η. S. Bailey, Jr., the Director of the Press, and Mr. R. S. Snedeker, who has supervised the project at the Press have been of great help. The figures were prepared by Mr. Zane Anderson. Special mention is also due Mrs. E. W. Wetterau of this office who has handled the bulk of the detailed editorial work for the program. Coleman duP. Donaldson General Editor

PREFACE TO VOLUME VIII This volume of the Princeton Series has as its intention the completion of the present discussion of the aerodynamics of high speed aircraft and missiles which starts in Volume VI and is carried further in Volume VII. The first portion of the volume is concerned with aircraft design prob­ lems that are particularly associated with high speeds. An introductory discussion of the type of performance calculation peculiar to high speed aircraft is followed by a section devoted to the problems of stability and control at high speeds. The final section of this portion is a rather com­ plete treatise on the problems of aeroelasticity which are encountered with high speed aircraft. The second portion of the volume is concerned with the techniques of aerodynamic testing in wind tunnels and shock tubes. Consideration is given not only to testing techniques in the three speed ranges—tran­ sonic, supersonic, and hypersonic—but also to facilities for testing in the regime of rarified gas dynamics. The final portion of the volume is devoted to a discussion of free flight test techniques. A discussion of the basic techniques of instrumented rocketry is followed by a section on the testing of piloted aircraft. The final section is a treatise on free flight range techniques. It is unfortunate, although perhaps not unexpected, that in producing a volume such as, this, the rapid advances in experimental techniques that have taken place in the last few years coupled with the heavy commit­ ments of many of the authors to high priority national projects have made it almost impossible for all sections of this volume to be of equal currency. For this the volume editors hope that the individual contributors to this volume will not be blamed. With this in mind, the volume editors would like to take this opportunity to thank the authors for their cooperation and to express their appreciation for the assistance they have received from the General Editor and his staff. Our thanks are extended to all. A. F. Donovan H. R. Lawrence R. R. Gilruth

CONTENTS P A R T 1. HIGH SPEED PROBLEMS OF AIRCRAFT EDITORS: A . F. DONOVAN A N D H. R. LAWRENCE

A. Methods of Performance Calculation at High Speed

3

I. L. Ashkenas, Control Specialist, Incorporated, Englewood, California Introduction Basic Relations Point Performance Path Performance Application of Path Performance Methods Approximate Methods Cited References and Bibliography

3 5 11

B. Stability and Control of High Speed Aircraft

57

1. 2. 3. 4. 5. 6. 7.

20

34 49 56

Clarence L. Gillis and Thomas A. Toll, Langley Aeronautical Laboratory, National Aeronautics, and Space Administration Langley Field, Virginia 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Introduction Elementary Concepts Real Flows in Relation to Stability Definitions of Stability Derivatives Analytical Evaluation of Derivatives Analytical and Experimental Trends Controls Equations of Motion for Small Disturbances Longitudinal Motions Lateral Motions Small Amplitude Continuous Motions Special Dynamic Problems of Missiles Large Disturbance Cases Automatic Stability and Guided Flight Problems in Hypersonic Flight Problems of Take-off and Landing Cited References

57 60 67 71 75 94 104 108 112

126

131 135 138 145 149 152 155

CONTENTS

C. Aeroelasticity and Flutter

161

Alexander H. Flax, Cornell Aeronautical Laboratory, Incor­ porated, Buffalo, New York 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Introduction Early History The Two-Dimensional Wing Static Aeroelasticity—The Uniform Cantilever Wing Strain Energy and the Principle of Virtual Work Deformation Analysis of Structures Steady Aerodynamic Forces Algebraic Formulation of Aeroelastic Problems Matrix Formulation of Aeroelastic Problems Some Results from the Theory of Differential Equations IntegralEquationsofAeroelasticity The Nonuniform Unswept Wing of High Aspect Ratio. Strip Theory Static Aeroelastic Theory for Straight Wings of High Aspect Ratio Static Aeroelastic Problems of Swept and Low Aspect Ratio Wings Aerodynamic Forces in Flutter Flutter in One and Two Degrees of Freedom General Theory of Flutter of Continuous Structures Cited References

161 162 165 174 184 187 193 207 217 220 228 232 248 276 327 336 369 417

PART Z. WIND TUNNEL TECHNIQUES EDITOR: F. E. GODDARD

D. Principles of Model Testing

427

Albert E. von Doenhoff, Langley Aeronautical Laboratory, National Aeronautics and Space Administration, Langley Field, Virginia 1. Introduction 2. Dimensional Analysis 3. Application to Incompressible Frictionless Fluid with No Free Surfaces 4. Effects of Viscosity 5. Effects of Compressibility 6. Problems Peculiar to Tests at Very Large Mach Numbers 7. Problems Associated with Tests at Very Low Densities 8. Gravity Effects

427 427 428 429 433 436 437 438

CONTENTS

9. 10. 11. 12. 13.

Dynamic Model Testing Static Aeroelastic Effects Dynamic Aeroelastic Effects. Flutter Convective Heat Transfer Cited References

E. Introductory Discussion of High Speed Test Techniques

440 443 444 446 453 454

Frank E. Goddard, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 1. 2. 3. 4. 5.

Transonic Wind Tunnels Supersonic Wind Tunnels Hypersonic Wind Tunnels Low Density Wind Tunnels The Shock Tube

F. Transonic Wind Tunnels

455 456 457 458 459 460

Josiah E. Smith, Ramo-Wooldridge Corporation, Los Angeles, California 1. 2. 3. 4. 5.

Introduction Performance Requirements Types of Throats Conclusion Cited References

G. Supersonic Tunnels

460 461 464 486 486 491

Frank E. Goddard, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 1. 2. 3. 4. 5. 6. 7. 8.

General Types and Operational Requirements Supply-Section Design Nozzle Design Test-Section Design Diffuser Design Performance Calculations Typical Model and Balance Installations Cited References and Bibliography

H. Hypersonic Experimental Facilities

491 495 496 520 520 526 530 531 533

Henry T. Nagamatsu, Research Laboratory, General Electric Company, Schenectady, New York Chapter 1. Hypersonic Wind Tunnels

1. Introduction and Description of Several Tunnels 2. Flow Conditions in the Test Section

534 537

CONTENTS

3. 4. 5. 6. 7.

Condensation of the Working Medium The Diffuser Starting Problem in the Diffuser Hypersonic Wind Tunnel Instrumentation Methods of Designing Hypersonic Nozzles

540 544 547 548 553

Chapter 2. Intermittent Hypersonic Wind Tunnel with Compression Heater

8. Introduction 9. Convection and Compression Heaters 10. General Arrangement and Types of Aerodynamic Problems

555 556 557

Chapter 3. Hypersonic Helium Tunnel

11. Introduction 12. Operative Characteristics 13. General Arrangement

560 560 563

Chapter 4. Hypersonic Shock Tunnel

14. 15. 16. 17. 18.

Introduction Components of a Hypersonic Shock Tunnel Instrumentation for the Shock Tunnel Investigations in the Hypersonic Shock Tunnel Cited References

I. Low Density Wind Tunnels

1. 2. 3. 4. 5. 6. 7. 8.

E. D. Kane, California Research and Development Corpora­ tion, San Francisco, California G. J. Maslach and Samuel A. Schaaf, Department of Mechani­ cal Engineering, University of California, Berkeley, California Introduction Types of Low Density Wind Tunnel Available Drive Equipment Tunnel Element Selection Instrumentation Mechanical Design Other Low Density Flow Equipment Cited References

J. The Shock Tube

566 567 571 572 573 576

576 577 579 581 585 594 594 595 596

Walker Bleakney, Department of Physics, Princeton Uni­ versity, Princeton, New Jersey R. J. Emrich, Department of Physics, Lehigh University, Bethlehem, Pennsylvania 1. Introduction 596 2. Elementary Theory 598

CONTENTS

3. 4. 5. 6.

Construction and Technique of Operation Comparison of Performance with Theory Examples of Uses of Shock Tubes Cited References and Bibliography

K. Wmd Tunnel Measurements

615 626 642 644 648

H. Julian Allen, J. H. Spiegel, John Dimeff, Benjamin H. Beam, Jackson R. Stalder, and Alvin Seiff, Ames Aeronautical Laboratory, National Aeronautics and Space Administration, Moffett Field, California Chapter 1. Test-Section Flow Uniformity

1. 2. 3. 4. 5. 6. 7. 8.

Introduction Effects of Flow Nonuniformity Effects of Wall Interference Equations of Subsonic Wall Interference Boundary Conditions Blockage and Lift Interference in a Circular Tunnel Discussion of Slotted and Porous Boundaries Theory of Supersonic Wall Interference

648 648 656 660 663 664 667 672

Chapter 2. Methods of Model Support

9. Introduction 10. Wall Support Methods 11. Strut and Sting Support Methods

683 683 685

Chapter 3. Static Force Measurements

12. 13. 14. 15. 16.

Introduction The Electric Straingauge Electrical Circuits The Internal Spring Balance Calibration of Balances

691 693 710 715 737

Chapter Jt. Dynamic Force Measurements

17. Measurement of Dynamic Forces

739

Chapter 5. Convective Heat Transfer Measurements

18. 19. 20. 21. 22.

Introduction Simulation Methods of Measurement Summary Cited References

751 752 758 762 763

CONTENTS PART S . FREE FLIGHT TECHNIQUES EDITOR: R. R. GILRUTH

L. Objectives of Free Flight Research

771

Robert R. Gilruth, Langley Aeronautical Laboratory, Na­ tional Aeronautics and Space Administration, Langley Field, Virginia M. Instrumented Models in Free Flight

775

Joseph A. Shortal and William J. O'Sullivan, Jr., Langley Aeronautical Laboratory, National Aeronautics and Space Administration, Langley Field, Virginia 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction Propulsion Launching and Boosting Required Physical Measurements Free Flight Model Instrumentation Simple Drag Testing Simple Control Measurements Measurement of Damping in Roll Complete Model Testing Tests of Flutter and Other Aeroelastic Effects Aerodynamic Heating and Friction Drag Propulsive System Testing Cited References

N. Piloted Aircraft Testing

775 777 781 784 788 796 799 800 802 816 819 824 830 833

Frederick J. Bailey, Jr., John A. Zalovcik, William H. Phillips, and Wilber B. Huston, Langley Aeronautical Laboratory, National Aeronautics and Space Administration, Langley Field, Virginia Chapter 1. Basic Flight Measurements

1. 2. 3. 4. 5. 6. 7. 8.

Measurement of Air Speed Measurement of Air Temperature Measurement of Altitude Measurement of Lift Coefficient Angle-of-Attack Measurement Measurement of Thrust Measurement of Drag Boundary Layer and Wake Characteristics

836 843 844 846 846 847 851 853

CONTENTS Chapter 2. Flight Measurements of Stability and Controllability of Airplanes

9. Objectives of Stability Measurements

855

HANDLING QUALITIES INVESTIGATIONS

10. 11. 12. 13. 14.

Introduction Instrumentation Longitudinal Stability and Control Lateral Stability and Control Stalling Characteristics

855 856 857 863 873

DYNAMIC STABILITY INVESTIGATIONS

15. Introduction 16. Instrumentation 17. Dynamic Stability Characteristics of an Automatically Con­ trolled Airplane 18. Determination of Stability Derivatives from Flight Tests

873 874 875 877

Chapter 3. Flight Measurements of the Loads on Airplanes ACCELEROMETER MEASUREMENTS

19. Application of Accelerometers to Loads Research

879

STRAINGAUGE MEASUREMENTS

20. Application of Straingauges to Loads Research 21. Calibration Procedure 22. Evaluation of Flight Data

882 883 895

PRESSURE MEASUREMENTS

23. Pressure CoeiScients 24. Measurements of Pressure

896 897

ELASTIC DEFORMATION MEASUREMENTS

25. Methods of Measuring Elastic Deformation 26. Cited References 0. Free Flight Range Methods

900 901 905

A. C. Charters, Ames Aeronautical Laboratory, National Aeronautics and Space Administration, Moffett Field, California Chapter 1. The Free Flight Range

1. 2. 3. 4. 5.

Introduction The Range as a Flight Test Facility Current Status of Free Flight Range Facilities Experimental Procedure Model Measurement and Launching Techniques

905 906 909 910 910

CONTENTS

6. Spark Photography Apparatus 7. Chronographs

911 918

Chapter 2. Theory of Motion

Fundamental Considerations Geometry Conditions of Testing Differential Equations for Rotation Differential Equations for Translation Differential Equations for Rotation and Translation in Terms of Complex Angles and Displacements 14. Integrated Equations of Motion

8. 9. 10. 11. 12. 13.

920 922 925 927 933 935 937

Chapter 8. Reduction of Flight Data

15. Determination of Parameters of Motion 940 16. Determination of Aerodynamic Coefficients from Parameters 948 of Motion 17. Accuracy of Experimental Data 951 18. Reduction of Data for Missiles with Aerodynamic Asym­ 951 metries and Nonlinearities 19. Illustrative Example: Reduction of Data from BRL Aero­ 953 dynamics Range Round 2594 956 20. Definition of Symbols 961 21. Cited References Index

963

PART ONE

HIGH SPEED PROBLEMS OF AIRCRAFT

SECTION A

METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED I. L. ASHKENAS A,l· Introduction. The prediction or calculation of aircraft performance serves a twofold purpose: (1) to fix the airframe configuration best suited to a given job, and (2) to outline, in whatever detail is desired, the operational utility of any given machine. Thus the conception of any new aircraft must be based on an adequate solution of the performance problem. The final configuration will be additionally affected, of course, by considerations of stability and control, but in the last analysis the preliminary design problem reduces to one of how far or how fast or how high an aircraft will fly with how much volume or weight available for payload. Once the aircraft has been laid down in a preliminary fashion and the detailed design has begun, continued performance analysis is required, not only as an aid to making design decisions which may affect the possibility of accomplishing the craft's primary mission, but, further, to establish the environment in which the machine must be capable of functioning and surviving. For example, maximum rates of climb and dive, and the speeds and attitudes associated with each, although not expressly required for the proposed mission, never­ theless strongly influence the design of such items as the fuel system, the cabin conditioning system, and the aircraft's structure. Even after the aircraft has been accepted into service, the requirement for continued performance prediction exists. Now, such calculations are helpful in discover­ ing improvements in the efficiency of the basic operation, or to uncover new uses for the aircraft, or to indicate revisions in the operating technique due to possible changes in the power plant, the payload, or the equipment. The accurate calculation of airplane performance thus emerges as an omni­ present problem and, as such, it has naturally received considerable attention. In the past much of this attention was focused on the solution of performance problems for low speed airplanes. In the present and future, in addition to such problems as are common to both types, we have to consider the additional performance aspects of high speed airplanes. These two classifications are essentially relative, so that, to clarify our thinking, let us consider that a low speed airplane is (1) powered by a reciprocating engine-propeller combination, and (2) for all flight conditions

A · M E T H O D S OF PERFORMANCE CALCULATION A T HIGH SPEED

except the dive, it flies at Mach numbers below the critical. A high speed airplane on the other hand (1) flies for a great variety of important conditions at a Mach number greater than the critical and (2) is generally jet-propelled by either a gas turbine engine, a ramjet, a pulse jet, or a rocket motor. It is apparent, as a result of the above somewhat arbitrary definitions, that for the high speed airplane an additional primary variable, the Mach number, M, is immediately involved. Whereas for the low speed airplane a single GD(CL) relationship served over the entire flight regime for a given configura­ tion, now, for the high speed airplane the CD(CI) relationship is critically dependent on M, and a single drag "polar" no longer suffices to define the aircraft. Because of this dependence, performance characteristics for high speed airplanes can be more conveniently expressed in terms Qf pressure and temperature, rather than density. In general, too, the power plant types involved have characteristics and governing parameters greatly different from those used on low speed machines, generally including much higher specific fuel consumption. That this basic difference, while it is the most obvious change, particularly from the standpoint of the mechanics of making the performance calculations, can be a relatively minor effect compared to other differences is not, at first thought, too apparent. For one thing the high speed airplane is essentially also a high altitude airplane. It must, if it is to be operated efficiently and effectively, spend the major portion of its flight time above 35,000 feet, and in general the higher the better; whereas for a low speed machine, increasing the cruising altitude usually results in increased power plant complication, decreased range, increased cooling capacity and drag, etc. Because of this, climb and descent performance becomes of increasing importance, and for some high speed machines the choice of climb and descent paths can critically influence the effective accomplishment of the aircraft's mission. Another difficulty, associated with the large thrusts required to obtain high speeds, arises from the accelerations to which the airplane may be subjected. In general, inertia effects are of considerable importance and must be carefully examined in any formulation of the performance problem. Finally there is a large region of uncertainty, or at least an area of compromise, involved in a high speed machine which will normally result in the imposition of operative limitations due to such effects as aerodynamic heating, extreme aeroelastic effects, buffeting, maximum lift changes with M, and loss of control effectiveness, to name some of the more common ones. Such limitations must, of course, be considered in the prepara­ tion of the performance flight plan. With these differences in mind, let us consider briefly the interplay of conflicting requirements that may be specified for a typical high speed airplane. In the first place, it will be necessary to know the desired high speed so that the designer can properly choose the power plant type and approximate wing and body geometry. The choice of speed can even affect the type of

A,2 · BASIC RELATIONS

structure and the materials to be used, since, in general, higher Mach numbers will ordinarily involve increased skin temperatures and increasingly thinner wings. The required flight altitude or range of altitudes is also of considerable importance, particularly for machines which require aerodynamic lift to maintain and/or control their flight path. For such machines, increasing the altitude requires an increase in the area of the lifting surface or Mach number and, in either case, for air-breathing engines, an increase in power plant size. Reducing the altitude or requiring operation at low altitudes, as well as high, imposes large air loads on the airplane and power plant structures resulting in increased weight for these items. Other performance requirements can influence the basic design. Some of the more common items which may be specified are: take-off distance, landing distance, time to climb to design altitude, turning radius (and time) at design altitude, range, and cost. For high altitude airplanes, take-off and landing will normally not be critical because of the low wing and power loadings required for such designs: the converse is obviously also true. For long range machines the amount of fuel which can be carried is exceedingly critical, and this consideration can impose severe restrictions on the aerodynamic shapes of practical interest. Wings with extreme sweep or small thickness may become excessively heavy and thereby severely restrict the available range. The choice of design altitude may require a power plant which is excessively large and therefore heavy, so that the range again suifers. The foregoing examples of the aerodynamic, structural, and performance compromises which may be required to obtain a useful airplane illustrate some of the difficulties involved in the preliminary design of a high speed machine. For an airplane required to fly in a specified range of supersonic speeds, optimizations and compromises involving wing loading, power loading, aspect ratio, taper ratio, and fuel load would be required for a number of design Mach numbers, and possibly for more than one engine type, before a final configuration could logically be established. The large amount of labor and time involved in making such design studies seriously limits the number of variables usually considered and, hence, the validity of the final choice of configuration. It is highly desirable, therefore, from the standpoint of design performance to have available short-cut methods of computation which yield approximate, or at least comparatively correct, results in a minimum time. The performance of the chosen configuration can then be checked by a more rigorous calculation. In the sections to follow, both of these aspects of the performance problem for high speed airplanes will be covered. Reasonably exact perform­ ance equations and methods will be developed and discussed first. Useful approximations and short-cut methods will then be presented. A,2. Basic Relations. The equations defining the various performance items of interest are derived from simple considerations of mechanics and

A · M E T H O D S OF PERFORMANCE CALCULATION A T HIGH SPEED

kinematics. These items can be considered in general to be largely concerned with space-time relationships in the vertical plane. The one exception to this general interest is in the case of turning performance. The ability of an airplane or missile to execute sharp turns can be of extreme importance in any interception problem, but the plane in which a turn is required depends on the relative positions of the target and interceptor and the path of the target; the possibilities are infinite. It is common practice, therefore, to assess

Fig. A,2a.

Free body diagram. Vertical plane.

maneuverability by calculating the turning performance in the horizontal plane at maximum power, and in a spiral diving turn at maximum lift. Accordingly, the performance expressions to be derived and discussed will be concerned with relations in the vertical plane, except for the specific case of turning performance. Consider the free-body diagram shown in Fig. A,2a, where F = power plant thrust D = drag W = airplane weight L = lift

γ = angle of flight path to horizontal a. = angle of attack Also define F = velocity along flight path χ = horizontal distance along flight path ¾ = altitude = vertical distance along flight path re = radius of the earth

A,2 • BASIC

RELATIONS

Equating forces parallel and normal to the flight path, (2-1)

(2-2)

and by definition, (2-3) (2-4) (2-5) Eq. 2-1 and 2-2 can be considerably simplified, with little loss of accuracy, for all but a very few special problems. In the first place h is generally very small compared to r e , so that the term which accounts for the change in the earth's gravitational field with altitude, may be set equal to unity. Further simplification in the case where airplane weight is largely supported by aerodynamic lift is possible by assuming sin The error involved in this last assumption can be evaluated as follows: Assume that the drag due to lift may be represented by the expression where represents the forward inclination of the lift vector from a normal to the wing and is dependent on wing geometry and Mach number. Consider, now, the excess thrust in both cases; then for the general case,

and for the simplified

and the error in net thrust due to the simplification is

( 7 )

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

For the case of a wing with a supersonic leading edge, e = 0, whereas for a wing swept behind the Mach line, e ^ 0.25 and for a subsonic wing e = JRI(2 + .

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

For increasing values of L j d a a = W j d x , the excess thrust and the corresponding rate of climb are progressively decreased. Zero excess thrust finally occurs at the value of Ljdta which is just tangent to the pertinent Fjdaa curve. The absolute ceiling or the altitude corresponding to zero rate of climb is then obtained by solving for the dx, corresponding to the maximum value of Ljdx above, and the values of η and W corresponding to the problem at hand

oo)max

4. The speeds corresponding to a maximum rate of climb per pound of fuel are obtained by applying the construction illustrated in Fig. A,3b to the thrust corresponding to a constant fuel flow. Points 4 in Fig. A,3a are such speeds for an Ljd = 80,000 lb and the indicated fuel flows. For power plants operating in a region where the net propulsive efficiency is not rapidly dropping off, an increase in thrust will give a proportional increase in fuel consumption. Since rate of climb depends, however, on excess thrust (F — D), the relative increase in climb for a given thrust increase will be greater than the relative increase in fuel flow. Therefore, the rate of climb per pound of fuel steadily increases with increasing thrust, and the speed corresponding to the maximum occurs at the maximum thrust, along the line of points 4. These intersection points are shown as an asterisk in Fig. A,3a. In cases where the propulsive efficiency drops off with increasing power or thrust, the speed for maximum climb per pound does not necessarily correspond to the maximum thrust point. Such cases can usually be recognized by the crowding of the constant fuel flow lines, and the best speed and thrust must be determined by sampling. 5. From the discussion under item 3 above, it is clear that the maximum climb angle corresponds to the speed for maximum excess thrust. These speeds can be determined by direct inspection (Fig. A,3a, points 5). 6. The cruise speed for maximum range at a given W j d 0 0 corresponds to the speed at which the drag curve is tangent to lines of constant reduced fuel economy (mi/lb), and the range factor is given by the corresponding value of the fuel economy (Fig. A3a, points 6). 7. The cruise speed for maximum time at a given W j d a o corresponds to the speed at which the drag curve is tangent to lines of constant reduced fuel flow, and the fuel flow is read at this point (Fig. A,3a, points 7). 8. The speed for maximum-load-factor horizontal turns is the speed at which maximum Ljdao can be obtained in level flight. This corresponds to the point at which the Ljd00 family becomes tangent to the maximum thrust line, and the load factor is given by the ratio of the corresponding L j d 0 0 to the pertinent value of W j d 0 0 . ao

A,3 • POINT

PERFORMANCE

9. Minimum radius and minimum time horizontal steady turns can be obtained by considering the basic turn relations

and setting the derivatives with respect to M equal to zero, whereby

or The points corresponding to the above conditions may be obtained from an auxiliary plot of the square of the maximum lift obtainable in steady horizontal flight vs. M, as indicated in Fig. A,3c, point 1 corresponding to

Fig. A,3c.

Construction for minimum radius and minimum time turns.

the speed for minimum radius turn, point 2 corresponding to the speed for a minimum time turn. The foregoing items of point performance are of some interest in themselves, but they represent the performance capabilities of a given design in only an approximate way. For a more complete picture of the potential of a machine, it is necessary to solve the general problem of path performance, which to some extent we can do by using point performance items. Consider, ( 15

>

A · M E T H O D S OF P E R F O R M A N C E C A L C U L A T I O N A T H I G H S P E E D

for example, the problem of calculating the minimum time to climb between two altitudes. From Eq. 2-4 and 2-7, and setting d V j d t = G d V / d h , F

c =

D

-

C

WdV

V

g dh

~ w ~=- - - c

(F-D)V

C'

/

VdV\

\

g dh)

TFlH

IH

VdV 0 dh

where C denotes the point performance climb ( d V j d t performance forward speed. Then,

/ dh \ dt = — = C

—

0) and

V

the point

V d V

\jj, g dh)

d

A,3 • POINT

PERFORMANCE

or in the stratosphere then the condition for best range becomes

and the range may be written

(3-9)

where the exponent a has the two values, p — 0.19 or p, for flight in the troposphere and stratosphere respectively. The initial cruising altitude will be determined by calculating If this altitude is above 35,000 feet, then the entire range will be flown in the stratosphere and the corresponding equations apply. If it is less than 35,000 feet, then the first portion of the range will be flown in the troposphere until the weight has been sufficiently reduced to bring the cruise altitude up to 35,000 feet. For the remainder of the cruise, then, flight will be in the stratosphere. The maximum endurance problem is handled in a manner exactly analogous to the range problem. From the data obtained as in item 7, (in the general case), optimized from a speed standpoint, is obtained as a function of only Remembering that can be represented by and optimizing with respect to altitude,

The conditions for maximum endurance thus correspond to the point on the ( 19 >

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

log-log plot of W x I d p c a O t x versus W j d a s when the geometric slope is equal to a . The endurance can now be calculated: CiWIdJa dW

rdW

J =^rf - -J (Wf/o W^ t =

(Wldaa)I

iw\- a ~w\-%

(W^)c I

1

—

a

/

A,4. Path Performance. The preceding items of path performance are very simple examples in that the basic problem statement requires only a maximization with regard to a single parameter, the other variables being completely unrestrained. Further, the optimized path that results represents only a portion of the flight plan; and thete is no information available as to how best to arrive at the conditions representing this optimum from an initial completely arbitrary state. The general problem, wherein we seek to define a maximum performance path between two sets of arbitrary conditions, will now be approached by considering again Eq. 2-3 through 2-7, previously expressed, which govern the flight of an aircraft.1 From these we can itemize the variables, which eventually must be determined, as ten in number, namely L, D, W, F, t, x, wt, h, V, and γ. If we consider that one of these will be the path independent variable we are left with nine dependent variables to be determined. Now, in addition to these basic performance equations, we must necessarily know something about the airframe and power plant with which we are concerned, and so we write two additional functional relationships expressing this knowledge: D = D(L, h, V)

(4-1)

F = F{wt, h, V)

(4-2)

To complete the system of equations, two other independent relations can be found in the control exercised over lift and fuel consumption. These relations are, in general, undetermined functions of the path independent variable (say f) which have the form L = L{t) (4-3) wt = Wf(t)

(4-4)

As will be evident, these "control functions," when determined, never involve other variables than those appearing in the other relations, and they therefore complete the system of equations. In a few cases, the control functions arid the drag and thrust relations can be reduced to simple analytical expressions without great errors, although in many practical problems these relations must be represented graphically or 1 The author is indebted to G. S. Raetz for much of the mathematical development of this article.

A,4 · PATH PERFORMANCE

numerically. On the other hand, the basic performance equations constitute a difficult nonlinear system of differential equations which, even for the most elementary forms of the other path equations, cannot be treated analytically in a simple manner. Indeed, these differential equations cannot be integrated in quadratures, and even integration in series is impractical because the resulting terms are complex and difficult. Therefore the only practical method of solving the path equations is numerical integration. A simple numerical solution would consist of dividing the path into segments of small time intervals and, for each interval, performing the following sequence of calculations, where for steps 1 through 6 the increments summed are those of steps 16 through 21 of all previous segments: Table A,4

Step 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Quantity

Step

t = t t + Σ(Δί)

Σ(Δ®) W = W 1 + Σ(ΔΜ0 X = X1 +

h = Ii 1 + X ( A h ) V = V1 + E(AV) V = Vi + Ζ(Αγ) Eq. 4-3 L = L(t) Eq. 4-4 Wt = wt(t) Eq. 4-1 D = D(L, h, V) F = F(wu h, V) X = V cos γ

Eq. 4-2 Eq. 2-3

Quantity

12. W = — Wt 13. h = V sin γ 14. V = ( g / W ) ( F - D - W sin γ ) 15. V = ( g / W V ) ( L - W cos γ ) 16. A t = chosen 17. Ax = χ • (At) 18. A W = W • ( A t ) 19. Ah = h • (At) 20. A V = V • ( A t ) 21. Αγ = γ • (At)

Eq. 2-5 Eq. 2-4 Eq. 2-7 Eq. 2-7

In this procedure, the path performance is represented by the relationship of the values t, x, W, h, V, and γ appearing in steps 1 through 6. While any degree of accuracy can be attained by choosing sufficiently small time inter­ vals, it is often necessary, in order to avoid diverging errors, to keep the time intervals small even when accuracy is not critical. For this reason it is generally preferable to use one of various modifications of this procedure such as are commonly employed in numerical integration. The above procedure therefore is of value mainly as a simple illustration of the principles involved. Of course, automatic calculating machines ordinarily should be used when a number of paths must be determined, although in some cases the nature of the relations defining the airframe, the power plant, and their control and the limitations of such machines may warrant or require manual calculation. At this point, we need only to determine the control functions in order to compute a path. But this, of course, is the crux of the problem, because what we would like to do is to determine these functions so that the resulting path represents an "optimum." We regard an "optimum path" as one in which the net change in one path performance variable, termed "optimized variable," is a maximum or minimum as specified, while the net change in one or more other path variables, termed "constrained variables," equals specified values. The optimized variable is usually time, distance, or weight of fuel, and the

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

constrained variables ordinarily are time, distance, weight, height, speed, and path angle. There are several reasons why optimum paths are significant. A primary reason, of course, is that the associated performance exactly represents the ultimate capability of a machine. Another important reason, however, is that this performance also approximately represents the capa­ bility of the craft along a usually wide band of neighboring paths, and there­ fore indicates well its behavior in actual operation where variations in control occur. In fact, because of this tolerance, the path itself constitutes an excellent set of control instructions. Now the net changes in the optimized and constrained variables are given by integrals taken along the path, and the problem at hand is that of finding, from the infinity of possible functions relating L and W{ to the path variables, those particular functions yielding an extreme relationship among these integrals. Because functions extremizing an integral rather than values extremizing a function are sought, this problem is one of the calculus of variations rather than one of ordinary maxima and minima. In the past, particularly for low speed flight, this problem has been greatly simplified by introducing several approximations, such as equating some of the derivatives W, h, V, and γ to zero. However, in high speed performance, especially in some problems in this field, these previous approximations are often unreliable or entirely invalid, and a more exact solution generally is necessary. Further­ more, a knowledge of the exact solution always is desirable, for then the approximations permitted in a particular problem are easily found and checked. Therefore the ensuing analysis is devoted mainly to seeking the control functions for exactly optimum paths, although a few useful approxi­ mations are discussed briefly. A few elementary principles of the calculus of variations are used without discussion of the underlying theory, and therefore reference to the literature on this calculus, some of which is listed at the end of this section, may be helpful. Of this literature, the most applicable is that on the problem of Bolza since this problem contains the one concerned here as a special case. In deriving the control functions, the optimized variable is regarded as the independent path variable. Then, denoting the optimized variable as v, the undetermined control functions take the form

L — L(v) = Wf(v)

Wi

Also, denoting each constrained variable as μν where the index t is a number identifying the particular constrained variable considered, the net change in each constrained variable takes the form

where the superimposed dot is used to denote total differentiation with respect

A,4 • PATH PERFORMANCE

to the independent path variable, and the integrand /Ii appearing in each such expression is related to the path variables through the basic performance equations. For example, if weight and speed are the optimized and con­ strained variables respectively, Δ 7 =

Jw 1 dt dW

Jw 1

WWf

In all ordinary cases, as may be seen by inspection of the basic performance and the drag and thrust relations, the integrand μί is dependent only on some group of the variables L, W{, W, h, V, and γ, giving Δμι the general form Α μ = J*' μΙΣ, «*, W , h, V , γ) dv

(4-5)

The problem of determining the optimum can be simplified by replacing the constrained variables by a single linear function, termed "collective function," having the constraints as independent variables and the undeter­ mined constants as coefficients. Denoting this function as ψ and each undeter­ mined constant as Xi, where the index { is to be the same number as that of the associated constrained variable, the collective function has the form

i

and its first derivative may be expressed as ψ = 2 Ki 1 IL, W i , W, h,

V, γ )

i

or, except where evaluation of the undetermined constants is concerned, as ψ = I 1 b(L, W{, W, h, V , γ)

(4-6)

Corresponding to this latter form, the second derivative of ψ may be expressed as

But, as evident from inspection of the path equations, each of the derivatives Iir, h, Ϋ, and γ is in general dependent only on some group of the variables L, wt, W, h, V, and γ, giving ψ the general form ψ = y)(L, Wf, L, wt, W, h, V, γ)

(4-8)

Now suppose that ψ is a unique variable, i.e. that a unique set of A/s for the undetermined constants is chosen. Then, in general, there exist unique control functions yielding an extreme change in ν for a given change in ψ or vice versa. But the changes in ψ correspond to a particular set of changes, AjMi, among the individual constraints, for which the change in ν is

A · M E T H O D S OF P E R F O R M A N C E C A L C U L A T I O N A T H I G H S P E E D

still an extremum. Therefore the control functions corresponding to the chosen values X 1 are optimum for the resulting changes Δ μ . As the set of values X is varied, the resulting set of changes Δμ also varies, but the change in υ is an extremum provided that the control functions are optimum for the particular values of X . We see now that the concept of the collective function breaks a difficult problem into two relatively simple parts: first, the determination of the control functions for a single variable ψ depending on undetermined constants, and second, the evaluation of these constants to provide a par­ ticular set of changes among the individual constrained variables. In the ensuing analysis, this substitution is made, dividing the problem into two such parts. Although the optimized variable is regarded as the path independent variable, an expression giving the net change in the optimized variable in terms of the first and second derivatives of the collective function affords a direct approach to finding the control functions and will be used. This expression is obtained from the definition ψ = dip/dv as {

i

ί

i

(4-9)

where the subscripts 1 and 2 denote any two particular points along the path; and the conditions noted on ψ, with the further conditions that ψ and ψ not be indeterminate, are to be satisfied. In general, a path may be divided into segments differing in the fulfilment of these required conditions. Those segments along which all required conditions are fulfilled we call "ordinary segments;" those along which either ψ = 0 or ψ = ±oo we call "simple segments;" and those along which ψ and ψ are un­ defined we call "singular segments." Along a simple segment characterized by the condition ψ — Ο , ψ is a constant of maximum or minimum magnitude and the simple expression V

2

-V

1

=

is valid subject to the condition noted. If, as seldom happens, this condition is not fulfilled and ψ = 0, the condition ψ2 — ψ1 = 0 exists for all finite changes in v, which are the only physically possible changes in this variable. In contrast, along a simple segment characterized by the condition ψ = ^oo, only the condition V — V = 0 can exist if this segment is other than an isolated point of the path, since continuation of the condition ψ = ±co over a finite interval of υ would lead to the condition ψ2 — ψ1 = ±οο, which is physically impossible. A singular segment, along which ψ and ψ are undefined, has meaning only when equations of the control functions, which do not define the relationship of L and wj, are under consideration. Such equations can, and often do, come 2

1

A,4 · PATH PERFORMANCE

under consideration when the ordinary segments are continued to conditions where L and Wt are undefined. For such conditions, the equations under consideration have meaning only when a singular solution is admitted. With the relations now available, a fundamental property of optimum paths can be established. For this purpose, consider two paths starting at a particular set of initial conditions (I , X , W1, A1, V1, yx) and conforming to particular physical limitations, and let the undetermined constants have particular though not necessarily known values, making the collective function a unique variable. Along each path between the initial point 1 and a final point 2, the net change in the optimized variable, to be denoted as Δυ, can be expressed from Eq. 4-9 as 1

1

with the understanding, to be assumed tacitly hereafter, that this equation and all other relations based on it are to hold exactly along ordinary segments but are to admit special solutions along other segments. Next, suppose that one of the paths is an optimum path yielding, for the particular conditions considered, an extreme change in υ for a given change in ψ. At every point of this path, all variables and their derivatives have particular values, and therefore the control functions of this path are particular functions. Also suppose that another path, to be called the "varied path," is generated by control functions differing only slightly from those of the optimum path so that between the two paths the differences in L, wt, L, and Wf, and therefore in all other variables and their derivatives, are small. This change between the two adjacent paths is denoted by the term "variation" and the symbol δ as opposed to the normal "differential" operator d, which represents a change along a particular path. Each type of change is small by definition and each operator is entirely independent of the other, allowing the order of applying both operators to the same variable to be reversed (dd = dd). The variation of between the two paths, noting that ψ is the independent variable in this relation, may be expressed as the difference |>i+*«

άψ

Vi+ίψχ ψ + δ ψ

f*« dy> Jyi 1

ψ

which, after expansion of the integrand of the first integral in a Taylor's series and elimination of all terms beyond the first order, reduces to (4-10) The variations

δψχ, δψ 2 ,

and

δψ

appearing above may be expressed from

A • METHODS

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

Eq. 4-6 and 4-8, using an expansion in a Taylor's series and considering that there is no variation in the initial conditions, as (4-11) (4-12)

(4-13) provided, as will be assumed hereafter, that all partial derivatives involved are continuous and defined. The variations and included in these expressions are all dependent on the variations and since the arguments of these variations depend on L and through the path equations and the definitions Therefore the variations and and hence the variation are all dependent on the relationship of and to the path independent variable, regarded as for the moment. Moreover, the varied path can always be chosen so that the variations and everywhere have small finite values approaching zero as a limit. Along this type of varied path, which is the only type considered hereafter, L and ws are restricted in essentially the same manner, and all remaining quantities have essentially the same values, as along the optimum path. From all these facts, it is evident that for such varied paths the sign of can be reversed merely by reversing the signs of and at each value of , But along the optimum path, is an extreme and cannot be both increased and decreased as would be possible if were finite. Therefore the variations must vanish when evaluated along the optimum path with respect to sufficiently close neighboring paths conforming to the same conditions. Further, it is clear that in a sufficiently small neighborhood of the optimum path the value of is independent of the relationship of L and to the path independent variable, except possibly where these variables are restricted by physical limitations. Indeed, the values of L and can often differ from those of the optimum path by large amounts without changing the value of from the extremum by an unacceptable amount, a fact that often is convenient in approximating optimum paths. Among the possible varied paths of the type just described, some always can be found for which the variations , and all are continuous and defined along an arbitrary small interval and this interval can be so small that the change in and hence the change in v, is small compared to the variations, even though by definition all variations are small. Varied paths ( 26 )

A , 4 • PATH

PERFORMANCE

of this sort directly reveal a fundamental requirement on the form of the control functions and therefore will be examined in detail. Consider that across each interval to be explored, the variations and are small compared to the variations since the former initially are zero and are restrained by the basic performance equations to small changes when the change in v is small. Using this fact, the variation given exactly by Eq. 4-13, becomes, to a first order of accuracy, (4-14) and the variation

obtained from Eq. 4-6 becomes, to the same accuracy, (4-15)

Substituting into the relation (Eq. 4-14) the expressions found from Eq. 4-7

and the identities, based on the previously discussed properties of variations,

A • METHODS

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CALCULATION

AT

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SPEED

and the resulting expression into Eq. 4-10, remembering that the expression for becomes

Since the change in v is small compared to all variations which in turn are small, the interval under consideration actually is very small, allowing the quantities and appearing in the above integral to be treated as constants in the evaluation of this integral. Using this fact together with the observation that according to Eq. 4-15 the second term in the curly brackets is the above relation becomes

(4-16) But along the optimum path, the quantity and hence the difference must vanish, which condition can occur only if the integral in the above expression is, to a first order of accuracy, identically zero. Furthermore, although the varied paths under consideration are of a special nature, the variations and nevertheless are arbitrary to a large extent, except where their arguments are restricted on the optimum path by physical limitations. Therefore this integral will be zero for all possible variations of the type considered, only if the coefficients of these variations vanish wherever the variations themselves do not vanish due to the physical limitations. From these facts it is evident that, along the optimum path, either or both of the equations (4-17)

(4-18) must be satisfied individually, within the accuracy considered, wherever either or both L and respectively are unrestricted on the optimum path by physical limitations. Actually there is no difficulty in letting the interval under consideration approach a point as a limit, in which case all approximation errors become vanishingly small and the above equations become exact for that point, nor in locating this interval along any part of the optimum path. Therefore, Eq. 4-17 and 4-18 together with the restrictions < 28 )

A,4 · PATH PERFORMANCE

imposed by physical limitations must be satisfied exactly and at all points on the optimum path. This requirement when supplemented by other simpler requirements actually defines the control functions sought here. The foregoing equations can be expressed more conveniently by expanding the derivatives of these expressions using Eq. 4-6 and 4-7, whereupon several terms cancel out, leaving the simpler equations Bw dJV

I

BW BL

I T

Bw dfi Bw BV L ι ι dh B L ^ BV

ι

dw By L -LBy BL

— η

Βψ BW

Βψ Bh

Βψ ΒΫ

Bip By

BW Bwf

Bh Bwt

BV Bwi

Βγ Bwt

(4-19} K

'

to be satisfied except for the restrictions imposed by physical limitations. In particular applications, one or more terms in each of these "ordinary control equations" is usually zero, further simplifying the expressions. Since each of the derivatives, ψ, W, h, V, and γ is dependent only on some group of the variables L, Wf, W, h, V, and y, all partial derivatives in the above equations and hence the equations themselves are algebraic expressions in these variables, and the applicable physical limitations can ordinarily also be expressed algebraically in terms of these variables. Therefore, at every point of a path, the ordinary control equations, or whatever physical limita­ tions apply, constitute two simultaneous algebraic equations in the two unknowns sought here, namely L and Wf. However, since either or both of these equations may be of a degree greater than one, these equations generally may be satisfied by more than one set of values of L and w%, or, since they are not valid explicitlyfor conditions permitting simple or singular segments, they may be replaced altogether by equations applying to such segments. Hence, before the control functions can be established, the particular equations and roots applying to the optimum path sought must be selected from among these several possibilities and, in particular, the transition points between different equations and roots must be located. The selection of the applicable equations and roots can always be accom­ plished by the simple process of eliminating improbable solutions by careful inspection of the particular problem concerned and then trying the remaining solutions. Also, any solution or sequence of solutions can always be checked for extremizing properties merely by comparing it with closely neighboring but otherwise arbitrary relationships of L and Wf to v. In fact, these methods probably are as practicable as any other kind, especially after an initial path of the general type involved has become known by any means, and therefore they are assumed here to be suitable. In general, the ordinary control equations first apply explicitly but later may become invalid and thereby admit a solution for a nonordinary segment, which solution also may later admit still another one. Therefore, recognition of invalid or transition points

A · METHODS OF PERFORMANCE CALCULATION A T HIGH SPEED

is important, and the invalid points of the ordinary control equations will now be discussed. The points where ψ becomes zero or infinity are easily recognized from a plot of ψ versus v, since these points appear on this plot as those having horizontal or vertical slopes. In fact, such a plot is helpful otherwise in ascertaining the applicable solutions, since on this plot these solutions furnish the curves enclosing an extreme area. The points where ψ and ψ become undefined by the ordinary control equations, and a singular segment is possible, commonly but not necessarily occur at those conditions where these equations yield three or more real roots, two or more of equal value, for either or both L and Wf. To illustrate, suppose that Eq. 4-19 reduces to a relation in CL and γ, only, having the form represented in Fig. A,4, and that Wf is

CL Fig. A,4.

Typical C i , γ relationship.

restricted by some physical limitation. Further suppose that the path begins at point a and continues beyond the conditions corresponding to point g, so that branch a-g alone cannot define the entire path. For any point e, beyond g, the equations yield an oscillating path, cycling about the points e, /, c, and g and lacking an extremum trend. In fact, as point e is moved toward point g, the oscillations become more frequent until, when point e reaches point g as a limit, CL and hence ψ and ψ become entirely undefined. In contrast, for the particular region of conditions concerned, some sequence of values of CL, maintaining a defined path at the path angle ys corresponding to points g and c, necessarily exists between these points, and this sequence does not violate the ordinary control equation since the latter is here invalid. Therefore a singular segment corresponding to this sequence and hence defined by the equation, γ = ys = const, can occur after point g is reached along the path.

A,4 · PATH PERFORMANCE

In seeking the applicable solutions, it is important to remember that the formulation of the problem may be inadequate to give the path expected or in fact any optimum path whatever. For example, for a problem setting forth fuel and distance as the optimized and constrained variables respectively, the minimal solution is merely a power-off glide, and none of the control equations is required to have a different solution, such as for a power-on climb and cruise as might be expected. However, by formulating this problem differ­ ently, such as by constraining height in addition to distance, and by con­ sidering all possible simple and singular solutions, other types of minimal solutions certainly can be obtained. The original problem of finding undetermined functions has now been reduced to one of evaluating the undetermined constants appearing in the collective function (and hence in each possible equation and root) so as to provide a specified set of changes among the individual constrained variables. Unfortunately these constants cannot, in general, be determined explicitly from the specification of the problem. Rather the inverse situation exists, where a set of constants is assumed and the corresponding changes in the constrained variables are calculated as already outlined. By assuming a number of sets of constants and calculating a corresponding number of paths, it should be possible to obtain the desired net changes, Δ/Ii, for the optimum Av from the resulting path family. Such a procedure can be discouragingly time-consuming, but by proper consideration of the physical aspects of the problem it appears possible to narrow the choice of undetermined constants considerably, so that, for some cases, only a single path calculation is required. Some of the examples considered later will illustrate to some degree the physical reasoning required to properly set up these problems. At this time, however, we can point out that the units of a given X i are the inverse of those of the corresponding constrained variable μit and that setting a particular X1 equal to zero removes all constraint on the corre­ sponding μν Further, some problems can be simplified or improved by replacing certain constrained variables by others, as for example, the constraint of specific kinetic energy F2/2g rather than speed V. Also, the constraints imposed must be properly chosen to make the problem physically significant, so that, for example, in the case of a least fuel path for a given distance including climb, cruise, and descent phases, height as well as distance must be constrained. To illustrate the functional relations involved and also to establish specific equations for several important paths, the significant derivatives and ordinary control equations will now be expressed for paths of extreme time, distance, and fuel successively. For each of these paths the variables time, distance, weight, height, speed, and path angle will all be regarded as constrained, giving the collective function the common form ψ = Xjt 4- X?x + X3W -j- XJi -f- X i V + Χ 6 γ

(4-21)

A • METHODS

OF PERFORMANCE

CALCULATION

AT HIGH

SPEED

For the path of extreme time, the derivatives of the constrained variables are

and therefore the derivative xp is

Using these relations, the ordinary control equations (Eq. 4-19 and 4-20) for the path of extreme time become

(4-22)

(4-23)

Similarly, for the path of extreme distance, the derivatives of the constrained variables are

( 32 >

A,4 • PATH

and the derivative

PERFORMANCE

is

Using these relations, the ordinary control equations for the path of extreme distance become

(4-24)

(4-25)

Finally, for the path of extreme fuel, the derivatives of the constrained variables are

and the derivative

is

( 33 )

A • METHODS

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

Using these relations, the ordinary control equations for the path of extreme fuel become

(4-27) As seen by inspection, several denominators and numerators of the ordinary control equations are identical, although the form of each pair of these equations is significantly different. If, instead of the constraints imposed, fewer constraints are desired, the corresponding expressions are easily obtained by setting the appropriate of the above expressions equal to zero. Also, the number of undetermined constants can always be reduced by one, merely by dividing through by one of the Usually, the partial derivatives in the above expressions can be expressed either exactly or empirically in sufficiently simple algebraic forms to permit although of course these expressions can algebraic solutions for L and always be solved graphically. A,5.

Application of Path Performance Methods.

Examples, To demonstrate the application of the foregoing principles and also the nature of optimum paths, some of the equations just found will now be applied to the determination of some typical paths. For simplicity, the drag is assumed to be defined by Eq. 2-11

and the thrust by the parabolic expression (5-1) where / and w denote the ratio and u refers to a limiting value) and the quantities < 34 )

respectively (the asterisk and

A,5 • APPLICATION

OF PATH

PERFORMANCE

METHODS

are all constants, independent of M. Letting we denote the value of w at zero thrust, the relation (Eq. 5-1) also may be expressed as

This relationship is illustrated in Fig. A,5a.

Fig. A,5a.

Assumed engine thrust, fuel flow characteristic.

As further simplifications, and will be assumed constant; and, unless otherwise indicated, the numerical values listed below will be used for all paths:

to stratosphere)

From these assumptions and data, the simple relations below are obtained (5-2)

(5-3)

< 35 )

A • METHODS

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AT

HIGH

SPEED

(5-4)

(5-5)

As a matter of interest, the values of w and / at the point of minimum specific fuel consumption, denoted as and / c respectively, are 0.6 and 0.75 respectively. Although these assumed characteristics may not be closely representative of most actual problems, they nevertheless should be adequate for the demonstrations to be made here. The first example will be a path of least time for given distance and height changes, assuming that the weight remains constant. In this case, maximum thrust obviously is optimum, leaving only the value of lift to be found. From Eq. 4-22 and 5-3, setting and equal to zero and letting , the equation defining lift is obtained as

Along normal paths of the type concerned, both fi and y are of the order of 0.1 or less, permitting the simple but reliable approximation (5-6) In order to fix the proper value of fi, consider that in the final phases the craft, because of thrust limitations, will approach a constant altitude, constant velocity flight path, in which case 0. Accordingly the lift control (Eq. 5-6) approaches the relation

and the drag is equal to the power plant thrust, so that

For the assumed values of

and W, this becomes

and, substituting the asymptotic control ( 36 ) equation above, further reduces to

A,5 · APPLICATION OF PATH PERFORMANCE METHODS

It is clear now that, for the problem as formulated, the value chosen for μ will determine directly the asymptotic flight path altitude and that there are two values of μ which result in the same final altitude. For the example shown here a final altitude of 57,500 feet was chosen yielding the values μ — 0.080, 0.049. Using the path basic equations (Eq. 5-6), setting F = F*, and assuming the values CiJo = 0.014 - 0.07 A1 = 25,000 ft V1 — 300 ft/sec Vi = O the optimum paths of Fig. A,5b result. For comparison, a "conventional" 1300 1100

900 •2000

700 £,

1600 a> 500 X

•1200 800

μ = 0.08 μ = 0.049 Conventional poth

0

400

800

1200

1600

400

2000

x, ft χ 10

-3

Fig. A,5b.

Example I. Least time; distance and height constrained.

path consisting of a level flight acceleration to, and a climb at, the speed of maximum excess power, and then a level flight acceleration and cruise at the final altitude, are included in this figure. As expected, to reach a given distance and height, the time required along the optimum path is 91 per cent for point A and 87 per cent for point B of the time required along a conven­ tional path. For the particular conditions assumed, the type of path resulting is characterized by a slight initial dive, a generally higher climbing speed than used in a "conventional" path, and a slight zoom past the steady flight ceiling. The second example is a path of least time with distance, height, and

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

velocity constrained. Again, maximum thrust is obviously optimum and, from Eq. 4-22 and 5-3, setting A6 equal to zero and letting μ — A4/A2 and V — ?s9/h' a n d defining G w = W/qS:

For the paths of interest this equation may conveniently be approximated by

As before, in order to fix values of μ and η consistent with the physical problem, consider that the final path corresponds to flight at a constant altitude and velocity. Then, since γ = 0 and F j W = ^ = C j C , 00

2gD gL l

δ

D

L

— μ —

D

G

= —5

ρ

In order, now, to obtain consistency in the choice of μ and η , it is necessary to specify both a final altitude and a final velocity. Thus, for a final altitude of 57,000 feet and a final speed of 1115 ft/sec, μ

1115

= 2 X 0.07 X 0.377 = 0.0528

Notice here that this relationship does not determine unique values of μ and η, but allows some choice in the matter. Accordingly two cases were computed, assuming two sets of constants, μ

— 0.08, 0.10,

μ=

η η

= 30 = 52.6

and the numerical constants of the first example, and the resulting paths are shown in Fig. A,5c. An examination of these two paths shows that the larger values of the constants resulted in somewhat steeper dives and climbs, but that, on the whole, the paths are very nearly identical. This relative insensitivity to changes in the arbitrary constants is taken as an indication that the paths shown are very nearly optimum. The third case is one of least time with height and speed constrained.

A,5 • APPLICATION

OF PATH

PERFORMANCE

METHODS

As before, maximum thrust is used and theordinary lift control equation is obtained from Eq. 4-22 and 5-3, setting 0 and letting

This equation unfortunately cannot be simplified by any approximation

Fig. A,5c.

Example 2. Least time; distance, height, and speed constrained.

without sacrificing the solution of interest, and its roots are therefore determined as:

where

< 39 )

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

Examining the equation further, we see that at V / μ = 1, three roots exist, one of them zero, the other two real and of equal magnitude but opposite in sign; and a plot of CL VS. VJΜ for V]Μ ^ 1, has the characteristic form shown in Fig. A,5d. We can deduce from this figure that as F exceeds

ι V / μ increasing

Fig. A,5d.

Greneral form of C 1 , V/μ relation for example 3.

μ , the airplane will start to pull out and to slow down, and that after it has slowed down to the point where V is less than μ, the control will call for a push-over and consequent acceleration, and the cycle will repeat. In order to avoid such an oscillating path which is of little physical significance, we must set up an asymptotic flight path that has meaning for the formulated problem. This we can do by considering the simple segment. Remembering that along a simple segment, ψ = constant = maximum, we write

w=

dh dt

1-

μ dV g dt

= max

or, substituting the basic expression for dh/dt,

) y - d r < ' •(F — μ ) = max 9

Neglecting the second term in this equation under the assumption that it is

A,5 . APPLICATION OF PATH PERFORMANCE METHODS

small with respect to the first, and setting the derivative of the first term with respect to C equal to zero, to obtain the maximum, l

F C1 C2l + ^ L ^ W C D l

3C d -° = 0 Cdl

For the conditions corresponding to this "simple" segment (viz. sin γ = (F — D)IW, cos γ ^ 1, C = C ) the "ordinary" equation reduces to: V \ 1 F CLlV / V \ Cn 1 /F C\[ 2 > μ) W C Dl μ \ μ) C 2 ¾ w

1

D f

-)

To obtain compatibility between the "ordinary" and "simple" path segments it is clear now that μ should equal Fclimb. The assumption made above therefore appears permissible. Considering the initial conditions and numerical constants to be the same as those of the first example, and taking the climb speed at the initial altitude, we set μ = 859. Before proceeding with the path calculations we should additionally take note of the singularity occurring in the ordinary control equation when cos γ = 0. At this condition which corresponds to a vertical descent, we again have three roots, one zero, the other two real, equal in magnitude and opposite in sign. In this case the root C = 0 makes sense physically and is accordingly used to define a singular segment. This segment of the path must be terminated in such a way that the transition between it and the ensuing ordinary segment is smooth. This requires that the pull-out be started at the right instant so that the maximum speed attained in the pull-out, under conditions corresponding to the ordinary control equations, does not exceed the value of μ. This can be done only by a trial and error solution. The path resulting from the foregoing considerations is shown in Fig. A,5e,. along with a constant altitude acceleration to the same final speed and height. It will be noted that the "optimum" path achieves the required changes in speed and height in 85 per cent of the time corresponding to the more conventional path. The fourth example will be a path of greatest distance for given height and speed changes. This path will be assumed to start at a high level of energy (potential plus kinetic) and to be flown with power off, as would be typical of a rocket-propelled craft after a burning period. In this particular example, specific kinetic energy F2/2g is constrained in place of speed V , giving the first derivative of the collective function the form l

,i , FF D + W sin γ ψ = λ J i + A5 = X 1 tan γ - A3 — — g W cos γ

A • METHODS

OF PERFORMANCE

CALCULATION

AT HIGH

SPEED

From Eq. 4-19, 5-3, and 5-4, l e t t i n g t h e expression defining lift is obtained as

Fig. A,5e.

Example 3. Least time; height and speed constrained. Inset: lift coefficient control function.

Using the conventional formulas for the roots of a cubic equation, the real values of are obtained as

where

( 42 )

A,5 · APPLICATION OF PATH PERFORMANCE METHODS

To simplify the study of the path, the constant η is assumed to be unity, thereby eliminating all terms containing Gw and making Gl dependent on tan γ only. For this particular value of η, the collective function actually is equivalent to specific energy, and the relationship of Cl to tan γ can be represented by a single curve, shown together with the physical Umitation

0.4-

tan γ 0.2-

- 0.8

0.4

- 0.Ϊ

0.8

-0.2-

- 0.4Fig. A,5f. C 1 , γ relationship for example 4. C l = C* in Fig. A,5f. Using the path basic equations, the fact that F •

and assuming the values G ub =0.02

-O C DL = 0.10

A1 = 80,000 ft V1 = 4000 ft/sec

Case 1

Vi =\rad

the optimum path of Fig. A,5g is obtained. The pertinent values of C l are assumed to lie along the segments a, g, c, and d of Fig. A,4, as explained above, in order to avoid excessive path oscillations which appear to be of little physical significance. For this particular example, Cl jumps from the value at point g to that at point c, completely avoiding the constant-angle singular segment g-c. For other values of η and/or the initial conditions, a

A • METHODS

OF PERFORMANCE

CALCULATION

AT HIGH

SPEED

singular segment may be required to effect a smooth transition between the push-over and dive conditions represented by branch a-g and the pull-out and climb conditions represented by branch c-d. To illustrate such cases, paths for the same value of (unity) but for each of the sets of values

are shown in Fig. A,5g along with the path just found. It may be noted that after 150 miles of travel the height and speed, and therefore the energy of

Fig. A,5g.

Example 4. Maximum distance; energy constrained.

case 3, is less than that of case 1 in spite of the fact that its initial energy is higher. This difference is due to the choice of initial flight path angles and also to the physical limitation on which prevents case 3 from turning as rapidly as the control equations require. The simple segment control equation for this problem reduces to the requirement that

A,5 • APPLICATION

Setting

OF PATH

PERFORMANCE

METHODS

and assuming y is small, therefore W = L,

and the corresponding range increment may be written

where, to repeat, For the conditions pertinent to our particular problem is constant, having been assumed independent of altitude or speed, and the range is therefore (5-7) For case 1, comparing the results of this equation with the calculations corresponding to Fig. A,5g for the glide portion of the path, we find that both computations yield essentially the same answer.

From Fig. A,5g

From Eq. 5-7

For other values of rj and particularly for cases involving very high initial energies, the resulting path will consist of a series of skips, with the craft being subjected to a violent pull-out at low altitudes followed by essentially a ballistic path in the upper rarified atmosphere, the cycle repeating at ever-decreasing altitude and energy level until a final glide is reached. The final example is a path of least fuel with distance, height, and speed constrained. From Eq. 4-26, 4-27, 5-3, and 5-5, setting and letting < 45 )

A • METHODS

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

the lift and fuel control equations for the "ordinary" segment of the path become

The "simple" segment of the path, it can readily be shown, corresponds to the conditions for long range cruise at Expressing V and in terms of and and setting yields the lift and fuel control equations for the "simple" segment as (5-8) (5-9) So that the flight path will asymptotically approach the long range cruise, and compatible with these final conditions. it is necessary to make Accordingly, substituting the long range conditions in the "ordinary" control equations, we obtain

where long range cruise speed, and finally, the equations defining lift and fuel flow for the "ordinary" segment become

< 46 )

A,5 • APPLICATION

OF PATH

PERFORMANCE

METHODS

where

I t may be seen from the form of the lift-control function that a singularity will exist at Unlike previous cases, the flight path corresponding to this "singular" segment is not obvious and we must now develop the lift equation corresponding to this segment if we are to obtain a nonoscillatory, physically plausible path. This is done by recognizing that the singularity corresponds to the condition which yields the lift-control function:

where

( 47

)

A · METHODS OF PERFORMANCE CALCULATION AT HIGH SPEED

Assuming the previous set of constants except for CDe = 0.02, Gd = 0.1, and considering the minimum thrust and fuel flow to correspond to / = 0 and w = We respectively, the flight path of Fig. A,5h is computed. A path corresponding to a more conventional control (viz. acceleration and climb at maximum thrust and cruise at optimum long range conditions) is also shown. In this case, the conventional path apparently yields more optimum results than the so-called optimum because, as the figure shows, it achieves the same end conditions and expends some 25 lb less fuel in so doing. 10,000

T 70 \\\ O Λ X 60

r\

L-

Γ

ε

>

bO

40 30

CO

O •—

9800

ν

If I

\\ \N

\\ N>

1/) Xl

\

I I 1

I

I

/

/

f

f

χ ^ • 3C N. r Si / J X

/

X

/ /

/

9600

"Optinium" path *V

Ifc

Ν»

20 / /

X 4M-

V

\

•N.

Conve ntional path

—"

/

'"s. κ»

10

9400

"Ν. ' -Η»

-C

"Ν*

0 40

80

120

160

200

9200

x, miles Fig. A,5h.

Example 5. Least fuel; distance, height, and speed constrained.

This result points up a general difficulty that exists whenever the applicable control equations are limited by physical considerations. In such cases, continuity no longer exists, the control equations are invalid, and the corresponding path segment is not necessarily optimum, although it may be. Further, even when the conditions corresponding to the physical limitation are past and the control equations once more yield usable solutions, the corresponding path segment is not necessarily optimum because now this segment will not compensate for whatever losses may have occurred in the physically limited portion. The magnitude of this difficulty depends on the relative length of the limited segment and, for the cases previously shown, apparently has not invalidated the solutions obtained. In the present case, however, in the dive portion of the path, the minimum fuel flow is limited to that corresponding to We == 0.2. Removing this limitation will eliminate the difference in fuel weight previously found and make the two paths equivalent in the amount of fuel used although, as is

A,6 · APPROXIMATE METHODS

apparent from the velocity plot of Fig. A,5h, the "optimum" path takes approximately 10 per cent less time than the conventional. Since these two widely differing paths result in essentially the same fuel consumption, we can conclude that this particular problem is relatively insensitive to fairly large path variations. A,6. Approximate Methods. The foregoing examples of optimum path performance each involve considerable computational time (20-40 hours) in spite of the relatively simple thrust and drag functions assumed. Increasing experience with the procedure, and the employment of automatic computing machines, will doubtless greatly reduce the time required to obtain useful solutions. For many purposes, however, it is sufficient to substitute for the optimum path a more easily calculated flight plan which results in an approximately optimum answer. This is illustrated by considering some of the examples encountered above. In the case of least time for given altitude and distance changes, we found that a conventional path took approximately 10 per cent longer to achieve the required end conditions. Now, if we want to study the effect of changes in the power plant or airframe on the time required to effect a given change in distance and height, we can easily do so, approximately, by making our calculations for the conventional path and applying a 10 per cent correction factor. The same procedure may be followed for most of the other examples shown. In the case of a complicated radius mission, involving warm-up, take-off, climb, cruise to a target area, combat, return cruise, descent, loiter, and landing, it may be possible after one or two sample calculations to relate the actual mission radius to the hypothetical still-air range by a suitable factor. The simple segment expressions for range (Eq. 5-7, 5-8, and 5-9) which are well-known relations [2,2] can be used as the basis for making such approximate calculations. In some cases, however, Eq. 5-8 and 5-9 will not be compatible; that is, the engine-airframe com­ bination will be mismatched, and it will be impossible to satisfy both equations simultaneously. If the match is fairly poor it may be advisable to reconsider the entire design. Also, in many cases, the optimum CL represented by Eq. 5-8 will occur at a Mach number greater than the drag-rise critical, thus making the expression invalid. To avoid such difficulties the range may be calculated using the expression: (6-1) max

The quantity in parentheses will usually be a maximum just below .Mcr and near (L/Z))max, but in any case should be maximized by a cut-and-try procedure. In general, then, increasing experience with optimum path performance calculations allows a large variety of close approximations to the optimum

A • METHODS

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

by more easily calculated conventional paths. From the standpoint of preliminary performance analyses, such approximations are particularly useful, since it is often desirable to study the influence of a large number of airplane and/or power plant parameters on the resulting performance; and short-cut methods invariably increase the scope of the investigation and the validity of the final compromise. Similarly, whereas the use of a graphical analysis based on "corrected" quantities is important in the calculation of a large variety of point performance conditions for a well-established design, for many preliminary design studies an abbreviated procedure is indicated. It is much faster, for instance, when calculating altitude performance for a given weight and thrust condition, to use Eq. 2-3 through 2-8, directly, to obtain the performance potential at a series of discrete Mach numbers. This potential can be readily represented by simply plotting the absolute ceiling vs. the Mach number. The significance of such a plot can be established by the following development. F r o m t h e basic equations, and again expressing the drag coefficient as we may write

where A s s u m i n g t h e absolute ceiling may be written:

pressure ratio at

(6-2)

where and is evaluated in the region, Substituting this expression in the basic equation above, we find (6-3) where thrust coefficient The radical term on the righthand side of this equation is, for air-breathing engines, primarily a function of Mach number, whereas the term is, for a standard atmosphere, largely a function of only Ah, the altitude difference between the absolute ceiling and the desired flight altitude as shown in Fig. A,6a. This function furthermore may be approximated by the relation, (6-4) with a maximum error of 5 per cent for the applicable range of The acceleration, climb, and turn performance at altitude are thus directly dependent on the height of the machine above or below its absolute < 50 >

A,6 • APPROXIMATE

METHODS

ceiling; and the variation of the absolute ceiling at a given Mach number with variations in airframe and engine parameters is

(6-5)

where, again, is largely a function of as shown in Fig. A,6a. If we consider now a plot of absolute ceiling vs. Mach number, as in Fig. A,6b, we see that the contours for 2g and 3g flight are displaced below corresponding to values of of 1 that for \g by a constant and 2 respectively. The climb and acceleration performance at any altitude, speed, and load factor may be obtained by taking the height above or below the ceiling at the load factor being considered, using the chart of Fig. A,6a to obtain and using Eq. 6-3 to calculate the performance item in question. I t should be noted in Fig. A,6b that the buffet boundary shown for turning flight is displaced by the same altitude increment as in the normal ceiling plot. We can see this by considering that buffeting conditions may be represented by a given vs. M relationship. Then at a given const, whereby For most high speed machines, the best climb for altitudes above 35,000 feet is obtained at a constant Mach number corresponding to a value somewhat below the drag-rise critical. Accordingly, for air-breathing engines, the rate of climb will be approximately linear with altitude as shown by Eq. 6-3 and 6-4. The fuel flow, on the other hand, is governed by the relation and is not nearly so linear with h, as is shown by the pertinent curve in Fig. A,6a. I n spite of this last discrepancy, the fuel used and the time required to climb may be approximated, discounting optimum path considerations, by assuming linear variations of rate < 51 )

A • METHODS

Fig. A,6a.

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

Variation of pressure ratio parameters with altitude below ceiling.

of climb and fuel flow between sea level and 35,000 ft, and between 35,000 f t and the absolute ceiling. The expressions giving these time and fuel increments are obtained by integrating the expressions whereby:

(6-6)

A,6 · APPROXIMATE METHODS

Fig. A,6b.

Typical ceiling plot.

for the increments from sea level to 35,000 ft, and Δί :

, — 35,000 C 35 Vf

AW = C 35

35,000\

In

-¾

!

(6-7)

(h — 35,000) + Wfa Δί

for the increments from 35,000 ft to altitude h, where the subscripts 0, 35, and a refer to conditions at sea level, 35,000 ft, and absolute ceiling respectively. For rocket-propelled vehicles, intended to attain extreme altitudes and speeds, the performance at zero lift with drag neglected can be used to show important trends. The following abbreviated analysis of this problem parallels the material in [2] which may be consulted for more detail. Assume that the rocket-motor characteristics are constant with altitude and speed, so that the thrust may be written *•=*>< = const g

(6-8)

where Vj is the effective velocity of the exhaust jet relative to the rocket. For vertical ascent, we may write, after Eq. 2-7, F

dZ

w =l

a dt

A • METHODS

OF PERFORMANCE

CALCULATION

AT

HIGH

SPEED

and considering the change in weight due to fuel consumption,

Integrating, we obtain for the velocity at any time t, (6-9) Denoting conditions at the end of the fuel-burning process by the subscript b and recognizing that we may write for the velocity at the end of burning (6-10)

To obtain the altitude at the end of burning we use Eq. 2-4, recognizing that sin 6 = 1, and substituting therein the value of V given by Eq. 6-9; then

and finally, applying the limit (6-11)

We may rewrite Eq. 6-10 and 6-11, in terms of the fundamental rocket characteristics (Eq. 6-8) whereby, (6-12)

(6-13) We see, now, that other parameters being equal, the value of Vj will determine to a large degree the performance capability of a given installation, and that, to attain high performance, the jet velocity should be as high as possible. If it were possible to completely convert all of the fuel heat of combustion into jet kinetic energy, then the jet velocity would be related to the average heating value of a pound of fuel mixture , by the relation (6-14) In actual practice there are many factors, such as incomplete combustion, viscous losses, radiation, dissociation, and low operating pressures, which