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Handbook of Compressible Aerodynamics

Book Description

This book is dedicated to compressible aerodynamic flows in the context of the inviscid fluid hypothesis. Each chapter offers a simple theoretical presentation followed by an overview of practical calculation methods based on recent results, in order to make theoretical understanding easier and present current applications. Chapters 1 through 8 introduce the fundamental principles of theoretical aerodynamics and continue with vital reminders for understanding the discussions in the following chapters. Chapters 9 through 17 present the theory of steady unidimensional flows and breach surfaces such as shock waves and flow lines. This is central to gas dynamics. Chapters 18 through 24 develop the theory of characteristics applied to the study of supersonic flows as well as unsteady flows. The final chapter describes specific properties of transonic flows.

Table of Contents

  1. Coverpage
  2. Dedication
  3. Titlepage
  4. Copyright
  5. Table of Contents
  6. List of Symbols
  7. Acknowledgment
  8. General Introduction
  9. Chapter 1. Aerodynamics: Aims, Areas of Application and Current Issues
  10. Chapter 2. Aerodynamic Forces: Basic Definitions
    1. 2.1. Main aerodynamic coefficients
    2. 2.2. The lift-over-drag ratio and its practical implications
    3. 2.3. Curves of lift, drag, and polar
    4. 2.4. Other coefficients characterizing the effects of a fluid on a body
      1. 2.4.1. Effect of pressure
      2. 2.4.2. The effect of skin friction
      3. 2.4.3. Heat flux and heating
    5. 2.5. Evaluation of the action exerted by a fluid on a body
      1. 2.5.1. General relations
      2. 2.5.2. Lift and drag: drag decomposition
      3. 2.5.3. Distribution of pressure on an airfoil and lift
      4. 2.5.4. Aerodynamic stall of an airfoil
      5. 2.5.5. Return to the aerodynamic moment: center of pressure
    6. 2.6. The complete aircraft case: aerodynamic torsor
  11. Chapter 3. Review of Thermodynamics
    1. 3.1. Macroscopic systems, basic state variables, and absolute temperature
    2. 3.2. Energy exchanges
    3. 3.3. Postulates of thermodynamics
      1. 3.3.1. Postulate of the total energy or first law of thermodynamics.
      2. 3.3.2. Internal energy
      3. 3.3.3. Enthalpy
      4. 3.3.4. Postulate on entropy and the second law of thermodynamics
    4. 3.4. Expression of the entropy
    5. 3.5. Equation of state and the perfect gas model
    6. 3.6. Isentropic relations for a calorically perfect gas
    7. 3.7. Comments on the thermodynamics of studied gases
      1. 3.7.1. Introductory remarks
      2. 3.7.2. Variation of the specific heats of a thermally perfect gas.
      3. 3.7.3. Dissociation phenomena
      4. 3.7.4. Mollier diagram
      5. 3.7.5. Differential parameters of a dissociated gas in equilibrium
  12. Chapter 4. Fundamental Equations of Fluid Mechanics
    1. 4.1. Continuous regime and molecular regime
    2. 4.2. The continuous regime and Navier–Stokes equations
    3. 4.3. Law of mass conservation and continuity equation.
    4. 4.4. Equation of motion
    5. 4.5. Expression of the stress tensor
    6. 4.6. Energy equation
    7. 4.7. Entropy equation and expression of the heat flux
    8. 4.8. Summary of basic equations
    9. 4.9. Boundary conditions
    10. 4.10. Acoustic analogy and Lighthill’s equation
  13. Chapter 5. First Applications of the Conservation Equations
    1. 5.1. Theorem of the dynalpy
    2. 5.2. Action of a flow through an engine nacelle: thrust and captation drag
    3. 5.3. Drag of a vehicle in uniform flow: origin of drag and the Oswatitsch relation
    4. 5.4. The Betz method for determining drag
  14. Chapter 6. Dimensionless Equations: Similarity Parameters
    1. 6.1. Normalization of equations: Reynolds, Froude, and Prandtl numbers
    2. 6.2. Other notable numbers
      1. 6.2.1. Temporal similarity and the Strouhal number.
      2. 6.2.2. Reactive flows and the Darnköhler number
      3. 6.2.3. Multi-species flows and the Lewis number
    3. 6.3. Similarity conditions and the importance of the Reynolds number
    4. 6.4. Simulation of the Reynolds number in wind tunnels
    5. 6.5. Conclusion
  15. Chapter 7. Classification of Flows
    1. 7.1. Viscous and non-viscous flows: Euler equations
    2. 7.2. Compressible and incompressible flows
    3. 7.3. Subsonic and supersonic flows
    4. 7.4. Physical and mathematical approximations
  16. Chapter 8. Fundamental Concepts of Fluid Mechanics
    1. 8.1. Trajectories, streamlines and streaklines
      1. 8.1.1. Trajectories in a moving fluid
      2. 8.1.2. Streamlines of a moving fluid
      3. 8.1.3. Streak lines of a moving fluid
      4. 8.1.4. The case of steady flow: stream surface and stream tube
    2. 8.2. Vorticity, circulation, stream function and velocity potential
      1. 8.2.1. Vorticity and Crocco’s theorem
      2. 8.2.2. Circulation: Kelvin, Lagrange and Helmholtz theorems
      3. 8.2.3. Stream function
      4. 8.2.4. Potential function for the velocity or velocity potential
  17. Chapter 9. One-Dimensional, Non-Viscous and Adiabatic Steady Flows
    1. 9.1. Definition and basic hypotheses
    2. 9.2. The fundamental equations
      1. 9.2.1. Mass conservation: continuity equation
      2. 9.2.2. Equation of motion or Euler equation
      3. 9.2.3. Energy equation
      4. 9.2.4. Isentropy of the flow
    3. 9.3. The Hugoniot theorem
    4. 9.4. Consequences of the Hugoniot theorem: sonic throat and starting a supersonic nozzle
    5. 9.5. Isentropic expansion or compression of a flow
      1. 9.5.1. Limit velocity concept
      2. 9.5.2. Isentropic relations and the area-Mach number relation
    6. 9.6. Incompressible case: the Bernoulli relation and application
      1. 9.6.1. The Bernoulli relation
      2. 9.6.2. Measurement of the velocity: the Pitot probe
      3. 9.6.3. The Venturi
  18. Chapter 10. Application of the One-Dimensional Theory to the Calculation of Supersonic Nozzles
    1. 10.1. General definition of a supersonic nozzle
    2. 10.2. Calculation of the flow in the nozzle
    3. 10.3. Calculation of the nozzle mass flow
    4. 10.4. Calculation of the nozzle thrust
    5. 10.5. Conventional thrust at the pressure Pa and adaptation of a nozzle
    6. 10.6. Concluding remarks
  19. Chapter 11. One-Dimensional Flows with Friction and Heat Transfer
    1. 11.1. Friction force and heat transfer on a wall
    2. 11.2. One-dimensional theory and the generalized Hugoniot relation
    3. 11.3. Flow with friction without heat input: stagnation pressure drop in a pipe
    4. 11.4. Flow with heat transfer without friction: thermal choking
    5. 11.5. Calculating method for a one-dimensional flow with friction and heat transfer
  20. Chapter 12. Application of the One-Dimensional Theory to the Calculation of Supersonic Ejectors
    1. 12.1. Introduction
    2. 12.2. Technological and aerodynamic characteristics
    3. 12.3. Physical analysis ofthe supersonic ejector
    4. 12.4. Physical analysis of the mixed regime
    5. 12.5. Calculation of an ejector with a cylindrical mixing chamber
      1. 12.5.1. General calculation conditions
      2. 12.5.2. The case ofthe supersonic regime
      3. 12.5.3. Algorithm for the calculation of the characteristic of the supersonic regime
      4. 12.5.4. Algorithm for the calculation of the characteristic of the mixed regime
    6. 12.6. Application examples
  21. Chapter 13. Discontinuity Surfaces: Shock Wave and Slip Line
    1. 13.1. Existence of shock waves
    2. 13.2. Discontinuity relations: Rankine–Hugoniot equations
    3. 13.3. Shock wave type solution
    4. 13.4. Slip line type solution
    5. 13.5. Normal shock wave case
      1. 13.5.1. General relations
      2. 13.5.2. Case of the calorically perfect gas
      3. 13.5.3. Normal shock wave with real gas effects and chemical kinetics
    6. 13.6. Shock structure: the role of viscosity
  22. Chapter 14. Oblique Shock Wave and Shock Polar
    1. 14.1. General solution
    2. 14.2. Shock polar
      1. 14.2.1. Polar in the plane (pressure – deflection angle)
      2. 14.2.2. Polar in the plane (deflection angle – shock angle)
    3. 14.3. Other properties of the shock polar (pressure – deflection angle)
    4. 14.4. The oblique planar shock wave reflection problem
    5. 14.5. Transition from regular reflection to Mach reflection
    6. 14.6. External, internal and isentropic compressions
    7. 14.7. Conical shock wave
    8. 14.8. Indication on the effects of relaxation or non-equilibrium
  23. Chapter 15. Shock Intersections or Shock–Shock Interferences
    1. 15.1. Introduction
    2. 15.2. Type I interference
    3. 15.3. Type II interference
    4. 15.4. Type III interference
    5. 15.5. Type IV interference
    6. 15.6. Type V interference
    7. 15.7. Type VI interference
    8. 15.8. Lambda shock structure
    9. 15.9. Transition between type VI and type V interferences
  24. Chapter 16. Application of the Shock Wave Theory to Supersonic Air Intakes
    1. 16.1. The role of air intake
    2. 16.2. Pitot type air intake
      1. 16.2.1. Arrangements and definitions
      2. 16.2.2. Operation in subsonic flight
      3. 16.2.3. Operation in supersonic flight
      4. 16.2.4. An important relation between the efficiency and the mass flow coefficient
    3. 16.3. Two-dimensional supersonic air intake
      1. 16.3.1. Design and operation regimes
      2. 16.3.2. Two-dimensional supersonic air intake with multiple ramps
    4. 16.4. Air intake examples
  25. Chapter 17. Supersonic, Steady, Two-Dimensional Flows and the Theory of Characteristics
    1. 17.1. The theory of characteristics: introduction
    2. 17.2. Intrinsic equations of the planar or axisymmetric flows
    3. 17.3. The Cauchy problem and characteristic relations
    4. 17.4. Characteristic relations for the two-dimensional planar flow of a calorically perfect gas
    5. 17.5. Applications of the theory of characteristics
      1. 17.5.1. The progressive expansion wave
      2. 17.5.2. The limit case of the Prandti–Meyer expansion
      3. 17.5.3. The limit expansion angle
      4. 17.5.4. The progressive compression wave
  26. Chapter 18. The Numerical Method of Characteristics
    1. 18.1. Introductory remarks
    2. 18.2. The basic operators
      1. 18.2.1. Normal point in a flow: operator N
      2. 18.2.2. Point on a wall: operator W
      3. 18.2.3. Point on a jet boundary: operator J
      4. 18.2.4. Point on an axisymmetry axis: operator A
      5. 18.2.5. Origin of a centered expansion: operator Q
    3. 18.3. Calculation of a supersonic field
      1. 18.3.1. Domains of dependence and influence
      2. 18.3.2. Extension of the calculation: taking into account a boundary condition
    4. 18.4. Some examples of calculation
  27. Chapter 19. Application of the Method of Characteristics to the Calculation of Supersonic Nozzles
    1. 19.1. General principle
    2. 19.2. Calculation of the transonic domain
      1. 19.2.1. The potential equation in compressible flow
      2. 19.2.2. The Carriere development method
      3. 19.2.3. The Sauer small perturbation method
    3. 19.3. Calculation of the flow in a nozzle with a given contour: direct method
    4. 19.4. Calculation of the contour of a nozzle producing a given flow: inverse or design method
    5. 19.5. Example of the definition of the contour of a two-dimensional planar nozzle
    6. 19.6. Mach rhombus of a nozzle
  28. Chapter 20. Flows with Shock Waves: Rotational Method of Characteristics
    1. 20.1. Shock wave and rotational flow
    2. 20.2. The origin of a shock caused by a wall discontinuity
    3. 20.3. Origin of a shock caused by a pressure discontinuity
    4. 20.4. The point on a shock in the vicinity of its origin
    5. 20.5. Origin of a focalization shock
    6. 20.6. Normal point of a shock
    7. 20.7. Calculation examples of flows with shock formation
    8. 20.8. Application to the analysis of the structure of a supersonic jet
      1. 20.8.1. Case of the under-expanded jet
      2. 20.8.2. Case of the over-expanded jet
    9. 20.9. Drag of the symmetric bump in supersonic flow
  29. Chapter 21. One-Dimensional, Non-Viscous and Adiabatic Unsteady Flows
    1. 21.1. Introduction
    2. 21.2. General equations
    3. 21.3. Unsteady isentropic flow in a cylindrical tube
      1. 21.3.1. Equation of motion
      2. 21.3.2. Cauchy problem and characteristic curves
      3. 21.3.3. Characteristic relation: Riemann invariants
      4. 21.3.4. Case of a calorically perfect gas
      5. 21.3.5. Initial and boundary conditions
      6. 21.3.6. The expansion wave
    4. 21.4. The compression wave
    5. 21.5. Intersection of compression waves: formation of a shock wave
    6. 21.6. Distortion of a periodic wave and nonlinear acoustics
    7. 21.7. Reflection–refraction of a wave on a contact surface: acoustic impedance
    8. 21.8. Back on the boundary conditions
  30. Chapter 22. Unsteady Shock Wave, Contact Surface, and Wave Reflections
    1. 22.1. The shock wave equations and relations
    2. 22.2. Formation of a shock wave
    3. 22.3. Contact surface
    4. 22.4. Reflection of a wave
    5. 22.5. Multiple reflections
    6. 22.6. Collision of two shock waves
    7. 22.7. Impact of a shock wave on a fixed wall
  31. Chapter 23. Shock Tube
    1. 23.1. General theory
    2. 23.2. Reflected shock tube
    3. 23.3. Tailoring the contact surface
    4. 23.4. Numerical simulation of the functioning of a shock tube
    5. 23.5. Shock tube arrangement: the shock tunnel
  32. Chapter 24. Numerical Methods for Calculating Unsteady Flows
    1. 24.1. Integration by the method of characteristics
    2. 24.2. Integration by finite differences
      1. 24.2.1. Dimensionless equations in conservative form
      2. 24.2.2. Explicit numerical integration method
    3. 24.3. Stability condition and CFL criterion (Courant–Friedrich–Lewy)
    4. 24.4. Equations in spherical coordinates: spherical shock wave
    5. 24.5. Starting and unstarting a supersonic nozzle
  33. Chapter 25. Some Properties of Transonic Flows
    1. 25.1. Introduction
    2. 25.2. Potential equation for small disturbances
    3. 25.3. Compressibility correction: the Prandti–Giauert rule
    4. 25.4. Linearized theory for supersonic flows
    5. 25.5. The small perturbation potential equation in transonic
    6. 25.6. Final comments on the potential equation
    7. 25.7. Critical Mach number of an airfoil
    8. 25.8. Transonic flow around an airfoil
    9. 25.9. Drag divergence Mach number
    10. 25.10. Supercritical airfoils
  34. General Conclusion
  35. Appendix 1. Review of Mathematical Notations and Relations
  36. Appendix 2. Table of Useful Relations for the Fundamental Concepts of Aerodynamics
  37. Appendix 3. Table of Useful Relations for Stationary One-dimensional Flows and Discontinuity Surfaces
  38. Appendix 4. Table of Useful Relations for Applications of the Theory of Characteristics and Transonic Flows
  39. Bibliography
  40. Index