16.120 | Spring 2003 | Graduate

Compressible Flow

Pages

LEC # TOPICS ASSIGNMENTS
  Gas Dynamic Discontinuities (4 Lectures by Prof. Wesley Harris)  
14 Shock Waves Problem Set 3 (PDF)
Problem Set 4 (PDF)
15 Deflagrations  
16 Detonations  
17 Crocco’s Theorem Problem Set 5 (PDF)
  Two-Dimensional, Compressible, Linear Flows (4 Lectures by Prof. Wesley Harris)  
18-20 Constant Entropy Flows Problem Set 6 (PDF)
21-22 Slender, Axially Symmetric Flows  
  Two-Dimensional, Compressible, Non-Linear Flows (4 Lectures by Prof. Wesley Harris)  
23-24 Inviscid, Unsteady Transonic Flows Problem Set 7 (PDF)
25-26 Inviscid Hypersonic Flows Problem Set 8 (PDF)

LEC # TOPICS
  Basics (3 Lectures by Prof. Zoltan Spakovszky)
1 Continuity and Momentum Equations (Seen Before so no Detailed Derivation)
Derivation of Energy Equation for a Compressible, Viscous, Heat Conducting Fluid
2 Thermo - and Mechanical Processes, Splitting the Energy Equation
Entropy Changes and Thermodynamic Context
Physical Meaning of the Terms, 2nd Law
Exact Solution of the Navier-Stokes Equations for a One-Dimensional Flow
Physical Insight into Shock Thickness and the “Origin” of Losses for a Shock Wave
3 Integral Forms of Equations of Motion
Non-dimensional Parameters Characterizing Compressible Flow
“Appropriate” Scaling-what are the Relevant Nondimensional Parameters?
When is a Flow Compressible [For Both Steady and Unsteady Flow]?
  Quasi 1-D Compressible Flow and Extensions (8 Lectures by Prof. Zoltan Spakovszky)
4 Physical Basis for the 1-D Flow Assumption, Role of 1-D Flow Descriptions (Insight, Overall Flow Characterization), Regimes of Relevance
1-D Isentropic Flow and Choking, Maximum Flow Per Unit Area
“Corrected Flow” and Flow Functions
5 1-D Flow in Converging-diverging Nozzles
Shock Waves, Behavior of Nozzles (Brief Description)
Starting of Supersonic Flows
Transitions from Supersonic to Subsonic Flow in Constant Area Ducts
6-7 Generalized One-dimensional Flow With Mass, Momentum, and Energy Interchange
Effect of Friction (Movement of Choke Point, Reduction in Max Flow)
Effect of Heat Addition
Analogies Between Mass, Momentum, and Heat Addition
8 General Compressible Flow Processes in Propulsion Systems
Representation of Flow Processes in H-K Diagrams
Applications to Ramjets and Scramjets
9-10 Axisymmetric Compressible Swirling Flow
Derivation of Equations
Examples: (i) Behavior of Static Pressure, (ii) Effects of Swirl on Maximum Mass Flow, (iii) Effect of Mass Addition on Total Pressure
11 Compound-compressible Flow (Combined Subsonic and Supersonic Streams)
Compound Flow Regimes and the Compound Mach Number
Compound Choking and Effect on Mass Flow
Examples: Nozzles, Mixing Ducts
  Disturbance Behavior in a Compressible Flow (2 Lectures by Prof. Zoltan Spakovszky)
12 Types of Perturbations in Compressible Flow: Entropy, Vorticity and Pressure Disturbances
Coupling of Disturbance Types in Compressible Flow
13 Characteristic Disturbance Velocities and Behavior
Unsteady Flow and Upstream Influence in Compressible Flow
  Gas Dynamic Discontinuities (4 Lectures by Prof. Wesley Harris)
14 Shock Waves
15 Deflagrations
16 Detonations
17 Crocco’s Theorem
  Two-Dimensional, Compressible, Linear Flows (4 Lectures by Prof. Wesley Harris)
18-20 Constant Entropy Flows
21-22 Slender, Axially Symmetric Flows
  Two-Dimensional, Compressible, Non-Linear Flows (4 Lectures by Prof. Wesley Harris)
23-24 Inviscid, Unsteady Transonic Flows
25-26 Inviscid Hypersonic Flows

Both the midterm and final exams in 16.120 were conducted orally. Students were given the problems in advance, and alloted time to devise solutions. Each student would then present their solutions to the instructors. In contrast to a written exam, the format allows for dialog between faculty and student, often allowing more accurate evaluation.

Final Oral Exam (also available in PDF)

Do both problems.

Problem 1.

Contrast small disturbance, linear, inviscid external supersonic flow with small disturbance, nonlinear, inviscid external hypersonic flow (no credit is given without clear explanations, perhaps bolstered by some equations).

Problem 2.

Consider a row of guide vanes in a compressible internal flow as shown below. There is a pressure probe at a measurement location a distance L upstream of the blade row. At time t=t0, the setting of the blades is changed suddenly such that they assume the position denoted by the solid blades, perturbing the mass flow in a one-dimensional sense.

a) What are the governing equations representing the flow field in this situation?

b) Discuss the creation of flow field perturbations encountered in this situation and describe their attributes (an explanation is expected perhaps bolstered by some equations).

c) At what time does the pressure probe respond to the flow disturbance associated with the change in angle setting of the blades if: (1) the flow is subsonic, and (2) the flow is supersonic?

d) Sketch the propagation of the disturbances in an x-t diagram for both cases (1) and (2).

Additional references are given below to either supplement the discussion in the text by Anderson or, in the sections listed, to provide a source for material that he does not cover.

Basic Equations

Many textbooks have derivations of the equations of motion. We list some below. The discussion in Reference 3 (indeed the whole book) is very clear.

  1. Anderson, J. D. Modern Compressible Flow with Historical Perspective. McGraw-Hill Publishers, 1990.
  2. Emmons, H. W. Fundamentals of Gas Dynamics, Chapter 1, “The Equations of Gas Dynamics”, by H. S. Tsien, Volume III in the Princeton Series of Volumes on “High Speed Aerodynamics and Jet Propulsion”, Princeton University Press, 1958 (Be warned: there is a mistake in the derivation of the energy equation in differential form!)
  3. Liepmann, H., and A. Roshko. Elements of Gas Dynamics. John Wiley Publishers, 1957 (see chapters 7 and 13).
  4. Schlichting, H. Boundary Layer Theory. 7th ed. McGraw-Hill Publishers, 1979. Chapter 3.
  5. Thompson, P. A. Compressible Fluid Dynamics. Maple Press Company, 1984.
  6. White, F. M. Viscous Fluid Flow. 2nd ed. McGraw-Hill Publishers, 1991. Chapter 2.
  7. Greitzer, E.M., Tan, C.S, Graf, M.B. and J.E. Ffowcs Williams (Ed.). Internal Flow: Concepts and applications. Cambridge University Press, 2004. Chapter 1.

Quasi-One-Dimensional Compressible Flow

The basics of this topic are also in many texts. Reference 7 is the classic text on the subject. References 5 and 6 give very readable introductory discussions. Reference 3 could also be entitled as “everything you wanted to know (or might ever conceive of wanting to know!) about quasi one-dimensional flow”. References 1, 2, and 10 give discussion of the extensions to classical one-dimensional gas dynamics that we will discuss.

  1. Anderson, L. R., W. H. Heiser, and J. C. Jackson. “Axisymmetric One-Dimensional Compressible Flow—Theory and Applications.” ASME Transactions, Journal of Applied Mechanics (1970): pp. 917-923.
  2. Bernstein, A., W. H. Heiser, and C. Hevenor. “Compound-Compressible Nozzle Flow.” AS_ME Transactions, Journal of Applied Mechanics_ (1967): 548-554.
  3. Emmons, H. W. Fundamentals of Gas Dynamics, Chapter 2, “The Equations of Gas Dynamics,” by L. Crocco, Volume III in the Princeton Series of Volumes on “High Speed Aerodynamics and Jet Propulsion”. Princeton University Press, 1958.
  4. Heiser, W. H., and D. T. Pratt. Hypersonic Airbreathing Propulsion. AIAA Education Series, 1994. Chapter 2.
  5. Hill, P., and C. Peterson. Mechanics and Thermodynamics of Propulsion. Addison Wesley Publishers, 1992. Chapter 3.
  6. Li, W.-H., and S. -H. Lam. Principles of Fluid Mechanics. Addison-Wesley Publishing Company, 1964. Chapters 14 and 15.
  7. Shapiro, A. H. The Dynamics and Thermodynamics of Compressible Fluid Flow. Vol. 1. Ronald Press. Chapters 1 to 8.
  8. Tew, D. E., B. S. Teeple, and I. A. Waitz. “Mixer-Ejector Noise Suppressor Model.” J. Propulsion and Power 14. No. 6 (1998): 941-950.
    (This shows a recent application and extension of the compound compressible flow ideas in reference 2)
  9. Thompson, P. A. Compressible Fluid Dynamics. Maple Press Company, 1984.
  10. Greitzer, E.M., Tan, C.S, Graf, M.B. and J.E. Ffowcs Williams (Ed.). Internal Flow: Concepts and applications. Cambridge University Press, 2004. Chapters 2, 10, 11.

Disturbance Behavior in a Compressible Flow

  1. Dowling, A. P., and J. E. Ffowcs Williams. Sound and Sources of Sound. Ellis Horwood Press, 1983.
  2. Lighthill, J. Waves in Fluids. Cambridge University Press, 1978.
  3. Morse, P. M., and Ingard. Theoretical Acoustics. Princeton University Press, 1986.
  4. Thompson, P. A. Compressible Fluid Dynamics. Maple Press Company, 1984.
  5. Greitzer, E.M., Tan, C.S, Graf, M.B. and J.E. Ffowcs Williams (Ed.). Internal Flow: Concepts and applications. Cambridge University Press, 2004. Chapter 6.

Gas Dynamic Discontinuities

  1. Hayes, W. D. Gasdynamic Discontinuities. No. 3. Princeton Aeronautical Paperbacks. Princeton, NJ: Princeton University Press, 1960.

Two-Dimensional, Compressible, Linear Flows

  1. Ashley, H., and M. Landahl. Aerodynamics of Wings and Bodies. Reading MA: Addison-Wesley Publishing Co., Inc., 1965.
  2. Katz, J. and A. Plotkin. Low-Speed Aerodynamics: From Wing Theory to Panel Methods. New York, NY: McGraw-Hill, Inc., 1991.

Two-Dimensional, Compressible, Non-Linear Flows

  1. Landahl, M. T. Unsteady Transonic Flow. New York, NY: Pergamon Press, 1961.
  2. Chernyi, G. G. Introduction to Hypersonic Flow. New York, NY: Academic Press, 1961.

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Concepts Addressed

Fundamentals of compressible fluid dynamics and application to external and internal flows. Quasi-one-dimensional channel flow, extensions, and analysis of multi-dimensional flows in nozzles, diffusers, and inlets. Forces, moments, and loss generation resulting from compressible fluid flow interactions with aerodynamic shapes in subsonic, supersonic, transonic, and hypersonic flight, shock waves, and vortices. Disturbance behavior in unsteady compressible flow.

Learning Objectives

Students will be able to:

  1. Describe assumptions, physical meaning of terms and to utilize key relationships for compressible flow, speed of sound, isentropic and non-isentropic flows, and potential and rotational flows;
  2. Calculate the effect of area change, shaft work, heat addition, mass addition and friction on flow states in a compressible channel flow, including effects on mass flow capacity and flow regime;
  3. Characterize quantitatively the behavior of velocity and density non-uniformities in an unsteady compressible flow including their evolution in fluid system components;
  4. Estimate the lift and drag for basic aerodynamic shapes in compressible, inviscid flows.

Measurable Outcomes

Students will be able to:

  1. Describe the assumptions and physical meaning of terms in the equations of motion for continuum flow;
  2. Define quantitatively the regimes of applicability of quasi-one-dimensional channel flow theory;
  3. Define quantitatively what it means for a flow to be considered “compressible”;
  4. Define quantitatively effects of swirl and flow non-uniformity on maximum flow per unit area in a channel;
  5. Explain the starting behavior of supersonic diffusers and inlets;
  6. Characterize quantitatively the links between flow angle and pressure changes in a supersonic flow and the differences with subsonic flows;
  7. Define the conditions for, and effect on flow state of the different types of discontinuities that occur in a compressible flow;
  8. Describe expressions for and explain the link between vorticity generation and gradients in shock strength;
  9. Use the Method of Characteristics to analyze a variety of internal and external flows: airfoils, inlets, nozzles, and jet flows;
  10. Use the Method of Characteristics to design optimum inlets, nozzles, and wind tunnels;
  11. Define the similarity principles that apply to compressible flows;
  12. Calculate the lift and drag over simple aerodynamic shapes in compressible, inviscid flows.

Quizzes/Problem Sets

There are approximately six problem sets, an oral mid term exam, and an oral final exam.

Lecturers

Prof. Z. S. Spakovszky
Prof. W. L. Harris

Class Exercises

The lectures are held twice a week for 1.5 hours per session.

Course Info

Instructor
As Taught In
Spring 2003
Level
Learning Resource Types
Problem Sets
Exams