1. Advanced fluid mechanics (II) Course content: 1.Compressible Fluid Mechanics Textbook: Modern Compressible Flow, 2 nd ed., by John D Anderson, Jr. Reference Book 1. Gas Dynamics, 2 nd ed., by James. E. A. John 2. Compressible – Fluid Dynamics by Philip A. Thompson 3. Elements of Gasdynamics by H. W. Liepmann and A. Roshko 4. Compressible Fluid Flow., 2 nd ed., by Michel A. Saad Grading: 1. Homework 60% 2. Final Project 40%

2. Chapter I 1. Introduction and Review of Thermodynamics What is Compressible Flow? 1. 2. Energy transformation and temperature change are important considerations → Importance of Thermodynamics e.q Flow of standard sea level conditions, Specific internal energy Specific kinetic energy

3. 1.1 Definition of Compressible Flow Incompressible flow → compressibility effect can be ignored. ν is the specific volume & Compressibility of the fluid Physical meaning: the fractional change in volume of the fluid element per unit change in pressure Chapter I Note: dp(+) → dv(-)

4. …. Isothermal compressibility …..isentropic compressibility (speed of sound) Compressibility is a property of the fluid Liquids have very low values of e.g for water = at 1atm Gases have high e.g for air =10 -5 m 2 /N at 1 atm, Alternate form of Chapter I

5. General speaking M a >0.3 → Compressible effect can not be ignored M a < 0.3 → Incompressible flow For most practical problem compressible

6. 1.2. Regimes of compressible flow Streamline deflected far upstream of the body Flow is forewarned of the presence of the body Subsonic flow Chapter I

7. Transonic flow is less than 1, but high enough to produce a pocket of locally supersonic slow

8. Loosely Defined as the “ Transonic regime” If is increased to slightly above 1, the λ shock will move to the trailing edge of the airfoil, and bow shock appears upstream of the leading edge. (Highly unstable)

9. Chapter I Supersonic Flow Everywhere Behind the shock + Parallel the free stream flow is not forewarned of presence of the body until the shock is encountered + Both flow of upstream of the shock and downstream of the shock are supersonic + Dramatic physical and mathematical difference between subsonic and supersonic flows. (We will mostly focus on this regimes)

10. High enough to excite the internal modes of energy dissociate or even ionize the gas. Chapter I Hypersonic Flow Real gas effect !!! Chemistry comes in

11. Chapter I Incompressible flow is a special case of subsonic flow limiting case Trivial, no flow For incompressibility Flow can be also be classified as Viscous inviscid Viscous flow: + Dissipative effects : Viscosity, thermal conduction, mass diffusion…. + Important in regions of large gradients of V, T and C i e.g. Boundary layer Flows

12. Chapter I Inviscid flows: - ignore dissipative effects outside of B.L (We will treat this kind of flow ) Also consider the gas to be “ Continuum ” Mean free path

13. R - specific gas constant 1.3 A Review of Thermodynamics 1.3.1 Ideal gas – intermolecular force are negligible 8314 (J/kg.mole.k) Molecular weights For air at standard conditions Boltzmann constant =

14. Isothermal compressibility L d L > 10d, for most compressible flows

15. Chapter I -Translational -Rotational No of collisions > 5 → equilibrium -Vibration : No of collisions > 0 (100 ) → equilibrium Add one more time scale or length scale -Electronic excitation + nuclear 1.3.2. Internal Energy and Enthalpy If the particles of the gas (called the system) are rattling about their state of “maximum disorder”, the system of particle will be in equilibrium. Statistical Thermodynamics + Quantum mechanics

16. Let be specific internal energy Let be specific enthalpy For both a real gas and a chemically reacting mixture of perfect gases. Thermally perfect gas Chapter I Return to macroscopic view continuum

17. Chapter I Calorically perfect gas are const → Will be assumed in the discussion of this class Ratio of specific heat, γ =1.4 for a diatomic gas γ =5/3 for a monatoinic gas Air, T<1000 K – Calorically perfect gas O 2, N 2, 1000<T<2500 – Thermally perfect gas Vibrational excited O 2 dissociate 2500<T<4000 K N 2 dissociate T>4000K

18. Consider caloriacally perfect gas + thermally perfect gas Note: specific heat at constant pressure specific heat at constant volume

19. Chapter I Ideal gas Perfect gas

20. -Conservation of Energy Consider a system, which is a fixed mass of gas separated from the surroundings by a flexible boundary. For the time being, assume the system is stationary, i.e., it has no directed kinetic energy e is state variable, de is an exact differential depends only on the initial and final states of the system 1.3.3. First law of the thermodynamics An incremental amount of heat added to the system across the boundary The work done on the system by the surrondings

21. Chapter I For a given, there are in general an infinite different ways (processes) of We will be primarily concerned with 3 types of processes: 1.Adiabatic process 2.Reversible process – no dissipative phenomena occur, i.e,. Where the effects of viscosity, thermal conductivity, and mass diffusion are absent (see any text on thermodynamic) 3. Isentropic process - both adiabatic & reversible 2nd law of thermodynamic

22. A contribution from the irreversible dissipative phenomena of viscosity thermal conductivity, and mass diffusion occurring within the system Chapter I Define a new state variable, the entropy, The actual heat added/T, These dissipative phenomena “ always” increase the entropy For a reversible process If the process is adiabatic, 2 nd law In summary, the concept of entropy in combination with the 2 nd law allow us to predict the direction that nature takes. 1.3.4 Entropy and the Second Law of Thermodynamic or

23. Chapter I Assume the heat is reversible, 1 st law becomes For a thermally perfect gas, If the gas also obey the ideal gas equation of state Integrate Note

24. 1.3.5. Isentropic realtions For an adiabatic process and for a reversible process Hence, from eq,i.e., the entropy is constant. Chapter I If we further assume a calorically perfect gas,

25. Chapter I

26. Outside B.L-Isentropic relations prevail e.g. T=1350K P=? T=2500 K P=15atm M=12, Cp=4157 J/kg.K

27. Chapter I 1.3.6. Aerodynamic forces on a Body Main concerns : Lift & drag Forces on a body of airfoil -Surface forces: pressure shear stress -Body forces : gravity ; electric-magnetic Sources of aerodynamic force, resultant force and its resolution into lift and drag

28. Drag D is the component of parallel In our plot. L//, D// Lift L is the component of perpendicular to the relative wind Chapter I Let be unit vectors perpendicular and parallel, respectively to the element ds, inviscid

29. Chapter I Pressure drag -> wave drag, e.g slender supersonic shapes with shock waves Skin friction drag -We consider only inviscid flows and both pressure and skin- friction drags are important -In the most cases, we can not predict the drag accurately For blunt bodies, D p dominates For streamlined bodies, D skin dominates with shock wave, D wave drag dominate and D skin can be neglected D can be predicted reasonably