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Introduction to Compressible Flow

Introduction to Compressible Flow. Introduction and Review of Thermodynamics What is Compressible Flow? 1. 2. Energy transformation and temperature change are important considerations → Importance of Thermodynamics

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Introduction to Compressible Flow

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  1. Introduction to Compressible Flow

  2. 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. 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 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 m2/N at 1 atm, Alternate form of

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

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

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

  8. SHOCK WAVES For some back pressure values, abrupt changes in fluid properties occur in a very thin section of a converging–diverging nozzle under supersonic flow conditions, creating a shock wave. We study the conditions under which shock waves develop and how they affect the flow. Normal Shocks Normal shock waves: The shock waves that occur in a plane normal to the direction of flow. The flow process through the shock wave is highly irreversible.

  9. Oblique Shocks When the space shuttle travels at supersonic speeds through the atmosphere, it produces a complicated shock pattern consisting of inclined shock waves called oblique shocks. Some portions of an oblique shock are curved, while other portions are straight.

  10. EXAMPLES OF SUPERSONIC WAVE DRAG F-104 Starfighter

  11. Upstream and Downstream of Shock wave Upstream: 1 M1 > 1 V1 p1 r1 T1 s1 p0,1 h0,1 T0,1 Downstream: 2 M2 < 1 V2 < V1 P2 > p1 r2 > r1 T2 > T1 s2 > s1 p0,2 < p0,1 h0,2 = h0,1 T0,2 = T0,1 (if calorically perfect, h0=cpT0) Typical shock wave thickness 1/1,000 mm

  12. Variation of flow properties in subsonic and supersonic nozzles and diffusers.

  13. 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. Loosely Defined as the “ Transonic regime” (Highly unstable)

  14. Supersonic Flow Everywhere (We will mostly focus on this regimes) 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.

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

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

  17. 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

  18. 1.3 A Review of Thermodynamics 1.3.1 Ideal gas – intermolecular force are negligible 8314 (J/kg.mole.k) R - specific gas constant Molecular weights Boltzmann constant = For air at standard conditions

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

  20. 1.3.2. Internal Energy and Enthalpy -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 Statistical Thermodynamics + Quantum mechanics 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.

  21. Return to macroscopic view continuum 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

  22. Calorically perfect gas 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 are const → Air, T<1000K – Calorically perfect gas O2, N2, 1000<T<2500 – Thermally perfect gas Vibrational excited O2 dissociate 2500<T<4000 K N2 dissociate T>4000K

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

  24. Chapter I Perfect gas Ideal gas

  25. 1.3.3. First law of the thermodynamics • 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 The work done on the system by the surrondings An incremental amount of heat added to the system across the boundary

  26. For a given , there are in general an infinite different ways (processes) of • We will be primarily concerned with 3 types of processes: • Adiabatic process • 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

  27. 1.3.4 Entropy and the Second Law of Thermodynamic Define a new state variable, the entropy, A contribution from the irreversible dissipative phenomena of viscosity thermal conductivity, and mass diffusion occurring within the system or The actual heat added/T, These dissipative phenomena “ always” increase the entropy For a reversible process If the process is adiabatic, 2nd law In summary, the concept of entropy in combination with the 2nd law allow us to predict the direction that nature takes.

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

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

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

  31. 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

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

  33. 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, Dp dominates For streamlined bodies, Dskin dominates with shock wave, Dwave drag dominate and Dskin can be neglected D can be predicted reasonably

  34. Topics Covered • Stagnation properties • Speed of sound and Mach number • One-dimensional isentropic flow • Variation of fluid velocity with flow area • Property relations for isentropic flow of ideal gases • Isentropic flow through nozzles • Converging nozzles • Converging–diverging nozzles • Shock waves and expansion waves • Normal shocks • Oblique shocks • Prandtl–Meyer expansion waves • Duct flow with heat transfer and negligible friction (Rayleigh flow) • Property relations for Rayleigh flow • Choked Rayleigh flow • Steam nozzles

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