Note for Compressive Flow and Gas Dynamics - CFG by Engineering Kings

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Introduction to Gas Dynamics All Lecture Slides Teknillinen Korkeakoulu / Helsinki University of Technology Autumn 2009 Gasdynamics — Lecture Slides 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Compressible flow Zeroth law of thermodynamics First law of thermodynamics Equation of state — ideal gas Specific heats The “perfect” gas Second law of thermodynamics Adiabatic, reversible process The free energy and free enthalpy Entropy and real gas flows One-dimensional gas dynamics Conservation of mass — continuity equation Conservation of energy — energy equation Reservoir conditions On isentropic flows Gasdynamics — Lecture Slides

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16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Euler’s equation Momentum equation A review of equations of conservation Isentropic condition Speed of sound — Mach number Results from the energy equation The area-velocity relationship On the equations of state Bernoulli equation — dynamic pressure Constant area flows Shock relations for perfect gas — Part I Shock relations for perfect gas — Part II Shock relations for perfect gas — Part III The area-velocity relationship — revisited Nozzle flow — converging nozzle Gasdynamics — Lecture Slides 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Nozzle flow — converging-diverging nozzle Normal shock recovery — diffuser Flow with wall roughness — Fanno flow Flow with heat addition — Rayleigh flow Normal shock, Fanno flow and Rayleigh flow Waves in supersonic flow Multi-dimensional equations of the flow Oblique shocks Relationship between wedge angle and wave angle Small angle approximation Mach lines Weak oblique shocks Supersonic compression by turning Supersonic expansion by turning The Prandtl-Meyer function Gasdynamics — Lecture Slides

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46 Detached shocks 47 Shock-expansion theory 48 Reflection and intersection of oblique shocks 49 Cones in supersonic flow 50 Derivation of perturbation equation 51 Irrotational flow 52 Governing equations for small perturbation flows — Part I 53 Governing equations for small perturbation flows — Part II 54 Pressure coefficient 55 Boundary conditions 56 Flow past a wave-shaped wall — an example 57 Flow past a wave-shaped wall — subsonic case 58 Flow past a wave-shaped wall — supersonic case Gasdynamics — Lecture Slides Compressible flow In a nutshell, the term compressible flow refers to the fluids of which there can be found significant variation of density in the flow under consideration. Compressibility is strongly related to the speed of the flow itself and the thermodynamics of the fluid. A good grasp of thermodynamics is imperative for the study of compressible flow. For low-speed flow, the kinetic energy is often much smaller than the heat content of the fluid, such that temperature remains more or less constant. On the other hand, the magnitude of the kinetic energy in a high-speed flow can be very large, able to cause a large variation in the temperature. Some important phenomena strongly associated with compressibility are the flow discontinuity and choking of the flow. Gasdynamics — Lecture Slides

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Compressible flow To illustrate, consider a car at sea-level, 1 atm and 15 ◦ C, going at a speed of 90 km/h. The density is 1.225 kg/m3 . At a stagnation point, the density there is found to be 1.228 kg/m3 , a mere 0.27 % difference. The temperature rises by 0.311 ◦ C and the pressure changes by 0.38 %. Here, the incompressible assumption can be applied. Now, consider a typical air flow around a cruising jetliner at 10 km altitude. The speed is now 810 km/h, while the ambient conditions are 0.413 kg/m3 , 0.261 atm and −50 ◦ C. At the stagnation point the temperature rises by over 25 ◦ C, while density and pressure changes by more than 30 % and 45 %, respectively. It is clear that compressibility must now be taken into account. Gasdynamics — Lecture Slides Compressible flow Figure 1: Breaking the sound barrier. . . ? An extreme example of compressible flow in action is the re-entry flow. Another is shown here on the left as a jet fighter seemingly punches through the “sound barrier”. However, more daily mundane applications can also be found in flows through jet engines, or around a transport aircraft. Gasdynamics — Lecture Slides