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1 I. Review of Fundamental Fluid Mechanics and Thermodynamics 1. 1 Some fundamental aerodynamic variables htt://en.wikiedia.org/wiki/hurricane_ivan_(2004) 1) Pressure: the normal force er unit area exerted on a surface due to the time rate of change of momentum of the gas molecules imacting on that surface. Pressure at one oint is defined as: df lim( ) da 0 da df force on one side of da due to ressure da elemental area at the oint Generally to say ressure is a function of sace and time. ( x, y, z, t) Units: si (ound er inch); Pa (N/m 2 ); mmhg etc. Problem1: convert si to Pa. 2) Density: the mass er unit volume Density at one oint is defined as: dm ρ lim dv 0 dv dv dm elemental volume around the oint mass of fluid inside dv Density is also a function of sace and time. ρ ( x, y, z, t). Comressible flow (focus of this class) and incomressible flow. 1

2 3) Temerature: is roortional to the average kinetic energy of the molecules of the fluid. 3 KE kt, where k is the Boltzmann constant, x JK -1 Units: British R; S.I. K Degrees Fahrenheit ºF, (develoed in the early 1700's by G. Daniel Fahrenheit), are used to record surface temerature measurements by meteorologists in the United States. However, since most of the rest of the world uses degrees Celsius (develoed in the 18th Century), it is imortant to be able to convert from units of degrees Fahrenheit to degrees Celsius: C ( F 32) /1.8 F 1.8 C + 32 Kelvin is another unit of temerature that is very handy for many scientific calculations, since it begins at absolute zero, meaning it has no negative numbers. (the word "degrees" is NOT used with Kelvin. NO for Kelvin). The conversion from degrees Celsius to Kelvin is: K C Rankine ºR R F htt:// 4) Velocity: The most imortant variable of aerodynamics. Velocity of flow can be defined as: the velocity of a flowing gas at any fixed oint in sace is the velocity of an infinitesimally small fluid element as it swees through that oint. Velocity is a vector. Suersonic, subsonic, and transonic flows. 2

3 1. 2 Aerodynamic flows Block diagram categorizing the tyes of aerodynamic flows: Incomressible versus comressible flows: ρ is constant: incomressible flow. Liquid flow hydrodynamics For M (Mach number) < 0.3, we assume ρ is constant. High seed flow should be treated as comressible because ρ varies in large magnitude. Mach number regime: Mach number is local flow velocity over local seed of sound. Subsonic if M < 1 Sonic is M 1 Suersonic if M > 1 3

4 1. 3 Review of thermodynamics 1) Prefect gas Defined as a gas in which the intermolecular forces are neglected. For erfect gas equation of state: Or by using ρrt 1 ν, we get ν RT ρ 4

5 Where ν is secific volume. Where is the secific gas constant, which is a different value for different gases. For Air at standard conditions, R287 J/(kg.K)1716 (ft.lb)/(slug.ºr). 2) Internal Energy and Enthaly Secific internal energy deends on the state of the gas, thus so, e e( ν, T ) ν is secific volume de ( ) dν + ( ) dt T ν for erfect gas e is a function of temerature only: e e(t ); ( ) 0 T de ( ) dt c ( T ) dt ν v or e c T v Secific enthaly h, secific internal energy e has the relationshi: h e + ν dh dt so, ( ) ( ) + ν ( ) + ( ) at constant ressure, ( ) 0 dh dt Thus, ( ) ( ) + ( ) 5

6 Because By definition: ν ( ) R, dh c ( ) dh c ( T ) dt dt c R + c ν C v secific heat at constant volume; C secific heat at constant ressure. When air T <1000K C v, C can be assumed as constants. In this case, we can integrate to get: and h c T. For a secific gas, C and Cv are related through the equation: c c ν R if we define γ c c v, we get c γr γ 1 and c ν R γ 1 γ 1.4 for air at standard conditions. Problem 2: Calculate the C and Cv for air at standard conditions. 6

7 3) First law of thermodynamics Surroundings System (fixed mass) Boundary δq + δw de δw δq Note: 1) e is a state variable and de is differential between two states; while δw, δq deend on the rocess. 2) Adiabatic rocess: δq0; reversible rocess: no energy dissiation; isentroic: both adiabatic and reversible; Problem 3: Give an examle of adiabatic rocess. 7

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