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##### One-dimensional Flow

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**One-dimensional Flow**• 3.1 Introduction Normal shock In real vehicle geometry, The flow will be axisymmetric One dimensional flow**The variation of area A=A(x) is gradual**Neglect the Y and Z flow variation**3.2 Steady One-dimensional flow equation**Assume that the dissipation occurs at the shock and the flow up stream and downstream of the shock are uniform Translational rotational and vibrational equilibrium**The continuity equation**L.H.S of C.V (Continuity eqn for steady 1-D flow) • The momentum equation**Remember the physics of momentum eq is the time**rate of change of momentum of a body equals to the net force acting on it.**The energy equation**Physical principle of the energy equation is the energy is the energy is conserved Energy added to the C.V Energy taken away from the system to the surrounding**3.3 Speed of sound and Mach number**Mach angle μ Wave front called “ Mach Wave” Always stays inside the family of circular sound waves Always stays outside the family of circular sound waves**1**2 A sound wave, by definition, ie: weak wave ( Implies that the irreversible, dissipative conduction are negligible) Wave front • Continuity equation**Momentum equation**No heat addition + reversible General equation valid for all gas Isentropic compressibility**For a calorically prefect gas, the isentropic relation**becomes For prefect gas, not valid for chemically resting gases or real gases Ideal gas equation of state**Form kinetic theory**a for air at standard sea level = 340.9 m/s = 1117 ft/s Mach Number The physical meaning of M Subsonic flow Kinetic energy Sonic flow Internal energy supersonic flow**3.4 Some conveniently defined parameters**Inagine: Take this fluid element and Adiabaticallyslow it own (if M>1) or speed it up (if M<1) until its Mach number at A is 1. For a given M and T at the some point A associated with Its values of and at the same point**Note: are sensitive to the reference coordinate**system are not sensitive to the reference coordinate In the same sprint, image to slow down the fluid elements isentropically to zero velocity , total temperature or stagnation temperature total pressure or stagnation pressure Stagnation speed of sound Total density (Static temperature and pressure)**3.5 Alternative Forms of the 1-D energy equation**= 0(adiabatic Flow) calorically prefect B If the actual flow field is nonadiabatic form A to B → A Many practical aerodynamic flows are reasonably adiabatic**Total conditions - isentropic**Adiabatic flow isentropic Note the flowfiled is not necessary to be isentropic If not → If isentropic → are constant values**= 1 if M=1**< 1 if M < 1 > 1 if M > 1 If M → ∞ or**1**Known 2 To be solved adiabatic 3.6 Normal shock relations ( A discontinuity across which the flow properties suddenly change) The shock is a very thin region , Shock thickness ~ 0 (a few molecular mean free paths) ~ cm for standard condition) Ideal gas E.O.S Calorically perfect Continuity Momentum Energy Variable : 5 equations**Prandtl relation**Note: 1.Mach number behind the normal shock is always subsonic 2.This is a general result , not just limited to a calorically perfect gas**Infinitely weak normal shock . ie: sound wave or a Mach wave**Special case 1. 2.**Note : for a calorically perfect gas , with γ=constant**are functions of only Real gas effects**Mathematically eqns of hold for**Physically , only is possible The 2nd law of thermodynamics Why dose entropy increase across a shock wave ? Large ( small) Dissapation can not be neglected entropy**To is constant across a stationary normal shock wave**To ≠ const for a moving shock Note: 1 2. The total pressure decreases across a shock wave Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7**Hugoniot equation**It relates only thermodynamic quantities across the shock General relation holds for a perfect gas , chemically reacting gas, real gas Acoustic limit is isentropic flow 1st law of thermodynamic with**For a calorically prefect gas**In equilibrium thermodynamics , any state variable can be expressed as a function of any other two state variable Hugoniot curve the locue of all possible p-v condition behind normal shocks of various strength for a given**For a specific**Straight line Rayleigh line Note ∵supersonic ∴ Isentropic line down below of Rayleigh line In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope**as function (weak) shock strength for general flow**Shock Hugoniot For fluids**Coefficient**For gibbs relation**Let**For every fluid “Normal fluid “ “Compression” shock if if “Expansion “shock p p s=const s=const u u**q**A 3.8 1-D Flow with heat addition e.q 1. friction and thermal conduction 2. combustion (Fuel + air) turbojet ramjet engine burners. 3. laser-heated wind tunnel 4. gasdynamic and chemical leaser +E.O.S Assume calorically perfect gas**The effect of heat addition is to directly change the total**temperature of the flow Heat addition To Heat extraction To**Given: all condition in 1 and q**To facilitate the tabulation of these expression , let state 1 be a reference state at which Mach number 1 occurs.**Table A.3.**For γ=1.4**Adding heat to a**supersonic flow M ↓**To gain a better concept of the effect of heat addition on**M→TS diagram**At point A**B A 1.0 Momentum eq. Continuity eq. Rayleigl line ∴ At point A , M=1**B(M<1)**lower jump Heating cooling M<1 heating cooling M>1 At point B is maximum A (M=1) ds=(dq/T)rev →addition of heat ds>0 MB subsonic**q**1 2**For supersonic flow**Heat addition → move close to A M → 1 → for a certain value of q , M=1 the flow is said to be “ choked ” ∵ Any further increase in q is not possible without a drastic revision of the upstream conditions in region 1**For subsonic flow**heat addition → more closer to A , M →1 → for a certain value of the flow is choked → If q > , then a series of pressure waves will propagate upstream , and nature will adjust the condition is region 1 to a lower subsonic M → decrease E.X 3.8**3.9 1-D Flow with friction**Fanno line Flow • In reality , all fluids • are viscous. • - Analgous to 1-D flow with heat addition.**Momentum equation**Good reference for f : schlicting , boundary layer theory