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One-dimensional Flow 3.1 Introduction In real vehicle geometry, The flow will be axisymmetric Normal shock One dimensional flow

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The variation of area A=A(x) is gradual Neglect the Y and Z flow variation

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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

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The continuity equation L.H.S of C.V The momentum equation (Continuity eqn for steady 1-D flow)

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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.

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The energy equation Physical principle of the energy equation is the energy is the energy is conserved Energy added to the C.VEnergy taken away from the system to the surrounding

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3.3 Speed of sound and Mach number Wave front called “ Mach Wave” Mach angle μ Always stays inside the family of circular sound waves Always stays outside the family of circular sound waves

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Wave front A sound wave, by definition, ie: weak wave ( Implies that the irreversible, dissipative conduction are negligible) Continuity equation 12

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Momentum equation No heat addition + reversible General equation valid for all gas Isentropic compressibility

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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

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Form kinetic theory a for air at standard sea level = m/s = 1117 ft/s Mach Number Subsonic flow Sonic flow supersonic flow The physical meaning of M Kinetic energy Internal energy

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3.4 Some conveniently defined parameters Inagine: Take this fluid element and Adiabatically slow 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 Its values of and at the same point associated with

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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)

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3.5 Alternative Forms of the 1-D energy equation = 0(adiabatic Flow) A B If the actual flow field is nonadiabatic form A to B → Many practical aerodynamic flows are reasonably adiabatic calorically prefect

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Total conditions - isentropic Adiabatic flow isentropic Note the flowfiled is not necessary to be isentropic If not → If isentropic → are constant values

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= 1 if M=1 < 1 if M < 1 > 1 if M > 1 or If M → ∞

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EX. 32

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3.6 Normal shock relations The shock is a very thin region, Shock thickness ~ 0 (a few molecular mean free paths) ~ cm for standard condition) 1 Known adiabatic 2 To be solved Continuity Momentum Energy ( A discontinuity across which the flow properties suddenly change) Ideal gas E.O.S Calorically perfect Variable : 5 equations

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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

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Special case Infinitely weak normal shock. ie: sound wave or a Mach wave

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Note : for a calorically perfect gas, with γ=constant are functions of only Real gas effects

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The 2nd law of thermodynamics Why dose entropy increase across a shock wave ? Mathematically eqns of hold for Physically, only is possible Large ( small) Dissapation can not be neglected entropy

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Note: 1 2. Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7 To is constant across a stationary normal shock wave To ≠ const for a moving shock The total pressure decreases across a shock wave

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3.7 Hugoniot Equation

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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

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In equilibrium thermodynamics, any state variable can be expressed as a function of any other two state variable For a calorically prefect gas Hugoniot curve the locue of all possible p-v condition behind normal shocks of various strength for a given

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For a specific Straight line Note Rayleigh line ∵ supersonic ∴ Isentropic line down below of Rayleigh line In acoustic limit ( Δs=0) u 1 →a insentrop & Hugoniot have the same slope

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as function (weak) shock strength for general flow Shock Hugoniot For fluids

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Coefficient For gibbs relation

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u p s=const u p For every fluid “Normal fluid “ “Compression” shock “Expansion “shock Let s=const if

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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 Assume calorically perfect gas +E.O.S q A

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The effect of heat addition is to directly change the total temperature of the flow Heat addition T o Heat extraction T o

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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.

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Table A.3. For γ=1.4

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Adding heat to a supersonic flow M ↓

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To gain a better concept of the effect of heat addition on M→TS diagram

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1.0 B A At point A ∴ At point A, M=1 Rayleigl line Momentum eq. Continuity eq.

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At point Bis maximum A (M=1) M B subsonic B(M<1) M<1 Heating cooling heating M>1 jump cooling lower ds=(dq/T) rev →addition of heat ds>0

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12 q Supersonic flow M 1 >1 subsonic flow M 1 <1 M (M 2

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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

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→ 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 For subsonic flow E.X 3.8 heat addition → more closer to A, M →1

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3.9 1-D Flow with friction - In reality, all fluids are viscous. - Analgous to 1-D flow with heat addition. Fanno line Flow

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Momentum equation Good reference for f : schlicting, boundary layer theory

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∵ adiabatic, To = const

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Analogous to 1-D flow with heat addition using sonic reference condition.

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Table A.4 IF we define are the station where, M = 1 F: average friction coefficient

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P At point P T high u low above P, M < 1 T low u high below P, M > 1 Fanno line ds > 0 ds < 0 chocked

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Supersonic flow M 1 >1 Subsonic flow M 1 <1 M (M 2

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