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

Continuity equation A sound wave, by definition, ie: weak wave 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

The physical meaning of M 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 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 associated with Its values of and at the same point

total temperature or stagnation temperature 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 Note: are sensitive to the reference coordinate system are not sensitive to the reference coordinate (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

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

EX. 32

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) 1 Known adiabatic 2 To be solved 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 ≠ const for a moving shock 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 Note: 1 2. The total pressure decreases across a shock wave Ex.3.4 Ex.3.5 Ex 3.6 Ex 3.7

3.7 Hugoniot Equation

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 Note Rayleigh line ∵supersonic ∴ Isentropic line down below of Rayleigh line In acoustic limit (Δs=0) u1→a insentrop & Hugoniot have the same slope

For fluids Shock Hugoniot 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

3.8 1-D Flow with heat addition q A 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 Momentum eq. Continuity eq. Rayleigl line B A 1.0

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

M P T T0 P0 u q 1 2 (M2>M1) (P2>P1) (T2>T1) (T02>T01) Supersonic flow M1>1 subsonic flow M1<1 M (M2<M1) (M2>M1) P (P2>P1) T (T2>T1) T0 (T02>T01) P0 (P02<P01) u (U2<U1)

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

∵ adiabatic , To = const

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

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

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

1-D adiabatic flow with friction Supersonic flow M1>1 Subsonic flow M1<1 M (M2<M1) (M2>M1) P (P2>P1) T (T2>T1) T0 unchanged P0 (P02<P01) u (u2<u1) ρ