One-dimensional Flow

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# One-dimensional Flow - PowerPoint PPT Presentation

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.

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Presentation Transcript
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

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 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

General equation valid for all gas

Isentropic compressibility

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

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

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

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

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

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

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 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

supersonic flow

M ↓

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

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

IF we defineare the station where , M = 1

F: average friction coefficient

Table A.4

Fanno line

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