one dimensional flow l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
One-dimensional Flow PowerPoint Presentation
Download Presentation
One-dimensional Flow

Loading in 2 Seconds...

play fullscreen
1 / 54

One-dimensional Flow - PowerPoint PPT Presentation


  • 111 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'One-dimensional Flow' - rodd


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
one dimensional flow
One-dimensional Flow
  • 3.1 Introduction

Normal shock

In real vehicle geometry, The flow will be axisymmetric

One dimensional flow

slide2

The variation of area A=A(x) is gradual

Neglect the Y and Z flow variation

3 2 steady one dimensional flow equation
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

slide4

The continuity equation

L.H.S of C.V

(Continuity eqn for steady 1-D flow)

  • The momentum equation
slide5

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.

slide6
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

slide7

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

slide8

1

2

A sound wave, by definition,

ie: weak wave

( Implies that the irreversible,

dissipative conduction are negligible)

Wave front

  • Continuity equation
slide9

Momentum equation

No heat addition + reversible

General equation valid for all gas

Isentropic compressibility

slide10

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

slide11

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

slide12

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

slide13
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)

slide14

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

slide15

Total conditions - isentropic

Adiabatic flow

isentropic

Note the flowfiled is not necessary to be isentropic

If not →

If isentropic → are constant values

slide17

= 1 if M=1

< 1 if M < 1

> 1 if M > 1

If M → ∞

or

slide19

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

slide20

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

slide23

Note : for a calorically perfect gas , with γ=constant

are functions of only

Real gas effects

slide25

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

slide26

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

slide28

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

slide29

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

slide30

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

slide32

Coefficient

For gibbs relation

slide33

Let

For every fluid

“Normal fluid “

“Compression” shock

if

if

“Expansion “shock

p

p

s=const

s=const

u

u

slide34

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

slide35

The effect of heat addition is to directly change the total temperature of the flow

Heat addition To

Heat extraction To

slide38

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.

slide39

Table A.3.

For γ=1.4

slide40

Adding heat to a

supersonic flow

M ↓

slide43

At point A

B

A

1.0

Momentum eq.

Continuity eq.

Rayleigl line

∴ At point A , M=1

slide44

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

slide45

q

1

2

slide46

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

slide47

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

slide48

3.9 1-D Flow with friction

Fanno line Flow

  • In reality , all fluids
  • are viscous.
  • - Analgous to 1-D flow with heat addition.
slide49

Momentum equation

Good reference for f : schlicting , boundary layer theory

slide52

IF we defineare the station where , M = 1

F: average friction coefficient

Table A.4

slide53

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