Shock Tube Nptel

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Experimental Aero(Gas) dynamics Prof. Job Kurian Chapter-3 1 Dept. of Aerospace Engg., Indian Institute of Technology, Madras Module 3 Lectures 13 - 18 Shock Tube Keywords: Moving shocks, shock tube, Rankine Hugoniot equations, shock velocity, gas particle velocity in shock tube, diaphragm pressure ratio, reflected shock, shock tube boundary layer, observation time in shock tubes. Topics 3.0 Introduction 3.1 The shock tube 3.2 Shock tube equations 3.3 Comparison between shock heating and isentropic heating 3.4 Particle velocity behind moving shock 3.5 Dependence of shock strength on diaphragm pressure ratio 3.6 Reflected shocks Reflected shock parameters 3.7 Viscous effects and the shock tube boundary layer 3.8 Observation time in shock tube 3.9 Interaction of reflected shock and the contact surface 3.10 Shock tube diaphragm and bursting techniques 3.11 Measurement of shock speed

description

SHOCKTUBE

Transcript of Shock Tube Nptel

Page 1: Shock Tube Nptel

Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

1 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Module 3 Lectures 13 - 18

Shock Tube

Keywords: Moving shocks, shock tube, Rankine Hugoniot equations, shock

velocity, gas particle velocity in shock tube, diaphragm pressure ratio, reflected shock,

shock tube boundary layer, observation time in shock tubes.

Topics

3.0 Introduction

3.1 The shock tube

3.2 Shock tube equations

3.3 Comparison between shock heating and isentropic heating

3.4 Particle velocity behind moving shock

3.5 Dependence of shock strength on diaphragm pressure ratio

3.6 Reflected shocks

Reflected shock parameters

3.7 Viscous effects and the shock tube boundary layer

3.8 Observation time in shock tube

3.9 Interaction of reflected shock and the contact surface

3.10 Shock tube diaphragm and bursting techniques

3.11 Measurement of shock speed

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

2 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

3.0 Introduction

The shock tube is a device in which a plane propagating shock is produced in a long tube by

the abrupt rupturing of the diaphragm separating one section of the tube at higher pressure than

that in the other section. This device, developed after a series of studies and experiments in mid

1800s started with the realization that waves from explosions travel at a velocity faster than the

sound waves. In 1948 Sir George Stokes discussed the instability of finite amplitude sound

waves. Paul Vieille (1899) measured the speed of the pressure pulse generated in a glass tube

in which a thin diaphragm across a pressure ratio of 27atm was burst. This pioneering work on

bursting diaphragm shock tube led others to follow similar experiments with different aims.

Payman, Shepherd and other colleagues of Mines Research Board did a series of investigations

on the ignition of explosive gas mixtures by shock waves. The first research paper making use

of shock tube was published by the Royal Society and authored by Payman and Shepherd

(1946). During the post 2nd

world war period, led by many Universities around the globe the

shock tube was developed as a tool for aeronautical research.

3.1 The shock tube

The shock tube (Fig 3.1a) is a long tube of uniform cross section and with uniform internal

dimensions. The diaphragm separates the high pressure driver section from the low pressure

driven or test section. The material of the diaphragm and its thickness are dictated by the

pressure ratio between the sections. On abrupt rupturing of the diaphragm, pressure waves

emanating from the diaphragm station coalesce to form the shock front which propagates in to

the low pressure section. As the shock front moves in to the low pressure section, a contact

surface which is an imaginary line of separation between the driven and driver gases follows

the shock front. The wave diagram of the shock tube after the diaphragm rupture is given in

Figure 3.1b

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

3 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Fig.3.1 (a) The shock tube (b) The shock tube wave diagram

(c) Temperature history in the shock tube at time t1

(d) Pressure history in the shock tube at time t1

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

4 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Simultaneously an expansion fan travels in to the driver section. The wave diagram in Fig 3.1b

shows these wave systems. The regions in the wave diagram are designated as 1, 2, 3 and 4.

Region 1 represents the initial conditions in the driven section. The conditions of gas in this

region are denoted by subscript 1.The Region 2 corresponds to the shocked gas conditions. The

region between the contact surface and the expansion fan is referred to as 3. Region 4

represents the conditions of the initial high pressure driver gas.

The shock tube is a versatile experimental facility for the study of gaseous phenomena at

elevated temperature and pressure. Along with the wave diagram, which is in the (x-t) plane,

the diagrams showing the temperature and pressure history in the shock tube along its length

are given in Fig.3.1c and 3.1d at time. The temperature and pressure history corresponds to a

particular time t1 on the wave diagram. As shown in the figures, the experimental gas is

brought almost instantly to a known and high temperature in region 2 and held at steady

temperature and pressure for a few hundred micro sec. Usually there is some mixing and inter

diffusion of gases at the contact surface so that the temperature fall is less sudden, than in the

ideal case.

3.2 Shock tube equations

The equations applicable in the case of shock tube which are the moving normal shock

equations may be derived considering the shock fixed coordinate system. In the figures, Ws

represents the wave speed which is the speed at which the shock front propagates in the shock

tube and v1 and v2 are the gas particle velocities ahead and behind the moving shock

respectively. To convert the Laboratory fixed coordinates to shock fixed, wave speed Ws is

applied in the opposite direction of the propagating shock.

1 S 1u = W -v ………… 3.1

2 S 2u = W -v ………… 3.2

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

5 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

u1 and u2 are the relative velocities in the shock fixed candidates

Fig.3.2 (a) Laboratory fixed coordinates (b) Shock fixed coordinates

Consider the conservation equations and the other thermodynamic equations as follows:

continuity 1 1 2 2ρ u = ρ u ….……3.3

momentum 2 2

1 1 1 2 2 2p + ρu = p +ρ u ……….3.4

energy 2 2

1 1 2 2

1 1h + u = h + u

2 2 ……….3.5

TPh = c

R

= T-1

p=

-1ρ

………3.6

Substituting in energy equation

2 21 21 2

1 2

p 1 p 1+ u = + u

-1ρ 2 -1ρ 2

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

6 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Using equations 3.3 to 3.6 u1 and u2 are eliminated from the equations 1 2 2 1

2 2 1 1

ρ p - ρ p -1=

ρ p - ρ p +1

Rearranging this equation we get

1

2 2

11

2

-1 ρ1-

p +1ρ=

ρ -1p -ρ +1

………… 3.7

2

2 1

1 2

1

-1 p+

ρ +1 p=

ρ -1 p+1

+1 p

……………. 3.8

The equations 3.7 and 3.8 are the Rankine - Hugoniot equations

From continuity and momentum equations

2

2 1 1 1

1 1 2

p ρ u ρ= 1+ 1-

p p ρ ……………… 3.9

Define S1

1

1 1

WuM = =

c c ……………. 3.10

Equation 3.10 is applicable when v1 the particle velocity ahead of the moving shock is zero, in

other words when shock is propagating in to still air

11 1

1

pc = RT =

ρ

11

1

uM =

c

Using equation 3.6 and 3.10

M1

1

21

1

1

ρ= u

p

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

7 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

22 1 11

1

u ρM =

p

2 2 11 1

1

ρM = u

p ………….. 3.11

Substitute equation 3.11 in 3.9 to write

22 11

1 2

p ρ= 1+ M 1-

p ρ

……………….. 3.12

Combining equations 3.8 and 3.12 gives

2

12

1

2 M - -1p=

p +1

……………. 3.13

2

12

2

1 1

+1 Mρ=

ρ -1 M +2

………………. 3.14

Combining the equations 3.13 and 3.14 with the perfect gas equation gives

2 2

1 1

22

1 2

1

-1 -1M - M +1

T 2 2=

T +1M

2

…………………3.15

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

8 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Fig.3.3 Theoretical ratios of shocked gas properties

3.3 Comparison between shock heating and isentropic heating

For a given pressure change (or density change) in a gas, the temperature produced by a strong

shock wave is greatly in excess of those in the isentropic case. Shock wave is an irreversible

adiabatic compression

2 2

1 1 2 2

1 1h + v = h + v

2 2

Conservation of energy derived for flow where a resistance to motion occurs. In the case of

shock wave, the abrupt pressure rise across the front presents a resistance to flow and as such

implies an irreversible conversion of kinetic energy (K.E) into heat.

2 2

1 2

1 1v - v

2 2

is a measure of this change in K.E for unit mass of gas.

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

9 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Across the shock front there is an increase of entropy, for an adiabatic process.

2 2

1 1

T T

V

T T

dQ dT dvΔS = = c + p

T T T

From this, for a perfect gas

2 1p -1

2 1

T /TΔS = C ln

p /p

-1

2 2

1 1 P

T p Δs= exp

T p c

…………………… 3.16

From this it can be seen that (T2/T1) across a shock front ΔS > 0 exceeds that for an isentropic

compression ΔS 0 by a factor P

ΔSexp

c

between the same limits. Compression by a

piston across which differential pressure is kept small.

Table 3.1 Comparison between isentropic and shock heating

2

1

p

p

Shock

Wave 2T K

Isentropic

Compression

2T K

M1

2 420 357 1.32

10 895 566 2.95

25 1753 735 4.45

50 3177 897 6.55

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

10 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

3.4 Particle velocity behind the moving shock wave

Refer to Fig.3.7 (b) of shock fixed coordinates

1 S 1

2 S 2

u = W -v

u = W -v

1 2 1 2

2 1

u -u = - v - v

= v - v

2

12 1 S 1

2

1 1 2 S 2

+1 Mρ u W -v= = =

ρ -1 M +2 u W -v

2

2 1

211

ρ +1 M=

-1ρ 2 1+ M2

1 1 1u = M c

2

1

2 1 2

1

-1 M +2u = u

+1 M

u2

2

1

1 1 2

1

-1 M +2= M c

+1 M

2

1

2 1 1 1

1

-1 M +2u -u = c -M

+1 M

22 2 2 21 11 1 1 1

1

1 1

c 2 1-MM -M +2- M -M= c =

+1 M +1 M

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

11 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

2

1 11 2

1

2c M -1u -u =

+1 M

= 1

1

1

2c 1M -

+1 M

12 1 1

1

2c 1v = M - +v

+1 M

…………….3.17

Illustration of calculation of particle velocity (v2) and Mach number

behind the moving shock

Calculations are done assuming Air to be the test gas at 300K

1For M = 8

T2/T1 = 13.387 (from Gas Tables)

2T = 4016.1 2c = 1270m/s

12 1

1

2c 1v = M - = 2280m/s

+1 M

2M = 1.79

1For M = 10

2

1

T 20.388 

T (From Gas Tables)

T2 = 6116.4 2c = 1567m/s

2

2×347.5v = 10-0.1 = 2866m/s = 1.829

2.4

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

12 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

1For M = 25

2 2

1 1

22

1 2

1

-1 -1M - M +1

T 2 2=

T +1M

2

= 1.4 625-0.2 0.2 625+1

1.44 625

874.8 126

= = 122.47900

T2 = 36741

2T = 191.68×20.04

C2 = 3842.21m/s

2

2×347.5 1v = 25- ×24.96

2.4 25

7228

= 1.883842

What is shown above is that increase of shock velocity (Ws) and consequently the shock Mach

number will not correspondingly raise the Mach number after the shock (M2) based on v2.

Understandably, the reason is the increase in the post shock temperature (T2) and the value of

speed of sound (c2) after the shock to high values.

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

13 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

3.5 Dependence of shock strength on diaphragm pressure ratio

Fig.3.4 Wave diagram and the gas properties along the shock tube at time t1

Driver gas at a high pressure p4 in region 4 is being expanded through the expansion fan to a

lower pressure p3 behind the contact surface. Limits of expansion fan are formed by the head

and tail of rarefaction wave.

[Ref: The production of High Temperature Gases in Shock Tube Rester, Lin, Kantowitz

Th. App. Phy. Vol 29 (Dec 1952) PP 1390-1399]

Expansion is an isentropic process. Applying Newton’s 2nd law, F = m dv

dt to a one

dimensional fluid element of thickness dx. Under the action of a sound wave travelling at a

velocity c, we get

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

14 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

dv

- dp = ρdx = ρcdv1

dxc

……………… (3.18)

ρdx m

1dx dt

c

Fig.3.5 Elemental fluid volume subjected to forces

If an ideal gas with constant specific heats undergoes a change of state the entropy change ΔS

of the gas is

f fP

i i

T pΔS = c ln - Rln

T p

In differential form P

dT dpdS = c - R

T p

For an isentropic process ds = 0

Hence, Pdp c dT=

p R T ……………… (3.19)

2c = RT ie T = 2c

R

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

15 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

2cdc

dT =R

2

dT 2cdc 2dc= =

cT cR

R

………………………………………...(3.20)

Combining (3.19) and (3.20)

Pdp 2c dc=

p Rc ……………………………………….. (3.21)

From Equation (3.18) - dp = c dV

From (4) Pcdv 2c dc- =

p Rc

2

P

c Rdv ρ pRdv2c dc = =

p ρ p

P2c dc = R

2 dc = - Rdv-1

P

Rc =

-1

ie

2dc+dv = 0

-1

Thus, for an isentropic process

2

c+v = constant-1

For the shock tube problem

4 4 3 3

4 3

2 2c +v = c +v

-1 -1

On each side of the contact surface, the particle velocities and pressures must be continuous,

since there is no gas flow across the interface.

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

16 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

v3 = v2 and p3 = p2

As v4 = 0 and 4 = 3

4 3 2

4 4

2 2c = c +v

-1 -1

4-1

3 4 2c = c - v2

From adiabatic gas law

4

4-1

2

4 4 4

3 3 2

p c p= =

p c p

p3 = p2

4

4-1

2

4 4

4-124 2

p c=

pc - v

2

……………………………………. (3.22)

12 1

1+1 1

2c 1v = M -

M

………………………………………. (3.23)

Combining these equations with equation for 2

2 1

1

p 2 M - -1=

p +1

leads to

4

4-1

2-

24-11 1 14 1

1

1 1 1+1 4 1

2 M - -1p c 1= 1- M -

p +1 c M

……………………… (3.24)

Strongest shocks are obtained when

4

1

p

p and then

4-1 11

1+1 4 1

c 11- M - 0

c M

( 1)

( 1)

1 41

4 1

cM

c

Strongest shocks are obtained for driver with

1) High speed of sound

2) Low specific heat ratio

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

17 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

3.6 Reflected shocks

When the primary shock wave reaches the end of the shock tube it gets reflected back in to the

medium which is already heated by the incident shock. Higher temperature and pressure are

obtained behind the reflected shock. Additionally the gas behind the reflected wave is at rest

relative to the shock tube.

Summarizing:

a) Much higher temperatures are obtained that too at high pressures. Dissociation is

prevented because of increased pressures.

b) Study on a fixed group of molecules can be conducted. Behind the incident shock, the

gas particles are moving with a velocity v2 and it is impossible to follow a fixed volume

of gas.

3.6.1 Reflected shock parameters

(a) (b)

(c) (d)

Fig.3.6 Incident and reflected shocks: (a&c) Laboratory fixed coordinates,

(b&d) Shock fixed coordinates

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

18 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

The subscript 5 refers to the reflected gas properties. The gas flows into the shock front with a

relative velocity 2u where

2 R 2u = W +v

In the reflected shock region v5 = 0. This means that the gas gives all its kinetic energy (K.E)

on passing through the front thus increasing the properties of the state of the gas. This can be

seen as an ever extending column which advances out from the end wall of the tube and

contains a gas at high temperature; density and pressure. The particle velocities relative to the

reflected shock at different sections can be written.

5 R 5 Ru = W -v W

1

2 R 2u = W +v

If Mach no. of the reflected shock is defined as

2

R

2

uM =

c

Equations relating to reflected shock may be written analogy with those with incident shock.

For incident shock

11 2 2 1 1

1

2c 1u -u = v -v = M -

+1 M

1 Su = W

2 S 2u = W -v

0

2 5 2 5u -u = v + v

2R

R

2c 1= M -

+1 M

1 22 1 R

1 R

2c 1 2c 1v = M - = M -

+1 M +1 M

…..…………………… (3.25)

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

19 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

From perfect gas equation and equation of the speed of sound

2

2 2 1

1 1 2

c p ρ= ×

c p ρ ………………………………… (3.26)

From Rankine -Hugoniot equations

2

2 1

1 2

1

-1 p+

ρ +2 p=

ρ -1 p+1

+1 p

T1

1

22

2

p= RT

ρ

p= RT

ρ

2 -1

2 2 2 2

1 1 1 1

c p -1 p -1 p= 1+ +

c p +1p +1 p

………………………(3.27)

Substitute in the incident shock equations

2

12

1

2 M - -1p=

p +1

By analogy, we may write for reflected shock

2

R5

2

2 M - -1p=

p +1

……………………….. (3.28)

Solving equations (3.25) to (3.28)

1

5 2

12

2

+1 p+2-

p -1 p=

+1 pp 1+-1 p

………………………. (3.29)

5

2

ρ

ρcan be written by analogy

5

5 2

2 5

2

-1 p+

ρ +1 p=

ρ -1 p+1

+1 p

……………………….(3.30)

Similarly,

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

20 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

5

5 5 2

2 2 5

2

+1 p+

T p -1 p=

T p +1 p1+

-1 p

……………… (3.31)

These ratios can also be written as:

5

1

2 2

1 1

2

1

p 2 M - -1 3 -1 M -2 -1=

p +1 -1 M +2

2 2

1 15

2

1 1

2 -1 M + 3- 3 -1 M -2 -1T=

T +1 M

The ratio of speeds can be written as

1

R 2

1S

2

2 p2+

W -1p=

+1 pW --1 p

Table 3.2 Reflected shock parameters in terms of incident parameters

M1

2

1

p

p 5

2

p

p 2

1

T

T 5

2

T

T R

S

W

W

1.4 2.95 10 4.95 2.62 1.76 0.423

6.56 50 7.12 9.31 2.28 0.351

1.66 2.87 10 4.22 3.42 1.94 0.589

6.34 50 5.54 13.4 2.34 0.517

3.7 Viscous effects and the shock tube boundary layer

Due to the viscous nature of the flow in the shock tube, boundary layer is formed. Its thickness

will be zero at the shock front and increases back through the shock heated region and the

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

21 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

contact surface in to the expanding driver gas and becomes zero again at the head of the

rarefaction fan. Schematic of the boundary layer formed is shown in Figure 3.13. Important

effects of formation of the boundary layer are the following.

1. Kinetic energy is dissipated as heat in the retarding layer of the boundary layer and this

is conveyed to walls as by heat transfer.

2. Deceleration of the shock front.

3. Acceleration of contact surface.

One of the important effects of the formation of boundary layer is its reducing effects of the

useful observation time in shock tube experiments.

Fig.3.7 Schematic of the growth of boundary layer in a shock tube

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

22 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

3.8 Observation time in shock tube

Observation time is decided by

a) length of observation station from diaphragm

b) length of driver section [due to the influence reflected rarefaction fan]

c) length of end plate from diaphragm [ influence of the reflected shock]

d) growth of the boundary layer

Fig.3.8 (x,t) diagram of the shock tube indicating the different waves

3.9 Interaction of reflected shock and contact surface

The reflected shock, after going through the shocked gas in the region 2 encounters the contact

surface and the reaction will be dependent on the speed.

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

23 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Fig. 3.9 (a)

Fig.3.9 (b)

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

24 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Time available for measurements in the reflected shock region can be improved.

Fig.3.9 (c)

Fig.3.9 Interaction of reflected shock with the contact surface

(a) c2 > c3 (b) c2<c3 (c) c2= c3

Speed of sound in regions 2 and 3. The schematic of the wave systems generated based on the

values of c2 and c3 is shown in Figures 3.9 (a) and (b). As shown in Fig.3.9a when c2 > c3, the

reflected shock after passing through the contact surface enters a region of higher Mach

number. The resultant properties in the region 3 would be greater than those in regions 2.This

is physically not feasible and in order to make the pressures on either side of the contact

surface an additional shock system is generated in region 5. The opposite effect is there

corresponding to the situation in Fig.3.9b where c2<c3.The resultant wave system in the

reflected shock region is an expansion wave. In both the cases, flow in the reflected shock

region is disturbed by the additional waves generated. In order to have larger observation times

behind the reflected shock c2 is designed to be same as c3.This is called ‘tailoring’ of the shock

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

25 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

tube contact surface. The wave system when c2=c3 is shown in Fig.3.9c.Tailoring enhances the

observation time behind the reflected shock.

3.10 Shock tube diaphragm and bursting techniques

Diaphragms vary from 0.025mm cellophane sheet to 6mm thick steel plate. For the same

material the diaphragm rupturing pressure increases directly as the pressure and inversely as

the exposed area.

1) pressure driving

2) Mechanical piston drive

3) Combustion driving

4) Heating of driver gas

3.11 Measurement of shock speed

Conventionally, the time distance method is used for the measurement of shock speed. Sensors

of pressure or temperature are mounted flush with the interior of the low pressure section of the

shock tube at known distance between them. Time taken by the propagating shock to

transverse the known distance is used to calculate the shock speed. Though this is a simple

method to implement, the speed obtained is an average value between the two stations. Other

methods such as that making use of Doppler principle could give higher spatial resolution.

************************************

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Experimental Aero(Gas) dynamics Prof. Job Kurian

Chapter-3

26 Dept. of Aerospace Engg., Indian Institute of Technology, Madras

Exercises

Answer the following

1. With the help of the wave diagram explain the waves in the shock tube on rupture of its

diaphragm.

2. Graphically show the variation of the ratios of temperature, pressure and density across

the moving shock against the shock Mach number.

3. Derive an equation for the gas particle velocity behind the moving shock.

4. The gas particle velocity behind the moving shock increases with increasing shock

speed, but the corresponding Mach number does not. Why?

5. What is the equation for the diaphragm pressure ratio in terms of the shock Mach

number

6. For the given pressure ratio across the diaphragm of a shock tube, how can shocks of

different strengths be generated?

7. What the feature of gas behind the reflected shock?

8. Sketch the boundary layer developed in the shock tube on diaphragm rupture.

9. What factors decide the observation time in a shock tube?

10. What is the effect of boundary layer on the observation time?

11. Unless special care is taken, the reflected shock does not pass through the contact

surface without additional waves. Elucidate.

12. Why is shock velocity measurement very important in shock tube experiments? How is

the velocity measured conventionally?

Work out the following numerical problem

1. In a shock tube experiment using air at 5 21× 10 N/m and 310K as the test gas, a pressure

ratio of 29 across the wave was observed. Find the stagnation temperature and pressure

behind the propagating shock wave.