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International Conference on Mechanical and Industrial Engineering114

Numerical Simulation and Validation of Inviscid

Transient Flow in Shock Tube

Rachan D Shekar1*, Rajanna D2,

14th Sem MTech, Thermal Power Engg.,2 Assistant Professor, Dept of Mechanical Engg., Adichunchanagiri Institute of Technology, Chikmagalur,

[email protected]; [email protected]

Abstract : The shock tube has found widespread use as an experimental device to investigate the chemical kinetic behaviour in reactive gas mixtures. A shock tube consists of a high pressure driver section and low-pressure driven section initiallyseparated by a diaphragm. The driver section is pressurized high enough to cause the diaphragm to rupture and as a result, a

shock wave is generated and travels down the driven tube, the shock developed will cause elevation of temperature andcorrespondingly pressure, velocity and density. Here analysis is being made on the mixture of Helium-Argon gases as it hasa wide spread applications today in all the elevated temperature-pressure working environments like bulbs, welding etc., asthese gases are very scarce and difficult and costly to extract it and the experimentation on shock is too costly, a tube modelwill be developed to simulate the shock-wave propagation and reflection flows. Computations were carried out in the CFDsolver FLUENT based on the finite volume approach and instead of experimentation and the robustness of the numericalmodel and the accuracy of the simulations will be assessed through validation with the analytical ideal shock-tube theory.

Keywords : Shock Tubes, diaphragm, driver section, driven section, Fluent

1. INTRODUCTION

The shock tube is a device in which a normal shock wave

is produced by the sudden bursting of a diaphragm

separating a gas at high pressure from one at lower

pressure. The simplest form of a shock tube is illustrated

in Figure 1 where the high pressure and low pressure

sections are commonly referred to, respectively, as the

driver and driven sections of the tube. A Shock Tube is

an experimental device used to study the chemical kinetic

behaviour of gases. A shock tube consists of a high

pressure driver section and low-pressure driven section

initially separated by a diaphragm. The driver section is

pressurized to energy high enough to cause the

diaphragm to rupture and as a result, a shock wave is

generated and travels down the driven tube.Simultaneously, an expansion fan propagates through the

high-pressure side. Both waves reflect off the shock tube

end walls [1].

Chemical kinetics includes investigations of how

different experimental conditions can influence the speed

of chemical reaction and yield information about the

reaction’s mechanism and transition states, as well as the

construction of mathematical models that can describe

the characteristics of a chemical reaction. The different

experimental conditions here refer to the tube diameter,

pressure and the shock Mach number.

A typical shock tube consists of: driven and driver

section.

Fig. 1 A typical Shock Tube

The mechanisms that are used to burst the

diaphragm that separates the different gaseous mixtures

or the pressure regions the shock tubes are:

1. A mechanically-driven plunger is used to pierce the

diaphragm or an explosive charge may be used to burst

it.

ISBN: 978-93-81693-89-6

mailto:[email protected]




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International Conference on Mechanical and Industrial Engineering

115

Numerical Simulation and Validation of Inviscid Transient Flow in Shock Tube

2. Use of diaphragms of plastics or metals to define the

bursting pressures. Plastics are used for the lowest burst

pressures, aluminum and copper for high levels and

mild steel and stainless steel for highest burst pressures.

3. Use of a mixture of combustible gases, with an initiator

to produce a detonation within it.

Driver Section Driven Section

Fig. 2 An idealized shock tube. The plot shows different

waves which are formed in the tube once the diaphragm

is ruptured

Appl ications of Shock tube:

1 Used for the measurement of the Chemical kinetic

behavior of the various gas and gas mixtures.

2. Used as a wind tunnel allowing higher temperatures

and pressures therein replacing conditions in the turbine

sections of jet engines.

3. Used to simulate and analyze the re-entry of spacecraftor hypersonic crafts into the atmosphere.

Shock tube is an experimental device used to

evaluate the chemical kinetics of various gases at

elevated temperatures and pressures. The process that

takes inside the shock tube will takes place in a time span

of micro seconds to nanoseconds which is a very short

time span and needs special measuring devices like

Oscilloscope for measuring velocities, pressure and

temperature probes which is capable of sensing the

minute changes in pressures and temperatures in the

order or micro to nanoseconds. The above saidinstruments are very expensive and there is a possibility

of containing errors in calibration. We have divided the

shock tube into five parts in this project work to evaluate

the temperature and pressure at various time steps, so if

experimental results are opted for validation, it becomes

too expensive and may be filled with errors [1-4].

Shock tube is an isolated system where in it must be

totally insulated to prevent any loss of temperature into

the atmosphere, so it must be perfectly insulated for this

purpose and as the pressures builds up to about 1MPa it

must be pressure tight to withstand this condition, if there

are any loss of pressure and temperature the entire

purpose of this work may be wasted. Again care must be

taken to avoid explosion of the shock tube due to the

build up of high pressures and temperatures. Shock tube

must be leak proof, both for avoiding the leakage of

gases that is being tested for and also to maintain the

elevated pressures and temperatures. So the process of

fabrication of this setup is very tedious [5-9].

The main objective of this present work is to analyse

a typical Shock tube of 5m length with 2.5m as driver

section and 2.5m as the driven section, with argon-

helium mixture as the working fluid and for three

different pressure ratios across the diaphragm namely

20Bar, 100Bar and 50Bar and validating these results

with the analytical calculations using the Standard Ideal

Shock tube theory.

2. SHOCK TUBE ANALYTICAL

CALCULATIONS

a. The suffixes

b. Refers to Driven Section

c. Refers to point of Diaphragm rupture

d. Refers to the contact surface

e. Refers to the Driver sectionf. refers to the reflected zone

V= Velocity of the gaseous mixture

P= Pressure

= Density

T = Temperature

= ratio of specific heats

M = Mach number

R = gas constant

All gases are assumed to be ideal gases, i.e. p = RT.

The strength of the normal shock is dependant on the

initial pressure ratio of the shock tube. A higher initial

pressure ratio will result in a higher strength normal

shock. A temperature discontinuity also occurs behind

the normal shock.

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The derivation of the analytical solution given

below, the purpose of this work is to evaluate how

effectively CFD can predict high speed transient flow

with discontinuities. The following analytical shock tube

solution is based on the solution process from: Gas

Dynamics, section “Shock Tube and One-Dimensional

Unsteady Flow”.

The Sod problem (Sod, 1978) is an

essentially one- dimensional flow discontinuity problem

which provides a good test of a compressible code's

ability to capture shocks and contact discontinuities with

a small number of zones and to produce the correct

density profile in a rarefaction. The problem spatial

domain is 0 ≤ x ≤ 1.

The initial solution of the problem consists of two

uniform states, termed as left and right states,

separated by a discontinuity at the origin, xo = 2.5. The

fluid is initially at rest on either side of the interface, and

the density and pressure jumps are chosen so that all

three types of flow discontinuity (shock, contact, and

rarefaction) develop.

This solution assumes the system is initially

isothermal, thus T1=T4. Additionally the solution

assumes one dimensional, in viscid and frictionless flow.

For any given ini tial conditions in the shock tube

Universal gas constant R u=8.314 J/mol K

1atm=1.013X105 Pa

Dr iven gas – Argon (1)

Density ρ1=1.72 Kg/m3

Molar mass M1= 39.948g/mol = 0.039948Kg/mol

Ratio of Sp. Heats γ1= 1.67

Let the initial pressure be P1=10X103 Pa

Temperature T1 = 300K

Dr iver gas – Heli um (4)

Density ρ4=0.1786 Kg/m3 (Not constant increases as

pressurized to around 23Kg/m3 to produce shock)

Molar mass M4= 4.002g/mol = 4.002 X 10-3 Kg/mol

Ratio of Sp. Heats γ4= 1.67

Pressure across the incident shock-

1/2

1

2111

1

2

4

14

24

44

1P

P)1(22

1P

P

R

R )1(

1PP

Pressure ratio across the diaphragm =P4/P1

Temperature across the incident shock

1

2

1

1

1

2

1

1

1

2

P

P

1

1

1

P

P

1

1

T

T

Density across the incident shock-

1

2

1

2

1

1

1

2

1

1

P

P

1

1

P

P

1

11

Velocity of shock and gas ahead of the shock tube-

2

1S2112

1MTR V

I ncident shock properti es – I ncident shock number Ms-

11PP

21M

1

2

1

1S

Reflected Mach No M r-

Consider a factor MR,

1M

MMR

2

r

r

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2

1

22

s

1

2

s

2

s 1

12M

1)1M(

11M

MsMR

On solving the above equation the value of M r is

obtained

Temperature of the reflected shock T 5 -

2

r

2

1

1

2

r 11

2

r 1215

M1

12M133M12TTT

Pressure of the ref lected shock P 5 -

2M1

12M13

1

1MPP

2

r 1

1

2

r 1

1

1

2

r 115

Velocity of the refl ected shock V 5 -

2

14415 1TR Mr V

At Contact Sur face-

P3 = P2

V3 = V2

ρ3 = ρ2

TABLE I Pressure result table

TrailNo.

P4/P1

Pressures (Bar)

P1 P2 P3 P4 P5

1 20 1 7.7 7.7 20 1.97

2 50 1 13.1 13.1 50 2.58

3 100 1 18.6 18.6 100 2.68

TABLE III Temperature result table

TrailNo.

P4/P

1

Temperatures (K)

T1 T2 T3 T4 T5

1 20 300 110.64 110.64 300 619.68

2 50 300 96.34 96.34 300 627.49

3 100 300 90.32 90.32 300 617.11

TABLE IIIII velocity result table

TrailNo.

P4/P1Velocities (m/sec)

V1 V2 V3 V4 V5

1 20 0 311.81 311.81 0-600 459.02

2 50 0 409.38 409.38 0-600 505.13

3 100 0 486.56 486.56 0-600 512.25

The suffixes

Refers to Driven Section

Refers to point of Diaphragm rupture

Refers to the contact surface

Refers to the Driver section

Refers to the reflected zone

CFD Analysis of Shock Tubes Modell ing-

Fig. 3 Computational Domain (not to scale)

The model used for the analysis is drawn in Design

Modeller of Ansys-13 Work Bench

Meshing

Fig.4 CFD Mesh Shock tube

All the meshes consist of hexahedral elements and

are created with ANSYS Meshing. A typical mesh is

shown in Fig. 4. A uniform grid spacing with the number

N = 500 is used. The boundary conditions of the problem

are held fixed as a short time span of the unsteady flow is

considered. The wave pattern of this problem consists of

a rightward moving shock wave, a leftward moving

rarefaction wave and a contact discontinuity separating

the shock and rarefaction waves and moving rightward.

3. DOMAIN SETUP-

The working fluid used for the simulations is Air IdealGas. The domain reference pressure is set to various

pressure levels for various trails. Since kinetic energy

effects are significant, the Total Energy Heat Transfer

Model is used. Also, the flow is assumed to be laminar.

Even though the analytical solution assumes inviscid

flow, the laminar assumption is valid for the numerical

computations because symmetry boundary conditions are

5m

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used instead of no slip walls (so there is no boundary

layer) and viscous effects tend to not significantly effect

the interior flow in the presence of shocks

Procedure followed for the Pressure ratio of 20 is

shown below.

Boundary conditi ons-

In order to remove the impact of any boundary layer

growth on the simulations and enforce a one-dimensional

solution, symmetry planes are used on the cross-stream

walls of the domain. The end walls of the domain are

modelled as free slip walls.

Fig. 5 Initial Condition Fluent

I niti al conditions-

The shock tube domain is centred at the origin with

the stream wise direction aligned along the z-axis.

The initial pressure of the shock tube is specified with

20 bar in left side of the tube shown in Fig.5, and the

right side absolute pressure to become 1Bar. The

initial velocity is set to zero and the initial temperature

is set to 300K.

The processing of the above meshed Shock tube for

a pressure ratio of 20using the FLUENT solver is as

following:

Fig. 6 Velocity Contours 0.5 ms to 2 ms

Fig. 7 Temperature Contours 0.5 ms to 2 ms

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4. RESULTS AND DISCUSSION

Figure 8 to 16 shows comparisons between the present

results for the pressure, velocity and temperature at a

time t = 0.2 ms and the exact solutions. It can be

observed that the present solver is capable of capturing

the different types of discontinuities accurately.

Fig. 8 Comparison of Pressure Results shock for the

Pressure ratio of 20 at t = 2 ms

Fig. 9 Comparison of Velocity Results shock for the


Fig. 10 Comparison of Temperature Results shock for the




Fig. 12 Comparison of Velocity Results shock for the Pressure ratio of 50 at t = 2 ms

Fig. 13 Comparison of Temperature Results shock for the


0.00E+00

5.00E+05

1.00E+06

1.50E+06

2.00E+06

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

P r e s s u r e P a

Length m

Analytical CFD

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

8.00E+02

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

V e l o c i t y m / s

Length m

Analyt ical CFD

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

8.00E+02

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

T e m p e r a t u r e K

Length m

Analytical CFD

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Fig. 15 Comparison of Velocity Results shock for the Pressure ratio of 100 at t = 2 ms

Fig. 16Comparison of Velocity Results shock for the


From the above graphs it is can be observed that the

analytical values obtained is equal to the results of the

CFD analysis of the Shock tubes. It can be observed that

the present solver is capable of capturing the different

types of discontinuities accurately.

5. CONCLUSIONS

From the results obtained and the comparisons made

between them it can be concluded that the Numerical

simulation or the CFD analysis and its validation using

the analytical results can be preferred over the

experimental analysis thereby overcoming the hurdles of

cost and the difficulties involved in conducting the

experimentation of shock tubes. An extension of this

work may be carried out by varying the cross section of

the shock tube instead of a diaphragm for producing

shocks.

ACKNOWLEDGMENT

We express our humble pranamas to his holiness

JAGADGURU Sri Sri Sri PADMABHUSHANA

Dr.BALAGANGADHARANATHA MAHASWAMIJI

who has showered his blessings on us. Our heartiest

gratitude to our honourable Principal Dr. C.K. Subbraya

for creating the right kind of Milieu. We express sincere

thanks to our beloved H.O.D, Prof. T.N Krishnaiah,

during the work and his moral support. We also take this

opportunity to thank Mr. T. G. Girish , Associate

professor and P G coordinator, Department of Mechanical Engineering, for his valuable suggestions.

REFERENCES

[1]. Al-Falahi Amir, Yusoff M Z, N H Shuaib and

Yusaf T., Flow instability in shock tube due to shock wave- boundary layer – contact surface

interactions, a numerical study at EuropeanJournal of Scientific Research ISSN 1450-216X

Vol.30 No.1 (2009), pp.164-176..[2]. Mouna Lamnaouer., Numerical modeling of the

shock tube flow fields before and during ignition

delay time experiments at practical conditions (A

dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of

Philosophy in Mechanical Engineering in the

Department of Mechanical, Materials, and

Aerospace Engineering in the College of

Engineering and Computer Sciences at theUniversity of Central Florida Orlando, Florida)

pp.194-200

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121


[3]. Muniswamy Reddy.,Shock-tube flow analysis with

a dimensionless Velocity number N.,in Ames

Research Center, Mofett Field, California

(National Aeronautics and Space Administration.,

Washington D C)

[4]. Khaled Alhussan, Application of Computational Fluid Dynamics in Discontinuous Unsteady Flow

with Large Amplitude Changes; the Shock Tube

Problem Research for Science and Technology,

P.O. Box 6086 Riyadh 11442, KSA

[5]. Erheng Wang, and Arun Shukla., Energy and

Impulse Evaluation during a Shock Tube

Experiment (Proceedings of the IMPLAST 2010

Conference October 12-14 2010 Providence,

Rhode Island USA © 2010 Society for Experimental Mechanics, Inc.)

[6]. Numerical Heat Transfer and Fluid Flow by S V

Patankar., Hemisphere Pub. Corp., 1980. pp-5

[7]. Versteeg H.K., Malalasekera W. Introduction to

computational fluid dynamics. The finite volume

method, Longman, 1995.

[8]. Ferziger, Joel H. and Peric, Milovan,Computational Methods for Fluid Dynamics, third

ed., 2001, Springer-Verlag, Berlin . pp-29-152

[9]. Hirsch, C., Numerical Computation of Internal

and External Flows, Volume 1: The Funda-

mentals of Computational Fluid Dynamics, 2007,

Butterworth-Heinemann, 2nd ed. Pp-17-39

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