37

Engineering e-Transaction (ISSN 1823-6379)

Vol. 6, No. 1, June 2011, pp 37-42

Online at http://ejum.fsktm.um.edu.my

Received 5 October, 2010; Accepted 30 December, 2010

COMPUTATIONAL STUDY OF SUPERSONIC FLOW THROUGH A CONVERGING

DIVERGING NOZZLE

M.S.U. Chowdhury1, J.U. Ahamed

2, P.M.O. Faruque

1 and M.M.K. Bhuiya

1,

1Department of Mechanical Engineering

Chittagong University of Engineering and Technology (CUET)

Chittagong - 4349, Bangladesh 2Department of Mechanical Engineering

University of Malaya, 50603 Kuala Lumpur, Malaysia

Email: jamal293@yahoo.com

ABSTRACT

Computational solution has been obtained for

Supersonic Flow through a Converging Diverging

Nozzle. Various characteristics of compressible fluid

flow through nozzle is analyzed and determined. The

nozzle geometry is assumed as circular and axi-

symmetric and flow as two dimensional flow.

Discredited equation is formed by dividing the

geometry into 20×50 meshes. Ideal gas is assumed as

the working fluid & Iteration is done until

convergence. The variations in static pressure are

decreasing gradually and consequently the velocity of

the flow is increased. The variation of velocity,

Pressure, Temperature is determined along the length

of the nozzle and plotted its contour. Two dimensional

double precision (2-DPP) is used for the analysis of the

geometry.

Key words: Converging diverging nozzle; Control

volume; Back pressure; Mach number; Velocity

vector, Contour.

1. INTRODUCTION

Computational Fluid Dynamics (or CFD) is the

analysis of systems involving fluid flow, heat transfer

and associated phenomena such as chemical reactions

by means of computer-based simulation (Ferziger and

Peril, 2002; Sriram, 2009; Cherrared et al., 2008). It is

becoming very popular to solve flow problems without

doing any experiment. So it is very economical and

time saving. By the development of high speed

computer, there has been phenomenal growth in use of

computer for the simulation of the flow system. We

can now dispense with experimental methods in many

cases even in aerospace design. Before, we have to

analyze the flow system by manual discretization and

solved it by coding in computer language. But now it is

easier to simulate the flow problem by software

package very easily. Converging diverging nozzle is

one of the most important devices used in all

supersonic vehicles (Chima, 2010). Moreover we are

very much interested to increase the speed more and

more so it is important to carry out research on

converging diverging nozzle. CFD made the research

easier. The control volume, nozzle material, initial

velocity of fluid, geometry is to be selected first for the

analysis of back pressure, velocity vector, properties,

Mach number, pressure distribution of supersonic flow

through nozzle. Computational study for only U and V

components are calculated here. The converging –

diverging geometry is shown in Fig.1.

Fig.1 Converging-diverging nozzle geometry

2. CO-ORDINATE SYSTEM

Cartesian co-ordinate system is used for drawing the

geometry of the nozzle. Moreover the geometry is

assumed as two dimensional and x- direction y-

direction velocity u and v respectively (Rahman et al.,

2010). To draw the geometry of the nozzle some

parameter is to be assumed. For analysis, the whole

structure is to be divided into small fragment called

cell. The smaller the cell size the finer the analysis.

Each cell is composed of six faces in case of three

dimensional analyses or one face in case of in case of

Converging Diverging

38

two dimensional analyses. Again each face is

composed of four edges and each edge is created by

joining point of two nodal points as shown in Figure 2

(Anderson, 1995, 1990).

Fig. 2 Process of meshing the geometry

3. GOVERNING EQUATIONS

CFD is based on the fundamental governing equations

of fluid dynamics. The Equations are continuity,

momentum, and energy equations.

To find out the properties of the flow system the

governing equations are applied in every node (Ahmed

et al., 2010; Zafar, 2003). Using governing equation

the whole control volume is discretized and then the

governing equations are solved using boundary

conditions. ‘Finite volume’ method is used to solve the

discretized equation then this flow system is solved on

the basis of density, because the density-based

formulation may give it an accuracy (i.e. shock

resolution) advantage over the pressure-based solver

for high-speed compressible flows (Labworth, 2002).

Applying the mass, momentum and energy

conservation, assuming two-dimensional, steady,

compressible flow of varying density the governing

equations are:

Continuity Equation:

0i

j

x

U

Dt

D

------------------------ (1)

Momentum Equation:

j

j

ij

ji

j

i

jg

xx

P

x

UU

t

U

--------- (2)

I II III IV V

Where,

k

k

ijj

i

i

j

ijx

U

x

U

x

U

3

2

I. Local change with time

II. Momentum convection

III. Surface force

IV. Molecular - dependent momentum exchange

V. Mass- force

Energy Equation

i

j

ij

ij

i

i

ix

U

x

T

x

Up

x

TUc

t

Tc

2

2

---- (3)

Where,

I. Local energy change with time

II. Convective term

III. Pressure work

IV. Heat flux (diffusion)

V. Irreversible transfer of mechanical energy into heat

These above governing equations are applied over the

control volume and then solved by proper selection of

input conditions: flow system, solver selection,

boundary conditions, grid size, edge division etc

(Labworth, 2002).

4. METHODOLOGY

Since the nozzle has a circular cross-section, it is

reasonable to assume that the flow is axi-symmetric

and the geometry created to be two-dimensional

(Moinier et al., 2002; Cusdin and Muller, 2005).

Now 2rA ; Where, xr is the radius of the

cross-section at a distance x

39

And 21.0 xA is assumed for generating the

nozzle geometry, then for the given nozzle geometry,

we get

5.05.0;1.0

5.02

x

xxr

---------- (4)

This is the equation of the curved wall. GAMBIT is

used for generating the geometry. The points of the

curve are connected by ‘NURBS’ arc (Gambit, 2004).

The Nozzle geometry is drawn by using the following

data shown in Table 1. Meshed edges, control volume

and curved edges of nozzle are shown in Figures 3, 4

and 5.

Table 1 Data for Nozzle geometry

x -0.5 0.3337

-0.4 0.2876

-0.3 0.2458

-0.2 0.2111

-0.1 0.1870

0 0.1787

0.1 0.1870

0.2 0.2111

0.3 0.2458

0.4 0.2876

0.5 0.3337

Fig. 3 Curved edge of the nozzle.

Fig. 4 Meshed edges

Fig. 5 Mesh control volume

Boundary types for each of the edges are specified in

the Table 2. The input conditions have to be defined

one by one as: defining solver, defining the viscosity

effect, defining Energy Equation, defining Fluid

Properties, (let ideal gas), defining Operating

Conditions, defining Boundary conditions, defining

Equation Type, Initializing inlet properties (Pressure =

99298.5 Pa; Axial velocity = 58.90128; Temperature =

298.2764), defining convergence criteria(let the

solution will be conversed at 10e-6

), defining number

of iteration( let iteration number is 500) etc. is to be

done.

Table 2 Boundary types of edges

The residuals of the iteration are printed out as well as

plotted in the graphics window as they are calculated.

Since the iteration converges within 10-6

value shown

in Fig. 6. So our criteria for solving the problem are

correct.

5. RESULTS AND DISCUSSIONS

5.1 Centerline Velocity

The variation of the axial velocity is plotted along the

centerline as shown in the Fig. 7. Fig. 7 shows that the

velocity of the centerline is increasing gradually and it

is maximum at the exit of the nozzle i.e. the velocity

turning from subsonic to supersonic gradually.

Edge

Position Name Type

Left inlet PRESSURE_INLET

Right outlet PRESSURE_OUTLET

Top wall WALL

Bottom centerline AXIS

40

5.2 Centerline Pressure

The variation of the axial pressure is plotted along the

centerline and presented in the Fig. 8. The figure

shows that the pressure of the centerline is decreasing

gradually and it is the minimum at the exit of the

Fig. 6 Convergence diagram with iterations

Fig. 7 Centerline velocity change with respect to position.

41

Fig. 8 Static pressure changes with respect to position.

5.3 Vector Display

Fig. 9 shows the change of velocity along the flow

direction and red color represents the supersonic flow.

The scale on the left of the Fig. 9 represents the value

of temperature, pressure, velocity in the respective

figure along left to right of the nozzle.

Fig. 9 Velocity vectors vs. position colored by velocity.

6. CONCLUSIONS

CFD reduces time as well as cost of production of fluid

dynamics related products. It abates our experimental

cost. Any fluid flow and can be analyzed very easily.

All the aircrafts (whose velocity is more than the

velocity of sound) requires this type of simulation else

the cost of production of supersonic aircrafts will be

high and very much complicated.

Based on the simulation of supersonic flow through

nozzle the following conclusion may be drawn:

1. The range of horizontal axis is taken from -

0.5 to +0.5 because of simplifying the

drawing as well as simulation.

42

2. Solution is conversed at around 140 iteration,

residuals is ignored after 10-6

value.

3. The velocity profile for the flow is sketched

in X-Y plane. The profile gives us

information about the increment of the

velocity in the right side of the nozzle.

4. The pressure profile shows that it reduces

along the right side and the back pressure is

around 1500 Pascal. Which means that the

pressure at exit of the nozzle must be 1500

else it will not act as a nozzle i.e. the flow will

not be supersonic.

5. If Mach number exceeds the value of 5, it

creates high temperature which causes the

chemical change of the fluid.

The flow through a converging-diverging nozzle is the

important problems used for modeling the

compressible flow for computational fluid dynamics.

Occurrence of shock in the flow field shows one of the

most prominent effects of compressibility over fluid

flow

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