Contribution to the Study of the Numerical Simulation of

Compressible Flow in a Convergent-divergent Nozzle

A. Ederouich

, M. Essahraoui

, D. Zejli

and A. Saad

2d,*

SEALAB, Advanced Systems Engineering Laboratory. Kenitra, Morocco

National School of Applied Sciences, Ibn Tofail University, Kenitra, Morocco

Keywords: CFD, Finite Volume, Compressible Flow, Convergent-Divergent Nozzle.

Abstract: The use of a convergent-divergent nozzle meets several needs, namely the study of the performance of wind

turbines, turbine engine blades, aeroplane wings, and propellants. In this work, we have been studying

compressible flow of an ideal gas in a one-dimensional convergent-divergent nozzle essentially. The study

aims at determining the evolution of all the parameters of the flow along with the Nozzle (the pressure,

temperature, Mach number, and density). Thus, we studied the influence of the nozzle geometry on the flow

by the variation of the diverging angle. The numerical simulation of the flow has been carried out using the

ANSYS Fluent software, which uses the finite volume method to solve the various partial differential

equations modelling the physical phenomenon. The results obtained from this model have been compared

with the theoretically calculated results. Good agreement was observed.

1 INTRODUCTION

The nozzle is widely used in various areas, from

rocket propulsion to fuel sprayer. It has been applied

in industrial, aerospace, automobile, and other

sectors. The nozzle is a major part of any high-

performance engine or rocket motor. It is used to

control the velocity, direction, and required

parameters of the flow. Nozzles are designed to

operate in all flow regions like subsonic, sonic,

supersonic, and hypersonic. The design of the

supersonic nozzle remains a challenging task in fluid

mechanics. In a supersonic nozzle, not only do the

physical parameters of the nozzle play an essential

role, but the thermodynamic parameters of the flow

also play a crucial role in defining the design of a

nozzle. The Converging-Diverging Nozzle known as

de Laval nozzle is the most common and converts

high pressure, high temperature, and low velocity

(subsonic) gases into low pressure, low temperature,

and high velocity (supersonic) gases, hence

producing high thrust (Khalid and Ahsan, 2020).

CFD (computational fluid dynamics) a branch

which is widely used for solving governing

equations of fluid dynamics. Today we can find its

applications for all disciplines such as heat transfer,

fluid dynamics and even for natural science etc.

Problems which are very complicated to solve by

means of general analytical method can be easily

solved by CFD. Since the set of equations of

continuity, momentum, energy of fluid dynamics is

called as Navier-stokes equation.

There are many approaches in CFD through

which we can obtain the appropriate result, but the

standard method used is finite volume method. For

every bit of volume, the equations are solved and

results are obtained. Thus after the completion of

iterations each point specific some value. Thus

through these results we can make a point on the

behavior of fluid flow (Maddu et al., 2018).

For the present study, we are using ANSYS to

determine the evolution of the

flow parameters

(the

pressure, temperature, Mach number, and density) in

the convergent-divergent nozzle. Thus, we put focus

on the influence of the nozzle geometry on the flow

by the variation of the diverging angle



2 GOVERNING EQUATIONS OF

FLUID FLOW

To understand the physics of the fluid in motion

related to any engineering problem, it’s important

that we develop a accurate relationship among the

360

Ederouich, A., Essahraoui, M., Zejli, D. and Saad, A.

Contribution to the Study of the Numerical Simulation of Compressible Flow in a Convergent-divergent Nozzle.

DOI: 10.5220/0010734400003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 360-365

ISBN: 978-989-758-559-3

variations of the fluid flow properties such pressure,

temperature, velocity, density etc at discrete points

in space and time. The fluid governing equations

proves a theoretical solution to how these flow

properties are related to each other by either integral,

differential or algebraic equations. The following

three fundamental laws known as the conservation

laws are used to establish the governing equations of

the fluid flow [3].

 Conservation of Mass [4]:

∂ρ

∂t

∇



ρv

⃗



S



1

 Conservation of momentum equation is:

∂ρ

∂t



ρv

⃗



∇



ρv

⃗

v

⃗



∇p∇



τ



ρg

⃗

F



⃗

2

 Conservation of energy equation is :

∂ρE

∂t

div



ρEv

⃗

σv

⃗

q

⃗



ρf

⃗

v

⃗

 ρw 3

To study compressible flow in CD nozzles, the

conservation equations for mass, momentum and

energy must be solved. Note that these equations are

partial differential equations whose resolution is not

known a priori. Put as such, the calculation is done

based on the following assumptions:

 Perfect gas.

 Flow is unidirectional and steady.

 Flow is adiabatic and reversible.

DESCRIPTION OF THE

PROBLEM

We propose to study the flow of air assumed to be

ideal gas in a convergent -divergent nozzle (De

Laval), the basic dimensions of which are given in

the following table:

Table 1 : Nozzle Dimensions.

Parameters Dimensions (mm)

ozzle len

th 200

ozzle exit radius 70

ozzle inlet radius 50

Convergent angle

Divergent angle

The convergent and the divergent are connected

by an arc of a circle.

4 CFD ANALYSIS

Designing, modelling, and mesh generation of the

nozzle were carried out in Design Modeler and

ANSYS Mesh.

4.1 Geometry and Modelling



Figure 1: The Geometry of the Half-Nozzle Studied.

4.2 Meshing and Boundary Conditions

One of the most important steps in numerical

simulations is to achieve a mesh appropriate to the

problem being dealt with.

Figure 2: Mesh of the Half-Nozzle (Unstructured).

Contribution to the Study of the Numerical Simulation of Compressible Flow in a Convergent-divergent Nozzle

361

Figure 3: The Final Mesh of the Half-Nozzle (Structured).

Figure 4:Limit Boundaries.

4.3 "Fluent" Solver

Table 2:General Setup.

Setup

Solve

Densit

ase

2D Space axis

mmetric

Time stead

Pressure at inle

100000 Pa

Temperature at inle

300 K

Pressure at outlet 2814 Pa

Temperature at outle

300 K

5 RESULTS AND

INTERPRETATIONS

5.1 Evolution of Flow Parameters

5.1.1 Variation of Static Pressure

In the convergent-divergent nozzle the gas

undergoes a large expanding operation to transform

the thermal energy and the pressure energy of the

gases into kinetic energy. Figures (5 and 6) show the

drop in static pressure in the nozzle. The gas

expands from pressure 100911.4 (Pa) to pressure

799.6063 (Pa) at the outlet of the nozzle. This given

fact is logical since in a supersonic flow the pressure

is inversely proportional to the section (referring to

the formula of Hugoniot), and the pressure graph

does not represent any disturbance or fluctuating,

which

corresponds to a completely isentropic flow

along of the divergent.

Figure 5: Contours of Static Pressure.

Figure 6: Evolution of the Static Pressure along the Axis

of the Nozzle.

5.1.2 Variation of Static Temperature

As the flow is completely isotropic in the nozzle,

then the temperature change is proportional to the

pressure, referring to the ideal gas law. This can be

seen in Figures (7 and 8); the temperature in the

middle of the nozzle is subjected to a uniform and

continuous descent of the outlet. It varies from

299,667 K at the inlet of the nozzle to 75,383 K at

the outlet.

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

362



Figure 7: Contours of the Static Temperature.

Figure 8: Evolution of the Static Temperature along the

Axis of the Nozzle.

5.1.3 Variation of Mach Number

Figures (9 and 10) display the distribution of the

Mach number in the nozzle. It is observed that the

speed at the inlet of the nozzle remains almost

constant or invariable up to its throat. We can say

that the regime is subsonic (Ma 1). We can neglect

the compressibility of air for Mach numbers less

than 0.3 because the value of the Mach number is

strictly less than unity. The flow in the throat is

transonic (0.8 Mach 1.2). In the divergent flow, the

flow becomes supersonic and reaches a maximum

value equal to 3.863 at the outlet of the nozzle. Thus,

the convergent-divergent profile of the nozzle allows

the gas to accelerate from a subsonic speed to a

supersonic speed.

This evolution follows Hugoniot's law which

states that the speed is proportional to the section of

a supersonic flow, and varies in the opposite

direction of the section for a subsonic flow.

Figure 9: contours of Mach number.

Figure 10: Evolution of the Mach number along the Axis

of the Nozzle.



5.1.4 Variation of Density

Figures (11 and 12) show the variation of the density

along the nozzle. We observe that the density almost

constant at the convergent part at its maximum value

(1.173 𝐾𝑔/𝑚



).On the divergent part the density

undergoes a decrease until the exit of the nozzle

(0.037𝐾𝑔/𝑚



).This is normal since the pressure

decreases.

Figure 11: Contours of Density.

Contribution to the Study of the Numerical Simulation of Compressible Flow in a Convergent-divergent Nozzle

363

Figure 12: Evolution of the Density along the Axis of the

Nozzle.

5.2 Study of the Influence of the Nozzle

Geometry on the Flow

In this part, the numerical study is carried out at

different angles of the divergent to find out the effect

of the diverging angle on the Mach number and the

static pressure. The different diverging angles used

for the analysis are 17 °, 20 °, 23 ° and 26 °. The

boundary conditions are the same in the four cases.

Four different dimensional designs are modelled

by changing the diverging angle, then the CFD

analysis is performed for four different models and

the variation in Mach number and static pressure is

observed in each case.

Figure 13: Static pressure Contours for Different Angles

of the Diverging Part: a (17 °), b (20 °), c (23 °) and d

(26 °).

Figure 14: Mach number Contours for Angles a, b, c and

5.2.1 Exit Conditions

Table 3: The Conditions at the Outlet of the Nozzle for the

Four Cases.

Case

Diverging

angle

Mach

number

Static pressure

(Pa)

1 17 ° 3 .676 1032.146

2 20 ° 3.863 799.606

3 23 ° 4.089 589.508

4 26 ° 4.362 413 .517

The results of the analysis on the nozzle with a

variable diverging angle are as follows:

 In the divergent section, the velocity

distribution increases with increasing

divergent angle.

 Static pressure decreased with increasing

divergent angle.

5.3 Comparison with Analytical

Results

We choose case 2 (the diverging angle is 20 °).Table

(4) shows the comparison of the values of pressure,

temperature and velocity at the throat obtained from

this simulation and the analytical results calculated

from the following relations:







γ1

0.8316 4

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

364



















⁄



γ1









⁄

0.5275 5







γrT







2γ

γ1



6

The simulation results obtained from this model

have been showed a good agreement with the

analytical results.

Table 4: The Parameters of the Flow at the Throat.

Analytical This

model

Static pressure

at throat (Pa)

53 448.94 53500

Static

temperature at

throat (K)

249.48 250

velocity at

throat (m/s)

316 318.75

6 CONCLUSION

This study project allowed us to discover a topical

and research topic that concerns the field of

turbomachines, aeronautics and space. The primary

motivation for this work has been the study and

understanding of physical phenomena encountered

in practical fields such as turbines, supersonic wind

tunnels, supersonic airplanes and space launchers.

Our project is mainly interested in the processing by

numerical simulation of compressible flows in

convergent-divergent nozzles.

The results obtained from this model were

compared with the theoretically calculated results.

Good agreement was observed.

REFERENCES

Muhammad Waqas Khalid & Muhammad Ahsan, 2020.

Computational Fluid Dynamics Analysis of

Compressible Flow Through a Converging-Diverging

Nozzle using the k-ε Turbulence Model. Engineering,

Technology & Applied Science Research Vol. 10, No.

1, 5180-5185. http://www.etasr.com/.

Yesu Ratnam.Maddu, Shaik. Saidulu, Md. Azeem & S.

Jabiulla. Design and Fluid Flow Analysis of

Convergent-Divergent Nozzle. International Journal of

Engineering Technology Science and Research

IJETSR, www.ijetsr.com, ISSN 2394 – 3386, Volume

5, Issue 4 April 2018.

Ekanayake, E. M. Sudharshani. Numerical Simulation of a

Convergent Divergent Supersonic Nozzle Flow.

School of Mathematical and Geospatial Sciences,

RMIT University, Melbourne, Australia July 15, 2013.

https://researchrepository.rmit.edu.au/discovery/search

?vid=61RMIT_INST:ResearchRepository.

Liu, Yong Qi, and Xiang Chun Chen. "Numerical Study

on the Effect of Dislocation Relationship on

Resistance Characters of the Ceramic Oxidation Bed

Based on Fluent Software", Advanced Materials

Research, 2011.

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Contribution to the Study of the Numerical Simulation of Compressible Flow in a Convergent-divergent Nozzle

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