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ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

GRADUATION PROJECT

JUNE 2021

DESIGN OF A SUPERSONIC NOZZLE USING METHOD OF

CHACTERISTICS

Thesis Advisor: Dr. Öğr. Üyesi Duygu ERDEM

Yunus Emre ÖZKAN

Department of Astronautical Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : Herhangi Program

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JUNE 2021

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

DESIGN OF A SUPERSONIC NOZZLE USING METHOD OF

CHACTERISTICS

GRADUATION PROJECT

Yunus Emre ÖZKAN

110160560

Department of Astronautical Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : Herhangi Program

Thesis Advisor: Dr. Öğr. Üyesi Duygu ERDEM

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Thesis Advisor: Dr. Öğr. Üyesi Duygu ERDEM

İstanbul Technical University

Jury Members:

İstanbul Technical University

İstanbul Technical University

Yunus Emre Özkan, student of ITU Faculty of Aeronautics and Astronautics

student ID 110160560, successfully defended the graduation entitled “DESIGN OF

A SUPERSONIC NOZZLE USING METHOD OF CHACTERISTICS”

which he prepared after fulfilling the requirements specified in the associated

legislations, before the jury whose signatures are below.

Date of Submission : 14 June 2021

Date of Defense : 28 June 2021

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To my family,

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TABLE OF CONTENTS

Page

TABLE OF CONTENTS ................................................................................. v ABBREVIATIONS .......................................................................................... vi LIST OF TABLES .......................................................................................... vii

LIST OF FIGURES ....................................................................................... viii SUMMARY ...................................................................................................... ix 1. INTRODUCTION ........................................................................................ 1

1.1 Background to the Study ........................................................................... 1 1.2 Purpose of the Study ................................................................................. 2

1.3 Scope of the Study .................................................................................... 2

2. METHOD OF CHARACTERISTICS ........................................................ 3

2.1 Prandtl-Meyer Waves ............................................................................... 3

2.2 Characteristic Lines .................................................................................. 5

2.3 The Compatibility Relation ...................................................................... 8

2.4 Application of the MOC ........................................................................... 9 2.5 Initial Data Line ..................................................................................... 11

2.6 Types of Nozzles .................................................................................... 12

2.6.1 Minimum Length Nozzle(MLN) ..................................................... 12

2.6.2 Axisymmetric Nozzle ...................................................................... 13

3. NOZZLE CONTOUR DESIGN ................................................................ 14 3.1 Theoretical Background .......................................................................... 14

3.2 Requirements and Condition of Design .................................................. 15

3.3 Nozzle Geometry .................................................................................... 18

4. NOZZLE CONTOUR VALIDATION ..................................................... 19 4.1 Meshing and Defining Boundaries ........................................................ 20

4.2 Running SU2 CFD .................................................................................. 21

4.2 Results ..................................................................................................... 22

5. ANALYSIS OF RESULTS ........................................................................ 25 REFERENCES ............................................................................................... 27

APPENDICES ................................................................................................. 28 APPENDIX A .............................................................................................. 29 APPENDIX B .............................................................................................. 34

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ABBREVIATIONS

MOC : Method of Characteristic

1-D : 1 Dimensional

2-D : 2 Dimensional

PDE : Partial Differantial Equation

MLN : Minimum Length Nozzle

CFD : Computational Fluid Dynamics

SU2 : Stanford University Unstructured

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LIST OF TABLES

Page

Table 3.1: Inlet and outlet conditions ........................................................................ 16

Table 3.2: X and y coordinates of nozzle contour..................................................... 17

Table 5.1: Quasi 1-D values and CFD results ........................................................... 25

Table 5.2: Percentage error ....................................................................................... 25

viii

LIST OF FIGURES

Page

Figure 2.1 : Ideal expansion fan .................................................................................. 4

Figure 2.2 : Streamline geometry. ............................................................................... 7

Figure 2.3 : Characteristic curves. ............................................................................... 8 Figure 2.4 : Characteristic curves .............................................................................. 10 Figure 2.5 : Domain of dependence and region of influence. ................................... 11

Figure 2.6 : Minimum length nozzle. ........................................................................ 12 Figure 3.1 : Nozzle geometry .................................................................................... 16 Figure 4.1 : SU2 V5.0.0 Raven version. ................................................................... 19 Figure 4.2 : Boundary Conditions. ............................................................................ 20 Figure 4.3 : Solver Preprocessing. .......................................................................... 21

Figure 4.4 : Mach number along the nozzle .............................................................. 22 Figure 4.5 : Coloured map of Mach number. ............................................................ 22 Figure 4.6 : Pressure [Pa] along the nozzle ............................................................... 23

Figure 4.7 : Coloured map of pressure values. .......................................................... 23 Figure 4.8 : Temperature [K] along the nozzle. ........................................................ 24 Figure 4.9 : Coloured map of temperature values. .................................................... 24

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SUPERSONIC NOZZLE DESIGN WITH METHOD OF CHACTERISTICS

SUMMARY

In this study, design of a supersonic nozzle under the assumptions of 2-D, steady,

inviscid, isentropic, irrotational flow has been realized with using very famous and

reliable method of characteristic. Nozzle wall coordinates are obtained in 50

divisions with assuming air ideal gas. Analysis performed by using SU2 CFD

software to validate the nozzle contour. Quasi 1-D solution and CFD results expected

to be compatible with each other.

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ÖZET

Bu çalışmada, oldukça yaygın ve güvenilir olan karakteristik metod kullanılarak 2

boyutlu, zamandan bağımsız, viskoz olmayan, izentropik, irrotasyonel akış

varsayımları altında süpersonik bir lüle tasarımı gerçekleştirilmiştir. Lüle

koordinatları havayı ideal gaz kabul ederek 50 noktada elde edilmiştir. Lüle

tasarımını doğrulamak için SU2 CFD yazılımı kullanılarak elde edilen sonuçlarla

sanki 1 boyutlu çözüm sonuçlarının birbiriyle uyumlu olması beklenmiştir.

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1.INTRODUCTION

In this study it is aimed to get wall contours of supersonic nozzle for described

conditions and analysing of these wall contours with CFD. It is used very famous

and reliable method of characteristic which solves the governing equations of 2-D,

supersonic, steady, inviscid, irrotational flow for designing wall contours. This

method can be applied to design diverging section of a supersonic nozzle.

1.1 Background to the Study

Combining the continuity and Euler’s equations, under the assumption of 2-D

irrotational we can derive the velocity potential equation. For a superonic flow this

equation becomes hyperbolic in type, since the square of the velocity magnitude

divided by the speed of of is larger than 1. A hyperbolic equation presents particular

direction in the space called characteristics. Along the characteristics, the flow

properties are continuous and the derivatives are determinate and can be

discontinuous. Also, the velocity potential equation satisfies the compatibility

equation. The velocity potential equation can be combined with the potential

derivatives to obtain a system of three lineear algebraic equations. This sytem can be

solved using Cramer’s rule. It can be obtained the characteristic lines where the

system equals zeros in denominator and numerator. Then relations can be found in

terms of velocity components. The slope of characteric lines reduces to very simple

equation with mathematical manipulation. Two characteristic lines passes through an

arbitrary velocity particle is simply two Mach lines. Depending on the position in the

flow, the fluid could have different Mach number and velocity vector orientation.

This causes the characteristic line to have a different orientation based on the

position in the flow. Along the characteristic lines the governing equations that

describe the fluid, reduce to compatibility equations. They can simply obtained

setting the numerator equal to zero. Sum or difference between the flow direction

angle and the value of the local Prandtl-Meyer function is equal to a constant.

Combining the two relations we can also obtain simple expressions to calculate

angles based on the two constants at a point in the domain.

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1.2 Purpose of the Study

It is aimed to solve the design problem wall contours for a converging-diverging

nozzle to allow shock-free isentropic expansion of a gas from rest to a given

supersonic Mach number at the exit. For the convergent section there is not a

particular contour that gives better results than others. Experimental studies would

guide better to determine such a nozzle. For the diverging section of the nozzle the

method of characteristics should be used.

1.2 Scope of the Study

In this study, moc is used to determine the wall contours of nozzle. Based on the exit

pressure which can be calculated based on isentropic flow, pressure ratios can be

obtained. This ratio helps to get pressure, temperature and velocity both at the throat

and exit. Prandtl-Meyer expansion function which is used for expansion waves gives

the maximum wall angle. Based on the the number of divisions, wall positions can be

calculated along the centerline. These wall contours must be in coincidence with the

CFD results.

In second chapter, the theoritical background on nozzle design is presented.

Governing equations were taken from the very basics and discussed in detail. In the

third chapter, the requirements and condition of wall design is described and nozzle

geometry is presented. After then, CFD analysis is realized based on the described

conditions. This chapter also include information about the setup procedure for open-

source CFD analysis tool SU2. In the last chapter, the analysis of results are

evaluated and compared with the referance data from quasi 1-D theory. Important

conclusions are presented and the recommendations for future work are given.

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2. METHOD OF CHARACTERISTIC

The design of the supersonic nozzle from the inlet section to the throat is relatively

simple when compared to the designing of the nozzle from the throat section until the

test section. Nozzle contour design techniques can be categorized generally into two

types: direct design or design by analysis.[1] Because design by analysis requies the

optimization of a known nozzle and goes beyond the subject of this study, the direct

design techniques which gives nozzle contour as an output by using most frequently

utilized and reliable method of characteristics was prefered. The method of

characteristics is a numerical procedure appropriate for solving among other things

2-D compressible flow problems. By using this technique flow properties such as

direction and velocity can be calibrated at distant points throughout a flow field. The

basis of the method of characteristics, which is the most crucial part of the project.

The basis of the MOC design technique whose foundations were laid by Prandtl and

Busemann starts from the expansion of steady supersonic flow through Mach

waves.[2]

2.1 Prandtl-Meyer Waves

Prandtl-Meyer expansion fan which is 2-D simple wave occurs when a supersonic

flow turn around a convex corner. The fan consists of a infinite number of Mach

waves or infinitely weak normal shock waves between a leading or forward Mach

wave and a trailing or reward Mach wave. Prandtl-Meyer waves can occur as a

gradual expansion or an abrupt expansion such as a sharp corner as illustrated in

figure 2.1. The Mach angle µ defined as

µ = 𝑠𝑖𝑛−11

𝑀 (2 − 1)

Supersonic flow properties and its direction change by an infinitesmall amount as

there many Mach wave between leading and trailing ones. Across the expansion fan,

the flow accelerates and the Mach number increases, while the static pressure,

temperature and density decrease. Since the process is isentropic, the stagnation

properties remain constant across the fan.

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Figure 2.1: Ideal expansion fan

This isentropic process simplifies the calculation of flow properties significantly.

Following Prandtl-Meyer function determines the Mach number:

𝑣(𝑀) = ∫√𝑀2 − 1

1 +𝛾−1

2𝑀2

𝑑𝑀

𝑀 (2 − 2)

This integral form of the Prandtl-Meyer function can be simplified to an algebraic

form by choosing an integration constant that function corresponds to zero at Mach 1

After integration, the Prandtl-Meyer function becomes,

𝑣(𝑀) = √𝛾 + 1

𝛾 − 1𝑡𝑎𝑛−1√

𝛾 − 1

𝛾 + 1+ (𝑀2 − 1) − 𝑡𝑎𝑛−1√𝑀2 − 1 (2 − 3)

The Mach number after the turn M2 is related to the initial Mach number M1 and the

turn angle θ2 by,

𝑣(𝑀2) = 𝜃2 + 𝑣(𝑀1) (2 − 4)

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One dimensional flow turns into the two dimensional flow as it passes through

Prandtl-Meyer expansion fan. This function would be very practical for analysing

two dimensional supersonic flow in terms of quantifying the properties of more

compilcated flow fields.

2.2 Characteristic Lines

Characteristics are lines in a supersonic flow oriented in specific directions along

which disturbances (pressure waves) are propagated. Fore a more deterministic

approach to identifying characteristic lines, nonlinear equations of 2-D, irrotational

flow must be solved. The governing equation of flow:

(𝑢2 + 𝑎2)𝜕𝑢

𝜕𝑥+ 𝑢𝑣 (

𝜕𝑢

𝜕𝑦+

𝜕𝑣

𝜕𝑥) + (𝑣2 + 𝑎2)

𝜕𝑣

𝜕𝑦 (2 − 5)

𝜕𝑣

𝜕𝑥−

𝜕𝑢

𝜕𝑦= 0 (2 − 6)

Equation 2-6 is the curl of velocity and irrotationality condition which is known as

the vorticity. Substituting equation 2-5 into equation 2-6 and dividing the result by

the negative square of the speed of sound gives,

(1 −𝑢2

𝑎2)

𝜕𝑢

𝜕𝑥− 2

𝑢𝑣

𝑎2

𝜕𝑢

𝜕𝑦+ (1 −

𝑣2

𝑎2)

𝜕𝑣

𝜕𝑦= 0 (2 − 7)

Equation 2-7 is the 2-D velocity potential equation. The 2-D velocity potential can

be written as a function of x and y where the following relations hold,

𝜕𝜙

𝜕𝑥= 𝜙𝑥 = 𝑢

𝜕𝜙

𝜕𝑦= 𝜙𝑦 = 𝑣 (2 − 8)

The solution of second-order partial differantial equation which gained by

substituting velocity potential derivatives into equation 2-7 can be obtained through

exact numerical solution whre there is no general solution of the equation. The

method of characterics is an example of a such type solution. Solution of velocity

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potential equation reveals flowfield whereas method of characteristics help identify

the characteristic lines within the flowfield. The 2-D velocity potential equation can

be written as a system of equation as follows,

(1 −𝑢2

𝑎2)𝜙𝑥𝑥 −

2𝑢𝑣

𝑎2𝜙𝑥𝑦 + (1 −

𝑣2

𝑎2) + 𝜙𝑦𝑦 = 0 (2 − 9)

𝑑𝑥𝜙𝑥𝑥 + 𝑑𝑦𝜙𝑥𝑦 = 𝑑𝑢 (2 − 10)

𝑑𝑥𝜙𝑥𝑦 + 𝑑𝑦𝜙𝑦𝑦 = 𝑑𝑣 (2 − 11)

These equations can be written in a matrix form,

[ 1 −

𝑢2

𝑎2−

2𝑢𝑣

𝑎21 −

𝑣2

𝑎2

𝑑𝑥 𝑑𝑦 00 𝑑𝑥 𝑑𝑦 ]

[

𝜙𝑥𝑥

𝜙𝑥𝑦

𝜙𝑦𝑦

] = [0𝑑𝑢𝑑𝑣

] (2 − 12)

Using Cramer’s Rule for the solution of variable ϕxy,

𝜙𝑥𝑦 =𝜕𝑢

𝜕𝑦=

|1 −

𝑢2

𝑎2 0 1 −𝑣2

𝑎2

𝑑𝑥 𝑑𝑢 00 𝑑𝑣 𝑑𝑦

|

|1 −

𝑢2

𝑎2 −2𝑢𝑣

𝑎2 1 −𝑣2

𝑎2

𝑑𝑥 𝑑𝑦 00 𝑑𝑥 𝑑𝑦

|

(2 − 13)

Since there is a physical limitation of finiteness, setting the denomitor to zero and

arranging into quadratic form gives,

(1 −𝑢2

𝑎2) (

𝑑𝑦

𝑑𝑥)

2

+2𝑢𝑣

𝑎2(𝑑𝑦

𝑑𝑥) + (1 −

𝑣2

𝑎2) = 0 (2 − 14)

The slope of characteristic line,

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𝑑𝑦

𝑑𝑥= 𝑀2 − 1 =

−𝑢𝑣

𝑎2± √

𝑢2+𝑣2

𝑎2− 1

1 −𝑢2

𝑎2

(2 − 15)

The equation is hyperbolic since Mach number is greater than 1, two characteristic

curves exist as solution which is illustrated in figure 2.2 presents 2-D flow field

streamline geometry.

Figure 2.2: Streamline geometry

The slopes of characteritics line can be written also using the geometric relationships

as follows,

𝑑𝑦

𝑑𝑥=

−𝑀2𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 ± √𝑀2 − 1

1 − 𝑀2𝑐𝑜𝑠2𝜃 (2 − 16)

Using the equation 2-1 and trigonometric substitution, the characteristic equation

becomes,

𝑑𝑦

𝑑𝑥= tan (𝜃 ± µ) (2-17)

A graphical interpolation of equation 2-17 is given in figure 2.3. There are two

characteristics passing through point A where streamline parallel to the x axis. One

of them at the angle µ above the streamline, and the other at the angle µ below the

streamline. Hence, the characteristic line are Mach lines. The characteristics at an

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angle θ + μ is called the left-running characteristics (C+) while the characteristic line

of the θ – μ is a right-running characteristic (C-).

Figure 2.3: Characteristic curves

2.3 The Compatibility Relation

Equation 2-13 represents a combination of the continuity, momentum and energy

equations for 2-D, steady, adiabatic, irrotational flow. Equation 2-17 doesn’t describe

any flow properties although it identifies the characteristic lines. There is

compatibility relation which originates from the theory of hyperbolic equations

between θ angle and Prandtl-Meyer function v on the characteristic lines. When the

numeration of equation 2-13 becomes zero the determinant yields,

𝑑𝑣

𝑑𝑢=

−(1 −𝑢2

𝑎2)

1 −𝑣2

𝑎2

𝑑𝑦

𝑑𝑥 (2 − 18)

Keep in mind that N is set to zero only when D = 0 in order to keep the flowfield

derivatives finite, avoiding from indeterminate form. Equation 2-18 substituted into

equation 2-18,

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𝑑𝑣

𝑑𝑢=

−(1 −𝑢2

𝑎2)

1 −𝑣2

𝑎2(

−

2𝑢𝑣

𝑎2± √

𝑢2+𝑣2

𝑎2− 1

1 −𝑢2

𝑎2)

(2 − 19)

Equation 2-19 reduces an equation in terms of inclination angle, Mach number and

the velocity after some algebraic manipulation,

𝑑𝜃 = ±√𝑀2 − 1𝑑𝑉

𝑉 (2 − 20)

Equation 2-20 is the compatibility equation which describes the variation of flow

properties along the characteristic lines. This can be integrated to give the Prandtl-

Meyer function v(M) as displayed in equation 2-3. Therefore equation 2-20 replace

by the algebraic compatibility equations:

𝜃 + 𝑣(𝑀) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝐾− (2 − 21)

𝜃 − 𝑣(𝑀) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝐾+ (2 − 22)

The constants K+ and K- are also known as Riemann invariants of the solution.

Compatibility equations doesn’t include terms of coordinates x and y. Therefore,

they can be solved without requiring knowledge of the geometric location of the

characteric lines.

2.4 Application of the MOC

It is discussed the methodology of method of characteristics in order that

determination of the characteristic lines, determination and solution of compatibility

equations. The process that applying the method of characteristics is unit process.

Nozzle contour design is a result of the solution of this process for internal flow and

flow at a wall. [3]

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Figure 2.4: Characteristics at internal and wall

If we know the properties and location of two points in the flow, then we can find the

conditions at a third point. Since the Riemann invariants are constant along the

characteristic of the same group, it can be written that:

𝐾𝑎− = 𝐾𝑐

− = 𝜃𝑐 + 𝑣𝑐(𝑀) (2 − 23)

𝐾𝑏+ = 𝐾𝑐

+ = 𝜃𝑐 − 𝑣𝑐(𝑀) (2 − 24)

This system of equations yields for the inclination and Prandtl-Meyer angle:

𝜃𝑐 =1

2(𝐾𝑐

− + 𝐾𝑐+) (2-25)

𝑣𝑐(𝑀) =1

2(𝐾𝑐

− − 𝐾𝑐+) (2 − 26)

The location of point c is determined by the intersection of the C- characteristic

through point a and the C+ characteristic through point b, as shown in figure 2.4. It is

gained that the knowlodge of only direction of these curved characteristics at point a

and b. The problem to locate the c point can be solved by assuming the

characteristics are straight-line segments between the grip points, with slopes that are

avarege values. These segment slopes are:

𝑚𝑎𝑐 = tan(1

2((𝜃 − µ)𝑎 + (𝜃 − µ)𝑐)) (2 − 27)

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𝑚𝑏𝑐 = tan(1

2((𝜃 + µ)𝑏 + (𝜃 + µ)𝑐)) (2 − 28)

If the conditions near a wall has been known, it would be possible to find the flow

variables. However, the procedure of the characteristics for the wall is slightly

different. Since the flow directon at the wall tangent to itself Prandtl-Meyer function

for the intersection of the characteristic with the wall is calculated as follows,

𝑣𝑚(𝑀) = 𝜃𝑤 − 𝐾𝑖+ = 𝜃𝑤 − 𝜃𝑏 − 𝑣𝑏(𝑀) (2 − 29)

It is sufficient to make these processes for the upper wall since the lower wall can be

symmetrically designed.

2.5 Initial Data Line

So far, the discussion is based on knowing the properties of previous points to

calculate the third point. In order to implement the method of characteristics for the

unknown flowfield, we must have a line in the locally supersonic flow along the

flowfield properties are known. The process must begin in a supersonic region since

the nature of hyperbolic partial differantial equations to carry out the downstream-

marching from the initial data line.

Figure 2.5: Domain of dependence and region of influence

12

It should be examined that two regions described in figure 2.5 to understand the

nature of hyperbolic pdes. Properties at point A depend on any information in the

flow within this upstream region and properties of region of influence influenced by

any action that is going on at point A. Evidently, point A don’t propagate upstream,

where downstream propagated by any distubances is going on point A

2.6 Types of Nozzles

In this section, minimum length and axisymmetric nozzle examined for the design

conditions.

2.6.1 Minimum Length Nozzle(MLN)

Minimum lenth nozzles allow for the supersonic flow to expand to its maximum

Mach number in a reduced distance. [4] If the nozzle contour is not proper, shock

waves may occur inside the nozzle.

Figure 2.6: Minimum length nozzle

The method of characteristics provides a technique for properly designing the

contour of a supersonic nozzle for shock free, isentropic flow, taking into account the

multidimensional flow inside the nozzle. Rocket nozzles are short in order to

13

minimize weight. Also, in cases where rapid expansions are desirable, such as the

non-equilibrium flow in modern gas dynamic lasers, the nozzle length is as short as

possible In such minimum length nozzles, the expansion section is shrunk to a point,

and the expansion takes place through a centered Prandtl Meyer wave emanating

from a sharp-corner throat with an angle θ max, MLN as sketched in figure 2.6. The

length of the supersonic nozzle, denoted as L in figure 2.6 is the minimum value

consistent with shock free, isentropic flow. If the contour is made shorter than L,

shocks will develop inside the nozzle. A fluid element moving along a streamline is

constantly accelerated while passing through these multiple reflected waves. For the

minimum length nozzle, the expansion contour is sharp corner at point a. There are

no multiple reflections and a fluid element encounters only two systems of waves,

the right-running waves emanating from point a and the left-running waves

emanating from point d.

𝜃𝑚𝑎𝑥 =1

2𝑣(𝑀𝑒) (2 − 30)

Equation 2-30 demonstrates that, for a minimum length nozzle the expansion angle

of the wall downstream of the throat is equal to one-half the Prandtl-Meyer function

for exit Mach number.

2.6.2 Axisymmetric Nozzle

Axisymmetric refers to nozzle cross section. This types of nozzles are symmetric

about its axis where they become three-dimensional. Axisymmetric nozzles are

commonly used in rocket propulsion applications since the difficulties in production

are eliminated. [5]

14

3. NOZZLE CONTOUR DESIGN

A nozzle designed to expand a gas from rest to supersonic speeds must have a

convergent-divergent shape. It is required to design a convergent-divergent nozzle to

expand a gas from rest to a given supersonic Mach number at the exit. For the

convergent, subsonic section, there is no specific contour which is better than any

other. There are rules based on experience and guided by subsonic flow theory.

3.1 Theoretical Background

We derive the equations which explain and describe why a supersonic flow

accelerates in the divergent section of the nozzle while a subsonic flow decelerates in

a divergent duct. Conservation of mass equation and its differantiation:

m = ρVA (3 − 1)

dρVA + ρdVA + ρVdA = 0 (3 − 2)

Dividing equation 3-2 by ρVA yields,

dρ

ρ+

dV

V+

dA

A= 0 (3 − 3)

Conservation of momentum and isentropic relation:

ρVdV = −dp (3 − 4)

dp

p= γ

dρ

ρ dp = a2dρ (3 − 5)

Combining this equation for the change in pressure with the momentum equation,

−𝑀2𝑑𝑉

𝑉=

𝑑𝜌

𝜌 (3 − 6)

Substitute this value of (dρ/ρ) into the mass flow equation to get,

15

(1 − 𝑀2)𝑑𝑉

𝑉= −

𝑑𝐴

𝐴 (3 − 7)

Equation 3-7 tells posivite rate of change in area produces decrease in velocity in

subsonic case. Negative rate of change in area produces increase in velociy for

supersonic case. To conserve both mass and momentum in a supersonic flow, the

velocity increases and the density decreases as the area is increased.

3.2 Requirements and Condition of Design

The relations from quasi 1-D theory for sonic condition assuming air is an ideal gas,

with values specific gas constant 𝑅 = 287𝐽

𝐾𝑔𝐾 and specific heat ratio 𝛾 = 1.4 is

given below:

𝑃∗ = 𝑃0 + (2

𝛾 + 1)

𝛾

𝛾+1

(3 − 8)

𝑇∗ = 𝑇0 ∗ (2

𝛾 + 1) (3 − 9)

𝑎∗ = √𝛾𝑅𝑇∗ (3 − 10)

𝑃𝑒 = 𝑃0

(1 +(𝛾−1)

2𝑀𝑒

2)𝛾

𝛾−1

(3 − 11)

𝑇𝑒 =𝑇0

(1 +(𝛾−1)

2𝑀𝑒

2) (3 − 12)

For the stagnation pressure P0 = 7000000 [Pa] and T0 = 3600 [K] inlet and outlet

flow properties can be obtained.

16

For the inlet condition from the equation 3-8 and 3-9 P*=3697972.514 [Pa]

T* = 3000 [K]. Outlet condition is Pe = 331090.842 [Pa] Te = 1505.525 [K] from

equation 3-11 and 3-12. Sound velocity at throat a* = 1097.907 [m/s] from equation

3-10.

(𝐴

𝐴∗)2

=1

𝑀2 (2

𝛾+1(1 +

𝛾+1

2𝑀2))

𝛾+1

𝛾−1

(3-13)

It can be obtaind from equation 3-13 using the area Mach number relation for the exit

exit condition 2.6374. They are presented in the table 3.1.

Table 3.1: Inlet and outlet conditions

P* 3697972.514 [Pa]

T* 3000 [K]

Pe 331090.842 [Pa]

Te 1505.525 [K]

a* 1097.907 [m/s]

Me 2.6374

Wall contours can be determined from firstly getting θmax = 42.0851 from equation

2-30. For 50 division, x and y coordinates are presented in table 3.2.

17

Table 3.2: X and y coordinates of nozzle contour

x y x y

1 1 0 26 2.4900 4.7741

2 1.1512 0.39144 27 2.5218 4.9443

3 1.5009 1.30678 28 2.5530 5.1189

4 1.6095 1.59767 29 2.5836 5.2981

5 1.6758 1.77937 30 2.6136 5.4821

6 1.7321 1.93714 31 2.6430 5.6711

7 1.7819 2.08045 32 2.6717 5.8653

8 1.8275 2.21476 33 2.6997 6.0649

9 1.8718 2.34845 34 2.7269 6.2702

10 1.9143 2.47998 35 2.7532 6.4813

11 1.9553 2.6105 36 2.7787 6.6985

12 1.9953 2.74086 37 2.8031 6.9220

13 2.0343 2.8717 38 2.8266 7.1520

14 2.0725 3.00352 39 2.8489 7.3881

15 2.1100 3.13674 40 2.8700 7.6320

16 2.1469 3.27171 41 2.8899 7.8832

17 2.1833 3.40874 42 2.9084 8.1420

18 2.2192 3.5481 43 2.9255 8.4087

19 2.2546 3.69005 44 2.9411 8.6837

20 2.2896 3.83481 45 2.9549 8.9671

21 2.3241 3.9826 46 2.9670 9.2593

22 2.3582 4.13363 47 2.9772 9.5607

23 2.3919 4.2881 48 2.9854 9.8716

24 2.4251 4.4462 49 2.9915 10.1924

25 2.4578 4.6081 50 2.9952 10.5234

26 2.4900 4.7741 51 2.9965 10.865

18

3.3 Nozzle Geometry

Nozzle geometry is presented in figure 3.1 for the values in table 3.2.

Figure 3.1: Nozzle geometry

19

4. NOZZLE CONTOUR VALIDATION

Design conditions for the nozzle contour must be consistent with the CFD analysis of

it to validate our case. SU2 is a open-source CFD tool written in C++ for the

numerical solution of the partial differantial equations. Installing procedure of SU2

on linux operating system is consisted of simple steps. Official website of SU2

presents all the packages and files. Configuration file located in SU2 files will scan

the sytem for the necessary prequisites and generate the appropriate makefiles. Make

command compiles all the files in the head folder. Subsequently, make install

command installs and makes SU2 ready to run. [6] Solution is realized with SU2

Release 5.0.0 Raven version as seen in figure 4.1.

Figure 4.1: SU2 V5.0.0 Raven version

20

For our steady case compressible solver configurations must be based on the Euler

equations since flow is inviscid. Mathematical problem selected as “direct” for the

solution of sparse matrices which has been created for the getting results of

governing pdes.[7] Mesh and configuration file must be located at the same folder to

start the solution.

4.1 Meshing and Defining Boundaries

Gmsh is a open source meshing generator. It was created to provide a fast, light and

user-friendly meshing tool with parametric input and advanced visualization

capabilities. [8]

Figure 4.2: Boundary Conditions

In figure 4.2 inlet and outlet marked. Marking the bottom line as symmetry saves

compuational power for getting the results in less time. Upper curve is marked as

nozzle. Mesh is created in unstructured type with 15083 points, 4 surface markers as

stated, 60, 218, 21, 237 boundary elements in order at marker outlet, symmetry, inlet

and nozzle. 29627 triangle created in result for 2-D problem.

Inlet

Nozzle

Symmetry

Outlet

21

4.2 Running SU2 CFD

SU2 has been configured to compute the density based on free-stream temperature

and pressure using the ideal gas law for inviscid flow. Input conditions presented in

figure 4.3.

Figure 4.3: Solver Preprocessing

4.3 Results

Solution converged after 202 seconds. Expected exit Mach number was 2.6374.

Graph along the nozzle and the exit presented in figure 4.4. Pressure decreases with

the increment in the x axis as it is expected in Fig 4.6. Pe becomes 326146 [Pa] at the

exit.

22

Figure 4.4: Mach number along the nozzle

Figure 4.5: Coloured map of Mach number

23

Figure 4.6: Pressure [Pa] along the nozzle

Figure 4.7: Coloured map of pressure values

Figure 4.8 presentes the change in temperature. It is expected that a flow with lower

Mach number and higher pressure, temperature values expands along the nozzle.

24

This expansion provides higher Mach number and lower pressure, temperature

values.

Figure 4.8: Temperature [K] along the nozzle

Figure 4.9: Coloured map of temperature values

25

5. ANALYSIS OF RESULTS

Quasi 1-D solution and CFD results are compared in this section. Table 5.1 presents

the inlet and outlet conditions for Mach number, pressure and temperature.

Table 5.1: Quasi 1-D values and CFD results

Quasi 1-D Values CFD Results

P* 3697972.514 [Pa] P* 3697972.514 [Pa]

T* 3000 [K] T* 3000 [K]

Pe 331090.842 [Pa] Pe 326146 [Pa]

Te 1505.525 [K] Te 1509.9 [K]

Me 2.6374 Me 2.63232

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =|𝐶𝐹𝐷 𝑟𝑒𝑠𝑢𝑙𝑡| − |𝑄𝑢𝑎𝑠𝑖 1 − 𝑑 𝑣𝑎𝑙𝑢𝑒|

|𝑄𝑢𝑎𝑠𝑖 1 − 𝑑 𝑣𝑎𝑙𝑢𝑒|∗ 100 (5 − 1)

Percentage errors presented in table 5.2.

Table 5.2: Percentage error

Percentage Error

Pe % 1.49

Te % 0.29

Me % 0.19

Quasi 1-D theory provides ideal results for steady, 2-D, isentropic, inviscid,

irrotational flow. Difference in solution for the same assumption caused by nature of

computation. It is is impossible take into account all the decimal numbers in every

case. Properties of flow that exit one mesh cell and enters another mesh cell is

rounded and truncated based on the precision of the calculation. Also number of

division can be increased to design more accurate wall contour for lower errors. As a

resut, it is aimed to design MLN and validate the obtained contour with CFD analysis

26

in this study. Variation of Mach number with temperature and pressure observed.

SU2 configuration is compatible with the results for CFD analysis. For future

studies, boundary layer can be taken into account for more realistic results since our

model was inviscid. 2-D steady problem with ideal gas can be extent to a problem

with chemically reacting gas mixtures and gas-particle mixtures both in equilibrium

and nonequilibrium type. [9]

27

REFERENCES

[1] Shope, Fredrick L., “Contour Design Techniques for Super/Hypersonic Wind

Tunnel Nozzles” AIAA-2006-3665, 24 th Applied Aerodynamics Conference, 5-8

June 2006, San Francisco, California as cited in Adams, S.E (2016) “The Design and

Computational Validation of a Mach 3 Wind Tunnel Nozzle Contour”

[2] Sivells, J.C. (1978), “Computer Program for the Aerodynamic Design of

Axisymmetric and Planar Nozzles for Supersonic and Hypersonic Wind Tunnels”

[3] Anderson, J. D. (2003). Modern compressible flow: With historical perspective

[4] Preliminary Design Document (2016) “Supersonic Air-Breathing Redesigned

Engine Nozzle Conceptual Document”, University of Colorado

[5] Adams, S.E (2016) “The Design and Computational Validation of a Mach 3

Wind Tunnel Nozzle Contour”

[6] Retrieved from: “https://su2code.github.io/”, June 2021

[7] Fletcher, C.A.J (1991), 2nd Edition, “Computational Techniques for Fluid

Dynamics 1 Fundamental and General Techniques”

[8] Retrieved from: “https://gmsh.info/, June 2021

[9] Zucrow, J.M, Hoffman, J.D. (1977), Gas Dynamics, Vol 2

.

28

APPENDICES

APPENDIX A: Nozzle Design Codes

APPENDIX B: Code for creating .geo file for Gmsh

29

APPENDIX A

clc,clear;

G = 1.4;

Me = 2.6374;

n = 50;

%{

G is gamma

Me is the design exit mach number

n is the num of characteristics

%}

%% Initialize datapoint matrices

Km = zeros(n,n); % K- vlaues (Constant along right running characteristic lines)

Kp = zeros(n,n); % K- vlaues (Constant along left running characteristic lines)

Theta = zeros(n,n); % Flow angles relative to the horizontal

Mu = zeros(n,n); % Mach angles

M = zeros(n,n); % Mach Numbers

x = zeros(n,n); % x-coordinates

y = zeros(n,n); % y-coordinates

%% Find NuMax (maximum angle of expansion corner)

[~, B, ~] = PMF(G,Me,0,0);

NuMax = B/2;

%% Define flow of first C+ line

y0 = 1;

x0 = 0;

dT = NuMax/n;

30

Theta(:,1) = (dT:dT:NuMax);

Nu = Theta;

Km = Theta + Nu;

Kp = Theta - Nu;

[M(:,1) Nu(:,1) Mu(:,1)] = PMF(G,0,Nu(:,1),0);

%% Fill in missing datapoint info along first C+ line

y(1,1) = 0;

x(1,1) = x0 - y0/tand(Theta(1,1)-Mu(1,1));

for i=2:n;

s1 = tand(Theta(i,1)-Mu(i,1));

s2 = tand((Theta(i-1,1)+Mu(i-1,1)+Theta(i,1)+Mu(i,1))/2);

x(i,1) = ((y(i-1,1)-x(i-1,1)*s2)-(y0-x0*s1))/(s1-s2);

y(i,1) = y(i-1) + (x(i,1)-x(i-1,1))*s2;

end

%% Find flow properties in characteristic

for j=2:n;

for i=1:1+n-j;

Km(i,j) = Km(i+1,j-1);

if i==1;

Theta(i,j) = 0;

Kp(i,j) = -Km(i,j);

Nu(i,j) = Km(i,j);

[M(i,j) Nu(i,j) Mu(i,j)] = PMF(G,0,Nu(i,j),0);

s1 = tand((Theta(i+1,j-1)-Mu(i+1,j-1)+Theta(i,j)-Mu(i,j))/2);

x(i,j) = x(i+1,j-1) - y(i+1,j-1)/s1;

y(i,j) = 0;

31

else

Kp(i,j) = Kp(i-1,j);

Theta(i,j) = (Km(i,j)+Kp(i,j))/2;

Nu(i,j) = (Km(i,j)-Kp(i,j))/2;

[M(i,j) Nu(i,j) Mu(i,j)] = PMF(G,0,Nu(i,j),0);

s1 = tand((Theta(i+1,j-1)-Mu(i+1,j-1)+Theta(i,j)-Mu(i,j))/2);

s2 = tand((Theta(i-1,j)+Mu(i-1,j)+Theta(i,j)+Mu(i,j))/2);

x(i,j) = ((y(i-1,j)-x(i-1,j)*s2)-(y(i+1,j-1)-x(i+1,j-1)*s1))/(s1-s2);

y(i,j) = y(i-1,j) + (x(i,j)-x(i-1,j))*s2;

end

end

end

%% Find wall datapoint info

xwall = zeros(1,n+1);

ywall = zeros(1,n+1);

xwall(1,1) = x0;

ywall(1,1) = y0;

walls = tand(NuMax);

webs = tand(Theta(n,1)+Mu(n,1));

xwall(1,2) = ((y(n,1)-x(n,1)*webs)-(ywall(1,1)-xwall(1,1)*walls))/(walls-webs);

ywall(1,2) = ywall(1,1)+(xwall(1,2)-xwall(1,1))*walls;

for j=3:n+1;

walls = tand((Theta(n-j+3,j-2)+Theta(n-j+2,j-1))/2);

webs = tand(Theta(n-j+2,j-1)+Mu(n-j+2,j-1));

32

xwall(1,j) = ((y(n-j+2,j-1)-x(n-j+2,j-1)*webs)-(ywall(1,j-1)-xwall(1,j-

1)*walls))/(walls-webs);

ywall(1,j) = ywall(1,j-1) + (xwall(1,j)-xwall(1,j-1))*walls;

end

%% Provide wall geometry to user and plot

assignin('base','xwall',xwall)

assignin('base','ywall',ywall)

grid=1;

if grid == 1

plot(xwall,ywall,'-')

axis equal

axis([0 ceil(xwall(1,length(xwall))) 0 ceil(ywall(1,length(ywall)))])

hold on

for i=1:n

plot([0 x(i,1)],[1 y(i,1)])

plot([x(n+1-i,i) xwall(1,i+1)],[y(n+1-i,i) ywall(1,i+1)])

end

for i=1:n-1

plot(x(1:n+1-i,i),y(1:n+1-i,i))

end

for c=1:n

for r=2:n+1-c

plot([x(c,r) x(c+1,r-1)],[y(c,r) y(c+1,r-1)])

end

end

xlabel('Length [x/y0]')

33

ylabel('Height [y/y0]')

end

% PMF FUNCTION

function [ M nu mu ] = PMF(G,M,nu,mu)

Gp=G+1;

Gm=G-1;

% for known M

if M~=0;

nu = sqrt(Gp/Gm).*atand(sqrt(Gm*(M.^2-1)/Gp))-atand(sqrt(M.^2-1));

mu = asind(1./M);

% for known nu

elseif norm(nu)~=0;

% Find M

%Nu = @(Mg)sqrt(Gp/Gm)*atand(sqrt(Gm*(Mg.^2-1)/Gp))-atand(sqrt(Mg.^2-

1))-nu;

for i=1:length(nu(1,:))

for j = 1:length(nu(:,1))

M(j,i) = fzero(@(Mg)sqrt(Gp/Gm)*atand(sqrt(Gm*(Mg.^2-1)/Gp))...

-atand(sqrt(Mg.^2-1))-nu(j,i),[1 100]);

end

end

mu = asind(1./M);

34

% for known mu

elseif mu~=0;

M=1./sind(mu);

nu=sqrt(Gp/Gm)*atand(sqrt(Gm*(M.^2-1)/Gp))-atand(sqrt(M.^2-1));

end

APPENDIX B

xNoz = [ ]; % Define nozzle boundary X-points

yNoz = [ ]; % Define nozzle boundary Y-points

% Number of nozzle points and lines

numNozPts = length(xNoz); % Number of nozzle

points

numNozLns = length(xNoz)-1; % Number of nozzle

lines

% Clear any old variables

clearvars Point;

% ===== FILE PROPERTIES =====

fid = fopen('Output_Mesh.geo','w');

% ===== POINTS =====

% Nozzle bounday

Point(:,1) = xNoz;

Point(:,2) = yNoz;

35

% Additional points

addPts = [xNoz(end) 0;

0 0;

xNoz(1) yNoz(1)];

Point = [Point; addPts];

Point(:,3) = 0.0;

Point(:,4) = 1.0;

% Number of points

numPts = size(Point,1);

% Write points to a file

for i = 1:1:numPts

fprintf(fid,'Point(%i) = {%g, %g, %g, %g};\r\n',...

i,Point(i,1),Point(i,2),Point(i,3),Point(i,4));

end

% ===== LINES =====

% Number of lines

numLns = numPts;

% Write lines to file

for i = 1:1:numPts

if (i ~= numLns)

fprintf(fid,'Line(%i) = {%i, %i};\r\n',i,i,i+1);

else

fprintf(fid,'Line(%i) = {%i, %i};\r\n',i,i,1);

end

end

% ===== LINE LOOP =====

36

% Create string with all lines

lineStr = '1';

for i = 2:1:numLns

lineStr = [lineStr ', ' num2str(i)];

end

% Write to file

fprintf(fid,'Line Loop(%i) = {%s};\r\n',numPts+1,lineStr);

% ===== PLANE SURFACE =====

fprintf(fid,'Plane Surface(%i) = {%i};\r\n',numPts+2,numPts+1);

% ===== PHYSICAL LINE =====

phyLineNozzle = '1';

for i = 2:1:numNozLns

phyLineNozzle = [phyLineNozzle ', ' num2str(i)];

end

fprintf(fid,'Physical Line("Outlet") = {%i};\r\n',numLns-3);

fprintf(fid,'Physical Line("Symmetry") = {%i};\r\n',numLns-2);

fprintf(fid,'Physical Line("Inlet") = {%i};\r\n',numLns-1);

fprintf(fid,'Physical Line("Nozzle") = {%s};\r\n',phyLineNozzle);

% ===== CHARACTERISTIC LENGTH =====

charVal = 0.2;

charLength = '1';

for i = 2:1:numPts

charLength = [charLength ', ' num2str(i)];

end

37

fprintf(fid,'Characteristic Length {%s} = %f;\n',charLength,charVal);