nesh thesis

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NUMERICAL COMPUTATION OF INTERNAL

FLOW IN A HYBRID PROPELLANT MOTOR

A Project Report

submitted by

NITESH SHAH

in partial fulfilment of the requirements

for the award of the degree of

MASTER OF TECHNOLOGY

DEPARTMENT OF AEROSPACE ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, MADRAS.

May 2010


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THESIS CERTIFICATE

This is to certify that the thesis titled NUMERICAL COMPUTATION OF INTER-

NAL FLOW IN A HYBRID PROPELLANT MOTOR, submitted by Nitesh Shah,

to the Indian Institute of Technology, Madras, for the award of the degree of Master of

Technology, is a bona fide record of the research work done by him under our supervi-sion. The contents of this thesis, in full or in parts, have not been submitted to any other

Institute or University for the award of any degree or diploma.

Prof. M. Ramakrishna

Project Guide

Professor

Dept. of Aerospace Engineering

IIT-Madras, 600 036

Prof. P. Sriram

Professor

Head Of Department

Dept. of Aerospace Engineering

IIT-Madras, 600 036

Place: Chennai

Date: 3rd May 2010


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ACKNOWLEDGEMENTS

It is my privilege to acknowledge the people who have helped in successfully finishing

this project report. I owe my greatest gratitude to my project guide Prof. M. Ramakr-

ishna for his guidance and his unique way to boost my confidence throughout the course

of the project.

I thank Dr. P. A. Ramakrishna and Rajiv Kumar, who provided the details of the geom-

etry used for the computation from their experiment setup. I am thankful to Prof. S.

C. Rajan who always inspired me to learn new things and to be a good human being.

I thank Ramesh, Rajesh, Manimaran, Shivaprasad and Ashok for their valuable inputs

during the project. I also thank Kartik to provide some useful references to develop the

solver.

I am grateful to Dr. Prabhu Ramachandran for developing scientific data visualizer,

MayaVi2 and providingLATEXtemplate for thesis submission. I thank Nitin Chaturvedi

and Mrs. Ritu Chaturvedi who always considered me as a part of their family and al-

ways motivated me to do things with passion. I extend my thank to Sandip Roy and all

MTech batch mates for their help on different academic and non academic issues.

I am grateful to my friend Gouripriya, who was always there to boost my confidence

with her constant support and understanding. She played a major role to make this

project successful. I am grateful to my parents, brother and sister for their love and

affection which always enlighten me with full of energy. Lastly I would like to thank

the nature and the beautiful campus of IIT Madras which refreshed me every time afterhours of work.

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ABSTRACT

KEYWORDS: Hybrid Propellant Motor, Internal Flow; Euler Equation; HLL; Ru-

sanov; Riemann Solver; Runge Kutta; FVM; Multi-Block; Local

Time Stepping; Roe Average.

HLL and Rusanov Riemann flow solvers using Finite Volume Method have been de-

veloped to analyze the flow in a hybrid propellant motor with a bluff body near the

inlet. The validation of the solver is done using benchmark internal flow problems. A

Multi Block, structured grid is used to discretized the computational domain using al-

gebric and elliptic grid generation. First order forward time stepping and Runge Kutta

time stepping with second and fourth order, are used as time marching schemes and

their comparative study is performed on the basis of processing time to reach the steady

state. The convergence rate to reach the steady state is increased using the local time

stepping. The steady state numerical solution for the three dimensional Euler equation

is obtained for the rocket motor with a body at inlet. The results obtained are analyzed

and a comparison of the Rusanov and HLL Riemann solvers is performed.

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

ABSTRACT ii

LIST OF TABLES vi

LIST OF FIGURES viii

ABBREVIATIONS ix

NOTATION x

1 INTRODUCTION 1

2 Grid Generation 4

2.1 Geometrical Details of Computational Domain . . . . . . . . . . . 4

2.2 Two Dimensional Grid Generation . . . . . . . . . . . . . . . . . . 6

2.2.1 Grid Generation of Different Blocks . . . . . . . . . . . . . 6

2.2.2 Issues Related to Quality of Grid . . . . . . . . . . . . . . . 8

2.3 3 D Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Euler Solver 11

3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Time Marching Schemes . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 First Order Forward Time Stepping . . . . . . . . . . . . . 13

3.3.2 Second Order Runge Kutta time stepping . . . . . . . . . . 13

3.3.3 Fourth Order Runge Kutta Time Stepping . . . . . . . . . . 14

3.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4.1 Artificial Dissipation . . . . . . . . . . . . . . . . . . . . . 15

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3.4.2 Computation of2Q and 4Q . . . . . . . . . . . . . . . 163.5 Riemann Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5.1 Godunovs Method . . . . . . . . . . . . . . . . . . . . . . 17

3.5.2 HLL Approximate Riemann Solver . . . . . . . . . . . . . 18

3.5.3 Rusanov Approximate Riemann Solver . . . . . . . . . . . 20

3.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.6.1 Inlet Boundary Condition . . . . . . . . . . . . . . . . . . 21

3.6.2 Exit Boundary Condition . . . . . . . . . . . . . . . . . . . 22

3.6.3 Inviscid Wall Boundary Condition . . . . . . . . . . . . . . 23

3.6.4 Transfer Boundary Condition . . . . . . . . . . . . . . . . 23

3.7 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.8 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.9 Class Diagram of Solver . . . . . . . . . . . . . . . . . . . . . . . 25

4 Performance Analysis of Different Schemes on Euler Solver using Pipe

Flow Problem 27

4.1 Numerical Schemes to Approximate the flux at Faces . . . . . . . . 27

4.1.1 Second Order Artificial Dissipation . . . . . . . . . . . . . 27

4.1.2 Fourth Order Artificial Dissipation . . . . . . . . . . . . . . 28

4.1.3 Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.4 Performance Analysis of Schemes . . . . . . . . . . . . . . 29

4.1.5 Residue Plots . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Time Marching Schemes . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Performance Analysis of Schemes . . . . . . . . . . . . . . 30

4.2.2 Residue Plots . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Code Validation 32

5.1 Test Case A : Supersonic Wedge Flow . . . . . . . . . . . . . . . . 32

5.1.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1.2 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 34

5.1.4 Residue Plots . . . . . . . . . . . . . . . . . . . . . . . . . 34

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5.1.5 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Test Case B : Flow in a Channel with a Forward Facing Step . . . . 38

5.2.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.2 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.3 Boundary Condition . . . . . . . . . . . . . . . . . . . . . 39

5.2.4 Computational Strategy . . . . . . . . . . . . . . . . . . . 39

5.2.5 Test Result . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Test Case C : Flow in a Channel with a Bump . . . . . . . . . . . . 41

5.3.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.2 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 41

5.3.4 Residue Plots . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3.5 Test Result . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Results 46

6.1 Two dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1.1 2D Results . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1.2 Residue plots . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 Three Dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2.1 Grid Detail and Processing time . . . . . . . . . . . . . . . 51

6.2.2 3D Results . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2.3 Residue plots . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Conclusion and Future Scope 57

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A Class Descriptions 58


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LIST OF TABLES

3.1 Inlet conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Initial condition for the 2D and 3D supersonic flow in channel with step 25

4.1 Performance of different schemes for pipe flow with 25x10x16 cells 29

4.2 CPU time per Iteration per grid point . . . . . . . . . . . . . . . . . 29

4.3 Performance of different time marching schemes for pipe flow with

25x10x16 cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 CPU time per iteration per grid point for time marching schemes . . 31

5.1 Initial condition for the 2D and 3D supersonic wedge flow . . . . . 34

5.2 Boundary condition for the 2D and 3D supersonic wedge flow, AW :

Adiabatic Wall, TBC: Transfer boundary condition . . . . . . . . . 34

5.3 Inlet boundary condition for supersonic wedge flow . . . . . . . . . 34

5.4 Initial condition for the 2D and 3D supersonic flow in channel with step 39

5.5 Inlet boundary condition for supersonic channel flow with a step . . 395.6 Initial condition for the 2D and 3D subsonic flow over a bump . . . 41

6.1 Grid detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Processing time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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LIST OF FIGURES

1.1 Schematic diagram of Hybrid Rocket Motor . . . . . . . . . . . . . 1

2.1 Dimensions of the computational domain . . . . . . . . . . . . . . 5

2.2 Different blocks in 2D grid . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Different regions in BlockB1 . . . . . . . . . . . . . . . . . . . . . 7

2.4 Issues in grid quality and their remedies . . . . . . . . . . . . . . . 9

2.5 Three dimensional grid of Rocket Motor . . . . . . . . . . . . . . 10

3.1 Ghost cell layers in 2D . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Inter block transfer boundary condition in 2D case . . . . . . . . . . 24

3.3 Class diagram of the solver . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Residue plots for different schemes to obtain the flux at the faces . . 30

4.2 Residue plots for different time stepping schemes . . . . . . . . . . 31

5.1 Test case A : grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Residue plots for supersonic flow over 2D, 3D ramp and 3D cone . 35

5.3 Pressure contours for 2D ramp using HLL and Rusanov . . . . . . . 36

5.4 Pressure contours for cone using HLL and Rusanov . . . . . . . . . 37

5.5 2D grid for the test case of foraward facing step in a channel flow . . 38

5.6 Test case : density contours for 2 D case of supersonic flow in a channel

with a forward facing step using HLL and Rusanov solver . . . . . . 40

5.7 Test case: density contours for supersonic flow in a channel with a for-

ward facing step using HLL and Rusanov solver . . . . . . . . . . . 40

5.8 Grids for the test case of subsonic flow in a channel with a bump . . 42

5.9 Residue plots for subsonic flow in a channel with a bump . . . . . . 43

5.10 Test Case: Pressure contours for subsonic flow in a 2D channel with a

bump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.11 Test Case: Pressure contours for subsonic flow in a 3D channel with a

bump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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6.1 Pressure contours for 2D geometry using HLL solver . . . . . . . . 46

6.2 Pressure contours for 2D geometry using Rusanov solver . . . . . . 47

6.3 Mach number contours for 2D geometry using HLL solver . . . . . 47

6.4 Mach number contours for 2D geometry using Rusanov solver . . . 48

6.5 Various regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.6 Velocity profile for 2D geometry using HLL and Rusanov solver . . 50

6.7 Residue plots for 2D geometry using HLL and Rusanov solver . . . 51

6.8 Pressure contours along the axis using HLL and Rusanov solver . . 53

6.9 Mach number contours along the axis using HLL and Rusanov solver 54

6.10 Mach number cut plane near inlet and near the starting of the body using

HLL and Rusanov solver . . . . . . . . . . . . . . . . . . . . . . . 55

6.11 Residue plots for 3D geometry using HLL and Rusanov solver . . . 56

A.1 Point, Vector, Mesh, Solver Object . . . . . . . . . . . . . . . . . . 58

A.2 Cell, Face, GhostCell, PhysicalCell Object . . . . . . . . . . . . . . 59

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ABBREVIATIONS

PDE Partial Differential Equation

FVM Finite Volume Method

BC Boundary Condition

IC Initial Condition

TBC Transfer Boundary Condition

AW Adiabatic Wall

HLL Harten, Lax and van Leer

CFD Computational Fluid Dynamics

OOD Object Oriented Design

B1 Block Number 1

B2 Block Number 2

B3 Block Number 3

B4 Block Number 4

STP Standard Temperature and Pressure

GC Ghost Cell

P C Physical Cell

Avg Average

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NOTATION

P PressureP0 Total Pressure Density0 Total DensityT TemperatureT0 Total TemperatureR Real Gas Constant of Air

Et Specific Energyu x-component of velocityv y-component of velocityw z-component of velocityc speed of soundV Velocity VectorCFL Courant Friedrich Lewy number

n unit normal vectorCp specific heat coefficient at constant pressureCv specific heat coefficient at constant volume

ratio of specific heat coefficients at constant pressure and volume

Control Volume Control Surfacet timet time stepK Number of FacesSk Area of face K{I} Set of all Cells Computational domainResi Residue ofi

th Cell

H Total Enthalpy2 Second order artificial dissipation factor

4 Fourth order artificial dissipation factorA Flux Jacobian Matrix Azimuth angle in cylindrical coordinates systemr radial distance in cylindrical coordinate system

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CHAPTER 1

INTRODUCTION

Hybrid rockets have low regression rate, which makes it impractical even though it

combines the technical advantages of both liquid and solid rockets, such as simplicity,

safety, restart ability and low cost of development and operation. The configuration of

hybrid motor (Lin, 2009) is shown in fig 1.1. If the flame can be stabilized near the head

end of the motor then the regression rate may improve further. It is observed by Dr. P.

A. Ramakrishna and Rajiv Kumar that the burning inside rocket motor is improved if

a bluff body is placed at head end of the motor. The research work on the effect of

bluff body on the combustion is being pursued in Aerospace Engineering Department,

IIT Madras by Prof. P. A. Ramakrishna and Rajiv Kumar. The details of the geometry

used in experiment are given by Rajiv Kumar through personal communication. The

objective of this project is to develop a solver to visualize the internal flow in rocket

motor without combustion using Finite Volume Method (FVM). A hemispherical bluff

body is placed at the inlet of the motor and its effect on the flow is considered. The

presence of body has been shown to improve the regression rate, hence the performance

of rocket motor can be improved. To design the solver, 3D Euler equation is considered

as the flow is assumed to be inviscid and compressible. The solid propellant is treated

as solid adiabatic wall.

Liquid Oxidizer

Injector Solid Fuel

Valve

Gas Generator

Figure 1.1: Schematic diagram of Hybrid Rocket Motor

This project is split into five stages to obtain the results for the problem considered.

In the first stage, grid generation part is considered. In FVM, the governing integral

equations are solved within a control volume by discretization of the equations in each

cell. This process of creating cells is known as grid generation. To solve the Euler equa-tion, FVM requires geometrical data like volume of each cell and the areas of all the


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faces, that define a cell. Different grid generation methods can be considered as given

in Chung (2002). Here the basic algebric and elliptic grid methods are used to create a

structured grid. As the computational domain is complex, a multi block, structured grid

is generated to solve the Euler equation in problem domain. Physical domain is divided

into four blocks and each block is divided into cells.

The second stage corresponds to the method/schemes used to design a solver. An

FVM solver was introduced by Jameson (Jameson et al., 1981). The addition of second

order artificial dissipation makes the solution to converge for a range of CFL values.

The addition of the fourth order dissipation factor reduces the spurious oscillation near

the steady state and it increases the convergence rate as well. As multi block grid is

used, the solver needs to be robust, where the solution can converge for different di-

mensions and shapes of individual blocks. In explicit artificial dissipation methods, the

second and fourth order dissipation factors are added by trial and error, hence some

methods are required where the dissipation factor can be incorporated implicitly in the

scheme. Riemann solvers are being used in CFD for a long time. Toro (1999) has con-

sidered the details of the Riemann solvers while Kitamura and Nishikawa (Nishikawa

and Kitamura, 2008) developed simple approximate Riemann solvers. HLL and Ru-

sanov are the simplest Riemann solvers, which are used in present work. First order

forward time, second and fourth order Runge-Kutta time stepping schemes are imple-

mented in the solver in order to march in time. The Initial and boundary conditions are

incorporated in the solver according to the flow problem considered.

The third stage deals with design and implementation of the method considered, on

computers. The solver is inclined towards an Object Oriented Design (OOD) and is

implemented in C++.

Code validation is the fourth stage of this project. In this stage some well known test

problems e.g., supersonic flow over a ramp, supersonic flow in a channel with a for-

ward facing step and subsonic flow in a channel with bump, are considered. The results

obtained for these problems are compared with the results given in literature (Schwane,

2003; Galanin et al., 2009; Sun et al., 2009).

In the fifth and final stage, the results for the internal flow in a propellant motor is

obtained using the solver developed.

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The outline of this report is as follows, the grid generation part is discussed in chapter

2. Chapter 3 deals with the methods and schemes to solve the Euler equations. The de-

sign and implementation of Euler solver is considered in chapter 4, the results obtained

for various methods/schemes are discussed in chapter 5 on pipe flow problem. Code

validation in performed in chapter 6. The result for the internal flow of solid propellant

motor is considered in chapter 7. The thesis is concluded with the scope for future work

in chapter 8.

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CHAPTER 2

Grid Generation

To solve the governing PDE in computational domain using FVM, domain needs to be

discretized into finite size volumes called as cells. A structured grid is generated using

algebric and elliptic methods to construct cells.

2.1 Geometrical Details of Computational Domain

Geometry Details:

Inlet Diameter Din 7.5 mm

Diameter of Inner domain of motor Dmotor 15 mm

Diameter of sphere(body) Dbody 6 mm

Diameter of the Hole in body Dhole 2 mm

Exit Diameter Dexit 14 mm

Distance of the body from inlet L1 2mm

Distance between inlet and Nozzle L3 124 mm

Length of the Nozzle section L4 12 mm

Length of constant area section L5 10 mm

Angle of Nozzle surface with axis 35o


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Solid Body

154

132

All dimensions are in mm

Inlet

12

7

2

15

Figure 2.1: Dimensions of the computational domain

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The computational domain is shown in Fig. 2.1. As the computational domain is

complex, multi-block approach is used to create the structured grid. The whole domain

is divided into four blocks so that the grid can be generated for each block easily. To

create a three dimensional grid for the computational domain, a two dimensional grid

is generated and it is rotated by 360o about the axis of symmetry.

2.2 Two Dimensional Grid Generation

The 2D grid is generated by using linear interpolation. The quality of grid is improved

by using the elliptic grid near the body. For detail explanation of the grid generation

methods Chung (2002) cab be reffered. The process of creating grid for each block and

issues related to the grid quality are mentioned below.

2.2.1 Grid Generation of Different Blocks

Different blocks used in generation of 2D are shown in fig. 2.2.

Body

BlockB2

L3L4

BlockB3 BlockB4

BlockB1

Dmotor2

L5L1 L2

Figure 2.2: Different blocks in 2D grid

Block B1

To create the grid for blockB1, the block is divided in three regions as shown in fig. 2.3.

The region where blockB1 interfaces with blockB2 is named as Region-1. In Region-2,

blockB1 interfaces with the body while Region-3 is considered where the B1 interfaces

with B3 behind the body. The radius of sphere (body) is 6 mm and it is placed 2 mm

ahead from inlet with a cylindrical hole of 2 mm diameter. In two dimension case the

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Figure 2.3: Different regions in BlockB1

sphere will be circle. To create the grid in Region-2, the equation of the circle needs

to be identified. To find the equation of circle, the center of the circle is considered

as (xc, yc ). The computational domain is symmetrical about the x-axis hence yc = 0.

Now to obtain the value ofxc we write the equation of the circle as

(x xc)2 + (y yc)2 = 62, (2.1)

where yc = 0. As the distance of the body from the inlet is 2 mm with a hole of 2 mmdiameter, it passes through the point (2,1) where origin is assumed at the inlet point of

the axis. Hence,

(2 xc)2 + (1 0)2 = 62 (2.2) xc = 2 +

35 (2.3)

So the variation of y with x can be given as,

y =

36 (x 2 35). (2.4)

Here L1 =2 mm, L2 =

35 mm and L3 = 124 mm, where L2 is the length of the

body along the x-axis. So Region-1, Region-2 and Region-3 are defined from inlet to

L1, L1 to L2 and L2 to L3 respectively. For simplicity during the grid generation, the

value ofL2 is approximated to 6 mm. While creating the grid points, for the Region-1,

equal spacing was implemented in x-direction and same spacing is used for Region-2 as

well. For Region-3 the spacing in y-direction was find by distributing the whole length

along y-direction equally. As the length of the Region-3 is much higher compared tothe length of Region-2 in x-direction, hence the spacing between grid points is more

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than the spacing in Region-2. In y-direction for Region-1, uniform grid spacing is used.

While for region-2, the total length in computational domain is (Dmotor/2 y) and herey varies with x, hence y changes with x till Region 2. By using linear interpolationthe grid points for Region-2 are obtained. For Region-3 the length in y-direction is same

as 0.5 (Dmotor Dbody) = 1.5 mm, hence a uniform grid is generated for Region-3.

Block B2

BlockB2 is also divided in three regions similar to block B1, Region 1 where it is

interfacing with B1, Region 2 where it is interfacing with body and the last region

where B2 interfaces with B3. y, the distance between two grid points in y-direction,to create the grid points, is kept same.

x is kept same as of blockB1 for the grid

points corresponding with same value ofx along the axis.

Block B3

A uniform grid is created for blockB3 such that at interface the grid is overlapping with

blockB1 and B2 properly at the interfaces.

Block B4

For B4 the nozzle and constant area duct parts are kept withequal spacing in x-direction.

The grid points in y- direction are obtained by linear interpolation from the points ob-

tained at the interface of this block with other blocks.

2.2.2 Issues Related to Quality of Grid

Skewness of the Grid Near the Body in Block B1

During the Grid Generation for B1, high skewness of the grid is observed near body. To

solve this issue in Region 1 and Region 2, elliptic grid is generated as shown in fig. 2.4.

Abrupt Change in Point Spacing

While creating the grids of different blocks, it is observed that the grid size from one

grid point to another, increases abruptly at the interface of two blocks. As a result afterrunning the solver on such a grid, abrupt jump in the properties between two near cells is

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Figure 2.4: Issues in grid quality and their remedies

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observed. To resolve this issue, the point spacing is implemented such that grid spacing

should not change abruptly. To create such a grid, a factor of 1.217759 is used for Block

B3 and for Region-3 of blockB1 and B2 while using 33 grid points along x-direction

from the length L2 to L3 to create a nonuniform grid. First the factor is multiplied with

the x of the region 2 of blockB1 till first 20 grid points are obtained in the areasmentioned and then the factor is inverted to decrease the size to make the spacing of

the order ofx compatible with B4. This factor is obtained by using trial and error.Similarly the spacing in y direction is also changed non-uniformly by using suitable

factors using trial and error. Fig. 2.4 shows the issues mentioned. The optimized grid

options are used to create the finalized 2D grid.

2.3 3 D Grid

After getting the grid on 2-D, the grid is rotated by 360o about x-axis to get the 3-D

grid. The final grid is shown in fig. 2.5.

Figure 2.5: Three dimensional grid of Rocket Motor

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CHAPTER 3

Euler Solver

3.1 Governing Equations

The three dimensional Euler equation in integral form can be written as,

d

dt

Qd +

F.n dS = 0 (3.1)

Where is the control volume and is the control surface.

F and n are defined as,

F = Ei + Fj + Gk (3.2)

n = nxi + nyj + nzk (3.3)

Where Q, E, F and G are defined as,

Q =

u

v

w

Et

=

q1

q2

q3

q4

q5

(3.4)

E =

u

u2 + P

uv

uw

(Et + P)u

=

q2

q22/q1 + P

q2q3/q1

q2q4/q1

(q5 + P)q2/q1

(3.5)

F =

v

uv

v2 + P

vw

(Et + P)v

=

q3

q2q3/q1

q32/q1 + P

q3q4/q1

(q5 + P)q3/q1

(3.6)


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G =

w

uw

vw

w2 + P(Et + P)w

=

q4

q2q4/q1

q3q4/q1

q42/q1 + P(q5 + P)q4/q1

(3.7)

From state equation and enthalpy relation,

P = R T (3.8)

Et = CvT +(u2 + v2 + w2)

2(3.9)

P = ( 1)(Et 2

(u2 + v2 + w2)) (3.10)

or P = ( 1)(q5 q22 + q32 + q42

2 q1) (3.11)

3.2 Finite Volume Method

A cell-centered FVM has been used to solve the governing system of equations. The

computational domain is divided into finite number of sub volumes called as cells. The

discretized equation is solved for each cell separately. Consider ith cell with control

volume ofi with Knumber of faces. The discretized equation can be represented as,

iQ

t=

k=Kk=1

(F.n)kSk (3.12)

(F.n) = Enx + F ny + Gnz (3.13)

Qn = Qin+1 Qin (3.14)

where Sk is the area of the cell corresponding to kth face ofith cell. nx, ny, nz are the

components of the unit normal at corresponding face. For each cell i {I}, having aset of faces {Ki} the cell-residual Resi is defined as a numerical approximation of theintegral over the cell of the spatial operator, {Ki} divided by the cell volume i. Hencethe Resi can be written as

Resi = k=K

k=1 (F.n)kSk

i(3.15)

The cell-averaged value, Qi, is evolved by the non-zero residual in the form

dQidt

= Resi (3.16)

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3.3 Time Marching Schemes

Three types of time stepping schemes were incorporated into the solver. The first order

time accurate is the easiest to implement. In this scheme the t will be minimum com-

pared to higher order time accurate schemes because of the limitation on CFL number.

By using second and fourth order Runge Kutta schemes, the time step can be increased

as higher values of CFL number can be used for convergence. In the one-dimensional

case, the CFL condition for grid spacing x, wave speed u and speed of sound c isgiven by equation. 3.17 and similarly the equivalent three dimensional constraint on

time spacing can be given as,

t1D CFL(x)

|u|

+ cmax(3.17)

t3D CFL iK

k=1 |k|maxSk(3.18)

Here |k|max is the maximum characteristic wave speed normal to the cell face k andSk is the area of the face k.

3.3.1 First Order Forward Time Stepping

In first order forward time stepping scheme, the value of the Qi at any time tn+1 can be

given as

Qn+1i = Qni + t Resi(Qni ) (3.19)

3.3.2 Second Order Runge Kutta time stepping

The two stage second order Runge Kutta method can be implemented as given below.

Q(0)i = Q

ni

Q(1)i = Q(0)i + t Resi(Q(0)i )

Q(2)i =

1

2Q(0)i +

1

2(Q

(1)i + t Resi(Q(1)i ))

Qn+1i = Q(2)i

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3.3.3 Fourth Order Runge Kutta Time Stepping

The four stage fourth order Runge Kutta method is given below.

Q(0)i = Q

ni

Q(1)i = Q

(0)i +

t2

Resi(Q(0)i )

Q(2)i = Q

(0)i +

t2

Resi(Q(1)i )

Q(3)i = Q(0)i + tResi(Q(2)i )

Q(4)i = Q

(0)i +

t6

(Resi(Q(0)i ) + 2Resi(Q

(1)i ) + 2Resi(Q

(2)i ) + Resi(Q

(3)i ))

Qn+1i = Q(4)i

3.4 Numerical Methods

To solve the discretized equation 3.12, the flux term (F.n)k at kth face has to be eval-

uated. The term (F.n)k can be implemented in two ways. Lets consider, cell i and j

share a common face k with area vector as Sk. Then

(F.n)k = Avg(Fi , Fj) . Sk (3.20)

or (F.n)k = F(Avg(Qi , Qj)) . Sk (3.21)

The Avg function is defined on the basis of different schemes to be applied. For the

schemes related to the equation (3.21), the Avg function can be defined in two ways,

simple and Roes averaging. Simple averaging is defined as

Avg(Qi , Qj) =(Qi + Qj)

2(3.22)

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In Roe averages, different variables at the face are approximated in terms ofith and jth

cell properties taken from Toro (1999) are given below,

=

ii (3.23)

u = uii + ujji +

j

(3.24)

v =vi

i + vj

j

i +

j(3.25)

w =wi

i + wj

j

i +

j(3.26)

H =Hi

i + Hj

j

i +

j(3.27)

c =

( 1)(Hu2 + v2 + w2

2 ) (3.28)

V = V .n = unx + vny + wnz (3.29)

Here q correspond to the property q at face, hence Q can be evaluated using primitive

variables at face k and the flux at face k can be calculated as a function of Q. Detailed

explanation on Roes averaging is given in Toro (1999).

3.4.1 Artificial Dissipation

As suggested by Jameson (Jameson et al., 1981), to suppress the tendency for even-odd

point decoupling, and to prevent the appearance of oscillations in regions containing

severe pressure gradients in the neighborhood of shock waves or stagnation points, it is

necessary to augment the finite volume scheme by the addition of artificial dissipative

terms. Hence equation (3.12) is replaced by adding dissipative term in it and can be

given as,

Qt

=k=K

k=1 (F

.n)kSki

+ D(Q) (3.30)

D(Q) = 22Q + 44Q (3.31)

with 2 > 0 and 4 < 0, where 2 and 4 are the second and fourth order artificial

dissipation factors respectively.

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3.4.2 Computation of2Q and 4Q

The theorem of Gauss is used to compute 2Q and 4Q. The Gausss theorem statesthat,

. A d =

A.n dS

where A is any arbitrary vector.

Let the axes, xi form a right-handed Cartesian coordinate system and suppose A = ei,

where ei is a unit vector in xi direction. Then the theorem of Gauss for such a vector

field yields,

xid =

ni dS

The above equation takes the below form when it is discretized for a cell of volume ,

xi=

1

ni dS

The term

ni dS is approximated for a cell with K faces as,

ni dS =k=K

k=1

(i + k)

2niSk

Where k denotes the corresponding neighboring cell for face k and i represents the

value at the cell center. Ski is the ith component of the area at face k. Similarly for any

scalar function , the divergence theorem can be written as,

d =

n dS (3.32)

Hence the higher derivatives (m + 1) can also be computed using the values ofmth

derivative as given below,

m+1

xim+1=

1

m

ximni dS

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3.5 Riemann Solver

To implement the solver using the artificial dissipation, the right amount of artificial

dissipation (2 and 4) is required, with a limiting range of CFL number. If the right

amount of viscosity is not added the solution diverges. Sometimes code diverges after

running a number of iterations, hence some suitable scheme has to be used where the

right amount of artificial viscosity can be added without any trial and error. To solve

this issue Riemann solvers are incorporated in the Euler solver requiring no explicit

artificial dissipation.

The flux at faces are equated to Godunovs flux. To get the Godunovs flux at the

face, the approximate Riemanns solvers, HLL and Rusanov, are used. For the formu-

lation of Riemann solvers, Toro (1999) is referred. Some basic formulations are given

below,

3.5.1 Godunovs Method

Lets consider the one dimensional spatial domain [0, L] along x-axis. We divide this

domain into N cells with Ii = [xi 12, xi+ 12

] of regular size x = xi+ 12 xi 12 = L/N,with i = 1, 2,..,N. For a given cell Ii the location of the cell center xi and the cell

boundaries xi 12 ; xi+ 12 are given by,

xi 12= (i 1) x , xi = (i 1

2) x , xi+ 12 = i x (3.33)

The temporal domain is given by [0, T], where T is some output time. The discretization

of the time interval [0, T] is generally done in time steps t of variable size as the wavespeeds vary in space and time, the ch hence the choice oft is carried out as marchingin time proceeds. Now the Euler equation can be written as

Qt + F(Q)x = 0 (3.34)

In Godunov method, a piece wise constant distribution ofQ is assumed for all cells.

Lets assume Q(x, tn) is the value of the data at time t = tn. The cell averages can be

defined as

Qni =1

xx

i+12

xi 12

Q(x, tn)dx (3.35)

Hence the piecewise constant distribution Q(x, tn) can be given as,

Q(x, tn) = Qni , for x in each cell Ii = [xi 12

, xi+ 12

] (3.36)

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In Godunovs method the new average value Qn+1i at time tn+1 = tn + t is given

by,

Qn+1i = Qni +

tx [Fi+ 12 Fi 12 ] (3.37)

Where the inter-cell numerical flux is given by,

Fi+ 12= F(Qi+ 12

(0)) (3.38)

if the time step is given by,

t xSnmax

(3.39)

Where Snmax denotes the maximum wave velocity present throughout the domain at time

tn.

3.5.2 HLL Approximate Riemann Solver

From Godunov method, the inter-cell numerical flux is given as equation (3.50). Here

Qi+ 12(0) is the exact similarity solution Qi+ 12

(x/t) of the Riemann problem given as,

t Q + x F(Q) = 0

Q(x, 0) ={

Qni , x < xi+1/2

Qni+1, x > xi+1/2

Qi+1/2(x/t)

Fi+1/2 =1t

t0

F(Qi+1/2(0)) dt = F(Qi+1/2(0))

Consider the case where the whole wave structure arising from the exact solution of

the Riemann problem is contained in the control volume [xi, xi+1] x [0, T], that is

xi T Si , xi+1 T Si+1 (3.40)

where Si and Si+1 are the slowest and fastest signal velocities perturbing the initial datastates Qi and Qi+1 respectively, and T is a chosen time. The integral average of the

exact solution of the Riemann problem between the slowest and fastest signals at time

T is a known constant, provided that the signal speeds Si and Si+1 are known; as given

such constant is given by

Qhll =(Si+1Qi+1 SiQi + Fi Fi+1)

(Si+1 Si) (3.41)

WhereFi = F(Qi)

andFi+1 = F(Qi+1)

. Harten, Lax and van Leer put forward the

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following approximate Riemann solver

Q(x, t) =

Qi ifxt

SiQhll if Si xt Si+1

Qi+1 ifxt Si+1

(3.42)

where Qhll is the constant state vector and the speeds Si and Si+1 are assumed to be

known. The flux corresponding to the state Qhll along the t-axis is given by,

Fhll =Si+1Fi SiFi+1 + SiSi+1(Ui+1 Ui)

Si+1 Si (3.43)

The corresponding inter-cell flux at the face can be given by,

Fhlli+ 12

=

Fi if 0 SiFhll if Si 0 Si+1

Fi+1 if 0 Si+1(3.44)

or Fhlli+ 12

=S+i+1Fi Si Fi+1 + SiS+i+1(Ui+1 Ui)

S+i+1 Si(3.45)

Where S+i+1 = max(0, Si+1), Si = min(0, Si).

Given an algorithm to compute the speeds Si and Si+1 we have an approximate value

of flux at the face. In case of three dimensional Euler equation a numerical flux across

each interface k, k, is determined by solving Riemann problem approximately, based

on the one dimensional Euler equation in the direction of the face normal n. Hence the

3D Euler equation equation can be written as,

Q

t+

F(Q)

n= 0, (3.46)

where F(Q) = (F

.n), is the flux at the face k (equation 3.13) and here the initial data,Qi and Qj are separated by the cell interface. These initial data simply taken as the cell

averaged states in the adjacent cells. Here F(Q) is the flux corresponding to the kth

face is given as,

F(Q) =

V

uV + P nx

vV + P ny

wV + P nz

(Et + P)V

(3.47)

where V = V .n = unx + vny + wnz and the normal vector on the surface is n = nxi +

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nyj + nzk. The left and right speeds, SL and SR ( Si and Si+1 in one dimension), are set

to be minimum and maximum eigenvalues of the Flux Jacobian Matrix A (i.e.F(Q)Q

),

SL = min(VL cL, V c), SR = min(VR + cR, V + c), (3.48)

where (V)L,R are propagation speed normal to the surface corresponding to the left and

right cell.

3.5.3 Rusanov Approximate Riemann Solver

As in HLL Riemann solver, two wave speeds SL and SR are considered, it is a two

wave approximate Riemann solver. Rusanov suggested a one wave approximate Rie-

mann solver, by reducing the number of waves. Now Assume a single wave speed

estimate:S+ > 0. We define a second wave speed:SL = S+; SR = +S+. Substitu-tion ofSL and SR values into HLL flux gives the Rusanov flux as,

FRusanov =1

2(FL + FR) S+

2(QR QL) (3.49)

Choice of the wave speed S+ considered by Davis(Davis, 1988) is,

S+ = max

{|VL

cL

|,

|VR

cR

|,

|VL + cL

|,

|VR + cR

|}(3.50)

The above speed is bounded by,

S+ = max{|VL| + cL, |VR| + cR} (3.51)

The wave speed S+ can be considered as maximum eigenvalue of Jacobian flux

matrix as well given by Kitamura and Nishikawa (Nishikawa and Kitamura, 2008).

S+ = |V| + c (3.52)

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3.6 Boundary Conditions

To apply boundary conditions ghost cells are used. A layer of ghost cells is added at

individual boundary, corresponding to i = 0, i = Imax, j = 0, j = Jmax, k = 0 and k = Kmax,

where i, j and k are used as indices in structured grid with i along the axis, j radial and

k corresponding to azimuth angle for rotational grid and along x,y,z respectively for

regular Cartesian grid. In a ghost cell, we need to update the value ofQ on the basis of

boundary conditions. Fig. 3.1 shows to create the ghost cell layers in 2D case. Different

kind of boundary conditions are incorporated in following subsections.

Physical CellGhost Cell

I0Layer

Imax

Layer

Jmax Layer

J0 Layer

Figure 3.1: Ghost cell layers in 2D

3.6.1 Inlet Boundary Condition

At inlet, the total stagnation temperature and pressure are prescribed. Subsonic flow

is considered at the inlet hence in case of 3D Euler equation, one numerical and four

physical boundary conditions are required at inlet. The inlet condition at the ghost cell

layer J0 is given in table 3.1

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Table 3.1: Inlet conditions

Property Value

P0 (P a) 121590.0T0(K) 300.0

v 0

w 0

0 can be find out using equation of state hence 0 =P0RT0

, where R is the gas

constant for air. The ghost cell values are updated as follows

uGC = uPC (3.53)

vGC = 0wGC = 0

TGC = T0 (u2GC + v

2GC + w

2GC)

2Cp

GC = 0(T

T0)

1(1) ,

where GC subscript corresponds to Ghost Cell and PC corresponds to Physical Cell.

After getting primitive variables (, u, v, w and T) in ghost cells the value ofQ is up-

dated.

3.6.2 Exit Boundary Condition

At exit plane, the static pressure is specified. Being subsonic flow, one physical and

four numerical boundary conditions are required. The values transferred to ghost cell

layer, Imax, at exit plane are given as,

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PGC = Pexit (3.54)

(u)GC = (u)PC

(v)GC = (v)PC

(w)GC = (w)PC

TGC = TPC

GC =PGC

RTGC

uGC = (u)GCGC

vGC =(v)GC

GC

wGC =(w)GC

GC

3.6.3 Inviscid Wall Boundary Condition

The flow is assumed to be inviscid and solid adiabatic walls are considered. Being solid

walls no penetration condition (V .n = 0 and Pn

= 0 ) is applied . Adiabatic wall gives

constraint on Temperature as Tn

= 0 at wall. Using first order boundary condition the

values of P, T and are extrapolated to the ghost cells. Hence the inviscid wall boundary

conditions can be applied as

PGC = PPC (3.55)

TGC = TPC

GC = PC

VGC = VPC 2(VPC.n)n

where VGC is the velocity vector in ghost cell and n is the normal to the boundary orface where boundary condition is applied.

3.6.4 Transfer Boundary Condition

At the block level, the ghost cell layers are created for all boundaries for each block.

At interface of two blocks, the data from the one cell is copied to the ghost cells of the

near block and similarly the near block data is transferred to its ghost cell layer. The

basic idea of inter block transfer is taken from Blazek (2001). The inter-block transfer

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is shown in fig. 3.2 between two blocks A and B. The data from the boundary cells of

physical domain are transferred to the corresponding ghost cell layer of other block. In

case of rotated grid the ghost cell layers are created for k=0 and k=Kmax as well and

the data are exchanged between these two layers from their corresponding neighboring

physical cells. Hence the data for k=0 and k=Kmax ghost cell layers are obtained by

copying the data from physical cells of k=Kmax and k=K0 layers respectively.

BA

Exchange of data from

Exchange of data from

physical cell of A to ghost cell of B

physical cell of B to ghost cell of A

Figure 3.2: Inter block transfer boundary condition in 2D case

3.7 Initial Condition

Generally fluid is considered stationary for the initial condition but as the geometry con-

sidered in problem is axisymmetric, the solution of 2D flow is a better approximation to

get the steady state solution for 3D case. For 2D case the initial condition of stationary

fluid is taken. Hence, the initial condition for 2D case is given in table 3.2. To obtain

the initial values for 3D case from the solution of 2D, all vector quantities like velocity

have to be transformed accordingly. Scalar quantities like pressure or density, which

are invariant with respect to coordinate rotation, remain unchanged. As the 3D grid is

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created by rotating the 2D grid about x-axis, the rotation matrixR becomes,

R =

1 0 0

0 cos sin

0 sin cos

(3.56)

where is the angle of rotation and R is the rotation matrix.

Table 3.2: Initial condition for the 2D and 3D supersonic flow in channel with step

Property Value

(Kg/m3) 1.225T(K) 287.70

u 0

v 0

3.8 Convergence Criteria

The convergence of the scheme is defined on the basis of the residue. The residue is

obtained for set of

{I

}cells is given by following formulation,

Relative Residue =

i{I}

5m=1

Qn+1i (m) Qni (m)

Qni (m)

2i

(ti)2 if Qni (m) > 10

6

Absolute Residue =

i{I}

5m=1

Qn+1i (m) Qni (m)

2 i(ti)2 if Q

ni (m) 106

The Residue value was dependent on the scheme and the case considered. The steady

state was considered to be obtained whenever the Residue value was dropped from

higher values (order of106) to significant value (order of107).

3.9 Class Diagram of Solver

To develop the solver, programming language C++ is used. The class diagram is shown

in fig. 3.3. Here the Mesh object is passed to Solver and different functions of Solver

class can be used through the instance of the solver. Different objects used are:

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1. Point

2. Vector

3. Face

4. Cell

5. Mesh

6. Solver

Class diagram with their attributes and operations are given in appendix A.

Figure 3.3: Class diagram of the solver

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The value oft is not calculated as mentioned in equation 3.20 as in this scheme, theeigen values are not calculated to obtain the flux at faces. Hence by keeping conserva-

tive approach the maximum eigen value is considered as (|u| + |v| + |w| + c) hence

t can be given as,

t = CFLlmin|u| + |v| + |w| + c (4.2)

The value oflmin is obtained as,

l = iKk=1 Sk

(4.3)

The solution diverges in case of local time stepping with CFL number and 2 value used

for global time stepping; hence only global time stepping is considered in this scheme.

In Roe averaging schemes, the Q value is taken as Roes averages given in equations

3.35 to 3.41. The solution obtained in this scheme is similar to the solution obtained in

simple averaging with same CFL and 2 values.

4.1.2 Fourth Order Artificial Dissipation

To improve the quality of solution, second and fourth order artificial dissipation factors

are added. In this scheme, the convergence rate is also improved compared to the case

where only second order dissipation is considered. The dissipation factor 4 is taken as

-0.0001, which is obtained by trial and error.

4.1.3 Riemann Solvers

The problem considered has multi block grid, hence some scheme is required where the

value of artificial dissipation factor is independent on the computational domain. Using

Riemann solver, the flux value at the face is approximated such that the amount of dis-

sipation is incorporated in the scheme implicitly. The HLL and Rusanov approximate

Riemann solvers are employed on pipe flow problem. In these cases, CFL is 0.5 and

local time stepping is incorporated. Riemann solvers require less number of iterations

to reach the steady state than the scheme with 2nd and 4th order artificial dissipation

considered.

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4.1.4 Performance Analysis of Schemes

Total time taken by individual scheme to reach a certain value of residue is given in

table 4.1. The performance of the schemes on the basis of the time taken per iteration

per grid point is considered in table 4.2. It is observed that the Riemann solvers give

high performance compared to the schemes with artificial dissipation.

Table 4.1: Performance of different schemes for pipe flow with 25x10x16 cells

Scheme CFL Residue Number Of Iteration Processing time(sec)

2nd Order Simple Avg 0.3 1.18x1005 1045082 21946.72

2nd Order Roe Avg 0.3 1.18x1005 1046413 21974.67

4th Order Simple Avg 0.3 1.18x1005 562438 23059.95

HLL Solver 0.5 9.15x1006 259084 2072.672Rusanov Solver 0.5 1.18x1005 249980 1749.86

Table 4.2: CPU time per Iteration per grid point

Scheme CPU Time/Iteration/Grid Point(sec)2nd Order Simple Avg 5.25

2nd Order Roe Avg 5.50

4th

Order Simple Avg 10.25HLL Solver 2.00

Rusanov Solver 1.75

4.1.5 Residue Plots

Residue plots for different schemes are given in fig. 4.1.

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1e-10

1e-08

1e-06

0.0001

0.01

1

100

10000

1e+06

0 100000 200000 300000 400000 500000 600000 700000

Residue

Number Of Time Steps

Residues of different FVM schemes

2nd Order2nd Order Roe

4th orderHLLRusanov

Figure 4.1: Residue plots for different schemes to obtain the flux at the faces

4.2 Time Marching Schemes

First order forward time, second order Runge-Kutta and fourth order Runge Kutta

schemes are implemented to the solver. HLL Riemann solver is used to obtain the

flux at faces on pipe flow problem with grid size of 26x11x17.

4.2.1 Performance Analysis of Schemes

The results are compared on the basis of number of iterations, residue and processing

time as given in table 4.3. The performance of the schemes on the basis of the time

taken per iteration per grid point is given in table 4.4.

4.2.2 Residue Plots

Residue plots corresponds to different time marching schemes are shown in fig. 4.2.

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Table 4.3: Performance of different time marching schemes for pipe flow with

25x10x16 cells

Time Stepping Scheme CFL Residue Number Of time steps CPU time(sec)

First Order Forward Time 0.5 1.00005x105 177230 1772.302nd Order Runge Kutta 1.0 1.00005x 105 88619 1329.284th Order Runge Kutta 2.5 1.00022x 105 35447 1063.41

Table 4.4: CPU time per iteration per grid point for time marching schemes

Scheme CPU Time/Iteration/Grid Point(sec)First Order Forward Time 2.0

2nd Order Runge Kutta 3.754th Order Runge Kutta 7.5

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

0 20000 40000 60000 800001000001200001400001600001800

Residue

Number Of Iterations

Residue Plot for Different Time Marching schemes

First Order Forward TimeRunge Kutta 2nd OrderRunge Kutta 4nd Order

Figure 4.2: Residue plots for different time stepping schemes

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CHAPTER 5

Code Validation

The code validation of the solver is done by comparing the results on test problems

considered with the results available in literature. To achieve the steady state, marching

in time is performed using local time-stepping until the L2 residual has leveled off.

CFL is used as 0.5 for both Rusanov and HLL solver. First order forward time stepping

method is used to march in time.

5.1 Test Case A : Supersonic Wedge Flow

In this test, supersonic flow past a wedge, with an inclination of 15 degrees, is consid-

ered with upstream Mach number of 3.

5.1.1 Grid

For two dimensional case, the computational domain with 2m length in x-direction and

1m in y-direction is considered. The wedge is placed from x = 1m to x = 2m with an

inclination of 15 degrees with the x-axis. A single-block, two-dimensional, structured

grid is generated for the flow domain using algebric grid generation. The grid size of

81x81 is considered. For three dimensional test two different cases are considered. In

case1 the 3D grid is generated by stacking the xy grid along z-direction and a grid of

51x51x26 is created. In case2, flow over a cone is considered and grid is generated by

rotating the 2D grid about x-axis. For 3D cone test double blocks are used, one blockfrom inlet to the starting of the cone and other the conical shape obtained after rotating

the wedge about x-axis. Hence, both the blocks have structured grid with the grid size

of 21x41x37 corresponding to (x,r,) direction. The grids used are shown in fig. 5.1.

5.1.2 Initial condition

Initial conditions for 2D and 3D cases are considered as shown in table 5.1 for whole

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Figure 5.1: Test case A : grids

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1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

1e+10

0 2000 4000 6000 8000 10000 12000 14000 1600

Residue


Test Case A: Supersonic Wedge flow

2D HLL2D Rusanov

3D Ramp HLL3D Ramp Rusanov3D Cone HLL

3D Cone Rusanov

Figure 5.2: Residue plots for supersonic flow over 2D, 3D ramp and 3D cone

5.1.5 Test Results

The pressure contours are plotted and it is observed that the Rusanov solver is more

dissipative than HLL. The results obtained in 2D case are very near to the theoretical

results for the flow over a ramp. In the case of 3D case1 the results obtained are same

as obtained in 2D case while in 3D case2 the change is pressure after shock is less

compared to the value obtained in the case of 2D. Here rather than a ramp it is a cone in

3D hence the 3D effects are there. The pressure variation can be seen in fig. 5.4 for all

cases considered. The results are comparable with the results mentioned in reference

(Schwane, 2003).

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Figure 5.3: Pressure contours for 2D ramp using HLL and Rusanov

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Figure 5.4: Pressure contours for cone using HLL and Rusanov

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5.2 Test Case B : Flow in a Channel with a Forward

Facing Step

In this test the supersonic flow is considered and the shock reflection is observed be-cause of a forward facing step in the channel. The result is compared with the available

numerical results in references (Galanin et al., 2009).

5.2.1 Grid

For the two dimensional case, the computational domain is considered of 3m in x-

direction and 1m in y-direction. The step of 0.2m height is placed at x=0.6m. A single-

block, two-dimensional, structured grid was generated for the flow domain and the

solver is applied on the domain where the flow exists.The cells which resides between

step and the boundary y=0 are not considered in computational domain. The grid size

of 101x101 was considered for 2D case and the solver is run for two regions of 21x101

and 81x81. For three dimensional case the code was tested on the grid size of 51x51x21,

which is obtained by stacking the x-y grid of size 51x51 along z-direction. The grids

corresponding to 2D is shown in fig. 5.5.

Figure 5.5: 2D grid for the test case of foraward facing step in a channel flow

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5.2.2 Initial condition

Initial conditions for 2D and 3D cases were considered as shown in table 5.4.

Table 5.4: Initial condition for the 2D and 3D supersonic flow in channel with step

Test Case (Kg/m3) Et N/m2 u v w

2D Case 0.5 0.125 0.0 0.0 -

3D Case 0.5 0.125 0.0 0.0 0.0

5.2.3 Boundary Condition

The inlet boundary condition is applied as mentioned in table 5.5. The adiabatic wall

boundary condition is applied in the domain corresponding to J0, Jmax, K0 and Kmax

boundaries. The cells near the step, which are not used in computations are used as

ghost cells to apply the boundary condition. Being a supersonic flow the value of Q

is specified at inlet without extrapolating any variable from computational domain. At

exit, supersonic exit boundary condition is applied i.e. all variables are extrapolated

from the computational domain to ghost cells.

Table 5.5: Inlet boundary condition for supersonic channel flow with a step

(Kg/m3) Et N/m2 u v w

1.0 6.286 3.0 0.0 0.0

5.2.4 Computational Strategy

In this test case, the results mentioned in literature (Galanin et al., 2009) are given

at time t=4 sec, hence to obtain the result for HLL and Rusanov solver, global time

stepping is used.

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5.2.5 Test Result

The pressure contours are shown in fig. 5.6 for 2D cases and in fig. 5.7 for 3D cases. It

is observed that the Rusanov scheme is very dissipative comparative to HLL and as a

result the region near shock is wider in case of Rusanov than HLL.

Figure 5.6: Test case : density contours for 2 D case of supersonic flow in a channel

with a forward facing step using HLL and Rusanov solver

Figure 5.7: Test case: density contours for supersonic flow in a channel with a forward

facing step using HLL and Rusanov solver

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5.3 Test Case C : Flow in a Channel with a Bump

In this test, the subsonic flow is considered through a channel with a bump and the

pressure contours near the bump are compared with the available results (Sun et al.,

2009).

5.3.1 Grid

For the two dimensional case the computational domain is considered of 2m length in

x-direction and 0.5m in y-direction. The bump curvature is considered of 1m which

intersect the channel at x=0.5m and x=1.5m. The maximum height of the bump is

considered as 0.1m at x=1.0 m. A single-block, two-dimensional, structured grid was

generated for the flow domain using elliptic grid generation of the size 51x51. To test

in 3D, two cases are considered, case1 where the grid is obtained after stacking in z-

direction and case2 where the grid is created after rotation of the 2D grid.In case2, 3

blocks are considered and the boundary conditions are applied accordingly. The case2

has three blocks, structured grid with grid size of 51x51x17 along (x,r,) direction

while case1 is single block, structured grid size of 101x26x11. The grids used are

shown in fig. 5.8

5.3.2 Initial Condition

Initial conditions used for 2D and 3D cases are given in table 5.6, where all cells are

initialized with same values in computational domain.

Table 5.6: Initial condition for the 2D and 3D subsonic flow over a bump

Test Case Density(Kg/m3) Pressure(Pa) u v w

2D Case 1.225 101325.0 0.0 0.0 -3D Case 1.225 101325.0 0.0 0.0 0.0

5.3.3 Boundary Conditions

At inlet of the channel, the inlet boundary condition is applied with stagnation pres-

sure and temperature of 107853.3984 Pa and 300K respectively and exit pressure of

101325.0 Pa, which corresponds to the flow of M=0.3 in a channel without bump. The

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Figure 5.8: Grids for the test case of subsonic flow in a channel with a bump

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adiabatic inviscid wall boundary condition is applied in the domain corresponding to

J0, Jmax boundaries in case of 2D and in case of 3D, adiabatic wall is considered for

Jmax, K0, Kmax boundaries for all three block and for block 2, J0 is also considered.

5.3.4 Residue Plots

The residue plots obtained for all cases for HLL and Rusanov are given in fig. 5.9.

1e-08

1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

0 20000 40000 60000 80000 1000

Residue


Test Case C: Subsonic Flow in a Channel with Bump

2D HLL2D Rusanov

3D Bump Case1 HLL3D Bump Case1 Rusanov

3D Bump Case2 HLL3D Bump Case2 Rusanov

Figure 5.9: Residue plots for subsonic flow in a channel with a bump

5.3.5 Test Result

The pressure contours obtained are compared with results available in literature (Sun

et al., 2009). The HLL solver gives very similar results mentioned in literature while in

case of Rusanov, being more dissipative scheme, slight change in the results is observed

as a effect of dissipation. The results are given in fig. 5.11. 3D case2 is not considered

in references mentioned. In this case, it is observed that the results obtained from the

HLL and Rusanov solvers are are very similar. As in this case, 3D effect is there, the

contours lines are not exactly similar to the result obtained in 3D case1.

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Figure 5.10: Test Case: Pressure contours for subsonic flow in a 2D channel with a

bump

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Figure 5.11: Test Case: Pressure contours for subsonic flow in a 3D channel with a

bump

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CHAPTER 6

Results

Before obtaining the results for three dimensional flow in computational domain, results

for two dimension are obtained and analyzed. The solution for three dimension is found

to be axisymmetric.

6.1 Two dimensional Case

In two dimensional case, different grids were considered before finalizing the grid to be

used to create three dimensional grid. The results for finalized grid are discussed below.

6.1.1 2D Results

The result for the two dimensional case is obtained using HLL as well as Rusanov

solver. The pressure contours near the body, nozzle and whole computational domainare shown in fig. 6.1 using HLL and in fig. 6.2 using Rusanov solver. The contours for

Mach number are shown in fig. 6.3 using HLL and in fig. 6.4 using Rusanov solver.

Figure 6.1: Pressure contours for 2D geometry using HLL solver


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Figure 6.2: Pressure contours for 2D geometry using Rusanov solver

Figure 6.3: Mach number contours for 2D geometry using HLL solver

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Figure 6.4: Mach number contours for 2D geometry using Rusanov solver

Figure 6.5: Various regions

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Here some regions are shown in fig. 6.5 as given below

Region A : Sharp corner at Inlet

Region B : Near the starting of the bluff body in block B1

Region C : Near the starting of the bluff body in block B2

Region D : Behind body at the upper left corner in block B3

Region E : Behind body at the lower left corner in block B3

Region F : Sharp Corner at the start of nozzle in block B1

Region G : Sharp Corner in Nozzle at the start of duct in block B4

In region A, C, D, E, F and G, locally low pressure region creates as the flow does

not bend sharply at all these corners. Region B corresponds to the stagnation point,

as the flow hit the bluff body directly, it comes to rest. In region D and E, the results

corresponds to HLL and Rusanov solvers are different. In case of HLL, recirculation

region is created while in case of the Rusanov solver this phenomena is not observed as

the maximum Mach number in the case of Rusanov solver is of the order of 0.25 while

in case of HLL it is 0.42, hence in Rusanov solver it behaves like potential flow while in

HLL the recirculation region is created. The Mach number range for both the schemes

shows the dissipative nature of Rusanov scheme compared to HLL.The velocity profile

at different portions of the domain are shown in fig. 6.6.

6.1.2 Residue plots

The residue plots are shown in fig. 6.7.

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Figure 6.6: Velocity profile for 2D geometry using HLL and Rusanov solver

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1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

0 50000 1000001500002000002500003000003500004000

Residue


2D Main Problem

HLLRusanov

Figure 6.7: Residue plots for 2D geometry using HLL and Rusanov solver

Table 6.1: Grid detail

Block Name Grid Size Number Of Cells

BlockB1 65x51x37 115200BlockB2 65x11x37 23040

BlockB3 33x41x37 46080BlockB4 23x121x37 95040

Total - 279360

6.2 Three Dimensional case

The results for 3D case shows that the flow is axisymmetric. The dissipative nature of

Rusanov scheme compared to HLL scheme can be observed in different contour plots.

Different regions observed in 2D case are present in 3D solution as well.

6.2.1 Grid Detail and Processing time

The detail of individual block with number of cells used per block is given in table 6.1.

Runge-Kutta fourth order scheme is used to march in time. The detail of processing

speed is given in table 6.2.

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Table 6.2: Processing time

Scheme Number Of Cells CFL CPU Time/Iteration/Cell

HLL(Runge Kutta 4th order) 279360 2.5 14.31844 secRusanov(Runge Kutta 4th order) 279360 2.5 14.31844 sec

HLL(First Order in time) 279360 0.5 3.5796 secRusanov(First Order in time) 279360 0.5 3.5796 sec

6.2.2 3D Results

The pressure contours for HLL andRiemann solvers along the axis are shown in fig. 6.8.

The contours of Mach number are given in fig. 6.9. It is observed that the flow is

axisymmetric. In case of 3D, the Mach number range of Rusanov scheme is increased

compared to the value obtained in 2D case. Still the dissipative nature of Rusanov

scheme compared to HLL scheme can be observed in different contour plots. Different

regions observed in 2D case are present in 3D solution as well, denoting the same

phenomena mentioned earlier. It can be observed from the contours of Mach number

that the presence of the body accelerate the flow near the wall and its direction is radial.

Higher velocity near the wall increases the burning and hence the regression rate can be

increased.

The variation of Mach number and pressure at different locations along the axis is

shown by taking the scalar cut planes. The cut planes of Mach number at the starting of

the inlet and at the starting of the body are shown in fig. 6.10.

It is observed that in case of 3D, Rusanov and HLL solvers are giving similar re-

sults. The maximum Mach number achieved in HLL and Rusanov are 0.345 and 0.338

respectively. It is observed that in 2D case the maximum Mach number in physical

domain is 0.42 in case of HLL solver while it is 0.25 for Rusanov. In fig. 6.10, thedissipative nature of Rusanov compared to HLL can be observed.

6.2.3 Residue plots

The residue plots are shown in fig. 6.11.

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Figure 6.8: Pressure contours along the axis using HLL and Rusanov solver

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Figure 6.9: Mach number contours along the axis using HLL and Rusanov solver

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Figure 6.10: Mach number cut plane near inlet and near the starting of the body usingHLL and Rusanov solver

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1e-06

0.0001

0.01

1

100

10000

1e+06

0 20000 40000 60000 80000 100000 120000 140000

Residue


3D Main Problem

HLLRusanov

Figure 6.11: Residue plots for 3D geometry using HLL and Rusanov solver

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CHAPTER 7

Conclusion and Future Scope

7.1 Conclusion

Euler solver is implemented successfully using structured grid for HLL and Rusanov

Riemann solvers. The solver has been tested for benchmark problems of internal flow.

The performance analysis of different schemes to approximate the flux value at the

faces is done. Different time marching schemes are incorporated into the solver and it is

found that the Runge Kutta fourth order time stepping scheme gives better performance

compared to other time marching schemes employed in the solver. It is observed that

the Rusanov solver is more dissipative compared to HLL solver. The numerical solution

of 2D and 3D Euler equations are obtained for the rocket motor witha bluff body at inlet

and it is find out that the radial component of velocity are higher near the body which

increases the regression rate and hence the performance of the Hybrid Rocket Motors.

7.2 Future Scope

In the solver implemented, the convergence rate to reach the steady state can beincreased by using some acceleration scheme e.g Multigrid Methods.

The solver can be extended to Navier Stokes equations and the combustion phe-nomena can also be considered into account.

The solver can be modified to work with unstructured grid files.

Other than Riemann solvers and artificial schemes, other schemes can be incor-porated in the solver.


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APPENDIX A

Class Descriptions

The description of the classes used in the solver are shown below.

Figure A.1: Point, Vector, Mesh, Solver Object


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Figure A.2: Cell, Face, GhostCell, PhysicalCell Object

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REFERENCES

1. Blazek, J., Computational Fluid Dynamics: Principles and Applications. Elsevier,

2001.

2. Chung, T. J., Computational Fluid Dynamics. Cambridge University Press, 2002.

3. Davis, S. F. (1988). Simplified second-order godunov-type methods. SIAM Journal on

Scientific and Statistical Computing, 9(3), 445473.

4. Galanin, M. P., E. B. Savenkov, and S. A. Tokareva (2009). Solving gas dynam-

ics problems with shock waves using the rungekutta discontinuous galerkin method.

Mathematical Models and Computer Simulations, 1(5), 635645.

5. Jameson, A., W. Schmidt, and E. Turkel (1981). Numerical solution of the euler

equations by finite volume methods using runge-kutta time-stepping schemes. AIAA,

(1259).

6. Lin, J. L. (2009). Two-phase flow effect on hybrid rocket combustion. Acta Astronau-

tica, 65, 1042 1057.

7. Nishikawa, H. and K. Kitamura (2008). Very simple, carbuncle-free, boundary-layer-

resolving, rotated-hybrid riemann solvers. Journel of Computational Physics, 227,

25602581.

8. Schwane, R. (2003). A multi-dimensional solver for the steady euler equations. Com-

puters & Fluids, 32, 109119.

9. Sun, Y., Z. L. Wang, and Y. Liu (2009). Efficient implicit non-linear lu-sgs approach

for compressible flow computation using high-order spectral difference method. Com-

munications in Computational Physics, 5(2-4), 760778.

10. Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynanics. Springer,

1999.

nesh thesis

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