EGFD 637 – Computational Fluid Dynamics

Project 2

Fluent Simulation for the flow through a Convergent-Divergent Nozzle with

Subsonic, Supersonic andTransonic Flow

BySalah Soliman

Sowmya KrishnamurthySantosh Konangi

Bhaskar Chandra KonalaMohamed Abdelaal

InAerospace Engineering

Under Supervision of

Dr. Kirti Ghia

College of EngineeringUniversity of Cincinnati

A research project submitted to the Universityof Cincinnati as the 1st project for the

Computational Fluid Dynamics Course

University of CincinnatiCincinnati, Ohio, USA

Winter 2009

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Table of ContentsSubject PAGETable of Contents 2ABSTRACT 5CHAPTER 1 INTRODUCTION 6CHAPTER 2- Problem Description and Mathematical Formulation 112.1 Problem Description and Boundary conditions 112.2 Initial And Inlet Conditions 122.3 Governing Equations 12CHAPTER 3- Solution Mesh (Generated using Gambit) 143.1 Summarized Procedures for Mesh Generation Using Gambit 143.2 Grid Size, Zones and Quality 15CHAPTER 4- Fluent Solver Setup, Solution Strategy and Conversion Criterion. 164.1 Fluent Solver Setup and Solution Strategy 164.2 Convergence Criterion 18CHAPTER 5 -Fluent Results 205.1 Subsonic Flow through the CD nozzle (Case 1, Pback=585 kPa) 205.2 Supersonic Flow through the CD nozzle (Case 2, Pback=35 kPa) 205.3 Transonic Flow (shock wave) through the CD nozzle (Case 3,Pback=300 kPa) 21CHAPTER 6- Problem Exact Solution 326.1 Quasi 1D-Flow: Characteristics and Implications 326.2 Governing Equations for Quasi-1D Flow 326.3 Case 1: Subsonic Flow throughout the Nozzle (Pback=585 kPa) 346.4 Case 2: Supersonic Flow throughout the Nozzle (Pback=35 kPa) 356.5 Case 3: Normal Shock in the Diverging Section of the Nozzle (Pback=300 Kpa) 366.6 Summary of Exact Solution 37CHAPTER 7 -Comparison between Exact and CFD (Fluent) Results 43CHAPTER 8- Summary and Conclusions 46REFERENCES 49APPENDICES 50APPENDIX A – Height Versus Displacement 50APPENDIX B - Time Step calculation 51APPENDIX C - Convergence criteria 52

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LIST OF TABLES

No. Description Page

1- Summarized boundary conditions defined using Gambit. 14

2- Exact solution Results for Case 1 (Pback = 585 KPa) 38

3- Exact solution Results for Case 2 (Pback = 35 KPa). 39

4- Exact solution Results for Case 3 (Pback = 300 KPa). 40

A-1 Height of C-D nozzle as a function of displacement x. 49

B-1 Table B.1- Time step calculation. 50

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LIST OF FIGURESNo. Description Page1- Diagram of a de Laval nozzle, showing approximate flow velocity (v),

together with the effect on temperature (t) and pressure (p) 72- Convergent Divergent Nozzle Configuration 83- Flow regimes in CD nozzle for different Pback/Po 94- The Convergent Divergent nozzle and boundary conditions. 115- Grid 1 used for the subsonic case. 156- Grid 1 used for the supersonic and transonic cases. 157- Mach Contours for Pback=585 kPa (subsonic flow). 22 8- Static pressure Contours for Pback=585 kPa (subsonic flow). 22 9- Static temperature Contours for Pback=585 kPa (subsonic flow). 2310- Density Contours for Pback=585 kPa (subsonic flow). 2311- Stream function Contours for Pback=585 kPa (subsonic flow). 2412- Mach Contours for Pback=35 kPa (supersonic flow). 2413- Static pressure Contours for Pback=35 kPa (supersonic flow) 2514- Static temperature Contours for Pback=35 kPa (supersonic flow). 2515- Density Contours for Pback=35 kPa (supersonic flow) 2616- Stream function Contours for Pback=35 kPa (supersonic flow) 2617- Mach Contours for Pback=300 kPa (transonic flow). 2718- Static pressure Contours for Pback=300 kPa (transonic flow). 271 9- Static temperature Contours for Pback=300 kPa (transonic flow). 2820- Density Contours for Pback=300 kPa (transonic flow). 2821- Stream function Contours for Pback=300 kPa (transonic flow). 2922- Fluent result summery of Mach number for the three cases 2923- Fluent result summery of P/P0 for the three cases 3024- Fluent result summery of T/T0 for the three cases 3025- Fluent result summery of 𝛒/𝛒0 for the three cases 3126- Mach Number Profile – Exact Solutions. 4127- Pressure Profile– Exact Solutions. 4128- Temperature Profile – Exact Solutions. 4229- Density Profile – Exact Solutions. 4230- Mach Number vs. Non-dimensional Nozzle Length. 4431- Density Ratio vs. Non-dimensional Nozzle Length. 4432- Pressure Ratio vs. Nozzle Length. 4533- Temperature vs. Nozzle Length. 45C.1- Convergence monitor (scaled residuals) for case 1 51C.2- Convergence monitor (scaled residuals) for case 2 51C.3- Convergence monitor (scaled residuals) for case 3 52C.4 - Convergence monitor (mass flow rate) for case 1 52C.5 - Convergence monitor (mass flow rate) for case 2 53C.6 - Convergence monitor (mass flow rate) for case 3 53C.7 - Convergence monitor (Heat transfer rate) for case 1 54C.8 - Convergence monitor (Heat transfer rate) for case 2 54C.9 - Convergence monitor (Heat transfer rate) for case 3 55

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ABSTRACT

Flow through Convergent Divergent nozzle is solved using Fluent. The nozzle runs under a total inlet pressure (P0) and temperature (T0) of 600 kPa and 300 K respectively. The nozzle cross-section area (A) is specified as a function of the axial distance (x). The flow through the nozzle is investigated for three different back pressures that will result in subsonic (Pback=585 kPa), supersonic (Pback=35 kPa) and transonic flow (Pback=300 kPa). The grid used for the subsonic case was a coarse grid of 31 nodes in the axial directions and 11 nodes in the normal direction. To capture the thin shock in the transonic case a relatively finer mesh is used (121*41 nodes), also the finer grid was used in the second case with Pback=35KPa . A density based 2nd order implicit, inviscid and unsteady solver is selected in Fluent. Air is assumed as an ideal gas with a constant value of specific heat. A time step of 0.000143 seconds is used based on the CFL (courant) number of 0.5. The convergence criterion of continuity, x and y velocities is set to 10-4, while energy convergence criterion is set to 10-6. The flow field values obtained from fluent are averaged at each axial location over the area and compared with the exact 1D quasi-steady solver. The variation Mach, density, pressure and Temperature along the nozzle axis show good agreement with the exact 1D solution. For the subsonic case the relative difference between CFD results and the 1D results ranges between 5-21%, 1.1-17%, 0.1-4.6%, 0.14-11% for the Mach, P/Po, T/To and ρ/ρo respectively (depending on axial location). Also, for the supersonic case difference ranges between 11-25%, 4.8-25%, 1.4-19.6%, 3.5-15% for the Mach, P/Po, T/To and ρ/ρo respectively (depending on axial location). Finally, for the transonic case (pback=300kPa), the variation of Mach, P/Po, T/To and ρ/ρo along the nozzle axis also has a very good agreement with the exact 1D-quasi steady solution up to the point were the bow shock forms. A bow shock was expected because the real flow in the nozzle is a 2D (nozzle area increases sharply with axial distance). The relative difference between CFD results and the 1D results (up to that point) ranges between 0.13-17%, 0.5-15%, 0.6-14%, 1.3.14.6% for the Mach, P/Po, T/To and ρ/ρo respectively.

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CHAPTER 1INTRODUCTION

The development of high speed digital computers along with the achievement of efficient numerical algorithms has enormously affected Computational Fluid Dynamics (CFD) to be advanced the last decades.

In this project work, the flow through a converging diverging nozzle will be computed through the commercial CFD program (FLUENT) in 3 different flow cases. Results will be compared with well established exact solutions of such cases.

In Chapter two, cd nozzle problem is defined and the formulation of the problem is presented. In chapter Three, Generation of mesh is discussed for all cases of flow. In Chapter Four, FLUENT solver setup will take place. The Fluent results will be presented and discussed in Chapter Five. In Chapter Six, the problem will be solved using the 1-D approach in exact form. Next Chapter, which is Seven, will deal with the comparison between the results and the exact solution. Finally, Summary and conclusion for this work are presented.

1.1 CONVERGENT- DIVERGENT NOZZLE:

Any fluid-mechanical device designed to accelerate a flow is called a nozzle. A de Laval nozzle (or convergent-divergent nozzle, CD nozzle ) is a tube that is pinched in the middle, making an hourglass-shape. It is used as a means of accelerating the flow of a gas passing through it to a supersonic speed. It is widely used in some types of steam turbine and is an essential part of the modern rocket engine and supersonic jet engines.

Its operation relies on the different properties of gases flowing at subsonic and supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is isentropic (gas entropy is nearly constant). At subsonic flow the gas is compressible; sound, a small

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pressure wave, will propagate through it. At the "throat", where the cross sectional area is a minimum, the gas velocity locally becomes sonic (Mach number = 1.0), a condition called choked flow. As the nozzle cross sectional area increases the gas begins to expand and the gas flow increases to supersonic velocities where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0).

Fig.1- Diagram of a de Laval nozzle, showing approximate flow velocity (v), together with the effect on temperature (t) and pressure (p)

The Mach number is a non-dimensional velocityand it is equal to the velocity of the fluid relative to the local speed of sound. 

(1)

The regimes of flow depending on the value of Mach number are:

Subsonic: M < 1 Sonic: Ma=1 Transonic: 0.8 < M < 1.2

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Supersonic: 1.2 < M < 5 Hypersonic: M > 5

1.2 CD Nozzle Characteristics:

The configuration of a converging diverging nozzle (CD) is shown in the figure. Gas flows through the nozzle from a region of high pressure (usually referred to as the chamber) to one of low pressure (referred to as the ambient or tank). The chamber is usually big enough so that any flow velocities here are negligible. The pressure here is denoted by the symbol pc. Gas flows from the chamber into the converging portion of the nozzle, past the throat, through the diverging portion and then exhausts into the ambient as a jet. The pressure of the ambient is referred to as the 'back pressure' and given the symbol Pback.

Fig. 2- Convergent- Divergent Nozzle Configuration

To understand the flow behavior in a CD nozzle let’s assume that the pressure at the exit of the nozzle is reduced than the inlet total pressure. Consequently, the mass flow increases through the nozzle. But if the back pressure is lowered too much then the flow rate suddenly stops increasing all together. This condition is called ‘Choking’. The reason for this behavior has

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to do with the way the flow behaves at Mach 1, i.e. when the flow speed reaches the speed of sound. A good number to keep in mind while conducting experiments is that approximately 50% pressure ratio will result in a chocked nozzle.

The flow regime of a convergent divergent nozzle depends on the pressure ratio across the nozzle. That is, the value of Pback/Po (back pressure to total pressure ratio) will imply wither the flow will be supersonic or subsonic. A common plot that shows different flow configuration based on the pressure ratio is shown in Fig. 3 [3].

Fig. 3- Flow regimes in CD nozzle for different Pback/Po [3].

Analysis of above figure we can summarize the following:

• Case a: When the nozzle isn't choked, the flow in both sections of the nozzle is subsonic.

• Case b: As the back pressure is lowered the flow speed increases everywhere in the nozzle and eventually reaches the sonic speed

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(Mach 1) at the throat, but continues to be subsonic in the divergent part.

• Case c: further reduce in pressure will result in a supersonic flow in the divergent part, but as the back pressure is not low enough the supersonic acceleration is terminated by a normal shock wave after which the flow will be subsonic.

• Case d: further decrease in the back pressure will result in the movement of the normal shock to the nozzle exit.

• Case e: That is an under expanded nozzle.

• Case f: The design point of the nozzle. The back pressure is low enough such that the flow will be fully expanded and supersonic flow is generated.

• Case g: That is an over expanded nozzle.

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Chapter Two

Problem Description and Mathematical Formulation.

2.1- Problem description and boundary conditions.

A quasi 1-D inviscid compressible flow through a converging-diverging nozzle

(CD) is assumed to take place. The area of the nozzle varies according to equation (2).

The nozzle throat is located at x = 1.5 and the convergent section occurs for x <1.5 and

the divergent section occurs for x > 1.5. The nozzle is shown in Fig. 4

(2)

Three cases discussed in this project are:(a) Subsonic flow at the exit, using back pressure of 585kPa.

(b) Supersonic flow at the exit, using back pressure of 35kPa.

(c) Supersonic flow with a shock leading to subsonic flow at the exit,

using back pressure of 300kPa

Fig. 4- The Convergent Divergent nozzle and boundary conditions.

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The Grid is to have 31 point in the x direction (dx=0.1) and 11 points in the y

direction. The problem is to be solved as an unsteady problem with the initial values of ρ,

T and u defined by equations 3, 4 and 5 respectively. The problem is to be solved as a

symmetrical problem (to reduce computation al time). The total pressure and temperature

are known at the inlet are:

(3)

(4)

(5)

The time step (dt) is to be evaluated from the Courant number (C) condition using

equation (6);

(6)

Where: a is the speed of sound and u is the velocity in x direction velocity

2.2 Initial And Inlet Conditions:

The following are the inlet conditions applied to the CD nozzle-

P0= 600kPa

T0= 300K

The above parameters are used for modeling using GAMBIT and solving the

problem using FLUENT software.

Using the above mentioned parameters, the nozzle was modeled using the popularly

used software GAMBIT, and eventually all parameters in the flow field were determined

using the commercially available code, FLUENT.

2.3- Governing Equations.

The continuity equation (7), Navier stokes equations (8, 9) along with the energy equation (10) will be solved using FLUENT codes

(7)

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(8)

(9)

(10)

The governing equations for steady quasi one dimensional flow are:ρ1u1 A1=¿ ρ2u2 A2¿ (11)ρuA=ρ¿u¿ A¿=m (12)

Where, 1 and 2 denote the inlet and the outlet sections of the nozzle and conditions denoted with an * at sonic speed i.e. at M=1.Area Mach number relation:

( AA ¿ )

2= 1M 2 [ 2

γ+1 (1+ γ−12M 2)]

γ +1γ−1

(13) The above equation tells us that the Mach number M is a function of the ratios of the areas (A/A*). i.e. the Mach number at any location is the ratio of the local duct area to the sonic duct area. Other relations used to determine the required values analytically are given below.

pop2

=(1+ γ−12M 2

2)γγ−1

(14)a=√γ RTT 2

T o=( pepo )

γ−1γ

(15)

(16)

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Chapter Three

Solution Mesh

GAMBIT is a software package designed to help analysts and designers build and

mesh models for computational fluid dynamics (CFD) and other scientific applications.

GAMBIT receives user input by means of its graphical user interface (GUI). The

GAMBIT GUI makes the basic steps of building, meshing, and assigning zone types to a

model simple and intuitive, yet it is versatile enough to accommodate a wide range of

modeling applications.

3. 1- Summarized Procedures for Mesh Generation using Gambit.

Procedure for creating the mesh is briefed as follows and is detailed in Appendix A:

In order to create the geometry, the equation

is solved using Excel.

The number of data points used is 61 and the unit cross sectional area at each step

along the nozzle (x-direction) is calculated. Data points are also given in

Appendix A.

Mesh is divided into 30 divisions in the x-axis and 10 divisions in the y-axis.

Boundary conditions are specified as summarized in Table 1.

2D mesh is exported.

Edge Position Name TypeLeft inlet PRESSUR_INLETRight outlet PRESSURE_OUTLETTop wall WALLBottom centerline SYMMETRY

Table 1- Summarized boundary conditions.

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3. 2 Grid Size, Zones and Quality.

Two grids were used in the current research. First, a coarse gird (30*10) for the

simulation of the subsonic flow case 1. Second, a relatively finer grid for the simulation

of the supersonic and transonic case (120*40). Both grids are shown in Fig. 5 and 6

respectively.

Fig. 5- Grid 1 used for the subsonic case.

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Fig. 6- Grid 2 used for the supersonic and transonic case.

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Chapter FourSolver Setup, Strategy and Convergence Criterion

We are required to solve the problem defined in Chapter two for three different

cases. Namely; subsonic, supersonic and flow with a normal shock. The solver setup

will be the same for all three cases, but the flow field initialization will be different. The

reason is that the flow field characteristics are different. The convergence criterion that

will be used is similar for all three cases as we will discuss later.

4.1- Fluent Solver Setup and Solution Strategy.

The setup of the solver is summarized as follows:

Precision: a 2D double precision is used as it permits a higher precision than 2d

single precision solver, but larger memory is required [1].

Solver: Density based solver is recommended as the flow is compressible.

Time: Unsteady formulation. A time step of 0.000143 seconds based on a CFL of

0.5 using the equation 6. A complete calculation of time step is shown in

Appendix B.

Space: 2D problem. The center line of the CD nozzle is taken as symmetric to

reduce computational time.

Solver Formulation: Implicit formulation is used which is more stable than the

explicit formulation. It depends on time step and it is well known that implicit

formulation is less sensitive to the time step selected than Explicit formulation.

Unsteady Formulation: 2nd Order Implicit which is more accurate than 1st order

implicit.

Energy: Energy equation is involved in the computations as the flow is

compressible and temperature will change in a sensitive manner, and

consequently the there will be an effect the density. (Pressure, density and

temperature are related through the ideal gas equation).

Viscous: Inviscid formulation. It is a good approximation when solving CD

nozzles. That means that we are dropping the viscous terms from the Navier

Stoke’s equation .

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Materials: Air is chosen as an ideal gas.

Operating Conditions: Operating pressure in the CD nozzle to be zero as

recommended by the Fluent Manuals [1].

Boundary Conditions: For all three cases the inlet total pressure is set as 600,000

Pascal. The inlet static pressure is 585,000 Pascal. The inlet total temperature is

300 K. However at the exit we define the following for the three different cases:

Case one (Subsonic) Back pressure is 585,000 Pascal.

Case two (Super Sonic) Back pressure is set to 35,000 Pascal. Once Fluent

determines that the flow is Supersonic it will ignore the defined exit back

pressure and will calculate the pressure at the exit from adjacent cells [2].

Case Three (Normal shock) Back pressure is set to 300,000 Pascal.

Solution Initialization: The flow should be initialized in a manner which is

physically correct. For example, a supersonic flow through the CD nozzle will

have the values of pressure, density and temperature continuously decreasing

throughout the nozzle. Then, it is advisable to use an initial profile that is

consistent with the expected profiles to reduce the convergence time steps

required [2]. While in the case of subsonic flow that is not the case and initializing

the flow in the same manner will result in a solution that needs a long period of

time to converge and reach steady solution. Based on that the flow field was

initialized as follows:

Subsonic Case and Transonic case:

The solution was initialized from the inlet conditions. Fluent computes the

inlet velocity and temperature based on the defined total pressure (600

KPa), static pressure (585 KPa) and total temperature (300 K).

Supersonic:

The solution was initialized using the equations defined previously in

chapter 1. However, Fluent doesn’t enable the initialization of the density

in the flow field, but rather enables pressure initialization (using ideal gas

equation). Also, Fluent uses dimensional values and not dimensionless

values. The way we can initialize the flow filed using equations like 1.2,

1.3 and 1.4 is by using the Patch function in Fluent. However the first step

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will be defining the equations using User Defined Function (UDF) in

Fluent. In summary here are the steps to initialize the flow field:

A. We define the following equations using UDF:

(17)

(18)

(19)

B. We use Patch, Zone, and User Defined Function to initialize

temperature, velocity and pressure respectively.

4.2 Convergence Criterion.

As it is usually the case in Fluent three convergence Criterion are used to ensure

convergence of the solution:

4.2.1- Residuals:

Usually Residuals plots are used to monitor how the solution is conversing. Also, a

certain value is assigned for residuals such that once this value is reached that is an

indication that solution converged. The first question is what are the residuals? To

answer that question we choose for example the velocity residuals to talk about.

When we are solving a problem using CFD variables like velocity change changes as

iterations proceed. An indication of convergence is that this change keeps on

decreasing until it becomes insignificant. Therefore, Residuals of velocity is the

average value, though the domain, by which the velocity is changing from one

iteration to the other. The values we used to ensure convergence are 1*10 -4 for

continuity and velocity. We used a value of 1*10-6 for energy as recommended by

Fluent for the density based solver [3]. Convergence monitors for the three cases are

presented in figures listed in Appendixes C.

4.2.2 – Surface variables:

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Usually variables at different surfaces are monitored through the solution iterations to

ensure that they will reach a steady value. Usually that is done to derived values as

forces, but we will use in our case velocities at inlet and exit.

4.2.3 - Flux balance:

It is important to check the mass flow rate balance and energy balance through the

control volume (or solution domain) to ensure convergence. Indeed, if convergence is

reached the mass flow rate in should be equal to the mass flow rate out (steady state is

reached) and same applies to energy.

To sum up, the three different convergence criterion discussed previously will be

used in combination to ensure that we do have a converged solution. As listed in

Appendixes C.

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Chapter Five

Fluent Results

As it was mentioned previously we are solving the CD nozzle for three different

back pressures. Firstly, a back pressure of 585 kPa will result in a subsonic flow through

the nozzle. Secondly, a back pressure of 35 kPa will result in a supersonic flow through

the nozzle with complete expansion. Finally, a back pressure of 300 kPa will result in a

flow with a shock wave. The 30*10 grid was good enough for the subsonic case, but it

was required to have a finer mesh to capture the shock for the transonic case and we use

it also for the second case (supersonic flow) because of the dramatic changes in flow

parameters. The mesh was adapted for that reason and we used a 120*40 grid. However,

we would like to mention that the mesh we are solving for is a coarse mesh and that will

be the main reason that the solution contours are not smooth enough.

5. 1 Subsonic flow through the CD nozzle (Case 1, Pback=585 kPa).

The Mach, static pressure, density and static temperature contours are shown in

Fig. 7, 8, 9 and 10 respectively. As expected, the flow accelerates in the convergent

section until it reaches the maximum velocity at the throat and then decelerates again in

the divergent section. Consequently, pressure decreases until the throat is reached and

then increases again in the divergent section until the exit is reached. Temperature and

density behaves in the same manner as pressure. Both of them are inversely proportional

to the fluid velocity. The total pressure, density, and temperature are almost constant

through the flow for the shock free inviscid flow configuration we are solving. The

Stream function is plotted in Fig. 11. Again, particles path and regions of high and low

velocity are easily observed. Appendix C shows the convergence history of the Fluent

simulation.

5. 2 Supersonic flow through the CD nozzle (Case 2, Pback=35 kPa).

The Mach, static Pressure, Density and static temperature contours are shown in

Fig. 12, 13, 14 and 15 respectively. In the convergent part the flow behaves in the same

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manner as the previous case, but reaches sonic at the throat. Then, through the divergent

the flow continues as a supersonic flow were Mach number and velocity keep increasing

until the flow exits from the CD nozzle. Pressure, temperature and density will be

decreasing in the divergent part because of the increase in velocity, The Stream function

is plotted in Fig. 16. The flow field looks as expected!

5. 3-Transonic flow (shock wave) through the CD nozzle (Case 3,

Pback=300 kPa).

The Mach, static Pressure, Density and static temperature contours are shown in

Fig. 17, 18, 19 and 20 respectively. The shock region is noticed in all figures (region of

sudden change from super sonic to subsonic). The mesh we used to capture the shock

has 120 intervals (dx=0.025 m) in the x axis direction and 40 intervals in the y axis

direction. To produce a better solution for the normal shock case we even need a finer

mesh, but the computational time will increase significantly.

From the figures we notice that the Mach number increase until it reaches the

shock region and then changes to a subsonic flow. We can easily note that pressure,

temperature, velocity and density jumps around the shock region as the flow changes

from subsonic to supersonic.

There are two reasons for why we see a thick bow shock rather than a thin normal

shock in the simulated flow. First, the mesh we used is not fine enough. If the mesh size

was to be reduced less than 1 mm we would have been able to obtain a better thin shock.

However, as the mesh becomes finer the simulation time will increase drastically.

Second, the nozzle we are simulating has a profile that is increasing sharply with the x

axis (area gradient is sharp). From the previous results we notice that the stream lines are

not straight, but rather they do follow the contours of the nozzle. Then, the shock is still

normal to the flow, but not the axis of the nozzle. That is clear from the stream function

plot Fig. 21.

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Contours of Mach Number (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

6.85e-016.55e-01

6.24e-015.94e-01

5.63e-015.32e-015.02e-01

4.71e-014.41e-014.10e-01

3.79e-013.49e-013.18e-01

2.87e-012.57e-012.26e-01

1.96e-011.65e-01

1.34e-011.04e-017.31e-02

Fig. 7- Mach Contours for Pback=585 kPa (subsonic flow).

Contours of Static Pressure (pascal) (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

5.98e+055.90e+05

5.82e+055.73e+05

5.65e+055.56e+055.48e+05

5.40e+055.31e+055.23e+05

5.14e+055.06e+054.97e+05

4.89e+054.81e+054.72e+05

4.64e+054.55e+05

4.47e+054.38e+054.30e+05

Fig. 8- Static pressure Contours for Pback=585 kPa (subsonic flow).

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Contours of Static Temperature (k) (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

3.00e+022.98e+02

2.97e+022.96e+02

2.95e+022.93e+022.92e+02

2.91e+022.90e+022.88e+02

2.87e+022.86e+022.84e+02

2.83e+022.82e+022.81e+02

2.79e+022.78e+02

2.77e+022.75e+022.74e+02

Fig. 9- Static temperature Contours for Pback=585 kPa (subsonic flow).

Contours of Density (kg/m3) (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

7.08e+007.00e+00

6.92e+006.84e+00

6.76e+006.67e+006.59e+00

6.51e+006.43e+006.35e+00

6.27e+006.19e+006.11e+00

6.03e+005.95e+005.87e+00

5.79e+005.71e+00

5.63e+005.55e+005.46e+00

Fig. 10- Density Contours for Pback=585 kPa (subsonic flow).

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Contours of Stream Function (kg/s) (Time=2.8000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

6.08e+02

5.48e+02

4.87e+02

4.26e+02

3.65e+02

3.04e+02

2.43e+02

1.83e+02

1.22e+02

6.08e+01

0.00e+00

Fig. 11- Stream function Contours for Pback=585 kPa (subsonic flow).

Contours of Mach Number (Time=1.7374e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

2.79e+00

2.65e+00

2.52e+00

2.38e+002.25e+00

2.11e+00

1.98e+00

1.84e+001.70e+00

1.57e+00

1.43e+00

1.30e+001.16e+00

1.03e+00

8.92e-01

7.57e-016.22e-01

4.87e-01

3.51e-01

2.16e-018.05e-02

Fig. 12- Mach Contours for Pback=35 kPa (supersonic flow).

26

Contours of Static Pressure (pascal) (Time=1.7374e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

5.98e+05

5.69e+05

5.40e+055.11e+05

4.82e+05

4.52e+05

4.23e+05

3.94e+053.65e+05

3.36e+05

3.07e+05

2.78e+052.49e+05

2.19e+05

1.90e+05

1.61e+051.32e+05

1.03e+05

7.39e+04

4.47e+041.56e+04

Fig. 13- Static pressure contours for Pback=35 kPa (supersonic flow).

Contours of Static Temperature (k) (Time=1.7374e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

3.00e+02

2.91e+02

2.81e+02

2.72e+02

2.63e+02

2.54e+02

2.45e+02

2.36e+02

2.26e+02

2.17e+02

2.08e+02

1.99e+02

1.90e+02

1.81e+02

1.71e+02

1.62e+02

1.53e+02

1.44e+02

1.35e+02

1.26e+02

1.16e+02

Fig. 14- Static temperature contours for Pback=35 kPa (supersonic flow).

27

Contours of Density (kg/m3) (Time=1.7374e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

7.09e+00

6.76e+00

6.43e+00

6.09e+00

5.76e+00

5.43e+00

5.10e+00

4.77e+00

4.44e+00

4.11e+00

3.78e+00

3.44e+00

3.11e+00

2.78e+00

2.45e+00

2.12e+00

1.79e+00

1.46e+00

1.12e+00

7.94e-01

4.62e-01

Fig. 15- Density contours for Pback=35 kPa (supersonic flow).

Contours of Stream Function (kg/s) (Time=1.7374e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 22, 2008

7.04e+02

6.34e+02

5.63e+02

4.93e+02

4.22e+02

3.52e+02

2.82e+02

2.11e+02

1.41e+02

7.04e+01

0.00e+00

Fig. 16- Stream function for Pback=35 kPa (supersonic flow).

28

Contours of Mach Number (Time=3.5000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 24, 2008

2.53e+002.40e+00

2.28e+002.15e+00

2.02e+001.90e+001.77e+00

1.64e+001.52e+001.39e+00

1.27e+001.14e+001.01e+00

8.88e-017.61e-016.35e-01

5.09e-013.83e-01

2.57e-011.31e-014.56e-03

Fig. 17- Mach contours for Pback=300 kPa (transonic flow).

Contours of Static Pressure (pascal) (Time=3.5000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 24, 2008

6.00e+055.72e+05

5.44e+055.16e+05

4.88e+054.60e+054.32e+05

4.04e+053.76e+053.48e+05

3.20e+052.92e+052.64e+05

2.36e+052.08e+051.80e+05

1.52e+051.24e+05

9.57e+046.77e+043.96e+04

Fig. 18- Static pressure contours for Pback=300 kPa (transonic flow).

29

Contours of Static Temperature (k) (Time=3.5000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 24, 2008

3.03e+022.94e+02

2.86e+022.78e+02

2.69e+022.61e+022.52e+02

2.44e+022.35e+022.27e+02

2.19e+022.10e+022.02e+02

1.93e+021.85e+021.76e+02

1.68e+021.59e+02

1.51e+021.43e+021.34e+02

Fig. 19- Static temperature contours for Pback=300 kPa (transonic flow).

Contours of Density (kg/m3) (Time=3.5000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 24, 2008

6.98e+006.68e+00

6.38e+006.08e+00

5.78e+005.49e+005.19e+00

4.89e+004.59e+004.29e+00

3.99e+003.69e+003.39e+00

3.09e+002.79e+002.49e+00

2.20e+001.90e+00

1.60e+001.30e+009.99e-01

Fig. 20- Density contours for Pback=300 kPa (transonic flow).

30

Pathlines Colored by Particle ID (Time=3.5000e-01)FLUENT 6.3 (2d, dp, dbns imp, unsteady)

Jan 24, 2008

3.90e+01

3.51e+01

3.12e+01

2.73e+01

2.34e+01

1.95e+01

1.56e+01

1.17e+01

7.80e+00

3.90e+00

0.00e+00

Fig. 21- Stream function contours for Pback=300 kPa (transonic flow).

Fig. 22- Fluent result summery of Mach number for the three cases

31

Fig. 23- Fluent result summery of P/P0 for the three cases

Fig. 24- Fluent result summery of T/T0 for the three cases

32

Fig. 25- Fluent result summery of 𝛒/𝛒0 for the three cases

33

Chapter Six

PROBLEM EXACT SOLUTION

Current chapter will discuss the exact solution, the calculations and

equations used to obtain it.

6.1- Quasi 1-D Flow: Characteristics and Implications:

Inviscid, compressible flow in a nozzle is the example of quasi-one-

dimensional flow encountered most often. In general, a steady 1-D flow is

one in which the properties of the fluid are functions of x only. Strictly

speaking, a 1-D flow must be inviscid (to avoid the 2-D effects of the

formation of a boundary layer at the walls) and must be a constant-area flow.

This second requirement ensures there are no velocity components along the

y and z directions, and consequently, that the flow properties only change

along the x-axis.

For the case of quasi 1-D flows, the flow area does change. But if this

change is gradual, it can be assumed that the properties are uniform at a

certain flow station x. In quasi 1-D flow, it is assumed that velocity

components in directions other than x can be neglected, and thus will be a

more realistic assumption for a nozzle if the wall slopes are small.

6.2-Governing Equations for Quasi 1-D Flow

An extensive discussion and derivation of the governing equations for

quasi 1-D flow as well as for normal shocks is can be found in the NACA

Report 1135, titled “Equations, Tables, and Charts for Compressible Flow.”

The report includes equations for numerous parameters as functions of Mach

34

number for isentropic flow at constant ratio of specific heats. Since our

problem setup defines our flow area, the most important equation required is

one that relates the area ratio to the local Mach number, which is the

following:

A∗¿A

=( γ+12 )

γ+12( γ−1)M (1+ γ−1

2M 2)−

γ+12( γ−1 ) ¿

(17)

Where A* denotes the throat area assuming sonic flow. Thus, this is a

expression of the ratio between the sonic throat area and the current location

as a function of the Mach number. Note that there is both a subsonic and

supersonic Mach number that matches any given area ratio. Also note that if

the flow is not sonic at the throat, the throat area is not equivalent to A*.

Regarding equation 17, it is important to note that our geometry provides

us with the area ratio, and that we need to find the Mach number

corresponding to it. An inspection of equation 17 reveals the problems in

solving for Mach number as a function of area ratio.

Once the Mach number is found at a certain location, additional

relationships help calculate the static to stagnation pressure, temperature,

and density ratios:

PP0

=(1+ γ−12M 2)

−γγ−1

(18)

TT0

=(1+ γ−12M2)

−1

(19)

35

ρρ0

=(1+ γ−12M 2)

−1γ−1

(20)

For the case with the normal shock within the nozzle, we need additional

equations. The first equation relates the flow Mach number after the shock to

the flow Mach number ahead of the shock:

M 22=

( γ−1 )M 12+2

2 γM 12−( γ−1 ) (21)

The static pressure ratio across the shock is given by the following relationship:

P2

P1=1+ 2 γ

γ+1 (M 12−1)

(22)

Once these two values are know, the new stagnation pressure can be

computed from equation 18 above. The rest of the parameters can be

computed via equations 18-22 and with the ideal gas law:

P= ρRT (23)

Finally, as verification, we calculate mass flow as follows:

m=ρVA , where V=Ma=M √γ RT (24)

6.3-Case 1: Subsonic Flow throughout Nozzle:

The first step for this case is to verify if the flow is actually subsonic. To do so, we

first assume supersonic flow throughout, when the flow reaches sonic speed and then

decelerates back to subsonic (which corresponds to the point with a sonic throat speed

and the highest possible back pressure). Using equation 17 and the area ratio at the exit

(2.65/1), we iterate to a subsonic Mach number of 0.225. At this Mach number, the

36

pressure ratio P/P0 = 0.963, resulting in an exit pressure of 579.2 kPa, which is lower than

Pback. Thus, the flow is subsonic.

Now that we verified the flow is subsonic, we used the following procedure:

1. Assume nozzle exit pressure is 585 kPa. Using equation 18 for P/P0 we solve

for Mach number at the exit and obtain Me = 0.191.

2. Using Me = 0.191, we calculate A*/A at exit with equation 17. Multiplying by

the exit area (3.2 m2) gives us the theoretical sonic throat area of 0.854 m2

which understandably is smaller than our current throat area.

3. We now calculate Mach at each point from new A/A* value, using the

calculated A* and the actual area at every station x, using Excel’s solver to

iterate on Mach number for equation 17 until A/A* is matched.

4. Once we have Mach number for all stations, we can use equations 18, 19, and

20 to obtain the pressure, temperature, and velocity ratios throughout the

nozzle.

6.4-Case 2: Supersonic Flow throughout Nozzle:

We expect the flow for this case to be supersonic, as specified in the problem

statement. However, we verify this assumption before proceeding. Again using equation

17 and the area ratio at the exit (2.65/1), we iterate to a supersonic Mach number of

2.505. Using equation 18, P/P0 = 0.058, resulting in an exit pressure of 34.8 kPa, which is

slightly lower than our back pressure. However, we assume that this is just a round-off

issue when specifying the exit conditions. Even if this were not the case, the solution

would still correspond to supersonic flow with a slight over-expansion. Now that we have

verified subsonic flow, we proceed as follows:

1. Use equation 17 and the calculated area ratios, with A* = 1 m2, to iterate for

Mach number at all points using Excel’s solver.

2. Use equations 18, 19, and 20 to obtain the pressure, temperature, and velocity

ratios throughout the nozzle.

37

6.5- Case 3: Normal Shock in the Diverging Section of the Nozzle:

As shown in Fig. 26 through Fig. 29, case 3 should be identical to the supersonic case

until at some point in the diverging section, a normal shock appears, converting the flow

back into subsonic and making the rest of the diverging section behave as a diffuser,

further creating a rise of static pressure. We thus need to find the location of the shock

such that the static pressure behind it is slightly less than our exit pressure of 300 kPa.

This is an iterative process, changing the shock location until matching exit pressure. The

procedure is as follows:

1. Using normal shock tables, which essentially are a tabulation of equations 21

and 22, among others, and using case 2 results to obtain Mach number in the

diverging section, we make an initial guess of M = 2.3, which gives an static

pressure after the shock of about 6 times the pressure from before the shock.

As seen in the results below, this Mach number occurs somewhere between

x = 2.7 and x = 2.8 meters. This would result in the static pressure after the

shock to be at around 300 kPa.

2. We now choose a guess value for the shock location in the vicinity of x = 2.8.

We calculate the area at this location, and using equation 17, we obtain the

supersonic Mach number, which will be the Mach number ahead of the shock.

3. With equation 18, we determine static pressure ahead of the shock.

4. With equations 21 and 22, we calculate the Mach number and static pressure

after the shock.

5. With the post-shock Mach number and static pressure, we can use equation 18

to calculate the total pressure after the shock, which will be lower than the

initial stagnation pressure due to shock losses.

38

6. Because stagnation temperature does not change across a shock, we can use

equation 19 with the post shock Mach number to calculate static temperature

after the shock.

7. Using the ideal gas law (equation 24), we can obtain static density from static

pressure and temperature, and using this density in conjunction with equation

23 to calculate stagnation density.

8. We now are in flow conditions reminiscent of the subsonic case. We need to

use the post-shock Mach number and the area at the shock location to

compute a new A* value, which will be used to compute area ratio at each

location downstream of the shock.

9. Equation 22 to find the new Mach number at all locations after the shock. This

Mach number should be decreasing.

10. Equations 17, 18, and 19 with the new stagnation conditions to calculate static

pressure, temperature and density.

11. Compare exit pressure to the specified back pressure of 300 kPa. If it does not

match, select a new value for shock location and repeat steps 2 through 10.

After iterating, we find the location of the shock to be x = 2.8 m, with a Mach

number of 2.33 before the shock and 0.52 after the shock. Static pressure after the shock

rises to 281 kPa and continues to rise throughout the rest of the divergent section until

reaching a value of 300 kPa,. Stagnation pressure after the shock decreases to 341 kPa,

Stagnation density decreases from 6.97 kg/m3 to 3.96 kg/m3.

6.6-Summary of Exact Solutions

The exact solution results are tabulated in Table 2, 3, and 4 while Fig. 26, 27, 28,

and 29 below shows the Mach number, Pressure, Temperature and Density respectively :

39

40

x M P/Po T/To r/ro0 0.1905 0.9750 0.9928 0.9821

0.1 0.2080 0.9703 0.9914 0.97870.2 0.2275 0.9646 0.9898 0.97460.3 0.2494 0.9577 0.9877 0.96960.4 0.2737 0.9493 0.9852 0.96350.5 0.3008 0.9392 0.9822 0.95620.6 0.3307 0.9271 0.9786 0.94730.7 0.3636 0.9127 0.9742 0.93680.8 0.3993 0.8959 0.9691 0.92450.9 0.4375 0.8768 0.9631 0.9104

1 0.4772 0.8557 0.9564 0.89461.1 0.5169 0.8334 0.9493 0.87801.2 0.5542 0.8117 0.9421 0.86151.3 0.5857 0.7928 0.9358 0.84711.4 0.6071 0.7796 0.9313 0.83711.5 0.6148 0.7749 0.9297 0.83351.6 0.6071 0.7796 0.9313 0.83711.7 0.5857 0.7928 0.9358 0.84711.8 0.5542 0.8117 0.9421 0.86151.9 0.5169 0.8334 0.9493 0.8780

2 0.4772 0.8557 0.9564 0.89462.1 0.4375 0.8768 0.9631 0.91042.2 0.3993 0.8959 0.9691 0.92452.3 0.3636 0.9127 0.9742 0.93682.4 0.3307 0.9271 0.9786 0.94732.5 0.3008 0.9392 0.9822 0.95622.6 0.2737 0.9493 0.9852 0.96352.7 0.2494 0.9577 0.9877 0.96962.8 0.2275 0.9646 0.9898 0.97462.9 0.2080 0.9703 0.9914 0.9787

3 0.1905 0.9750 0.9928 0.9821

Subsonic Case: Pback = 585 kPa

Table 2- Exact solution Results for Case 1 (Pback = 585 KPa).

41

x M P/Po T/To r/ro0 0.225085 0.96533 0.989969 0.975111

0.1 0.246171 0.958711 0.988025 0.9703310.2 0.269886 0.950641 0.985641 0.9644890.3 0.296589 0.940785 0.982711 0.9573370.4 0.326681 0.928748 0.979102 0.9485710.5 0.360599 0.91406 0.974653 0.9378320.6 0.398811 0.896192 0.969171 0.9246990.7 0.441805 0.87456 0.962428 0.9087010.8 0.490067 0.84857 0.954168 0.8893290.9 0.544054 0.817671 0.94411 0.866076

1 0.604157 0.781446 0.931965 0.8384931.1 0.670658 0.739721 0.917468 0.8062641.2 0.743681 0.692678 0.900404 0.7692971.3 0.823151 0.64095 0.880657 0.7278091.4 0.908769 0.585645 0.858242 0.6823781.5 1 0.528282 0.833333 0.6339381.6 1.096103 0.470621 0.806264 0.5837051.7 1.196195 0.414427 0.777498 0.5330261.8 1.299278 0.361268 0.747594 0.4832421.9 1.40436 0.312316 0.717131 0.435508

2 1.510497 0.268288 0.686662 0.3907132.1 1.61684 0.229465 0.65667 0.3494362.2 1.722673 0.195771 0.627541 0.3119652.3 1.827414 0.166884 0.59956 0.2783442.4 1.930621 0.142336 0.572915 0.2484422.5 2.031963 0.1216 0.547713 0.2220152.6 2.131217 0.104146 0.523994 0.1987542.7 2.22824 0.089478 0.501753 0.1783312.8 2.322954 0.077152 0.480949 0.1604162.9 2.415331 0.066782 0.461518 0.144701

3 2.505382 0.05804 0.443383 0.130902

Supersonic Case: Pback = 35 kPa

Table 3- Results for Case 2 (Pback = 35 KPa).

42

x M P/Po T/To r/ro0 0.225085 0.96533 0.989969 0.975111

0.1 0.246171 0.958711 0.988025 0.9703310.2 0.269886 0.950641 0.985641 0.9644890.3 0.296589 0.940785 0.982711 0.9573370.4 0.326681 0.928748 0.979102 0.9485710.5 0.360599 0.91406 0.974653 0.9378320.6 0.398811 0.896192 0.969171 0.9246990.7 0.441805 0.87456 0.962428 0.9087010.8 0.490067 0.84857 0.954168 0.8893290.9 0.544054 0.817671 0.94411 0.866076

1 0.604157 0.781446 0.931965 0.8384931.1 0.670658 0.739721 0.917468 0.8062641.2 0.743681 0.692678 0.900404 0.7692971.3 0.823151 0.64095 0.880657 0.7278091.4 0.908769 0.585645 0.858242 0.6823781.5 1 0.528282 0.833333 0.6339381.6 1.096103 0.470621 0.806264 0.5837051.7 1.196195 0.414427 0.777498 0.5330261.8 1.299278 0.361268 0.747594 0.4832421.9 1.40436 0.312316 0.717131 0.435508

2 1.510497 0.268288 0.686662 0.3907132.1 1.61684 0.229465 0.65667 0.3494362.2 1.722673 0.195771 0.627541 0.3119652.3 1.827414 0.166884 0.59956 0.2783442.4 1.930621 0.142336 0.572915 0.2484422.5 2.031963 0.1216 0.547713 0.2220152.6 2.131217 0.104146 0.523994 0.1987542.7 2.22824 0.089478 0.501753 0.1783312.8 2.322954 0.077152 0.480949 0.160416

2.812 0.530411 0.825642 0.94673 0.8720982.9 0.477582 0.855452 0.956373 0.894476

3 0.428044 0.881649 0.964651 0.913956

Normal Shock Case: Pback = 300 kPa

Table 4- Results for Case 3 (Pback = 300 KPa).

43

Fig. 26- Mach Number Profile – Exact Solutions.

Fig. 27- Pressure Profile– Exact Solutions.

44

Fig. 28- Temperature Profile – Exact Solutions.

Fig. 29- Density Profile – Exact Solutions.

45

Chapter Seven

Comparison between Exact and CFD (Fluent) Solution.

In this chapter, we will compare the results obtained from the theoretical

calculation with the results obtained from the Fluent computations. For comparison

purposes, we will not only compare the mach numbers and density ratio but also the

pressure and temperature ratios. Fluent results at each cross section were averaged in

order to compare with the exact solution. This will inevitably induce errors, since we

have already seen that the flow has strong 2D characteristics. We will see that for the

subsonic and supersonic cases, the results show fairly good agreement with the exact

solution. However, for the transonic case, the results are markedly different around the

shock region. This was expected because, as we pointed out earlier (Chapter 5), the

nozzle area changes sharply along the axis and the flow is 2D rather than 1D, as shown

clearly by the streamlines. This results in a markedly curved shock, which is in strong

disagreement with the quasi-1D assumption. The flow field values obtained from fluent

are averaged at each axial location over the area and compared with the exact 1D quasi-

steady solver.

Analysis of Fig. 30 through 33 show that the variation Mach, density, pressure and

Temperature along the nozzle axis show good agreement with the exact 1D solution. For

the subsonic case the relative difference between CFD results and the 1D results ranges

between 5-21%, 1.1-17%, 0.1-4.6%, 0.14-11% for the Mach, P/Po, T/To and ρ/ρo

respectively (depending on axial location). Also, for the supersonic case difference

ranges between 11-25%, 4.8-25%, 1.4-19.6%, 3.5-15% for the Mach, P/Po, T/To and

ρ/ρo respectively (depending on axial location). Finally, for the transonic case

(pback=300kPa), the variation of Mach, P/Po, T/To and ρ/ρo along the nozzle axis also has

a very good agreement with the exact 1D-quasi steady solution up to the point were the

bow shock forms. A bow shock was expected because the real flow in the nozzle is a 2D

(nozzle area increases sharply with axial distance). The relative difference between CFD

results and the 1D results (up to that point) ranges between 0.13-17%, 0.5-15%, 0.6-14%,

1.3.14.6% for the Mach, P/Po, T/To and ρ/ρo respectively.

46

Fig. 30- Mach Number vs. Non-dimensional Nozzle Length.

Fig. 31- Density Ratio vs. Non-dimensional Nozzle Length.

47

Fig. 32- Pressure Ratio vs. Nozzle Length.

Fig. 33- Temperature vs. Nozzle Length.

48

Chapter Eight

Summary and Conclusions.

The flow through a CD nozzle was simulated using Fluent code for three different

back pressures. There types of flow took place for the different three back pressures

which are subsonic, supersonic and transonic flows. Results were compared to the exact

quasi-1D steady solution. The area of the nozzle was defined to be a function of the axial

location. Total pressure and temperature at the inlet of the nozzle were 600 kPa and 300

K respectively.

For case one, subsonic flow, a coarse grid (generated using Gambit) of 31 nodes in the

axial direction and 11 nodes in the normal direction was used. The computational time

was around 1 hour using an AMD Turion 64 X2 processor. For the Transonic case and

supersonic case finer mesh was used. The mesh has 121 nodes in the axial direction and

41 nodes in the normal direction. The computational time was around 36 hours on the

same machine.

A time step of 0.000143 seconds was used based on the Courant number value of 0.5. A

unsteady, density based, 2nd order implicit and inviscid solver was used. The air was

treated as an ideal gas and the specific heat was assumed to be constant. CFD solution

convergence was set to 10-4, for continuity, x and y velocity, but 10-6 for the energy

equation. Also, the mass balance and energy balance were checked at the end of the

simulation to ensure convergence for obtained results.

In case three, transonic flow, the shock will appear in the divergent part of the nozzle.

The observed shock is curved because it is normal to the pathlines or stream lines that

follow the nozzle contour. Thus, in the general area of the shock, there are locations

along the nozzle in which data is averaged from both ahead of the shock and behind the

shock, which explains the strong variations in that area when compared to the quasi-1D

solution.

49

For the subsonic case (pback=585 KPa), the variation of Mach number, r/ro, p/po and T/To

along the nozzle axis has very good agreement with the exact-quasi 1D steady solution.

The relative difference between CFD results and the 1D results ranges between 5-21%,

1.1-17%, 0.1-4.6%, 0.14-11% for the Mach, , p/po, T/To and r/ro respectively (depending

on axial location).

For the supersonic case (pback=35kPa), the variation of Mach number, r/ro, p/po and T/To

along the nozzle axis also has a very good agreement with the the exact-quasi 1D steady

solution. The relative difference between Fluent results and the 1D results ranges

between 11-25%, 4.8-25%, 1.4-19.6%, 3.5-15% for the Mach, P/Po, T/To and ρ/ρo

respectively (depending on axial location).

For the transonic case (pback=300kPa), the variation of Mach number, r/ro, p/po and T/To

along the nozzle axis also has a very good agreement with the the exact-quasi 1D steady

solution up to the point were the bow shock forms. The relative difference between CFD

results and the 1D results (up to that point) ranges between 0.13-17%, 0.5-15%, 0.6-14%,

1.3.14.6% for the Mach, P/Po, T/To and ρ/ρo respectively (depending on axial location).

50

REFERENCES

1- John D. Anderson, “Computational Fluid Dynamics: The Basics with Applications”, McGraw-Hill, 1995.

2- http://en.wikipedia.org/wiki/De_Laval_nozzle.3- http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html.4- Fluent 6.3.26 Documentation/ Fluent Inc.5- http://en.wikipedia.org/wiki/Mach_(speed)6- Fluent.Inc/help/html/ug/node374.htm7- Fluent.Inc/help/html/ug/node403.htm8- Fluent.Inc/help/html/ug/node1047.htm9- http://courses.cit.cornell.edu/fluent/index.htm

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Appendix A

Height versus displacement for C-D nozzle

x H x h x h0 2.284523 1 1.428174 2 1.428174

0.05 2.241706 1.05 1.385357 2.05 1.4709920.1 2.198888 1.1 1.34254 2.1 1.513809

0.15 2.156071 1.15 1.299722 2.15 1.5566270.2 2.113253 1.2 1.256905 2.2 1.599444

0.25 2.070436 1.25 1.214087 2.25 1.6422620.3 2.027619 1.3 1.17127 2.3 1.685079

0.35 1.984801 1.35 1.128452 2.35 1.7278970.4 1.941984 1.4 1.085635 2.4 1.770714

0.45 1.899166 1.45 1.042817 2.45 1.8135310.5 1.856349 1.5 1 2.5 1.856349

0.55 1.813531 1.55 1.042817 2.55 1.8991660.6 1.770714 1.6 1.085635 2.6 1.941984

0.65 1.727897 1.65 1.128452 2.65 1.9848010.7 1.685079 1.7 1.17127 2.7 2.027619

0.75 1.642262 1.75 1.214087 2.75 2.0704360.8 1.599444 1.8 1.256905 2.8 2.113253

0.85 1.556627 1.85 1.299722 2.85 2.1560710.9 1.513809 1.9 1.34254 2.9 2.198888

0.95 1.470992 1.95 1.385357 2.95 2.2417063 2.284523

Table A.1- Height of C-D nozzle as a function of displacement x.

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Appendix B

Calculation of Time Stepx height Area rho T V Dt

0 1.325 2.65 1 1 1.732051 0.0001430.1 1.2187 2.437333 0.96854 0.97686 3.577858 0.0001440.2 1.1197 2.239333 0.93708 0.95372 5.378958 0.0001450.3 1.028 2.056 0.90562 0.93058 7.13453 0.0001460.4 0.9437 1.887333 0.87416 0.90744 8.843708 0.0001470.5 0.8667 1.733333 0.8427 0.8843 10.50558 0.0001480.6 0.797 1.594 0.81124 0.86116 12.1192 0.000150.7 0.7347 1.469333 0.77978 0.83802 13.68354 0.0001510.8 0.6797 1.359333 0.74832 0.81488 15.19755 0.0001520.9 0.632 1.264 0.71686 0.79174 16.6601 0.0001541 0.5917 1.183333 0.6854 0.7686 18.06998 0.000155

1.1 0.5587 1.117333 0.65394 0.74546 19.42594 0.0001571.2 0.533 1.066 0.62248 0.72232 20.7266 0.0001581.3 0.5147 1.029333 0.59102 0.69918 21.97054 0.000161.4 0.5037 1.007333 0.55956 0.67604 23.1562 0.0001621.5 0.5 1 0.5281 0.6529 24.28194 0.0001641.6 0.5037 1.007333 0.49664 0.62976 25.34598 0.0001661.7 0.5147 1.029333 0.46518 0.60662 26.3464 0.0001681.8 0.533 1.066 0.43372 0.58348 27.28112 0.0001711.9 0.5587 1.117333 0.40226 0.56034 28.14792 0.0001742 0.5917 1.183333 0.3708 0.5372 28.94433 0.000176

2.1 0.632 1.264 0.33934 0.51406 29.66769 0.0001792.2 0.6797 1.359333 0.30788 0.49092 30.31506 0.0001832.3 0.7347 1.469333 0.27642 0.46778 30.88321 0.0001862.4 0.797 1.594 0.24496 0.44464 31.36857 0.000192.5 0.8667 1.733333 0.2135 0.4215 31.76712 0.0001942.6 0.9437 1.887333 0.18204 0.39836 32.0744 0.0001992.7 1.028 2.056 0.15058 0.37522 32.28536 0.0002042.8 1.1197 2.239333 0.11912 0.35208 32.39422 0.000212.9 1.2187 2.437333 0.08766 0.32894 32.39439 0.0002163 1.325 2.65 0.0562 0.3058 32.2782 0.000223

Table B.1- Time step calculation.

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Appendix CConvergence Criteria

Fig C.1 Convergence monitor (scaled residuals) for case 1

Fig. C.2 - Convergence monitor (scaled residuals) for case 2

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Fig. C.3 - Convergence monitor (scaled residuals) for case 3

Fig C.4 - Convergence monitor (mass flow rate) for case 1

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Fig. C.5- Convergence monitor (mass flow rate) for case 2

Fig. C.6 - Convergence monitor (mass flow rate) for case 3

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Fig. C.7 - Convergence monitor (Heat transfer rate) for case 1

Fig. C.8 - Convergence monitor (Heat transfer rate) for case 2

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Fig. C.9 - Convergence monitor (Heat transfer rate) for case 3

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1

2

3