EGFD 637 – Computational Fluid Dynamics
Project 2
Fluent Simulation for the flow through
a Convergent-Divergent Nozzle with
Subsonic, Supersonic and
Transonic Flow
By
Salah Soliman
Sowmya Krishnamurthy
Santosh Konangi
Bhaskar Chandra Konala
Mohamed Abdelaal
In
Aerospace Engineering
Under Supervision of
Dr. Kirti Ghia
College of Engineering
University of Cincinnati
A research project submitted to the University
of Cincinnati as the 1st project for the
Computational Fluid Dynamics Course
University of Cincinnati
Cincinnati, Ohio, USA
Winter 2009
1
Table of Contents
Subject
PAGE
2
Table of Contents
ABSTRACT
CHAPTER 1 INTRODUCTION
5
6
CHAPTER 2- Problem Description and Mathematical Formulation
2.1 Problem Description and Boundary conditions
2.2 Initial And Inlet Conditions
2.3 Governing Equations
CHAPTER 3- Solution Mesh (Generated using Gambit)
3.1 Summarized Procedures for Mesh Generation Using Gambit
3.2 Grid Size, Zones and Quality
CHAPTER 4- Fluent Solver Setup, Solution Strategy and Conversion
Criterion.
4.1 Fluent Solver Setup and Solution Strategy
4.2 Convergence Criterion
CHAPTER 5 -Fluent Results
5.1 Subsonic Flow through the CD nozzle (Case 1, Pback=585 kPa)
5.2 Supersonic Flow through the CD nozzle (Case 2, Pback=35 kPa)
5.3 Transonic Flow (shock wave) through the CD nozzle (Case 3,Pback=300 kPa)
CHAPTER 6- Problem Exact Solution
6.1 Quasi 1D-Flow: Characteristics and Implications
6.2 Governing Equations for Quasi-1D Flow
6.3 Case 1: Subsonic Flow throughout the Nozzle (Pback=585 kPa)
6.4 Case 2: Supersonic Flow throughout the Nozzle (Pback=35 kPa)
6.5 Case 3: Normal Shock in the Diverging Section of the Nozzle
(Pback=300 Kpa)
6.6 Summary of Exact Solution
CHAPTER 7 -Comparison between Exact and CFD (Fluent) Results
CHAPTER 8- Summary and Conclusions
REFERENCES
APPENDICES
APPENDIX A – Height Versus Displacement
APPENDIX B - Time Step calculation
APPENDIX C - Convergence criteria
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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
3
LIST OF FIGURES
No.
Description
1- Diagram of a de Laval nozzle, showing approximate flow velocity (v),
together with the effect on temperature (t) and pressure (p)
2- Convergent Divergent Nozzle Configuration
3- Flow regimes in CD nozzle for different Pback/Po
4- The Convergent Divergent nozzle and boundary conditions.
5- Grid 1 used for the subsonic case.
6- Grid 1 used for the supersonic and transonic cases.
7- Mach Contours for Pback=585 kPa (subsonic flow).
8- Static pressure Contours for Pback=585 kPa (subsonic flow).
9- Static temperature Contours for Pback=585 kPa (subsonic flow).
10- Density Contours for Pback=585 kPa (subsonic flow).
11- Stream function Contours for Pback=585 kPa (subsonic flow).
12- Mach Contours for Pback=35 kPa (supersonic flow).
13- Static pressure Contours for Pback=35 kPa (supersonic flow)
14- Static temperature Contours for Pback=35 kPa (supersonic flow).
15- Density Contours for Pback=35 kPa (supersonic flow)
16- Stream function Contours for Pback=35 kPa (supersonic flow)
17- Mach Contours for Pback=300 kPa (transonic flow).
18- Static pressure Contours for Pback=300 kPa (transonic flow).
1 9- Static temperature Contours for Pback=300 kPa (transonic flow).
20- Density Contours for Pback=300 kPa (transonic flow).
21- Stream function Contours for Pback=300 kPa (transonic flow).
22- Fluent result summery of Mach number for the three cases
23- Fluent result summery of P/P0 for the three cases
24- Fluent result summery of T/T0 for the three cases
25- Fluent result summery of 𝛒/𝛒0 for the three cases
26- Mach Number Profile – Exact Solutions.
27- Pressure Profile– Exact Solutions.
28- Temperature Profile – Exact Solutions.
29- Density Profile – Exact Solutions.
30- Mach Number vs. Non-dimensional Nozzle Length.
31- Density Ratio vs. Non-dimensional Nozzle Length.
32- Pressure Ratio vs. Nozzle Length.
33- Temperature vs. Nozzle Length.
C.1- Convergence monitor (scaled residuals) for case 1
C.2- Convergence monitor (scaled residuals) for case 2
C.3- Convergence monitor (scaled residuals) for case 3
C.4 - Convergence monitor (mass flow rate) for case 1
C.5 - Convergence monitor (mass flow rate) for case 2
C.6 - Convergence monitor (mass flow rate) for case 3
C.7 - Convergence monitor (Heat transfer rate) for case 1
C.8 - Convergence monitor (Heat transfer rate) for case 2
C.9 - Convergence monitor (Heat transfer rate) for case 3
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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%, 0.1411% 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.
5
CHAPTER 1
INTRODUCTION
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
6
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
7


1.2
Supersonic: 1.2 < M < 5
Hypersonic: M > 5
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
8
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
9
(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.
10
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
A x  1
2.2
2
 x  1.5
3
0x3
(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.
11
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:
initial  1  0.3146x
(3)
Tinitial  1  0.2314x
(4)
1
u initial   0.1  1.09x  T 2
(5)
The time step (dt) is to be evaluated from the Courant number (C) condition using
equation (6);
dt  C
dx
au
(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 nozzleP0= 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
   u    v 


0
t
x
y
(7)
12
 u
u
u 
p
  u  v   
x
y 
x
 t
(8)
 v
v
v 
p
  u  v   
x
y 
y
 t
(9)
 T
T
T    T    T 
Cp 
u
 v   K
 K 
x
y  x  x  y  y 
 t
(10)
The governing equations for steady quasi one dimensional flow are:
𝜌1 𝑢1 𝐴1= 𝜌2 𝑢2 𝐴2
(11)
∗ ∗ ∗
𝜌𝑢𝐴 = 𝜌 𝑢 𝐴 = 𝑚̇
(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:
 1
1  2    1 2   1
 A
M 
 *  2 
1 
2
M   1 
A 

2
(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.

po    1 2   1
 1 
M2 
p2 
2

(14)
a  RT
T2  p e 
 
To  p o 
 1

(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
A x   1
to
create
2.2
2
 x  1.5
3
the
geometry,
the
equation
0  x  3 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
Left
Right
Top
Bottom
Name
inlet
outlet
wall
centerline
Type
PRESSUR_INLET
PRESSURE_OUTLET
WALL
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.
Fig. 6- Grid 2 used for the supersonic and transonic case.
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Chapter Four
Solver 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).
16

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 .

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
17
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
will be defining the equations using User Defined Function (UDF) in
Fluent. In summary here are the steps to initialize the flow field:
A.
B.
We define the following equations using UDF:
Tinitial  300* 1 0.2314x 
(17)
u initial   0.1 1.09x  T
(18)
Pinitial  287*T*6.5* 1  0.3146x 
(19)
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.
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4.2.2 – Surface variables:
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.
19
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
20
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.
21
6.85e-01
6.55e-01
6.24e-01
5.94e-01
5.63e-01
5.32e-01
5.02e-01
4.71e-01
4.41e-01
4.10e-01
3.79e-01
3.49e-01
3.18e-01
2.87e-01
2.57e-01
2.26e-01
1.96e-01
1.65e-01
1.34e-01
1.04e-01
7.31e-02
Fig. 7- Mach Contours for Pback=585 kPa (subsonic flow).
Contours of Mach Number (Time=2.8000e-01)
5.98e+05
FLUENT 6.3
5.90e+05
5.82e+05
5.73e+05
5.65e+05
5.56e+05
5.48e+05
5.40e+05
5.31e+05
5.23e+05
5.14e+05
5.06e+05
4.97e+05
4.89e+05
4.81e+05
4.72e+05
4.64e+05
4.55e+05
4.47e+05
4.38e+05
4.30e+05
Fig. 8- Static pressure Contours for Pback=585 kPa (subsonic flow).
Contours of Static Pressure (pascal) (Time=2.8000e-01)
FLUENT 6.3 (2
22
3.00e+02
2.98e+02
2.97e+02
2.96e+02
2.95e+02
2.93e+02
2.92e+02
2.91e+02
2.90e+02
2.88e+02
2.87e+02
2.86e+02
2.84e+02
2.83e+02
2.82e+02
2.81e+02
2.79e+02
2.78e+02
2.77e+02
2.75e+02
2.74e+02
Fig. 9- Static temperature Contours for Pback=585 kPa (subsonic flow).
Contours of Static Temperature (k) (Time=2.8000e-01)
FLUENT 6
7.08e+00
7.00e+00
6.92e+00
6.84e+00
6.76e+00
6.67e+00
6.59e+00
6.51e+00
6.43e+00
6.35e+00
6.27e+00
6.19e+00
6.11e+00
6.03e+00
5.95e+00
5.87e+00
5.79e+00
5.71e+00
5.63e+00
5.55e+00
5.46e+00
Fig. 10- Density Contours for Pback=585 kPa (subsonic flow).
Contours of Density (kg/m3) (Time=2.8000e-01)
23
FLUENT 6.
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
Contours of Stream Function (kg/s) (Time=2.8000e-01)
kPa (subsonic flow).
FLUENT 6.3 (2d, dp, d
2.79e+00
2.65e+00
2.52e+00
2.38e+00
2.25e+00
2.11e+00
1.98e+00
1.84e+00
1.70e+00
1.57e+00
1.43e+00
1.30e+00
1.16e+00
1.03e+00
8.92e-01
7.57e-01
6.22e-01
4.87e-01
3.51e-01
2.16e-01
8.05e-02
Fig. 12- Mach Contours for Pback=35 kPa (supersonic flow).
Contours of Mach Number (Time=1.7374e-01)
24
FLUENT 6.3 (2d
5.98e+05
5.69e+05
5.40e+05
5.11e+05
4.82e+05
4.52e+05
4.23e+05
3.94e+05
3.65e+05
3.36e+05
3.07e+05
2.78e+05
2.49e+05
2.19e+05
1.90e+05
1.61e+05
1.32e+05
1.03e+05
7.39e+04
4.47e+04
1.56e+04
Fig. 13- Static pressure contours for Pback=35 kPa (supersonic flow).
Contours of Static Pressure (pascal) (Time=1.7374e-01)
FLUENT 6.3 (2d, d
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).
Contours of Static Temperature (k) (Time=1.7374e-01)
FLUENT 6.3 (2d, dp
25
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 Density (kg/m3) (Time=1.7374e-01)
FLUENT 6.3 (2d,
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).
Contours of Stream Function (kg/s) (Time=1.7374e-01)
FLUENT 6.3 (2d,
26
2.53e+00
2.40e+00
2.28e+00
2.15e+00
2.02e+00
1.90e+00
1.77e+00
1.64e+00
1.52e+00
1.39e+00
1.27e+00
1.14e+00
1.01e+00
8.88e-01
7.61e-01
6.35e-01
5.09e-01
3.83e-01
2.57e-01
1.31e-01
4.56e-03
Fig. 17- Mach contours for Pback=300 kPa (transonic flow).
Contours of Mach Number (Time=3.5000e-01)
6.00e+05
FLUE
5.72e+05
5.44e+05
5.16e+05
4.88e+05
4.60e+05
4.32e+05
4.04e+05
3.76e+05
3.48e+05
3.20e+05
2.92e+05
2.64e+05
2.36e+05
2.08e+05
1.80e+05
1.52e+05
1.24e+05
9.57e+04
6.77e+04
3.96e+04
Fig. 18- Static pressure contours for Pback=300 kPa (transonic flow).
Contours of Static Pressure (pascal) (Time=3.5000e-01)
FLUENT
27
3.03e+02
2.94e+02
2.86e+02
2.78e+02
2.69e+02
2.61e+02
2.52e+02
2.44e+02
2.35e+02
2.27e+02
2.19e+02
2.10e+02
2.02e+02
1.93e+02
1.85e+02
1.76e+02
1.68e+02
1.59e+02
1.51e+02
1.43e+02
1.34e+02
Fig. 19- Static temperature contours for Pback=300 kPa (transonic flow).
Contours of Static Temperature (k) (Time=3.5000e-01)
FLUEN
6.98e+00
6.68e+00
6.38e+00
6.08e+00
5.78e+00
5.49e+00
5.19e+00
4.89e+00
4.59e+00
4.29e+00
3.99e+00
3.69e+00
3.39e+00
3.09e+00
2.79e+00
2.49e+00
2.20e+00
1.90e+00
1.60e+00
1.30e+00
9.99e-01
Fig. 20- Density contours for Pback=300 kPa (transonic flow).
Contours of Density (kg/m3) (Time=3.5000e-01)
28
FLUE
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).
Pathlines Colored by Particle ID (Time=3.5000e-01)
F
Fig. 22- Fluent result summery of Mach number for the three cases
29
Fig. 23- Fluent result summery of P/P0 for the three cases
Fig. 24- Fluent result summery of T/T0 for the three cases
30
Fig. 25- Fluent result summery of 𝛒/𝛒0 for the three cases
31
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-onedimensional 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
32
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:
 1
A *    1  2( 1)    1 2 

M 1 
M 

A  2 
2



 1
2 ( 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:

P    1 2   1
 1 
M 
P0 
2

T   1 2 
 1 
M 
T0 
2

(18)
1
(19)
1
    1 2   1
 1 
M 
0 
2

(20)
33
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 
  1M 12  2
2M 12    1
(21)
The static pressure ratio across the shock is given by the following
relationship:
P2
2
 1
M 12  1
P1
 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  RT
(23)
Finally, as verification, we calculate mass flow as follows:
m  VA , where V  Ma  M 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
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:
34
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.
35
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.
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.
36
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 :
37
Subsonic Case: Pback = 585 kPa
x
M
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
0.1905
0.2080
0.2275
0.2494
0.2737
0.3008
0.3307
0.3636
0.3993
0.4375
0.4772
0.5169
0.5542
0.5857
0.6071
0.6148
0.6071
0.5857
0.5542
0.5169
0.4772
0.4375
0.3993
0.3636
0.3307
0.3008
0.2737
0.2494
0.2275
0.2080
0.1905
/o
P/Po
T/To
0.9750
0.9928
0.9821
0.9703
0.9914
0.9787
0.9646
0.9898
0.9746
0.9577
0.9877
0.9696
0.9493
0.9852
0.9635
0.9392
0.9822
0.9562
0.9271
0.9786
0.9473
0.9127
0.9742
0.9368
0.8959
0.9691
0.9245
0.8768
0.9631
0.9104
0.8557
0.9564
0.8946
0.8334
0.9493
0.8780
0.8117
0.9421
0.8615
0.7928
0.9358
0.8471
0.7796
0.9313
0.8371
0.7749
0.9297
0.8335
0.7796
0.9313
0.8371
0.7928
0.9358
0.8471
0.8117
0.9421
0.8615
0.8334
0.9493
0.8780
0.8557
0.9564
0.8946
0.8768
0.9631
0.9104
0.8959
0.9691
0.9245
0.9127
0.9742
0.9368
0.9271
0.9786
0.9473
0.9392
0.9822
0.9562
0.9493
0.9852
0.9635
0.9577
0.9877
0.9696
0.9646
0.9898
0.9746
0.9703
0.9914
0.9787
0.9750
0.9928
0.9821
Table 2- Exact solution Results for Case 1 (Pback = 585 KPa).
38
Supersonic Case: Pback = 35 kPa
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
M
0.225085
0.246171
0.269886
0.296589
0.326681
0.360599
0.398811
0.441805
0.490067
0.544054
0.604157
0.670658
0.743681
0.823151
0.908769
1
1.096103
1.196195
1.299278
1.40436
1.510497
1.61684
1.722673
1.827414
1.930621
2.031963
2.131217
2.22824
2.322954
2.415331
2.505382
P/Po
0.96533
0.958711
0.950641
0.940785
0.928748
0.91406
0.896192
0.87456
0.84857
0.817671
0.781446
0.739721
0.692678
0.64095
0.585645
0.528282
0.470621
0.414427
0.361268
0.312316
0.268288
0.229465
0.195771
0.166884
0.142336
0.1216
0.104146
0.089478
0.077152
0.066782
0.05804
T/To
0.989969
0.988025
0.985641
0.982711
0.979102
0.974653
0.969171
0.962428
0.954168
0.94411
0.931965
0.917468
0.900404
0.880657
0.858242
0.833333
0.806264
0.777498
0.747594
0.717131
0.686662
0.65667
0.627541
0.59956
0.572915
0.547713
0.523994
0.501753
0.480949
0.461518
0.443383
/o
0.975111
0.970331
0.964489
0.957337
0.948571
0.937832
0.924699
0.908701
0.889329
0.866076
0.838493
0.806264
0.769297
0.727809
0.682378
0.633938
0.583705
0.533026
0.483242
0.435508
0.390713
0.349436
0.311965
0.278344
0.248442
0.222015
0.198754
0.178331
0.160416
0.144701
0.130902
Table 3- Results for Case 2 (Pback = 35 KPa).
39
Normal Shock Case: Pback = 300 kPa
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.812
2.9
3
M
0.225085
0.246171
0.269886
0.296589
0.326681
0.360599
0.398811
0.441805
0.490067
0.544054
0.604157
0.670658
0.743681
0.823151
0.908769
1
1.096103
1.196195
1.299278
1.40436
1.510497
1.61684
1.722673
1.827414
1.930621
2.031963
2.131217
2.22824
2.322954
0.530411
0.477582
0.428044
P/Po
0.96533
0.958711
0.950641
0.940785
0.928748
0.91406
0.896192
0.87456
0.84857
0.817671
0.781446
0.739721
0.692678
0.64095
0.585645
0.528282
0.470621
0.414427
0.361268
0.312316
0.268288
0.229465
0.195771
0.166884
0.142336
0.1216
0.104146
0.089478
0.077152
0.825642
0.855452
0.881649
T/To
0.989969
0.988025
0.985641
0.982711
0.979102
0.974653
0.969171
0.962428
0.954168
0.94411
0.931965
0.917468
0.900404
0.880657
0.858242
0.833333
0.806264
0.777498
0.747594
0.717131
0.686662
0.65667
0.627541
0.59956
0.572915
0.547713
0.523994
0.501753
0.480949
0.94673
0.956373
0.964651
/o
0.975111
0.970331
0.964489
0.957337
0.948571
0.937832
0.924699
0.908701
0.889329
0.866076
0.838493
0.806264
0.769297
0.727809
0.682378
0.633938
0.583705
0.533026
0.483242
0.435508
0.390713
0.349436
0.311965
0.278344
0.248442
0.222015
0.198754
0.178331
0.160416
0.872098
0.894476
0.913956
Table 4- Results for Case 3 (Pback = 300 KPa).
40
Fig. 26- Mach Number Profile – Exact Solutions.
Fig. 27- Pressure Profile– Exact Solutions.
41
Fig. 28- Temperature Profile – Exact Solutions.
Fig. 29- Density Profile – Exact Solutions.
42
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 quasisteady 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.
43
Fig. 30- Mach Number vs. Non-dimensional Nozzle Length.
Fig. 31- Density Ratio vs. Non-dimensional Nozzle Length.
44
Fig. 32- Pressure Ratio vs. Nozzle Length.
Fig. 33- Temperature vs. Nozzle Length.
45
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.
46
For the subsonic case (pback=585 KPa), the variation of Mach number, /o, 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 /o respectively (depending
on axial location).
For the supersonic case (pback=35kPa), the variation of Mach number, /o, 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, /o, 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).
47
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.htm
7- Fluent.Inc/help/html/ug/node403.htm
8- Fluent.Inc/help/html/ug/node1047.htm
9- http://courses.cit.cornell.edu/fluent/index.htm
48
Appendix A
Height versus displacement for C-D nozzle
x
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
H
2.284523
2.241706
2.198888
2.156071
2.113253
2.070436
2.027619
1.984801
1.941984
1.899166
1.856349
1.813531
1.770714
1.727897
1.685079
1.642262
1.599444
1.556627
1.513809
1.470992
x
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
h
1.428174
1.385357
1.34254
1.299722
1.256905
1.214087
1.17127
1.128452
1.085635
1.042817
1
1.042817
1.085635
1.128452
1.17127
1.214087
1.256905
1.299722
1.34254
1.385357
x
2
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3
h
1.428174
1.470992
1.513809
1.556627
1.599444
1.642262
1.685079
1.727897
1.770714
1.813531
1.856349
1.899166
1.941984
1.984801
2.027619
2.070436
2.113253
2.156071
2.198888
2.241706
2.284523
Table A.1- Height of C-D nozzle as a function of displacement x.
49
Appendix B
Calculation of Time Step
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
height
1.325
1.2187
1.1197
1.028
0.9437
0.8667
0.797
0.7347
0.6797
0.632
0.5917
0.5587
0.533
0.5147
0.5037
0.5
0.5037
0.5147
0.533
0.5587
0.5917
0.632
0.6797
0.7347
0.797
0.8667
0.9437
1.028
1.1197
1.2187
1.325
Area
rho
T
2.65
1
1
2.437333 0.96854 0.97686
2.239333 0.93708 0.95372
2.056 0.90562 0.93058
1.887333 0.87416 0.90744
1.733333
0.8427
0.8843
1.594 0.81124 0.86116
1.469333 0.77978 0.83802
1.359333 0.74832 0.81488
1.264 0.71686 0.79174
1.183333
0.6854
0.7686
1.117333 0.65394 0.74546
1.066 0.62248 0.72232
1.029333 0.59102 0.69918
1.007333 0.55956 0.67604
1
0.5281
0.6529
1.007333 0.49664 0.62976
1.029333 0.46518 0.60662
1.066 0.43372 0.58348
1.117333 0.40226 0.56034
1.183333
0.3708
0.5372
1.264 0.33934 0.51406
1.359333 0.30788 0.49092
1.469333 0.27642 0.46778
1.594 0.24496 0.44464
1.733333
0.2135
0.4215
1.887333 0.18204 0.39836
2.056 0.15058 0.37522
2.239333 0.11912 0.35208
2.437333 0.08766 0.32894
2.65
0.0562
0.3058
Table B.1- Time step calculation.
50
V
1.732051
3.577858
5.378958
7.13453
8.843708
10.50558
12.1192
13.68354
15.19755
16.6601
18.06998
19.42594
20.7266
21.97054
23.1562
24.28194
25.34598
26.3464
27.28112
28.14792
28.94433
29.66769
30.31506
30.88321
31.36857
31.76712
32.0744
32.28536
32.39422
32.39439
32.2782
Dt
0.000143
0.000144
0.000145
0.000146
0.000147
0.000148
0.00015
0.000151
0.000152
0.000154
0.000155
0.000157
0.000158
0.00016
0.000162
0.000164
0.000166
0.000168
0.000171
0.000174
0.000176
0.000179
0.000183
0.000186
0.00019
0.000194
0.000199
0.000204
0.00021
0.000216
0.000223
Appendix C
Convergence Criteria
Fig C.1 Convergence monitor (scaled residuals) for case 1
Fig. C.2 - Convergence monitor (scaled residuals) for case 2
51
Fig. C.3 - Convergence monitor (scaled residuals) for case 3
Fig C.4 - Convergence monitor (mass flow rate) for case 1
52
Fig. C.5- Convergence monitor (mass flow rate) for case 2
Fig. C.6 - Convergence monitor (mass flow rate) for case 3
53
Fig. C.7 - Convergence monitor (Heat transfer rate) for case 1
Fig. C.8 - Convergence monitor (Heat transfer rate) for case 2
54
Fig. C.9 - Convergence monitor (Heat transfer rate) for case 3
55