2d mln nozzle
DESCRIPTION
characteristics nozzleTRANSCRIPT

1
Chapter one
Introduction
11 Introduction:
The supersonic nozzle is divided in two parts. The supersonic portion is
independent of the upstream conditions of the sonic line. We can study this part
independently of the subsonic portion. Because the subsonic portion is used to give
a sonic flow to the throat. There are two categories for supersonic nozzle according
to the sonic line. If the sonic line is a straight line, the wall at the throat generates
centered and divergent waves. The second category has a curved sonic line. In this
case, the flow inside the nozzle has not centered Mach lines. Each type exists for
twodimensional and axisymmetric flow [14].
12 Objectives:
The basic objectives of this research are to investigate numerical approaches to
design and study the supersonic nozzle and develop an efficient and accurate code
for performing such simulations. These objectives have been split into the
following:
1 Design the Supersonic Nozzle by calculating the nozzle contour using the
method of characteristics and creating program code.
2 Analysis of nozzle by using FLUENT software to analyze the flow inside
the designed nozzle so as to check its performance.

2
13 Thesis Overviews:
This thesis designs and analyzes supersonic wind tunnel nozzle contour. The
nozzle will be designed using the 2D Method of Characteristic.
Chapter 1 gives an introduction to the subject and objectives of this research.
In chapter 2, the methods and results of others research will be discussed. This
section will also give background of wind tunnels types.
Chapter 3 gives details of the characteristic equations and development of the grid
points. This process are written as a generic computer code in MATLAB so
multiple nozzle contours is calculated for different user inputs such as desired exit
Mach number, working fluids ratio of specific heats and predetermined throat
height and number of characteristic lines. These codes are available in Appendix
B. The development of the characteristic and compatibility equations for 2D flow
is available in Appendix A. Also Chapter 3 outline the procedure and techniques
used in running the previously mentioned nozzle designs through the wellknown
CFD program FLUENT. This was done to verify that the nozzle reach their desired
exit Mach number.
Chapter 4 discusses the checks performed to verify the accuracy of the code
developed in chapter 3 by three ways, comparison the nozzle designed with
Brittons code, GA10 supersonic wind tunnel nozzle and FLUENT simulation.
Also will examines the variation analysis of designed code for multiple ratios of
specific heats, exit Mach numbers and number of characteristic lines. This section
also discusses the results given by the FLUENT simulations.
Chapter 5, Conclusions of the major findings of this thesis. Also in this chapter
will discuss improvements needed in the computer code outlined in chapter 3 to
enhance the codes ability to produce nozzle wall contours for nonisentropic,
viscid supersonic flow as recommendations.

3
Chapter two
Background and Literature review
21 Historical background:
In order to better understand the process of designing and analyzing a supersonic
wind tunnel nozzle, it is highly important to understand the properties of the flow
through this nozzle in order to successfully achieve supersonic flow as well as to
ensure that the flow is uniform.
211 Wind Tunnels:
A wind tunnel is a device designed to generate air flows of various speeds through
a test section. Wind tunnels are typically used in aerodynamic research to analyze
the behavior of flows under varying conditions, both within channels and over
solid surfaces. Aerodynamicists can use the controlled environment of the wind
tunnel to measure flow conditions and forces on aircraft models as they are being
designed [16].
212 Supersonic Wind Tunnel Types:
In order to better understand wind tunnel operation, three types of wind tunnels
were researched. continuous, blowdown, and indraft.
Continuous wind tunnels are essentially a closedcircuit system and can be
used to achieve a wide range of Mach numbers [16]. They are designed so that the
air that passes through the tunnel does not exhaust to the atmosphere; instead, it
enters through a return passage and is cycled through the test section repeatedly as
shown in Fig. 2.1. This type of wind tunnel is beneficial because the operator has
more control of the conditions in the test section than with other approaches since
the tunnel is cut off from the environmental conditions once running.

4
In comparison to other wind tunnel types, continuous wind tunnels have
superior flow quality due to the different facets of the tunnel's construction. The
turning vanes in the corners and flow straighteners near the test section ensure that
relatively uniform flow passes through the test section [3]. Continuous tunnels also
operate relatively quietly. Finally, the testing conditions can be held constant for
extended periods of time [16].
Fig. 2.1: Continuous Wind Tunnel [16]
Blowdown tunnels (see Fig. 2.2) have a variety of different configurations and are
generally used to achieve high subsonic and midtohigh supersonic Mach numbers
[16, 4].
Blowdown tunnels use the difference between a pressurized tank and the
atmosphere to attain supersonic speeds. They are designed to discharge to the
atmosphere, so the pressure in the tank is greater than that of the environment in
order to a create flow from the tank out of the tunnel. In one configuration, known
as "a closed" blowdown tunnel two pressure chambers are connected to either side
of the tunnel [4].

5
Fig. 2.2: Blowdown Wind Tunnel [16]
In this configuration, one chamber would contain a high pressure gas and the other
chamber would be at a very low pressure. At the beginning of a run, valves are
opened at each chamber, and the pressure differential causes air flow in the
direction of the lower pressure until the two chambers have reached equilibrium.
The test section is positioned at the end of the supersonic nozzle. Many blowdown
tunnels have two throats, with the second throat being used to slow supersonic
flow down to subsonic speeds before it enters the second chamber.
In other types of blowdown wind tunnels, the low pressure chamber is removed,
and the tunnel discharges directly into the atmosphere, as with Fig. 2.2. There are
several advantages of blowdown tunnels: they start easily, are easier and cheaper to
construct than other types, and have superior design for propulsion and smoke
visualization" [4]. Blow down tunnels also has smaller loads placed on a model as
a result of the faster start time. These tunnels, however, have a limited test time. As
a consequence, faster, more expensive measuring equipment is needed. They can
also be noisy.

6
Indraft wind tunnels use the difference between a low pressure tank and the
atmosphere to create a flow. A vacuum tank is pumped down to a very low
pressure, and the other end of the tunnel is open to the atmosphere. When the
desired vacuum pressure is reached, a valve is opened, and air rushes from outside
the tunnel, in through the test section, into the vacuum chamber. The end of the run
occurs when the pressure differential is no longer great enough to drive the tunnel
at the desired test section Mach number [16]. One of the benefits of an indraft
tunnel is that the stagnation temperature can be considered constant throughout a
run. Additionally, the flow is free of contaminants from equipment used by other
wind tunnel types. For example, there is no need for the pressure regulators
required by blowdown tunnels. In comparison to other types of tunnels, indraft
tunnels can operate at higher Mach numbers before a heater is necessary to prevent
flow liquefaction during expansion. Lastly, using a vacuum is safer than using high
pressures. High pressure tanks face the risk of exploding, while the reversed
pressure differential of a vacuum chamber only results in the risk of an implosion.
One of the major disadvantages of indraft wind tunnels is that they can be up to
four times as expensive as their blowdown counterparts. Additionally, the
Reynolds number for a particular Mach number can be varied over a greater range
with a blowdown tunnel [16].
Fig. 2.3: Indraft Wind Tunnel [16]

7
213 Method of Characteristics:
The Method of Characteristics (MOC) is a numerical procedure appropriate for
solving twodimensional compressible flow problems. By using this technique,
flow properties such as direction and velocity, can be calculated at distinct points
throughout a flow field. The method of characteristics, implemented in computer
algorithms, is an important element of supersonic computational fluid dynamics
software. These calculations can be executed manually, with the aid of spreadsheet
programming or technical computing software as the number of characteristic lines
increase, so do the data points, and the manual calculations can become
exceedingly tedious [9].
The method of Characteristics was developed by the mathematicians Jaques
Saloman Hadamard in 1903 and by Tullio LeviCivita in 1932 [8]. The method of
characteristics uses a technique of following propagation paths in order to find a
solution to partial differential equations.
The physical conditions of a twodimensional, steady, isentropic, irrotational
flow can be expressed mathematically by the nonlinear differential equation of the
velocity potential. The method of characteristics is a mathematical formulation that
can be used to find solutions to the aforementioned velocity potential, satisfying
given boundary conditions for which the governing partial differential equations
(PDEs) become ordinary differential equations (ODEs).
The name comes from a method used to solve hyperbolic partial differential
equations: Find "characteristic lines" (combinations of the independent variables)
along which the partial differential equation reduces to a set of ordinary differential
equations, or even, in some cases, to algebraic equations which are easier to solve.
The applications of the method of characteristics for nozzle flows are not
limited to the design of contours. The method may also be used to analyze the flow

8
field inside a known contour as well. The method is also not limited to the flow
within the nozzle. The approach can be extended to analyze the exhaust plume for
both under expanded and over expanded nozzle flow using the free pressure
boundary of the exhaust plume [22].
22 Literature review:
221 Contoured Nozzle Design:
It can easily be shown that in order to expand flow through a duct from subsonic
flow to supersonic flow the area of the passage that the fluid is passing must first
decrease in area and then increase in area. This area relationship is the basis in
nozzle design given in Eq. (2.1). The relationship between local Mach number and
the local area ratio was found through the study of quasionedimensional flow.
Although this relationship provides no information for the contour of such a duct
or the losses that are associated with a multidimensional flow field [1].
=
1
2
+1 1 +
1
22
+1
2 1 (2.1)
The first successful implementation of method of characteristics for nozzle
design was performed by Ludwig Prandtl and Adolf Busemann in 1929 [1]. Since
the implementation by Prandtl and Buseman, the method of characteristics has
become a fundamental basis in nozzle design. This is because the method allows
for physical boundaries to be located. Prandtl and Busemann implemented the
method graphically to solve twodimensional nozzle problems. A comprehensive
presentation on both the graphical and the analytical approach to twodimensional
method of pharmacogenetics was completed by Shapiro and Edelman in 1947[1].

9
Foelsch (1959, [8]) proposed a method for developing solutions to axis
symmetric supersonic streams using a method of characteristics approach .Antonio
Ferri extendedthe approach to axissymmetric flows in 1954, through a theoretical
adaptation to the mathematics [5].
Guerntert and Netmann (1959, [7, 8]) implemented the analytical approach
for the development of supersonic wind tunnels with desired mass flows. This
implementation developed a solution based on initial conditions along the nozzle
centerline. The Guerntert and Netmann considered this approach for wind tunnel
design had no length requirement but required uniform exit flow. Their solution
resulted in difficulties designing short length and large expansion ratio nozzles.
Their work also showed that truncation of nozzle lengths resulted in only a small
reduction of vacuum specific impulse from the uniform flow case.
The work of G.V.R. Rao used the method of characteristics as part of his
solution in developing contour nozzle designs. Rao developed a method of
designing a contoured exhaust nozzle for optimum thrust of a fixed length nozzle.
Rao's solution used a combination of Lagrangian multipliers and method of
characteristics [17, 18].
Allman and Hoffman (1978, [2]) presented a procedure for the design of a
maximum thrust contours by a direct optimization method. The contour used was a
seconddegree polynomial fitted to a prescribed initial expansion contour. The
contours produced were similar to that of a Rao nozzle. Allman and Hoffman
showed that a polynomial could be used to develop the nozzle boundary with
comparable results to a Rao nozzle, however the flow field was solved in much the
same way, and the solution differed only slightly in the formation of the nozzle
boundary. Essentially, their solution was a Rao nozzle with a polynomial fitted
boundary instead of a boundary determined by the solution.

10
223 Transonic Flow Zone:
Transonic flow is the flow regime where the fluid transitions from subsonic
to supersonic velocities. The transonic flow regime has been intensely studied. The
work of Sauer detailed the complexities and the mathematical treatment of such
flows, especially as applied to the passage of flow through Laval nozzles. The flow
in the throat region of a converging diverging nozzle under choked flow conditions
is transonic [21].
The work of Sauer has been the primary basis for the treatment of the
transonic flow zone in supersonic nozzle design, because Sauer's method is a
closed form solution for the flow field in the nozzle throat, and it can produce
excellent approximate solutions for nozzles with a large subsonic radius of
curvature relative to the throat radius.
The transonic solution is important to the method of characteristics solution.
It allows to determination of a subsonic radius of curvature that allows for
substantially supersonic flow at the nozzle wall at the minimum area point and also
locates the position where like flow is on the axis of symmetry. In determining this
line of constant substantially supersonic Mach number, an initial value line can be
formed so that the flow satisfies the wall boundary condition at the throat exactly.
224 CFD Nozzle Analysis:
K.M. Pandey and A.P. Singh [10] worked on the topic of CFD Analysis of
Conical Nozzle for Mach 3 at Various Angles of Divergence with Fluent
Software and they found that the variation in the Mach number, pressure ratio.
K. M. Pandey et.al [12] worked on the topic of Studies on Supersonic
Flows in the De Laval Nozzle at Mach No. 1.5 and its flow Development into a
Suddenly Expanded Duct and there findings are Solution of supersonic flow
fields of flow development in 2D De Laval nozzle with a duct. The study is aimed

11
with 1.5 Mach numbers for various L/D, into a duct. The nature of the flow is
smooth when the flow gets attached and streamlined. The suddenly expanded
cavity not only causes head losses but also is accompanied by flow oscillations due
to phenomenon called vortex shedding near the nozzle exit region.
K. M. Pandey et.al [11] worked on the topic of Study on Rocket Nozzles
with Combustion Chamber Using Fluent Software at Mach 2.1 and there findings
are The pressure and Temperature parameter depend upon airfuel ratio. Loss of
pressure and temperature above two fuel inlet for same quantity of air fuel ratio.
K. M. Pandey et.al [13] worked on the topic of Study on Supersonic Free
Single Jet Flow: A Numerical Analysis with Fluent Software and there findings
are to review the basic aspects of free jet flow and to contribute additional data
concerning effects on the free jet flow map of efflux Reynolds number and orifice
geometry.
K.M. Pandey and A.P. Singh [14] worked on the topic of Design and
Development of De Laval nozzle for Mach 3 & 4 using methods of Characteristics
with Fluent Software and there findings are gas flows in a De Laval nozzle using
2D axisymmetric models, which solves the governing equations by a control
volume method. The throat diameter is same for both nozzles and designed using
method of characteristics. Detailed flow characteristics like the centerline Mach
number distribution and Mach contours of the steady flow throughthe converging
diverging nozzle are obtained.

12
Chapter three
Design of Supersonic Nozzle and CFD Setups
31 Introduction:
There are several types of supersonic/hypersonic wind tunnel nozzles. The
most common type is axisymmetric with a circular cross section at every station.
A few facilities use twodimensional (2D) nozzles, and the primary focus here,
where two opposite walls are contoured in a convergingdiverging shape but are
bounded by parallel walls, giving a rectangular cross section at every station. The
2D configuration admits the use of flexible plates driven by jacks to alter the
contour as needed to vary the flow speed in the test section. A single flexible plate
nozzle can produce many flow speeds in the test section. However, the 2D nozzle
is generally not used above Mach number 5 or 6 for two reasons [6]:
1 The complexity of water cooling a flexible structure.
2 The very small slitthroat height needed to achieve the large nozzle exit area
ratio.
With thermal deflections and pressure loading, variations in the slit height lead to
unacceptable flow nonuniformity in the test section and might result in missing the
target flow speed. A third type of hypersonic nozzle has an exit cross section with
a shape tailored to a specific application, such as directconnect scramjet
combustor testing. For such an application, four sides of a nearrectangular cross
section nozzle may be contoured to deliver uniform flow. A fourth type of nozzle,
used at least once in an archeated facility, has a circular throat and a semicircular
exit, its purpose being to maximize exposure of wedge surface area to the hot jet
for material testing. Finally, there is the concept of a minimumlength nozzle in

13
which the throat is sharp, and thus causes the flow to turn suddenly to the nozzle
inflection angle (the maximum angle of the contour) [6].
Although the goal of contour design is to deliver a perfectly uniform flow, there
are limitations to what can be achieved through contouring alone. The following
conditions are proposed as sufficient (in the mathematical sense) for a contour to
exist that will deliver uniform flow:
Continuum flow
Steady flow
Inviscid flow
Isentropic flow
2D flow
Uniform entrance flow at the nozzle inlet station
Equation of state is perfect, or in thermochemical equilibrium.
An important fact to realize regarding rigorous contour design is that there
are an infinite number of contours that will deliver uniform flow at specified
conditions. Parameters that influence the shape of the contour include nozzle
length and height, inflection angle, and specifications made for the various
boundary conditions, particularly on the nozzle centerline for some design
techniques. While any contour among the infinity of choices that would yield
uniform flow might be chosen, some choices are better than others. This fact is
illustrated below.
Of the many factors that can influence flow quality, some effects can be
significantly ameliorated by the choice of design options. In general, long nozzles
with small inflection angles (as a rule, those of less than about 12 deg) yield the
most uniform flow, which is a primary criterion for aerodynamic testing. On the
other hand, for highenthalpy facilities, such long nozzles produce large losses of

14
the often elaborately achieved total enthalpy. For archeated and combustion
heated facilities, very short designs are usually chosen with the cognizance that
some flow quality is being sacrificed. Short nozzles often have large inflection
angles, which increase the concern about disastrous flow separation. Short nozzles
also tend to have a small wall radius of curvature at the throat, perhaps as small as
the throat radius, which is said to make accurate machining more difficult. Short
nozzles with large flow expansion rates tend to exacerbate the effects of
nonequilibrium on flow quality, while long nozzles give the flow more transit time
to relax toward equilibrium. For many nonequilibrium nozzle flows, even absurdly
long nozzles are not sufficient for relaxation to occur [6].
Most operational nozzle design techniques can be placed in one of two
categories: direct design or design by analysis (DBA). In direct design, the nozzle
contour is computed as the primary output of the computation with only a single
sweep through the flow field using some numerical procedure. Nearly all direct
design methods are based on the classical method of characteristics (MOC). In
design by analysis, a computational fluid dynamics (CFD)based analysis flow
solver, to which the nozzle contour is an input, is coupled with a numerical
optimization technique. The optimization technique alters the contour to drive the
exit flow toward better uniformity and may require a flowfield solution for each
contour perturbation

15
32 Methodology:
To expand a gas from rest to supersonic speed, a convergent divergent nozzle
should be used. Quasione dimensional analyses predict the flow properties as a
function of x through a nozzle of specified shape. Although quasione dimensional
analysis represents the properties at any cross section as an average of the flow
over a given nozzle crosssection, it cannot predict both the three dimensional flow
and the proper wall contour of the convergent divergent nozzle. Therefore, quasi
two dimensional analysis is used to predict the proper contour for different
conditions.
Therefore, the steady, inviscid supersonic flow is governed by hyperbolic equations,
sonic flow by parabolic equations, and subsonic flow by elliptic equations (Eq.A.11).
Moreover, because two real characteristics exist through each point in a flow
where M > 1, the method of characteristics becomes a practical technique for
solving supersonic flows. In contrast, because the characteristics are imaginary for
M < 1, the method of characteristics is not used for subsonic solutions. (An
exception is transonic flow, involving mixed subsonicsupersonic regions, where
solutions have been obtained in the complex plane using imaginary
characteristics.).
321 Method of Characteristics: There are several ways to derive a method of characteristics. In one approach, the
2D, or axisymmetric, Euler equations are transformed to directions along which
the partial differential equations reduce to ordinary differential equations. The
finding is that:
= 2 1
(3.1)
+ = = (3.2) = = + + (3.3)

16
Where is the PrandtlMeyer function for a perfect gas given by
= +1
11
1
+1(2 1) 1 2 1 (3.4)
In Eq. (3.1) the + corresponds to rightrunning characteristics and the to left
running characteristics. Mach lines emanate from a point at an angle of for
the right characteristic and + for the left characteristic. Eq. (3.1) applies along
the Mach lines defined by
= (3. 5)
Where corresponds to right characteristics and + to left characteristics.
322 Grid generation:
Grid points used in calculation of method of characteristics are of two types:
1 Internal points which are away from wall
2 Wall points.
Lines can be classified into the four parts shown in fig. 3.1:
1. The initial curve.
2. The reflection about the symmetry line.
3. The intersection of characteristic lines.
4. The wall contour.
4
2
3
1
Fig. 3.1: Design steps in the method of characteristic

17
3221 Initial Line:
The Initial Curve is the convex portion (before the inflexion point) of the
expansion curve. PrandtlMeyer shows that the magnitude of supersonic flow
(Mach number) increases over a convex expanding surface and in doing so creates
a series of Mach (expansion or characteristic) waves, as illustrated in Fig. 3.2.
In our MATLAB program, properties of point 1 (theta, nu) can be found from
equation (3.5) and equation (3.6), where 1 equal to max
max =(Me )
2 (3.6)
And then the properties of point 2 (theta,nu) can be found by using equation of
leftrunning characteristic C eq. (3.2) where theta at all centerline points are equal
zero then we need to calculate nu () for each point.
To find properties of point 3 we use equation of leftrunning characteristic C
equation(3.2) between point 1 and 3, and equation of Rightrunning characteristic
C+ equation(3.3) between point 3 and 2, points 4, 5, ., (num+1) found as point 3.
To draw characteristic lines (12), (13),(1(num+1)) use eqs. (3.7), (3.8) to
find x,y coordinate [3].
=(11tan ())( 11tan ())
tan tan () (3.7)
1
2 3
4 5
num+2
Wall
Centerline
First characteristic line
Fig. 3.2: First characteristic line points

18
= 1 + 1 tan = 2,3,4 + 1 (3.8)
Where
=( )1+( )
2 And =
(+)1+(+)
2 = 1 sin() [Ref.1] (3.9)
Mach number (M) for all points can be found by using numerical method (Newton
Raphson Method) for eq. (3.4).
3222 Centerline Points:
For All centerline points, is equal to zero then we need to calculate v for each
point as shown in fig. 3.3. Here we will use leftrunning characteristic C eq. (3.2)
only because; there is no Rightrunning characteristic C+
To draw characteristic lines (a(num+3)) (q1) use eq. (3.10) to find x
coordinate, y at all centerline points is equal to zero. As example x for point
(num+3)
+3 = + +3
tan () (3.10)
=( ) +( ) +3
2 [Ref.1]
1
2
num+2
Wall
Centerline
num+3 q1
q
Figure 3.3: Centerline points
a

19
3223 Interior Points:
In this case, interior points are coming from intersection the leftrunning
characteristic C and Rightrunning characteristic C+ then eqs. (3.2) (3.3) are used
to find properties as shown on fig. 3.4.
To draw characteristic lines between all points eqs. (3.11), (3.12), (3.13) can be
used to find x,y coordinate.
=( tan ())( tan ())
tan tan () (3.11)
= + tan (3.12)
=( )+( )
2 And =
(+)+(+)
2 [Ref.1] (3.13)
1
2
num+2 Wall
Centerline
q1
q
a b
c
Fig. 3.4: Interior points
b
a
C+
C
Straight Line
1
2 + +
1
2( + )
1
2 +
1
2( + )
c

20
3224 Wall Points:
Wall points are very important points, because they represent the most important
design element and the main objective of the design. These points connected in
straight lines, but in fact they are connecting as curves so that the higher number of
points gives high accuracy. The final shape of the output from the wall connecting
points by straight lines represents the nozzle contour.
Properties of wall points are same the properties at previous points on Right
running characteristic C+ as shown on fig. 3.5, This means [1]:
+2 = +1 +2 = +1
To draw characteristic lines between wall points eqs. (3.14), (3.15), (3.16) can be
used to find x,y coordinate. As example x,y for point (num+2)
+2 =(11tan ())( +1 +1tan ())
tan tan () (3.11)
+2 = +1 + +3 +1 tan (3.12)
=1+ +2
2 And =
(+) +1+(+) +2
2 [Ref.1] (3.13)
Fig. 3.5: Wall points
1
2
num+2 Wall
Centerline
q1
q
num+1

21
Finally with above procedure and calculations of points and wall contour, and
using equations and concept of numerical method and iteration, computer program
written in MATLAB software with so many number of statement as will be
discussed in section 3.5.
33 Subsonic Portion:
Method of characteristic is not applicable for subsonic portion there for, we need
another method to design it, so many equation used to design subsonic contour
analytically. One of them, method used by Frederick L.Shope in his paper
Contour Design Techniques for Super/Hypersonic Wind Tunnel Nozzle [6].
+ = + =52
12+ (3.14)
= 2 + ++
=
(3.15)
=3
2 =
2
5
8 (3.16)
0 = !
+ =
2
3
2
(3.17)
+ + + = + +
2 (3.18)
Figure 3.6: Construction of subsonic contour

22
+ + = 2
12 6
2
+ (3.19)
The variables are defined in Fig.3.6. The designer specifies RI, R*/r* ,r*, , a, and L.
34 CFD Setups:
The supersonic nozzle program designed produces a set of points which define the
nozzles contour. These points are imported into Gambit. Gambit is a mesh
generating program used to mesh the fluid domain of the simulation. All points are
connected to produce a 2D symmetry virtual geometry. Fig. 3.7 shows the typical
geometry and boundary conditions used to simulate the nozzle.
Fig. 3.7: Typical Supersonic Nozzle CFD Boundary Conditions
Once the geometry of the nozzle has been virtually created, the fluid region can be
meshed. Fig. 3.10 is a typical the meshed geometry of the supersonic nozzle.
Produced Table 3.1 gives the meshing inputs used for this particular mesh.
Wall
Pressure outlet
Symmetry
Pressure inlet

23
Table 3.1: meshing inputs
Mesh Conditions: Scheme: Elements: Quad
Type: Map
Smother: None
Spacing: Interval Size: 0.0003
Total Number of Nodes 12831
Total Number of Elements 12512
Fig. 3.8 Typical Supersonic Nozzle Mesh
Now that the geometry has been meshed, it can be imported into FLUENT, the
fluid flow simulation program. Once imported, the solver type, material and
properties, operating conditions and boundary conditions must all be defined.
Table 3.2 defines the conditions used in the simulations for the supersonic nozzle.
To validate the designed code, one quantity is checked once the simulations
converge, Mach number at the exit of the nozzle. A shock change flow to subsonic;
therefore, Mach number plot will show shock if they exist in the flowfield. The
simulation results are discussed in chapter 4.
Table 3.3 contains the variable that will use in FLUENT as input conditions used
for Three Nozzles Simulation for different exit Mach numbers.

24
Table 3.2 : FLUENT Input Conditions Used for Supersonic Nozzle Simulations Solver: Solver: Density Based Space : 2D
Velocity Formation: Absolute Gradient Option: GreenGauss Cell Formulation: Implicit Time: Steady Porous Formulation: Superficial Velocity Energy Equation: Checked Viscous Model: kepsilon Checked
Material: Name: air
Properties: Density: Ideal Gas Cp: 1006.43 J/kg*K
Molecular Weight: 28.966 kg/kmol Operating Conditions: Pressure: Operating Pressure: 0 Pa Gravity: Not Checked
Reference Pressure Location: X(m): 0.028 Y(m): 0
Pressure Inlet: Gauge Total Pressure: 577500 Pa Constant
Supersonic/Initial Gauge Pressure:
555370 Pa Constant
Total Temperature: 300 K Constant Direction Specification Method: Normal to Boundary Intensity and Hydraulic Diameter Turbulent Intensity % 10
Hydraulic Diameter 0.00513 m Pressure Outlet: Gauge Pressure: 15722 Pa Constant
Backflow Total Temperature: 300 K Constant Backflow Direction Specification
Method: Normal to Boundary
Non reflecting Boundary: Not Checked Target Mass flow Rate: Not Checked Intensity and Hydraulic Diameter
Backflow Turbulent Intensity % 10 Backflow Hydraulic Diameter 0.00658 m
Solution Controls: Discretization Second Order Upwind
Solver Parameter: Courant Number: 5 Solution Initialization: Compute From: Pressure Inlet
Reference Frame: Relative to Cell Zone Initial Values: Automatically Set by Compute From
Monitors: Residual
Plot Checked Monitors Convergence Criteria
Continuity 1x10^6
X velocity 1x10^6
Y velocity 1x10^6
Energy 1x10^6
k 1x10^6
Iterate: Number of Iteration 4000

25
Table 3.3 : FLUENT Input Conditions Used for Three Nozzles Simulation
Mach
Number
Reference
Pressure
Location
Gauge
Total
Pressure
Supersonic/Initial
Gauge
Pressure(Pa)
Gauge
Pressure
(Pa)
Hydraulic
Diameter (m)
X(m) Y(m) Inlet Outlet
3 0.028 0 577500 555370 15722 0.00513 0.00658 2.5 0.027 0 325000 297980 19021 0.00556 0.00668 3.1 0.0292 0 635000 614690 14890 0.00519 0.00665
Table 3.5 defines the conditions used in the simulations for the supersonic nozzle
with reservoir at the exit of nozzle for study the effect of back pressure on flow
through nozzle.
Table 3.4 contains the variable that will use in FLUENT as input conditions used
for three cases simulation for different back pressure.
Table 3.4: FLUENT Input Conditions Used for Four Nozzle Conditions
Mach Number
= 2.5
Reference
Pressure
Location
Operating
Pressure
(pa)
Gauge
Pressure(Pa)
Absolute
pressure(Pa Hydraulic
Diameter (m)
X(m) Y(m) Inlet Outlet
Designed Case 0.027 0 0 19021 19021 0.00556 0.00882
Case 1 0.027 0 0 0 0 0.00556 0.00882
Case 2 0.027 0 0 41535 41535 0.00556 0.00882
Case 3 0.027 0 101325 19021 120346 0.00556 0.00882

26
Table 3.5 : FLUENT Input Conditions Used for Supersonic Nozzle with Reservoir Simulations Solver: Solver: Density Based Space : 2D
Velocity Formation: Absolute Gradient Option: GreenGauss Cell Formulation: Implicit Time: Steady Porous Formulation: Superficial Velocity Energy Equation: Checked Viscous Model: kepsilon Checked
Material: Name: air
Properties: Density: Ideal Gas Cp: 1006.43 J/kg*K
Molecular Weight: 28.966 kg/kmol Operating Conditions: Pressure: Operating Pressure: 0 Pa Gravity: Not Checked
Reference Pressure Location: X(m): 0.0270 Y(m): 0
Pressure Inlet: Gauge Total Pressure: 325000 Pa Constant
Supersonic/Initial Gauge Pressure:
297980 Pa Constant
Total Temperature: 300 K Constant Direction Specification Method: Normal to Boundary Intensity and Hydraulic Diameter Turbulent Intensity % 10
Hydraulic Diameter 0.00556m Pressure Outlet: Gauge Pressure: 19021 Pa Constant
Backflow Total Temperature: 300 K Constant Backflow Direction Specification
Method: Normal to Boundary
Non reflecting Boundary: Not Checked Target Mass flow Rate: Not Checked Intensity and Hydraulic Diameter
Backflow Turbulent Intensity % 10 Backflow Hydraulic Diameter 0.00882m
Solution Controls: Discretization Second Order Upwind
Solver Parameter: Courant Number: 5 Solution Initialization: Compute From: Pressure Inlet
Reference Frame: Relative to Cell Zone Initial Values: Automatically Set by Compute From
Monitors: Residual
Plot Checked Monitors Convergence Criteria
Continuity 1x10^6
X velocity 1x10^6
Y velocity 1x10^6
Energy 1x10^6
k 1x10^6

27
35 Supersonic Nozzle Program:
Firstly, three hypotheses was used to estimate the increase in the tendency lines
characteristics for supersonic portion, after that a high accuracy was selected.
First hypothesis:
This formula is used in Ref [15]
=
1 (3.20)
Second hypothesis:
This formula is used in Ref [19]
=
(3.21)
Third hypothesis:
This formula is used in Ref [1]
1 = ( ) 2 = ( )
1 (3.22)
( ) Means value of without fraction.
The program begins by asking the user for all necessary design variables that the
program will need to calculate the nozzle contours. The list of variables required is
described in Table 3.6 with description below:
Table 3.6
Program Variable Description
num Number of Characteristic lines
gamma Ratio of Specific Heats of the working fluid Cp/Cv
Po Total Pressure
M_e Mach Number At Exit
h_th Throat Height (meters)
width
Nozzle width (meters)

28
The program then passes the necessary input variables to the script file that needs
them. All input variables are passed to script files (PG_nozzle, nozzle_plot1,
nozzle_plot2, nozzle_plot3, nozzle_plot4, Subsonic_part and nozzle_CFD). These
script files calculate the contour of supersonic nozzle.
PG_nozzle, the script file that calculates the nozzle properties (Riemann
Invariants(k_p, k_m), Streamline Angle with x axis(theta), PrandtlMeyer
Function(nu), Mach number at any x, Mach angle (mu), and x,y coordinate), where
requires input variables (num, gamma, Po, M_e, and h_th).
A second script file, nozzle_plot1, is used to Plot the interior point, a third script
file, nozzle_plot2 is used to plot the points at axis. A fourth script file,
nozzle_plot3, is used to plot first characteristic line points, A fifth script file,
nozzle_plot4, is used to plot wall contour points, A sixth script file, Subsonic_part,
is used to calculating and plot the subsonic part and a seventh script file,
nozzle_CFD is used to Calculate Isentropic 1D Qusi flow parameters (Throat
Area, Exit Area, Area ratio, Nozzle length, Total Pressure Ratios, Total
Temperature Ratios and calculating the Mach numbers of the points in the
centerline and mass flow rate). Once all script files run their solutions is nozzle
contour and properties of flow inter and out of the nozzle. Fig. 3.9 is an example of
a nozzle solution plot.
Fig. 3.9 Supersonic Nozzle Contour (M=3 at Exit)
0.15 0.1 0.05 0 0.05 0.1 0.15 0.20.06
0.04
0.02
0
0.02
0.04
0.06
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines

29
351 Flow Chart:
Fig. 3.10 shows the flow chart for script files of supersonic nozzle contour. The
sources are available in Appendix B.
num Po M_e gamma h_th width
for T=1:1000
No If T==1
Yes
P(T)=3
else
P(T)=P(T1)+T+1
q=P(num)
for i=1:q
k_p(i)=0, k_m(i)=0 nu(i)=0, mu(i)=0
theta(i)=0 x(i)=0 y(i)=0
theta_max d_theta
A
For Q=1:1000 B C End

30
A
for i=1:q
if i

31
End
End
C
A=(gamma1)/(gamma+1)
for i=1:q
f f_d M(j+1)
M_x(i)=M(j)
for j=1:100
For Q=1:1000 F D
End
1 2 3 4 5 6
Supersonic Nozzle
Contour
END

32
End
E
j=num n=num+3
n
for i=1:q
if i==2
C_m C_p x y
elseif i==num+3
C_m C_p x y
Yes
Yes
No
No
elseif i==n+j No
C_m C_p x y n=n+1 j=j1
Yes
End
End
End
D
L=num+4 M=num*2+1 j=num
for i=L:M
C_m C_p x y
for o=1:num
C_m C_p x y
for i=((L+j):(M+j1))
L=L+j M=M+j1 j=j1
End for i=1:q
if i

33
End for i=1:q
if i

34
R,r_st,x1,z,theta,a,X
for i=1:10
L,y,R1,d,x_pluse,r_pluse,e,b,c
if x>=0 & x=a & x=a+b & x

35
M=num+2,j=num,k=3
for i=1:q
if i==m
Yes
x_char , y_char, M_char
Yes
No
6
elseif i==m+j
x_char , y_char, M_char
m=m+j, k=k+1, j=j1
elseif i==1
x_char , y_char, M_char
Yes
No
End
table_Supersonic
G
No
A_star, To_T, Po_P, rwo_ew, Ax_Ast
plot(x_char,M_char,'*') plot(x_char,(1./Po_P),'*r')

36
Fig. 3.10: flow chart for script files of supersonic nozzle contour
a
for i=1:a
for j=1000
f, f_d, M(j+1)
G
End
6
End
M_Subsonic(i)=M(j+1
P_Po_Exit, P_Exit, A_Astar_Exit,
A_Astar_Inlet, M_Inlet, Po_P_Inlet,
P_Po_Inlet, P_Inlet, M_all

37
Chapter four
Results and Discussion
41 Introduction:
This chapter discusses the checks performed to verify the accuracy of the code
developed in chapter 3, by using three ways: to compare with Brittons code,
GA10 supersonic wind tunnel nozzle and FLUENT simulation. As mentioned in
chapter 3, a combination of theoretical and CFD simulations were employed to
verify the accuracy of the code for design the supersonic nozzle.
42 Accuracy of Present Code with Britton Code:
Britton Jeffrey Olson develops a computer program using MATLAB in his Ph.D.
thesis from Stanford University [20]. Britton was used the method of characteristic
to design supersonic nozzle contour for perfect gas flow from combustion chamber
where no effect of specific heat change.
Table 4.1 contains the variables that used by Britton. Same values were used in our
code for comparison.
Table 4.1: Program Variable Description
Program Variable Value
gamma 1.4
M_e 3
h_th 0.025
width 0.1

38
Fig. 4.1 shows the configurations of supersonic nozzle divergent part that obtained
by Britton code and for different d formula Increment of leftrunning
characteristics angles. This figure indicates the effect of d on length, exit height
and contour of nozzle.
Fig. 4.1: Comparison between two codes and effect of d
From fig. 4.1 one can observe that, a good agreement between the results obtained
by Britton [20] and the results for = . This result indicates that the
present code has high ability to get the contour of divergent part. Also fig. 4.1
indicates that the hypothesis = is high accuracy, also the present
code gives very good results according to Brittons code where the difference does
not exceed 10% of Brittons code results.
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0 0.05 0.1 0.15 0.2 0.25
d=_maxnum
d_1=_maxfix(_max) d_2=(fix(_max))/(num1)
d= (_max_i)/(num1)
Britton's code

39
43 Accuracy of Present Code with GA10 Nozzle:
GA10 was selected as case study to validate the present code, as explained
in Appendix D, GA10 has constant exit height (30mm) and width (25mm) and
divergent part length equal to 191mm but its throat height is variable. As
explained in table 4.2 the mach numbers and mass flow rate are changed according
to the variation in throat height.
Table 4.2 shows a comparison between the dimensions of GA10 supersonic nozzle
and supersonic nozzle designed by present code. From this table, it is clearly
appeared that the present code, gives dimension values very close to those
available for GA10 supersonic nozzle where the percentage of error does not
exceed 2% for all exit heights, and percentage of error does not exceed 5% for all
mass flow rate. But length obtained by present code is more less than that for
GA10 because present code used minimum length nozzle (MLN).

40
Ta
ble
4.2
: C
om
pari
son
Bet
wee
n P
rese
nt
Cod
e and
GA
10
No
zzle
Len
gth
Of
Sup
erso
nic
Par
t(m
m)
(pre
sen
t co
de)
40
.0
43
.4
46
.5
49
.6
52
.6
55
.7
58
.7
Acc
ura
cy
of
Mas
s
Flo
w
Rat
e
%
95
.4
95
.4
95
.26
95
.4
95
.4
95
.44
95
.4
Mas
s Fl
ow
Rat
e
(Kg/
s)
(pre
sen
t co
de)
0.1
95
3
0.2
03
5
0.2
15
7
0.2
26
5
0.2
36
4
0.2
38
7
0.2
37
9
Mas
s Fl
ow
R
ate
(Kg/
s)
(GA
10
)
0.2
04
7
0.2
13
3
0.2
26
1
0.2
37
4
0.2
47
8
0.2
50
1
0.2
49
3
Acc
ura
cy
of
Exi
t H
eigh
t
%
98
.3
98
98
98
98
.3
98
.66
99
Exit
H
eigh
t (m
m)
(pre
sen
t
cod
e)
29
.5
29
.4
29
.4
29
.4
29
.5
29
.6
29
.7
Ae/
A*
(pre
sen
t
cod
e)
1.8
03
3
2.1
49
2
2.5
82
7
3.1
20
6
3.7
82
5
4.5
91
6
5.5
75
0
Ae/
A*
(GA
10
)
1.8
369
2.1
931
2.6
367
3.1
830
3.8
498
4.6
573
5.6
278
Tota
l P
ress
ure
(bar
)
(GA
10
) &
(p
rese
nt
cod
e)
2.0
5
2.5
5
3.2
5
4.1
2
5.2
6.3
5
7.6
5
Thro
at
Hei
ght
(m)
0.0
163
32
0.0
136
79
0.0
113
78
0.0
094
25
0.0
077
93
0.0
064
42
0.0
053
3
Exit
M
ach
No
.
2.1
2.3
2.5
2.7
2.9
3.1
3.3

41
44 Effect of Number of Characteristic Lines on Nozzle Geometry:
Table 4.3 shows the variables input and output from present code. Present code
used the GA10 variables as input variable.
Table 4.3: Program Variable Description and Outputs
Input Output
Program Variable Value Variable Value
Gamma 1.4 M_Inlet 0.2133
M_e 3.1 P_Inlet [pa] 615190
h_th [m] 0.006442 mmax [kg/s] 0.2387
Width [m] 0.025 P_Exit [pa] 14890
Po [pa] 635000 Hexit [m] 0.0288
In Fig. 4.2 we made the tracing of three cases for different exit Mach number. Fig.
4.2.a presents a large grid, for Num=5. We notice that the nozzle wall is badly
presented in the vicinity of the throat, as well as a broad space between the sonic
line and first regular C. Fig. 4.2.b contains a large grid with moderated refinement
for Num=10, the wall shape in the vicinity of the throat is badly introduced still.
But the bad presentation is less compared to the Fig. 4.2.a. Here the distance
between the sonic line and the first regular C is decreased a little but remains large
compared to the other distance between successive C . Fig. 4.2.c contains a grid
with moderated refinement for N=20. We always notice, in spite of the number of
point is raised enough; the wall shape in the vicinity of the throat is badly
introduced still. Also the distance between the sonic line and the first regular C is
decreased a little but remains large compared to the other distances between
successive C .

42
In conclusion, it is clear that, if the number of wall points is large, we will have a
good wall contour.
(a): Grid for Num=5. (b): Grid for Num=10. (c): Grid for N=20.
Fig. 4.2: MLN contour & Grids Characteristics Line.
In fig. 4.3.a, the solid curve shows the minimum length supersonic nozzle (MLN)
contour and the thin lines showing characteristic lines (twenty lines), which are
found on the side of supersonic because the method of characteristic is not used in
subsonic region. Fig. 4.3.b represents the variation of Mach number and pressure
ratio along supersonic part of nozzle. Fig. 4.4.a show the distribution of
characteristic lines and grid points and fig. 4.4.b illustrates the intersection of
0.04 0.02 0 0.02 0.04 0.06 0.080
0.005
0.01
0.015
0.02
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.060
0.005
0.01
0.015
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.060
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.060
0.005
0.01
0.015
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.060
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
(a)
(b)
(c)

43
characteristic line in interior grid points region, it is clear that the distribution of
grid points are uniform.
Fig. 4.3: MLN contour & Mach numbers & P/Po (Num=20)
Fig. 4.4: characteristic lines distribution (Num=20)
0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.060.015
0.01
0.005
0
0.005
0.01
0.015
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.060
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 0.05 0.060
0.005
0.01
0.015
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.060
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
4 2 0 2 4 6 8 10 12 14 16
x 103
0
0.5
1
1.5
2
2.5
3
3.5
x 103
Nozzle length (m)
Nozzle
heig
ht
(m)
Supersonic Nozzle Design
Char. Lines
0 0.01 0.02 0.03 0.04 0.05 0.060
0.5
1
1.5
2
2.5
3
3.5Supersonic Nozzle Design
Nozzle length (m)
Mach n
um
ber
and P
/Po
Mach Number
P/Po
(b)
(a)

44
45 Effect of Exit Mach number on Nozzle Geometry:
Fig. 4.5 illustrates the effect of three exit Mach numbers on nozzle geometry for
perfect gas and constant throat height for present code. This figure indicates that a
change on both exit area and length when exit Mach number change. From this
figure we can see that increasing in Mach number results increase in both exit area
and length.
Fig. 4.5: Effect of exit Mach number on nozzle geometry
46 Effect of Specific Heat ratio on Nozzle Geometry:
Fig. 4.6 illustrates the effect of three specific heat values for perfect gas and
constant throat height on nozzle geometry. This figure indicates that a change on
both exit area and length when specific heat ratio is change. From this figure we
can see that, each increase of specific heat ratio results decrease in exit area and
length.
0.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
0 0.1 0.2 0.3 0.4
Mach=2
Mavh=2.4
Mach=3

45
Fig. 4.6: Effect of specific heat ratio on nozzle geometry
47 CFD Analysis of Designed Nozzle:
The CFD program was used to simulate the flow and produce Mach contour plots
to evaluate if the flow was shock free and the desired exit Mach was reached.
Using the setup configurations outlined in Section 3.4 and tables 3.2, 3.3, Fig. 4.7
show the typical Mach contours of supersonic nozzle designed for three exits Mach
numbers.
The exit Mach number of code is checked by the Mach contours of the simulation
as well as having FLUENT calculates the Mach number at the exit plane of the
nozzle. The Mach number calculated by FLUENT is compared to the Mach
number calculated by the present code.
0.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
0 0.1 0.2 0.3 0.4
Gamma=1.4
Gamma=1.3
Gamma=1.2

46
Fig. 4.7: Typical Mach Contours for Supersonic Nozzle
(a)
(b)
(c)

47
From Fig. 4.7.a, it is clear that the Mach number at the exit has a maximum of 3.09
and in fig. 4.7.b, the Mach number at the exit has a maximum of 2.52 also from fig.
4.7.c, it is clear that the Mach number at the exit has a maximum of 3.19. % errors
of exit Mach numbers for supersonic nozzle designed by the code when compared
to FLUENT result are illustrated in table 4.4.
From Fig. 4.7, it is clear that there is no shock wave, because Mach lines are not
changed from supersonic to subsonic. Mach number plot will show shock if they
exist in the flow field.
Table 4.4: Comparison between Code and FLUENT Results
Case Exit Mach Number for
% Error Code FLUENT
a 3 3.09 2.9
b 2.5 2.52 0.8
c 3.1 3.19 2.8
Table 4.4 shows the comparison between results of code and FLUENT simulation
for three exits Mach number, the error for each case does not exceed 3% as shown
in the table 4.4.

48
48 Effect of Back Pressure on Flow:
Using the setup configurations outlined in Section 3.4 and table 3.5, Fig. 4.8 shows
the typical Mach contours of supersonic nozzle have a reservoir at the end for
Mach number equal 2.5. It is clear that, there was no shock inside the nozzle and
Mach number at the outlet quite satisfactory so that not less than 2.45.
Fig. 4.8: Typical Mach Contours for Supersonic Nozzle with Reservoir
Using the setup configurations outlined in Section 3.4 and table 3.5, Fig. 4.9 shows
the Mach contours for designed nozzle with reservoir at the exit, for different back
pressures as explained in table 3.4.

49
Fig. 4.9: Typical Mach Contours for Supersonic Nozzle with Reservoir
(c)
(b)
(a)

50
From fig. 4.9.a shows the Mach contours for case one. It is clear that, Mach
number at the outlet differed slightly from the designed case. Fig. 4.9.b shows the
Mach contours for case two. In this case, there is no shock inside the nozzle, but
weak shock formed after exit nozzle. Fig. 4.9.c shows the Mach contours for case
three. In this case, it is clear that, there is a shock formed at the exit of the nozzle.
As explained above, we find that when the back pressure exceeds design pressure
shock is formed at outside of the nozzle as shown in Fig. 4.9.b, when it grew more
pressure shock is formed at the nozzle exit, as shown in Fig. 4.9.c.

51
Chapter five
Conclusion and Recommendations
51Conclusion:
MATLAB code developed in this thesis proves to be a useful tool in creating
supersonic nozzle contours for isentropic, irrotational, inviscid flow.
The contour of nozzle designed shows good agreement with three
assessment methods, the first was Brittons code, the second was GA10
supersonic wind tunnel nozzle and the third was FLUENT simulation.
The present code gives very good results according to Brittons code where
the difference does not exceed 10% of Brittons code results.
The present code gives dimension values very close to those available for
GA10 supersonic nozzle, where the percentage of error does not exceed 2%
for all exit heights, and percentage of error does not exceed 5% for all mass
flow rates.
Increasing in exit Mach number results increase in both exit area and length
but if specific heat ratio increase, exit area and length will decrease.
FLUENT simulation used to check the desired exit Mach number and shock
inside the nozzle for designed nozzle contour.
FLUENT simulation was used for three nozzles (three exits Mach number),
results gives percentage error does not exceed 3% for each case, and there is
no shock wave.
The code developed in this thesis will enable to create other types of nozzle
if we need change the exit Mach number and specific heat ratio.
FLUENT analysis was carried out to investigate the flow field of the nozzle
with reservoir at exit of nozzle for exit Mach number of 2.5. The flow

52
behaviors were analyzed to assist in understanding the change in the flow
conditions.
52 Recommendations:
Upon error analysis of the code developed in this thesis, it is evident that for
isentropic, irrotational, inviscid flow the code is valid and accurate. It is
recommended that:
1. Expanded the code to include the effects of viscosity, entropy change and
rotation in its calculation of a supersonic nozzle contour. This will increase
the codes ability to accurately predict real world flowfields and ultimately
produce even higher efficient nozzle contours.
2. Expanded the code to include the effect of real gas.
3. Further research is devoted to better characterizing the flowfield of
supersonic nozzles.
4. To have a good wall presentation in the vicinity of the throat or even on the
horizontal axis in the vicinity of the throat, use additional inserted C
between the sonic line and the first regular C.

53
References:
[1] Anderson, J., Modern Compressible Flow: with Historical Perspective,
McGrawHill, 2003
[2] Allman, J. G. and Ho_man, J. D., \Design of Maximum Thrust Nozzle
Contours by Direct Optimization Methods," AIAA/SAE Joint Propulsion
Conference, Vol. 14, Aug. 1978.
[3] Benson T. Closed Return Wind Tunnel. National Aero nautics and Space
Administration; 2009 May 07 [cited 2009 October 07]. Website:
http://www.grc.nasa.gov/WWW/K12/airplane/tuncret.html
[4] Benson T. Blowdown Wind Tunnel. National Aeronautics and Space
Administration; 2009 May 07 [cited 2009 October 07]. Website:
http://www.grc.nasa.gov/WWW/K12/airplane/tunblow.html
[5] Ferri, A., \The Method of Characteristics," General Theory of High Speed
Aerodynam ics, edited by W. Sears, The Macmillan Co., 1951
[6] Frederick L. Shope, Contour Design Techniques for Super/Hypersonic Wind
Tunnel Nozzles, AIAA 20063665, AIAA 14th Applied Aerodynamics
Conference, San Francisco, California, June 2006
[7] Guentert, E. C. and Neumann, H. E., \Design of Axisymmetric Exhaust
Nozzles By Method of Characteristics Incorporating A Variable Isentropic
Exponent," NASA TR R33, 1959.
[8] Hart_eld, R. J. and Ahuja, V., \The Closing Boundary Condition for the Axis
Symmetric Method of Characteristics," Joint Propulsion Conference, AIAA, San
Diego, CA, 2011
[9] John, J. E., & Keith, T. G. Gas Dynamics: Third Edition. Upper Saddle River,
NJ: Pearson Prentice Hall, 2006

54
[10] K.M. Pandey and A.P. Singh., CFD Analysis of Conical Nozzle for Mach 3 at
Various Angles of Divergence with Fluent Software, International Journal of
Chemical Engineering and Applications, Vol. 1, No. 2, August 2010, pp.179185.
[11] K. M. Pandey, Surendra Yadav, and A.P.Singh., Study on Rocket Nozzles
with Combustion Chamber Using Fluent Software at Mach 2.1, The 10th Asian
Symposium on Visualization, SRM University, Chennai, March15, 2010, pp. 171
177.
[12] K. M. Pandey, Prateek Shrivastava, K.C.sharma and A.P.Singh., Studies on
Supersonic Flows in the De Laval Nozzle at Mach No. 1.5and its Base Pressure
into a suddenly Expanded Duct, The 10th Asian Symposium on Visualization,
SRM University, Chennai, March15, 2010, pp.624631.
[13] K. M. Pandey, Virendra Kumar, and A.P.Singh., Study on Supersonic Free
Single Jet Flow: A Numerical Analysis with Fluent Software, The 10th Asian
Symposium on Visualization, SRM University, Chennai, March15, 2010, pp.179
184.
[14] K.M. Pandey and A.P. Singh., Design and Development of De Laval nozzle
for Mach 3 & 4 using methods of Characteristics with Fluent Software, ISST
Journal of Mechanical Engineering, Vol.1, Issue 1, 2010, pp. 6172.
[15] Michael R. Vanco and Louis J.Goldman, Computer program for design of
twodimensional supersonic nozzle with sharp edge throat, NASA,
Washington,D.C.,Jan 1968
[16] Pope A., Goin K. High Speed Wind Tunnel Testing. New York: John Wiley
& Sons; 1965.
[17] Rao, G. V. R., \Exhaust Nozzle Contour for Optimum Thrust," Jet Propulsion,
Vol. 28,No. 6, June 1958, pp. 377382.
[18] Rao, G., \Contoured Rocket Nozzles," Proc 9th, 1958.

55
[19] T. Zebbiche and Z. Youbi, Supersonic TwoDimantional Minimum Length
Nozzle Design at High Temperature, EJER, 11 (1), 91102 (2006)
[20] Website: http://www.mathworks.com/matlabcentral/fileexchange/146822d
nozzledesign
[21] Zucrow, M. J. and Ho_man, J. D., Gas Dynamics, Vol II., John Wiley and
Sons, 1977.
[22] Zucrow, M. J. and Ho_man, J. D., Gas Dynamics, Vol II., John Wiley and
Sons, 1977

56
Appendix A
Derivation of the Characteristic and Compatibility Equations
A1 Determination of the Characteristic Lines:
Consider a xy coordinate space that is divided into a rectangular grid, as sketched
in Fig. A.1. The solid circles denote grid points at which the flow properties are
either known or to be calculated. The points are indexed by the letters i in the x
direction and j in the y direction. For example, the point directly in the middle of
the grid is denoted by (i , j), the point immediately to its right is (i + 1, j), and so
forth . It is not necessary to always deal with a rectangular grid as shown in
Fig.A.1, although such grids are preferable for finitedifference solutions. For the
method of characteristics solutions, we will deal with a nonrectangular grid. They
are predicated on the ability to expand the flowfield properties in terms of a
Taylor's series. For example, if ui,j denotes the x component of velocity known at
point (i, j), then the velocity ui+1,j at point (i+ 1, j) can be obtained from
+1, = , +
,
+ 2
2 ,
2
2+ (A.1)

57
Fig. A.1: Rectangular finitedifference [1].
Let us begin to obtain a feeling for the method of characteristics by considering
Fig. A.1 and Eq. (A.1). Neglect the secondorder term in Eq. (A.1), and write
+1, = , +
,
+ (A.2)
The value of the derivative / can be obtained from the general conservation
equations. For example, consider a twodimensional irrotational flow, so that the
velocity potential equation (Eq. (A.3)) yield, in terms of velocities (Eq.(A.4)).
1
2
2 + 1
2
2 + 1
2
2
2
2
2
2
2
2 = 0 (A.3)
1 2
2
+ 1
2
2
2
2
= 0 (A.4)
Solve Eq.(A.4) for /
=
2
2
1
2
2
12 2 (A.5)

58
A line that makes a Mach angle with respect to the streamline direction at a point
as shown in Fig.A.2 is also a line along which the derivative of u is indeterminate,
and across which it may be discontinuous. We have just demon started that such
lines exist, and that they are Mach lines. The choice of u was arbitrary. The
derivatives of the other flow variables, , P, T, v, etc., are also indeterminate along
these lines. Such lines are defined as characteristic lines.
Fig. A.2: Illustration of the characteristic direction [1].
With this in mind, we can now outline the general philosophy of the method of
characteristics. Consider a region of steady, supersonic flow in x y space. This
flowfield can be solved in three steps, as follows:
Step 1. Find some particular lines (directions) in the xy space where flow variables
( , p , T, u, v, etc.) are continuous, but along which the derivatives (/, /,
etc.) are indeterminate, and in fact across which the derivatives may even
sometimes be discontinuous. As already defined, such lines in the xy space are
called characteristic lines.

59
Step 2. Combine the partial differential conservation equations in such a fashion
that ordinary differential equations are obtained that hold only along the
characteristic lines. Such ordinary differential equations are called the
compatibility equations.
Step 3. Solve the compatibility equations step by step along the characteristic lines,
starting from the initial conditions at some point or region in the flow. In this
manner, the complete flowfield can be mapped out along the characteristics. In
general, the characteristic lines (sometimes referred to as the "characteristics net")
depend on the flowfield, and the compatibility equations are a function of
geometric location along the characteristic lines; hence, the characteristics and the
compatibility equations must be constructed and solved simultaneously, step by
step. An exception to this is twodimensional irrotational flow, for which the
compatibility equations become algebraic equations explicitly independent of
geometric location.
To begin the determination of the characteristic line, consider steady,
adiabatic, twodimensional, irrotational supersonic flow. The governing nonlinear
equation is Eqs.(A.3). For twodimensional flow, Eq.(A.3) becomes
1
2
2 + 1
2
2
2
2 = 0 (A.6)
Note that is the full velocity potential
Hence,
= = = +
Recall that = f (x, y); hence
=
+
= + (A.7)

60
=
+
= + (A.8)
Recopy ing these equations,
From Eq. (A.6) 1 2
2 + 1
2
2
2
2 = 0
From Eq. (A.7) () + () =
From Eq. (A.8) () + () =
These equations can be treated as a system of simultaneous, linear, algebraic
equations in the variables , . For example, using Cramer's rule, the
solution for is
=
1
2
20 1
2
2
00
1
2
2
2
21
2
2
00
=
(A.9)
To calculate the equations of the characteristic lines, from Eq.(4.9) set D =0. This
yields:
1 2
2 2 +
2
2 + 1
2
2 2 = 0
1 2
2
2
+
2
2
+ 1 2
2 = 0 (A.10)
In Eq. (A.10), is the slope of the characteristic lines. Using the
quadratic formula, Eq. (A.10) yields

61
=
2 2 2 2 2 4 1 2 2 1 2 2
2 1 2 2
= 2 2+2 2 1
1 2 2 (A.11)
Equation (A.11) defines the characteristic curves in the physical xy space.
Examine Eq. (A.11) more closely. The term inside the square root is
2 + 2
2 1 =
2
2 1 = 2 1
Hence, we can state
1. If M > 1, there are two real characteristics through each point of the f1owfield.
Moreover, for this situation, Eq. (A.6) is defined as a hyperbolic partial differential
equation.
2. If M = 1, there is one real characteristic through each point of the flow. By
definition, Eq. (A.6) is a parabolic partial differential equation.
3. If M < 1, the characteristics are imaginary, and Eq.(A.6) is an elliptic partial
differential equation.
Therefore, we see that steady, inviscid supersonic flow is governed by hyperbolic
equations, sonic flow by parabolic equations, and subsonic flow by elliptic equations.

62
Moreover, because two real characteristics exist through each point in a flow
where M > 1, the method of characteristics becomes a practical technique for
solving supersonic flows. In contrast, because the characteristics are imaginary for
M < 1, the method of characteristics is not used for subsonic solutions. (An
exception is transonic flow, involving mixed subsonicsupersonic regions, where
solutions have been obtained in the complex plane using imaginary
characteristics.).
Concentrating on steady, twodimensional supersonic flow, let us examine the real
characteristic lines given by Eq. (A.11). Consider a streamline as sketched in Fig.
A.3. At point A, = and = .Hence, Eq.(A.11) becomes
=
2
2
2
2 2+ 2 1
12
2 2
(A.12)
Fig. A.3: Streamline geometry [1].
Recall that the Mach angle is given by = 1 1 , or = 1 . Thus,
2 2 = 2 = 1 2 , and Eq. (A.12) becomes
=
2
2+ 2
21
1 2
2
(A.13)

63
From trigonometry,
2+ 2
2 1 =
1
2 1 = 2 1 = 2 =
1
Thus, Eq. (4.13) becomes
= 21
1 2 2 (A.14)
After more algebraic and trigonometric manipulation, Eq. (A.14) reduces to
= (A.15)
A graphical interpretation of Eq.(A.15) is given in Fig.A.4, which is an elaboration
of Fig.A.3. At point A in Fig.A.4, the streamline makes an angle with the x axis.
Equation (A.15) stipulates that there are two characteristics passing through point
A, one at the angle above the streamline, and the other at the angle below the
streamline. Hence, the characteristic lines are Mach lines. Also, the characteristic
given by the angle + is called a C+ characteristic; it is a leftrunning. The
characteristic in Fig.A.4 given by then angle is called a C characteristic; it is
a rightrunning characteristic. Note that the characteristics are curved in general,
because the flow properties (hence ) change from point to point in the
flow.

64
A2 Determination of the Compatibility Equations:
In essence, Eq. (A.9) represents a combination of the continuity, momentum, and
energy equations for twodimensional, steady, adiabatic, irrotational flow. In above
section, we derived the characteristic lines by setting D = 0 in Eq. (A.9). In this
section, we will derive the compatibility equations by setting N = 0 in Eq. (A.9).
When N = 0, the numerator determinant yields
1 2
2 + 1
2
2 = 0
=
1 2 2
1 2 2 (A.16)
=
Substituting Eq. (A.11) into (A.16), we have
=
12
2
12
2
2
2+2
21
12
2
This simplifies to:
=
2
2+2
21
12
2
(A.17)
Recall that, u = V cos and v = V sin.Then, Eq. (A.17) becomes
=
2 21
12 2
This, after some algebraic manipulations, reduces to
= 2 1
(A.18)
Equation (A.18) is the compatibility equation, i.e., the equation that describes the
variation of flow properties along the characteristic lines. From a comparison with
Eq. (A.15), we note that

65
= 2 1
( (A.19)
= + 2 1
( + (A.20)
Eq. (A.18) can be integrated to give the PrandtlMeyer function ( ) as will
display in Eq. (A.31).Therefore, Eqs. (A.19) and (A.20) are replaced by the
algebraic compatibility equations:
+ = = (A.21)
= = + + (A.22)
Integration of Eq. (A.18)
= 2 1
21
21
(A.23)
The integral on the righthand side can be evaluated after dV/V is obtained in terms
of M, as follows. From the definition of Mach number,
= (A.24)
Hence ln = ln + ln
Differentiating Eq. (4.24)
=
+
(A.25)
Specializing to a calorically perfect gas, the adiabatic energy equation can be
written as:
2=
= 1 +
1
22
or, solving for a,
= (1 +1
22)1 2 (A.26)
Differentiating Eq.(A.26)
=
1
2 (1 +
1
22)1 (A.27)

66
Substituting Eq. (A.27) into (A.25), we obtain
=
1
1+1
22
(A.28)
Equation (A.28) is the desired relation for in terms of M; substitute it into
Eq. (A.23):
= 2 0 = 21
1+1
22
21
21
(A.29)
In Eq. (A.29), the integral term
= 21
1+1
22
(A.30)
is called the PrandtlMeyer function, and is given the symbol . Performing the
integration, Eq. (A.30) becomes
= +1
11
1
+1(2 1) 1 2 1 (A.31)
Finally, we can now write Eq. (A.29), combined with (A.30), as
2 = 2 1
= (A.32)
Method of Characteristics analysis for this project used above equations; all
equations taken from chapter 11 of Modern Compressible Flow with Historical
Perspective  Third Edition  John D. Anderson [1].

67
A3 Angle of Sharp Corner:
Figure A.4: MinimumLength Nozzle [1]
Let VM be the PrandtlMeyer function associated with the design exit Mach
number. Hence, along the C+ characteristic cb in Fig.A.4, = = = Now
consider the C_ characteristic through points a and c. At point c, from Eq. (A.21),
+ = () (A.33)
However, = 0 and = . Hence, from Eq. (A.33),
() = (A.34)
At point a, along the same C_ characteristic ac, from Eq. (A.21),
wmax ,ML + a = (K)a (A.35)
Since the expansion at point a is a PrandtlMeyer expansion from initially sonic
conditions, we know that a = wmax ,ML Hence, Eq. (A.35) becomes
wmax ,ML =1
2(K)a (A.36)
However, along the same C_ characteristic, (K_ )a = ( K _ )c; hence, Eq.(A.36)
becomes

68
wmax ,ML =1
2(K)c (A.37)
Combining Eqs. (A.34) and (A.37), we have
wmax ,ML =M
2 (A.38)

69
Appendix B
MATLAB Supersonic Nozzle Design Codes
B1 PG.m Program:
Figure B.1: PG.m MATLAB Source Code

70

71

72

73

74
B2 nozzle_plot1.m Program:
Figure B.2: nozzle_plot1.m MATLAB Source Code

75
B3 nozzle_plot2.m Program:
Figure B.3: nozzle_plot2.m MATLAB Source Code
B4 nozzle_plot3.m Program:
Figure B.4: nozzle_plot3.m MATLAB Source Code

76
B5 nozzle_plot4.m Program:
Figure B.5: nozzle_plot4.m MATLAB Source Code

77
B6 Subsonic_part.m Program:
Figure B.6: Subsoic_part.m MATLAB Source Code

78
B7 nozzle_CFD.m Program:
Figure B.7: nozzle_CFD.m MATLAB Source Code

79

80
Appendix C
Tables
Table C.1
Brittons Code Designed Code
x y x y 0 0.0125 0 0.0125
0.0086 0.0165 0.0274 0.0252
0.0284 0.0249 0.0361 0.0289
0.0344 0.0272 0.0418 0.0311
0.0418 0.0299 0.0479 0.0332
0.0497 0.0324 0.0546 0.0353
0.0583 0.0349 0.062 0.0374
0.0678 0.0373 0.0702 0.0394
0.0784 0.0396 0.0793 0.0415
0.0904 0.0419 0.0896 0.0434
0.1039 0.044 0.1011 0.0453
0.1193 0.0459 0.1142 0.047
0.1369 0.0475 0.1289 0.0485
0.157 0.0488 0.1457 0.0497
0.1801 0.0495 0.1648 0.0505
0.2078 0.0499 0.1865 0.0508
0 0.0125 0 0.0125
0.0086 0.0165 0.0274 0.0252
0.0284 0.0249 0.0361 0.0289
0.0344 0.0272 0.0418 0.0311
0.0418 0.0299 0.0479 0.0332
0.0497 0.0324 0.0546 0.0353
0.0583 0.0349 0.062 0.0374
0.0678 0.0373 0.0702 0.0394
0.0784 0.0396 0.0793 0.0415
0.0904 0.0419 0.0896 0.0434
0.1039 0.044 0.1011 0.0453
0.1193 0.0459 0.1142 0.047
0.1369 0.0475 0.1289 0.0485
0.157 0.0488 0.1457 0.0497
0.1801 0.0495 0.1648 0.0505
Table C.1 shows the data used in figure 4.1 (Comparison between two codes)

81
Table C.2
M = 2 M = 2.4 M = 3
x y x y x y 0 0.0125 0 0.0125 0 0.0125
0.0191 0.0179 0.0235 0.0223 0.0297 0.0309
0.0231 0.0189 0.0298 0.0247 0.041 0.0374
0.0256 0.0195 0.0339 0.0261 0.0488 0.0413
0.028 0.02 0.038 0.0274 0.0574 0.0452
0.0303 0.0205 0.0424 0.0286 0.0671 0.0491
0.0328 0.0209 0.0471 0.0298 0.0782 0.0532
0.0352 0.0213 0.052 0.031 0.0908 0.0573
0.0378 0.0216 0.0574 0.032 0.1054 0.0615
0.0405 0.022 0.0632 0.033 0.1222 0.0656
0.0432 0.0222 0.0695 0.0339 0.1415 0.0696
0.0461 0.0225 0.0763 0.0348 0.1639 0.0734
0.0491 0.0227 0.0837 0.0354 0.1899 0.0768
0.0523 0.0228 0.0918 0.036 0.22 0.0795
0.0556 0.0229 0.1006 0.0363 0.255 0.0815
0.0591 0.0229 0.1102 0.0364 0.2957 0.0822
0 0.0125 0 0.0125 0 0.0125
0.0191 0.0179 0.0235 0.0223 0.0297 0.0309
0.0231 0.0189 0.0298 0.0247 0.041 0.0374
0.0256 0.0195 0.0339 0.0261 0.0488 0.0413
0.028 0.02 0.038 0.0274 0.0574 0.0452
0.0303 0.0205 0.0424 0.0286 0.0671 0.0491
0.0328 0.0209 0.0471 0.0298 0.0782 0.0532
0.0352 0.0213 0.052 0.031 0.0908 0.0573
0.0378 0.0216 0.0574 0.032 0.1054 0.0615
0.0405 0.022 0.0632 0.033 0.1222 0.0656
0.0432 0.0222 0.0695 0.0339 0.1415 0.0696
0.0461 0.0225 0.0763 0.0348 0.1639 0.0734
0.0491 0.0227 0.0837 0.0354 0.1899 0.0768
0.0523 0.0228 0.0918 0.036 0.22 0.0795
0.0556 0.0229 0.1006 0.0363 0.255 0.0815
0.0591 0.0229 0.1102 0.0364 0.2957 0.0822
Table C.2 shows the data used in figure 4.5 (Effect of exit Mach number on nozzle
geometry)

82
Table C.3
Gamma = 1.4 Gamma = 1.3 Gamma = 1.2
x y x y x y 0 0.0125 0 0.0125 0 0.0125
0.0274 0.0252 0.0284 0.0275 0.0297 0.0309
0.0361 0.0289 0.0381 0.0323 0.041 0.0374
0.0418 0.0311 0.0446 0.0351 0.0488 0.0413
0.0479 0.0332 0.0517 0.0378 0.0574 0.0452
0.0546 0.0353 0.0596 0.0406 0.0671 0.0491
0.062 0.0374 0.0684 0.0434 0.0782 0.0532
0.0702 0.0394 0.0782 0.0462 0.0908 0.0573
0.0793 0.0415 0.0894 0.049 0.1054 0.0615
0.0896 0.0434 0.1021 0.0517 0.1222 0.0656
0.1011 0.0453 0.1165 0.0543 0.1415 0.0696
0.1142 0.047 0.1329 0.0567 0.1639 0.0734
0.1289 0.0485 0.1517 0.0589 0.1899 0.0768
0.1457 0.0497 0.1733 0.0606 0.22 0.0795
0.1648 0.0505 0.198 0.0618 0.255 0.0815
0.1865 0.0508 0.2265 0.0623 0.2957 0.0822
0 0.0125 0 0.0125 0 0.0125
0.0274 0.0252 0.0284 0.0275 0.0297 0.0309
0.0361 0.0289 0.0381 0.0323 0.041 0.0374
0.0418 0.0311 0.0446 0.0351 0.0488 0.0413
0.0479 0.0332 0.0517 0.0378 0.0574 0.0452
0.0546 0.0353 0.0596 0.0406 0.0671 0.0491
0.062 0.0374 0.0684 0.0434 0.0782 0.0532
0.0702 0.0394 0.0782 0.0462 0.0908 0.0573
0.0793 0.0415 0.0894 0.049 0.1054 0.0615
0.0896 0.0434 0.1021 0.0517 0.1222 0.0656
0.1011 0.0453 0.1165 0.0543 0.1415 0.0696
0.1142 0.047 0.1329 0.0567 0.1639 0.0734
0.1289 0.0485 0.1517 0.0589 0.1899 0.0768
0.