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  • 1

    Chapter one

    Introduction

    1-1 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

    two-dimensional and axisymmetric flow [14].

    1-2 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

    1-3 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 well-known

    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 non-isentropic,

    viscid supersonic flow as recommendations.

  • 3

    Chapter two

    Background and Literature review

    2-1 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.

    2-1-1 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].

    2-1-2 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 closed-circuit 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 mid-to-high 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

    2-1-3 Method of Characteristics:

    The Method of Characteristics (MOC) is a numerical procedure appropriate for

    solving two-dimensional 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 Levi-Civita 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 two-dimensional, 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].

    2-2 Literature review:

    2-2-1 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 quasi-one-dimensional 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 two-dimensional nozzle problems. A comprehensive

    presentation on both the graphical and the analytical approach to two-dimensional

    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 axis-symmetric 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

    second-degree 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

    2-2-3 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.

    2-2-4 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 air-fuel 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 axi-symmetric 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

    3-1 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 two-dimensional (2-D) nozzles, and the primary focus here,

    where two opposite walls are contoured in a converging-diverging shape but are

    bounded by parallel walls, giving a rectangular cross section at every station. The

    2-D 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 2-D 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 slit-throat 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 direct-connect scramjet

    combustor testing. For such an application, four sides of a near-rectangular cross-

    section nozzle may be contoured to deliver uniform flow. A fourth type of nozzle,

    used at least once in an arc-heated 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 minimum-length 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

    2-D 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 high-enthalpy facilities, such long nozzles produce large losses of

  • 14

    the often elaborately achieved total enthalpy. For arc-heated 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 flow-field solution for each

    contour perturbation

  • 15

    3-2 Methodology:

    To expand a gas from rest to supersonic speed, a convergent- divergent nozzle

    should be used. Quasi-one dimensional analyses predict the flow properties as a

    function of x through a nozzle of specified shape. Although quasi-one dimensional

    analysis represents the properties at any cross section as an average of the flow

    over a given nozzle cross-section, 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 subsonic-supersonic regions, where

    solutions have been obtained in the complex plane using imaginary

    characteristics.).

    3-2-1 Method of Characteristics: There are several ways to derive a method of characteristics. In one approach, the

    2-D, 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 Prandtl-Meyer 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 right-running 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.

    3-2-2 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

    3-2-2-1 Initial Line:

    The Initial Curve is the convex portion (before the inflexion point) of the

    expansion curve. Prandtl-Meyer 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

    left-running 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 left-running characteristic C-

    equation(3.2) between point 1 and 3, and equation of Right-running characteristic

    C+ equation(3.3) between point 3 and 2, points 4, 5, ., (num+1) found as point 3.

    To draw characteristic lines (1-2), (1-3),(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).

    3-2-2-2 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 left-running characteristic C- eq. (3.2)

    only because; there is no Right-running characteristic C+

    To draw characteristic lines (a-(num+3)) (q-1) 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 q-1

    q

    Figure 3.3: Centerline points

    a

  • 19

    3-2-2-3 Interior Points:

    In this case, interior points are coming from intersection the left-running

    characteristic C- and Right-running 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

    q-1

    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

    3-2-2-4 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

    q-1

    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.

    3-3 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.

    3-4 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: Green-Gauss Cell Formulation: Implicit Time: Steady Porous Formulation: Superficial Velocity Energy Equation: Checked Viscous Model: k-epsilon 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: Green-Gauss Cell Formulation: Implicit Time: Steady Porous Formulation: Superficial Velocity Energy Equation: Checked Viscous Model: k-epsilon 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

    3-5 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), Prandtl-Meyer

    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 1-D 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 0.2-0.06

    -0.04

    -0.02

    0

    0.02

    0.04

    0.06

    Nozzle length (m)

    Nozzle

    heig

    ht

    (m)

    Supersonic Nozzle Design

    Char. Lines

  • 29

    3-5-1 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(T-1)+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=(gamma-1)/(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=j-1

    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+j-1))

    L=L+j M=M+j-1 j=j-1

    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=j-1

    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

    4-1 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.

    4-2 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 left-running

    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=_max-fix(_max) d_2=(fix(_max))/(num-1)

    d= (_max-_i)/(num-1)

    Britton's code

  • 39

    4-3 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

    4-4 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 0.06-0.015

    -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 10-3

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    x 10-3

    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

    4-5 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

    4-6 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

    4-7 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

    4-8 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

    5-1Conclusion:

    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.

    5-2 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,

    McGraw-Hill, 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/K-12/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/K-12/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 2006-3665, 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 R-33, 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.179-185.

    [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, March1-5, 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, March1-5, 2010, pp.624-631.

    [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, March1-5, 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. 61-72.

    [15] Michael R. Vanco and Louis J.Goldman, Computer program for design of

    two-dimensional 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. 377-382.

    [18] Rao, G., \Contoured Rocket Nozzles," Proc 9th, 1958.

  • 55

    [19] T. Zebbiche and Z. Youbi, Supersonic Two-Dimantional Minimum Length

    Nozzle Design at High Temperature, EJER, 11 (1), 91-102 (2006)

    [20] Website: http://www.mathworks.com/matlabcentral/fileexchange/14682-2-d-

    nozzle-design

    [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

    A-1 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 finite-difference 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 finite-difference [1].

    Let us begin to obtain a feeling for the method of characteristics by considering

    Fig. A.1 and Eq. (A.1). Neglect the second-order 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 two-dimensional 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 two-dimensional 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, two-dimensional, irrotational supersonic flow. The governing nonlinear

    equation is Eqs.(A.3). For two-dimensional 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 subsonic-supersonic regions, where

    solutions have been obtained in the complex plane using imaginary

    characteristics.).

    Concentrating on steady, two-dimensional 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 left-running. The

    characteristic in Fig.A.4 given by then angle is called a C- characteristic; it is

    a right-running 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 two-dimensional, 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 Prandtl-Meyer 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 right-hand 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 Prandtl-Meyer 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

    A-3 Angle of Sharp Corner:

    Figure A.4: Minimum-Length Nozzle [1]

    Let VM be the Prandtl-Meyer 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 Prandtl-Meyer 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

    B-1 PG.m Program:

    Figure B.1: PG.m MATLAB Source Code

  • 70

  • 71

  • 72

  • 73

  • 74

    B-2 nozzle_plot1.m Program:

    Figure B.2: nozzle_plot1.m MATLAB Source Code

  • 75

    B-3 nozzle_plot2.m Program:

    Figure B.3: nozzle_plot2.m MATLAB Source Code

    B-4 nozzle_plot3.m Program:

    Figure B.4: nozzle_plot3.m MATLAB Source Code

  • 76

    B-5 nozzle_plot4.m Program:

    Figure B.5: nozzle_plot4.m MATLAB Source Code

  • 77

    B-6 Subsonic_part.m Program:

    Figure B.6: Subsoic_part.m MATLAB Source Code

  • 78

    B-7 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.