American Institute of Aeronautics and Astronautics1

Supersonic Plug Nozzle Design

Toufik Zebbiche *

Department of Aeronautics, University of Blida, Blida, B.P. 270 Ouled Yaich, Algeria

The aim of this work is to calculate the flow parameters and to trace suitably the profilesof the supersonic nozzle, so as to obtain an uniform and parallel flow at the exit section, andto have for the same exit Mach number, several shapes by changing the repport of specificheats γ. The nozzle chosen in this study is the plug nozzle with a central body. Theperformances obtained like the length and the weight are better compared to the other typesin particular the Minimum Length Nozzle. Our field of study is limited in the supersonicregim not to have the dissociation ofthe molecules. The design method is based on the use ofthe Prandtl Meyer function of a gas thermally and calorifiquement perfect. The flow is nothorizontal at the col, and consequently is inclined comparaed to the horizontal. Thecomparaison will be made with the minimum length nozzle type.

Nomenclatureθ = flow deviation compared to the horizontal.M = Mach number.γ = specific heats repport.µ = angle of Mach.φ = polar angle of a Mach wave.λ = polar ray of a Mach wave.x, y = cartesian co-ordinates of a point.Ψ = deviation of the Lip compared to the vertical.ν = Prandtl Meyer function.T = temperature.P = static pressure.ρ = density.A = aire of a section.tM = thickness of structural material of a plug nozzle.ρM = density of a structural material of a plug nozzle.L = length of a nozzle.xplug = distance between the exit section and the Lip.Fx = axial pressure force exerced on the wall of the central body.ε = relative error given by the computation.N = number of the discretization points of the nozzle wall.σ = interpolation coefficient of the pressure of the wall. For the application, wa take σ=0.5.α, β = angles respectively at tops A and i of the triangle connecting the points A, i and i+1 of the fig. 6.l = unit of nozzle depth.0 = index for chamber condition.* = index for critical condition.E = index for exit section.i, (i) = indexs respectively for nodes and segments of the nozzle wall.

I. IntroductionN the aeronautical construction industry, one is always interested to design engines having minimum length andweight. For the manufacture of the missiles, the large part of the mass is devoted for nozzle. An improvement in

mass wants to say a minimization of the weight of the machine and consequently, we can use this profit to embark

* Assisstant professor, Department of Aeronautics, University of Blida, Algeria, z_toufik270169@yahoo.fr

I

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit10 - 13 July 2005, Tucson, Arizona

AIAA 2005-4490

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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Figure 1. Geometry of the plug nozzle.

Symmetryaxis

Exit (Tip)y

Central

Body

Lip

x

ColFlow space

Figure 5. Discretisation of the expansion zone.

N

B

E

12

34

i-1i i+1

xy

LipA

other useful apparatuses or to increase the mass of the fuel which gives us a chance to increase the time of autonomyof the flight. From this study, we will conceive the shape of very recent supersonic nozzle, which answers well ourwish of improvement of the performances. This type of nozzle is named by plug nozzle with central body, presentedin the fig. 1. The shape of the central body will be obtained in such manner to obtain an uniform and parrallel flowat the exit section1-5. The application will be for various values of the repport γ of the perfect gas for goal to designnozzles of blower and rocket motors. The application will be for different type of gas for goal to design nozzles ofsouffelerie and rocket motors.

The nozzle is consisted of convergent and divergentdependent between them by the sonic col. Our study is basedfor the divergent part. The subsonic part is used to give a sonicflow to the col. The difference between our model and theother models is that the flow at tthe col is inclined of anangle θ * compared to the horizontal as indicates the fig. 2 what is not the case for other type, where the flow ishorizontal4. Consequently, the Lip must be tilted of an angle Ψ compared to the vertical as the fig. 3 shows it. Forthe other forms, the flow is horizontal at the col. The flow is permanent, two-dimensional, and irrotational. Theapplication is limited in the supersonic field until Mach number ME≤5.00 not to have the dissociation of themolecules. The stream line determined by calculation will be replaced by a rigid surface limiting the flow field andwhich represents the shape of the central body.

II. Mathematical formulation of the problemThe flow at the col and exit section is one directional, the repot of critical sections remains always valid and it is

taken to compare the numerical calculations found by our model and the theory. The calculation of the flow is ratherdelicate, since the shape of the nozzle is unknown a priori. The required form of the central boby accelerates theflow of a Mach number M=1.00 at the col until Mach number ME at the exit section. As the flow deviation is notnull at the col, the flow through the central body redresse only from θ=θ* to θ=0. The calculation of the flow and thedetermination of the central body wall for a perfect gaz are based on the Prandtl Meyer function shown in Réf. 1, by:

21221221]1arctg[)]1)(1)/(1arctg[()]11[(

///MMγγ)/(γγν(M) −−−+−−+= (1)

The angle ν is measured compared to the velocity vector of the col.On the fig. 4, line AB and AE respectively presents the Mach waves of the col and exit section. These lines are

inclined respectively by the angles µB=90 degree and µE=arcsin[1/ME] < 90 degree.

Figure 2. Flow at the col and the exit section.

θE = 0ME>1

Col

θ* > 0M* = 1

Wall

Exit

Figure 3. Presentation of the angle Ψ.

Flow deviation at the col

θ*>0

Lip

θE=0

Ψ

yx

Flow deviationat the exit section

AA

Figure 4. Angle of Mach at the col and exit section.

θE=0

θ*=θB

µB=90

yx

Lip

A

BE

νEµE

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Figure 6. Parameters of an intermediate Machline connecting the points A and i (i=2, 3, …, N).

Between these two lines, there is an infinity of Mach waves, centered, exit of point A as shown in fig. 5. Eachline gives a Mach number, which we can easily deduct a point on the wall. As the gas is perfect, the velocity vectoris tangent with a stream line, which will be regarded as the contour of the central body wall to required.

The Proprieties of the flow like, Mach number, flow deviation, thermodynamics reports, of pressures,temperatures and density are constant along each line of Mach exits of the point A, which will be absorbed by thewall of central body2. The fig. 6 presents the parameters of an intermediate Mach line connecting the points A and i.The angle θB is unknown a priori. The design is considered on the basis of the exit Mach number. To have anuniform and parallel flow at the exit, it is necessary to incline the flow at the col of an angle θB by:

θB=νE=ν(ME) (2)

The slope of a Lip compared to the vertical will be calculated by:

Ψ = 90 - νE (3)

A. DiscretizationLet us share the zone of expansion ranging between lines AB and AE to N waves, including the two ends, and

number these waves from the left towards the right, we obtains the diagram presented on the fig. 5. The referencemark of calculation is placed at point A. The calculation and the determination of the central body contour isindependent of the choise of positioning of the referencemark considering the nozzle is two-dimensional.

More number N of Mach waves is large, more weobtain a very good presentation of the central body. Weprefer in our study to begin the calculation of point A ofthe col towards the point E of exit.

The determination of the wall points is done in explicitway. If we know the position and the properties of a pointon the wall, we can easily determine those of the adjacentpoint on the right until where we arrive at the exit sectionpoint. Let us note here that the waves of Mach are straightlines.

To manage to design the nozzle, we choise thediscretization of the zone of variation of the Mach number1.00≤M≤ME by N values so that calcultations are fast.

The diagram of the model under the presence of a line of Mach is illustrated in the fig. 6. The Mach number Mi

at the point i is known. In that case we can write:

)M/(µ ii 1arcsin= (4)

)ν(Mν ii = (5)

iiiµνΨφ +−−= 90 (6)

iiiµφθ −= (7)

On the fig. 6, the properties Mi , θi , νi , xi ,and yi at the point i are known, and the problem becomes thedetermination of these properties at the point i+1. On the triangle connecting the points A, i and i+1, we can write :

iEiννφπα −+−= (8)

iEiννφ +−= +1

β (9)

( ) )sin(/)sin(//1 βαλλλλ BiBi =+ (10)

By analogy with Eqs. (4), (5), (6), and (7), we can deduce the relations for the point i+1 by changing index i byi+1. In this point, the Mach number Mi+1 is known. The position of point i+1, in adimensional form, is given by:

i+1

Ψθ φiµiνi

θiB

Ei

LipA

yE

Axis of symmetry

λi

American Institute of Aeronautics and Astronautics4

( ) )(cos// 111 +++ = iBiBix ϕλλλ (11)

( ) )(sin// 111 +++ = iBiBiy ϕλλλ (12)

Each point i on the wall has its own Mach number, different at the other points, and which all, are connectedwith point A by a Mach line. Then, at point A, there is an infinity of values of Mach number and consequently, thepoint A is a discontinuity point of properties M, et θ… etc.

B. Procedure de calculationThe first stage consists to determining some results necessary for the design:

1. Critical reportsFor a perfect gas, the critical reports T*/T0 , ρ*/ρ0 , and P*/P0 are presented in Réf.1.

2. Ratios thermodynamics at the exit sectionThe thermodynamics reports TE/T0 , ρE/ρ0 , and PE/P0 of a perfect gas corresponding to exit Mach number are

presented in Ref. 1.3. Theoretical report of critical sections

From the Ref. 2, it is given by the following relation :

( ) ]12[1211

*2111

)(γ)/(γ

EEEM)/(γ)(γM/AA

−+−−

−++= ][ (13)

This report will be useful to us like a source of comparaison of validation of our numerical calculations.4. Value of Prandtl Meyer function corresponding to the Mach number

The value of the of Prandtl Meyer function νE is given by Eq.(2).The deviation of the Lip compared to the vertical will be calculated by using Eq.(3).Such as the process of calculation is related to two successive points, it is necessary to give the results for the

starting point. The starting point is that the point B In this point, we have:1) The Mach number is equal to MB=1.00. Sonic entry.2) The angle of Mach is equal to µB=90.0 degree.3) The value of Prandtl Meyer function is equal to νB=0.0 degree.4) The polar angle is equal to φB=90-Ψ-νB+µB.5) The polar ray is equal to λB=1.00 (among the data).6) Position of the first point of the wall is given by :

)(/x BBB ϕλ cos= (14)

)(/y BBB ϕλ sin= (15)

7) The angle of deviation of the flow at the col is given by :

BBB µϕθθ −==* (16)

8) The theoretical adimensional ray of the exit section is given by using Eqs. (13) and (15) by:

)/A) (A/(y/y EBBBE *λλ = (17)

The same value will be computed by the numerical model by using Eq.(20) like the last point of calculation.The second stage consists in assigning the results obtained to the point B like the first point of numerical

calculation for i=1.For each line of Mach, it is necessary to know the Mach number in the center of expansion A which represents

also the Mach number on the wall. Thus, the lines of Mach represent the curves isoMach. Such as the number ofselected point is equal to N, we obtain N -1 of panels, from where, the Mach number as in point i is given by:

( ) , ..., N),,(iNMi -M Ei 321)1/()1(11 =−−+= ][ (18)

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By incrementing the meter of i=2 to N, we can determine the thermodynamic and physical properties along of allline of Mach selected. Consequently the shape of the central body will be obtained.

From the last point, we can fix the following results:1) The position of the point E of the exit section is given in adimensional form by:

BNBE xx λλ // = (19)

BNBE yy λλ // = (20)

2) The axial distance between the exit section and the Lip (point A) is given by:

BEBPlug xx λλ // = (21)

3) The length of the nozzle is measured as axial distance between the point B of the col and the point E of theexit section. It is given in adimensional form by :

)/()/(/ BBBEB xxL λλλ −= (22)

4) The report of the sections corresponding to the discretization of N points is given by:

BNBEE yycomputedAA λλ //)(/*

== (23)

C. Thermodynamics ParametersIn each point i of the wall, the thermodynamics report can be determined by the following relations:

120

]211[ −−+=ii

M)/(γ)(T/T (24)

)1/(120

]211[ −−−+= γρρii

M)/(γ)/( (25)

iiiTTPP )/()/()/(

000ρρ= (26)

The ratio of temperatures will be used to make the suitable choise of building material of the central bodyresistant to this temperature. The density repot will be used to evaluate the mass of gas existing at every moment inth espace of the flow, that of the pressures for the determination of the pressure force exerted on the wall. We cancalculate these reports during the determination of the contour of the central body.

D. Mass of the central bodyThe segment number (i) of the wall is illustrated on the fig. 7. To manage

to calculate the mass of structure of the central body, let us consider the twofollowing assumptions :

1) The shape of the wall between two successive points isapproximated by a straight line. This assumption gives good resultsif the number of points N is very high.

2) The central body is made up of the same material, and constant thickness.The calculation of the mass of the structure is dependent with the calculation curvilinear length of the wall of thecentral body.Then, per unit of depth and in adimensional form, we obtain:

∑=−=

= ++ −+−1Ni

1i

2/12

1

2

1 ][2)(/ )//()//( BiBiBiBiBMMyyxxltMasse λλλλλρ (27)

E. Pressure force exerted on the wallThe pressure exerted on the panel of the fig. 7 is approximated by the following interpolation:

Figure 7. Segment of à Plug nozzle.

(i)θ

i+1

i(i)Fx

(i)

American Institute of Aeronautics and Astronautics6

11

+−+=

ii(i) ) P(PP σσ (28)

The axial pressure force exerted on this panel is given by:

)(1 ii(i)(i) yyPFx −=

+(29)

The axial pressure force exerted on the central body, per unit of depth, is calculated as the sum of all the axialpressure forces exerted on all the panels. The central body is formed by two parts by reason of symmetry. Inadimensional form, we obtain:

∑=−=

= +

1

1 1)(002

Ni

i BiBiiB)/λy-/λ(y)(P/Pl)λ(P/Fx (30)

F. Mass gas in the divergentThe existing mass of gas in the divergent part of the central body between the lines of Mach AB and AE

including the uniform zone can be evaluated. We can consider the space of the nozzle, as the union of the trianglesplaced adjacent of the other as the fig. 5 shows it, including the uniform zone between the line of Mach AE andhorizontal. The number of the triangles is equal to N-1 by adding the triangle of the uniform zone. The mass of gas,per unit of depth, which is in the space ranging between two successive lines of Mach connecting the points A and iand points A and i+1 as the fig. 6 shows it, is approximated by:

l(i)iiGas

AρMass)()(

= (31)

with 21)()/ρ(ρρ

iii ++= (32)

211

)/yxy( xAiiii(i) ++

−= (33)

At point A of the triangle considered, there are two values of the density, one equal to the values of point i andthe other equal to the value of the point i+1. With this reason, we considered the average value of the densitybetween those two points as present Eq. (32). The position of point A is not illustrated in the Eq. (33), becausexA=yA=0.

In the uniform zone, the mass of gas, per unit of depth, is given by

lyxzoneuniformMass EEEGas 2/)()( ρ= (34)

The mass of gas in the divergent one including the symmetry of the central body, per unit of depth, will thus begiven, in adimensional form, by the following relation:

∑ −++=−=

= +++

1

1 111000

2

0])()()()][()()([5.0)]()[()(

Ni

i BiBiBiBiiiBEBEEBGas/λy/λx/λy/λxρ/ρρ/ρ/λy/λxρ/ρl)λ/(ρMass (35)

III. Results and commentsThe results presented are considered for three values of the ratio of the specific heats γ=1.30, 1.40, and 5/3. The

presentation of the shapes of the central body is deferred in a reference mark to adimensional axis.The results of design like the length, mass of the structure, the pressure force, mass of gas presented respectively

by the Eqs. (22), (27), (30), and (37), and the other parameters are presented in an adimensional form.

A. Effect of the discretization on the convergence of the problemIf we increase the number N of points of the discretization, we can show the convergence of the numerical

results towards the exact solution. We take an example for ME=2.50, λB=1.0 and γ=1.40. The theoretical report ofthe sections is given by AE/A*=2.6367188. The comment is valid for any example.The results presented in table 1 do

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Table 2 Design Parameters of the suggested exemple depend on the discretization.N AE / A* (computed) L / λB Mass / (ρM tM λB l) Fx / (P0 λB l) MassGas / (ρ0 λB

2 l) ε,%3 13.092047 30.6286 66.38397 4.09742 67.08608 79.86

10 3.152120 7.85340 16.75316 1.20082 6.56516 16.3550 2.718915 6.86081 14.58344 1.06895 5.29471 3.02100 2.676870 6.76447 14.37254 1.05540 5.17773 1.49200 2.656574 6.71797 14.27070 1.04873 5.12158 0.74500 2.644610 6.69055 14.21066 1.04473 5.08857 0.29

1000 2.640656 6.68150 14.19081 1.04339 5.07766 0.145000 2.637505 6.67427 14.17499 1.04232 5.06897 0.02

10000 2.637112 6.67337 14.17302 1.04219 5.06789 0.0120000 2.636915 6.67292 14.17203 1.04212 5.06735 7.45 10-3 50000 2.636797 6.67265 14.17144 1.04208 5.06702 2.98 10-3

100000 2.636758 6.67256 14.17124 1.04206 5.06691 1.49 10-3 200000 2.636738 6.67252 14.17114 1.04206 5.06686 7.45 10-4

500000 2.636726 6.67249 14.17108 1.04205 5.06682 2.98 10-4 1000000 2.636722 6.67248 14.17106 1.04205 5.06681 1.49 10-4 2000000 2.636720 6.67248 14.17106 1.04205 5.06681 7.45 10-5 3000000 2.636720 6.67247 14.17105 1.04205 5.06681 4.96 10-5 4000000 2.636719 6.67247 14.17105 1.04205 5.06681 3.72 10-5 5000000 2.636718 6.67247 14.17105 1.04205 5.06680 2.98 10-5

10000000 2.636718 6.67247 14.17105 1.04205 5.06680 1.49 10-5

Table 1 Design Parameters of the suggested exemple independentof nthe discretizaton.

Ψ (degree)φE

(degree)φB

(degree)xB / λB yB / λB θ* (degree)

50.87643 23.57817 130.85273 -0.65411 0.75639 40.85273

Table 4 Minimum values of N giing the error ε for ME=3.0.ε=1.0 ε=0.1 ε=0.01 ε=0.001 ε=0.0001 ε=0.00001

γ=1.30 330 3296 32959 329584 3295767 32950902γ=5/3 182 1814 18138 181380 1813769 18134834

Table 3 Minimum values of N giving the error ε for γ=1.40.ε=1.0 ε=0.1 ε=0.01 ε=0.001 ε=0.0001 ε=0.00001

ME=1.50 15 136 1349 13481 134803 1347936ME=2.00 63 621 6205 62040 620391 6203134ME=3.00 271 2706 27052 270515 2705097 27046001ME=4.00 591 5909 59094 590939 5909278 58973000ME=5.00 978 9789 97901 979028 9790076 97873419

not depend on the discretization. Table 2 has the various numerical results obtained of the parameters of design ofthe example suggested versu the number N of points. The problem is convergent with a given relative error ε, if theratio of the sections calculated numerically for a discretization and the theoretical report of the sections check therelation (36). The parameters also converge towards the precise solution.

%(computed)/A[A/al)(theoretic/AAε*E*E

100]][1 ×−= (36)

If N increases, the ratio of the sections andthe other parameters converge by adecreasing way, i.e. the computed value isalways higher than the theoretical value.The other reports mentioned in table 2convergence towards the precise solutionbefore the convergence of the ratio of thesections, which is an advantage, in order to control that the convergence of theratio of the sections. Thus for a givenvalue N, the error made by the length of the central body, the mass of the structure, the pressure force and the massof gas is lower than the error made by the report of the sections, given by the relation (36).

The error made for each selected discretization is presented in table 2. If we take N=1000 points, we can obtain anerror better to 0.15 %. and for N=10000 points we can obtain an error ε = 0.01%. We notice according to theresults obtained, that for two discretizations, we can check this equality.

2112 // εε≈NN (37)

We made calculations for N=10 millions, 50 millions and N=100 millions points, we found respectively an errorε=1.49 10 -5, 2.97 10 -6, and 1.48 10 -6. By comparing these results with those of N=1000000, N=5000000 and N=10millions of table 2, we can check the relation (37).

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-1.0 0.0 1.0 2.0 3.0

Adimensional X-coordinates

0.0

0.5

1.0

1.5

2.0

Adi

men

sion

alY

-coo

rdin

ates

1 2 3

Figure 9. Shapes of the plug nozzle when ME=2.00.

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Adimensional X-coordinates

0.00.51.01.52.02.53.0

Adi

men

sion

alY

-coo

rdin

ates

12

3

Figure 10. Shapes of the plug nozzle when ME=3.00.

-0.3 0.0 0.3 0.6 0.9 1.2 1.5

Adimensional X-coordinates

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Adi

men

sion

alY

-coo

rdin

ates 123

Figure 8. Shapes of the plug nozzle when ME=1.50.

Table 5 Numerical results of design of nozzlesgiving the exit Mach number ME=1.50.

γ=5/3 γ=1.40 γ=1.30Ψ (degree) 79.77782 78.09479 77.30715

yE / λB 1.14843 1.17616 1.18948xPlug / λB 1.28399 1.13499 1.32988xB / λB -0.17746 -0.20629 -0.21972yB / λB 0.98412 0.97849 0.97556L / λB 1.46145 1.52128 1.54961

Mass / (ρM tM λB l) 2.94617 3.07504 3.13662Fx / (P0 λB l) 0.13616 0.17804 0.19923

MassGas / (ρ 0 λB2 l) 1.48174 1.45453 1.44410

θ* (degree) 10.22217 11.90520 12.69284

Table 6 Numerical results of design of nozzlesgiving the exit Mach number ME=2.00.

γ=5/3 γ=1.40 γ=1.30Ψ (degree) 68.21321 63.62023 61.31914

yE / λB 1.53125 1.68750 1.77318xPlug / λB 2.65220 2.92283 3.07125xB / λB -0.37115 -0.44431 -0.47993yB / λB 0.92857 0.89586 0.87730L / λB 3.02335 3.36715 3.55118

Mass / (ρM tM λB l) 6.21429 6.99731 7.42381Fx / (P0 λB l) 0.41069 0.57514 0.66616

MassGss / (ρ 0 λB2 l) 2.73024 2.79898 2.84232

θ* (degree) 21.78678 26.37976 28.68085

Table 7 Numerical results of design of nozzlesgiving the exit Mach number ME=3.00.

γ=5/3 γ=1.40 γ=1.30Ψ (degree) 51.05755 40.24265 34.24157

yE / λB 3.00000 4.23457 5.15978xPlug / λB 8.48528 11.97718 14.59406xB / λB -0.62853 -0.76331 -0.82667yB / λB 0.77777 0.64602 0.56268L / λB 9.11382 12.74049 15.42073

Mass / (ρM tM λB l) 19.15348 27.28409 33.34269Fx / (P0 λB l) 0.97927 1.49746 1.81970

MassGas/ (ρ 0 λB2 l) 7.32008 9.01751 10.22928

θ* (degree) 38.94244 49.75734 55.75842

Table 4 Minimum values of N giving the error ε for ME=3.0.ε=1.0 ε=0.1 ε=0.01 ε=0.001 ε=0.0001 ε=0.00001

γ=1.30 330 3296 32959 329584 3295767 32950902γ=5/3 182 1814 18138 181380 1813769 18134834

The results illustrated in table 3 presents, the number minimal N of points required to obtain the error ε indicatedand that for some values of the exit Mach number ME when γ=1.40. Here we can check the relation (37). In table 4,we presented the effect of gas on the minimum number of points of the discretization to have the error ε. Theminimal number N required to have a ε error depends on the values of ME and γ.

B. Effect of gas on the contour of the central bodyThe figs. 8, 9 and, 10 presents the form of the central body for various values of the exit Mach number ME=1.5,

2.0, and 3.0. The presentation of the contour of the nozzle is given in real scale. The numerical results of design arepresented respectively in tables 5, 6, and 7 with an error ε=10 -5 In these tables, the goal of presentation of the xPlug

results is that, if the central body does not respect this distance, we will have a subsonic flow in the divergent part.

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-2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0

Adimensional X-coordinates

0

10

20

30

40

50

Flo

wde

viat

ion

angl

e,de

gree 1

2

3

Figure 12 Variation of the flow deviation anglealong the plug nozzle wall for air with γ=1.40.

0.0 2.0 4.0 6.0 8.0 10.0 12.

Adimensional X-coordinates

1.0

1.5

2.0

2.5

3.0

Mac

hnu

mbe

r

Figure 11 Variation of the Mach number alongthe plug nozzle wall for air with γ=1.40.

1 2 3 4 5

Exit Mach number

0306090

120150180210240

Adi

men

sion

alle

ngth

ofth

epl

ugno

zzle

1

2

3

Figure 13. Variation of the adimensional lengthof the nozzle versus exit Mach number

1 2 3 4 5

Exit Mach number

0102030405060708090

Dev

iati

onan

gle

ofth

eL

ip,d

egre

e

1

2

3

Figure 15. Variation of the deviation angle of theLip versus exit Mach number

1 2 3 4 5

Exit Mach number

0102030405060708090

Flo

wan

gle

devi

atio

nat

the

col,

degr

ee

1

2

3

Figure 16. Variation of the flow angle deviationat the col versus exit Mach number

1 2 3 4 5

Exit Mach number

0

100

200

300

400

500

Adi

men

sion

alm

ass

ofth

epl

ugno

zzle

1

2

3

Figure 14. Variation of the adimensional mass ofthe nozzle versus exit Mach number

C. Variation of the parameters through the wall of the central bodyThe figs. 11 and 12 presents the variation of the Mach number and the deviation of the flow along the wall of the

central body for various values of exit Mach number. We notice the expansion of gas by increase in the Machnumber and straightening of the flow from angle θ* at the col to θ=0 at the exit.

D. Variation of the design parametersThe fig. 13 represent the variation in the length of the central body versus to the exit Mach number. On the

figs.14, 15, 16, 17 and 18, we respectively presented the variation of the mass of the structure, the deviation of theLip compared to the vertical, the angle of deviation of the flow to the col, the pressure force and the variation of themass of gas according to the exit Mach number.

American Institute of Aeronautics and Astronautics10

0 2 4 6 8 10 12 14 16 18

Adimensional X-coordinates

0

1

2

3

4

5

Adi

men

sion

alY

-coo

rdin

ates

1 Plug Nozzle2 Minimum Length Nozzle

1 2

Lip

Figure 19. Comparaison between the shapes of plug nozzle and MLN Nozzle for ME=2.50 and γ=1.40.

1 2 3 4 5

Exit Mach number

0

50

100

150

200

250

300

350

Adi

men

sion

alm

ass

ofth

eno

zzle

1

21 Plug Nozzle2 Minimum Length Nozzle

Figure 21. Comparaison between the massstructure of the plug nozzle and ML N Nozzle.

We note that if we want to design a nozzle for missiles applications having the smallest possible length andconsequently having a small mass of the structure, it is necessary to choose a gas having a possible smallest ratio γ.In this case, the produced pressure force will be large. For the applications of the blowers, we make the design onthe basis to obtain the smallest possible temperature at the exit section, not to destroy the measuring instruments,and make cold the ambient conditions, and possible a largest ray of the exit section for the site of the instruments.For the blowers, we are interested to use a gas having possible largest value of report γ. We can deduce theseresults starting from the relations (13) and (24).

E. ComparaisonOn the fig. 19, we presented the form of the plug nozzle and the Minimum Length Nozzle when ME= 2.50 and

γ=1.40, for goal to make a comparison between the shapes of these types of nozzles. We clearly notices the profit inlength and consequently in mass.The two nozzles deliver same exit Mach number, since have same exit section.

1 2 3 4 5

Exit Mach number

020406080

100120140160

Adi

men

sion

alL

engt

hN

ozzl

e 1

2

1 Plug Nozzle2 Minimum Length Nozzle

Figure 20. Comparaison between the length ofthe plug nozzle and MLN Nozzle.

1 2 3 4 5

Exit Mach number

0

20

40

60

80

100

120

Adi

men

sion

alm

ass

ofth

ega

s

1

2

3

Figure 18. Variation of the adimensional massgas versus exit Mach number.

1 2 3 4 5

Exit Mach number

0

1

2

3

4

Adi

men

sion

alpr

essu

refo

rce

1

2

3

Figure 17. Variation of the adimensionalpressure force versus exit Mach number.

American Institute of Aeronautics and Astronautics11

The fig. 20 represent the comparison between the length of the two nozzles versus the exit Mach number. Onfig. 21, we presented the comparison of the mass of the structure and on fig. 22 we presented the comparison of thecompressive force produced by the divergent. In the case of the fig. 19, the absolute benefit on the length, the massof the structure and the force of the pressure can arriverespectively at 2.45, 4.53 and 0.56 and one relativebenefit respectively equal to 26%, 24%, and 55%.

According to this profit, we can see the advantagesof use of this type of nozzle. The results concerning theMinimal Length Nozzle are presented in severalreferences, quoting Réf. 5 here. The design of the MLNnozzle is done by the method of characteristics. Thefield of the flow inside the plug nozzle is divided intothe zone of transition simple ABE (simple region) and azone of an uniform flow, which is not the case for theMLN nozzle where it has in more one zone of nonsimple flow named by zone of Kernel.

IV. ConclusionFrom this study, we have to illustrate an improvement of the parameters of the supersonic nozzle of propulsioncompared to the MLN nozzle, which is often used in the aeronautical applications, by the new shape of nozzle calledplug nozzle. Thus a simple change of MLN nozzle by our nozzle, allows a new strategy of use of the missiles andspacecraft. The developed method can make the design until an error of 10 -6 , in a very reduced time although thediscretization requires a high number of point.

AcknowledgmentsI hold to thank professor ZineEddine Youbi for the scientific assistance that it gave me for the realization of thiswork and the the authorities of the university of Blida and the aeronautical institute for the support financial grantedto complete this work without forgetting to thank Mr Djamel Zebbiche for time that it gave me to make the seizureof this document.

References1John D. Anderson, Jr., Fundamentals of Aerodynamics, 2nd ed., Mc Graw-Hill Book Company, New York, 1988.2John D. Anderson, Jr., Modern Compressible Flow. With Historical Pers pective, 2nd ed., Mc Graw-Hill Book Company,

New York, 1982.3Hill. P. G. and Peterson C.R., Mechanical and Thermodynamics of Propulsion, Addition-Wesley Publishing Company Inc.,

New York, 1965.4Rao G.V.R., “Recent Developments in Rocket Nozzle Configuration,” ARS Journal, Nov. 1961, 1488-1494.5Shapiro, A. H., Compressible Fluid Flow, Vol. 1, New York, the Ronald Press, 1953

1 2 3 4 5

Exit Mach number

0

1

2

3

Adi

men

sion

alpr

essu

refo

rce

1

21 Plug Nozzle2 MLN Nozzle

Figure 22. Comparaison between the pressure forceof the plug nozzle and MLN nozzle for γ=1.40