When the stagnation temperature of the combustion chamber or ambient air increases,the specific heats and their ratio do not remain constant any more, and start to vary withthis temperature. The gas remains perfect, except, it will be calorically imperfect andthermally perfect. A new generalized form of the Prandtl Meyer function is developed, byadding the effect of variation of this temperature, lower than the threshold of dissociation.The new relation is presented in the form of integral of a complex analytical function, havingan infinite derivative at the critical temperature. A robust numerical integration quadratureis presented in this context. The classiacl form of the Prandtl Meyer function of a perfect gasbecomes a particular case of the developed form. The comparison is made with the perfectgas model for goal to determine a limit of its application. The application is for air.
Nomenclatureθ = flow angle deviation.M = Mach number.γ = specific heats ratioR = specific heats rationν = Prandtl Meyer function.a = sound velocityH = enthalpy except for a constant.CP = specific heat to constant pressure.Ε = relative error of computation.N = order of the Gauss Legendre quadrature.E = error of Gauss Legendre quadrature.ξj = positive roots of the Legendre polynomial of ordre 2n+1.σj = associated coefficnets of the ξj roots of Gauss Legendre quadrature of ordre 2n+1.HT = abreviation of the word High Temperature.PG = abreviation of the word PerfectGas.PM = abreviation of the word Prandtl MeyerC- = abreviation of downward characteristicsAB = last C- of the Kernal zone.
Subscripts
0 = indice for stagnation condition.* = indice for critical conditionj = indice for nodes.E = indice for nozzle exit.S = indice for supersonic section
Superscripts
* = downstream of expansion
* Assistant Professor, Department of Aeronautics, University of Blida, Algeria, [email protected]
24th Applied Aerodynamics Conference5 - 8 June 2006, San Francisco, California
American Institute of Aeronautics and Astronautics2
I. IntroductionHE Prandtl Meyer function plays a very significant role in the supersonic flows calcalation. If we wants todesign a supersonic nozzles giving a uniform and parallel flow at the exit section4, 6, 7, it is necessary to deviate
the nozzle at the throat of an initial expansion angle to have the desired exit Mach number.The design of this type of nozzle is based on the application of the method of characterisitcs6, 8, wich is
formulated on the basis od the PM function. A second application on external aerodynamic, it is to study thesupersonic flow around a dihedron, in particular around a pointed airfoil.
The supersonic results of a perfect gas flow presented in Réfs. 4, 8 are based on the callorically perfect gasassumptions, i.e., the specific heats are constant and do not depend on the temperature, wich is not valid in the realcase when the stagantion temperature grows.
The goal of this work is to develop a new form of the function of PM like a generalization of the function forperfect gas with γ constant, by adding effect of variation CP and γ with the temperature. The gasremains perfect,except it becomescallorically imperfect andthermically perfect calorifiquement or at High Temperature.
The new PM function is presetend in the form of a certain complex analytical form, where the analyticalprocedure of its integration is impossible. Thus, our interest is directed towards the determination of the approximatenumerical solutions.
The integration of integration contains always the critical temperature. The derivative of the function in thispoint is equal ot infinite.
The integration quadrature to constant step require a very high discretization to have a suitable precision. Let usquote the trapezoid and Simpson’s quadrature5.
A robust quadrature make a fast calculation is presented, answering the specificity of the function.With a very reduced number of points, the function can be calculate with a very high precision. The difficulty
arises in the application of this quadrature is that the points of integration are irrational number, to see table 4.The application is for air in the supersonic field lower than the molecules dissociation threshold, applicable as
long as the Mach number remains lower than approximately8 6.00.The problem encountered in the aeronautical experiments and applications, is that the use of the nozzle
dimensionned on the basis of perfect gas assumptions, degrades the performances desired by thid nozzle (pressureforce, exit Mach number)3. If measurements of an experiment are made, we can find values different to thosedetermined by calculation, especially if the stagnation temperature of the combustion chamber is high. Severalreasons are responsible for this change. Let us note here that the PG thery does not take account of this temperature.For goal to determine the limit of application of PG model, a study on the error given by this model compared to HTmodel will be presented.
The table of variation of the specific heat CP of air according to T0, between 55 K and 3550K is presented in theRéf. 1. A polynomial interpolation with the values of this table is applied, in order to find an analytical form9. Thepresented relations are valid in the general case, independently of the interpolation and the substance. Thecomparison is made with the PG model, for goal to determine the limit of application of this model.
II. Mathematical formulationThe supersonic flow deviation can generate an expansion or a compression. When the intensity of the shock
∆P/P tends towards zero, provide the necessary tool to study of this phenomenon8. The oblique shock wavebecomes a Mach wave. The normal speed with the wave is the speed of sound. A small flow deviation ∆θ downstream of the wave is related to a difference in speed ∆V by the following relation1:
VdV)(Mddν / 212 1−=−= θ (1)
The relation (1) is very significant to make the isntropic supersonic flow study. Let us note that the angle ∆θ iscounted positive when the flow moves away from the normal direction to the wave (compression wave), andnegative when the flow approaches the normal direction to the wave (expansion wave).
To make the integration of the relation (1), it is advisable to express dV/V and it Mach number by theirexpression3, 6, 7, we obtain the following relation:
(T)Fdν ν−= (2)
T
American Institute of Aeronautics and Astronautics3
with
122
2 −= (T)H(T)/aH(T)
(T)C(T)F Pν (3)
The terms intervenning in the expression (3) are given by3:
)
2
a(T
H(T)M(T)= (4)
γ(T) R Ta(T)= (5)
R(T)C
(T)Cγ(T)P
P
−= (6)
The expression of CP and H(T) are presented in the Réf. 3. It is noticed that the PM function is connceted directlywith the temperature.
The function ν ispurely defined in the supersonic mode. When M=1.0, we take ν=0.0. then, the value of ν for aMach number M>1.0 (T<T*), can be obtained by integration of the relation (3) gives:
∫= *TT ν dTTFTν )()( (7)
The calculation of the value of ν needs to integrate the function Fν(T), where the analytical procedure isimpossible, considering the complexity of this function. Therefore, our interest is directed towards numericalcalculation. We can obtain the relation of PM of perfect gas starting from the relation (7) by cancelling allcoefficients of the polynomial CP(T) except the first. Then, the relation for a perfect gas becomes a particular of thegeneral form (7). In this relation, the temperature corresponding to the Mach number must be determined by theresolution of the equation (4). Then, the relation for a perfect gas becomes a particular case of the general form (7).In this relation, the temperature T corresponding to the Mach number M must be determined by the resolution of theequation (4).
To make a comparison between the two models, we can recall the PM function given by the perfect gas theory8:
21221
2 1111
11 /
/
]arctg[M)(Mγγ
arctgγγν(M) −−−
−+
−+=
(8)
The relation (8] is connected explicilty with the Mach number, wich is the basic variable for perfect gas model,on the contrary for HT model, the basic variable is the temperature, because of the equation (4) connecting M and Twich cannot determine an analytical expression of his reverse.
III. Mathematical formulationThe function Fν(T) contains that positive terms, presented by CP(T), H(T), a(T) and the square root some is the
interval of integration.The function Fν(T) is a regular function does not have a singularity. In other words, the function to be integrated
is completely definedin theclosedinterval someis thevaluesof TS and T* wich is null for T=T*.The function Fν(T) has an infinite derivative at this temperature. Then, in this point, we have:
−∞===
*TT
ν*ν dT
dF.)(TF ,00 (9)
And consequenlty, the successive derivates of higher order present a singularity at point T=T*.
American Institute of Aeronautics and Astronautics4
, ...),,(ndT
Fd
*TTnν
(n)432
)(=∞−=
=
(10)
The numerical integration quadratures based on the calculation of the area of the function (3) requires a veryhigh discretization to have a suitable convergence, considering the result (9). Amajor disadvantage if thesequadratures are used is that no information on the error made by calculation will be given, considering thecalculation of this error is based on the maximum value of derived from the function Fν(T) and derived from higherorder5 in the interval of integration, considering the result (10). We can quote of trapezoid and Simpson’squadratures5.
The function Fν(T) has consequently a term known by weight function, that is responsible for the singularity ofderived and the higher orders derivatives from the function in point T=T*. Our interest is thus based on thedecomposition of the function, so removing the singularity and to consider remains to it function for the numericalcalculation of the integral.
The function under the sign square root in the expression (3) has a root T=T*. We can show this result startingfrom the relation (4), when M=1 (T=T*). Then this expression is divisible by (T*-T). This relation can be written in
the following form (here one has to multiply and divide at the same time by TT* − ) :
TT
(T)aH(T)TT
H(T) a(T)
(T)C(T)F
**
Pν −
−−=
222
(11)
In the expression (11), one did not prefer the result of Euclidean division for reason taking the general caseindependently of the interpolation of CP(T). The expression has a weight function of the form square.
Let us take the variable change following for aim to transform the interval [TS, T*] to the interval [0, 1] ofreference. Then
x)T(TTT S** −−= (12)
Then
dx)T(TdT S* −−=
When x=0, one has T=T*, and when x=1 one obtains T=TS..Consequently, the value νS can be obtained by the evaluation of the following integral in the interval of [0, 1]. Weobtain:
∫
−−= 1
02
2
223
dxTT
(T)aH(T)
H(T) a(T)
(T)C)-T(Txν
*
P/
S*S (13)
In this relation, the temperature is given by the relation (12). The obtaining of νS depends on two Mach numberand T0. We can write:
nullnotvaluefinite)(TCdTda)a(T
TT
(T)aH(T)
TTLim
*PTT
*
**
22
002
*
2
=+
==−−
→
=
(14)
One can considerthe relation (13) under the following form:
∫=1
0 dxf(x)w(x)νS (15)
with:
American Institute of Aeronautics and Astronautics5
xw(x) = (16)
and
TT(T)aH(T)
H(T) a(T)
(T)C)-T(Tf(x)
*
P/
S*−−=
2232
2(17)
In the relation (17), the developed integration quadrature does not need to know the valueof the function f(x)when x=0 (T=T*). The difficulty which arises it is during the evaluation of the error made by quadrature, since theexpression (17) will be maximum at the point x=0.
The function f(x) and the successivehigher order derivativesf΄(x), f ˝(x), f(3)(x), …, f(n)(x) do not present anysingularity in theclosed interval [0, 1] andin particularity at thepoint x=0. Thenwith x=0, one has:
, ...),,,nie (nvaleur fi(dx)
fd
xn
(n)
32100
===
(18)
Figures 1 and 2 respectively represent the form of the functions w(x)f(x) and f(x). The presented functions areselected for the Mach number MS=6.00. For the other values of MS, one obtains the same pace with different values.
On figure 1, one can view clearly that the function w(x) f(x) has a infinite derivative with x=0. Figure 2 shows usthat the function f(x) is regular in the interval [0, 1] some is T0 and MS.
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4 w(x) f(x)
43
2
1
Figure 1. Graph of the function w(x) f(x) for MS=6.00.
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0 f(x)
43
2
1
Figure 2. Graph of the function f(x) when MS=6.00.
Considering the form of the function f(x), one can say that the higher successive derivative |f(2n)(x)| (n=1, 2, 3,…) of an even nature reach the maximum value at the point x=0.
The suitable numerical integration quadrature is that of Gauss Legendre5 when the function to be integrated has a
weight function w(x) form x . The general form of the quadrature is given by:
American Institute of Aeronautics and Astronautics6
∑=∫=
=+=
nj
jnjj E)f(aαdxf(x)xI
1
10 (19)
The weight function does not intervence in the calculation of the right sum of the term of the relation (19). Theintegration points aj and the coefficientsαj of quadrature (19) until the ordre n=15 are presentedin table 4. Theyare givenstarting from thepositive rootsof theodd Legendrepolynomialuntil order31by the following relation5:
2,2 2 jjjjj ξσαξa == j=1, 2, …,n (20)
To determine the quadratures of order n, it is necessary intiallty to determine the roots and the correspondingcoeffiencts of the Legendre polynomial of order 2n+1, and used the relation (20) only for the positive roots todeterminetheconsideredquadrature.
The major disadvantage of this quadrature is that the points aj and the coefficients αi of the integral evaluationare irrational numbers. The error E of this quadrature is given by5:
)(2 ηn)(nn fKE = n=1 ,2, 3, … (21)
with
n)!()!]n[()n(
)!]n[(K
n
n22434
1222
434
+++=
+
The relation of recurrence connecting two successive values of the coefficient K is given by the following form :
320 0 == Kn , nn K
)n()n)(n)(n(
)n)(n(K
74543412
32222
2
1++++
++=+
The expression (21) gives for the first three values of n, the following results:
1754
1=K ,130977
162 =K ,
37268302596
3=K
0.0 0.2 0.4 0.6 0.8 1.0x
0.0
1.0
2.0
3.0
4.0
5.0
6.0
12
3
4
Figure 3. Second derivative of f(x) for MS=6.00.
The value of η in the relation (21) represents the point in the interval [0, 1] for which the derivative of the functionf(2n)(x) is maximum. The maximum value of functions f(2n)(η) (n=0, 1, 2, …) is reached at the point x=0 (T=T*) someis the order of derived from the function f(x) and the parametrs T0. and MS. This relation requires that the successivederivative of an even nature. On figure 3, one took the form of the function f"(x) for some values of T0 when
American Institute of Aeronautics and Astronautics7
MS=6.00. It is clearly viewed that the maximum of the values of the successives derivative functions is reached atthe critical point. Then:
)(xf(ηf,η n)(n)( 0)max0 22 === (22)
The calculation of the successive derivative of f(x) must be done numerically and not anallytically. As thisfunction does not give its value to x=0, it was brought closer to it calculation with the value for x=ε with ε=10-6 considering the function is regular and the value with x=0 is finite.
IV. ApplicationIn we wants to obtain a supersonic nozzle giving a uniform and parallel flow at the exit section (M=ME , et
θ=θE=0), it is necessary to deviate the wall for certain type of nozzle of an angle θ* at the throat by6:
2/*sνθ = (23)
This situation is presented in figure 4. The relation (23) is valid for the 2D Minimum Length Nozzle.For the Plug Nozzle, the deviation of the nozzel at the throat is given by7 θ*=νS.The term νS is equal to the value of the PM function corresponding to the exit Mach number of the nozzle.The Mach number M* just after the expansion can be consequently given . It corresponds to the PM function
when ν=θ*. For HT model, it is necessary to determine the temperature T*, as a root of the equation (7), then replaceT=T* in the equation (4) to determine M*. for the PG model, it is necessary to determine M=M* like root of theequation (8) when θ*-ν(M*)=0.
Figure 4. Expansion center of MLN configuation.
Consequently, all the design parameters of the nozzle (length, mass of the structure, pressure force) of these twotypes of nozzles depends primaly on thestagnation temperature T0, especialy, if it starts to exceed 1000K , see Réfs.6 and 7.
V. Error given by Perfect Gas ModelThe PG model is developed on the basis to consider the specific heats and the ratio γ as constants and not
depends on the temperature, wich gives good results for low temperature. From this study, we can remark adifference between the relatrions given by the PG and HT developed model.
The relative error given by the PM function of PG model compared to HT model can be calculated. Then foreach (T0, M), the relative error ε can be evaluated by the following relation:
10010
00 , M)(T
, M)(T, M)(Tε
HT
GP ×−=νν
ν (24)
VI. Results and Comments
A. Prandtl Meyer function
Zone deKernel
Axe de symétrie de la tuyère
θ=0.0M=1.0
M*
θ*
T0
MS
Col
Paroi
Zone detransition Zone
uniforme
A
x
B
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Table 1 presents the effect of the developed Gauss Legendre Quadrature on the convergence of the problem. Theselected exemple is for T0=3550K and MS=6.00. This exemple request for a quadrature of raided order compared tothe other values (T0, MS) for same desired precision (the most unfavourable case). The result given quadratures isalways lower that the exact solution, i e. theconvergenceof thesolution will takeplacein a monotonousway.
Table 1. Effect of Gauss Legendre quadrature on convergencen ν n ν1 84.63802 6 97.565152 95.15011 7 97.568173 96.96984 8 97.568774 97.47855 9 97.568895 97.55014 10 97.56891
The control of fixing ofthe exact decimal digits in saying that, to have to precision ε=10-5, one needs to thequadrature of order 10 to have the result. Then, some is (T0,MS), one can use the quadrature of order n=10 to themaximum to have the better precision with ε=10-5. As information, for the same exemple, the trapezoid andSimpson’s quadraures with constant step request for a minimum number of points presented in table 2.
Table 2. Effectiveness of quadratures for a precision ε=10-5.Quadrature Trapezoid Simpson Our quadrature
n 390975 78954 10
For trapezoid and Simpson’s quadratures, one has to control the fixing of the deciamal digits to wantedprecision, since the error analysis relations of these quadratures does not give any information on the minimumnumber of points which it is necessary to obtain the wished precision for the integration of the relation (3),considering the properties (9) and (10).
One can have the same precision ε by using the trapezoid and Simpson’s quadratures with a number of pointslower than indicated in table 2, if the condensation of nodes3, 9 is used towards the two ends of the interval ofintegration [TS,T*] . In this case, the Simpson’s quadrature requires 2656 points and the trapezoid quadrature requires24534 points. One can obtain even values lower than those indicated, if one selected of other parameters of thecondensation function3.
Figure 5 presents the variation of the PM function according to the Mach number, for some values of T0
including the case of perfect gas. It is noticed clearly that if one takes into account, the variation of CP(T), thestagnation temperature influences the size of this function. The numerical values for some values of M and T0 arepresented in table 3.
Table 3. Numerical results of the PM function at High Temperature.γ=1.402 T0=1000 K T0=2000 K T0=3000K
It is noticed that the four curves are almost confounded until approximately M=2.00, which is interpreted by thepossibility of using GP model as long as M<2.00.
Figure 6 presents the variation of the PM function at high temperature according to T0 when M=3.00. Oneclearly notices that at low temperature, the gas can be regarded as calorically perfect and that until approximately240K with an error ε=0.0, considering in this interval the function CP(T) is constant. More T0 increases, the value ofν increases and moves away considerably from GP model, from where, need for using HT model to correct theresults.
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1 2 3 4 5 6
Machnumber
0
10
20
30
40
50
60
70
80
90
100
1234
Figure 5. Variation of the PM function according to M.
1000 2000 3000 4000Stagnation temperature(K)
40
42
44
46
48
50
52
54
56
58
60
Figure 6. Variation of PM function versus T0 for M=3.0.
Figure 7 introduces l`error given by the PM function of PG model compared to our HT model for differentstagnation temperature T0. It is clear that the error varies according to T0 and the Mach number. For example ifT0=2000K and M=3.00, the error given by model PG is equal to ε=9.70%..
1 2 3 4 5 6
Machnumber
0
2
4
6
8
10
12
14
1
2
3
Curve 1 Error comparedto HT model for (T0=3000 K)Curve 2 Error comparedto HT model for (T0=2000 K)Curve 1 Error comparedto HT model for (T0=1000 K)
Figure 7. variation of the error given by the PM function PG model compared to HT model versus the Machnumber
American Institute of Aeronautics and Astronautics10
The numerical results founded show that there is a difference in values in spite of at low temperature. Forexample, when T0=298.15 K and M=3.00, one obtain ν=49.648.In let us compare this value with that of perfect gasν=49.651, one finds ε=0.006%.
Considering the PM function is equal to zero when M=1 for the two models, the error analysis poses a problemof indetermination (zero out of zero). The error value in this point is equal to:
=
=
=
=×−
=×−==→→
KT
KT
KT
T
M)(Mε
HT
GP
)T(TM *
3000lorsqe%4.949
2000lorsqe%4.510
1000lorsqe%3.104
100001
100)(
)(1lim1
0
0
0
1 νν
ν
(25)
These values represent the intersection of curves 1, 2 and 3 of figure 6 with the vertical axis of the errors.
B. Expansion center of a supersonic nozzleFigure 8 presents the variation of the expansion initial angle θ * at high temperature of the 2D MLN.
On figure 9, one presented the variation of θ* versus T0 when MS=3.00 for goal to clearly illustrate the effect ofthe temperature T0 on the expansion initial angle. Thus the more the value of T0 increases, more there is opening ofthe wall at the throat.The curves on figure 8 are almost confounded until approximately MS=2.00, then start todifferentiate. Between curves 4 and 3, one can notice a small difference between the values of a PG model and ourmodel at high temperature.
On figure 9, one presented a horizontal straight line concerning PG model to prove that this model does not takeinto account of T0. The two curves merges until T0=240 K.
1 2 3 4 5 6Exit Machnumber
0
10
20
30
40
50
4321
Figure 8. Variation of θ* versus ME.
0 1000 2000 3000 400020
21
22
23
24
25
26
27
28
29
30
American Institute of Aeronautics and Astronautics11
Figure 9. Variation of θ* versus T0
1 2 3 4 5 6Exit Machnumber
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
4321
Figure 10. Variation of M* versus ME.
0 1000 2000 3000 40001.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2.00
Figure 11. Variation of θ* versus T0.
On figure 10, one presented the variation at High Temperature of the Mach number M* at point A of the throatversus ME of the nozzle, and figure 11 represents the variation of M* versus T0 of the combustion chamber for anozzle delivering even exit Mach number (Here ME=3.00). Then, figure 8 shows that there is an discontinuedexpansion centered at the point A which increases the Mach number of M=1 to M=M*.
Figure 9 shows that the PG model does not depend on T0, and that the value of M* depends on T0 for HT model,which influences the nozzle design. One notices a portion of superposition of the two models until T0=240 K. Theondulated variation of M* with T0 is the same one some is the value of ME. Calculations were made, where onechanged ME, and each time one traced the same graph as figure 10, and one found the same onduled variation. Ourus interest of this variation is that M* take account of the variation of T0 and that this variation increases if T0
increases6.
VII. Conclusion
From this study, one has to prove the following points:For an error lower than 5 % what is generally the case for the aerodynamic applications one can study a
supersonic flow by using the PG relation of Prandtl Meyer if T0<1000K for any value of supersonic Mach number,or when M<2.00 for does not import T0.
The PG model is represented by simple and explicit relations and do not request a time raised to makecalculation, which is not the case for HT model, where it is represented by the resolution of a nonlinear algebraicequation (4) and numerical integration of a complex analytical function (3) requiring a data-processingprogramming.
The relations presented in this study are valid for any interpolation type chosen for the function CP(T). Theessential one is that the selected interpolation error is small and acceptable.
American Institute of Aeronautics and Astronautics12
One can choose another substance instead of air. It is necessary to have in this case the table of CP variationwith the temperature and to make a suitable interpolation.
One can use the PM function to solve problems of the external flows in a hot medium. In particular, the flowaround a supersonic pointed airfoil.
The new form of the PM function leads us to develop a new mathematical model of the method of characteristicsto design and concept various shapes of the supersonic nozzles at high temperature, especially the MLN and plugnozzle.
AcknowledgmentsI would like to thank the authorities of the university SAAD Dahleb of Blida and the Department of Aeronautics
for the financial support granted for the completion of this research, without forgetting to thank Mr DjamelZEBBICHEand Mm FettoumMEBREKfor time that they gave to me for the seizure of this manuscript.
References1Peterson C. R. and Hill. P. G., Mechanics and Thermodynamics of Propulsion,Eddition-Wesley Publishing Company Inc.,
1965.2Zebbiche T. et Youbi Z., Fonction de Prandtl Meyer A Haute Température, Conférences Internationales sur la Mécanique
Avancée, CIMA’04, Boumerdès (Algérie), 30 Nov., 02 Déc. 2004.3Zebbiche T. and Youbi Z., Supersonic Flow Parameters at High Temperature.Application for Air in Nozzles,DGLR-05-
0256, German Aerospace Congress 2005, Friendrichshafen, 26-29 Sep., 2005, Germany.4Zucker R. D. And Bilbarz O., Fundamentalsof GasDynamics,John Wiley & Sons, New Jersey, 2002.5Raltson A. and Rabinowitz A., A First Course in Numerical Analysis, McGraw Hill Book Company, 1985.6Zebbiche T. and Youbi Z., Design of Two-Dimensional Supersonic Minimum Length Nozzle at Higth Temperature.
Application for Air, DGLR 2005-257, German Aerospace Congress 2005, 26-29 Sep. 2005, Friendrichshafen, Germany.7Zebbiche T. and Youbi Z., Supersonic Plug Nozzle Designat High Temperature.Application for Air, AIAA 2006-0592,
44th Aerospace Sciences Meeting and Exhibit, 9-12 Jan. 2006, Reno Nevada, Reno Hilton, USA.8Anderson J. D. Jr., Modern Compressible Flow: With Historical Perspective, Mc Graw-Hill Book Company, New York,
1982.9Fletcher C. A. J. Computational Techniques for Fluid Dynamics: Specific Techniques for Different Flow Categories, Vol.
II, Springer-Verlag, Berlin, Heidelberg, 1988.
VIII. Appendix
Table 4. X-coordinates and coefficients associated of the Gauus Legendre quadrature until ordre 15.n ai αi
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