CALCULATION AND OPTIMIZATION OF AERODYNAMIC COEFFICIENTS FOR LAUNCHERS ...

104 Pages·2014·4.23 MB·English

CALCULATION AND OPTIMIZATION OF AERODYNAMIC
COEFFICIENTS FOR LAUNCHERS AND RE-ENTRY VEHICLES
Alberto Ferrero
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisor: Prof. Paulo Jorge Soares Gil
Examination Committee
Chairperson: Prof. Fernando José Parracho Lau
Supervisor: Prof. Paulo Jorge Soares Gil
Member of the Committee: Prof. Carlos Frederico Neves Bettencourt da Silva
June 2014
Abstract
This work develops a procedure to calculate the aerodynamic coe(cid:30)cients for hypersonic
(cid:29)ight condition. The coe(cid:30)cients are obtained analytically, based on the Newton theory of
the hypersonic (cid:29)ow, and numerically by the solution of the Navier-Stokes equations. Along
the chapters, the process to obtain the analytic solution and its results are presented and
commented.
Several shapes are analyzed and an open source code for the evaluation of the coe(cid:30)cients
is developed. The code is implemented in the FreeFem++[31] environment, an open source
software that also allows the solution of Partial Di(cid:27)erential Equations (PDE) by the (cid:28)nite
elements method. Due to this capability, the hypersonic (cid:29)ux is even simulated solving nu-
merically the Euler’s equations. These equations are adopted for computational reasons: in
fact they permit a relatively rapid simulation of the (cid:29)ux, but they are generally not valid
for the hypersonic (cid:29)ux. The advantages and the disadvantages of the two methods, analytic
and numerical, are also analyzed.
The analytic expressions of the aerodynamic coe(cid:30)cients allow the implementation of an
optimizationalgorithm, basedonthesolutionofaconstrainedproblem. Theanalyticsolution
of this problem, obtained with the software Mathematica, obtains the optimal geometric
con(cid:28)gurations of the hypersonic shape, in order to reach a minimal value of drag, in case
of studying a launcher, or a minimal value of the ballistic coe(cid:30)cient, in case of studying a
re-entry vehicle.
Keywords
Hypersonic (cid:29)ight regime, aerodynamic coe(cid:30)cients, analytic calculation, shape optimization.
2
Abstract - Portuguese
Este trabalho desenvolve um procedimento para o cÆlculo dos coe(cid:28)cientes aerodin(cid:226)micos
para a condi(cid:231)ªo de v(cid:244)o hipers(cid:244)nico. Os coe(cid:28)cientes sªo obtidos analiticamente, com base
na teoria do escoamento hipers(cid:244)nico de Newton, e numericamente atravØs das equa(cid:231)ıes de
Navier-Stokes. Ao longo dos cap(cid:237)tulos, o processo para obter a solu(cid:231)ªo anal(cid:237)tica e os seus
resultados sªo apresentados e comentados.
VÆrias formas sªo analisadas e foi desenvolvido um c(cid:243)digo-fonte aberto para a avalia(cid:231)ªo
dos coe(cid:28)cientes. O c(cid:243)digo Ø executado no ambiente FreeFem++[31], um software open
source que permite tambØm a solu(cid:231)ªo de Equa(cid:231)ıes Diferenciais Parciais (PDE), pelo mØtodo
dos elementos (cid:28)nitos. Devido a esta capacidade, o (cid:29)uxo hipers(cid:244)nico Ø mesmo simulado
resolver numericamente as equa(cid:231)ıes de Euler. Estes equa(cid:231)ıes sªo adotados para razıes
computacionais: na verdade, eles permitem uma relativamente rÆpida simula(cid:231)ªo do (cid:29)uxo,
mas geralmente nªo sªo vÆlidos para o (cid:29)uxo hipers(cid:244)nico. As vantagens e as desvantagens dos
dois mØtodos, anal(cid:237)tico e numØrico, tambØm sªo analisados.
As expressıes anal(cid:237)ticas dos coe(cid:28)cientes aerodin(cid:226)micos permite a implementa(cid:231)ªo de um
algoritmo de otimiza(cid:231)ªo, com base na solu(cid:231)ªo de um problema restrito. A solu(cid:231)ªo anal(cid:237)tica
deste problema, obtida com o software Mathematica, obtØm as melhores con(cid:28)gura(cid:231)ıes ge-
omØtricasdaformahipers(cid:244)nico,a(cid:28)mdeatingirumvalorm(cid:237)nimoderesistŒnciaaerodin(cid:226)mica,
em caso de anÆlise de um lan(cid:231)ador, ou um valor m(cid:237)nimo do coe(cid:28)ciente bal(cid:237)stico, em caso de
estudo de um ve(cid:237)culo de reentrada.
Palavras-chave
Regimedev(cid:244)ohipers(cid:243)nico,coe(cid:28)cientesaerodin(cid:226)micos,cÆlculoanal(cid:237)tico,otimiza(cid:231)ªodeforma.
3
Abstract - Italian
Questo lavoro sviluppa una procedura per il calcolo dei coe(cid:30)cienti aerodinamici per una
condizione di volo ipersonico. I coe(cid:30)cienti sono ottenuti analiticamente, basando lo studio
sulla teoria del (cid:29)usso ipersonico di Newton, e numericamente utilizzando le equazioni di
Navier-Stokes. Nel corso dei capitoli, vengono presentati e discussi il processo per risalire a
una formulazione analitica e i risultati ottenuti.
Sono analizzate varie forme ed Ł stato sviluppato un software open source per la valu-
tazione dei coe(cid:30)cienti. Il codice viene eseguito in un ambiente FreeFem++[31], un codice
open source che permette anche la soluzione di equazioni di(cid:27)erenziali parziali (PDE), at-
traverso il metodo degli elementi (cid:28)niti. Dovuto a questa capacit(cid:224), il (cid:29)usso ipersonico viene
anche simulato numericamente attraverso la soluzione delle equazioni di Eulero. Queste
equazioni sono adottate per ragioni di calcolo: infatti, esse permettono una simulazione del
(cid:29)usso relativamente veloce, ma non sono generalmente valide per il (cid:29)usso ipersonico. Sono
inoltre analizzati i vantaggi e gli svantaggi dei due metodi, analitico e numerico.
In(cid:28)ne,avereleespressionianalitichedeicoe(cid:30)cientiaerodinamicipermettel’implementazione
di un algoritmo di ottimizzazione basato sulla soluzione di un problema vincolato. La
soluzione analitica di questo problema, ottenuta con il software Mathematica, permette di
trovare la con(cid:28)gurazione geometrica ottimale della forma del pro(cid:28)lo ipersonico, al (cid:28)ne di
raggiungere un minimo valore di resistenza aerodinamica, in caso di studiare un lanciatore,
o un valore minimo del coe(cid:30)ciente balistico, in caso di studiare un veicolo di rientro.
Parole chiave
Voloinregimeipersonico, coe(cid:30)cientiaerodinamici, calcoloanalitico, ottimizzazionediforma.
4
Contents
1 Introduction 12
1.1 Research motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 S/HABP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 CBAERO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 OpenFOAMfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Considerations on the sate of the art . . . . . . . . . . . . . . . . . . 16
2 Theoretical Background 17
2.1 Launch and re-entry aerodynamic . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Inviscid (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Inviscid hypersonic (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Earth atmosphere model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 Karush-Kuhn-Tucker (KKT) method . . . . . . . . . . . . . . . . . . 28
2.5.2 Newton-like interior point method . . . . . . . . . . . . . . . . . . . . 29
2.5.3 Sensitivity of the parameters . . . . . . . . . . . . . . . . . . . . . . . 33
3 Calculation of aerodynamic coe(cid:30)cients 34
3.1 Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 Navier-Stokes method . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 In(cid:29)uence of Knudsen number . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Shapes de(cid:28)nition 41
4.1 Cone family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Spherical nose family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Ogive families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Parabolic families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 More general shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.1 BØzier curves of revolution . . . . . . . . . . . . . . . . . . . . . . . . 47
5
CONTENTS 6
4.6 Superposition of the shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Shape optimization 50
5.1 Cone family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Ogive family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Parabolic family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 BØzier family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 Software description and results 60
6.1 Software structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Shape initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.1 Shadowed area de(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.2 Partially shadowed condition . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Aerodynamic coe(cid:30)cients calculation . . . . . . . . . . . . . . . . . . . . . . . 71
6.3.1 Simple conic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3.2 Simple parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.3 Simple ogive nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.4 Simple BØzier nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.5 Study cases analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4.1 Conic nose optimization results . . . . . . . . . . . . . . . . . . . . . 85
6.4.2 Parabolic nose optimization results . . . . . . . . . . . . . . . . . . . 88
6.4.3 Ogive nose optimization results . . . . . . . . . . . . . . . . . . . . . 89
6.4.4 BØzier nose optimization results . . . . . . . . . . . . . . . . . . . . . 91
7 Conclusions 93
List of Figures
1.1 Re-entry vehicles with shapes simple to describe analytically[1]. . . . . . . . 13
2.1 Body-axes reference frame[15] . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 β −θ diagram for M = [2,∞] . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Sonic (a) and hypersonic (b) shock layers for a 20(cid:176) wedge . . . . . . . . . . . 21
2.4 Typical variation of the pressure coe(cid:30)cient with Mach number[26] . . . . . . 23
2.5 Momentum of a gas particle in Newton assumption[2] . . . . . . . . . . . . . 24
2.6 ICAO atmosphere: pressure, density, temperature, speed of sound evolution
with the altitude[5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Flow regimes with typically re-entry events[19] . . . . . . . . . . . . . . . . . 27
3.1 Domain for the control problem and boundary conditions . . . . . . . . . . . 38
4.1 Side and front view of cone parametrization . . . . . . . . . . . . . . . . . . 42
4.2 Side and front view of spherical parametrization . . . . . . . . . . . . . . . . 43
4.3 Side and Front View of ogive parametrization . . . . . . . . . . . . . . . . . 44
4.4 Side and front view of parabolic parametrization . . . . . . . . . . . . . . . . 46
4.5 Side and front view of BØzier curve of II degree parametrization . . . . . . . 48
4.6 Side and front view of sphere-cone . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Constrains on the cargo volume (a), nose radius (b), nose mass (c) . . . . . . 51
5.2 Non-ablativepeakheatingversusvelocityforpastandplannedplanetaryentry
vehicles[24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.1 Geometrical input for the bi-conic nose . . . . . . . . . . . . . . . . . . . . . 61
6.2 Free stream de(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Mesh dimension de(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.4 Summarize of the algorithm structure . . . . . . . . . . . . . . . . . . . . . . 63
6.5 Bi-conic nose in blunted and unblunted con(cid:28)guration . . . . . . . . . . . . . 64
6.6 Ogive nose in blunted and unblunted con(cid:28)guration . . . . . . . . . . . . . . 65
6.7 Parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.8 BØzier nose in blunted and unblunted con(cid:28)guration . . . . . . . . . . . . . . 67
6.9 Shadowed condition for a conic nose in x−zand x−y planes . . . . . . . . 68
6.10 Unshadowed portion in y −z plane . . . . . . . . . . . . . . . . . . . . . . . 69
6.11 Partially shadowed condition for a bi-conic and ogive nose . . . . . . . . . . 69
6.12 II degree BØzier curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7
LIST OF FIGURES 8
6.13 Velocity and pressure (cid:28)elds for cone nose with α = 0(cid:176),10(cid:176),20(cid:176) . . . . . . . . . 72
6.14 Diagram of C ∝ α for the three di(cid:27)erent methods, conic nose . . . . . . . . 73
D
6.15 Accuracy on the solution and Cpu time for increasing mesh or simulation steps 75
6.16 Velocity and pressure (cid:28)elds for parabolic nose with α = 0(cid:176),10(cid:176),20(cid:176) . . . . . . 77
6.17 Diagram of C ∝ α for the three di(cid:27)erent methods, parabolic nose . . . . . . 78
D
6.18 Velocity and pressure (cid:28)elds for ogive nose with α = 0(cid:176),10(cid:176),20(cid:176) . . . . . . . . 80
6.19 Diagram of C ∝ α for the three di(cid:27)erent methods, ogive nose . . . . . . . . 81
D
6.20 Velocity and pressure (cid:28)elds for BØzier nose with α = 0(cid:176),10(cid:176) . . . . . . . . . . 82
6.21 Diagram of C ∝ α for the three di(cid:27)erent methods, ogive nose . . . . . . . . 83
D
6.22 Diagram of C ∝ M for the two di(cid:27)erent methods, simple conic nose . . . . 84
D
6.23 Blunted bi-conic nose with drag coe(cid:30)cient C optimized, in design (a) and
D
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.24 Blunted bi-conic nose with ballistic coe(cid:30)cient B∗ optimized, in design (a) and
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.25 Blunted parabolic nose with drag coe(cid:30)cient C optimized, in design (a) and
D
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.26 Blunted parabolic nose with ballistic coe(cid:30)cient B∗ optimized, in design (a)
and optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.27 Blunted ogive nose with drag coe(cid:30)cient C optimized, in design (a) and op-
D
timized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.28 Blunted ogive nose with ballistic coe(cid:30)cient B∗ optimized, in design (a) and
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.29 Blunted BØzier nose with drag coe(cid:30)cient C optimized, in design (a) and
D
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.30 Blunted ogive nose with ballistic coe(cid:30)cient B∗ optimized, in design (a) and
optimized con(cid:28)guration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
List of Tables
2.1 Sea Level values for ICAO atmosphere[5] . . . . . . . . . . . . . . . . . . . . 26
2.2 Stratosphere values for ICAO atmosphere at 20 km[5] . . . . . . . . . . . . 27
4.1 Table of common shapes for parabolic nose . . . . . . . . . . . . . . . . . . . 46
5.1 Default mass of the body, density and thickness of the nose . . . . . . . . . . 50
5.2 Optimization parameters for the cone family . . . . . . . . . . . . . . . . . . 53
5.3 Optimization parameters for the ogive family . . . . . . . . . . . . . . . . . . 55
5.4 Optimization parameters for the parabolic family . . . . . . . . . . . . . . . 56
5.5 Optimization parameters for the BØzier curve family . . . . . . . . . . . . . . 58
6.1 Design parameters for bi-conic con(cid:28)guration . . . . . . . . . . . . . . . . . . 64
6.2 Design parameters for tangent ogive con(cid:28)guration . . . . . . . . . . . . . . . 65
6.3 Design parameters for parabolic con(cid:28)guration . . . . . . . . . . . . . . . . . 66
6.4 Design parameters for II degree BØzier con(cid:28)guration . . . . . . . . . . . . . . 67
6.5 Geometrical features of the conic nose . . . . . . . . . . . . . . . . . . . . . . 72
6.6 Standard deviation of the analytic dates, compared with the numerical results
for the conic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.7 Geometrical features of the parabolic nose . . . . . . . . . . . . . . . . . . . 76
6.8 Standard deviation of the analytic dates, compared with the numerical results
for the parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.9 Geometrical features of the ogive nose . . . . . . . . . . . . . . . . . . . . . . 79
6.10 Standard deviation of the analytic dates, compared with the numerical results
for the ogive nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.11 Geometrical features of the ogive nose . . . . . . . . . . . . . . . . . . . . . . 82
6.12 Standard deviation of the analytic dates, compared with the numerical results
for BØzier nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.13 Expect accuracy on the Euler’s results, respect to Newton theory . . . . . . 85
6.14 Results of optimized drag coe(cid:30)cient for conic nose family . . . . . . . . . . . 86
6.15 Results of ballistic coe(cid:30)cient optimization for conic nose family . . . . . . . 87
6.16 Results of drag coe(cid:30)cient optimization for parabolic nose family . . . . . . . 88
6.17 Results of ballistic coe(cid:30)cient optimization for parabolic nose family . . . . . 88
6.18 Results of drag coe(cid:30)cient optimization for ogive nose family . . . . . . . . . 89
6.19 Results of drag coe(cid:30)cient optimization for ogive nose family . . . . . . . . . 90
6.20 Results of drag coe(cid:30)cient optimization for BØzier nose family . . . . . . . . 91
6.21 Results of ballistic coe(cid:30)cient optimization for BØzier nose family . . . . . . . 92
9
List of Symbols and Acronyms
Symbols
α angle of attack [deg]
β side-slip or shock de(cid:29)ection angle [deg]
γ the ratio of speci(cid:28)c heat capacity
δ half-cone angle [deg]
c
θ surface to free-stream angle [deg]
ρ air density [kg/m3]
a speed of sound [m/s]
B position vector for BØzier control point i
i
C drag coe(cid:30)cient
D
C lift coe(cid:30)cient
L
C lateral force coe(cid:30)cient
S
C roll moment coe(cid:30)cient
l
C pitch moment coe(cid:30)cient
m
C jaw moment coe(cid:30)cient
n
C pressure coe(cid:30)cient
p
I
identity tensor
J BØzier basis function
Kn Knudsen number
M Mach number
nˆ surface outward-normal vector
p air pressure [Pa]
Re Reynolds number
s arc-length for BØzier curve
b
T
stress tensor
ˆt surface tangent vector
V velocity vector
V nose volume [m3]
10