calculation and optimization of aerodynamic

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 CommitteeChairperson: Prof. Fernando José Parracho Lau

Supervisor: Prof. Paulo Jorge Soares GilMember of the Committee: Prof. Carlos Frederico Neves Bettencourt da Silva

June 2014

Abstract

This work develops a procedure to calculate the aerodynamic coecients for hypersonicight condition. The coecients are obtained analytically, based on the Newton theory ofthe hypersonic ow, and numerically by the solution of the Navier-Stokes equations. Alongthe chapters, the process to obtain the analytic solution and its results are presented andcommented.

Several shapes are analyzed and an open source code for the evaluation of the coecientsis developed. The code is implemented in the FreeFem++ [31] environment, an open sourcesoftware that also allows the solution of Partial Dierential Equations (PDE) by the niteelements method. Due to this capability, the hypersonic ux is even simulated solving nu-merically the Euler's equations. These equations are adopted for computational reasons: infact they permit a relatively rapid simulation of the ux, but they are generally not validfor the hypersonic ux. The advantages and the disadvantages of the two methods, analyticand numerical, are also analyzed.

The analytic expressions of the aerodynamic coecients allow the implementation of anoptimization algorithm, based on the solution of a constrained problem. The analytic solutionof this problem, obtained with the software Mathematica, obtains the optimal geometriccongurations of the hypersonic shape, in order to reach a minimal value of drag, in caseof studying a launcher, or a minimal value of the ballistic coecient, in case of studying are-entry vehicle.

Keywords

Hypersonic ight regime, aerodynamic coecients, analytic calculation, shape optimization.

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Abstract - Portuguese

Este trabalho desenvolve um procedimento para o cálculo dos coecientes aerodinâmicospara a condição de vôo hipersônico. Os coecientes são obtidos analiticamente, com basena teoria do escoamento hipersônico de Newton, e numericamente através das equações deNavier-Stokes. Ao longo dos capítulos, o processo para obter a solução analítica e os seusresultados são apresentados e comentados.

Várias formas são analisadas e foi desenvolvido um código-fonte aberto para a avaliaçãodos coecientes. O código é executado no ambiente FreeFem++ [31], um software opensource que permite também a solução de Equações Diferenciais Parciais (PDE), pelo métododos elementos nitos. Devido a esta capacidade, o uxo hipersônico é mesmo simuladoresolver numericamente as equações de Euler. Estes equações são adotados para razõescomputacionais: na verdade, eles permitem uma relativamente rápida simulação do uxo,mas geralmente não são válidos para o uxo hipersônico. As vantagens e as desvantagens dosdois métodos, analítico e numérico, também são analisados.

As expressões analíticas dos coecientes aerodinâmicos permite a implementação de umalgoritmo de otimização, com base na solução de um problema restrito. A solução analíticadeste problema, obtida com o software Mathematica, obtém as melhores congurações ge-ométricas da forma hipersônico, a m de atingir um valor mínimo de resistência aerodinâmica,em caso de análise de um lançador, ou um valor mínimo do coeciente balístico, em caso deestudo de um veículo de reentrada.

Palavras-chave

Regime de vôo hipersónico, coecientes aerodinâmicos, cálculo analítico, otimização de forma.

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Abstract - Italian

Questo lavoro sviluppa una procedura per il calcolo dei coecienti aerodinamici per unacondizione di volo ipersonico. I coecienti sono ottenuti analiticamente, basando lo studiosulla teoria del usso ipersonico di Newton, e numericamente utilizzando le equazioni diNavier-Stokes. Nel corso dei capitoli, vengono presentati e discussi il processo per risalire auna formulazione analitica e i risultati ottenuti.

Sono analizzate varie forme ed è stato sviluppato un software open source per la valu-tazione dei coecienti. Il codice viene eseguito in un ambiente FreeFem++ [31], un codiceopen source che permette anche la soluzione di equazioni dierenziali parziali (PDE), at-traverso il metodo degli elementi niti. Dovuto a questa capacità, il usso ipersonico vieneanche simulato numericamente attraverso la soluzione delle equazioni di Eulero. Questeequazioni sono adottate per ragioni di calcolo: infatti, esse permettono una simulazione delusso relativamente veloce, ma non sono generalmente valide per il usso ipersonico. Sonoinoltre analizzati i vantaggi e gli svantaggi dei due metodi, analitico e numerico.

Inne, avere le espressioni analitiche dei coecienti aerodinamici permette l'implementazionedi un algoritmo di ottimizzazione basato sulla soluzione di un problema vincolato. Lasoluzione analitica di questo problema, ottenuta con il software Mathematica, permette ditrovare la congurazione geometrica ottimale della forma del prolo ipersonico, al ne diraggiungere un minimo valore di resistenza aerodinamica, in caso di studiare un lanciatore,o un valore minimo del coeciente balistico, in caso di studiare un veicolo di rientro.

Parole chiave

Volo in regime ipersonico, coecienti aerodinamici, calcolo analitico, ottimizzazione di forma.

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Contents

1 Introduction 121.1 Research motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.1 Analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 S/HABP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 CBAERO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.3 OpenFOAM® . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.4 Considerations on the sate of the art . . . . . . . . . . . . . . . . . . 16

2 Theoretical Background 172.1 Launch and re-entry aerodynamic . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Inviscid ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Inviscid hypersonic ow . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Earth atmosphere model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1 Karush-Kuhn-Tucker (KKT) method . . . . . . . . . . . . . . . . . . 282.5.2 Newton-like interior point method . . . . . . . . . . . . . . . . . . . . 292.5.3 Sensitivity of the parameters . . . . . . . . . . . . . . . . . . . . . . . 33

3 Calculation of aerodynamic coecients 343.1 Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Navier-Stokes method . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Inuence of Knudsen number . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Shapes denition 414.1 Cone family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Spherical nose family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Ogive families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Parabolic families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 More general shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5.1 Bézier curves of revolution . . . . . . . . . . . . . . . . . . . . . . . . 47

5

CONTENTS 6

4.6 Superposition of the shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Shape optimization 505.1 Cone family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Ogive family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Parabolic family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Bézier family optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Software description and results 606.1 Software structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2 Shape initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2.1 Shadowed area denition . . . . . . . . . . . . . . . . . . . . . . . . . 676.2.2 Partially shadowed condition . . . . . . . . . . . . . . . . . . . . . . . 69

6.3 Aerodynamic coecients calculation . . . . . . . . . . . . . . . . . . . . . . . 716.3.1 Simple conic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3.2 Simple parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.3 Simple ogive nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3.4 Simple Bézier nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.5 Study cases analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4 Shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.4.1 Conic nose optimization results . . . . . . . . . . . . . . . . . . . . . 856.4.2 Parabolic nose optimization results . . . . . . . . . . . . . . . . . . . 886.4.3 Ogive nose optimization results . . . . . . . . . . . . . . . . . . . . . 896.4.4 Bézier 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] . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 β − θ diagram for M = [2,∞] . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Sonic (a) and hypersonic (b) shock layers for a 20° wedge . . . . . . . . . . . 212.4 Typical variation of the pressure coecient with Mach number [26] . . . . . . 232.5 Momentum of a gas particle in Newton assumption [2] . . . . . . . . . . . . . 242.6 ICAO atmosphere: pressure, density, temperature, speed of sound evolution

with the altitude [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 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 . . . . . . . . . . . . . . . . . . 424.2 Side and front view of spherical parametrization . . . . . . . . . . . . . . . . 434.3 Side and Front View of ogive parametrization . . . . . . . . . . . . . . . . . 444.4 Side and front view of parabolic parametrization . . . . . . . . . . . . . . . . 464.5 Side and front view of Bézier curve of II degree parametrization . . . . . . . 484.6 Side and front view of sphere-cone . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Constrains on the cargo volume (a), nose radius (b), nose mass (c) . . . . . . 515.2 Non-ablative peak heating versus velocity for past and planned planetary entry

vehicles [24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1 Geometrical input for the bi-conic nose . . . . . . . . . . . . . . . . . . . . . 616.2 Free stream denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Mesh dimension denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Summarize of the algorithm structure . . . . . . . . . . . . . . . . . . . . . . 636.5 Bi-conic nose in blunted and unblunted conguration . . . . . . . . . . . . . 646.6 Ogive nose in blunted and unblunted conguration . . . . . . . . . . . . . . 656.7 Parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.8 Bézier nose in blunted and unblunted conguration . . . . . . . . . . . . . . 676.9 Shadowed condition for a conic nose in x− zand x− y planes . . . . . . . . 686.10 Unshadowed portion in y − z plane . . . . . . . . . . . . . . . . . . . . . . . 696.11 Partially shadowed condition for a bi-conic and ogive nose . . . . . . . . . . 696.12 II degree Bézier curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7

LIST OF FIGURES 8

6.13 Velocity and pressure elds for cone nose with α = 0°, 10°,20° . . . . . . . . . 726.14 Diagram of CD ∝ α for the three dierent methods, conic nose . . . . . . . . 736.15 Accuracy on the solution and Cpu time for increasing mesh or simulation steps 756.16 Velocity and pressure elds for parabolic nose with α = 0°, 10°,20° . . . . . . 776.17 Diagram of CD ∝ α for the three dierent methods, parabolic nose . . . . . . 786.18 Velocity and pressure elds for ogive nose with α = 0°, 10°,20° . . . . . . . . 806.19 Diagram of CD ∝ α for the three dierent methods, ogive nose . . . . . . . . 816.20 Velocity and pressure elds for Bézier nose with α = 0°, 10° . . . . . . . . . . 826.21 Diagram of CD ∝ α for the three dierent methods, ogive nose . . . . . . . . 836.22 Diagram of CD ∝M for the two dierent methods, simple conic nose . . . . 846.23 Blunted bi-conic nose with drag coecient CD optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.24 Blunted bi-conic nose with ballistic coecient B∗ optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.25 Blunted parabolic nose with drag coecient CD optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.26 Blunted parabolic nose with ballistic coecient B∗ optimized, in design (a)

and optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . 896.27 Blunted ogive nose with drag coecient CD optimized, in design (a) and op-

timized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.28 Blunted ogive nose with ballistic coecient B∗ optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.29 Blunted Bézier nose with drag coecient CD optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.30 Blunted ogive nose with ballistic coecient B∗ optimized, in design (a) and

optimized conguration (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

List of Tables

2.1 Sea Level values for ICAO atmosphere [5] . . . . . . . . . . . . . . . . . . . . 262.2 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 . . . . . . . . . . 505.2 Optimization parameters for the cone family . . . . . . . . . . . . . . . . . . 535.3 Optimization parameters for the ogive family . . . . . . . . . . . . . . . . . . 555.4 Optimization parameters for the parabolic family . . . . . . . . . . . . . . . 565.5 Optimization parameters for the Bézier curve family . . . . . . . . . . . . . . 58

6.1 Design parameters for bi-conic conguration . . . . . . . . . . . . . . . . . . 646.2 Design parameters for tangent ogive conguration . . . . . . . . . . . . . . . 656.3 Design parameters for parabolic conguration . . . . . . . . . . . . . . . . . 666.4 Design parameters for II degree Bézier conguration . . . . . . . . . . . . . . 676.5 Geometrical features of the conic nose . . . . . . . . . . . . . . . . . . . . . . 726.6 Standard deviation of the analytic dates, compared with the numerical results

for the conic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.7 Geometrical features of the parabolic nose . . . . . . . . . . . . . . . . . . . 766.8 Standard deviation of the analytic dates, compared with the numerical results

for the parabolic nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.9 Geometrical features of the ogive nose . . . . . . . . . . . . . . . . . . . . . . 796.10 Standard deviation of the analytic dates, compared with the numerical results

for the ogive nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.11 Geometrical features of the ogive nose . . . . . . . . . . . . . . . . . . . . . . 826.12 Standard deviation of the analytic dates, compared with the numerical results

for Bézier nose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.13 Expect accuracy on the Euler's results, respect to Newton theory . . . . . . 856.14 Results of optimized drag coecient for conic nose family . . . . . . . . . . . 866.15 Results of ballistic coecient optimization for conic nose family . . . . . . . 876.16 Results of drag coecient optimization for parabolic nose family . . . . . . . 886.17 Results of ballistic coecient optimization for parabolic nose family . . . . . 886.18 Results of drag coecient optimization for ogive nose family . . . . . . . . . 896.19 Results of drag coecient optimization for ogive nose family . . . . . . . . . 906.20 Results of drag coecient optimization for Bézier nose family . . . . . . . . 916.21 Results of ballistic coecient optimization for Bézier nose family . . . . . . . 92

9

List of Symbols and Acronyms

Symbols

α angle of attack [deg]

β side-slip or shock deection angle [deg]

γ the ratio of specic heat capacity

δc half-cone angle [deg]

θ surface to free-stream angle [deg]

ρ air density [kg/m3]

a speed of sound [m/s]

Bi position vector for Bézier control point i

CD drag coecient

CL lift coecient

CS lateral force coecient

Cl roll moment coecient

Cm pitch moment coecient

Cn jaw moment coecient

Cp pressure coecient

I identity tensor

J Bézier basis function

Kn Knudsen number

M Mach number

n surface outward-normal vector

p air pressure [Pa]

Re Reynolds number

sb arc-length for Bézier curve

T stress tensor

t surface tangent vector

V velocity vector

V nose volume [m3]

10

LIST OF TABLES 11

Subscripts

I ,II with respect to rst, second, etc., shape segment

∞ with respect to free-stream condition

BF with respect to body-axes frame

b with respect to Bézier shape

c with respect to conic shape

o with respect to ogive shape

p with respect to parabolic shape

s with respect to spherical shape

Acronyms

CBAERO Conguration Based Aerodynamic toolCFD Computational Fluid DynamicsCpu Computational power unitKKT Karush-Kuhn-Tucker methodICAO International Civil Aviation OrganizationISA International Standard AtmosphereLSI Local Surface Inclination methodsOpenFOAM® Open Field Operation and ManipulationPDE Partial Dierential EquationS/HABP Supersonic/Hypersonic Arbitrary Body ProgramSL Sea Level

Chapter 1

Introduction

1.1 Research motivation and goals

The primary goal of this work is to develop a procedure to calculate the aerodynamic co-ecients, both analytically and numerically for the hypersonic ight regime and possiblyother regimes, eventually demonstrating the advantage of this approach. The comparisonbetween both is made in order to validate the results and to emphasize their advantages anddisadvantages.

After having calculated the aerodynamic coecients, it is possible to optimize the shapeof the studied vehicle: for launchers the optimization is done minimizing the drag, for there-entry vehicles maximizing the air resistance, respecting the structural and thermal limitsand other constrains.

Contrary to commercial codes, this work will be developed for being free and user-friendly,in order to create a fast and easy instrument for helping the engineer in the preliminarydesign of hypersonic vehicles. For this reason the work will be developed in the FreeFem++environment. This is a completely open source software that uses nite elements to solveintegrals. Due to its capacity of solving partial dierential equations (PDE), it will be usedfor obtaining a direct comparison between the numerical and the analytical calculation of theaerodynamic coecients. The analysis is made simulating the hypersonic ux by the Euler'sequations, an approximation of the Navier-Stokes equations.

The result of this work will be a code in which the future users can obtain analyticallythe aerodynamic coecients, eventually validate the results with numerical simulations, andknow the optimal conguration for the chosen shape.

1.1.1 Analytic approach

An important purpose of this work is to help the designers of rockets or re-entry vehicles ina preliminary phase of the project. Basing the calculation on the work of Grant [1], it willbe demonstrated that an analytical instead of a numerical approach can save computationaltime. This is especially important in the rst steps of a design. Some dierent shapes areanalyzed and implemented in the program but other shapes can be added easily.

Considering launchers and re-entry vehicles, one of the most critical part of the mission isthe hypersonic ight. The study of the shapes of these kinds of vehicles is a multidisciplinary

12

CHAPTER 1. INTRODUCTION 13

phase of the project that usually involves several teams. Due to the complexity of theproblem, this phase can be takes a long time to be developed [1].

Generally, computational uid dynamics (CFD) is used to dene the aerodynamic param-eters of the vehicles. Important parameters for the good accomplishment of the missions aregenerally chosen, such as L/D and the ballistic coecient and then the vehicles are designedto meet these performance constrains [1].

This process is usually very complex and during conceptual design, some methods are typ-ically employed to improve computational speed at the expense of delity [1]. For exampleas will be dened in the Chapter 3, panel methods can be used with Newtonian ow the-ory to obtain hypersonic aerodynamic characteristics quicker instead of with Navier-Stokesequations.

Although this method results quite rapid, the approximation made are sometimes notnegligible. So an analytical approach that can give more precise results could be preferredin the early phases of the project [1]. Moreover it is important to underline that the geom-etry of many common hypersonic vehicles of interest are basically made by simple shapeslike sphere-cones, blunted bi-conics and spheres and it is relatively simple to express thesecurves analytically. Considering the hypersonic ow simplications given by the Newton'saerodynamic theory, the corresponding aerodynamic coecients can also be obtained ana-lytically [1].

Figure 1.1: Re-entry vehicles with shapes simple to describe analytically [1].

Figure 1.1 shows some applications in re-entry vehicles with shapes described analytically,with simple equations in a good approximation. They are generally solids of rotation, havingplanes of symmetry around the x-axis in which the calculation of the surface integrals areoften possible analytically.

Having an analytic approach can be useful because it allows to have exact calculationsof the aerodynamic coecients currently approximated by numerical methods. Additionally,these relations eliminate the large aerodynamic tables enabling the designer to do rapidsimulations of hypersonic ight. This is essential above all in conceptual design and for globaloptimization, where the phase space is often large [1]. From this point of view, obtaining ananalytic expression for the aerodynamic coecients is essential in order to be able to solve


the optimization process without having to solve Navier-Stokes equations numerically. Infact this can be studied as a constrained optimization problem, whose solution can be foundanalytically or with a simple numerical algorithm.

1.2 State of the art

In the last decades the computation of the the aerodynamic coecients have been donemostly numerically. The improving in computation power of allowed engineers to have verygood results in a reasonable time [1].

Unfortunately the numerical codes are often commercial and expensive. And in orderto achieve high precision on the results, the computational time is not negligible anymore.However, in a preliminary phase of the project it is often important to have fast results, evenif not very accurate, in order to help the design process. An approximate analytic solutioncan thus be useful in this content [1].

Some codes dedicated to the study of the hypersonic conditions are presented in the nextsection. Some are based on the hypothesis of hypersonic ow, others are more general CFD(computational uid dynamics) solvers.

1.2.1 S/HABP

The rst software which presented the concept of approximate hypersonic ow was the Su-personic/Hypersonic Arbitrary Body Program - S/HABP - developed by Douglas Aircraftcompany in 1964, and modied various times until a nal version was released in 1980 [14].

The main concept of this software is to dene an arbitrary geometry using quadrilateralsor panels and to determine with approximate methods the inviscid ow properties. Thisset of methods, which are known as Local Surface Inclination methods, or LSI, are used toevaluate the inviscid pressure on a panel knowing only the local angle between the panel andthe free-stream ow, and obviously the free-stream ow properties. This approach is directlyderivated from the Newtonian theory that will be presented in section 2.3.

When the inviscid ow eld is resolved, streamlines are calculated and a viscous approx-imation based on this information is attempted. The approximate viscous methods used arehowever quite obsolete. Another limitation of S/HABP is that a real consideration of hightemperature eects is missing, and these phenomena are reconstructed using only experi-mental correlations, instead of performing a thermochemical equilibrium calculation, thusreconstructing the actual properties of the mixture.

The other main advantage of S/HABP is that - at present day - it is released as freesoftware, FORTRAN 77 source code included. For these reasons it is still used today as apreliminary design and optimization tool [2]. On the other hand this could be a disadvantagefor all the users not used to this kind of language. Who has the necessity of obtaining resultswithout entering in a new programming language, should prefer an other approach.


1.2.2 CBAERO

The Conguration Based Aerodynamic tool, or CBAERO, represents the present day evo-lution of S/HABP software [2]. Developed at the Ames Research center at the beginningof this century, it is made from the same intuitions of S/HABP. So it uses LSI methods incombination with streamline calculation as a basis for the viscous and thermal analysis butwith present day implementation concepts.

It uses a triangular unstructured mesh, which can provide lower accuracy but a fastersimulation in describing complex shapes with respect to a panel, and can be directly linked toother solvers parametrization. The advantages are that an immediate comparison with CFDsolution can be easily made, and it is also possible, for example, to link an Euler equationsinviscid solver to the aerothermal viscous analyzer of CBAERO, bypassing the LSI methodsand using a more accurate viscous solution.

Another interesting concept in CBAERO is the denition of search trees to order the meshpoints in order to perform faster searches of panel points in the integration of the velocityeld on the surface.

CBAERO is surely a modern hypersonic preliminary analysis tool, but it still shows somelimitations. The rst and most evident one is that the program is not freely available to thepublic, and also its availability for purchase is very limited. The other fact is that the viscoussolution doesn't take into account the most recent developments of the reference enthalpymethod for shear stress and heat ow calculation.

However, for the purpose of this work, the thermal evolution of the ux over the body isnot taken into account, so an higher level of details is not required.

1.2.3 OpenFOAM®

The OpenFOAM® (Open Field Operation and Manipulation) CFD Toolbox is a free, opensource CFD software package which has a large user base across most areas of engineeringand science, from both commercial and academic organizations [17]. OpenFOAM has anextensive range of features to solve anything from complex uid ows involving chemicalreactions, turbulence and heat transfer, to solid dynamics and electromagnetic. It includestools for meshing, for the initialization of complex CAD geometries, and for pre- and post-processing. Almost everything (including meshing, and pre- and post-processing) runs inparallel as standard, enabling users to take full advantage of computer hardware at theirdisposal.

By being open source, OpenFOAM oers users complete freedom to customize and extendits existing functionality, either by themselves or through support from OpenCFD. It followsa highly modular code design in which collections of functionality are each compiled intotheir own shared library. Executable applications are then created that are simply linked tothe library functionality.

OpenFOAM can be adapted to solve hypersonic ow due to its exibility and the pres-ence of several included examples. OpenFOAM is rst and foremost a C++ library, usedprimarily to create executables, known as applications. The applications fall into two cate-gories: solvers, that are each designed to solve a specic problem in continuum mechanics;and utilities, that are designed to perform tasks that involve data manipulation.


At the same time being a C++ library can be a big disadvantage for users that are notcondent with this type of language. Especially in a preliminary phase of the design it isimportant to obtain results easily. OpenFOAM can be used in a second phase in which moredetails are required.

1.2.4 Considerations on the sate of the art

Although several codes for the hypersonic ux simulation exist, in the previous sections onlythree are presented in order to have a quick introduction of the state of the art. It can bedone a rst big separation of the existing codes: the open-source and the commercial.

The rsts use often programming languages as FORTRAN or C++ and the utilizationof them can result eventually hard for users that are not used to these languages. On theother hand, they often have the possibility to be implemented and an expert user has theopportunity to modify the code in order to adapt it for his scopes.

Otherwise the commercial codes have usually an user friendly interface but they aregenerally very expensive or hardly available. They oer a big level of detail on the resultsthat generally an open-source code don't take into account.

Starting from these considerations, it is written a code completely open-source that tryto be user-friendly, with the possibility to be improved or adapted to dierent cases. Havingas a goal the creation of an instrument for the calculation of the aerodynamic coecients ina conceptual phase of the design, the level of the detail of the analysis remain at a high level.

Chapter 2

Theoretical Background

In this chapter it is introduced all the theoretical background that has been analyzed forthis work. First the laws that characterize the dynamic of a vehicle are introduced, then thefeatures of the hypersonic motion, nally the theory of optimal control is presented, in orderto understand the shape optimization process.

2.1 Launch and re-entry aerodynamic

In order to evaluate the aerodynamic coecients of the noses of the vehicles, a body-axesreference frame is chosen. This frame is set in a longitudinal symmetry plane of the nose, atthe beginning of the nose as shown in the Figure 2.1. This reference frame will be used in allcalculations.

Figure 2.1: Body-axes reference frame [15]

From the classical aerodynamic theory, the forces and the moments acting on the aircraftare dened as [11]:

D =1

2ρ∞ArefV

2∞CD (2.1)

17

CHAPTER 2. THEORETICAL BACKGROUND 18

S =1

2ρ∞ArefV

2∞CS (2.2)

L =1

2ρ∞ArefV

2∞CL (2.3)

L =1

2ρ∞Aref lrefV

2∞Cl (2.4)

M =1

2ρ∞Aref lrefV

2∞Cm (2.5)

N =1

2ρ∞Aref lrefV

2∞Cn (2.6)

in which the Aref and lref are the reference surface and length of the nose, ρ∞ and p∞ arethe air density and pressure referred to the innity.

Evaluating the six coecients for drag CD, lateral force CS, lift CL, roll moment Cl, pitchmoment Cm and yaw moment Cn, it will be possible to calculate all the forces and momentsacting on the vehicle.

The speed of the airow can be referred as the body-axes reference frame with the attackangle α and the side-slip angle β, obtaining

V∞ = V∞ · [− cos(α) cos(β) − sin(β) − sin(α) cos(β)]T (2.7)

One important parameter in the aerodynamics of rockets and re-entry vehicles is theballistic coecient, B∗ [kg/m2] dened as

B∗ =m

CDAref(2.8)

This engineering parameter gives an estimate of the deceleration that the aircraft have, dueto drag. This can be an optimization parameter. In fact, for launchers it should be as low aspossible in order to reduce the thrust, so the fuel. On the other hand for the re-entry vehicles,it should be high in order to dissipate the kinetic energy of the spacecraft and landing withlow speed. The shape can be optimized taking into account all the structural and thermallimitations and other constrains.

Finally it will be taken into account the heating rate exchanged with the aircraft and theEarth atmosphere q in [W/m2] that can be expressed with the empirical formulation [12]:

q = 1.83 · 10−4V 3∞

√ρ∞rn

(2.9)

in which rn the vehicle's nose radius. This approximation is only valid in hypersonic ightregime and it is generally used for the rst estimations of the heat ux.


2.2 Inviscid ow

An inviscid ow is the ow of an ideal uid that is assumed to have no viscosity. In uiddynamics there are problems that are easily solved by using the simplifying assumption ofan inviscid ow. The ow of uids with low values of viscosity agrees closely with theinviscid ow everywhere except close to the uid boundary where the boundary layer plays asignicant role. The presence of a boundary layer, which enhances the eect of even a smallamount of viscosity, can often cause sensible eects on the ow, namely energy dissipation.

Nevertheless, in the present work it is used the inviscid ow model for three dierentreasons. First, according to the main goal of developing a tool for helping the design ofhypersonic vehicle in the preliminary phases of the project, the level of details in the analysisof the uid evolution can't be so high in order to achieve results in a faster way. Then in thehypersonic model, the eects of the boundary layer change, as will be explained in section2.2.2. Finally, to achieve analytical results some simplications are necessary.

2.2.1 Navier-Stokes equations

The Navier-Stokes equations describe the motion of uid substances. These equations arisefrom applying Newton's second law to uid motion, together with the assumption that thestress in the uid is the sum of a diusing viscous term (proportional to the gradient ofvelocity) and a pressure term.

They describe the conservation of mass, momentum and energy of all the particles in alimited system. They can be expressed in the general form as [4]:

∂ρ

∂t+∂(ρui)

∂xi= 0 (2.10)

∂ρui∂t

+∂(ρujui)

∂xj=

∂τij∂xj− ∂p

∂xi(2.11)

∂(ρE)

∂t+∂(ρujE)

∂xj=

∂

∂xj

(k∂T

∂xj

)+

∂

∂xj(τijui) (2.12)

in which the Eq.(2.10) denes the conservation of mass, Eq.(2.11) the conservation of mo-mentum, equation Eq.(2.12) the conservation of energy.

In the equation ρ and p are the density and the pressure of the uid, ui is the componentof the velocity eld in the direction i, E is the specic total energy made of the kinetic

energy and the initial energy, and for the Newtonian uid τij = µ[(

∂ui∂xj

+∂uj∂xi

)− 2

3∂uk∂xk

],

where µ is the viscosity of the uid. For studying the evolution of the velocity eld, only theEq.(2.10) and Eq.(2.11) are analyzed and approximated. In fact the conservation of massand momentum can be studied separately from the conservation of energy and due to thesimplications made in this work, the thermal eects that are governed by Eq.(2.12) areignored. The eects of this simplication are discussed in section 2.2.2.

In case of inviscid ow, the viscosity µ is supposed to be zero, so the conservation of themomentum in 3D reduces to:


ρ

(∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

)= ρgx −

∂p

∂x(2.13)

ρ

(∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

)= ρgy −

∂p

∂y(2.14)

ρ

(∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z

)= ρgz −

∂p

∂y(2.15)

in which the velocity eld is u =[u v w

]Tand the gravity eld is g =

[gx gy gz

]T.

These equations are known as Euler's equations.Note that the equations governing inviscid ow have been simplied compared to the

Navier-Stokes equations; however, they still cannot be solved analytically due to the com-plexity of the nonlinear terms u∂u

∂x, v ∂u

∂y, w ∂u

∂z, etc. Hence, in the study of uid mechanics,

numerical methods such as the nite element and nite dierence methods are often used toapproximate the uid ow problems. Considering an incompressible ux, ρ = const, Euler'sequations can be also written in vector form as [28]:

∇ · u = 0 (2.16)

ρ∇uDt

= ρg −∇p (2.17)

Eqs.(2.16-17) will be developed in the Chapter 3 for building the numerical model, usingthe nite element theory. Traditionally, computational methods solve the inviscid Euler'sequation [18], but they are generally not valid for the hypersonic ux. The estimated errorof this simplication will be discussed in Chapter 6.

2.2.2 Inviscid hypersonic ow

In the design of a launcher or of a re-entry vehicle, one of the most critical phase of themission is the hypersonic ight. In fact this is the phase in which the vehicle has the largervelocity, so the high solicitations. Moreover in this region the intense temperature rises behindthe shock wave, combined with the rarefaction of the free-stream ow, leads to phenomenasuch as molecular dissociation and ionization, and radiative heath transfer from the hightemperature gas to the surface.

Moreover it is not trivial to dene which is an hypersonic ow because there is nota denite boundary between the simply supersonic ow [2]. Generally the main propertydening the hypersonic ow is that when it encounters a body, it produces a shock which isvery close to the body. Calling the region between the shock and the body a shock layer, thehypersonic ows are characterized by a thin shock layer.

This phenomenon happens as a prosecution of the oblique shock layer theory, which statesthat increasing the free-stream Mach number, the shock is going to lie closer to the body [2].It can be easily dened by considering the relation between the wedge angle θ and the shockangle β, given by the equation [11]:


tan(θ) = 2 cot(β)

[M2

1 sin2(β)− 1

M21 (γ + cos(2β)) + 2

](2.18)

in which γ is the heat capacity ratio and M1 is the Mach number of the free-stream.From Eq.(2.18) it is possible to build the well known diagram that shows the trend of the

shock that tends to be closer to the body as M tends to innity.

Figure 2.2: β − θ diagram for M = [2,∞]

Figure 2.3 shows the dierence between a sonic and an hypersonic shock for a wedge of20: the hypersonic ow results very close to the body.

Figure 2.3: Sonic (a) and hypersonic (b) shock layers for a 20° wedge


This behavior of the hypersonic ows leads to one of the key hypotheses of the algo-rithm [2]. The ows tends to change almost instantaneously its direction from the free-streamorientation to a direction tangential to the surface and with this consideration it's possible tosimplify the study of the phenomena. In fact it is possible to write an approximate denitionof the velocity vector over the body surface, which is found by considering the velocity onthe body only with the tangential component of the free-stream velocity [2]:

Vbody−local = V‖ = V∞ −V⊥ = V∞ + (n ·V∞) · n

where Vbody−local = V‖ means that the velocity vector considered on the body has only thetangential direction and it is possible to consider the normal component of the velocity asV⊥ = − (n ·V∞) · n, for n the outward normal vector to the surface of the body.

With this hypothesis it is possible to make two assumptions:

it is necessary to know the velocity eld on the vehicle's surface for characterize theow, so the forces that act on the body [2]

it is possible to obtain the inviscid pressure on the vehicle's surface, simply by consider-ing the loss of normal momentum in the almost instantaneous change of ow directionfrom normal to tangential. This is the fundamental hypothesis of Newton method,which will be discussed in the section 2.3 [2].

Moreover, the hypersonic ow is characterized by a dramatic increasing of the temperature.This eect is given by the high friction between the air and the body.

The temperature can reach a level in which the gas changes its chemical proprieties,dissociates and can have molecular recombination in the boundary layer, or combinationwith ablative material chemical products, which produces complex chemical reactions nearthe wall.

For the purpose of this work, all this eects will be taken into account only in order toverify the structural and thermal limits. In fact, although the chemical reactions change theproperty of the uid and modify the interaction in term of forces between the body and theux, for the assumptions made in this work these eects are ignored [26]. For conceptualdesign phase, the temperature eects can be negligible: all the analysis is done only onthe dynamic eects of the ow that change the eld of pressure along the vehicle's surface.Adopting the Newton model for the hypersonic ow, described in section 2.3, for a perfectgas, the expected error between a more realistic model of the ow can be estimated up to10%, as shown in Figure 2.4. Increasing the Mach number, the real evolution of the free-owtends to the Newton model. In fact, even having a stronger eect of the high temperaturedue to the friction between the body and the ux, improving the velocity, the dynamic eectsbecome dominant, so closer to the Newton theory in which they are only considered.


Figure 2.4: Typical variation of the pressure coecient with Mach number [26]

2.3 Newton method

There are several models for analyzing this type of aerodynamic ow, some very based on thephysics of the problem as the Portland-Meyer theory, others that simplify more the problemas the Newton one. In this work the Newton method is used to evaluate analytically thepressure coecient along the dierent shapes of the noses. In fact in the high hypersonicmotion, the behavior of the ux perfectly agree with the Newton theory assumptions.

The origin of the Newton method is based upon a coincidence between the hypothesesthat sir Isaac Newton formulated in the 1687 about a uid ow and the conditions that arisewhen considering an hypersonic ow.

In fact Newton described a uid ow as a system of particles traveling in rectilinearmotion that in case of striking a rigid surface, lose all the momentum normal to the surfaceconserving only the momentum tangential to the surface [2], as it is shown in Figure 2.5.


Figure 2.5: Momentum of a gas particle in Newton assumption [2]

This assumption is generally wrong for normal ows, but it ts with the simplicationmade for the velocity vector of an hypersonic ow. In fact when the Mach number rises,the shock layer lies close to the body surface, so the resulting momentum of the particles istangent to the surface [2], as said in section 2.2.

Considering the plate in Figure 2.5, it is possible to dene the mass ux passing throughthe surface A as

m = ρ∞V∞A sin(θ) (2.19)

and after the impact, loosing the normal velocity, the ow has a momentum per unit massreduction of

∆V = V∞ sin(θ) (2.20)

So the time variation of ow momentum, that is equal to the force acting on the body forthe second principle of dynamics, becomes

F =dP

dt= m∆V = ρ∞V

2∞A sin2(θ) (2.21)

or

F

A= ρ∞V

2∞ sin2(θ) (2.22)

It is easy to notice that Eq.(2.22) is a denition for the pressure acting on the body [2].This pressure is not absolute, but it's the dierence between the absolute pressure and thefree-stream static pressure

F

A= p− p∞ = ρ∞V

2∞ sin2(θ) (2.23)

or, dividing for the dynamic pressure,

p− p∞12ρ∞V 2

∞= 2 sin2(θ) (2.24)

Eq.(2.24) is the simple denition of the pressure coecient in the Newton theory, that is:

Cp = 2 sin2(θ) (2.25)


As the particles are supposed to move straight and to change their direction only whenimpacting a surface, on the back side of a plate there would be no impact at all. So for theregion of the body which does not impact directly the hypersonic ow Cp = 0, that meansno modication on free-stream pressure. This is called shadow region.

Dening the outward-normal vector of the surface as n and the angle between this vectorand the free-stream velocity vector as φ, it's possible to dene the inclination of the surfaceas [2]:

θ = π − φ (2.26)

so the shadow region can be dened by the region in which θ < 0, from Eq.(2.26). Later inChapter 6 the analytic law that denes the shadow areas of each shape will be analyzed inorder to take into account on the calculation of the pressure coecient, only the parts of thebody wetted by the hypersonic ux.

2.4 Earth atmosphere model

In order to evaluate the forces and the moments acting on an aircraft ying in the Earthatmosphere, it's necessary to dene all the characteristics of this uid.

In this work a standard atmosphere is assumed, according to the International StandardAtmosphere (ISA) model [5].

Temperature, pressure, density and viscosity of the Earth's atmosphere are changing withthe altitudes and their evolution depends on the area of the atmosphere. In fact the ISAmodel divides the atmosphere into layers with linear temperature distributions up to analtitude of 120 km in the extended version of the International Civil Aviation Organization(ICAO).


Figure 2.6: ICAO atmosphere: pressure, density, temperature, speed of sound evolution withthe altitude [5]

Figure 2.6 shows the pressure, density, temperature, speed of sound evolution with thealtitude for the ICAO atmosphere.

ICAO sets the sea level values (SL) of this parameters as it is shown in Table 2.1.

Parameter Valuep [Pa] 101325ρ [kg/m3] 1.225TSL [K] 288.15aSL [m/s] 343

Table 2.1: Sea Level values for ICAO atmosphere [5]

In the simulation of the hypersonic ux, a Stratosphere condition of the free ow ischosen to x the characteristics of the air in the numerical simulations of the Navier-Stokesequations. In fact generally the velocity starts to be hypersonic in this area of the atmosphereas shown in Figure 2.7. For the simulation has been chosen the most critical area with theopen possibility to change the characteristics of the ow [19].


Figure 2.7: Flow regimes with typically re-entry events [19]

Parameter Valuep [Pa] 5474ρ [kg/m3] 0.088TSL [K] 270aSL [m/s] 249.41µ [kg/m/s] 1.42146·10−5

Table 2.2: Stratosphere values for ICAO atmosphere at 20 km [5]

Table 2.2 summarizes the characteristic of the air at 20km.

2.5 Constrained optimization

One of the goals of this work is to obtain analytical expressions for the aerodynamic coe-cients. Having these expressions, the optimization process can be done analytically.


In mathematical optimization, constrained optimization is the process of optimizing anobjective function with respect to some variables in the presence of constraints on thosevariables.

In this work the objective function is the drag coecient CD or the ballistic coecientB∗ = B∗(CD) which are to be minimized. Constraints can be either hard constraints whichset conditions for the variables that are required to be satised, or soft constraints whichhave some variable values that are penalized in the objective function if the conditions onthe variables are not satised.

Considering a problem of constrained optimization for an objective function f(x), gener-ally it is expressed as [20]:

find min f(x), x ∈ X ⊂ Rn

hj(x) = 0, j = 1, ..., l

gi(x) R 0, i = 1, ...,m

(2.27)

in which x ∈ X ⊂ Rn are the unknowns of the problem that are limited to the space X ⊂ Rn

by the constraints, h(x) and g(x) are respectively the equality and inequality constraintfunctions. So the space of the feasible solutions is dened as:

X = x ∈ R : gi(x) ≤ 0, i = i = 1, ...,m, hj(x) = 0, j = 1, ..., l (2.28)

Many ways of solving a constrained optimization exist, depending on the nature of thefunction f(x). For the case of having f(x) = CD(x), the objective function is often non-linear.

2.5.1 Karush-Kuhn-Tucker (KKT) method

One way to solve this problem is with the Karush-Kuhn-Tucker (KKT) method. For under-standing how to solve a constrained optimization problem with the KKT method, is necessaryto dene the KKT conditions [20]. In mathematics, the KarushKuhnTucker conditions arerst order necessary conditions for a solution in non-linear programming to be optimal, pro-vided that some regularity conditions are satised. Allowing inequality constraints, the KKTapproach to nonlinear programming generalizes the method of Lagrange multipliers, whichallows only equality constraints. The system of equations corresponding to the KKT condi-tions is usually not solved directly, except in the few special cases where a closed-form solutioncan be derived analytically. In general, many optimization algorithms can be interpreted asmethods for numerically solving the KKT system of equations.

In general, a point x∗ ∈ X is a local minimum if exists a neighborhood of x∗, I(x∗, ε),with radius ε > 0, that gives f(x)− f(x∗) ≥ 0 for every x ∈ X ∩ I(x∗, ε).

Having a constrained problem as dened by Eqs.(2.27), there are 4 KKT necessary con-ditions of the rst order, for having x∗ as optimal solution of the problem [20].

1. Condition of cancellation of the gradient of the Lagrangian function, associated to theproblem

∇f (x∗) +m∑i=1

λi∇gi (x∗) +l∑

j=1

νj∇hj (x∗) = 0 (2.29)

in which λi and νj are the Lagrangian multipliers for inequality and equality constrains.


2. Condition of feasibility of the optimal point x∗

gi (x∗) R 0 ∀i = 1, ...,m

hj (x∗) = 0 ∀j = 1, ..., l(2.30)

3. Condition of non-negativity of the Lagrangian multipliers associated to the inequalityconstrains

λ∗i ≥ 0 ∀i = 1, ...,m (2.31)

4. Condition of complementary slackness

λ∗i gi (x∗) = 0 ∀i = 1, ...,m (2.32)

that means that in case of having an inactive constrain, it has to be equal to zero. A constrainis called active if the Jacobin matrix of the constrain in the point x∗ has maximum range.

Since all the conditions have to be valid, it is necessary to solve a system in order toobtain the Lagrangian multiplicators and the optimal parameters x∗.

There are other KKT conditions of the second order, only sucient [21]. So the x∗ is aminimum of the problem if also the Hessian matrix of the optimal solution H (x∗, λ∗, ν∗) ispositive semi-denite on the tangent space T , that means

yTH (x∗, λ∗, ν∗) y ≥ 0 ∀y ∈ T (2.33)

dening the Hessian matrix as

H (x∗, λ∗, ν∗) = ∇2f (x∗) +m∑i=1

λ∗i∇2gi (x∗) +

l∑j=1

ν∗j∇2hj (x∗) (2.34)

and the tangent space as

T ≡y/∇Tgi(x)y, ∀i ∈ I(x) and ∇Thj(x)y = 0

(2.35)

2.5.2 Newton-like interior point method

As said in the section 2.5.1, it is very rare to obtain an analytical solution of the KKTconditions. The equality and inequality constraints will be dened in Chapter 5 but it ispossible to anticipate that they are all non-linear, due to the denition of the CD. Having anon-linear problem to solve, it is necessary to use an iterative algorithm.

A possible way of solutions are Newton-like algorithms. The Newton algorithm is basedon the idea of approximate the function with its derivative and looking at the sign of thederivative it's possible to nd a minimum or a maximum after some iterations [22]. Thealgorithms based of this approach require an initial estimate of x, at the interior of thefeasible region dened by the inequality constraints, and generate a sequence of points alsoat the interior of this set.

The Internal Point Non-Linear Programming (IPNLP) solver implements a primal-dualinterior point algorithm that shares several powerful features from recent state-of-the-art


algorithms. It uses a line-search procedure and a merit function to ensure convergence ofthe iterates to a local minimum. The term primal-dual means that the algorithm iterativelygenerates better approximations of the decision variables x, usually called primal variables,as well as the dual variables, referred to as Lagrange multipliers. At each iteration, thealgorithm solves a system of non-linear equations by using Newton's method. The solutionof that system provides the direction and the steps along which the next approximation of thelocal minimum will be searched. The algorithm also ensures that the primal iterates alwaysremain strictly within their bounds. The current version of the solver is particularly suitedfor problems that contain many dense nonlinear inequality constraints, and it is expected toperform better than other non-linear programming solvers.

They are feasible directions algorithms, since at each iteration they dene a search di-rection that is a feasible direction with respect to the inequality constraints and a descentdirection of the objective, or another appropriate function. When only inequality constraintsare considered, the objective is reduced at each iteration. The method of the interior pointis relatively simple to code, strong and ecient [21].

Let consider the non-linear inequality constrained problem:find min f(x), x ∈ X ⊂ Rn

gi(x) ≤ 0, i = 1, ...,m(2.36)

with the relatively KKT conditions dened from Eq.(2.29) to Eq.(2.32). It is a barrier methodand in particular the interior points algorithms try to nd a KKT point by solving a sequenceof problems, unconstrained with respect to the inequality constraints, of the form [27]

findxµ = arg minx∈Rn|g(x)=0

B(x, µ) (2.37)

where µ > 0 is a positive real number and the barrier function can be dened as

B(x, µ) = f(x)− µm∑i=1

ln(gi(x)) (2.38)

In order to respect the rst KKT condition, it must have

∇B(x, µ) = ∇f(x) + µ

m∑i=1

1

−gi(x)∇gi(x) = 0 (2.39)

in which it is possible to dene new Lagrangian multipliers λi = µ−gi(x)

of the Lagrangian

function [27]

L(x, λ) = f(x) +m∑i=1

λigi(x) = f(x)−mµ ≤ f(x) (2.40)

From that it is possible to notice that infxL(x, λ) ≤ f(x∗). Since that the point x

minimizes the Lagrangian function L(x, λ), it is valid that

f(x)−mµ ≤ f(x∗) ≤ f(x) (2.41)


so the point x is an optimal point only if µ→ 0.In order to solve this problem, a Newton's iteration method is adopted to solve the non-

linear system of the KKT conditions in (x, λ). It can be dened as [21]:[H(xk, λk

)∇g(xk)

Λk∇Tg(xk) G(xk)

] [xk+1 − xkλk+1

0 − λk]

= −[∇f

(xk)

+ λk∇g(xk)

λkG(xk)

](2.42)

where(xk, λk

)is the starting point of the iteration and

(xk+1, λk+1

0

)is the new estimate.

H(xk, λk

)is the Hessian matrix of the Lagrange equation, as dened in Eq.(2.34), G(xk) ≡

diag[g(xk)

]is the diagonal matrix such that Gii(x) = gi(x), Λ is a diagonal matrix with

Λii = λi.So it is possible to set the linear system in

(dk0, λ

k+10

), for dk0 = xk+1− xk is a direction in

the primal space:

Skdk0 + λk+10 ∇g

(xk)

= −∇f(xk)

(2.43)

Λk∇Tg(xk)dk0 + λk+10 G(xk) = 0 (2.44)

in which the matrix Sk is symmetric and positive denite and it can be taken as to H(xk, λk

),

a quasi-Newton approximation of H (x, λ), or the identity.It can be proved that d0 is a descent direction of f [21]. However, d0 is not useful as

a search direction since it is not necessarily feasible. This is due to the fact that as anyconstraint goes to zero, Eq.(2.44) forces d0 to tend to a direction tangent to the feasible set.In fact, Eq.(2.44) is equivalent to

λki∇Tgi(xk)dk0 + λk+1

0i gi(xk) = 0 i = 1, ...,m (2.45)

which implies that∇Tgi(xk)dk0 = 0 such that gi(x

k) = 0. To avoid this fact [21], it is necessaryto dene a new system in dk and λk+1:

Skdk + λk+1∇g(xk)

= −∇f(xk)

(2.46)

Λk∇Tg(xk)dk + λk+1G(xk) = −ρkλk (2.47)

obtained adding a negative vector to the right side of Eq.(2.44), with a positive scalar factorρ. In this case Eq.(2.45) is equivalent to

λki∇Tgi(xk)dk + λk+1

i gi(xk) = −ρkλki i = 1, ...,m (2.48)

and then ∇Tgi(xk)dk < 0 for the active constraints. Thus, dk is a feasible direction obtained

by adding the term −ρkλk. It produces a deection of dk0 into the feasible region, where thethe deection is proportional to ρk. As the deection of dk0 is proportional to ρk and d

k0 is a

descent direction of f , it is possible to nd bounds on ρk, in a way to ensure that dk is also

a descent direction. Since(dk0)T ∇f(xk) < 0, the bounds are dened by imposing [21](dk)T ∇f(xk) ≤ α

(dk0)T ∇f(xk) < 0 (2.49)


Eq.(2.49) means that dk is in a circular straight cone whose axis is ∇f(xk). In general,the rate of descent of f along dk will be smaller than along dk0 . This is a price to pay forobtaining a good feasible descent direction. Let consider the auxiliary linear system [21]:

Skdk1 + λk+11 ∇g

(xk)

= 0 (2.50)

Λk∇Tg(xk)dk1 + λk+11 G(xk) = −λk (2.51)

it is possible to deduce that dk = dk0 + ρdk1 and substituting this denition in Eq.(2.49) adenition of the parameter ρ is obtained,

ρ ≤(α− 1)

(dk0)T ∇f(xk)(

dk1)T ∇f(xk)

(2.52)

Finally, to get a new feasible primal point and a satisfactory decrease of the objectivefunction, an inaccurate constrained line search along dk is done. Dierent updating rules canbe adopted to dene a positive λk+1.

ALGORITHM DESCRIPTION [21] Setting parameters α ∈ (0, 1) and ϕ > 0. Initial-izing x0 ∈ X, λ0 > 0 and the matrix S. (for a more detailed description of the algorithmand of the other cases, see [21], [22]).

STEP 1. Computation of the search direction d.i) Solve linear system for (d0, λ0):

Sd0 + λ0∇g (x) = −∇f (x)Λ∇Tg(x)d0 + λ0G(x) = 0

(2.53)

and if d0 = 0, STOP.ii) Solve linear system for (d1, λ1):

Sd1 + λ1∇g (x) = −∇f (x)Λ∇Tg(x)d1 + λ1G(x) = 0

(2.54)

iii) Set:

ρ = infϕ||d0||2; (α−1)(d0)T∇f(x)

(d1)T∇f(x)

if (d1)T ∇f(x) > 0

ρ = ϕ||d0||2 otherwise(2.55)

iv) Compute:d = d0 + ρd1

λ = λ0 + ρλ1(2.56)

STEP 2. Line search.

Find a step length t satisfying a given constrained line search criterion on the objectivefunction and such that:

gi(x+ td) < 0 if λi ≥ 0gi(x+ td) ≤ gi(x) otherwise

(2.57)

STEP 3. Updates.


i) Set:x = x+ td (2.58)

and dene a new value for λ > 0 and evaluate a new S symmetric and positive denite.ii) Go back to STEP 1.

2.5.3 Sensitivity of the parameters

Often is important to know how each constraint inuences the objective function.Studying an active constraint gj, with the associated Lagrangian multiplier λj > 0, for

the second KKT condition in an optimal point is valid that gj(x∗) = 0. Introducing a little

perturbation ε > 0, it is possible to write [20]

gj(x) ≥ −ε ‖∇gj(x∗)‖ (2.59)

that indicates that the constraint has been improved of the ε quantity. So the optimal pointbecomes x∗(ε).

Using the developing on Taylor siree, it is possible to write

−ε ‖∇gj(x∗)‖ = gj(x∗(ε))− gj(x∗) ≈ (∇gj(x∗))T (x∗(ε)− x∗) (2.60)

while for the other constraints the fourth KKT condition

0 = gk(x∗(ε))− gk(x∗) ≈ (∇gk(x∗))T (x∗(ε)− x∗) (2.61)

is still valid. Dening the optimal solution f(x∗) and the perturbed solution f(x∗(ε)), by thedenition of Lagrange equation is possible to obtain [20]

f(x∗(ε))− f(x∗) ≈ (∇gj(x∗))T (x∗(ε)− x∗)=∑k∈Iλ∗k∇gk (x∗) (x∗(ε)− x∗) ≈ λ∗j (∇gj(x∗))T (x∗(ε)− x∗) (2.62)

being valid Eq.(2.61). From Eq.(2.62) is possible to dene the perturbed solution as

f(x∗(ε))− f(x∗) ≈ −ελ∗j ‖∇gj(x∗)‖ (2.63)

and nally the sensitivity of the function f(x) to the parameter j

df(x∗(ε))

dε

∣∣∣∣ε=0

= −λ∗j ‖∇gj(x∗)‖ (2.64)

If λ∗j ‖∇gj(x∗)‖ is big, the second term of Eq.(2.64) highly inuence the objective function.A small perturbation will induce a large change in the optimal solution [20].

Chapter 3

Calculation of aerodynamic coecients

This chapter is focused on the denitions of the aerodynamic coecients. Starting fromthe assumptions given by the Newton method, the following sections show the mathematicalapproach used for the analytically and numerically calculation of the aerodynamic coecients.

3.1 Analytical approach

As already described in section 2.3, the pressure coecient for a hypersonic ow is given byEq.(2.25). This simple relation is valid only for the hypersonic motion in which the shocklayer lies on the wall of the vehicles and so the pressure exerted on the vehicle is due to thetotal loss of momentum normal to the surface [1].

Thus, considering a dierential element of the surface dA, results in a dierential force, dfacting on the vehicle surface in the direction of the inward normal vector n. So it's possibleto dene df = CpndA [1].

Using conventional aircraft body axes shown in Figure 2.1 and the corresponding free-stream velocity vector V∞, dened in Eq.(2.7) as function of angle of attack α and side-slipβ, the aerodynamic force coecients along the body axes are [1] CD

CSCL

=1

Aref

¨S

df =1

Aref

¨S

Cp

nxnynz

dA (3.1)

and for the moments [1] ClCmCn

=1

Aref lref

¨S

r× df =1

Aref lref

¨S

Cp

(r×n)x(r×n)y(r×n)z

dA (3.2)

The position vector r denes all the points of the surface S by a parametrization it twovariables (u, v) [1],

r =[f(u, v) g(u, v) h(u, v)

]T(3.3)

where f(u, v), g(u, v), and h(u, v) describe the x, y, and z location of a point on the surfaceof the vehicle as a function of the surface parametrization (u, v).

34

CHAPTER 3. CALCULATION OF AERODYNAMIC COEFFICIENTS 35

One of the most important choice of the designer is on the type of parametrization used.In fact the selection of (u, v) parameters dramatically aects the possibility of having aclosed-form solution for the integrations [1]. Generally, given a function in the x − z planez(x) = f(x), the parameter functions for a surface of revolution with the x-axis of symmetryare

r =[x(u) z(u) cos(v) −z(u) sin(v)

]T(3.4)

Additionally, due to the convention adopted on the surface outward-normal n, the choiceof u and v also inuences the expression for the dierential area dA of the integrations. Sothe outward-normal vector to the surface can be dened as [1]

n =ru × rv‖ru × rv‖

(3.5)

where ru and rv are the partial derivative dened as

ru = ∂r∂u, rv = ∂r

∂v(3.6)

Then, it is possible to dene the dierential area by

dA = ‖n‖ = ‖ru × rv‖ (3.7)

Dened the parametrization of the surface, it is possible to evaluate the pressure coe-cient. It is possible to dene sin(θ) as a relation between the normal vector and the velocityof the free-stream.

sin(θ) = V∞ · n (3.8)

So recalling Eq.(2.7) and Eq.(2.25), it is possible to obtain a new denition of the pressurecoecient, substituting Eq.(3.5) in Eq.(3.8)

V∞ = [− cos(α) cos(β) − sin(β) − sin(α) cos(β)]T

Cp = 2 sin2(θ) = 2(V∞ · n

)2

= 2

(V∞ ·

ru × rv‖ru × rv‖

)2

(3.9)

In order to evaluate the shadowed region in which Cp = 0, it is necessary a study of thewetted surface, depending on the angle of attack and side-slip of the velocity. So it is requireda preliminary study for xing the limits of the integration, [umin, uMAX ] and [vmin, vMAX ].The solution of this problem is not trivial, especially if the surface is parametrized withtrigonometric functions [1]. For this region only convex shapes are supported and the shad-owed regions can be characterized by θ = π − φ < 0.

The reference area and reference length for each shape are computed based on theparametrization used. The reference area is computed as the projected area of the shapeon the y − z plane, considering the entire surface unshadowed α = β = 0. For composedshapes, Aref is always the projected area of the largest section. The reference length iscomputed as the maximum span of the vehicle in the x direction [1].


Finally, after all these considerations, it is possible to compute the integrals in order toobtain the aerodynamic coecients. CD

CSCL

=1

Aref

¨2

(V∞ ·


)2 nx

nynz

dudv (3.10)

ClCmCn

=1

Aref lref

¨2

(V∞ ·


)2 (r×n)x

(r×n)y(r×n)z

dudv (3.11)

Once it is dened the geometry of the problem, these integrals can be evaluated analyt-ically with the help of Mathematica. The code automatically evaluates also the shadowedregions and uses the results for the optimization.

3.2 Numerical approach

In order to evaluate the results obtained with the analytical approach, a numerical calculationis developed in parallel. This calculation conrms the analytic results and allows for thecomparison of the performance of the analytical approach.

3.2.1 Newton Method

In the current state of the art of Newton method, the surface integration is performed numer-ically using panel methods that approximate the shape of a vehicle using small at plates.This numerical calculation is performed directly in Mathematica, in order to validate theanalytical results.

Selecting an arbitrary number N of plates for the discretization, the integral surface isdivided in ∆Ai panels, with i ∈ [0, N ]. The choice of N inuences the rapidity and theprecision of the calculation.

Each panel can be studied as a at plate. The plate is parametrized by the half-angleof the blended wedge, δc that indicate the direction of the normal vector ni, for the plate i.According to the Newtonian ow theory, the pressure coecient of a plate is constant, so theglobal results can be expressed as summing the eects of every plate [1].

So having already dened the pressure coecient in Eq.(3.9), the aerodynamic coecientscan be approximated by:


CD = 1Aref

N∑i=0

Cp,inx∆Ai

CS = 1Aref

N∑i=0

Cp,iny∆Ai

CL = 1Aref

N∑i=0

Cp,inz∆Ai

Cl = 1Aref lref

N∑i=0

Cp,i(r×n)x∆Ai

Cm = 1Aref lref

N∑i=0

Cp,i(r×n)y∆Ai

Cn = 1Aref lref

N∑i=0

Cp,i(r×n)z∆Ai

(3.12)

This method is directly derivated from the Newton's theory. The results of the applicationof this approach are presented in Chapter 6 to validate the analytical results in terms ofnumerical values. The approximation is then compared with the numerical simulation of theux by the solution of the Navier-Stokes equations, as described in section 3.2.2.

3.2.2 Navier-Stokes method

The Navier-Stokes equations can be implemented using this tool obtaining directly the streamevolution along the bodies. On this purpose, a solution of the equations is obtained withFreeFem++, a free numerical integrator of PDE, using nite elements.

It is considered a body embedded in a 2D ow governed by the steady incompressibleNavier-Stokes. The uid is considered with density and viscosity, so it's the case of Eulerequations, dened in section 2.2.1. This case is considered for its relatively simplicity on theimplementation, but can be inadequate for the study of the hypersonic ux. The expectederror on the result respect to the Newton theory will be discussed in Chapter 6. However thesimulation is made in order to understand its order of magnitude in terms of computationaltime, compared with the analytical approach.

Let's consider an approximate model of the Stokes equations, namely the pseudo-compressibleapproximation where a pressure term is added to the continuity equation with a coecientε and introducing the kinematic viscosity ν = µ/ρ, Eqs.(2.16-17) can be written in a dimen-sionless form as [13]:

∇ · u + εp = 0 (3.13)

∂u

∂t+ u · ∇u− 1

Re4u +

1

ρ∇p = g (3.14)

in which the Reynolds number is Re =V∞lref

ν. Small Reynolds numbers lead to a laminar

ow dominated by Stokes eects, whereas large Reynolds numbers lead to a turbulent ow.For intermediate Reynolds, the ow is said to be in transition regime [13].


Figure 3.1: Domain for the control problem and boundary conditions

Let's consider the body as a sum of control boundaries Γ in a control domain Ω, in relativemotion with the ow, dened by the vector u, as shown in Figure 3.1. It is possible to writethe Navier-Stokes equation as [6]:

∇ · u + εp = 0 in Ω∂u∂t

+ u · ∇u− 1Re4u + 1

ρ∇p = 0 in Ω

u = V∞ on ΓIN

u · n = 0, u · t = 0 on ΓSYM

p = 0 on ΓOUT

u = 0 on ΓNS

(3.15)

where n and t are the inward directed normal and the tangential unit vectors on the boundaryΓi.

Inow conditions are imposed on the boundary: on ΓIN Dirichlet conditions (u is thecontrol variable), which corresponds to imposing the velocity V∞, symmetry conditions onΓSYM , no stress conditions on ΓOUT and no slip conditions on ΓNS. It's possible to noticethat ∪iΓi = ∂Ω and Γi ∩ Γj = ∅, ∀i, j [6].

It is important to notice that the system becomes non-linear due to the convection termu·∇u. So in order to solve the problem with nite elements method, it's necessary to multiplythe Eqs.(3.13-14) by a smooth test function v, that has the same dimension of the velocityeld u, and by a smooth test function q for the pressure p. Then considering the variationin time dt ≈ ∆t, the problem becomes semi-discrete in time according to the method ofcharacteristics [13]. So integrating in the domain Ω and applying the Green's theorem, it'spossible to write the variational or weak formulation of the problem:

ˆΩ

∇ · un+1qdΩ + ε

ˆΩ

pn+1qdΩ = 0 (3.16)


ˆΩ

un+1 − un Xn

∆t· vdΩ + ν

ˆΩ

∇un+1 · vdΩ− 1

ρ

ˆΩ

pn+1∇ · vdΩ =

ˆΩ

g · vdΩ (3.17)

For a given eld un, considering (un+1, pn+1) as the unknowns to be determinate, for all(v,q)∈ V , for V the space of the test functions [13].

Finding the eld of velocity and pressure, it is easy to evaluate the pressure coecient as

Cp = 2γM2∞

(pp∞− 1)for comparing with the one given by the Newton theory. For a better

comparison a value of the ratio of specic heat capacity, γ → 1, that better approximatesthe hypersonic conditions should be taken [11].

Using Eqs.(3.1-2) all the aerodynamic coecients can be evaluated with the eld of pres-sure given by the Navier-Stokes equations. For example, considering the drag coecients,its 2D value can be calculated, having from the simulation a value of Cp and due to thesymmetry of the problem, the 3D drag coecient is obtained rotating the 2D shape for 2π:

CD3D= 2πCD2D

=2π

lrefCp

ˆ∂Σ+

nxdu (3.18)

where ∂Σ+ is the upper boundary of the 2D shape.

3.3 Inuence of Knudsen number

Considering especially the re-entry of vehicles in the Earth atmosphere, the hypersonic ighttakes place in the higher parts of the atmosphere. So throughout the re-entry trajectory, thevehicle will be operating in one of the three ight regimes: free molecular ow, transition, orcontinuum ow.

These regimes are dependent upon the Knudsen number that is a non-dimensional pa-rameter which indicates the relative importance of the particulate nature of air [8]. It can bedened as

Kn =λ

lref(3.19)

in which λ is the mean free path of the particles, that is an indication of the relaxationdistance in a gas. The relaxation distance is a measure of the distance that a particle of gashave to go through before interacting with another particle. It's possible to estimate the freemolecular path knowing the type of uid

λ =m(√

2πσ2ρSLe− hH

) (3.20)

in which m is the mass of the particles [kg], σ2 is the eective diameter of gas particles [m],h is the altitude, H w 29.26 · T m/K is the scale height depending on the temperature [8].

So for high values of λ, the gas can be dened as a continuum in which the collisionsbetween the particles are very frequent. There is not a dened boundary between a continuumgas or a region with free molecular ow.


Generally the free molecular ow region is dened as a region whereKn 1, let's considerKn > 10, while the continuum region is dened where Kn 1, let's consider Kn < 0.1.The region between the two, 0.1 ≤ Kn ≤ 10 can be considered the rareed-ow transitionregion [8]. In this region the ow cannot be treated as a continuum, nor can the particularnature of the gas molecules be neglected. Summarizing the three dierent regions:

1. Kn 1, free molecular ow. Very high altitude, there are practically no collisionbetween molecules near the vehicle and each of them interacts individually with thevehicle wall according to the statistical theory of gases.

2. Kn 1, continuous ow. Low altitude, motion of the molecules is governed by theclassical uid mechanics theory, following the Euler's equations for the inviscid ow [18].

3. 0.1 ≤ Kn ≤ 10, rareed ow. This intermediate section can be approached either usingNavier-Stokes equations with slip conditions at the wall (ows close to continuous) orusing Boltzmann equations solved by direct methods of resolution or by direct MonteCarlo simulation (DSMC) methods [18]. This ow corresponds to altitude section wherethe aerodynamics forces are much lower than the weight, but the aerodynamic momentsmoments start to be signicant.

In this last region the expressions of the aerodynamic coecients need to be corrected withempirical factors. The analysis is done for the drag and lift coecients for obtaining [8]

C ′D = CD(1− CD) (3.21)

C ′L = CL(1− CL) (3.22)

in which the coecients CD and CL are corrected by the correction factors, expressed follow-ing the bridging relation as [8]

CD = exp[−0.29981 (1.3849− log(Kn))1.7128] if log(Kn) < 1.3849

CL = exp[−0.2262 (1.2042− log(Kn))1.8410] if log(Kn) < 1.2042

(3.23)

Chapter 4

Shapes denition

The geometry of the main part of the launchers and the ballistic vehicles is often simple andit can be well approximated by a geometric shape that can be expressed analytically[1].

In this chapter some basic shapes are analyzed and parametrized. The vehicle's nosesare built by linking two or more basic curves, always showing a blunted or an unbluntedconguration of the nose. Each basic shape is chosen to have symmetry along the bodyaxis dened in the Figure 2.1. With the help of the software Mathematica, it is possible toevaluate analytically the surfaces integrals, easily or not so easily, depending on the shapeand on the parametrization variable (u, v).

For all further calculations the most general case is analyzed. However, in case of theimpossibility of having an easy integration it has been considered only the unshadowed con-dition in which all the shape is wetted by the ux. This approximation ts well with rockettrajectories which have generally very small angles of attack or side-slip.

4.1 Cone family

The cones are not used alone for the hypersonic nose due to the relevant heating that wouldoccur on the sharp nose. In face for a radius rn → 0 on the nose, for Eq.(2.9), the heatingux q →∞. Rather they are used to compose fore-body of the hypersonic aircraft in severalcombinations.

The geometrical design parameters of the cone are dened by the length along the axisof revolution LcIi and LcIf , by the beginning radius RcIi and the ending radius RcIf , inwhich the notation cI is referred at the cone I. So the cone half-angle is easily dened as

δcI = arctan(RcIi−RcIfLcIf−LcIf

).

The surface of the cone is parametrized by:u = z(x) = r(x), r ∈ [RcIi , RcIf ]

v = ω, ω ∈ [0, 2π](4.1)

in which ω is the revolution angle as shown in Figure 4.1. The equation of the semi-cone inthe x− z plane is

z(x) = −x tan δc +RcIi (4.2)

41

CHAPTER 4. SHAPES DEFINITION 42

Figure 4.1: Side and front view of cone parametrization

So the position vector is dened as

r =[− u

tan δc+

RcIitan δc

u cos(v) −u sin(v)]T

(4.3)

Eq.(4.3) is used in Eq.(3.9) for evaluate the pressure coecient and then integrated alongthe surface obtaining the following aerodynamic coecients:

CD = − 1Aref

2πtanδc(RcIf−RcIi )(cos2β(2cos2αtan2δc+sin2α)+sin2β)

(tan2δc+1)3/2

CS = − 1Aref

4π cosα cosβ(RcIf−RcIi) sinβ tan δc

(1+tan2 δc)3/2

CL = − 1Aref

4π cosα cos2 β(RcIf−RcIi) sinα tan δc

(1+tan2 δc)3/2

Cl = 0

Cm = − 1Aref lref

2π cosα cos2 β(RcIf−RcIi) sinα(RcIf−RcIi+(RcIf+RcIi) tan2(δc))(1+tan2(δc))

3/2

Cn = 1Aref lref

2π cosα cosβ(RcIf−RcIi) sinβ(RcIf−RcIi+(RcIf+RcIi) tan2(δc))(1+tan2(δc))

2

(4.4)

and as expected due to the symmetry of the shape, the roll moment coecient is alwaysequal to zero. The integrations were performed with the help of Mathematica.

Then it could be useful to evaluate the angular coecient Cm,α = ∂Cm∂α

and Cn,β = ∂Cn∂β

,for an additional stability studies, considering small angle of attack and side-slip.

sinα ' α, cosα ' 1

sin β ' β, cos β ' 1(4.5)

So the angular coecients:

Cm,α = − 1Aref lref

2π(RcIf−RcIi)(RcIf−RcIi+(RcIf+RcIi) tan2(δc))(1+tan2(δc))

3/2

Cn,β = 1Aref lref

2π(RcIf−RcIi)(RcIf−RcIi+(RcIf+RcIi) tan2(δc))(1+tan2(δc))

3/2

(4.6)

Later in Chapter 6 it will be dened the shadowed surface so the extremes of the inte-gration, depending on the direction of the free-stream. It's easy to notice that the shadowoccurs when α,β ≥ δc.


4.2 Spherical nose family

To reduce the aeroheating, is often added a spherical segment as nose of the vehicle. Generallythe spherical nose is dened by the nose radius rn and the tangent half-angle δtan that isdened by the tangency condition between the nose and the previously part of the vehicle.Spherical noses are generally used to create blunted shapes.

The spherical surface is parametrized by:u = δ, δ ∈ [0, π

2− δtan]

v = ω, ω ∈ [0, 2π](4.7)

in which ω is the revolution angle and δ is the nose angle as shown in Figure 4.2. Theequation of the spherical segment in the x− z plane is

z2 + (x− xc)2 = r2n (4.8)

with xc = Ls − rn sin(δtan) is the coordinate of the center of the sphere.

Figure 4.2: Side and front view of spherical parametrization


r =[xc + rn cos(u) rn cos(v) sin(u) rn sin(v) sin(u)

]T(4.9)

Eq.(4.9) is used in Eq.(3.9) for evaluate the pressure coecient and than integrated alongthe surface obtaining the following aerodynamic coecients:

CD = 1Aref

13π sin(δsp)(cos(2δsp) (cos2β (2cos2α− sin2α)−−sin2β) + cos2β (10cos2α + sin2α) + sin2β)

CS = 1Aref

43πcosαcosβsinβ sin3(δsp)

CL = 1Aref

43πcosαcos2βsinα sin3(δsp)

Cl = 0Cm = − xc

Aref lref

43πcosαcos2βsinα sin3(δsp)

Cn = xcAref lref

43πcosαcosβsinβ sin3(δsp)

(4.10)

where δsp = π2− δtan and as aspected due to the symmetry of the shape, the roll moment

coecient is always equal to zero.


As in the previous case, the angular coecient Cm,α = ∂Cm∂α

and Cn,β = ∂Cn∂β

, can beevaluated for stability studies, considering small angle of attack and side-slip:

Cm,α = − xcAref lref

43π sin3 δsp

Cn,β = xcAref lref

43π sin3 δsp

(4.11)

Later in Chapter 6 the shadowed surface will be dened so the extremes of the integration,depending on the direction of the free-stream. It's easy to notice that the shadow occurs whenα,β ≥ δtan.

4.3 Ogive families

Next to a simple cone, the tangent ogive shape is the most familiar in rocketry. The proleof this shape is formed by a segment of a circle such that the rocket body is tangent to thecurve of the nose at its base; and the base is on the radius of the circle. In this work only atangent ogive is analyzed, with the open possibility to implement others types in the future.The popularity of this shape is largely due to the ease of constructing its prole [10].

Figure 4.3: Side and Front View of ogive parametrization

The radius of the circle that forms the ogive is called the ogive radius, ρ, and it is relatedto the length and base radius of the nose as [29]:

ρ =R2oi

+ L2o

2Roi

(4.12)

The nose cone length, Lo, must be less than or equal to ρ. If they are equal, then the shapeis a hemisphere. The equation of the ogive segment in the x− z plane [11] is:

z = Roi − ρ+√ρ2 − x2 (4.13)

The surface is parametrized by: u = x, x ∈ [0, Lo]

v = ω, ω ∈ [0, 2π](4.14)


in which ω is the revolution angle and Lo is the ogive segment length as shown in Figure 4.3.So the position vector is dened as

r =[u Roi − ρ+

√ρ2 − u2 cos(v) −

(Roi − ρ+

√ρ2 − u2

)sin(v)

]T(4.15)

The aerodynamic coecients are calculated similar to the previous cases:

CD = 1Aref

πL2o(2cosα2cosβ2L2

o−(L2o−2ρ2)(cosβ2

sinα2+sinβ2))2ρ3

CS = 1Aref

πcosαcosβsinβ(2L2oρ

2−L4o)

ρ3

CL = 1Aref

πcosαcosβ2sinα(2L2

oρ2−L4

o)ρ3

Cl = 0

Cm = 1Aref lref

πcosαcosβ2sinα(Roi−ρ)

(Lo(2L2

o−ρ2)√ρ2−L2

o+ρ4 tan−1

(Lo√ρ2−L2

o

))2ρ3

Cn = 1Aref lref

πcosαcosβsinβ(ρ−Roi )(Lo(2L2

o−ρ2)√ρ2−L2

o+ρ4 tan−1

(Lo√ρ2−L2

o

))2ρ3

(4.16)

and:

Cm,α = 1Aref lref

π(Roi−ρ)

(Lo(2L2

o−ρ2)√ρ2−L2

o+ρ4 tan−1

(Lo√ρ2−L2

o

))2ρ3

Cn,β = 1Aref lref

π(ρ−Roi )(Lo(2L2

o−ρ2)√ρ2−L2

o+ρ4 tan−1

(Lo√ρ2−L2

o

))2ρ3

(4.17)

Later in Chapter 6 the shadowed surface will be dened so the extremes of the integration,depending on the direction of the free-stream.It is easy to notice that the shadow occurs whenα,β ≥ maxδ(x), with δ(x) the local inclination of the ogive.

4.4 Parabolic families

The parabolic series nose shape are generated by rotating a segment of a parabola arounda line parallel to its latus rectum. This construction is similar to that of the tangent ogive.Just as it does on an ogive, this construction produces a nose shape with a sharp tip.


Figure 4.4: Side and front view of parabolic parametrization

The parabolic surface is parametrized by:u = z(x), z ∈ [Rpi , Rpf ]

v = ω, ω ∈ [0, 2π](4.18)

in which ω is the revolution angle and Rpi is the initial radius of the parabola as shown inFigure 4.4. The equation of the parabolic segment in the x− z plane is

z =Rpi

K

√1− x

Lp(4.19)

the constant 1/K can vary between 0 and 1. The most common values used for nose shapesare [29]:

1/K SHAPE0 cylinder0.5 1/2 parabola0.75 3/4 parabola1 full parabola

Table 4.1: Table of common shapes for parabolic nose

For the case of the full parabola (1/K = 1) the shape is tangent to the body at its base,and the base is on the axis of the parabola. Values of 1/K < 1 result in a slimmer shape.The shape is no longer tangent at the base, and the base is parallel to, but oset from, theaxis of the parabola [29].


r =[Lp − Lp

(K u

Rpi

)2

u cos(v) −u sin(v)]T

(4.20)

The aerodynamic and angular coecients are obtained in the same way as the previouscases. The analytic expressions obtained by the integrations made with Mathematica resultvery long. For space reasons the formulas are presented in Appendix A.


Later in Chapter 6 the shadowed surface will be dened so the extremes of the integration,depending on the direction of the free-stream. It is easy to notice that the shadow occurswhen α,β ≥ maxδ(x), with δ(x) the local inclination of the parabola.

4.5 More general shapes

Even if mostly of the the hypersonic bodies are built with basic gures, shapes with improvedaerodynamic performance outside of the range of basic shapes may be required to improvefuture missions. Therefore, it is desirable to be able to obtain the analytic aerodynamics ofmore general shapes. One possible solution of this problem is the use of low-order Béziercurves of revolution. The study of this kind of curves has been developed for total unshadowedcondition, in which the full body is exposed to the flow [1].

4.5.1 Bézier curves of revolution

The development of analytic relations for Bézier curves of revolution allows for rapid analysisof general bodies of revolution. The geometry of a Bézier curve is parametrized by a non-dimensional arc-length, sb, for 0 ≤ sb ≤ 1. Given n + 1 points, P0, P1, P2, . . . , Pn in space,called control points, the Bézier curve dened by these control points is [1]:

P (sb) =n∑i=0

BiJn,i(sb)

Jn,i(sb) =

(ni

)sib (1− sb)n−i(

ni

)= n!

i!(n−i)!

(4.21)

in which the location of the ith control node is specied in the vector Bi and the order of theBézier curve is given by n. Jn,i(sb) is the basis function.

The control nodes specify a control polygon P (sb), inside which the Bézier curve mustreside. As expected, each Bézier curve resides inside the control polygon and is connected tothe initial and nal control nodes [1].

Sometimes the domain of a Bézier curve is [a, b] rather than [0,1]. Thus, a change ofvariable is required. It's simply necessary to convert a sb in [a, b] to a new sb in [0, 1] andusing this sb in the basis functions. Converting sb to [0,1] can be done as sb = sb−a

b−a , soobtaining a new basis function

Jn,i(sb) =

(ni

)(sb − ab− a

)i(1− sb − a

b− a

)n−i(4.22)


Figure 4.5: Side and front view of Bézier curve of II degree parametrization

In this work a quadratic Bézier curve has been analyzed. Let's dene the control pointslike [B0 B1 B2] = [(x0, y0) (x1, y1) (x2, y2)] as shown in Figure 4.5, so the curve can be denedas

P (sb) = (1− sb)2B0 + 2sb(1− sb)B1 + s2bB2 (4.23)

The Bézier surface is parametrized by:u = sb, sb ∈ [0, 1]

v = ω, ω ∈ [0, 2π](4.24)

In order to reduce the number of unknowns, some geometrical constrains have been imposed:x0 = 0, y0 = Rbi

x1 = ax2, with 0 < a < 1

x2 = Lb, y2 = Rbf

obtaining the position vector:

r =

2a(1− t)tx2 + t2x2

cos(v) (t2y2 + (1− t)2y0 + 2t(1− t)y1)− sin(v) (t2y2 + (1− t)2y0 + 2t(1− t)y1)

(4.25)

The aerodynamic and angular coecients are obtained in the same way as the previouscases. The analytic expressions obtained by the integrations made with Mathematica resultvery long, like the case of the parabolic nose. For space reasons the formulas are presentedin Appendix A.

4.6 Superposition of the shapes

As demonstrated in this chapter, one fundamental result of the Newton ow theory is thatevery aerodynamic coecient is derived from the surface integral of the pressure coecient.So for more complex noses, the global coecients can be calculated superpositioning theeects of each basic shapes in which is possible to divide the nose.


As already introduced at the beginning of this chapter, common hypersonic vehicles canbe determined through superposition of basic shapes [1]. For example, sphere-cones can beconstructed using a spherical segment and a single conical frustum, and bi-conics can beconstructed using a spherical segment and two conical frustums ans so on.

Each basic shape used will likely have dierent reference areas and lengths. Therefore,the superpositioning of basic shapes cannot be performed by simply adding the aerodynamiccoecients from each shape. Rather, the aerodynamic coecients of each basic shape mustbe scaled to a common reference area and length.

Analyzing the simple sphere-cone nose shown in Figure 4.6, the drag coecient CDSC , forexample, is calculated using the superposition of the eects, where the reference area is thebase area of the sphere-cone and the reference length is the length of the entire nose [1].

Figure 4.6: Side and front view of sphere-cone

CDSC = CDCArefCASC

+ CDSPArefSPASC

(4.26)

The Eq.(4.26) is only valid if the ux is studied with the Newton model. In this caseonly shape that makes the nose add its contribution to the total value of the aerodynamiccoecient. This approach permits to create several hypersonic noses just composing a littlenumber of basic shapes.

In the Chapter 6 will be analyzed the results obtained with this approach, compared witha numerical one. The software will have already some default shapes, leaving to the usersthe possibility to implement the database.

Chapter 5

Shape optimization

Having introduced the problem of the constrained optimization in section 2.5, in this chapterthe constrains of the optimization will be dened. Generally the problem of the optimizationfor a rocket, Eqs.(2.27), can be written as:

find min CD(x), x ∈ X ⊂ Rn

hj(x) = 0, j = 1, ..., l

gi(x) R 0, i = 1, ...,m

(5.1)

or for the re-entry vehicles:find min B∗(x) = m(x)

CD(x)Aref, x ∈ X ⊂ Rn

hj(x) = 0, j = 1, ..., l

gi(x) R 0, i = 1, ...,m

(5.2)

in which the variables x are the free geometrical features of the shapes. As a design choice,three inequality constrains are selected, in order to solve the problem with the internal pointmethod described in section 2.5.2. So for the considered case l = 0 and m = 3.

It is important to notice that in case of minimizing the ballistic coecient B∗, it isnecessary to dene the mass of the entire re-entry vehicle. It can be divided in a constantpart, Mbody, and a part depending on the shape of the nose, m(x) = Slat(x)ρt, where Slat isthe lateral surface of the nose, ρ is a mean density of the nose, t is a mean thickness. Ofdefault, the values of common manned re-entry capsule like the Apollo Command Moduleare considered:

Parameter ValueMbody [kg] 5809 [30]ρ [kg/m3] 1200 [25]t [m] 0.02 [25]

Table 5.1: Default mass of the body, density and thickness of the nose

So the ballistic coecient can be expressed as

B∗ =Mbody +m(x)

CD(x)Aref=Mbody + Slat(x)ρt

CD(x)Aref(5.3)

50

CHAPTER 5. SHAPE OPTIMIZATION 51

Figure 5.1: Constrains on the cargo volume (a), nose radius (b), nose mass (c)

CARGO VOLUME It can be useful to dene a internal volume that has to be main-tained during the optimization process. In fact the noses of the launchers are often usedfor containing the payload, so it is necessary to maintain a certain volume for the cargo. Itis considered a cylindrical volume with height Lvol and radius Rvol, so V ol = πLvolR

2vol, as

shown in Figure 5.1(a)-(b). As design parameter is selected the height of the volume. Thenis possible to evaluate the correspondent radius of the nose, using the equation of the noseshape f(x),

Rvol = f(x = Lvol) (5.4)

where Lvol ≤ lref and with simple geometrical consideration is possible to dene the volumehigh as a function of the longitudinal dimension of the shape as

Lvol = Lvol(Lfig) (5.5)

So xed the cargo volume V ol∗, the optimization process has to change the shape of thenose maintaining or improving this internal volume.

g1 = πLvolR2vol ≥ V ol∗ (5.6)

and substituting in Eq.(5.6) the Eq.(5.4) and Eq.(5.5) it is possible to obtain the dependenceof the inequality constrain g1 from the geometrical variables Lfig,

g1 = V ol∗ − πLvol(Lfig) (f (Lvol(Lfig)))2 ≤ 0 (5.7)

MAXIMUM HEAT In case of having a blunted conguration, it is necessary to x themaximum heat rate qMAX that the material can aord. Recalling Eq.(2.9), it is easy to obtainthe second inequality constraint, that imposes that the heat ux have to be lower than themaximum,


g2 = 1.83 · 10−4V 3∞

√ρ∞rn− qMAX ≤ 0 (5.8)

A value of qMAX = 120 kWm2 is set as default. This is the limit value for a non-ablative

shield of a re-entry vehicle [24], as shown in Figure 5.2.

Figure 5.2: Non-ablative peak heating versus velocity for past and planned planetary entryvehicles [24]

Of course in the unblunted conguration this condition is not taken into account.

NOSE MASS The optimization process has to maintain or even reduce the mass of thenose m∗. As shown in Figure 5.1(c), the mass of the nose can be simply dened as

m∗ = Slatρt (5.9)

The lateral surface is a function of the geometrical features of the nose so it can be expressedas


Slat = Slat(Lfig, Rfig) = 2π

ˆ

Lfig

f(x)

√1 +

(df(x)

dx

)2

dx (5.10)

so the third inequality of the optimization problem can be written as

g3 = Slat(Lfig, Rfig)ρt−m∗ ≤ 0 (5.11)

Having dened the three inequality constraints of the optimization problem, it is impor-tant to notice that all the constraints and the objective function too are non-linear and forsolving the problem the recursive algorithm described in section 2.5.2 will be used.

In the following section the optimization process of each shape is analyzed.

5.1 Cone family optimization

The initial radius of the shape, so the Aref , is taken as constant, so the geometrical parametersfree for the optimization are described in the Table 5.2, for all the possible congurations(for a complete description of the geometrical parameters see Table 6.1).

Conguration Fixed parameters Free parametersConic (unblunted) RcIi ,RcIf = 0 L∗cIConic (blunted) RcIi L∗cI ,R

∗cIf

Bi-conic (unblunted) RcIi ,RcIIf = 0 L∗cI ,L∗cII ,R

∗cIf

= R∗cIIiBi-conic (blunted) RcIi L∗cI ,L

∗cII ,R

∗cIf

= R∗cIIi ,R∗cIIf

Table 5.2: Optimization parameters for the cone family

So it is possible to dene the three constraints. It is described the case of having a bi-conicnose with blunted conguration that is the most complete, in which the drag coecient canbe expressed as

CD(x) = CD

(L∗cI , L

∗cII , R

∗cIf

= R∗cIIi , R∗cIIf

)=

= CDcI

(L∗cI , R

∗cIi, R∗cIf

)+ CDcII

(L∗cII , R

∗cIIi

, R∗cIIf

)+ CDS

(L∗cII , R

∗cIIi

, R∗cIIf

)(5.12)

CARGO VOLUME As reference length is taken the 10% of the length of the secondcone,

Rvol = RcIIi − Lvol tan δcII = RcIIi − 0.1LcII tan δcII (5.13)

where Lvol = 0.1LcII and tan δcII =RcIIi−RcIIf

LcII.


So the cargo volume,

V ol∗ = πR2volLvoltot = π

(RcIIi − 0.1LcII

(RcIIi −RcIIf

LcII

))2

(LcI + 0.1LcII) (5.14)

and substituting in Eq.(5.6) it is possible to dene the rst inequality constraint, consideringthe free parameters dened in the Table 5.2,

g1 = V ol∗ − π(R∗cIIi − 0.1L∗cII

(R∗cIIi −R

∗cIIf

L∗cII

))2

(L∗cI + 0.1L∗cII) ≤ 0 (5.15)

MAXIMUM HEAT With a blunted conguration is valid the second inequality con-straint,

g2 = 1.83 · 10−4V 3∞

√ρ∞rn− qMAX ≤ 0 (5.16)

for the spherical radius rn =RcIIf

cos δcIIin which δcII = arctan

((RcIIi−RcIIf )

LcII

).

NOSE MASS The lateral surface is the sum of the surface of the two cones and thespherical termination,

S∗lat = SlatconeI + SlatconeII + Slatsphere (5.17)

where:

SlatconeI = π(

(RcIi +RcIf )√L2cI − (RcIi −RcIf )

2)

(5.18)

SlatconeII = π(

(RcIIi +RcIIf )√L2cII − (RcIIi −RcIIf )

2)

(5.19)

Slatsphere = 2πrnLs = 2πR2cIIf

(1 +

(RcIIf −RcIIi

)2

L2cII

)(5.20)

for the spherical length Ls = rn − rn sin δcII . Substituting the free parameters is possible towrite the third inequality constraint as

g3 = Slat(L∗cI , L

∗cII , R

∗cIf

= R∗cIIi , R∗cIIf

)− S∗lat ≤ 0 (5.21)

For geometrical reason it is necessary to add a fourth and a fth inequality constraintin order to maintain the shape to be only convex, so valid for the Newton theory for thehypersonic ux,

g4 = R∗cIf −RcIi ≤ 0 (5.22)

g5 = R∗cIIf −RcIIi ≤ 0 (5.23)


5.2 Ogive family optimization


Conguration Fixed parameters Free parametersOgive (unblunted) Roi ,Rof = 0 L∗oOgive (blunted) Roi L∗o,R

∗of

Table 5.3: Optimization parameters for the ogive family

So it is possible to dene the three constraints. It is described the case of having an ogivenose with blunted conguration that is the most complete, in which the drag coecient canbe expressed as

CD(x) = CD

(L∗o, R

∗of

)= CDo

(L∗o, R

∗of

)+ CDS

(L∗o, R

∗of

)(5.24)

CARGO VOLUME As reference length is taken the 10% of the total length of the ogive,

Rvol =

√ρ2 − (0.1Lo)

2 +Roi − ρ (5.25)

where Lvol = 0.1Lo.So the cargo volume,

V ol∗ = πR2volLvol = π

(√ρ2 − (0.1Lo)

2 +Roi − ρ)2

(0.1Lo) (5.26)


g1 = V ol∗ − π(√

ρ2 − (0.1L∗o)2 +Roi − ρ

)2

(0.1L∗o) ≤ 0 (5.27)

where by denition ρ =R2oi

+(L∗o)2

2Roi.


g2 = 1.83 · 10−4V 3∞

√ρ∞rn− qMAX ≤ 0 (5.28)

for the spherical radius rn =Rof

cos δtanin which δtan = arctan

(f ′ogive(x = LoF )

)is the angle at

the nal length of the ogive LoF = Lo(z = Rof ) =√ρ2 − (ρ+Rof −Roi)

2, that takes into

account the blunted termination.


NOSE MASS The lateral surface is the sum of the surface of the ogive and the sphericaltermination,

S∗lat = Slatogive + Slatsphere (5.29)

where:

Slatogive =π(L2

o+R2oi)

2R2oi

(R2oi− L2

o

)tan−1

2Roi

√(Roi−Rof )(L2

o+RofRoi)

Roi

L2o+2RofRoi−R2

oi

+

+2Roi

√(Roi−Rof )(L2

o+RofRoi)Roi

) (5.30)

Slatsphere = 2πrnLs = 2πRofA

RofA +2RofRoi

√(Roi−Rof )(L2

o+RofRoi)Roi(

L2o −R2

oi

)2

(5.31)

for the spherical length Ls = rn−rn sin δtan and the parameter A =

√1 +

4Roi (Roi−Rof )(L2o+RofRoi)

(L2o−R2

oi)2 .

Substituting the free parameters is possible to write the third inequality constraint as

g3 = Slat

(L∗o, R

∗of

)− S∗lat ≤ 0 (5.32)

For geometrical reason it is necessary to add a fourth inequality constraint in order tomaintain the shape to be only convex, so valid for the Newton theory for the hypersonic ux,

g4 = R∗of −Roi ≤ 0 (5.33)

5.3 Parabolic family optimization


Conguration Fixed parameters Free parametersParabolic (unblunted) Rpi ,Rpf = 0 L∗p,K

∗

Parabolic (blunted) Rpi L∗p,K∗,R∗pf

Table 5.4: Optimization parameters for the parabolic family


So it is possible to dene the three constraints. It is described the case of having aparabolic nose with blunted conguration that is the most complete, in which the dragcoecient can be expressed as

CD(x) = CD

(L∗p, K

∗, R∗pf

)= CDp

(L∗p, K

∗, R∗pf

)+ CDS

(L∗p, K

∗, R∗pf

)(5.34)

CARGO VOLUME As reference length is taken the 10% of the total length of theparabola,

Rvol =Rpi

K

√1− Lvol

Lp=Rpi

K

√1− 0.1LpCOMP

LpCOMP

(5.35)

where Lvol = 0.1LpCOMP.

So the cargo volume,

V ol∗ = πR2volLvoltot = π

(Rpi

K

√1− 0.1LpCOMP

LpCOMP

)2

(0.1LpCOMP) (5.36)

and substituting in Eq.(5.6) it is possible to dene the rst inequality constraint, consideringthe free parameters dened in the Table 5.4:

g1 = V ol∗ − π(Rpi

K∗√

1− 0.1

)2 (0.1L∗p

)≤ 0 (5.37)


g2 = 1.83 · 10−4V 3∞

√ρ∞rn− qMAX ≤ 0 (5.38)

for the spherical radius rn =Rpf

cos δtanin which δtan = arctan

(f ′parabola(x = Lp)

)is the angle at

the nal length of the parabola Lp = Lp(z = Rpf ) = LpCOMP

(1−

(KRpfRpi

)2), that takes into

account the blunted termination.

NOSE MASS The lateral surface is the sum of the surface of the parabola and the spher-ical termination,

S∗lat = Slatparabola + Slatsphere (5.39)

where:

Slatparabola =πRpiLpCOMP

6K

(√1− Lp

LpCOMP

(Lp

LpCOMP− 1)·

·(

R2pi

K2LpCOMP(LpCOMP

−Lp)+ 4)3/2

+(

R2pi

K2L2pCOMP

+ 4)3/2

) (5.40)


Slatsphere == 2πrnLs =

π

(4K4L2

pCOMPR2pf−K2LpCOMP

RpfR2pi

√R4pi

K4L2pCOMP

R2pf

+ 4 +R4pi

)2K4L2

pCOMP

(5.41)for the spherical length Ls = rn − rn sin δtan. Substituting the free parameters is possible towrite the third inequality constraint as

g3 = Slat(L∗p, K

∗, R∗pf )− S∗lat ≤ 0 (5.42)

For geometrical reason it is necessary to add a fourth inequality constraint in order tomaintain the shape to be only convex, so valid for the Newton theory for the hypersonic ux,

g4 = R∗pf −Rpi ≤ 0 (5.43)

5.4 Bézier family optimization


Conguration Fixed parameters Free parameters

Bézier (unblunted)x0 = 0 y0 = Rbi

y2 = 0x1 = a∗x∗2 = a∗L∗b y∗1

x2 = L∗b

Parabolic (blunted) x0 = 0 y0 = Rbi

x1 = ax2 = aL∗b y∗1x∗2 = L∗b y∗2 = R∗bf

Table 5.5: Optimization parameters for the Bézier curve family

In order to obtain close solution, the middle point (x1, y1) has to have one xed positionin the case of having the blunted conguration. In this work the parameter a is a designpoint, so only the parameter y∗1 is a free optimal parameter.

It is possible to dene the three constraints. It is described the case of having a Béziernose with blunted conguration that is the most complete, in which the drag coecient canbe expressed as

CD(x) = CD (y∗1, x∗2, y∗2) = CDb (y∗1, x

∗2, y∗2) + CDS (y∗1, x

∗2, y∗2) (5.44)


CARGO VOLUME As reference length is taken the 10% of the arc-length sb,

Rvol = (1− 0.1sb)2y0 + 2(0.1sb)(1− 0.1sb)y1 + (0.1sb)

2y2 (5.45)

where, if normalized, 0.1sb = 0.1 and Lvol = (1 − 0.1)2x0 + 2(0.1)(1 − 0.1)x1 + (0.1)2x2 inwhich x1 = ax2 and x0 = 0.

So the cargo volume can be simplied as

V ol∗ = πR2volLvol = π

((0.9)2y0 + 2(0.1)(0.9)y1 + (0.1)2y2

)2 (2(0.1)(0.9)ax2 + (0.1)2x2

)(5.46)


g1 = V ol∗ − π((0.9)2y0 + 2(0.1)(0.9)y∗1 + (0.1)2y∗2

)2 (2(0.1)(0.9)ax∗2 + (0.1)2x∗2

)≤ 0 (5.47)


g2 = 1.83 · 10−4V 3∞

√ρ∞rn− qMAX ≤ 0 (5.48)

for the spherical radius rn =Rpf

cos δtanin which δtan = arctan (y1/ (x2(1− a))) is the angle of

tangency of the Bézier curve in the third point.

NOSE MASS The lateral surface is the sum of the surface of the parabola and the spher-ical termination,

S∗lat = SlatBezier + Slatsphere (5.49)

where SlatBezier is dened in Appendix A, and for the sphere

Slatsphere = 2πrnLs = 2πy22

√y2

1

(a− 1)2x22

+ 1

(√y2

1

(a− 1)2x22

+ 1 +y1

(a− 1)x2

)(5.50)

for the spherical length Ls = rn − rn sin δtan. So substituting the free parameters is possibleto write the third inequality constraint as

g3 = Slat(y∗1, x

∗2, y∗2)− S∗lat ≤ 0 (5.51)

For geometrical reason it is necessary to add a fourth and a fth inequality constraintsin order to maintain the shape to be only convex, so valid for the Newton theory for thehypersonic ux,

g4 = y∗1 −Rbi ≤ 0 (5.52)

g5 = Rbf − y∗1 ≤ 0 (5.53)

that simply means y∗1 ∈ [Rbi , Rbf ].

Chapter 6

Software description and results

In this chapter the structure of the software is described. The instructions for the utilizationand the interpretation of the results are shown in a detailed way in order to allow the usersof the code to understand how it works.

The software needs FreeFem++ [31] to work, an open source program. The code wasdeveloped in FreeFem++3.26-3, to avoid possible problems of compatibility, please use thesame version.

6.1 Software structure

This section is dedicated to describe the structure of the code. Basically it is divided in 4sections:

1. SHAPE INITIALIZATION. The rst part of the code is dedicated to the denition ofthe geometry of the problem. The rst step to use this program is to select the noseshape of the vehicle that is going to be analyzed. The default shapes are an ensembleof the basic shapes described in the Chapter 4, connected following the superpositioncriteria. Generally are presented noses created with the connection of 2 basic shapes, inunblunted or in a blunted conguration. The blunted conguration is always obtainedadding a spherical segment at the end of the nose. Opening the program, the userhave to dene the geometrical parameters of the nose. A detailed description of theparameters is presented in section 6.2. Figure 6.1 shows the program for the bi-conicshape. The parameters that the user has to modify, are identied by the tag #USER.They are generally the geometrical features of the nose and the direction of the airspeed.

60

CHAPTER 6. SOFTWARE DESCRIPTION AND RESULTS 61

Figure 6.1: Geometrical input for the bi-conic nose

2. AIR SPEED DEFINITION. After the section dedicated to the geometric parameters,the characteristics of the air stream have to be dened. The user can set the direction ofthe velocity vector and the physical parameters of the air, according to the atmospheremodel described in section 2.4. Some air parameters are evaluated, like the Reynoldsnumber Re, the Knudsen number Kn and the cinematic viscosity of the air ν, in orderto evaluate the type of free-ow. As default is set a Mach=6 ight condition in theStratosphere.

Figure 6.2: Free stream denition


3. AERODYNAMIC COEFFICIENT CALCULATION. After the setting of the inputs,the program calculates the aerodynamic coecients for the selected shape, automat-ically evaluating the shadow condition, as described in the following section 6.2.1. Ifthe numerically evaluation of the Navier-Stokes equation is activated, setting the pa-rameter runNUM=true, the code try to evaluate the pressure coecient solving theEuler's equations, Eqs.(3.16-17). If it is impossible to achieve a numerical results, thenumber of the time steps simulation ∆t should be reduced, or should be changed thenumber of the points of the mesh for the boundary Nbnd and for the volume Nvol or thedimension of the control volume itself, Lx and Ly, as shown in Figure 6.3.

Figure 6.3: Mesh dimension denition

4. SHAPE OPTIMIZATION. The shape optimization is made in Mathematica. It isnecessary to select the shape by a list of the ones presented in Chapter 4. The codeevaluates the constraints and minimizes the drag or the ballistic coecient followingthe equations dened in Chapter 5. The results are presented in term of percentagegain respect to the value obtained before the optimization.

5. RESULTS. In the same folder in which the software is stored, a RESULTS.txt le issaved, containing the geometrical features of the shape, the airstream parameters andthe aerodynamic coecients.

Figure 6.4 summarizes the algorithm.


Figure 6.4: Summarize of the algorithm structure

6.2 Shape initialization

All the default shapes and their geometrical parameters are described in the Figures 6.5-6.8.

Bi-conic nose


Figure 6.5: Bi-conic nose in blunted and unblunted conguration

Table 6.1: Design parameters for bi-conic conguration

It is easy to notice that if the user puts LcI = 0, the program analyzes a conic congurationand if RcIIf = 0 is the case of an unblunted conguration.

Ogive nose


Figure 6.6: Ogive nose in blunted and unblunted conguration

Table 6.2: Design parameters for tangent ogive conguration

It is easy to notice that if the user puts Rof = 0, the program analyzes a simple ogiveconguration.

Parabolic nose


Figure 6.7: Parabolic nose

Table 6.3: Design parameters for parabolic conguration

It is easy to notice that if the user puts Rpf = 0, the program analyzes a simple parabolicconguration.

Bézier nose


Figure 6.8: Bézier nose in blunted and unblunted conguration

Table 6.4: Design parameters for II degree Bézier conguration

For the nature of the Bézier curves, the maximum and minimum angle are at the rootand at the tip of the gure. These angles are evaluated knowing the tangency of the segmentthat links the in one case the rst, in the other the last point of the curve.

It is easy to notice that if the user puts Rbf = 0, the program analyzes a simple parabolicconguration.

6.2.1 Shadowed area denition

An important issue of the Newton theory for the hypersonic ight is the denition of theshadowed area. In that area the ux doesn't interact with the shape, so consequently thepressure coecient is equal to zero, Cp = 0.

In order to evaluate analytically the shadowed areas of each surfaces, it's important tounderline again that a hypersonic ux interact with the body only with its normal direction.


So when the ux passes tangent to the body, there is no interaction that means no loss ofmomentum. Those are the areas called shadowed, because not eected by the hypersonic ux.Of course these areas present turbulence and eventually ux separation, but kind of eectsare not analyzed in the Newton theory. Later, section 6.3 will analyze these aerodynamiceects and how much they change the solution obtained with the Newton theory, comparingthe results of the Navier-Stokes equations.

So by denition, it's necessary to dene the tangency condition of the ux, for the givenV∞. First it is dened the slightest inclination of the shape, by the angle δmin. For example,in a conic nose, δmin = δc, being the inclination angle constant. Then there is a portion ofthe shape shadowed, only if it is satised this condition:

maxα, β > δmin (6.1)

Due to the double symmetry of the shapes, the angle of attack and the angle of side-slipact on the shape with the same eect, as shown in the Figure 6.9.

Figure 6.9: Shadowed condition for a conic nose in x− zand x− y planes

The Figure 6.9 shows that if the condition of shadowed area, Eq.(6.1), is satised, theux interacts only with an half of the shape. It is now important to dene which half of theshape is wetted by the hypersonic ux.

All the analyzed shapes are built by rotating a curve, or an ensemble of curves, arounda symmetry axis. Cutting the shape with a plane y − z, it's obtained a circle dened by theangle ω. This angle has been parametrized by the parameter v = ω, as described in Chapter4. Projecting the components of the air speed, Vy = − sin β and Vz = − cos(β) sin(α) on they − z plane, it is possible to identify the direction of the velocity in this plane. The velocityvector in the y − z plane can be dened with as:

ωi = arctan

(cos(β) sin(α)

sin(β)

)(6.2)

So nally the half of the shape wetted by the ux, so unshadowed for an arc of π, is givenby v ∈ [ωi − π

2; ωi + π

2], as shown in Figure 6.10.


Figure 6.10: Unshadowed portion in y − z plane

In case of having a blunted nose, so if the nose has a spherical segment at the end, an otherassumption is necessary. Due to the parametrization chosen for this surface, the parameteru = δ denes the half-angle of the spherical segment, as shown in Figure 4.2. For this segmentthe δmin is given by the tangency of the spherical segment with the backward part of the nose.So in case of that the shadow condition is satised, Eq.(6.1), the unshadowed surface of thespherical segment on the x− y or x− z plane, is given by u ∈ [0 ; max α, β].

6.2.2 Partially shadowed condition

In case of having a back part of the body with a dierent inclination, can happen that evenif one part of the nose has shadowed areas, not all the nose has, as shown in Figure 6.11.Figure 6.11(a) shows the possibility to have the second part of the nose unshadowed, dueto its higher inclination; instead Figure 6.11(b) shows the shadow area for gures with aprogressive lower inclination like parabolas or ogives.

Figure 6.11: Partially shadowed condition for a bi-conic and ogive nose


Bi-conic

In case of having a bi-conic nose, the unshadowed condition can occur if the second cone hasa higher inclination than the rst one, as shown in Figure 6.11(a). The unshadowed area canbe dened geometrically by the interaction of the velocity vector and the back part of thenose as:

xnoS =lref tan(maxα,β)−RcIi(tan(maxα,β)−tan δcI)

znoS = RcIi − xnoS tan δcI(6.3)

Of course the condition to have an unshadowed area is still valid, so the partially shadowedcondition only exists where maxα, β < δcI and if znoS < RcIi . It means that the ux hasto hit the back part of the nose.

Ogive

In this work a tangent ogive is considered. This kind of ogive is tangent with relative incli-nation of 0 with the back part of the hypersonic vehicle. This kind of condition means that,apart of having a ux with zero inclination, a partially shadowed condition always occursdue to the higher inclination that the shape has until the ending part.

Considering the tangent to the ogive tan δ(x) = x√ρ2−x2

, where x is the coordinate in

which the inclination is calculated, it is possible to dene the partially shadowed conditionas the point in which Eq.(6.1) is not more valid:

xnoS =√

tan(maxα,β)2+ρ2

1+tan(maxα,β)2

znoS = Roi − ρ+√ρ2 − x2

noS

(6.4)

Parabolic

There is partially shadow condition for a parabola if the ending inclination of the nose islower than the inclination of the velocity eld. Considering the tangent to the parabola astan δ(x) =

Rpi

2KLp√

1−x/Lp, it is easy to obtain the lower inclination of the parabola substituting

to the x = 0, the initial point of the nose. So if tan δ(x = 0) < tan(maxα, β), it possibleto dene the partially shadowed condition as:

xnoS = Lp

(1−

(Rpi

2KLp tan(maxα,β)

)2)

znoS =RpiK

√1− xnoS

Lp

(6.5)

It is nally important to underline that in case of blunted nose, the total segment lengthof the parabola Lp results shorter than a complete parabolic nose. In this case it is necessaryto use in the calculation the length of the entire parabola, LpCOMP

.

Bézier


Due to the real high complexity of the integration of Bézier curves in shadow condition,only the unshadowed condition in considered. For this reason it's important to dene thebeginning of the shadowed area, in order to obtain results only if the free-stream wets all thenose.

For denition the Bézier curve is tangent to the segment that connects the rst and thelast point, as shown in Figure 6.12.

Figure 6.12: II degree Bézier curves

So having dened the geometry in Table 6.4, it is necessary that the airstream has a lowerinclination than the minimum angle in order to maintain a condition of totally unshadowednose, so:

maxα, β ≤ δmin (6.6)

6.3 Aerodynamic coecients calculation

In this section are presented the results coming from the dierent methods presented inChapter 3. The goal of this work is to demonstrate the validity of the analytic approach ina preliminary phase of the design, in which a order of approximation can be accepted if theresults are gain in a faster way.

Some cases are analyzed in order to test the validity of the approach.

6.3.1 Simple conic nose

The rst shape analyzed is a simple cone. It is studied only in an unshadowed condition,varying the attack angle α, keeping constant and equal to zero the side-slip angle, due to the2D solution of the Navier-Stokes.

In the Table 6.5 the geometrical features of the cone are presented.


Lc [m] Rci [m] δc [°] Aref [m2]

4.5 2.5 29.05 19.63

Table 6.5: Geometrical features of the conic nose

In order to analyze the results from the dierent approaches, the drag coecients atseveral attack angle have been calculated. The results are obtained analytically as presentedin section 3.1, and numerically as presented in sections 3.2.1 and 3.2.2.

For the Navier-Stokes problem, an hypersonic ux at Mach=6 have been analyzed, in aStratosphere conditions of pressure, density and temperature, according to the ISA standardsas presented in Table 2.2. As shown in Figure 2.6, this is a typical ight condition for thehypersonic vehicles.

Figure 6.13 shows the velocity and pressure elds obtained by the implementation of theproblem in FreeFem++.

Figure 6.13: Velocity and pressure elds for cone nose with α = 0°, 10°,20°

Then the drag coecient is obtained with Eq.(3.18).


Figure 6.14: Diagram of CD ∝ α for the three dierent methods, conic nose

The standard deviation of each coecient has been calculated for testing the validity ofthe analytical approach. As it is shown in Table 6.6, the results obtained with the analyticalapproach are perfectly conrmed by the panel method, with a dierence less than 1.15·10−3%.This is because both are derived from the Newton theory for the hypersonic ux. A largerlocal deviation of the results has been registered comparing the analytical approach with thesolution given by the Navier-Stokes method. As aspected, with all the approximations donefor computing the Navier-Stokes equations, the results aren't extremely precises, so even thetting with the analytic results can't be perfect. Evaluating the drag coecient for a quitewide range of angle of attack, the standard deviation between the two approaches resultsglobally less than 26%.

This result can be due to the fact that the conical nose can be described in a 2D geometryas two plates. For this geometrical reason the Navier-Stokes equations give a similar result tothe panel method because in both of the approaches the shape analyzed can be reduced to asum of plates. However, since that the Newton method doesn't take into account turbulenceor eventual presence of ux separation, surely present with higher angle of attack, improvingthe free-stream inclination, the two methods give more dierent results. It can be even provedwith more accurate simulations that the eects of the turbulence is dominant for this ightregime.

Compared Methods Analytical/Panels Analytical/Navier-Stokes% 1.15·10−3 25.5

Table 6.6: Standard deviation of the analytic dates, compared with the numerical results forthe conic nose

Due to the geometric nature of the cone, that has the same behavior as a plate in theNavier-Stokes and in the analytical method, it is possible to analyze the with more detailsthe accuracy of the results and computational time of the approaches, for a zero angle of


attack conguration. In fact for its simple geometry, the conic conguration let the code runfor dierent situation.

Accuracy and computational time comparison

Considering the computational time, the panels methods is comparable with the analytic,but there are other advantages on using the latter. In fact the pannelization process could benot so easy with other type of less regular shapes. Moreover, having the equation of the aero-dynamic coecients permits to have a more precise calculation without the computationalerrors, surely present in a numerical calculation.

Instead, the dierence in computational time between the Navier-Stokes solution and theanalytic method it is really relevant. One takes some fractions of a second, the other needsa quantity of time one order of magnitude bigger, seconds or tens of seconds. A qualitativestudy has been done for understanding the numerical process in term of Cpu (computational)time and accuracy in the results. After some tests, the computational time and the accuracychanging have been evaluated, keeping constant once the number of steps ∆t of the timesimulation, once the dimension of the triangular mesh.


Figure 6.15: Accuracy on the solution and Cpu time for increasing mesh or simulation steps

Figure 6.15 shows the interpolation of the dates registered in some simulations. The rstdiagram represents the Cpu time and the accuracy of the solution depending on the numberof triangles of the mesh. The accuracy on the solution for this case, has been evaluated as thedierence between two consecutive results given by improving the number of the mesh points.Accuracy and Cpu time have been interpolated with a exponential function. As aspected,after a big improvement of the quality of the solution, the results tend to the certain value,so the dierence between the values tend to zero. On the opposite, having a bigger meshjust increases signicantly the computational time, without a signicant improving of theaccuracy.

The second diagram shows the accuracy and Cpu time depending on the number of steps∆t of the numerical simulation. Due to the semi-steady condition of the hypersonic ux, abig increase of the accuracy of the results has been registered for the rst steps ∆t. For thatreason, even in this case the accuracy has been interpolated with an exponential function. It


is important to notice that the solutions tends to have good accuracy, better than 1%, after6/7 ∆t. The computational time on the other hand registers a linear increment, due to thefact that every single steps of the simulation calculates the pressure and velocity eld in thesame way, maintaining the same mesh.

After this study it's possible to admit that it would not be useful to have more iterationsor a very high number of mesh triangles because the solutions obtained have very similarvalues, registering on the other hand a signicant increment on the computational time.

6.3.2 Simple parabolic nose

The second shape analyzed is a simple parabolic nose. It is studied only in an unshadowedcondition, varying the attack angle α, keeping constant and equal to zero the side-slip angle,due to the 2D solution of the Navier-Stokes.

In the Table 6.7 are presented the geometrical features of the parabola.

Lp [m] Rpi [m] δmin [°] Aref [m2]

3 2.5 22.62 19.63

Table 6.7: Geometrical features of the parabolic nose

In order to analyze the results from the dierent approaches, the drag coecients atseveral attack angle have been calculated. The results are obtained analytically as presentedin section 3.1, and numerically as presented in sections 3.2.1 and 3.2.2.




Figure 6.16: Velocity and pressure elds for parabolic nose with α = 0°, 10°,20°


The standard deviation of each coecient has been calculated for testing the validity ofthe analytical approach. As it is shown in Table 6.8, the analytical approach presents valuesperfectly coherent with the panel method. This is because both are derived from the Newtontheory for the hypersonic ux.

A bigger local deviation of the dates has been registered comparing the analytical ap-proach with the solution given by the Navier-Stokes method. As aspected, with all the ap-proximations done for computing the Navier-Stokes equations, the results aren't extremelyprecises, so even the tting with the analytic dates can't be perfect. Evaluating the dragcoecient for a quite wide range of angle, the standard deviation between the two approachesresults globally around 18%.

This result can be due to the high level of approximation done on the Navier-Stokesequation. Moreover, for simulating the hypersonic ux, the velocity eld often reaches a


Figure 6.17: Diagram of CD ∝ α for the three dierent methods, parabolic nose

critical value after few steps of simulation, ∆t. This kind of event blocks the simulationafter a number of steps not sucient to reach a high accuracy on the value of the pressurecoecient. The error can be qualitative estimated from Figure 6.15 as around 10%, for anumber of steps ∆t = 1−2. So the simulation can be inappropriate for this kind of situation.

As even shown in Figure 6.16, the presence of turbulence can badly inuence the resultsof the Navier-Stokes equations. In fact in the model of the hypersonic ow the presence ofthe turbulence is neglected.

Moreover, due to the more complexity of the shape, the ux has a more dierent evolutionduring the motion, instead of simply colliding with the nose as supposed in the Newton theory.On the contrary to the simple conic nose, in this case the two approaches dier even in thedenition of the geometry.

Compared Methods Analytical/Panels Analytical/Navier-Stokes% 5·10−5 17.9

Table 6.8: Standard deviation of the analytic dates, compared with the numerical results forthe parabolic nose

As already said in the section 6.3.1, the panels method has a computational time com-parable with the analytic one. There is a relevant dierence between Navier-Stokes solutionand the analytic method in terms of computational time.

6.3.3 Simple ogive nose

The third shape analyzed is a simple ogive nose. Due to the geometrical features of the nose,varying the attack angle α and keeping constant and equal to zero the side-slip angle, due tothe 2D solution of the Navier-Stokes, a partially shadowed condition is considered. In fact,considering a tangent ogive nose, even a small angle of attack let the back part of the nosebe shadowed.


In the Table 6.9 are presented the geometrical features of the ogive.

Lo [m] Roi [m] δtan [°]5.5 2.5 48.89

Table 6.9: Geometrical features of the ogive nose

In order to analyze the results from the dierent approaches, drag coecients at severattack angle have been calculated. The results are obtained analytically as presented insection 3.1, and numerically as presented in sections 3.2.1 and 3.2.2.




Figure 6.18: Velocity and pressure elds for ogive nose with α = 0°, 10°,20°


The standard deviation of each coecient has been calculated for testing the validityof the analytical approach. Even for the ogive, as it is shown in Table 6.10, the analyticalapproach presents values perfectly coherent with the panel method, with a dierence around6.5 10−5%. This is because both are derived from the Newton theory for the hypersonic ux.

Taking into account all the considerations made for the parabolic nose, a bigger dierencehas been registered between the analytic solutions and the results from the Navier-Stokesequations. Evaluating the drag coecient for a quite wide range of angle, the standarddeviation between the two approaches results globally in this case around 30%.


Figure 6.19: Diagram of CD ∝ α for the three dierent methods, ogive nose

Inappropriate approximation on the Navier-Stokes equations for the hypersonic ux, crit-ical value of the velocity eld after a few steps of simulation ∆t and a more complex geometrycan let the solutions be so dierent.

Moreover, due to the tangent ogive peculiarity, a simple unshadowed condition analysiscan't be done. In fact for each angle of attack α 6= 0, the nose has a shadowed part. Usingthe denition of partially shadowed condition presented in section 6.2.2 for the analyticalmethod, the part of the nose not wetted by the hypersonic ux increases, increasing the

angle of attack α, as xnoS =√

tan(α)2+ρ2

1+tan(α)2 . So progressively a big part of the nose present a

shadowed condition. On the other hand, the Navier-Stokes equations study the evolution ofthe ux around all the nose, independently on the possible unwetted part. This considerationlet the results be even more dierent. The nal gap of 30% can be explained because thetwo method analyzes the ux starting from very dierent assumptions.

Compared Methods Analytical/Panels Analytical/Navier-Stokes% 6.64·10−5 30

Table 6.10: Standard deviation of the analytic dates, compared with the numerical resultsfor the ogive nose


6.3.4 Simple Bézier nose

The last shape analyzed is a simple Bézier nose. Due to the impossibility of having shadowedareas, that give a complex solution of the integration, the results are obtained only for anunshadowed condition.

In the Table 6.11 are presented the geometrical features of the Bézier curve.


(x0, y0) [m] (ax2, y1) [m] (x2, y2) [m] δmin [°] δMAX [°] Aref [m2]

(0,2.5) (0.35x2,2.2) (4,0) 12.09 40.23 19.63

Table 6.11: Geometrical features of the ogive nose

In order to analyze the results from the dierent approaches, coecients at sever attackangle have been calculated, until reached the condition of partially shadowed. The resultsare obtained analytically as presented in section 3.1, and numerically as presented in sections3.2.1 and 3.2.2.


Figure 6.20 shows the velocity and pressure elds obtained by the implementation of theproblem in FreeFem++. A lower range of angle of attack α has been considered in order tomaintain a condition of total unshadowed nose. In fact only for this condition the analyticexpression of the aerodynamic coecients has been calculated.

Figure 6.20: Velocity and pressure elds for Bézier nose with α = 0°, 10°



Figure 6.21: Diagram of CD ∝ α for the three dierent methods, ogive nose

The standard deviation of each coecient has been calculated for testing the validity ofthe analytical approach. Even for the Bézier curve, as it is shown in Table 6.12, the analyticalapproach presents values perfectly coherent with the panel method, with a dierence less than2.5·10−5%. This is because both are derived from the Newton theory for the hypersonic ux.

Taking into account all the considerations made for the parabolic nose, a bigger dier-ence has been registered between the analytic solutions and the results from the Navier-Stokes equations. Evaluating the drag coecient, the standard deviation between the twoapproaches results globally around 40%.

Inappropriate approximation on the Navier-Stokes equations for the hypersonic ux, crit-ical value of the velocity eld after a few steps of simulation ∆t and a more complex geometrycan let the solutions be so dierent.

Compared Methods Analytical/Panels Analytical/Navier-Stokes% 2.5·10−5 38.9

Table 6.12: Standard deviation of the analytic dates, compared with the numerical resultsfor Bézier nose


6.3.5 Study cases analysis

The results obtained permit to make some considerations. In each case the validity of theanalytic formulation has been conrmed by the panel method. Both of the methods arederived from the Newton theory so they give high delity results for the high hypersonicmotion. In fact only with number of Mach larger than 10, the shock layer lies on the shape,as shown in Figure 2.2, and the assumption of the Newton theory are conrmed.

For lower Mach number it is necessary to compare the results from the numerical solutionof the Euler's equations in order to understand the order of magnitude of the approximationsmade. More the shape is complicated, less the results obtained form the two method are


similar. The presence of turbulence, possible ux separation, shadow areas are all possibleeects that let the results be so dierent, up to 40%. Moreover it is necessary to underlinethat the model of the Euler's equation doesn't t with the assumptions made for the Newtontheory. Figure 6.22 shows that more the Mach number is high, more the results have asmaller gap. The study has been done for a simple conic nose at zero angle of attack andside-slip.

Figure 6.22: Diagram of CD ∝M for the two dierent methods, simple conic nose

For this case a 35% mean quadratic error of the result respect to the Newton theory iscalculated. This is a demonstration that the model of Euler's equations don't t with theassumptions made for the hypersonic ux. For a high Mach, the results tend to be similarwith an expected minimum error of 6%, for high hypersonic motion. A qualitative evaluationof the numerical results can be done, taking into account all the possible causes of error.

δεtot = δε± δεacc (6.7)

Eq.(6.7) denes the global expected dierence between the Newton method and the nu-merical simulation of the Euler's equations. It can be considered as a sum of two terms. δεaccthat is the error given by the lack of accuracy on the results given by stopping the simulationafter a few steps ∆t or using an inappropriate mesh, and δε that is the relative error due tothe number of Mach and the angle of attack or side-slip.

A qualitative result can be obtained for the dierent shapes, in a ight condition ofMach=6 at zero angle of attack and side-slip. Due to the high velocity of the ux, thesimulation is stopped after ∆t = 2 steps, it is considered a mesh that gives a perfect accuracy.


Conguration ∆t = 2Mach = 6α, β = 0

δεtot

Cone 10% 18.5% 18.5±1.8%Parabola 10% 20.7% 20.7±2%Ogive 10% 45% 45±4.5%Bézier 10% 44% 44±4.4%

Table 6.13: Expect accuracy on the Euler's results, respect to Newton theory

Table 6.13 summarizes the qualitative evaluation of the accuracy of the results given bythe numerical simulation of the hypersonic ux governed by the Euler's equations, respectedto the values of the Newton theory. Even considering a simple case as the conic conguration,the gap registered is around 20%. As already said, more complex is the shape more eects liketurbulence, ux separation, etc. inuence the numerical results while they are not consideredin the Newton theory. This conrm that the approximations made for the Euler's equationsare inappropriate for describe the high hypersonic ight, which is well approximated by theNewton model for high number of Mach.

6.4 Shape optimization

Obtained an analytical equation for the aerodynamic coecients and tested its validity insection 6.3, it is possible to optimize the shape of the noses in order to minimize the drag incase of designing a rocket or reach a dened value of L/D in case of re-entry vehicles.

The optimization process follows the steps introduced in section 2.5. In fact, due to havingan analytical expression of the coecients, the solution of the Navier-Stokes equations in aproblem of optimal control becomes redundant. For this reason the optimal condition of theshape can be reached with the problem of the constrained optimization described in section2.5.

In the next sections are presented the results obtained implementing the optimizationproblem for each shapes, as dened in Chapter 5. The results of both the optimizations arepresented, reducing the drag or the ballistic coecient.

6.4.1 Conic nose optimization results

Table 6.14 denes the geometrical features and the values of the drag coecients after andbefore the optimization.


Conguration Fixed parameters [m] Free parameters [m] CD → CDopt % reduction

Conic (unblunted)RcIi = 2.5RcIf = 0

L∗cI = 5→ 5. 0.1431→ 0.1431 5.15·10−6

Conic (blunted) RcIi = 2.5L∗cI = 5→ 5.

R∗cIf

= 2→ 2.0.4269→ 0.4269 0

Bi-conic (unblunted)RcIi = 2.5RcIIf = 0

L∗cI = 2→ 2.070

R∗cIf

= R∗cIIi

= 2→ 1.965

L∗cII = 3→ 2.930

0.2230→ 0.2228 0.1

Bi-conic (blunted) RcIi = 2.5

L∗cI = 2→ 1.009

R∗cIf

= R∗cIIi

= 2→ 2.340

L∗cII = 3→ 2

R∗cIIf

= 1→ 1.866

0.4498→ 0.4304 4.3

Table 6.14: Results of optimized drag coecient for conic nose family

From the results it is possible to notice that higher is the number of the free parameters,more the optimization is eective. In fact imposing three or more constraints, the optimiza-tion of the problem can be blocked. That means that the constraints don't allow the shape tochange in order to minimize the drag coecient because all the limitations on the parametersare already reached.

In the case of having a bi-conic nose the optimization results more eective, obtaining areduction of the drag coecient up to around the 4.5%. It is obtained reducing the length ofthe nose in order to obtain a more cylindrical shape, respecting the limitation on the massand the cargo volume. Reducing the inclination of the cone, even the surface exposed to thenormal part of the velocity eld is reduced, so globally the drag can be lower. Noticing thatthe unblunted congurations have the lowest values of CD, the spherical termination is theone that gives the bigger contribution to the drag, but for supporting the heat ux, it cannot be reduced to zero.

Figure 6.23: Blunted bi-conic nose with drag coecient CD optimized, in design (a) andoptimized conguration (b)

Table 6.15 denes the geometrical features and the values of the ballistic coecients afterand before the optimization. For heat dissipation reasons, only the blunted congurationsare taken into account.


Conguration Fixed parameters [m] Free parameters [m] B∗ → B∗opt % reduction

Conic (blunted) RcIi = 5L∗cI = 3→ 3.157

R∗cIf

= 2→ 0.814446.36→ 250.8 43.8

Bi-conic (blunted) RcIi = 5

L∗cI = 1→ 0.394

R∗cIf

= R∗cIIi

= 4→ 5

L∗cII = 3→ 2.4

R∗cIIf

= 2.5→ 0.653

667.3→ 250.1 70.8

Table 6.15: Results of ballistic coecient optimization for conic nose family

It is important to underline that the optimization process does not always converge. Infact it is necessary to introduce geometrical parameters that are already close to the optimalshape in order to have a starting point that allows the convergence. Generally it is enoughto check that the radius of the blunted part is satisfying the constraint on the heat ux, soq(rn) < qMAX . Moreover a limitation on the reduction of not more than 80% on the lengthof the nose has been imposed.

The results conrm that the spherical termination gives the bigger contribute to the drag,but the conical part is the one that inuences more the result. With a study on the sensitivityof the geometrical parameters, using Eq.(2.63), it can be easily demonstrated that the conesparameters inuence more the nal solution. In order to minimize the ballistic coecient ormaximizing the drag, the spherical part is reduced until the heat ux constraint is valid, dueto maximizing the conical part.

A physical interpretation of this behavior is given by the Newton model itself, from whichthe coecients are calculated. In fact for improving the drag it is necessary a higher pressurecoecient. It is calculated from the loss of normal momentum of the free-steam particlesthat hit the shape. It is easy to notice that having a bigger conic portion of the nose, with anhigher angle δc, the normal collision of the particles along the nose results more eective. Infact for the relative angle θ that tends to π

2, the pressure coecient increases, Cp = 2 sin2 θ.

Figure 6.24: Blunted bi-conic nose with ballistic coecient B∗ optimized, in design (a) andoptimized conguration (b)


6.4.2 Parabolic nose optimization results



Parabolic (unblunted)Rpi = 2.5Rpf

= 0L∗p = 5→ 5

1/K∗ = 1→ 10.3880→ 0.3880 0.001

Parabolic (blunted) Rpi= 2.5

L∗p = 5→ 2.295

1/K∗ = 1→ 0.677R∗

pf= 1→ 2.499

0.4869→ 0.3554 27

Table 6.16: Results of drag coecient optimization for parabolic nose family

In this case it is impossible to optimize the unblunted conguration. For the bluntedconguration it is reached a 27% reduction in the drag coecient with a semi-cylindricalconguration. In fact for Rpf → Rpi , only the spherical termination gives a contribution tothe drag. Even in this case for heat reasons, this part can't be eliminated.

Figure 6.25: Blunted parabolic nose with drag coecient CD optimized, in design (a) andoptimized conguration (b)

Table 6.17 denes the geometrical features and the values of the ballistic coecients afterand before the optimization. For heat dissipation reasons, only the blunted conguration istaken into account.


Parabolic (blunted) Rpi = 2.5

L∗p = 5→ 2.9

1/K∗ = 1→ 0.761R∗

pf= 1.5→ 1.366

750→ 424.1 43.4

Table 6.17: Results of ballistic coecient optimization for parabolic nose family

As aspected the optimization process tends to reduce the total length of the nose. In this


way more surface is directly wetted by the free-ow, increasing the drag. So even in this casethe considerations made for the bi-conic nose are valid.

Figure 6.26: Blunted parabolic nose with ballistic coecient B∗ optimized, in design (a) andoptimized conguration (b)

6.4.3 Ogive nose optimization results



Ogive (unblunted)Roi = 2.5Rof = 0

L∗o = 5→ 5. 0.4096→ 0.4095 2.6·10−7

Ogive (blunted) Roi = 2.5L∗o = 5→ 4.398

R∗of

= 1→ 2.1110.5459→ 0.4205 23

Table 6.18: Results of drag coecient optimization for ogive nose family

For the ogive nose, in the unblunted conguration there is a very little reduction of thedrag, due to a slightly increment in the length. In general it can be notice that the reductionof the drag is made by having a bigger spherical termination, but it's important to underlinethat a simple ogive conguration presents a lower drag of an optimized blunted conguration.The spherical termination permits to dissipate more heat, but increments the drag.

Even in this case the optimization shape tend to be cylinder-like, and all the previousconsiderations are valid.


Figure 6.27: Blunted ogive nose with drag coecient CD optimized, in design (a) and opti-mized conguration (b)



Ogive (blunted) Roi = 2.5L∗o = 4→ 3.65

R∗of

= 1.5→ 0.527811.2→ 600.4 26

Table 6.19: Results of drag coecient optimization for ogive nose family

Even in this case the optimization is made reducing the spherical termination. In orderto maintain the internal volume, a limitation on reducing the ogive length has been imposed.The optimization tends to give a shape in which a bigger part of the nose is directly exposedto the hypersonic ux. In this way a bigger value of drag can be reached, being the sensitivityof the ogive parameter higher than the spherical one.


Figure 6.28: Blunted ogive nose with ballistic coecient B∗ optimized, in design (a) andoptimized conguration (b)

6.4.4 Bézier nose optimization results



Bézier (unblunted)x0 = 0

y0 = Rbi = 2.5y2 = Rbf = 0

x1 = a∗L∗b = 0.35L∗

b → 0.363L∗b

y∗1 = 2.2→ 1.944x2 = L∗

b = 4→ 4.0280.0919→ 0.0905 1.5

Bézier (blunted)x0 = 0

y0 = Rbi = 2.5x1 = aL∗

b = 0.35L∗b

y∗1 = 2.2→ 2.5x2 = L∗

b = 4→ 3.822y2 = R∗

bf= 1→ 1.174

0.4208→ 0.4024 4.4

Table 6.20: Results of drag coecient optimization for Bézier nose family

The blunted and the unblunted conguration, having the same number of optimizationparameter, present a similar reduction of the drag coecient. Having II order Bézier curvesthere is more exibility on the geometrical parameters that allows the optimization to bemore eective.

Similar to the ogive nose, the introduction of the spherical termination increases dramat-ically the drag coecient. Even after the optimization the values are one order of magnitudebigger. With these results it is clear that due to their exibility, the Bézier curves presentthe best aerodynamic characteristics, but they are also the most complicated in the buildingof the nose.


Figure 6.29: Blunted Bézier nose with drag coecient CD optimized, in design (a) andoptimized conguration (b)



Bézier (blunted)x0 = 0

y0 = Rbi = 2.5x1 = aL∗

b = 0.45L∗b

y∗1 = 2.2→ 1.847x2 = L∗

b = 5→ 5.299y2 = R∗

bf= 1→ 0.370

882.5→ 763.516 7

Table 6.21: Results of ballistic coecient optimization for Bézier nose family

The optimization is made reducing the spherical termination and increasing the wettedsurface of the nose. The shape tends to assume a conic form, that is the one that gives thebest ballistic coecient.

Figure 6.30: Blunted ogive nose with ballistic coecient B∗ optimized, in design (a) andoptimized conguration (b)

Chapter 7

Conclusions

Since that this work wants to reach various objectives, several subjects were analyzed. Focus-ing the study on the hypersonic ight condition, the theory and the models of the hypersonicow have been studied. A discussion between the model of evolution of the ux, governed bythe Navier-Stokes equations, and the hypothesis of the Newton theory has been done in orderto validate the results and underline the advantages and the disadvantages of each approach.Finally the constraint optimization problem was introduced in order to understand the op-timization procedure which can permit to reach the minimum value of the drag coecient,for rocket, or the minimum value of the ballistic coecient, for re-entry vehicles.

Noticing that generally the noses of the hypersonic vehicles can be often referred tosimple shapes of revolution, the aerodynamic coecients of some basic shapes were calculated,following the Newton theory for the hypersonic regime. Often, the integration of the pressurecoecient along the dierent surfaces is not a trivial problem. The integrations have beendone with the help of the software Mathematica. It is necessary to underline that the selectionof the parametrization of the shapes strongly inuences the possibility of obtaining an analyticsolution. Although Mathematica is a very powerful software, human insight was still criticalto obtain the formulation of the aerodynamic coecients. It was often necessary to addmathematical assumptions to the integrals and model the equations in order to obtain theanalytic expressions of the coecients.

The aerodynamic coecients have been calculated for conic, sphere, ogive and parabolicnose family. The calculation was also extended to more general shapes, dened by the Béziercurves. A model of superposition of the eects [1] was adopted in the case of the Newtontheory to determinate the coecient of composed shapes. With this model each shapegives its contribution to the total value of the aerodynamic coecient so conic, bi-conic,ogive, parabolic and Bézier noses have been analyzed in unblunted or blunted conguration,that is made adding a spherical termination to the nose. Validated the results from theanalytic approach with the panel method for every shape, the formulation of the aerodynamiccoecients was also implemented in FreeFem++.

The FreeFem++ environment allows to have a completely open-source code for the cal-culation of the aerodynamic coecients for hypersonic vehicles and allows to simulate theux along the shapes, using nite elements to solve the Navier-Stokes equations. The resultsgiven by the numerical simulation however present sometimes a sensible gap with the valuesgiven by the analytic calculation. This is due to many factors: rst of all, methods as nite

93

CHAPTER 7. CONCLUSIONS 94

dierences or nite volumes are usually preferred for solving the Navier-Stokes equations.Moreover the simplications made on the equations and on the ux model don't allow theresults to be very precise. In fact the density of the ow is considered constant (or semi-constant) and the eects of the viscosity µ were ignored. Also the eects of the temperatureare not considered in the equations, eects that are even more important when the velocityof the ux tends to hypersonic values.

Having as objective the developing of a procedure for the calculation of the aerodynamiccoecients in a preliminary phase of the design, the approximations given by the Newtonmodel are adopted. The results of the simulation conrm the validity of this model for highhypersonic ight condition. For lower hypersonic Mach number the solution given by the twomethods, analytic and numerical, have a dierence up to 40%, depending on the complexityof the shape, the angles of attack or side-slip, the presence of turbulence or ux separation.

However the study of the evolution of the ux by the numerical simulation of the Navier-Stokes equations proved the sensible advantage of having an analytic model in terms of Cputime, as expected. In fact the results of the analytic calculation are obtained in fractions ofsecond, while the ones given by the numerical simulation can take tens of seconds, with alower precision for the considered model of the ux.

Saving computational time is fundamentally important in a preliminary phase of thedesign of a hypersonic vehicle, in which is usually preferred obtaining faster results and inwhich is important to understand which are the best design choices, taking into account anhigher level of approximation. Moreover, by the implementation of the analytic model in theFreeFem++ environment, the code results to be completely free, instead of the commercialcodes, and there is the open possibility to improve the numerical model in order to obtainan higher level of details in the evolution of the hypersonic ux. In fact, the implementationof the Euler's equations results inappropriate to describe the hypersonic regime and a bettermodel for the evolution of the ux has to be developed to obtain higher delity results inthe numerical simulation. In fact after the results obtained, it is possible to admit that thenumerical model adopted doesn't describe properly the hypersonic ow regime, but it wasperformed in order to prove the advantage in terms of computational time of the analyticapproach.

Finally, the analytic expressions of the aerodynamic coecients allow to build a simple butvery eective optimization procedure. The analytic formulations permit to solve the problemof minimizing the drag or the ballistic coecient, solving a constrained optimization problem.Due to the high non-linearity of the equations, the solution was obtained using the Newton-like internal point method. This algorithm was implemented in Mathematica and can beadded in the FreeFem++ code too. The results of the optimization are in some cases veryeective, reaching generally up to 70% of reduction in the ballistic coecient, up to 25% ofin drag coecient.

In conclusion, this work creates an interesting approach on the hypersonic aerodynamiccoecients, obtained in an analytic way for several shapes that allows their optimization,depending on the goals. The advantage in terms of computational time is shown in compar-ison with the numerical simulation. The biggest advantage of having analytic expressionsconsist in the possibility of implementing directly the coecients in the study of the optimalcondition needed, obtaining good results.

As open future develops, following the analysis described along the chapters, it is possi-


ble to implement the existing database of the studied shapes, adding new or creating newcombinations. Having already a FreeFem++ code, it could be possible to improve the uidmodel governed by the Euler's equations, in order to obtain a numerical model that betterdescribe the evolution of the hypersonic ux. Finally, the Internal Point algorithm can beimplemented in the FreeFem++ environment, obtaining a complete open-source software forthe calculation of the aerodynamic coecients and the shape optimization for a wide rangeof hypersonic vehicles.

Appendix A

Aerodynamic coecients for parabolic nose

Are presented the aerodynamic coecients for a parabolic nose in the conditions presentedin section 4.4, obtained with Mathematica.

96


And the angular coecients:

Aerodynamic coecients for Bézier

As introduced in section 4.5.1, for a nose made by a Bézier curve, it's analyzed only theunshadowed case. For simplicity is presented the case in which the air stream has zero angleof attack and zero angle of side-slip. In this case only the drag coecient is dierent fromzero. In the implementation of the code in FreeFem++ the possibility to have small angle ofthe air is considered, but only in the case of maintaining a condition of unshadowed nose.


Lateral surface of second degree Bézier curve

The calculation has been made following the calculation algorithm presented in [23].

Appendix B

A sample FreeFem++ code is presented, regarding the bi-conic\conic nose. From the code ispossible to recognize the six dierent part in which is divided: SHAPE INITIALIZATION,AIR SPEED DEFINITION, AERODYNAMIC COEFFICIENT CALCULATION, SHAPEOPTIMIZATION, RESULTS. For reducing the number of pages, it is presented only the caseof having only unshadowed condition.

99

Bibliography

[1] Michael J Grant. Rapid simultaneous hypersonic aerodynamic and trajectory optimiza-tion for conceptual design. Phd thesis, Georgia Institute of Technology. Atlanta, 2012.

[2] Francesco Villa. Algebraic models for aerodynamic coecients calculation during theatmospheric re-entry. MSc thesis, Politecnico di Milano. Milano, 2011.

[3] John David Anderson. Hypersonic and high-temperature gas dynamics (Second Edition).AIAA series. 2006.

[4] Gianluca Iaccarino. Solution methods for the incompressible Navier-Stokes equations.Lecture notes, Standford University. Standford, 2008.

[5] Graham Getty. The Standard Atmosphere: a mathematical model of the 1976 U.S.Standard Atmosphere. NASA. 2011.

[6] Luca Dedè. Adaptative and reduced basis methods for optimal control problems in en-vironmental applications. Phd thesis, Politecnico di Milano. Milano, 2008.

[7] Bijan Mohammadi, Olivier Pironneau. Applied shape optimization for uids (2ed., OUP,2010). Oxford science publication. Oxford, 2010.

[8] Dennis J McNabb. Investigation of atmospheric reentry for the space maneuver vehicle.MSc thesis, Air Force Institute Of Technology. Wright-Patterson Air Force Base, Ohio,2004.

[9] Lester L. Cronvich, Ione D. V. Faro. Handbook of supersonic aerodynamics (section 8)- Bodies of revolution. NAVWEPS report 1488, Volume 4, Bareau of Navy Weapons.Maryland, 1961.

[10] M. G. Busato. Rappresentazione analitica dei puntali ogivali per proiettili. Lev-rotto&Bella. Torino, 2010.

[11] Jack Moran. An Introduction to Theoretical and computational aerodynamics. Doverpublication, New York, 1984.

[12] Dominic Dirkx, Erwin Mooij. Continuous aerodynamic modeling of entry shapes. AIAAAtmospheric Flight Mechanics Conference, 8-11 August. Portland, Oregon, 2011.

[13] Florian De Vuyst. Numerical modeling of transport problems using FreeFem++ software.Phd thesis, Department of Mathematics-ENS CACHAN. Paris, 2013.

102

BIBLIOGRAPHY 103

[14] Arvel E. Gentry et al. The mark IV: supersonic-hypersonic arbitrary body program -volume II. NTIS, U.S. department of commerce, Douglas Aircraft Company. Springeld,Virginia, 1973.

[15] Antonio Vitale, Federico Corraro, Guido De Matteis and Nicola de Divitiis. Identicationfrom ight data of the aerodynamics of an experimental re-entry vehicle. Dr. ChaoqunLiu (ed.). Arlington, Texas, 2012.

[16] Gianluigi Rozza. Controllo ottimale e ottimizzazione di forma in uidodinamica com-putazionale. MSc thesis, Politecnico di Milano. Milano, 2002.

[17] OpenFOAM overview, http://www.openfoam.org/dev.php. Silicon Graphics Interna-tional. San Jose, California, 2014.

[18] Patrick Gallais. Atmospheric re-entry vehicle mechanics. Springer. Berlin, 2007.

[19] Frank J. Regan, Satya M. Anandakrishnan. Dynamics of atmospheric re-entry. AIAAeducation series. Washington DC, 1984.

[20] Alessandro Agnetis. Introduzione all'ottimizzazione vincolata. Lecture notes, Universitàdi Siena. Siena, 2013.

[21] José Herskovits. A view on non-linear optimization. Advances in Structural Optimiza-tion, Volume 25, Pages 71-116, Ed. KLUWER ACADEMIC PUBLISHERS. Holland,1995.

[22] Andreas Wachter, Lorentz T. Biegler. On the implementation of a primal-dual interiorpoint lter line search algorithm for large-scale nonlinear programming. MathematicalProgramming Ser. A. Yorktown Heights, New York, 2005.

[23] Marcelo Walter, Alain Fournier. Approximate arc-length parametrization. Dept. of Com-puter Science - The University of British Columbia. Vancouver, British Columbia, 1996.

[24] Peter A. Gnoo. Planetary-entry gas dynamics. Annu. Rev. Fluid Mech, Volume 31,Pages 459-494. Hampton, Virginia, 1999.

[25] F. S. Milos, Y. K. Chen, T. H. Squire. Analysis of Galileo Probe heatshield ablationand temperature data. JOURNAL OF SPACECRAFT AND ROCKETS, Volume 36,Number 3. Moett Field, California, 1999.

[26] Neda Mansouri. Compressible uid ow. Lecture notes, Queen's University. Kingston,Ontario, 2013.

[27] D. Bauso, R. Pesenti. Programmazione non lineare, ottimizzazione vincolata. Lecturenotes, DINFO, Università di Palermo. Palermo, 2011.

[28] Leonardo Rubino. EQUAZIONI DI NAVIER-STOKES: la regina della uidodinamica.Scribd Inc. Torino, 2010.

[29] Various. NOSE CONE DESIGN. Nevada Aerospace Science Associates. Nevada, 2014.

BIBLIOGRAPHY 104

[30] Apollo 11 Command and Service Module (CSM).http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1969-059A/.

[31] F. Hecht. New development in freefem++. Journal of Numerical Mathematics, Volume20, Issue 3-4, Pages 251266. Paris, 2013.

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