Aerodynamics Modeling of

Sounding RocketsA Computational Fluid Dynamics Study

Kristoffer Hammargren

Engineering Physics and Electrical Engineering, master's level

2018

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Abstract

Any full scale rocket consists of four main components: the structural system(or frame), the payload system, the guidance system and the propulsionsystem. The guidance system of a rocket includes sophisticated sensors,on-board computers and communication equipment. The guidance systemprovides stability for the rocket and controls the rocket during maneuvers.

When developing guidance systems, reliable aerodynamic models of therockets are required. This report conducts a comparative study betweenCFD-simulations and the current aerodynamic models of two sounding rocketconfigurations used at RUAG Space in Linkoping; the Maxus configurationand the two-stage Terrier-Black Brant configuration.

Three parameters related to rocket stability are evaluated: the zero-liftdrag coefficient CD0 , the stability derivative CNα , and the center of pressureXcp. The parameters are evaluated in terms of behavior and parametric val-ues, ultimately determining how accurate the current aerodynamic modelsare compared to the CFD-simulations. Previous studies are used to deter-mine which data is most plausible in case of any deviations. All parametersare evaluated as a function of Mach number for complete configurations.

The CFD-simulations are performed using Ansys CFX 18.2. Ansys is anAmerican company that develops and markets engineering simulation soft-ware. Ansys is one of the leading commercial CFD-softwares.

The results of this report show that the current aerodynamic model forthe Maxus configuration corresponds well with the CFD-simulations, bothin terms of behavior and parametric values. Data obtained from the currentaerodynamic model are physically plausible and also corresponds well withearlier studies.

Data provided from the current aerodynamic model for the first stageof the Terrier-Black Brant configuration show a mainly identical behaviorto the CFD-simulations, but with varying accuracy in terms of parametric

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values. Although the behavior of the current model is physically plausible,the provided data predict the position of the center of pressure to be too farfrom the nose.

Data provided from the current aerodynamic model for the second stageof the Terrier-Black Brant configuration show an, at times, inconsistent andphysically implausible behavior. These deviations are, however, limited tocertain flight sequences, and not to the data as a whole. Apart from theinconsistencies shown at times, the provided data correspond well with theCFD-simulations in terms of parametric values.

The aerodynamic model of the Maxus configuration is concluded to beaccurate and sufficient for continued analysis. There are no indicators thecurrent aerodynamic model could be faulty in any way.

The aerodynamic model for the first stage of the Terrier-Black Brantconfiguration accurately represents the physical behavior of the rocket, butlacks accurate parameter value predictions in some cases. The current aero-dynamic model is concluded to be sufficiently accurate for continued analysis,provided the limitations of the model is taken into consideration.

The aerodynamic model for the second stage of the Terrier-Black Brantconfiguration corresponds well compared to the CFD-simulations in terms ofparametric values, but fails to predict the stabilization of CD0 at high Machnumbers and the rearward movement of the position of the center of pressureduring transonic flight. The current aerodynamic model is concluded to besufficiently accurate for continued analysis, provided the limitations of themodel is taken into consideration.

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Sammanfattning

Varje storskalig raket bestar av fyra huvudkomponenter: raketkroppen, nyt-tolasten, styrsystemet och framdrivningssystemet. Styrsystemet bestar avsofistikerade sensorer, datorer och kommunikationsutrustning. Styrsystemetavser halla raketen stabil och manovrera raketen under flygningen.

Nar man utvecklar och tillverkar styrsystem kravs palitliga modeller overraketernas aerodynamik. Den har rapporten utfor en komparativ studiemellan CFD-simuleringar och dagens modeller for aerodynamikmodelleringfor tva sondraketskonfigurationer vid RUAG Space i Linkoping. Konfigura-tionerna som undersokts ar Maxus och en tvastegs Terrier-Black Brant.

Tre parametrar relaterade till raketernas stabilitet har undersokts: mot-standskoefficienten da den effektiva anblasningsvinkeln ar noll CD0 , stabilitets-derivatan CNα , och tryckcentrum Xcp. Parametrarna ar utvarderade i formav beteende och parametervarden for att avgora hur noggranna dagens mod-eller ar jamfort med CFD-simuleringarna. Vid avvikelser jamfors data medtidigare studier for att avgora vad som ar mest fysikaliskt troligt. Allaparametrar ar utvarderade som funktion av Machtal for fullstandiga kon-figurationer.

CFD-simuleringarna ar utforda i Ansys CFX 18.2. Ansys ar ett amerikan-skt foretag som utvecklar och saljer programvara inom flera ingenjorsomraden.Ansys ar en av de storsta kommersiella programvarorna for CFD-simuleringar.

Resultaten i denna rapport visar att dagens metod for aerodynamikmod-ellering av Maxus overensstammer val med CFD-simuleringarna, bade vadgaller beteende och parametervarde. Data fran den nuvarande modellen arfysikaliskt rimliga och stammer aven val overens med tidigare studier.

Resultaten for forsta steget av Terrier-Black Brant uppvisar, for detmesta, ett identiskt beteende med CFD-simuleringarna, men med varierandenoggrannhet vad galler parametervarden. Aven om beteendet for den nu-varande modellen ar fysikaliskt rimligt, forutspar den en position for tryck-

4

centrum under den transsoniska fasen som ar orimligt langt bak.Resultaten for andra steget av Terrier-Black Brant uppvisar ett, i vissa

fall, inkonsekvent och fysikaliskt orimligt beteende. Avvikelserna ar emeller-tid begransade till vissa flygsekvenser, och inte till modellen som helhet.Bortsett fran de fa inkonsistenser den nuvarande modellen uppvisar stammerden nuvarande modellen val overens med CFD-simuleringarna vad galler pa-rametervarden.

Den nuvarande modellen for Maxus bedoms vara noggrann och kan fortsa-tta anvandas oforandrat inom analyser. Det finns inga indikatorer pa att dennuvarande modellen pa nagot vis skulle vara bristfallig.

Den nuvarande modellen for forsta steget av Terrier-Black Brant forutsparett korrekt fysikaliskt beteende, men med varierande precision vad gallerfaktiska parametervarden. Den nuvarande modellen bedoms kunna fortsattaanvandas inom analyser, forutsatt att detta gors med bristerna med modelleni atanke.

Den nuvarande modellen for andra steget av Terrier-Black Brant overens-stammer val med CFD-simuleringarna vad galler parametervarden, men mis-sar att forutspa stabiliseringen av CD0 for hoga Machtal samt den bakatforfly-ttning av tryckcentrum som sker under den transsoniska fasen. Den nu-varande modellen bedoms vara tillrackligt noggrann for fortsatta analyser,forutsatt att detta gors med bristerna med modellen i atanke.

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Acknowledgements

Many people have helped me during the course of this thesis work. Theseare my acknowledgements to these people, whom I am greatly thankful for.

First, I would like to thank Albert Thuswaldner at RUAG Space inLinkoping who was my initial contact at RUAG Space and who helped toactualize this thesis work. Thank you for initiating the process of actualizingthis work and for always being a familiar and kind face whenever I wouldrun into you.

Second, I would like to thank Anders Helmersson, my supervisor at RUAGSpace. Thank you for always taking time to answer my questions and foraiding me throughout this work.

Third, I would like to thank Gunnar Hellstrom, my supervisor at LuleaUniversity of Technology. Thank you for all the support, especially in theearly phase of this work when I had seemingly endless technical problems.

Fourth, I would like to thank my desk neighbours; Malin Thuswaldner,Johan Hedblom, Natasa Jankovic and Erik Sundberg. Thank your for makingevery day at RUAG Space a pleasant and fun day.

Finally, I would like to thank the rest of the personnel at RUAG Space inLinkoping for giving me such a warm welcome and for being overwhelminglykind to me since day one. You have given me a fantastic impression of RUAGSpace. I will always cherish my time here.

”Make it a habit to tell people thank you. To express your appreciation,sincerely and without the expectation of anything in return. Truly appreci-ate those around you, and you’ll soon find many others around you. Trulyappreciate life, and you’ll find that you have more of it.”

-Ralph Marston

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Contents

1 Introduction 101.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Disposition of the thesis . . . . . . . . . . . . . . . . . . . . . 11

2 Sounding rockets 122.1 Maxus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Terrier-Black Brant . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Fundamental concepts of fluid dynamics 163.1 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Laminar vs turbulent flow . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Relationship with Reynold’s number . . . . . . . . . . 173.3 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Compressible vs incompressible flow . . . . . . . . . . . . . . . 193.5 The no-slip condition . . . . . . . . . . . . . . . . . . . . . . . 203.6 Newtonian vs non-Newtonian fluids . . . . . . . . . . . . . . . 20

4 Aerodynamic forces 214.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Molecular definition of pressure . . . . . . . . . . . . . 224.2 Viscous shearing stresses . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Couette flow . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Parameters to be evaluated 315.1 Aerodynamic coefficients . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Drag coefficient . . . . . . . . . . . . . . . . . . . . . . 325.1.2 Lift coefficient . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Center of pressure . . . . . . . . . . . . . . . . . . . . . . . . . 37

7

6 Computational fluid dynamics 386.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 38

6.1.1 Conservation of mass . . . . . . . . . . . . . . . . . . . 396.1.2 Conservation of momentum . . . . . . . . . . . . . . . 396.1.3 Conservation of energy . . . . . . . . . . . . . . . . . . 41

6.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4 Fields of application . . . . . . . . . . . . . . . . . . . . . . . 43

7 Model implementation 447.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.1.1 Maxus . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.1.2 Terrier-Black Brant . . . . . . . . . . . . . . . . . . . . 45

7.2 Fluid domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.3.1 Yplus . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.3.2 Orthogonal quality . . . . . . . . . . . . . . . . . . . . 497.3.3 Aspect ratio . . . . . . . . . . . . . . . . . . . . . . . . 497.3.4 Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.4 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.4.1 Flow analysis . . . . . . . . . . . . . . . . . . . . . . . 507.4.2 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 507.4.3 Boundary conditions . . . . . . . . . . . . . . . . . . . 507.4.4 Solver control . . . . . . . . . . . . . . . . . . . . . . . 527.4.5 Output control . . . . . . . . . . . . . . . . . . . . . . 53

8 Results 548.1 Maxus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.1.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 558.1.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 578.1.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 59

8.2 Terrier-Black Brant (first stage) . . . . . . . . . . . . . . . . . 608.2.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 608.2.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 618.2.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 62

8.3 Terrier-Black Brant (second stage) . . . . . . . . . . . . . . . 638.3.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 638.3.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 64

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8.3.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 65

9 Discussion 669.1 Reliability of CFD-simulations and verifying the results from

such simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 669.2 Maxus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9.2.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 679.2.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 679.2.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 68

9.3 Terrier-Black Brant (first stage) . . . . . . . . . . . . . . . . . 699.3.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 699.3.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 699.3.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 70

9.4 Terrier-Black Brant (second stage) . . . . . . . . . . . . . . . 709.4.1 Zero-lift drag coefficient . . . . . . . . . . . . . . . . . 709.4.2 Stability derivative . . . . . . . . . . . . . . . . . . . . 719.4.3 Center of pressure . . . . . . . . . . . . . . . . . . . . . 71

10 Conclusions 7210.1 Maxus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7210.2 Terrier-Black Brant (first stage) . . . . . . . . . . . . . . . . . 7310.3 Terrier-Black Brant (second stage) . . . . . . . . . . . . . . . 74

Bibliography 75

Appendix 77

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Chapter 1

Introduction

The purpose of this master’s thesis work is to use CFD-simulations to eval-uate current methods for aerodynamics modelling of sounding rockets atRUAG Space in Linkoping. The current aerodynamics models are based onanalytic calculations and numerical methods. To determine whether thesemodels are sufficiently accurate, a comparative study between data obtainedfrom the current models and corresponding results from CFD-simulations isperformed.

Three aerodynamic parameters related to rocket stability are studied: thezero-lift drag coefficient CD0 , the derivative of the lift coefficient with respectto the angle of attack CNα , and the center of pressure Xcp. The results arecompared in terms of behavior and parametric values. Any deviations areanalyzed and compared with earlier studies to decide which result is moreplausible and physically accurate.

Two configurations of sounding rockets are evaluated: Maxus and Terrier-Black Brant (first and second stage). All parameters are evaluated as afunction of Mach number for complete configurations.

1.1 Background

RUAG Space is the space division of the Swiss technology group RUAG, andthe leading supplier of products for the space industry in Europe. RUAGSpace operates in Switzerland, Sweden, Finland, Austria, Germany and theUSA, with some 1 350 employees.[1]

RUAG Space in Linkoping is a Product Unit focusing on mechanical

10

systems. Part of their operation includes developing, manufacturing andrefurbishing guidance systems for sounding rockets. In order to achieve de-sired performance, reliable aerodynamic models of the sounding rockets arerequired. The aerodynamic models RUAG Space in Linkoping uses are basedon analytic calculations and numerical methods, and they want to investigatehow accurate these models are compared to CFD-simulations.

1.2 Disposition of the thesis

The following section describes the disposition of the thesis.

Chapter 2, Sounding rockets describes sounding rockets in generaland gives an introduction to the configurations evaluated in this thesis.

Chapter 3, Fundamental concepts of fluid dynamics describesfundamental concepts vital to the understanding of this thesis.

Chapter 4, Aerodynamic forces gives a theoretical background toaerodynamic forces and a mathematical derivation of how they arise.

Chapter 5, Parameters to be evaluated describes the aerodynamicparameters evaluated in this thesis and what role they play in rocketanalysis.

Chapter 6, Computational fluid dynamics introduces the con-cept of CFD-simulations and describes the governing equations CFD isbased on.

Chapter 7, Model implementation gives a step-by-step descriptionof the creation of the CFD-models for each configuration.

Chapter 9, Results presents data from the current models with theresults from the CFD-simulations.

Chapter 9, Discussion summarizes and compares the results.

Chapter 10, Conclusions concludes the results for each configura-tion.

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Chapter 2

Sounding rockets

Sounding rockets take their name from the nautical term ”to sound”, whichmeans to take measurements. They carry scientific instruments into space,typically to an altitude between 200 and 300 kilometers, and in some casesup to 1 000 kilometers. Sounding rockets provide researchers a low-cost andtime efficient method for conducting experiments in areas inaccessible toeither weather balloons or satellites. Common research applications includeresearch in aeronomy, X-ray astronomy and microgravity.

Sounding rockets consist of either a single-stage or multistage solid-fuelrocket motor. As the rocket has consumed all its fuel, it separates fromthe payload and falls back to Earth. The payload continues into space ina parabolic trajectory and begins conducting the experiments, during whichtime data is returned to Earth by telemetry links. The overall time in space istypically 5-20 minutes. As the payload re-enters the atmosphere, a parachuteis deployed to gently bring it back to Earth.

Figure 2.1 shows the different steps of a typical sounding rocket flightsequence.[2]

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Figure 2.1: Flight sequence of a typical sounding rocket.

2.1 Maxus

The Maxus sounding rocket is a single-stage rocket funded by the EuropeanSpace Agency and used in the MAXUS microgravity programme. It is ajoint venture between the Swedish Space Corporation and Airbus Defenceand Space. The first Maxus rocket was launched in May 1991 from EsrangeSpace Center in Kiruna, Sweden. Since then, another nine launches havebeen performed from Kiruna - the latest in April 2017.

The Maxus sounding rocket is Europe’s largest single-stage soundingrocket with an overall length of 15.5 meters and a weight of approximately12.5 tonnes, depending on the payload. It consists of an ogive nose, followedby cylindrical and conical shapes, with four fins attached at the tail.

Figure 2.2 shows Maxus 8 on the launcher at Esrange in Kiruna.[3]

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Figure 2.2: Maxus 8 on the launcher.

2.2 Terrier-Black Brant

The Black Brant is a family of Canadian-designed sounding rockets builtby aerospace firm Magellan Aerospace. The Black Brant originated froma demand by the Canadian government to develop a sounding rocket tocharacterize the ionosphere in order to improve military communications.The first test flight took place from the Churchill Rocket Research Rangein Churchill, Canada, in September 1959. Since then, over 800 launches ofdifferent versions of the Black Brant sounding rocket have been performed.[4]

Originally a single-stage rocket, today’s Black Brant sounding rocketscan utilize a range of multistage boosters. The Terrier-Black Brant soundingrocket is a two-stage rocket with a Terrier booster. Sounding rockets oftenuse excess military assets that have either become obsolete or have exceededproscribed shelf life. This is the case with the Terrier booster, which comesfrom Navy Terrier missiles.[5]

The Terrier-Black Brant consists of an ogive nose, followed by cylindrical

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and conical shapes. Three sets of fins are attached to the Terrier-Black Brantsounding rocket: triangular fins at the midsection of the payload, swept-backfins at the tail section of the Black Brant rocket, and cruciform fins at thetail of the Terrier booster.

Figure 2.3 shows a lunch of a Terrier-Black Brant sounding rocket atWhite Sands, New Mexico, USA, on April 3, 1996.[6]

Figure 2.3: Terrier-Black Brant launch.

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Chapter 3

Fundamental concepts of fluiddynamics

Fluid dynamics is a subdiscipline of fluid mechanics that studies the flowof fluids and how forces affect them. Fluid dynamics has, in turn, severalsubdisciplines, including aerodynamics.

This chapter describes a few fundamental concepts of fluid dynamics vitalto the understanding of this thesis.

3.1 Viscosity

The viscosity of a fluid is a measure of its resistance to flow. It can bedescribed as the ”thickness” of the fluid; for example, honey has a higherviscosity than water. Viscosity may be thought of as internal friction betweenthe fluid molecules, opposing the relative motion between two surfaces of thefluid that are moving at different velocities.

The viscosity of a fluid is defined as the ratio of the shearing stress to thevelocity gradient,

µ =F/A

∆vx/∆z. (3.1)

The viscosity defined in equation (3.1) is commonly called the dynamicviscosity. There is, however, another viscosity quantity. The kinematic vis-cosity, represented by ν, is the ratio of the dynamic viscosity of a fluid to itsdensity,

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ν =µ

ρ. (3.2)

Understanding the basic concept of viscosity and being able to distin-guish dynamic viscosity from kinematic viscosity will be of importance whendiscussing further fundamental concepts of fluid dynamics and, especially,aerodynamic forces arising from viscous shearing stresses (Section 4.2).

3.2 Laminar vs turbulent flow

A flow is said to be laminar when a fluid flows in parallel streamlines, i.e.each particle of the fluid follows a smooth path. In laminar flow, the velocity,pressure and other fluid properties remain constant. In contrast to laminarflow, turbulent flow is irregular and undergoes mixing. In turbulent flow, thevelocity of the fluid at a point is continuously undergoing changes in bothmagnitude and direction. A flow that alternates between being laminar andturbulent is called transitional.

Figure 3.1 shows the difference between laminar, transitional and turbu-lent flows.[7]

Figure 3.1: Laminar, transitional and turbulent flow regimes.

3.2.1 Relationship with Reynold’s number

The Reynold’s number is a dimensionless quantity used to predict flow pat-terns in different flow situations. A low Reynold’s number indicates laminar

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flow, whilst a high Reynold’s number indicates turbulent flow.The Reynold’s number expresses the ratio of inertial forces to viscous

forces,

Re =Inertial forces

Viscous forces. (3.3)

Expanding equation (3.3) yields

Re =Inertial forces

Viscous forces=

mass · acceleration

dynamic viscosity · velocitydistance

· area

=ρL3 ·

(ut

)µ(uL

)L2

=ρL3

(1t

)µ(

1L

)L2

=ρL2

(1t

)µ

=ρ(Lt

)L

µ=ρuL

µ=uL

ν,

(3.4)

where ρ is the density of the fluid, u is the velocity of the fluid relative tothe object, L is a characteristic length of the geometry, t denotes the time,µ is the dynamic viscosity, and ν is the kinematic viscosity.

Examining equation (3.4) shows the relation between a fluid’s viscosityand the Reynold’s number, and thus the flow regime. More importantly, itshows that after a certain length L of flow, a laminar boundary layer willbecome unstable and turbulent, thus drastically changing the aerodynamiccharacteristics of the object (see Section 5.1.1).

3.3 Boundary layer

A boundary layer is a thin layer of flow near a solid wall where viscous forcesare significant (see Section 4.2). The velocity of the flow within the boundarylayer varies from zero at the wall to the free stream velocity at the height ofthe boundary layer.

The thickness of the boundary layer is conventionally denoted δ(x) anddescribes the distance away from the wall at which the velocity componentparallel to the wall is 99% of the fluid speed outside the boundary layer.[8]The boundary layer thickness is not constant, but increases as the down-stream distance x along the wall increases.

Figure 3.2 shows the development of the boundary layer along a flat platein a fluid with free stream velocity u∞.[9]

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Figure 3.2: Velocity boundary layer development on a flat plate.

Boundary layers may be either laminar or turbulent, depending on theReynold’s number. As the distance x along the wall increases, the Reynold’snumber increases linearly with x (see Section 3.2.1). At some point, distur-bances in the flow begin to grow, and the boundary layer cannot remain lam-inar. The boundary layer then becomes transitional at a critical Reynold’snumber, and continues until the boundary layer is fully turbulent. This oc-curs at the transition Reynold’s number.

The aerodynamic forces generated within the boundary layer greatly af-fects the skin friction drag component of a body (see Section 5.1.1).

3.4 Compressible vs incompressible flow

The compressibility of a fluid is a measure of the relevant density change inresponse to a pressure or temperature change. All fluids are compressibleto some extent. However, in many situations the changes in pressure andtemperature are sufficiently small that any changes in density are negligible.The fluid is then said to be incompressible.

When changes in pressure and temperature are not sufficiently small thatchanges in density can be negligible, the fluid is said to be compressible.When analyzing rockets, spacecraft, and other systems that involve highspeed gas flows, the speed is often expressed in terms of the dimensionlessMach number,

M =V

c=

Speed of flow

Speed of sound. (3.5)

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A flow is called sonic when M=1, subsonic when M<1, and supersonicwhen M>1. In aerodynamics, flows are usually treated as compressible if thedensity changes are above 5%. This is usually the case when M<0.3.[10]

3.5 The no-slip condition

The no slip-condition states that a fluid will have zero velocity relative to thesurface boundary of an object immersed in the fluid. The physical justifica-tion for this assumption originates from adhesion forces between a fluid anda solid surface exceeding cohesion forces within the fluid, causing the fluidlayer along the solid boundary to attain the velocity of the boundary.[11]Conceptually, one can think of the outermost fluid molecules as stuck to thesurfaces past which it flows. The viscous forces around a body emanate fromthe no-slip condition.

3.6 Newtonian vs non-Newtonian fluids

A Newtonian fluid is a fluid for which the rate of deformation is linearlyproportional to the shear stresses. Most common fluids such as water, air,gasoline and oils are Newtonian fluids.

In contrast, non-Newtonian fluids do not follow Newton’s law of viscosity.Blood, liquid plastics and many salt solutions are examples of non-Newtonianfluids.

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Chapter 4

Aerodynamic forces

Aerodynamic forces affect a body immersed in a fluid medium. These forcesare due to the relative motion between the body and the fluid. When twosolid objects interact in a mechanical process, forces are transmitted at thepoint of interaction. In contrast, aerodynamic forces on a solid body im-mersed in a fluid act on every point on the surface of the body. These forcesare due to pressure distributions over the surface of the body and shearstresses arising from fluid viscosity.

4.1 Pressure

Pressure is defined as a normal force exerted by a fluid per unit area,[10]

P =F

A. (4.1)

We speak of pressure only when we deal with a fluid. The counterpart ofpressure in solids is normal stress. Since pressure is defined as force perunit area, the unit of pressure is newtons per square meter, which is called apascal ;

1 Pa = 1 N/m2. (4.2)

Understanding what pressure is and how it works is fundamental to un-derstanding aerodynamics. In order to understand pressure, one must studythe molecular definition of pressure.

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4.1.1 Molecular definition of pressure

The small scale action of individual air molecules is commonly explained bythe kinetic theory of gases (also known as the kinetic-molecular theory). Itexplains the behavior of a hypothetical ideal gas (i.e. a gas composed of manyrandomly moving point particles whose only interactions are perfectly elasticcollisions) within a container. According to this theory, gases are made up ofmolecules in random, straight line motion. The individual molecules possessthe standard physical properties of mass, momentum and energy.

The pressure of a gas is a measure of the linear momentum of the molecules.As the gas molecules collide with the walls of the container, the moleculesimpart momentum to the walls, producing a force that can be measured.

Mathematically, the expression for pressure is derived using Newton’slaws of motion. Newton’s second law of motion states that the rate of changeof momentum is directly proportional to the force applied,

F =dp

dt= m

d

dt(v). (4.3)

An impulse J occurs when a force F acts over an interval of time ∆t, accordingto

J =

∫∆t

Fdt. (4.4)

Since equation (4.3) states that force is the time derivative of momentum,the impulse can be written as

J = ∆p = m∆v. (4.5)

A gas molecule experiences a change in momentum when it collides with acontainer wall. During the collision, an impulse is imparted by the wall to themolecule that is equal and opposite to the impulse imparted by the moleculeto the wall, according to Newton’s third law of motion. The pressure is thesum of the impulses imparted by all molecules to the wall.

Consider a gas of N molecules, each of mass m, enclosed in a cube ofvolume V = L3. In order to arrive at an expression for the pressure, acalculation will be made of the impulse imparted to the wall perpendicularto the x-axis by a single impact. Although the molecules are moving in alldirections, only those with a velocity component toward the wall can collidewith it. We call this component vx. Not all molecules have the same vx. To

22

find the total pressure, the contributions from molecules with all differentvalues of vx must be summed.

A molecule approaches the wall with an initial momentum mvx. Afterthe impact, the molecule bounces off the wall with an equal momentum inthe opposite direction, −mvx. Thus, the total change in momentum is

∆p = mvx − (−mvx) = 2mvx. (4.6)

Equation (4.6) equals the total impulse imparted to the wall.The number of impacts on a small area A of the wall in time t is equal

to the number of molecules that reach the wall in time t. The moleculesimpacts one specific side wall every

∆t =2L

vx, (4.7)

where L is the distance between opposite walls. The force due to this moleculeis

F =∆p

∆t=

2mvx2L/vx

=mv2

x

L. (4.8)

The total force on the wall is then

F =Nmv2

x

L, (4.9)

where the bar denotes an average over N molecules. Since the moleculesare in random motion and there is no bias applied in any direction, thisresult is independent of the choice of axis. By Pythagorean theorem in threedimensions, the total squared speed v is given by

v2 = v2x + v2

y + v2z . (4.10)

The gas is in equilibrium, meaning the average squared speed in each direc-tion is identical,

v2x = v2

y = v2z . (4.11)

Combining equations (4.10) and (4.11) gives

v2 = 3v2x, (4.12)

23

and thus

v2x =

v2

3. (4.13)

Inserting equation (4.13) into (4.9) yields the force as

F =Nmv2

3L. (4.14)

Combining equations (4.14) and (4.1) gives the expression for the pressureof the gas as

P =F

L2=Nmv2

3V⇔ PV =

1

3Nmv2. (4.15)

This expression can be expanded further in terms of temperature and thekinetic energy of the gas. Combining equation (4.15) with the ideal gas law,

PV = NkBT, (4.16)

gives

kBT =mv2

3, (4.17)

where kB is the Boltzmann constant and T the absolute temperature. Equa-tion (4.17) leads to the simplified expression of the kinetic energy per molecule,

1

2mv2 =

3

2kBT. (4.18)

Solving for the temperature T yields

T =mv2

3kB. (4.19)

The kinetic energy of the gas is N times that of a molecule,

K =1

2Nmv2. (4.20)

Combining equations (4.20) into (4.19) gives

T =2

3

K

NkB. (4.21)

24

Equation (4.21) is an important result from the kinetic theory of gases, asit concludes that the average molecular kinetic energy is proportional to theideal gas law’s absolute temperature. Simply put; the higher the temper-ature, the greater the motion and kinetic energy. Finally, equations (4.15)and (4.21) yield

PV =2

3K ⇔ P =

2

3

K

V. (4.22)

This final expression relates pressure to volume (density) and to kinetic en-ergy (velocity/temperature). The kinetic theory of gases can be expandedfurther to calculate the number of molecular collisions with a wall of thecontainer per unit area per unit time, as well as the speed of the molecules.However, since this section is merely aimed at giving the reader a basic the-oretical background to aerodynamic forces vital to the understanding of thisthesis, these expansions are not explored in detail.

Although the kinetic theory of gases is a simplified model within a con-tained environment, it sufficiently accounts for the properties of gases andhow they affect the impulses due to molecular collisions between bodies. Italso provides a valid physical explanation in terms of the laws of Newtonianmechanics.

The pressure distributions along the surface of a body arise due to themolecular collisions described in this section - whether it be inside an enclosedcontainer, or due to a rocket moving through the atmosphere. In the latercase, the normal direction changes from the nose of the rocket to the rear. Toobtain the net force arising from the pressure distributions, the contributionsfrom all small sections along the body must be summed,

F =∑

P · ndA, (4.23)

where n is the direction of the force normal to the surface, and dA theincremental area. For a fluid in motion, the velocity has different values atdifferent locations around the body. The local pressure is related to the localvelocity, so the pressure also varies around the closed surface. The net forceis calculated by summing the pressure perpendicular to the surface times thearea around the body,

F =

∮P · ndA. (4.24)

25

This concludes the section about pressure; a physical explanation as tohow and why forces due to pressure distributions arise and how these forcesare calculated.

4.2 Viscous shearing stresses

When two solid bodies in contact move relative each other, a friction forcedevelops at the contact surface in the direction opposite to motion. Themagnitude of the friction force depends on the friction constant. The situa-tion is similar when a fluid moves relative to a solid or when two fluids moverelative each other. The magnitude of the force opposite to motion dependson the viscosity of the fluid. The phenomena is most easily described by theidealized situation known as Couette flow, commonly taught to increase theunderstanding of how viscous shearing stresses arise, and the shearing forcesthey consequently induce.

4.2.1 Couette flow

Couette flow is the flow of a viscous fluid in the space between two surfaces,one of which is moving tangentially relative to the other. The simplest formof Couette flow is flow between two parallel plates separated by a distance h,where the lower plate is at rest and the upper plate is moving continuouslywith a velocity U. The fluid in contact with the upper plate sticks to theplate surface and moves with it at the same speed. The shear stress τ actingon this fluid layer is

τ =F

A, (4.25)

where A is the contact area between the plate and the fluid and F is theshear force. Just as for pressure, shear stress is defined as force per unitarea.

The fluid in contact with the lower plate assumes the velocity of thatplate, which is zero (according to the no-slip condition, see Section 3.5).For steady laminar flow, the fluid velocity between the plates varies linearlybetween zero at the lower plate and U and the upper plate. Figure 4.1 showsthe Couette configuration for such a flow.

26

Figure 4.1: Couette flow between two parallel plates.

The governing equations for Couette flow emanate from the Navier-Stokesequations for steady incompressible flow. The continuity equation for such aflow is

∂u

∂x+∂v

∂y+∂w

∂z= 0. (4.26)

Choosing x to be the direction along which all fluid particles travel (i.e.u 6= 0, v = w = 0) yields

∂u

∂x+

���∂v

∂y+

���∂w

∂z= 0, (4.27)

and thus

∂u

∂x= 0. (4.28)

Equation (4.28) means u = u(y, z). Assuming the plates are infinitely largein the z-direction (so the z-dependence can be disregarded) gives u = u(y).

The momentum equation in the x-direction for steady incompressible flowis

ρ

(∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

)= −∂P

∂x+ ρgx + µ

(∂2u

∂x2+∂2u

∂y2+∂2u

∂z2

).

(4.29)Assuming gravitational forces can be neglected yields

27

ρ

(���∂u

∂t+ u

���∂u

∂x+ v

���∂u

∂y+ w

���∂u

∂z

)= −∂P

∂x+��ρgx + µ

(���∂2u

∂x2+∂2u

∂y2+

���∂2u

∂z2

),

(4.30)and thus

∂P

∂x= µ

∂2u

∂y2. (4.31)

Integrating equation (4.31) and solving for u gives

u(y) =1

2µ· ∂P∂x

y2 + C1y + C2. (4.32)

Invoking the boundary condition (at y=0, u=0) yields C2 = 0. Invoking theboundary condition (at y=h, u=U) yields

C1 =U

h− 1

2µ· ∂P∂x

h. (4.33)

Inserting C1 and C2 into equation (4.32) gives

u(y) =y

hU − h2

2µ· ∂P∂x· yh

(1− y

h

). (4.34)

For simple Couette flow, there is no pressure gradient in the direction of theflow, reducing equation (4.34) to

u(y) =y

hU. (4.35)

Equation (4.35) is the velocity profile of the flow. Consequently, thevelocity gradient is

du

dy=U

h. (4.36)

During a differential time interval dt, the sides of fluid particles along avertical line MN rotate through a differential angle dβ while the upper platemoves a differential distance da = Udt, see Figure 4.2.[10]

28

Figure 4.2: The behavior of Couette flow between two parallel plates.

The angular displacement or deformation (or shear strain) can be ex-pressed as

dβ ≈ tan(dβ) =da

h=U dt

h=du

dydt. (4.37)

Rearranging equation (4.37), the rate of deformation becomes

dβ

dt=du

dy. (4.38)

Thus we conclude that the rate of deformation of a fluid element is equiv-alent to the velocity gradient du/dy. Furthermore, it can be verified experi-mentally that for most fluids the rate of deformation (and thus the velocitygradient) is directly proportional to the shear stress τ ,[10]

τ ∝ dβ

dtor τ ∝ du

dy. (4.39)

Including the proportionality constant between shear stress and the velocitygradient yields the final expression for the shear stress,

τ = µdu

dy, (4.40)

where µ is the dynamic viscosity of the fluid. Inserting equation (4.40) into(4.25) and solving for F gives the shear force acting on a Newtonian fluidlayer (or, by Newton’s third law, the force acting on the plate) as

29

F = τA = µAdu

dy. (4.41)

This concludes the section about viscous shearing stresses. Emanatingfrom the Navier-Stokes equations, it has been mathematically shown howthe rate of deformation is equivalent to the velocity gradient for a simpleCouette configuration. Furthermore, the relationship between viscous shear-ing stresses and the rate of deformation has been discussed, finally arrivingat an expression for the shear force.

Although Couette flow between two parallel plates assumes several sim-plifications, the mathematical derivation of the viscous shearing stresses forsuch flows provides a basic understanding of forces arising from shear-drivenfluid motion. Couette flow is frequently used in undergraduate physics andengineering courses, and the explanation provided in this section suffices atproviding the reader basic concepts of viscous shearing forces important forthe understanding of this thesis.

30

Chapter 5

Parameters to be evaluated

This thesis evaluates three aerodynamic parameters related to rocket stabil-ity: the zero-lift drag coefficient, the derivative of the lift coefficient withrespect to angle of attack, and the center of pressure. Data from the Maxusand the Terrier-Black Brant configurations have been compared with resultsfrom CFD-simulations in order to determine if they are sufficiently accurate.The pre-existing data is based on analytic calculations and numerical meth-ods. All parameters are evaluated as a function of Mach number for completeconfigurations.

This chapter describes the aerodynamic parameters evaluated and whatrole they play in rocket analysis.

5.1 Aerodynamic coefficients

Aerodynamic coefficients are dimensionless quantities that are used to deter-mine the aerodynamic characteristics of an object. Aerodynamic coefficientsare determined by a ratio of forces rather than just a simple force, thusmaking them an important engineering tool for comparing the efficiency fordifferent designs in terms of various aspects.

This thesis examines two aerodynamic coefficients: the drag coefficient,CD, and the lift coefficient, CN . More specific, the zero-lift drag coefficientand the derivative of the lift coefficient with respect to angle of attack areevaluated.

31

5.1.1 Drag coefficient

The drag coefficient is used to model the drag (resistance) of a body immersedin a fluid medium. It takes into account the complex dependencies of shape,inclination and flow conditions on a body. A low drag coefficient indicatesthat an object will have less aerodynamic or hydrodynamic drag.

The drag coefficient is defined as

CD =FD

ρAv2/2, (5.1)

where FD is the drag force, ρ the density of the fluid, A a reference area (inrocket analysis commonly the cross-sectional area of the midsection), and vthe flow speed. The drag force acts opposite to the relative motion of anybody moving with respect to the surrounding fluid.

Introducing the term for dynamic pressure,

q =ρv2

2, (5.2)

into equation (5.1) gives the drag coefficient as the ratio of the drag force tothe force produced by the dynamic pressure times the area,

CD =FDqA

. (5.3)

The aerodynamic forces on a body immersed in a fluid come primarilyfrom differences in pressure and viscous shearing stresses (see Chapter 4).The drag force on a body can be divided into two main components; pressuredrag and viscous drag. The net drag force can thus be decomposed as follows:

CD =FDqA

= cp + cf =1

qA

∫S

dA(p− p0)(n · i

)︸ ︷︷ ︸

cp

+1

qA

∫S

dA(t · i)τ︸ ︷︷ ︸

cf

, (5.4)

where cp is the pressure drag coefficient, cf the friction drag coefficient, pthe pressure at the surface dA, p0 the free-stream pressure, n the normaldirection to the surface with area dA, i the unit vector in direction normalto the surface dA, t the tangential direction to the surface with area dA, andτ the shear stress acting on the surface dA.

32

The zero-lift drag coefficient, CD0 , is the drag coefficient when the effectiveangle of attack is zero, i.e. when affects from lift-induced drag are excluded.In rocket analysis, the zero-lift drag coefficient is used to estimate the apogeeof flight. This thesis evaluates the zero-life drag coefficient.

As mentioned above, the main components of the drag force are pressuredrag and viscous drag. Expanding this discussion further, the different typesof drag are generally divided into parasitic drag, wave drag and lift induceddrag. Parasitic drag is, in turn, made up of skin friction drag, form drag andinterference drag. The following sections describe the different types of dragand how they arise.

Skin friction drag

Skin friction drag is drag resulting from viscous shearing stresses acting overthe surface of a body (see Section 4.2). Skin friction drag is caused by frictionbetween the fluid molecules and the surface of a body. As the fluid moleculesflow over the surface and past each other, viscous resistance slow additionalfluid molecules, generating a force which retards forward motion and causesthe boundary layer to grow in thickness. At some point, the boundary layerflow transits from laminar to turbulent.

Turbulent flow creates more friction drag than laminar flow due to itsgreater interaction with the surface of the body. A rough surface acceleratestransition between laminar and turbulent flow, causing the thickness of theboundary layer and the fluid flow disruption within the boundary layer toincrease. Skin friction drag is thus primarily dependent upon the smoothnessof the surface.

Form drag

Form drag (or pressure drag) is the drag resulting from the static pressureacting normal to the surface of a body. It is caused by the separation of theboundary layer from a surface and the wake created by that separation. Asa fluid flows around a body, a stagnation point with a high static pressureis generated in front of the body. Analogously, the wake behind the bodygenerates a low pressure area. The pressure differential causes the body tobe pushed in the direction of the relative fluid flow, resulting in a force whichretards forward motion. For a streamlined body, the streamlines follow theshape of the body. This results in a smaller wake, and consequently a smaller

33

low pressure area, than for a bluff body. Thus, the form drag is primarilydependent upon the shape of the body.

Interference drag

Interference drag is the drag resulting from when fluid flow around one partof a body is forced to occupy the same space as the fluid flow around anotherpart of the body (i.e. between the fins and the body of a sounding rocket).The intersection point between two bodies causes the flow from each body toaccelerate in order to pass through the restricted physical space. This resultsin turbulent mixing of the two flows and thus increased skin friction drag. Theeffects of interference drag become particularly substantial during transonicand supersonic flows, as the increased speeds at the intersection points willproduce local shock waves and create wave drag. To counter the effects ofinterference drag, the pressure distributions on the intersecting bodies shouldideally complement each others pressure distribution. In reality, however, thisis not always possible.

Wave drag

Wave drag arise during transonic and supersonic flight due to the formationof shock waves. As a rocket passes through the air, it creates a series ofpressure waves in front of it and behind it. These pressure waves travel at thespeed of sound. As the speed of the rocket increases, the pressure waves arecompressed. Eventually, the pressure waves merge into a single shock wavewhich travels at the speed of sound. This occurs at Mach 1, equivalent toapproximately 343 m/s, and generates a sonic boom. The sudden increase indrag at Mach 1 is due to the boundary layer being strongly separated, thuscausing a sharp increase in skin friction drag, and also the static pressurepoint in front of the rocket sharply increasing, thus also increasing the formdrag.

5.1.2 Lift coefficient

The lift coefficient is, in conformity with the drag coefficient, defined as

CN =FN

ρAv2/2=FNqA

, (5.5)

34

where FN is the lift force, ρ is the density of the fluid, A is a reference area,v is the flow velocity, and q is the dynamic pressure. The lift coefficientexpresses the ratio of the lift force to the force produced by the dynamicpressure times area.

Lift force is the force perpendicular to the oncoming flow direction. Liftis generated when a moving flow of fluid is turned by a solid object. Thetheory of flight is often explained by Bernoulli’s principle or Newton’s lawsof motion. The following sections describe the theory of flight according toboth Bernoulli and Newton.

Theory of flight according to Bernoulli

Bernoulli’s principle can be derived from the principle of conservation of en-ergy. The law of conservation of energy states that the energy of an isolatedsystem remains constant. In fluid dynamics, Bernoulli’s principle states thatan increase in speed of a fluid occurs simultaneously with a decrease in pres-sure or a decrease in the fluid’s potential energy. The air passing over anairfoil must travel further and hence faster than the air travelling the shorterdistance below the airfoil, but the energy must remain constant at all times.Consequently, the air above the wing will consist of a lower pressure regionthan the air below the wing, thus generating lift.

Figure 5.1 illustrates the theory of flight according to Bernoulli’s princi-ple.[12]

Figure 5.1: Lift of an airfoil according to Bernoulli.

35

Theory of flight according to Newton

The theory of flight according to Newton is based on Newton’s laws of motion.Newton’s second law states that the force on an object is equal to its masstimes its acceleration or, equivalently, to its rate of change of momentum,

F = ma = md

dt(v). (5.6)

In other words, when there is a change in momentum there must be aforce causing it. Furthermore, Newton’s third law states that to every actionthere is an equal and opposite reaction. That means that when the force ofan airfoil pushes the air downwards - creating the downwash - there is anan equal and opposite force from the air pushing the airfoil upwards, thusgenerating lift.

Figure 5.2 illustrates the theory of flight according to Newton’s laws ofmotion.[12]

Figure 5.2: Lift of an airfoil according to Newton.

In contrast to an airplane - which mainly uses lift to overcome its weight- a rocket uses thrust to overcome its weight. Many rockets instead use liftto stabilize and control the direction of flight.

The derivative of the lift coefficient with respect to the angle of attack,CNα , is called a stability derivative. Stability derivatives are measures ofhow particular forces and moments on a body change as other parametersrelated to stability change (such as airspeed, altitude, angle of attack, etc.).In rocket analysis, CNα indicates the magnitude of the lateral acceleration.

36

5.2 Center of pressure

The center of pressure, Xcp is the average location of all the pressure actingupon a body moving through a fluid. As a rocket moves through the atmo-sphere, the velocity of the air varies around the surface of the rocket. Thisvariation of air velocity produces a variation in the local pressure at variousplaces on the rocket. In the same way that the weight of all components of abody acts through the center of gravity, the total aerodynamic force can beconsidered to act through the center of pressure.

In rocket analysis, the center of pressure is important for predicting theflight stability of a rocket. For positive stability in rockets, the center ofpressure must be further away from the nose than the center of gravity, seeFigure 5.3.[13] This ensures that any increased forces resulting from increasedangle of attack results in increased restoring moment to drive the rocket backto the trimmed position. In rocket analysis, a positive static margin impliesthat the complete rocket makes a restoring moment for any angle of attackfrom the trim position.

Figure 5.3: Rocket stability condition.

Mathematically, the center of pressure is determined by characterizingthe pressure variation around the surface of a body as P(x), which indicatesthat the pressure depends on the distance x from a reference line (usuallytaken as the rear or leading edge of the object),

Xcp =

∫x · p(x)dx∫p(x)dx

. (5.7)

37

Chapter 6

Computational fluid dynamics

Computational fluid dynamics (CFD) is a branch of fluid mechanics thatuses numerical analysis and data structures to solve and analyze problemsthat involve fluid flows.

This chapter introduces computational fluid dynamics; mainly governingequations, history, and fields of application.

6.1 Governing equations

Fluid flows are governed by partial differential equations which representconservation laws for the mass, momentum, and energy. These equationsare called the Navier-Stokes equations, named after Claude-Louis Navier andGeorge Gabriel Stokes.

The Navier-Stokes equations describe how the velocity, pressure, temper-ature, viscosity, and density of a moving fluid are related. A mathematicalmodel of the physical case and a numerical method are used in a softwaretool to analyze the fluid flow. The complexity of these mathematical modelsvary depending on what kind of flow is being analyzed (such as compressiblevs incompressible flows or Newtonian flows vs non-Newtonian flows).

Parts of the Navier-Stokes equations in Cartesian coordinates were refer-enced in Section 4.2.1. As mentioned above, the Navier-Stokes equations areexpressed based on the principles of conservation of mass, momentum andenergy. The equations can be expressed in Cartesian coordinates, cylindri-cal coordinates and spherical coordinates and are adjustable regarding whatkind of flow problem is being analyzed. For the fullness of this report, and

38

to give the reader a greater appreciation of the Navier-Stokes equations, thederivation of the three-dimensional Navier-Stokes equations for incompress-ible, irregular viscous flow in Cartesian coordinates is included below.

6.1.1 Conservation of mass

In accordance with physical laws, the mass in a control volume can be neithercreated nor destroyed. The conservation of mass (also called the continuityequation) states that the mass flow difference throughout a systems’ inlet-and outlet-section is zero,

Dρ

Dt

+ ρ(∇ · V

)= 0, (6.1)

where ρ is the fluid density, V is the fluid velocity and ∇ is the gradientoperator;

∇ = i∂

∂x+ j

∂

∂y+ k

∂

∂z. (6.2)

For incompressible flow (see Section 3.4), the continuity is simplified to

Dρ

Dt

= 0 → ∇ · V =∂u

∂x+∂v

∂y+∂w

∂z= 0. (6.3)

6.1.2 Conservation of momentum

The momentum in a control volume is constant. The description is set up inaccordance with Newton’s second law of motion,

F = m · a, (6.4)

where F is the net force applied to any particle, m is the mass, and a is theacceleration. If the particle is a fluid, it is convenient to divide the equationto volume of the particle to generate a derivation in terms of density:

ρDV

Dt

= f = fbody + fsurface. (6.5)

In equation (6.5), f is the force exerted on the fluid particle per unit volume,and fbody is the applied force on the whole mass of fluid;

39

fbody = ρ · g, (6.6)

where ρ is the fluid density and g is the gravitational acceleration. Theexternal forces through the surface of fluid particles, fsurface, is expressed bypressure and viscous forces;

fsurface = ∇ · τij =∂τij∂xi

= fpressure + fviscous, (6.7)

where τij is the stress tensor,

τij = −Pδij + µ

(∂ui∂xj

+∂ujxi

)+ δijλ∇ · V. (6.8)

Thus, Newton’s second equation of motion can be expressed as

ρDV

Dt

= ρ · g +∇ · τij. (6.9)

Inserting equation (6.8) into (6.9) yields the the conservation of momentumof a Newtonian viscous fluid in one equation:

ρDV

Dt︸ ︷︷ ︸I

= ρ · g︸︷︷︸II

− ∇P︸︷︷︸III

+∂

∂xi

[µ

(∂ui∂xj

+∂uj∂xi

)+ δijλ∇ · V

]︸ ︷︷ ︸

IV

, (6.10)

I: Momentum convection,II: Mass force,III: Surface force,IV: Viscous force.

D/Dt is the material derivative,

D()

Dt

=∂()

∂t+ u

∂()

∂x+ v

∂()

∂y+ w

∂()

∂z= V · ∇(). (6.11)

For incompressible flow, the equations are greatly simplified in which theviscosity coefficient µ is assumed constant and ∇ · V = 0. Thus, the conser-vation of momentum equation for an incompressible three-dimensional flowcan be expressed as

40

ρDV

Dt

= ρg −∇P + µ∇2V. (6.12)

Expanding equation (6.12) for each dimension when the velocity is V (u, v, w)yields:

ρ

(∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

)= ρgx−

∂P

∂x+µ

(∂2u

∂x2+∂2u

∂y2+∂2u

∂z2

), (6.13)

ρ

(∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

)= ρgy−

∂P

∂y+µ

(∂2v

∂x2+∂2v

∂y2+∂2v

∂z2

), (6.14)

ρ

(∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z

)= ρgz −

∂P

∂y+ µ

(∂2w

∂x2+∂2w

∂y2+∂2w

∂z2

),

(6.15)where P , u, v and w are unknowns where a solution is sought by apply-ing both the continuity equation and boundary conditions. If any thermalinteraction is considered, the energy equation also has to be considered.

6.1.3 Conservation of energy

The conservation of energy equation is the first law of thermodynamics. Itstates that the sum of the work and heat added to the system will result inthe increase of energy of the system,

dEt = dQ+ dW, (6.16)

where dEt is the increment in the total energy of the system, dQ is the heatadded to the system, and dW is the work done on the system. One commontype of the energy equation is:

ρ

∂h

∂t︸︷︷︸I

+∇ · (hV )︸ ︷︷ ︸II

= −∂P∂t︸ ︷︷ ︸

III

+∇ · (k∇T )︸ ︷︷ ︸IV

+ φ︸︷︷︸V

, (6.17)

I: Local change with time,II: Convective term,

41

III: Pressure work,IV: Heat flux,V: Heat dissipation term.

6.2 Approximation

To obtain an approximate solution numerically, a discretization method isused to discretize the differential equations by a system of algebraic equa-tions, which can then be solved using a computer. The approximations areapplied to small domains in space and/or time so the numerical solutionprovides results at discrete locations in space and time. The accuracy of anumerical solution is highly dependent on the quality of the discretizationsused.

The differential equations based on the Navier-Stokes equations could besolved for a given problem by using methods from calculus. However, inpractice, these equations are too difficult and too time-consuming to solveanalytically. In the past, engineers made rough approximations and simpli-fications to the equation set until they had a group of equations that theycould solve. But since the introduction of high-speed computers - thus greatlyimproving numerical accuracy and lowering computational time - CFD hasbecome an important engineering tool when solving fluid flow problems.

6.3 History

This section gives a brief history of computation fluid dynamics.[14]

• Until 1910: Improvements on mathematical models and numericalmethods.

• 1910-1940: Integration of models and methods based to generate nu-merical solutions based on hand calculations.

• 1940-1950: Transition to computer-based calculations with early com-puters.

• 1950-1960: Initial study using computers to model fluid flow based onthe Navier-Stokes equations. First implementation for 2D, transient,incompressible flow.

42

• 1960-1970: First scientific paper “Calculation of potential flow aboutarbitrary bodies” was published about computational analysis of 3Dbodies by Hess and Smith in 19675. Generation of commercial codes.

• 1970-1980: Codes generated by Boeing, NASA and some have un-veiled and started to use several yields such as submarines, surfaceships, automobiles, helicopters and aircrafts.

• 1980-1990: Improvement of accurate solutions of transonic flows inthree-dimensional case. Commercial codes have started to implementthrough both academia and industry.

• 1990-Present: Thorough developments in informatics: worldwide us-age of CFD virtually in every sector.

6.4 Fields of application

Computational fluid dynamics has a wide range of fields of applications.Basically, where there is fluid, there is CFD. Whether you are working to im-prove the aerodynamic characteristics of a vehicle or studying the dynamicsof blood flow (hermodynamics), CFD is a great engineering tool with vastcapabilities. Other application areas include:

• Aerospace,

• Architecture,

• Nuclear thermal hydraulics,

• Chemical engineering,

• Turbomachinery,

• Process industry,

• Semiconductor industry,

and more!

43

Chapter 7

Model implementation

This chapter gives a step-by-step description of the creation of the CFD-models for each configuration. The CFD-simulations in this thesis are per-formed using Ansys CFX. Everything else - from CAD-modeling, meshingand setup - is performed in Ansys 18.2.

7.1 Geometry

The first step when creating a CFD-model is to create the geometries youwish to simulate. In this case that meant creating CAD-models of theMaxus and Terrier-Black configurations, which was done using the computer-aided design tool Ansys 18.2 DesignModeler. The CAD-models are based onsketches of the rockets. All geometries were created in full-scale, in threedimensions, and with each configuration fixed to its frame of reference.

7.1.1 Maxus

The sketch of the Maxus configuration from which the Maxus geometry isbased on is attached to the appendix. The sketch is an extract from a reportsubmitted by Per Elvemar[15] at Saab. The sketch is complemented withdetailed dimension of the nose cone from a report by T. Andersson[16] atSwedish Space Corporation, also included in the appendix.

Figure 7.1 shows the CAD-model of the Maxus configuration.

44

Figure 7.1: Maxus.

7.1.2 Terrier-Black Brant

Due to confidentiality concerning the exact dimensions of the Terrier-BlackBrant configuration, the sketches used for said configuration are not includedin this thesis.

It should be noted that although the sketch of the payload for the Terrier-Black Brant configuration was detailed, sketches of other parts of said config-uration were inadequate. Among other things, the sketches lacked dimensionsfor the fins. Parts of the geometry for the Terrier-Black configuration hasthus been approximated.

Figures 7.2 - 7.3 show the CAD-models of the Terrier-Black Brant, firstand second stage, respectively.

45

Figure 7.2: Terrier-Black Brant (first stage).

Figure 7.3: Terrier-Black Brant (second stage).

46

7.2 Fluid domain

The second step is to define a fluid domain. Both the Maxus and Terrier-Black Brant geometries were enclosed by a cylindrical fluid domain. In orderto achieve meshing flexibility, the fluid domain was divided into differentsections with a high cell density approximate the rockets, and a graduallylower cell density further away from the rockets. To assert the boundaryconditions were far enough from the geometries, a sensitivity analysis wasperformed in each direction.

Figure 7.4 shows the fluid domain for the Maxus configuration.

Figure 7.4: Maxus fluid domain.

7.3 Mesh

A high-quality mesh is of vital importance to the accuracy and stability ofa numerical simulation. In order to attain reliable results, the mesh mustbe constructed to properly represent the physical properties of the domain.Also, the aspect of accuracy versus CPU time must be considered.

47

Since the free-stream flow will be in one direction, the meshing cellsshould ideally be parallel to the flow to avoid numerical inaccuracies. Thisis achieved by implementing a hexahedral dominant mesh. However, thegeometries of the rockets require a more flexible meshing method to avoidhighly skewed cells. Consequently, the fluid domain has been meshed usinga combination of hexahedral dominant mesh in regions of free-stream flow,and a tetrahedral mesh in the region proximate the rockets.

As mentioned in section 7.2, the fluid domain was divided into differentsections with a high cell density approximate the rockets and a graduallylower cell density further away from the rockets. To avoid skewed elements,and to ensure the interlocking of nodes, small sections of tetrahedral domi-nant meshes were created between the hexahedral dominant free-stream sec-tions. Figure 7.5 shows the transition between such sections.

Figure 7.5: Mesh sections.

Any numerical simulation designed to determine the drag requires a finemesh representation of the boundary layer (see Section 3.3), often resultingin complex meshing. This becomes particularly substantial when performingtransonic and supersonic simulations, as the required wall distance of the firstcell typically needs to be very small. In order to determine if the boundarylayer is adequately represented, the Yplus (y+) value has been used as a

48

reference.

7.3.1 Yplus

y+ is a dimensionless distance from the wall which is used to check thelocation of the first node away from the wall. The distance will have asignificant impact on the model’s ability to represent the boundary layer.The desired y+ varies with turbulence model. For the turbulence model usedfor these simulations, the SST model (see Section 7.4.2), the required y+

should be y+ < 1 for accurate simulations.Apart from acquiring a satisfactory y+, there are several other mesh pa-

rameters that determine the quality of a mesh. The primary mesh qualityparameters are typically divided into orthogonal quality, aspect ratio andskewness.

7.3.2 Orthogonal quality

Mesh orthogonality relates to how close the angles between adjacent elementfaces or element cells are to being equiangular. Orthogonality measures de-pend on several parameters including orthogonality angle, orthogonality fac-tor, orthogonal angle minimum, etc. All these parameters are summarizedinto an orthogonal quality factor between 0 and 1, with good cells having anorthogonal quality closer to 1.

7.3.3 Aspect ratio

The aspect ratio is a measure of the stretching of a cell. It is calculated asthe ratio of the maximum value to the minimum vale of the distance betweenthe cell centroid and nodes. For stable simulations, the aspect ratio shouldideally be around 20.

7.3.4 Skewness

Skewness is defined as the difference between the shape of the cell and theshape of an equilateral cell of equivalent volume. Highly skewed cells candecrease accuracy and destabilize the solution. A general rule is that themaximum skewness for a triangular/tetrahedral mesh in most flows shouldbe kept below 0.95, with an average value that is significantly lower.

49

7.4 Setup

The following section describes simulation setup. Settings not referenced areset as default.

7.4.1 Flow analysis

Test runs of both transient and steady-state analysis types were performed todetermine which type was most suitable. Steady-state rendered satisfactoryresidual and force profiles and was therefore chosen as an appropriate analysistype.

7.4.2 Domain

Under Basic settings, the material was set to air ideal gas for speeds aboveMach 0.3 in order to account for compressibility effects. Dynamic viscositywas set to 1.789 · 10−5 N·/m2 and reference temperature to 15◦C. For speedsless than Mach 0.3, the material was set to air at 15◦C with constant density1.225 kg/m3 and dynamic viscosity 1.789 · 10−5 N·/m2.

Under Fluid models, the heat transfer option was set to Total energy. Theturbulence model was set to Shear stress transport (SST).

Under Initialization, Cartesian velocity components were set to zero in alldirections except in the flow direction. The velocity component in the flowdirection was set to whatever speed was being simulated. Static pressure wasset to 0 Pa, Temperature to 15◦C and Turbulence to low intensity.

7.4.3 Boundary conditions

Rocket body

The rocket body is defined as a no-slip wall, i.e. the fluid will have zerovelocity relative to the surface boundary. Since the rocket is fixed in itsframe of reference, the velocity at the solid boundary is zero.

Inlet

The inlet is defined in front of the rocket, at the face perpendicular to theflow. The flow regime option is set to supersonic for speeds equal to orgreater than Mach 1. For speeds less than Mach 1, the subsonic option is

50

selected. The cartesian velocity component in the flow direction is set towhatever speed is being simulated.

Outlet

The outlet is defined at the opposite face to the inlet. The outlet is positionedfar enough behind the rocket to prevent it from intersecting the recirculationarea behind the rocket. Just as for the inlet, the flow regime option is set tosupersonic for speeds equal to or greater than Mach 1. For speeds less thanMach 1, the subsonic option is selected.

Opening

The cylinder wall parallel to the free stream flow is set as an opening. Theopening option is preferred over the outlet option since the free stream flowwill be along the cylinder wall, without much outflow expected. Turbulenceoption is set to zero gradient, i.e. the velocity gradient perpendicular to theboundary is zero. In order for this to be an appropriate option, the cylinderenclosure is assured to be wide enough for this condition to be valid.

Thrust

Figure 7.6 shows a simple schematic of a rocket engine.[17] Stored fuel andoxidizer are ignited in a combustion chamber, producing great amounts ofexhaust gas at high temperature and pressure. The hot exhaust is passedthrough a nozzle which accelerates the flow, pushing the gases out of thenozzle in one direction and propelling the rocket in the opposite direction.

Figure 7.6: Rocket engine schematic.

The amount of thrust produced by a rocket engine depends on the massflow rate through the engine, the exit velocity of the exhaust, and the pressure

51

at the nozzle exit. These variables, in turn, depend on the design of thenozzle.

The generalized thrust equation,

F = mVe + (pe − p0)Ae, (7.1)

describes the thrust of the system. In equation (7.1), m is the mass flow rate,Ve the exit velocity of the exhaust, pe the pressure at the nozzle exit, p0 thefree stream pressure, and Ae the exit area of the exhaust.

In order to determine appropriate boundary conditions for the exhaust,data of the thrust, mass flow rate and exit velocity of Maxus 8 and Terrier-Black Brant MARK70 (MOD2) have been used. Since the geometry of theexhaust is know, equation (7.1) can be rearranged to express the unknownexit pressure as

pe =F − mVe

Ae+ p0. (7.2)

The actual thrust of a rocket will, however, vary from launch to launch andalso during the different flight sequences.

The simulated rockets are fixed in its frame of reference, meaning there isno propulsion. Instead, the CFD-model is created as a virtual wind tunnel.In order to include entrained flow and drag effects generated by the thrustfrom the rocket engine, the exhaust is set as an inlet boundary with theflow conditions of the exhaust specified. Mean values from data providedfrom each configuration has been used to determine the exit velocity of theexhaust and to calculate the exit pressure.

7.4.4 Solver control

For advection scheme and turbulence numerics, upwind and first order werechosen, respectively. To determine an appropriate time scale, several testsimulations were performed. Given the high turbulence gradients generatedat supersonic speeds, a small time scale was necessary to prevent the numer-ical solution from diverging. By trial and error it could be determined thata maximum time scale of 1 · 10−5 was appropriate.

52

7.4.5 Output control

In order to monitor the progression of aerodynamic parameters during sim-ulations, monitor points were added. The monitor points were based on theexpression for each parameter and derived using function calculators.

53

Chapter 8

Results

This chapter presents the results from the CFD-simulations along with thepre-existing data.

The pre-existing data for Maxus is presented as hand-drawn graphs ob-tained from [15]. The corresponding results from the CFD-simulations arepresented as Matlab plots.

The pre-existing data for Terrier-Black Brant (first and second stage)are presented as Matlab plots along with the corresponding results from theCFD-simulations.

54

8.1 Maxus

8.1.1 Zero-lift drag coefficient

Figure 8.1: Pre-existing data for CD0 , Maxus.

55

0 2 4 6 8 10 12

Mach number

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cd

0

Zero-lift drag coefficient

Figure 8.2: CFD-simulations for CD0 , Maxus.

56

8.1.2 Stability derivative

Figure 8.3: Pre-existing data for CNα , Maxus.

57

0 2 4 6 8 10 12

Mach number

0

2

4

6

8

10

12C

N

Derivative of the lift coefficient w.r.t. angle of attack

Figure 8.4: CFD-simulations for CNα , Maxus.

58

8.1.3 Center of pressure

Figure 8.5: Pre-existing data for Xcp, Maxus.

59

0 2 4 6 8 10 12

Mach number

2

3

4

5

6

7

8X

cp

Center of pressure

Figure 8.6: CFD-simulations for Xcp, Maxus.

8.2 Terrier-Black Brant (first stage)

8.2.1 Zero-lift drag coefficient

0 0.5 1 1.5 2 2.5 3

Mach number

0

0.5

1

1.5

2

2.5

Cd

0

Zero-lift drag coefficient, first stage

Pre-existing data

CFD-simulations

Figure 8.7: CD0 for pre-existing data vs CFD-simulations, Terrier-BlackBrant (first stage)

60

8.2.2 Stability derivative

0 0.5 1 1.5 2 2.5 3

Mach number

0

5

10

15

20

25

30

CN

Derivative of the lift coefficient w.r.t. angle of attack

Pre-existing data

CFD-simulations

Figure 8.8: CNα for pre-existing data vs CFD-simulations, Terrier-BlackBrant (first stage)

61

8.2.3 Center of pressure

0 0.5 1 1.5 2 2.5 3

Mach number

0

1

2

3

4

5

6

7

Xcp

Center of pressure, first stage

Pre-existing data

CFD-simulations

Figure 8.9: Xcp for pre-existing data vs CFD-simulations, Terrier-Black Brant(first stage)

62

8.3 Terrier-Black Brant (second stage)

8.3.1 Zero-lift drag coefficient

0 1 2 3 4 5 6 7 8 9

Mach number

0

0.5

1

1.5

Cd

0

Zero-lift drag coefficient, second stage

Pre-existing data

CFD-simulations

Figure 8.10: CD0 for pre-existing data vs CFD-simulations, Terrier-BlackBrant (second stage)

63

8.3.2 Stability derivative

0 0.5 1 1.5 2 2.5 3

Mach number

0

5

10

15

20

25

30

CN

Derivative of the lift coefficient w.r.t. angle of attack

Pre-existing data

CFD-simulations

Figure 8.11: CNα for pre-existing data vs CFD-simulations, Terrier-BlackBrant (second stage)

64

8.3.3 Center of pressure

0 0.5 1 1.5 2 2.5 3

Mach number

0

1

2

3

4

5

Xcp

Center of pressure, second stage

Pre-existing data

CFD-simulations

Figure 8.12: Xcp for pre-existing data vs CFD-simulations, Terrier-BlackBrant (second stage)

65

Chapter 9

Discussion

This chapter compares the results obtained from the CFD-simulations withthe provided data. The results are compared in terms of behavior and para-metric values. Any deviations are discussed and motivated in order to deter-mine which data is more physically plausible.

9.1 Reliability of CFD-simulations and veri-

fying the results from such simulations

As mentioned in Chapter 7, the quality of the mesh is of high importanceto the accuracy and stability of a numerical simulation. Different aspects ofthe mesh must be emphasized in regard to the problem at hand to ensurethe reliability of the simulations. Thus, numerous sensitivity analysis havebeen performed for several mesh quality parameters (as discussed in Section7.3) to ensure the accuracy of the CFD-simulations in this thesis. Also,the simulation setup has been implemented with great care (as discussed inSection 7.4) to ensure the physical properties of the simulations correspondto the problem at hand.

Although every precautionary measure mentioned above have been taken,rather than assuming the CFD-simulations are more accurate, any inconsis-tencies between the provided data and the CFD-simulations have been eval-uated based on the theories presented in Chapter 3 and Chapter 4, as wellas earlier studies.

As a general discussion regarding the verification of CFD-simulations, itshould also be mentioned that complementary measurements, such as wind

66

tunnel testing, can provide diagnostic information to help verify the resultsfrom CFD-simulations.

9.2 Maxus

9.2.1 Zero-lift drag coefficient

The zero-lift drag coefficient from the provided data, Figure 8.1, and theresults from the CFD-simulations, Figure 8.2, correspond well, both in termsof coefficient value and behavior. The results from the CFD-simulations are,for most parts, well within range of the 10% margin of error assumed for theprovided data. There are, however, a few disparities.

First of all, the provided data seems to underestimate the drag divergenceMach number, i.e. the Mach number at which the aerodynamic drag beginsto increase rapidly. The provided data seems to estimate the drag divergenceMach number to exactly Mach 1, whilst the CFD-simulations estimate thisnumber to be within the interval of Mach 0.5 and Mach 0.8.

In fact, the drag divergence Mach number is expected to be lower thanMach 1. As discussed in section 5.1.1, there will be regions around the rocketwhere the fluid flow is forced to accelerate. During transonic flight, the fluidflow around these regions will reach supersonic speed, even though the flowspeed around the rest of the rocket is still subsonic. As a result, local shockwaves will form at these regions, consequently increasing the drag coefficient.This is something the provided data does not regard.

Secondly, the provided data underestimates the maximum value of CD0

compared to the CFD-simulations. The provided data estimates the max-imum value to approximately 0.62, whilst the CFD-simulations estimatesa maximum value of 0.67. This would effectively mean the provided dataoverestimates the apogee for the Maxus configuration.

Finally, as the Mach number increases, both the provided data and theCFD-simulations predict CD0 to begin stabilize slightly below 0.2.

9.2.2 Stability derivative

The derivative of the lift coefficient with respect to the angle of attack fromthe provided data, Figure 8.3, and the results from the CFD-simulations,Figure 8.4, show three apparent disparities.

67

First of all, the provided data estimates a sharper decrease of CNα afterthe transonic phase compared to the CFD-simulations. This would effec-tively mean the provided data underestimates the magnitude of the lateralacceleration during this phase.

Secondly, the provided data predicts CNα to decrease in a somewhat os-cillating manner, whilst the CFD-simulations predicts a smooth decrease. Itis difficult to motivate why CNα would have an oscillating behavior. Earlierstudies [18] indeed predicts CNα to have a smooth decrease, in conformitywith the results obtained from the CFD-simulations. Thus, based on theresults from the CFD-simulations and on earlier studies, it can be concludedthat the oscillating behavior of CNα predicted by the provided data mostlikely is an incorrect prediction.

Thirdly, comparing the continuous graph of the provided data to theCFD-simulations show the provided data somewhat underestimates the max-imum value of CNα . The provided data estimates a maximum value of ap-proximately 9.5, whilst the CFD-simulations estimates a maximum value of10.57. This would effectively mean the provided data underestimates themaximum lateral acceleration during transonic flight.

It should be noted that in addition to the continuous graph in the provideddata, there is another ”dash-dotted” hand-drawn graph. There is no informa-tion as to where this second graph originates from. Most likely, it was addedto the report after its initial filing. Comparing the original continuous graphand the second graph to the CFD-simulations show the maximum value ofCNα from the simulations is located between these two graphs. Perhaps PerElvemar realized, in retrospect, that the initial estimation of the maximumvalue of CNα was too low, and corrected it afterwards. However, since there isno information where this second graph originates from, this thesis will focuson the original continuous graph, which, as mentioned above, underestimatesthe maximum value of CNα compared to the CFD-simulations.

Finally, as the Mach number increases, both the provided data and theCFD-simulations predict CNα to begin stabilize slightly above 0.3.

9.2.3 Center of pressure

The center of pressure from the provided data, Figure 8.5, and the resultsfrom the CFD-simulations, Figure 8.6, correspond well, both in terms ofposition and behavior. Both predict the center of pressure moving rearwardduring transonic flight due to local shock waves forming at the fins. Once the

68

rocket has reached supersonic flight, and the affects of the local shock wavesbecome recessive, the center of pressure instead begins moving forward.

Examining the actual position of the center of pressure, there are, how-ever, small disparities between the provided data and the CFD-simulations.

For low Mach numbers (up to about Mach 2), the provided data predictsthe center of pressure to be located slightly further away from the nose thancompared to the CFD-simulations. This would effectively mean the provideddata predicts the Maxus configuration to be more stable during this sequence.

For higher Mach numbers, the provided data predicts the center of pres-sure to be located slightly closer to the nose than compared to the CFD-simulations. This would, correspondingly, effectively mean the provided datapredicts the Maxus configuration to be less stable for higher Mach numbers.The variations are, however, less than 0.5 meters in both cases, thus onlyhaving a marginal impact on the stability margin.

9.3 Terrier-Black Brant (first stage)

9.3.1 Zero-lift drag coefficient

Comparing the zero-lift drag coefficient from the provided data to the CFD-simulations, Figure 8.7, shows that the provided data overestimates CD0 forall Mach numbers. This would effectively mean the provided data underesti-mates the apogee for the first stage of the Terrier-Black Brant configuration.Furthermore, comparing the predicted behavior of CD0 for the provided datato the CFD-simulations show two clear deviations.

First, the provided data predicts a plateau between Mach 1.2 and Mach1.5. This is inconsistent with the CFD-simulations and with the results fromthe Maxus configuration. It is also inconsistent with previous studies [18],[19], [20].

Second, the provided data predicts CD0 to stabilize already at Mach 2.This is also inconsistent with the CFD-simulations, earlier results and earlierstudies.

9.3.2 Stability derivative

Comparing CNα from the provided data to the CFD-simulations, Figure 8.8,shows a similar behavior, but with varying deviations in terms of coefficient

69

values. Even though the maximum value of CNα from the provided dataand the CFD-simulations overlap, the provided data underestimates CNα forall other Mach numbers. This would effectively mean the provided dataunderestimates the lateral acceleration for these Mach numbers.

9.3.3 Center of pressure

Comparing the center of pressure from the provided data to the CFD-simulations,Figure 8.9, shows an identical behavior. Both predict the location of thecenter of pressure moving rearward during transonic flight (as discussed inSection 9.2.3), before moving forward once transitioned to supersonic flight.

There is, however, a distinct difference in the actual position of the cen-ter of pressure between the provided data and the CFD-simulations. Theprovided data predicts the location of the center of pressure to be, in mostcases, around 2 meters further from the nose than the CFD-simulations. Theprovided data predicts the center of pressure to be located less than 2 metersfrom the nozzle exit plane throughout subsonic and transonic flight - an im-plausible location for a rocket that measures over 17 meters tall. Althoughthe Terrier-Black Brant configuration utilizes two large sets of fins - one setat the midsection of the Black Brant rocket, and one set at the rear of theTerrier-booster - thus effectively generating a stable configuration, the pres-sure distributions along the cylindrical sections of varying dimensions at thepayload, and the long rocket body, would inevitably generate a location forthe center of pressure further away from the nozzle exit plane than what ispredicted by the provided data.

9.4 Terrier-Black Brant (second stage)

9.4.1 Zero-lift drag coefficient

Examining the zero-lift coefficient from the CFD-simulations, Figure 8.10,shows that the results in terms of behavior is consistent with earlier results;CD0 increases rapidly during transonic flight, and eventually begins to stabi-lize for higher Mach numbers. The provided data in the same figure, however,shows two distinct deviations.

First, the provided data once again predicts a plateau - this time aroundMach 1, before CD0 sharply increases. As discussed in Section 9.3.1, this is

70

not a behavior we would expect.Second, and most distinguishable, the provided data predicts CD0 to

sharply increase after Mach 5. This contradicts all previous results, pro-vided data and earlier studies. It has previously been shown time and timeagain how CD0 eventually begins to stabilize as the Mach number increases.Nowhere else has the behavior predicted from the provided data for the sec-ond stage of the Terrier-Black Brant configuration been observed.

Although the provided data manages to predict a maximum value nottoo far from the CFD-simulations (although a slight underestimation), theprovided data is inconsistent in terms of behavior and coefficient values forhigher Mach numbers.

9.4.2 Stability derivative

Comparing the provided data to the CFD-simulations for CNα , Figure 8.11,shows, once more, how the maximum value between the both overlap. Thebehavior is again near identical, although this time the provided data predictsa slight overestimation of the coefficient values for other Mach numbers thanMach 1.2. This would effectively mean that the provided data overestimatesthe magnitude of the lateral acceleration for these Mach numbers.

9.4.3 Center of pressure

Comparing the center of pressure between the provided data and the CFD-simulations, Figure 8.12, shows overlapping predictions or similar estimationsfor most Mach numbers. The provided data, however, fails to predict therearward movement of the center of pressure during transonic flight, whichwas observed in both previous cases (see Section 9.2.3 and Section 9.3.3).

Although failing to predict the rearward movement of the center of pres-sure would not effectively impact the estimated stability in any negative way(as the reward movement actually increases the stability margin), it still givesevidence of the shortcomings of the provided data.

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Chapter 10

Conclusions

This chapter concludes the results discussed in Chapter 9. The final conclu-sions are based on the overall results from comparing all parameters fromthe provided data to the CFD-simulations.

The results are evaluated per configuration, ultimately determining howwell the provided data for each configuration corresponds to the CFD-simulations,and whether the provided data is physically plausible.

10.1 Maxus

The deviations in the provided data with the performed CFD analysis in-clude:

• Overestimation of the drag divergence Mach number,

• An approximate 7.8% underestimation of the maximum value of CD0 ,

• An approximate 10.7% underestimation of the maximum value of CNα ,

• Predicting a sharper decrease of CNα after transonic flight,

• Marginal deviations for the position of the center of pressure.

Overall, the provided data for all parameters corresponded well with theCFD-simulations and with earlier studies in most aspects. The provideddata is physically plausible, and most deviations are within the 10% marginof error assumed for the provided data.

72

In conclusion, the provided data for the Maxus configuration is assessedto be sufficiently accurate and can safely be used for future analysis.

10.2 Terrier-Black Brant (first stage)

The deviations in the provided data with the performed CFD analysis in-clude:

• Perpetual overestimation of CD0 ,

• A 12.9% overestimation of the maximum value of CD0 ,

• An inexplicable plateau for CD0 between Mach 1.2 and Mach 1.5,

• A too early prediction of the stabilization of CD0 ,

• Substantial underestimation of CNα during supersonic flight (at themost 75.4%),

• Predicting the center of pressure to be located 1.4 - 2.8 meters furtheraway from the nose.

Apart from the plateau of CD0 between Mach 1.2 and Mach 1.5, theprovided data and the CFD-simulations exhibits an, at large, identical be-haviour. The provided data is thus considered to capture the physical be-haviour of the first stage of the Terrier-Black Brant configuration sufficiently.However, in terms of actual parameter values, the provided data has vary-ing precision compared to the CFD-simulations. Although parameter valuessometimes are overlapping (or close to overlapping), there are several devia-tions that are too great to oversee. Mainly, the substantial underestimationof CNα during supersonic flight and the position of the center of pressuretestify to inaccuracies of the provided data.

In conclusion, the provided data gives an accurate prediction of the phys-ical behavior, but at times lacks parameter value accuracy. The overall eval-uation is that the provided data can neither be discarded as insufficient ordeemed satisfactorily accurate. The provided data is, however, sufficient tobe used for continued analysis, provided the limitations of the provided datais taken into consideration.

73

10.3 Terrier-Black Brant (second stage)

The deviations in the provided data with the performed CFD analysis in-clude:

• An inexplicable plateau for CD0 during transonic flight,

• A 6.5% underestimation of the maximum value of CD0 ,

• Physically implausible increase of CD0 after Mach 5,

• Varying precision of CNα (at the most 21.8%),

• Failure to predict the rearward movement of the center of pressureduring transonic flight.

Once more, the provided data for the Terrier-Black Brant configura-tion exhibits varying precision compared to the CFD-simulations. However,whilst the provided data for the first stage predicted a behavior that, formost parts, was identical to the CFD-simulations and to what is physicallyplausible, the provided data for the second stage exhibits a behaviour that,at times, contradicts both the CFD-simulations and earlier studies and thatis physically implausible. Concretely, the sudden increase of CD0 and thefailure to predict the rearward movement of the position of the center ofpressure testify to the physical limitations presented by the provided data.However, apart from these apparent physical limitations, the provided datacorresponds well with the CFD-simulations, both in terms of behavior andparameter values.

In conclusion, the provided data gives, for most parts, an accurate predic-tion of the parameter values, but fails to predict some key physical behaviors.Consequently, the provided data predicts false parameter values for CD0 athigh Mach numbers and position pf the center of pressure during transonicflight. The overall evaluation is that the provided data can neither be dis-carded as insufficient or deemed satisfactorily accurate. The provided data is,however, sufficient to be used for continued analysis, provided the limitationsof the provided data is taken into consideration.

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Bibliography

[1] RUAG Space. URL: https://www.ruag.com/en/about-ruag/

organisation/divisions/ruag-space, Accessed 25 May 2018.

[2] California Science Center. Orion Sounding Rocket. URL: https:

//californiasciencecenter.org/exhibits/air-space/mission-

to-the-planets/orion-sounding-rocket, Accessed 12 April 2018.

[3] Sven Grahn. MAXUS-raketen. URL: http://www.svengrahn.pp.se/

histind/Swefirst/Soundingrocketdetails/MAXUS/MAXUS.html, Ac-cessed 10 April 2018.

[4] Encyclopedia Astronautics. Black Brant. URL: http://www.

astronautix.com/b/blackbrant.html, Accessed 19 Mars 2018.

[5] White Sands Missile Range Museum. Terrier-Black Brant ResearchRocket. URL: http://www.wsmr-history.org/terrierblackbrant.

htm, Accessed 19 Mars 2018.

[6] Beggs Aerospace. Black Brant Rocket Photos. URL: http://www.

postwarv2.com/BBIX/photos.html, Accessed 19 Mars 2018.

[7] NinetyEast. How Liquids Flow. URL: http://www.ninetyeast.net/

physics/university/rheology/how-liquids-flow-laminar-flow,Accessed 31 May 2018.

[8] Thermopedia. Boundary Layer. URL: http://www.thermopedia.com/

content/595/, Accessed 29 May 2018.

[9] QUORA.URL: https://www.quora.com/Could-you-explain-the-

mechanism-of-the-formation-of-the-boundary-layer-in-a-

fluid-flowing-over-a-flat-plate, Accessed 14 May 2018.

75

[10] Cengel, Y. and Cimbala, J. (2006). Fluid mechanics. 3rd ed. Boston:McGraw-HillHigher Education.

[11] Taha Sochi (2011). Slip at Fluid-Solid Interface.

[12] The Electropedia. Aerodynamic Lift and Drag and the Theory ofFlight. URL: https://www.mpoweruk.com/flight_theory.htm, Ac-cessed 5 June 2018.

[13] Space Flight Systems. Glenn Research Center. Rocket Stability Condi-tion. URL: https://spaceflightsystems.grc.nasa.gov/education/

rocket/rktstabc.html, Accessed 31 May 2018.

[14] SimScale. Computational Fluid Dynamics. URL: https:

//www.simscale.com/docs/content/simwiki/cfd/whatiscfd.html,Accessed 4 June 2018.

[15] Elvemar, P. (1990). Document number: TURLA-Ry-2.46 M5.

[16] Andersson, T (1990). Document number: RSYO.

[17] NASA Glenn Research Center. Rocket Thrust Equation. URL: https://www.grc.nasa.gov/www/k-12/airplane/rockth.html, Accessed 16Mars 2018.

[18] Burke, J.D. (1949). Sounding Rocket Performance Analysis. CaliforniaInstitute of Technology. Pasadena, California.

[19] Olalekan, A.H. Attah, O.C. Bisola, O. Jimoh, H. Atinuke, Ojo. (2014).Flight Trajectory Simulation of a Single Stage Sounding Rocket. Interna-tional Journal of Scientific & Engineering Research, Volume 5, Issue 11,November 2014. ISSN 2229-5518.

[20] Jan-Erik Rønningen. (2008). Rocket Aerodynamics and Stability.URL: https://www.mn.uio.no/fysikk/english/research/doctoral-degree/research-schools/workshop-arr/JanRonningen-Rocket_

Aerodynamics_and_Stability_V.1.40_2008.ppt, Accessed 6 February2018.

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Appendix

This appendix includes sketches of the Maxus configuration and its nosedimensions used to create the CAD-model of Maxus.

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Figure 10.1: Maxus configuration.

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Figure 10.2: Maxus nose dimension.

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