Aerodynamics Modeling of Sounding Rockets A 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 propulsion system. The guidance system of a rocket includes sophisticated sensors, on-board computers and communication equipment. The guidance system provides stability for the rocket and controls the rocket during maneuvers. When developing guidance systems, reliable aerodynamic models of the rockets are required. This report conducts a comparative study between CFD-simulations and the current aerodynamic models of two sounding rocket configurations used at RUAG Space in Link¨oping;the Maxus configuration and the two-stage Terrier-Black Brant configuration. Three parameters related to rocket stability are evaluated: the zero-lift drag coefficient CD0 , the stability derivative CNα , and the center of pressure Xcp. The parameters are evaluated in terms of behavior and parametric val- ues, ultimately determining how accurate the current aerodynamic models are compared to the CFD-simulations. Previous studies are used to deter- mine which data is most plausible in case of any deviations. All parameters are evaluated as a function of Mach number for complete configurations. The CFD-simulations are performed using Ansys CFX 18.2. Ansys is an American 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 for the Maxus configuration corresponds well with the CFD-simulations, both in terms of behavior and parametric values. Data obtained from the current aerodynamic model are physically plausible and also corresponds well with earlier studies. Data provided from the current aerodynamic model for the first stage of the Terrier-Black Brant configuration show a mainly identical behavior to the CFD-simulations, but with varying accuracy in terms of parametric

2 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 far from the nose. Data provided from the current aerodynamic model for the second stage of the Terrier-Black Brant configuration show an, at times, inconsistent and physically implausible behavior. These deviations are, however, limited to certain flight sequences, and not to the data as a whole. Apart from the inconsistencies shown at times, the provided data correspond well with the CFD-simulations in terms of parametric values. The aerodynamic model of the Maxus configuration is concluded to be accurate and sufficient for continued analysis. There are no indicators the current aerodynamic model could be faulty in any way. The aerodynamic model for the first stage of the Terrier-Black Brant configuration accurately represents the physical behavior of the rocket, but lacks 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 Brant configuration corresponds well compared to the CFD-simulations in terms of parametric values, but fails to predict the stabilization of CD0 at high Mach numbers and the rearward movement of the position of the center of pressure during transonic flight. The current aerodynamic model is concluded to be sufficiently accurate for continued analysis, provided the limitations of the model is taken into consideration.

3 Sammanfattning

Varje storskalig raket best˚arav fyra huvudkomponenter: raketkroppen, nyt- tolasten, styrsystemet och framdrivningssystemet. Styrsystemet best˚arav sofistikerade sensorer, datorer och kommunikationsutrustning. Styrsystemet avser h˚allaraketen stabil och man¨ovreraraketen under flygningen. N¨arman utvecklar och tillverkar styrsystem kr¨avsp˚alitligamodeller ¨over raketernas aerodynamik. Den h¨arrapporten utf¨oren komparativ studie mellan CFD-simuleringar och dagens modeller f¨oraerodynamikmodellering f¨ortv˚asondraketskonfigurationer vid RUAG Space i Link¨oping.Konfigura- tionerna som unders¨okts¨arMaxus och en tv˚astegsTerrier-Black Brant. Tre parametrar relaterade till raketernas stabilitet har unders¨okts:mot- st˚andskoefficienten d˚aden effektiva anbl˚asningsvinkeln ¨arnoll CD0 , stabilitets- derivatan CNα , och tryckcentrum Xcp. Parametrarna ¨arutv¨arderadei form av beteende och parameterv¨ardenf¨oratt avg¨orahur noggranna dagens mod- eller ¨arj¨amf¨ortmed CFD-simuleringarna. Vid avvikelser j¨amf¨orsdata med tidigare studier f¨oratt avg¨oravad som ¨armest fysikaliskt troligt. Alla parametrar ¨arutv¨arderadesom funktion av Machtal f¨orfullst¨andigakon- figurationer. CFD-simuleringarna ¨arutf¨ordai Ansys CFX 18.2. Ansys ¨arett amerikan- skt f¨oretagsom utvecklar och s¨aljerprogramvara inom flera ingenj¨orsomr˚aden. Ansys ¨aren av de st¨orstakommersiella programvarorna f¨orCFD-simuleringar. Resultaten i denna rapport visar att dagens metod f¨oraerodynamikmod- ellering av Maxus ¨overensst¨ammerv¨almed CFD-simuleringarna, b˚ade vad g¨allerbeteende och parameterv¨arde.Data fr˚anden nuvarande modellen ¨ar fysikaliskt rimliga och st¨ammer¨aven v¨al¨overens med tidigare studier. Resultaten f¨orf¨orstasteget av Terrier-Black Brant uppvisar, f¨ordet mesta, ett identiskt beteende med CFD-simuleringarna, men med varierande noggrannhet vad g¨allerparameterv¨arden. Aven¨ om beteendet f¨orden nu- varande modellen ¨arfysikaliskt rimligt, f¨orutsp˚arden en position f¨ortryck-

4 centrum under den transsoniska fasen som ¨arorimligt l˚angtbak. Resultaten f¨orandra steget av Terrier-Black Brant uppvisar ett, i vissa fall, inkonsekvent och fysikaliskt orimligt beteende. Avvikelserna ¨aremeller- tid begr¨ansadetill vissa flygsekvenser, och inte till modellen som helhet. Bortsett fr˚ande f˚ainkonsistenser den nuvarande modellen uppvisar st¨ammer den nuvarande modellen v¨al¨overens med CFD-simuleringarna vad g¨allerpa- rameterv¨arden. Den nuvarande modellen f¨orMaxus bed¨omsvara noggrann och kan forts¨a- tta anv¨andasof¨or¨andratinom analyser. Det finns inga indikatorer p˚aatt den nuvarande modellen p˚an˚agotvis skulle vara bristf¨allig. Den nuvarande modellen f¨orf¨orstasteget av Terrier-Black Brant f¨orutsp˚ar ett korrekt fysikaliskt beteende, men med varierande precision vad g¨aller faktiska parameterv¨arden. Den nuvarande modellen bed¨omskunna forts¨atta anv¨andasinom analyser, f¨orutsattatt detta g¨orsmed bristerna med modellen i ˚atanke. Den nuvarande modellen f¨orandra steget av Terrier-Black Brant ¨overens- st¨ammerv¨almed CFD-simuleringarna vad g¨allerparameterv¨arden,men mis- sar att f¨orutsp˚astabiliseringen av CD0 f¨orh¨ogaMachtal samt den bak˚atf¨orfly- ttning av tryckcentrum som sker under den transsoniska fasen. Den nu- varande modellen bed¨omsvara tillr¨ackligt noggrann f¨orfortsatta analyser, f¨orutsattatt detta g¨orsmed bristerna med modellen i ˚atanke.

5 Acknowledgements

Many people have helped me during the course of this thesis work. These are my acknowledgements to these people, whom I am greatly thankful for. First, I would like to thank Albert Thuswaldner at RUAG Space in Link¨opingwho was my initial contact at RUAG Space and who helped to actualize this thesis work. Thank you for initiating the process of actualizing this work and for always being a familiar and kind face whenever I would run into you. Second, I would like to thank Anders Helmersson, my supervisor at RUAG Space. Thank you for always taking time to answer my questions and for aiding me throughout this work. Third, I would like to thank Gunnar Hellstr¨om,my supervisor at Lule˚a University of Technology. Thank you for all the support, especially in the early 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 making every day at RUAG Space a pleasant and fun day. Finally, I would like to thank the rest of the personnel at RUAG Space in Link¨opingfor giving me such a warm welcome and for being overwhelmingly kind to me since day one. You have given me a fantastic impression of RUAG Space. 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. Truly appreciate life, and you’ll find that you have more of it.” -Ralph Marston

6 Contents

1 Introduction 10 1.1 Background ...... 10 1.2 Disposition of the thesis ...... 11

2 Sounding rockets 12 2.1 Maxus ...... 13 2.2 Terrier-Black Brant ...... 14

3 Fundamental concepts of fluid dynamics 16 3.1 Viscosity ...... 16 3.2 Laminar vs turbulent flow ...... 17 3.2.1 Relationship with Reynold’s number ...... 17 3.3 Boundary layer ...... 18 3.4 Compressible vs incompressible flow ...... 19 3.5 The no-slip condition ...... 20 3.6 Newtonian vs non-Newtonian fluids ...... 20

4 Aerodynamic forces 21 4.1 Pressure ...... 21 4.1.1 Molecular definition of pressure ...... 22 4.2 Viscous shearing stresses ...... 26 4.2.1 Couette flow ...... 26

5 Parameters to be evaluated 31 5.1 Aerodynamic coefficients ...... 31 5.1.1 Drag coefficient ...... 32 5.1.2 Lift coefficient ...... 34 5.2 Center of pressure ...... 37

7 6 Computational fluid dynamics 38 6.1 Governing equations ...... 38 6.1.1 Conservation of mass ...... 39 6.1.2 Conservation of momentum ...... 39 6.1.3 Conservation of energy ...... 41 6.2 Approximation ...... 42 6.3 History ...... 42 6.4 Fields of application ...... 43

7 Model implementation 44 7.1 Geometry ...... 44 7.1.1 Maxus ...... 44 7.1.2 Terrier-Black Brant ...... 45 7.2 Fluid domain ...... 47 7.3 Mesh ...... 47 7.3.1 Yplus ...... 49 7.3.2 Orthogonal quality ...... 49 7.3.3 Aspect ratio ...... 49 7.3.4 Skewness ...... 49 7.4 Setup ...... 50 7.4.1 Flow analysis ...... 50 7.4.2 Domain ...... 50 7.4.3 Boundary conditions ...... 50 7.4.4 Solver control ...... 52 7.4.5 Output control ...... 53

8 Results 54 8.1 Maxus ...... 55 8.1.1 Zero-lift drag coefficient ...... 55 8.1.2 Stability derivative ...... 57 8.1.3 Center of pressure ...... 59 8.2 Terrier-Black Brant (first stage) ...... 60 8.2.1 Zero-lift drag coefficient ...... 60 8.2.2 Stability derivative ...... 61 8.2.3 Center of pressure ...... 62 8.3 Terrier-Black Brant (second stage) ...... 63 8.3.1 Zero-lift drag coefficient ...... 63 8.3.2 Stability derivative ...... 64

8 8.3.3 Center of pressure ...... 65

9 Discussion 66 9.1 Reliability of CFD-simulations and verifying the results from such simulations ...... 66 9.2 Maxus ...... 67 9.2.1 Zero-lift drag coefficient ...... 67 9.2.2 Stability derivative ...... 67 9.2.3 Center of pressure ...... 68 9.3 Terrier-Black Brant (first stage) ...... 69 9.3.1 Zero-lift drag coefficient ...... 69 9.3.2 Stability derivative ...... 69 9.3.3 Center of pressure ...... 70 9.4 Terrier-Black Brant (second stage) ...... 70 9.4.1 Zero-lift drag coefficient ...... 70 9.4.2 Stability derivative ...... 71 9.4.3 Center of pressure ...... 71

10 Conclusions 72 10.1 Maxus ...... 72 10.2 Terrier-Black Brant (first stage) ...... 73 10.3 Terrier-Black Brant (second stage) ...... 74

Bibliography 75

Appendix 77

9 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 at RUAG Space in Link¨oping.The current aerodynamics models are based on analytic calculations and numerical methods. To determine whether these models are sufficiently accurate, a comparative study between data obtained from the current models and corresponding results from CFD-simulations is performed. Three aerodynamic parameters related to rocket stability are studied: the zero-lift drag coefficient CD0 , the derivative of the lift coefficient with respect to the angle of attack CNα , and the center of pressure Xcp. The results are compared in terms of behavior and parametric values. Any deviations are analyzed and compared with earlier studies to decide which result is more plausible and physically accurate. Two configurations of sounding rockets are evaluated: Maxus and Terrier- Black Brant (first and second stage). All parameters are evaluated as a function of Mach number for complete configurations.

1.1 Background

RUAG Space is the space division of the Swiss technology group RUAG, and the leading supplier of products for the space industry in Europe. RUAG Space operates in Switzerland, Sweden, Finland, Austria, Germany and the USA, with some 1 350 employees.[1] RUAG Space in Link¨opingis a Product Unit focusing on mechanical

10 systems. Part of their operation includes developing, manufacturing and refurbishing guidance systems for sounding rockets. In order to achieve de- sired performance, reliable aerodynamic models of the sounding rockets are required. The aerodynamic models RUAG Space in Link¨opinguses are based on analytic calculations and numerical methods, and they want to investigate how 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 general and gives an introduction to the configurations evaluated in this thesis.

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

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

Chapter 5, Parameters to be evaluated describes the aerodynamic parameters evaluated in this thesis and what role they play in rocket analysis.

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

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

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

Chapter 9, Discussion summarizes and compares the results.

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

11 Chapter 2

Sounding rockets

Sounding rockets take their name from the nautical term ”to sound”, which means to take measurements. They carry scientific instruments into space, typically to an altitude between 200 and 300 kilometers, and in some cases up to 1 000 kilometers. Sounding rockets provide researchers a low-cost and time efficient method for conducting experiments in areas inaccessible to either weather balloons or satellites. Common research applications include research in aeronomy, X-ray astronomy and microgravity. Sounding rockets consist of either a single-stage or multistage solid-fuel rocket motor. As the rocket has consumed all its fuel, it separates from the payload and falls back to Earth. The payload continues into space in a parabolic trajectory and begins conducting the experiments, during which time data is returned to Earth by telemetry links. The overall time in space is typically 5-20 minutes. As the payload re-enters the atmosphere, a parachute is deployed to gently bring it back to Earth. Figure 2.1 shows the different steps of a typical sounding rocket flight sequence.[2]

12 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 European Space Agency and used in the MAXUS microgravity programme. It is a joint venture between the Swedish Space Corporation and Airbus Defence and Space. The first Maxus rocket was launched in May 1991 from Esrange Space Center in Kiruna, Sweden. Since then, another nine launches have been performed from Kiruna - the latest in April 2017. The Maxus sounding rocket is Europe’s largest single-stage sounding rocket with an overall length of 15.5 meters and a weight of approximately 12.5 tonnes, depending on the payload. It consists of an ogive nose, followed by 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]

13 Figure 2.2: Maxus 8 on the launcher.

2.2 Terrier-Black Brant

The Black Brant is a family of Canadian-designed sounding rockets built by aerospace firm Magellan Aerospace. The Black Brant originated from a demand by the Canadian government to develop a sounding rocket to characterize the ionosphere in order to improve military communications. The first test flight took place from the Churchill Rocket Research Range in Churchill, Canada, in September 1959. Since then, over 800 launches of different versions of the Black Brant sounding rocket have been performed.[4] Originally a single-stage rocket, today’s Black Brant sounding rockets can utilize a range of multistage boosters. The Terrier-Black Brant sounding rocket is a two-stage rocket with a Terrier booster. Sounding rockets often use excess military assets that have either become obsolete or have exceeded proscribed shelf life. This is the case with the Terrier booster, which comes from Navy Terrier missiles.[5] The Terrier-Black Brant consists of an ogive nose, followed by cylindrical

14 and conical shapes. Three sets of fins are attached to the Terrier-Black Brant sounding rocket: triangular fins at the midsection of the payload, swept-back fins at the tail section of the Black Brant rocket, and cruciform fins at the tail of the Terrier booster. Figure 2.3 shows a lunch of a Terrier-Black Brant sounding rocket at White Sands, New Mexico, USA, on April 3, 1996.[6]

Figure 2.3: Terrier-Black Brant launch.

15 Chapter 3

Fundamental concepts of fluid dynamics

Fluid dynamics is a subdiscipline of fluid mechanics that studies the flow of fluids and how forces affect them. Fluid dynamics has, in turn, several subdisciplines, including aerodynamics. This chapter describes a few fundamental concepts of fluid dynamics vital to the understanding of this thesis.

3.1 Viscosity

The viscosity of a fluid is a measure of its resistance to flow. It can be described as the ”thickness” of the fluid; for example, honey has a higher viscosity than water. Viscosity may be thought of as internal friction between the fluid molecules, opposing the relative motion between two surfaces of the fluid that are moving at different velocities. The viscosity of a fluid is defined as the ratio of the shearing stress to the velocity gradient,

F /A µ = . (3.1) ∆vx/∆z The viscosity defined in equation (3.1) is commonly called the dynamic viscosity. There is, however, another viscosity quantity. The kinematic vis- cosity, represented by ν, is the ratio of the dynamic viscosity of a fluid to its density,

16 µ ν = . (3.2) ρ Understanding the basic concept of viscosity and being able to distin- guish dynamic viscosity from kinematic viscosity will be of importance when discussing 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 laminar flow, turbulent flow is irregular and undergoes mixing. In turbulent flow, the velocity of the fluid at a point is continuously undergoing changes in both magnitude and direction. A flow that alternates between being laminar and turbulent 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

17 flow, whilst a high Reynold’s number indicates turbulent flow. The Reynold’s number expresses the ratio of inertial forces to viscous forces, Inertial forces R = . (3.3) e Viscous forces Expanding equation (3.3) yields

Inertial forces mass · acceleration Re = = Viscous forces dynamic viscosity · velocity · area distance (3.4) ρL3 · u
ρL3 1
ρL2 1
ρ L
L ρuL uL = t = t = t = t = = , u
2 1
2 µ µ µ ν µ L L µ L L where ρ is the density of the fluid, u is the velocity of the fluid relative to the 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 viscosity and the Reynold’s number, and thus the flow regime. More importantly, it shows that after a certain length L of flow, a laminar boundary layer will become unstable and turbulent, thus drastically changing the aerodynamic characteristics 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 forces are significant (see Section 4.2). The velocity of the flow within the boundary layer varies from zero at the wall to the free stream velocity at the height of the boundary layer. The thickness of the boundary layer is conventionally denoted δ(x) and describes the distance away from the wall at which the velocity component parallel 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 plate in a fluid with free stream velocity u∞.[9]

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

Boundary layers may be either laminar or turbulent, depending on the Reynold’s number. As the distance x along the wall increases, the Reynold’s number 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’s number, 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 in response to a pressure or temperature change. All fluids are compressible to some extent. However, in many situations the changes in pressure and temperature 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 that changes in density can be negligible, the fluid is said to be compressible. When analyzing rockets, spacecraft, and other systems that involve high speed gas flows, the speed is often expressed in terms of the dimensionless Mach number, V Speed of flow M = = . (3.5) c Speed of sound

19 A flow is called sonic when M=1, subsonic when M<1, and supersonic when M>1. In aerodynamics, flows are usually treated as compressible if the density 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 the surface boundary of an object immersed in the fluid. The physical justifica- tion for this assumption originates from adhesion forces between a fluid and a solid surface exceeding cohesion forces within the fluid, causing the fluid layer along the solid boundary to attain the velocity of the boundary.[11] Conceptually, one can think of the outermost fluid molecules as stuck to the surfaces past which it flows. The viscous forces around a body emanate from the no-slip condition.

3.6 Newtonian vs non-Newtonian fluids

A Newtonian fluid is a fluid for which the rate of deformation is linearly proportional 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-Newtonian fluids.

20 Chapter 4

Aerodynamic forces

Aerodynamic forces affect a body immersed in a fluid medium. These forces are due to the relative motion between the body and the fluid. When two solid objects interact in a mechanical process, forces are transmitted at the point 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 forces are due to pressure distributions over the surface of the body and shear stresses arising from fluid viscosity.

4.1 Pressure

Pressure is defined as a normal force exerted by a fluid per unit area,[10] F P = . (4.1) A We speak of pressure only when we deal with a fluid. The counterpart of pressure in solids is normal stress. Since pressure is defined as force per unit area, the unit of pressure is newtons per square meter, which is called a pascal;

1 P a = 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 study the molecular definition of pressure.

21 4.1.1 Molecular definition of pressure The small scale action of individual air molecules is commonly explained by the kinetic theory of gases (also known as the kinetic-molecular theory). It explains the behavior of a hypothetical ideal gas (i.e. a gas composed of many randomly moving point particles whose only interactions are perfectly elastic collisions) within a container. According to this theory, gases are made up of molecules in random, straight line motion. The individual molecules possess the 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 molecules impart momentum to the walls, producing a force that can be measured. Mathematically, the expression for pressure is derived using Newton’s laws of motion. Newton’s second law of motion states that the rate of change of momentum is directly proportional to the force applied, dp d F = = m (v). (4.3) dt dt An impulse J occurs when a force F acts over an interval of time ∆t, according to Z J = Fdt. (4.4) ∆t 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 a container wall. During the collision, an impulse is imparted by the wall to the molecule that is equal and opposite to the impulse imparted by the molecule to the wall, according to Newton’s third law of motion. The pressure is the sum of the impulses imparted by all molecules to the wall. Consider a gas of N molecules, each of mass m, enclosed in a cube of volume V = L3. In order to arrive at an expression for the pressure, a calculation will be made of the impulse imparted to the wall perpendicular to the x-axis by a single impact. Although the molecules are moving in all directions, only those with a velocity component toward the wall can collide with 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 different values of vx must be summed. A molecule approaches the wall with an initial momentum mvx. After the impact, the molecule bounces off the wall with an equal momentum in the 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 molecules impacts one specific side wall every 2L ∆t = , (4.7) vx where L is the distance between opposite walls. The force due to this molecule is

∆p 2mv mv2 F = = x = x . (4.8) ∆t 2L/vx L The total force on the wall is then

Nmv2 F = x , (4.9) L where the bar denotes an average over N molecules. Since the molecules are in random motion and there is no bias applied in any direction, this result is independent of the choice of axis. By Pythagorean theorem in three dimensions, the total squared speed v is given by

2 2 2 2 v = vx + vy + vz . (4.10) The gas is in equilibrium, meaning the average squared speed in each direc- tion is identical,

2 2 2 vx = vy = vz . (4.11) Combining equations (4.10) and (4.11) gives

2 2 v = 3vx, (4.12)

23 and thus

v2 v2 = . (4.13) x 3 Inserting equation (4.13) into (4.9) yields the force as

Nmv2 F = . (4.14) 3L Combining equations (4.14) and (4.1) gives the expression for the pressure of the gas as

F Nmv2 1 P = = ⇔ PV = Nmv2. (4.15) L2 3V 3 This expression can be expanded further in terms of temperature and the kinetic energy of the gas. Combining equation (4.15) with the ideal gas law,

PV = NkBT, (4.16) gives

mv2 k T = , (4.17) B 3 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 3 mv2 = k T. (4.18) 2 2 B Solving for the temperature T yields

mv2 T = . (4.19) 3kB The kinetic energy of the gas is N times that of a molecule, 1 K = Nmv2. (4.20) 2 Combining equations (4.20) into (4.19) gives 2 K T = . (4.21) 3 NkB

24 Equation (4.21) is an important result from the kinetic theory of gases, as it concludes that the average molecular kinetic energy is proportional to the ideal 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 2 2 K PV = K ⇔ P = . (4.22) 3 3 V This final expression relates pressure to volume (density) and to kinetic en- ergy (velocity/temperature). The kinetic theory of gases can be expanded further to calculate the number of molecular collisions with a wall of the container 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 this thesis, 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 and how they affect the impulses due to molecular collisions between bodies. It also provides a valid physical explanation in terms of the laws of Newtonian mechanics. The pressure distributions along the surface of a body arise due to the molecular collisions described in this section - whether it be inside an enclosed container, or due to a rocket moving through the atmosphere. In the later case, the normal direction changes from the nose of the rocket to the rear. To obtain the net force arising from the pressure distributions, the contributions from all small sections along the body must be summed, X F = P · ndA, (4.23) where n is the direction of the force normal to the surface, and dA the incremental area. For a fluid in motion, the velocity has different values at different locations around the body. The local pressure is related to the local velocity, so the pressure also varies around the closed surface. The net force is calculated by summing the pressure perpendicular to the surface times the area around the body, I F = P · ndA. (4.24)

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

4.2 Viscous shearing stresses

When two solid bodies in contact move relative each other, a friction force develops at the contact surface in the direction opposite to motion. The magnitude 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 move relative each other. The magnitude of the force opposite to motion depends on the viscosity of the fluid. The phenomena is most easily described by the idealized situation known as Couette flow, commonly taught to increase the understanding of how viscous shearing stresses arise, and the shearing forces they 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 form of 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 continuously with a velocity U. The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same speed. The shear stress τ acting on this fluid layer is F τ = , (4.25) A where A is the contact area between the plate and the fluid and F is the shear force. Just as for pressure, shear stress is defined as force per unit area. The fluid in contact with the lower plate assumes the velocity of that plate, which is zero (according to the no-slip condition, see Section 3.5). For steady laminar flow, the fluid velocity between the plates varies linearly between zero at the lower plate and U and the upper plate. Figure 4.1 shows the 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-Stokes equations for steady incompressible flow. The continuity equation for such a flow is ∂u ∂v ∂w + + = 0. (4.26) ∂x ∂y ∂z Choosing x to be the direction along which all fluid particles travel (i.e. u 6= 0, v = w = 0) yields

∂u ∂v¡ ∂w + ¡ +  = 0, (4.27) ∂x ¡∂y ∂z and thus ∂u = 0. (4.28) ∂x Equation (4.28) means u = u(y, z). Assuming the plates are infinitely large in the z-direction (so the z-dependence can be disregarded) gives u = u(y). The momentum equation in the x-direction for steady incompressible flow is

∂u ∂u ∂u ∂u ∂P ∂2u ∂2u ∂2u ρ + u + v + w = − + ρg + µ + + . ∂t ∂x ∂y ∂z ∂x x ∂x2 ∂y2 ∂z2 (4.29) Assuming gravitational forces can be neglected yields

27 ∂u¡ ∂u¡ ∂u¡ ∂u¡  ∂P ∂2u ∂2u ∂2u ¡ ¡ ¡ ¨   ρ + u + v ¡ + w = − + ¨ρgx + µ 2 + 2 + 2 , ¡∂t ∂x¡ ¡∂y ¡∂z ∂x ∂x ∂y ∂z (4.30) and thus

∂P ∂2u = µ . (4.31) ∂x ∂y2 Integrating equation (4.31) and solving for u gives 1 ∂P u(y) = · y2 + C y + C . (4.32) 2µ ∂x 1 2

Invoking the boundary condition (at y=0, u=0) yields C2 = 0. Invoking the boundary condition (at y=h, u=U) yields U 1 ∂P C = − · h. (4.33) 1 h 2µ ∂x

Inserting C1 and C2 into equation (4.32) gives y h2 ∂P y  y  u(y) = U − · · 1 − . (4.34) h 2µ ∂x h h For simple Couette flow, there is no pressure gradient in the direction of the flow, reducing equation (4.34) to y u(y) = U. (4.35) h Equation (4.35) is the velocity profile of the flow. Consequently, the velocity gradient is du U = . (4.36) dy h During a differential time interval dt, the sides of fluid particles along a vertical line MN rotate through a differential angle dβ while the upper plate moves 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 da U dt du dβ ≈ tan(dβ) = = = dt. (4.37) h h dy Rearranging equation (4.37), the rate of deformation becomes dβ du = . (4.38) dt dy 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 velocity gradient) is directly proportional to the shear stress τ,[10] dβ du τ ∝ or τ ∝ . (4.39) dt dy Including the proportionality constant between shear stress and the velocity gradient yields the final expression for the shear stress, du τ = µ , (4.40) dy 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 fluid layer (or, by Newton’s third law, the force acting on the plate) as

29 du F = τA = µA . (4.41) dy This concludes the section about viscous shearing stresses. Emanating from the Navier-Stokes equations, it has been mathematically shown how the rate of deformation is equivalent to the velocity gradient for a simple Couette configuration. Furthermore, the relationship between viscous shear- ing stresses and the rate of deformation has been discussed, finally arriving at 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 for such flows provides a basic understanding of forces arising from shear-driven fluid motion. Couette flow is frequently used in undergraduate physics and engineering courses, and the explanation provided in this section suffices at providing the reader basic concepts of viscous shearing forces important for the 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 with respect to angle of attack, and the center of pressure. Data from the Maxus and the Terrier-Black Brant configurations have been compared with results from 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 complete configurations. This chapter describes the aerodynamic parameters evaluated and what role 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 coefficients are determined by a ratio of forces rather than just a simple force, thus making them an important engineering tool for comparing the efficiency for different 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 coefficient and the derivative of the lift coefficient with respect to angle of attack are evaluated.

31 5.1.1 Drag coefficient The drag coefficient is used to model the drag (resistance) of a body immersed in a fluid medium. It takes into account the complex dependencies of shape, inclination and flow conditions on a body. A low drag coefficient indicates that an object will have less aerodynamic or hydrodynamic drag. The drag coefficient is defined as F C = D , (5.1) D ρAv2/2 where FD is the drag force, ρ the density of the fluid, A a reference area (in rocket analysis commonly the cross-sectional area of the midsection), and v the flow speed. The drag force acts opposite to the relative motion of any body moving with respect to the surrounding fluid. Introducing the term for dynamic pressure,

ρv2 q = , (5.2) 2 into equation (5.1) gives the drag coefficient as the ratio of the drag force to the force produced by the dynamic pressure times the area, F C = D . (5.3) D qA The aerodynamic forces on a body immersed in a fluid come primarily from differences in pressure and viscous shearing stresses (see Chapter 4). The drag force on a body can be divided into two main components; pressure drag and viscous drag. The net drag force can thus be decomposed as follows:

Z Z FD 1  ˆ 1  ˆ CD = = cp + cf = dA(p − p0) ˆn · i + dA ˆt · i τ, (5.4) qA qA S qA S | {z } | {z } cp cf where cp is the pressure drag coefficient, cf the friction drag coefficient, p the pressure at the surface dA, p0 the free-stream pressure, ˆn the normal direction to the surface with area dA, ˆi the unit vector in direction normal to the surface dA, ˆt the tangential direction to the surface with area dA, and τ the shear stress acting on the surface dA.

32 The zero-lift drag coefficient, CD0 , is the drag coefficient when the effective angle 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 apogee of flight. This thesis evaluates the zero-life drag coefficient. As mentioned above, the main components of the drag force are pressure drag and viscous drag. Expanding this discussion further, the different types of drag are generally divided into parasitic drag, wave drag and lift induced drag. Parasitic drag is, in turn, made up of skin friction drag, form drag and interference drag. The following sections describe the different types of drag and how they arise.

Skin friction drag Skin friction drag is drag resulting from viscous shearing stresses acting over the surface of a body (see Section 4.2). Skin friction drag is caused by friction between the fluid molecules and the surface of a body. As the fluid molecules flow over the surface and past each other, viscous resistance slow additional fluid molecules, generating a force which retards forward motion and causes the boundary layer to grow in thickness. At some point, the boundary layer flow transits from laminar to turbulent. Turbulent flow creates more friction drag than laminar flow due to its greater interaction with the surface of the body. A rough surface accelerates transition between laminar and turbulent flow, causing the thickness of the boundary layer and the fluid flow disruption within the boundary layer to increase. Skin friction drag is thus primarily dependent upon the smoothness of the surface.

Form drag Form drag (or pressure drag) is the drag resulting from the static pressure acting normal to the surface of a body. It is caused by the separation of the boundary layer from a surface and the wake created by that separation. As a fluid flows around a body, a stagnation point with a high static pressure is generated in front of the body. Analogously, the wake behind the body generates a low pressure area. The pressure differential causes the body to be pushed in the direction of the relative fluid flow, resulting in a force which retards forward motion. For a streamlined body, the streamlines follow the shape 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 primarily dependent upon the shape of the body.

Interference drag Interference drag is the drag resulting from when fluid flow around one part of a body is forced to occupy the same space as the fluid flow around another part 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 to accelerate in order to pass through the restricted physical space. This results in turbulent mixing of the two flows and thus increased skin friction drag. The effects of interference drag become particularly substantial during transonic and supersonic flows, as the increased speeds at the intersection points will produce local shock waves and create wave drag. To counter the effects of interference drag, the pressure distributions on the intersecting bodies should ideally complement each others pressure distribution. In reality, however, this is not always possible.

Wave drag Wave drag arise during transonic and supersonic flight due to the formation of shock waves. As a rocket passes through the air, it creates a series of pressure waves in front of it and behind it. These pressure waves travel at the speed of sound. As the speed of the rocket increases, the pressure waves are compressed. Eventually, the pressure waves merge into a single shock wave which travels at the speed of sound. This occurs at Mach 1, equivalent to approximately 343 m/s, and generates a sonic boom. The sudden increase in drag at Mach 1 is due to the boundary layer being strongly separated, thus causing a sharp increase in skin friction drag, and also the static pressure point in front of the rocket sharply increasing, thus also increasing the form drag.

5.1.2 Lift coefficient The lift coefficient is, in conformity with the drag coefficient, defined as F F C = N = N , (5.5) N ρAv2/2 qA

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 coefficient expresses the ratio of the lift force to the force produced by the dynamic pressure times area. Lift force is the force perpendicular to the oncoming flow direction. Lift is generated when a moving flow of fluid is turned by a solid object. The theory of flight is often explained by Bernoulli’s principle or Newton’s laws of motion. The following sections describe the theory of flight according to both 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 isolated system remains constant. In fluid dynamics, Bernoulli’s principle states that an 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 an airfoil must travel further and hence faster than the air travelling the shorter distance below the airfoil, but the energy must remain constant at all times. Consequently, the air above the wing will consist of a lower pressure region than 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 mass times its acceleration or, equivalently, to its rate of change of momentum, d F = ma = m (v). (5.6) dt In other words, when there is a change in momentum there must be a force causing it. Furthermore, Newton’s third law states that to every action there is an equal and opposite reaction. That means that when the force of an airfoil pushes the air downwards - creating the downwash - there is an an equal and opposite force from the air pushing the airfoil upwards, thus generating lift. Figure 5.2 illustrates the theory of flight according to Newton’s laws of motion.[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 lift to 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 of how particular forces and moments on a body change as other parameters related 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 acting upon 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. This variation of air velocity produces a variation in the local pressure at various places on the rocket. In the same way that the weight of all components of a body acts through the center of gravity, the total aerodynamic force can be considered to act through the center of pressure. In rocket analysis, the center of pressure is important for predicting the flight stability of a rocket. For positive stability in rockets, the center of pressure must be further away from the nose than the center of gravity, see Figure 5.3.[13] This ensures that any increased forces resulting from increased angle of attack results in increased restoring moment to drive the rocket back to the trimmed position. In rocket analysis, a positive static margin implies that the complete rocket makes a restoring moment for any angle of attack from the trim position.

Figure 5.3: Rocket stability condition.

Mathematically, the center of pressure is determined by characterizing the pressure variation around the surface of a body as P(x), which indicates that the pressure depends on the distance x from a reference line (usually taken as the rear or leading edge of the object),

R x · p(x)dx X = . (5.7) cp R p(x)dx

37 Chapter 6

Computational fluid dynamics

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to solve and analyze problems that involve fluid flows. This chapter introduces computational fluid dynamics; mainly governing equations, history, and fields of application.

6.1 Governing equations

Fluid flows are governed by partial differential equations which represent conservation laws for the mass, momentum, and energy. These equations are called the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes. The Navier-Stokes equations describe how the velocity, pressure, temper- ature, viscosity, and density of a moving fluid are related. A mathematical model of the physical case and a numerical method are used in a software tool to analyze the fluid flow. The complexity of these mathematical models vary depending on what kind of flow is being analyzed (such as compressible vs 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 are expressed based on the principles of conservation of mass, momentum and energy. The equations can be expressed in Cartesian coordinates, cylindri- cal coordinates and spherical coordinates and are adjustable regarding what kind 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, the derivation 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 neither created nor destroyed. The conservation of mass (also called the continuity equation) states that the mass flow difference throughout a systems’ inlet- and outlet-section is zero, D ρ + ρ ∇ · V
= 0, (6.1) Dt where ρ is the fluid density, V is the fluid velocity and ∇ is the gradient operator; ∂ ∂ ∂ ∇ = i + j + k . (6.2) ∂x ∂y ∂z For incompressible flow (see Section 3.4), the continuity is simplified to D ∂u ∂v ∂w ρ = 0 → ∇ · V = + + = 0. (6.3) Dt ∂x ∂y ∂z

6.1.2 Conservation of momentum The momentum in a control volume is constant. The description is set up in accordance 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 the acceleration. If the particle is a fluid, it is convenient to divide the equation to volume of the particle to generate a derivation in terms of density: DV ρ = f = fbody + fsurface. (6.5) Dt 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. The external forces through the surface of fluid particles, fsurface, is expressed by pressure and viscous forces;

∂τij fsurface = ∇ · τij = = fpressure + fviscous, (6.7) ∂xi where τij is the stress tensor,   ∂ui ∂uj τij = −P δij + µ + + δijλ∇ · V. (6.8) ∂xj xi Thus, Newton’s second equation of motion can be expressed as DV ρ = ρ · g + ∇ · τij. (6.9) Dt Inserting equation (6.8) into (6.9) yields the the conservation of momentum of a Newtonian viscous fluid in one equation:

    DV ∂ ∂ui ∂uj ρ = ρ · g − ∇P + µ + + δij λ∇ · V , (6.10) Dt |{z} |{z} ∂xi ∂xj ∂xi | {z } II III | {z } I IV I: Momentum convection, II: Mass force, III: Surface force, IV: Viscous force.

D/Dt is the material derivative, D() ∂() ∂() ∂() ∂() = + u + v + w = V · ∇(). (6.11) Dt ∂t ∂x ∂y ∂z For incompressible flow, the equations are greatly simplified in which the viscosity coefficient µ is assumed constant and ∇ · V = 0. Thus, the conser- vation of momentum equation for an incompressible three-dimensional flow can be expressed as

40 DV ρ = ρg − ∇P + µ∇2V. (6.12) Dt Expanding equation (6.12) for each dimension when the velocity is V (u, v, w) yields:

∂u ∂u ∂u ∂u ∂P ∂2u ∂2u ∂2u ρ + u + v + w = ρg − +µ + + , (6.13) ∂t ∂x ∂y ∂z x ∂x ∂x2 ∂y2 ∂z2

∂v ∂v ∂v ∂v  ∂P ∂2v ∂2v ∂2v  ρ + u + v + w = ρg − +µ + + , (6.14) ∂t ∂x ∂y ∂z y ∂y ∂x2 ∂y2 ∂z2

∂w ∂w ∂w ∂w ∂P ∂2w ∂2w ∂2w ρ + u + v + w = ρg − + µ + + , ∂t ∂x ∂y ∂z z ∂y ∂x2 ∂y2 ∂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 thermal interaction 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. It states that the sum of the work and heat added to the system will result in the 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 heat added to the system, and dW is the work done on the system. One common type of the energy equation is:   ∂h ∂P   ρ  + ∇ · (hV ) = − + ∇ · (k∇T ) + φ , (6.17) ∂t | {z } ∂t | {z } |{z} |{z} II | {z } IV V I III 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 is used to discretize the differential equations by a system of algebraic equa- tions, which can then be solved using a computer. The approximations are applied to small domains in space and/or time so the numerical solution provides results at discrete locations in space and time. The accuracy of a numerical solution is highly dependent on the quality of the discretizations used. The differential equations based on the Navier-Stokes equations could be solved for a given problem by using methods from calculus. However, in practice, these equations are too difficult and too time-consuming to solve analytically. In the past, engineers made rough approximations and simpli- fications to the equation set until they had a group of equations that they could solve. But since the introduction of high-speed computers - thus greatly improving numerical accuracy and lowering computational time - CFD has become 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 numerical methods.

• 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 on the Navier-Stokes equations. First implementation for 2D, transient, incompressible flow.

42 • 1960-1970: First scientific paper “Calculation of potential flow about arbitrary bodies” was published about computational analysis of 3D bodies 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, surface ships, automobiles, helicopters and aircrafts.

• 1980-1990: Improvement of accurate solutions of transonic flows in three-dimensional case. Commercial codes have started to implement through 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 dynamics of blood flow (hermodynamics), CFD is a great engineering tool with vast capabilities. 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, meshing and setup - is performed in Ansys 18.2.

7.1 Geometry

The first step when creating a CFD-model is to create the geometries you wish to simulate. In this case that meant creating CAD-models of the Maxus and Terrier-Black configurations, which was done using the computer- aided design tool Ansys 18.2 DesignModeler. The CAD-models are based on sketches of the rockets. All geometries were created in full-scale, in three dimensions, 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 is based on is attached to the appendix. The sketch is an extract from a report submitted by Per Elvemar[15] at Saab. The sketch is complemented with detailed dimension of the nose cone from a report by T. Andersson[16] at Swedish 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-Black Brant configuration, the sketches used for said configuration are not included in 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 dimensions for the fins. Parts of the geometry for the Terrier-Black configuration has thus been approximated. Figures 7.2 - 7.3 show the CAD-models of the Terrier-Black Brant, first and 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 order to achieve meshing flexibility, the fluid domain was divided into different sections with a high cell density approximate the rockets, and a gradually lower cell density further away from the rockets. To assert the boundary conditions were far enough from the geometries, a sensitivity analysis was performed 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 of a numerical simulation. In order to attain reliable results, the mesh must be 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 cells should ideally be parallel to the flow to avoid numerical inaccuracies. This is achieved by implementing a hexahedral dominant mesh. However, the geometries of the rockets require a more flexible meshing method to avoid highly skewed cells. Consequently, the fluid domain has been meshed using a 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 different sections with a high cell density approximate the rockets and a gradually lower 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 fine mesh representation of the boundary layer (see Section 3.3), often resulting in complex meshing. This becomes particularly substantial when performing transonic and supersonic simulations, as the required wall distance of the first cell typically needs to be very small. In order to determine if the boundary layer 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 the location of the first node away from the wall. The distance will have a significant impact on the model’s ability to represent the boundary layer. The desired y+ varies with turbulence model. For the turbulence model used for 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 quality parameters are typically divided into orthogonal quality, aspect ratio and skewness.

7.3.2 Orthogonal quality Mesh orthogonality relates to how close the angles between adjacent element faces 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 summarized into an orthogonal quality factor between 0 and 1, with good cells having an orthogonal quality closer to 1.

7.3.3 Aspect ratio The aspect ratio is a measure of the stretching of a cell. It is calculated as the ratio of the maximum value to the minimum vale of the distance between the cell centroid and nodes. For stable simulations, the aspect ratio should ideally be around 20.

7.3.4 Skewness Skewness is defined as the difference between the shape of the cell and the shape of an equilateral cell of equivalent volume. Highly skewed cells can decrease accuracy and destabilize the solution. A general rule is that the maximum skewness for a triangular/tetrahedral mesh in most flows should be 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 are set as default.

7.4.1 Flow analysis Test runs of both transient and steady-state analysis types were performed to determine which type was most suitable. Steady-state rendered satisfactory residual and force profiles and was therefore chosen as an appropriate analysis type.

7.4.2 Domain Under Basic settings, the material was set to air ideal gas for speeds above Mach 0.3 in order to account for compressibility effects. Dynamic viscosity was set to 1.789 · 10−5 N·/m2 and reference temperature to 15◦C. For speeds less than Mach 0.3, the material was set to air at 15◦C with constant density 1.225 kg/m3 and dynamic viscosity 1.789 · 10−5 N·/m2. Under Fluid models, the heat transfer option was set to Total energy. The turbulence model was set to Shear stress transport (SST). Under Initialization, Cartesian velocity components were set to zero in all directions except in the flow direction. The velocity component in the flow direction was set to whatever speed was being simulated. Static pressure was set 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 zero velocity relative to the surface boundary. Since the rocket is fixed in its frame 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 the flow. The flow regime option is set to supersonic for speeds equal to or greater 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 to whatever speed is being simulated.

Outlet The outlet is defined at the opposite face to the inlet. The outlet is positioned far enough behind the rocket to prevent it from intersecting the recirculation area behind the rocket. Just as for the inlet, the flow regime option is set to supersonic for speeds equal to or greater than Mach 1. For speeds less than Mach 1, the subsonic option is selected.

Opening The cylinder wall parallel to the free stream flow is set as an opening. The opening option is preferred over the outlet option since the free stream flow will be along the cylinder wall, without much outflow expected. Turbulence option is set to zero gradient, i.e. the velocity gradient perpendicular to the boundary is zero. In order for this to be an appropriate option, the cylinder enclosure 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 and oxidizer are ignited in a combustion chamber, producing great amounts of exhaust gas at high temperature and pressure. The hot exhaust is passed through a nozzle which accelerates the flow, pushing the gases out of the nozzle 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 mass flow 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 the nozzle. The generalized thrust equation,

F =mV ˙ e + (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 the free 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 the exhaust is know, equation (7.1) can be rearranged to express the unknown exit pressure as

F − mV˙ e pe = + p0. (7.2) Ae The actual thrust of a rocket will, however, vary from launch to launch and also during the different flight sequences. The simulated rockets are fixed in its frame of reference, meaning there is no propulsion. Instead, the CFD-model is created as a virtual wind tunnel. In order to include entrained flow and drag effects generated by the thrust from the rocket engine, the exhaust is set as an inlet boundary with the flow conditions of the exhaust specified. Mean values from data provided from each configuration has been used to determine the exit velocity of the exhaust and to calculate the exit pressure.

7.4.4 Solver control For advection scheme and turbulence numerics, upwind and first order were chosen, respectively. To determine an appropriate time scale, several test simulations were performed. Given the high turbulence gradients generated at supersonic speeds, a small time scale was necessary to prevent the numer- ical solution from diverging. By trial and error it could be determined that a 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 the expression for each parameter and derived using function calculators.

53 Chapter 8

Results

This chapter presents the results from the CFD-simulations along with the pre-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 are presented 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 the CFD-simulations.

54 8.1 Maxus

8.1.1 Zero-lift drag coefficient

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

55 Zero-lift drag coefficient

0.8

0.7

0.6

0.5 0

Cd 0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 Mach number

Figure 8.2: CFD-simulations for CD0 , Maxus.

56 8.1.2 Stability derivative

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

57 Derivative of the lift coefficient w.r.t. angle of attack 12

10

8

N 6 C

4

2

0 0 2 4 6 8 10 12 Mach number

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 Center of pressure 8

7

6

cp 5 X

4

3

2 0 2 4 6 8 10 12 Mach number

Figure 8.6: CFD-simulations for Xcp, Maxus.

8.2 Terrier-Black Brant (first stage)

8.2.1 Zero-lift drag coefficient

Zero-lift drag coefficient, first stage 2.5 Pre-existing data CFD-simulations 2

1.5 0 Cd 1

0.5

0 0 0.5 1 1.5 2 2.5 3 Mach number

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

60 8.2.2 Stability derivative

Derivative of the lift coefficient w.r.t. angle of attack 30 Pre-existing data CFD-simulations 25

20

N 15 C

10

5

0 0 0.5 1 1.5 2 2.5 3 Mach number

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

61 8.2.3 Center of pressure

Center of pressure, first stage 7 Pre-existing data CFD-simulations 6

5

4 cp X 3

2

1

0 0 0.5 1 1.5 2 2.5 3 Mach number

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

Zero-lift drag coefficient, second stage 1.5 Pre-existing data CFD-simulations

1 0 Cd

0.5

0 0 1 2 3 4 5 6 7 8 9 Mach number

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

63 8.3.2 Stability derivative

Derivative of the lift coefficient w.r.t. angle of attack 30 Pre-existing data CFD-simulations 25

20

N 15 C

10

5

0 0 0.5 1 1.5 2 2.5 3 Mach number

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

64 8.3.3 Center of pressure

Center of pressure, second stage 5 Pre-existing data CFD-simulations 4

3 cp X 2

1

0 0 0.5 1 1.5 2 2.5 3 Mach number

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

65 Chapter 9

Discussion

This chapter compares the results obtained from the CFD-simulations with the 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 importance to the accuracy and stability of a numerical simulation. Different aspects of the mesh must be emphasized in regard to the problem at hand to ensure the reliability of the simulations. Thus, numerous sensitivity analysis have been performed for several mesh quality parameters (as discussed in Section 7.3) to ensure the accuracy of the CFD-simulations in this thesis. Also, the simulation setup has been implemented with great care (as discussed in Section 7.4) to ensure the physical properties of the simulations correspond to 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 well as earlier studies. As a general discussion regarding the verification of CFD-simulations, it should also be mentioned that complementary measurements, such as wind

66 tunnel testing, can provide diagnostic information to help verify the results from 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 the results from the CFD-simulations, Figure 8.2, correspond well, both in terms of 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 the provided data. There are, however, a few disparities. First of all, the provided data seems to underestimate the drag divergence Mach number, i.e. the Mach number at which the aerodynamic drag begins to increase rapidly. The provided data seems to estimate the drag divergence Mach number to exactly Mach 1, whilst the CFD-simulations estimate this number 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 than Mach 1. As discussed in section 5.1.1, there will be regions around the rocket where the fluid flow is forced to accelerate. During transonic flight, the fluid flow around these regions will reach supersonic speed, even though the flow speed around the rest of the rocket is still subsonic. As a result, local shock waves 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 estimates a maximum value of 0.67. This would effectively mean the provided data overestimates the apogee for the Maxus configuration. Finally, as the Mach number increases, both the provided data and the

CFD-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 from the 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 after the transonic phase compared to the CFD-simulations. This would effec- tively mean the provided data underestimates the magnitude of the lateral acceleration 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. It is difficult to motivate why CNα would have an oscillating behavior. Earlier studies [18] indeed predicts CNα to have a smooth decrease, in conformity with the results obtained from the CFD-simulations. Thus, based on the results from the CFD-simulations and on earlier studies, it can be concluded that the oscillating behavior of CNα predicted by the provided data most likely is an incorrect prediction. Thirdly, comparing the continuous graph of the provided data to the CFD-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 of 10.57. This would effectively mean the provided data underestimates the maximum lateral acceleration during transonic flight. It should be noted that in addition to the continuous graph in the provided data, 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 added to the report after its initial filing. Comparing the original continuous graph and the second graph to the CFD-simulations show the maximum value of

CNα from the simulations is located between these two graphs. Perhaps Per Elvemar realized, in retrospect, that the initial estimation of the maximum value of CNα was too low, and corrected it afterwards. However, since there is no information where this second graph originates from, this thesis will focus on the original continuous graph, which, as mentioned above, underestimates the maximum value of CNα compared to the CFD-simulations. Finally, as the Mach number increases, both the provided data and the

CFD-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 results from the CFD-simulations, Figure 8.6, correspond well, both in terms of position and behavior. Both predict the center of pressure moving rearward during 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 waves become 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 predicts the center of pressure to be located slightly further away from the nose than compared to the CFD-simulations. This would effectively mean the provided data 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 data predicts the Maxus configuration to be less stable for higher Mach numbers. The variations are, however, less than 0.5 meters in both cases, thus only having 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 for all 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 data to the CFD-simulations show two clear deviations. First, the provided data predicts a plateau between Mach 1.2 and Mach 1.5. This is inconsistent with the CFD-simulations and with the results from the 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 earlier studies.

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 data and the CFD-simulations overlap, the provided data underestimates CNα for all other Mach numbers. This would effectively mean the provided data underestimates 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 the center of pressure moving rearward during transonic flight (as discussed in Section 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. The provided data predicts the location of the center of pressure to be, in most cases, around 2 meters further from the nose than the CFD-simulations. The provided data predicts the center of pressure to be located less than 2 meters from the nozzle exit plane throughout subsonic and transonic flight - an im- plausible location for a rocket that measures over 17 meters tall. Although the Terrier-Black Brant configuration utilizes two large sets of fins - one set at the midsection of the Black Brant rocket, and one set at the rear of the Terrier-booster - thus effectively generating a stable configuration, the pres- sure distributions along the cylindrical sections of varying dimensions at the payload, and the long rocket body, would inevitably generate a location for the center of pressure further away from the nozzle exit plane than what is predicted 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 around

Mach 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 time again 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 not too far from the CFD-simulations (although a slight underestimation), the provided data is inconsistent in terms of behavior and coefficient values for higher 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. The behavior is again near identical, although this time the provided data predicts a slight overestimation of the coefficient values for other Mach numbers than Mach 1.2. This would effectively mean that the provided data overestimates the 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 estimations for most Mach numbers. The provided data, however, fails to predict the rearward movement of the center of pressure during transonic flight, which was 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 gives evidence of the shortcomings of the provided data.

71 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 from the provided data to the CFD-simulations. The results are evaluated per configuration, ultimately determining how well 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 the CFD-simulations and with earlier studies in most aspects. The provided data is physically plausible, and most deviations are within the 10% margin of error assumed for the provided data.

72 In conclusion, the provided data for the Maxus configuration is assessed to 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 the most 75.4%),

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

Apart from the plateau of CD0 between Mach 1.2 and Mach 1.5, the provided 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 values sometimes are overlapping (or close to overlapping), there are several devia- tions that are too great to oversee. Mainly, the substantial underestimation of CNα during supersonic flight and the position of the center of pressure testify 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 or deemed satisfactorily accurate. The provided data is, however, sufficient to be used for continued analysis, provided the limitations of the provided data is 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 pressure during 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, for most parts, was identical to the CFD-simulations and to what is physically plausible, the provided data for the second stage exhibits a behaviour that, at times, contradicts both the CFD-simulations and earlier studies and that is physically implausible. Concretely, the sudden increase of CD0 and the failure to predict the rearward movement of the position of the center of pressure testify to the physical limitations presented by the provided data. However, apart from these apparent physical limitations, the provided data corresponds well with the CFD-simulations, both in terms of behavior and parameter 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 at high Mach numbers and position pf the center of pressure during transonic flight. 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 limitations of the provided data is taken into consideration.

74 Bibliography

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76 Appendix

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

77 Figure 10.1: Maxus configuration.

78 Figure 10.2: Maxus nose dimension.

79