Aerodynamics Modeling of Sounding Rockets: A Computational Fluid Dynamics Study

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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

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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 X_cp. The parameters are evaluated in terms of behavior and parametric values, ultimately determining how accurate the current aerodynamic models are compared to the CFD-simulations. Previous studies are used to determine 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 software. 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

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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 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 aerodynamic 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 C_D₀ 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.

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Sammanfattning

Varje storskalig raket best˚ar av fyra huvudkomponenter: raketkroppen, nyt- tolasten, styrsystemet och framdrivningssystemet. Styrsystemet best˚ar av sofistikerade sensorer, datorer och kommunikationsutrustning. Styrsystemet avser h˚alla raketen stabil och man¨ovrera raketen under flygningen.

När man utvecklar och tillverkar styrsystem krävs p˚alitliga modeller över raketernas aerodynamik. Den här rapporten utför en komparativ studie mellan CFD-simuleringar och dagens modeller för aerodynamikmodellering för tv˚a sondraketskonfigurationer vid RUAG Space i Linköping. Konfigura- tionerna som undersökts är Maxus och en tv˚astegs Terrier-Black Brant.

Tre parametrar relaterade till raketernas stabilitet har undersökts: mot- st˚andskoefficienten d˚a den effektiva anbl˚asningsvinkeln är noll C_D₀, stabilitets- derivatan CNα, och tryckcentrum Xcp. Parametrarna är utvärderade i form av beteende och parametervärden för att avgöra hur noggranna dagens modeller är jämfört med CFD-simuleringarna. Vid avvikelser jämförs data med tidigare studier för att avgöra vad som är mest fysikaliskt troligt. Alla parametrar är utvärderade som funktion av Machtal för fullständiga kon- figurationer.

CFD-simuleringarna är utförda i Ansys CFX 18.2. Ansys är ett amerikan- skt företag som utvecklar och säljer programvara inom flera ingenjörsomr˚aden.

Ansys är en av de största kommersiella programvarorna för CFD-simuleringar.

Resultaten i denna rapport visar att dagens metod för aerodynamikmodellering av Maxus överensstämmer väl med CFD-simuleringarna, b˚ade vad gäller beteende och parametervärde. Data fr˚an den nuvarande modellen är fysikaliskt rimliga och stämmer även väl överens med tidigare studier.

Resultaten för första steget av Terrier-Black Brant uppvisar, för det mesta, ett identiskt beteende med CFD-simuleringarna, men med varierande noggrannhet vad gäller parametervärden. Även om beteendet för den nuvarande modellen är fysikaliskt rimligt, förutsp˚ar den en position för tryck-

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centrum under den transsoniska fasen som ¨ar orimligt l˚angt bak.

Resultaten för andra steget av Terrier-Black Brant uppvisar ett, i vissa fall, inkonsekvent och fysikaliskt orimligt beteende. Avvikelserna är emeller- tid begränsade till vissa flygsekvenser, och inte till modellen som helhet.

Bortsett fr˚an de f˚a inkonsistenser den nuvarande modellen uppvisar stämmer den nuvarande modellen väl överens med CFD-simuleringarna vad gäller pa- rametervärden.

Den nuvarande modellen för Maxus bedöms vara noggrann och kan fortsä- tta användas oförändrat inom analyser. Det finns inga indikatorer p˚a att den nuvarande modellen p˚a n˚agot vis skulle vara bristfällig.

Den nuvarande modellen för första steget av Terrier-Black Brant förutsp˚ar ett korrekt fysikaliskt beteende, men med varierande precision vad gäller faktiska parametervärden. Den nuvarande modellen bedöms kunna fortsätta användas inom analyser, förutsatt att detta görs med bristerna med modellen i ˚atanke.

Den nuvarande modellen för andra steget av Terrier-Black Brant överens- stämmer väl med CFD-simuleringarna vad gäller parametervärden, men mis- sar att förutsp˚a stabiliseringen av CD0 för höga Machtal samt den bak˚atförfly- ttning av tryckcentrum som sker under den transsoniska fasen. Den nuvarande modellen bedöms vara tillräckligt noggrann för fortsatta analyser, förutsatt att detta görs med bristerna med modellen i ˚atanke.

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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¨oping who 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¨oping for 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 appreciate 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

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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 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 C_D₀, the derivative of the lift coefficient with respect to the angle of attack C_N_α, and the center of pressure X_cp. 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¨oping is a Product Unit focusing on mechanical

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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¨oping uses 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 concept 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 configuration.

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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]

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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 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]

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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 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

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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.

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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,

µ = F /A

∆v_x/∆z. (3.1)

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

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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 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 turbulent 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,

R_e = Inertial forces

Viscous forces. (3.3)

Expanding equation (3.3) yields

R_e= Inertial forces

Viscous forces = mass · acceleration

dynamic viscosity · _distance^velocity · area

= ρL³· ^u_t

µ _L^u L² = ρL³ ¹_t

µ _L¹ L² = ρL² ¹_t

µ = ρ ^L_t L

µ = ρuL

µ = uL ν ,

(3.4)

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]

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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 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 laminar. The boundary layer then becomes transitional at a critical Reynold’s number, and continues until the boundary layer is fully turbulent. This occurs at the transition Reynold’s number.

The aerodynamic forces generated within the boundary layer greatly affects 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,

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 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.

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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 immersed 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]

P = F

A. (4.1)

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 understanding aerodynamics. In order to understand pressure, one must study the 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 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,

F = dp

dt = md

dt(v). (4.3)

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

J = Z

∆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 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 v_x. Not all molecules have the same v_x. To

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find the total pressure, the contributions from molecules with all different values of v_x 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, −mv_x. Thus, the total change in momentum is

∆p = mv_x− (−mv_x) = 2mv_x. (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

∆t = 2L

v_x, (4.7)

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

F = ∆p

∆t = 2mv_x

2L/v_x = mv_x²

L . (4.8)

The total force on the wall is then

F = N mv_x²

L , (4.9)

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

v² = v²_x+ v²_y+ v_z². (4.10) The gas is in equilibrium, meaning the average squared speed in each direction is identical,

v²_x = v²_y = v²_z. (4.11) Combining equations (4.10) and (4.11) gives

v² = 3v_x², (4.12)

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and thus

v²_x = v²

3. (4.13)

Inserting equation (4.13) into (4.9) yields the force as F = N mv²

3L . (4.14)

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

P = F

L² = N mv²

3V ⇔ P V = 1

3N mv². (4.15) 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,

P V = N k_BT, (4.16)

gives

k_BT = mv²

3 , (4.17)

where k_B 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

2k_BT. (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

2N mv². (4.20)

Combining equations (4.20) into (4.19) gives T = 2

3 K

N k_B. (4.21)

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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 temperature, the greater the motion and kinetic energy. Finally, equations (4.15) and (4.21) yield

P V = 2

3K ⇔ P = 2 3

K

V . (4.22)

This final expression relates pressure to volume (density) and to kinetic energy (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 theoretical 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,

F =X

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,

F = I

P · ndA. (4.24)

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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 situation 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

τ = F

A, (4.25)

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.

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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

∂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 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

ρ ∂u

∂t + u∂u

∂x + v∂u

∂y + w∂u

∂z

= −∂P

∂x + ρg_x+ µ ∂²u

∂x² +∂²u

∂y² + ∂²u

∂z²

. (4.29) Assuming gravitational forces can be neglected yields

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ρ

∂u

∂t + u

∂u

∂x + v

∂u

∂y + w

∂u

∂z

= −∂P

∂x +

ρg_x+ 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

∂xy²+ C₁y + C₂. (4.32) Invoking the boundary condition (at y=0, u=0) yields C₂ = 0. Invoking the boundary condition (at y=h, u=U) yields

C1 = U h − 1

2µ ·∂P

∂xh. (4.33)

Inserting C₁ and C₂ into equation (4.32) gives u(y) = y

hU − h² 2µ · ∂P

∂x · y h

1 − y

h

. (4.34)

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

u(y) = y

hU. (4.35)

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

du dy = U

h. (4.36)

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 = U dt, see Figure 4.2.[10]

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

The angular displacement or deformation (or shear strain) can be expressed 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 equivalent 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]

τ ∝ dβ

dt or τ ∝ du

dy. (4.39)

Including the proportionality constant between shear stress and the velocity gradient 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 fluid layer (or, by Newton’s third law, the force acting on the plate) as

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F = τ A = µAdu

dy. (4.41)

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 shearing 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.

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Chapter 5 Parameters to be evaluated

This thesis evaluates three aerodynamic parameters related to rocket stability: 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 methods. 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 determine 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, C_D, and the lift coefficient, C_N. More specific, the zero-lift drag coefficient and the derivative of the lift coefficient with respect to angle of attack are evaluated.

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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 CD = FD

ρAv²/2, (5.1)

where F_D 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, q = ρv²

2 , (5.2)

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,

C_D = F_D

qA. (5.3)

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:

C_D = F_D

qA = c_p+ c_f = 1 qA

Z

S

dA(p − p₀) ˆ n · ˆi

| {z }

cp

+ 1 qA

Z

S

dA ˆt · ˆi

τ

| {z }

cf

, (5.4)

where c_p is the pressure drag coefficient, c_f 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.

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The zero-lift drag coefficient, C_D₀, 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

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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 C_N = FN

ρAv²/2 = FN

qA, (5.5)

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where F_N 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 energy. 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 pressure 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 principle.[12]

Figure 5.1: Lift of an airfoil according to Bernoulli.

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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,

F = ma = md

dt(v). (5.6)

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, C_N_α, 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, C_N_α indicates the magnitude of the lateral acceleration.

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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 atmosphere, 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),

X_cp= R x · p(x)dx

R p(x)dx . (5.7)

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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, temperature, 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, cylindrical coordinates and spherical coordinates and are adjustable regarding what kind of flow problem is being analyzed. For the fullness of this report, and

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to give the reader a greater appreciation of the Navier-Stokes equations, the derivation of the three-dimensional Navier-Stokes equations for incompressible, 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 Dt

+ ρ ∇ · V = 0, (6.1)

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

∇ = i ∂

∂x + j ∂

∂y + k ∂

∂z. (6.2)

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

D_t = 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 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:

ρDV

D_t = f = f_body+ f_{surf ace}. (6.5) In equation (6.5), f is the force exerted on the fluid particle per unit volume, and f_body is the applied force on the whole mass of fluid;

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f_body = 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, f_{surf ace}, is expressed by pressure and viscous forces;

f_{surf ace} = ∇ · τ_ij = ∂τ_ij

∂xi

= f_pressure+ f_viscous, (6.7) where τ_ij is the stress tensor,

τ_ij = −P δ_ij + µ ∂u_i

∂x_j + ∂u_j x_i

+ δ_ijλ∇ · V. (6.8) Thus, Newton’s second equation of motion can be expressed as

ρDV

D_t = ρ · g + ∇ · τ_ij. (6.9) Inserting equation (6.8) into (6.9) yields the the conservation of momentum of a Newtonian viscous fluid in one equation:

ρDV D_t

| {z }

I

= ρ · g

|{z}

II

− ∇P

|{z}

III

+ ∂

∂x_i

µ ∂u_i

∂x_j + ∂u_j

∂x_i

+ δ_i_jλ∇ · V

| {z }

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 the viscosity coefficient µ is assumed constant and ∇ · V = 0. Thus, the conservation of momentum equation for an incompressible three-dimensional flow can be expressed as

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ρDV

D_t = 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 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,

dE_t= 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:

ρ







∂h

∂t

|{z}

I

+ ∇ · (hV )

| {z }

II





= −∂P

∂t

| {z }

III

+ ∇ · (k∇T )

| {z }

IV

+ φ

|{z}

V

, (6.17)

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

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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 equations, 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 numerical solutions based on hand calculations.

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

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

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• 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 improve 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!

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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 performed 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.

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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 configuration 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.

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

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

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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.

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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 dominant meshes were created between the hexahedral dominant free-stream sections. 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

Aerodynamics Modeling of Sounding Rockets: A Computational Fluid Dynamics Study