LES Simulation of Hot-wire Anemometers

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LES Simulation of Hot-wire Anemometers

Assiye Süer

Space Engineering, masters level

2017

Luleå University of Technology

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Final Degree Project

LES simulation of hot-wire anemometers

Author: Assiye S¨uer Supervisor: Manel Soria

Supervisor: Lars-G¨oran Westerberg

A thesis submitted in partial fulfillment of the requirements for the masters degree in Space Engineering

at the

Department of Computer Science, Electrical and Space Engineering Lule˚a University of Technology

Sweden

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UNIVERSITAT POLITECNICA DE CATALUNYA

Abstract

Escola Superior d’Enginyeries Industrial, Aeron`autica i Audiovisual de Terrassa ESEIAAT - UPC

Aeronautical Engineering

LES simulation of hot-wire anemometers by Assiye S¨uer

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Acknowledgements

I would like to thank my supervisor, Manel Soria, for the guidance and help he offered me during this master’s thesis, as well as letting me attend his classes. Also, I would like to thank Ph.D. student Arnau Mir´o for lending a hand and providing me with help in a pedagogical manner. Lastly, I would like to thank my supervisor at LTU, Lars-G¨oran Westerberg for giving me necessary information related to the thesis.

Besides my supervisors, I would like to thank my family members and friends that have been there during my period in Spain and UPC. I appreciate all the advice and guidance I received from you.

Assiye S¨uer

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List of Figures

1.1 Difference in polar caps due to seasonal changes . . . 2

1.2 Difference in size between Mars and Earth. . . 3

1.3 The atmospheric structure of Mars . . . 6

1.4 The basic design of a hot-wire and hot-film anemometer . . . 8

2.1 Laminar and Turbulent flow. . . 22

3.1 Phoenix lander’s Telltale anemometer . . . 25

3.2 Mars Science Laboratory rover . . . 27

3.3 An illustration of the two booms as well as the UV sensor onboard MSL . 28 3.4 The sensors and their positions . . . 28

3.5 Closeup of boom 1 and the sensor . . . 31

3.6 Trend interpolation graph for the species . . . 37

3.7 Dimensionless wall profile . . . 41

3.8 Differentially square cavity . . . 42

3.9 The 40x40 structured mesh. . . 43

3.10 Temperature and Velocity contour lines. . . 44

3.11 Local Nu number over the hot and cold wall. . . 46

3.12 The average, maximum, and minimum Nusselt number versus mesh size with reference value. . . 47

3.13 The schematics of the square cylinder . . . 51

3.14 The mesh used in the square cylinder case. . . 52

3.15 Temperature map of the fluid flow . . . 53

3.16 Velocity map of the fluid flow. . . 53

3.17 Contour lines of the fluid flow. . . 54

3.18 The total Nusselt Number. . . 55

3.19 The surface averaged Nusselt Number over time for each face. . . 56

3.20 The time averaged local Nusselt Number. . . 57

3.21 Convergence of average Nusselt number . . . 58

3.22 Plots for the drag repective lift coefficients. . . 59

3.23 Convergence of the Drag Coefficient. . . 60

3.24 Convergence of Coefficient RMS values . . . 60

3.25 The Shedding Frequency . . . 61

3.26 Convergence of Frequency Shedding. . . 62

3.27 The surface averaged Nusselt Number over time for each face. . . 63

3.28 The time averaged local Nusselt Number. . . 63

3.29 The time averaged Local Nusselt Number. . . 64

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List of Figures vii

3.31 The Lift Coefficient . . . 65

3.32 The Shedding Frequency . . . 65

3.33 The mesh with the two points of interest. . . 68

3.34 The points on the anemometer. . . 69

3.35 Maps of the fluid depending on Temperature, Velocity, and Pressure . . . 70

3.36 Q-criterion of the temperature and velocity, as well as the temperature contours with respect to the velocity. . . 72

3.37 The square cylinder seen from the side. . . 73

3.38 Nusselt number at both point 1 and point 2 . . . 74

3.39 Surface and total Nu number for LES . . . 75

3.41 Time averaged local Nu number for LES. . . 76

3.42 RMS of lift coefficient as well as the drag coefficient and frequency shedding. 77 3.43 Temperature map of RANS k-omega results at the last time instant. . . . 78

3.44 Temperature map of RANS k-omega results at the last time instant. . . . 78

3.45 Nu number at point 1 for RANS k-omega. . . 79

3.46 Nu number at point 2 for RANS k-omega. . . 80

3.47 FFT of the Nu number for RANS at point 1. . . 80

3.48 Time-averaged local Nu numbers for RANS. . . 81

3.49 RMS of lift coefficient , Drag coefficient, and Frequency shedding for RANS k-ω . . . 82

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List of Tables

1.1 Orbital parameters and data of Mars and Earth . . . 3

1.2 Comparison between conditions on Earth and Mars. . . 4

1.3 Mars’ atmosphere composition . . . 5

3.1 Anemometers sent to Mars . . . 26

3.2 Table of Wind Sensor Measuring Characteristics[1]. . . 30

3.3 REMS dimensions and mass Components[2] _{. . . 32}

3.4 REMS and its Operational Requirements[3] . . . 33

3.5 Atmospheric composition . . . 35

3.6 Thermal conductivity of Mars’ atmosphere at different temperatures. . . . 36

3.7 Kinematic Viscosity . . . 38

3.8 The post-processing results of the Mars atmospheric conditions. . . 38

3.9 Papers used for guidance during the process. . . 44

3.10 The physical fluid properties. . . 44

3.11 Relevant problem parameters in relation with the mesh size. . . 45

3.12 Values attained compared with mentioned papers. . . 46

3.13 The dimensions of the square cylinder. . . 50

3.14 Our results compared with two references. . . 55

3.15 Summary of the results. . . 84

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Abbreviations

CFD Computational Fluid Dynamics DNS Direct Unsteady Simulaions EUV Extra Ultra Violet

FFT Fast Fourier Transform HWA Hot Wire Anemometer JPL Jet Propultion Laboratory KH Kelvin Helmholtz

LES Large Eddy Simulations MSL Mars Sience Laboratory

NASA National Aeronautics and Space Administration NS Navier Stokes

PBL Planetary Boundary Layer RANS Reynold Averaged Navier Stokes SST Shear-Stress Transport

UV Ultra Violet VK Von K´arm´an

VMIS Viking Meteorology Instrument System WS Wind Sensor

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Physical Constants

Day/night cycle on Mars sol = 24.62 hours One Astronomical Unit AU = 149.597 × 109 m

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Symbols

T temperature K

P pressure P a

q heat power W

L length of the computational domain m

V volume m3 m mass kg ˙ q/ ˙Q heat flux W/m2 g gravity acceleration m/s2 k thermal conductivity W/mK cp heat capacity J/kgK

h superficial heat transfer coefficient W/m2K

R specific gas constant J/kgK

H scale height m

ρ density kg/m3

µ dynamic viscosity P a s

β thermal expansion coefficient at constant pressure K−1

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Symbols xii P r Prandtl number − Ra Rayleigh number − Re Reynolds number − Ri Richardson number − St Strouhal number − y+ − Cd drag coefficient − Cl lift coefficient −

Cl,rms root mean square of lift coefficient −

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This thesis is dedicated to my beloved mother.

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Introduction

A hot-wire anemometer is an instrument widely used to measure the instantaneous velocities of a fluid flow and it is based on the dependence of the sensor heat transfer on the fluid temperature, velocity, and composition. Besides being used on Earth, hot-wire anemometers have also been sent to, for instance, Mars. One example is the type of anemometer onboard the Mars Science Laboratory (MSL) rover, which has a hot-film configuration. That is, instead of a wire, we have a piece of silicon heated by a fine, conducting film is covering the sensor.

The topic of this master thesis is to estimate turbulent fluctuations and the effect they may have on the sensor readings by using various turbulence models. We will focus on the hot-film anemometer onboard MSL and see how the fluid behaves past it.

Computational Fluid Dynamics (CFD) is used to numerically solve the governing equa-tions of the flow. One of the conventional problems in fluid mechanics is the study of the flow field past a bluff body characterised by a square or circular cylinder. The flow past square cylinders has been studied many times since it has an importance in numer-ous practical purposes, such as heat exchangers. Therefore, we will create a simplified model of the anemometer by treating it as a square cylinder. This will be possible by calculating a characteristic length (or hydraulic diameter) of the square cylinder while we consider the dimensions of one of the anemometers on MSL (see chapter 3).

In scientific terms, the flow around square cylinders embraces a diversity of fluid dynam-ics phenomena, such as vortex shedding, separation, and the transition from laminar to turbulent flow.

The mechanisms of vortex shedding and its dominance in the flow have great effects on the different fluid-mechanical properties of interest, which are forces that are flow-induced. Some of them are pressure gradients, drag and lift coefficient.

The RANS (Reynolds-Averaged Navier-Stokes) equations are time-averaged equations for fluid flow and are commonly used to model turbulent flows. A drawback with RANS is that turbulent fluctuations, which have important impact on the flow velocity and

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Symbols xv

pressure changes, are lost due to the time-averaging. By testing cases with conditions that are already known (e.g. fluid composition, velocity, or temperature) and using LES, one can take these important fluctuations into account. Here too, the case that is often used for this process is the flow past a square cylinder.

This thesis is divided as follows:

• Chapter 1 covers the facts about planet Mars and compares it with Earth in terms of size, physical properties, and other conditions. One special aspect of the planet is it’s atmosphere, such as the composition and structure of it. These are two topics that will be introduced here as well since the thesis involves the atmosphere of Mars. This chapter also explains the basics about the mechanism behind heat transfer, hot-wire and hot-film anemometers, where examples of those that have been sent to Mars are presented briefly.

• Chapter 2 is an introduction to CFD, where the steps used during and after a CFD analysis are presented. A short introduction to grid generating is brought up as well due to its importance.

• Chapter 3 explains more about the dimensionless numbers often used in CFD, such as Nusselt, Reynolds, Prandtl, and Rayleigh number. A non-dimensionalization of Navier-Stokes equation is performed, where the purpose is to show where the non-dimensional numbers comes from and why they are relevant in the case that is being studied. Lastly, the topic of Boussinesq approximation and natural convec-tion is covered since the chapter after will involve a case with a buoyancy driven flow.

• Chapter 4 is going more into the theory behind the hot-film anemometer on MSL, as well as a brief overview of the other instruments onboard the rover. Estimations of parameters related to the Martian atmospheric conditions is covered too since they are necessary for the simulations. This chapter is also governing two CFD cases: the square cylinder and thermal driven cavity, as validation cases for the post-processing tools.

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Objectives

The objectives of this master thesis are the following:

• To investigate the influence of turbulent fluctuations in the measurement of tem-perature, and hence the velocity, in a hot-film anemometer located on Mars by:

– Assessing the value of Re, P r and Ra numbers for the sensor in Martian conditions.

– Taking the atmospheric composition of Mars into account.

– Performing LES simulations of a simplified hot-film anemometer under Mar-tian conditions.

– Investigating the influence of the vortex shedding frequency with respect to the frequency of the sensor’s temperature signal.

– Comparing the LES with RANS simulations of the same simplified hot-film anemometer under the same conditions.

• To validate the CFD code ”Code Saturne” and the necessary subroutines needed in order to compute natural and mixed convection flows, such as the Thermal driven cavity and low Reynolds Square cylinder cases.

• To develop and validate the post-processing tools needed in order to obtain the Nusselt number (representative of the heat transfer) in both 2D and 3D simulations for the thermal driven cavity and low Reynolds square cylinder.

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Scope

The scope of this work is to estimate fluctuations due to turbulence in the measure of temperature in a simplified hot-film anemometer found on NASA’s Curiosity rover by taking into account that:

• The flow is an incompressible, Newtonian fluid with constant properties (i.e., con-stant pressure, temperature, velocity etc.).

• The computational domain studied are both two- and three-dimensional. • The effect of buoyancy is taken into account in at least one of the simulations. • The turbulence models considered are LES and RANS. DNS is out of the scope of

this work.

• Martian physical and atmospheric properties are implemented.

• Only incidence angle zero is considered. Changing the angle of incidence is out of the scope of this work.

It is also part of the scope of this work to briefly describe the flow around a simplified hot-film anemometer without a detailed analysis of the outcome. Moreover, one addi-tional task of this thesis is to develop and validate all the post-processing codes needed in order to obtain the expected results (which can be found in papers). Additionally, it is part of the scope of this work to validate the behaviour of Code Saturne when simulating buoyancy driven flows.

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Chapter 1 State-of-the-Art

This chapter begins with an overview of planet Mars. The atmospheric conditions of the planet will be brought up, where we will look at its composition and structure. To get an insight of the difference between Mars and Earth, the same numbers related to our planet will be presented next to its values.

1.1 Planet Mars

Planet Mars is the fourth planet counted from our Sun and it is one of the bodies of great interest at the moment, especially since the plans are to send human beings to it very soon, which will turn us into an interplanetary specie.

It has a distance of 1.524 AU in average, an equatorial radius of 3396.2 km (about half of Earth’s), and a year equal to 689.9 Earth days (about 669 sols)[4]. What makes it more interesting is that it has, similarly to our planet, seasons and weather phenomena. One particular and noticeable feature is the growth of the polar caps during it’s winters. This growth can be seen on both the northern and southern areas, an event that resembles what occurs here. The polar caps are mainly made of CO2 ( also known as dry ice),

whereas here on Earth, they are made of H2O.

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Chapter 1. State-of-the-Art 2

Figure 1.1: Difference in polar caps due to seasonal changes[5].

A theory states that Mars was once covered with liquid water, just like Earth.

One of the discoveries, which were made by the Mars Curiosity rover, was the presence of liquid water beneath Mars’ surface. This was identified while the rover dug on the Martian soil, a founding that did most certainly strengthened this theory about the fluid water. The modern-day Mars is a noticeably dry, dusty, and cold planet. In any case, the surface shows evidence of apparent water flow, especially during its younger days, when the ”floods” shaped networks of major channels on its surface. This lead to the lowland plains on its northern province, which is a feature that have been observed by satellites. These channels are the single major line of proof for water that might have once been present on the planet’s surface.

1.1.1 Earth and Mars - an overview and comparisons

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Figure 1.2: Difference in size between Mars and Earth[6].

Table 1.1: Orbital parameters and data of Mars and Earth

Mars Earth Earth-Mars ratio Unit

Mass, 1024 0.64185 5.9736 0.107 kg

Mean radius 3389.50 6371.13 0.532 km

Semi-major axis, 106 227.92 149.60 1.524 km

Aphelion, 106 249.23 152.10 1.639 km

Perihelion,106 206.62 149.09 1.405 km

Sidereal orbit period 686.980 365.256 1.881 days

Orbit inclination 1.850 0 - degrees

Orbit eccentricity 0.0935 0.0167 0.599

When it comes to the atmosphere of Mars and Earth, there are some similarities; Mars’ is thin and fairly translucent to sunlight.

The axial tilt and spin rate of Mars are also relatively Earth-like. This combination puts the Martian atmosphere into the class known as a fast revolving, differential warmed atmosphere with a compact lower BL (Boundary Layer). Yet, there are also significant differences between them as well. Mars’ atmosphere is mainly made of CO2,

with a much lower surface pressure than here on Earth.

Table 1.2 gives a better insight and a summary on how different the atmospheric condi-tions are on both planets. The transfer of heat between the surface and the atmosphere is the key driver of the thermal structure of the PBL, leading the atmosphere into greater instability.

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Table 1.2: Comparison between conditions on Earth and Mars[4].

Mars Earth Unit

Gravity, g 3.7 9.8 m/s2

Atmospheric gas constant, R 188 287 J/kgK

Typical surface pressure, p 7 1015 hPa

Typical surface density, ρ 1.5 × 10−2 1 kg/m3 Typical surface temperature, T 220 300 K Specific heat constant, cp 730 1010 J/kgK

Kinematic viscosity, ν 10−3 1.5 × 10−5 m2/s

The small impact on the atmosphere and its impact on the surface heat budget is am-plified by another aspect of the atmosphere, which is the lack of significant amounts of water.

Additionally, Mars’ low atmospheric density leads to a much higher kinematic viscosity, ν, and heat diffusivity than here on Earth. This is due to the fact that the kinematic val-ues entering the image of the atmospheric motion described earlier divided the dynamic characteristics.

The great value of the kinematic viscosity for the Martian atmospheric BL in turn has an impact on the parameters used to describe the fluid flow of its atmosphere.

The overall empirical characteristics are that, close to the surface, we find typical wind speeds to be much as we know them on Earth. See Table 1.2 for characteristic numbers for the unstable BL on both Mars and Earth.

1.2 Atmosphere of Mars

The atmosphere of Mars differs from Earth’s in both composition and volume. Even though it is very thin, it still allows weather phenomena on the planet to happen, such as high-speed winds and seasonal variations.

This section will focus on the atmosphere of Mars since is has a great part of the thesis as we will se further on.

1.2.1 Atmospheric conditions

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the gas CO2 was a key component of the atmosphere. This is based on the large number

of absorption in the 2-4 µm section[7].

The Spacecrafts Mariner 4, 6, and 7 later came to the conclusion that CO2 was the

main component of the atmosphere (Table 1.3), but found that the current atmosphere is very thin and has a pressure that lies in an average of only 500 Pa[8]. The pressure of the surface can vary up to 20% as an outcome of the volumes of atmospheric H2O and

CO2 due to seasonal changes. This prevents the preservation of solar heat, which leads

to large temperature changes. This in turn results into great temperature fluctuations between the day and night sides of the planet.

The average pressure of 500 Pa and 210 K[8] as an average surface temperature prevents the existence of liquid water on most of the surface of Mars today.

Table 1.3: The composition of Mars’ atmosphere[8].

Constituent Abundance CO2 95.32 % N2 2.7% 40_Ar _1.6% O2 0.13 % CO 0.07 % H2O 0.03 % (variable) Ne 2.5ppm Kr 0.3ppm Xe 0.08 ppm O3 0.04-0.2 (variable)

Dust 0 to 5 (visible optical depth)

There are numerous heating sources that begins the atmospheric motions on Mars. One of the most significant ones when it comes to atmospheric movement is solar heating. This is something that is caused by the absorption of visible wavelength solar photons by the surface and within atmospheric layers, with moderately large optical depths (i.e., near cloud layers). The surface of the planet and particles of atmospheric dust radiate the visible wavelength energy they absorb, usually at IR wavelengths, which increase heat the atmosphere.

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1.2.1.1 Atmospheric structure

Mars’ atmosphere is structured in different layers, which is based on temperature, com-position etc. similar to Earth. Continuous observations by orbiting satellites, using measurements with radio occultation and IR sounding, have provided understandings of the atmospheric structure over greater areas and times. The results showed that the atmosphere of Mars is sectioned into three layers: lower, middle, and upper.

The first layer, which is the lower part of the atmosphere, ranges from the surface to an altitude of approximately 40 km[7]_{. The temperature and pressure decrease with rising}

altitude through the lower atmosphere and energy transport is dominated by convection within the first 10 km. At night, the convection stops and a strong temperature inversion progresses close to the surface.

The middle atmosphere, also known as the mesosphere, ranges between about 40 to 100 km over the surface of Mars[7]. The temperature here can vary considerably with time and these differences are a result from the near-IR absorption and emission of solar radiation by CO2. It is also a result from atmospheric waves started in the lower

atmosphere and altered by thermal drifts between day and night hemispheres. The

Figure 1.3: The atmospheric structure of Mars[9].

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nm[7]. The EUV output from the Sun changes due to the solar activity cycle, which leads in great difference in thermosphere temperatures above an altitude of 120 km[7]. During the periods when the solar activity is lower, the temperatures are cooler and rise as the sunspot cycle reaches its maximum. The region above the altitude of 130 km is called the ionosphere[7] because solar radiation ionises the gases in the atmosphere above this level. Most of the electrons in Mars’ ionosphere are derived from CO2 and

peak photoelectron densities are being reached on the dayside. More about the effects of dayside and nightside effects will be presented further on.

1.3 Hot-wire and Hot-film anemometers

Thermal anemometry is the most widely used technique to measure instant fluid velocity. The method is dependent on the convective heat loss to the fluid that is surrounding it from an electrically heated sensing element or probe[10]. If just the velocity of the fluid alters, then the heat loss can be translated as a measure of that specific variable. In this thesis, the Nusselt number will be that variable.

1.3.1 General principle

Hot-wire anemometers are of great interest in CFD simulations since they can quantify the fluctuations of the Nusselt number due to turbulence. There are two kinds of thermal wind sensors; it either include a hot-film or a hot-wire, both function with the same principles and have been used effectively on previous missions to Mars.

The operation of this method is rather straightforward. Hot-wire anemometry is a technique used to measure velocities based on forced convective heat transfer from a thin, heated wire, which is immersed in a fluid flow.

The instrument has a wire that is made of a material with resistivity that is temperature dependent. When an electric current is passing through the wire, it becomes heated and then exceeds the temperature of the fluid. Thus, the heat transfer from the wire relies on the fluids flow rate that it is exposed to.

As a sum, a hot-wire (or any type of anemometer actually) are necessary when one is searching for accurate relations between velocity and the Nusselt number in order to obtain the velocity (from this dimensionless number that is).

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Although they are reliable, low mass methods for sensing wind speed on Mars (or any other place), they need correct discrete measurements and are prone to invalid results triggered by changing ambient temperatures and further thermal loads.

An anemometer is often very little and the small dimensions of it gives a very fast response, and allows it to measure local velocities as well.

1.3.2 Hot-film anemometers

As mentioned, the hot-film anemometer functions in the same manner as the hot-wire one. The difference lies on their respectively design. While a hot-wire anemometer are using a wire for measurements, a hot-film type consists of a conducting film.

This sensor is basically a conducting film rested on a substrate made of quartz or ceramic. The most usual forms of the substrates are wedges, cones, or cylinders. This project will consider an anemometer of the shape of a square cylinder. The thickness of the metal film is typically a couple of µm. Hence, the thermal conductivity and thermal strength of the anemometer are defined almost completely by the material of the substrate. This covering protects the material of the film from rough particles that may affect it in negative ways.

Figure 1.4 below[11]shows the comparison of a basic hot-wire and hot-film anemometer. It is worth to mention that a hot-wire sensor can consists of more than one wire. We are only presenting the basic, single wire design below.

Figure 1.4: The basic design of a hot-wire and hot-film anemometer[12].

1.3.2.1 Heat transfer

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and the wire. Even the fluids physical properties, together with the physical properties and dimensions of the wire, are some affecting parameters that can have an impact on the collected data.

Usually, the velocity of the flow, and the dimensions and physical properties of the wire are recognised. If the fluid’s physical properties are know or held continuous, the velocity can be defined. And the opposite, if the velocity is know or continuous, those properties can be measured.

Let us begin by considering a thin and long, heated wire of a circular cross section, cooled by a surrounding fluid that is flowing through it. Let us also assume a velocity, V , affecting it by being constant over the entire length of the wire. With this kept in mind, the total heat loss of the wire depends on the following parameters[13] :

• The absolute value of the fluid velocity V, and the angle between the wire and the velocity vector. The angle of attack has an effect, that is.

• The temperature difference between the wire and the fluid, ∆T = Tw− Tf.

• The physical properties of the fluid; density ρ, viscosity µ, heat conductivity k, specific heats cp and so on.

• The dimensions of the wire (diameter d and length l) and its physical properties, density ρ, specific heat cv, heat conductivity k and so on.

Normally, number (2) and (4) in the list are already known. If then the temperature is known or kept constant, number (1) can be measured too. These are the points that this project is based on, but the angle of attack is not altered.

From the definition of heat transfer coefficient (h), the heat loss is[13] ˙

Q = hπdl∆T, (1.1)

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Chapter 2 Turbulent heat transfer

This section is covering turbulent heat transfer and turbulence models in CFD, together with an overview of the dimensionless numbers that will be in focus during the thesis. We will introduce the Navier-Stokes equation and how to turn it into a non-dimensional form, where each term emerges as a product of a non-dimensional distinctive number with the non-dimensional physical term of O. In this way, it is possible to tell which effect that is more significant in the flow problem before the simulation.

2.1 Conduction, Convection and Radiation

The mechanism, where heat energy is transmitted from or to an object, is called heat transfer. In other words, heat transfer alters the internal energy of both systems that are involved according to the First Law of Thermodynamics. There are three types of heat transfer; convection, radiation, and conduction When heat transfer happens between substances (such as fluid) that are in direct contact with each other we have conduction. Convection happens when hotter areas of a gas or liquid rise to cooler area, a case that will be presented later in this thesis. The gas or liquid that is cooler then takes the place of the regions that are warmer, which have risen higher. This ends in a constant circulation pattern as the heat exchange continues. Lastly, when it comes to radiation, it is explained as a heat transfer method that does not depend on any contact between the heated body and the heat source, which is the opposite to conduction and convection. For fluids, the mechanisms of heat transfer are not similar as for solids. When a fluid is moving, we are looking at convection. If it so happens that no bulk fluid motion exist, conduction by heat transfers are the leading event. The rule is that the higher the velocity of the specified fluid, the higher heat transfer rate. The reason behind

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Chapter 2 11

this is because contact between cooler and hotter amounts of fluid, which increases the conduction rate. Between these three, convection is the one phenomenon that is most complex. Not only does it depend on the properties of the fluid, but also on the shape of the surface and the type of fluid that it is interacting with. The relationship for heat transfer by convection, which agrees with Newton’s law of cooling, is

˙

Q = hAs(Ts− T0), (2.1)

where ˙Q = hA(Ts − T0) is the heat transferred per time unit, A is the geometry’s

area, h is the heat transfer coefficient, Tsis the object’s surface temperature and T0

is the temperature of the fluid. Re-arranging the equation gives us the heat transfer coefficient as

h = Qconvective As(Ts− T0)

. (2.2)

The general heat transfer coefficient is used to estimate total rate of heat transfer between a solid and a fluid. This coefficient relies on the fluid and its properties (e.g., density, specific heat and so on) on both sides of the wall, the properties of the wall and the transmission surface[14].

We are neglecting conduction and radiation and the reason is that we firstly assume that the surface temperature is constant and that the fluid has the same temperature as the cylinder (or very close).

2.2 Suitability of continuum approach in Mars

One thing to consider is if there is a possibility of a continuous flow in the atmosphere of Mars or not. We are aware of continuous, or viscous, flow here on Earth, but why is it significant?

In theory, a flow like this reveals that the atmosphere is in movement and the molecules are in collision with each other.

On Mars, on the other hand, since the composition and density differs from our planet’s, this might not be the case. Therefore, we need to look at one parameter that is respon-sible of continuous flows, namely the Knudsen number.

2.2.1 Knudsen Number

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Chapter 2 12

in a fluid is an example of a length scale. The equation describing the Kn number is Kn = λ

L. (2.3)

The mean free path is the average distance every particle travel between collisions and is given for a hard sphere by

λ = √ 1 2πd2_n

v

, (2.4)

where nv the number density per unit volume. 2d stands for the effective collision area

of a molecule while treating the target molecules as point masses.

Equation 2.4 , reveals that at low densities, the mean free path (and thus Kn) becomes great. It is commonly accepted that kinetic, non-equilibrium effects turn out to be important when Kn > 0.01 .

In the case of continuous, or viscous, flow, a distinction is made between flows that are laminar and turbulent.

In the laminar flow (or layered), the fluid particles stay in the similar displaced layers with a constant, parallel relation to each other. If the velocity of the flow starts to increase, these layers will broke up and the particles of the fluid will collide in a fully disordered manner. In other word, we have a turbulent flow.

The Reynolds number can express the boundary between these two areas of viscous flow. This dimensionless constant will be presented further on in this thesis.

One significant point of this dimensionless number is that, when Kn 1, the gas behaves as a continuum fluid on the length scale L.

An odd fact about the atmosphere of Mars is that the molecular free path of it lies in the order of 10µm, a size that is comparable to the size of dust grains. Studies by Caroline de Beule et al.[15] shows that this can result in a continuous gas flow with a force strong enough to move particles that can be found in the soil on the surface.

2.3 Turbulence models

There are diverse turbulence models in CFD and the choice depends on your preferences and how accurate results you want.

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Chapter 2 13

2.3.1 The RANS model

It is possible to simulate flows, such as the flow of air over a plane, by re-writing the Navier-Stokes equations into the Reynolds-Averaged Nusselt number (RANS) equations instead.

The idea behind the RANS equations (also called the Reynolds decomposition) is that the time-dependent turbulent velocity oscillations are separated from the average veloc-ity of the flow. This transformation in turn presents a number of unknowns termed as Reynolds stresses, which are functions of the velocity oscillations, and which require a turbulence model (e.g., the k-ω model) to generate equations that are solvable.

As a consequence of this transformation, the reduced computational requirements for the RANS equations are less costly to run (in terms of simulation time) compared with an LES simulation. This is one benefit of using this model.

One additional benefit of using the RANS equations for a flow that is steady is that the velocity of the mean flow is calculated as a direct result without the requirement to average the instant velocity over a sequence of time steps.

For this thesis, three turbulence models were used. These are the LES and RANS k-ω (SST) model. Below is a short introduction to the two latter models.

• The k-ω (or k-omega) model: The k-ω model is similar to k-, but it solves for omega - the specific rate of dissipation of kinetic energy, and also k, which is the turbulent kinetic energy. It also uses wall functions and therefore has comparable memory requirements. Additionally, it has more difficulty converging and is quite sensitive to the initial guess at the solution. Hence, the k-epsilon model is often used first to find an initial condition for solving the k-omega model. The k-omega model is useful in many cases where the k-epsilon model is not accurate, such as internal flows or flows that exhibit strong curvature.

2.3.2 The LES model

The LES (Smagorinsky) model may be seen as a compromise between DNS and the use of RANS. The largest vortices, which naturally interact most intensely with the mean flow (due to it having the most energy), the LES method become a good model of the turbulence and its key effects.

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Chapter 2 14

coarse and fine grids are still needed. This since the computational constraint is still high.

One can imagine a flow with a lot of scales. The behaviour of smallest scales is similar no matter the what the Reynolds number is, and it is therefore we model that part. The production part is case dependent and the way to separate both is the mesh. This means that one filter with the generated mesh, what is being solved and what is being modeled.

2.4 Dimensionless form of the NS equations

In CFD calculations, Navier-Stokes equations are one of the main used equations. They basically govern the motion of a fluid and can be considered as Newton’s second law of motion used for fluids.

The following are essential laws that can be used to derive governing differential equa-tions that are solved in a CFD study:

• conservation of mass

• conservation of linear momentum (can be seen as Newton’s second law) • conservation of energy (can be seen as the first law of thermodynamics)

The incompressible Navier-Stokes equations are generally given as

∇ · u = 0, (2.5) ∂u ∂t + (u · ∇) u − µ ρ∇ 2_u ₌ ₋1 ρ∇P + g, (2.6)

where ρ represents the density,g the gravitational acceleration, P the pressure, and µ the dynamic viscosity.

Developing from the vectorial form, equations 2.7 is the continuity equation, whereas equations 2.8 to: 2.10 are the momentum equations ( for each vector)

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Chapter 2 15

Note that in Eqs. 2.7 to 2.10, the terms u, v and w are the three components of the velocity.Finally, fx,fy and fz are volumetric forces. For example, the gravity force is

represented as f = ρg.

The following non-dimensional variables are considered for the non-dimensionalization of the NS equations ¯ x = _Lx y =¯ _Ly z =¯ _Lz u =¯ _Uu 0 ¯v = v U0 w =¯ w U0 ¯ t = tU0 L g =¯ g g0 ¯ P = _ρUP2 0

where L is a reference length, U0 a reference speed, g0 the gravity acceleration at Earth.

The velocity for the x, y, and z components are marked as ¯u, ¯v, and ¯w. These can be re-arranged as the following form

x = L¯x y = L¯y z = L¯z u = U0u¯ v = U0¯v+ w = U0w¯ t = _U¯tL₀ g = g0g¯ P = U02ρ ¯P .

and the derivatives that can be seen on Eqs. and can be expressed as ∂u ∂t = ∂ ¯u ∂¯t ∂u ∂ ¯u ∂¯t ∂t, (2.11) ∂v ∂t = ∂ ¯v ∂¯t ∂v ∂ ¯v ∂¯t ∂t, (2.12) ∂w ∂t = ∂ ¯w ∂¯t ∂w ∂ ¯w ∂¯t ∂t. (2.13)

By using the chain rule, one can express the derivatives from equation 2.7 to 2.10as presented below. Note that ∇, which is the vectorial operator, has a general distance r considered. With this pointed out, the values of the derivatives can be calculated as

∇ = ∂ ∂r ∂ ¯r ∂¯t, (2.14) ∇P = ∂P ∂r = ∂ ¯P ∂ ¯r ∂P ∂ ¯P ∂ ¯r ∂r. (2.15)

The derivatives, together with the non-dimensional variables, can now be expressed as

∂u ∂ ¯u = U0 ∂v ∂ ¯v = U0 ∂w ∂ ¯w = U0 ∂¯t ∂t = U0 L ∂ ¯r r = 1 L ∂P ∂ ¯P = ρU 2 0. (2.16)

The next step is to insert these into Eqs. 2.8 to 2.10. The NS equations can now be written in the following form

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Chapter 2 16

Finally, after some re-arrangement, the non-dimensionalised NS equations can be written as ∂ ¯u ∂¯t + ¯u · ¯∇ ¯u = − ¯∇P + µ ρU0L ¯ ∇2_{u +}_¯ Lg0 U₀2 gx, (2.20) ∂ ¯v ∂¯t + ¯v · ¯∇ ¯v = − ¯∇P + µ ρU0L ¯ ∇2_{v +}_¯ Lg0 U2 0 gy, (2.21) ∂ ¯w ∂¯t + ¯w · ¯∇ ¯w = − ¯∇P + µ ρU0L ¯ ∇2_{w +}_¯ Lg0 U₀2 gz. (2.22) Let us now define the following non dimensional groups

• Froude number: F r2₌ U

g0L, which measures the flow inertia with respect to the

external field.

• Reynolds number: Re = ρU0L

µ , which measures the ratio between the inertial

and viscous forces.

Which will now give us the non-dimensional form of the NS equation ∂ ¯u ∂¯t + ¯u · ¯∇ ¯u = − ¯∇ ¯P + 1 Re ¯ ∇2_{u +}_¯ 1 F r2¯g. (2.23)

This is the final, non-dimensional form of the Navier-Stokes equations. An equation like this one allows for several interpretations that are of particular interest for the flow that is being studied.

As mentioned, the NS equations are also including the energy equation and therefore, we need to consider it as well. This equation can be expressed as follows (in non-dimensional form) ∂T ∂t + uj ∂T ∂xj = 1 ReP r + ∂2_T ∂2_x jxj , (2.24)

where Pr is the Prandtl number and T the temperature.

2.5 Natural Convection

Natural convection, also know known as buoyancy-driven flow, is a type of heat transport in which the motion of the fluid is not caused by any external source. A fan or heater that is cooling or warming the air is two examples of external sources that are related to a forced convection instead (the opposite to natural convection). They are basically forcing great movements in the fluid, in e.g. the air in a room.

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Chapter 2 17

In one of them, a density gradient occurs in a way that it is parallel to the vector of the gravity or opposite to it. Situations like these can lead to density stratifications that are ”unstable” or ”stable” in the fluid.

In a situation where the stratification is stable, fluids that are less dense will move to the top, and fluids that are denser will move to the bottom. When other effects are not affecting and are absent, convection of the fluid will be absent as well, and one can treat the heat transfer problems as one of conduction.

In contrary, when we have unstable stratification in which less dense fluid is at the bottom and the opposite at the top, if the gradient of the density is sufficiently large, spontaneous convection will begin and noticeably mixing of the fluid will begin.

2.6 Governing parameters for convection problems

In the case of natural convection, a central dimensionless constant is the Grashof number. To be able to deliver some physical meaning to this constant, we use the following to approximate velocity of the natural convection in the square cavity mentioned earlier. When a fluid, with ρ as a density moves with a velocity V , the kinetic energy per unit volume can be written in the following form:

Ek=

1 2ρV

2_. _(2.25)

Over a vertical distance L, the potential energy difference between the lesser dense fluid in the boundary layer of the cavity and the denser fluid outside it can be roughly explained as g∆ρL, where g is the gravitational acceleration, and ∆ρ is a characteristic difference in density between the fluid next to the boundary layer and that far away (e.g. in the middle of the cavity). We can compare these two order of magnitude estimations, and neglect that it is divided by 2, since this is only an order of magnitude analysis:

ρV2 ≈ g∆ρL. (2.26)

Therefore, a characteristic magnitude of the velocity starting from natural convection is:

V = s

∆ρ

ρ gL. (2.27)

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Chapter 2 18

which means that it can be rewritten as

Re2_L=

∆ρ ρ gL

3

ν2 . (2.29)

This is a dimensionless number that is often mentioned in problems regarding natural convection, and is known as the Grashof Number, Gr. We can now write the equation as:

Gr =

∆ρ

ρ gL

3

ν2 . (2.30)

The volumetric expansion coefficient of a fluid, β, can be defined as

β = 1 V ∂V ∂T = ρ ∂ ∂T 1 ρ = −1 ρ ∂ρ ∂T (2.31)

where T is the fluid’s temperature, V it’s volume, and P is the fluid’s pressure. Thus, we ca write ∆ρ ρ = − ∆T ρ 1 ρ = β∆T. (2.32)

As can be seen, there is a minus sight in linking ∆ρ to ∆T . This is due to both of them being defined as being positive. As the temperature rises, density will decrease as a consequence.

We can lastly rewrite the Grashof, Gr, number as follows

Gr = β∆T gL

3

ν2 . (2.33)

The Grashof number is associated with Reynolds number that was brought up earlier, and in heat transfer, the Prandtl number is an important part as well. Hence, in heat transfer due to natural convection, we face another dimensionless number, called the Rayleigh number, Ra, which is the product of the Prandtl and Grashof numbers.

Ra = Gr · P r = β∆T gL 3 να , (2.34) where P r = ν α. (2.35)

Here, α is the thermal diffusivity of the fluid. The Nusselt number, N u, in natural convection heat transfer situations is typically a function of the Rayleigh number, the Prandtl number, and aspect ratio parameters.

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Chapter 2 19

is calculated using the following equation

N u(t) = ∇T

∂T · S, (2.36)

here, S are the total surface of the square cavity.

Surface average heat transfer at both the hot and cold wall can be found by integrating the local Nusselt number alongside the walls

N uM(t) = B

Z

A

N u(t)dt, (2.37)

where A and B are the faces of the square cavity.

The time dependent average heat transfer rate from the square cavity (N utotal(t)) is the

total of the average rate of heat transfer (N uM(t)) at each wall. The total heat transfer

can be found by using

N utotal(t) = CD

X

AB

N uM(t), (2.38)

here, A,B,C, and D are each face of the geometry. The time average local rate of heat transfer ¯N u is obtained by integrating the local Nusselt number over a large period of time.

2.7 Boussinesq approximation

It is important to consider Boussinesq applications to a case where the temperature of the fluid that is being studied varies, especially since this is forcing heat transfer and fluid movement in the fluid. In the Boussinesq approximation, differences in the fluid other than density ρ are ignored, and the density of the fluid exists only when g is multiplied by it.

The NS equations are concerning the motion of fluids. The Boussinesq approximation says that the variation of the density is only significant in the buoyancy term, ρg, and can be neglected in the rest of the NS equation.

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Chapter 2 20

The parameter F is the gravitational acceleration force times the density of the fluid, that is F = ρg, (2.40) g = 0 gy 0 and gy=g0=9.81 m/s2.

In this equation, the variations in the density are presumed to have a fixed part and another part that is linearly dependent on temperature

ρ = ρ0(1 − β∆T ) . (2.41)

Since we are considering the gravitational force as well, the density can be re-written as

F = ρ0g − βgρ0∆T, (2.42)

which in turn, with the aid of equation (2.39), the NS equations with the Boussinesq approximation becomes ∂u ∂t + (u · ∇) u − µ ρ0 ∇2u = − 1 ρ0 ∇P − βg∆T. (2.43)

The following non-dimensional components were presented earlier, and we can use them in this section as well

V2 0 L, βLg∆T U2 0 .

With these kept in mind, the equation for Reynolds number and Grashof number can now be re-arranged and inserted in the NS equation

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Chapter 2 21

which finally gives us ∂ ¯u ∂¯t + ¯u · ¯∇ ¯u − 1 Re2∇¯2u = − ¯¯ ∇ ¯P − Gr Re2∆ ¯T . (2.49)

The combination between the Grashof number and Reynolds number, _ReGr2, is a term

known as Richardson number. It is shortly a parameter that can be used to guess if there will occur any turbulence in the fluid.

2.8 Dimensionless numbers - Nu, Re, Pr, and Ra numbers

Below are some of the main, dimensionless numbers used for CFD cases.

Only four of them are considered since the calculations, which will be introduced further on, involves these only.

2.8.1 Nusselt Number

The Nusselt number (N u), is dimensionless number used in the study of forced con-vection which gives a measure of the ratio of the total heat transfer to conductive heat transfer, and is equal to the heat-transfer coefficient times a characteristic length divided by the thermal conductivity such as

N u =hL

k , (2.50)

where h is the convective heat transfer coefficient of the flow, L is the characteristic length, k is the thermal conductivity of the fluid.

2.8.2 Prandtl Number

Prandtl number (Pr) is a dimensionless number approximating the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It can be expressed as

P r = v α =

µcp

k , (2.51)

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Chapter 2 22

2.8.3 Reynolds Number

Reynolds number (Re) is dimensionless number or quantity that is used to help pre-dict similar flow patterns in different fluid flow situations. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterised by smooth, constant fluid motion; turbulent flow occurs at high Reynolds numbers and is domi-nated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities. The equation is Re= inertial forces/viscous force, or

Re = ρvL µ =

vL

ν , (2.52)

where v is the maximum velocity of the object relative to the fluid, L the linear dimension is e.g. the travelled length of the fluid , and is given in meters. µ is the dynamic viscosity of the fluid, ν, which is kinematic viscosity and lastly, ρ, which is density of fluid. The figure below is a great representation of how the fluid behaves when it is in a laminar and/or turbulent state.

Figure 2.1: Laminar and Turbulent flow[16].

2.8.4 Rayleigh Number

This dimensionless number is linked to flows that are buoyancy-driven, also known as natural convection (or free convection).

When this number is below a critical value for the fluid that is being observed, the primarily form of conduction is heat transfer. When it exceeds the critical value, heat transfer is primarily in the form of convection, which is heat transfer due to fluid move-ment.

The relation for this number is

Ra = gβ να

∆T

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Chapter 2 23

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Chapter 3 Turbulent fluctuations of the N u

number in Martian wind sensors

There have been a number of rovers sent to Mars and each of them have been equipped with diverse meteorological instruments that have revealed more about the climate and atmospheric conditions on Mars.

One instrument that has previously been sent to Mars is the wind-sensor, also known as an anemometer. The anemometer has the ability to tell more about the BL and the flow that is surrounding the rover that it is placed on. By examining this further it is possible to learn more about for example heat distribution and how it works there (Nusselt number becomes interesting here).

This chapter will cover more about the hot-film anemometer onboard MSL, as well as the details about the case where we study the turbulent fluctuations of the Nu number past REMS and the anemometer on it.

3.1 Wind sensors for Martian atmosphere

Many Mars missions have been developed since the 60’s and five of them have been on the surface of Mars with wind sensors, sending data to the ground station here on Earth. Both Viking 1 and 2 carried the Viking Meteorology Instrument System (VMIS) as a part of their payloads. This system included a meteorology boom, which held wind velocity and wind direction sensors that extended out and up from the top of one of the legs of the lander. The direction of the wind was measured with a quadrant sensor (photo detector with active photodiode areas), while the wind speed was obtained by an overheat hot-film sensor array.

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Chapter 3 Turbulent fluctuations of N u number in Martian wind sensors 25

The Mars Pathfinder, which was a lander that were more of a demonstration mission (to show that rovers could be made cheaper), was also equipped with an anemometer, a hot-wire configuration that is. Even though it suffered from calibration failures, it still provided limited data about the boundary layer profiles of the Martian atmosphere. For the Phoenix lander, instead of a hot-film anemometer, a Telltale sensor/indicator was placed on it instead.

The Telltale wind indicator is a mechanical anemometer that was designed to operate on the Martian surface as part of the meteorological package.

It consists of a lightweight cylinder suspended by Kevlar fibres and is defected under the action of wind.

Imaging of the Telltale deflection allows the wind speed and direction to be quantified, and image blur caused by its oscillations provides information about wind turbulence. Figure 3.1 shows how this kind of anemometer looks like since it differs from the previous sensors in the sense of design and function.

Figure 3.1: Phoenix lander’s Telltale anemometer[17].

Lastly, we have the hot-film anemometer onboard the Mars Science Laboratory lander, which we will go into detail in the next section.

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Table 3.1: Anemometers sent to Mars

Rover Type of wind sensor

MSL Hot-film

Phoenix Telltale

Mars Pathfinder Hot-wire (6-wire configuration) Viking 1 Hot-film (3-film configuration) Viking 2 Hot-film (3-film configuration)

3.1.1 Mars Science Laboratory

There have been many rovers studying the conditions on Mars with anemometers. Two examples are the Viking mission and Mars Pathfinder mission. The latest mission is the Mars Science Laboratory, which is planned to be replaced by Mars 2020 in a couple of years.

Mars Science Laboratory, also known as the Curiosity rover, was designed to give infor-mation on whether or not Mars had an environment to support small life form during its younger days. To be able to answer such a complex question, the rover carries some of the most sophisticated number of instruments for scientific studies ever sent to the surface of Mars.

Since 2012, it has provided us with daily information regarding the Martian environment, such as soil and air temperature, thanks to a number of sensors onboard the rover. Even sample analyses have been possible with Curiosity from the day it landed thanks to the onboard laboratory. It can either scoop the soil or drill rocks surrounding it, which gives us more information about the chemical components of the neighbouring material.

The other reason behind the investigation about the geological setting is to detect chem-ical building blocks of life, for example carbon. Carbon carries a great portion of infor-mation, especially if one wants to learn more about a planet’s past.

The operation time for MSL is still uncertain, but NASA extended it in October 2016 for two additional years.

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Figure 3.2: Mars Science Laboratory rover[18].

3.1.1.1 The Environmental Sensor Suite for the Mars Science Laboratory Rover

Onboard the Mars Curiosity rover is the Rover Environmental Monitoring Station, or shortly REMS, which was designed to measure and feed us with daily and seasonal reports regarding the atmospheric humidity, pressure, wind direction, and other condi-tions at the surface of Mars.

The REMS was designed to save data from six atmospheric parameters, namely air- and ground temperature, wind speed and direction, UV radiation, and relative humidity. Onboard the rover are also, as mentioned, the Wind Sensors (WS). These are based on hot-film anemometry and are composed of three recording points; two of them being booms attached to the Remote Sensing Mast (RSM) and the Ultraviolet Sensor (UVS) located on the rover’s body.

Lastly, we have the Instrument Control Unit (ICU) inside the body of the rover.[2]. The two small booms on the rover registers wind data from the vertical and horizontal components of the speed. The thought behind this configuration was to describe the flow of the air near the BL of Mars from dust devils, dust storms, and winds from different directions.

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Figure 3.3: An illustration of the two booms as well as the UV sensor onboard MSL[19].

3.1.1.2 An overview of the Instruments onboard

The rover, which is almost the size of a car, has various instruments with different tasks. Below is a short presentation for each of them.

Figure 3.4: The sensors and their positions[20].

• Ultraviolet Sensor: On the deck of MSL is the UV sensor (UVS), which is a cluster of six photodiodes, which covers a diverse spectral region to extend the UVS spectrum.

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the deposition of dust. The individual diodes cover a combined spectrum of 200 to 380nm[2].

• Air Temperature Sensor: The Air temperature sensor offers temperature anal-yses at a precision of 5K and a resolution of 0.1K[2]_.

It is required to have a resolution this high for the air temperature during mea-surements of subtle temperature variations across active features. On example of these are the dust devils that can appear on the surface.

• Pressure Sensor: The pressure sensor, which is placed REMS-ICU box, works at a pressure range of 1 to 1150 Pa[2] with an end of life precision of 20 Pa. This sensor offers surface pressure data at a resolution of 0.5 Pa.

• Ground Temperature Sensor: The Ground Temperature Sensor is placed on the side, facing Boom 1. It’s objective is to collect data about the surface temper-ature on Mars, which is crucial for determination of physical activities regarding the planet’s surface (e.g., the water cycle that is happening on the planet). The range of temperatures that the instrument can sense lies between 250 to 300 K, with a resolution of 2K and an precision of 10 K[2]_{. This range covers the whole}

spectrum of temperatures occurring on the planet.

• Wind Sensor: The speed of the wind and its direction is determined from data given by three 2D wind sensors on both booms.

Placed at a separation of 120◦ around the mast (in the front, center of the rover) axis each boom registers local wind speed and direction in the plane of the sensor[2]. These data points are enough to decide speed as well as pitch and yaw angles of both booms in relation with the direction of the air flow.

• Humidity Sensor: The humidity Sensor (HS) of REMS is positioned in the boom 2, which is directed in the rover’s driving direction. The device is attached on the boom offering ventilation with the atmosphere surrounding it through a filter that is shielding the sensor from dust that is airborne.

The results of the final comparative humidity seem to be substantial and are aligned with previous observations (indirect ones) of the overall atmospheric visible water content.

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3.1.2 Thermal anemometry

The most common technique used in order to measure instant fluid velocity is by using thermal anemometry, which was mentioned earlier. The method relies on the heat loss due to convection to the neighbouring fluid from an electrically heated sensing probe or element. If only the velocity of the fluid changes, then the loss of heat can be understood as a measure of that variable. MSL has a thermal anemometer onboard, namely a hot film anemometer. We will go into more detail regarding the sensor in this section.

3.1.2.1 MSL Wind Sensor (WS)

The wind sensor (WS) that is on MSL measures the wind speed in the vertical and horizontal courses, as well as the wind direction. WS is founded on hot film anemometry. There are two units of REMS wind sensor and both of them are attached on individual booms. The booms are designed to hold the units of the wind sensor in order to lessen aerodynamics effects, but also to reduce the weight.

The position of both boom 1 and boom 2 was mentioned earlier. The different positions of them let MSL to calibrate data from the surrounding wind from many directions, and in this way it can be guaranteed that the perturbation of the mast (which the booms are placed on) affects just one unit of sensor at a time.

Table 3.2: Table of Wind Sensor Measuring Characteristics[1].

Range Resolution Sampling Horizontal Wind Speed 0-70 m/s 0.5 m/s 1 Hz Horizontal Wind Direction 0-360◦ 30◦ 1 Hz Vertical Wind Speed 0-10 m/s 0.5 m/s 1 Hz

Figure 3.5 is showing how the sensor looks like.

We can see four hot dice transducer boards, both of them are built on the concept of hot film anemometry. The dices are gathered in a square formation and are held at a fixed temperature variance with respect to a cold (or reference) die. This, by using a control loop that feeds the hot die with power.

The power that is being delivered to the hot die is being measured by the circuit, and since the change of the temperature is known, the hot die’s thermal conductance to the surrounding CO2 can then be possible to compute.

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Figure 3.5: Closeup of boom 1 and the sensor[2].

information regarding the thermal bit. This is important in order to evaluate losses of radiation.

The thought behind the boom and hot-film anemometer configuration is rather straight-forward. They are placed on booms in order to be exposed to as much wind as possible. To add, the 120 degrees separation between them insures that at least one of them will record clean wind data for any given wind direction[2].

By installing them on a mast, a bit over the surface, the team can be sure that the sensors are exposed to less surface material, such as dust particles that can destroy the anemometers. There is also a 50 mm difference in height between the booms to minimize mutual wind perturbation[2].

Additional reasons to why scientists chose to use a chip (key piece of the anemometer) to measure the wind on Mars is that it is more efficient in terms of energy than those previously developed, which gives more accurate readings[21].

And lastly, no moving parts can assure that nothing breaks on the way to Mars, as well as on Mars. Moving part increases this chance further.

3.1.2.2 REMS operational Requirements

As mentioned, there are two booms on the rover. Boom 1, which faces towards the side and slightly to the back of Curiosity, has a set of IR sensors. These are studying the intensity of IR radiation that is emitted by the Martian surface, which provides an approximation of the ground temperature.

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The separation between them is 120◦in azimuth[2] and this is to assure that at least one of the booms are saving wind data for any known direction of it. Figure 3.3 illustrates the relative point of location of the booms. The height difference between them is 5 cm and the reason behind this is to reduce mutual wind perturbation[2].

A couple months after the rover’s descend on Mars, one of the two instruments that are recording the wind speed began to send data with errors. It was later concluded that the instrument might have been hit by a rock on Mars and thus leaving it broken, which explains the errors. Even though this happened, Curiosity is still able to send data regarding the wind speed and other conditions due to the many sensors onboard it.

Table 3.3: REMS dimensions and mass Components[2]

Component Dimensions L × W × H(mm3) Mass(kg) Boom 1 151.27 × 56.18 × 92.4 0.154 Boom 2 151.08 × 56.16 × 93.77 0.147

3.1.2.3 REMS Wind Sensor’s Operational requirements

Below is a list with the operational requirements for REMS[2]

• Speed of the wind shall be based on info given by three 2D wind sensors on each boom.

• The sensors shall be 120 ◦ apart on the boom axis and they will gather data from the direction of the wind as well as the local speed of the wind in the same plane of the sensor.

• It shall determine horizontal wind speed with an accuracy of 1m/s, in the range between 0-70 m/s. The resolution shall be 0.5 m/s.

• The directional accuracy is expected to be better than 30 ◦.

• For wind range in the vertical direction it shall be 0-10 m/s, and the resolution and accuracy shall match the one for the horizontal wind.

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Table 3.4: REMS and its Operational Requirements[3]

Parameter Range Resolution Accuracy Vertical speed 0 − 10m/s 0.5m/s 1m/s Horizontal speed 0 − 70m/s 0.5m/s 1m/s

Direction 360◦ 1◦ 30◦

3.1.3 Estimation of parameters

Before the simulation can be run, some estimations and preparations are necessary to be done in order to get reasonable results in the end. Therefore, we need to look at the parameters that are desired in order to perform LES and RANS simulations.

3.1.3.1 Justification of chosen atmospheric values

By looking at the Equations 3.1 to 3.1, we see that there are some values (e.g. velocity) that are necessary to be known in order to understand the fluid-anemometer interaction. Therefore, during the pre-processing stage, one important task was to find values that can be set as our parameters before the simulation.

The following numbers were used when calculating the dimensionless constants that we pointed out earlier

T= 210 K , ∆T =30 K, v=10 m/s.

Let us look at each parameter separately and justify why these were used.

First, the temperature T = 210K, which is the global surface temperature on Mars[4]. This is a value that has been averaged after numerous observations done by satellites and rovers through the years. Therefore, we found it as one of the reasons to why this should be used as our temperature parameter.

Another reason to why this value were chosen, besides it being the global average, has to do with the atmospheric pressure of the planet. The pressure of the atmosphere on the surface, at zero ground level, is roughly 600 Pa. The temperature of the regolith on the surface, at this pressure, lies between 133 K to 300 K, with an average of circa 210 K[22].

LES Simulation of Hot-wire Anemometers

LES Simulation of Hot-wire Anemometers

Assiye Süer

Space Engineering, masters level

2017

Final Degree Project

LES simulation of hot-wire anemometers

Abstract

Acknowledgements

Contents

List of Figures

List of Tables

Abbreviations

Physical Constants

Symbols

This thesis is dedicated to my beloved mother.

Introduction

Objectives

Scope

Chapter 1

State-of-the-Art

1.1

Planet Mars

1.2

Atmosphere of Mars

1.3

Hot-wire and Hot-film anemometers

Chapter 2

Turbulent heat transfer

2.1

Conduction, Convection and Radiation

2.2

Suitability of continuum approach in Mars

2.3

Turbulence models

2.4

Dimensionless form of the NS equations

2.5

Natural Convection

2.6

Governing parameters for convection problems

2.7

Boussinesq approximation

2.8

Dimensionless numbers - Nu, Re, Pr, and Ra numbers

Chapter 3

Turbulent fluctuations of the N u

number in Martian wind sensors

3.1

Wind sensors for Martian atmosphere