Master thesis : Implementation in OpenFoam of a thermal ... · the sensitivity analysis for the turbulent intensity at inlets shows that a 1% variation of the turbulent intensity

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Master thesis : Implementation in OpenFoam of a thermal-fluid analysis for

thermal internal flows

Auteur : Martinez Carrascal, Jon

Promoteur(s) : Terrapon, Vincent

Faculté : Faculté des Sciences appliquées

Diplôme : Master en ingénieur civil en aérospatiale, à finalité spécialisée en "aerospace engineering"

Année académique : 2016-2017

URI/URL : http://hdl.handle.net/2268.2/3315

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University of Liege

Faculty of Applied Sciences

Graduation studies conducted for obtaining the master’sdegree in aerospace engineering by Jon Martínez

Implementation in OpenFoam ofthermal-fluid analysis for internal flows

Industrial Tutor: Salvador LucasAcademic advisor: Terrapon VincentJury members: Ponthot Jean-Philipe, Terrapon Vincent, Salvador

Lucas, Dewallef Pierre

February-August 2017

Abstract

A CFD analysis of the flow inside of the VMU MkII in micro-gravity conditions is pre-sented. In the context of complementing the different calculations made by CSL pro-fessional using the ESATAN-TMS software, this thesis will contribute to support theexisting data of the unit regarding the airflow and the components of the VMU.

After considering the characteristics and conditions of the flow inside this unit, themathematical formulation of the problem is proposed. Then, the numerical implementa-tion is presented and for this task, the finite volume method OpenFOAM software is used.

A CAD model of the VMU MkII is been loaded and re-built using the SALOME soft-ware. After the model is meshed using the snappyHexMesh OpenFOAM utility, a meshconvergence study has been performed defining the mesh where the final results will beobtained.

The results of the thesis display an impact of the bottom rails of the FPIU of 60% inthe velocity field and a maximum discrepancy in velocity of 22.37% between the k − ωand k − ω SST turbulence models. On the other hand, it is observed that the meantemperatures of the components surpass the thermal requirements of the VCU by 4.2K

for the VCU, by 53.38K for the CPU and by 97.95K for the SA50-120 modules. Also,the sensitivity analysis for the turbulent intensity at inlets shows that a 1% variationof the turbulent intensity at the inlets gives rise to an average variation of the velocitymagnitude of 0.07%.

As a conclusion, it is important to underline that the inclusion of conduction in themodeling and a different power distribution may be the reason why the mean tempera-tures of the components are overestimated. Also, due to the lack of solid experimentaldata it is not possible to confirm which turbulence model is more suitable for the case ofstudy. Finally, the small variations of the solution due to the sensitivity analysis may bean indicator that the boundary condition for the turbulent kinetic energy at the inlets iswell posed.

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Contents

Symbols, Abbreviations and Constants viii

1 Introduction 11.1 Thermal control of spacecraft . . . . . . . . . . . . . . . . . . . . . . . . 21.2 VMU MkII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Main components . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Power breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Forced convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Thermal-fluid conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Mathematical formulation 122.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Quasi-steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Incompressibility and continuity equation . . . . . . . . . . . . . . . . . . 15

2.4.1 Pressure unsteadiness compressibility . . . . . . . . . . . . . . . . 172.4.2 Compressibility of large scale flows . . . . . . . . . . . . . . . . . 172.4.3 Compressiblity caused by the coefficient of volumetric expansion . 182.4.4 Compressibility due to temperature gradient . . . . . . . . . . . . 182.4.5 Derivation of the continuity equation . . . . . . . . . . . . . . . . 18

2.5 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7.1 Two equation turbulence models . . . . . . . . . . . . . . . . . . . 242.7.2 k − ω turbulence model . . . . . . . . . . . . . . . . . . . . . . . 252.7.3 k − ω SST turbulence model . . . . . . . . . . . . . . . . . . . . 25

2.8 Final formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.9 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9.1 Boundary conditions at inlets . . . . . . . . . . . . . . . . . . . . 282.9.2 Boundary conditions at outlet . . . . . . . . . . . . . . . . . . . . 292.9.3 Boundary conditions at walls . . . . . . . . . . . . . . . . . . . . 292.9.4 Near wall treatment of the turbulent parameters . . . . . . . . . . 31

2.10 Heat transfer mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 372.10.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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3 Numerical approach 393.1 Discretization of the computational domain . . . . . . . . . . . . . . . . 40

3.1.1 Mesh generation and importation . . . . . . . . . . . . . . . . . . 413.1.2 snappyHexMesh utility . . . . . . . . . . . . . . . . . . . . . . . . 433.1.3 Mesh convergence study . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Discretization of equations . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.1 Discrezitation of the convective terms . . . . . . . . . . . . . . . . 493.2.2 Discrezitation of the diffusive terms . . . . . . . . . . . . . . . . . 503.2.3 Discrezitation of the source terms . . . . . . . . . . . . . . . . . . 523.2.4 Final form of the discretized transport equation . . . . . . . . . . 533.2.5 Under-relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.6 Solution convergence . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 SIMPLE algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Implementation of the boundary conditions . . . . . . . . . . . . . . . . . 58

4 Results 614.1 Default simulation case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Thermal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1.3 Turbulence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.4 Impact of the VCU, mid-stiffener and FPIU rails . . . . . . . . . 73

4.2 Comparison of turbulence models . . . . . . . . . . . . . . . . . . . . . . 764.3 Regimes comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Final conclusions 87

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List of Figures

1.1 CAD model of the VMU from the outside [6]. . . . . . . . . . . . . . . . 31.2 VMU MkII CAD model (exploded) [6]. . . . . . . . . . . . . . . . . . . . 31.3 Exploded view of the CAD model of the VCU [9]. . . . . . . . . . . . . . 41.4 CAD model of the FPIU [6]. . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 CAD model of the CPU. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 CAD model of the HDD cards, Enet board and FPGA. . . . . . . . . . . 61.7 Simplified sketch of the forced convection phenomenon. . . . . . . . . . . 82.1 Shear stress over strain rate for different fluid models. . . . . . . . . . . . 142.2 Laminar and turbulent flow around a cylinder. . . . . . . . . . . . . . . . 222.3 Channel flow for DNS, LES and RANS. . . . . . . . . . . . . . . . . . . . 232.4 Percentages of the dissipated power assigned to each element of the CPU

slot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Sketch of the various regions of the turbulent boundary layer in inner and

outer variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Law of the wall displaying the viscous sublayer and the log region. . . . . 363.1 Structured and unstructured mesh for the surroundings of a blunt body. . 413.2 Example of high aspect ratios for triangular and quadrilateral cells. . . . 423.3 Unstructured mesh generated by the meshing module of SALOME based

on a simplified CAD model of the VMU including inlets, outlet, mid-stiffener and walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Schematic example of the initial mesh generation displaying the base meshand the CAD model (the car) [1]. . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Refinement levels from 0 to 3. The level 0 of refinement represents theoriginal size of the control volume of the base mesh [1]. . . . . . . . . . . 44

3.6 Schematic example of the outer surface refinement [1]. . . . . . . . . . . . 443.7 Schematic example of the cell removal [1]. . . . . . . . . . . . . . . . . . 443.8 Schematic example of the surface snapping [1]. . . . . . . . . . . . . . . . 453.9 Structured mesh generated with the snappyHexMesh utility. . . . . . . . 453.10 Mean heat flux of the bottom wall over the number of steps (1 step = 100

iterations) for four different meshes. . . . . . . . . . . . . . . . . . . . . . 473.11 Mean temperature of the second HDD over the number of steps (1 step =

100 iterations) for four different meshes. . . . . . . . . . . . . . . . . . . 473.12 Central scheme for φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.13 Upwind scheme for φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.14 Orthogonal correction in a non-orthogonal mesh. . . . . . . . . . . . . . . 51

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3.15 Final residuals (in logarithmic scale) for the fields U , p, T , k and ω as afunction of the number of iterations for the final model of the VMU MkII. 56

3.16 Flow chart of the SIMPLE algorithm. . . . . . . . . . . . . . . . . . . . . 573.17 y+ values at walls over the number of steps for the mesh used in the

convergence study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 Front 2D plot of the velocity field in m/s inside the FPIU displaying the

vector field (left) and isolines (right). . . . . . . . . . . . . . . . . . . . . 624.2 Velocity profile of the flow through the HDD slots. Counting from 1 to 4

from left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Left side 2D plot of the velocity field in m/s inside the FPIU (CPU slot)

displaying the vector field (left) and isolines (right). . . . . . . . . . . . . 644.4 Transversal cut of the VMU showing the velocity field in m/s displaying

the vector field (left) and isolines (right). . . . . . . . . . . . . . . . . . . 644.5 Velocity profile of the flow through the 11 fins of the VCU. . . . . . . . 654.6 3D view of the streamlines of the airflow inside the VMU MkII. General

view (left) and top view (right). . . . . . . . . . . . . . . . . . . . . . . . 654.7 3D view of the temperature distribution (in K) of the CPU and the hea-

texchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.8 Temperature field (in K) and velocity vector field (in m/s) of a transversal

cut of the CPU slot of the FPIU. From left to right: FPIU suppliers 1, 2and VCU supplier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.9 3D view of the temperature distribution (in K) of the SA50-120 modules. 684.10 Middle cross-section of the temperature profile of the SA50-120 modules.

From left to right: VCU supplier, FPIU supplier 2 and 1. . . . . . . . . 694.11 2D plot of the temperature distribution (in K) and velocity vector field

(in m/s) of the SA50-120 modules. . . . . . . . . . . . . . . . . . . . . . 694.12 2D plot of the temperature distribution (in K) and velocity vector field

(in m/s) around the VCU. . . . . . . . . . . . . . . . . . . . . . . . . . . 704.13 Temperature profile of the flow (in K) through the 11 fins of the VCU. . 704.14 Specific turbulent kinetic energy field (in m2/s2) and velocity field isolines

(in m/s) inside the FPIU. . . . . . . . . . . . . . . . . . . . . . . . . . . 724.15 Specific turbulent kinetic energy field (in m2/s2) of a transversal cut of

the VMU MkII featuring the outlet, VCU, mid-stiffener, left inlet andproximity of the FPIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.16 Specific turbulent dissipation rate profile (in 1/s) across the HDD slots 2-4from left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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4.17 Transversal cut of the VMU displaying the velocity vector field (in m/s)with (right) and without (left) the VCU component. . . . . . . . . . . . . 74

4.18 2D cut of the flow near the outlet displaying the velocity vector field (inm/s) and the temperature field (in K) with (right) and without (left) theVCU component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.19 Transversal cut of the VMU displaying the velocity vector field (in m/s)with (right) and without (left) the mid-stiffener. . . . . . . . . . . . . . . 75

4.20 2D cut of the FPIU displaying the velocity isolines (in m/s) with (right)and without (left) the FPIU bottom rails. . . . . . . . . . . . . . . . . . 75

4.21 Velocity profile (in m/s) of the flow across the HDD boards with andwhitout the FPIU bottom rails. . . . . . . . . . . . . . . . . . . . . . . . 76

4.22 Velocity profile (in m/s) of the flow in the cross-section of the mid-stiffenerfrom the left wall to the right wall of the VMU for the k−ω and k−ω SSTturbulence models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.23 Velocity profile (in m/s) of the flow across the HDD slots from 1-4 fromleft to right, for the k − ω and k − ω SST turbulence models. . . . . . . 78

4.24 Velocity profile (in m/s) of the flow across left duct, for the k − ω andk − ω SST turbulence models. . . . . . . . . . . . . . . . . . . . . . . . . 78

4.25 Temperature profile (in K) of the flow across the HDD slots from 1-4 fromleft to right, for the k − ω and k − ω SST turbulence models. . . . . . . 79

4.26 Turbulent kinetic energy profile (in m2/s2) of the flow across the HDDslots from 1-4 from left to right, for the k − ω and k − ω SST turbulencemodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.27 Specific turbulent dissipation rate profile (in 1/s) of the flow across theHDD slots from 1-4 from left to right, for the k−ω and k−ω SST turbu-lence models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.28 Velocity profile (in m/s) at the left duct for the low, medium and highregimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.29 Turbulent kinetic energy profile (in m2/s2) at the left duct for turbulentintensities of 1%, 2%, 3%, 4% and 5%. . . . . . . . . . . . . . . . . . . . 84

4.30 Velocity profile (in m/s) at the left duct for turbulent intensities of 1%,2%, 3%, 4% and 5%. General plot (left) and zoomed plot (right). . . . . 84

4.31 Specific turbulent dissipation rate profile (in 1/s) across the HDD slots1-4 from left to right, for turbulent intensities of 1%, 2%, 3%, 4% and 5%.General plot (left) and zoomed plot (right). . . . . . . . . . . . . . . . . 85

4.32 Turbulent kinetic energy profile (in m2/s2) across the HDD slots 1-4 fromleft to right, for turbulent intensities of 1%, 2%, 3%, 4% and 5%. . . . . . 85

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4.33 Velocity profile (in m/s) across the HDD slots 1-4 from left to right, forturbulent intensities of 1%, 2%, 3%, 4% and 5%. . . . . . . . . . . . . . . 86

5.1 Mean heat flux at the right back wall over the number of steps for fourmeshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Mean heat flux at the left wall over the number of steps for four meshes. 905.3 Mean temperature at the CPU over the number of steps for four meshes. 915.4 Mean temperature at the heat exchanger over the number of steps for four

meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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List of Tables

1 Power breakdown of the components of the FPIU ordered from 1 to 12starting to count from left to right [6]. . . . . . . . . . . . . . . . . . . . 7

2 Power breakdown of the components of the VCU and SA50-120 modulesv[6]. 73 Properties of the air in the Columbus cabin [6]. . . . . . . . . . . . . . . 104 Thermal-fluid properties of the air in the VMU. . . . . . . . . . . . . . . 105 Boundary conditions at left and right inlets. . . . . . . . . . . . . . . . . 286 Boundary conditions at the outlet. . . . . . . . . . . . . . . . . . . . . . 297 Boundary conditions at walls. . . . . . . . . . . . . . . . . . . . . . . . . 308 Temperatures at the outer walls of the VMU. . . . . . . . . . . . . . . . 319 Used grids for the mesh convergence study. . . . . . . . . . . . . . . . . . 4610 Field and equation under-relaxation factors. . . . . . . . . . . . . . . . . 5511 Implementation in OpenFOAM of the boundary conditions at the outlet

and inlets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812 Implementation in OpenFOAM of the boundary conditions at walls. . . . 5913 Set of boundary conditions for the high and low Reynolds approaches. . . 6014 Mean temperatures, convective heat transfer coefficients of the components

and the mean velocities of the free-stream flow around them for the defaultsimulation case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

15 Mean temperatures of the components for low, medium and high flow rateregimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

16 Mean convective heat transfer coefficients of the components for low, mediumand high flow rate regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

17 Heat transported by the flow for low, medium and high flow rate regimes. 82

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Naming and Abbreviations

q Heat fluxh Convective heat transfer coefficientTs Temperature at the surfaceT∞ Temeperature of the free-stream flowκ Thermal conductivityTin Temperature at the inletTout Temperature at the outletρ Densityµ Dynamic viscosityν Kinematic viscosityc Specific heat capacity~u Velocity field~g Gravity¯τ Viscous stress tensorΦ Viscous dissipationγ Strain rateL Characteristic lenghtUc Characteristic velocitys Entropyβ Coefficient of volumetric expansiona Speed of soundα Thermal diffusivityk Turbulent kinetic energyω Specific turbulent disspation rateνT Turbulent viscosityI Turbulent intensityl Turbulent lenght scaleτω Wall shear stressu∗ Friction velocityη Turbulent length scale near wallsδ Boundary layer thicknessy+ Dimensionless vertical coordinate near wallsK Von Karman constantA Areaγ Under-relaxation factorr Normalized residual

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Gr Grashof numberSt Strouhal numberM Mach numberFr Froude numberRe Reynolds numberPr Prandtl numberRa Rayleigh numberGn Generation numberNu Nusselt numberVMU Video Management UnitFSL Fluid Science LaboratoryISS International Space StationFPIU Front Panel Interface UnitOPTP Overhead Pass Through-PanelFPGA Field Programmable Gate ArrayDNS Direct Numerical SimulationRANS Reynolds-Averaged Navier-StokesLES Large-Eddy SimulationFVM Finite Volume MethodFEM Finite Element MethodCV Control Volume

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1 Introduction

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1.1 Thermal control of spacecraft

Among all the disciplines involved in spacecraft design, thermal control is the one thatassures that all spacecraft component systems are within an acceptable temperatureranges during all mission phases. Therefore, space thermal control is required for twomain reasons:

1. Electronic and mechanical equipment usually operate efficiently and reliably onlywithin a relatively narrow temperature ranges

2. Most materials have non-zero coefficients of thermal expansion and hence temper-ature changes will imply thermal distortions

Moreover, spacecraft equipment is designed to operate most effectively at or aroundroom temperature (usually 20◦C) because most of the components used in it, whetherelectronic or mechanical, were originally designed for terrestrial use. Also, it is easierand cheaper to perform equipment development and, eventually, qualification and flightacceptance testing at room temperature.

Heat is generated both within the spacecraft and by the environment. Componentsproducing heat include rocket motors, electronic devices and batteries. On the otherhand, heat from the space environment is largely the result of solar radiation, albedo andIR loads. Heat is lost from the spacecraft by radiation (mostly IR), mainly to deep space.The balance between heat gained and heat lost will determine the spacecraft tempera-tures.

Among all those sources of heat in space, the area covered in this thesis will be theheat produced by electronic components inside the spacecraft.

All electronic devices and circuitry generate excess heat and thus require thermalmanagement to improve reliability and prevent premature failure and it is the task of athermal designer to discover the means of cooling such components. In this context, onecan find many solution to the same problem such as heat sinks, thermoelectric coolers,forced air systems (our case of study), fans or heat pipes.

1.2 VMU MkII

The VMU or Video Management Unit is an electronic box located in the InternationalSpace Station that provides videos of experiments on board the Fluid Science Laboratory

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(FSL). Its size is 2.4× 4.2× 6.4 dm.

Figure 1.1: CAD model of the VMU from the outside [6].

Similarly, the VMU MkII is located inside of a rack in the FSL situated in the Colum-bus cabin at a certain pressure, temperature and humidity conditions.

1.2.1 Main components

The components of greater relevance can be classified into dissipative or non-dissipative.On the one hand, the non-dissipative components (such as the stiffener and card rails)perform a mechanical function and on the other hand, the dissipative elements representthe different electronic components that altogether conduct the duty of the VMU.

Figure 1.2: VMU MkII CAD model (exploded) [6].

• Inlets and outletThe VMU MkII has two inlets (left and right) and one outlet. The inlets are rect-angular apertures located at the front of the box and cover an area of 0.0046m2.They incorporate a fine metallic grid in order to control turbulent phenomena at

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the entrances.

At the back, the outlet is placed. It has a circular geometry and covers an area of0.0019m2. The outlet is directly connected to an airflow loop where the fan thatgenerates the motion of the flow resides. It is important to note that the outlet isnot located on the middle of the back wall (it is displaced 1.2 dm to the left wall)which implies an asymmetric geometry.

• Mid-stiffenerThe mid-siffener is a rectangular metallic non-dissipative component. It is placedin the middle of the VMU and its function is to withstand vibrations, specially atlaunch. Its size is the same as the front area of the unit, 4.2× 2.4 dm.

• SA50-120 modulesThe SA50-120 modules are three elements suppliers of electricity to the other dis-sipative components. The first one (starting to count from the closest one to theback wall), provides current to the VCU (VMU Control Unit) and the other twosupply to the FPIU (Front Pannel Interface Unit) and the electronic cards. All ofthem have an approximated size of 5.2× 7.7× 1.1 cm

• VMU Control UnitThe VMU Control Unit or VCU in short controls the operation of the whole unit.It can be divided into three parts [9]: cover, P400k Ethernet slice and power supplyslice. The cover is a non-dissipative element that has mechanical purposes and also,it includes fins in order to enable a better heat transfer. In the other two slices thereare different electronic components and boards that dissipate heat.

Figure 1.3: Exploded view of the CAD model of the VCU [9].

The VCU is located at the back of the VMU next to the outlet and has a size of1.3 × 0.6 × 2.01 dm. Due to its location inside the unit, it will critically affect thestructure of the flow close to the outlet.

4

• FPIUThe FPIU or Front Panel Interface Unit is the element located at the front of theVMU that contains the card rails along with the electronic components inside. Thecard rails are a mechanical element directly connected to both inlets containing theaperture for the cards and causing asymmetry at the front of the unit. They presenta double purpose: guide the airflow from the inlets to the electronic cards in orderto transport the heat dissipated by them and provide a mechanical support for suchelectronic cards.

Figure 1.4: CAD model of the FPIU [6].

The air ducts are not symmetrical, being the left duct longer than the right onewhich will impact the airflow. It contains a stiffener that fulfills the same functionas the mid-stiffener but in a more localized way. Inside the card rails 12 slots areplaced, 5 of which are empty boards (with no electronics) or free space and theother 7 are the following: 4 HDD cards, CPU, Enet board and FPGA.

• HDD cardsThe HDD cards are electronic boards used for storage purposes. They occupy thefirst four cards starting to count from left to right, all four of them are identicalcopies with a dissipative element in the middle and three connectors.

• CPUThe CPU of the VMU is located at the fifth slot and it manages the video of theunit. Its slot is composed by a heat exchanger that helps transferring the heat tothe convective flow, the CPU itself located below the heat exchanger and severalelectronic components in the proximity. It is the most critical part among all theother electronic cards having the highest heat loads.

• Enet board

5

Figure 1.5: CAD model of the CPU.

The Enet board takes care of the communication between the camera and electron-ics. It is arranged in the same way as the HDD boards, a card with a dissipativeelement in the middle and three connectors. It occupies the seventh place in therail.

• FPGAThe Field Programmable Gate Array occupies the eleventh slot of the rail andcontains an array of programmable logic blocks, and a hierarchy of reconfigurableinterconnects that allow the blocks to be wired together. Its arrangement is thesame used for the HDDs or the Enet board.

Figure 1.6: CAD model of the HDD cards, Enet board and FPGA.

• OPTPThe Overhead Pass-Through Panel or OPTP in short is located at the front of theVMU and provides the astronauts access to the cards when an emergency happensor a card replacement is required.

6

1.2.2 Power breakdown

The card rails of the FPIU contains 12 slots for the electronic components explainedin the previous section. Once the steady state of the VMU is achieved, these elementsdisplay the following dissipated powers:

Slot Board name Steady power [W] Max. transient power [W]H1 CPS3101 HDD1 2.4 6.1H2 CPS3101 HDD2 2.4 6.1H3 CPS3101 HDD3 2.4 6.1H4 CPS3101 HDD4 2.4 6.1S CPS3003-SA (SBC) 31 48P1 Empty board 0 0P2 Enet board 4.5 4.5P3 Empty board 0 0P4 Free 0 0P5 Free 0 0P6 FPGA 1.19 3.2P7 Empty board 0 0

Table 1: Power breakdown of the components of the FPIU ordered from 1 to 12 startingto count from left to right [6].

The other elements outside the FPIU such as the VCU and the SA50-120 moduleshave the following associated steady state dissipated powers:

Name Steady power [W] Max. transient power [W]SA50-120 modules

FPIU supplier 1 8.17 14.14FPIU supplier 2 8.17 14.14VCU supplier 3.65 7.38

VCUPower supply 3.92 11.971553 Board 2.98 4.44P400k Board 11.6 18.93

SGMII / Ethernet Board 0.33 4.3LUP + LTC Board 1.22 1.4

Table 2: Power breakdown of the components of the VCU and SA50-120 modulesv[6].

It is important to note that the transient power of the components will not be usedin this study. Also, the thermal requirements of the VMU MkII [6] state that all theelectronic components presented above must diplay a mean temperature inside a thermalrange of [273, 333]K except for the CPU that must operate under 378K. In addition to

7

that, the power requirements state that the VMU MkII heat load to FSL air loop shallbe 82% of the total power dissipation, for a maximum of 140 W which results in 114.8W .

1.3 Forced convection

Convection is the mechanism of heat transfer through a fluid in the presence of bulk fluidmotion. Convection is classified as natural (or free) and forced convection depending onhow the fluid motion is initiated. In natural convection, any fluid motion is caused bynatural means such as the buoyancy effect, i.e. the rise of warmer fluid and fall the coolerfluid. Whereas in forced convection, the fluid is forced to flow over a surface or in a tubeby external means such as a pump or fan.

Convection heat transfer is complicated since it involves fluid motion as well as heatconduction. The fluid motion enhances heat transfer (the higher the velocity the higherthe heat transfer rate). The rate of convection heat transfer is expressed by Newton’slaw of cooling:

˙qconv = h(Ts − T∞) (1.1)

Where ˙qconv is the power heat dissipated by a surface, h represents the convective heattransfer coefficient, Ts the temperature of the surface and T∞ is the temperature of theupstream flow. The convective heat transfer coefficient h strongly depends on the fluidproperties and roughness of the solid surface, and the type of the fluid flow (laminar orturbulent).

Figure 1.7: Simplified sketch of the forced convection phenomenon.

If it is assumed that the velocity of the fluid is zero at the wall, this assumption iscalled no-slip condition. As a result, the heat transfer from the solid surface to the fluidlayer adjacent to the surface is by pure conduction, since the fluid is motionless. Thus, theconvection heat transfer coefficient, in general, varies along the flow direction. The mean

8

or average convection heat transfer coefficient for a surface is determined by (properly)averaging the local heat transfer coefficient over the entire surface.

˙qconv = ˙qcond = −κfluid∂T

∂y

∣∣∣∣y=0

(1.2)

Where ˙qcond is the heat power transferred by conduction and κfluid represents theconductivity of the fluid. Therefore, combining the two equations presented above, onecan isolate h obtaining:

h =−κfluid ∂T

∂y|y=0

(Ts − T∞)(1.3)

One can spot a clear relation between the convective heat transfer coefficient and theconductivity of the fluid, the thermal gradient of the surface, the temperature of thesurface and the temperature of the fluid.

In our case of study, it is of great interest to determine the convective heat transfercoefficients (h) since the electronic components of the VMU are restricted to a certaintemperature range (Ts). As the flow inside the Video Management Unit is driven byforced convection generated by a fan outside the unit, it is important to note the differentrotating regimes of the fan since they will establish the velocity profiles of each regime.In order to simplify all the endless flow regimes that the VMU could perform, it has beendecided to divided them into three characteristic regimes that are represented by the flowrate at the outlet:

• Low airflow regime: It is the minimum flow rate at which the VMU can fulfillits goal and it is equal to 820 l/min.

• Medium airflow regime: It represents an average case of the typical performanceof the VMU with a flow rate of 830 l/min.

• High airflow regime: It is the maximum flow rate 850 l/min.

1.4 Thermal-fluid conditions

The Video Management Unit is located in the Columbus chamber inside the InternationalSpace Station. There, the air is at certain humidity, pressure and temperature conditionsthat will impact on the properties of the air coming in the VMU.

Those properties determine the thermal-fluid properties of the air that will criticallyaffect the performance of the VMU.

9

Property ValuePressure 986 hPa

Mean temperature 20.8 ◦CRelative humidity 47.5 %

Temperature left inlet 28 ◦CTemperature right inlet 25.3 ◦C

Table 3: Properties of the air in the Columbus cabin [6].

Property ValueConductivity (κ) 0.0267 W/mK

Density (ρ) 1.13 kg/m3

Dynamic viscosity (µ) 2 ·10−5 kg/smKinematic viscosity (ν) 1.77 ·10−5 m2/sSpecific heat capacity (c) 1.4 kJ/kg KThermal diffusivity (α) 2.1 ·10−5 m2/s

Table 4: Thermal-fluid properties of the air in the VMU.

1.5 Objective of the thesis

The CSL (Centre Spatial de Liège) has made different calculations to deduce the thermalbehaviour of the VMU MkII using ESATAN-TMS. This software usually displays a goodperformance when it comes to radiative heat transfer computations, but when it facesthermal-hydraulic problems it only produces simple estimations (with a few fluid nodes).In this framework, a realistic CFD analysis of the unit was required and among all theCFD available softwares, OpenFOAM was chosen due to its flexibility and open-sourceadvantage.

In order to implement the CFD study, several aspects need to be covered:

• Properly create, import and mesh the CAD models of the Video Management Unit

• Choose the appropriate OpenFOAM solver and modify it to meet the characteristicsof the problem

• Perform convergence studies to quantify the impact of the mesh in the solution andobserve convergence in the discretization of the equations

• Account the source of error of the turbulence model

• Develop different post-processing utilities to calculate heat fluxes, average temper-atures of components and the convective heat transfer coefficients

10

The information obtained from the study will complement the other thermal com-putations of the CSL. This information comprises: general and localized (in the criticalparts) structure of the flow for different regimes, temperature of electronic components,convective heat transfer coefficients, heat fluxes at walls and a sensitivity analysis of theturbulent intensity.

Among all the mentioned data, the convective heat transfer coefficients are of greatimportance since they will be used in a classic thermal analysis performed with the ESA-TAN software.

Concretely, the simulation cases that are carried out are all in microgravity conditions(emulating a flight model) in which certain parameters are varied:

• Flow rate (fan regime): low, medium and high.

• Impact of certain elements of the VMU : mid-stiffener, VCU and rails of the FPIU.

• Comparison of two turbulence models: k − ω and k − ω SST .

• Sensitivity analysis of the turbulence intensity at inlets covering Iε[1%, 5%]

In these simulations, both experimental and computational test data is used as sup-port and not for rigorous correlation purposes.

To sum up, the aim of the thesis will be to conduct a CFD analysis of the VMU MkIIimplemented in OpenFOAM that contributes to complete all the other numerical results(that will not be discussed in this study) done by CSL employees.

11

2 Mathematical formulation

12

2.1 Governing equations

The equations that describe the behaviour of a fluid are a set of non-linear partial differ-ential equations. These equations govern in any phenomenon where Newtonian fluids areinvolved. They are three equations, the continuity equation that represents the conserva-tion of mass in a domain, the momentum or Navier-Stokes equation that represents thesecond law of Newton applied to a fluid and the energy equation that allows to analyzethe heat transference in the fluid. Here below the equations are shown:

∂ρ

∂t+∇ · (ρ~u) = 0 (2.1)

∂(ρ~u)

∂t+ ρ~u · ∇(~u) = −∇p+ ρ~g +∇ · ¯τ (2.2)

ρc∂T

∂t+ ρc~u · ∇(T ) = ∇ · (κ∇T ) + Φ (2.3)

Where ρ is the density of the fluid, t represents the temporal variable, p is the pres-sure, ~g represents the acceleration of gravity, ¯τ is the shear-stress tensor, κ represents theconductivity of the fluid, T the temperature, c the specific heat capacity and Φ representsthe energetic dissipation.

2.2 Newtonian fluid

The first hypothesis that is going to be made is that the concerning fluid is a Newtonianfluid. A fluid is said to be Newtonian if its viscous stresses are, at every point, linearlyproportional to the local strain rate (the rate of change of its deformation over time).

τ︸︷︷︸Shear stress

= µ︸︷︷︸Viscosity

× γ︸︷︷︸Strain rate

(2.4)

This relationship is known as the Newton’s Law of Viscosity, where the proportional-

13

ity constant µ is the viscosity of the fluid.

Figure 2.1: Shear stress over strain rate for different fluid models.

As a result, the viscosity will remain constant over time. If the fluid is also isotropic,that is to say, with the same mechanical properties along any direction, the viscosity willbe constant over space as well. Some examples of Newtonian fluids include water, organicsolvents, honey and air (our case of study).

2.3 Quasi-steady flow

One of the most important, and often easiest to recognize, distinctions is that associatedwith steady and unsteady flow. In the most general case all flow properties depend ontime; for example the functional dependence of pressure at any point (x, y, z) at anyinstant might be denoted p(x, y, z, t). But, if all properties of a flow are independent oftime, then the flow is steady; otherwise, it is unsteady [7].

Real physical flows essentially always exhibit some degree of unsteadiness, but inmany situations the time dependence may be sufficiently weak (slow) to justify a steady-state analysis, which in such a case would often be termed a quasi-steady analysis. Oneway of quantifying the steadiness or unsteadiness of a flow is estimating the Strouhaldimensionless number. This number can be expressed as:

St =ωL

Uc∼ trt0

with tr ∼L

Uc, t0 ∼

1

ω(2.5)

Where ω is the characteristic pulsation frequency, L the characteristic length, Uc rep-

14

resents the characteristic velocity, tr the time of residence and t0 the characteristic timeof pulsation.

The Strouhal number can be important when analyzing unsteady, oscillating flowproblems. This dimensionless number represents a measure of the ratio of the inertialforces due to the unsteadiness of the flow or local acceleration to the inertial forces due tochanges in velocity from one point to an other in the flow field [12]. Therefore, in orderto have a quasi-steady flow, the Strouhal number must meet the following condition:

St� 1 (2.6)

Thus, taking into account the dimensions of the VMU MkII mentioned in Chapter 1L ∼ 10−1m and having a characteristic velocity of Uc ∼ 1m/s, the critical characteristicpulsation frequency ωc can be calculated:

St ∼ ω · 10−1

1� 1 (2.7)

Which gives a critical characteristic pulsation frequency of ωc ∼ 10Hz. That is to say,for frequencies much lower than ωc the flow of study can be considered as quasi-steadywhich is presumed by CSL professionals.

As demonstrating the quasi-steadiness of a flow is a more complex task that wouldrequire a profound study of the flow, CSL professionals assume the quasi-steady conditionbased on the fact that measurements of the VMU (pressure, room temperatures, axialfan velocity...) show very small variations around a mean value. Therefore, for simplicitythe quasy-steady state is simulated.

2.4 Incompressibility and continuity equation

According to Thermodynamics the differential of the density of a fluid ρ can be writtenas a function of two other thermodynamic variables, such as pressure p and entropy perunit mass s. Then, at first order:

dρ =∂ρ

∂p

∣∣∣∣s

dp+∂ρ

∂s

∣∣∣∣p

ds (2.8)

Both coefficients multiplying dp and ds are known as Thermodynamic coefficients andthey correspond to:

15

∂ρ

∂p

∣∣∣∣s

=1

a2(2.9)

∂ρ

∂s

∣∣∣∣p

=Tρβ

c(2.10)

Where a is the speed of sound and β is the coefficient of volumetric expansion. Takingthe material derivative of the density and using the Thermodynamic coefficients, oneobtains:

Dρ

Dt=

1

a2

Dp

Dt+T 2β

c

Ds

Dt(2.11)

And from the definition of material derivative:

Dp

Dt=∂p

∂t+ ~u · ∇p (2.12)

Also, the differential entropy conservation yields:

Ds

Dt=

Φ

ρT+ κ∇2T

ρT(2.13)

Where the equation states that change in entropy in a fluid volume is caused only byviscous dissipation Φ and the divergence of the heat flux. Now, if we take the continuityequation and apply the conditions of steady and isotropic flow, one gets:

Dρ

Dt= ρ∇ · ~u = 0 (2.14)

Hence, inserting the material derivative of the density, the divergence of the velocityfield will be equal to:

∇ · ~u =1

ρ

[1

a2

(∂p

∂t+ ~u · ∇p

)+Tβ

c

(Φ

T+ κ∇2T

T

)](2.15)

In order to obtain∇·~u ∼ 0 the four terms of the equation must tend to zero. Therefore,one can define four different cases for which a fluid can be compressible. Thus, we aregoing to look at the order of magnitude expected of each term of the right hand side ofthe equation, relative to the order of magnitude of the left hand side. Then, the order ofmagnitude of the right hand side is:

∇ · ~u ∼ O

(UcL

)(2.16)

Where Uc and L represent the characteristic velocity and length respectively.

16

2.4.1 Pressure unsteadiness compressibility

Despite having steady flow and thus, knowing that this term can be neglected, it can beeasily estimated. The term ∂p/∂t represents the unsteadiness of the pressure field and itis usually associated to pulses of pressure translated across the fluid generating pulses ofdensity.

1ρa2

∂p∂t

∇ · ~u∼O(ω∆pρa2

)O(UcL

) = St ·M2 (2.17)

Where St is the Strouhal number and M is the Mach number that represents theratio between the velocity of the flow and the speed of sound.

M =U

a(2.18)

As the order of magnitude of the characteristic velocity is Uc ∼ 1m/s and a ∼ 102m/s

leading toM2 � 1 and having St� 1 due to the quasi-steady flow condition, one derives:

St ·M2 � 1 (2.19)

2.4.2 Compressibility of large scale flows

The second term of the right hand side of the equation ~u · ∇p represents the transport ofthe pressure by the inertia of the fluid. This transport is important specially where largescales are involved:

~u·∇pρa2

∇ · ~u∼O(gUa2

)O(UcL

) ∼ gL

a2=M2

Fr2(2.20)

Where Fr is the Froude dimensionless number that expresses the ratio between theinertia forces and the gravity forces affecting a fluid:

Fr =U2c√gL

(2.21)

As M2 � 1 and due to microgravity conditions g � 1 which yields to Fr2 � 1, thefollowing inequality is satisfied, enabling to neglect the second term:

M2

Fr2� 1 (2.22)

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2.4.3 Compressiblity caused by the coefficient of volumetric expansion

The third term of the right hand side of the equation contains the viscous dissipationfunction Φ which is proportional to the contracted product of the strain tensor with itself.

ΦTβcρ

∇ · ~u∼O(µU2

L2βcρ

)O(UcL

) ∼ Re−1βU2c

c(2.23)

Where β is the coefficient of volumetric expansion of the fluid and Re is the Reynoldsnumber. Taking into account that the Reynolds number is approximately Re ∼ 105, thecoefficient of volumetric expansion is β ∼ 10−3 and the specific heat capacity is c ∼ 103,we can estimate this term and assure the following inequality:

Re−1βU2c

c� 1 (2.24)

2.4.4 Compressibility due to temperature gradient

The last term of the right hand side of the equation represents the irreversible source ofentropy due to heat transfer.

Tβκ∇2TcT

∇ · ~u∼O(βκ∆TcρL2

)O(UcL

) ∼ Re−1Pr−1β∆T (2.25)

Where Pr is the Prandtl number and expresses the relation between the momentumdiffusivity (viscosity) and the thermal diffusivity α. For air, it takes values around Pr ∼0.7 :

Pr =µc

κ(2.26)

Having a characteristic temperature difference of ∆T ∼ 10K between inlets and outlet(measured in tests and calculated by CSL professionals using the ESATAN software), onecan have an estimation of the last term and neglect it for this case of study:

Re−1Pr−1β∆T � 1 (2.27)

2.4.5 Derivation of the continuity equation

According to the analysis developed above, the right hand side of the equation becomesvanishingly small compared to the left hand side under the stated conditions. Therefore,if we call:

18

Λ = St ·M2 , Π =gL

a2, Γ = Re−1βU

2c

cand Θ = Re−1Pr−1β∆T (2.28)

Being,

Λ,Γ,Π,Θ� 1 (2.29)

Making an asymptotic expansion of the velocity field in terms of these small param-eters:

~u

U= ~u0 + (Λ + Γ + Π + Θ) ~u1 +O(Λ2,Γ2,Π2,Θ2) (2.30)

And if we substitute this expansion into the continuity equation letting all these fournon-dimensional parameters tend to zero, then at first order, the equation yields:

∇ · ~u = 0 +O(Λ + Γ + Π + Θ) (2.31)

2.5 Momentum equation

According to the simplifications developed in the previous sections, the momentum equa-tion could be written as:

ρ~u · ∇~u = −∇p+ ρ~g +∇ · ¯τ (2.32)

As it has been stated in Chapter 1, the air inside VMU MkII will be under micro-gravity conditions g ≈ 0 which could let us neglect the second term on the right handside of the equation. However, when the CSL tests the unit there will be a gravitationalfield, so the effect of gravity in the momentum equation in that case must be consideredand quantified.

In the experimental case of study, in addition to forced convection, the air is goingto be flowing subjected to temperature gradients under the effect of gravity. In thissituation, natural convection or Rayleigh-Bénard convection must be considered. Thedensity of a fluid can decrease with the increment of temperature, and when it faces agravitational field, such differences in density can create buoyancy forces or Archimedesforces.

In natural convection the dimensionless Rayleigh number is defined:

Ra =Archimedes forcesViscous forces

=gβ∆T

ν∆~u=gβ∆TL3

να= GrPr (2.33)

19

Where Gr is the Grashof number and Pr represents the Prandtl number. The Garshofnumber is used to determine the coefficient of convection in natural convection:

Gr =gβ∆TL3

ν2(2.34)

As the equation (2.33) expresses the Rayleigh number, Ra, represents the ratio be-tween the pulling force due to density differences and viscous forces. Depending on theorder of magnitude of both terms, three scenarios may occur:

• Ra� 1: The viscous forces are larger when compared with the Archimedes forces.In this situation there will not be convection heat transfer and conduction willdominate.

• Ra � 1: The buoyancy forces dominate over the viscous forces. In this case, theconvective flow is allowed.

• Ra ∼ 1: The viscous and buoyancy forces are the same order. Therefore, theconvective flow will exist but slowed down compared to the Ra� 1 case.

Estimating Ra for the experimental case of study, we have g ∼ 10m/s2, β ∼ 10−3 1/K,a characteristic thermal difference between the top and bottom of the VMU of ∆T ∼ 3K,L ∼ 10−1m, ν ∼ 10−5m2/s and α ∼ 10−5m2/s, giving a Rayleigh number of Ra � 1.Therefore, the buoyancy forces will be an important term in the equation when perform-ing the pertinent experiments on Earth and they must be modeled. However, as the caseof study is in micro-gravity conditions it will not be considered.

2.6 Energy equation

The equation for the temperature field is derived from the conservation of energy [8]and once all the simplifications and hypothesis have been applied, it can be expressed asfollows:

ρc~u · ∇T = ∇ · (κ∇T ) + Φ (2.35)

The viscous dissipation Φ represents the mechanical energy of the fluid that is trans-formed into internal energy [11]. It can be a term of great importance in areas such asaerodynamic heating where the thin boundary layer around high speed aircraft raises thetemperature of the skin or in polymer processing flows. In order to see if that is our case,one must define the viscous dissipation as:

20

Φ = ¯τ : ∇~u =1

2µ(γ : γ) (2.36)

Where ¯τ is the shear-stress tensor and γ represents the shear-rate. Now, one canconveniently rewrite the energy equation in the dimensionless form and see the order ofmagnitude of the viscous dissipation compared to the other terms. To do so, we are goingto scale such equation with the factor L2/κ∆T :

ρcUcL

κ~u∗ · ∇∗T ∗ = ∆∗T ∗ +

1

2

U2c µ

κ∆T︸︷︷︸Generation number, Gn

(γ∗ : γ∗) (2.37)

Where the new dimensionless variables are equivalent to: T ∗ = (T − T0)/∆T ,∇∗ = L∇, γ∗ = γL/Uc. With T0 being the reference temperature, L and U the charac-teristic length and velocity respectively.

The generation number Gn can adopt different definitions and names in the literatureand it expresses the relation between viscous dissipation and heat conduction. A largegeneration number implies that viscous dissipation cannot be neglected in comparison toheat conduction. However, we should realize that the product γ∗ : γ∗ might locally adoptvery large values affecting the temperature even if the generation number is smaller thanone. A safe value for neglecting the effects of Φ seems to be Gn � 0.1 [11]. So, for thecase of study with κ ∼ 10−2W/mK , Uc ∼ 1m/s, µ ∼ 10−5 kg/ms and ∆T ∼ 10K, onecan assure the fulfillment of the following inequality and neglect the viscous dissipationterm.

Gn� 0.1 (2.38)

Which leads to the final form of the energy equation:

~u · ∇T = α∆T (2.39)

Where α represents the thermal diffusivity of the fluid.

2.7 Turbulent flow

Turbulent flow is a type of fluid flow in which the fluid undergoes irregular fluctuations,or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers.Turbulence is a flow property that is characterized by chaotic changes in pressure and flowvelocity, it encompasses large range of scales, involves 3D vorticity and dissipation. Thedimensionless Reynolds number is useful to determine the turbulent or laminar nature of

21

a flow and it can be defined as:

Re =Inertia forcesViscous forces

=ρUcL

µ=UcL

ν(2.40)

Where ν = µ/ρ represents the kinematic viscosity. Depending on the value that Readopts, one can distinguish three general types of fluid flows:

• Laminar flow when Re < 2300

• Transient flow when 2300 < Re < 4000

• Turbulent flow when 4000 < Re

Figure 2.2: Laminar and turbulent flow around a cylinder.

So, it can be observed that for a turbulent flow the inertial term of the momentumequation dominates over the viscous term. In our case of study, having ρ = 1.13 kg/m3

, µ = 2 · 10−5 kg/ms, a characteristic length of L ∼ 0.43m and a characteristic velocityof Uc ∼ 1m/s, one gets a Reynolds number of Re ∼ 24295 which classifies our flow as aturbulent flow.

There exist many ways of approaching turbulent flows and the three most commonones are:

• Direct Numerical Simulation (DNS): It resolves all scales, solves Navier-Stokesequations and provides an exact but computationally more expensive solution.

• Reynolds-Averaged Navier-Stokes (RANS): It resolves only the mean flow,models the fluctuations, solves averaged equations, requires a turbulence model andprovides a cheap but inexact solution.

22

• Large-Eddy Simulations (LES): It resolves larger scales, models small scales,solves filtered equations and provides a relatively good fidelity with a cost that canbe approximated to Re9/4.

Figure 2.3: Channel flow for DNS, LES and RANS.

As it has been stated, unsteady flow is where, over a large time scale, the flow ischanging at each spatial location with time. Therefore, turbulent flows would be inher-ently unsteady because over a small time scale (at each spatial location), the velocitycomponents are varying rapidly with time, but, when averaged over relatively small timeintervals, the averaged velocity components may vary much more slowly or not at all.In this last scenario, the turbulent flow is consider steady or quasi-steady. As we areinterested in the steady solution of our problem, and due to the impossibility of disposingmore powerful computational resources the RANS approach has been chosen which willrequire a turbulence model.

In Reynolds averaging, the solution variables in the Navier-Stokes equations are de-composed into the mean (or time-averaged) and fluctuating components. For the velocitycomponents:

ui = Ui + u′i (2.41)

Where Ui and u′i are the mean and fluctuating velocity components for i = 1, 2, 3.Likewise, for pressure and other scalar quantities:

φ = Φ + φ′ (2.42)

Where φ denotes a scalar such as pressure or energy.

23

2.7.1 Two equation turbulence models

Two equation turbulence models are one of the most common types of turbulence models.Models like the k-epsilon model and the k-omega model have become industry standardmodels and are commonly used for most types of engineering problems. By definition,two equation models include two extra transport equations to represent the turbulentproperties of the flow. This allows a two equation model to account for history effectslike convection and diffusion of turbulent energy. The basis for all two equation modelsis the Boussinesq eddy viscosity assumption, which postulates that the Reynolds stresstensor, τij, is proportional to the trace-less mean strain rate tensor, S∗ij, , and can bewritten in the following way:

τij = 2µt S∗ij −

2

3ρkδij (2.43)

Where µt is a scalar property called the eddy viscosity which is normally computedfrom the two transported variables. The last term is included for modelling incompressibleflow to ensure that the definition of turbulence kinetic energy (k) is obeyed:

−ρu′iu′j = µt

(∂Ui∂xj

+∂Uj∂xi

)− 2

3ρkδij (2.44)

The Boussinesq assumption is both the strength and the weakness of two equationmodels. This assumption is a huge simplification which allows one to think of the effectof turbulence on the mean flow in the same way as molecular viscosity affects a laminarflow. The assumption also makes it possible to introduce intuitive scalar turbulence vari-ables like the turbulent energy and dissipation and to relate these variables to even moreintuitive variables like turbulence intensity and turbulence length scale.

The weakness of the Boussinesq assumption is that it is not in general valid. Thereis no real proof that states that the Reynolds stress tensor must be proportional to thestrain rate tensor. It is true in simple flows like straight boundary layers and wakes, butin complex flows, like flows with strong curvature, or strongly accelerated or deceleratedflows the Boussinesq assumption is simply not valid. This gives two equation models in-herent problems to predict strongly rotating flows and other flows where curvature effectsare significant.

Despite the disadvantages that the two equation models present, among all the tur-bulence models that OpenFOAM provides the ones for which there is more informationavailable are the two equation models. Also, this kind of models are the ones that exhibitthe least amount of inconveniences when implementing.

24

Two different turbulence models have been chosen which are: k − ω and k − ω SST .

2.7.2 k − ω turbulence model

The k − ω model incorporates modifications for low-Re number effects and shear flowspreading. It predicts free shear flow spreading rates that are in close agreement withmeasurements for far wakes, mixing layers, and plane, round, and radial jets, and is thusapplicable to wall-bounded flows and free shear flows. The k−ω turbulence model offerstwo extra transport equations: one for the turubulent energy k and another one for theturbulent specific dissipation ω. Among all the different k−ω models the Wilcox’s k−ωmodel has been chosen (1988).

∂k

∂t+ Uj

∂k

∂xj= τij

∂Ui∂xj− βkkω +

∂

∂xj

[(ν + σkνT )

∂k

∂xj

](2.45)

∂ω

∂t+ Uj

∂ω

∂xj= α

ω

kτij∂Ui∂xj− βωω2 +

∂

∂xj

[(ν + σωνT )

∂ω

∂xj

](2.46)

Where νT = k/ω represents the kinematic turbulent viscosity and the closure co-efficients have constant values of : α = 5/9, βk = 9/100, βω = 3/40, σk = 1/2 andσω = 1/2.

2.7.3 k − ω SST turbulence model

The SST k − ω turbulence model [Menter 1993] is a two-equation eddy-viscosity model.The shear stress transport (SST) formulation combines the best of two models. The useof a k−ω formulation in the inner parts of the boundary layer makes the model directlyusable all the way down to the wall through the viscous sub-layer, hence the SST k − ωmodel can be used as a low-Re turbulence model without any extra damping functions.The SST formulation also switches to a k − ε behaviour in the free-stream and therebyavoids the common k−ω problem that the model is too sensitive to the inlet free-streamturbulence properties. Authors who use the SST k − ω model often merit it for its goodbehaviour in adverse pressure gradients and separating flow. The SST k− ω model doesproduce a bit too large turbulence levels in regions with large normal strain, like stagna-tion regions and regions with strong acceleration.

The SST k − ω is similar to the standard k − ω model, but includes the followingrefinements:

• The standard k−ω model and the transformed k− ε model are both multiplied by

25

a blending function and both models are added together. The blending function isdesigned to be one in the nearwall region, which activates the standard k−ω model,and zero away from the surface, which activates the transformed k − ε model.

• The SST model incorporates a damped cross-diffusion derivative term in the ωequation.

• The definition of the turbulent viscosity is modified to account for the transport ofthe turbulent shear stress.

• The modeling constants are different.

The equations of the model are:

∂k

∂t+ Uj

∂k

∂xj= Pk − βkkω +

∂

∂xj

[(ν + σkνT )

∂k

∂xj

](2.47)

∂ω

∂t+ Uj

∂ω

∂xj= αS2 − βωω2 +

∂

∂xj

[(ν + σωνT )

∂ω

∂xj

]+ 2(1− F1)σω2

1

ω

∂k

∂xi

∂ω

∂xi(2.48)

Where the kinematic turbulent viscosity νT can be expressed as:

νT =a1k

max(a1ω, SF2)(2.49)

Where a1 = |u′ν ′|/k is generally a1 ≈ 0.3 for many flows with no adverse pressuregradients.The blending functions F1 and F2 are critical to the success of the method andtheir function is to restrict the limiter to the wall boundary layer:

F2 = tanh

[max

(2√k

βkωy,500ν

y2ω

)]2 (2.50)

F1 = tanh

{min

[max

( √k

β∗ωy,500ν

y2ω

),

4σω2k

CDkωy2

]}4 (2.51)

CDkω = max(

2ρσω21

ω

∂k

∂xi

∂ω

∂xi, 10−10

)(2.52)

Where it is important to note that their formulation is based on the distance to thenearest surface y and on the flow variables.

26

One of the disadvantages of standard two-equation turbulence models is the excessivegeneration of turbulence energy Pk, specially in the vicinity of stagnation points. In orderto avoid the build-up of turbulent kinetic energy in stagnation regions, the productionterm is added:

Pk = min(τij∂Ui∂xj

, 10βkkω

)(2.53)

A coefficient of the k − ω SST model φ is a linear combination of the correspondingcoefficients of the underlying models (φ1 and φ2):

φ = φ1F1 + φ2(1− F1) (2.54)

Finally, the closure coefficients are: α1 = 5/9 , α2 = 0.44, βω = 3/40, β2 = 0.0828,βk = 9/100, σk1 = 0.85, σk2 = 1, σω1 = 0.5 and σω2 = 0.856.

2.8 Final formulation

Taking into account the mathematical formulation developed in the previous sections,the final formulation of the equations for the k−ω SST turbulence model can be writtenin Cartesian tensor form as:

∂Ui∂xi

= 0 (2.55)

ρ∂UiUj∂xi

= −∂P∂xi

+ µ∆Ui −∂

∂xj(ρ ¯u′iu

′j) (2.56)

With:−ρ ¯u′iu

′j = µT

(∂Ui∂xj

+∂Uj∂xi

)− 2

3

(ρk + µT

∂Ui∂xi

)δij (2.57)

Ui∂T

∂xi= α∆T (2.58)

∂k

∂t+ Uj

∂k

∂xj= Pk − βkkω +

∂

∂xj

[(ν + σkνT )

∂k

∂xj

](2.59)

∂ω

∂t+ Uj

∂ω

∂xj= αS2 − βωω2 +

∂

∂xj

[(ν + σωνT )

∂ω

∂xj

]+ 2(1− F1)σω2

1

ω

∂k

∂xi

∂ω

∂xi(2.60)

The last two equations may vary according to the turbulence model in use.

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2.9 Boundary conditions

The set of additional constraints added to the partial differential equations are known asboundary conditions. As their name states, they are imposed at the boundaries which inthe case of study can be divided in three groups: inlets, outlet and walls. Moreover, itis important to underline that a problem will be well-posed if the solution depends in acontinuous way on the boundary conditions, that is to say, a small perturbation of theseconditions should give rise to a small variation of the solution at any point of the domainat a finite distance from the boundaries [10].

2.9.1 Boundary conditions at inlets

The boundary conditions for both inlets are:

Field Boundary conditionLeft inlet

~u ∇~u = 0p p = p0

T T = Tlk k = 3/2(UI)2

ω ω = 0.09k/νTRight inlet

~u ∇~u = 0p p = p0

T T = Trk k = 3/2(UI)2

ω ω = 0.09k/νT

Table 5: Boundary conditions at left and right inlets.

Where p0 represents the ambient pressure at the Columbus cabin which is equal to986hPa, Tl = 28◦ and Tr = 25.3◦C are the temperature of the incoming air a the leftand right inlets respectively, I represents the turbulence intensity and l is the turbulentlength scale.

The turbulence intensity or turbulence level I expresses the ratio between velocityfluctuations and the mean velocity and can be estimated according to the Reynoldsnumber, geometry of the problem and speed of the flow. Depending on those aspects,three cases can be distinguished:

• High-turbulence case: High-speed flow inside complex geometries like heat-exchangers and flow inside rotating machinery (turbines and compressors). Typi-cally the turbulence intensity is between 5 % and 20 %.

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• Medium-turbulence case: Low-speed flows with Low-Re. Typically the turbu-lence intensity is between 1 % and 5 %.

• Low-turbulence case: : Flow originating from a fluid that stands still, like exter-nal flow across cars, submarines and aircrafts. Very high-quality wind-tunnels canalso reach really low turbulence levels. Typically the turbulence intensity is verylow, well below 1 % .

Having a complex geometry, a rather low speed flow and a low Reynolds number,a turbulent intensity of I = 2.5% has been chosen as default. However, a sensitivityanalysis of this parameter will be performed covering different value ranges.

2.9.2 Boundary conditions at outlet

The boundary conditions at the outlet are:

Field Boundary condition~u Q = Q0

p ∇p = 0T ∇T = 0k ∇k = 0ω ∇ω = 0

Table 6: Boundary conditions at the outlet.

Where Q represents the flow rate and Q0 is the constant flow rate that depends onthe regime of the fan outside of the VMU MkII. As it was stated in Chapter 1, Q0 can beequal to three values depending on the working regime of the unit: 820 l/min , 830 l/min

and 850 l/min.

The rest of the boundary conditions are set to zero gradient which is a Neumannboundary condition whose meaning is that the quantity is developed in space and itsgradient is equal to zero in direction perpendicular to the patch (perpendicular to theboundary). Its purpose is especially numerical.

2.9.3 Boundary conditions at walls

The rest of the boundaries of the problem are considered walls which can be classified asdissipative (electric components), non-dissipative (mechanical component) or boundarywall (walls conforming the outer shape of VMU).

29

Field Boundary conditionNon-dissipative

~u ~u = 0p ∇p = 0T ∇T = 0k k = 0

ω ω =√ω2vis + ω2

log

Dissipative~u ~u = 0p ∇p = 0T ∂T/∂n = q/κk k = 0


log

Boundary~u ~u = 0p ∇p = 0T T = Twk k = 0


log

Table 7: Boundary conditions at walls.

It can be observed that the no-slip condition is used which assumes that at a solidboundary, the fluid will have zero velocity relative to the boundary. It is a fair assumptionsince at the fluid-solid interface, the force of attraction between the fluid particles andsolid particles (adhesive forces) is greater than that between the fluid particles (cohesiveforces).

For the temperature field the boundary condition varies according to the type of thewall. The non-dissipative walls are considered as adiabatic because only the heat transferthrough the fluid is being studied. On the other hand, for the dissipative walls thedissipated power P (indicated in the power breakdown in Chapter 1) has been dividedby the area of the wall A obtaining the heat flux q = P/A. Then, due to the non-slipcondition and the Fourier Law one can assume:

qcond = qconv = κ∂T

∂n(2.61)

Where κ is the thermal conductivity of the fluid and ∂T/∂n the thermal gradient.As the velocity of the flow is zero at the boundary, we can equal the heat transportedby conduction and convection at the boundary and therefore, impose a fixed thermal

30

gradient as a function of the heat flux q and thermal conductivity κ.

It is important to underline that for the CPU a modeling of the dissipated power hasbeen applied. A certain percentage of the dissipated power is assigned to the proximityelectronics (10%), the heatexchanger (72%) and the CPU itself (13.5% and 4.5% modules1 and 2 respectively). The division of power is indicated in the image below:

Figure 2.4: Percentages of the dissipated power assigned to each element of the CPU slot.

Finally, for the boundary walls a constant temperature is imposed Tw that representsthe temperature of the wall extracted from the experimental data provided by the CSL.Each wall has a different temperature and some of them are divided into two areasseparated by the mid-stiffener.

Wall Temperature [K]Front 304Back 305

Top back 308Top front 304Bottom 305

Right front 305Right back 311.5

Left 305

Table 8: Temperatures at the outer walls of the VMU.

2.9.4 Near wall treatment of the turbulent parameters

Due to the complexity involved in the modelling of the behaviour of turbulent flows nearwalls or also known as wall-bounded turbulent flows, it is important to study their treat-ment separately. Without the presence of walls or surfaces, turbulence in the absence ofdensity fluctuations could not exist. This is because it is only at surfaces that vorticitygenerated by an incoming flow is brought to rest to satisfy the no-slip condition. More-over, it is known we cannot satisfy the no-slip condition unless we can figure out how to

31

keep a viscous term in the governing equations. And we know that there can be such aterm only if the mean velocity near the wall changes rapidly enough so that it remains,no matter how small the viscous terms become. In other words, we need a length scalefor changes in the y-direction very near the wall which enables us keep a viscous term inour equations. This new length scale, η, is going to be much smaller than δ, the boundarylayer thickness. Therefore, it is necessary to go back and take a look to a 2D sample ofour equations an re-scale them for the near-wall region.

x component:

U∂U

∂x+ V

∂U

∂y= −1

ρ

∂P

∂x− ∂ 〈u2〉

∂x− ∂ 〈uv〉

∂y+ ν

∂2U

∂x2+ ν

∂2U

∂y2(2.62)

y component:

U∂V

∂x+ V

∂V

∂y= −1

ρ

∂P

∂y− ∂ 〈uv〉

∂x− ∂ 〈u2〉

∂y+ ν

∂2V

∂x2+ ν

∂2V

∂y2(2.63)

Two dimensional mean continuity:

∂U

∂x+∂V

∂y= 0 (2.64)

In order to do this re-scaling, first it is required to decide how the mean and turbu-lence velocities scale near the wall scale. The flow of study is so close to the wall thatthe velocity drops drastically (to fulfill the no-slip condition) that trying to characterizevelocities by Uc (the characteristic velocity in the outer region of the turbulent boundarylayer) is no longer useful. However, we do not dispose any means of knowing what thisscale should be, so it will be named uw and be defined later on. Also, it is known fromdifferent experimental studies that turbulence intensity near the wall is relatively high(30 % or more). So, distinguishing between a turbulence scale and the mean velocity hasno practical effect and uw can be used for both. Finally, L will be used to characterizechanges in the x-direction, since these will vary no more rapidly than in the outer bound-ary layer above the near wall region we are interested in.

The complete x-momentum equation can be estimated:

U∂U

∂x︸︷︷︸uw

uwL

+ V∂U

∂y︸︷︷︸(uw η

L)uwη

= −1

ρ

∂P

∂x︸︷︷︸∆PρL

− ∂ 〈u2〉

∂x︸︷︷︸u2wη

− ∂ 〈uv〉∂y︸︷︷︸u2wL

+ ν∂2U

∂x2︸︷︷︸ν uwL2

+ ν∂2U

∂y2︸︷︷︸ν uwη2

(2.65)

Where the continuity equation has been used to estimate V ∼ uwη/L near the wall.As the aim of the re-scaling and estimation is to keep the viscous term the equation is

32

going to be divided by νuw/η2 since it is the largest between the two viscous terms.

(uwην

) ηL

+(uwην

) ηL∼ ∆p

ρL−(uwην

) ηL− uwη

ν+η2

L2+ 1 (2.66)

Now, depending on the value assigned to η, the Reynolds shear stress term may beneglected or conserved in the equation by picking uwη/ν → 0 or uwη/ν ∼ 1. The maingeneral choice (supported by many authors in the literature) is to pick η ∼ ν/uw so theReynolds shear stress term remains and by doing so, all the other terms vanish, except forthe viscous term. In fact, if the same kind of analysis is applied to the y-mean momentumequation, it can be shown that the pressure in the near wall layer is also imposed fromthe outer turbulent boundary layer.

So, in first order the mean momentum equation for the near wall region is reduced to:

0 ≈ ∂

∂y

[−〈uv〉+ ν

∂U

∂y

](2.67)

Which can be integrated from the wall location (y = 0) to the location y obtaining:

0 = −{〈uv〉 − 〈uv〉 |y=0}+ ν

{∂U

∂y− ∂U

∂y

∣∣∣∣y=0

}(2.68)

From the kinematic and no-slip boundary conditions at the wall one observes that〈uv〉 |y=0 = 0. Also, it is known that the wall shear stress is given by:

τw = µ∂U

∂y

∣∣∣∣y=0

(2.69)

Substituting this expression in the simplified equation:

τwρ

= −〈uv〉+ ν∂U

∂y(2.70)

It is important to note that this result is only valid in the limit of infinite Reynoldsnumber which leads to confirm that the total stress in the wall layer is constant. At finiteReynolds numbers the total stress is almost constant, but never quite so because of theterms that have been neglected.

Now, the conditions to define the inner velocity scale (uw) can be satisfied. As it iscustomary to define the friction velocity denoted as u∗ by:

u2∗ =

τwρ

(2.71)

33

And rewriting the previous equation, one gets:

u2∗ = −〈uv〉+ ν

∂U

∂y(2.72)

Therefore, it can be seen that the direct choice for the inner velocity scale is uw = u∗,in other words, the friction velocity is the appropriate scale velocity for the wall region.Moreover, the inner length scale can be defined as:

η = ν/u∗ (2.73)

Finally, we can use our new length scale to define the vertical coordinate in this nearwall layer. In fact, we can introduce a whole new dimensionless coordinate called y+

defined as:

y+ ≡ y

η=yu∗ν

(2.74)

Similarly, we can adimensionalize the main equation using inner variables and rewriteit as:

1 = ri +dfidy+

(2.75)

Where, fi(y+) = U/u∗ and ri(y+) = −〈uv〉 /u2

∗. According to the value that y+

adopts one can differentiate various viscous sublayers:

• Linear sublayer: For y+ < 3 − 5 the Reynolds stress is negligible reducing themain equation to :

u2∗ ≈ ν

∂U

∂y(2.76)

Or in inner variables:

1 ≈ dfidy+

(2.77)

It can be integrated to obtain the leading term in an expansion of the velocity aboutthe wall as:

fi(y+)

= y+ (2.78)

In physical variables:

34

Figure 2.5: Sketch of the various regions of the turbulent boundary layer in inner andouter variables.

U (x, y) =u2∗y

ν(2.79)

It is important to note that some authors consider the extent of the linear sublayerto be y+ = 5 but by y+ = 3 the Reynolds stress has already begun to evolve,making the approximations above invalid. Therefore, the linear approximation isonly valid about 10% at y+ = 3 and it deteriorates rapidly outside this value.

As the linear region in the mean velocity at the wall is one of the very few exactsolutions in all turbulence, it can be used with great advantage as a boundarycondition in numerical solutions.

• Sublayers of the constant stress region: As y+ increases out of the linear regionvery close to the wall, the Reynolds stress rapidly develops until it becomes greaterthan the viscous stress. Also, as we move outward, the mean velocity gradientslowly drops until the viscous stress is negligible compared to the Reynolds shearstress. In fact, by y+ = 30, the viscous shear stress is approximately less than apercent of the Reynolds shear stress. Thus, outside this point we have only:

u2∗ = −〈uv〉 (2.80)

35

The constant stress layer can be divided in two: a viscous sublayer and an inertialsublayer. Similarly, the viscous sublayer has two identifiable regions: the linearsublayer where only viscous shear stresses are important, and a buffer layer inwhich both viscous and Reynolds shear stresses are important.

• Inertial sublayer: The inertial sublayer is one of the most discussed and debatedsubjects in Fluid Mechanics. There is one whole school of thought that claims thatthe mean profile (in inner variables) is logarithmic and independent of the externalflow.

u+ =1

Kln(y+) +B (2.81)

Where K ≈ 0.41 is the von Karman’s constant and B ≈ 5.1 is another constant.Close to the wall, in the viscous sublayer one still obtains u+ = y+.

Figure 2.6: Law of the wall displaying the viscous sublayer and the log region.

It can be deduced that close to a wall the flow behaves in one way but as we moveout of the wall the behaviour of the flow changes. As the distances one has to move inorder to spot this changing in behaviour are not very large, if we want to capture all thephenomena explained above a fine mesh will be required. Therefore, in the setting ofthe boundary conditions of the turbulent parameters there are both mathematical andnumerical approaches implied.

So, the dimensionless turbulent energy at the wall k+ for both k−ω and SST k−ω issaid to be approximately k+ ∼ (y+)m, being m = 3.23. Therefore, at y = 0 the turbulentenergy must be k = 0.

For the specific turbulent dissipation ω an automatic wall treatment has been chosen.The purpose of automatic wall treatments is to make results insensitive with respect towall mesh refinement. The one used in this study takes advantage of the fact that thesolution to ω equations is known for both viscous and log layer:

36

ωvis =6ν

βωy2and ωlog =

u∗

C1/4µ ky

(2.82)

Where y is the cell centroid distance from the wall. The value at the wall for ω willbe calculated as:

ω =√ω2

vis + ω2log (2.83)

It is important to note that for low y values the ωvis will dominate over ωlog. Con-versely, for larger values of y the ωlog term will be dominant making ω adopt the valueat the log layer.

2.10 Heat transfer mechanisms

It is commonly known as heat transfer, the transition of thermal energy from one bodyof greater temperature to another of lower temperature. Given the case of a body with ahigher temperature than its surroundings, the heat transfer or exchange happens in suchway that the body and the environment reach thermal equilibrium. The direction of theheat flux is always from the hotter body to the colder one, as the second principle ofThermodynamics dictates.

The heat transfer modes are defined as the different processes of heat transport andthey are classified in three groups: radiation, conduction and convection. As the simula-tions represent a worst case scenario of the performance inside the VMUMkII, conductionbetween components and panels is not considered, as well as radiation. Only conductionand convection of the fluid are the considered heat transfer mechanisms, particularly,convection is predominant which is beneficial since convection is the most efficient wayto cool down this kind of systems.

2.10.1 Conduction

It is known as thermal conduction to the heat transfer through microscopic collisionsof particles and electron movements in a body. The particles that collide are usuallymolecules, atoms and electrons that transfer in a disorganized way kinetic and potentialenergy, that combined is known as internal energy. Conduction occurs in all states ofmatter, but it is more efficient in solid phases.

The physical property of the materials that determines their capacity to conduct heatis denominated thermal conductivity and it is defined as:

37

κ =dQ/dt

A(dT/dx)(2.84)

Which represents the coefficient that controls the heat transfer dQ/dt through an areaA, due to a thermal gradient dT/dx. If the expression above is reformulated, the FourierLaw is obtained which is a characteristic law in conductive phenomenon:

~q = −κ∇T (2.85)

An important dimensionless number in the analysis of conductive and convectivephenomena is the Nusselt number, which compares the order of magnitude of the heattransferred by convection with the heat transferred by conduction:

Nu =hL

κ(2.86)

Where h represents the convective heat transfer coefficient, L is the characteristiclength and κ the thermal conductivity. The Nusselt number will indicate which heattransfer mode will dominate in our problem or in which areas the conduction or convectionis larger.

38

3 Numerical approach

39

In order to get a solution of the majority of the partial differential equations a dis-cretization method is required. Currently, the commonly used methods are: finite differ-ences (FD), finite element method (FEM), spectral methods and finite volume method(FVM). The finite volume method is one of the most popular in the present and one ofthe most used in the utilization of Navier-Stokes equations. This method is based onthe formulation of the equations in the weak or integral form allowing the user to obtainthe solution of the transport equations in the volumes that conform the computationaldomain. This characteristic gives the FVM geometric flexibility due to the fact that thesolution can be obtained in arbitrary volumes, facilitating the meshing operations in morecomplex geometries.

One of the differences between FVM and FEM is the fulfillment of the equation ofthe conservation of mass and the energy equation. Using FVM, the amount of mass andenergy at the beginning and end of the problem remains strictly constant. However, usingFEM, the equation of conservation of mass could not be met with total accuracy.

As OpenFOAM is a software that uses the finite volume method, that will be thediscretization method used in this thesis.

3.1 Discretization of the computational domain

Discretising the computational domain consists on creating a mesh that will approximatethe geometry of the problem and will be used for the resolution of the discretized equa-tions. This domain is divided into non-overlapped control volumes (CV) whose unionconforms the computational domain. Each CV has a centroid or computational point Pwhere the solution is obtained. In this point the following condition is fulfilled:∫

VP

(~x− ~xP )dV = ~0 (3.1)

Where ~x is a position vector of a point inside the CV, ~xP is the position vector of thecentroid P and VP is the volume of the CV. Such CV is limited by a number of faces,each of which is only shared with another CV. These faces can be classified into internalfaces (those that belong to two CVs) and boundary faces (those that only belong to oneCV), that conform the boundaries of the domain. ~Sf is the surface vector of each faceand its direction is normal to the face. This kind of discretization enables the FVM touse arbitrary control volumes.

40

3.1.1 Mesh generation and importation

Generally, two types of meshes can be created: structured and unstructured meshes. Onthe one hand, the structured meshes are characterized by a regular connectivity and thecontrol volumes are usually generated by the intersection of families of curves. On theother hand, the unstructured meshes are identified by an irregular connectivity.

Figure 3.1: Structured and unstructured mesh for the surroundings of a blunt body.

The quality of a mesh can be defined by the rate of convergence, solution precision andthe required CPU time (which also depends on the available computational resources).Similarly these three factors depend on some mesh properties, some of which are:

• Skewness: The skewness of a grid is an apt indicator of the mesh quality andsuitability. Large skewness compromises the accuracy of the interpolated regions.For quadrilateral cells skewness should not pass 0.9 in order not to compromise themesh quality. The meshes used for the simulations present an average skewness of0.2.

• Smoothness: The change in size should also be smooth. There should not besudden jumps in the size of the cell because this may cause erroneous results atnearby nodes.

• Aspect ratio: It is the ratio of longest to the shortest side in a cell. For multi-dimensional flow, it should be near to one. Also local variations in cell size shouldbe minimal, i.e. adjacent cell sizes should not vary by more than 20 %. Having alarge aspect ratio can result in an interpolation error of unacceptable magnitude.In our mesh the average aspect ratio is of 1.2.

41

Figure 3.2: Example of high aspect ratios for triangular and quadrilateral cells.

In order to assure a good mesh quality, the checkMesh OpenFOAM utility is usedensuring that these three properties and all the other mesh properties are inside of safevalue ranges.

All the meshes created in this thesis are based on an imported CAD model built us-ing the SALOME software. SALOME is an open-source software that provides a genericplatform for pre and post-processing purposes. It contains various modules but onlythe geometry and meshing modules have been used. The geometry module is used tocreate the CAD model of the VMU MkII where each component was either generatedor imported and then assembled. Besides that, the meshing module allows to mesh theassembled CAD model generating an unstructured mesh.

Figure 3.3: Unstructured mesh generated by the meshing module of SALOME based ona simplified CAD model of the VMU including inlets, outlet, mid-stiffener and walls.

Different meshes were created, imported and used which led to discover that as thecomplexity of the mesh increased, the compatibility with OpenFOAM of the mesh gen-erated with SALOME decreased. For complex meshes including the FPIU and electronic

42

boards, the meshes generated using the SALOME meshing module did not display con-vergence in the solution as the number of iteration increased. Also, the results obtainedfrom those simulations did not match the physical representation of the flow inside theVMU MkII. The most complex geometry in which OpenFOAM simulations could be runis the one shown in the image (3.3).

3.1.2 snappyHexMesh utility

Taking into account this scenario, snappyHexMesh, the OpenFOAM mesh generator util-ity was chosen to create the mesh as an alternative to the SALOME meshing module.The methodology to properly import and create the mesh is the following one [5]:

1. Inside the constant directory of the simulation case the triSurface directory is cre-ated where all the elements imported from the geometry module of SALOME willbe placed in STL format. It is important to remark that the name of the elementwill be the name of the patch (boundary) where the boundary condition will beimposed.

2. A cubic mesh of 2 × 2 × 2 m3 is created using the blockMesh OpenFOAM utilitythat allows to create simple grids. The size of the cube must be bigger than thegeometry (the VMU) and the number of cells of the block will influence the finalrefinement of the mesh. Several base meshes are created with a varying number ofcells of: 91125, 125000, 216000 and 274625.

Figure 3.4: Schematic example of the initial mesh generation displaying the base meshand the CAD model (the car) [1].

3. The boundaries of the geometry are intersected (outer panels of the VMU in ourcase) with the base mesh by using surfaceFeatureExtract. That will create anotherdirectory in the constant directory called extendedFeatureEdgeMesh.

4. After executing snappyHexMesh the surface and volume refinement of the outerboundaries start for which 6 levels of refinement can be selected and the refinementpattern is shown in the figure (3.5) (only 3 levels displayed).

43

Figure 3.5: Refinement levels from 0 to 3. The level 0 of refinement represents the originalsize of the control volume of the base mesh [1].

Figure 3.6: Schematic example of the outer surface refinement [1].

5. The unused cells of the intersection between the geometry and the cubic mesh areremoved by selecting a point whose location will determine if the inside or outsideof the intersection is kept. In our case, the point has coordinates in (0, 0, 0) froma coordinate system located in the middle of the VMU and therefore, selecting tokeep the inside of the unit.

Figure 3.7: Schematic example of the cell removal [1].

6. A last phase of refinement starts once the unused cells are removed. In this phase,the faces of each element are smoothed, merged and snapped following again the 6levels of refinement.

44

Figure 3.8: Schematic example of the surface snapping [1].

7. The quality of the mesh is checked according to some quality parameters previouslyselected and if the quality conditions are not met a scale back displacement isexecuted. This process is repeated automatically until the quality prerequisites arefulfilled.

8. The mesh is copied to the polyMesh directory.

The steps from 4 to 8 are automatically executed based on the code of the snappy-HexMesh dictionary. The codes to generate the mesh: blockMeshDict, snappyHexMesh-Dict and surfaceFeatureExtractDict are all located inside the system directory inside thesimulation case.

Figure 3.9: Structured mesh generated with the snappyHexMesh utility.

Due to the approach taken of the numerical implementation of the boundary condi-tions for the turbulent parameters (that will be explained later on), the criterion chosenfor the refinement of the meshes created with the snappyHexMesh utility is to set themaximum level of refinement (6 level) on the small components and inner walls (all elec-tronic boards except for the VCU and SA50-120 modules), a level 3 of refinement for the

45

VCU and SA50-120 modules, a level 1 of refinement for the outer walls and a level 4 forthe inlets and outlet. The reason behind this selection is to try to have the suitable y+

value ranges for the low-Re approach (explained in the next sections), without having anumber of cells that would derive in very long computational times.

Finally, having chosen the snappyHexMesh utility, all the meshes used in this thesisare created and used assuring a complete compatibility with the OpenFOAM software.

3.1.3 Mesh convergence study

One of the issues that affects the accuracy of the solution is mesh convergence. Thisrefers to the smallness of the elements required in a model to ensure that the results ofan analysis are not affected by changing the size of the mesh.

Mesh convergence studies can be performed by increasing the number of cells in thedomain. Increasing the meshing refinement causes two consequences:

• The accuracy of the solution increases because the solution is obtained for a greateramount of points.

• The simulation delays to obtain the solution, due to the fact that the equationsmust be solved in more points.

In order to perform the mesh convergence study, four meshes with an increasingnumber of cells are used. Also, mean temperature of components and mean heat fluxesat the walls are calculated for each mesh:

Mesh Average y+ Number of cells1 5.76 12487682 5.4 16019863 4.54 23761804 4.34 2730171

Table 9: Used grids for the mesh convergence study.

T =1

A

∫T dA =

1

A

∑i

TiAi (3.2)

q =1

A

∫κ∇TdA =

1

A

∑i

κ∇TiAi (3.3)

46

Where T and q represent the mean temperature and heat flux of the patch, A is thetotal area of the patch, Ai is the area of the cell and Ti represents the temperature ateach cell.

Figure 3.10: Mean heat flux of the bottom wall over the number of steps (1 step = 100iterations) for four different meshes.

Figure 3.11: Mean temperature of the second HDD over the number of steps (1 step =100 iterations) for four different meshes.

It can be seen that as the number of cells increases both the mean temperatures andheat fluxes converge to a constant value. It has been decided to opt for the mesh number

47

3 (with a number of cells of 2376180) due to the fact that there is only an incrementof 0.67% in mean temperatures and an increment of 1.75% in heat fluxes in precisionwith respect to the finest mesh. On the other hand, the chosen mesh presents a shortercomputation time (of the order of days) becoming the most balanced mesh in terms ofcomputation time and accuracy of the solution.

In the appendix, mean temperatures and heat fluxes calculated for other walls andcomponents can be found.

3.2 Discretization of equations

In order to characterize the transport of a variable inside a continuum the followingpartial differential equation is used:

∂(ρφ)

∂t+∇ · (ρ~uφ) = ∇ · (Γφ∇φ) + Sφ (3.4)

∫VP

∂(ρφ)

∂tdV +

∫VP

∇ · (ρ~uφ)dV =

∫VP

∇ · (Γφ∇φ)dV +

∫VP

SφdV (3.5)

Where φ can be any scalar or vectorial variable, ρ is the density of the continuum, Γφ

represents the diffusivity of the variable in the continuum and Sφ is a source term.

The integral formulation consists on integrating the transport equation in each CV. Ifthe mentioned equation is integrated in an arbitrary CV the weak or integral formulationis obtained. Moreover, using the Gauss theorem and assuming that the time derivativecan be moved out of the integral, one obtains the final transport equation in its weakformulation:

∂

∂t

∫VP

(ρφ)dV +

∫S

(ρ~uφ) ~dS =

∫S

(Γφ∇φ) ~dS +

∫VP

SφdV (3.6)

It can be observed that in the weak formulation there are four different terms, whosediscretization must be treated differently:

• Time dependent terms ∂∂t

∫VP

(ρφ)dV (not discussed)

• Convective terms∫S(ρ~uφ) ~dS

• Diffusive terms∫S(Γφ∇φ) ~dS

• Source terms∫VPSφdV

48

3.2.1 Discrezitation of the convective terms

In the convective terms, the Gauss theorem is applied and it is required to particularizedthe variable φ on the surfaces of the CV. Therefore, the centroid of each face f is definedas: ∫

S

(~x− ~xf ) ~dS = 0 (3.7)

Hereafter, a Taylor expansion of φ on the faces (φf ) is developed:

φ = φf + (~x− ~xf ) · ∇φf (3.8)

Substituting the expression in the convective terms and appliying the definition of thecentroid of each face: ∫

S

(ρ~uφ) ~dS =∑f

∫f

(ρ~uφ) ~dS =∑f

(ρ~uφ)f ~Sf (3.9)

The term (ρ~uφ)f represents the flux and it is written as Ff . The variable φ is obtainedfrom the centroid φP , and the convective terms need the value of such variable in thecentroids of the faces φf . Therefore, a reconstruction of the value of φ will be required inorder to obtain its value in the centroids of the faces, using the values of the computationalpoints of neighbour cells. To do so, the following schemes are presented:

Figure 3.12: Central scheme for φ.

• Central scheme: In the image (3.12), P represents the centroid of a cell and Eand W the centroids of neighbour cells. On the other hand, e and w represent thefaces of the cells P and E and P and W respectively. This scheme assumes thatbetween the centroids of two neighbour cells the variable φ can be approximatedby a straight line, being the value of φe:

49

φf = fxφP + (1− fx)φe (3.10)

Where fx = || ~eE|||| ~PE

. The central scheme is a second order scheme, but it can generateoscillations in the solution and it is unstable.

Figure 3.13: Upwind scheme for φ.

• Upwind scheme: The image (3.13) shows the approximation and configurationof the upwind scheme. In the image, P , E, W , e and w have the same meaning asin the image (3.12). The upwind scheme assumes that the value of φe depends onthe value of the flux ~F = (ρ~u)e on the face e:

φe =

φP if F ≥ 0

φE if F < 0

(3.11)

If the speed has the same direction as the normal of the face e the value of φe isthe value of the centroid P , φP . In the opposite case, the value of φe is the valueof φ in the centroid E, φE. The upwind scheme retains first order errors, in otherwords, it is very diffusive but it is a stable scheme that does not generate numericaloscillations.

Having tried both schemes, the upwind scheme presents a greater convergence rateand consequently, it is set as a default scheme in the fvSchemes dictionary inside thesystem directory.

3.2.2 Discrezitation of the diffusive terms

The diffusive terms are discretized in a similar way when compared to the convectiveterms, the Gauss theorem and the definition of the centroid on the faces is used:

50

∫S

(Γφ∇φ) ~dS =∑f

(Γφ∇φ)f ~Sf (3.12)

In order to calculate (∇φ)f approximations are used that enable to obtain its valuefrom the value φ in the centroids of the neighbour CVs. Such approximations requirecorrections when the cells are not orthogonal.

Figure 3.14: Orthogonal correction in a non-orthogonal mesh.

As shown in the image (3.14) the centroids P and N of two non-orthogonal neighbourcells share the face f . The modulus of the vector ~Sf (referred as ~S in the image 3.14) isthe surface of such face and it is normal to the face. If the vector ~PN is parallel to thevector ~Sf , it is considered that the face is orthogonal with respect to the centroids P andN . In that case, the expression to calculate the gradient would be:

~Sf (∇φ)f = || ~Sf ||φN − φP|| ~PN ||

(3.13)

Using the equation above, the gradient of the variable φ on the faces can be derivedfrom the values of the centroids of the cells that share the face f . Another valid approxi-mation in order to obtain the gradient of φ on the faces consist on obtaining the gradientin the centroid P from the values of φ on the faces of the cell:

(∇φ)P =1

VP

∑f

~Sfφf (3.14)

And then, interpolating the value of the gradient on the face f as a function of thegradient in the centroids P and N , one obtains:

(∇φ)f = fx(∇φ)P + (1− fx)(∇φ)N (3.15)

Where fx = || ~fN |||| ~PN ||

. However, in the generic case posed, the vector ~PN is not parallel

to the vector ~Sf and therefore, the CVs are not orthogonal to the face f . In this case, theerror caused when using the approximations explained above is not acceptable. In order

51

to minimize the error the following expression is used:

~Sf (∇φ)f = ~∆ · (∇φ)f + ~k · (∇φ)f (3.16)

Where ~∆ · (∇φ)f represents the orthogonal contribution and ~k · (∇φ)f the non-orthogonal contribution. The two vectors ~∆ and ~k must fulfill the following condition:

~Sf = ~∆ + ~k (3.17)

The vector ~∆ is chosen to be parallel to the vector ~PN , while the vector ~k representsthe lack of orthogonality of the face. The commonly used method to obtain the value of~∆ is known as the orthogonal correction method:

~∆ =~PN

|| ~PN |||| ~Sf || (3.18)

Consequently, the orthogonal part of the non-orthogonal correction can be expressedas:

~∆ · (∇φ)f = ||~∆||φN − φP|| ~PN ||

(3.19)

For the non-orthogonal term is necessary to interpolate the gradient on the face ffrom the equation (3.15). Therefore, the final expression for the orthogonal correction is:

~Sf (∇φ)f = ||~∆||φN − φP|| ~PN ||

+ ~k · (fx(∇φ)P + (1− fx)(∇φ)N) (3.20)

The non-orthogonal meshes generate stability problems and oscillations in the dif-fusive terms. In order to avoid these problems, is necessary to apply limiters to thenon-orthogonal corrections increasing the error in the solution. Therefore, it is of greatimportance that the mesh used to solve the problem has the least amount of cells withnon-orthogonal faces in order to reduce the error in the solution and improve the stabilityof the simulations. In our mesh, the non-orthogonality parameter has an average of 11.49.The OpenFOAM checkMesh utility considers to be a critically non-orthogonal mesh foraverage values of 60 and as our mesh is under that value no non-orthogonal correctionshave been applied.

3.2.3 Discrezitation of the source terms

The source terms are linearized with the following expression:

Sφ = Su + SPφ (3.21)

52

Enabling to express the source terms in the integral formulation as:∫VP

SφdV = SuVP + SPVPφP (3.22)

3.2.4 Final form of the discretized transport equation

Developing each term of the transport equation as indicated in the previous sections:

∑f

(ρ~uφ)f ~Sf =∑f

(Γφ∇φ)f ~Sf + SuVP + SPVPφP (3.23)

As it can be observed in the last equation, the solution to the problem φP is obtainedby resolving an implicit system that depends on the values of the variables and its gradi-ents on the faces. Using the discretization schemes of the different developed terms thatenable to obtain the values of the variable φ and its gradient on the faces from the valuesof the variable in the centroids, the following system of equation is obtained:

aPφP +∑nb

anbφnb = bP (3.24)

Where P represents the centroid of the cell, bP is a value that depends on the values ofφ from previous iterations and from other source terms of the equation and nb representsthe centroids of neighbour cells. This system of equations if fulfilled for each centroidof each cell P , therefore for each CV, the variable φP depends on the values of theneighbouring cells allowing to create a system of equations that enables to obtain thevalue of φP :

¯A~φ = ~b (3.25)

The solution of this linear system can be obtained with iterative methods such as:

• Multigrid methods

• Method of the conjugated gradient

• Method of the biconjugated gradient

In this work for the pressure field the Geometric Agglomerated algebraic Multi-Grid(GAMG) solver is used with the GaussSeidel smoother. For all the other fields, thesmoothSolver wiht the GaussSeidel smoother is used.

53

3.2.5 Under-relaxation

The under-relaxation of the solution is a method used to improve the stability of thesolution in the simulations. It is based on the idea of limiting the change that a variablecan experience per iteration. The parameter representing this under-relaxation quantitywill be referred as γ and takes values between [0, 1]. Assuming that the known value ofany variable for the iteration n is φn, and the solution given by the finite volume methodof the next iteration is φ∗, then the relaxation method determines that the value for then+ 1 iteration is:

φn+1 = γφ∗ + (1− γ)φn (3.26)

It can be observed that for values of γ = 1 the solution of the iteration n+ 1 matchesthe solution obtained by the finite volume method. However, for values of γ = 0 thesolution of the iteration n + 1 corresponds to the solution of the previous iteration andthere is no change between iterations. As γ decreases the solution under-relaxation in-creases slowing down convergence and increasing stability. It is important to keep inmind that imposing under-relaxation can lead to non-physical results because it forcesthe convergence of of the variable φ as a weighing of values between iterations and there-fore, the original formulation of the finite volumes method is not necessary fulfilled whenconvergence is reached.

Two methods are distinguished when applying under-relaxation: field under-relaxationand equation under-relaxation. In the first case, the equation of the variable φ is not mod-ified and the coefficients aP , anb and bP do not change. However, in the second case,φ∗ is isolated from equation (3.26) and it is introduced in the final discretized transportequation (3.24) . In this case, the coefficients aP , anb and bP are modified.

The under-relaxation factors used in this work are included in the chart below. Af-ter running several simulations for different values of γ, the selected criterion for therelaxation factors shown is to choose the combination of relaxation factors that bring theresiduals of the solution to their minimal value.

54

Field γField under-relaxationp 0.3

Equation under-relaxation~u 0.7T 0.7k 0.5ω 0.5

Table 10: Field and equation under-relaxation factors.

3.2.6 Solution convergence

Once the equations are discretized and the relaxation of the solution is specified it isimportant to account for the impact of the discretization of the equations in the solution.As the number of iterations increases, the solution of the simulation should converge toa value and a way to quantify this, is to monitor the residuals of the solution. Residualsindicate how well the obtained solution fits the theoretical solution (Ax = b).

r = Ax− b (3.27)

Where r is the residual vector. In OpenFOAM, the residual calculation is solver-specific but the general approach is the same [2] and it applies residual scaling using thefollowing normalisation procedure:

n =∑

(|Ax− A|+ |b− A|) (3.28)

Where x is the average of the solution vector. Finally the scaled residual is given by:

r =1

n

∑|b− Ax| (3.29)

55

Figure 3.15: Final residuals (in logarithmic scale) for the fields U , p, T , k and ω as afunction of the number of iterations for the final model of the VMU MkII.

It is assured for all the other simulation cases that the residuals go down to 10−5 orbelow as it is shown in the figure (3.15).

3.3 SIMPLE algorithm

When the RANS approach for turbulence is used, a stationary problem arises. RANSand many other methods for steady problems in computational fluid dynamics can beregarded as unsteady problems until a steady state is reached. If an implicit method isused in time, the discretized momentum equations at the new time step are non-linear[3]. Due to this and that the underlying differential equations are coupled, the equationssystem resulting from discretization cannot be solved directly. Iterative solution methodsare the only choice.

Given this scenario, the simpleFoam iterative solver is used. This solver, besides beingfit for the characteristics of our problem, uses the SIMPLE algorithm which is computedas follows [4]:

1. An initial guess (un, vn, wn, pn, φn) is taken from boundary and initial conditionsspecified in the 0 directory.

2. Gradients of velocities and pressure are computed.

3. The discretized momentum equation is solved (for each component) to compute theintermediate velocity field.

56

4. The uncorrected mass fluxes at faces are computed.

5. The pressure correction equation is solved to produce cell values of the pressurecorrection (p∗).

6. The pressure field is updated pn+1 = γpn+(1−γ)p∗. Where γ is the under-relaxationfactor.

7. Mass fluxes are corrected.

8. Velocity field is corrected ~un+1 = ~un + V∇p∗/ ~aP . Where V is the volume of thecell, ~aP is the vector of central coefficients for the discretized linear system of thevelocity equation and ∇p∗ is the gradient of the pressure corrections.

9. All other discretized equations are solved obtaining φn+1.

Figure 3.16: Flow chart of the SIMPLE algorithm.

57

The steps explained above are computed until the convergence criterion is met whichin our case is when the residuals of all fields are equal or below to 10−5.

A small modification was made to the simpleFoam solver adding the equation for thetemperature field creating the my_simpleFoam solver. The equation for T is added inthe step 9 before computing the turbulent parameters.

3.4 Implementation of the boundary conditions

In OpenFOAM, the boundary conditions are specified in the 0 directory inside the di-rectory of the case. The boundary conditions explained in Chapter 2 are implemented inOpenFOAM as follows:

Field ImplementationInlets

~u flowRateInletVelocityp zeroGradientT constantValuek turbulentIntensityKineticEnergyInletω turbulentMixingLengthFrequencyInlet

Outlet~u zeroGradientp constantValueT zeroGradientk zeroGradientω zeroGradient

Table 11: Implementation in OpenFOAM of the boundary conditions at the outlet andinlets.

When implementing the boundary conditions as stated in the mathematical formu-lation, certain inconveniences were found. These inconveniences comprised a lack ofconvergence and a lack of physical results (flow coming out of the inlets for example)when implementing the turbulence model. In order to solve those problems, it was de-cided to impose constant flow rates at the inlets instead than at the outlet. Assigningthe appropriate flow rate values at both inlets, the value of the flow rate at the outletwould still be the same constant value Q0 in the three regimes. Similarly, we are forcedto change the boundary condition for the pressure field and specify a constant value atthe outlet that gives the constant pressure at both inlets p0 = 986hPa .

For the other parameters, the boundary conditions and values expressed in Chapter 2

58

are conserved, having to specify the turbulent intensity, I, for the turbulent kinetic energy.

Field ImplementationDissipative wall

~u noSlipp zeroGradientT fixedGradientk constantValueω omegaWallFunctionNon-dissipative wall~u noSlipp zeroGradientT zeroGradientk constantValueω omegaWallFunctionWalls of the VMU~u noSlipp zeroGradientT fixedValuek constantValueω omegaWallFunction

Table 12: Implementation in OpenFOAM of the boundary conditions at walls.

First, the noSlip condition sets to zero the velocity field at all walls. Also, the ze-roGradient condition extrapolates the quantity to the patch from the nearest cell valueand its gradient is equal to zero in direction perpendicular to the patch. On the otherhand, for the temperature field a constant gradient is imposed using fixedGradient in alldisspative electronic components.

In order to treat the turbulent parameters two approaches are valid: the high and lowReynolds approaches. Each approach determines a different set of boundary conditionsand each one is appropriate to a certain kind of meshes.

By keeping values between 30 < y+ < 300 the high-Re approach takes advantage ofthe known solution at the log-region. But, as the geometry of our problem contains ele-ments of small size a fine meshing is required.In order to keep y+ at that range, the smallelements need to be removed (such as HDD boards, CPU, Enet board and FPGA) fromthe geometry simplifying the model. This, led to discard the high-Reynolds approach asvalid approach when implementing the wall treatment of the turbulent parameters.

59

Field Boundary conditionHigh-Re approach - 30 < y+ < 300k kqRWallFunctionω omegaWallFunctionLow-Re approach - 0 < y+ < 5k constantValue = 10−12

ω omegaWallFunction

Table 13: Set of boundary conditions for the high and low Reynolds approaches.

Therefore, the low-Reynolds approach is selected which requires an approximated av-erage value of y+ at walls of 0 < y+ < 5. Also, the k field is set to 10−12 due to theimpossibility of imposing an exact zero for numerical reasons. On the other hand, theomegaWallFunction is explained in Chapter 2 and as a rather low y+ is presented, theviscous-layer term ωvis will dominate over the term of the log-region ωlog.

Figure 3.17: y+ values at walls over the number of steps for the mesh used in the conver-gence study.

When performing the mesh convergence study it is important to monitor the y+

whenever the mesh refinement is changed because that will lead to a change in y+ thatcould put out of the valid range. Also, it is key to note that the presented value rangesare approximated ranges that only pretend to indicate more or less where the y+ shouldplace.

60

4 Results

61

In the previous chapters a theoretical basis of the characteristics of the flow inside theVMU MkII has been developed, as well as the the implementation in OpenFOAM of themesh, discretization of equations and boundary conditions.

Several simulations have been run varying different parameters and the result thatis going to be presented comprise simulations for three different flow rate regimes, twoturbulence models (k−ω and k−ω SST ), the impact of some elements in the flow (VCU,mid-stiffener and FPIU rails) and a sensitivity analysis of the turbulence intensity (I) atthe inlets.

4.1 Default simulation case

The simulation case taken as default is the case with a low airflow rate regime of 820 l/min

at the outlet, a turbulent intensity at both inlets of I = 2.5% , the k−ω SST turbulencemodel and microgravity conditions. It displays the same conditions as for the mesh con-vergence study and represents a worst case scenario of the performance of the VMU inflight mode. Varying one of the parameters of this simulation case and leaving the restconstant is the way the conditions for the other simulation cases are created.

4.1.1 Flow analysis

First, the flow comes inside the VMU MkII from the inlets passing through the ductsof the FPIU. Then, it goes up crossing the air vents of the bottomw rails of the FPIUand it goes through slots of the Front Panel Interface Unit cooling down the electroniccomponents of the different slots.

Figure 4.1: Front 2D plot of the velocity field in m/s inside the FPIU displaying thevector field (left) and isolines (right).

62

It can be observed in the figure (4.1) some peaks (of approximately 4.3m/s) in thevelocity field located in each of the air vents of the bottom rails. The bottom rails haveonly seven apertures located below the slots that need to be cooled down. Also, all theair vents are of the same size except the aperture for the CPU slot which is larger. As theheat loads associated with the CPU are larger in comparison with the other electroniccomponents of the FPIU, its aperture is larger to enable a greater flow rate to enhancethe cooling down of the CPU.

Figure 4.2: Velocity profile of the flow through the HDD slots. Counting from 1 to 4from left to right.

Due to the large aperture of the CPU slot, some part of the flow passes to the fourthHDD slot increasing the velocity in that slot. This can be verified in the figure (4.2)where it can be seen an increase of the velocity profile in the fourth HDD.

Taking a look of the flow inside the CPU slot from the left side of the VMU (figure4.3), it can be seen how the flow ascends an interacts with the different components of theCPU. It is interesting to note the way that the flow divides itself in three paths: one thatinteracts with the connectors where vortexes are created in between connectors, anotherone that interacts with the CPU itself cooling it down and a third one that interactsmainly with the proximity electronics of the slot.

Also, it can be observed that when the flow leaves the FPIU, it is separated into twowhere one part of the flow goes down heading towards the outlet and the other one goesto the front part of the VMU generating vortexes.

63

Figure 4.3: Left side 2D plot of the velocity field in m/s inside the FPIU (CPU slot)displaying the vector field (left) and isolines (right).

The part of the airflow that leaves the FPIU and heads towards the outlet faces firstthe mid-stiffener which separates the flow in three parts according to its geometry. Theupper flow hits the stiffener and descends until it reaches the outlet, the central flowis directly directed towards the outlet and the lower flow faces the VCU which makesredirect the flow up and then turn to the outlet.

Figure 4.4: Transversal cut of the VMU showing the velocity field in m/s displaying thevector field (left) and isolines (right).

Near the outlet one can find the highest speed peak in the VMU of 8.93m/s whichonly occurs in the middle of the outlet. In this simulation case the outlet presents anaverage velocity of 6.86m/s which gives the low flow rate regime of 820 l/min.

The lower part of the flow that encounters the VCU interacts with the fins that thisunit incorporates on the top, giving the velocity profile shown in the figure (4.5). Theplot displays a tendency to gradually increment the velocity peak as we move to the leftof the plot (left of the VMU), and hence, closer to the outlet. As the proximity of theoutlet is where the maximum velocities appear, the velocity peak tends to increase as itgets closer. Moreover, between the left panel of the VCU ant the left wall of the VMUthere a wall effect is produced increasing the velocities in that area.

64

Figure 4.5: Velocity profile of the flow through the 11 fins of the VCU.

Finally, trying to give an overall view of the flow inside the VMU MkII the streamlinesof the airflow are presented in the figure (4.6):

Figure 4.6: 3D view of the streamlines of the airflow inside the VMU MkII. General view(left) and top view (right).

4.1.2 Thermal analysis

In order to identify the peaks in temperature of the element that conform the VMUMkII the mean temperatures and mean convective heat transfer coefficients (h) of eachcomponent have been calculated as shown in the chart below. As stated in chapter one,the convective heat transfer coefficients can be calculated as follows:

h =1

A

∫ −κfluid ∂T∂y|y=0

(Ts − T∞)dA =

1

A

∑i

−κfluid ∂Ti∂y|y=0

(Ti − T∞)Ai (4.1)

Where κfluid represents the thermal conductivity of the airflow, Ts the temperatureof the surface, T∞ is the temperature of the stream far from the surface which is selectedto be equal to 300K (being that the mean temperature of the free-stream flow inside the

65

unit), Ti is the temperature of each cell, A is the area of the surface and Ai the area ofthe respective cell.

Component Mean temperature [K] h [W/m2K] Mean velocity [m/s]FPIU

HDD1 309.63 19.31 2.01HDD2 307.62 24.76 2.51HDD3 307.78 24.52 2.47HDD4 307.29 26.05 2.74

CPU Module 1 480.18 44.66 3.42CPU Module 2 382.59 48.40 3.82

CPU Proximity electronics 317.20 23.16 2.48CPU Heat exchanger 319.13 35.75 1.42

Enet board 318.85 28.29 2.93FPGA 304.95 29.41 3.07

SA50-120 modulesFPIU supplier 1 443.82 8.88 0.2FPIU supplier 2 460.19 8.02 0.12VCU supplier 388.85 6.17 0.03

VCUCover 335.17 8.34 0.41

Power supply 337.45 10.93 0.32P400K Ethernet 338.99 5.03 0.11

Other patchesOutlet 303.98 - 6.86

FPIU walls 301.55 - 0.1Mid-stiffener 301.50 - 0.5

WallsBottom 305 6.58 -Left 305 8.88 -

Right back 311.5 8.79 -Right front 305 14.16 -Top back 308 17.18 -Top front 304 10.13 -Front 304 2.05 -Back 305 0.79 -

Table 14: Mean temperatures, convective heat transfer coefficients of the componentsand the mean velocities of the free-stream flow around them for the default simulationcase.

As stated in Chapter 1, the thermal requirements of the VMU MkII claim that thetemperature range of the electronic boards is [273, 333]K except for the CPU that mustoperate below 378K. Taking this information into account, it can be seen how there are

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three critical parts where these requirements are not fulfilled: CPU, VCU and the SA50-120 modules. One of the main reasons for these high temperatures could be the fact thatconduction through the components to the solid panels of the VMU is not modeled whichin reality conduction would carry a part of the heat load decreasing the temperature of thecommponents. Also, the power distribution for some components (such as the CPU andVCU) affects the temperature field and could contribute to have these temperature peaks.

First the highest temperature peak is found at the CPU and its distribution is shownin the figure (4.7). It is observed that the highest temperature peak corresponds to thefirst module of the CPU (on the right), being the second one considerably lower. On theother hand, the heat exchanger is inside of the thermal requirements range.

Figure 4.7: 3D view of the temperature distribution (in K) of the CPU and the heatex-changer.

The combination of a very high dissipated power and a small area gives that highpeak in temperature in the CPU. If we a take a closer look into the flow around this slot(figure 4.8), we can see how the back part of the module 1 of the CPU is not in contactwith the faster flow being that the reason for its high temperature. On the other hand,as the second module dissipates a lower power, the mean temperature that displays issmaller.

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Figure 4.8: Temperature field (in K) and velocity vector field (in m/s) of a transversalcut of the CPU slot of the FPIU. From left to right: FPIU suppliers 1, 2 and VCUsupplier.

Secondly, the temperature distribution of the SA50-120 modules is presented in figure(4.9). We can observe a similar temperature distribution (figure 4.10) between the twoFPIU suppliers presenting the highest temperature peak of 614K in the side surfaces infront of each other. Furthermore, the temperature distribution presented by the VCUsupplier is averagely 63K smaller than the one for the FPIU suppliers.

Figure 4.9: 3D view of the temperature distribution (in K) of the SA50-120 modules.

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Figure 4.10: Middle cross-section of the temperature profile of the SA50-120 modules.From left to right: VCU supplier, FPIU supplier 2 and 1.

Displaying the same dissipated power, the small differences of the temperature distri-bution of the FPIU suppliers is due to the structure of the flow around them as shownin figure (4.11). The first FPIU supplier faces the upper part of the flow divided by themid-stiffener resulting in a low velocity area behind it. As the velocity in between theFPIU suppliers is low, the airflow is unable to carry the heat dissipated by the secondsupplier by convection deriving in a high peak of temperature in between FPIU supplierand in a higher average temperature in the second supplier.

Figure 4.11: 2D plot of the temperature distribution (in K) and velocity vector field (inm/s) of the SA50-120 modules.

Finally, the temperature field and flow around the VCU is presented in figure (4.12).The average temperature of the VCU (averaging the three parts) is of 337.20K exceedingthe thermal requirements in 4.2K. As the VCU is close to the outlet where we find thehighest speeds in the VMU, a great part of the heat dissipated by the VCU is taken away

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by convection.

Figure 4.12: 2D plot of the temperature distribution (in K) and velocity vector field (inm/s) around the VCU.

Moreover, by monitoring the temperature profile in the top surface of the VCU (figure4.13), we find a profile with peaks an valleys. Each peak represents one of the fins ofthe cover of the VCU which are hotter than the flow and the valleys represent the flowin between fins which is at a lower temperature. Also, we can observe a tendency ofdecreasing temperature as we get closer to the outlet where there are higher velocitiesand hence, a greater convective transport which derives in a lower temperature in thatarea.

Figure 4.13: Temperature profile of the flow (in K) through the 11 fins of the VCU.

Another interesting property we could look at is the heat transported by the airflow(Qpower) from the inlets to the outlet. This quantity can be expressed as:

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Qpower = mc(Tout − Tin) = 93.68W (4.2)

Where m = 0.015 kg/s represents the mass flow rate, c = 1.4 kJ/kgK is the specificheat capacity, Tout = 303.98K the temperature at the outlet and Tin = 299.65 the tem-perature at the inlets for which the average of the temperatures of both inlets has beentaken. The thermal and power requirements state that the VMU MkII heat load to FSLair loop shall be 82 % of the total power dissipation, for a maximum of 140 W whichresults in 114.8 W . Therefore, for the simulated low fan regime the power requirementis not fulfilled which could be due to the fact in this simulations all the heat is taken bythe flow ulike in reality where conduction and radiation also play a significant role.

4.1.3 Turbulence analysis

As the RANS approach was taken, the small fluctuations in the velocity field are modeledby the added transport equations of the k−ω SST model. It is of great interest to knowin which areas of the VMU MkII this fluctuations are larger and to do so, the turbulentkinetic energy is been plotted in certain areas of the unit. As it has been stated in chapter2 the turbulent kinetic energy per unit mass k is associated with the turbulent eddiesand it is expressed as:

k =1

2(u′2 + v′2 + w′2) (4.3)

Where u′, v′ and w′ represent the fluctuations of the three components of the velocityfield. Therefore, in the zones of the VMU where the kinetic energy is higher these fluctu-ations will be higher as well. There are several peaks of k in two zones: inside the FPIUand in the proximity of the outlet.

Taking a look to the k field inside the FPIU (figure 4.14) it can be observed that thereare peaks of k of 0.13m2/s2 near the air vents and specially in the fourth HDD slot. Itis interesting to note how the increments of turbulent kinetic energy are related to theareas with higher speeds.

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Figure 4.14: Specific turbulent kinetic energy field (in m2/s2) and velocity field isolines(in m/s) inside the FPIU.

Another zone of the VMU with k peaks is the outlet nearby area as shown in figure(4.15). Close to the outlet the maximum turbulent kinetic energy is of 0.018m2/s2. Also,the inlets display k peaks of 0.02m2/s2 as well as in the area in between the FPIU and theleft wall. The correlation between high speed areas and turbulent kinetic energy peakscan be observed again.

Figure 4.15: Specific turbulent kinetic energy field (in m2/s2) of a transversal cut of theVMU MkII featuring the outlet, VCU, mid-stiffener, left inlet and proximity of the FPIU.

In the mentioned zones, the fluctuations of the velocity field can be estimated to be of0.36m/s in the air vents of the FPIU and 0.13 near the outlet. Excluding the presentedzones, in the rest of the VMU the k displays really low values below 10−6 where theturbulent phenomena is not relevant.

The other turbulent parameter calculated with the k−ω SST turbulence model is the

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specific turbulent dissipation rate ω which represents the ratio of turbulent kinetic energythat is converted into thermal energy per unit volume and time (also, sometimes referredas mean frequency of turbulence). Inside the VMU, the simulations display an overallconstant value or with smooth variations for ω away from the walls, but close to thewalls its value increases rapidly several orders of magnitude, as shown in picture (4.16).Depending on the wall and the geometry, ω can rise between 2-4 order of magnitudes inproximity to a wall.

Figure 4.16: Specific turbulent dissipation rate profile (in 1/s) across the HDD slots 2-4from left to right.

4.1.4 Impact of the VCU, mid-stiffener and FPIU rails

Three additional simulations have been run in order to represent the impact of someelements of the VMU MkII, particularly, the VCU, mid-stiffener and the bottom rails ofthe FPIU. In order to analyze the impact that these components make in the flow, thedefault mesh has been re-meshed removing the pertinent elements and then, it has beenrun comparing the results with the standard case.

The impact in velocity field of the VCU is presented in figure (42) displaying how thelower flow no longer faces the walls of the VCU and smoothly heads towards the outlet.Also, there is no significant change in the magnitude of the velocity field.

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Figure 4.17: Transversal cut of the VMU displaying the velocity vector field (in m/s)with (right) and without (left) the VCU component.

Taking a look to a cut of the flow parallel to the outlet (figure 4.18), it can be observedhow there is no longer a wall effect that increases the velocity magnitude at the left ofthe VCU which caused an increasing velocity profile of the flow around the fins of theVCU.

Figure 4.18: 2D cut of the flow near the outlet displaying the velocity vector field (inm/s)and the temperature field (in K) with (right) and without (left) the VCU component.

The impact of the mid-stiffener in the flow is presented in figure (4.19) where theabsence of the mid-stiffener unifies the flow. Without the mid-stiffener the flow is nolonger directed to the outlet or the VCU, it displays a single direction towards the topwall. As the flow does not face directly the VCU, it results in an increase of temperatureof 4K for the VCU board.

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Figure 4.19: Transversal cut of the VMU displaying the velocity vector field (in m/s)with (right) and without (left) the mid-stiffener.

Finally, the impact of the rails is presented in figure (4.20) where it can be seen howwithout the rails the magnitude of the velocity field decreases averagely 2.5m/s and itis less directional. The rails display small apertures below each board that needs coolingdown in order to present higher speed flows and decrease the temperature of the compo-nents. Without it, the flow lacks directionality and the fastest parts of the flow are usedto cool down areas of the FPIU that do not need cooling, such as the empty boards.

Figure 4.20: 2D cut of the FPIU displaying the velocity isolines (in m/s) with (right)and without (left) the FPIU bottom rails.

As the velocity profile is smaller without the presence of the rails (figure 4.21), thecapability of the flow to transport the heat dissipated by the electronic components byconvection is significantly lower which leads to an average increase of temperature of theelectronic boards of 7K.

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Figure 4.21: Velocity profile (inm/s) of the flow across the HDD boards with and whitoutthe FPIU bottom rails.

4.2 Comparison of turbulence models

In all CFD studies where turbulence models are used, the source of error coming fromthe usage of such turbulence model must be taken into account. Ideally, after runningsimulations with several turbulence models, the data obtained should be compared withexperimental data in order to decide which turbulence model is more suitable for theproblem. In our case, as we are using the experimental data only as support and not toconfirm rigorous correlations, it has been decided to choose two turbulence models andcompare the results between them in order to obtain an estimation of the discrepancy inthe solution between models.

Two turbulence models are going to be compared: k−ω and k−ω SST . Despite thefact that these two turbulence models are based on the same turbulent parameters (kand ω), there are several differences between them which are explained in chapter 2. Ingeneral, one could say that the main difference between these turbulence models residesin the behaviour far from a wall. The k − ω SST model uses the k − ω formulation inthe inner parts of the boundary layer makes the model directly usable all the way downto the wall through the viscous sub-layer and then, switches to a k − ε behaviour in thefree-stream avoiding the common k − ω problem that the model is too sensitive to theinlet free-stream turbulence properties.

In order to see if that difference is observed in our study, the modulus of the velocityfield in the cross section of the mid-stiffener is been plotted for both models, as presentedin figure (4.22). As a result, it can be observed a practical coincidence in the profilein the proximity to the left and right walls except for a small difference of 11.04% near

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the left wall and a varying difference in the middle far from the wall with a maximumdiscrepancy of 22.37%. Also, in the free-stream far from the wall the shape of the profiledisplays differences in shape.

Figure 4.22: Velocity profile (in m/s) of the flow in the cross-section of the mid-stiffenerfrom the left wall to the right wall of the VMU for the k − ω and k − ω SST turbulencemodels.

Also, it has been notices how in the areas of an increment of turbulent kinetic energy,the difference in solution from one model to another increases, as shown in figure (4.23).It was shown in the subsection 4.1.3 that k increased from the first to the fourth slot ofthe HDD boards inside the FPIU and consequently, there is a greater difference betweenmodels in the fourth board than in the first one displaying a discrepancy of 6.7% in thefourth slot while in the first one the solution only differs in 1.03%.

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Figure 4.23: Velocity profile (in m/s) of the flow across the HDD slots from 1-4 from leftto right, for the k − ω and k − ω SST turbulence models.

However, in the areas where the turbulent kinetic energy is rather low the solution ofboth models matches with a maximum difference of 0.8% as displayed in figure (4.24).

Figure 4.24: Velocity profile (in m/s) of the flow across left duct, for the k − ω andk − ω SST turbulence models.

For the temperature field the same observations apply, having a matching profile nearwalls as shown in figure (4.25) with a maximum difference of 1.1% .

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Figure 4.25: Temperature profile (in K) of the flow across the HDD slots from 1-4 fromleft to right, for the k − ω and k − ω SST turbulence models.

For the turbulent parameters k and ω, despite having different transport equation,as close to the wall the SST formulation switches to a standard k − ω formulation, thesolutions match in the proximity to the wall as presented in figures (4.26) and (4.27).However, in the free-stream the SST formulation switches to the k − ε formulation andthe results display great differences specially for k which presents a maximum discrepancyof 86.85% while ω displays a maximum difference in the solution of 2.03%.

Figure 4.26: Turbulent kinetic energy profile (in m2/s2) of the flow across the HDD slotsfrom 1-4 from left to right, for the k − ω and k − ω SST turbulence models.

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Figure 4.27: Specific turbulent dissipation rate profile (in 1/s) of the flow across the HDDslots from 1-4 from left to right, for the k − ω and k − ω SST turbulence models.

4.3 Regimes comparison

The fan regime of the VMU MkII is directly linked to the flow rate at the outlet, as itwas explained in chapter 1. The flow rate value ranges at which the unit operates is:[820, 850] l/min. Consequently, it was decided to simulate at the same conditions as inthe default case but changing the flow rate at the outlet for three different regimes: low(820 l/min), medium (830 l/min) and high (850 l/min). The aim of this section is topresent a worst, medium and best case scenario of the performance of the VMU MkII.

Theoretically, as the flow rate increases, the airflow is able to take away a greateramount of the heat dissipated by the components decreasing their temperature. There-fore, the chart below presents the mean temperatures for the components of the VMU:

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Mean temperatures [K]Component Low regime Medium regime High regime

FPIUHDD1 309.63 309.55 309.46HDD2 307.62 307.58 307.50HDD3 307.78 307.74 307.67HDD4 307.29 307.24 307.13



Enet board 318.85 318.68 318.39FPGA 304.95 304.88 304.77


VCUCover 335.17 335.07 334.98


Other patchesOutlet 303.98 303.93 303.85

FPIU walls 301.55 301.52 301.47Mid-stiffener 301.50 301.43 301.34

Table 15: Mean temperatures of the components for low, medium and high flow rateregimes.

It can be seen how the decrease of the temperatures is really small ( ∼ 0.1K). Evenat the highest flow rate the thermal requirements are not fulfilled for the VCU, CPUand SA50-120 modules. Similarly, the convective heat transfer coefficients (h) have beencalculated for the three regimes, as shown in the chart below:

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Mean convective heat transfer coefficients [W/m2K]Component Low regime Medium regime High regime

FPIUHDD1 19.31 19.42 19.62HDD2 24.76 24.86 25.07HDD3 24.52 24.62 24.80HDD4 26.05 26.22 26.52



Enet board 28.29 28.51 28.87FPGA 29.41 29.74 30.36


VCUCover 8.34 8.38 8.46


Table 16: Mean convective heat transfer coefficients of the components for low, mediumand high flow rate regimes.

The increment in h fore each regime with respect one another is ∼ 0.01W/m2K. Asthe increment in flow rate per regime is not very high, high increments in the mean con-vective heat transfer coefficients are not expected. The velocity profile for each regimeat the left duct is shown in figure (4.28) displaying an increment of ∼ 0.01m/s per regime.

Moreover, the power transported by the airflow (Qpower) from the inlets to the outletis shown in the chart below:

Regime Power [W ]Low 93.35

Medium 93.95High 94.31

Table 17: Heat transported by the flow for low, medium and high flow rate regimes.

The power requirement is not fulfilled for any of the regimes since the heat transportedis lower than 114W . However, as mentioned before, the possible reason behind it couldbe the absence of conduction and radiation in the modeling along with the modeling of

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the power distribution of the components.

Figure 4.28: Velocity profile (in m/s) at the left duct for the low, medium and highregimes.

4.4 Sensitivity analysis

The aim of the sensitivity analysis is to analyze the sensitivity of the CFD results to thevariation of parameters [10]. In this case, the boundary conditions are going to be varied,specifically, the turbulent intensity (I) at both left and right inlets. To proceed withsensitivity analysis of the turbulent intensity, the default case is going to be selected andall the conditions and parameters are going to remain constant except for the turbulentintensity at the inlets.

The turbulent intensity at the inlets is linked with the turbulent kinetic energy (k) asfollows:

k =3

2(UI)2 (4.4)

Where U represents the magnitude of the velocity field. So, it can be seen how k is go-ing to be directly affected by the change in I. However, the relation between the turbulentintensity at the inlets and the other parameters of the simulation is not straightforward.

Five simulation cases have been run covering a range of Iε[1%, 5%] with an incre-ment of ∆I = 1% per simulation. In figure (4.29), the k profile through the left ductis presented for five different turbulent intensities where the direct relation presented in

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equation (4.4) can be observed.

Figure 4.29: Turbulent kinetic energy profile (in m2/s2) at the left duct for turbulentintensities of 1%, 2%, 3%, 4% and 5%.

On the other hand, as the velocity field has no direct relation with I, it has been plot-ted has shown in figure (4.30) displaying small decreases in velocity of ∆U = 0.001m/s

per ∆I.

Figure 4.30: Velocity profile (in m/s) at the left duct for turbulent intensities of 1%, 2%,3%, 4% and 5%. General plot (left) and zoomed plot (right).

Another interesting side of performing a sensitivity analysis is to find out if the bound-ary condition (equation 4.4) is well posed. As explained in chapter 2, a boundary condi-tion is well posed if the solution depends in a continuous way on the boundary conditions,that is to say, if a small perturbation of these conditions gives rise to small variations ofthe solution at any point of the domain at a finite distance from the boundaries [10], theboundary condition is said to be well posed.

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Monitoring other parameters in other locations of the VMU, one finds the same trend,as shown in image (4.31) where increments of ∆ω = 2 s−1 per ∆I are displayed. Also, itis observed how in the locations with a higher value of k the differences in between pa-rameters is also higher for different turbulent intensities as displayed in the figure (4.31)in the fourth HDD board where the turbulent kinetic energy is higher (as mentioned insection 4.3).

Figure 4.31: Specific turbulent dissipation rate profile (in 1/s) across the HDD slots 1-4from left to right, for turbulent intensities of 1%, 2%, 3%, 4% and 5%. General plot (left)and zoomed plot (right).

This last idea is displayed in figure (4.32) and (4.33). As the turbulent kinetic energyis significantly higher in the fourth board, the differences in the velocity profiles are higherfor each turbulence intensity.

Figure 4.32: Turbulent kinetic energy profile (in m2/s2) across the HDD slots 1-4 fromleft to right, for turbulent intensities of 1%, 2%, 3%, 4% and 5%.

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Figure 4.33: Velocity profile (in m/s) across the HDD slots 1-4 from left to right, forturbulent intensities of 1%, 2%, 3%, 4% and 5%.

However, the generated differences for each turbulence intensity level in the param-eters can be considered small which might indicate that the boundary condition is wellposed.

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5 Final conclusions

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After considering the results exhibited in chapter 4 providing the structure of the flowinside of the VMU in micro-gravity conditions, a study of the performance of the unitfor three flow rate regimes, a comparison of two turbulence models, observations of theimpact of certain components and a sensitivity analysis of the turbulence intensity atboth inlets, several conclusions have been made.

First, the VCU affects mainly the structure of the flow near the outlet without modi-fying its magnitude. Furthermore, the mid-stiffener not only provides mechanical supportto the unit but also directs the flow towards the outlet and the VCU enhancing convectivephenomena in this area and decreasing the mean temperature of the VCU. On the otherhand, the bottom rails of the FPIU play a crucial role in the cooling down of the compo-nents of this unit enabling higher speeds around the electronic boards and consequently,improving the cooling down of the FPIU.

Secondly, after studying the CFD results of the simulations for the low, medium andhigh flow rate regimes, it is concluded that an overestimation of the mean temperaturesof the components is made leading not to fulfill the thermal requirements of the VMUMkII in the VCU, CPU and SA50-120 modules. The cause of this overestimation maybe due to the fact that the conduction through the solid is not being modeled whichin reality would bring down the mean temperatures of the components leading to morerealistic results.

The comparison between the k − ω and k − ω SST turbulence models underlines theimportance of accounting for the error committed when using a turbulence model. Withthe absence of solid experimental data to compare with the CFD results of both models,it is not possible to conclude which one is more suitable for the problem of study. How-ever, a maximum discrepancy between models of 22.37% in the velocity field is estimatedfor the free-stream flow.

Additionally, the sensitivity analysis of the turbulence intensity at inlets indicates thata variation in the turbulent intensity of 1% gives rise to an average variation of 0.07% forthe velocity field. However, in the areas with large variations of turbulent kinetic energy amaximum change in velocity of 2.76% is observed for the same variation of the turbulentintensity. In any case, all this changes due to the variation of the boundary condition canbe considered small which may be an indicator that the boundary condition is well posed.

Finally, as a recommendation for future work lines, in addition to including the mod-

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eling of heat conduction through the solid panels of the VMU MkII, it would be of greatinterest to measure the impact of gravity in the CFD results. Since the experimental dataused as support in this thesis was obtained under gravity conditions, performing a CFDstudy of the experimental conditions of the unit and compare them with the solutionunder micro-gravity conditions would be highly recommended.

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Appendix

A - Mesh convergence study graphs

Figure 5.1: Mean heat flux at the right back wall over the number of steps for four meshes.

Figure 5.2: Mean heat flux at the left wall over the number of steps for four meshes.

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Figure 5.3: Mean temperature at the CPU over the number of steps for four meshes.

Figure 5.4: Mean temperature at the heat exchanger over the number of steps for fourmeshes.

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[4] Eric Furbo. Evaluation of rans turbulence models for flow problems with significantimpact of boundary layers. December 2010.

[5] James E. Hansen. A comprehensive tour of snappyhexmesh. 2012.

[6] T. Janke. Fsl vmu mkii thermo-hydraulic analysis report. January 2017.

[7] J. M. McDonough. LECTURES IN ELEMENTARY FLUID DYNAMICS: Physics,Mathematics and Applications. 2009.

[8] C. C. Mei. Notes on Advanced Environmental Fluid Mechanics. 2002.

[9] Prasant Murtyy. Thermal analysis space applications (cdr). July 2015.

[10] D. Lamalle P. Carlotti. Sensitivity to boundary conditions for simulations of fireplumes in enclosures. August 2013.

[11] H. H. Winter. Viscous dissipation term in energy equations. 2011.

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