Abstract The behaviour of gas flows in microscale systems cannot be adequately repre-sented by

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  • O'Hare, Lynne (2008) Continuum Simulation of Fluid Flow and Heat Transfer in Gas Microsystems. PhD thesis, University of Strathclyde. http://eprints.cdlr.strath.ac.uk/6501/ This is an author-produced version of an unpublished thesis. Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (http://eprints.cdlr.strath.ac.uk) and the content of this paper for research or study, educational, or not-for-profit purposes without prior permission or charge. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website. Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk

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  • University of Strathclyde

    Department of Mechanical Engineering

    Continuum simulation of fluid flow andheat transfer in gas microsystems

    Lynne OHare M.Eng.

    2008

    A thesis presented in fulfilmentof the requirements of the degree of

    Doctor of Philosophy

  • Declaration of Authors Rights

    The copyright of this thesis belongs to the author under the terms of the United

    Kingdom Copyright Acts as qualified by University of Strathclyde Regulation

    3.49. Due acknowledgement must always be made of the use of any material

    contained in, or derived from, this thesis.

    Declaration of Authors Rights i

  • Abstract

    The behaviour of gas flows in microscale systems cannot be adequately repre-

    sented by the Navier-Stokes-Fourier (N-S-F) equations of macroscale fluid dy-

    namics. The key flow features that cannot typically be replicated by continuum-

    based methods are discontinuities of energy and momentum at system boundaries,

    known as velocity slip and temperature jump, and the Knudsen layer, a region

    of flow close to boundary interfaces where the gas and the surface are not in

    local thermodynamic equilibrium. In this thesis, fluid flow and heat transfer in

    gas microsystems are simulated numerically using an extended form of the N-S-F

    relations, which incorporates these non-equilibrium effects.

    Constitutive scaling is a phenomenological approach that alters the linear

    constitutive relationships of the N-S-F shear stress and heat flux closures to in-

    corporate Knudsen layer effects in microflow simulations. This method has been

    implemented here in a 3D finite-volume numerics package. The aim of the present

    work is to make use of the constitutive scaling method to produce a computa-

    tional tool suitable for analysing microscale flows. Both incompressible and com-

    pressible numerical solvers featuring constitutive scaling models and a range of

    appropriate boundary conditions have been developed to this end. Verification

    and validation processes have been undertaken, comparing the performance of

    the numerical models to analytical solutions, discrete molecular simulations and

    experimental results for key engineering case studies.

    A detailed assessment of the implications of extending the constitutive scaling

    method to fully compressible flows has also been carried out. As a result of this

    study, a new methodology for defining constitutive scaling functions empirically

    has been produced. The methodology has shown, for a simple test case, that

    Knudsen layer features can be incorporated in continuum simulations using scal-

    ing functions based on the local features of a flow configuration, rather than a

    global scaling function curve-fit to theoretical data for a single type of case.

    Abstract ii

  • Acknowledgements

    The author would like to thank supervisors J.M. Reese and T.J. Scanlon for guid-

    ance and support throughout this project. Work carried out by D.A. Lockerby,

    Y. Zheng and C.J. Greenshields that has facilitated this research is also gratefully

    acknowledged. The work was funded by the Engineering and Physical Sciences

    Research Council under Grant No. GR/S77196/01.

    Acknowledgements iii

  • Contents

    Abstract ii

    Contents iv

    List of Figures viii

    List of Tables xiv

    Nomenclature xv

    1 Introduction 1

    1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Rarefied flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.2.1 Continuum-equilibrium . . . . . . . . . . . . . . . . . . . . 3

    1.3 Scope of this research . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.3.1 Availability of data . . . . . . . . . . . . . . . . . . . . . . 7

    1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    2 Modelling rarefied gases 12

    2.1 Characterising gas rarefaction . . . . . . . . . . . . . . . . . . . . 12

    2.2 Approaches to modelling rarefied flows . . . . . . . . . . . . . . . 16

    2.2.1 Kinetic theory of gases . . . . . . . . . . . . . . . . . . . . 16

    2.2.2 Discrete molecular models DSMC . . . . . . . . . . . . 18

    2.2.3 The Chapman-Enskog expansion . . . . . . . . . . . . . . 20

    2.2.4 Moment models . . . . . . . . . . . . . . . . . . . . . . . . 21

    Table of Contents iv

  • 2.2.5 Extending the Navier-Stokes-Fourier equations . . . . . . . 22

    2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    3 Physics of rarefied flows 26

    3.1 Interfacial phenomena . . . . . . . . . . . . . . . . . . . . . . . . 26

    3.1.1 Maxwells phenomenological model . . . . . . . . . . . . . 28

    3.1.2 Phenomenological model vs. physical behaviour . . . . . . 33

    3.1.3 Boundary effects on temperature . . . . . . . . . . . . . . 35

    3.1.4 Alternative slip and jump models . . . . . . . . . . . . . . 38

    3.2 The Knudsen layer . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    4 Constitutive scaling 44

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    4.1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . 47

    4.2 OpenFOAM: a CFD framework . . . . . . . . . . . . . . . . . . . 52

    4.3 Incompressible solver: icoFoam . . . . . . . . . . . . . . . . . . . 52

    4.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 54

    4.4 OpenFOAM for non-equilibrium flows . . . . . . . . . . . . . . . . 54

    4.4.1 Implementing effective viscosity . . . . . . . . . . . . . . . 56

    4.4.2 Maxwells slip and constitutive scaling . . . . . . . . . . . 57

    4.5 Progress summary . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    5 Incompressible flows 60

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    5.2 Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    5.2.1 Verifying numerical results . . . . . . . . . . . . . . . . . . 64

    5.3 Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    5.3.1 Verifying numerical results . . . . . . . . . . . . . . . . . . 74

    5.3.2 Validating numerical results . . . . . . . . . . . . . . . . . 77

    5.4 Constricted channels . . . . . . . . . . . . . . . . . . . . . . . . . 79

    5.4.1 Validating numerical results . . . . . . . . . . . . . . . . . 82

    Table of Contents v

  • 5.4.2 Extending the analysis . . . . . . . . . . . . . . . . . . . . 88

    5.5 Cylindrical Couette flow . . . . . . . . . . . . . . . . . . . . . . . 91

    5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    6 Compressible flows 96

    6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    6.2 Compressible solvers in OpenFOAM . . . . . . . . . . . . . . . . 97

    6.2.1 Compressible solver: rhopSonicFoam . . . . . . . . . . . . 98

    6.2.2 Modified solver: rhopEsonicFoam . . . . . . . . . . . . . . 100

    6.2.3 Compressible microflows solver . . . . . . . . . . . . . . . 105

    6.3 Constitutive scaling models . . . . . . . . . . . . . . . . . . . . . 106

    6.3.1 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    6.3.2 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    6.4 Half-space problems . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    6.4.1 Kramers problem . . . . . . . . . . . . . . . . . . . . . . . 110

    6.4.2 The temperature jump problem . . . . . . . . . . . . . . . 112

    6.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    6.5 Compressible micro-Couette flow . . . . . . . . . . . . . . . . . . 115

    6.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    6.5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    7 A new approach to constitutive-scaling 128

    7.1 Identifying a scaling function . . . . . . . . . . . . . . . . . . . . . 129

    7.2 Extracting effective viscosity . . . . . . . . . . . . . . . . . . . . . 135

    7.3 Incorporating Kn . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

    7.4.1 Scope for future work . . . . . . . . . . . . . . . . . . . . . 144

    7.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

    8 Conclusions 148

    8.1 Research contributions . . . . . . .