VARIATIONAL MULTISCALE METHOD FOR STOCHASTIC

THERMAL AND FLUID FLOW PROBLEMS

A Dissertation

Presented to the Faculty of the Graduate School

of Cornell University

in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

by

Badrinarayanan Velamur Asokan

May 2006

c 2006 Badrinarayanan Velamur AsokanALL RIGHTS RESERVED

VARIATIONAL MULTISCALE METHOD FOR STOCHASTIC THERMAL

AND FLUID FLOW PROBLEMS

Badrinarayanan Velamur Asokan, Ph.D.

Cornell University 2006

Engineering transport phenomena are physically characterized at the continuum

level as systems of nonlinear advection-diffusion -reaction equations with multiscale

parameters. Any computational technique for the solution of such systems should

be able to resolve the smallest length and time scales in the problem while pro-

viding accurate characterization of the process conditions viz. material properties,

constitutive laws, initial and boundary conditions. In practice, description of the

process conditions involves experiments and theoretical models that invariably yield

inadequate information due to gappy data, theoretical assumptions. This manifests

as uncertainty that has to be considered as an integral part of the computational

model. Thus a practical computational framework should be able to capture the

complex interplay of uncertainty across various length scale and simulate the process

using a very coarse discretization and yet retaining sufficient small scale information.

In addition to the above complications, significant interest lies in characterizing the

effect of uncertainty on ill-posed inverse problems. The work in this thesis can be

divided into three stages. In the first stage, a mathematical framework for represen-

tation of uncertainty was developed. In particular, the generalized polynomial chaos

method and the support-space method were developed. In the second stage, a vari-

ational multiscale framework with algebraic subgrid scale modelling was developed

for the finite element solution of stochastic advection-diffusion and Navier-Stokes

equations. An interesting application of uncertainty analysis for capturing unstable

equilibrium in natural convection was addressed, wherein, it was shown that the gen-

eralized polynomial chaos method fails is capturing discontinuous/highly nonlinear

input-output uncertainty propagation whereas, the support-space method provides

accurate solution statistics. In the final stage, the variational multiscale framework

is extended for stochastic upscaling of a transient multiscale diffusion equation. This

involved the coupling of operator upscaling techniques with uncertainty represen-

tation techniques. A summary of achievements and suggestions for future research

are given at the end of the thesis.

Biographical Sketch

The author was born in Madras, India in August, 1979. After completing his high

school education from Hindu Senior Secondary School in Madras, the author was

admitted into the Bachelors program at the Indian Institute of Technology, Madras

in 1997, from where he received his Bachelors in Technology degree in June, 2001. In

August 2001, the author was admitted into the doctoral program at the Sibley School

of Mechanical and Aerospace Engineering, Cornell University and was awarded a

special Masters degree in January 2005.

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This thesis is dedicated to my mother S. Kamala for her unwavering support and

belief in me.

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Acknowledgements

I would like to thank my thesis advisor, Professor Nicholas Zabaras, for his constant

support and guidance over the last 5 years. I would like to thank Professor Subrata

Mukherjee and Professor Shane Henderson for serving on my special committee and

for their encouragement and suggestions at various times during the course of this

work.

The financial support for this project was provided in part by NASA, Office

of Biological and Physical Sciences Research (grant NAG8-1671), the Computa-

tional Mathematics program of the Air Force Office of Scientific Research (grant

FA9550-04-1-0070) and the Design and Integration Engineering Program of the De-

sign Manufacture and Industrial Innovation Division of the National Science Founda-

tion (grant DMI-0113295).I would like to thank the Sibley School of Mechanical and

Aerospace Engineering for having supported me through a teaching assistantship for

part of my study at Cornell. The computing for this project was supported by the

Cornell Theory Center during 2001-2006.

The algorithms developed as a part of this thesis have been implemented in a

high-performance computing environment using C++ and PETSc [79, 80, 81]. The

author wishes to acknowledge the PETSc development team. I am particularly

indebted to the present and former members of the MPDC group. Thanks are also

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due to my family and friends for their company and support. Finally, my thanks

are extended to Elsevier, Ltd. for granting permission to reproduce figures from our

papers [7, 8]

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Table of Contents

List of Tables viii

List of Figures ix

1 Introduction 1

2 Mathematical representation of uncertainty 14

2.1 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Random variable . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Mathematical expectation and definition of Lp() . . . . . . 16

2.2 Reduced modeling of stochastic processes . . . . . . . . . . . . . . . . 172.2.1 Karhunen-Loe`ve expansion - KLE . . . . . . . . . . . . . . . . 182.2.2 Generalized Polynomial Chaos Expansion - GPCE . . . . . . . 192.2.3 Pitfalls in the GPCE and the support-space representation . . 212.2.4 Support space method . . . . . . . . . . . . . . . . . . . . . . 222.2.5 Comparison between support-space and Wiener-Haar approach 24

3 Stochastic variational multiscale formulation for advection-diffusion

and Navier-Stokes equations 26

3.1 VMS for linear advection-diffusion equation . . . . . . . . . . . . . . 273.1.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Variational multiscale method . . . . . . . . . . . . . . . . . . 293.1.3 Models for - intrinsic subgrid time scale . . . . . . . . . . . 323.1.4 Intrinsic time scale models and induced constraints - A one-

dimensional case study . . . . . . . . . . . . . . . . . . . . . . 353.2 The stochastic incompressible Navier-Stokes equations . . . . . . . . 38

3.2.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Variational multiscale hypothesis . . . . . . . . . . . . . . . . 40

3.3 Finite element implementation . . . . . . . . . . . . . . . . . . . . . . 443.3.1 Implementation of the stochastic advection-diffusion problem . 443.3.2 Implementation of the stochastic Navier-Stokes equations . . . 46

3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Steady advection skew to a mesh . . . . . . . . . . . . . . . . 49

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3.4.2 Transient advection of a cosine hill in rotating flow field . . . . 573.4.3 Internal channel flow: Poiseuille flow . . . . . . . . . . . . . . 613.4.4 Driven cavity flows - Lid driven square cavity problem . . . . 643.4.5 Flow past a circular cylinder - wake flow . . . . . . . . . . . . 68

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Capturing unstable equilibrium in natural convection using sto-

chastic VMS analysis 78

4.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1.1 Boussinesq assumption and simplified equations of fluid motion 80

4.2 Application of the variational multiscale method to Boussinesq equa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.1 Definition of function spaces . . . . . . . . . . . . . . . . . . . 834.2.2 Variational multiscale method . . . . . . . . . . . . . . . . . . 844.2.3 Apriori scale decomposition of solution . . . . . . . . . . . . . 874.2.4 Scale decomposed variational formulation for energy equation 874.2.5 Scale decomposed variational formulation for mass and mo-

mentum conservation equations . . . . . . . . . . . . . . . . . 914.3 NUMERICAL EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6.1 Derivation of equation for energy . . . . . . . . . . . . . . . . 1104.6.2 Derivation of equation for momentum . . . . . . . . . . . . . . 111

5 Variational multiscale methods for stochastic diffusion in heteroge-

neous random media 113

5.0.3 Problem definition and variational formulation . . . . . . . . . 1145.0.4 Additive scale decomposition and variational multiscale method1155.0.5 Subgrid modeling . . . . . . . . . . . . . . . . . . . . . . . . . 1165.0.6 C2S map and multiscale basis functions . . . . . . . . . . . . . 1175.0.7 Boundary conditions for subgrid basis functions . . . . . . . . 1215.0.8 Affine correction term . . . . . . . . . . . . . . . . . . . . . . 1235.0.9 Modified coarse-scale formulation . . . . . . . . . . . . . . . . 124

5.1 Computational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.1 Special structure of subgrid problems . . . . . . . . . . . . . . 1255.1.2 Quasistatic subgrid solution . . . . . . . . . . . . . . . . . . . 1265.1.3 Post-processing: Fine-scale solution reconstruction . . . . . . . 127

5.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2.1 Example I: Transient diffusion in a functionally graded material1305.2.2 Example II: Transient diffusion in a two-phase microstructure 136

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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6 Fractional-step solvers for stochastic Navier-stokes equations 145

6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.2 Mathematical model and the fractional-time step algorithm . . . . . . 147

6.2.1 Fractional-time step algorithm . . . . . . . . . . . . . . . . . . 1486.2.2 Stochastic finite element formulation . . . . . . . . . . . . . . 1506.2.3 Special case: uncertainty only from boundary conditions . . . 152

6.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.3.1 Lid driven square cavity problem . . . . . . . . . . . . . . . . 153

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7 Suggestions for future research 158

7.1 Uncertainty representation schemes . . . . . . . . . . . . . . . . . . . 1597.2 Variational multiscale method . . . . . . . . . . . . . . . . . . . . . . 1607.3 Extensions to robust design . . . . . . . . . . . . . . . . . . . . . . . 160

Bibliography 161

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List of Tables

1.1 Different uncertainties encountered in the characterization of an en-gineering system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Various uncertainty analysis techniques in current literature . . . . . 6

2.1 Relation between type of Askey-polynomial chosen for the GPCEand the underlying probability density function of the inputs . . . 20

5.1 Computational parameters used in Example I . . . . . . . . . . . . . 1315.2 Computational parameters used in Example II . . . . . . . . . . . . 139

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List of Figures

2.1 A schematic of the importance based gridding approach. Note therefinement of the mesh in regions with high-probability density func-tion values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Schematic of problem definition with mesh details for steady advec-tion skew to a mesh example. . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Whole domain solutions for the (a) mean and (b) the coefficient ofLegendre chaos corresponding to 1() for a second-order Legendrechaos approximation of the solution. . . . . . . . . . . . . . . . . . . 51

3.3 Comparison between solution mean and standard deviation of solu-tion obtained by 100, 000 Monte Carlo iterations and a fourth-orderLegendre chaos solution approximation at various y values: (a) meanat y = 0.2 (b) standard deviation at y = -0.2 (c) mean at y = 0(d) standard deviation at y = 0 (e) mean at y = 0.2 (f) standarddeviation at y = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Whole domain solutions for the (a) mean and (b) the standard de-viation obtained for a fourth-order Legendre chaos approximation ofthe solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Comparison between solution mean and standard deviation of solu-tion obtained by 100, 000 Monte Carlo iterations and a fourth-orderLegendre chaos solution approximation at various y values: (a) meanat y = 0.2 (b) standard deviation at y = 0.2 (c) mean at y = 0(d) standard deviation at y = 0 (e) mean at y = 0.2 (f) standarddeviation at y = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Whole domain solutions for the (a) mean and (b) the standard de-viation obtained for a fourth-order Legendre chaos approximation ofthe solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7 Problem definition for transient advection of a cosine hill in a rotatingflow field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.8 Pure advection of a cosine hill in a rotating flow field: comparisonof mean and standard deviation for various orders of Legendre chaosapproximation of solution versus the exact solution (computed with100, 000 MC realizations). . . . . . . . . . . . . . . . . . . . . . . . . 60

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3.9 Whole domain solutions for the (a) mean and (b) the standard de-viation obtained for a fourth-order Legendre chaos approximation ofthe solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.10 Schematic of the computational domain and mesh details for thePoiseuille flow example. . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.11 Poiseuille flow: (a) Mean axial velocity (b) First term in GPCE ofaxial velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.12 Problem definition for the lid driven cavity example. . . . . . . . . . 653.13 Comparison between mid-plane quantities with deterministic results

and plots of higher order coefficients in Legendre chaos expansion ofsolution at the midplane: (a) mid-plane mean pressure, (b) secondand third term in LCE expansion of mid-plane pressure, (c) mid-plane mean x-velocity, (d) second and third term in LCE expansionof mid-plane x-velocity, (e) mid-plane y-velocity, (f) second and thirdterm in LCE expansion of mid-plane y-velocity. . . . . . . . . . . . . 66

3.14 Steady state contours for (a) the mean pressure, (b) the determin-istic pressure, (c) the mean streamlines and (d) the deterministicstreamline pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.15 Problem definition and computational domain for the flow past acircular cylinder example. . . . . . . . . . . . . . . . . . . . . . . . . 68

3.16 (a) Mean pressure and (b) First-order term in Legendre chaos ex-pansion of pressure at t = 79.2 s for the flow pass a circular cylinderexample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.17 Mean pressure contours at t = 144 s for the flow pass a circular cylin-der example: (a) Stochastic simulation (b) Deterministic simulation . 73

3.18 Mean streamline pattern t = 144 s for the flow pass a circular cylinderexample: (a) Stochastic simulation (b) Deterministic simulation . . . 74

3.19 Higher order Legendre chaos terms for stochastic pressure solutionat t = 144 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.20 Comparison of deterministic velocity components with means of thestochastic velocity components along the centerline in the cylinderwake region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.21 Mean velocity spectrum (dc component not shown) . . . . . . . . . . 76

4.1 a. Schematic of the computational domain for example I, b. Meshfor example I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 a. Mean non-dimensional temperature at t = 0.1, b. First orderterm in Legendre chaos expansion of non-dimensional temperatureat t = 0.1 (Example I). . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 a. Mean non-dimensional x-velocity component at t = 0.1, b. Firstorder term in Legendre chaos expansion of non-dimensional x-velocitycomponent at t = 0.1 (Example I). . . . . . . . . . . . . . . . . . . . 98

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4.4 a. Mean non-dimensional y-velocity component at t = 0.1, b. Firstorder term in Legendre chaos expansion of non-dimensional y-velocitycomponent at t = 0.1 (Example I). . . . . . . . . . . . . . . . . . . . 98

4.5 a. Deterministic non-dimensional temperature at steady state, b.Mean non-dimensional temperature at steady state (Example I). . . 99

4.6 a. Deterministic non-dimensional x-velocity component at steadystate, b. Mean non-dimensional x-velocity component at steady state(Example I). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.7 a. Deterministic non-dimensional y-velocity component at steadystate, b. Mean non-dimensional y-velocity component at steady state(Example I). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.8 a. First order term in Legendre chaos expansion of non-dimensionaltemperature at steady state, b. Second order term in Legendre chaosexpansion of non-dimensional temperature at steady state (ExampleI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.9 a. First order term in Legendre chaos expansion of non-dimensionalx-velocity component at steady state, b. Second order term in Legen-dre chaos expansion of non-dimensional x-velocity component atsteady state (Example I). . . . . . . . . . . . . . . . . . . . . . . . . 101

4.10 a. First order term in Legendre chaos expansion of non-dimensionaly-velocity component at steady state, b. Second order term in Legen-dre chaos expansion of non-dimensional y-velocity component atsteady state (Example I). . . . . . . . . . . . . . . . . . . . . . . . . 101

4.11 a. Mean non-dimensional x-velocity component at steady state usingGPCE approach, b. Mean non-dimensional y-velocity component atsteady state using GPCE approach (Example II). . . . . . . . . . . . 104

4.12 a. Mean non-dimensional temperature at steady state for determin-istic simulation at Ra=1530, b. Prediction of support-space methodfor non-dimensional temperature at steady state at Ra=1530 (Ex-ample II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.13 a. Mean non-dimensional x-velocity component at steady state fordeterministic simulation at Ra=1530, b. Prediction of support-spacemethod for non-dimensional x-velocity component at steady state atRa=1530 (Example II). . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.14 a. Mean non-dimensional y-velocity component at steady state fordeterministic simulation at Ra=1530, b. Prediction of support-spacemethod for non-dimensional y-velocity component at steady state atRa=1530 (Example II). . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.15 a. Mean non-dimensional temperature at steady state for determin-istic simulation at Ra=1870, b. Prediction of support-space methodfor non-dimensional temperature at steady state at Ra=1870 (Ex-ample II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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4.16 a. Mean non-dimensional x-velocity component at steady state fordeterministic simulation at Ra=1870, b. Prediction of support-spacemethod for non-dimensional x-velocity component at steady state atRa=1870 (Example II). . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.17 a. Mean non-dimensional y-velocity component at steady state fordeterministic simulation at Ra=1870, b. Prediction of support-spacemethod for non-dimensional y-velocity component at steady state atRa=1870 (Example II). . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1 A. Schematic of the time integration framework: t is the coarse-time step and t is the local time coordinate. The integration para-meters A(t) and B(t) are shown in the figure. Also, uCs and u

Cs

are identified as the coarse solution fields at the start and end ofthe coarse time step, respectively. B. Schematic of a typical coarseelement sub-domain: The coordinates normal and tangential to theelement edges are denoted by the letters n and , respectively. . . . . 119

5.2 Example I - Decay of a sine hill (results at time = 0.05): A, B andC: Coefficients u0, u1 and u2 obtained from the GPCE of the fully-resolved stochastic finite element solution. D, E and F: Coefficientsu0, u1 and u2 obtained from the GPCE of the fine-scale reconstruc-tion of the VMS solution with a quasistatic subgrid assumption. G,H and I: Coefficients u0, u1 and u2 obtained from the GPCE of thefine-scale reconstruction of the VMS solution with a dynamic subgridassumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3 Example I - Decay of a sine hill (results at time = 0.2): A, B andC: Coefficients u0, u1 and u2 obtained from the GPCE of the fully-resolved stochastic finite element solution. D, E and F: Coefficientsu0, u1 and u2 obtained from the GPCE of the fine-scale reconstruc-tion of the VMS solution with a quasistatic subgrid assumption. G,H and I: Coefficients u0, u1 and u2 obtained from the GPCE of thefine-scale reconstruction of the VMS solution with a dynamic subgridassumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.4 Example I - Decay of a sine hill: Plot of L2 error in GPCE coefficientsvs time: (Quasistatic subgrid case) A. For a 1010 coarse-mesh witha 20 20 subgrid mesh, B. For a 20 20 coarse-mesh with a 10 10subgrid mesh. (Dynamic subgrid case) C. For a 10 10 coarse-meshwith a 20 20 subgrid mesh, D. For a 20 20 coarse-mesh with a10 10 subgrid mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.5 A gray scale plot of a two-phase (-) microstructure. The intensitiesare scaled to the interval [0,1] with zero representing the pure -phaseand one representing the pure -phase. . . . . . . . . . . . . . . . . . 138

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5.6 Example II - Diffusion in a microstructure (results at time 0.05):Coefficients u0, u1 and u2 in the GPCE of the solution for the fol-lowing: A, B and C: Fully-resolved stochastic finite element solution,D, E and F: Fine-scale reconstruction of the VMS solution with aquasistatic subgrid assumption (2020 coarse-mesh, 1010 subgridmesh), G, H and I: Coarse-scale solution (20 20 coarse-mesh), J,K and L: Coarse-scale solution (10 10 coarse-mesh). . . . . . . . . 141

5.7 Example II - Diffusion in a microstructure (results at time 0.2): Co-efficients u0, u1 and u2 in the GPCE of the solution for the following:A, B and C: Fully-resolved stochastic finite element solution, D, Eand F: Fine-scale reconstruction of the VMS solution with a qua-sistatic subgrid assumption (20 20 coarse-mesh, 10 10 subgridmesh), G, H and I: Coarse-scale solution (20 20 coarse-mesh), J,K and L: Coarse-scale solution (10 10 coarse-mesh). . . . . . . . . 142

5.8 Example II - A, B and C: Fully-resolved stochastic FEM simulation:Coefficients u3, u4 and u5 in the GPCE expansion of the solution. D,E and F: Fine-scale reconstruction of the stochastic solution using aquasistatic subgrid assumption in the VMS simulation: Coefficientsu3, u4 and u5 in the GPCE expansion of the solution obtained attime 0.2 (non-dimensional). . . . . . . . . . . . . . . . . . . . . . . . 144

6.1 Comparison between the mean X-velocity obtained using the fractional-time step method with a fifth order GPCE expansion for velocitywith the benchmark results obtained by Ghia et al. [40] . . . . . . . 154

6.2 Plots of midplane velocity GPCE coefficients for two successive or-ders of GPCE: (a) mean X-velocity, (b) first GPCE coefficient in X-velocity, (c) second GPCE coefficient in X-velocity, (d) third GPCEcoefficient in X-velocity. . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3 Plots of midplane velocity GPCE coefficients for two successive or-ders of GPCE: (a) mean Y-velocity, (b) first GPCE coefficient in Y-velocity, (c) second GPCE coefficient in Y-velocity, (d) third GPCEcoefficient in Y-velocity. . . . . . . . . . . . . . . . . . . . . . . . . . 156

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Chapter 1

Introduction

Multiscale transport processes are ubiquitous in engineering and allied fields viz.

geophysics, computational biology, meteorology etc. Typical examples involve flow

in porous media, fluid mechanics and diffusion in composites. The behavior of such

processes can be described in continuum sense using systems of coupled partial dif-

ferential equations (PDEs) of the advection-diffusion-reaction type. Two critical

commonalities emerge while designing a computational technique to address the

above problems: i. They are highly computation intensive and can rarely be solved

to an acceptable degree of tolerance using a single mesh technique like the finite el-

ement/difference method. ii. The model parameters in the governing equations for

such problems involve parameters that are inherently statistical e.g. material poros-

ity. This statistical nature should be directly incorporated in the computational

model by modelling the governing equations as stochastic PDEs. In addition to

these considerations, computational modelling of multiscale transport processes us-

ing finite element methods involve complexities associated with advection-dominant

systems (whenever, the advective processes dominate the diffusion and/or the re-

active processes). This thesis develops the stochastic variational multiscale (VMS)

1

2method [7, 8]as a common platform on which computational techniques can be devel-

oped for addressing multiscale stochastic problems with applications involving fluid

mechanics, convection-diffusion processes and multiscale diffusion in heterogeneous

random media. The stochastic VMS is built on mathematically rigorous spectral

stochastic framework [38, 19, 89] and the deterministic VMS method [50, 51, 52, 53].

Historically, the research on computational techniques for multiscale problems

have been restricted to multigrid techniques, fast multipole expansions, wavelets

and the like. The main drawback in the approaches is that their cost is that they

resolve all the relevant physical scales and hence their cost is comparable to that

of a fully resolved simulation. Recent research in multiscale methods involve tech-

niques like the variational multiscale method (VMS), the heterogeneous multiscale

method (HMM) and the multiscale finite element method (MsFEM) that strive at

the derivation of a coarse scale equation that includes modelled effects from the

fine scale. Another commonality in these approaches is that their cost scales sub-

linearly in comparison to a fully resolved simulation. The heterogeneous multiscale

method (HMM) [26, 27] was proposed as a general framework for designing mul-

tiscale methods, wherein, heterogeneous emphasizes that the governing models

may be different at each length scale (e.g. for a microfluidics problem, HMM will

involve a lattice-Boltzmann model at mesoscale and molecular dynamics model at

microscale). The effectiveness of HMM lies in the fact that it allows the extraction

of maximal information about the problem at all length and time scales. However,

application of HMM to specific problems is non trivial and still sparsely researched,

thus leaving a lot to guesswork.

On the other hand, The deterministic variational multiscale (VMS) method was

introduced as a paradigm for constructing computational methods for multiscale

3problems. The main idea behind the VMS approach is to decompose the exact so-

lution of the governing equations into a coarse scale and a fine scale solution. Using

a suitable function space decomposition, the variational formulation corresponding

to the governing PDEs are also decomposed into a coarse and a fine scale equation.

The fine scale solution is then solved for approximately using one of the following

models: i. Algebraic models (the resulting formulation explains the origins of the

stabilized finite element methods) [50, 51, 21, 11] and ii. Explicit models (the re-

sulting formulation falls under a generalized operator upscaling method) [2, 3, 4] .

The approximate fine scale solution is then used to eliminate the effect of fine scales

on the coarse scale equation. This yields a modelled equation that is completely

defined on the coarse scale yet having information from the fine scales. The deter-

ministic VMS method has been successfully applied in deriving stabilized techniques

for the finite element solution of Navier-Stokes equations, advection-diffusion equa-

tions; operator upscaling techniques for addressing flow in heterogeneous porous

media; diffusion in porous media, etc. The main disadvantage of the VMS method

is that it is essentially a single governing equation system model i.e the same system

of PDEs are assumed to describe the governing physics at all relevant length scales.

Parallel to the development of these multiscale techniques, there has been a pro-

liferation in efficient stochastic representation and uncertainty analysis techniques.

Uncertainty in engineering systems can be sub-divided into two broad categories as

shown in Table(1.1). Of these, the extrinsic uncertainties are effects that are exter-

nal to the system and can be inferred only by indirect means. For e.g. Uncertainty

in numerical code arises due to the internal precision of the computer workstation,

byte ordering, compiler based optimizations, basic linear algebra calculations, trun-

cations (round-off Vs truncation) errors etc. The effect of such uncertainties cannot

4Nature of uncertainty Examples

Extrinsic Effect of surroundings (interactions)

Numerical code, machine precision

Data inconsistencies

Intrinsic (model-form) Constitutive laws

Unresolved dynamics

Intrinsic (parametric) Material data

Initial, boundary conditions

Other process inputs

Table 1.1: Different uncertainties encountered in the characterization of an engi-

neering system

be readily quantified and have to be minimized by conducting several numerical ex-

periments across different computing platforms (serial, parallel, different operating

systems).

On the other hand, the intrinsic uncertainties are inherent to the statistical

nature of the governing physics and material data in the process at hand. These

uncertainties can be quantified using one of the several techniques e.g. sampling,

probability density function modelling, correlations, sensitivity based analysis etc.

For e.g. a material property like diffusion coefficient can be modelled as a spatial

random process with a specified probability distribution and a covariance function.

For all the analysis in this thesis, we assume that such models exist and are known

to us (this thesis does not address derivation of uncertainty models for physical

quantities). The focus here is restricted to capturing the propagation of uncertainties

from the inputs (material data, process conditions) to the output (solution).

5To this end, classical uncertainty analysis methods can be classified into two

types: statistical and non-statistical. The non-statistical techniques involve the

perturbation method, method of moments and analysis of sensitivity derivatives

[42] for investigating free-convection in corn syrup [16], k- turbulence model and

supersonic flow and other. These methods are however, limited to small fluctuations

and do not provide detailed information about higher order- statistics of the solution.

Also, the derivation of a suitable perturbation method for statistical moments of

order greater than two can rapidly become intractable.

In the statistical approaches, The Monte Carlo method has been used extensively

but its is computationally expensive and can become unrealistic even for simple flow

problems. Other techniques involving improved Neumann expansions are also lim-

ited to small fluctuations in the input. To overcome these disadvantages, a more

effective approach (polynomial chaos expansion) based on the spectral representa-

tion of uncertainty was introduced by Ghanem and Spanos [38, 41], wherein, the

stochastic solution is represented as a sum of projections on an appropriate Hilbert

space of random variables. Such a representation was shown to converge to the

exact stochastic solution in the L2 sense [19] and hence in probability distribution.

The polynomial chaos expansion (PCE) in its original context uses Gauss-Hermite

polynomials in standard normal random variable as the trial basis in the Hilbert

space of second-order random variables. In this context it is a Fourier-like polyno-

mial series representation. Though exponentially convergent for a multi-dimensional

random input (the joint PDF of material data, process condition possesses a normal

distribution), the original polynomial chaos does not possess optimal convergence

for other types of inputs e.g. Gamma distributed, uniformly distributed etc. Xiu

and Karniadakis [58, 59] introduced an extension to the PCE called as the gen-

6eralized polynomial chaos expansion (GPCE) that employs hypergeometric Askey

polynomials as the trial basis [86]. Numerical studies indicate that the exponential

convergence behavior is extended to other kinds of input probability distributions as

well. The main drawback of the GPCE based representation methods is that they

fail when the input-output relationship is highly nonlinear /discontinuous due to a

phenomena called as Gibbs effect [8]. This limitation is not encountered in the case

of the stochastic Galerkin method [13] that employs finite element discretization of

the support space of inputs (regions of strictly positive input joint probability den-

sity). The common disadvantage of both approaches is that they are susceptible to

curse of dimensionality i.e. the computational cost of both approaches increases ex-

ponentially as the inputs approach white noise [37]. A summary of the uncertainty

analysis techniques is provided in Table(1.2).

Classification Name of the method

Non-statistical Analysis of sensitivity derivatives

Perturbation analysis

Method of moments

Statistical Monte-Carlo

Neumann expansion

Generalized polynomial chaos expansion

Stochastic Galerkin

Table 1.2: Various uncertainty analysis techniques in current literature

The first part of this thesis deals with derivation of stabilized finite element meth-

ods for stochastic Navier-Stokes and stochastic advection-diffusion equations. The

governing continuum model for these problems are depicted as systems of coupled

7stochastic partial differential equations of the advection-diffusion type. Numerical

solution of these equations often exhibit unphysical behavior in regions characterized

by sharp gradients e.g. presence of boundary or internal layers, discontinuities due

to interfaces, material property variations [48, 49]. Such behavior is more prevalent

in advection dominant systems that are characterized by high advection/diffusion

ratio (indicated by large Pe`cle`t numbers and Reynolds numbers). Further errors are

introduced due to phase, amplitude errors and excess diffusion. Historically, efforts

in computational modelling of such advection-dominant systems has been restricted

to addressing the issues of numerical stability (prevention of spurious oscillations)

[48, 49]. Hence, to simplify the context, the input to such systems were assumed

to be deterministic (precise inlet boundary conditions, initial conditions, material

properties and computational domain). In practice, this is hardly the case and the

input is always polluted with uncertainties (imprecise boundary conditions, statisti-

cal material property variations, model effects arising from inexact governing model

constitutive laws) [59]. Any new development in computational techniques for above

should incorporate an uncertainty analysis framework.

Stabilized finite elements for fluid flow and transport have grown in popularity

over the last two decades starting from the initial techniques like Streamline up-

wind Petrov Galerkin (SUPG) methods, Galerkin least squares (GLS), the SUPG-

Pressure stabilized Petrov Galerkin (PSPG) method [48, 49] upto the more recent

residual free bubbles, subgrid scale models and the variational multiscale framework

[50, 57, 11]. The variational multiscale method has also been used for of large eddy

simulation for turbulent flows [51, 53, 53]. The variational multiscale method also

provides a mathematically rigorous framework for stabilization of convection dom-

inant flows. It unifies the concepts of residual free bubbles, subgrid scale models

8and Greens functions and encompasses other stabilization techniques like SUPG,

SUPG-PSPG and GLS. Herein, we integrate the variational multiscale method with

the spectral stochastic approach to derive the stabilized stochastic FE formulations.

An introduction to mathematical techniques for representation of uncertainty is pro-

vided in Chap.(2) followed by the derivation of stabilized finite element formulations

for fluid-flow and advection-diffusion equations in Chap.(3). A brief introduction to

the definition of a stochastic convection-diffusion equation with specified material

data, initial and boundary conditions is provided with motivation for stabilization

of the equation. The VMS approach is then used to derive stabilization terms for

the case of a stochastic convection-diffusion equation. Extension of the approach is

provided for the case of Navier-Stokes equation with a brief discussion on handling

the nonlinear advection term. Stochastic extensions of standard benchmark numer-

ical examples are considered to validate the accuracy, convergence and efficiency of

the VMS based stabilized FE method.

The second part of this thesis deals with an interesting application of stochastic

analysis in capturing unstable equilibrium in convection systems. Typically, flow

and temperature (output) patterns occurring in the process of natural convection

in enclosures often exhibit high sensitivity to inputs conditions like flow geometry,

boundary conditions, initial perturbations and material data (e.g. viscosity, per-

meability, etc.). Hence, characterization of the effect of input uncertainties on the

output patterns in these systems is a research problem of significant current interest.

For the purpose of addressing this problem, we can identify three major categories

of natural convection systems based on transient behavior and increasing complex-

ity: (i) systems that reach a steady equilibrium state, (ii) systems that undergo

transition between two or more equilibrium states based on the values taken by the

9inputs and (iii) completely transient systems that do not attain equilibrium and

are characterized for example by turbulent flow and heat transfer.In particular, nat-

ural convection exhibits several unstable equilibrium regimes e.g. Flow is started

when the fluid buoyancy effect due to temperature gradients exceeds the stabilizing

viscous effect [20]. The state where these opposing effects neutralize each other is

called the first critical point (here denoted as CP1). Below CP1, the viscous effect

dominates, the fluid flow is absent and heat transfer takes place predominantly by

conduction. Above CP1, the buoyancy effect dominates, fluid-flow is initiated and

heat transfer is by conduction and convection.

Considering a natural convection system with uncertainty in initial conditions,

boundary conditions and fluid properties (collectively called here as inputs). If the

fluctuations in the inputs are such that finite probabilities can be assigned to the

system being both above and below the first critical point, then the propagation

of uncertainty from inputs to output patterns is highly nonlinear and possibly dis-

continuous. This is because of the drastic change in the governing dynamics of the

system above CP1 (convection and diffusion) and below CP1 (pure diffusion). This

poses some interesting questions: Can we capture the effect of input uncertainty

on the flow and temperature patterns in natural convection (i) far from any critical

points (Problem I) and (ii) when the input fluctuations are such that the system

can be above and below CP1 with finite probabilities (Problem II).

In the first part of the thesis, we used the VMS approach in conjunction with

the generalized polynomial chaos for deriving stabilized finite element formulations

for stochastic Navier-Stokes and stochastic advection diffusion equations. We in-

corporated assumptions that the time-integration procedure does not affect the

stabilization and that the subgrid scale/unresolved solution components are qua-

10

sistatic stochastic processes and hence need not be tracked in time. Here, we relax

a few of these assumptions and derive a stabilized stochastic finite formulation for

the natural convection equations. It will be shown that GPCE fails in addressing

problems with discontinuous input-output dependance (Problem type II). We will

further address the problem using the support-space method. The algorithms and

results pertaining to this part of the thesis will be presented in Chap.(4). We start

with the definition of the natural convection problem with a brief discussion of the

Boussinesq assumption. The VMS method is then used to derive the stabilized fi-

nite element formulation. We then discus implementation issues with reference to

the GPCE and the support-space approach. This is followed by two test examples

corresponding to Problems I and II.

In the penultimate part of this thesis, we address a variational multiscale (oper-

ator upscaling) framework for modelling transient diffusion in a multiscale random

medium i.e. the diffusion coefficient of the medium is characterized by spatial vari-

ations across multiple length scales. The potential applications for the problem

include heat transfer in composites [45, 76] and flow in porous media [5]. The cur-

rent state-of-the-art techniques are limited to deterministic description of inherently

statistical material properties like porosity. In reality however, these properties are

characterized by gappy measurements and model constitutive relations that intro-

duce uncertainty. Also, the multiscale nature of variations in material properties

pose the requirement that fully-resolved transient computations should resolve the

smallest length scales in the material data (here, the diffusion coefficient) and time

scales in the solution. This creates a computational challenge.

In practice, however, most engineering decisions are based on coarse scale infor-

mation e.g. net flow, pressure drop, temperature distribution etc. It is thus desirable

11

to develop computational techniques that solve for a coarse-scale solution by defin-

ing an appropriate coarse-scale problem that captures the effect of the fine-scales

[2]. This forms the backbone of most upscaling methods [84].

Current upscaling techniques (more specifically termed as multiscale methods)

aim at the derivation of a coarse-scale formulation that has the following charac-

teristics: i. it contains adequate fine-scale information, i.e. the fine-scale solution

can be reconstructed from the coarse solution and subgrid results, ii. the computa-

tion cost for the coarse-scale problem should scale sub-linearly with respect to the

fully-resolved computation, and iii. it should avoid assumptions of special problem

structures like periodicity and scale separation. However, when these structures are

present, we should be able to exploit them for increasing computation speed.

Using deterministic upscaling techniques for statistical analysis of the solution

typically involves the use of computationally expensive Monte Carlo methods. Re-

cently, there has been a considerable interest in stochastic homogenization, upscal-

ing methods [28, 29]. These methods however employ restrictive assumptions on the

problem structure viz. periodicity and scale separation. Considering the advances

in deterministic upscaling methods and stochastic analysis approaches, it is time to

address direct incorporation of the inherent randomness in material data and the

effect of modelling assumptions in the design of upscaling methods. In this part of

the thesis, the VMS method [50], the MsFEM method [46, 47] and deterministic

operator upscaling technique [1, 2] are combined with the GPCE/polynomial chaos

approach [59, 41] to derive a variationally consistent upscaling technique for the

stochastic transient diffusion equation. Though all of the above techniques are not

original contributions to the thesis, this is the first time these techniques are tied

together to achieve a framework for designing stochastic upscaling techniques. Fur-

12

ther, this work addresses operator upscaling in the context of a transient multiscale

diffusion equation, a work that has not been done even for the deterministic case.

Since randomness is effectively seen as an additional dimension in the problem [38],

the VMS based upscaling method essentially performs upscaling for a class of prob-

lems corresponding to various realizations of the random material data (here, the

diffusion coefficient).

The choice of VMS as the upscaling method and GPCE as the stochastic analysis

method is motivated by a number of reasons. VMS and operator upscaling methods

have emerged in recent years as computational paradigms for development of multi-

scale analysis [56]. The VMS approach essentially involves splitting the variational

formulation for the governing equations into a coarse and a fine-scale part. The

fine-scale part is then solved approximately using localized finite element problems

to obtain the fine-scale solution model, that is substituted in the coarse-scale part

of the variational formulation to obtain an operator upscaled problem. The work

related to this part of the thesis is summarized in Chap.(5). We start with the def-

inition of the transient multiscale stochastic diffusion problem. This is followed by

the definition of the VMS (operator upscaling approach), wherein, model equations

are derived for the fine-scale solution. Finally, we derive the upscaled coarse for-

mulation. Algorithmic issues involved in the implementation of the stochastic VMS

approach are described and numerical examples for validation and error analysis.

Convergence studies and comparisons with a fully-resolved stochastic finite element

solution are provided wherever possible.

Meaningful research aims at generating an equal amount of questions and an-

swers. In this spirit, the last chapter in this thesis is designed to summarize several

open problems involving variational multiscale technique, coupling statistics across

13

length scales and finally, the prospect of a fully multiscale formulation utilizing dif-

ferent governing equations and models at various length scales will be presented as

avenues for future research.

Chapter 2

Mathematical representation of

uncertainty

This chapter presents a function analytic mathematical framework for representa-

tion of uncertainty encountered in description of physical systems. In particular,

this chapter provides information about ideas used extensively in the later parts of

this thesis viz. functional interpretation of a random variable, the Karhunen-Loe`ve

expansion technique, the Generalized polynomial chaos expansion technique and the

support-space method for uncertainty representation. For further detailed informa-

tion about uncertainty representation theory and probability preliminaries, readers

are encouraged to consult [82], [38], [74], [86], [68]

2.1 Preliminary concepts

In this section, basic probability concepts necessary for the development of truncated

series representation techniques for stochastic processes are described.

14

15

2.1.1 Random variable

A probability space [82] is a triple (,F ,P), where, is the sample space corre-sponding to outcomes of some experiment, F is the -algebra of subsets of (thesesubsets are called events) and P is a probability measure, that is, P is a functionwith domain F and range [0, 1] such that the following axioms hold:

P(A) 0 for all A F .

P is -additive.

P() = 1.

Consider two sets and B, with B representing the real line or a subset of the

real line. Then, a measurable function X : 7 B is called a real-valued randomvariable. We can further define the following relation for all A B

[X A] := X1(A) = { : X() A} F . (2.1)

Also, for A = (, x], the function P(X1(A)) is referred to as the inducedprobability distribution of the random variable X.

P(X1(A)) = P[X x] =: FX(x), (2.2)

where, the function FX(x) is a monotone non-decreasing function known as the

cumulative probability distribution function (CDF) of the random variable X. In

addition, if we assume X to be a continuous random variable and hence FX(x) is

differentiable for all x B, we can define the probability density function PDF ofX as

fX(x) := dFX(x)/dx.

16

The above assumption is quite general and can be used to model most of the con-

tinuum quantities of interest such as velocity, material data and other.

In the remaining of this paper, a random variable X will be denoted as X().

The independent variable denotes association with a probability space and is used

to denote randomness in X.

2.1.2 Mathematical expectation and definition of Lp()

The expectation ofX() can be defined using any of these mathematically equivalent

forms

E(X) :=

X()dP() =

X()P(d) =

B

xFX(dx) =

B

xfX(x)dx, (2.3)

where, E denotes the mathematical expectation operator.

Definition of Lp() - For the notion of convergence and distance between random

variables, we use the Lp() function spaces. The space of all random variables with

E(|X()|p) < is referred to as Lp(). The associated norm of any functionX Lp() is defined as

||X||Lp := (E(|X|p))1/p. (2.4)

A sequence of random variables {Xn} converges in Lp to X() if and only if

XnLp X, E(|Xn X|p) 0, as n. (2.5)

Special case of L2() - The most important case of the above discussion is for

n = 2, wherein L2() is a Hilbert space with the inner product defined as follows:

If X, Y L2(), then

(X, Y ) := Cov(X, Y ) = E({X E(X)}{Y E(Y )}

). (2.6)

17

Also, convergence in L2() implies convergence in probability and convergence in

distribution [82], i.e.

XnL2 X P[|Xn X| ] 0, FXn(x) FX(x), as n. (2.7)

This equation is of special significance since we will attempt to construct approxi-

mations to random variables and stochastic processes in the form of sequences that

converge uniformly according to Eq. (2.7).

Remark 1 - The definitions in Sections 2.1.1 and 2.1.2 can be generalized to

vectors of random variables denoted as X() := (X1(), . . . , Xm()) and spatio-

temporally varying random fields (stochastic processes) denoted as W (x, t, ).

2.2 Reduced modeling of stochastic processes

Theoretically, the stochastic process W (x, t, ) can be represented as a random

variable at each spatial and temporal location. Thus, we require an infinity number

of random variables to completely characterize a stochastic process. This poses a

numerical challenge in modeling uncertainty in physical quantities that have spatio-

temporal variations, hence necessitating the need for a reduced-order representation.

In this section, we will consider two most popular ways of approximating a L2

stochastic process using a truncated spectral expansion comprising of a few random

variables:

Approximation by Karhunen-Loe`ve expansion [74, 38].

Approximation by generalized polynomial chaos expansion [59, 68].

18

2.2.1 Karhunen-Loe`ve expansion - KLE

LetW (x, t, ) be a random function defined on the tensor product of a closed spatial

domain D and a closed time interval T . Let E(|W (x, t, )|2)

19

Remark 2 - In practice, N is taken to be a sufficiently small number and is referred

to as the KL dimension or in the context of this paper as the input dimensionality.

In order to define the KLE in Eq. (2.12), an apriori knowledge of the covariance

function is required. This information is not available for the outputs of a contin-

uum system (often solutions of a system of stochastic partial differential equations

(SPDEs)). This shortcoming is avoided by selecting a suitable trial basis compris-

ing of polynomials in L2-random variables. The stochastic process W (x, t, ) can

be represented as a sum of its projections on the trial basis.

2.2.2 Generalized Polynomial Chaos Expansion - GPCE

Polynomial chaos expansion technique for representation of L2-random processes

was originally described in [89]. This constituted representing the random process

as an expansion in terms of Hermite polynomials in the random space (a trial basis

for L2()). Cameron and Martin theorem [19] proved the convergence of such an

expansion to any L2 random process. As an extension to this scheme, the GPCE

was introduced in [59] as an extension to the original polynomial chaos expansion

technique. It uses hypergeometric orthogonal polynomials from the Askey series in

the random space as a trial basis for L2(). For concise definitions of hypergeometric

polynomials and limit relations for the Askey series, see Schoutens [86].

The truncated GPCE of an output random variable X() belonging to L2() as

a function of := (1(), . . . , N()) can be written as

XN()L2= X()

L2= a0I0 +N

i1=1

ai1I1(i1()) +

+N

i1=1

in1in=1

ai1i2...inIn(i1(), . . . , in()) + , (2.13)

where In(i1(), . . . , in()) denote the Wiener-Askey polynomial chaos of order n

20

in terms of the uncorrelated random vector := (1(), . . . , N()), where, N is the

KL dimension for the problem (see Remark 2). In the original polynomial chaos,

{In} are multi-dimensional Hermite polynomials and are orthonormal standardGaussian random variables. In the GPCE, however, {In} and are inter-relatedthrough the joint PDF of . The choice of Askey-polynomials for different types of

probability distributions are shown in Table 1.

Distribution of Askey-polynomials {In} SupportContinuous Gaussian Hermite (,)

Gamma Laguerre (0,)Beta Jacobi [a,b]

Uniform Legendre [a,b]

Discrete Poisson Charlier {0,1,. . . }Binomial Krawtchouk {0,1,. . . ,N}

Table 2.1: Relation between type of Askey-polynomial chosen for the GPCE and

the underlying probability density function of the inputs

For notational convenience, Eq. (2.13) can be rewritten as

X() =

j=0

ajj(), (2.14)

where, the equality is interpreted in the L2() sense and there is a one-to-one corre-

spondence between In(i1(), . . . , in()) and j(). Since each type of polynomial

in the Askey-series forms a complete basis for L2(), we can expect the GPCE to

converge to any L2 random process in the mean-square sense. Additionally, the

following orthogonality relation holds:

E

(i()j()

):=

B

i()j()f()d = E(i()

2)ij , (2.15)

21

where, ij is the Kronecker delta function and f() is the joint probability density

function for the random variables 1(), . . . , N(), i.e.

f() :=dN

d1 . . . dN

(F(1, . . . , N)

), (2.16)

where, F(1, . . . , N) is the generalization of Eq. (2.2) to a vector of random vari-

ables. It was found that by choosing Askey polynomials such that their weighting

function coincides with the joint PDF f() (see Table 2.1), the GPCE Eq. (2.13)

converges exponentially [59].

Remark 3 - The GPCE encompasses the original polynomial chaos/Wiener-

Hermite representation [89]. The convergence results proved by Cameron and Martin

[19] for the original polynomial chaos, however, are not yet extended for the case

of GPCE, which is motivated and supported solely by the exponential convergence

obtained for ODE systems, stochastic advection equation and fluid-flow problems.

2.2.3 Pitfalls in the GPCE and the support-space represen-

tation

From Eq. (2.13), the GPCE can be viewed as a Fourier-like expansion of X()

in terms of hypergeometric Askey polynomials in the random vector . Thus, if

X() as a function of possesses steep gradients or finite discontinuities, then the

GPCE approximation would contain spurious oscillations (Gibbs effect) that are

characteristic of Fourier expansions.

However, the behavior of the solution of a system of PDEs near a critical equilib-

rium point is characterized by sharp changes in the solution values (e.g. velocity or

temperature) and/or the governing physics for small variations in input data (mate-

rial data or boundary condition fluctuations). This nonlinearity is not well captured

22

by the GPCE. A localized representation scheme for output was developed along

the lines of the stochastic Galerkin method. This approach is referred in this thesis

as the support-space method.

2.2.4 Support space method

Let us assume that the inputs to the stochastic system are approximated in their

KLEs and that the input KL dimensionality is N . This means that the stochas-

tic inputs are represented in terms of N independent standard random variables

1(), . . . , N(). If the PDF of i() is given by fi(i), then the joint PDF of

inputs is given by

f() =

Ni=1

fi(i). (2.17)

We can now define the input support space as follows

A := {(1, . . . , N) : fi(i) > 0, for i = 1, . . . , N}, (2.18)

where we further assume that the PDF of each random variable i is bounded.

Using the Doob-Dynkin lemma [13], any solution of a stochastic system of inter-

est can be represented in terms of the inputs as follows

W (x, t, ) := W(x, t, 1(), . . . , N()

)= W (x, t, ). (2.19)

Now consider a discretization of the support space into finite element subdomains

A = Nele=1A(e). The mesh size h is defined as the maximum diameter of A(e). Thefinite element/support-space approximation of W (x, t, ) is constructed as a piece-

wise polynomial of degree q in each element A(e). Note that this approximation is

L2-convergent as follows

||W h(x, t, )W (x, t, )||2L2() :=A

(W h(x, t, ) W (x, t, )

)2f()d. (2.20)

23

Now using the assumption that the PDF of each random variable i is bounded, we

can derive from the above equation the following:

||W h(x, t, )W (x, t, )||2L2() MA

(W h(x, t, ) W (x, t, )

)2d, (2.21)

where M = max f(). The term under integration is now a standard finite-element

representation and hence can be simplified as

||W h(x, t, )W (x, t, )||2L2() MC(x, t)hq, (2.22)

where, C(x, t) is some deterministic function of space and time. The value of C(x, t)

depends on the nature of the stochastic process W (x, t, ) and q is the order of

interpolation.

From Eq. (2.22), it can be observed that the support space approximation is L2-

convergent to the stochastic solution as the mesh spacing parameter h goes to zero.

This type of expansion is highly generalized and encompasses the spectral expansion

techniques (GPCE) as a special case when the number of elements is unity.

Further, discontinuity capturing techniques used in finite elements can be ex-

tended to capture discontinuities/nonlinearities in the output uncertainty. How-

ever, the convergence rate of the support space method is not optimal and should

be avoided while modeling stochastic systems away from critical equilibrium points.

Remark 4 - From Eq. (2.20), we can observe that the error in representation

W h(x, t, ) W (x, t, ) is penalized severely in the regions with large values forthe input joint PDF f(). Hence, for computational purposes, we use a mesh

discretization that is refined at the high input PDF regions. This process is called

an importance sampled gridding (ISG) approach. Our initial investigations point

that ISG increases numerical accuracy for the same discretization level. A schematic

of the ISG approach shown for a two-dimensional input in Fig(). (KLE is truncated

24

Support-space of input Importance spaced grid

Figure 2.1: A schematic of the importance based gridding approach. Note the

refinement of the mesh in regions with high-probability density function values

to two random variables)

2.2.5 Comparison between support-space and Wiener-Haar

approach

We will now discuss in brief, the differences between the above approach and the

Wiener-Haar approach introduced in [36]. We can recall that Haar-wavelets neces-

sarily constitute a piecewise constant representation/orthogonal sampling of a sto-

chastic process. Support-space representation is a finite element kind of representa-

tion of a stochastic process and allows adaptive calculations. Further, Wiener-Haar

and the GPCE approaches complement each other (one can handle discontinuity in a

more robust manner than the other), whereas, the support-space method is actually

a super-set of the GPCE approach. It can be easily shown that the support-space

method using a single element to discretize the entire support-space of the joint PDF

of inputs is equivalent to the GPCE approach. We can choose the basis functions

25

to be the optimal polynomials from Askey-chaos and thereby obtain exponential

convergence rates characteristic of GPCE approach. This however is not possible

in a Wiener-Haar approach. This could imply that the convergence properties of

GPCE approach can carry over to the support-space approach (though more re-

search is needed on this). Furthermore, the solution stochastic process obtained

using the support-space method directly carries physical significance (it represents

the solution for a particular realization of the input).

Based on intelligent combinations of the GPCE, the support-space method, sev-

eral hybrid uncertainty representation techniques can be derived. One of the promis-

ing ones in the multi-element GPCE technique that uses a finite element mesh dis-

cretization of the support-space and describes the stochastic output in a localized

GPCE over each element in the mesh [61]. Such a representation yields the exponen-

tial convergence property of GPCE together with the ability of the support-space

method to capture nonlinearities and discontinuities in uncertainty propagation.

Chapter 3

Stochastic variational multiscale

formulation for advection-diffusion

and Navier-Stokes equations

In this chapter, the deterministic analog of the variational multiscale method is

extended using the spectral stochastic framework and algebraic subgrid modelling

techniques for deriving stabilized FEM formulations for the stochastic advection-

diffusion in Sec. 3.1 and the incompressible stochastic Navier-Stokes equations in

Sec. 3.2. The stabilized formulations are numerically implemented using the spec-

tral stochastic finite element method in Sec. 3.3 wherein, the generalized polynomial

chaos expansion is used for representation of the solution and Karhunen-Loe`ve ex-

pansion is used for representation of the random inputs (material data and boundary

conditions). The developed stabilized formulations are then tested against stochastic

extensions of various standard benchmark advection-diffusion and fluid flow exam-

ples in Sec. 3.4. Comparisons are drawn between the numerical solutions and Monte

Carlo/analytical solutions wherever possible. A summary of computational consid-

26

27

erations and issues pertaining to the proposed formulations are provided in Sec.

3.5.

3.1 VMS for linear advection-diffusion equation

Let D Rd, where d1 is the number of space dimensions, be an open, bounded,polyhedral domain with piecewise smooth boundary , T = {t : t [0, T ]} and(,F ,P) be a probability space. T is identified as the time interval for simulation.

The transient advection and diffusion of a stochastic scalar process in the pres-

ence of a divergence-free stochastic velocity field a(x, ) in a medium with random

non-negative diffusion coefficient () can be defined as follows:

Find (x, t, ) : (D T ) 7 R such that

t+ L = f, (x, t, ) (D T ), (3.1)

(x, t, ) = g(x, t, ), (x, t, ) ( T ), (3.2)

(x, 0, ) = 0(x, ), (x, ) (D ), (3.3)

where L(x, t, ) is the stochastic advective-diffusive operator defined as

L(x, t, ) = a (). (3.4)

f(x, t, ) : (D T ) 7 R is a source term and g(x, t, ) : ( T ) 7 R isthe specified stochastic Dirichlet boundary condition.

It should be emphasized here that the probability model used for the advec-

tion velocity a(x, ) should ensure the divergence-free constraint a = 0 and theprobability model used for the diffusion coefficient () should have a non-negative

support space.

Since Gaussian distribution assigns finite probability to negative values, the dif-

fusion coefficient cannot be modelled as a Gaussian random variable. This startling

28

result has been proved for the case of diffusion problems [62] wherein, it was shown

that a Gaussian thermal conductivity assumption leads to an ill-posed polynomial

chaos system of equations when implemented using the spectral stochastic finite

element method. The argument shall henceforth be dropped if it is clear by the

content that the quantities are random.

3.1.1 Variational formulation

Let E denote the trial solution space and E0 denote the weighting function spacedescribed as follows:

E = {u : u(x, t, ) L2(;L2(T ;H1(D))), u = g on }, (3.5)

E0 = {v : v(x, ) L2(;L2(T ;H1(D))), v = 0 on }. (3.6)

Where, L2() denotes the space of second-order (finite variance) random variables,

L2(T ) denotes the space of square integrable functions defined on the time interval TandH1(D) is the Sobolev space of square-integrable functions with square-integrablederivatives defined on the spatial domain D. The variational counterpart of thestrong system of Eqs. (3.1)-(3.3) reads as follows:

Find E such that w E0 and t T

(t, w) + b(, w) = (f, w). (3.7)

The bilinear form b(, w) introduced above is given by

b(, w) = (L, w) = (,Lw) = (a, w) + (,w)v, (3.8)

where the inner-product (g,h)v is defined as

(g,h)v =

D

E[g(x, t, ) h(x, t, )]dx =

D

g(x, t, ) h(x, t, )dPdx, (3.9)

29

and the inner-product (g, h) for a given t T is defined as

(g, h) =

D

E[g(x, t, )h(x, t, )]dx =

D

g(x, t, )h(x, t, )dPdx. (3.10)

It should be noted that the first two equalities in Eq. (3.8) require stronger regularity

conditions (viz differentiability) on the solution (x, t, ). The assumption that

g(x, t, ) and h(x, t, ) are second order random processes together with the Schwarz

inequality,

E[g(x, t, )h(x, t, )] (E|g(x, t, )|2)12 (E|h(x, t, )|2)12

30

The objective of the variational multiscale (VMS) method is to derive a variational

statement for that takes into account an approximate model for the subgrid scale

solution .

Typically, the large scale trial solution and weighting function spaces are asso-

ciated with finite element spaces and hence are finite-dimensional. In contrast, the

subgrid scale function spaces do not possess scaling information and are infinite-

dimensional. Using the multiscale framework developed above, Eq. (3.7) can be

split into two-scale problems as follows:

(t+ t, w) + b(, w) + b(, w) = (f, w), (3.16)

(t+ t, w) + b(, w) + b(, w) = (f, w). (3.17)

The main idea here is to use Eq. (3.17) to arrive at approximate model for (x, t, ).

This approximate model is then used to eliminate from Eq. (3.16).

Until now, Eqs. (3.16) and (3.17) are exact, highly coupled and hence extremely

tough to solve. Thus, several stages of modeling assumptions are introduced to

simplify the subgrid scale Eq. (3.17).

Assumption I. The subgrid scales are quasi-static i.e. t 0. The validity

of this assumption is discussed in [21],[22]. This assumption requires that the time

integration be accurate enough so that the large scale can capture the temporal vari-

ation of the solution. In problems involving high Reynolds number flows, one needs

to explicitly track the subscales [23], [24], [25]. Since this preliminary effort is to in-

tegrate the variational multiscale method with the stochastic finite element method

and to study the effects of stochastic modeling on the stabilization parameters, only

quasistatic subgrid scales are investigated herein.

Eq. (3.17) can now be simplified as

b(, w) = (f t, w) b(, w). (3.18)

31

The above equation together with strong regularity conditions for (see Eq. (3.8))

yields the following subgrid scale equation

For E , the following is satisfied for all w E 0

(L, w) = (R, w), (3.19)

where, R(x, t, ) = f tL is the large scale residual.Consider a finite element partition into elements indexed as {1, . . . , Nel}, divid-

ing the spatial domain D into subdomains D(e) with element boundaries (e). Thestrong form of Eq. (3.19) is then approximated over each element (e) as follows:

L = R, (x, t, ) (D(e) T ), (3.20)

= 0, (x, t, ) ((e) T ). (3.21)

The assumption that the subgrid scale solution vanishes at element boundaries is

a strong assumption and is intimately linked with the idea of residual-free bubble

functions [11], [83]. For computational simplicity, it is desirable to derive an ap-

proximate algebraic model for the subgrid scale solution. This leads to the second

modeling assumption.

Assumption II. The algebraic subgrid scale model is considered to be of the form

(x, t, ) = (x, )R(x, t, ), (3.22)

where, the parameter (x, ) is inherently stochastic and is interpreted as the in-

trinsic time scale for the stochastic subgrid solution.

With substitution of Eq. (3.22) into Eq. (3.16), the complete multiscale sta-

bilized variational statement for the stochastic advection- diffusion problem can be

written as

(t+ a, w + (x, )[aw + ()w]) +

32

((),w)v ((), (x, )[aw + ()w]) =

(f, w + (x, )[aw + ()w]), (3.23)

where denotes the Laplacian operator 2. Typically for linear finite elementsmost of the terms in the above stabilized formulation drop out leading to a SUPG

like formulation for stochastic advection-diffusion equation.

We shall now proceed to derive models for the intrinsic subgrid time scale. For

simplicity, in the ensuing derivation it will be assumed that the advection velocity is

constant in an element. The results however are general and can be used for velocity

and diffusion coefficient varying within an element.

3.1.3 Models for - intrinsic subgrid time scale

Models for the intrinsic subgrid time scale are not unique. Different models can

be suggested based on the level of subgrid characterization desired, phase lag and

transient behavior restrictions. All these models however should essentially possess

similar behavior in the limits of pure advection and pure diffusion.

Several techniques viz. Greens function methods, Fourier analysis, Taylor se-

ries expansion can be employed to arrive at different models for . In this work

however, we follow the Fourier analysis approach. This helps keep the derivation

of consistent for the stochastic advection-diffusion and stochastic Navier-Stokes

problems.

We begin by defining the Fourier transform of a generic stochastic function

g(x, ) defined on an element (e)

g(k, ) :=

D(e)

exp( ik x

h

)g(x, )dx, (3.24)

where h is an elemental length parameter, k denotes the wave number and denotes

33

association with the probability space.

In the definition of variational multiscale framework we interpreted the exact

solution as an overlapping sum of a resolved large scale component and an unre-

solvable subgrid scale component. Thus in the wave number space, the large scale

solution corresponds to the lower wave number modes and the subgrid scale solu-

tion corresponds to the larger wave number modes. This allows us to arrive at an

approximation for the spatial derivative of the generic stochastic function g(x, ).

g

xj(k, ) =

(e)

nj exp( ik x

h

)g(x, )d + i

kjhg(k, ) ikj

hg(k, ), (3.25)

where nj is the jth component of the outward normal to the element (e). The

Fourier transform of the subgrid scale Eq. (3.20) now yields

(k, t, ) (k, )R(k, t, ), (k, ) :=(()

||k||2h2

+ ik a

h

)1. (3.26)

Note that in the above expression the assumption of velocity being constant within

an element is required. Using Plancherals formula and the mean value theorem, we

arrive at

() [(c1()

()

h2

)2+(c2()

|a|h

)2]12, (3.27)

where |a| = |a()| is defined as

|a| = |a()| = a a. (3.28)

This choice of the intrinsic time scale makes ()R equal to the subgrid solution in the L2 sense.

Furthermore, the asymptotic behavior of in the diffusive limit is dominated

by the term h2

()and the asymptotic behavior in the advection limit is dominated

by the term h|a|. The intimate link between Eq. (3.27) and SUPG like stabilization

34

methods is seen in a particular choice of the random constants viz. c1() = 4 and

c2() = 2. This leads to an intrinsic subgrid time scale model of the form

() [(

4()

h2

)2+(2|a|h

)2]12. (3.29)

In this work, however, we chose a model having similar asymptotic properties

as the model in Eq. (3.29). The proposed model minimizes phase lag in transient

problems and extends to the more general case of spatially varying stochastic velocity

field and random diffusion coefficient. It is given as:

(x, ) =h

2|a(x, )|f(Pe(x, )), (3.30)

where, h is the elemental length and the function f(Pe) is defined for linear finite

elements as

f(Pe) =Pe

3I[Pe:03], (3.31)

where, IA is the indicator function for set {A} and Pe is the element Pe`clet number

Pe(x, ) =|a(x, )|h2(x, )

. (3.32)

Remark 1. (x, ) represents the intrinsic time scale for a real process viz. the

subgrid solution. Hence the model chosen for should ensure that the subgrid scale

solution has finite statistical moments (mean and variance). However, the statistical

behavior of depends on the kind of probability models chosen for the advection

velocity and diffusion coefficient.

Given a model for (x, ), the above conditions constrain the probability mod-

els available for a(x, ) and (). Typically, spurious oscillations are noticed in the

numerical solution when probability models with unbounded support space are spec-

ified for the advection velocity and diffusion coefficient. The models that fall under

35

this category are normal, gamma and lognormal distribution models. Most prob-

ability models with finite support space are usually compatible with the proposed

intrinsic time scale model. These include beta and uniform probability models.

We shall now elaborate on the remark using a simple one-dimensional advection-

diffusion case study.

3.1.4 Intrinsic time scale models and induced constraints -

A one-dimensional case study

Consider the one dimensional version of the stochastic convection-diffusion problem

defined by Eqs. (3.1)-(3.3) with the spatial domain D = [0, L]. The norm ofthe advection velocity then simply is |a|, the absolute value of a(x, ). We donot use boldface for the advection velocity since it has a single component. The

intrinsic subgrid time scale for this problem is as defined in Eq. (3.30). We now

consider different probability models for advection velocity and diffusion coefficient

and analyze the behavior of the subgrid scale solution.

Case I: Pure advection, no source term

In this case, the expression for simplifies to

(x, ) =h

2|a(x, )| , (3.33)

and the subgrid scale solution can be written as

(x, t, ) =h

2|a(x, )|(t+ a

x

). (3.34)

Since, represents the subgrid scale for a physical quantity, the statistical moments

for upto second order should be finite [since H1(D) L2(T ) L2()].

36

Expanding in a truncated generalized polynomial chaos expansion, we obtain

the following:

(x, t, ) =

Pi=0

h

2|a(x, )|(tii() + a

ii()

x

), (3.35)

where the polynomials {i()}Pi=0 belong to the Askey series of orthogonal polyno-mials and form an orthogonal basis of L2().

If we consider the advection velocity to be a normal random variable withN(, )

distribution, the mean of can be written as

E =

Pi=0

h

2|+ |(tii() + a

ii()

x

)12

exp(122)d. (3.36)

Note that, in the above equations, dependence is shown viz and that all poly-

nomial chaos are functions of (here Hermite polynomials).

In order for the above integral to converge, it can be shown that the following

term needs to be finite (note that the coefficients of polynomial chaos 0, 1, . . .,

are deterministic):

h

2|+ |(t0 +

0x

)12

exp(122)d. (3.37)

However, the behavior of the above integral is governed by the divergent integral

1

2|+ |12

exp(122)d. (3.38)

Hence, a normal distribution is not an appropriate model for the advection velocity

under the proposed choice of . However, if a uniform distribution model is chosen

for a with

a = + (), d

=U [1, 1], , > 0, < , (3.39)

then it can be shown that the first two statistical moments of are finite. The

exact expressions are very complicated and hence are not supplied here. However,

37

the counterpart of Eq. (3.37) can be written as

+

h

2|a|(t0 +

0x

)1

2da. (3.40)

This integral converges and has the value

(t0 +

0x

){log( + ) log( )} 1

2. (3.41)

Most probability distributions with finite support behave in a similar manner under

the proposed assumption of . This case study is not to discourage the use of

probability distributions with infinite support. The study is to point to the fact that

caution has to be exerted to ensure that the model for and the input uncertainty

models are compatible.

Case II: Diffusion dominant regime, no source term

In this case, the expression for simplifies to

() =h2

4(). (3.42)

Analysis of the behavior of for this case proceeds along similar lines as in the

previous case. In this case, even distributions with infinite support like Gamma,

chi-squared and shifted log-normal ensure L2().However, one should be careful in selecting distributions with infinite support.

Choosing () to be of the form |X()| where X() d=

N(, ), leads to problems

very similar to the previous case. This is because |X()| attributes non vanish-ing probability for values near zero. Since ()1 is unbounded near = 0, the

expectation diverges. This leads to an important observation summarized below.

Remark 2. The selection of distributions for advection velocity and diffusion

coefficient is constrained upon the definition of the intrinsic time scale. There is

38

a great scope for defining appropriate models for that are consistent with the

probability models for a and .

3.2 The stochastic incompressible Navier-Stokes

equations

Let D Rd, where d1 is the number of space dimensions, be an open, boundeddomain with piecewise smooth boundary , T = {t : t [0, T ]} be the time intervalof analysis and (,F ,P) be a probability space.

The strong form of stochastic Navier-Stokes problem consists of finding the sto-

chastic velocity v(x, t, ) and pressure p(x, t, ) such that

tv + vv ()v +p = f(x, t, ), (x, t, ) (D T ), (3.43)

v = 0, (x, t, ) (D T ), (3.44)

v = vg(x, t, ), (x, t, ) ( T ), (3.45)

v(x, 0, ) = v0(x, ), (x, ) (D ), (3.46)

where, () is the random kinematic viscosity and f(x, t, ) is a stochastic forcing

term. The uncertainty in this problem comes from (), f(x, t, ), initial and

boundary conditions. In this work, we consider constant property flows, hence

the kinematic viscosity is considered to be a random variable with a non-negative

support space (regions of strictly positive probability density).

3.2.1 Variational formulation

Let V and V0, the trial solution and weighting function spaces for velocity and letQ and Q0 denote the trial and weighting function spaces for pressure.

39

V = {v : v(x, t, ) [L2(;L2(T ;H1(D)))]d, v = vg on }, (3.47)

V0 = {w : w(x, ) [L2(;L2(T ;H1(D)))]d, w = 0 on }, (3.48)

Q = {p : p(x, t, ) L2(;L1(T ;H1(D)))}, (3.49)

Q0 = {q : q(x, ) L2(;L1(T ;H1(D)))}}, (3.50)

where L1(T ) denotes functions of bounded variation in time. It should be notedhere that the function spaces used for velocity and pressure do not have the same

regularity conditions (velocity should be twice differentiable whereas pressure need

only be once differentiable), hence the weighting function spaces also differ. The

variational formulation counterpart for the strong system of Eqs. (3.43)(3.44) reads

as follows:

Find (v, p) (V,Q) such that (w, q) (V0,Q0), the following is satisfied t T

(tv,w) + (()v,w)v + (vv,w) (p,w) = (f,w), (3.51)

(q,w) = 0, (3.52)

where, the inner-product (g,h) is defined as

(g,h) :=

D

E[g h]dx, (3.53)

and the inner-product (g,h)v is defined as

(g,h) :=

D

E[g : h]dx. (3.54)

It is assumed that the initial condition is satisfied in a weak sense.

40

3.2.2 Variational multiscale hypothesis

Consider an overlapping sum decomposition for velocity and pressure v = v+v and

p = p+ p. Consider similar decomposition for the weighting functions w = w+w

and q = q + q. This induces a multiscale decomposition for the function spaces of

the form V = V V , V0 = V0 V 0, Q = Q Q and Q0 = Q0 Q0, where the barindicates reference to large resolved scales and the dash indicates reference to the

subgrid scales.

Unlike the stochastic advection-diffusion equation, the presence of a nonlinear

convection term necessitates an apriori assumption in the derivation of a multiscale

stabilized formulation.

Assumption III. Assuming the large scales are sufficiently resolved, the subgrid

scale solution can be considered to be small compared to the resolved large scale

solution. This justifies a one step Picards linearization for the nonlinear advection

term

vv vv + vv. (3.55)

This assumption is valid for low to moderate Reynolds numbers.

At high Reynolds numbers, adequate grid resolution is computationally highly

demanding. Hence often the large scales are only partially resolved. As a conse-

quence, the kinetic energy held in subgrid scales becomes substantial (> 20% of

energy in the system). Further, the nonlinear subgrid convection term vv as-sumes importance. Thus, a coupled subgrid scale and resolved scale equation has

to be solved at each time step. Hence such high Reynolds number flows are not

addressed in this preliminary work.

The variational form given by Eqs. (3.51)(3.52) now reads as follows:

(tv + tv, w) + (v + v,w)v + (vv + vv, w) (3.56)

41

(p + p, w) = (f, w), (3.57)

( v + v, q) = 0, (3.58)

(tv + tv,w) + (v + v,w)v + (vv + vv,w)

(p+ p,w) = (f,w), (3.59)

( v + v, q) = 0. (3.60)

Assumption IV. Consider subgrid scale velocity and pressure to be quasistatic

random processes tv 0 and tp 0 [21, 22]. If the time scales of subgrid solu-

tions are different from those of the large scale solutions as in the case of turbulent

flows, this assumption is not valid. We would then need to explicitly track the

subgrid solution evolution in time. By assuming linearity of subscales, we tacitly

assume that the time scales of subgrid solution and large scale are nearly the same

and that we capture the complete time evolution of the solution through tv. Fur-

ther, the following relations hold under assumption of stronger regularity conditions

on velocity and pressure (twice differentiability).

The subgrid scale variational fo