Evaluation of Discrete Explicit Filtering for an Approximate … · 2013-10-17 · Evaluation of...

Evaluation of Discrete Explicit Filtering for anApproximate Deconvolution Approach to LES

by

Sintia Bejatovic

A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science

Graduate Department of Department of Applied Science and EngineeringUniversity of Toronto

Copyright © 2011 by Sintia Bejatovic

Abstract

Evaluation of Discrete Explicit Filtering for an ApproximateDeconvolution Approach to LES

Sintia Bejatovic

Masters of Applied Science

Graduate Department of Department of Applied Science and Engineering

University of Toronto

2011

In the study of computational turbulence, the success of Large Eddy Simulation (LES) is largely

determined by the quality of the sub-filter scale (SFS) model and the properties of the filter used

to introduce resolved and unresolved length scales. Additionally, the SFS model and filter are

strongly linked, in such a way that the filter process cannot be too contaminated with aliasing

errors, as this would cause an inaccurate representation of the flow. One way to alleviate this

issue is to use explicit filtering, so that better control over the filter may be achieved, and filter

operator errors can be then controlled to a desired order of accuracy. One large advantage to

using an explicit filter is that the mathematical definition of the filter may be exploited when

considering various SFS models or even different LES techniques. Approximate deconvolution

is a technique used in LES, which performs an inverse filtering operation to partly restore

the original unfiltered solution. This thesis considers a form of structural modeling, known

as approximate deconvolution, to perform a large-eddy simulation of homogeneous, isotropic

turbulence. In particular, the discrete explicit filtering technique will be used to perform the

deconvolution, and numerical results will show how the approximate solution may be used to

perform LES.

ii

Acknowledgements

I would first like to acknowledge Prof. C.P.T. Groth and Prof. O.L.G.Gulder, for providing

me with an opportunity, to study under their supervision, and in particular to Prof. C.P.T.

Groth, who introduced me to an incredibly interesting area in LES research. I would also like

to extend gratitude to the University of Toronto and Prof. C.P.T. Groth, for providing funding

for this research experience.

I am thankful to the students in the UTIAS CFD lab who have taken the time and patience

to help me become comfortable with the hurdles of computing. I would also like to take this

opportunity to express my appreciation to Prof. U. Piomelli, who’s insights and discussion have

made the completion of this work possible.

Toronto, November 2010 Sintia Bejatovic

iii

Contents

Abstract iii

Acknowledgments iii

Contents vi

List of Tables vii

List of Figures ix

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

List of Symbols 1

2 Large Eddy Simulation in Turbulence 6

2.1 Nature of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Statistical Turbulence and Kolmogorov’s Hypothesis . . . . . . . . . . . . 8

iv

2.1.3 The Energy Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Turbulence in Fourier Space . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Methods in Computational Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Reynolds Averaged Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Initializing Turbulence in a Periodic Box . . . . . . . . . . . . . . . . . . . 18

2.3 Techniques in Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Filtering of the Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . 19

2.3.2 The Unresolved Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 A Priori and a Posteriori in LES . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Fundamentals of Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.5 Errors in LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Explicit Filtering Techniques 29

3.1 Analytical Explicit Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Standard Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Differential Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 Commutation Error of Analytical Filters . . . . . . . . . . . . . . . . . . . 33

3.2 The Discrete Explicit Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Formulation of the Discrete Filter . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Least-squares (LS) Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Discrete Commutation Error . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Approximate Deconvolution Methods . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Approximate Deconvolution by Iteration . . . . . . . . . . . . . . . . . . . 43

3.3.2 Approximate Deconvolution by Series Expansion for a Discrete Filter . . 43

v

4 Approximate Deconvolution Approach and Numerical Solution Method 46

4.1 Scale Similarity Approximate Deconvolution . . . . . . . . . . . . . . . . . . . . . 46

4.2 Refined Approximate Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Results and Discussion 53

5.1 Approximate Deconvolution for Homogeneous

Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1 Initial Conditions of u and u∗ . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.2 Transfer Function and Commutation Error of QG . . . . . . . . . . . . . 58

5.1.3 Time Advanced Solution of u and u∗ . . . . . . . . . . . . . . . . . . . . . 61

5.1.4 Approximate Deconvolution Error Term . . . . . . . . . . . . . . . . . . . 64

5.1.5 Computational Cost of Deconvolution . . . . . . . . . . . . . . . . . . . . 68

6 Conclusions and Future Work 72

6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

References 77

vi

List of Tables

5.1 Parameters of LS filter and initial turbulent spectrum . . . . . . . . . . . . . . . 55

5.2 Computational cost of approximate deconvolution at t = 2 ms for grid size 32 x

32 x 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Computational cost of approximate deconvolution at t = 2 ms for grid size 64 x

64 x 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

vii

List of Figures

2.1 Wavenumber spectrum of turbulent length scales on a logarithmic scale. . . . . . 11

2.2 Simplified-idealistic turbulent energy cascade. (logarithmic scales). . . . . . . . . 12

2.3 Real turbulent energy cascade (logarithmic scales). . . . . . . . . . . . . . . . . . 12

2.4 The resolved and modeled (unresolved) portions of the turbulent kinetic energy

spectrum in a RANS simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 The resolved and modeled (unresolved) portions of the turbulent kinetic energy

spectrum in a LES simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Multiple scale interaction on the turbulent kinetic energy spectrum in LES. . . . 18

2.7 Illustration of the convolution operation used in filtering with kernel, G. . . . . . 26

3.1 Sharp Fourier cut-off filter. Convolution kernel G depicted in physical space. . . 31

3.2 Top-hat filter. Convolution kernel G depicted in physical space. . . . . . . . . . . 32

3.3 Gaussian filter. Convolution kernel G depicted in physical space. . . . . . . . . . 32

5.1 Turbulent initial field on 64 x 64 x 64 grid. Spectrum of x-directional velocity. . 56

5.2 Q-criterion of solutions, u∗ and u on 64 x 64 x 64 mesh at t = 0 ms. . . . . . . . 57

5.3 Turbulent kinetic energy spectrum of filters, QG and G on 64 x 64 x 64 mesh. . . 57

5.4 QG(κ) and G(κ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5 QG(κ) for Van Cittert series truncations, N = 1,2,3,4,5,6. . . . . . . . . . . . . 60

5.6 L2 norm of commutation errors of filters G and QG for desired orders of com-

mutation, Ec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

viii

5.7 Turbulent field on 64 x 64 x 64 grid at t = 2ms. . . . . . . . . . . . . . . . . . . . 63

5.8 Turbulent field on 64 x 64 x 64 grid at t = 7 ms. . . . . . . . . . . . . . . . . . . 64

5.9 Decay of turbulent kinetic energy at t = 2 ms for solutions filtered with QG and

G on grid resolution of 64 x 64 64. . . . . . . . . . . . . . . . . . . . . . . . . . . 65


G on grid resolution of 32 x 32 x 32. . . . . . . . . . . . . . . . . . . . . . . . . . 66


G on grid resolution of 32 x 32 x 32. . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.12 Decay of isotropic homogeneous turbulence using traditional LES and refined

approximate deconvolution for grid resolution of 32 x 32 x 32. . . . . . . . . . . . 68

5.13 Solution, u∗, for Van Cittert series truncations, N = 3, 6, on grid sizes 643 and

853. The solution in (a) and (b) corresponds to t = 2 ms and in (c) and (d) the

solution is at t = 0.5 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ix

Chapter 1

Introduction

1.1 Introduction

Turbulence is a complex phenomena characterized by non-deterministic behaviour, whose study

must lend itself to numerical and/or statistical techniques as rigorous analytical solution tech-

niques have not yet been successful for the full range of problems. Turbulence is viewed as

a complex system in the most general sense, and thus carries the attributes needed for study

in fields ranging from pure mathematics to applied science and engineering. In applications

commonly found in engineering, such as aircraft propulsion and biological fluid flow for in-

stance, turbulent fluid motion can create undesirable affects which, in order to be controlled,

first need to be better understood. In the field of pure applied mathematics, turbulence is

studied from the perspective of dynamical systems, with the aim of finding weak solutions to

the Navier-Stokes equations, usually under certain restrictive circumstances (for instance the

incompressible case). In the study of dynamical systems, the problem of turbulence is stated

as a problem of asymptotic regimes of solutions to the Navier-Stokes equations as ν → 0 [1].

Alternatively, in engineering, turbulent fluid motion is described by the governing equations

of fluid motion (Navier-Stokes equations), which are most commonly grouped as continuity,

momentum and energy. In such applications, the interest is not to obtain any form of exact

solution to the governing equations, but rather to use discrete techniques to approximate the

solution [2].

An attempt to describe turbulent fluid motion in such applications today is seen in the area

of numerical methods, which provide a discrete solution to fluid dynamics problems associated

with some magnitude of error. One goal in the field of computational fluid dynamics (CFD)

is concerned then with ways of controlling and reducing this error using numerical methods.

1

Chapter 1. Introduction 2

However, the global goal of CFD is to understand the physical phenomena of fluid motion

both in and around the presence of various geometries [2]. One could, then further consider

the field of computational turbulence as a sub-field of CFD, in cases where the ultimate goal

is to simply understand turbulence, and the numerical method associated with progressing

the discrete solution in time, is not the primary concern. It then suffices to be reminded

that studying turbulence is a daunting task from a theoretical aspect, such that in academic

turbulent studies, the primary goal lies in understanding the phenomena itself.

Numerical simulations of turbulence have been divided into three categories, commonly termed

Direct numerical simulation (DNS), Reynolds averaged Navier-Stokes (RANS), and Large eddy

simulation (LES). Most certainly, DNS provides the most accurate solution to simulations in

fluid dynamics, and whenever computational resources permit, this approach is most desirable

[3]. The largely varying alternative to DNS, is the RANS simulation technique, which can pro-

vide practical solutions when a detailed study of turbulence phenomena is not the primary goal

[4]. It is presently agreed, that LES is the most promising approach to studying turbulence [5],

since it offers the best compromise between solution accuracy and reduction of computational

resources, which are the two key issues presently being addressed in CFD. Although LES is

promising for the aforementioned reasons, the slightly unfortunate (or perhaps fortunate in the

author’s opinion) issue is the increased complexity introduced in LES techniques. An attempt

to describe the previous remark can proceed as follows. First, the complexity of the system

can be broken down into theoretical complexity, referring to the chaotic nature of the dynam-

ical system itself and the numerical complexity, resulting from the various space-time scales

present. Certainly, it is desirable that more of the former complexity be captured by the LES

solution, as this would give a more accurate representation of the flow. However, present LES

techniques, are such that, there is an introduction of multiple length scales, resulting from the

mathematical operators acting on the Navier-Stokes equations. This latter form of complexity

is not undesirable only if a distinction can be made between the nonlinear interactions [6] of

such scales, which is presently not well understood. Another shortcoming of LES, and one that

is typically not addressed in LES studies, is that, unlike the unfiltered Navier-Stokes equations,

the filtered LES equations, are not Galilean invariant [7]. This fact however, has not prevented

the progression of some of the most successful models used today.

1.2 Motivation

The study of LES has progressed rapidly in the recent past, to overcome the limitations of more

primitive simulation techniques, and as a result has permitted better insight into the physics


of turbulence. In practical settings, less rigorous forms of filtering have been used with some

success, however to increase the capability of LES for a larger class of turbulent flows, there

is a requirement for precisely a more rigorous approach to filtering. To reflect on the contrast

between more and less rigorous forms of filtering, we can first distinguish between implicit and

explicit filtering [3, 8, 9]. Implicit filtering, is not a direct approach to filtering, since the filtering

is considered inherent in the numerical discretization scheme. In other words, the discretization

scheme itself, acts as a filter by removing high frequency content in the solution according to

the grid resolution. In many cases however, a sub-filter scale model is used in combination

with an implicit filtering approach, which adds dissipation when necessary. Although quite

successful [10], the direct dependence of the grid on the filter, does not really allow one to

perform a true LES [8], which in this work is understood to be one in which the filtering and

numerical discretization procedures are distinct.

Alternatively, a stronger approach to distinguishing the grid and filter, is to introduce a filter

operator which acts on the Navier-Stokes equations, such that this operator is endowed with

desired a priori properties. This explicit operator, then allows one to understand the filtered

quantities more rigorously. An important remark is made here regarding grid-indepedence and

LES. One needs to realize that although an explicit filter decouples the filtering operation from

the grid-discretization, this does not imply a complete grid-independent LES. The reason for

this, is that the length scale associated with the filter, is still most likely dependent on the grid

spacing [11], so that using an explicit filter may be seen as a weak form of grid-independence.

Using explicit filtering as an alternative to grid-dependent, implicit filtering has allowed more

deeply rooted issues with turbulent simulations to surface. Such issues include the desire for

stronger grid-independent LES, control over aliasing errors and more complex SFS modeling.

Using explicit filtering techniques addresses all three issues by allowing for a more rigorous

approach to LES.

An explicit filter may be either of the analytical or discrete type, where the latter is more desir-

able when the control of commutation errors is necessary and implementation in CFD code is

often straightforward [12]. More specific advantages of explicit filtering, is that it allows for con-

trol over the filter width and in the presence of commutation errors between various operators,

using explicit filtering provides an opportunity to control this error. Perhaps an even greater

potential, which has already been explored, is the ability to create better SFS models by ex-

ploiting the explicit nature of these filters. The most clear example of this is in the class of SFS

models known as structural modeling, where approximate deconvolution methods have made

considerable progress [13, 14, 15]. Approximate deconvolution has already been used to simu-

late compressible flows in wall-bounded turbulence [15], and the results have been promising.


Specifically, the approximate deconvolution technique was found to quite accurately describe

shock-turbulent boundary-layer interaction and turbulence decay for low-order pseudo-spectral

LES [16]. It was found however, in the case of low-order schemes, adjustments had to be made

when prescribing the cutoff wavenumber to produce accurate turbulence decay.

1.3 Objectives

This work seeks to gain a more comprehensive understanding of how discrete explicit filtering

may be used to perform structural modeling, so that traditional sub-filter scale (SFS) modeling

will not be required. The class of structural models which involve performing approximate

deconvolution through forms of series expansions as suggested by Sagaut et al. and Mathew

et al. [17, 15], will be explored.

The primary focus will then be the analysis of how discrete explicit filtering, may be used to

perform more non-conventional LES using approximate deconvolution. The proposed struc-

tural model will be used to predict the solution of a homogeneous, isotropic turbulent flow on

a uniform mesh with periodic boundary conditions. The discrete explicit filtering procedure

of Deconinck [18] based on least-squares reconstruction is considered here. Components of

this study will cover both application of the approximate deconvolution model in addition to

theoretical implications of the proposed LES approach. More precisely, the structural model

investigated in this work is one, which uses series expansion to perform an approximate de-

convolution of the filtered variable. The approximate deconvolution approach studied in this

thesis, unlike common application of approximate deconvolution, will advance the deconvolved

variable in time, and treat the error associated with deconvolution as a corrective term, which

is also advanced in the LES equations [15].

1.4 Overview of Thesis

This thesis will be presented as follows. Chapter 2 will provide the reader with fundamental

concepts of turbulence and its study in CFD, which is really the field of computational tur-

bulence. Emphasis will be placed on describing fundamental concepts in LES. Chapter 3 will

demonstrate the concept of explicit filtering, which will cover common filters, differential filters

with a focus on the discrete explicit filter. The end of the chapter will introduce structural

modeling and the concept of approximate deconvolution. Chapter 4 will present the numerical


formulation for both the deconvolution model considered in this work and will conclude by

briefly describing the numerical spatial and temporal scheme used to perform the simulations

presented in this thesis. Chapter 5 will then present the results of the deconvolution model us-

ing a discrete filter for the decay of homogeneous, isotropic turbulence, followed by statements

on considerations for future work in Chapter 6.

Chapter 2

Large Eddy Simulation in

Turbulence

This chapter provides a brief overview of the nature of turbulence and current approaches used

in predicting turbulent flow structures. A general overview of statistical turbulence will be

presented along with how turbulence is studied in spectral space. This will provide preparation

for introducing the concept of LES.

2.1 Nature of Turbulent Flows

2.1.1 Description

There most obvious characteristic of turbulence is irregularity, first introduced by Kolmogorov

as the notion of chaos. Strictly speaking, chaos defines a dynamical system, whose behaviour is

unpredictable, due to the strong dependency on initial conditions of the system. Then, in the

study of dynamical systems, such as turbulence, statistical methods are the likely approach,

when direct numerical solutions of the governing equations are not possible [5]. Another at-

tribute of turbulent flows described in [19], are diffusivity and rotationality. Diffusivity carries

fluid properties throughout the flow and is created through the mixing that occurs in turbu-

lence, creating vortical structures, known as eddies. One last key feature of turbulent flows

mentioned here, is that turbulence obeys the law of continuum, in the sense that micro-scopic

and non-equilibrium effects are negligible [20].

One last detail worth noting, is to notice the significance of the universality of turbulence and

its self consistency. Understanding turbulence through universality is, and has been a goal

6

Chapter 2. Large Eddy Simulation in Turbulence 7

amongst researchers, as has ensuring a self-consistent description of turbulence. An instance

that reflects how these two concepts might contradict each other, is best seen by observing the

dynamic modeling approach used to model the small scales of turbulence [20]. Complete univer-

sality of turbulence would imply that a single model would be capable of capturing the universal

behaviour of the small scales. However, to achieve greater success, the dynamic model appears

to perform slightly better when model coefficients are computed to achieve self-consistency for

a particular turbulent flow, which shows that no such model (to date) is capable of capturing

complete universality.

Before proceeding with statistical definitions used to describe turbulent flows, the governing

equations of turbulent motion for a compressible flow will be introduced here. Later, in section

2.3.1, the LES form of the Navier-Stokes equations, derived by applying a filter operator, will

be presented.

The governing equations for a compressible, turbulent flow field, are then given below in tensor

form as

∂ρ

∂t+∂(ρui)∂xi

= 0 , (2.1)

D(ρui)Dt︸︷︷︸

I

− ∂σij

∂xj︸︷︷︸II

+∂p

∂xi︸︷︷︸III

= 0 , (2.2)

D(ρE)Dt︸︷︷︸IV

− ∂(σijuj)∂xj︸︷︷︸V

+∂qj

∂xj︸︷︷︸VI

= 0 . (2.3)

Equations (2.1)–(2.3) are the continuity, conservation of mass, and conservation of momentum

equations, respectively, assuming there are no body forces introduced into the system. The

above set of equations are presented using the substantial derivative, DDt . Expanding terms I

and IV, yields respectivelyD(ρui)Dt

=∂(ρui)∂t

+∂(ρuiuj)∂xj

, (2.4)

D(ρE)Dt

=∂(ρE)∂t

+∂((ρE + p)uj)

∂xj. (2.5)

Note that for an incompressible flow, equation (2.1) may be written as

∇ · u = 0 . (2.6)

There are several terms appearing in equations (2.1)–(2.3) which require further explanation.

The term σij , in term II, is the viscous fluid stress and represents the viscous diffusion of


momentum in the flow. The viscous stress is a function of the strain rate according to

σij = 2ρν

(Sij −

δij3Skk

), (2.7)

where the strain rate, Sij , is given by

Sij =12

(∂ui∂xj

+∂uj∂xi

), (2.8)

and where ui is the velocity vector and δij is the Kronecker-delta tensor. In term I, ρ is the

density and p is the pressure introduced in term III. In term, IV, E is the total energy (per

unit volume), and may be defined as

E =p

γ − 1+ρuiui

2. (2.9)

The term qj appearing in term VI is the heat flux defined vis Fourier’s law as a function of

temperature, T ,

qj = −kc∂T

∂xj. (2.10)

The constants, kc and γ appearing in equations (2.9) and (2.10) respectively, are the specific

heat ratio and thermal conductivity.

2.1.2 Statistical Turbulence and Kolmogorov’s Hypothesis

Before the theoretical nature of small eddies may be introduced, a basic background in statistical

turbulence is necessary. Then, to understand terminology appearing in later chapters of this

thesis, some definitions will be useful.

A common form of averaging in turbulence, is the ensemble averaging, which is an average

taken over n realizations of a single turbulent flow having infinitesimally different initial and/or

boundary data. Then let the set of all such measurements be un and define the ensemble

average to be [17]

〈u〉 = limn→∞

1n

n∑i=1

ui , (2.11)

which allows the turbulent solution, u, to be decomposed according to

u = 〈u〉+ u′ , (2.12)

where u′ is known as the fluctuating quantity, and by construction, it is true that 〈u′〉 = 0. A

statistical correlation can be computed between two points in both space and time, such that,


for instance, a two point correlation between two points in space, x and x′ at the same time, t,

is defined as

Rαi,αj (x,x′, t) = 〈αi(x, t)αj(x′, t)〉 . (2.13)

The decomposition of equation (2.13) is a measure of the degree of randomness in a turbulent

flow, found by measuring the statistical correlation between two arbitrarily selected points in

either only space, only time, or both space and time.

Two additional definitions that will be used frequently in this thesis, are isotropic and homogeneous.

Isotropic, is a local (in space and time) invariance of statistical moments in the presence of co-

ordinate rotation, and plane symmetry [17]. Homogeneous turbulence refers to the situation

when the statistical moments of a quantity are independent of its location in space. Formally

this may be expressed as∂

∂x〈u1, ...un〉 = 0 . (2.14)

A fundamental aspect of turbulence is understanding the behaviour of the small scales or eddies.

This is due to the fact that they are largely responsible for the behaviour of the energy cascade,

and were shown by Kolmogorov [20, 17], to be universal over a considerable range of their

spectrum. Kolmogorov developed three hypotheses (H1 – H3 below) regarding the behaviour

of the small scales and how they scale relative to each other. These three hypotheses will be

now be summarized briefly.

Local Isotropy [17]. The argument here states that at high Reynold’s numbers, the small scales

(eddies), l << l0 are statistically isotropic. Here l0 denotes the large eddies for some integral

length scale, denoted by, L, where then it is assumed, L >> l0.

H1. Similarity Hypothesis 1. At high-Reynolds number flows, the small scales of turbulence, l,

l << l0 << L, have a universal behaviour, which are uniquely determined by the viscosity and

dissipation ; ν and ε. These laws, may be derived using principles of dimensional analysis, and

are known to be

η ≡ (ν3

ε)1/4 , (2.15)

uη ≡ (νε)1/4 , (2.16)

τη ≡ (ν

ε)1/2 . (2.17)

Equations (2.15)–(2.17) are the Kolmogorov scales of length, velocity and time, respectively.

H2. Similarity Hypothesis 2. In all turbulent flows of high-Reynolds numbers, the turbulent

length scale, l, falling in the range lη << l << l0, has universal form which is uniquely deter-

mined by ε and is independent of ν. Here, lη is used to denote the smallest eddy, and is also

known as the Kolmogorov length scale.


Gathering the above statements, we can place the relevant length scales in an ordered sequence

by magnitude : lη, lDI , lEI , l0, L. Here, lDI is the upper bound on the length scales at which

turbulent kinetic energy dissipates, and lEI is the upper bound on the inertial range length

scale. These will be described in the next section containing the description of the energy cas-

cade.

To show the universality of the small scales, one can use the Kolmogorov scales in equations

(2.15) and (2.16), to non-dimensionalize the position vector, y and velocity difference, ∆u,

translated in space [21]. The non-dimensional form of these two quantities are

y ≡ (x− x0)/η , (2.18)

u(y) ≡ [u(x, t0)− u(x0, t0)]/uη . (2.19)

Then it can be said that universality is achieved by introducing equations (2.18) and (2.19),

which state that when u(y) is not too large, the velocity, u(y) is isotropic and homogeneous.

The last hypothesis regarding the small scales, which will be used in this thesis, is stated below.

H3. Over the small scales all turbulence velocity fields are statistically similar,

or identical when normalized by the Kolmogorov scales [20].

The behaviour of scales in the inertial subrange (lDI < l < lEI) is a consequence of H2 [20],

such that the time and velocity scales decrease with l in this range. By dimensional analysis

one can arrive at

u(l) = (εl)1/3 = uη(l/η)1/3 ≈ u0(l/l0)1/3 , (2.20)

and

t(l) = (l2/ε)1/3 = tη(l/η)2/3 ≈ t0(l/l0)2/3 . (2.21)

The length scale spectrum of turbulence is shown in spectral space in Figure 2.1, where for any

li, κi = 2πli

.

2.1.3 The Energy Cascade

The energy cascade of turbulence is a deterministic construction of how the energy associated

with turbulence is distributed throughout the flow. This energy spectrum is a primary technique

used in numerical simulation of turbulence for the validation of any numerical results, as it

exploits the existence of small-scale universality.

The energy spectrum can be described by once again looking at the distinct regions in Figure

2.1. The length scale spectrum then is the closed interval [lη, L], where η is the Kolmogorov


κη = 2πlη

κDI κEI κl0 κL

6

Dissipation ε

?

Production of TKE : E(κ)

Dissipation range Inertial subrangeEnergy-containing

range

Universal equilibrium range

-

Figure 2.1: Wavenumber spectrum of turbulent length scales on a logarithmic scale.

scale and L the integral length scale. All other scales fall within this interval, and energy is

passed down from the largest, l0, to the smallest length scale. Referring to Figure 2.1, the

region denoted, energy-containing range, includes the scales that contain the majority of the

energy of the flow. The inertial subrange is the range of length scales for which the turbulent

flow is dominated by momentum. The dissipation range is the range of scales dominated by

viscous effects, and thus responsible for removing, or dissipating, energy.

Although the energy spectrum in Figure 2.1 is simple to perceive, it is very much idealistic,

and is not often the case in real turbulence. Consider the quantity, Γ(l/u), which is the rate

at which energy is transferred from any larger scale to a smaller scale, and is a function of

length and time. Figure 2.2 and 2.3 illustrate how Γ is transmitted throughout the flow in

a more realistic manner. Figure 2.2, represents a disjoint energy transfer, which is typically

considered in the study of turbulence. Then the energy spectrum presented in Figure 2.2, may

be viewed as an idealistic energy spectrum, used to simplify the study of turbulence. Figure

2.3, illustrates, that energy transfer in more realistic turbulent flows, does not in fact occur in

a disjoint manner, but rather energy dissipates from a large scale to more than one small scale,

and so in a sense the energy is shared, or intersected. Note the horizontal axis in Figures 2.2

and 2.3, are obtained by mapping the regular logarithmic scale, to one where the constants, an,

create a spectrum where each eddy is the same length in spectral space and centered at ank

[22].


Figure 2.2: Simplified-idealistic turbulent energy cascade. (logarithmic scales).

Figure 2.3: Real turbulent energy cascade (logarithmic scales).


2.1.4 Turbulence in Fourier Space

Given the basic details of the nature of turbulent flows outline above, one can begin to study

any turbulent flow in spectral space (Fourier space), by using Fourier mode analysis. Once the

wavenumbers in spectral space are formulated according to the range of scales in physical space,

some of the theoretical concepts described in this section may be applied and a description of

the energy spectrum in spectral space may be formulated.

Consider the wavenumber given by, κ = 2π/l, and define the energy contained within the range,

[κa, κb] as

Ea,b =∫ κb

κa

E(κ)dκ . (2.22)

Similarly the dissipation, ε is [20]

εa,b =∫ κb

κa

2νκ2E(κ)dκ . (2.23)

Next it is in order to define E(κ), it is observed that as a consequence of hypotheses H1, the

energy spectrum within the equilibrium range is a universal function of ε and ν. In the inertial

subrange, κ ≡ 2π/lDI , which results in the following form for E(κ);

E(κ) = Cε2/3κ−5/3 , (2.24)

with C being a universal constant. Since equation (2.24) is a fundamentally important law used

in turbulence, the proof will be illustrated in words briefly.

To begin a power law is assumed for E(κ). The energy over the interval, [κ,∞), is integrated,

resulting in solution as a function of constants, m and p. Similarly the dissipation, ε, is inte-

grated over the interval, [0, κ], resulting in a solution containing constants p,m once again. In

the integral for E(κ) for some p ≤ 1 the integral diverges and converges otherwise, while in the

integral for ε, the integral diverges for some p ≥ 3. To ensure that both integrals converge, p

is set to equal 5/3. This construction forces the energy to decrease with decreasing κ since the

dissipation decreases as it tends to zero, satisfying the desired properties for the energy and

dissipation.

To end this section, Fourier representation of the turbulent velocity field will be introduced,

keeping in mind that only a basic representation will be shown here, where the details can be

found in [23]. The purpose of this section is to formalize turbulence for the case when it is

homogeneous and isotropic in a periodic domain, which is the domain of interest in this work.

Fourier analysis of fluids begins by mapping the solution domain into periodic space determined

by wavenumbers limited by the grid resolution. Then one can introduce an integral Fourier


representation, of the Fourier transform and its inverse, F and F−1, respectively, of solution u,

to be defined as

F(−→u (−→x , t)) =

(1

2π

)3 ∫Re−i−→κ ·−→x−→u (−→x , t)d−→x , (2.25)

F−1 =∫ei−→κ ·−→x u(−→κ , t)d−→κ . (2.26)

Note then, that the velocity field may be expanded as an infinite series to produce [23]

u(x, t) =

(1

2π

)3 +∞∑n1,n2,n3=−∞

e( 2iπL

)(n1x1 + n2x2 + n3x3)uB(n1, n2, n3, t) . (2.27)

In equation (2.27), −→u has been replaced by −→u (x, t), where x is the position vector in Rd. Next,

the discrete Fourier transform is required so that equations (2.25) and (2.26) may be used for

computational purposes. First one can define an elementary wavenumber to be 2π/L, where L

is the side length of the cubic box, so that the wavenumber is defined as

−→κ =

(2πLn1,

2πLn2,

2πLn3

), (2.28)

for some positive integers, n1, n2, and n3 [23]. After some manipulations, the Fourier transform

may be written as

u(κ, t) =

(1

2π

)3 ∫e−iκ·

−→x

(2πL

)3∑κ′

eiκ·xuB(κ′, t)dx . (2.29)

Equation (2.29) is a function of both continuous and discrete modes, so that to obtain a purely

discrete Fourier transform, Fd, one simply writes

Fd(−→u (−→x , t)) =

(1

2π

)3∑−→κ ′

e−i−→κ′ ·−→x−→u (−→x , t)d−→x . (2.30)

In equation (2.29), uB is equation (2.30), which is strictly a function of discrete wavenumbers, κ′

centered within a computational cell. Note that it is important to distinguish the wavenumbers

determined by κ′, from those that lie on the energy spectrum in Figure 2.1. The wavenumbers,

κ′, are dictated by the grid resolution, while the wavenumbers, κl, in Figure 2.1, are solely

functions of turbulent length scales.

2.2 Methods in Computational Turbulence

This section is devoted to providing an overview of the field of computational turbulence, which

studies turbulence through discrete approximations. Simulating turbulence in fluid motion is


difficult task, as it often times, requires different approaches than those used in typical CFD

problems. The chaotic nature of turbulence does not permit straightforward algorithms to

be applied, but rather mathematical models which are still not complete in their universality.

Three distinct approaches to simulating turbulence will be summarized in this section, with

emphasis on LES, which in general, always requires more lengthy discussion.

2.2.1 Direct Numerical Simulation

DNS simulation is a numerical simulation technique resolves all scales of a turbulent flow, from

the smallest to largest : η → l0, and thus is certainly the most accurate and reassuring approach

to simulating turbulence. The Navier-Stokes equations given by equations (2.4) – (2.6) are in the

ideal case, a complete description of a turbulent flow, however the extent to which the numerical

solution captures all the scale information is dependent on the grid resolution. Although a DNS

provides a solution closest to the true solution, it requires a great deal of computational time.

The true solution in this context, is one where in the limit that the grid spacing tends to

zero, a DNS would tend to an exact numerical solution of the Navier-Stokes equations. Then,

certainly as the grid spacing never approaches infinitely small values, a DNS in practice is not

a true solution, relative to the previous argument. For highly turbulent flows characterized by

disparate length scales in time and space, a DNS is often not possible and certainly not desirable

from a computational point of view. However, while DNS may not always be a realistic choice

for many flows of interest in computational turbulence, it has served other useful purposes,

primarily as a source of reference and validation. Various forms of testing that use DNS results

as a measure of simulation or model error, will be discussed in greater detail in section 2.3.

Instances that reflect the current capability of DNS, can be seen in works that demonstrate

how DNS has helped to achieve a better understanding of boundary-layer and transitional flows

[24, 5]. More specifically, great success was achieved by Moin et al. [24], in the investigation

of how DNS may be used to control wall-bounded flows. It still might be true however, in the

author’s opinion, that the real strength in DNS lies in its ability to assist in theoretical studies

of LES aimed at improving SFS models and filtering approaches. A recent instance of this

may be seen in [11], where LES convergence studies were used to construct stronger forms of

grid-independent LES.

2.2.2 Reynolds Averaged Navier-Stokes

Reynolds Averaged Navier-Stokes (RANS) approaches are typically found in settings where

details of turbulent structures and properties are not required. In other words, if one would like


Figure 2.4: The resolved and modeled (unresolved) portions of the turbulent kinetic energy

spectrum in a RANS simulation.

to gain a better understanding of the physics of turbulence, RANS modeling would not suffice

[4]. This form of simulating turbulence, has shown good results in certain flow configurations

and types of flows, and is primarily used to obtain a solution quickly and efficiently. In a RANS

simulation, no scales of the flow are resolved, but rather one uses time-averaged Navier-Stokes

equations, which are averaged over a long period of time, T , such that the solution variables

in the transport equations are those that have been averaged over several realizations of a

given flow, yielding, 〈φ〉. After applying an averaging procedure to the Navier-Stokes equations

which are functions of mean quantities, one then has to close the resultant equations in order to

represent the effect of the fluctuating quantities. Several models have been successful in RANS

simulations [4], and the most common ones include, algebraic models and k− ε models. More

rigorous models include writing an additional transport equation for the Reynolds stresses. In

many flows of application, RANS has been used as a technique to predict the fluid motion in

more complex geometries, however an immediate concern that arises when using RANS can

be observed quite easily. Since a RANS simulation has to capture the entire range of scales

present in a turbulent flow, as the range of scales increases, this becomes a greater challenge,

primarily from the point of view of the large scales [5]. The RANS simulation technique may

be understood by looking at the entire turbulent kinetic energy spectrum, and noticing that

the entire spectrum is modeled (Figure 2.4).

2.2.3 Large Eddy Simulation

We can now move our attention to the focus of this thesis which is LES. In the simplest phrasing,

LES resolves a portion of the turbulence scales and uses a turbulence model derived from

principles of universality, to model the unresolved scales. The distinction between resolved and

unresolved scales is made, by applying a low-pass filter which removes high-frequency content


Figure 2.5: The resolved and modeled (unresolved) portions of the turbulent kinetic energy

spectrum in a LES simulation.

from the solution content. The ratio of scales which are unresolved to those that are resolved,

is typically 1 : 2, however this really depends on the filter width at which the LES is performed.

In the context of the energy spectrum, this amounts to resolving approximately 80 percent of

the turbulent kinetic energy. Figure 2.5 shows how the resolved and unresolved portions of

the energy spectrum may be divided. While the original concept for LES was to reduce the

computational requirements of performing turbulent simulations as compared to DNS, as well

as reducing the modeling complexity of RANS-based methods, in the author’s opinion, LES

involves a greater level of complexity than any other numerical simulation primarily due to

the presence of multiple scale interaction. Multiple scale interaction refers to the interaction

of multiple scales that are introduced as a result of applying a low-pass filter. The filtering

introduces a new solution of scale denoted by (·), such that in LES the scales that one encounters

is the tuple of scales, u, u, u. The first scale is a representation of the small unrepresented

scales, which are not even resolved by the grid. The second scale, u, is the scale which is

resolved by the grid, however not by the LES filter and u, represents scales which are resolved

by both grid and filter. The combination of the scales is represented in the turbulent kinetic

energy spectrum in Figure 2.6, where the shaded portion shows the region where multiple scale

interaction may occur. One difficulty in LES recognized by many LES authors [25], is that

to date very little can be said about the interaction of these scales. In general, in an LES

simulation, though these scales would like to be treated separately, the distinction is typically

ignored, as present understanding is insufficient [17].

To construct a physical understanding of how LES is performed, the energy spectrum is intro-

duced here again. Typically the filter is applied to the solution field, u, such that 80 percent

of the energy spectrum is resolved, which corresponds to κcutoff on the logarithmic scale in

Figure 2.6. On a uniform mesh, κcutoff = 2π∆

, where ∆ is the filter width. To summarize, the

LES filter is applied in such a way that the ratio, ∆∆ , will resolve approximately 80 percent


Figure 2.6: Multiple scale interaction on the turbulent kinetic energy spectrum in LES.

of the turbulent kinetic energy decay spectrum, while the remainder is modeled. Then one

could introduce the set of relevant wavenumbers essential in LES, such that they are related

by, κ∆ > κ∆ > κl0 > κL. Note here that ∆ is the grid spacing, so that κgrid in Figure 2.6 is 2π∆ .

2.2.4 Initializing Turbulence in a Periodic Box

An important aspect in performing a large eddy simulation is the initialization of turbulence.

The turbulence is generated artificially throughout the grid, using some of the basic knowledge

of statistical turbulence outlined in the previous sections. The method used in this thesis

to initiate turbulence is the method suggested by Rogallo [26]. Initially a arbitrary velocity

distribution is imposed on the solution domain, where in the studies presented in this work,

this distribution will either be a radial cosine or some uniform distribution. Then the procedure

that follows, is performed entirely in Fourier space and requires that the velocity field satisfies

an isotropic state, is attributed with the proper energy spectrum and satisfies continuity [26].


First, to satisfy continuity, one must define the velocity u in spectral space such that [26]

u = ui · ei = α(−→κ ) · e1 + β(−→κ )e2 , (2.31)

where ei is the algorithmic basis vector (typically the unit vector in R3), and ei any basis vector

such that e3 is parallel to −→κ . Then to ensure that the initial field models the desired energy

spectrum, the only constraints imposed on the complex variables, α and β are

E(−→κ ) = (αα∗ + ββ∗)∫DdA(−→κ ), (2.32)

where E(κ) is the desired energy spectrum, which will be described in Chapter 5. The choice

of α and β suggested in [26] are

α =

(E(κ)4πκ2

)1/2

eiθ1 cosφ , β =

(E(κ)4πκ2

)1/2

eiθ2 sinφ , (2.33)

where θ1, θ2 and φ are uniformly distributed random numbers on the open interval, (0, 2 π).

The constraint on α and β in equation (2.32) ensures that the energy associated with each

wavenumber will have the desired value. The last key idea that is significant is the relation

between the algorithmic (computational) basis vector, ei, and its corresponding basis vector,

ei, which will be taken as suggested by Rogallo as, e1 · e3 = 0. Note that this is merely one way

to satisfy the required constraint that

κe3 = κ1e1 + κ2e2 + κ3e3 = κ . (2.34)

Doing so will leads to the final result below

u =

(α|κ|κ2 + βκ1κ2

|κ|(κ12 + κ2

2)1/2

)· e1 +

(βκ3κ2 − α|κ|κ1

|κ|(κ12 + κ2

2)1/2

)· e2 +

(β(κ2

1 + κ22)1/2

|κ|(κ12

)· e1 . (2.35)

The above proposed form for the initial spectral turbulent velocity field is not completely

characteristic of real turbulence, since it lacks anisotropic characteristics [26], however, the

above initialization may be used for the case of homogeneous, isotropic turbulence.

2.3 Techniques in Large Eddy Simulation

2.3.1 Filtering of the Navier-Stokes Equations

To begin a thorough discussion on even the most basic techniques used in LES, an illustration of

how the Navier-Stokes equations are filtered is a preliminary step. Then consider an LES filter


or convolution operator, G, acting on the compressible Navier-Stokes (NS) equations introduced

in (2.1)–(2.3), as follows:

G ?

[∂ρ

∂t+∂(ρui)∂xi

]= 0 , (2.36)

G ?

[∂(ρui)∂t

+∂(ρuiuj)∂xj

− ∂σij

∂xj+∂p

∂xi

]= 0 , (2.37)

G ?

[∂(ρE)∂t

+∂((ρE + p)uj)

∂xj− ∂(σijuj)

∂xj+∂qj

∂xj

]= 0 . (2.38)

All the terms in equations (2.36)–(2.38) were discussed in section 2.1.1 and so will not be

repeated here. Then considering only the momentum equation for simplicity, and applying G

yields

∂(ρui)∂t

+∂ρuiuj

∂xj− ∂σij

∂xj+∂p

∂xi= 0 . (2.39)

When the filter operator G, is applied, only the individual solution components are filtered,

and since, ρuiuj 6= ρ uiuj, an additional term, τij is added, such that the affect of advancing

ρuiuj as opposed to ρuiuj is considered. The addition of this term is the closure of equation

(2.39) such that equation (2.37) becomes

∂(ρui)∂t

+∂(ρ uiuj)∂xj

= − ∂p∂xi

+∂σij

∂xj− ∂τij∂xj

. (2.40)

The resulting operation will yield the Navier Stokes equations expressed in terms of the filtered

solution field, which implies that the affect of G on the NS equations is a change of variables

from solution vector φ to φ. In the compressible transport equations, a Favre-filter is used to

reduce the additional number of unknowns that are introduced in the filtering procedure. The

Favre-filter is simply an averaging involving the filtered density

φ =ρφ

ρ. (2.41)

Then the final form of the momentum equation for a compressible flow is

∂(ρui)∂t

+∂(ρuiuj)∂xj

− ∂σij

∂xj+∂p

∂xi= −∂τij

∂xj+∂(σij − σij)

∂xj. (2.42)

The first closure term on the left-hand side of (2.42) is the sub-filter scale (SFS) stress tensor,

and requires modeling, since its direct computation given below

τij = ρuiuj − ρuiuj , (2.43)


is not possible. The second closure term on the right-hand side of equation (2.42) appears since

the Favre-filter operation does not commute with differentiation, and due to the fact that σijis a non-linear term, since, ν = ν(x, t). A priori tests have shown that the assumption that

σij− σij ≈ 0, is a good one, and will be adopted in this work. Note that the energy conservation

equation for compressible flow is not shown here, however, the situation is similar to that

found when filtering the momentum equation. Once again, the filtering operator introduces a

change of variables from φ to φ in the energy equation, which introduces new terms, due to

presence of non-linear terms, and thus also requires closure models.

2.3.2 The Unresolved Scales

This section is devoted to one of the key concepts in LES which is concerned with the techniques

used to represent the small scales that were removed by the LES filter. Recall that after the

filtering operator is applied, the turbulent flow field is divided into resolved and unresolved

scales. Then u(κ∆ < κ < κη), is the range of small scales which are not represented in the

solution, U , and for the solution to be reliable, information about these small scales must be

placed back into the solution as U is advanced in time. This small scale representation is known

as sub-filter scale (SFS) modeling, and will be described below in some detail.

The first approach for constructing an SFS model, is to understand the reasoning that allows

one to perform an LES. In particular, the universality of the small scales allows us to adopt

a model that may be used for any turbulent flow. In other words, the concepts discussed in

section 2.3.2 allow us to construct a model that represents how the small scales behave, without

ever needing to resolve them. Several SFS models exist, and appropriate selection of a model

for a given flow situation, also contributes to a successful LES.

Smagorinsky Model

The Smagorinsky model is the most commonly used model in LES, due to its success in pre-

dicting turbulent flows. This model is based on the assumption that energy transfer from the

large scales to the small scales, is analogous to viscous diffusion transport, and is known as the

Boussinesq assumption [20]. The Smagorinsky model is given by [17, 27]

τij −13τkkδij = νT

(∂ui∂xj

+∂uj∂xi

)= −2νTSij . (2.44)

In (2.44), νT is known as the eddy viscosity, and is computed as

νT = Cs2∆2|Sij | . (2.45)


The Smagorinsky model, suffers occasionally from one, not immediately obvious, fact. The

parameter, Cs predicted by turbulence hypotheses, does not address the self-consistency of the

model. The following dynamic approach was constructed to address this issue.

Dynamics Smagorinsky Model. The word dynamic generally refers to a specific procedure which

may be applied to any SFS model, however will be demonstrated here using the Smagorinsky

model without loss of generality. For simplicity, the SFS terms presented will be those for

incompressible flow and application for compressible flows easily follows. What is sought is a

self-consistent model, in the sense that filtering at various levels will allow for use of the same

value for Cs. Then given the familiar expression for the SFS model

τij = uiuj − uiuj , (2.46)

a secondary filter, ξ, may be constructed, such that it is not necessarily true (recalling original

LES filter, G) that G = ξ. The operation of ξ will be denoted with a (). Then applying ξ yields

τ ′ij = uiuj − uiuj . (2.47)

Using the second filter, ξ, we can construct another SFS expression that reflects the large scales,

and is called the Leonard stress [22]

Lij = uiuj − uiuj . (2.48)

Then introducing the Germano identity [28]

Lij = τ ′ij − τij , (2.49)

leads to the Smagorinsky model which can be defined for both filters as

τij −13τkkδij = −C∆2|S|Sij , (2.50)

and

τ ′ij −13τ ′kkδij = −C∆

2|S|Sij . (2.51)

Applying the Germano identity to equations (2.50) and (2.51) and using a scalar representation

for the right hand side [22], results in a definition of C computed based on the LES itself;

C = − LklMkl

MklMkl. (2.52)

In this way, C may be computed to reflect the filtering operation itself, increasing the accuracy of

the SFS model. There are still limitations in the dynamic approach, which will not be discussed

here, however many of these issues are not difficult to overcome with a strong understanding


of LES.

The SFS models introduced above are known as functional modeling techniques and are a

small portion of the SFS models that have been developed. Various other forms include forms

of structural SFS modeling, which uses the filter operator itself to reverse the filtering process,

recover the information lost in filtering, and use this recovered solution for τij itself. This

approach will be discussed in greater detail in Chapter 4.

Scale Interactions Although a complete knowledge of how multiple scale interactions in LES

behave, has not been formalized, some hypotheses have been constructed. The multi-scale in-

teraction in LES is complex and even a largely incomplete discussion is beyond the scope of this

thesis, however some important remarks, in the author’s opinion, require brief consideration.

To fully understand the results obtained from LES and in order to improve SFS models, re-

quires comprehensive study of not just the scales individually, but the interaction among them.

Consider the interaction between the resolved, u and unresolved scales, u < u(κ∆), which are

referred to as [6] triadic interactions. These may be further divided into two subclasses ;

Local triads. Let the set of wavenumbers, p, q, k denote three scales in Fourier space such

that 1c ≤ max

pk ,

qk ≤ c, for some c = O(1). This represents the interaction of modes of similar

size such that p, q from a closed triangle with k.

Non Local Triads These triads are characterized by p << k, q or q << k, p. Another way to

define non-local triads is simply to say that they are the sets of p, q, k that do not satisfy1c ≤ max

pk ,

qk ≤ c.

2.3.3 A Priori and a Posteriori in LES

In this brief section, the concept of testing the validity of LES models will be discussed. Cer-

tainly, it is required that once a theoretical model has been developed to represent the small

scales, any such model must be compared against either experimental data, or possibly DNS

data. The two techniques in LES, which fundamentally refer to validation of results based on

prior knowledge and those based on obtained knowledge, are known as a priori and a posteriori

testing, respectively. An a priori approach will refer to the testing of a model by applying it

to the solution of a problem and comparing the model’s prediction with that of true data. We

can define an a posteriori approach as follows. Let the results of a DNS represent the exact

solution in the sense that, |u − udns| < ε, where u is the exact solution not attainable in nu-

merical simulation, and udns, the accepted exact solution within computational limitations. If


for instance the LES model would like to be used for a homogeneous flow study, then we would

consider udns of a homogeneous flow realization. Then let the true (DNS) velocity field at an

instant in time be ui, representing a realization of the flow. Upon application of the filtering

operator, G, we would obtain the large scale solution, ui. Since we have the true DNS solution,

one can easily compute the (true) SFS stress tensor, τdns = uiuj − uiuj . Then we would really

like to compute a normed measure between the SFS predicted by a turbulence model used in

the LES, τles, given by χ;

χ =〈τdnsτles〉

(〈τdns2〉〈τles2〉)1/2. (2.53)

Typically, a priori studies have shown that SFS models, such as the Smagorinsky model, are

not very accurate, however a possible reason for this may be due to the fact that the principal

axis of the two stress tensors, τdns and τles are not in strong agreement [22]. This would imply

that models such as the Smagorinsky model do not necessarily suffer from poor representation

of small scale structures. This might also indicate the need for different forms of SFS model

testing.

2.3.4 Fundamentals of Filtering

The next two sections will give an informal introduction to filtering in LES. Here the concept of

implicit filtering will be quickly illustrated and for the remainder of this work filtering discussion

will immediately imply the use of explicit filtering.

Implicit filtering has been the common approach to filtering in the past, and still today is the

most practical widely used approach. Implicit filtering may be seen as being embedded within

the numerical scheme of a solution discretization. More specifically, the discretization of the

NS equations results in an immediate filtering of the solution field, allowing for a sub-grid scale

(SGS) model to be used. Notice then that, an SGS model refers to the case when one uses

an implicit filter, since there is no existence of an operator, G. This is slightly different than

the implications of using an SFS model. Then in implicit filtering, the numerical discretization

possibly in combination with the SGS model, acts as a filter operator, implying that the filter

is dependent on the grid resolution and order of numerical discretization of the solution [9].

One extremely desirable attribute of LES is that the filtering achieve grid-independence, even

in a weak sense. Then considering the previous argument, clearly implicit filtering is not

characterized by any form of grid independence, which is a large motivation to study explicit

filtering approaches.

Explicit filtering is an alternative to implicit filtering, and has many appealing attributes not

offered by an implicit filtering approach. Primarily, explicit filtering requires a explicit filtering


operation, G to be applied to an arbitrary field, φ. The operator, G is a spatial function, when

the filtering is performed in the space, as is the case in the present work. The filtering operator,

G may be either analytical or discrete in nature, implying that the filtering kernel is continuous

or discrete, respectively. Then, proceeding with some general definitions, the filter kernel, or

operator, G, is defined as

φ(x) =∫ +∞

−∞φ(x′)G(x− x′; ∆)dx′ . (2.54)

The above expression is nothing more than the definition of convolution, for which the following

simplified form

φ(x) = [G ? φ](x) , (2.55)

will be used.

To gain a more comprehensive insight into how the convolution operation acts locally, we can

briefly recall the more formal definition of convolution. A simple illustration in R may be

given, by defining two functions, f(x) and g(y), shown in Figure 2.7. The convolution of some

function f with g, results in a new function defined by

(f ? g)(x) =∫y∈R

f(x− y)g(y)dy , (2.56)

which in words, is the product of integrating f and g, after moving g(y) to the point, x. Note

that by commutativity of the convolution operator, we can also write

(f ? g)(x) =∫y∈R

g(x− y)f(y)dy , (2.57)

so that one may interpret Figure 2.7 as they choose. From the figure and equations (2.56)

and (2.57) it is clear that the convolution may be seen as a local averaging of f . Furthermore,

assuming the kernel is g, the smaller the domain of the support of g becomes, the less that f

is averaged, which, in the context of filtering, would correspond to resolving more of the local

solution.

The last key relation in explicit filtering, is to note that in Fourier space the the filtering

operation is multiplication and no longer convolution, so that we may write

φ(κ) = G(κ) · φ(κ) . (2.58)

The kernel, G(κ) in (2.58) is called the transfer function of the filter kernel, G, and will be used

frequently in Chapter 5. The last key feature of filtering is related to the properties of the filter

operator, G. In order to consider the Navier-Stokes equations after the application of a filter,

we require G to have certain properties as follows. Consider an arbitrary operator, ψ acting on


Figure 2.7: Illustration of the convolution operation used in filtering with kernel, G.


function, f; ψ(f). Then we would like the following to necessarily be true;

1. ψ(a) = a. (for some constant, a).

ψ(a) = a = a ⇒ ψ = I . (2.59)

2. ψ is a linear operator.

f + g = f + g . (2.60)

3. Commutation of ∂(·)∂x with ψ.

ψ

(∂f

∂x

)=∂(ψ(f))∂x

. (2.61)

In general condition 3 is not trivially satisfied for any LES with a reasonable level of geometric

complexity, however we require that the LES filter operator, G, inherit properties (1) – (3).

The last basic definition of explicit filtering in this section is the commutator, [ψ, ξ], of two

operators, ψ and ξ, acting on φ expressed as

[ψ, ξ]φ = ψ ξ(φ)− ξ ψ(φ) = ψ(ξ(φ))− ξ(ψ(φ)) . (2.62)

The definition of the commutator allows for a simpler expression of condition 3,[G ?

∂

∂x

]f = 0 . (2.63)

The above commutator in (2.63) has the following properties ;

[ψ, ξ] = −[ξ, ψ] (skew - symmetry) , (2.64)

[ψ ξ, η] = [ψ, η] ξ + ψ [ξ, η] , (2.65)

and

[ψ, [ξ, η]] + [ξ, [η, ψ]] + [η, [ψ, ξ]] = 0 (Jacobi’s Identity) . (2.66)

The last general remark on filtering operators, is that a filter operator, ψ, does not act as

Reynolds operator. In general a Reynolds operator, Γ, may be applied, by function composition,

a countably infinite number of times, such that

Γn(f) = Γ(f) . (2.67)

The above is not true of ψ, so that

ψ(ψ(f)) = f 6= f . (2.68)


2.3.5 Errors in LES

Aliasing Errors. Aliasing errors occur when one studies properties in spectral space, and is the

result of undesirable frequency content entering the solution information. In particular, in LES,

the non-linear terms in τij as defined by

ρuiuj = ρuiuj + (ρuiuj − ρuiuj)︸︷︷︸τij

, (2.69)

introduce frequencies that are not characteristic of the frequency that defines, ui [3]. Thus the

separation of resolved and unresolved scales by the low-pass filter may not be a completely

accurate distinction. Controlling aliasing errors is extremely difficult when implicit filtering is

used, since one cannot simply control how the filter removes frequency content. This is another

motivation for studying explicit filtering in LES.

Filter Truncation Error. Truncation errors are the most intuitive and natural way to define

the local error associated with filtering. They are essentially the difference between the filtered

solution and the otherwise, non-filtered solution;

Et = φi − φi . (2.70)

Commutation Errors. Study surrounding commutation errors in explicit filtering has evolved

significantly, and much of the detail is beyond the scope of this thesis. Fundamentally, commu-

tation error occurs when the commutator is not zero;[G ?

∂

∂x

]f 6= 0, (2.71)

for some function, f , which is generally the case for many flows of interest in computational

turbulence. While this is not entirely desirable, control over such commutation errors is possible,

so that we would like to explore the possibility of

‖Ec‖ =

[G ?

∂

∂x

]f ≤ ε , (2.72)

for some ε = O(∆), O(∆), and filter widths and grid spacings, ∆ and ∆, respectively. If

the commutation error, ‖Ec‖, may be bounded from above by some small magnitude of error,

less than the order of error of the numerical discretization, than its affects may be considered

insignificant [29].

Chapter 3

Explicit Filtering Techniques

This chapter is designed to provide the reader with a more detailed illustration of explicit

filtering, by first introducing the family of analytical filters, followed by the focus of this thesis,

the discrete filter. More rigorous approaches will be introduced when developing expressions

for the analytical and discrete commutation errors, and the section will close with a description

of the family of structural modeling, known as approximate deconvolution methods. It is worth

pausing at this point and noting that filtering may be applied both in the time and space

domains, where typical application of the former is seen in signal processing. It should be clear

from hereon, that the filtering operators used in the present LES studies are strictly spatial

filtering operators.

3.1 Analytical Explicit Filters

The most commonly used explicit filters fall into the category of analytical filters, which may

be classified as those filters whose kernels are continuous functions in space. The most obvious

advantage of analytical filters, is that its differentiability allows methods involving series ex-

pansions, to be studied in a straightforward manner. However, while the smoothness properties

of such filters lend themselves nicely to analysis in the development of theoretical study, their

use in CFD implies that discrete methods are again imposed and/or required.

3.1.1 Standard Filters

There exist three analytical filters which are predominantly used in LES due to the simplicity

of their kernel functions, and effective use in both Fourier and physical space.

29

Chapter 3. Explicit Filtering Techniques 30

Spectral Filter

The spectral filter is a analogous to the box-filter (see below) in physical space, however the

sharp-cutoff is performed in Fourier space. In other words, the value of the kernel, G, is unity

over a desired range of retained wavenumbers, and zero otherwise. Then the kernel, G, is

defined as follows in Fourier space

G(x− x′) =sin(κc(x− x′))κc(x− x′)

, with, κc =π

∆, (3.1)

and

G(κ) =

1 if|κ| ≤ κc0 otherwise .

(3.2)

Once again, the filter width in physical space is denoted, ∆. Recall from Chapter 2, the

definition of G to be G(κ) = φ(κ)

φ(κ), which is referred to as the transfer function of kernel, G.

Box Filter

The Box filter operates in physical space on the solution content, the same way the spectral

filter operates on the solution in Fourier space. The box filter’s kernel and transfer function are

defined as, respectively,

G(x− x′) =

1∆

if|x− x′| ≤ ∆2

0 otherwise(3.3)

G(κ) =sin(κ∆/2)κ∆/2

. (3.4)

The last analytical filter shown in this section is the one whose kernel is based on the Gaussian

function.

Gaussian Filter

The Gaussian filter, is the most commonly used filter for analytical studies of filtering oper-

ations, particularly in mathematical literature. It is clear that the reason for this lies in the

continuous nature of the Gaussian function, which makes analysis more straightforward. The

kernel, G, of the Gaussian filter may be defined as

G(x− x′) =

(γ

π∆2

)1/2

exp

(γ|x− x′|2

∆2

), (3.5)

G(κ) = exp

(−∆2

κ2

4γ

). (3.6)


-10 -8 -6 -4 -2 0 2 4 6 8 10

Δ G

(x)

-0.2

0

0.2

0.4

0.6

0.8

1.0

Δ x

Figure 3.1: Sharp Fourier cut-off filter. Convolution kernel G depicted in physical space.

In the above expressions, γ is a constant which is typically chosen to have a value near 6 [17].

Graphs of the kernels, G, of the three above filters, are illustrated in physical space in Figures

(3.1) – (3.3) [18] below.


-1.0 -0.5 0 0.5 1.0

Δ G

(x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Δ x

Figure 3.2: Top-hat filter. Convolution kernel G depicted in physical space.

-1.0 -0.5 0 0.5 1.0

Δ G

(x)

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Δ x

Figure 3.3: Gaussian filter. Convolution kernel G depicted in physical space.


3.1.2 Differential Filters

We can now turn attention to the branch of analytical filters, known as differential filters.

Differential filters are characterized by the introduction of inverse linear differential operators

acting on the filtered field, φ [25]. Formally

φ = F (G ? φ) = F (φ) , (3.7)

where F is the linear inverse operator, such that

φ = φ+ ts∂φ

∂t+ ∆i

∂φ

∂xi+ ∆i ∆j

∂2φ

∂xixj+ ...., . (3.8)

Here, ts and ∆ are appropriately selected time and length scales respectively, associated with

the flow solution being filtered. Two such filters are presented below in slightly more detail.

Elliptic Filter

If one considers (3.8), the elliptic filter is of the form

φ = φ−∆2 ∂2φ

∂xixi. (3.9)

Then from (3.9), we can see that the filter width, ∆, is a reasonable length scale, so that the

filter may be expressed as a product of convolution by

φ = G ? φ =1

4π∆2

∫Rd

φ(x′)|x− x′|

exp

(− |x− x

′|∆

)dx′ . (3.10)

More specifically, the above elliptic filter is a second order elliptic operator.

Parabolic Filter

The parabolic filter, then is clearly a parabolic linear operator acting on φ, such that

φ = φ+ ts∂φ

∂t−∆2 ∂2φ

∂xixi, (3.11)

which may be integrated to obtain an expression for φ expressed as a convolution in both time

and space. Although it will not be illustrated here, it is not difficult to show that the both the

elliptic and parabolic filters satisfy properties (2.59)–(2.61) discussed in Chapter 2.

3.1.3 Commutation Error of Analytical Filters

Although analytical filters are not used in the present study, to understand general filter at-

tributes, such as the filter moment, it is desirable to illustrate the formal expression for the


commutation error associated with the filtering operator. This will allow for an easier under-

standing of the discrete commutation error, when a description of discrete filters is given in the

next section.

Before we can begin to develop expressions for the commutative errors, as will be shown,

commutative errors only occur in the presence of a varying filter width, ∆, which is most likely

the case when filtering over non-uniform and irregular mesh topologies [12]. Then first consider

the definition of the commutation error in terms of the commutator, acting on φ, such that[G ?

∂

∂x

]φ 6= 0 . (3.12)

The commutation error will be defined as follows in R ;[dφ

dx

]:=

dφ

dx− dφ

dx. (3.13)

The following derivation will proceed in one dimension, to allow demonstration of how the filter

operator and commutation error may be manipulated in higher dimensions. First, we will need

to make use of the convolution operation in (2.45) using the solution variable φ as given by

φ(x) =1

∆(x)

∫ b

aφ(x′)G

(x− x′

∆(x)

)dx′ . (3.14)

One can note that (3.14) is a truncation of (2.54) such that the supp(G) ∈ [a, b]. Here, the

terminology, supp() is used to define the support of a function. To obtain a more rigorous

understanding of the commutation error, the formal expression will be constructed in a sequence

of 7 steps, which follow closely from Haselbacher et al. [30]. First assume the solution is

discretized over an arbitrary periodic domain, such that one can construct a mapping of the

domain onto a uniform domain of spacing, ∆. A typical example of this are non-uniform meshes

defined by stretching functions, for instance. Then we can proceed as follows :

(1) Define the solution, φi(x), to be defined on a closed interval, [a, b], which is simply the

support of the kernel, G, of cell i and it will be necessary to make use of the following assumption.

A1. ∃ a monotonic and differentiable function, f(x), such that, f maps φ from [a, b] to [α, β],

where the grid spacing, ∆, is uniform in [α, β].

(2) Recognizing that f(x) = y, where y is the coordinate in the uniformly spaced domain, we

can used the filtering operation in (3.14) to define

φ(x) =1∆

∫ b

aφ(x′)f ′(x′)G

(f(x)− f(x′)

∆, f(x)

)dx′ . (3.15)

However, we would like to define the filtered solution in [α, β]. Then it is not difficult to show

that

ϕ(y) =1∆

∫ β

αG

(y − η

∆, y

)ϕ(η)dη . (3.16)


(3) To eventually make use of the Taylor expansion, we will introduce a change of variables,

γ = y−η∆ , which simplifies (3.16) to

ϕ(y) =1∆

∫ y−α∆

y−β∆

G (γ, y)ϕ(y − γ∆)dγ . (3.17)

(4) Next, we express, ϕ, as a Taylor series in y − γ∆, to arrive at

ϕ(y − γ∆) =+∞∑k=0

(−1)k

k!∆kγl

dkϕdyk

. (3.18)

Before we proceed with the next step, it will only be mentioned here, however not proven, that

(3.18) is uniformly convergent when we assume that the solution contains only wavenumbers

that are larger than some κmax. It is clear that this is certainly a fair assumption in numerical

simulations, since this follows immediately from the grid’s discretization. Vasilyev et al.[31]

provide an interesting proof of the radius of convergence of (3.18) by bounding the series from

above with the turbulent kinetic energy of the flow, and the interested reader is encouraged

to read this discussion. Then assuming uniform convergence, we can write the following by

combining (3.17) and (3.18)

ϕ(y) =+∞∑k=0

(−1)k

k!

(∫ y−α∆

y−β∆

γkG (γ, y) dγ

)∆k dkϕ

dyk, (3.19)

and if we further define the filter moment, Mk, at y,

Mk(y) =∫ y−α

∆

y−β∆

γkG(γ, y)dγ , (3.20)

then we can write (3.19) as

ϕ(y) =+∞∑k=0

(−1)k

k!∆kMk(y)

dkϕdyk

. (3.21)

(5) At this point we can note the relationship

dφ

dx=dϕ

dyf ′(x) . (3.22)

We can differentiate (3.21) and substitute the result into (3.22) to obtain

dφ

dx= f ′(x)

+∞∑k=0

(−1)k

k!∆k

(dMk

dy

dkϕ

dy+Mk d

k+1ϕ

dy

)ϕ(y) . (3.23)

(6) Then to obtain a final expression for [dφdx ] we need to perform the filtering operation on dφdx .

Since a derivation of this term follows similarly from steps (2)–(5), the details will be omitted,

however one can obtain in the end

dφ

dx=

1∆

∫ β

αG(y − γ

∆, y)

(+∞∑k=0

(−1)k

k!∆kγk

dk+1ϕ

dyϕ(y)

)f ′(x′)dγ . (3.24)


(7) The last step remaining is to subtract (3.23) from (3.24) so that we have the final expression

for the commutation error as a function of the uniform spacing, ∆;[dφ

dx

]=

+∞∑k=0

AkMk∆k +

+∞∑k=0

BkdMk

dy∆k . (3.25)

It is noted that prior to step (7), more extensive algebra is required to obtain the form of equa-

tion (3.25), and these have been omitted for brevity. As required, the commutation error is a

function of the mapping function, f , embedded in the coefficients Ak, and the filter moments,

Mk. The case when the shape of the kernel, G, is independent of position, corresponds to the

case where the filter width is constant, and in this case, Ak = 0 for all k [31]. In this same

case the filter moments are independent of position, so that dMk

dy = 0 for all k, so that the

commutation error becomes identically zero everywhere. Marsden [32] introduced a form of the

commutation error as a function of the filter width, ∆, which is not difficult to construct after

observing steps (2)–(7).

The steps presented above are merely a simplification, since only periodic domains are consid-

ered, and furthermore, the existence of f is essential. A more general proof may be found in

the detailed texts by Berselli et al. [19] and [33].

3.2 The Discrete Explicit Filter

This section will now introduce discrete filtering, which is the form of filtering implemented

in the structural modeling to be discussed in Chapter 4. Discrete filtering, in the author’s

opinion, carries significant advantages from a computational perspective, however, when the

kernel, G is required for LES techniques of a more rigorous nature, discrete filtering may not be

so desirable. An example of the previous remark, is the construction of theoretical SFS models

based on deconvolution by direct inversion.

Computationally, discrete filters have the potential to save computational work by reusing data

structures implemented in the existing numerical scheme. In particular, the computation of the

weights is one that is identical to least squares reconstruction in finite volume schemes [34], and

so computing the filter is nicely linked to the numerical method, particularly for finite-volume

techniques.

3.2.1 Formulation of the Discrete Filter

The fundamental concepts of filtering were introduced in Chapter 2, and so this section will

primarily be concerned with introducing the functions required for discrete filtering, without


too much emphasis on general filter concepts. For instance, the notion of a filter moment was

defined in section 3.1.3, and so here it will simply be stated. The discrete filter uses the concept

of a Riemann sum to approximate a continuous filter kernel defined using a weighted sum.

Then considering the closed interval, [a, b] ∈ R for simplicity, and introducing the dirac delta

distribution [35], δ, the kernel, G, of a discrete filter defined locally is

Gi(xi − x′) =Li∑

l=−Ki

wilδ(x

′ − xi+l) , (3.26)

where wil are cell weights, and making use of the following property of the dirac delta∫ b

aφ(x′)δ(x′ − x)dx′ = φ(x) , with x ∈ [a, b] . (3.27)

yields, after placing (3.26) into (3.14) and using (3.27), one to obtain

φi =Li∑

l=−Ki

wilφi+l . (3.28)

The notation used in (3.28) implies that φi = φ(xi) such that points in the neighbourhood of

xi have solutions, φ(xi+1). A more general form of notation will be used from hereon, which

expresses (3.28) as

φi =∑

xj∈N (xi)

wijφj , (3.29)

where, N (xi) denotes the neighbourhood of points of xi. The transfer function, G(κ), intro-

duced in (2.58), is important to define for a discrete filter, and is computed, simply by taking

the Fourier transform of (3.26), such that F(Gi) is

Gi(κ) =∑

xj∈N (xi)

wije−ıκ∆xij , (3.30)

with ∆xij = xj − xi. Eventually one would like to make use of the discrete filter moments,

computed by substituting (3.26) into (3.20) so that

Mk(x)i =∑

xj∈N (xi)

(xj − xi)kwij . (3.31)

To construct a filter in R3, one simply takes the product of filters in R [30] such that, if we

define the support of G as the product of close intervals determined by the filter stencil in each

direction, x1, x2, x3, we have

φ[xa1 ,xb1]×[xa2 ,x

b2]×[xa3 ,x

b3] = φ[xa1 ,x

b1] × φ[xa1 ,x

b1] × φ[xa1 ,x

b1] . (3.32)

Certainly if the supp(G) was not symmetric than a, b would be different for each direction,

x1, x2 and x3. In this work, a symmetric stencil implies that the mesh is uniform, however the


definition given in (3.26)–(3.28) may be used, without loss of generality, for both structured

and unstructured meshes, however, the following notation is constructed for structured grids.

Then the filtered solution in R3 is determined by taking the product of equation (3.28) and

this yields

φijk =Li∑

l=−Ki

Lj∑m=−Kj

Lk∑n=−Kk

wijklmnφi+l,j+m,k+n . (3.33)

The transfer function and filter moment follow by the same argument to yield, respectively,

Gijk(κ) =Li∑

l=−Ki

Lj∑m=−Kj

Lk∑n=−Kk

wijklmne

−ıκ∆rijklmn , (3.34)

M qrs(x)ijk =Li∑

l=−Ki

Lj∑m=−Kj

Lk∑n=−Kk

(xi − xl)q(yj − ym)r(zk − zn)swijklmn . (3.35)

It is clear now, the reasoning for the introduction of the notation in (3.29), so that (3.33)–(3.35)

will be referred to as

φi =∑

xj∈N (xi)

wijφj , (3.36)

Gi(κ) =∑

xj∈N (xi)

wije−ıκ∆rij , (3.37)

M qrs(x)i =∑

xj∈N (xi)

(xi1 − xj1)q(xi2 − x

j2)r(xi3 − x

j3)swi

j . (3.38)

3.2.2 Least-squares (LS) Filtering

The form of discrete filtering used in the present study is analogous to reconstructing the solu-

tion within a cell in finite-volume methods, and is therefore called, least-squares filtering [12].

The entirely same procedure follows as in formulating cell reconstruction, however it is the

filtered quantity now, which must be computed instead of the reconstructed solution. Recalling

that linear least-squares reconstruction may be expressed as

φi = φ0 + (∇φ)0 ·∆r0i , φi ∈ N (φ0) . (3.39)

where we are trying to reconstruct the solution in cell, i, in terms of an expansion of the

solution of a neighbouring cell, 0. Haselbacher et. al. [12] constructed the LS filter by arguing

that interchanging of φ0 with φ0 such that

φi = φ0 + (∇φ)0 ·∆r0i , φi ∈ N (φ0) , (3.40)


would yield a desirable filtered quantity. Since equation (3.40) is strictly for linear least-squares

reconstruction, one can consider (3.40) for a general high-order reconstruction of filtering, by

using a Taylor series to obtain

φi = φ0 +l∑

k=1

∑n1+n2+n3=k

an1,n2,n3

k!∂kφ0

∂xn11 ∂xn2

2 ∂xn33

∆xn110i∆x

n220i∆x

n330i

, φi ∈ N (φ0). (3.41)

where an1,n2,n3 are the trinomial coefficients defined by

an1,n2,n3 =(n1 + n2 + n3)!n1! n2! n3!

. (3.42)

It is clear then, that we are trying to solve for φ0 in terms of its neighbours, φi. To construct

a filter that resembles, (3.29) we need to compute the weights, wi, that appear in the weighted

sum. In the case of the linear least-squares filter, one can use a pseudo-inverse approach [12]

for computing the weights, by noting that equation (3.40) can be written in matrix form (for

the case of k = 1) to obtain1 (x1 − x0) (y1 − y0) (z1 − z0)

1 (x2 − x0) (y2 − y0) (z2 − z0)...

......

...

1 (xd0 − x0) (yd0 − y0) (zd0 − z0)

·

φ0

(∇xφ)0

(∇yφ)0

(∇zφ)0

=

φ1

φ2

...

φd0

. (3.43)

Typically expression (3.43) results in an overdetermined system, so that to solve for φ0, we

compute the pseudo-inverse of the matrix on the far left of the left-hand side. Calling the

matrix on the far left, A, it is not too difficult to arrive at

φ0 =∑

xi∈N (x0)

w0i φi , (3.44)

where we can take the set of wi0’s to be the first column of Ad, the latter being the pseudo-inverse

of A.

The LS filter is convenient to use for a few primary reasons. First, its application in three

dimensions is straightforward, as the filter may be directly applied in multiple dimensions

without applying it in one dimension and then taking the product. Implementation in finite-

volume codes is also convenient, since the reconstruction principles of the filter are precisely

those of least-squares reconstruction used to reconstruct the solution within a cell. This implies

that any computer code used to produce the numerical method in the LES, may be reused in

the implementation of the LS filter. Another desirable property of this filter, is that it may also

achieve a specified or desired order of commutation, which will be shown in section 3.2.3.


There are a two ways in which better spectral behaviour of the transfer function in equation

(3.30) can be achieved, and they will be listed here briefly.

(i) Relaxation Factor. The relaxation factor is designed to provide a sharper cutoff of wavenum-

bers [36]. It is formulated by applying a weight to the solution of the cell being filtered, so that

equation (3.44) becomes

φ0 = w0φ0 + (1− w0)∑

xi∈N (x0)

w0i φi . (3.45)

(ii) Geometric Weighting. Additional geometric weighting was suggested by Deconinck [18], to

remove possible oscillation in the transfer function at higher frequency solution content. This

may be done by applying a diagonal matrix, W, to the overdetermined system, such that

x = (WA)dW , (3.46)

where wi0 would now be the first column of x. Several geometric weights may be used to

introduce a weighting based on distance from the cell being filtered. The method used in

this work is one based on the Gaussian distribution. It is simply a function of distance and

parameter, D, defined by

Wi =

√6

πD2exp

(−6 |∆r0i|2

D2

),∀ xi ∈ N (x0) , (3.47)

where D is a parameter that may be used to control the filter width, ∆ [18]. In this thesis both

(i) and (ii) are used as weighting techniques in the definition of kernel G.

3.2.3 Discrete Commutation Error

In section 3.1, a detailed derivation of the commutation error for a general filter kernel, G,

was presented. Here a commutation error specifically derived for a discrete filter kernel will be

illustrated, and it will be shown that, like the commutation error presented in section 3.1.3,

[dφdx ] will be zero when the filter width, ∆, is constant.

First it is important to introduce the discrete derivative δφδx , defined as [12](

δφ

δx

)x0

=∑

xi∈N (x0)

w0i ∆φ0i , (3.48)

where ∆φ0i is equal to φi−φ0, and we seek to obtain an expression for the discrete commutation

error given by [δφ

δx

]=δφ

δx− δφ

δx. (3.49)


In equation (3.49) the first term on the right-hand side may be expressed as∑xi∈N (x0)

w0i

δφiδx

, (3.50)

and the second term on the right-hand side of (3.49) follows immediately from (3.48) and is

defined as ∑xi∈N (x0)

w0i ∆φ0i . (3.51)

It is noted that although the steps in this chapter follow similarly from Haselbacher et al., the

weights used in the discrete filter in this thesis are not identical in [12]. Then we have[δφ

δx

]=

∑xi∈N (x0)

w0i

δφiδx−

∑xi∈N (x0)

w0i ∆φ0i . (3.52)

To understand the present derivation, one must see that the weights, w0i , presented in (3.52),

have units of inverse length for linear least-squares reconstruction. This implies that the weight,

w0i , is proportional to 1

∆x0i, where ∆x0

i is the distance between cells i and 0. Before proceeding,

an important remark needs to be made regarding the reconstruction of the simulation solution

and the filter weights of the discrete filter. In this thesis the finite-volume reconstruction is of

second order, which immediately restricts that the weights be computed by solving the system

of equations in (3.43). The weights are slightly modified by introducing geometric weighting,

even prior to computing the pseudo-inverse of A in equation (3.46). Then the effect of the

modification, (WA)dW, on the relationship between w0i and ∆x0

i , is not completely certain, so

the author presents the discrete derivative from hereon, under the assumption that no geometric

weighting is applied. In other words, the discrete derivative definition in (3.48) assumes that

we are solving, Ad and not (WA)dW. Equation (3.52) may be further simplified, by noting that

∆φ0i is simply defined as

∆φ0i = φi − φ0 = φi + (−φ0) , (3.53)

by the linearity property of G ((2.50)), and the definition of the discrete derivative. We then

have

∆φ0i =∑

xj∈N (xi)

wijφj −

∑xi∈N (x0)

w0i φi . (3.54)

Then the entire error may be expressed as[δφ

δx

]=

∑xi∈N (x0)

w0i

∑xj∈N (xi)

wij∆φij −

∑xi∈N (x0)

w0i

( ∑xj∈N (xi)

wijφj −

∑xi∈N (x0)

w0i φi

), (3.55)

and after simple manipulation, the remaining terms are[δφ

δx

]=

∑xi∈N (x0)

w0i

( ∑xi∈N (x0)

w0i φi

)−

∑xi∈N (x0)

w0i

( ∑xj∈N (xi)

wijφi

). (3.56)


It is trivial to see that wij = w0

i on a uniform mesh, resulting in zero commutation error.

Since the LS filter is the only discrete filter considered in this work, it is important to note its

commutation properties. Without proceeding with a detailed derivation here, it was shown in

[12], that the discrete commutation error can be expressed in a more general form by introducing

a series approximation for neighbouring cell solutions. Then in such a case, the final form of

this discrete commutation error becomes a function of an infinite series if one introduces[δφ

δx

]x0

=

[d

dx+

+∞∑k=p

1k!

( ∑xi∈N (x0)

w0i ∆xk0i

)dk

dx

]φ0 . (3.57)

Note, the simplification used in (3.56). Then the order of commutation of the LS filter is

dictated by p, q, such that [12] [δφ

δx

]= min(p− 1, q) , (3.58)

where p is the last order of derivative used to reconstruct the solution, φ0, and q is the order

of accuracy such that if q > p + 1, this would imply that cancellation occurred in derivatives

of order larger than p. On non-uniform meshes which use asymmetric stencils, q = p + 1, and

so the order of commutation is of order q, while for a symmetric stencil this might not be the

case.

3.3 Approximate Deconvolution Methods

Approximate deconvolution methods are a class of SFS LES models known as structural models

that make use of the explicit filter itself to recover an approximate representation of the unfil-

tered numerical solution. There are several approaches for the ways in which one can proceed

to perform the LES, once the approximately deconvolved solution has been obtained. Only one

approach will be taken here, the approximate deconvolution method of Mathew et al. [15], and

the arguments that allow for such an approach will be presented in this section. We can now

turn our attention to the family of SFS modeling, known as structural modeling.

Structural modeling is different from the models introduced in Chapter 2, for the reason that the

model is constructed solely based on the filtered field, φ, not the universal small scales. There

are several classes of structural models, however only one will be illustrated in this section.

Sagaut and Meneveau [17] suggested that the structural modeling family can be divided into

the following 6 subclasses:


• models based on formal series expansion;

• models based on transport equations for the SFS terms;

• models constructed from deterministic models for the SFS structures;

• models which make use of the scale similarity concept;

• models based on reconstruction of the unresolved velocity fluctuations on an auxiliary grid;

and

• models based on numerical algorithms, such that the algorithm errors mimic the SFS terms.

This work only considers approximate deconvolution formulated from the class of formal series

expansion.

3.3.1 Approximate Deconvolution by Iteration

Although this form of deconvolution is referred to as iterative, it uses the Van Cittert series [13]

to approximate the inverse operator defined below as

φ∗ = Q ? φ . (3.59)

Here φ∗ is used to denote the recovered solution when the inverse operation is approximated

by the operator, Q, with

Q := G−1 + ε , (3.60)

such that ε is taken to be as small as we like by increasing the order of truncation, N , of the

Van Cittert series. Then we arrive at

Q =N∑k=0

(I −G)N + ε . (3.61)

It is interesting to note that recovering the solution, such that ε > 0, can be done by repeated

filtering, so that (3.28) when expanded becomes

φ∗ = φ+ (φ− φ) + (φ− 2φ+ φ) + (φ− 3φ+ 3φ− φ) + ...... . (3.62)

3.3.2 Approximate Deconvolution by Series Expansion for a Discrete Filter

The following argument for performing deconvolution by series expansion follows closely from

Sagaut et al. and so for details omitted here, the reader is referred to their text [17]. Before

proceeding, it should be mentioned the following expressions are derived for the case of a

solution in R, however using equation (3.32) for discrete filter easily allows extension to R3.


The starting point for constructing a structural model based on a series expansion, starts with

an expansion in the support of the kernel, G, as follows

φ(x) =∫ +∞

−∞φ(x′)G(

x− x′

∆)dx′ . (3.63)

Then we can expand φ(x′) about x to arrive at

φ(x′) = φ(x) + (x′ − x)∂φ

∂x+

12

(x′ − x)2∂2φ

∂x2+ ..... . (3.64)

Note that (3.64) assumes that an infinitely differentiable solution exists (φ(x) ∈ C∞). We will

assume here that any kernel, G, is one with properties (2.59)–(2.61), so that we can proceed

with the below expression by substituting (3.64) into (3.63), and obtain

φ(x) = φ(x) +12∂2φ

∂x2

∫ +∞

−∞(z)2G(z)

+∞∑k=1

+....+1n!∂nφ

∂xn

∫ +∞

−∞znG(z)dz + ... , (3.65)

where z = x′ − x. Then writing (3.65) in a more compact form and re-introducing the filter

moment, Mk, one can find that

φ(x) = φ(x)−+∞∑k=1

Mk

k!∂kφ

∂xk. (3.66)

For a discrete filter, and from section 3.2.1, we can write the discrete filter moment, Mk, at cell

centre, x0, as

Mk0 =

∑xi∈N (x0)

(xi − x0)kw0i . (3.67)

Note that in (3.66), φ∗, is not used when the series expansion is expressed with sum to ∞, as

such a sum would imply perfect devonvolution. Then (3.66), truncated at N and solved for φ

becomes

φ∗ =

(I +

N∑k=1

Mk

k!∂k

∂xk

)−1

φ . (3.68)

The formulation is not yet complete, since equation (3.68) contains the continuous derivative.

For a discrete filter a discrete derivative must be introduced, which will add another magnitude

of error. We will use the discrete derivative introduced in section 3.2.3, assuming this will not

significantly impact the results in (3.68). Then we can derive a discrete second-order derivative

defined byδ2φ

δx2=

∑xi∈N (x0)

w0i

∑xj∈N (x0)

w0jD

2f (φ) . (3.69)

Here the second order forward finite difference operator acting on φ was introduced so that

D2f (φi) = φi+2 − 2φi+1 + φi . (3.70)


Note that (3.70) is strictly for the presence of a uniform mesh, which is considered in the present

work. Then using the definition of the discrete filter operator acting on φi, (3.69) becomes(δ2φ

δx2

)x0

=∑

xi∈N (x0)

w0i

∑xj∈N (x0)

w0j

( ∑xk∈N (xj)

wjkφk +

∑xj∈N (xi)

wijφj +

∑xi∈N (x0)

w0i φi

). (3.71)

One can quickly see that higher-order derivatives force the number of terms to grow rapidly.

We can further introduce the Taylor approximation, to second-order, for some arbitrary α, as

(1 + α)−1 = 1− α+O(α2) . (3.72)

Then equation (3.66) may be re-written as

φ∗ =

(I −

N∑k=1

Mk

k!∂k

∂xk

)φ+O(α2) , (3.73)

with alpha equal to the portion of the sum truncated. Then substituting the discrete derivative

into (3.73) for N = 2, one obtains

φ∗ = I −M1 δφ

δx− M2

2δ2φ

δx2. (3.74)

The step just performed is considered possible since the series was truncated using α. A strong

hypotheses is made here by the author, in regards to placing the discrete derivative into equation

(3.73), so that error analysis and computational testing are required to verify whether or not

(3.74) would properly approximate the required deconvolution. It is simply claimed that it

may be possible to approximate equation (3.73) using the notion of a discrete derivative for

a discrete filter. Substituting the first two filter moments given by equation (3.31) and the

discrete derivatives, we can express the solution, φ∗i , using

φ∗i = 1−∑

xi∈N (x0)

(xi−x0)w0i

∑xi∈N (x0)

w0i ∆φ0i−

∑xi∈N (x0)

(xi−x0)2w0i

∑xi∈N (x0)

w0i

∑xj∈N (x0)

w0jD

2f (φ) .

(3.75)

The expression in (3.75) still requires further analysis and computational testing to ensure

that it is indeed a good approximation to G−1. Thus, from hereon it is understood that the

deconvolution by iteration approach introduced in section 3.3.1, is adopted in this thesis and

used on all of the numerical simulations.

Chapter 4

Approximate Deconvolution

Approach and Numerical Solution

Method

This brief chapter introduces the numerical approach of a different LES technique based on

the approximate deconvolution method described of Chapter 3. The application of this model

will be presented here to show how the algorithm may be constructed, and will be a good

predecessor to the results in Chapter 5. This latter part of this chapter will provide a quick

overview of the numerical scheme that was used to perform the LES in this thesis.

4.1 Scale Similarity Approximate Deconvolution

Traditionally, in any LES technique one advances the filtered solution in time, regardless of the

SFS model applied. This thesis considers a deviation from this traditional approach, however

to understand the next section, it may be wise to illustrate the basic idea of using approximate

deconvolution in LES.

Then consider one step of deconvolution dictated by equation (3.59), such that we obtain,

u∗ = Q ∗ u. In this basic approach, the role of u∗ will be, to construct τij so that one does not

need to use an SFS model, such as the Smagorinsky model, for instance. Then the same filtered

Navier-Stokes equations apply as in (2.36)–(2.38), however at every time step as the solution

proceeds, we perform deconvolution so that τij may be computed directly to yield

τij = ρu∗iu∗j − ρuiuj . (4.1)

46

Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method47

Then where we would typically use a model for the first term on the right-hand side of (4.1),

we can now compute directly using u∗. This results one performing the following sequence of

steps after every time step, as the solution is integrated in time [13]:

(1) deconvolve→ u∗n = Q ∗ un.

(2) integrate in t→ un+1 = un −∆t[f ′(un)|x].

(3) filter with G→ un+1 = G ? un+1.

Note that steps (1)–(3) demonstrate one of the concerns with filtering the solution at every

time step; when step (3) follows step (2), repeated filtering of the solution occurs over a certain

number of time steps, such that aliasing errors may occur.

This approach is simply the scale similarity model approach [8] applied to the approximate

deconvolution technique, and is the premise for understanding deconvolution in LES. Note that

steps (1)–(3) really depend on the filtering approach taken. One may choose to filter various

components of the solution field, in which case the solution may not be directly filtered after

every time step, but rather every n time steps, and filtering of the residual terms, may occur

every time step instead, to avoid build-up of high-frequency content, which are the cause of

aliasing errors.

4.2 Refined Approximate Deconvolution

The refined approximated deconvolution was suggested by Mathew et al.[15], however the au-

thors only studied a series of Pade filters on a mesh discretized using a sixth-order finite-

difference scheme. In this section, the refined approximate deconvolution method will be used

to advance the deconvolved solution in time, using the LS filter described in Chapter 3. The

methodology of this LES technique is as follows. One may consider the compressible Navier-

Stokes equations, (2.36)–(2.38), to be written in a more compact form as

∂U

∂t+∂−→F (U)∂x

= 0 , (4.2)

where,

U = [u, ρu, ρE] , (4.3)

with u = [u, v, w]. Then vector,−→F (U) has components in R3 of

−→F (U) =

(F Ix (U) + F Vx (U)

)i+

(F Iy (U) + F Vy (U)

)j +

(F Iz (U) + F Vz (U)

)k . (4.4)


Here F I and F V are used to denote the inviscid and viscid flux, respectively. More detail will

be given on the computation of the fluxes in equation (4.4) shortly. For simplicity, only Fx(U)

will be decomposed below, so that Fx(U) is then

Fx = F Ix + F Vx =

ρu

ρu2 + p

ρuv

ρuw

(ρE + p)u

+

0

σxx

σxy

σxz

u(σxx) + v(σxy) + w(σxz)− qx

, (4.5)

and Fy(U) and Fz(U) follow similarly. Typically, when the Navier-Stokes equations are filtered,

the closure of the resultant system is obtained by introducing the term, τij , which is placed in

the viscous flux, F Vx , as U is advanced in time. Then in such a case, Fx becomes (notice that

Fx(U) will be denoted Fx from hereon)

Fx = F Ix+F Vx =

ρu

ρu2 + p

ρuv

ρuw

(ρE + p)u

+

0

σxx − τxxσxy − τxyσxz − τxz

u(σxx − τxx) + v(σxy − τxy) + w(σxz − τxz)− qx

,

(4.6)

where, () once again denotes the Favre filter operation, and () denotes terms, σij and qx in

the Navier-Stokes equations, evaluated using filtered quantities. As will be shown below, in

the refined model, it is not U that will be advanced in time, but rather Q ∗ G ∗ (U). Then,

closure of the Navier-Stokes equations is the error associated with the approximate deconvo-

lution approximation which assumes that |u − u∗| < ε, and by doing so, the following term is

introduced

F (U∗)− F (U) = F (Q ? G ? U)− F (U) ≈ F (Q ? G ? U∗)− F (U∗) . (4.7)

Observe that applying the operator, Q on U from equation (3.59) results in U∗, which is

equivalent to

U∗ = Q ∗G ∗ U . (4.8)

Then if we replace U with U∗ in equation (4.2) and introduce a new closure term, Eadm,

equation (4.2) becomes∂U∗

∂t+∂F (U∗)∂x

=∂Eadm∂x

, (4.9)

and Eadm is defined by

Eadm = F (Q ∗G ∗ U∗)− F (U∗) . (4.10)


It may not be immediately clear that Eadm should really be understood as a measure of the

error associated with advancing U∗ as opposed to U . This is trivial if one argues that if

QG = I, we would have a perfect deconvolution and Eadm would be zero. Then in approximate

deconvolution, we do not require a model term, τij so that τij = 0 in equation (4.6) as suggested

by equation (4.5).

Although term (4.10) is analogous to the role of τij in the closure problem, it is not entirely

an SFS term in the same sense as is τij . This is due to the fact that it does not contain any

physical information of the flow. Rather, this term is strictly computed based on the operators

applied to the field, U . The term, does however, bare some resemblance to the term, τij , in the

sense that both terms appear based on scale similarity arguments.

Filtering Approach. As with any filtering approach in LES, there is no precise and unique way

to filter the turbulent solution field. Instead, one still must rely on some level of trial and error,

for each new flow and flow geometry encountered, in order to achieve the best filtering possible.

The filtering approach taken in the refined model proposed by Mathew et al. [15], follows the

sequence of steps below for each time step.

(1) deconvolve with Q→ u∗n = Q ∗ un.

(2) integrate in t→ u∗n+1 = u∗n −∆t∂f(u∗n)∂x .

(3) filter with G→ un+1 = G ? u∗n+1.

It is quickly observed, that between time integration steps, operators, Q and G are applied

sequentially, and so collapsing them into a single operator, QG, seems a logical approach. Then

after each time step, the solution field is filtered with the combined filtering operation, QG, and

computation of Eadm is obtained by filtering yet again, and evaluating equation (4.10), using

the nonlinear terms in equation (4.5). It should be noted that the numerical procedure above

advances the deconvolved solution, u∗, forward in time and the corresponding filtered solution

at any instance in time is, u = G ? u∗.

4.3 Numerical Method

For completeness of presenting the numerical approach used in this thesis, the numerical method

used to discretize the Navier-Stokes equations of Gao and Groth [37, 18] requires some discus-

sion. The way that this will be presented in this section, is through a quick description of the

second-order spatial and temporal numerical schemes, which immediately follow.

Spatial Scheme


Recalling that the compact form of the Navier-Stokes equations in equations (2.36)–(2.38), and

ignoring any source terms, we have

∂U∂t

+∂F (U)∂x

= 0 , (4.11)

where, U is the grid-independent exact solution, to the partial differential equations. Applying

the integral form of the conservation law on equation (5.1), yields∫V

[dUdt

+∂F (U)∂x

]dV = 0 . (4.12)

One can define an averaging of U over the volume (in R3), V , by introducing

U =1V

∫VUdV , (4.13)

where the averaging is taken over all cells on a uniformly spaced grid. In general, the finite-

volume scheme used here has been developed for a multi-block body-fitted mesh [37], however

the simulations presented in this thesis are restricted to multi-block uniform Cartesian grids.

Combining (5.2) and (5.3) will give the change in time of the average solution at cell i as

dUi,j,kdt

= − 1Vi,j,k

Nf∑k=1

[nk · Fk Ak]i,j,k , (4.14)

where Nf denotes a cell face and Ak the area of such a face.

Equation (5.4) requires the computation of the flux normal to face, Nf , be decomposed into

a viscous flux, F V , and an inviscid flux, F I , in the following manner. The inviscid fluxes,

F I , are computed using a Riemann solver, for which many various forms exist. The particular

Riemann solver that is used in the numerical scheme of the LES solution in this work, is called,

AUSM+ [38], and is typically used in more complex flow studies. A Riemann solver computes

the flux at the face, Nf , by using the solution state to the left (L) and right (R) of that face.

The general Riemann problem is posed as an initial value problem (in R) such that

U =

UL if x < x0

UR if x ≥ x0 .

(4.15)

In (5.5), x0 is the point at which wave propagation initiates and allows the solution to evolve

in time. Then, applying this concept to a solution domain in R, the cell interface location may

be seen as located at x0 and letting, Rx denote the Riemann solver operator in the x-direction,

the flux at face i+ 12 is

Fi+ 12,j,k = F (R(Ui,j,k, Ui+1,j,k)) . (4.16)


To determine the inviscid fluxes, F I , at the cell faces , linear least-squares reconstruction is used

within each solution cell, relative to a face, Ni, such that for a cell, Ui+ 12, we would compute

ULi+1/2,j,k = Ui,j,k + φi,j,k∆ri,j,k

∂

∂xUi,j,k . (4.17)

Equation (5.7), introduces the concept of a slope limiter, φ, which is required to control the so-

lution gradient and avoid solution instabilities. Several slope limiters are available [39], however

the one considered in this work is the Barth and Jespersen slope limiter [40].

The viscous flux evaluation, F V , cannot be evaluated using a Riemann solver, since these fluxes

are elliptic and do not have a hyperbolic nature. The technique used to evaluate the viscous

fluxes is the hybrid average gradient-diamond-path approach [41, 42], and uses the solution,

Ui+ 12

and its gradient, ∂∂xUi+ 1

2to evaluate the flux, F v

i+ 12

. The gradient is computed according

to

∇Ui+ 12,j,k =

Ui+1,j,k −Ui,j,k

ds

nn · es

+(∇U−∇U · es

nn · es

), (4.18)

with, ∇U computed by

∇U = α ∇Ui,j,k + (1− α) ∇Ui+1,j,k , (4.19)

and α,

α =Vi,j,k

Vi,j,k + Vi+1,j,k. (4.20)

In equation (5.8), n and es, are the vectors normal to a cell face, and the unit vectors, respec-

tively.

Temporal Scheme

To advance the solution in time, an explicit two-stage Runge-Kutta scheme is used on the set

of ODE’s resulting from the spatial discretization described previously. Since an N = 2 (two-

stage) scheme is used, the implies the time marching scheme, along with the spatial scheme, are

accurate to second order on the uniform grid considered in this work. Typically, it makes sense

that if one chooses an nth-order accurate spatial scheme, that the time marching scheme also

be at least of nth-order, to preserve a global definition of numerical error. The general form of

a M -stage explicit Runge Kutta scheme is

Un+1 = Un + ∆tM∑k=1

akkk , (4.21)

where k1, k2, ..., kM, are functions such that

ki = ki((Un + ∆tak1k1 + ∆tak2k2,+...+ ∆taki−1ki−1), tn + ci−1∆t) . (4.22)


Then a two-stage explicit Runge Kutta scheme would present as

Un+1 = Un + ∆t(a1k1 + a2k2) , (4.23)

with

k1 = f(tn, Un) , (4.24)

and

k2 = f(Un + ∆tak1k1, tn + c1∆t) . (4.25)

More detail on explicit time marching schemes may be found in [2, 43, 44]. The time step, ∆t

is computed based on a specified choice of CFL number, which is taken as 0.1 in the present

studies. The CFL number is computed at every time integration step, n, of equation (4.21)

and is equal to, a∆t∆x , for uniform grid spacing, ∆x.

Chapter 5

Results and Discussion

This chapter will focus on presenting a preliminary analysis of a refined model technique in-

troduced in [15], however for the case of a discrete filter. The results of the model, applied to

a homogeneous, isotropic turbulent flow, are presented and some discussion is provided. The

purpose of this section is to observe how the new proposed form of LES behaves, due to the

presence of a new SFS-like term, with appealing properties. The section is divided into ob-

serving the initial conditions of the approximate deconvolution solution, followed by results on

the energy spectrum at three different times. Brief studies are also performed on the transfer

function of a composite filter, in addition to assessing the computational feasibility of the new

LES approach.

5.1 Approximate Deconvolution for Homogeneous

Isotropic Turbulence

In the following sections, and for the remainder of this chapter, some results will be presented, in

the application of the refined model to a homogeneous, isotropic turbulent field on a uniformly

spaced grid. The purpose of this study is to understand how discrete filtering may be used

in approximate deconvolution methods. To observe differences between the solutions, u and

u∗, first the initial conditions are investigated, followed by a comparison between the transfer

functions of QG and G. To end this chapter some results will be shown in regards to the

behaviour of the term, Eadm and observations will be made regarding the CPU time of the

approximate deconvolution method.

53

Chapter 5. Results and Discussion 54

5.1.1 Initial Conditions of u and u∗

Before a time integrated solution is even observed, it is important to see how the turbulence is

initialized in the periodic domain. To create the turbulence in the periodic box, the procedure

described in Chapter 2 was used. Recalling the procedure, equations (2.32) and (2.33) require

a function, E(κ) to be assigned, and in this thesis the function used is the one suggested by

Pope [20], which sets the turbulent kinetic energy at time, t = 0, as

E(κ) =322

(2π

)1/2urms

2

κ0

(κ

κ0

)4

exp

(− 2

(κ

κ0

)2). (5.1)

The value of the variables that appear in equation (5.1) to initialize the turbulent field explored

in this work are summarized in Table 5.1.

Another interesting turbulent quantity to investigate in any turbulent simulation is the Q criterion.

The Q criterion, Qc, is used to observe the coherent structures of turbulent flow, and is really a

measure of the symmetry or asymmetry of the velocity gradient, ∂ui∂xj

. Coherent structures may

be identified by a positive iso-value [45] where Qc is given in the expression

Qc =12

(ΩijΩij − SijSij) , (5.2)

and where Sij and Ωij are the symmetric and antisymmetric parts of ∂ui∂xj

, respectively, such

that

Ω−ij , S+ij =

(∂ui∂xj∓ ∂uj∂xi

). (5.3)

For clarity and brevity, the approximate deconvolution solution advanced using the refined

model, will simply be referred to as the solution filtered with filter operator, QG. Then this

suggests that the solution filtered with the LS filter using the parameters in Table 5.1, is simply

the solution filtered with operator G. Three new variables, FGR,Ec, and Nr, also appear in

Table 5.1, and require some further explanation. The variable FGR is used to denote the filter

to grid ratio, which is simply locally defined as

FGRi =∆i

∆i, (5.4)

and translates to the ratio between the filter width and cell width at a cell, i. When meshes

are non-uniform and unstructured, there are many variations used to define this parameter,

and the selection of the best one, will depend on the gradient of cell width in each direction.

However, in this work the mesh is uniform, and so (5.4) is trivial to compute. The variable, Ecis notation for the specified order of commutation of the LS filter. It has already been shown

that the LS filter commutes on a stretched mesh, so that the parameter, Ec can be set to any


number. Last, Nr, refers to the number of rings used to construct the neighbourhood, N , of

the cell being filtered. It is important to notice, that the concept of rings work quite nicely for

structured grids in the presence of locally small gradients, such that

∣∣∣∣∂∆i

∂xi

∣∣∣∣ < ε , (5.5)

where ∂∆i∂xi

is the gradient of the cell width in direction xi. Then if this gradient is small

for some small ε, rings may be used to construct the filtered cell neighbourhood, however for

unstructured grids, this may become a trial and error procedure and new methodologies should

be adopted.

The reference solution filtered by G has been assessed by Deconinck [18] against a tested

implicitly filtered field, and so this reference solution is taken to be the true solution throughout

the remainder of this chapter.

parameter value

TKE 12000 m2/s2

urms 100 m/s

l0 1.96 m

κ0 5.0 m−1

L 2 π m

κ∆N2 ·

2πL

FGR ∆∆ = 2

Ec 2

Nr 3

Table 5.1: Parameters of LS filter and initial turbulent spectrum

Initial solution

It may be observed in Figure (5.1) that the solution, u is the least resolved, and furthermore, u∗

and u are in fairly good agreement. Figure 5.2 shows the Q-criterion for both the QG-filtered

and G-filtered solutions. The Q-criterion, reveals very similar turbulent structures in both, u

and u∗, however, more smaller structures may be observed in Figure 5.2(a). Figure 5.3 shows

the turbulent energy decay spectrum at time, t = 0 ms, for the three solutions, u, u∗, and u.

The spectrum confirms that the filter, QG preserves more higher-frequency modes than does

the solution filtered by G, creating, in affect, a larger cutoff wavenumber, κ∆.


(a) Refined approximate deconvolution, u∗ (b) Explicitly filtered solution, u

(c) Initial solution, u

Figure 5.1: Turbulent initial field on 64 x 64 x 64 grid. Spectrum of x-directional velocity.


(a) Filtered with QG (b) Filtered with G

Figure 5.2: Q-criterion of solutions, u∗ and u on 64 x 64 x 64 mesh at t = 0 ms.

Figure 5.3: Turbulent kinetic energy spectrum of filters, QG and G on 64 x 64 x 64 mesh.


5.1.2 Transfer Function and Commutation Error of QG

Transfer function. To understand how the filter QG appears to filter the solution in spectral

space, one should refer to its transfer function, QG(κ). One can immediately see that since

the filter QG has a weaker averaging effect on the solution, u, than does G, QG(κ) should

preserve more high-frequency content. This is observed in Figure 5.4, which is not surprising,

after observing Figures 5.1 and 5.2.

To derive the transfer function of filter, QG, a more formal definition of QG(x) is in order.

Then QG may be derived simply by taking the product of filters, Q and G, to yield

Q ·G =

[N∑k=0

(I −G)N]·G , (5.6)

which, after some simple algebra, may be shown to be equal to

QG = (I − (I −G)N+1) . (5.7)

The transfer function of QG is then computed as

Q(κ) · G(κ) =

[N∑k=0

(I −G(κ))N]G(κ) . (5.8)

It is also not surprising to note, that increasing the order of the Van Cittert series truncation,

N , results in greater preservation of high-frequency solution content, and this is demonstrated

in Figure 5.5, which shows the effect of increasing, N , on the transfer function QG.

Discrete commutation error. This brief section is devoted to some observations about the com-

mutation error which may occur in the refined approximate deconvolution method if the mesh

were to be non-uniform. The LES approach in this section was tested on uniform meshes, and

this would yield a commutation error of zero, as shown in Chapter 3. Suppose we seek an

expression for the discrete commutation error of the operator QG, as opposed to G, so that we

are now trying to compute [δφ

δx

]= QG ?

(δφ

δx

)+

δ

δx(QG ? φ) . (5.9)

Then one could easily compute the discrete commutation error of operator, QG by computing

the following expanded form of (5.9)[QG ?

δ

δx

]φ =

[(δφ

δx

)+

(δφ

δx− δφ

δx

)+

(δφ

δx− 2

δφ

δx+δφ

δx

)+ ...,

]−[ ∑

xi∈N (x0)

w0i (φ+ (φ− φ) + (φ− 2φ+ φ) + ...)

], (5.10)


Figure 5.4: QG(κ) and G(κ).

for some choice of N . Equation (5.10) may further be simplified, by applying the definition of

operator, G, and introducing compact notation, however for purposes of this section the above

form suffices. It also seems intuitive to claim that

[QG ?

∂

∂x

]<

[G ?

∂

∂x

]. (5.11)

This might seem to be the case since QG acts nearly as an identity approximation, particularly

in comparison to G, however this is not entirely true. Consider equation (5.10) expanded for a

general derivative, so that it becomes

[QG ?

∂

∂x

]φ =

[(∂φ

∂x

)+

(∂φ

∂x− ∂φ

∂x

)+

(∂φ

∂x− 2

∂φ

∂x+∂φ

∂x

)+ ...,

]−[

∂φ

∂x+

(∂φ

∂x− ∂φ

∂x

)+ .....,

]. (5.12)


Figure 5.5: QG(κ) for Van Cittert series truncations, N = 1,2,3,4,5,6.


Equation (5.12) may be written as function of the commutator of ∂∂x and G, to yield[

QG ?∂

∂x

]φ =

[G ?

∂

∂x

]φ+

[∂φ

∂x− ∂φ

∂x

]+

[∂φ

∂x− ∂φ

∂x

]+ ......,

±

[Gn

(∂φ

∂x

)− ∂(Gnφ)

∂x

]. (5.13)

Thus it appears that QG has the effect of possibly increasing the commutation error. To verify

this, one has to study equations (5.12) and (5.13) in greater detail, and in particular, the

magnitude of terms of the form

Gn

(∂φ

∂x

)−

(∂(Gnφ)∂x

). (5.14)

Figure 5.6 demonstrates that this lack of commutation may be true, by noticing that the slope

of the L2 norm of the QG-filtered solution is less than 2, for a desired order of commutation,

Ec = 2. In Figure 5.6, a solution was filtered both with QG and G, using the LS filter, on

a mesh stretched by a factor of α in all directions, x1, x2, and x3. What this brief study

shows, is that the commutation error should not be understood in terms of the operation of

given operators, QG and G, themselves, but the number of times such operators are applied.

Additionally, if equation (5.13) has the effect of increasing the commutation error compared to

filtering with G, than this error might also be larger for greater Van Cittert truncations, N .

Further investigation of these errors is certainly warranted.

5.1.3 Time Advanced Solution of u and u∗

To asses the approximate deconvolution method proposed in this thesis, the solution was ad-

vanced in time using the traditional LES approach and the structural model approach. All the

structural model approach results in this section were obtained using the Van Cittert series

method with N = 5.

The solution obtained by refined approximate deconvolution is compared against a solution

filtered in the following manner, which has been tested [18] for varying mesh resolutions and

filtering parameters, and so its success in correctly predicting homogeneous, isotropic turbulence

will be assumed. Then the traditionally filtered solution uses the parameters outlined in Table

5.1, and filters only the solution residual every time step, where the reference numerical solution,

u is filtered only every 100 iterations, to avoid aliasing errors.

Regularization. It has been suggested [14], that regularization may be necessary to achieve a

stable solution, when using approximate deconvolution scale similarity models. In the refined


Figure 5.6: L2 norm of commutation errors of filters G and QG for desired orders of commuta-

tion, Ec.

model, it has been suggested that regularization is not necessary, however it remains to be

determined whether the term, Eadm provides the necessary amount of correction needed for a

given mesh [15].

The solution at t = 2 ms is shown in Figure 5.7 for the density of the turbulent field. The

difference in solution resolution is quite large, as it is expected that the solution, ρ∗ resolves

higher frequency modes according to Figure 5.3. The solutions presented in this section are

studied at three different points in the decay of turbulent kinetic energy, specifically at 2, 4,

and 7 ms. To verify that similar structure is observed, Figure 5.8 compares the solutions of the

x-directional velocity, u∗, and u at a time of 7 ms. The solution of u appears to contain similar

structure to the solution, u∗, however, once again the structures are more resolved.

The decay of turbulent kinetic energy is presented at a time of 7 ms in Figure 5.11, against

the spectrum of the G-filtered field. It appears that the solution obtained by filtering with QG

every time step decays faster than does the solution filtered with kernel, G. This may indicate

the the term, Eadm is causing the turbulent kinetic energy to dissipate too fast. In this method,

one has to be reminded that Eadm is a numerical expression, and acts as a correction term

to correct the numerical assumption that u∗ ≈ u. There is no physical information contained

within Eadm as is the case with traditional SFS models. One way to alleviate this issue is


(a) Refined approximate deconvolution, ρ∗ (b) Explicitly filtered solution, ρ

Figure 5.7: Turbulent field on 64 x 64 x 64 grid at t = 2ms.

to introduce a physical scaling parameter within Eadm, so that the correction term contains

direct physical information of the flow as opposed to just numerical. This implies that although

the corrective term resulting from the application of a mathematical operator may contain

influence of the flow physics, this is not the most direct approach to modeling flow physics.

This is interesting, as it may support the notion that a model may never be complete, when

only considering numerical quantities.

To gain some more insight into how well the term Eadm performs the energy spectrum is

displayed at a time of t = 2 ms and t = 4 ms in Figures 5.9 and 5.10. Once again the solution,

filtered with QG appears to be decaying faster than the solution filtered with G. Another

point to observe in Figure 5.9, is the difference between the spectrum curves of the solutions

u∗ and u. In the case presented, the G filtered field was filtered by filtering the residuals at

every time step, however no additional filtering of the solution was performed at any point in

time as originally suggested by Deconinck [18]. In the case at t = 4, 7 ms, the G filtered

field was controlled by filtering the residuals every time step and filtering the solution every

100 iterations. Although the QG-filtered solution appears to be decaying more rapidly, it may

be more robust in the sense that the spectrum looks the same for both time cases studied

without any manipulation of changes in the filtering approach. In Figures 5.12(a) and 5.12(b),

the individual decay spectrums are presented for both the traditional LES approach and the

refined approximate deconvolution approach. It appears that, while the rate of energy decay in

Figure 5.12(b) is larger than that of Figure 5.12(a), the refined model, still decays only slightly

faster relative to its own rate. Then this may show, that the refined model correctly mimics the


(a) Refined approximate deconvolution, u∗ (b) Explicitly filtered solution, u

Figure 5.8: Turbulent field on 64 x 64 x 64 grid at t = 7 ms.

behaviour of isotropic energy decay, however the term, Eadm may not be enough to correctly

represent the decay rate of turbulent kinetic energy. One has to keep in mind, when comparing

Figures 5.12(a) and 5.12(b), that the LES methodology is quite different in each case. In Figure

5.12(a), the Smagorinsky model is used to model the small scales, while in Figure 5.12(b) the

numerical term, Eadm, which is a numerical correction term, is used in place of a traditional

SFS model. At this point, an exact explanation for the faster rate of decay of the approximate

deconvolution model is beyond the scope of this thesis; however, the low order of the numerical

spatial scheme may provide some insight [16].

5.1.4 Approximate Deconvolution Error Term

In section 4.2 it was mentioned that, unlike an SFS model used in traditional LES approaches,

the new term, Eadm, in equation (4.9), does not act as a physical quantity as does a sub-filter

scale stress model, such as the Smagorinsky model, for instance. Strictly speaking the term, τijis represented by an expression such as equation (2.43), which introduces physical quantities

dependent on the state of the flow (νT is a clear example). In contrast, Eadm, is computed

directly using the linear operator, QG, without any dependency on flow physics. An even

stronger argument, is that if one considers that for some small ε > 0, ∃; δ > 0 such that

|u∗ − u| < ε, whenever |QG− I| < δ, (5.15)

and we would have, as δ → 0, that |u∗−u| → 0. This simply illustrates how one should perceive

the term, Eadm, and note that the statement in (5.14) is an idealization, but demonstrates the


Figure 5.9: Decay of turbulent kinetic energy at t = 2 ms for solutions filtered with QG and G

on grid resolution of 64 x 64 64.


Figure 5.10: Decay of turbulent kinetic energy at t = 4 ms for solutions filtered with QG and

G on grid resolution of 32 x 32 x 32.


Figure 5.11: Decay of turbulent kinetic energy at t = 7 ms for solutions filtered with QG and

G on grid resolution of 32 x 32 x 32.


(a) Decay of turbulent kinetic energy for filter G (b) Decay of turbulent kinetic energy for filter QG

Figure 5.12: Decay of isotropic homogeneous turbulence using traditional LES and refined

approximate deconvolution for grid resolution of 32 x 32 x 32.

difference between τij and Eadm. Then u may be seen as a change of variables in the Navier-

Stokes equations, from u to u, resulting in the term, τij .

The term Eadm is a very interesting term, and, in the author’s opinion, largely in contrast to

the role of SFS modeling terms in traditional LES. This is an interesting source of possible

future work in approximate deconvolution methods.

5.1.5 Computational Cost of Deconvolution

This section investigates the computational cost associated with performing approximate de-

convolution. Certainly, since it is argued that u∗ ≈ u, this gain in solution accuracy, will bring

about an increase in computing time. Unfortunately, this increase in computational cost is con-

siderable, and so one should consider alternative approaches to performing the deconvolution,

as opposed to an iterative approach, if minimizing computing time is essential. One possible

suggestion, for a discrete filter, is to consider the series derivation outlined in section 3.3.2. Here

is might be said that discrete filtering has disadvantages since its kernel, G is not continuous,

and so constructing nice operators based on the kernel of a discrete filter, might be a tedious

process.

It may be observed from Table 5.2 that there is an increase in CPU time (minutes) as one moves

from N = 3 to N = 6, which is nearly doubled. The results in Table 5.2 are performed for a 32 x


N CPU ‖Eadm‖L2

0 11. 79 −−−−−3 25.8771 8.18955 ·103

4 30.9527 8.15021 ·103

5 35.8318 8.12370 ·103

6 40.8161 8.10445 ·103

Table 5.2: Computational cost of approximate deconvolution at t = 2 ms for grid size 32 x 32

x 32

N CPU ‖Eadm‖L2

0 174.519 −−−−−3 435.163 3.62421 ·104

4 524.078 3.49947 ·104

5 608.903 3.46956 ·104

6 694.674 3.44750 ·104

Table 5.3: Computational cost of approximate deconvolution at t = 2 ms for grid size 64 x 64

x 64


(a) Solution, u∗ : N = 3 : 643 (b) Solution, u∗ : N = 6 : 643

(c) Solution, u∗ : N = 3 : 853 (d) Solution, u∗ : N = 6 : 853

Figure 5.13: Solution, u∗, for Van Cittert series truncations, N = 3, 6, on grid sizes 643 and

853. The solution in (a) and (b) corresponds to t = 2 ms and in (c) and (d) the solution is at

t = 0.5 ms .


32 x 32 mesh resolution, which although may be too course for even practical purposes, suffices

for the purposes of this section. The same results are provided for a finer mesh resolution of 64

x 64 x 64 (Table 5.3), which shows that increasing N , on a finer mesh, requires more computing

time, as one would expect. Thus, if one would like to use an approximate deconvolution

technique on a finer mesh, it appears that using, even, N = 3, in the Van Cittert series would

suffice for a mesh resolution of 643.

Immediate differences are not obvious, however if one looks closely at Figures 5.13(c) and

5.13(d), the solution of u∗ for N = 3, 6 for a grid resolution of 853 at t = 0.5 ms, shows the

formation of more distinct structures. The appearance of a solution difference as one moves

to higher order of series truncation, shows that for very fine grid resolutions, the Van Cittert

series truncation may start to have a greater affect on the solution. However, observing Figures

5.13(a) – 5.13(b), reveals that according to the increase in CPU time, there is no need to

consider higher order (N > 3) Van Cittert series truncations. The last column, in Table 5.2

and Table 5.3, is intended to show how the L2 norm of the x-directional Eadm term, is affected

by increasing the Van Cittert series truncation. The L2 norm presented in Table 5.2 is computed

using,

‖Eadm‖ = ‖Fx(Q ? G ? U∗)− Fx(U∗)‖ , (5.16)

where the L2 norm is taken over the solution quantity, ρ∗u∗. The quantity ‖Eadm‖ decreases

as N increases, not significantly, however this is simply to show that Eadm should not really

be considered an SFS model, but rather a correction advanced in time, due to the assumption

that, u∗ ≈ u.

Additionally, in Figure 5.9, a simulation at t = 2 ms was performed without advancing Eadm, to

investigate its relative effectiveness. It may be observed, that advancing the solution, u∗, with-

out any corrective term, results in a faster-decaying energy spectrum, which deviates even fur-

ther from the anticipated solution. Thus, the affect of introducing the corrective term certainly

assists in predicting an appropriate turbulent energy decay rate, however to truly understand

whether or not this term is purely numerical in nature, requires further study.

Chapter 6

Conclusions and Future Work

6.1 Concluding Remarks

The intent of this research was to explore the potential of approximate deconvolution methods

for discrete explicit filters. The primary benefit for using discrete filters, in particular, the

least-squares filters used in this work, is the computational efficiency associated with using

data structures already employed in the numerical method of the simulation. A branch of LES

techniques which are based on structural modeling has been investigated, and shows that the

discrete filtering operation may be reversed within some order of error, to recover a solution

close to the exact numerical solution. The computational cost associated with advancing the

filtered solution in time versus the approximate solution computed by deconvolution, was found

to be roughly on the order of 4 times the computational cost of the traditional LES. However,

the the solution obtained with deconvolution better resembles the exact numerical solution, at

least after observing the initial conditions of the velocity field. The new LES approach is very

different from traditional approaches, in that instead of using an SFS model, or term, based on

some physical relationship of the turbulent flow, a corrective term is used instead. Furthermore,

although commutation errors were not heavily discussed in the context of approximate decon-

volution, some results revealed, that the commutation error associated with the new composite

filter operator, may be larger than when using the original filter operator.

72

Chapter 6. Conclusions and Future Work 73

6.2 Future Work

The primary purpose of this work was simply to observe how a discrete filter may be used to

perform deconvolution in the presence of a new LES approach. One of the most significant areas

of this research left to investigate why perhaps, advancing the numerical corrective term is not

enough to achieve proper turbulence decay. It appears that for a second-order finite-volume

scheme, some adjustments to the model may be required. Furthermore, the Van Cittert series

approach seems to work nicely for a discrete filter, however a series expansion approach based

on discrete derivatives should also be tested, particularly if it reduces the computational cost

associated with deconvolution. Another area that requires further investigation, is the possible

presence of commutation errors associated with the new composite filter operator, of the LES

approach studied in this work.

Due to the computational cost associated with performing approximate deconvolution, even at

short times, simulations of complete turbulence decay were not performed. Then the study of

LES solutions of homogeneous isotropic turbulence, at long times should be studied using the

approximate deconvolution technique, once an explanation for the fast decay rate, is obtained.

Additionally, the efficiency of the approximate deconvolution algorithm could be improved to

perhaps yield faster CPU times, than those reported in Chapter 5.

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