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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
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. . . . . . . . . . . . . . . . . . . . . . . . . . 66
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. . . . . . . . . . . . . . . . . . . . . . . . . . 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
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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
Chapter 1. Introduction 5
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
Chapter 2. Large Eddy Simulation in Turbulence 8
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,
Chapter 2. Large Eddy Simulation in Turbulence 9
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.
Chapter 2. Large Eddy Simulation in Turbulence 10
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
Chapter 2. Large Eddy Simulation in Turbulence 11
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].
Chapter 2. Large Eddy Simulation in Turbulence 12
Figure 2.2: Simplified-idealistic turbulent energy cascade. (logarithmic scales).
Figure 2.3: Real turbulent energy cascade (logarithmic scales).
Chapter 2. Large Eddy Simulation in Turbulence 13
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
Chapter 2. Large Eddy Simulation in Turbulence 14
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
Chapter 2. Large Eddy Simulation in Turbulence 15
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
Chapter 2. Large Eddy Simulation in Turbulence 16
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
Chapter 2. Large Eddy Simulation in Turbulence 17
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
Chapter 2. Large Eddy Simulation in Turbulence 18
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].
Chapter 2. Large Eddy Simulation in Turbulence 19
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
Chapter 2. Large Eddy Simulation in Turbulence 20
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)
Chapter 2. Large Eddy Simulation in Turbulence 21
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)
Chapter 2. Large Eddy Simulation in Turbulence 22
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
Chapter 2. Large Eddy Simulation in Turbulence 23
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
Chapter 2. Large Eddy Simulation in Turbulence 24
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
Chapter 2. Large Eddy Simulation in Turbulence 25
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
Chapter 2. Large Eddy Simulation in Turbulence 26
Figure 2.7: Illustration of the convolution operation used in filtering with kernel, G.
Chapter 2. Large Eddy Simulation in Turbulence 27
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)
Chapter 2. Large Eddy Simulation in Turbulence 28
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)
Chapter 3. Explicit Filtering Techniques 31
-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.
Chapter 3. Explicit Filtering Techniques 32
-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.
Chapter 3. Explicit Filtering Techniques 33
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
Chapter 3. Explicit Filtering Techniques 34
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)
Chapter 3. Explicit Filtering Techniques 35
(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)
Chapter 3. Explicit Filtering Techniques 36
(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
Chapter 3. Explicit Filtering Techniques 37
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
Chapter 3. Explicit Filtering Techniques 38
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)
Chapter 3. Explicit Filtering Techniques 39
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.
Chapter 3. Explicit Filtering Techniques 40
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)
Chapter 3. Explicit Filtering Techniques 41
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)
Chapter 3. Explicit Filtering Techniques 42
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:
Chapter 3. Explicit Filtering Techniques 43
• 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.
Chapter 3. Explicit Filtering Techniques 44
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)
Chapter 3. Explicit Filtering Techniques 45
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)
Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method48
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)
Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method49
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
Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method50
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)
Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method51
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)
Chapter 4. Approximate Deconvolution Approach and Numerical Solution Method52
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
Chapter 5. Results and Discussion 55
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∆.
Chapter 5. Results and Discussion 56
(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.
Chapter 5. Results and Discussion 57
(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.
Chapter 5. Results and Discussion 58
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)
Chapter 5. Results and Discussion 59
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)
Chapter 5. Results and Discussion 60
Figure 5.5: QG(κ) for Van Cittert series truncations, N = 1,2,3,4,5,6.
Chapter 5. Results and Discussion 61
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
Chapter 5. Results and Discussion 62
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
Chapter 5. Results and Discussion 63
(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
Chapter 5. Results and Discussion 64
(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
Chapter 5. Results and Discussion 65
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.
Chapter 5. Results and Discussion 66
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.
Chapter 5. Results and Discussion 67
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.
Chapter 5. Results and Discussion 68
(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
Chapter 5. Results and Discussion 69
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
Chapter 5. Results and Discussion 70
(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 .
Chapter 5. Results and Discussion 71
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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