Evaluation of Discrete Explicit Filtering for an Approximate Deconvolution Approach to LES by Sintia Bejatovic A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Graduate Department of Department of Applied Science and Engineering University of Toronto Copyright 2011 by Sintia Bejatovic
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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
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
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
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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