a localized dynamic model for large-eddy simulation of the
TRANSCRIPT
A Localized Dynamic Model for Large-Eddy Simulation of the Neutrally Buoyant
Atmospheric Boundary Layer
by
WILLIAM ANDERSON
A MASTER THESIS
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
Dr. Sukanta BasuCommittee Chairman
Professor Chris W. LetchfordAssociate Chair
John Borrelli
Dean of the Graduate School
August, 2007
Copyright c© 2007, William Anderson
Texas Tech University, William Anderson, August 2007
ACKNOWLEDGMENTS
I thank Sukanta Basu for patience, encouragement and tenacity in helping me to
learn about this challenging topic, and for much valuable career advice. As your
first graduate student, I appreciate that this has been a learning experience for us
both. Thank you also to Chris Letchford, who made this incredible experience
possible. I acknowledge the WISE center and The Atmospheric Sciences Group,
especially Andy Swift, Ann Wheeler and John Schroeder, for academic support and
valuable guidance.
I am deeply grateful to my family for being a constant and unconditional source of
support. However, being so far from home, I have had to rely on friends more than
at any other time in my life. When I arrived in Lubbock in January, 2006, I could
never have imagined making so many great friends, and the adventures that would
follow. To my closest friends and my loving girlfriend, Brittney, thank you for
making this experience at Texas Tech such an unforgettably good time.
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CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Large-Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . 2
1.2.1 Subgrid-Scale Parameterization . . . . . . . . . . . . . . . 4
1.2.2 Stability Regimes . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 10
2 LARGE-EDDY SIMULATION: NUMERICS . . . . . . . . . . . . . 12
2.1 Introduction to Numerical Method and MATLES . . . . . . . 12
2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Numerical filtering . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Manipulation of Equations of Motion . . . . . . . . . . . . . . 15
2.5 Subgrid-Scale Parameterization . . . . . . . . . . . . . . . . . 17
2.6 Boundary Conditions and Computational Domain . . . . . . . 18
2.7 Grid-Structure – Vertical Treatment . . . . . . . . . . . . . . . 22
2.8 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9 Time-Advancement . . . . . . . . . . . . . . . . . . . . . . . . 25
2.10 Pressure Solution . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.11 Algorithm and Flowchart . . . . . . . . . . . . . . . . . . . . . 27
2.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 LARGE-EDDY SIMULATION: SUBGRID-SCALE MODELING . . 32
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Smagorinsky Model . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2 Kolmogorov Scaling . . . . . . . . . . . . . . . . . . . . . . 33
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3.1.3 TKE-Based Model . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Dynamic Model Formulation . . . . . . . . . . . . . . . . . . . 33
3.2.1 Scale-Invariant Models . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Scale-Invariant Dynamic Smagorinsky Model . . . . . . . . 34
3.2.3 Scale-Dependent Dynamic Smagorinsky Model . . . . . . . 35
3.2.4 Scale-Invariant Dynamic Wong and Lilly Model . . . . . . 37
3.2.5 Scale-Dependent Dynamic Wong and Lilly Model . . . . . 38
3.3 LDTKE Formulation . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 LDTKE Algorithm Structure . . . . . . . . . . . . . . . . 41
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 CASE STUDY AND SIMULATION DETAILS . . . . . . . . . . . . 44
4.1 Intercomparison Study . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 INTERCOMPARISON OF SG-S MODELS . . . . . . . . . . . . . . 46
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Temporal Evolution . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 First-Order Statistics . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Second-Order Statistics . . . . . . . . . . . . . . . . . . . . . . 51
5.5 SG-S Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.6 Turbulent Kinetic Energy (TKE) . . . . . . . . . . . . . . . . 53
5.7 Visualizations and Energy Spectra . . . . . . . . . . . . . . . . 53
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . 68
6.1 Summary of Completed Work . . . . . . . . . . . . . . . . . . 68
6.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Longitudinal Coherent Structures . . . . . . . . . . . . . . . . 69
6.4 Higher-Resolution Simulations . . . . . . . . . . . . . . . . . . 71
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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ABSTRACT
The combination of geostrophic forcing and an atmospheric boundary layer with an
approximate thickness of 1500 m leads to a Reynolds number on the order of 108. A
Reynolds number of this magnitude indicates turbulence with an excessive number
of scales of motion, or degrees of freedom. The computational power required for
explicit representation of all scales (to the Kolmogorov scale) in such flows is far
beyond that which is currently available (even when massively parallel computing
facilities are employed); from this, the method of large-eddy simulation (LES) has
emerged. In this methodology, a filtering operation separates the scales of motion
into resolved and subgrid-scale (SG-S) motions. The resolved motions are typically
large and anisotropic (owing to their interaction with the boundary conditions),
whilst the SG-S motions are small. Resolved motions are solved explicitly using the
filtered Navier-Stokes equations – SG-S motions are parameterized.
Parameterization of the SG-S fluid motions has been, and remains, the topic of a
considerable research effort.
A new SG-S model is presented, namely the LDTKE model (localized dynamic
computation of turbulent kinetic energy). The model is applied to LES of the
neutrally buoyant atmospheric boundary layer. Many highly sophisticated dynamic
(tuning-free) SG-S models have recently been developed, however we still observe ad
hoc averaging/clipping. TKE-based SG-S parameterizations have been extensively
used, although often they are based on constant coefficients – the variant presented
here combines a completely dynamic modeling procedure with point-by-point
computations. The only constraint imposed on LDTKE is that the eddy-viscosity
may not be negative.
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LIST OF TABLES
2.1 MATLES variables and leveling . . . . . . . . . . . . . . . . . . . . . 29
2.2 MATLES subroutines and function . . . . . . . . . . . . . . . . . . . 31
3.1 MATLES subroutines and function (with LDTKE) . . . . . . . . . . 43
4.1 SG-S models and grid resolutions . . . . . . . . . . . . . . . . . . . . 44
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LIST OF FIGURES
1.1 Qualitative representation of LES, DNS and RAN-S accuracy-expense
relations (image: Geurts [34]) . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Computational domain for LES of N-ABL . . . . . . . . . . . . . . . 18
2.2 Node-element relations in the horizontal and vertical directions . . . . 23
2.3 Flow variables and corresponding levels for analyses in (computational)
domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 MATLES Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1 Temporal evolution of friction velocity for: (a) C-SM (403); (b) LDTKE;
(c) LASDD-SM; and (d) LASDD-WL . . . . . . . . . . . . . . . . . . 48
5.2 Non-dimensional velocity gradient for: (a) C-SM (403); (b) LDTKE;
(c) LASDD-SM; and (d) LASDD-WL . . . . . . . . . . . . . . . . . . 49
5.3 Non-dimensional velocity gradient from the Andren et al. (1994) in-
tercomparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 C-SM (403) Simulated vertical fluxes of: (a) x-component momentum,
Cs = 0.17; (b) x-component momentum, Cs = 0.24; (c) y-component
momentum, Cs = 0.17; and (d) y-component momentum, Cs = 0.24 . 52
5.5 LDTKE Simulated vertical fluxes of x-component momentum at: (a)
163, (c) 403 and (e) 643; and y-component momentum at: (b) 163, (d)
403 and (f) 643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6 LASDD-SM Simulated vertical fluxes of x-component momentum at:
(a) 163, (c) 403 and (e) 643; and y-component momentum at: (b) 163,
(d) 403 and (f) 643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7 LASDD-WL Simulated vertical fluxes of x-component momentum at:
(a) 163, (c) 403 and (e) 643; and y-component momentum at: (b) 163,
(d) 403 and (f) 643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.8 C-SM (403) simulated velocity variances for: (a) Cs = 0.17 and (b)
Cs = 0.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
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5.9 LDTKE simulated velocity variances for: (a) 163, (b) 403 and (c) 643 60
5.10 LASDD-SM simulated velocity variances for: (a) 163, (b) 403 and (c)
643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.11 LASDD-WL simulated velocity variances for: (a) 163, (b) 403 and (c)
643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.12 Temporal-averaged profiles for: (a) C∗, (b) Ck and (c) Cs . . . . . . . 63
5.13 Turbulent Kinetic Energy (TKE) from LDTKE simulations . . . . . . 64
5.14 C-SM predictions of longitudinal velocity fields (Cs = 0.17, top) and
(Cs = 0.24, middle) and spectra (bottom) at: z = 0.1zi (left) and
z = 0.5zi (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.15 LDTKE simulations of longitudinal velocity fields (top) and spectra
(bottom) at: z = 0.1zi (left) and z = 0.5zi (right) . . . . . . . . . . . 66
5.16 LASDD predictions of longitudinal velocity fields (LASDD-SM, top)
and (LASDD-WL, middle) and spectra (bottom) at: z = 0.1zi (left)
and z = 0.5zi (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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CHAPTER 1
INTRODUCTION
1.1 Background
This work addresses one of the fundamental problems of contemporary turbulence
research: parameterization of the subgrid-scale (SG-S) motions in large-eddy
simulation (LES) of turbulent flows. Here, we focus on land-atmosphere interactions
under the neutrally buoyant atmospheric boundary layer (ABL) regime, a
geophysical fluid dynamics problem, although LES has been widely used in
mechanical engineering flows [34]. Due to the excessive number of scales of motion
in ABL flows, explicit numerical representation is impossible. From an experimental
perspective, we note that the neutrally buoyant ABL thickness is approximately
1500 m, making experimental investigation of ABL flows exceptionally challenging;
many fascinating ABL phenomena are subsequently poorly understood (e.g.,
turbulent streaky structures in the ABL, as discussed by Drobinski et al. [26]). In
this thesis we consider a numerical domain with longitudinal, transverse and vertical
dimensions of 4000 m, 4000 m and 1500 m, respectively; experimental data for a
comparable physical domain would be unrealistically difficult to obtain.
Land-atmosphere interaction refers to the temporal and spatial exchange of water
vapor, heat and momentum at the land-atmosphere interface (i.e. the earth’s
surface). As will be discussed later, the ABL is generally categorized into
convective, neutral and stable (based on the net transfer of radiation at the
land-atmosphere interface). Although the case considered here (and in many other
LES works) is for a simplistic computational domain with homogeneous surface
roughness and no topographic features (i.e. no vertical change of the earth’s
surface), analyses of more complex (and relevant) problems including flow around
objects (bluff bodies), over terrain, during severe weather events (e.g., tornadoes),
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and for simulation of dispersion plume transport through the ABL, can also
potentially be performed with LES (provided the code is sufficiently generalized).
Of course, computational power is the governing factor for analysis of such problems
(existing work on this topic is discussed briefly in a subsequent section).
Early works by Deardorff [21, 22, 23, 24] were the impetus for development of the
LES methodology; although others had used direct numerical simulation (DNS) for
analysis of, for example, boundary layer engineering flows (e.g. Kline, et al. [44]),
Deardorff was the first to apply LES to planetary boundary layer turbulence. Based
on early experimental works at the National Center for Atmospheric Research
(NCAR), he developed numerical constants which remained in use for many years
(Moeng [55] and Sullivan et al. [82]). Following – and in addition to – these works,
many others have made notable contributions to the LES methodology.
1.2 Large-Eddy Simulation (LES)
LES allows compromise between the computational expense of DNS (where the
scales of motion are resolved to the finest turbulent length scale, the Kolmogorov
length scale, η), and the accuracy lost with Reynolds Averaged Navier-Stokes
(RAN-S) simulation (where ensemble-averaged flow variables are used in the
governing equations, and Reynolds stresses are parameterized with an
eddy-viscosity); in Figure 1.1 we illustrate the previous sentence.
This compromise is achieved through a spatial filtering operation of the input
velocity and scalar fields, resulting in a resolved-scale (R-S) velocity and scalar field.
Scales of motion removed from the initial fields by the filtering operation are known
as the SG-S motions, and are parameterized through use of a SG-S model;
parameterization of the SG-S motions allows relatively simplistic mathematical
representation of the small-scale flow physics. The rationale behind LES is that
large, energy-containing and anisotropic, scales of motion (i.e., those whose form are
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Figure 1.1: Qualitative representation of LES, DNS and RAN-S accuracy-expenserelations (image: Geurts [34])
dictated partially by interaction with the physical boundary conditions) shall be
modeled with the filtered Navier-Stokes (N-S) equations; the SG-S motions, which
are assumed to be isotropic (universal), can presumably be represented with a
parameterization. Vis-a-vis the energy transfer, the R-S motions are associated with
production- and inertial-range energy, while SG-S motions are associated with scales
smaller than the cutoff filter (incorporating part of the inertial subrange and down
toward the dissipation scale). Energy cascades through the energy spectrum, to be
dissipated by the SG-S model (Batchelor [10]). At this juncture it is appropriate to
review the now-famous quote from Richardson [72], in which he says:
Big whorls have little whorls
which feed on their velocity
and little whorls have lesser whorls
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and so on to viscosity (in the molecular sense).
This quote refers to the idealized transfer of energy from production range eddies
(resolved in LES of the ABL), down to the inertial subrange eddies, and finally to
the Kolmogorov length scale (at which scale energy is destroyed by viscosity). The
net transfer of energy is always from large to small scales, although there is evidence
to suggest energy may locally transfer from small to large scales (a phenomena
known as backscatter); in mechanical engineering flows, this has been notably
explored by Piomelli et al. [62], Davidson [20], Carati et al. [16] and Kim [42], while
Mason and Thomson [50] present a SG-S model for ABL LES which is generalized
to include stochastic backascatter.
The SG-S parameterization remains the focus of a considerable research effort
[77, 86, 33, 1, 45, 67, 18, 8], and has at times received criticism (Rodi et al. [71]).
With the flows separated, the filtered 3-dimensional N-S equations are solved (the
SG-S component is included by the spatial gradient of a three-dimensional stress
tensor). The filter width (Δf ) is commonly taken as the horizontal grid resolution
(Δg), such that (Δf
Δg= 1). The LES code used throughout this work is called
MATLES (MATLAB for LES), a pseudospectral code which computes derivatives in
the horizontal and vertical planes in spectral and real space, respectively. In the
horizontal directions, we prescribe periodic lateral boundary conditions (necessary
due to the spectral analysis); prescription of the upper and (especially) lower
boundary conditions is more complicated and is further discussed in Chapter 2.
1.2.1 Subgrid-Scale Parameterization
The original Smagorinsky model [77], using a constant (non-dynamic) coefficient, is
well-known to be over-dissipative of energy. Geurts [34] shows energy spectra from
LESs using various SG-S models – this work illustrates the over-dissipative nature
of the Smagorinsky model for SG-S stress predictions; such a result is also shown in
this work. The following characteristics impose serious deficiencies on the model:
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(a) requirement that a constant coefficient be prescribed for the simulation (various
researchers have offered values for the constant, however these values are based on
testing of specific flow cases and therefore are flow dependent); (b) prediction of
incorrect asymptotic behavior near a wall or in laminar flows [87]; (c) the model
does not allow for backscatter of energy to the resolved scales; and (d) the incorrect
assumption that the principal axes of the SG-S stress tensor and strain-rates are
aligned.
The dynamic SG-S modeling approach of Germano et al. [33] has been quite
successful in large-eddy simulations (LESs) of various engineering flows [66]. In this
approach, one dynamically computes the values of the unknown SG-S coefficients at
every time and position in the flow. By looking at the dynamics of the flow at two
different resolved scales, and assuming scale similarity as well as scale invariance of
the SG-S coefficients, these values are optimized. Thus, the dynamic modeling
approach avoids the need for a priori specification and tuning of the SG-S
coefficients. A recent study [54] based on extensive database analysis further
suggests that the dynamic modeling approach closely reproduces the minimal
simulation error strategy (termed as optimal refinement strategy), which is highly
desirable in turbulence modeling. In a seminal work by Ghosal et al. [35], the
theoretical foundations for the dynamic model of Germano et al. [33] are verified.
In ABL turbulence, where shear and stratification and associated flow anisotropies
are (almost) ubiquitous, the inherent scale-invariance assumption of the original
dynamic modeling approach breaks down. Porte-Agel et al. [67] relaxed this
assumption and introduced a scale-dependent dynamic modeling approach in which
the SG-S coefficients are assumed to vary as powers of the LES filter width (Δf ).
The unknown power-law exponents, and subsequently the SG-S coefficients, can be
determined in a self-consistent manner by filtering at three levels [67, 68]. In
simulations of the N-ABL, the scale-dependent dynamic SG-S model was found to
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exhibit appropriate dissipation behavior and more accurate spectra in comparison
to the original (scale-invariant) dynamic model [67, 68]. Recently the
scale-dependent dynamic modeling approach was modified and extended by
incorporating a localized averaging technique in order to simulate intermittent,
patchy turbulence in the S-ABL flows [8, 9]. In parallel, scale-dependent dynamic
SG-S models based on Lagrangian averaging over fluid flow path lines were
developed by Bou-zeid et al. [13] and Stoll and Porte-Agel [80] to simulate N-ABL
flows over heterogeneous surfaces.
The scale-dependent dynamic modeling approach and its variants so far have always
used the popular eddy-viscosity formulation of Smagorinsky [77] as the SG-S base
model. However, some of the the shortcomings of this model are inherent and
cannot be negated with dynamic generalizations (i.e. localized balance of energy
production and dissipation). In order to avoid this strong assumption, Wong and
Lilly [86] proposed a new SG-S model based on Kolmogorov’s scaling hypothesis. A
dynamic version of the Wong-Lilly SG-S model to some extent outperformed the
dynamic Smagorinsky model in simulations of the buoyancy-driven Rayleigh-Benard
convection [86]. Furthermore, the dynamic Wong-Lilly SG-S model is
computationally inexpensive in comparison to the dynamic Smagorinsky SG-S
model. The combination of lesser assumptions and cheaper computational cost
certainly make the Wong-Lilly model an attractive SG-S base model for LES.
Anderson et al. [4] show that, for LES of the N-ABL, the scale-dependent dynamic
variation of the Wong-Lilly model is over-dissipative of small-scale energy at higher
locations, indicated by prediction of large-scale coherent structures; this is
evidenced with correlation functions, energy spectra and visualizations (similar
outputs are also used in this work for presenting statistics).
In Chow et al. [18], they utilize the classical Smagorinsky (1963) eddy-viscosity
model for SG-S stress tensor predictions, as a control for comparison with their own
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model, the approximate deconvolution model (ADM) – a dynamic Wong-Lilly
(DWL) model (note that this model adopts the dynamic model of Wong and Lilly,
1994, for modeling the unclosed SG-S term). They perform explicit filtering of the
velocity field, where the filter is applied such that the subfilter-scale (SFS) motions
are divided into resolvable subfilter-scale (RSFS) motions and SG-S motions.
Kosovic [45] developed a SG-S model which accounts for nonlinear interactions
[63, 29] and anisotropy due to shear and the reverse flow of turbulent kinetic energy
(TKE) at under-resolved scales. For comparison, Kosovic tests the new SG-S model
for LES of a shear driven N-ABL, with physical characteristics identical to those
from the intercomparison study [1]. In doing this, he observes the principles of
simple fluid modeling such as the principle of determinism, the principle of local
action, and the principle of material frame indifference as outlined by Truesdell [84].
The principle of Kosovic’s (1997) SG-S model is that, if the turbulence production
mechanism is known and quantifiable with resolved variables, the nonlinear SG-S
interactions can justifiably be modeled with a deterministic or stochastic mechanism
(i.e. numerically solving and representing the perceived randomness of a complex
dynamic turbulent system); the SG-S model is also consistent with the constitutive
relations for conservation of mass, momentum, angular momentum, energy balance
and the second law of thermodynamics.
Kosovic (1997) discussed the developments of nonlinear constitutive fluid models
[73, 74], and how the Truesdell [84] principles for SG-S stress tensor frame
dependence and the second law of thermodynamics (i.e. real and positive entropy)
effects may be relaxed. Kosovic adopts a reduced and amended version of the
nonlinear constitutive relation, presented by Speziale [79]. Unlike conventional
linear SG-S models which rely on one parameter, a nonlinear SG-S model relies on
three parameters; Kosovic formulates these values accordingly and provides
illustrations of how the properties of the Smagorinsky parameter and a nonlinear
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model parameter differ.
The similarity model was first presented in Bardina et al. [6]. In this formulation,
scales less than the filter width, Δf , are assumed to have similar physical
characteristics to those above the filter width (this rationale seems to imply that
SG-S turbulence may not be isotropic). In Meneveau and Katz [52] a succinct
discussion of similarity models is presented. In Moeng [55], a LES model was
presented which followed from earlier work by Deardorff [22], although it exploits
the Fast Fourier Transform (FFT) for computations in the horizontal planes and
finite differencing in the vertical planes (demonstrating how computational power
has improved our understanding of turbulent flows). In resolving the SG-S motions,
Moeng [55] solves a prognostic equation for the TKE, contingent upon coefficients
suggested by Deardorff [24]. This model has since been revised by scientists at
NCAR (Moeng and Sullivan [57]; Sullivan et al. [82]; Sullivan et al. [83]).
Davidson [20] developed the 1-equation (1-E) model, in which the prognostic TKE
equation is dynamically computed at each time-step. The TKE equation relies on
two coefficients (for production and dissipation), which are also dynamically
computed. The production coefficient is computed following from Germano et al.
[33] and Ghosal et al. ghos95, although two filtering levels are used; the dissipation
coefficient is dynamically stepped forward using a modified relation, partially based
on the results in Piomelli and Juhnui [64]. The model presented in this thesis has
evolved from the 1-E model [20], however there are dramatic differences between the
Reynolds number’s (Re’s), physical problems considered, and numerical details.
Most notably: [20] considers Re’s of O(5000), flows around bluff bodies and
domain-averaged production coefficient (for momentum equation); in contrast, we
consider Re’s of 0(108), flows through the ABL over homogeneous surfaces, and
point-by-point computation of the production and dissipation coefficients.
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1.2.2 Stability Regimes
The ABL is categorized into one of three stability regimes: Neutral (N)-, Convective
(C)- and Stable (S)-ABL. The stability regime is most closely related to the diurnal
cycle, which corresponds with the net radiation flux to and from the earth’s surface.
In order of numerical complexity, the regimes are typically considered: C-, N- and
S-ABL. As of 2006, each regime has now been the topic of an inter-comparison
study, in which various contemporary numerical methods have been used to
simulate a specified case study (C-ABL: Nieuwstadt et al. [58]; N-ABL: Andren et
al. [1]; S-ABL: Beare at al. [11]).
In the C-ABL regime, turbulence is generated through two mechanisms: mechanical
shear at the lower boundary and buoyant convection of latent and sensible heat
from the earth’s surface (which is hotter than the overlying air). In this regime, the
vertical scales of fluid motion (i.e. eddies) can exceed the boundary layer height;
these structures often have large vertical and horizontal length scales, which is
attributed to the convective forcing. It follows that the C-ABL is most commonly
associated with day-time.
In contrast, during night-time and in some rare locations (e.g. polar climates), the
S-ABL occurs. In this regime, the earth’s surface is cooler than the overlying air; air
closest to the surface is subsequently cooler and heavier. Turbulence is generated by
mechanical shear (convection is not present), and suppressed (destroyed) by
gravitational forcing (negative buoyancy) and viscous dissipation (Arya [5]). The
S-ABL is typically the most challenging ABL regime to analyze with LES, because
the scales of motion are small (often smaller than the grid/filter width) and
concentrated in the so-called near-wall region (i.e. very near the surface); it follows
also that the influence of the SG-S model is most pronounced at the lower boundary.
The N-ABL occurs when there is no buoyancy forcing from above (due to cool
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Texas Tech University, William Anderson, August 2007
overlying air, as with the S-ABL) or below (due to convective force of
sensible/latent heat rising from the surface, as with the C-ABL). In this regime
turbulence is generated only by mechanical shear at the lower boundary, there is no
gravitational suppression of turbulence. The N-ABL entails a wide range of scales,
from those in the near-wall region smaller than the grid/filter width, to much larger
structures which scale with height above the surface. The N-ABL temporally occurs
at the sunrise and sunset events of the diurnal cycle; despite its relatively infrequent
occurrence, the N-ABL is commonly used for testing LES models because turbulence
production is concentrated in the near-wall region (increasing reliance on the SG-S
model) without vertical stimulation by (positive or negative) buoyancy effects.
1.3 Problem Statement
It is generally agreed upon that, in comparison to buoyancy-driven flows (e.g.
Rayleigh-Benard convection), LESs of shear-driven boundary layer flows are far
more challenging (owing to mechanical mixing at the fluid-boundary interface, and
subsequently the much larger number of motion scales). Accordingly, in the present
study, we focus on shear-driven N-ABL flow. In order to realistically account for the
near-wall shear effects and SG-S motions, we first formulate a point-by-point version
of the 1-E model presented in [20], which we call LDTKE (localized dynamic
computation of turbulent kinetic energy). In this model, we solve the prognostic
TKE equation (not a new concept in SG-S modeling). The argument for LDTKE is
that: (a) the prognostic TKE equation relies on two coefficients – for dissipation
and production – which are dynamically computed point-by-point at each time-step;
and (b) in the momentum equations, we use the point-by-point coefficients for all
computations. In other works, plane (or even small-scale) averaging is used to retain
numerical stability. Any type of averaging acts to “smooth” a data field.
In Chapter 2, the numerical framework and algorithm structure of MATLES is
extensively discussed, although in-depth discussion of the SG-S details are excluded.
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Texas Tech University, William Anderson, August 2007
Chapter 3 features derivation of the locally-averaged scale-dependent dynamic
(LASDD) model for the Smagorinsky (LASDD-SM) and Kolmogorov-based
(LASDD-WL) SG-S models, followed by derivation of the LDTKE SG-S model.
Furthermore, the very simplistic details of the original Smagorinsky [77] are shown
with constant coefficient (C-SM) are shown. MATLES is used for LESs of a
well-known (similar to the inter-comparison study of Andren et al. [1]), using
LDTKE, LASDD-SM, LASDD-WL, C-SM (Cs = 0.17) and C-SM (Cs = 0.24); note
that Cs is a numerical parameter discussed extensively in Chapter 3. Performance
of the LASDD-SM and -WL models have recently been compared in [4]. Details of
the numerical case study are discussed in Chapter 4, while in Chapter 5 we show
statistics including temporal evolution of friction velocity, non-dimensional velocity
gradients, velocity variance, vertical momentum flux, energy spectra and flow
visualizations. This work is summarized, with some future perspective discussions,
in Chapter 6.
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Texas Tech University, William Anderson, August 2007
CHAPTER 2
LARGE-EDDY SIMULATION: NUMERICS
2.1 Introduction to Numerical Method and MATLES
In Chapter 2 we outline numerical details of the MATLES algorithm. We begin by
presenting the governing equations for transport of momentum and scalars within
the numerical domain. We explain how the governing equations are manipulated to
obtain a closed (and solvable) system of N-S equations, and the numerical
functionality methods exploited to increase numerical accuracy while allowing for a
realistic computational expense (here, we note the benefits MATLAB offers for such
analyses, given its in-built functions for spectral analyses and efficiency for
vectorized analyses, and how these attributes typically allow a dramatic reduction
in software lines of code, SLOC; Kepner and Ahalt [40, 41]). In the final component
of Chapter 2 we present a simple flowchart, Figure 2.4, which graphically
demonstrates the LES algorithm; we discuss the flowchart and explain where the
LES subroutines are located. Throughout this work tensor notation is utilized for
representation of equations – see Panton [61] for further explanation of this.
2.2 Governing Equations
In this work we assume that the fluid is incompressible, a standard and reasonable
assumption in ABL flows [34]. We employ the Boussinesq approximation for the
pressure term, subtracting the mean hydrostatic balance (∂3 〈P 〉 + 〈ρ〉 g = 0) from
the vertical momentum equation, and subsequently compute pressures as deviations
from this value (i.e. P ′′ = P− < P >, where P ′′ represents fluctuations and 〈A〉indicates planar-averaging of flow variable A, whereby the average of all values in
each horizontal plane is assumed to represent the value at that height); this follows
from Stull [81] and Albertson [2]. To compute the velocity field, we employ the
equations for conservation of mass and momentum, as shown,
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Texas Tech University, William Anderson, August 2007
∂ui
∂xi
= 0, (2.1)
∂ui
∂t+ uj
∂ui
∂xj
= − 1
ρo
∂P ′′
∂xi
− ρ′′
ρo
gδi3 + fcεij3uj (2.2)
where ρo= hydrostatic density; δi3 is the standard Kronecker delta; ρ′′= fluctuations
from the hydrostatic density; ui= instantaneous velocity in the xi direction; ν=
kinematic viscosity; fc= Coriolis parameter; εij3= alternating unit tensor; g=
acceleration due to gravity; Equations 2.1 and 2.2 are based on the cartesian
coordinate system, such that x1 = x, x2 = y and x3 = z, where x, y and z
correspond with longitudinal, transverse and vertical (wall-normal) fluid flow
directions, respectively; and standard tensor notation is used, such that
∂∂xj
∂ui
∂xj= ∂
∂x1
∂ui
∂x1+ ∂
∂x2
∂ui
∂x2+ ∂
∂x3
∂ui
∂x3. The relative densities may be substituted for
relative potential temperatures, ρ′′
ρo= − θ
θo, where θ and θo are the potential and
reference potential temperatures, respectively; this relation subsequently gives the
force due to buoyancy, β, so Equation 2.2 may be written as:
∂ui
∂t+ uj
∂ui
∂xj
= − 1
ρo
∂P ′′
∂xi
+ βδi3 + fcεij3uj. (2.3)
Note that, in including the Coriolis parameter with Equations 2.2 and 2.3, our
numerical method differs slightly from that of [2], Porte-Agel et al. [67] and
Porte-Agel [68]. MATLES computes transport of potential temperature, θ, and
water vapor, q, and has capability to compute transport of other passive, active or
reactive scalars. Here we consider transport of a general non-reactive passive scalar,
S, as:
∂S
∂t+ uj
∂S
∂xj
= DS ∂
∂xj
∂S
∂xj
, (2.4)
where DS is a numerical constant for adiabatic diffusion of the scalar, S.
Solution of Equations 2.3 and 2.4 yield, respectively, the total velocity and scalar
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Texas Tech University, William Anderson, August 2007
fields – however these equations cannot be solved for ABL flows of the type
considered in this work, due to excessive number of scales of motion in these fields,
and our next step is to impose a filtering operation on the velocity and scalar fields
– such filtering removes small-scale (i.e. SG-S) motions from the field.
2.3 Numerical filtering
For an extensive discussion of the filtering operation, and filtering techniques
applicable for LES, see Geurts [34], Part 6, Leonard [47] and Aldama [3]. We use a
homogeneous and explicit filter, subsequently allowing the process to commute with
differentiation. Our filtering process removes the Nyquist frequency – the
significance of this is discussed later in Section 2.8. Mathematically, the filtering
operation is expressed with Equation 2.5, as:
ui (x1, x2, x3) =
∫ui (x
′
1, x′
2, x′
3) G (x1 − x′
1, x2 − x′
2, x3 − x′
3) dx′
1dx′
2dx′
3, (2.5)
where ui= the filtered (resolved) field in the xi-direction; and G is a general filtering
operation. It follows that the total field is the sum of the resolved and fluctuating
components, i.e.,
ui = ui + u′
i, (2.6)
where u′
i= is the fluctuating velocity component in the xi-direction. MATLES filters
the velocity and scalars in spectral space; the filter size is a variable nominated by
the user, although is typically equal to the horizontal grid resolution, Δx. Knowing
that the filtering and differentiation processes commute, Equations 2.1, 2.3 and 2.4
become:
∂ui
∂xi
= 0, (2.7)
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Texas Tech University, William Anderson, August 2007
∂ui
∂t+
∂ujui
∂xj
= − 1
ρo
∂P ′′
∂xi
− ∂τij
∂xj
+ βδi3 + fcεij3uj, (2.8)
∂s
∂t+ uj
∂s
∂xj
= DS ∂
∂xj
∂s
∂xj
− ∂πSj
∂xj
, (2.9)
where we employ continuity (Equation 2.6) by substituting uj∂ui
∂xjfor
∂uiuj
∂xj.
Equations 2.8 and 2.9 illustrate the defining characteristic of LES, as opposed to the
DNS or RAN-S methodologies: that we have unresolved terms. We group the
unresolved terms in Equations 2.8 and 2.9 with τij and πSj , respectively, expressing
the deviatoric component as:
τij = uiuj − uiuj, (2.10)
πSj = ujs − uj s. (2.11)
Equations 2.10 and 2.11 demonstrate the closure problem. The unresolved (SG-S)
quantities cannot be explicitly resolved, and we subsequently parameterize these
motions. This parameterization has been the topic of contention (Rodi et al. [71];
Pope [66]) in the turbulence research community, and we are yet to achieve a SG-S
closure of universal application. Equations 2.7, 2.8 and 2.10 constitute the general
equations of momentum transport, while Equations 2.9 and 2.11 constitute the
general equations of scalar transport. Equations 2.10 and 2.11 require further
attention, due to the presence of unresolved terms. Additional manipulation of the
equations of motion (momentum and scalar) is employed to simplify numerical
treatment.
2.4 Manipulation of Equations of Motion
Like [2], we acknowledge the convenience of removing the SG-S stress tensor from
τij and adding it to the pressure term, such that:
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Texas Tech University, William Anderson, August 2007
τij − 1
3τkkδij = −2νtSij, (2.12)
where νt is the eddy-viscosity (as computed with a SG-S model), and
P 1 =P ′′
ρo
+1
3τkk. (2.13)
Following Orszag and Pao [60], Ferziger and Peric [28] and [2], we substitute the
convective term for its rotational form based on the incompressible flow identity:
∂uiuj
∂xj
= uj
(∂ui
∂xj
− ∂uj
∂xi
)+
1
2
∂ujuj
∂xi
, (2.14)
and we now group the gradient of resolved kinetic energy, 12
∂uj uj
∂xi, with the pressure
gradient, to obtain a new pressure term:
P 0 = P 1 +1
2ujuj =
P ′′
ρ0
− 1
3τkk +
1
2ujuj. (2.15)
This yields the following revised version of Equation 2.8:
∂ui
∂t+
∂ujui
∂xj
= −∂P 0
∂xi
+ ν∂
∂xj
∂ui
∂xj
− ∂τij
∂xj
+ βδi3 + fcεij3uj. (2.16)
We have neglected the nonlinear viscous term, ν ∂∂xj
∂ui
∂xj; the effect of this term is
negligible relative to that of other loss terms, and this rationale is widely accepted in
the literature (Porte-Agel et al. [67]; Basu and Porte-Agel [8]; [2]). Finally, a general
forcing term, Fi, is introduced into Equation 2.16. This is derived by separating the
constant mean pressure gradient from the total pressure gradient, thus isolating the
driving force in the flow. In its final form, the momentum equation is:
∂ui
∂t+ uj
(∂ui
∂xj
− ∂uj
∂xi
)= −∂P
∂xi
+ Fi − ∂τij
∂xj
+ βδi3 + fcεij3uj (2.17)
where we have rewritten the pressure term, P 0 − Fixi, as, P . Equation 2.7 is
unchanged. For scalar transport (Equation 2.9), we omit the molecular transport
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Texas Tech University, William Anderson, August 2007
term, DS ∂∂xj
∂s∂xj
, to have the final equation for scalar transport (Porte-Agel et al.
[67]; Basu and Porte-Agel [8]; [2]):
∂s
∂t+
∂ (uj s)
∂xj
= −∂πSj
∂xj
. (2.18)
The closure problem is now addressed. Without values for τij and πSj , Equations
2.17 and 2.18 cannot be solved.
2.5 Subgrid-Scale Parameterization
Solution for the momentum (Equation 2.17) and scalar (Equation 2.18) fields are
achieved through parameterization of the SG-S terms. The most popular closure
method was developed by Smagorinsky [77], and is discussed briefly in Chapter 3.
The SG-S models are engaged to compute the eddy-viscosity (Chapter 3), while the
stress tensor is directly proportional to the eddy-viscosity, as:
τij = −2νtSij, (2.19)
where Sij is a strain-rate tensor,
Sij =1
2
(∂ui
∂xj
+∂uj
∂xi
). (2.20)
In its traditional form the [77] closure performs very poorly [34], owing to its
simplistic assumptions about the SG-S physics and a constant numeric coefficient
(Chapter 1, Section 1.2.1). The dynamic modeling approach of Germano et al. [33]
represented a major progression in analysis of turbulent flows. In this method,
additional filtering of the velocity and scalar fields allows for the coefficient to be
dynamically computed at all locations in the domain, based on the flow physics. In
using constant coefficients, as done with the traditional Smagorinsky [77] model, it
is implied that the SG-S contribution has lesser dependence on scales larger than
the filter width, unlike the dynamic modeling procedure. Numerous independent
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Texas Tech University, William Anderson, August 2007
research groups have developed and presented new SG-S models. In the following
chapter we outline the scale-dependent dynamic (SDD) formulation for
eddy-viscosity models, and show how this derivation is applied to the Smagorinsky
and Kolmogorov-scaling SG-S models (MATLES options); we further show
derivation of the LDTKE model. Note that MATLES has various SG-S model
options (e.g., SDD-Smagorinsky, SDD-Kolmogorov and the non-backscatter model
of the United Kingdom Meteorological Office) for LES of the ABL. The stability
regime is also a user-input; in this work MATLES is employed for simulation of the
N-ABL.
2.6 Boundary Conditions and Computational Domain
We impose boundary conditions at all six sides of the (cubic) computational
domain, which is shown in Figure 2.1.
x
z
y
Lower boundary
(surface)
Mean Flow
xL
yL
zL
Figure 2.1: Computational domain for LES of N-ABL
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Texas Tech University, William Anderson, August 2007
In the horizontal and vertical directions, MATLES evaluates derivatives using
spectral and finite-differencing techniques, respectively. In using spectral methods,
we must prescribe periodic boundary conditions (at the four vertical domain
surfaces) which is beneficial for turbulence modeling (because we attain greater
numerical accuracy [2, 15, 34]). By using pseudospectral analyses (x and y
directions), the following is true for flow property, f :
f (x, y, z) = f (x + kxLx, y + kyLy, z) (2.21)
where Lx and Ly are, respectively, the longitudinal and transverse domain
dimensions; and kx and ky are signed integers [34]. Sufficiently large horizontal
dimensions are provided to ensure the flow becomes uncorrelated between entering
and leaving the domain; this condition is satisfied by using horizontal domain
dimensions exceeding twice the lag value at which the longitudinal velocity
autocorrelation terminates, a value ranging between 200 and 1000 m.
Pseudospectral methods are popular for LES, although others (Chow et al. [18];
Menon and Kim [53]) utilize finite-difference methods in all three directions. For
vertical computations we use finite-differencing methods – this is necessitated by the
solid lower-boundary. The top-boundary is the least complicated, as we impose (like
[2]) a condition of vanishing vertical gradients (i.e. no stress) and no flow through
the boundary, mathematically:
∂u1
∂x3
=∂u2
∂x3
=∂θ
∂x3
=∂q
∂x3
= u3 = 0; at z = top. (2.22)
Note that our vertical dimension, Lz, is chosen to ensure that the boundary layer
profile is completely included in the domain (appropriate selection of this dimension
is based on experience; the boundary layer height is not constant and is directly
related to the stratification). At the lower boundary, where the flow is characterized
by small-scale turbulent motions (due to fluid-surface interaction and, for the C-
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Texas Tech University, William Anderson, August 2007
and S-ABL, buoyancy forces), prescription of the boundary conditions is very much
more complicated. Our approach follows strongly from [2] and Basu and Porte-Agel
[8]. We quantify the shear stress (SG-S stress) at the lower boundary based on the
Monin-Obokhov similarity theory, with local surface roughness length, zo, as shown
with the following:
τxz = −u2∗
[u (z)
M (z)
], (2.23)
τxz = −u2∗
[v (z)
M (z)
], (2.24)
where τxz and τyz are the instantaneous shear stresses in the longitudinal and
transverse directions, respectively; M (z) =
⟨(u2 + v2)
1/2⟩
at the first vertical
(computational) level above the wall (z = Δz/2); and u∗ is the friction velocity,
computed with Equation 2.25.
u∗ =M (z) κ
log(
zzo
)+ χi + χio
. (2.25)
Where κ is the von Karman constant (= 0.4); and χi and χio are non-dimensional
parameters based on the stability conditions. Prescription of the surface heat flux
depends on the stability regime. In this work, we consider only the N-ABL, and
thus do not prescribe a heat flux.
These stresses are defined at the first vertical (computational) cross-section above
the lower boundary. In later sections, we see how (and why) different analyses are
performed on different levels, and the importance of being consistent with these
computations.
Figure 2.1 shows the physical domain. The vertical dimension ensures geostrophic
flow is always achieved at the top-boundary (i.e. no further significant variations in
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Texas Tech University, William Anderson, August 2007
the boundary layer profile). Equation 2.26 shows the 2nd-order finite-difference
method used:
∂A
∂z(x, y, z) =
A(x, y, z + Δz
2
) − A(x, y, z − Δz
2
)Δz
, (2.26)
where A is a numerical variable. In a later section, we show how Equation 2.26
changes depending on the flow variable. Being a MATLAB-based code, we exploit
the FFTW toolbox – a variety of spectral analyses (Fourier) tools. Mathematically,
the flow variable, ui, is expressed as:
ui (x, y, z) =
′∑kx
′∑ky
ui (kx, ky, z) ei(kxx+kyy), (2.27)
where ui is the complex Fourier amplitude associated with (physical space) flow
variable, ui; kx and ky are wave-numbers in the x- and y-directions, and defined over
the integer wave-numbers, Nx/2 + 1 ≤ kx ≤ Nx/2, and, Ny/2 + 1 ≤ ky ≤ Ny/2,
respectively; and′∑
kall
indicates summation over all wavenumbers except the Nyquist
frequency (discussed in the following section). Pseudospectral derivatives are
computed with:
∂ui (x, y, z)
∂x=
′∑kx
′∑ky
[ui (kx, ky, z) (ikx)] ei(kxx+kyy)
, (2.28)
∂ui (x, y, z)
∂y=
′∑kx
′∑ky
[ui (kx, ky, z) (iky)] ei(kxx+kyy)
. (2.29)
Further clarification about the Fourier series, and Equations 2.28 and 2.29, is offered
in Press at al. [69]; in addition, for discussion about the benefits of spectral analyses
for fluid mechanics problems, see Canuto et al. [15]. The computational domain is
shown in Figure 2.1. The ABL depth is less than the vertical domain dimension; it
has been shown that, for the C-ABL, the largest turbulent eddies in the flow exceed
the boundary depth (Schmidt and Schuman [75]), and we prescribe the domain
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Texas Tech University, William Anderson, August 2007
height accordingly. Further, the horizontal dimensions of the domain must exceed
the autocorrelation length of the turbulent case being considered [2]. In this work,
the computational domain (Lx, Ly, Lz) is discretized uniformly into Nx, Ny and Nz,
respectively. Due to use of pseudospectral methods, it is beneficial to prescribe the
horizontal dimensions with respect to the vertical dimensions, as:
Lx = Ly = 2πLz. (2.30)
It follows that the computational domain is of volume LxLyLz and contains
NxNyNz grid points. MATLES defines the uniform grid-spacing, based on the
dimensions and discretization, such that:
Δx =Lx
Nx
, Δy =Ly
Ny
, Δz =Lz
Nz − 1, (2.31)
where Δx, Δy and Δz are the grid-spacing values in the longitudinal, transverse and
vertical directions, respectively. In addition to the dimensions shown above, we have
non-dimensional grid-spacing values based on the global dimension, zs, such that:
ndΔx =2π
Nx
, ndΔy =2π
Ny
, ndΔz =2π
(Lz/Lx
)Nz − 1
, (2.32)
where the super-script nd indicates non-dimensional. The pseudospectral approach
has been used with success by many for ABL turbulence studies (Orszag [59];
Moeng [55]; Mason [49]; and Schumann [76]) and the benefits of this approach are
well known. It is appropriate now to discuss the grid structure of the computational
domain; in MATLES numerical treatment is applied to flow variables at different
computational levels, as necessitated by the pressure solution.
2.7 Grid-Structure – Vertical Treatment
Due to the mixed numerical approach (spectral and finite-difference analyses), and
subsequent boundary conditions, our grid-structure must also be considered
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Texas Tech University, William Anderson, August 2007
differently. Periodic boundary conditions imply the “entry” information is
equivalent to the “exit” information; or that, in the horizontal direction, the number
of elements and nodes are equal. This differs from the vertical direction, in which
the number of vertical nodes exceeds the number of vertical elements by one. Figure
2.2 illustrates this. This figure pertains to numerical treatment due to the imposed
boundary conditions – in addition we analyze flow variables at different levels in the
vertical direction. We analyze flow variables (u, v, p, θ, q) and w at intermediate
computational levels, spaced with Δz/2 (this is true also for quantities related to
these variables, i.e. d (u, v, p, θ, q)/d (x, y, z) is treated on the u-level nodes). This is
clarified with Figure 2.3, shown below. In this approach we are consistent with
previous LES methods (Porte-Agel et al. [67]; Porte-Agel [68]; Basu and Porte-Agel
[8]; [2]).
Node
Horizontal
Element
Vertical
Elements
2
3
4
1
Nodes
1
2
3
1
14321
2 3 4
4
4
Figure 2.2: Node-element relations in the horizontal and vertical directions
The numerical value of flow variables at z = L + Δz/2 are equivalent to the values at
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Texas Tech University, William Anderson, August 2007
1Nk z −=
1Nk z −=
zNk =
zNk =
k = 2
k = 2
k = 1
k = 1
k = 3
Lz =
Top Boundary
2zLz Δ+=
zLz Δ−=
2zLz Δ−=
zz Δ=
2zz Δ=
z = 0
Bottom Boundary
2z3z Δ=
z2z Δ=
θ,q,p,v,u
w
Figure 2.3: Flow variables and corresponding levels for analyses in (computational)domain
z = L − Δz/2 – this implies the vertical gradients are zero between these two levels
(known as the stress-free upper boundary condition). Figure 2.3 illustrates the
association of flow variables with different computational levels; note, however, that
the imposed surface temperature and moisture content are prescribed at the z = 0
level (in the case of non-neutrally stratified boundary layer simulations). With
Figure 2.3 explained, we now show how Equation 2.26 (differentiation in the vertical
direction) is modified depending on the flow variable:
∂A
∂z(i, j, k) =
A (i, j, k) − A (i, j, k − 1)
Δz; for A = u, v, p, θ, q, (2.33)
∂A
∂z(i, j, k) =
A (i, j, k + 1) − A (i, j, k)
Δz; for A = w, (2.34)
where (i, j, k) is used to indicate nodal positions. i and j correspond with x and y,
respectively. k is shown also in Figure 2.3. Use of spectral methods allows
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Texas Tech University, William Anderson, August 2007
significant increase in accuracy, although such accuracy must be achieved using the
expensive aliasing technique.
2.8 Aliasing
After transforming the horizontal velocity fields into spectral space, the amplitude
of the highest wavenumber, the Nyquist amplitude, has only a non-zero real part.
Thus, when the momentum field is transformed from spectral space back to real
space, the (real) values corresponding with the Nyquist amplitude will possess an
imaginary component. Because of this we simply impose that both the imaginary
and real component of the Nyquist amplitude shall be zero. This approach is
commonly used in LES models.
For resolution of the extended inertial-range energy spectra (Chapter 5), and for
accurate analysis of the inherently nonlinear velocity fields, we use 3/2 padding of
the Fourier transforms. Padding removes errors generated in the aliasing process. In
real space, this is equivalent to reduction or removal of truncation errors.
2.9 Time-Advancement
In MATLES, the 2nd-order Adams-Bashforth method is used (Canuto et al. [15]) to
advance the momentum and scalar fields forward in time. Equation 2.35, below,
demonstrates this, where A is the independent variable and may be considered as a
momentum parameter (i.e. ui) or scalar parameter (i.e. θ) at the next time step
(t + Δt), and φ contains information about the dependent variables from previous
time step(s), such that:
At+Δt = At + Δt
(3
2φt − 1
2φt−Δt
). (2.35)
In this method, it follows that numerical values at t and t − Δt are known – this
leaves a single equation with a single unknown (value at t + Δt), thus we can step
the system forward in time. There are many time-stepping methods available; the
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Texas Tech University, William Anderson, August 2007
2nd-order Adamas-Bashforth is popular in ABL turbulence simulation, owing to its
robust numerical stability (Gao and Leslie [32]; [34, 67, 2]) and favorable damping
characteristics (Haltiner and Williams [37]).
2.10 Pressure Solution
The pressure solution is a numerical technique to solve for the pressure field, which
is subsequently used in the momentum equation. The pressure field is an adiabatic
and dynamic variable, which maintains a divergence free (incompressible) velocity
field [2]. It follows that the pressure solution is contingent upon vanishing the
velocity divergence – this is achieved by taking the divergence of the momentum
equation and applying continuity to the resultant expression. Following this, we are
left with a Poisson equation for the pressure field. Note that our pressure solution is
extremely similar to that of [2]. Recall Equation 2.17 – the momentum equation (in
continuous form); Equation 2.36 below is the discretized version of Equation 2.17,
ut+Δti − ut
i
Δt=
3
2˜RHS
t
i −1
2˜RHS
t−Δt
i , (2.36)
where ˜RHSi (indicating right-hand side) is the discrete component of Equation
2.17, as:
˜RHSi = −˜
uj
(∂ui
∂xj
− ∂uj
∂xi
)− ∂P
∂xi
+ Fi − ∂τij
∂xj
+ βδi3 + fcεij3uj. (2.37)
We next separate the pressure gradient term from the remaining components of
Equation 2.37, such that:
˜RHSt
i = −∂P t
∂xi
+ Γti, (2.38)
where
Γti =
˜
uj
(∂ui
∂xj
− ∂uj
∂xi
)+ Fi − ∂τij
∂xj
+ βδi3 + fcεij3uj. (2.39)
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Texas Tech University, William Anderson, August 2007
Rearranging Equation 2.36, we group the unknown and known terms on the left and
right, respectively, to have:
(2
3Δt
)ut+Δt
i +∂P t
∂xi
=
(2
3Δt
)ut
i + Γti −
1
3˜RHS
t−Δt
i . (2.40)
Equation 2.40 is the momentum equation, of the form used to advance the velocity
field. In the pressure solution, we seek to prescribe a pressure field such that ∂P t
∂xi, in
coordination with the known uti, Γt
i and RHSt−Δt
i values, will yield a divergence free
ut+Δti . Thus, we first take the divergence of Equation 2.40, obtaining:
(2
3Δt
)∂ut+Δt
i
∂xi
+∂
∂xi
∂P t
∂xi
=∂
∂xi
(2
3Δt
)ut
i + Γti −
1
3˜RHS
t−Δt
i . (2.41)
The velocity field at the next timestep, ∂ut+Δti , is set to zero, thus allowing solution
for the approximate pressure field. With this term set to zero, and with
Λi ≡(
23Δt
)ut
i + Γti − 1
3˜RHS
t−Δt
i , the Poisson equation for pressure is subsequently
expressed with Equation 2.42 as:
∂
∂xi
∂P t
∂xi
=∂
∂xi
Δi. (2.42)
Now, the known term, Δi, is used to solve for the pressure field, Pt, based on
prescription of appropriate boundary conditions. We neglect additional details for
solution of the pressure solution. The interested reader is referred to Albertson [2]
for a cogent discussion. The pressure solution is generally considered to be the most
complicated component of MATLES (and other LES codes).
2.11 Algorithm and Flowchart
The above sections explain the physics behind MATLES; it is appropriate now to
explain the code architecture, and how the solution algorithm is structured.
MATLES may be be considered as a master code, which links (calls) to a series of
smaller subroutines for analyses (profiling analyses demonstrate the computational
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Texas Tech University, William Anderson, August 2007
expense of different subroutines, although such statistics are not shown in this
work). Figure 2.4 is a flowchart of MATLES; Table 2.1 lists MATLES variables and
the corresponding node levels on which they are analyzed.
If scalars enabled Initialization:
-from file
-from codeCalculate Obukov
length & set temp.
gradient
At specific intervals
compute statistics
If scalars enabled
Compute buoyancy
Compute scalar RHS
SG-S diffusivity coeff.
optimization
Compute statistics
Update initialization files
SG-S viscosity coeff.
optimization
Calculate initial gradients
Enforce boundary conditions
Calculate convective terms
SG-S models
Compute divergence of SG-S
stresses
Calculate initial RHS
Solve Poisson pressure equation
Calculate pressure gradients
Calculate final RHS
Calculate forcing
Step momentum and
scalar fields forward in
time
End time-loop and output final vel.out
Figure 2.4: MATLES Flowchart
Table 2.2 lists all subroutines required for MATLES to run; here we discuss the
general processes for a time-step and include reference to some relevant subroutines.
1. The velocity and scalar fields are converted into spectral space, and these
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Texas Tech University, William Anderson, August 2007
Table 2.1: MATLES variables and levelingParameter Node Level (U – U, V, P nodes; W – Wnodes)u, v, P, θ, q U
w W∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y U
∂u/∂z,∂v/∂z W
∂w/∂x, ∂w/∂y W
∂w/∂z U∂P/∂x, ∂P/∂y U
∂P/∂z W∂θ/∂x, ∂θ/∂y, ∂q/∂x, ∂q/∂y U
∂θ/∂z,∂q/∂z W
τxx, τxy, τyy, τzz U
τxz, τyz W
πθx, π
θy, π
qx, π
qy U
πθz , π
qz W
∂τxj/∂xj, ∂τyj/∂xj
U
∂τzj/∂xjW
fields are filtered based on the user inputs for filter width (ML Flt).
2. These filtered fields are differentiated in the horizontal and vertical directions
(ML Derivxy; ML Derivz).
3. Wall stresses are computed for lower boundary conditions (ML Surfflux).
4. The convective terms in the horizontal and vertical directions are computed
(ML Convec).
5. The SG-S stress terms and spatial derivatives of these terms are computed.
6. The pressure field and subsequent horizontal and vertical derivatives of this
field are computed.
7. The components of the governing equations (i.e. convective, SG-S, pressure,
etc) are summed on the right-hand side.
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Texas Tech University, William Anderson, August 2007
8. Time-stepping is used to compute the momentum and scalar fields at the next
time-step.
9. Compute plane-averages of selected output fields to view output statistics.
10. Output and store simulation statistics.
Note that this process is for a Smagorinsky-based eddy-viscosity SG-S model.
However, in the following chapter, we discuss the algorithm structure for the
LDTKE model.
2.12 Summary
Chapter 2 has outlined the algorithm structure of the MATLES code. We have
shown how the N-S equations are manipulated, to accommodate the LES
methodology (i.e. separation of the SG- and R-S motions), and how we compute
individual components of the 3-dimensional N-S equations. We intentionally
excluded a detailed description of the SG-S computations in Chapter 2, this topic is
discussed in Chapter 3.
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Texas Tech University, William Anderson, August 2007
Table 2.2: MATLES subroutines and functionSubroutine Name Subroutine FunctionML Openfiles Initialize MATLES variables, and create corresponding
data filesML Flt Filter velocity field – resultant field is the R-S fieldML Derivxy Obtain horizontal derivatives of MATLES variables
(spectral differentiation)ML Surfflux Compute surface variables, based on imposed surface
fluxesML Derivz Differentiate MATLES variables in vertical direction
(finite differencing)ML Dealias1 Dealiasing of MATLES momentum and scalar vari-
ables, used in conjunction with ML Dealias2ML Dealias2 Dealiasing of MATLES momentum and scalar vari-
ables, used in conjunction with ML Dealias2ML DerivzX Second-order vertical differentiationML Convec Compute convective terms in x, y and z directionsML Sgs Compute SG-S stress tensor termsML ScalarRHS Components of the scalar solution equation are
grouped, allowing temporal advancement of scalarfield
ML Buoyancy Compute buoyancy forcing termML Divstress Compute derivative of stress tensors from ML SgsML Pressure Compute pressure term and subsequent spatial deriva-
tives for governing equationsML Stepuvwt Use the above (momentum) values to compute the ve-
locity field at the next time-stepML StepC Use the above (scalar) values to compute the velocity
field at the next time-stepML Average Compute plane-averages of various MATLES outputs
for viewingML Output Output and store simulation outputs
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Texas Tech University, William Anderson, August 2007
CHAPTER 3
LARGE-EDDY SIMULATION: SUBGRID-SCALE MODELING
3.1 Introduction
In Chapter 2 we briefly introduced the SG-S parameterization, and demonstrated
how the stress tensor (Equation 2.19) is a function of the eddy-viscosity (νt) and
strain-rate tensor (Sij). Parameterization of the eddy-viscosity has been the topic of
a considerable research effort, and is the topic of this work. In Chapter 3 we
introduce eddy-viscosity parameterizations for three well-known SG-S formulations,
namely the Smagorinsky [77], Wong and Lilly [86] and TKE based [20, 82] models.
Following this, derivation of the scale-dependent and -invariant dynamic
formulations (Porte-Agel et al. [67]) is discussed, with particular application to the
original [77] and [86] models. Finally, derivation of the LDTKE model is presented,
in addition to an amended algorithm flowchart (relevant to the LDTKE model).
3.1.1 Smagorinsky Model
The traditional C-SM model [77] computes the eddy-viscosity with Equation 3.1 as:
νt = (CsΔf )2∣∣S∣∣ , (3.1)
where
∣∣S∣∣ =(2SijSij
)1/2 , (3.2)
where Cs is the Smagorinsky coefficient; Δf is the filter width (often equal to the
grid width); and∣∣S∣∣ is the magnitude of the resolved strain-rate tensor. As
mentioned, the C-SM model is criticized for being over-dissipative of small-scale
energy. Dynamic generalizations of this model [8, 33, 67, 68], discussed in this
chapter, have led to dramatic improvements in its performance. Statistics from the
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Texas Tech University, William Anderson, August 2007
C-SM model are shown in Chapter 5.
3.1.2 Kolmogorov Scaling
Wong and Lilly [86], citing the shortcomings of the Smagorinsky approach,
developed an alternative eddy-viscosity model, based on Kolmogorov’s scaling laws,
νt = C2/3Δ4/3f ε1/3 = CεΔ
4/3f , (3.3)
where C (and Cε) are model coefficients (determined dynamically or prescribed);
and ε is the dissipation rate of energy.
3.1.3 TKE-Based Model
These models have been used extensively for various problems (mechanical
engineering and geophysical fluid dynamics), with varying success. In Moeng [55]
and later in Sullivan et al. [82], numerical coefficients from Deardorff [21, 22, 23, 24]
are employed for solution of the prognostic TKE equation (and computation of τij).
Equation 3.4 below shows the TKE-based eddy-viscosity expression:
νt = CkΔk1/2Δ , (3.4)
where Ck is a model coefficient (determined dynamically [20] or prescribed); Δ is a
length scale (often taken as the grid resolution); and kΔ is the TKE corresponding
with length scale, Δ. Later in this work, we show derivation of the LDTKE model,
in which Ck and C∗ (another coefficient required for solution of the prognostic TKE
equation) are point-by-point dynamically computed at every time-step in the
simulation.
3.2 Dynamic Model Formulation
We show statistical results for LASDD [8] versions of the Smagorinsky and Wong
and Lilly models. The LASDD model relies on the scale-dependent dynamic
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Texas Tech University, William Anderson, August 2007
formulation [67], the details of which are shown here for both the Smagorinsky and
Wong and Lilly models. We do not show statistics for the so-called scale-invariant
dynamic models, however derivation of the scale-dependent dynamic models relies
on derivation of the scale-invariant models. For a complete discussion on
scale-invariant SG-S models the reader is emphatically referred to Meneveau and
Katz [52] or Germano et al. [33].
3.2.1 Scale-Invariant Models
In conventional SG-S models (Equations 3.1 and 3.3), the velocity fields are filtered
based on the so-called “filter-to-grid ratio” (FGR),Δf
Δg, where Δf is the filter width
and Δg is the grid-spacing. Germano et al. [33] proposed using a second filtering
operation, at the “test-filter level” (TFL), αΔf , where α is a number greater than 1
(often taken as 2). In this approach, Cs and Cε are dynamically computed at all
locations in the domain. In using this second level of filtering, a new turbulent
stress tensor, Tij, is obtained at the Δf level as:
Tij = uiuj − ui uj, (3.5)
where the operation (...) indicates filtering at the TFL level. Equation 3.5 combines
with the definition, τij = uiuj − ui uj, to obtain the Germano identity:
Tij − τ ij = Lij = ui uj − ui uj. (3.6)
This relation allows dynamic computation of the model coefficient. At this juncture,
derivation of the LASDD model (for Smagorinsky and Wong and Lilly) become
different, and we subsequently consider the two derivations separately.
3.2.2 Scale-Invariant Dynamic Smagorinsky Model
With the dynamic procedure applied to Equation 3.6, we have:
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Texas Tech University, William Anderson, August 2007
Lij − 1
3Lkkδij =
(C2
s
)Δf
Mij, (3.7)
where
Mij = 2 Δ2f
(|S| Sij − α2
((C2
s )α Δf
(C2s )Δf
)|S| Sij
). (3.8)
In the original Germano et al. [33] formulation, they assume invariance between the
scales, such that:
(C2s )α Δf
(C2s )Δf
= 1, (3.9)
and with this assumption one can easily obtain the the coefficient using the error
minimization procedure of Lilly [48], as:
(C2
s
)Δf
=〈LijMij〉〈MijMij〉 . (3.10)
However, Porte-Agel et al. [67] were able to show that the assumption of invariance
between the scales (Equation 3.9) may not always be satisfied; note this was
explored with LESs of a N-ABL. In plotting (Cs)Δf, for varying values of z
Δf,
Porte-Agel et al. [67] were able to demonstrate a strong scale-dependence of the
coefficients. They pointed out that such a result is not entirely surprising,
considering that near the wall the grid scale approaches the integral scale, Li. They
also demonstrate that the need for generalization of the dynamic model [33] is most
pronounced in the near-wall region, where the traditional dynamic model is
under-dissipative. Field experiments from Kleissl et al. [43] support this finding.
3.2.3 Scale-Dependent Dynamic Smagorinsky Model
Generalization of the dynamic model [33] is achieved by use of a second test filtering
operation at scale α2 Δf . The Germano identity is again applied, although now at
the second test filtering operation, such that:
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Texas Tech University, William Anderson, August 2007
Qij − 1
3Qkkδij =
(C2
s
)Δf
Nij, (3.11)
where
Qij = ui uj − uiuj, (3.12)
and
Nij = 2 Δ2f
(|S| Sij − α4
((C2
s )α2 Δf
(C2s )α Δf
) |S| Sij
). (3.13)
The (...) indicates a second test filtering operation. Equation 3.13 leads to an
expression for the coefficient, as a function of values at the first and second test
filtering level, as:
(C2
s
)Δ f
=〈Qij Nij〉〈Nij Nij〉 . (3.14)
In Equation 3.9, scale invariance is assumed, however [67] relax this for
scale-dependence, with the introduction of Equation 3.15 (a much weaker
assumption):
β =(C2
s )α Δf
(C2s )Δf
=(C2
s )α2 Δf
(C2s )α Δf
. (3.15)
Now, with Equations 3.10 and 3.14, β is solved, subsequently allowing computation
of the coefficient. Solving for β generally relies on solution of a 5th-order polynomial
(with the exception of rare cases in which the polynomial solution yields complex or
unphysically large roots – in such cases β defaults to 1). We do not discuss solution
of the polynomial here.
In the LASDD formulation [8], more localized small-scale flow representation is
achieved by reducing the reliance on averaging operations. In the scale-dependent
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Texas Tech University, William Anderson, August 2007
dynamic formulation [67], Equations 3.10 and 3.14 rely on planar-averaging. To
account for localized patchy turbulence (inherent in S-ABL, the regime for which
the LASDD formulation was developed), LASDD averages in the horizontal plane
locally over a three-by-three stencil. Statistics from the LASDD-SM model are
shown in Chapter 5.
3.2.4 Scale-Invariant Dynamic Wong and Lilly Model
Derivation of the LASDD-WL model is remarkably similar to that of the
LASDD-SM model. Equations 3.5 and 3.6 are identical, with the dynamic
procedure (at the first test filter) giving:
Lij − 1
3Lkkδij = (CWL)Δf
Mij, (3.16)
where
Mij = 2 Δ4/3f
(1 − α4/3
(CWL)α Δf
(CWL)Δf
)Sij. (3.17)
(CWL)Δfand (CWL)αΔf
are the model coefficients computed at the filter width and
first test filtering level, respectively. Now, again, under the scale-invariance
assumption of Germano et al. [33], the following relation is imposed:
(CWL)α Δf
(CWL)Δf
= 1, (3.18)
and under this assumption the [33] formulation yields:
(CWL)Δf=
〈LijMij〉〈MijMij〉 . (3.19)
As with the Smagorinsky-base model, we are compelled to relax the assumption of
Equation 3.18, to allow for scale-dependence between the scales of motion.
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Texas Tech University, William Anderson, August 2007
3.2.5 Scale-Dependent Dynamic Wong and Lilly Model
The rationale for using a second test filter is equivalent for the LASDD-SM and
-WL models; see the previous section for discussion of this. With a second level of
test filtering, the Germano identity yields:
Qij − 1
3Qkkδij = (CWL)Δf
Nij, (3.20)
where Qij is shown in Equation 3.12, and
Nij = 2 Δ4/3f
(1 − α8/3
(CWL)α2 Δf
(CWL)Δf
)Sij. (3.21)
Resulting in:
(CWL)Δ f=
〈Qij Nij〉〈Nij Nij〉 . (3.22)
Following from [67], the scale-dependence assumption yields:
β =(CWL)α Δf
(CWL)Δf
=(CWL)α2 Δf
(CWL)α Δf
. (3.23)
Under the scale-invariance assumption, β is 1. However, in [67] this assumption is
relaxed to allow scale-dependence (which has been shown to exist). Solution of β for
LASDD-WL still requires solution of a 5th-order polynomial (the details of which
are shown in [4]). In Porte-Agel et al. [67], they use planar-averaging over the entire
horizontal cross-section; however in the LASDD methodology [8, 4], averages are
computed over localized three-by-three stencils. Statistics from LASDD-WL are
shown in Chapter 5.
3.3 LDTKE Formulation
LDTKE, like other TKE-based SG-S methodologies [55, 82], requires solution of the
prognostic TKE equation, given by:
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Texas Tech University, William Anderson, August 2007
∂kΔ
∂t= −C∗
Δk
1/2Δ +
∂
∂xj
(2ν
∂kΔ
∂xi
)− ui
∂kΔ
∂xj
+ 2ΔCkkΔ
(∂ui
∂xj
). (3.24)
Equation 3.24 relies on the two coefficients for production, Ck, and dissipation, C∗.
Sullivan et al. [82], in solving Equation 3.24, used numerical values proposed earlier
by Moeng and Wyngaard [56], with Ck = 0.1 and C∗ = 0.93. In earlier work by [55],
Moeng uses values proposed in [24], that C∗ = 0.19 +(0.51 l
Δs
)at all heights,
except at the lowest cross-section, where C∗ is 3.9; l and Δs are length scales. In
LDTKE, we use the dynamic approach of Davidson [20], such that these coefficients
are dynamically computed at every time-step and at every location in the
computational domain.
LDTKE relies on two levels of filtering, denoted by (...) and (...). The SG-S stress
tensor is computed with:
τij = −2CkΔk1/2Δ Sij, (3.25)
where the coefficient, Ck, is dynamically computed with:
Ck =LijMij
MijMij
, (3.26)
and Mij and Lij (the dynamic Leonard stresses) are computed with:
Mij = −1
2
(ΔK1/2 Sij − Δ
k1/2Δ Sij
), (3.27)
and
Lij = uiuj − uiuj. (3.28)
K is TKE at the second filtering level, Δ, and computed with Equation 3.29, as:
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Texas Tech University, William Anderson, August 2007
K = kΔ +1
2Lii. (3.29)
Equations 3.25 through 3.29 are more closely related to computation of the
production coefficient – the dissipation coefficient is computed with the following
time-dependent equation:
Cn+1∗
=
(PK − PkΔ
+1
Δ
Cn∗k
3/2Δ
)Δ
K3/2, (3.30)
where production at the first and second filter levels, PK and PΔ, are respectively
computed with Equations 3.31 and 3.32:
PK = −2CΔΔK1/2 Sij∂ ui
∂xj
, (3.31)
and
PΔ = −2CΔΔk1/2Δ Sij
∂ui
∂xj
. (3.32)
Although LDTKE is based on the 1-E [20] model, there are significant differences.
1. 1-E allows negative values of CΔ (i.e. TKE backscatter). In LDTKE we clip
negative CΔ values, to avoid numerical instabilities. We note that the Re’s
addressed with LDTKE are of O(108), while in [20] the Re is of O(5000).
2. In the momentum equations, [20] uses a domain-averaged CΔ, while we apply
no averaging to either coefficients in any of the computations – the coefficients
are always used for point-by-point computations.
Some additional features of LDTKE:
1. The dissipation coefficient, C∗, is clipped between 0 and 10. The large positive
limit is purely for numerical stability at early stages in the simulation, and
rarely (if ever) occurs after approximately 1500 time-steps.
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Texas Tech University, William Anderson, August 2007
2. The production coefficient, CΔ, is clipped only at 0 – there is no upper limit.
3. The TKE is clipped only at zero. This clipping is for numerical stability only,
and is activated extremely rarely (if ever) in early stages of the simulation.
The above restrictions are relatively loose, in comparison with other dynamic
models. Further, due to the use of only 2 filtering levels (compared with [67, 4, 8]),
computational time is expected to reduce.
3.3.1 LDTKE Algorithm Structure
We do not offer a flowchart diagram, such as that shown in Figure 2.4, although the
iterative steps which MATLES follows (using the LDTKE SG-S model) are outlined
below (we ask the reader to forgive repetition – aspects of the entire MATLES
algorithm structure vary due to use of the TKE-based closure, and we subsequently
provide the algorithm here for clarity):
1. The velocity, scalar and TKE fields are initialized.
2. The velocity and scalar fields are converted into spectral space, and these
fields are filtered based on the user inputs for filter width (ML Flt).
3. These filtered fields are differentiated in the horizontal and vertical directions
(ML Derivxy; ML Derivz).
4. Wall stresses are computed for lower boundary conditions (ML Surfflux).
5. The convective terms in the horizontal and vertical directions are computed
(ML Convec).
6. The strain-rate tensors (at both filter levels), filtered TKE, and CΔ are
computed.
7. The SG-S stress terms are computed using the production coefficients
(Equation 3.25).
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Texas Tech University, William Anderson, August 2007
8. Production at both filter levels (Equations 3.31 and 3.32) are computed,
allowing the dissipation coefficient to be dynamically computed (Equation
3.30).
9. The pressure field and subsequent horizontal and vertical derivatives of this
field are computed.
10. The components of the governing equation (i.e. convective, SG-S, pressure,
etc) are summed on the right-hand side.
11. Time-stepping is used to compute the momentum and scalar fields at the next
time-step.
12. The components of the prognostic TKE equation (Equation 3.25) are
assembled, and the TKE is temporally advanced forward.
13. Plane-averages of some output fields are computed for output statistics.
14. Simulation statistics are outputted and stored.
Table 3.1 shows the subroutines MATLES uses (while running the LDTKE SG-S
model).
3.4 Summary
Derivation of the LDTKE, and corresponding algorithm structure, has been
introduced. In addition, we have also shown derivations for the LASDD-WL and
-SM models. These models, in addition to the C-SM model (with various values for
the constant coefficient), are simultaneously used for LES of the N-ABL, the results
of which are used to discuss the strength of the LDTKE model. In Chapter 4 the
case-study is briefly outlined.
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Texas Tech University, William Anderson, August 2007
Table 3.1: MATLES subroutines and function (with LDTKE)Subroutine Name Subroutine FunctionML Openfiles Initialize MATLES variables, and create corresponding
data filesML Flt Filter velocity field – resultant field is the R-S fieldML Derivxy Obtain horizontal derivatives of MATLES variables
(spectral differentiation)ML Surfflux Compute surface variables, based on imposed surface
fluxesML Derivz Differentiate MATLES variables in vertical direction
(finite differences)ML Dealias1 Dealiasing of MATLES momentum and scalar vari-
ables, used in conjunction with ML Dealias2ML Dealias2 Dealiasing of MATLES momentum and scalar vari-
ables, used in conjunction with ML Dealias1ML DerivzX Second-order vertical differentiationML Convec Compute convective terms in x, y and z directionsML Ck DYN Compute strain-rate tensors and CΔ
ML SgsNCAR DYN Compute SG-S stress tensor terms using productioncoefficients from ML Ck DYN
ML StepC DYN Production at both filter levels are computed, allowingdynamic computation of the dissipation coefficient
ML ScalarRHS Components of the scalar solution equation aregrouped, allowing temporal advancement of scalarfield
ML Buoyancy Compute buoyancy forcing termML Divstress Compute derivative of SG-S stress tensorsML Pressure Compute pressure term and subsequent spatial deriva-
tives for governing equationsML EnrRHS DYN Components of the prognostic TKE equation are as-
sembled, with the values at the current time-stepgrouped
ML Stepuvwt Use the above (momentum) values to compute the ve-locity field at the next time-step
ML StepENR DYN Compute TKE at the next time-stepML Average Compute plane-averages of various MATLES outputs
for viewingML Output Output and store simulation outputs
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Texas Tech University, William Anderson, August 2007
CHAPTER 4
CASE STUDY AND SIMULATION DETAILS
4.1 Intercomparison Study
In this work, we perform large-eddy simulations of a turbulent Ekman layer (i.e.,
pure shear flow with a neutrally stratified environment in a rotating system)
utilizing the LASDD-SM [8, 9], LASDD-WL, C-SM and LDTKE SG-S models.
These simulations are identical in terms of initial conditions, forcing, and numerical
specifications (e.g., time integration, grid spacing). Technical details of our LES
code and the SG-S modeling approaches have been described in detail in Chapters 2
and 3, and will not be repeated here for brevity. The selected case study is
intentionally similar to that of the LES intercomparison study by Andren et al. [1].
4.2 Simulation Details
The simulated boundary layer is driven by an imposed geostrophic wind of
(Ug, Vg) = (10, 0) ms−1. The Coriolis parameter is equal to fc = 10−4 s−1,
corresponding to latitude 45◦ N. The computational domain size is: Lx = Ly = 4000
m and Lz = 1500 m. We consider three grid spacing configurations whereby the
domain is divided into Nx ×Ny ×Nz = 16× 16× 16, 40× 40× 40, and 64× 64× 64,
nodes (i.e., Δx = Δy = 250, 100, and 62.5 m, and Δz = 100, 38.5, and 23.8 m).
Table 4.1 summarizes the SG-S models and corresponding resolutions tested.
Table 4.1: SG-S models and grid resolutionsSG-S Model 16×16×16 40×40×40 64×64×64LASDD-SM Yes Yes YesLASDD-WL Yes Yes YesC-SM (0.17) No Yes NoC-SM (0.24) No Yes NoLDTKE Yes Yes Yes
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Texas Tech University, William Anderson, August 2007
We chose these resolutions for three inter-related reasons: (i) primarily, it allows us
to perform a direct comparison with the statistical results from [1] (for 40×40×40,
or 403, case), which used almost the same grid-resolutions; (ii) coarse grid-resolution
enables us to identify the strengths and/or weaknesses of the different SG-S models,
as well as, to underscore their impacts on LESs; and (iii) varying the resolution
allows sensitivity analyses. The simulations are run for a period of 10 × f−1c (i.e.,
100,000 s); the time step is varied depending on the resolution, although is typically
2 s. The last 3 × f−1c interval is used to compute statistics. The lower boundary
condition is based on the Monin-Obukhov similarity theory with a surface roughness
length of z◦ = 0.1 m.
We neglect 643 simulations (Table 4.1) for the C-SM models – for the purpose of
this work, 403 simulations are sufficient to demonstrate that a dynamical modeling
procedure (whether based on Smagorinsky, Kolmogorov scaling, or TKE) is
markedly more accurate for LES of the N-ABL.
45
Texas Tech University, William Anderson, August 2007
CHAPTER 5
INTERCOMPARISON OF SG-S MODELS
5.1 Overview
In Table 4.1, we show which SG-S models are tested and the resolutions at which we
run these LESs. In Chapter 5 we show a variety of statistics from these simulations,
all of which are familiar and well-known in turbulence literature (although the
statistics presented here have a particular resonance in atmospheric turbulence
studies). Statistical results presented here include: (a) temporal evolution of
simulated statistics; (b) first-order statistics of turbulent velocity fields; (c)
second-order statistics of turbulent fields; (d) energy spectra for the longitudinal
velocity field; and (e) characteristics of the SG-S (dynamically calculated)
coefficients. Elements of the aforementioned statistics are focused to the lower
spatial portions of the boundary layer (i.e. closer to the surface). It is in this region
that the flow energy is heavily distributed to small-scale motions and, subsequently,
where the importance of the SG-S model is most prominent. We run LESs with
grid-resolutions of 163, 403 and 643 (Table 4.1), with a particular emphasis on the
403 simulations. This approach is not uncommon ([1] used an intentionally coarse
grid-resolution to demonstrate the behavior of SGS models). Our comparative
statistics illustrate the performance of the LDTKE SG-S model.
5.2 Temporal Evolution
Figure 5.1 illustrates the temporal evolution of the friction velocity, u∗. It is evident
that the u∗ evolutions are generally quite similar, with average values during the
last 3 × f−1c interval approximately in the range of 0.435 – 0.460 m s−1. The
corresponding values found in [1] are: 0.425 m s−1 (Moeng), 0.448 m s−1 (Mason –
backscatter), 0.402 m s−1 (Mason – non-backscatter), 0.402 m s−1 (Nieuwstadt) and
0.425 m s−1 (Schumann). The nonstationarity parameters, Cu and Cv, computed
46
Texas Tech University, William Anderson, August 2007
with Equations 5.1 and 5.2, respectively:
Cu = − fc
uws
∫ Lz
0
(V − VG) dz, (5.1)
and
Cv =fc
vws
∫ Lz
0
(U − UG) dz, (5.2)
are not shown in this work, although we report that they agree with established
values. As we approach steady-state conditions the nonstationarity parameters
approach unity. Our simulation is close to – although does not satisfy conditions for
– steady-state; accordingly, an approximate phase agreement between Cu and Cv is
observed. The inertial oscillation period, 2πfc
, is evident.
In general we observe agreement between the statistics presented in Figure 5.1. We
note also agreement between statistics in the literature [4, 1, 8, 67] and those shown
in this figure.
5.3 First-Order Statistics
In this work we omit predictions of the normalized mean velocity, M =⟨√
u2 + v2⟩.
It is known that, in the surface layer, under neutrally buoyant conditions, this
profile follows a logarithmic profile. Statistics from LES with the C-SM (0.17 and
0.24), LASDD-SM, LASDD-WL and LDTKE SG-S models, for prediction of the
non-dimensional velocity gradient, φM , are shown here. The non-dimensional
velocity gradient (Equation 5.3) is widely considered a benchmark test for LES
models and the statistics presented here are encouraging. This is especially true in
the near-wall region, where the influence of the SG-S model receives maximum
exposure. Note that the non-dimensional scalar gradient, φC , is also a crucial LES
statistic, although not shown here because our simulations do not include scalar
transport. Figure 5.2 shows φM , as computed with:
47
Texas Tech University, William Anderson, August 2007
0 2 4 6 8 100.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
tfc
u * (m/s
)
163
403
643
0 2 4 6 8 100.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
tfc
u * (m/s
)
163
403
643
0 2 4 6 8 100.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
tfc
u * (m/s
)
163
403
643
(d)(c)
(b)0 2 4 6 8 10
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
tfc
u * (m/s
)
C
s = .17
Cs = .24
(a)
Figure 5.1: Temporal evolution of friction velocity for: (a) C-SM (403); (b) LDTKE;(c) LASDD-SM; and (d) LASDD-WL
φM =κz
u∗
√(∂U
∂z
)2
+
(∂V
∂z
)2
. (5.3)
It is well-known that the traditional Smagorinsky model is over-dissipative in the
near-surface region and gives rise to excessive mean gradients in velocity and scalar
fields [1, 67]. In the framework of the Monin-Obukhov similarity theory, the
non-dimensional velocity gradient is indisputably equal to one (vertical dotted line
in Figure 5.2).
In addition to Figure 5.2, we show Figure 5.3, which is a summary of the φM
profiles from the intercomparison study [1]. This figure demonstrates large φM
48
Texas Tech University, William Anderson, August 2007
0 0.5 1 1.5 20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Phim
zf/u
*
C
s = .17
Cs = 0.24
0 0.5 1 1.5 20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Phim
zf/u
*
163
403
643
0 0.5 1 1.5 20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Phim
zf/u
*
163
403
643
0 0.5 1 1.5 20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Phim
zf/u
*
163
403
643
(b)
(c) (d)
(a)
Figure 5.2: Non-dimensional velocity gradient for: (a) C-SM (403); (b) LDTKE; (c)LASDD-SM; and (d) LASDD-WL
gradients in the surface layer (indicating excessive energy dissipation, [67]), and
demonstrates the strengths of the SG-S models presented in this work, including the
simplistic C-SM models. We notice that for the C-SM statistics (Figure 5.2, a), the
largest φM value occurs for the case when Cs = 0.24, which indicates that the larger
Smagorinsky coefficient is dissipating more energy.
Figure 5.2 demonstrates that the C-SM model performs well compared to the other
more sophisticated (and computationally expensive) SG-S models. Although φM is
critical only in the surface layer, it is nevertheless interesting to see that this value is
closer to one above the surface layer (zfu∗
≈ 0.05) for the C-SM models than the
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Texas Tech University, William Anderson, August 2007
Figure 5.3: Non-dimensional velocity gradient from the Andren et al. (1994) inter-comparison study
other models. As will be shown in subsequent figures, the SG-S parameterization is
most active in locations where the flow energy is distributed strongly to small-scale
motions (i.e. in the surface layer).
The sensitivity of the LDTKE, LASDD-SM and LASDD-WL models to
grid-resolution is demonstrated in Figure 5.2 (b) – (d); for the 403 and 643
simulations the φM profiles agree closely, especially in the near-wall region,
indicating that these LES models are relatively insensitive to resolution changes
between 403 and 643, although for 163 the φM profile is more unstable and
fluctuates notably between (model) levels, indicating poor resolution of flow physics
at intermediate model levels. It follows that for progressive resolution increases (to
the extent that all scales of motion – to the Kolmogorov scale – are resolved), the
φM profile would approach one, although such resolution requires computational
power vastly exceeding that which is currently available.
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Texas Tech University, William Anderson, August 2007
5.4 Second-Order Statistics
Figures 5.4 through 5.7 show the normalized vertical flux of x- and y-component
momentum for the C-SM, LDTKE, LASDD-SM and LASDD-WL models,
respectively. In these plots, we present the total and SG-S contributions to these
quantities. These plots confirm the importance of the SGS closure method in the
near-wall region: the total and SGS momentum values are almost equivalent at the
wall; the SGS contribution dramatically decreases with height. Note that Figures
5.4 through 5.7 agree with the literature [45, 1].
In neutrally stratified boundary layer flows, the peak normalized velocity variances,
σu, σv and σw, are of magnitude: ≈ 0.205 – 0.287, ≈ 0.123 – 0.164 and ≈ 0.041 –
0.082, respectively [36, 57]; statistics presented in Figures 5.8 through 5.11 agree
with these values.
Figures 5.4 and particularly 5.8 illustrate that, as we increase the Smagorinsky
coefficient, Cs, numerical values of the flux and variance reduce (especially in the
transverse and vertical directions). This is attributed to smoothing or damping of
the turbulence, which translates to reduction of the variance of the velocity fields.
The grid-resolution sensitivity is again shown in Figures 5.5 – 5.7 and 5.9 – 5.11; it
is again evident that the models are relatively insensitive to grid-resolution
variations between 403 and 643, although differences are observed when the
resolution is reduced to 163.
5.5 SG-S Dynamics
Coefficient profiles from simulations with the LASDD-SM (Cs, Equation 3.14) and
LASDD-WL (CWL, Equation 3.22) models are well-known in the literature [4], and
we neglect to report them in this work. Coefficient profiles from LDTKE are shown
in Figure 5.12.
We present plots (Figure 5.12, c) for the Smagorinsky coefficient, Cs; this is
51
Texas Tech University, William Anderson, August 2007
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
(a) (b)
(c) (d)
Figure 5.4: C-SM (403) Simulated vertical fluxes of: (a) x-component momentum, Cs= 0.17; (b) x-component momentum, Cs = 0.24; (c) y-component momentum, Cs =0.17; and (d) y-component momentum, Cs = 0.24
computed using Equation 5.4 (or Equation 14 in Sullivan et al. [82]) as:
Cs =
(Ck
(Ck
C∗
))( 1
2). (5.4)
The Cs profile shows values larger than those observed in similar works [67, 8],
indicating that perhaps the LDTKE model is over-dissipative of SG-S energy. Here,
we observe peak coefficients no larger than ≈ 0.17 (163) , 0.23 (403) and 0.24 (643)
(we do, however, note similarity of the profile shapes). Interestingly, it is for
lower-resolution simulations (i.e. 163) that the coefficient becomes closer to the
previously obtained statistics in [67]. The Ck and C∗ profiles are shown also. In
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Texas Tech University, William Anderson, August 2007
Sullivan et al. [82], they use constant values for Ck and C∗ of 0.1 and 0.93,
respectively. Our values do not agree with those, although considering Figure 5.12
(a), it seems we could confidently prescribe a constant value for the dissipation
coefficient, C∗, of ≈ 2.5. Considering that our model dynamically responds to the
flow physics, one could assume that simplistic prescription of a constant production
coefficient does not allow particularly accurate computations.
5.6 Turbulent Kinetic Energy (TKE)
TKE profiles from the LDTKE simulations are shown in Figure 5.13. These profiles
agree with those shown in Sullivan et al. [82], although our result shows a slight rise
in the TKE profile at the top of the domain. When running the LESs, we choose
the vertical height such that the ABL is completely included in the simulations; and
this is achieved. However, above the ABL height there will still be small quantities
of TKE, due to small amounts of unresolved shear in the velocity field, which were
not included in the numerical domain (i.e. the LES). Therefore, the effect of this
unresolved TKE will manifest in the statistics as a small peak in the TKE profile at
the top of the boundary layer (such peaks are also observed in variance plots).
Grid-resolution effects are illustrated in Figure 5.13, and we again observe relative
insensitivity of the LDTKE model between 403 and 643, whilst for 163 the LES
indicates much larger TKE. It is interesting here to recall the Cs profile for 163,
shown in Figure 5.12 (c), in which the coefficient is smallest for the 163 simulation.
As we have seen, the amount of dissipated energy increases with Cs, and so the large
TKE (Figure 5.13) and smaller Cs coefficient from 163 LESs are strongly related.
5.7 Visualizations and Energy Spectra
In this section we show flow visualizations and energy spectra for the C-SM (Cs =
0.17 and 0.24), LDTKE, LASDD-SM and LASDD-WL models. These statistics are
shown at z = 0.1zi and z = 0.5zi, which allows for better understanding of the ABL
53
Texas Tech University, William Anderson, August 2007
structure, and also to see how the SG-S model influences the statistics (at z = 0.1zi
the LES is strongly reliant on the SG-S model, due to the presence of an excessive
number of scales of motion; while at z = 0.5zi the number of small-scale motions is
less, and the SG-S model should subsequently have a lesser influence in the LES).
Figures 5.14 through 5.16 show the flow visualization and energy spectra plots. The
visualizations are from the final time-step in the simulation; the spectra are based
on the velocity fields (temporally-averaged over the last 3 × f−1c interval). In Figure
5.14 we see the C-SM visualizations and spectra for Cs ≡ 0.17 and 0.24. It is known
that the traditional Smagorinsky model is over-dissipative of small-scale energy. In
the near-wall region (z = 0.1zi), the visualizations clearly illustrate (qualitatively)
that, although the LES has resolved large-scale streaks (streaky structures are
discussed in Chapter 6), there is a definite lack of small-scale turbulent activity.
Furthermore, as we increase Cs (i.e. increase small-scale energy dissipation), this is
more evident.
Quantitatively, we consider the energy spectra for understanding of the
energy-dissipation characteristics of the SG-S models. Over-dissipative SG-S models
are identified by spectral slopes steeper than Kolmogorov’s spectral scaling laws
(approximately k−1 and k−5
3 in the production and inertial ranges, respectively),
particularly in regions of the flow dominated by small-scale isotropic turbulence (i.e.
z = 0.1zi). For the C-SM models, at both heights, we observe large deviation
between Kolmogorov’s inertial range spectrum and energy spectra. Further, as Cs
increases, the amount of small-scale turbulence reduces (visualizations), and the
spectral slope steepens.
Figure 5.15 shows visualizations for the more sophisticated LDTKE SG-S model, in
which a larger number of small-scale coherent structures at z = 0.1zi are observed,
indicating that the SG-S model is better-representing the flow physics. The spectra
at this height agree with this inference, and we observe both the production range
54
Texas Tech University, William Anderson, August 2007
and extended inertial range spectra. At z = 0.5zi, however, we observe large-scale
coherent longitudinal structures and the spectral slope is subsequently steeper than
that of the scaling law. It is slightly dissapointing to observe that LDTKE mainly
predicts these large-scale coherent longitudinal structures at z = 0.5zi; in
comparison, LASDD-SM shows fewer and weaker large-scale coherent structures at
this height. The LDTKE, LASDD-SM and LASDD-WL models produce similar
spectra at z = 0.1zi, which is satisfying.
It is important to note that the visualizations seen in Figures 5.14 through 5.16 are
at single time-steps (the last time-step) only, and so do not represent the flow field
at all time throughout the simulation. Although it is interesting to observe the
general agreement between the visualizations and spectra. Note also that we show
spectra and visualizations for the 403 simulations only – statistics from the 643
simulation would not be expected to offer any deeper understanding of the
statistics; as we saw in previous statistics, the models are relatively insensitive to
resolution changes between 403 and 643.
5.8 Conclusion
In Chapter 5 we have shown important statistics from simulations of the N-ABL
with the C-SM, LDTKE, LASDD-SM and LASDD-WL models. The
non-dimensional velocity gradient, φM , is indisputably equal to one in the near-wall
region, and we have seen how the models tested here approach this condition (with
the LDTKE SG-S model performing well). In addition, we have seen how all the
models presented here (including the C-SM model) perform relative to those
considered in the Andren et al. [1] intercomparison study. For the C-SM model, we
have varied the Smagorinsky coefficient, and seen (both qualitatively and
quantitatively) how the value of this coefficient affects dissipation of small-scale
energy; when the coefficient is larger, we observe a reduction of velocity variances,
smoothed velocity fields, and steeper spectral slope, all of which indicate
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Texas Tech University, William Anderson, August 2007
over-dissipative SG-S computations.
The influence of grid-resolution variations is illustrated in the statistics, for 163, 403
and 643 simulations with the LDTKE, LASDD-SM and LASDD-WL models.
Statistics from these models for the 403 and 643 simulations are quite similar,
however the model statistics are consistently varied for the 163 grid-resolution.
The LDTKE model is shown to perform well, relative to the sophisticated
LASDD-SM and LASDD-WL models; this is especially true in the near-wall region.
It is, however, disappointing to observe the models over-dissipative characteristics of
LDTKE at z = 0.5zi. At this height, the model predominantly resolves large-scale
coherent structures, as opposed to the LASDD-SM model which predicts a more
isotropic (in the horizontal plane) velocity field. As a result, the velocity field
contains more production-range than inertial-range scales of motion. Interestingly,
we refer to Figure 5.12 (c), and see that Cs is much larger higher in the domain
than has been found in Porte-Agel et al. (2000), for example; in the near-wall region
the difference between the coefficients is not quite so great (although LDTKE does
predict larger coefficients).
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Texas Tech University, William Anderson, August 2007
−0.3 −0.2 −0.1 0 0.1 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.3 −0.2 −0.1 0 0.1 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
(a) (b)
(d)(c)
(e) (f )
Figure 5.5: LDTKE Simulated vertical fluxes of x-component momentum at: (a) 163,(c) 403 and (e) 643; and y-component momentum at: (b) 163, (d) 403 and (f) 643
57
Texas Tech University, William Anderson, August 2007
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.10
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.10
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
(a) (b)
(c) (d)
(e) (f )
Figure 5.6: LASDD-SM Simulated vertical fluxes of x-component momentum at: (a)163, (c) 403 and (e) 643; and y-component momentum at: (b) 163, (d) 403 and (f) 643
58
Texas Tech University, William Anderson, August 2007
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.10
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (v)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
1.2
1.4
Mom. Flux (u)
z/H
ResolvedSG−STotal
(a) (b)
(c) (d)
(e) (f )
Figure 5.7: LASDD-WL Simulated vertical fluxes of x-component momentum at: (a)163, (c) 403 and (e) 643; and y-component momentum at: (b) 163, (d) 403 and (f) 643
59
Texas Tech University, William Anderson, August 2007
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
(a) (b)
Figure 5.8: C-SM (403) simulated velocity variances for: (a) Cs = 0.17 and (b)Cs = 0.24
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
(a) (b)
(c)
Figure 5.9: LDTKE simulated velocity variances for: (a) 163, (b) 403 and (c) 643
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0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
(a) (b)
(c)
Figure 5.10: LASDD-SM simulated velocity variances for: (a) 163, (b) 403 and (c)643
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0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
1.2
1.4
Variance(u,v,w)
z/H
σ
u
σv
σw
(a) (b)
(c)
Figure 5.11: LASDD-WL simulated velocity variances for: (a) 163, (b) 403 and (c)643
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0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
Cs
z/H
163
403
643
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
C*
z/H
163
403
643
0 0.05 0.1 0.15 0.2 0.250
0.2
0.4
0.6
0.8
1
1.2
1.4
Ck
z/H
163
403
643
(a) (b)
(c)
Figure 5.12: Temporal-averaged profiles for: (a) C∗, (b) Ck and (c) Cs
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
1.2
1.4
TKE (kS G S
)
z/H
163
403
643
Figure 5.13: Turbulent Kinetic Energy (TKE) from LDTKE simulations
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Texas Tech University, William Anderson, August 2007
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
8
8.5
9
9.5
10
10.5
11
11.5
12
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
8
8.5
9
9.5
10
10.5
11
11.5
12
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
6
6.5
7
7.5
8
8.5
9
9.5
10
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
6
6.5
7
7.5
8
8.5
9
9.5
10
(a) (b)
(c) (d)
(e) (f )
Figure 5.14: C-SM predictions of longitudinal velocity fields (Cs = 0.17, top) and(Cs = 0.24, middle) and spectra (bottom) at: z = 0.1zi (left) and z = 0.5zi (right)
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Texas Tech University, William Anderson, August 2007
10−3
10−2
10−1
10−4
10−3
10−2
10−1
100
k1
Eu
(k1
)
LDTKE −5/3
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
8
8.5
9
9.5
10
10.5
11
11.5
12
10−3
10−2
10−1
10−4
10−3
10−2
10−1
100
k1
Eu
(k1
)
LDTKE −5/3 −1
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
6
6.5
7
7.5
8
8.5
9
9.5
10
(a) (b)
(c) (d)
Figure 5.15: LDTKE simulations of longitudinal velocity fields (top) and spectra(bottom) at: z = 0.1zi (left) and z = 0.5zi (right)
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Texas Tech University, William Anderson, August 2007
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
6
6.5
7
7.5
8
8.5
9
9.5
10
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
8
8.5
9
9.5
10
10.5
11
11.5
12
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
8
8.5
9
9.5
10
10.5
11
11.5
12
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
40
6
6.5
7
7.5
8
8.5
9
9.5
10
(a) (b)
(c) (d)
(e) (f )
Figure 5.16: LASDD predictions of longitudinal velocity fields (LASDD-SM, top) and(LASDD-WL, middle) and spectra (bottom) at: z = 0.1zi (left) and z = 0.5zi (right)
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CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 Summary of Completed Work
In Chapter 5 we show results from the C-SM simulations, in which Cs = 0.17 and
0.24. In both cases, it is demonstrated that these models are over-dissipative of
small-scale energy. Dynamic SG-S models were first introduced in Germano et al.
[33], and are now relatively common in LES-related turbulence research. In this
work, we presented a fully dynamic SG-S model based on solution of the prognostic
TKE equation (and in which all computations are point-by-point), namely the
LDTKE model. Results from simulations with this model, in addition to those with
the C-SM, LASDD-SM and LASDD-WL [4], are compared in order to gauge the
strength of the LDTKE model relative to other state-of-the-art SG-S models.
Results from LDTKE are generally encouraging; the non-dimensional velocity
gradient is evidently close to one in the surface-layer (Figure 5.2). The velocity field
variances and fluxes are comparable to those published in the literature. In the
near-wall region (z = 0.1zi), the energy spectra are comparable to those predicted
with LASDD-SM, and in close agreement with the inertial- and production-range
scaling laws. Unfortunately, the same is not true higher in the ABL (z = 0.5zi): the
LDTKE model is over-dissipative of small-scale energy.
Development of the LDTKE model, to its current form, has been completed only
recently. Preliminary results from the model are encouraging, although
improvements are required.
6.2 Future Perspectives
The LDTKE model was adopted from the 1-E model presented by Davidson [20],
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which was developed for mechanical engineering flows (with Reynolds numbers very
much less than those encountered in the N-ABL). We generalized the 1-E to be
completely dynamic with point-by-point computations. In the 1-E model [20], the
production coefficient, Ck, is allowed to be both negative and positive, for energy
transfer by backscatter and forwardscatter, respectively. The very existence of
energy backscatter (i.e. negative eddy-viscosity) remains a controversial topic in
contemporary fluid mechanics. TKE-based SG-S models have been used for ABL
turbulence studies [82, 55], although to our knowledge LDTKE is the most general
of these TKE-based SG-S models. Our attempts to allow energy backscatter in
LDTKE consistently failed due to numerical instabilities. In future work we plan to
review backscatter capability for the LDTKE model, perhaps by using artificial
clipping of (negative) Ck values. In any case, it is anticipated energy backscatter in
the LDTKE model will be deterministic, as opposed to the stochastic backscatter
approach of Mason and Thomson [50].
In this work LDTKE has been applied only to the relatively simple N-ABL. In
subsequent works it is planned that LDTKE will be applied to the S-ABL [8, 9].
This atmospheric regime is extremely challenging to simulate, owing to the negative
buoyancy which suppresses the turbulent activity (especially its vertical
component). Because LDTKE is based on TKE (i.e. momentum transported by the
subgrid-scales), this model may possess greater stability than other models.
6.3 Longitudinal Coherent Structures
Explanation for the streaky structure phenomena in ABL flows have been offered
[26, 27, 30] – they conclude, broadly, that: “streaks are a three-dimensional coherent
organization of the mean flow that transports smaller-scale turbulent eddies”.
Streaks are transient in nature, following a (re)generation, growth, and decay
life-cycle, with one cycle generally taking O(10 minutes); in addition, streaks are
characterized by a sweep (vertical downward) and ejection (vertical upward motion)
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[30]. It follows that sweeps and ejections correspond with the growth and decay
phases, respectively. The above life-cycle is merely an approximation – parameters
such as surface roughness and flow velocity may influence the streaky structure
characteristics.
We should note that, especially in the ABL, observation of streaky structures is
very much limited to numerical results, due to the complexity of experimental
observation of streaky structures. As Drobinski et al. [27] say, on the topic of
limited experimental data for ABL streaks: “while streaks are well known in LES
modeling, their occurrence and effects in the near-surface region of the PBL
(planetary boundary layer) have not received much attention”, and that: “we suspect
that the reasons streaks have escaped significant study is that they are difficult to
observe using conventional instrumentation”. In [34], statistical results for the
spatial distribution of streaky structures are offered, although such results are based
on observational (numerical) analyses from [70, 78], rather than rigorous
mathematical investigation.
Recent research suggests that the production range is (likely) related to elongated
streaky velocity structures (Figures 5.14 – 5.16), and from this perspective the
tested SG-S models could be considered near-successful. A few previous LES studies
have reported the existence of elongated streaky structures in the neutral surface
layers [45, 82, 57, 25, 17]. Evidences of these structures in various laboratory
experiments are undeniable (for example, see Hutchins and Marusic [39] and the
references therein). The link between experimentally observed long production
range (k−1 scaling) in the streamwise spectra of the longitudinal velocity and the
elongated streaky structures has recently been discussed in depth by Carlotti [17].
Moreover, strong correlations between these streaky structures and large negative
momentum flux were earlier reported by [57]. From Figures 5.14-top – 5.16-top, it is
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Texas Tech University, William Anderson, August 2007
clear that all the SG-S models in the present study show streaky structures, roughly
parallel to the mean wind direction, in the surface layer (at z = 0.1zi). However,
significant morphological differences are noticeable. For example, the C-SM models
(Cs ≡ 0.17 and 0.24) produce very long streaky structures and inadequate
small-scale structures (Figure 5.14). This can be directly associated with the
over-dissipative nature of the C-SM SG-S models, as previously discussed. In other
words, the existence of morphological characteristics in N-ABL flows are strongly
dependent on the choice of SG-S parameterizations, especially for coarse-resolutions
simulations. A few previous studies somewhat support this inference. For instance,
the nonlinear SG-S model [45], and the modified Smagorinsky SG-S model [25]
barely produced any elongated streaky structures. Thus, that the LDTKE model
has resolved streaky structures in the near-wall region should add to its appeal and
reliability for LES of the N-ABL.
6.4 Higher-Resolution Simulations
In this work LDTKE was applied only to solution of the momentum fields. In future
works it will be necessary to generalize the LDTKE model to include scalars, in
particular water vapor, q, and potential temperature, θ. We reiterate that
development of LDTKE has only recently been completed, however if this model is
to be considered favorably by the research community, scalar transport must be
added to its capabilities (such an addition has practical benefits). In Chapter 2 we
presented the basic equations for scalar transport.
In this work the sensitivity of the LDTKE, LASDD-SM and LASDD-WL models
was tested, for grid-resolutions of 163, 403 and 643. We found stronger agreement
between the 403 and 643, than between the 163 and 403 results. In future works we
may run LESs at simulations of 803 and 1283. It is of course a trivial outcome that
increasing the grid-resolution will enhance the LES results. As was shown in the
momentum flux results (Figures 5.4 – 5.7), the contribution of the SG-S model is
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Texas Tech University, William Anderson, August 2007
most pronounced in the surface layer region, where the majority of the flow energy
is distributed to the small-scale motions (Figures 5.8 – 5.11); moreover, the SG-S
model will be less active as the grid-resolution is increased (due to higher explicit
resolution of flow physics).
Thus, as Andren et al. [1] note, for SG-S modeling research it is desirable to expose
the deficiencies and strengths of the SG-S model, and this is best achieved with
coarse grid-resolution simulations.
6.5 Conclusion
The SG-S modeling approach significantly influences LES results, especially in the
surface layer region and for coarse-resolution simulations. Development of a robust
and universal SG-S model for LES of turbulent flows has not been completed. A
TKE-based SGS model, LDTKE, has been developed for simulations of the N-ABL.
We explained the numerical details of the LES model, and subsequently show
derivation of the LDTKE model, and required numerical constraints. The model is
compared with the traditional Smagorinsky model, and with other state-of-the-art
SG-S models. LDTKE performs well in the surface layer region, although we
identify over-dissipative characteristics higher in the boundary layer.
In future works, we will revisit the LDTKE model numerics, to locate the
over-dissipative characteristic (which perhaps is related to our omission of the
backscatter capability). In addition the model will be generalized for scalar
transport, and will again be used for LES of the N-ABL.
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