A Localized Dynamic Model for Large-Eddy Simulation of the ...

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

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

vii


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

viii


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),

1


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

2


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

3


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:

4


(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

5


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

6


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

7


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.

8


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

9


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.

10


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.

11


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,

12


∂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

13


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)

14


∂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:

15


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


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

16


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

17


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

18


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-

19


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

20


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

21


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

22


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

23


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

24


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

25


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


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


26


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

27


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

28


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.

29


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.

30


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

31


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

32


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

33


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:

34


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:

35


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

36


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.

37


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:

38


∂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:

39


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.

40


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).

41


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.

42


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

43


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

44


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


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


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


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


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

49


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.

50


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


−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

52


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


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


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

55


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).

56


−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


−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


−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


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

60


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

61


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

62


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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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)

65


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)

66


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)

67


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],

68


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)

69


[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

70


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

71


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.

72


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