CFD boundary conditions, turbulence models and dispersion study for flows around obstacles

118 Pages·2017·16.38 MB·English

CFD boundary conditions, turbulence models and
dispersion study for flows around obstacles
Alessandro Parente, Riccardo Longo, Marco Ferrarotti
Universit´e Libre de Bruxelles, Belgium
Politecnico di Milano, Italy
March 26, 2017
Contents
1 Introduction 7
2 ABL turbulence modelling 10
2.1 Inlet conditions and turbulence model . . . . . . . . . . . . . . . . . . . . . 11
2.2 Wall treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Generalization of the ABL model . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 BIA based on local turbulent properties deviation . . . . . . . . . . 20
2.3.2 BIA based on local deviation from a parallel shear flow . . . . . . . 20
2.3.3 Transition formulations . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.4 Inside the BIA: Non-Linear Eddy-Viscosity models . . . . . . . . . 22
2.4 Flow-field and turbulence applications . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Empty fetch at wind-tunnel and full scale . . . . . . . . . . . . . . 25
2.4.2 Flow around a ground-mounted building . . . . . . . . . . . . . . . 31
2.4.3 Flow around an array of obstacles . . . . . . . . . . . . . . . . . . 54
2.4.4 Flow over complex terrains, wind-tunnel and full-scale hills . . . . 61
3 Atmospheric dispersion modelling 68
3.1 Dispersion modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Gaussian models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.2 Integral models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1.3 CFD models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Dispersion Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 Dispersion in the wake of a ground-mounted building . . . . . . . . 85
4 Influence of stability classes 94
4.1 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2 Turbulence in homogeneous ABL . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Inlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
VKI - 1 -
CONTENTS CONTENTS
4.5 Validation Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.5.1 Simplified 3D hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6.1 3D computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6.2 Undisturbed ABL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6.3 Validation over simplified hill . . . . . . . . . . . . . . . . . . . . . 104
The present lecture notes are based on the research work performed by several Authors in the field
of ABL flows at the von Karman Institute of Fluid Dynamics, at the Universit´e Libre de Bruxelles and
at Politecnico di Milano. In particular, the contribution by Prof. Carlo Benocci, Prof. Catherine Gorl´e,
Prof. Jeroen van Beeck, Prof. Marco Derudi, Dr. Miklos Balogh and Dr. Clara Garcia Sanchez should
be acknowledged.
VKI - 2 -
LIST OF FIGURES LIST OF FIGURES
List of Figures
1 Computational domain with building models in ABL flows . . . . . . . . . 15
2 Law of the wall for smooth and sand-grain roughened surfaces . . . . . . . 15
3 Turbulentkineticenergyprofilesattheinletandoutletsectionsofanempty
computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Rough law of the wall implementation . . . . . . . . . . . . . . . . . . . . 19
5 Configurations PS1 and PS2 for the definition of the building influence area 19
6 Variation of the turbulence model parameter C for the single building test
µ
case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Computational domain of the unperturbed ABL at wind tunnel and full scale 25
8 Profiles of velocity, turbulent kinetic energy and turbulent dissipation rate
at inlet and outlet section of the computational domain . . . . . . . . . . . 27
9 Wall shear stress as a function of the axial coordinate . . . . . . . . . . . . 28
10 Profiles of velocity, turbulent kinetic energy, turbulent dissipation rate and
non-dimensional velocity gradient at inlet and outlet section of the compu-
tational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11 Profiles of velocity, turbulent kinetic energy and turbulent dissipation rate
at inlet and outlet section of the computational domain . . . . . . . . . . . 30
12 Wall shear stress as a function of the axial coordinate . . . . . . . . . . . . 31
13 Cedval A1-1 3D view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
14 Computational grid for A1-1 . . . . . . . . . . . . . . . . . . . . . . . . . . 32
15 Contour plots of non-dimensional velocity for PS1/PS2 configurations . . . 35
16 Contour plots of non-dimensional turbulent kinetic energy for PS1/PS2
configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
17 Experimental and numerical profiles of non-dimensional velocity for PS1
and PS2 configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
18 Experimental and numerical profiles of non-dimensional turbulent kinetic
energy for PS1 and PS2 configurations . . . . . . . . . . . . . . . . . . . . 36
19 Experimental and numerical profiles of non-dimensional velocity for PS,
UABL and ASQ configurations . . . . . . . . . . . . . . . . . . . . . . . . 37
20 Experimental and numerical profiles of non-dimensional turbulent kinetic
energy for PS, UABL and ASQ configurations . . . . . . . . . . . . . . . . 38
21 Localhitratesfornon-dimensionalturbulentkineticenergyapplyingASQ/UABL
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
22 Comparison of experimental and numerical predictions of non-dimensional
velocity using the standard k −(cid:15) model and Craft closure for the wake in
sinusoidal blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
23 Comparison of experimental and numerical predictions of non-dimensional
turbulent kinetic energy using the standard k −(cid:15) model and Craft closure
for the wake in sinusoidal blending . . . . . . . . . . . . . . . . . . . . . . 42
24 Comparison of experimental and numerical predictions of non-dimensional
velocity using Craft, Lien and Ehrhard NLEV models for the wake in sinu-
soidal blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
VKI - 3 -
LIST OF FIGURES LIST OF FIGURES
25 Comparison of experimental and numerical predictions of non-dimensional
turbulent kinetic energy using Craft, Lien and Ehrhard NLEV models for
the wake in sinusoidal blending . . . . . . . . . . . . . . . . . . . . . . . . 44
26 Comparison of experimental and numerical predictions of non-dimensional
velocity using Ehrhard in sinusoidal pure blending and Ehrhard in polyno-
mial hybrid blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
27 Comparison of experimental and numerical predictions of non-dimensional
turbulent kinetic energy using Ehrhard in sinusoidal pure blending and
Ehrhard in polynomial hybrid blending . . . . . . . . . . . . . . . . . . . . 46
28 Comparison of experimental and numerical predictions of non-dimensional
velocity and turbulent kinetic energy using three different marker defini-
tions and Ehrhard for the wake . . . . . . . . . . . . . . . . . . . . . . . . 47
29 Comparison of experimental and numerical predictions of non-dimensional
velocity and turbulent kinetic energy using both Hybrid configuration and
marker v.3 with Ehrhard for the wake against the standard k −(cid:15) . . . . . 48
30 Cedval A1-1 contour plots of velocity magnitude . . . . . . . . . . . . . . . 50
31 Cedval A1-1 contour plots of turbulent kinetic energy . . . . . . . . . . . . 51
32 Statistical evaluation of the models’ performance using MG and VG . . . . 52
33 Comparison of experimental and numerical predictions of non-dimensional
velocity and turbulent kinetic energy using both Hybrid configuration and
marker v.3 with non-zero and zero NLEV coefficients . . . . . . . . . . . . 53
34 Cedval B1-1 schematic view . . . . . . . . . . . . . . . . . . . . . . . . . . 55
35 Cedval B1-1 3D view and location of the measurement lines . . . . . . . . 55
36 Comparison of experimental and numerical predictions of non-dimensional
velocity for the B1-1 test case . . . . . . . . . . . . . . . . . . . . . . . . . 57
37 Comparison of experimental and numerical predictions of non-dimensional
turbulent kinetic energy for the B1-1 test case . . . . . . . . . . . . . . . . 57
38 B1-1 horizontal contours of x-velocity . . . . . . . . . . . . . . . . . . . . . 59
39 B1-1 vertical contours of turbulent kinetic energy . . . . . . . . . . . . . . 59
40 Comparison of experimental and numerical predictions of non-dimensional
velocity for the B1-1 test case using both Hybrid configuration and marker
v.3 with non-zero and zero NLEV coefficients . . . . . . . . . . . . . . . . 60
41 Comparison of experimental and numerical predictions of non-dimensional
turbulent kinetic energy for the B1-1 test case using both Hybrid configu-
ration and marker v.3 with non-zero and zero NLEV coefficients . . . . . . 60
42 Section of the geometry for the 3D lab-scale hill case [3] . . . . . . . . . . . 61
43 Stream-wise velocity and turbulent kinetic energy profiles in the symmetry
plane against measurements obtained on the 3D hill at laboratory scale . . 63
44 Simulated wall shear stress along the symmetry of the domain against the-
oretical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
45 Computational domain for the Askervein hill test case . . . . . . . . . . . . 65
46 Comparison of simulated and measured horizontal and vertical stream ve-
locity (U and W) and turbulent kinetic energy (k) for the Askervein hill
h
test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
47 Comparison of simulated and measured vertical profiles (U and k) at the
hill summit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
VKI - 4 -
LIST OF FIGURES LIST OF FIGURES
48 Relevant spatial scales in pollutant dispersion . . . . . . . . . . . . . . . . 68
49 Framework of a continuous release . . . . . . . . . . . . . . . . . . . . . . . 69
50 Framework of an instantaneous release . . . . . . . . . . . . . . . . . . . . 69
51 Evolution of the atmospheric boundary layer . . . . . . . . . . . . . . . . . 70
52 Schematic of the flow through and over an urban area . . . . . . . . . . . . 71
53 Typical characteristics of a continuous dispersion phenomenon . . . . . . . 72
54 Wind gradient: wind speed is influenced by ground features . . . . . . . . 73
55 Atmospheric temperature profiles . . . . . . . . . . . . . . . . . . . . . . . 74
56 Pasquill stability classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
57 Schematic of a Gaussian plume . . . . . . . . . . . . . . . . . . . . . . . . 77
58 Influence of the averaging time . . . . . . . . . . . . . . . . . . . . . . . . . 78
59 Transition in the Gaussian dispersion stage . . . . . . . . . . . . . . . . . . 80
60 Schematic representation of isolated building, isolated street canyon and
array of buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
61 Schematic view of the interaction between gas cloud/plume and obstacle . 82
62 Location of concentration measurement points . . . . . . . . . . . . . . . . 85
63 View of the building’s sources . . . . . . . . . . . . . . . . . . . . . . . . . 87
64 A1-1 Profiles of non-dimensional concentration using a constant turbulent
Schmidt equal respectively to 0.7 and 0.4 . . . . . . . . . . . . . . . . . . . 89
65 A1-1 Profiles of non-dimensional concentration using a variable turbulent
Schmidt number and a constant turbulent Schmidt number = 0.4 . . . . . 89
66 Contours of non-dimensional concentration of SO with variable turbulent
2
Schmidt formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
67 A1-1 Profiles of non-dimensional concentration . . . . . . . . . . . . . . . . 91
68 A1-1 Profiles of non-dimensional concentration . . . . . . . . . . . . . . . . 91
69 ContoursofvariableturbulentSchmidtnumbercoupledwithEhrhardNLEVM 93
70 Contours of variable turbulent Schmidt number coupled with Craft NLEVM 93
71 Measurement points for 3D lab-scale [71] . . . . . . . . . . . . . . . . . . . 101
72 Profiles of Velocity, TKE and Wall Shear Stress for stable ABL . . . . . . 102
73 Profiles of Velocity, TKE and Wall Shear Stress for unstable ABL . . . . . 103
74 Profiles of dimensionless turbulent kinetic energy for stable ABL . . . . . . 105
75 Profiles of dimensionless velocity for stable ABL . . . . . . . . . . . . . . . 105
76 Profiles of dimensionless turbulent kinetic energy for unstable ABL . . . . 106
77 Profiles of dimensionless velocity for unstable ABL . . . . . . . . . . . . . 106
78 Wall shear stress distribution obtained with different models for stable ABL107
79 Wall shear stress distribution obtained with different models for stable ABL108
80 Wall shear stress distribution obtained with different models for unstable
ABL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
81 Contour plots of non-dimensional iso-velocity on the planes y = 0 and z =
H/2 for stable ABL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
82 Contour plots of non-dimensional iso-velocity on the planes y = 0 and z =
H/2 for unstable ABL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
VKI - 5 -
LIST OF TABLES LIST OF TABLES
List of Tables
1 Inlet conditions and turbulence model formulation belonging to the so-
called ”comprehensive approach”. . . . . . . . . . . . . . . . . . . . . . . . 13
2 Formulation of the blending metric for the pure and hybrid blending ap-
proaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Yang and Parente . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
according to Richard and Hoxey and Brost and Wyngaard . . . . . . . . . 31
5 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the prescribed BIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Test cases and corresponding model settings for the numerical simulation
of the flow around a bluff-body using the prescribed BIA approach . . . . . 34
7 Test cases and corresponding model settings for the numerical simulation
of the flow around a bluff-body, using the automatic switch approach for
the BIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the automatic switch approach . . . . . . . . . . . . . . . . . . . . . . . 38
9 Turbulence models, wall functions and main configurations tested in the
Cedval A1-1 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10 Mesh different refinements and computational times . . . . . . . . . . . . . 54
11 Turbulence models, wall functions and main configurations tested in the
Cedval B1-1 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
12 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
and turbulent model parameters for the 3D hill simulation . . . . . . . . . 62
13 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the 3D hill simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14 Normalized errors in the prediction of the separation point, (cid:15) , and wake
SP
length, (cid:15) , for the 3D hill . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
WL
15 Fitting parameters for velocity and turbulent kinetic energy inlet profiles
and turbulent model parameters for the Askervein hill simulatio . . . . . . 65
16 Hit rate values for non-dimensional velocity and turbulent kinetic energy
for the Askervein hill simulation . . . . . . . . . . . . . . . . . . . . . . . . 66
17 Dispersion modelling approaches comparison . . . . . . . . . . . . . . . . . 76
18 Turbulence models, dispersion parameters and configurations tested for the
A1-1 dispersion study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
19 Coefficients for the determination of the boundary condition for the 3D-hill
in stable atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
20 Coefficients for the determination of the boundary condition for the 3D-hill
in unstable atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
21 Coefficients for blending in non-neutral flows . . . . . . . . . . . . . . . . . 104
VKI - 6 -
1 INTRODUCTION
1 Introduction
Computational Fluid Dynamics (CFD) is widely used to study flow phenomena in the
lower part of the atmospheric boundary layer (ABL), with applications to pollutant dis-
persion, risk analysis, optimization and ”siting” of windmills and wind farms. Numeri-
cal simulations of ABL flows can be performed either by solving the Reynolds-averaged
Navier-Stokes (RANS) equations or by conducting large-eddy simulations (LES). It is
generally acknowledged that LES, which explicitly accounts for the larger spatial and
temporal turbulent scales, can offer a more accurate solution for the turbulent flow field,
provided that the range of resolved turbulence scales is sufficiently large and that the
turbulent inflow conditions are well characterized [70, 44, 76, 77]. For example, Xie and
Castro [76] presented a comparison of LES and RANS for the flow over an array of uni-
form height wall-mounted obstacles: the Authors compared the results to available direct
numerical simulation (DNS) data, showing that LES simulations outperform RANS re-
sults within the canopy. Dejoan et al. [22] compared LES and RANS for the simulation of
pollutant dispersion in the MUST (Mock Urban Setting Test) field experiment and found
that LES performs better in predicting vertical velocity and Reynolds shear stress, while
the results for the stream-wise velocity component are comparable. However, LES simu-
lations are at least one order of magnitude computationally more expensive than RANS
[66] and the sensitivity to input parameters such as inlet conditions, imply that, as for
RANS,multiplesimulations areneededtoquantifythe resulting uncertaintyintheoutput
for realistic applications. Hence, practical simulations of ABL flows are still often carried
out solving the RANS equations in combination with two-equation turbulence models.
Consequently, the investigation of possible improvements to these models is worthwhile.
In RANS simulations, the effect of roughness on ABL flows is generally represented with
the so-called sand-grain based wall functions [20], based on the experiments conducted
by Nikuradze [50] for flow in rough, circular pipes covered with sand. Moreover, the up-
stream turbulent characteristics of a homogeneous ABL flows are generally modeled using
the profiles suggested by Richards and Hoxey [62] for mean velocity, turbulent kinetic
energy and turbulent dissipation rate. However, this modelling approach can result in an
unsatisfactory reproduction of the ABL for two main causes. The first cause of discrep-
ancy lies in the evident inconsistency between the fully developed ABL inlet profiles and
the rough wall function formulation [65, 27, 2, 11, 35, 28]. Furthermore, the inlet profile
for the turbulent kinetic energy, k, proposed by Richards and Hoxey [62], assumes a con-
stant value with height, in conflict with wind-tunnel measurements Leitl [42], Xie et al.
[78], Yang et al. [79], where a variation of k with height is observed. A remedial measure
to solve the inconsistency between the sand-grain based rough wall function and the fully
developed inlet profiles was proposed by Blocken et al. [11]. It consists in the modification
of the wall law coefficients, namely the equivalent sand-grain roughness height k and the
s
roughness constant C , in order to ensure a proper matching with the velocity boundary
s
conditions. This approach ensures the desired homogeneity of velocity distribution for the
fully developed ABL, but it is code dependent and does not provide a general solution to
the problem. Moreover, the standard law of the wall for rough surfaces poses limitations
concerning the level of grid refinement that can be achieved at the wall. This restriction
becomes particularly relevant for applications requiring a high resolution near the wall
boundaries. An additional complicating factor is the necessity to apply different wall
VKI - 7 -
1 INTRODUCTION
treatments when a combination of rough terrains and smooth building walls must be sim-
ulated. Concerning the inlet profile for turbulent kinetic energy, Yang et al. [79] derived a
new set of inlet conditions, with k decreasing with height. However, the application of this
specific profile at the inlet boundary only provides an approximate solution for the system
of equations describing a fully developed ABL. In a recent work, Gorl´e et al. [30] proposed
a modification of the constant C , and of the turbulent dissipation Prandtl number, σ ,
µ (cid:15)
to ensure homogeneity along the longitudinal ABL direction, when the k profile of Yang
et al. [79] is applied. Parente and Benocci [54], Parente et al. [57] proposed a modification
of the k−(cid:15) turbulence model compatible with the set of inlet conditions proposed by Yang
et al. [79]. Such a modification consisted in the generalization of the model coefficient C ,
µ
which becomes a local function of the flow variables, and in the introduction of two source
terms in the transport equations for k and (cid:15), respectively. The limitation of this approach
lied in the inlet profile adopted for turbulent kinetic energy, which does not satisfy all
the governing simulations involved in the problem [56]. Parente et al. [56] addressed the
aforementioned aspects by proposing a comprehensive approach for the numerical simu-
lation of the neutral ABL. First, a new profile for turbulent kinetic energy was derived
from the solution of the turbulent kinetic energy transport equation, resulting in a new
set of fully developed inlet conditions for the neutral ABL, which satisfies the standard
k −(cid:15) model. This was accomplished through the introduction of a universal source term
in the transport equation for the turbulent dissipation rate, (cid:15), and the re-definition of the
k −(cid:15) model coefficient C as a function of the flow variables. Secondly, with the aim of
µ
solving the flow around obstacles, the modelling approach derived for the homogeneous
ABL was generalized with an algorithm for the automatic identification of the building
influence area (BIA). As a consequence, the turbulence model formulation is gradually
adapted moving from the undisturbed ABL to the region affected by the obstacle. In-
side the building influence area specific and more performing turbulent treatments are
applied, in order to improve the performance in the vicinity of the obstacles. Longo et al.
[45] implemented Non-Linear Eddy-Viscosity models (NLEVM), often referred to as cubic
closures [21], inside the BIA for further improving the accuracy of the predictions. Merci
et al. [46] further investigated cubic closures in the impingement heat transfer context,
introducing a new non-linear formulation. Moreover he claimed C to be the only relevant
µ
parameter for all the flows characterized by reduced swirl and vorticity. As a consequence,
in this typology of flow-fields the higher order terms of the Reynolds stresses can be ne-
glected just by setting the non-linear coefficients equal to zero. Parente et al. [55, 57]
also proposed a novel implementation of a wall function, which incorporates both smooth
and rough-wall treatments, employing a screening algorithm to automatically select the
desired formulation, i.e. rough or smooth, depending on the boundary surface properties.
Balogh et al. [5] extended the approach in [55, 57] to the simulation of flows above com-
plex terrains, i.e. wind-tunnel scale 3D hill model and Askervein Hill. Longo et al. [45]
further investigated and improved the Building Influence Area concept by introducing a
hybrid blending approach and developing a marker formulation, resulting in a more effec-
tive detection of the disturbed flow-field. As for the simulation of a gaseous dispersion,
in the last decades different methodologies have been intensively developed and tested.
According to Rota and Nano [68], the most relevant approaches available in the disper-
sion panorama include: Gaussian models, Integral models and three-dimensional models
(CFD). Among these, the most performing, flexible and promising are the CFD models
VKI - 8 -
1 INTRODUCTION
[68] [59] [23], implementing the cardinal momentum eq. of the fluids as well as the mass
and energy conservation. In the dispersion phenomenon, a crucial rule is typically played
by the turbulent viscosity ν and, especially, by the turbulent Schmidt number Sc , Gorl´e
t t
et al. [32]. Both these terms need to be properly computed as deeply affecting the con-
centration field. Recently, Gorl´e et al. [32] proposed a new formulation for the turbulent
Schmidt number, which consists in computing it locally, depending on the C turbulent
µ
constant and on Re (Reynolds at the Taylor micro-scale). The dispersion phenomenon
λ
is deeply affected also by the stability classes characterizing the lower atmospheric layers.
The correct representation of their effects is nowadays a target of focal resonance (Pontig-
gia et al. [59]), especially when dealing with classes which are far away from the neutral
one (notably the stable classes). In this regard, remarkable attempts were made by Pon-
tiggia et al. [59] and Ferrarotti [25], taking into account the Monin-Obukhov similarity
theory and testing the models both in in undisturbed ABL and in presence of complex
orography. Meroney [47] proposed a comprehensive review both of wind tunnel and CFD
simulations of micro-scale pollutant dispersion in the context of hybrid wind tunnel - CFD
modelling. Canepa [17] re-visioned a great number of approaches for building wake down-
wash. Focusing on the dispersion of heavy hazardous gases in the risk assessment field,
Derudi et al. [23] proposed a general criterion to forecast the potential role played on the
hazardous distance by the presence of an obstacle on the cloud pattern. This criterion is
based on the comparison of the characteristic dimensions of the obstacles with those of the
cloud in open field conditions. It can be suitable in order to identify which obstacles can
be safely neglected when reproducing the real environment’s geometry in a computational
domain without compromising the simulation reliability and to easily estimate the mini-
mum size of a proper mitigation barrier. The present notes are organized as follows. The
modelling approach for the numerical simulation of neutral ABL flows is firstly presented,
by discussing the turbulence model formulation, the different inlet profiles and the wall
function. Applications are presented and discussed for the flow over flat terrain, around
ground mounted bluff bodies and over hills. Subsequently, after having selected the most
accurate turbulence modelling approaches, the dispersion field is intensively analyzed and
studied, both through theoretical background and modelling applications on the single
building test case. In conclusion, an introduction to the stability classes theory and its
main issues are discussed, supported by a proper testing session.
VKI - 9 -
2 ABL TURBULENCE MODELLING
2 ABL turbulence modelling
The standard k−(cid:15) model remains the most common option for the numerical simulation
of the homogeneous ABL. This specific family of models solves a transport equations for
turbulent kinetic energy, k, and for turbulent dissipation rate, (cid:15):
∂ ∂ ∂ µ ∂k
t
(ρk)+ (ρku ) = µ+ +G −G −ρ(cid:15)−Y (1)
i k b M
∂ ∂x ∂x σ ∂x
i j k j
(cid:20)(cid:18) (cid:19) (cid:21)
∂ ∂ ∂ µ ∂(cid:15) (cid:15) (cid:15)2
t
(ρ(cid:15))+ (ρ(cid:15)u ) = µ+ +C (G +C G )−C ρ . (2)
i (cid:15)1 k (cid:15)3 b (cid:15)2
∂ ∂x ∂x σ ∂x k k
i j (cid:15) j
(cid:20)(cid:18) (cid:19) (cid:21)
In Equations (1)-(2), u is the ith velocity component, ρ is the density, C , C and C are
i (cid:15)1 (cid:15)2 (cid:15)3
modelconstants,σ ,andσ aretheturbulentPrandtlnumbersfork and(cid:15),respectively,G
k (cid:15) b
istheturbulentkineticenergyproductionduetobuoyancy,Y representsthecontribution
M
of the fluctuating dilatation in compressible turbulence flows to the overall dissipation
rate, G is the generation of turbulent kinetic energy due to the mean velocity gradients,
k
calculated from the mean rate-of-strain tensor, S , as:
ij
1 ∂u ∂u
G = µ S2 S = 2S S S = i + j . (3)
k t ij ij ij
2 ∂x ∂x
j i
(cid:18) (cid:19)
(cid:112)
For a steady ABL, under the hypothesis of incompressibility, zero vertical velocity, con-
stant pressure along vertical (z) and longitudinal (x) directions, constant shear stress
throughout the boundary layer and no buoyancy effects, the transport equations for tur-
bulent kinetic energy k, and turbulent dissipation rate (cid:15), simplify to:
∂ µ ∂k
t
+G −ρ(cid:15) = 0 (4)
k
∂z σ ∂z
k
(cid:18) (cid:19)
∂ µ ∂(cid:15) (cid:15) (cid:15)2
t
+C G −C ρ = 0 (5)
(cid:15)1 k (cid:15)2
∂z σ ∂z k k
(cid:15)
(cid:18) (cid:19)
∂u 2
G = µ . (6)
k t
∂z
(cid:18) (cid:19)
The model is completed by the momentum equation, which takes the form:
∂u
µ = τ = ρu2 (7)
t∂z w ∗
where τ is the wall shear stress and u is the friction velocity
w ∗
τ
w
u = . (8)
∗
ρ
(cid:114)
In Equations (4), (5) and (7) the laminar viscosity has been neglected with respect to the
turbulent one, µ , expressed as:
t
k2
µ = ρc . (9)
t µ
(cid:15)
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