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Physical modeling of atmospheric boundary layer flows Part I: Overview of modeling concepts and techniques
Part II: Modeling neutrally stratified boundary layer flows Evgeni Fedorovich
School of Meteorology, University of Oklahoma, Norman, USA
Outline • Place of physical modeling in the triad of approaches to study
atmospheric boundary layer flows • Concept of physical modeling: prototype versus model • Commonly employed laboratory facilities and techniques • Wind tunnel modeling of neutrally stratified boundary layers • Methodology of generating neutral boundary layer flows in wind tunnels • Review of turbulent flow properties over rough and smooth surfaces • Similarity requirements; comparisons with atmospheric data • Tracer dispersion from a line source: numerical model evaluation
• Summarizing remarks
Triad of approaches in atmospheric boundary layer studies
I. Field observations/measurements • In situ/contact measurements
• Remote sensing techniques
II. Physical/laboratory models • Laboratory tank (thermal and saline) models
• Water channel/tunnel/flume models
• Wind tunnel (stratified and neutral) models
III. Theoretical/numerical techniques • Theoretical/analytical models
• Numerical models/parameterizations
• Numerical simulations (direct and large-eddy)
I. Field observations/measurements In situ/contact measurements and remote sensing techniques
Single global asset: it is real!
Hard or impossible to • separate different contributing forcings/mechanisms • match temporal/spatial requirements for retrieval of statistics • control external forcings and boundary conditions • obtain accurate and complete data at low cost
II. Physical/laboratory models Laboratory tanks, water channels, wind tunnels
Pros: • High level of complexity of
modeled flows • Controlled external/boundary
parameters • Repeatability of flow regimes • Possibility to generate well-
documented data sets for evaluation of numerical models/simulations
Hard or impossible to • reproduce several contributing
forcings in combination • sufficiently match scaling/similarity
requirements in order to relate the modeled flow to its atmospheric prototype
• find a reasonable balance between the value of results and cost of facility
III. Theoretical/numerical techniques Analytical models, numerical models/parameterizations, numerical simulations
2
31 ' , 0i ji i i
ij i j j i
u uu u up bt x x x x x
δ νρ
∂∂ ∂ ∂∂+ = − + + =
∂ ∂ ∂ ∂ ∂ ∂
Pros: • Availability at a relatively low cost • Capability to generate instantaneous flow fields • Accounting for processes within relatively broad ranges of
temporal and spatial scales
Hard or impossible to • reproduce flow regimes with realistic environmental settings • evaluate precisely effects of subgrid/subfilter/ensemble
turbulence closures • separate numerical artifacts from actual physical features of the
modeled/simulated flows
Wind tunnel modeling of neutral atmospheric BL flows
turbulencegenerators
flowstraightener
freestream
outerlayer
logarithmiclayer
bloweru(z)
ceiling boundary layer
adjustment distance- test section-City models
inletdiffuser
windprofil
δ
do
~0,15 δ
canopy layer
Design features of neutral boundary layer wind tunnels
AAuslaß
Teststrecke
Meßstrecke Anlaufstrecke(Grenzschichterz.)
Einlaufdüse
10,5 mGebläse
Bodenrauhigkeite
L B u(z)
Wirbelgeneratoren
x
H Sägezahnschwelle
z
y
Basic properties of a wall-bounded turbulent flow Consider turbulent flow that is parallel and horizontally homogeneous in x direction (an idealization of a wind tunnel flow far away from the inlet) with mean (in Reynolds sense) velocity in this direction u(z).
Prandtl concept of mixing distance/length l': particle that carries momentum between flow levels separated by distance l' instantly attributes momentum to surrounding air as it arrives to the destination level. First-order approximation: ( ') ( ) ( / ) 'u z l u z u z l+ = + ∂ ∂ ,
( ') ( ) ( / ) 'u z l u z u z l− = − ∂ ∂ , or, in terms of velocity fluctuations, ( ') ( ) '('( ') / )u z l uz z zu l l u= + − = ∂ ∂+ ,
'( ') ( ') ( ) '( / )u z l u z l u zu z l = − − = − ∂ ∂− . Prandtl also supposed: ' '( / )sign( ')w l u z u= − ∂ ∂ . Multiplying 'w with 'u and averaging, we come to ' 'u w 2 2( / )l u z= − ∂ ∂ ,
where l=1/ 2
2'l is the mixing length at level z, which may be interpreted as characteristic integral turbulence length scale for momentum exchange at level z.
Friction velocity and logarithmic wind profile Another hypothesis/finding by Prandtl: l z∝ . Vertical kinematic momentum flux is therefore: ' 'u w 2 2( / )z u z∝ − ∂ ∂ . Von Kármán constant κ is a proportionality coefficient between l and z:
l=κ z, ' 'u w 2 2 2( / )z u zκ= − ∂ ∂ . From the flux-profile parameterization (Boussinesq analogy):
' 'u w ( / )k u z= − ∂ ∂ , where k is the eddy viscosity (turbulent exchange coefficient for momentum:
1/ 2( ' ')k u w zκ= − = 2 2 ( / )z u zκ ∂ ∂ . Near the wall ' 'u w is approximately height-constant and may be conveniently represented through the velocity scale u∗=
1/ 2( ' ')u w− called the friction velocity. Therefore, k u zκ ∗= or k u l∗= . Also: / /( )u z u zκ∗∂ ∂ = , which provides the logarithmic velocity profile in the near-wall region of the neutral boundary layer:
( / ) lnu u z Cκ∗= + .
Aerodynamically smooth and rough surfaces Based on Reynolds-number criterion Re= / 1lu vδ∗ ∼ for laminarization
of the flow close to the wall, one may expect that at distances from the wall of the order and less than /l v uδ ∗∼ , the molecular shear stress ultimately dominates the turbulent stress: ' ' ( / )u w v u z− ∂ ∂ .
Experimental data show: 5 /l v uδ ∗≈ . The layer defined in this manner is called the viscous sublayer.
Smooth surface: surface roughness elements of characteristic size rh are deployed in the viscous sublayer: r lh δ .
Rough surface: r lh δ .
Laboratory experiments show that surface may be considered aerodynamically smooth for
5 /rh v u∗≤ ,
and aerodynamically rough when
75 /rh v u∗≥ .
Wind profile over smooth and rough surfaces Smooth-wall case: developed turbulent flow with ( / ) lnu u z Cκ∗= + is
realized at distances considerably larger than the length scale /l v uδ ∗∼ . Rough-wall case: flow is turbulent already in the immediate vicinity of
surface roughness elements, with mean flow velocity vanishing (u=0) at some level close to rh .
One may consider a reference level 0z close to the surface, where u=0,
0( / ) ln( / )u u z zκ∗= . Quantity 0z is called the aerodynamic surface roughness length or the surface roughness length for momentum.
Smooth-wall 0z is retrieved from / (1/ ) ln[ /( / )] su u z v u Cκ∗ ∗= + , where parameter 0(1/ ) ln[( / ) / ]sC v u zκ ∗= is about 5 (experiment):
0 / 0.1 /sCz e v u v uκ−∗ ∗= ≈ .
Rough-wall 0z is a function of the surface geometry, involving rh as one of parameters; generally speaking, 0z is growing with rh .
Wind profile approximations used in wind tunnel studies Velocity profile above the rough surface starts to follow the logarithmic law only at some distance away from the surface, at z>> 0z . In this sense, the surface roughness length is an asymptotic parameter rather than a limiting point of the observed wind profile.
In order to make logarithmic wind profile applicable in a broader range of z close to the surface, it is often used with another parameter, the so-called displacement height 0d :
0
0
ln z duuzκ
∗ −= .
Along with log law, another analytical representation of wind profile is used (primarily, in wind engineering), the so-called power law form:
0ref
ref 0
( ) ,z du z uz d
α⎛ ⎞−
= ⎜ ⎟−⎝ ⎠
where refu is u value at z= refz and 1α < is an empirical exponent.
Similarity criteria for wind tunnel modeling of neutral BL flows
Length scales: 1L = 0z , 2L = 0d , 3L =δ , ...
Criteria: model( / )i kL L = nature( / )i kL L
Wind profile: 0
ref ref 0
( ) ,fu z z dSu z d
α⎛ ⎞−
= = ⎜ ⎟−⎝ ⎠ 0
* 0
( ) lnlu z z dSu z
κ −= = .
Criteria: model naturef fS S= , model naturel lS S=
Turbulence intensity: /i i iI uσ= , and spectra: 2( ) /ni ii iS kS k σ= .
Criteria: model naturei iI I= , model natureni niS S=
Surface roughness in the model: * 00Re 1u z
ν= >> ,
where ( )1/ 2
* ' 's
u u w= − .
Scaled mean wind profiles in WOTAN
Umean [m/s]0 2 4 6 8 10
10-1
100
101
102
z0 ≈ 0.20 ± 0.01 m
Umean [m/s]
Z FS[m
]
0 2 4 6 8 100
50
100
150
200
250
300
campaign IIcampaign I
power law exponentα ≈ 0.175 ± 0.05
Lateral homogeneity of mean flow in WOTAN
Yms [mm]
Um
ean/U
ref
[-]
-1500 -1200 -900 -600 -300 0 300 600 900 1200 15000
0.2
0.4
0.6
0.8
1
1.2
Intensities of turbulent velocity fluctuations in WOTAN
IU [ - ]
Z FS[m
]
0 0.1 0.2 0.30
50
100
150
200
250
300measuredmod. roughrough
IV [ - ]
Z FS[m
]
0 0.1 0.2 0.30
50
100
150
200
250
300measuredmod. roughrough
IW [ - ]
Z FS[m
]0 0.1 0.2 0.3
0
50
100
150
200
250
300measuredmod. roughrough
Longitudinal velocity component spectrum in WOTAN
f z / U [-]
fSU
U(f,
z)/σ
U2[-]
10-3 10-2 10-1 100 10110-3
10-2
10-1
100
frequency averaged spectrumSimiu & Scanlan (1986)Kaimal (1972)
zfs = 34.5 m
Vertical turbulent kinematic momentum flux in WOTAN
fs [m]z
0 0.5 1 1.50
50
100
150
200
250
300
AF 2003AF 2006AF 2007
2
*' ' /u w u−
Dispersion of passive scalar from a ground line source
Schematic of the source (red line) Design of the line source after deployed in the UniKA neutral WT Meroney et al. (1996)
-600 -400 -200 0 200 400 600
y in mm
20
40
60
80
100
norm
aliz
ed c
once
ntra
tion
c* ← Normalized concentration
* ref ref
t
cu zc
Q= , where tQ is in [L2 T-1],
at z = 10 mm and x = 90 mm (solid lines) and x = 180 mm (dashed lines) for four test cases with different wind velocities
and source flow rates.
Longitudinal and vertical profiles of normalized concentration
0 200 400 600 800 1000
x in mm
0
20
40
60
80
100
norm
aliz
ed c
once
ntra
tion
c*
at z=10 mm
at x=45 mm0 50 100 150 200 250
normalized concentration c*
0
5
10
15
20
25
30
z in
mm
0 50 100
normalized concentration c*
0
10
20
30
40
50
60
70
80
z in
mm
at x=180 mm
Numerical model of dispersion from a ground line source Balance equation for concentration c of a passive tracer is solved in a x-z plane perpendicular to the source located at x=0, z=0:
( ) ( )c sc cu z K z Ix z z∂ ∂ ∂
= +∂ ∂ ∂
.
Mean velocity profile is assumed to be logarithmic: *
0
( ) lnu zu zzκ
= .
Eddy diffusivity linearly depends on height as *( ) Scc tK z κu z= , where Sct is the turbulent Schmidt number.
Boundary conditions: / 0c z∂ ∂ = at 0z z= and c=0 at lz δ= .
Friction velocity is determined from * 0(ln / )l lu u zκ δ= .
1 1/( )s sI Q x z= Δ Δ is the source function, where 1 1x zΔ Δ is the cross-section area of the numerical grid cell surrounding the source. Elsewhere in the model domain outside this cell: 0sI = .
Numerical solution: implicit integration over x and factorization over z.
Model verification against the wind tunnel data Ground-level concentration (left plot)
10 100 1000x in m
0
10
20
30
40
c 0* (x)
0 200 400 600x in m
0
50
100
150
200
B(x
)
Wind tunnel data are gray symbols and lines. Dashed-dotted line shows numerical data for Sc 1t = . Other lines represent different analytical solutions considered in Kastner-Klein and Fedorovich (2002).
Concentration profiles at x = 45 m (left), x = 90 m (center), and x = 180 m (right)
0 4 8 12 16c*
0
0.2
0.4
0.6
0.8
z/z re
f
0 2 4 6 8c*
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4c*
0
0.4
0.8
1.2
1.6
References Garratt, J. R., 1994: The Atmospheric Boundary Layer, Cambridge
University Press, 316pp. Kastner-Klein, P., and E. Fedorovich, 2002: Diffusion from a line source
deployed in a homogeneous roughness layer: interpretation of wind tunnel measurements by means of simple mathematical models. Atmospheric Environment, 36, 3709-3718.
Meroney, R. N., M. Pavageau, S. Rafailidis, and M. Schatzmann, 1996: Study of line source characteristics for 2-D physical modelling of pollutant dispersion in street canyons. Journal of Wind Engineering and Industrial Aerodynamics, 62, 37-65.
Sorbjan, Z., 1989: Structure of the Atmospheric Boundary Layer, Prentice Hall, 317 pp.
Thanks go to Petra Klein, Bernd Leitl, and Michael Schatzmann
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