Post on 31-Jan-2016
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Omduth Coceal Dept. of Meteorology, Univ. of Reading, UK.
Email: o.coceal@reading.ac.uk
and
A. Dobre, S.E. Belcher (Reading)T.G. Thomas, Z. Xie, I.P. Castro (Southampton)
Urban turbulence - flow statistics, dynamics and
modellingA numerical study using direct numerical simulations (DNS) over groups of idealized buildings
Seminar given at UK Met Office, Exeter, 1 May 2007
Motivation - why DNS?
Field measurements within urban areas are difficult to interpret
Dispersion/ventilation depend on unsteady flow; short timescales
We need a better understanding of turbulence dynamics in urban areas
LES/DNS is a useful tool for obtaining detailed spatial and temporal information
But how can results be useful for simplified modelling?
Outline
Description of numerical simulations
Validation with wind-tunnel data
Flow visualization
Spatial averaging of the data
Statistics for different types of building arrays
Unsteady effects and organized structures
Flow dynamics
Direct numerical simulations
• Parallel LES/DNS code developed by T.G. Thomas (Southampton)
• Domain size: 16h x 12h x 8h
• Resolution: 32 x 32 x 32 gridpoints per cube (also 643 gridpoints per cube on a smaller domain)
• Total of 512 x 384 x 256 ~50 million gridpoints
• Runs took ~3 weeks on 124 processors on SGI Altix supercomputer
Direct numerical simulations• Boundary conditions
free slip at top of domain
no slip at bottom and on cube surfaces
periodic in horizontal
• Reynolds number
5800 based on Utop and h
Re = 500
• Flow driven by constant body force
• Turbulent scales are sufficiently resolved
dissipation captured
good agreement with experiment even without SGS model
Coceal et al., BLM (2006)Coceal et al., BLM, to appearCoceal et al., IJC, to appear
Compared with wind-tunnel data from Cheng and Castro (2003)
and Castro et al. (2006)
Comparisons with experiment
velocity
stresses
pressure
spectra
Instantaneous windvectors in y-z plane
Mean flow is out of screen
643 gridpoints per cube
Instantaneous windvectors in y-z plane
Streamwise circulations visualised in y-z plane (mean flow is out of screen)
Streamwise vorticity in y-z plane
Counter-rotating vortex pairs
Mean flow structure
Staggered array Square array
Horizontal slices at z = 0.5 h
Mean flow structure is highly dependent on the building layout
(I) Flow statistics
Spatial averagingE.g. Urban canopy models (e.g. Martilli et al. 2002; Coceal & Belcher 2004, 2005)
- Compute these spatially-averaged quantities from the DNS data
- Don’t resolve horizontal heterogeneity at the building/street scale- Take horizontal averages: resolve vertical flow structure
h
y
x
€
D u
Dt+
1
ρ
∂ p
∂x=
∂
∂zu'w' +
∂
∂z˜ u ˜ w − D
€
u = u + ˜ u + u'
Spatial average of Reynolds-averaged momentum equation
Triple decomposition of velocity field
Spatially averaged statistics - uniform arrays
Different building layouts, same density(Detailed explanation of this plot in Coceal et al., 2006)
Arrays with random building heights (same density)
0.5hm
Compare results with LES performed by Zhengtong Xie (Southampton)
Same building density and staggered layout as in uniform array
Spatial averages - mean velocity
Velocities are smaller over the random array. The random array exerts more drag. Spatially-averaged velocities are very similar within arrays.Inflection is much weaker in random array.
Spatial averages - stresses
In the random array, the peaks are less strong, but still quite pronounced. They occur at the height of the tallest building, not at the mean or modal building height.
Spatial averages - dispersive stresses
Profiles of uw component of dispersive stress are very similar below z=h_m.
Spatial fluctuations
Qualitatively similar behaviour in the two arrays
Energy partitioning
(i) mke dominates above the canopy, but rapidly becomes a negligible fraction of the total k.e. within the canopy, while the fraction of dke and tke both increase. (ii) the fractions of mke, dke and tke for the two arrays are very similar below z=h_m; energy is partitioned roughly in the same proportion.(iii) above the canopy, the tke fraction over the random array is roughly twice as large as that over the regular array.
Buildings of variable heights - TKE
TKE from shear layers shed from vertical edges of tallest building dominates above half the mean building height.
Buildings of variable heights - Umag
The effect of the tallest building is more pronounced w.r.t. the total velocity magnitude.
Buildings of variable heights - Drag profiles (I)
Tallest building (1.72 times the mean building height) exerts 22% of the total drag! The 5 tallest buildings (out of 16) are together responsible for 65% of the drag.
Buildings of variable heights - Drag profiles (II)
The shapes of the drag profiles are in general similar for many of the tallest buildings (17.2m, 13.6m, 10.0m) except when they are in the vicinity of a taller building. The profile shapes of the shortest buildings (6.4m and 2.8m) are very different - but these buildings do not exert much drag.
Summary (I): Effect of building geometry on statistics
Effects of building layout
Mean flow structure and turbulence statistics vary substantially with layout
Effect of packing density still needs to be properly documented
Effects of random building heights:
Less strong shear layer on average
Inflection in spatially-averaged mean wind profile much less pronounced
Larger drag/roughness length
Below the mean building height, spatial averages are very similar to regular array
Effects of tall buildings:
Strong shear layers associated with tall buildings - high TKE
They exert a large proportion of the drag
They cause significant wind speed-up lower down the canopy
(II) Unsteady dynamics
Quadrant analysisDecompose contributions to shear stress <u’w’> according to signs of u’, w’
u’
w’
u’ > 0
w’ > 0
u’ < 0
w’ > 0
u’ < 0
w’ < 0
u’ > 0
w’ < 0
Which quadrants contribute most to the Reynolds stress <u’w’> ?
Ejections (Q2)
Sweeps (Q4)
Quadrant analysis
Profiles of fractional frequency and fractional contribution of each quadrant
Ejections and sweeps dominateThey are associated with turbulent organized motions
Quadrant analysis - ExuberanceExuberance
€
Ex =S1 + S3
S2 + S4
Exuberance is a measure of how disorganized the turbulence is
Magnitude of Exuberance is smallest near canopy top in DNS (uniform building heights)
Increases slowly above building canopy, rapidly within canopy
Real field data (Christen, 2005)From DNS
Quadrant analysis - Q2 vs Q4 (I)
Indicates character of the organized motions
Ejection dominance well above the canopy
Sweep dominance close to/within the canopy. Cross-over point is at z = 1.25 h
€
ΔS0 = S4 − S2
Real field data (Christen, 2005)
DNS
Fluctuating velocity vectors in x-z plane
Ejections and sweeps are associated with eddy structures
Mean flow from left to right. Local mean subtracted from velocity vectors.
Spatial distribution of ejections and sweeps
Fluctuating windvectors
Unsteady coupling of flow within and above canopy
Red = sweep eventsBlue = ejection events
Give information on lengthscale and spatial structure of organized motionsCorrelation lengthscale increases with height of reference point Small at z = h and within canopyStructures above canopy are inclined; inclination angle is a function of height
Two-point correlations Ruu
Instantaneous structures above buildings in 3d
Lower Reynolds number of 1200 (Re = 125, still fully rough flow)
Clearly reveals vortex structures (red) and low momentum regions (blue)
Vortex cores identified using isosurfaces of negative 2
3d structure of the conditional vortex
Hairpin-like conditional vortex obtained by conditional averaging of a large number of instantaneous realisations
Role of canopy-top shear layer
y
Intermittent impinging of shear layer on downstream buildings drives a recirculation.
cf Louka et al. (2000).
Space-time correlation Ruu with negative time delay of -0.4T; ref is at (8, 0.75).
T is an eddy turnover time of the largest eddies shed by the cubes.
Effect of shear layer on flow within canopy
z at z = 0.5 h x at y = 0.5 h
Interacting vertical shear layers Vortex tilting and stretching
Shear layers within the canopy
Small-scale circulations within canopy
Instantaneous windvectors in y-z plane within a cavity (flow is out of screen )
Summary (II): A conceptual model of the unsteady dynamics
Three flow regimes:
Flow well above canopy is a classical rough wall flow and its structure resembles that over a smooth wall boundary layer, although there are quantitative differences.
Flow near the canopy top is dominated by shear layer shed off top of cubes and by larger boundary layer eddies.
Flow within canopy is complicated by interaction of above with shear layers shed off vertical faces of the buildings, vortex stretching and tilting and distortion by roughness.
THE END
EXTRA SLIDES
Vortex identification methods(Jeong & Hussain, JFM 1995 )
Failures of intuitive criteria:
• Closed or spiral streamlines not Galilean invariant
• Vorticity magnitude fails in a shear flow if background shear is appreciable; necessary but not sufficient condition
• Local pressure minimum
could also exist in an unsteady irrotational flow without a vortex
vortex could exist without a pressure minimum, due to viscous term
hence, pressure minimum is neither a necessary nor a sufficient condition for existence of a vortex
Vortex identification methods
Positive second invariant of the velocity gradient tensor u (Hunt et al. 1988 )
( )[ ] ( )ijij
ijijx
u
xu
x
u
xu
xu SSQ
i
j
j
i
i
j
j
i
i
i −ΩΩ=−=−≡ ∂
∂
∂∂
∂
∂
∂∂
∂∂
21
21
2
21
Additionally, the pressure must be lower than its ambient value
where S and are the symmetric and antisymmetric components of u
( )i
j
j
i
x
u
xu
ijS ∂
∂
∂∂ += 2
1 ( )i
j
j
i
x
u
xu
ij ∂
∂
∂∂ −=Ω 2
1
Hence, Q represents balance between shear strain rate and vorticity magnitude
The 2 vortex identification method(Jeong & Hussain, JFM 1995 )
ijjiitj uPuD ∂∂+∂∂−=∂ 21 νρ
Take gradient of the Navier-Stokes equation:
This equation may be decomposed into symmetric and antisymmetric parts to give:
ijjikjikkjikijt SPSSSD 21 ∂+∂∂−=++ νρ
ijkjikkjikijt SSD ∂=++ 2ν
Second equation is vorticity equation
The 2 vortex identification method(Jeong & Hussain, JFM 1995 )
First equation may be rewritten as:
PSSSSD jikjikkjikijijt ∂∂−=++∂− ρν 12
unsteady irrotational straining
viscous effects
Both ‘mask’ local pressure minimum, hence ignore their contributions
Local pressure minimum in a plane Hessian of pressure has two +ve eigenvalues Pji∂∂
needs to have two -ve eigenvalues 22 ÙS +
Hence, if eigenvalues are 1, 2, and 3, with 1< 2 < 3, then 2 < 0 ihiν voρex coρe
Buildings of variable heights - U
Wind speed-up around the tall building in relation to the background flow, especially at lower levels.
Ruu with fixed (xref,zref) for successive time delays of 0.1T
Time-delayed two-point correlations
Convection velocities
cf Castro et al. (2006)
Vortex visualisation by Galilean decomposition
Vortex structures visualised after subtracting convection velocity (cf Adrian et al. 2000)
Galilean decomposition
POD analysis
Head-up hairpin-like vortices are energetically dominant
Eigenvalue spectrum
Vortex reconstructed using first few terms
Roger Shaw Ned Patton
Quadrant analysis - Q2 vs Q4 (II)
Real field data (Christen, 2005)
€
γ0 =S2
S4
€
Q0 =N2
N4
DNS
Space-time correlation Rpw with positive time delay of 0.4T; ref. is at (8, 0.5).
Effect of shear layer on flow within canopy