Omduth Coceal Dept. of Meteorology, Univ. of Reading, UK. Email: [email protected] and A.Dobre, S.E. Belcher (Reading) T.G. Thomas, Z. Xie, I.P. Castro.

Omduth Coceal Dept. of Meteorology, Univ. of Reading, UK.

Email: [email protected]

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