yamada science & art corporation proprietary a numerical simulation of building and topographic...

Yamada 　 Science & Art Corporation

www.ysasoft.com PROPRIETARY

A Numerical Simulation of Building and Topographic Influence on Air Flows

Ted Yamada （ YSA Corporation ）

Objective:to develop a model to simulate air flows and turbulence in and around an urban area located in complex terrain.

Objective:to develop a model to simulate air flows and turbulence in and around an urban area located in complex terrain.



Models

HOTMAC is a three dimensional, primitive equation model to predict airflows over complex terrain and around buildings. Governing equations are conservation equations for momentum (U,V, and W), internal energy (potential temperature), mixing ratio of water vapor and turbulence.

Second-moment turbulence closure model (Mellor and Yamada Level 2.5) was used. Prognostic equations are for the turbulence kinetic energy and a length scale.

Non-hydrostatic pressure was computed based on the HSMAC pressure-velocity correction method (Hirt and Cox, 1972, J. of Computational Phys.,324-340)



Models (continued)

RAPTAD is a three dimensional model which is useful to predict transport and dispersion of pollutants over complex terrain and around buildings.

RAPTAD is based on the random walk theory and releases puffs to determine pollutant concentration distributions.

RAPTAD used wind and turbulence distributions predicted by

HOTMAC.



Model equations Coordinate transformation

g

g

zH

zzHz

*

where z* is vertical coordinate after transformation, z is vertical coordinate in the Cartesian coordinates, zg is ground elevation , is top of computational domain after transformation and H is top of computational domain in the Cartesian coordinates.

H

where zgmax is the maximum value of the ground elevation.

maxgzHH



Before the transformation After the transformation

maxgz

H

H

z*z



Equations of motion

x

P

x

z

H

zHgVVf

Dt

DU nonhydrog

v

vg

1

1*

uwzzH

H

y

UK

yx

UK

x gxyx

*

UUzaCUUG dob )(

nudging Canopy drag



y

P

y

z

H

zHgUUf

Dt

DV nonhydrog

v

vg

1*

vwzzH

H

y

VK

yx

VK

x gyxy

*

VVzaCVVG dob )(

nudging canopy drag

where U, V are wind components in x, y directions, respectively; Ug, Vg are geostrophic wind components in x, y directions, respectively; Uob, Vob are observed wind components; G is a nudging coefficient; f is Coriolis Coefficient; g is acceleration of gravity; Kx, Ky, Kxy are eddy viscosity coefficients. The last terms on the right hand side equations represent effect of forest drag, where is the fractional coverage of forest (0 for no forest and 1 for complete Coverage), Cd is the drag coefficient, and a (z) is the vertical distribution of leaf areas.



Continuity equation

01

*

*

y

zV

x

zU

zHz

W

y

V

x

U gg

g

y

zV

x

zU

zH

HzW

zH

HW

gg

gg

**

where



Turbulence energy equation

22

2 222 q

yK

y

q

xK

xDt

qDxx

2

2

**

2q

zqlS

zzH

Hq

g

lB

qgw

z

Vvw

z

Uuw

zH

Hv

g 1

3

**

33 VUzaCd

Mellor-Yamada second-moment turbulence-closure equations provide the following equation for turbulence energy:

①diffusion in x, y, and z directions

②mechanical production

③buoyancy production

④dissipation⑤canopy drag

production

1000 m

200 m

daytime nighttime

③④

②

①

②④

③



Turbulence length scaleEquation for the length scale l is

Terms on the right hand side of equation correspond to the counterparts of the turbulence energy equation

lqy

Ky

lqx

KxDt

lqDyz

222

lq

zqlS

zzH

Hl

g

2**

2

2

21

3

**1

11

kzF

B

qgw

z

Vvw

z

Uuw

zH

HlF v

g

33 2 VUlzaCd

1000 m

Blackadar (1962)

0 m



Equation for potential temperature

gyx zH

H

yK

yxK

xDt

D

z

Wz

R

Cw

z

vN

p**

1

Long-wave radiation flux is from Sasamori (1968).pN CR /

vvv

　　 indicates the mean value in the horizontal plane. A similar equation for the mixing ratio of water vapor.



An Example of Simulation

Computational domain of 368 x 252 km over Grand Canyon. Horizontal grid spacing of 4 km.

Yamada, T., 2000: Numerical Simulations of Airflows and Tracer Transport in the Southern United States, J. of Applied Meteorology, vol. 39, No. 3, 399-411.



Transport and Diffusion

(1) Eulerian model

ii

i xC

KxDt

DC

where C is concentration and Ki is eddy diffusivity.

(2) Lagrangian puff/particle model

,

,)(

iuiUpiU

tpiUtixttix

Upi is the turbulence velocity at a puff center, xi; Ui and ui are the mean and turbulence velocities, respectively.

Large numerical diffusion near a point source

Common in atmospheric chemistry model

No numerical diffusion

Simple atmospheric chemistry



Gaussian Plume Model

Transport + Dispersion

source Distance

For flat terrain, uniform flows, and steady state



Lagrangian Particle/Puff Models

For complex terrain, 3-d, and time variations



Examples of Lagrangian Dispersion Model Results3 a.m.



Examples of Lagrangian Dispersion Model Results2 p.m.



Horizontal scale

Synoptic scale ………………...> 2000 km

Mesoscale ………………2 km ~ 2000 km

Microscale ………………...< 2km

microscale 　　 mesoscale 　　 synoptic scale

2 km 2000 km



American Environmental Review



Simulations of building and terrain effects

1mm 1 cm 1 m 10 m 100 m 1 km 10 km 50 km 500 km

GCM

mesoscale

CFD

Wind tunnel Building Urban Storm Fronts Synoptic

gap(open area)

horizontal grid spacing



Definition: Mesoscale and CFD (computational fluid dynamics) Models

CFD models: DNS (direct numerical simulation) LES (large eddy simulation) RANS (Reynolds averaged Navier-Stokes)

Mesoscale models: RSM (Reynolds stress model)



Approaches to fill the gap

mesoscale

CFD gap

gap

Single model

combination



Features of mesoscale models•Winds, potential temperatures, turbulence, radiation, clouds,

rain•Diurnal variations•Horizontal grid spacing of a few km•Hydrostatic and non-hydrostatic pressure

Features of CFD models•Winds, temperature, turbulence•Steady state•Horizontal grid spacing of a few m•Non-hydrostatic pressure (separation)



Grid structure of mesoscale models

Grid structure of CFD models

Terrain following coordinate

Cartesian coordinate

A few km

A few m



Combination of Models

ensemble averaged

ensemble averagedensemble averaged

instantaneous

LES RSM

RSMRANS

not yet

done



Single Model Expansion

LES

RANS

RSM

started

started

started



Cities in complex terrain

Terrain

Cities

Many cities are located in a coastal area or in the vicinity of the area where topographic influence is significant



Computation of pressure for CFD models

MAC (marker and cell) ： C.W. Hirt, 1966

Simultaneous adjustments of continuity and winds

0

z

W

y

V

x

UD

Continuity equation



kjikjil

kji Px

tUU ,,,,

1,,

1

kjikjil

kji Px

tUU ,,,,1

1,,1

1

lkjiU ,,

kjiU ,,1

kjikji P ,,,, accelerationdeceleration

Adjustment of winds ( where is iteration index)1 ll UU l



Pressure adjustmentSubstituting wind adjustment eq. into the continuity eq., we obtain

}{)

11(2 22

,,,, z

W

y

V

x

U

yxt

P kjikji

z

W

y

V

x

UD

....,, kjiP

kjikjil

kji Px

tUU ,,,,

1,,

1

kjikjil

kji Px

tUU ,,,,1

1,,1

1

D

yesno

start



Airflows and dispersion around buildings

HOTMAC for airflows and RAPTAD for puff transport and diffusion were used.

Computational domain was 200 m x 200 m x 500 m.

Grid spacing was 4 m in the horizontal direction (51 x 51 points) and 4 m for the first 10 vertical levels and increased with height (31 vertical levels).



An L-shaped building

The modeled horizontal wind distributions at 10 m above the ground.

The front part of the building is 30 m high and the back part is 14 m high.



The modeled streamlines in the vertical cross section along the east-west axis.



3D and 2D arrays of blocks

0 m

100 m

200 m

0 m 200 m 400 m 0 m 200 m 400 m

Each block is 28 x 28 x 30 (H) m Each block is 28 x 200 x 30 (H) m

No recirculation in front of the first block

Recirculation in front of the first block

Reattachment distance is ~1H

Reattachment distance is ~3.5H



3D and 2D arrays of blocks (continued)

0 m

40 m

0 m 0 m100 m0 m 100 m

The area of upward motion is smaller than the area of downward motion

The areas of upward and downward motion are similar

recirculationno recirculation



Terrain Following Coordinate

Ground elevation

Building height



Cut and Fill of the ground

New elevation

Building height



Short-away Tall-close



A city in complex terrain

Topography was extracted from digitized ground elevation data around Kobe City in Japan.

Two-way nesting was used.Domain 1: 6560 m x 8960 m with a horizontal grid spacing of 160 m

Domain 2: 1280 m x 1440 m with a horizontal grid spacing of 40 m

Domain 3: 360 m x 400 m with a horizontal grid spacing of 10 m

Building were located in Domain3.

51 vertical levels to reach 2800 m asl.

Domain 1

Domain 2

Domain 3



Simulation

Simulation started at 8 a.m., July 20 and continued for 24 hours.

Initial wind speed was 3 m/s and wind direction was westerly.

Initially sea breezes and upslope flows developed independently. They merged together around 2 p.m.

A 24 hour simulation took approximately 100 hours using a PC with 2GHz CPU and Linux OS.



Upslope flows

Sea breeze front

The modeled wind distributions in Domain 1 at 2 m above the ground at 9 a.m.

Arrows indicate wind directions, and wind speeds are proportional to the lengths of arrows.

1. Westerly flows over the ocean

2. Sea breezes along the coastal line

3. Westerly flows over the plain

4. Upslope flows over the slope



The modeled wind distributions in Domain 2 at 2 m above the ground at 9:10 a.m.

Dashed lines indicate the boundaries of Domain 3 where buildings were located.

Sea breeze front was modified by buildings.



The modeled wind distributions in Domain 3 at 2 m above the ground at 2 p.m.

Blank areas are where buildings were located. Building heights varied from 14 m to 30 m.

Upstream winds diverged as approaching the city and converged in the downstream.

Wind speeds in the city were small.



Domain 2 Domain 3



The modeled wind distributions in Domain 1 at 2 m above the ground at 11 p.m.

Upslope flows began to change to drainage winds over the slopes.

Land breezes began to form.



The modeled wind distributions in Domain 1 at 2 m above the ground at 1a.m.

Down-slope flow and land breezes merged to form northerly flows in the entire domain.



Domain 2 Domain 3



Summary

CFD and mesoscale modeling capabilities were merged together.

Sea breeze fronts were modified by buildings. Winds diverged in the upstream and converged in the downstream sides of the city.

Wind speeds and wind directions in the city changed as the winds in the outer domains encountered diurnal variations.

Future work includes verification of the model results. Observations were conducted in Salt Lake City, Oklahoma City and additional observations are planned in New York City and Washington D.C. in the near future.

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