nato asi conference, kyiv-20041 modeling and simulation of turbulent penetrative convection and...

NATO ASI Conference, Kyiv-2004 1

MODELING AND SIMULATION OF TURBULENT PENETRATIVE

CONVECTION AND POLLUTANT DISPERSION ABOVE THE URBAN HEAT

ISLAND IN STABLY STRATIFIED ENVIRONMENT

A.F. KurbatskiyInstitute of Theoretical and Applied Mechanics SB RAS

Novosibirsk State UniversityNovosibirsk, RussiaL.I. Kurbatskaya

Institute of Computational Math. and Math. Geophysics SB RASNovosibirsk, Russia

NATO ASI Conference, Kyiv-2004

O u t l i n e Introduction Objectives Turbulent Transport Models for

Environmental Stratified Flows Modeling and Simulation of Urban

Heat Island Phenomenon and Pollutant Dispersion

Numerical Results Conclusion


IntroductionFor stratified atmospheric flows the LES models

and third-order closure models should be considered as fundamental research tools because of their large computer demands.

A growing need for detailed simulations of turbulent structures of stably stratified flows motivates the development and verification of computationally less expensive closure models for applied research in order to reduce computational demands to a minimum.


ObjectivesThe algebraic modeling techniques can be used

in order to obtain for buoyant flows the fully explicit algebraic models for turbulent fluxes of the momentum, heat and mass.

The principal object of this work is the development of three-four-parametric

turbulence model minimizes difficulties in simulating of turbulent transport in stably stratified environment and reduces efforts needed for the numerical implementation of model.

cE 2


Governing Equations

Governing equations describing the turbulent stratified environmental flows are being written down in the hydrostatic approximation at absence of the Coriolis force and radiation with use a Boussinesq approximation.

6NATO ASI Conference, Kyiv-2004

Governing Equations in RANS-approach

TgxPuu

xU

xtDUD

ii

jij

i

j

i

1

jjj

uxxtD

DPr

cu

xC

ScxtDCD

jjj


Transport Equations for heat and mass fluxes

cgxUcu

cuEx

UcuxCuu

tDcuD

gcxUuc

uE

cxUu

xuu

tDuD

icj

ijc

icj

ij

jji

tcu

i

ij

ij

ij

ij

jji

tu

i

)1(

)1(

22

1

222

1

D

D


Explicit Algebraic Expressions for Turbulent Fluxes

The explicit algebraic models for the turbulent heat flux vector and turbulent mass vector were derived by truncation of the closed transport equations for turbulent fluxes of heat and concentration by assuming weak equilibrium, but retaining all major flux production terms.

For turbulent stresses we applied eddy viscosity expression.


CLOSURE: full explicit turbulent fluxes models for active (heat) and passive (mass) scalars

2

21

222

1

2

21

2 1θ 2 2 (1 )2

11(1 ) 2 ,2

2 12 2 (1 )2

(1

jij T t t

i j i

j it i

i j j

jij D t c t

i c j i

UUE R Eu C R Cx C x x

U U C EC R gx x x C

UUE C R Eu c C R Dx x x

22

1

11) 2 .2

Turbulent stresses

/ / (2 / 3) .

j i cc t i

i j j c

i j t i j j i ij

U U C ED R g cx x x

u u U x U x E


CLOSURE: three-equation model for active (heat) scalar field 2E ε θ

E E

1 2

22

,

,

1 .

EDE P D GDtD D P P GDtD P DDt R E


CLOSURE : four-equation model for passive scalar field2 c E ε

.

1

,

,

3

22

cEtD

cDERtD

D

GtD

D

GtDED

ccc PD

PD

PPD

PD

21

EEE


Modeling of Urban Heat IslandThe ability of the proposed full explicit algebraic

models for turbulent fluxes of heat and mass to reproduce correctly the environmental flows with a strong thermal stratification was tested on a large-scale circulation flow above an urban heat island


Modeling of Urban Heat Island

In the phenomenon of the unsteady turbulent penetration convection above an urban heat island the two remarkable features are shown.

The first, due to heating from bellow the interactions between stable and unstable regions occur, because the mixed turbulent ground layer to grow into a stable region.


Modeling of Urban Heat IslandThe second, there is the entrainment of overlaying non-

turbulent fluid into mixed layer causing very step gradients at the interface.

These features explain why the phenomenon of urban heat island represents a very challenging test case for turbulent models.


ObjectivesThus, the principal aim

of this investigation is the modeling and simulation of large-scale turbulent circulation flow above the urban heat island and pollutant dispersion in the stably stratified environment.


Limitations of Laboratory

Measurements for Full-scale Simulation

There are important limitations utilized in the laboratory experiment and simulation of the real urban heat-island in the nighttime atmosphere:

Very large heat fluxes from the heater surfacesVery strong temperature gradients that required to

obtain the low aspect ratios (zi/D) and small Froude numbers.


Structure of heat-island circulation

The penetrative turbulent convection is induced by the constant heat flux H0 from the surface of a plate with diameter D. It simulates a prototype of an urban heat island with the low-aspect-ratio plume (zi / D « 1) under near calm conditions and stably stratified atmosphere.


NUMERICAL MODELING OF HEAT ISLAND CIRCULATION

The problem of development of circulation above a heat island is assumed to be axisymmetric.

The domain of integration is a cylinder of a given height .


Numerical Method

f

zF

rF

rtzr

1

cCTEUr ,,,,,,: 2

Fr , Fz – turbulent fluxes of momentum, heat and mass

Semi-implicit alternating direction scheme

nn

zn

rnji

nji

nn

zn

rnji

nji

fzF

rF

rt

fzF

rF

rt

12/1,

1,

2/1,

2/1,

12/

12/


MeshThe numerical

method uses a staggered mesh.

The difference equations are solved by the three-diagonal-matrix algorithm.

Staggered mesh

z

r

z

0

r/2

r

z/2

Ur Uz

E, , T, <2>, C, <c>


Main Results of SimulationThe results of simulation correspond to a

quasi-steady state of circulation over an area heat source in stable stratified environment.

Figure (c): shadowgraph picture at t = 240 sec when the full circulation is established.


Calculation of Normal Turbulent Stresses

In this problem a simple gradient transport model preserves certain anisotropy of the normal turbulent stresses

rUEu r

tr 2

322

zUEu z

tz 2

322

2Ect is turbulent viscosity.


RESULTS: Temperature profiles Calculated temperature profiles

inside the plume have characteristic “swelling”:

the temperature inside the plume is lower than the temperature outside at the same height creating an area of negative buoyancy due to the overshooting of the plume at the center.

This behavior indicates that the plume has a dome-shaped upper part in the form of a “hat”.

29

CONCLUSION CONCLUSION

The three-equation model of turbulent transport of heat reproduces structural features of the penetrative turbulent convection over the heat island in a stably stratified environment.

This model minimizes difficulties in describing the non-homogeneous turbulence in a stably stratified environment and reduces computational resources required for the numerical simulation.

nato asi conference, kyiv-20041 modeling and simulation of turbulent penetrative convection and...

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