Shallow Flows Symposium, TU Delft, 2003

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Modeling and Simulation of Turbulent Transport of Active and Passive

Scalars above Urban Heat Island in Stably Stratified Environment

Albert F. KurbatskiiInstitute of Theoretical and Applied Mechanics of Russian

Academy of Sciences, Siberian Branch

Russia, Novosibirsk

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Introduction

For 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.

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Objectives

The principal aim of this investigation is the development of turbulent transport model for the simulation of the urban-heat-island structure and pollutant dispersion in the stably stratified environment.

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ObjectivesThe algebraic modeling techniques can be used in order to

obtain for buoyant flows the fully explicit algebraic model 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 the turbulent transport in stably stratified environment and reduces efforts needed for the numerical implementation of model.

cE 2

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The mathematical model of the urban heat-island

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.

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The mathematical model of the urban heat-island

The mixing height, zi , is defined as a height where the maximum negative difference between the temperature in the center of the plume and the ambient temperature Ta is achieved.

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The mathematical model of the urban heat-island

The problem of the development and evolution turbulent circulation above a heat island is assumed to be axisymmetric.

At the initial moment the medium is at rest and it is stably stratified.

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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 surfaces

Very strong temperature gradients that required to obtain the low aspect ratios (zi/D) and small Froude numbers.

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Governing Equations

Governing equations describing the turbulent circulation above a heat island can be written in the hydrostatic approximation at absence of the Coriolis force and radiation with use a Boussinesq approximation.

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Governing Equations in RANS-approach

TgxP

uux

U

xtD

UDi

iji

j

i

j

i

1

jjj

uxxtD

DPr

cu

xC

ScxtDCD

jjj

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Transport Equations

for heat and mass fluxes

cgx

Ucu

cuEx

Ucu

xC

uutD

cuD

gcx

Uuc

uE

cx

Uu

xuu

tD

uD

icj

ijc

icj

ij

jji

tcu

i

ij

ij

ij

ij

jji

tu

i

)1(

)1(

22

1

222

1

D

D

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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.

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

scalars

.)3/2(//

.21

21

)1(

21

)1(22

2

,21

21

)1(

21

)1(22

2θ

1

22

21

2

2

1

22

21

2

ijijjit

ic

c

jj

i

i

jtc

i

j

j

itct

ciD

ijj

i

i

jt

i

j

j

itt

iT

ExUxU

cgRE

xC

x

U

x

UD

x

U

x

UD

ERxC

RE

C

gRE

C

C

xx

U

x

UC

x

U

x

UC

ECR

xR

EC

ji

j

j

uu

cu

u

stresses Turbulent

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Three-Equation ModelE - - 2

The closure of expressions for the turbulent stresses and heat flux vector is achieved by solving the equations for turbulent kinetic energy, its dissipation rate and temperature variance, resulting

in three-equation model E--2

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CLOSURE: three-equation model

for active (heat) scalar field 2θεE

.1

,

,

22

ERtDD

GtD

D

GtDED

PD

PPD

PD

21

EEE

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Four-equation modelE - - 2 - c

For the closure of expression for turbulent flux vector of a passive scalar is involved the equation for covariance concentration – temperature.  

 Thus, for the description of a concentration field is formulated the four-

equation model E--2-c  

 

 

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CLOSURE : four-equation model

for passive scalar field θεE c2

.

1

,

,

3

22

cEtD

cD

ERtDD

GtD

D

GtDED

ccc PD

PD

PPD

PD

21

EEE

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Boundary ConditionsThe domain of integration is a cylinder of a given height H. The heated circular disc of diameter D is located at the center of the cylinder bottom.

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HEAT TRANSFER BOUNDARY CONDITIONS

At the plume axis and at its outer boundary

symmetric conditions

(/r) = (/r) = (/r) (2/r) = 0 are prescribed.

(Ur=0 at r = 0 and at r =1.8R)

At the top boundary

the zero-flux condition

V/z = /z = /z =

= 2/z = 0 is enforced.

Domain of integration

is a cylinder

rz

s o u r c e

Top boundary

Heat Flux, H0

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HEAT TRANSFER BOUNDARY CONDITIONS

The surface heat source is placed on the bottom (z = 0) has the size 0 r / D 0.5.

Boundary conditions at the bottom are specified as

heat flux H0 is prescribedvalues of E, and 2 at the first level above surface

are chosen according to Kurbatskii

(JAM, 2001, vol.40, No.10)

Domain of integration

is a cylinder

Z0

Top boundary

s o u r c e

Heat Flux, H0

r

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MASS TRANSFER BOUNDARY CONDITIONS

At the plume axis and at outer boundary,

(C/r) = (c/r) = 0.

At the top,

Constant flux of mass,

is prescribed inside a source.

At the bottom and outside of a source

Domain of integrationis a cylinder

Z

r

Top boundary

mass source

L = 0.5 D

.0// zczC

cHzCD )/(

,0 c.0/ zC

MASS TRANSFER BOUNDARY CONDITIONS

The same boundary conditions are used for source of small length located at the center of a heat island.

Domain of integration

is a cylinder

Z

r

Top boundary

mass source

L=0.1D

MASS TRANSFER BOUNDARY CONDITIONS

and the same boundary conditions are used for source of small length located at the periphery of a heat island.

Domain of integration

is a cylinder

Z

r

Top boundary

mass source

L= 0.1D

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Numerical Method

fzF

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/

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Numerical Procedure

The 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>

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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.

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Calculation of Normal Turbulent Stresses

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

rU

Eu rtr 2

322

zU

Eu ztz 2

322

2Ect is turbulent viscosity.

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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”.  

  

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Pollutant Dispersion

Investigation of buoyancy effects on distribution of mean concentration in mixing and inversion layers of urban heat island was the main goal in modeling and simulation of pollutant dispersion from a continuous surface source.

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Pollutant Dispersion

Experimental measurements were not available for the quantitative validation of simulation data.

Instead, we present some preliminary results that illustrate interesting properties of pollutant dispersion from a continuous source located inside the urban heat island and on its periphery.

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Pollutant Dispersion

One characteristic feature is observed in all three cases. The contaminant penetrates not only into the inversion layer but even higher beyond its boundary.

This behavior was recently reproduced in the laboratory experiment of Snyder et al. (BLM,2002, vol.102, 335-413.). These experimental data clearly show penetration of the continuous buoyant plumes into inversion above the convective boundary layer.

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Difference in Turbulent Diffusion Between Active and Passive Scalars 1

Sometimes assumed that in the stratified atmospheric boundary layer the eddy diffusivity of heat (KH) is equal to the eddy diffusivity of contaminant (KC). However, the stratification causes a larger difference in the eddy diffusivities between active heat and passive mass.

Indeed, for the ratio of the vertical eddy diffusivities of heat and mass can be written the following expression

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Difference in Turbulent Diffusion

Between Active and Passive Scalars 2 [1g <cC/z)]

[1 g <2/z)]

If both mass and heat are passive additives (the buoyancy terms in this expression are negligible) then it is evident from this expression that KC=KH.

It appears that cases for which the largest deflection of KC/ KH

from unit will occur are when either T or C is acting as a passive additive.

In our case the mass is acting as a passive additive.

 

Hc KK /

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Turbulent Fluxes of Active and Passive Scalars

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Ratio of Eddy Diffusivities of Passive Mass to that Active Heat

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CONCLUSIONS 1CONCLUSIONS 1

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.

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CONCLUSIONS 2CONCLUSIONS 2

The four-equation model for the description of pollutant dispersion in the stable stratified atmospheric boundary layer is formulated.

Favorable comparison the numerical results of pollution dispersion from the continuous surface source above the urban heat island with laboratory measurements in the convective boundary layer showing penetration of the continuous buoyant plumes into inversion above the convective boundary layer is found.

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The friction velocity u(r) / wD

The friction velocity on the underlying surface can be obtained on the calculated mean velocity as u (r) = ( Ur / z ).

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Turbulent Velocity Scale

Turbulent velocity scale Uf was estimated as 1/30 of “a mean wind velocity ” – velocity scale wD of the mean inflow velocity: Uf 1/30wD. This value was used as characteristic scale of the turbulent velocity for boundary conditions for E1 and at the first level of a grid above an underlying surface.

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Numerical Procedure

It took about 2.8·104 time steps to drive the numerical solution to a quasi-steady state.

Computations were performed on a mesh with 25 (and 50) points in radial direction.

In vertical direction 116 (and 232) mesh points were used.

The time step was chosen so that the numerical accuracy was preserved.