Modeling and Simulation of Turbulent Transport of Active and Passive Scalars above Urban Heat Island in Stably Stratified Environment

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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. Kurbatskii
• Institute 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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Objectives

• ?The 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.

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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
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• Transport Equations
• for heat and mass fluxes

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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

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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

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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

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Boundary Conditions

• The 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

• Domain of integration
• is a cylinder

z

• ?At the plume axis and at its outer boundary
• symmetric conditions
• (??/?r) (??/?r) (??/?r) (???2?/?r) 0
  are prescribed.
• (Ur0 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.

Top boundary
Heat Flux, H0
r
s o u r c e
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HEAT TRANSFER BOUNDARY CONDITIONS
Z

• ?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 prescribed
• ?values 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

Top boundary
Heat Flux, H0
0
r
s o u r c e
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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

Z

• Domain of integration
• is a cylinder

Top boundary
L 0.5 D
r
mass source
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MASS TRANSFER BOUNDARY CONDITIONS
Z
Domain of integration is a cylinder
Top boundary
?The same boundary conditions are used for source
of small length located at the center of a heat
island.
L0.1D
r
mass source
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MASS TRANSFER BOUNDARY CONDITIONS
Z
Domain of integration is a cylinder
Top boundary
?and the same boundary conditions are used for
source of small length located at the periphery
of a heat island.
L 0.1D
r
mass source
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Numerical Method
Fr , Fz turbulent fluxes of momentum, heat and
mass
Semi-implicit alternating direction scheme
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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
?r/2
?z
?z/2
r
0
Ur
Uz
?E, ?, T, lt?2gt, C, ltc?gt
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Main Results of Simulation

• ?The 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

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

• 1-D1bg ltcqgt (?C/?z)-1
  1-D2 bg
  ltq2gt (?Q/?z)-1
• ?If both mass and heat are passive additives
  (the buoyancy terms in this expression are
  negligible) then it is evident from this
  expression that KCKH.
• ? 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.
•  

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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 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 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 ? 1/30?wD. This
  value was used as characteristic scale of the
  turbulent velocity for boundary conditions for
  E1 and e1 at the first level of a grid above an
  underlying surface.

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Numerical Procedure

• ?It took about 2.8104 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.