How to use CFD (RANS or LES) models for urban parameterizations

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How to use CFD (RANS or LES) models for urban
parameterizations and the problem of
averages Alberto Martilli CIEMAT Madrid, Spain 

Pieter Bruegel the Elder. The Tower of Babel.
1563. Oil on panel. Kunsthistorisches Museum,
Vienna, Austria
Martilli, Exeter, 3-4 May 2007.
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Mesoscale model numerical resolution (few
kilometers or (at best) several hundreds of
meters in the horizontal) implies a spatial
average over a volume comparable with the grid
cell.
Only features larger than the volume over which
the average is performed can be resolved.
Martilli, Exeter, 3-4 May 2007.
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But.. Spatial averages of what?
In mesoscale modelling literature I could find
two approaches
RANS approach (spatial average of non-random
fields)
LES approach (spatial filter)
Martilli, Exeter, 3-4 May 2007.
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RANS approach
Atmosphere is turbulent. Non-deterministic
behavior. The next state of the environment is
partially but not fully determined by the
previous state of the environment.
For a variable in a turbulent flow f , we define
the probability to get the value h as f(h)
(probability density function). Then the mean is
(from Pope, 2000)
This mean fulfill the Reynolds assumptions and
filters out the stochastic, random component
leaving only the deterministic component.
Parameterization of turbulent effects must be
indipendent than resolution.
Martilli, Exeter, 3-4 May 2007.
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Mesoscale models computes, then, spatial averages
of the mean variables
There is a double averaging
Martilli, Exeter, 3-4 May 2007.
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Over homogeneous terrain turbulent structures can
form randomly anywhere.
mean
The mean fields do not see horizontal variations.
So, we can reasonably assume that
Martilli, Exeter, 3-4 May 2007.
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Over heterogeneous terrain (e. g. cities), the
mean (deterministic) fields can show structure at
the scale of the heterogeneities.
The grid resolution of the mesoscale model is
bigger than the scale of heterogeneities.
There can be deterministic features that are
subgrid (not resolved, must be prameterized)
Martilli, Exeter, 3-4 May 2007.
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Some consequences of the double averaging.
Every variable can be seen as the sum of three
terms.
is the departure from the mean (turbulent part,
stochastic)
is the spatial variation of the mean field within
the spatial averaging scale
Mean deterministic structures smaller than the
averaging spatial scale.
Dispersive stress (it can be present also in
laminar flows)
Reynolds stress (turbulent)
Resolved stress
Martilli, Exeter, 3-4 May 2007.
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How to validate?
Homogeneous terrains
Where T is the integral time scale of the
turbulence (Garratt 1994).
mean
Point measurements can be used to validate the
simulations
Martilli, Exeter, 3-4 May 2007.
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How to validate?
Hetereogenous terrains
Point measurements cannot be compared with model
results. A spatial average is needed. How to
perform it?
Very dense set of measurements (difficult).
Use a CFD model (RANS or LES).
or
Martilli, Exeter, 3-4 May 2007.
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LES approach
The only average performed is a spatial average
over the grid cell and the time step (Pielke,
1984, Jacobson, 1999)
This average filters out all the features that
are smaller than the grid cell (no matter if they
are turbulent or not)
Martilli, Exeter, 3-4 May 2007.
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Advantages only one averaging process (no
dispersive stress, etc.)

• Disadvantages
• Reynolds assumption is not (strictly) valid
  (Galmarini and Thunis, 1999)
• When the grid cell size becomes close to the size
  of the most energetic eddies (hundreds of meters
  during daytime), turbulent stochastic motions
  start to be resolved (we are in Terra Incognita,
  Wyngaard, 2004).

Parameterizations should be resolution dependent
The model solution represents then, only one of
the possible states of the atmosphere (how to use
these results?? Time averages??).
Martilli, Exeter, 3-4 May 2007.
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In any case, spatial averages are also needed. As
for the previous approach, CFD (LES) models can
be used to perform such averages over
hetereogenous terrain.
Martilli, Exeter, 3-4 May 2007.
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An example with a CFD-RANS code.
FLUENT over a regular array of cubes (Santiago,
Coceal, Martilli, Belcher, about to be submitted)

• Simulations for steady state based on
  Reynolds-Averaged Navier-Stokes equations (RANS)
• Turbulence model k-? standard
• Governing equations solved by means of a
  collocated grid system using finite volume method
• Pressure-velocity coupling SIMPLE
• Advection-differencing scheme QUICK

Martilli, Exeter, 3-4 May 2007.
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• Aim use the CFD RANS simulations to perform a
  parameter study to
• Test the hypotesis of Martilli and Santiago
  (2007) on the modified drag parameterization.
• Derive values of drag coefficients (CDmod)for
  different packing densities

Introduce two new velocity scales from turbulent
and dispersive kinetic energy and define the drag
as
where
dkedispersive kinetic energy
Martilli, Exeter, 3-4 May 2007.
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Validation against DNS simulations (Coceal et al.
2006)
Periodic Boundary conditions at
inflow-outflow Symmetric on lateral.
Martilli, Exeter, 3-4 May 2007.
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Comparison DNS-RANS
(spatial averages)
Martilli, Exeter, 3-4 May 2007.
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Comparison DNS-RANS
(spatial averages)
Martilli, Exeter, 3-4 May 2007.
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RANS simulations compare worse with measurements
than DNS, (or even LES). But they are much faster
in terms of CPU (around 100 times faster than
LES).
To perform a parameter study, RANS is a better
tool.
Martilli, Exeter, 3-4 May 2007.
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Study for different lf
sparse
lf0.0625 lf0.11 lf0.16 lf0.25 lf0.33 lf0.44
dense
Martilli, Exeter, 3-4 May 2007.
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for
where A 1.0, B 6.4, C -29. and D 28..
Martilli, Exeter, 3-4 May 2007.
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Conclusions/Summary
The averaging technique chosen is important over
hetereogenous terrain (model validation,
interpretation of the results, paramterizations,
etc.)
In urban areas point measurements are not
representatives of spatial averaged values (in
particular in the urban canopy). CFD models can
be used to obtian such spatial averages
CFD-RANS are less precise than LES or DNS, but
much faster. They can be used for parameter
studies.
Using CFD-RANS simulations, a paramterization for
Cdmod have been derived as a function of the
packing density.
Martilli, Exeter, 3-4 May 2007.
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Thank you!
Martilli, Exeter, 3-4 May 2007.
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Martilli and Santiago, 2007, BLM
Different configuration alligned array
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Estimation of CD
Alternative
Introduce two new velocity scales from turbulent
and dispersive kinetic energy and define
CD(z)
where
and
kinetic energy of the time averaged structures
smaller than the grid cell
Cdmod constant with height
dkedispersive kinetic energy
Martilli, Exeter, 3-4 May 2007.
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Spatial variability of vertical profiles.
F
E
WIND
Martilli, Exeter, 3-4 May 2007.