how to use cfd (rans or les) models for urban parameterizations – and the problem of averages...

How to use CFD (RANS or LES) models for urban parameterizations – and the problem of averages

Alberto Martilli CIEMATMadrid, Spain

Martilli, Exeter, 3-4 May 2007.

Pieter Bruegel the Elder. The Tower of Babel. 1563. Oil on panel. Kunsthistorisches Museum, Vienna, Austria

Only features larger than the volume over which the average is performed can be resolved.

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.

cellVcell

dxdydzV

1


‘LES’ approach (spatial filter)

But.. Spatial averages of what?

In mesoscale modelling literature I could find two approaches:

‘RANS’ approach (spatial average of non-random fields)


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 , we define the probability to get the value as f() (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.

‘RANS’ approach

df )(

Parameterization of turbulent effects must be indipendent than resolution.


Mesoscale models computes, then, spatial averages of the mean variables

There is a double averaging


Over homogeneous terrain turbulent structures can form randomly anywhere.

The mean fields do not see horizontal variations.

mean

So, we can reasonably assume that:


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)

Some consequences of the double averaging.

Every variable can be seen as the sum of three terms.

uuu

uuu

uuuu

~

~

is the spatial variation of the mean field within the spatial averaging scale

is the departure from the mean (turbulent part, stochastic)

wuwuwuuw ~~

Resolved stress Reynolds stress (turbulent)Dispersive stress (it can be present also in laminar flows)

Mean deterministic structures smaller than the averaging spatial scale.


How to validate?

mean

Homogeneous terrains:

T

dtT

1

Where T is the integral time scale of the turbulence (Garratt 1994).

Point measurements can be used to validate the simulations


How to validate?

Hetereogenous terrainsT

dtT

1

Point measurements cannot be compared with model results. A spatial average is needed. How to perform it?

or

Very dense set of measurements (difficult).

Use a CFD model (RANS or LES).


LES approach

The only average performed is a spatial average over the grid cell and the time step (Pielke, 1984, Jacobson, 1999)

tt

t

/xx

/xx

/yy

/yy

/zz

/zz

dxdydzdt)t,z,y,x(tzyx

2

2

2

2

2

2

1

This average filters out all the features that are smaller than the grid cell (no matter if they are turbulent or not):


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).

The model solution represents then, only one of the possible states of the atmosphere (how to use these results?? Time averages??).


Parameterizations should be resolution dependent

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.


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


Aim: use the CFD RANS simulations to perform a parameter study to 1) Test the hypotesis of Martilli and Santiago (2007) on the modified

drag parameterization. 2) 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

222

2

1

2

wvutke

tkevtke

222 ~~~2

1

2

wvudke

dkevdke

k

k

dketkekkDk u

uvvuCD

kk

222mod

where

dke=dispersive kinetic energy

Validation against DNS simulations (Coceal et al. 2006)

Periodic Boundary conditions at inflow-outflowSymmetric on lateral.


Comparison DNS-RANS


(spatial averages)

To perform a parameter study, RANS is a better tool.

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).


Study for different f

f=0.0625f=0.11f=0.16f=0.25f=0.33f=0.44

sparse

dense


))((

2

xyt

ff ShSh

h

A

A

32ffffmodd )(C DCBA

where A = 1.0, B = 6.4, C = -29. and D = 28..

for 0.440.0625 fλ


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.


Thank you!


Pressure drag P1 P2

kSxk

obsi

dsPnV

D 1

Sink of momentum

How to parameterize the drag?

Usually for vegetation canopy flows:

kkkDk uuCD

k building density

Different configuration alligned array

Martilli and Santiago, 2007, BLM


Estimation of CD

kkk

kD

uu

DC

CD(z)

Alternative

Introduce two new velocity scales from turbulent and dispersive kinetic energy and define

222

2

1

2

wvutke

tkevtke

222 ~~~2

1

2

wvudke

dkevdke

k

k

dketkekkDk u

uvvuCD

kk

222mod

where

and

dke=dispersive kinetic energy

k

k

dketkekk

kD

u

u

vvu

DC

kk

222mod

Cdmod constant with height kinetic energy of the time averaged structures smaller than the grid cell


Spatial variability of vertical profiles.


WIND

E F

I HAB C D G

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