Status of component 4, “urbanLS”… testing the four optional Status of component 4, “urbanLS”… testing the four optional configurations (forward/backward, 0configurations (forward/backward, 0thth/1/1stst order) order)

John Wilson, 22 August/06John Wilson, 22 August/06

• Context

• Random Displacement Model• is it well-mixed?• does it agree with standard dispersion

data?

• Performance of variants (0f, 0b, 1f, 1b) of urbanLS relative to standard dispersion data

• Performance of urbanLS for Oklahoma City

Context:Context:

High resolution weather analysis/prediction: “Urban GEM-LAM”

Building-resolving k- turbulence model: “urbanSTREAM”(steady state, no thermodynamic equation, control volumes congruent with walls)

Provides upwind and upper boundary conditionsProvides upwind and upper boundary conditions

Lagrangian stochastic model “urbanLS” to compute ensemble of paths from source(s); now offers four options 0f, 0b, 1f, 1bnow offers four options 0f, 0b, 1f, 1b

Provides computational mesh over flow domain and Provides computational mesh over flow domain and these gridded fields: these gridded fields:

,/'','', kjijij xuuuuu

• A zeroth-order Lagrangian stochastic model, also called the “Random Displacement Model” (RDM), does not explicitly model particle velocity, and (by some criteria) is equivalent to an eddy-diffusion treatment… however it is a Lagrangian method, thus grid free

• It is far less demanding, computationally, than the 1st-order LS model… and in the far field of a source, the RDM/eddy diffusion treatment is acceptable… as I will demonstrate here

Why is the RDM of interest?Why is the RDM of interest?

The complexity of a 1The complexity of a 1stst - order order LS algorithm… - order order LS algorithm…

dU a dt C dt G

dX U u dt

d t T

i i

i i i

L

0

aR

xC R U R

R

xU u U U

T T U T U U

ii

ij j ji

kj k j k

i ij j ijk j k

12

12 0

1 1

0 1 2

1

2

The T’s involve the mean velocity field, TKE dissipation rate , and the stress tensor Rij . They are computed and stored on the grid prior to computing the ensemble of paths. At each timestep, use T’s from gridpoint closest to particle (ie. no interpolation to particle position)

( G a standardized Gaussian random variate)

Relative simplicity of the 0Relative simplicity of the 0thth - order LS algorithm - order LS algorithm

dX uK

xdt K dt Gi i

i

ii

( )2

and no requirement that LTdt K R Ti ii L ( )

where

Attractive to use 0th-order LS for its speed. Criteria:

• is the algorithm well-mixed?• does it replicate standard experiments?

Will address these questions in context of simplest real world (atmospheric) regime of flow, viz., neutral surface layer

“Well-mixedWell-mixed”?

uniform initial density “p”

t = 0 t > 0

Still uniform?

Reference dispersion data traceable back to Project Prairie GrassReference dispersion data traceable back to Project Prairie Grass

Ideal neutral surface layer(no horiz. gradients)

zuS

kKzu

c

v*0* ,,

zsrc = 0.46 m

100 m

Detect crosswind-integrated concentration

9 min

Q

uz *0 universal function of

0z

z

x/z0

z/z0

Form of the RDM (0Form of the RDM (0thth-order LS) for neutral surface layer-order LS) for neutral surface layer

.const*

z

Kzu

S

kK

c

v

dZ mk u

Sdt K dt G

dXu

k

Z

zd t

v

c

v

*

* ln

2

0

*

0.constu

zdt

Forward model m=1

Do we reverse this (m = -1) in a backward model?

Analytical and numerical tests for well-mixed property (nb! Analytical and numerical tests for well-mixed property (nb! grid-free LS algorithm)grid-free LS algorithm)

L

zr

dzzpztzptzp 111 )0,()0,|,(),(

Chapman-Kolmogorov equation:

Upper and lower reflection boundaries

Initial state

(= 1 for present test)

Final state

“transition density”

Analytical and numerical tests for well-mixed propertyAnalytical and numerical tests for well-mixed property

)()()0,|,( 1 nrr ppztzp

0rz

1z

z

no reflection: path length

reflection path has total length

tzK

zzz

tzKztzp r

)(4

2exp

)(22

1)0,|,(

1

21

1

1

|| 1zz

rzzz 21

where .const`

zzz

Kt

14.5 min

0 0.5 1 1.5 2

p(z, t)

1

10

100

z

Chapman-KolmogorovRandom FlightsRF (no drift)

100

*

z

ut

Upper reflection at L=4000 z0

Lower reflection at zr=0

RDM with surface reflection is not a well-mixed modelRDM with surface reflection is not a well-mixed model

Flow regime: ideal neutral surface layer

L =

… … but it (RDM) gives excellent simulation of reference dispersionbut it (RDM) gives excellent simulation of reference dispersion

0 0.001 0.00210

100

1000

z/z 0

RFRF (mu=0)RF (zr>0)

0.001 0.002

z0 u* / kv Q

10

100

1000dt

101

“True” normalized, crosswind-integrated concentration at x/z0=2000 is 1.48 x 10-3

0 0.004 0.008

z0 u* / kv Q

10

100

1000

z/z 0

m=10-1

And treatment of the drift term?....And treatment of the drift term?....

“Truth”… do not reverse drift term

Now test actual codeNow test actual code (urbanLS3.for) against reference dispersion by scaling PPG onto the urbanSTREAM grid for Oklahoma City: at its highest resolution, z=3 m

(x, y irrelevant since horizontal gradients vanish)

• urbanLS is not a grid-free Lagrangian model; unless the grid resolves the strong near-ground gradients of the ASL precise forward/backward 0/1-order consistency should not be expected; if one simulates (eg.) PPG57 with z0= 0.006 m on this grid at full scale , then the velocity statistics in the lowest layer (k=2) represent the flow in the range

50000

z

z

• therefore specify z0=zc(2)/2=0.75 m and a source-detector separation of 2000z0 scales to 1500 m. Discretization error is greatly reduced, because the wind statistics in the plume layer are represented by more than 50 layers

• perfect reflection at zrefl=z0

TimestepTimestep

0th-order simulations:

1st-order simulations: 1

LT

t

0

22

CT w

L

)(

)(

Ku

Ixt

x (I)

)(Ku

• forward-backward consistency of 1st order simulations• only modest sensitivity to timestep, but do need t/TL smaller than 0.1 to attain agreement within one std error with reference dispersion data• bigger impact of t on backward than forward simulation

0th-order forward simulation excellent; backward very sensitive to t

• forward simulation good, and not very sensitive to whether one sets drift term to zero in lowest layer• backward simulation very sensitive to whether drift term is reversed, to whether

it is zeroed in lowest layer, and to t

Focus on 0Focus on 0thth-order simulations:-order simulations:

• 1st-order forward and backward consistent, in good agreement with the reference dispersion data, only weakly sensitive to t

• 0th-order forward simulation in good agreement with the reference dispersion data, even with large t, and only weakly sensitive to inclusion or neglect of drift term in lowest layer

• 0th-order backward simulation demands drift term should not be reversed, but should be zeroed in lowest layer… else spurious vertical gradient arises

• remains to comprehend the 0th-order backward simulations, which for the time being I distrust

Recapitulate implications of tests of 0f, 0b, 1f, 1b against reference dispersion case:Recapitulate implications of tests of 0f, 0b, 1f, 1b against reference dispersion case:

26.5 min

Simulation of Gas Plume from a source in Oklahoma CitySimulation of Gas Plume from a source in Oklahoma City

continuous source1.9 m above ground

Intensive Observation Intensive Observation Period 9, July 27, 2003: Period 9, July 27, 2003: 0615-0630 0615-0630

wind

200 forward paths, displayed 200 forward paths, displayed only below 50 m heightonly below 50 m height

wind

350 m

(Not to scale)

#74

800 900 1000 1100

y [m]

2000

2100

2200

2300

2400

2500

2600

2000

8000

1400

0

600 700 800 900 1000 1100 1200

y [m] (Crosswind)

2000

2100

2200

2300

2400

2500

2600

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2800

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x [m

]

52 53 54 55 56

62 63 64 6566

72 73 74 76

8384 86

94 96

512 513514 515 516

517

N

22002400

26002800

x [m]

600800

10001200y [m]

0

5000

10000

15000

C [ppt]

0

5000

10000

15000

C [ppt]Sampler positions

source

Mean ground-level concentration [parts per trillion] from Mean ground-level concentration [parts per trillion] from forward LS simulationforward LS simulation

• fully 3D, Cfully 3D, C00=4.8=4.8• dt = 0.05 min (Tdt = 0.05 min (TLL, , x/u )x/u )• zzreflrefl=0.1 m=0.1 m• reset veloc. fluc if > 6reset veloc. fluc if > 6• 9 x 80,000 paths9 x 80,000 paths• detector half-widths 20 x 20 x 1.75 mdetector half-widths 20 x 20 x 1.75 m• execution time 44 hrsexecution time 44 hrs(3.68 min per 1000 paths)(3.68 min per 1000 paths)

Paths displayed only below z = 25 m

600 800 1000 1200

y [m]

2000

2200

2400

2600

2800

3000

x [m

]

52 53 54 55 56

62 63 64 6566

72 73 74 76

8384 86

94 96

512 513514 515 516

517

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x [m]

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y [m]

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C [ppt]

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15000

C [ppt]

Performance of (1f) Lagrangian model relative to experimentPerformance of (1f) Lagrangian model relative to experiment

52 53

54 55

56

62

63

64 65 66

7273 74

76

83

8486

94 96

513

515

517

52 53 54 55 56 62 63 64 65 66 72 73 74 76 83 84 86 94 96 51251351451551651710

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100000

C [

pp

t]

expt (06:15-06:30)urbanEUurbanLS (17apr/06)

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Comparison of performance of (1f) Lagrangian and Comparison of performance of (1f) Lagrangian and Eulerian solutions relative to experiment…Eulerian solutions relative to experiment…

Mean concentration [ppt]Mean concentration [ppt]

Backward simulation… from detectors #54, 55, 56, 64Backward simulation… from detectors #54, 55, 56, 64

• evidence suggests 0f option in urbanLS is fast and reliable

• puzzles remain relative to 0b option

• remains to repeat the 0f, 0b, 1f, 1b consistency tests in disturbed flow

• evidence suggests 0f, 1f, 1b options are all very satisfactory for realistic urban simulations

• however no evidence yet that implementing Lagrangian approach driven by urbanSTREAM wind statistics offers any greater (or lesser) accuracy than Eulerian approach available in urbanSTREAM

ConclusionConclusion

31.5 min