Large-eddy structure and low-dimensional modelling of canopy turbulence

or, “Desperately seeking Big Eddy”

John J. Finnigan1 and Roger H. Shaw2

1CSIRO, Atmospheric Research, Canberra, Australia2University of California, Davis, California

Contents

• Evidence for transport by coherent eddies in canopies

• Dynamics of the coherent structures-the Mixing Layer analogy

• An objective route to coherent structures-EOF analysis

• 3D structure of the eddy motion• A dynamic model for the coherent eddies

Joint pdf’s of w’u’ and w’c’ reveal both dominance by sweeps and intermittency of transport

Data from Rivox forest, Gardiner (1994)

Intermittency

In the upper canopy ~90% of the momentum is transferred in ~5% of the time

Ensemble u-w-T and momentum and heat flux fields obtained from wavelet transforms triggering on ramps in a forest canopy

(Collineau and Brunet, 1993)

Scalar ‘ramps’ correlated through the depth of the canopy show wholesale ‘flushing’ of the canopy airspace by large

scale gusts.

Data from Gao, Shaw and Paw U, (1989)

- 4 - 3 - 2 - 1 0 1 2 3 4

S tream w ise p o s itio n x /h

0

1

2

3

Hei

ght z

/h

S tream w ise v e lo c ity p e rtu rb a tio n

Composited velocity fields in the x-z planeLES data triggered on ramps

Composited velocity fields in the x-z planeLES data triggered on ramps

- 4 - 3 - 2 - 1 0 1 2 3 4

S tream w ise p o s itio n x /h

1

2

3

Hei

ght z

/h

V ertica l v e lo c ity

Composited velocity fields in the x-z planeLES data triggered on ramps

- 4 - 3 - 2 - 1 0 1 2 3 4

S tream w ise p o s itio n x /h

1

2

3

Hei

ght z

/h

S ca la r co n cen tra tio n p e rtu rb a tio n

What have we learned directly from measurements?

• We know a good deal about the time evolution of transport events at a point.• We know that the integral scales are large cf. the canopy height, that gusts of this size regularly ‘flush’ the canopy and that the flushing events transport large amounts of both momentum and scalars.• We know something about the shape of the gusts in the x-z plane from time-height plots of experimental data• From conditionally sampled LES output we know a little about the three dimensional structure of the large eddies.•We don’t have a model for their dynamics

Coherent structures and predictive models

• Closure models (1st order, 2nd order, etc.) can’t use information on eddy structure although it can be used to explain why they fail!

• Lagrangian transport models implicitly use the results on eddy scale but they can’t be used to model the wind field

• Large eddy simulations don’t need the information but can be tested against it

• To use our knowledge of coherent structure in predictive models we need a formulation where large eddies appear explicitly in the mathematics.

The Canopy-Mixing Layer

Analogy

Primary Instability: Kelvin-Helmholtz waves. Wavelength x is set by shear scale w.

Clumped (Stuart) vortices retain initial wavelength. Thin sheet of vorticity between rollers is rapidly strained.

Secondary instability in the vortex sheet leads to braids of streamwise vorticity that contain most of the total vorticity. Transverse spacing of braids is close to x .

Canopy-Mixing Layer Analogy (2). Linear perturbation models provide 3D eddy structure (eigenmodes) but

may not be applicable to the fully turbulent case.

An objective approach to eddy structure: Empirical Orthogonal Functions (EOF)

• We are used to expanding turbulent fields in Fourier modes- sines and cosines- which are the eigenmodes of a vibrating string.

• EOF’s are the eigenmodes (3D spatial patterns) that fit the actual turbulent flow as closely as possible in the sense that a smaller number of these eigenmodes must be added together to reproduce any given fraction of the turbulent kinetic energy than any other possible choice of spatial pattern.

EOF’s are the eigenfunctions of the 2-point covariance field

T h e E m p ir i c a l O r t h o g o n a l F u n c t io n s P h i a r e t h e s o lu t io n s t o t h e E ig e n v a lu e p r o b le m :

a re th e e ig e n v e c to rs , th e e ig e n v a lu e s

is th e 3 D d o m a in o f th e d a ta

* 3

x

x x r x r x r x

i

i j j i

D

u u d

D

EOF’s capture the spatial structure of the velocity field optimally in a least squares sense,

(EOF=POD=PCA)

The original velocity fields, the two-point covariances and the turbulent stresses can be

reconstructed from the EOF’s

x x

x x r x x r

x x x x

0

ni n i

n

n ni j n i j

n

n ni j n i j

n

nn m

u a

u u

u u

n ma a

n m

* denotes complex conjugate

(Lumley, 1981; Finnigan and Shaw, 2000)

Wind tunnel measurements of the 2-point covariance x x ri ju u

u x u x r

x x ru u

1 1

1 1

af a faf a f

RS|T|UV|W|

We obtained the 2-point covariance field

From measurements in an aeroelastic model canopy in a wind tunnel

In 1D just the first five EOF’s capture most of the TKE in the canopy layer but convergence is slower in the surface layer

In 3D the first 5 eigenmodes account for 90% of the TKE and most of the structure in the covariance fields.

The characteristic eddy

Because we constructed the empirical eigenvectors from time averaged covariances, we have lost information about the relative phases of the eigenvectors that make up the velocity patterns. That is we can reconstruct second moments but to reconstruct the velocity field that gave rise to them, we must add the information that was lost in the averaging process.

A simple hypothesis is that the relative phases are those that make the resulting velocity pattern or eddy as compact as possible in space.

With this simple assumption we can reconstruct the velocity and scalar fields of a ‘characteristic eddy”

We construct the 3D vector velocity field of the ‘characteristic eddy’ in the WT from the first five EOF’s with the weak assumption that an eddy

is a structure that is compact in space. On the x-z plane we get:

Rivox Forest Camp Borden

We have now repeated the EOF analysis on a detailed LES data set where u, v, w and scalar c were modelled Comparing the velocity

patterns on the x-z plane we get:

Wind tunnel

LES

- 4 - 3 - 2 - 1 0 1 2 3 4

rx /h

0

0.5

1

1.5

2

z/h

V ec to r p lo t o f s tream w ise an d v e rtica l v e lo c itie s o f ch a rac te ris tic ed d y

- 4 - 3 - 2 - 1 0 1 2 3 4

r y / h

0

0.5

1

1.5

2

z/h

V e c to r p lo t o f la te ra l a n d v e rtic a l v e lo c itie s o f c h a ra c te ris tic e d d y

Velocity vectors in the y-z plane

WT

LES

u -v e lo c ity

w -v e lo c ity

sca la r

- 8 - 6 - 4 - 2 0 2 4 6 8

rx /h

00.5

11.5

2

z/h

- 8 - 6 - 4 - 2 0 2 4 6 8

rx /h

00.5

11.5

2

z/h

- 8 - 6 - 4 - 2 0 2 4 6 8

rx /h

00.5

11.5

2

z/h

C o n to u rs o f th e ch a rac te ris tic ed d y in th e x z p lan e

characteristic eddy in the x-z plane from LES data

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

u -v e lo c ity

v -v e lo c ity

w -v e lo c ity

sca la r

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

C o n to u rs o f th e ch a ra c te ris tic e d d y in th e y z p lan e

The 3D eddy structure reveals that the sweep transfer of u’w’ and w’c on the plane of symmetry is flanked by ejections.

This pattern is a result of the double roller vortex structure of the characteristic eddy

Distinct sweeps and ejections occupy the y-z plane

C o n to u rs o f u 'w ' fo r th e ch a rac te ris tic ed d y

C o n to u rs o f w 'c ' fo r th e ch a rac te ris tic ed d y

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r x / h

0

0.5

1

1.5

2

z/h

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r x / h

0

0.5

1

1.5

2

z/h

In the x-z plane of symmetry momentum and scalar are transferred by the same part of the eddy

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6

r y / h

0

0.5

1

1.5

2

z/h

C o n to u rs o f u 'w ' fo r th e ch a rac te ris tic ed d y

C o n to u rs o f w 'c ' fo r th e ch a rac te ris tic ed d y

And similarly in the y-z plane

Using EOF’s to formulate a dynamic model for coherent structures in the canopy

u x a x u x t a t xi nn

in

i nn

in af af a f af af ; ,

RST

UVW

u

tU

u

xu

U

xu

u

xu

u

x

xp s s s s UC u u

u

x xC Uu u u u u u

i ii j

i

jj

i

j

ikj kj kj kj d k k

Ti

j jd i i i i

13

31

1 2

3

2

1

1

3

1

3

1

e j b g

b g b gc hn s

....

RST

UVW

u

tu

u

x ta t x a a Fi

ji

j

n n n m nj i

mn

af af c h

1

5

1

5

Conclusions

• There is strong experimental evidence for the importance of large coherent structures in canopy turbulence

• The analogy between canopy flow and that in plane mixing layers provides a qualitative explanation for the origin of the eddies and suggests ways to scale eddy dynamics

• EOF’s provide the 3D structure of canopy coherent structures without any a priori assumptions. We can see that they are double roller vortices with complex 3D structure

• Forcing the empirical eigenmodes to be compatible with the canopy flow equations generates a dynamic non-linear model for the coherent structures and suggests a new direction for canopy dynamics.