Requirements to Predict the Surface Layer with High Accuracy

at High Reynolds Numbers using Large-eddy Simulation*

James G. Brasseur & Tie Wei

Pennsylvania State University

NCAR TOY Workshop Geophysical Turbulence Phenomena

Turbulence Theory and Modeling29 February 2008

*supported by the Army Research Office

*m

z dUu dz

i

z

z

0

0.2

0.4

0.6

0.8

1.0

0

surface layer

Fundamental Errors in LES Predictions in the Surface Layer of the Atmospheric Boundary

Layer

1

what should be predicted

neutral boundary

layer

*

Uu

z/

neutral boundary layer

rough wall

U(z)

*

Uu

z/

neutral boundary layer

*m

z dUu dz

i

z

z

0

0.2

0.4

0.6

0.8

1.0

0

surface layer

Fundamental Errors in LES Predictions in the Surface Layer of the Atmospheric Boundary

Layer

what is actually

predicted

1

*m

z dUu dz

Chow, Street, Xue, Ferziger JAS 62, 2005

what should be predicted

neutral boundary

layer

neutral boundary layer

rough wall

U(z)

The Importance of the Overshoot

Khanna & Brasseur 1998, JAS 55

z

L

*m

z dUu dz

8iz

L

U

isosurfaces of vertical velocity:

up: w > 0 (yellow)down: w < 0 (white or green)

Moderately Convective

Atmospheric Boundary Layer

horizontal integral scale of w

z

zi

w

U

w

Why the Overshoot Alters Turbulence Structure

Khanna & Brasseur 1998, JAS 55

8z

L

u

Moderately Convective ABL

U

z/zi = 0.05 z/zi = 0.10

z/zi = 0.25 z/zi = 0.50

z/zi = 0.05 z/zi = 0.10

z/zi = 0.25 z/zi = 0.50

Consequences of the Overshoot

horizontal

integral scale

z

ziz

L

*m

z dUu dz

Over-prediction of mean shear in the surface layer produces poor predictions throughout the ABL of:

turbulence production

thermal eddying structure (e.g., rolls)

vertical transport, dispersion and eddy structure of momentum, temperature, humidity, contaminants, toxins, …

correlations, turbulent kinetic energies,…

cloud cover, CO2 transport, radiation, …

Moderately Convective ABL

16-year History of the Overshoot

1. Mason & Thomson 1992, JFM 242.2. Sullivan, McWilliams & Moeng 1994, BLM 71.3. Andren, Brown, Graf, Mason, Moeng, Nieuwstadt &

Schumann 1994 QJR Meteor Soc 120 (comparison of 4 codes: Mason, Moeng, Neiustadt, Schumann).

4. Khanna & Brasseur 1997, JFM 345.5. Kosovic 1997, JFM 336.6. Khanna & Brasseur 1998, JAS 55.7. Juneja & Brasseur 1999 Phys Fluids 11.8. Port-Agel, Meneveau & Parlange 2000, JFM 415.9. Zhou, Brasseur & Juneja 2001 Phys Fluids 13.10. Ding, Arya, Li 2001, Environ Fluid Mech 1.11. Reselsperger, Mahé & Carlotti 2001, BLM 101.12. Esau 2004 Environ Fluid Mech 4.13. Chow, Street, Xue & Ferziger 2005, JAS 6214. Anderson, Basu & Letchford 2007, Environ Fluid

Mech 7.15. Drobinski, Carlotti, Redelsperger, Banta, Masson &

Newson 2007, JAS 64.16. Moeng, Dudhia, Klemp & Sullivan 2007 Monthly

Weather Rev 135.

Relevant to any LES of boundary layers where the viscous sublayer is

unresolved or nonexistent.

… enhanced with direct exchange between inner and outer boundary

layer:

i

z

zz

L

*m

z dUu dz

• resolution3

1283 1923•

Clues from Previous Studies

1. The overshoot is tied to the grid

Juneja & Brasseur 1999, Phys. Fluids 11

l ~ z

zl ~ z

2. Inherent under-resolution at the first grid level

Khanna & Brasseur 1997, JFM 345

i

z

zz

L

*m

z dUu dz

••

Clues

1. The overshoot is tied to the grid

Juneja & Brasseur 1999, Phys. Fluids 11

Khanna & Brasseur 1997, JFM 345

3. The overshoot is sensitive to

the SFS model

Chow, Street, Xue & Ferziger 2005, JAS 62

4. Lack of grid independence

l ~ z

zl ~ z

2. Inherent under-resolution at the first grid level

not strictly a modeling issue.

Smag

Sullivan, McWilliams & Moeng 1994, BLM 71

Into the Future

1. Mason & Thomson 1992, JFM 242.2. Sullivan, McWilliams & Moeng 1994, BLM 71.3. Andren, Brown, Graf, Mason, Moeng, Nieuwstadt & Schumann 1994 QJR Meteor Soc

120 (comparison of 4 codes: Mason, Moeng, Neiustadt, Schumann).4. Khanna & Brasseur 1997, JFM 345.5. Kosovic 1997, JFM 336.6. Khanna & Brasseur 1998, JAS 55.7. Juneja & Brasseur 1999 Phys Fluids 11.8. Port-Agel, Meneveau & Parlange 2000, JFM 415.9. Zhou, Brasseur & Juneja 2001 Phys Fluids 13.10. Ding, Arya, Li 2001, Environ Fluid Mech 1.11. Reselsperger, Mahé & Carlotti 2001, BLM 101.12. Esau 2004 Environ Fluid Mech 4.13. Chow, Street, Xue & Ferziger 2005, JAS 6214. Anderson, Basu & Letchford 2007, Environ Fluid Mech 7.15. Drobinski, Carlotti, Redelsperger, Banta, Masson & Newson 2007, JAS 64.16. Moeng, Dudhia, Klemp & Sullivan 2007 Monthly Weather Rev 135.

What is known after 15 years:

1. The overshoot is fundamental to LES of shear-dominated surface layers.

2. The overshoot is somehow connected to the grid.

3. The overshoot is somehow connected to the SFS stress model.

Today’s Discussion:

1. We have (finally) found the source(s) of the overshoot and its consequences.

2. The solution is a framework in which high-accuracy LES can be developed.

3. The framework involves and interplay between: (1) grid resolution, (2) SFS model, (3) numerical algorithm, (4) grid aspect ratio.

The First Discovery: ScalingMean Smooth-Wall Channel Flow

z

Ttot= Tt + TTtT

friction-dominated

inertia-dominated

2* tot tT TuP

x z

T

z z

U /P x

stationary, fully developed, mean:

2* tTuS

z z

*2 2* *

m m

T

u

zS

u u

inertial scaling:

integrate 0 z: in friction-dominated laye1 r m t

zz T z

exceeds 1 when 2.5 0. ( 4 )m z !

~ 5 10z

' 'tT u w

2*

tt

TT

u

zz

*/ u

( )U

T S zz

Smooth-Wall Channel Flow

DNS data from Iwamoto et al., Jimenez et al..

z+ 2.5

m

in frictional layerm z

peak at z+ 10

zz

Tt

Tm

Ttot= Tt + T

Conclusions1. In the smooth-wall channel flow the overshoot in m is real.

2. The real overshoot in m arises from applying inertial scale z in a

frictional layer that has characteristic viscous scale */ u

The First Discovery: ScalingMean LES of high Re or Rough-Wall Channel Flow

z

Ttot= TR + TSTRTSinertia-dominatedthroughout

U

2* tot R SuP

x z

TT

z

T

z

/P x

stationary, fully developed, mean:

( ) ˆ1r r rr ri j

SFS

j

j

j

ii

i

u uu p

t x x x

2*

RR

TT

u

Inertial Scaling… integrate 0 z: near the surface1

( )1 m L

LESR

leS

sE R

z zT

zz T

*/LESLES

zz

u

( )lesles

LES

z

13

' 'r

S SS

rR

FT

T u w

under-resolution of integral scalesat first few grid levels

1

13

, (

( ( ))2

)LES lesr

Sles

Tz z

z

S

1

we fi

,

nd

2 rSFSij

L tES

t ijS

IF

Extract Viscous Content

of SGS Model:define “LES viscosity”for strong shear flow

The First Discovery: A Spurious Frictional Surface Layer

ConclusionThe overshoot in m arises from applying an inertial

scaling to a numerical LES “viscous” layer

near the first grid v l le em LESz in friction-dominat ed layer= m z

DNS: Smooth-wall Channel Flow

~10

zz

Ttot= Tt + T

*m

z dUu dz

Tt

Reynoldsstess

T

viscous

stress

i

zz

TR

resolvedstess

TS

SFS

stress~10

LESz

Ttot= TR + TS

*m

z dUu dz

LES: Rough-wall ABL

*/

zz

uv where parameterizes

real friction */LLES

ESv

zz

u where LES parameterizes

friction in the (inertial) SFS stress

The First Discovery: A Requirement to Eliminate the Overshoot

*LES

LES

u

a numerical LES “viscous” scale

To eliminate the overshoot the ratio TR/TS must exceed a

critical value ~ O(1) at the first grid level.

1

*

z

~ (1> )R

S

T

TO

The overshoot arises from “numerical-LES friction” at the surface

akin to the real frictional layer on smooth walls

TR

resolvedstess

TS

SFS

stress~10

Ttot= TR + TS

*m

z dUu dz

LES of the Rough-wall ABL

LES

LES

zz

LES is a “numerical-LES viscosity”

The Second Discovery: Relative Inertia to Friction in the Real BL

**, whereRe /

uu

* to support an

inertial surface layer

Re > Re

U /P x

DNS data from Iwamoto et al., Jimenez et al..

m

z/

m

z

The Second Discovery: Relative Inertia to LES Friction in the

Simulation

* =ReLES

LESLES

u

LES Reynolds Numberdefine

* /LES LES u *Re ReLES LES to support an

inertial surface layer

Scaling

1/ 2 2 1/ 2 2 *1

1 1

1Smag model: | 2 ( ) 2 ( )LES t s s

z

uUC C

z

22 , ( ) | |SFS rij t ij t sS C S

2 / 3( ) , where yx

z z z

AR AR

S

1

2LE

4/3

2 Re

( )S

N

C AR

resolution grid in vertical z

N

LES LES

2 4/3, 1/ ( )Re Re SC ARN

Numerical LES Viscous Effects at the Surface: Vertical Grid Resolution and TR vs. TS

LES, Eddy Viscosity (Smag):

increasing ReLES

by increasing N

m

z/zi

TRTS

z/zi

LES

1

2

4/3

2

( )Re

S

NC

N

AR

subcriticalReLES from insufficient resolution

Why the Overshoot is Tied to the Grid

1

LES

2 4 / 3

*

2 4 / 3

2

( )

, fixed ( )

LES S

S

z

z

v C AR

u

C AR

increasing ReLES

by increasing N

z/zi

the overshoot cannot be “solved” with resolution

10LES

LES

zz

Putting the two Discoveries Together

2. To remove the overshoot in mean gradient,

the resolved to SFS stress at the first grid level

must exceed a critical value1

( / )R S zT T

1. For the simulation to have the possibility

of producing a complete inertial surface

layer,

an LES Reynolds Number

must exceed a critical value,

requiring a minimum vertical resolution

*ReLES

LESLES

u

*N

*ReLES

* ~ (1)O

The Third Discovery

1R

S

T1e

TR LESN

The Parameter SpaceReLES

0

2

*

ReLES

*ReLES

0

0.2

0.4

0.6

0.8

1.0

i

z

z

0 1

*m

z dUu dz

increasing N

1

N

*δN

0

1ˆcos( , ) 0.9x

NS e

N

The LES must reside here to eliminate the overshoot and resolve a full surface layer

0

In general,

Designing High-Accuracy LES

1R

S

T1e

TR LESN

ReLESIn the Parameter Space

1 2 4 / 3( )R 2eLES

SC

N

AR

2 4 / 3

21 1

(

2

)SC AR

If using the Smagorinsky model:

For any SFS stress model: Moving the simulation

into the “High-Accuracy

Zone (HAZ):

1. Adjust resolution in the

vertical so that

2. Adjust AR + model constant together until

* * and Re ReLES LES

*N N

0

2

*

ReLES

*ReLES

*N

increasing N

1

N

0

HAZ

Numerical Experiments

0

1

*

ReLES

*ReLES

*N

N

1

N

1 2 4 / 3( )R 2eLES

SC

N

AR

2 4 / 3

21 1

(

2

)SC AR

1R

S

T1e

TR LESN

* from the literature

Nz = 32 Nz = 64

Nz = 96 Nz = 128

Numerical Experiments

*

ReLES*ReLES

*N

N

N0.2 6

N0.2 12

Nz = 64

N 30

Nz = 128

N 60

N0.2 9

N0.2

3

Nz = 32

N 15

Nz = 96

N 45

Resolution N

N0.2 6

N0.2 12

Nz = 64

N 30

Nz = 128

N 60

N0.2 9

N0.2

3

Nz = 32

N 15

Nz = 96

N 45

Resolution N

1.under-resolution alters the entire boundary layer structure

2.~ 9 points in surface layer required

* 45N

Resolution N

* 0.85

*Re 350LES

* 45N

Numerical Experiments

0

1

*

ReLES

*ReLES

*N

N

1

N

1 2 4 / 3( )R 2eLES

SC

N

AR

S

1RT 1

TReLESN

2 4 / 3

21 1

(

2

)SC AR

Designing High-Accuracy LES

S

1RT 1

TReLESN

ReLESThe Parameter Space

1 2 4 / 3( )R 2eLES

SC

N

AR

2 4 / 3

21 1

(

2

)SC AR

Nz = 128Nz = 96

Nz = 128

Nz = 96

A Current Issue: Numerical Instability

Conclusions

High-accuracy LES 1. removal of the overshoot in mean gradient2. sufficient resolution of the surface layer

We have created a framework for developing high-accuracy LES:

To create high-accuracy LES the simulation must move into a “High-Accuracy Zone” (HAZ) through variation of• vertical grid resolution• grid aspect ratio• friction in model (e.g., model constant) and algorithm

Instability arises as the simulation moves into the HAZ:• Tie will discuss next

Re parameter spacethe LES

Extra: Cs used in simulations

Extra: AR used in simulations

Note: For this plot, Tie used the effective AR based on explicit dealiasing filter. To get true AR, each of these should be divided by 1.5

Effective AR