Relaminarisation of turbulent stratified flow

Bas van de WielMoene, Steeneveld, Holtslag

Overview

1) Motivation

2) A simple Couette flow analogy

3) Pressure driven flow: comparison with DNS

4) Conclusion and perspectives

(1)

Motivation

Why does the wind drop in the evening?

Classical picture of continuous turbulent quasi-steady SBL:

z

pot. T

t=0t=3 t=2 t=1

(Nieuwstadt, 1984)

z

Tw

t

T

)''( Quasi-steady:

Shape profiles cst.0

)''(2

2

z

Tw

z

T

t

Linear heat flux profile

Central question:

what happens for low pressure gradients?

Continuous turbulent, quasi-steady nocturnal boundary layer only observed for strong pressure gradient conditions

(high geostrophic winds)

Observational example(Cabauw, KNMI, Netherlands):

-30

-20

-10

0

10

1800 1830 1900 1930 2000 2030 2100 2130 2200 2230

Time [hour]

Sen

sib

le h

eat

flu

x [W

/m2]

decoupling

Tran-sition

•Clear sky conditions

•Little wind near surface

0

0.05

0.1

0.15

0.2

1800 1900 2000 2100 2200

Time [hour]

Fri

ctio

n v

elo

city

[m

/s]

Collapse of turbulence→

decoupling of the surface from the atmosphere

0

10

20

30

40

-4 -3 -2 -1 0

T-T_40 [C]

hei

gh

t [m

]

time=1810time=1850time=2000time=2100time=2130time=2140time=2150time=2200

zTemperature profilesQuasi-steady

T

Rationale present work “Yet not every solution of the equations of motion, even if it is exact, can actually occur in nature. The flows that occur in nature must not only obey the equations of fluid dynamics but also be stable.” Landau and Lifschitz (1959)

We hypothesize that:

1) The continuous turbulent SBL is hydrodynamically stable for high pressure gradient and are therefore observed in nature.

2) The continuous turbulent SBL is hydrodynamically unstable for low pressure gradient and are therefore not observed in nature. Instead a SBL with collapsed turbulence is observed.

In fact we aim to find the transition T-L!

(2) A simple Coutte flow model

Some characteristics:

•First order turbulence closure based on Ri

•No radiative divergence

•Rough flow using Z0=0.1 [m]

BC’s:•Top: Wind speed and temperature fixed •Bottom: No slip and fixed surface heat flux

U_Top T_Top

δ

Ho

Van de Wiel et al. (2006)

Flows, Turbulence and Combustion, submitted

Turbulence closure First order closure:

Two major elements controlling dominant eddie size:

stratification and presence solid boundary

zt

U

1

z

H

ct

T

p

1

z

UKm

z

TK

c

HH

p

RifzUlK nmH )(2, zln

2)1(Rc

RiRif

0Rif

cRRi

cRRi 2)( zU

zT

T

gRi

ref

•Non-trivial in a sense that collapse of system as whole occurs way before Rc!

•Support locality of TKE in strongly stratified flow e.g.:

Nieuwstadt ’84, Lenshow, ’88, Duynkerke ’91(Observations)

Mason and Derbyshire ’90, Galmarini ’98, Basu ’05 (LES)

Coleman et al. 1992 (DNS); also recall presentation by Clercx

Results

0

0.1

0.2

0.3

0 2 4 6 8

Time [hr]

Fric

tion

velo

city

[m/s

]

H=-10.00 [W/m2]; d/L=0.15

H=-15.25 [W/m2]; d/L=0.52

H=-15.40 [W/m2]; d/L>0.52*

H=-18.00 [W/m2]; d/L>0.52*

g

T

Tw

uL

0

0

3*

)''(

Continuous turbulent case

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0

T -T_top[K]

z/d

[-]

Initial profile0.5 hr10.0 hrANALYTIC

Continuous turbulent case

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

U [m/s]

z/d

[-]Initial profile0.5 hr10.0 hrANALYTIC

Collapse case

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0

T -T_top[K]

z/d

[-]

Initial profile2.0 hr4.5 hr4.85 hr

Collapse case

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

U [m/s]

z/d

[-]

Initial profile2.0 hr4.5 hr4.85 hr

Positive feedback mechanism:

(following Van de Wiel et al. 2002, J. Atmos. Sc.).

Temp. gradient

do

wn

wa

rd h

ea

t tr

an

sp

ort

z

TK

c

HH

p

Increasing gradient:

Equilibrium solutions: bifurcation analysis

)ln( 0* z

Uu TOPN

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

H/Hmax [-]

u */u

*N [-

]

)(

)ln(

27

4ˆ0

03*max z

z

g

cuH refpN

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

H/Hmax [-]

u */u

*N [-

]

Linear stability analysis

(i.e. on logarithmic profiles e.g. not linear!)

z

T

zb

zbzzz

U

zb

zbzzt

U PPP

ˆ

ˆ

ˆ1

ˆˆ

ˆ2

ˆ

ˆ

ˆ1

ˆ42ˆ

ˆˆ

ˆ

z

T

zb

zbzzz

U

zb

zbzzt

T PPP

ˆ

ˆ

ˆ1

ˆ21ˆ

ˆˆ

ˆ

ˆ1

ˆ41ˆ

ˆˆ

ˆ

eqLb

0)ˆˆ(ˆ 0 zzu p

0)ˆˆ( 0 zzH p

0)1ˆ(ˆ zu p

0)1ˆ(ˆ zp

)ˆˆexp()(ˆ),(ˆ tzutzU pP )ˆˆexp()(ˆ),(ˆ tztzT pP Ansatz:

(1-D!)

BC’s

Criterion for instability

0

0

12

ln

z

z

LCRITICALeq

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 0.5 1 1.5

d/L-critical [-]

Z0/d

[-]

Marginal curveText example

Agreement between theory and numerical results!

0.55

Previous example:

CRITICALeqL

=0.52

Continuous turbulent cases

Relaminarised cases

Thus:

•Collapse of SBL turbulence explained naturally from a linear stability analysis on the governing equations

(assuming local closure)

•The crucial question:

how close is our model in comparison with reality (here say reality~DNS)

(3) Comparison with DNS results from Nieuwstadt (2005)

Pressure

force

Cooling

BC’s

Top: stress free, fixed T

Bottom: no slip, prescribed heat extraction

Smooth flow; Re*= 360

(3) Comparison with DNS results from Nieuwstadt (2005)

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30

U/u* [-]

z/h

[-] 1-D Model

AnalyticalDNS

*0 135.0

uZ

We used a priori: (smooth flow)

Remarkable in view of origin model

(3) Comparison with DNS results from Nieuwstadt (2005)

*0 135.05.1

uZ

A posteriori

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30

U/u* [-]

z/h

[-] 1-D Model

AnalyticalDNS

DNS shows collapse at h/L~1.23 [-]

Note:

TKE normalised with u*^2

Our model shows collapse at h/L~1.45 [-]

A priori threshold h/L~1.55

h/L=1.5

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

5.E-03

6.E-03

0 10 20 30

t/t* [-]

u* [m

/s]

h/L=1.4

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

5.E-03

6.E-03

0 20 40 60 80

t/t* [-]

u* [m

/s]

Predicting relaminarisation:Generalisation of the results

Note: 135.0

Re

135.0

* *

0

uh

Z

h

10

100

1000

10000

0 0.5 1 1.5 2 2.5

h/L [-]

Re

* [

-] Continuous turbulent cases

Relaminarised cases

Summary/conclusions:

•Relaminarization of turbulent stratified shear flows is predicted from linear stability analysis on parameterized equations

•In this way relaminarization critically depends on two dimensionless parameters: Re* (or Z0/h) and h/L

•The results seem to be confirmed by recent DNS results (at least in a qualitative sense)

zWind speed profiles Quasi-steady

U

0

20

40

60

80

100

120

140

0 2 4 6 8

U [m/s]

hei

gh

t [m

]

time=1810time=1850time=1950time=2100time=2130time=2140time=2150