Relaminarisation of turbulent stratified flow Bas van de Wiel Moene, Steeneveld, Holtslag
Jan 03, 2016
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