Direct numerical simulation study
of a turbulent stably stratified air flow
above the wavy water surface.
O. A. Druzhinin, Y. I. Troitskaya
Institute of Applied Physics, RAS, Nizhny Novgorod, Russia
and
S.S. Zilitinkevich
Finnish Meteorological Institute, Helsinki, Finland
OBJECTIVE
In this work the detailed structure and statistical
characteristics of a turbulent, stably stratified atmospheric
boundary layer over waved water surface are studied by
direct numerical simulation (DNS). Two-dimensional water
wave with wave slope up to ka = 0.2 and bulk Reynolds
number up to Re = 80000 and different values of the bulk
Richardson number Ri is considered. The shape of the water
wave is prescribed and does not evolve under the action of
the wind. The full, 3D Navier-Stokes equations under the
Boussinesq approximation are solved in curvilinear coordinates
in a frame of reference moving the phase velocity of the
wave. The shear driving the flow is created by an upper
plane boundary moving horizontally with a bulk velocity in the
x-direction.
Schematic of numerical experiment
Lx  6
L y  4
Lz  
GOVERNING EQUATIONS
2
U i  (U iU j )

Ui
P
1
~



  iz Ri T f (t )
t
x j
x j Re x j x j
U j
x j
 0.
~ (U T~)
2~
T
1
 T
j

Uz 
t
x j
Pr Re x j x j
Re =
~
T  1 z  T
U 0

T2  T1 
Ri = g
T1 U 02
f (t )  1  exp( t / 100)
CURVILINEAR COORDINATES
x    a exp(k ) sin k
z    a exp(k ) cos k
1 2
z b ( x)  a cos kx  a k (cos 2kx  1)
2
Mapping over η:
~
tanh



  0. 5  1 

tanh 1.5 

 1.5  ~  1.5
BOUNDARY CONDITIONS
U ( , y,0)  cka cos kx( , )  1
Bottom plane:
V ( , y,0)  0
W ( , y,0)  cka sin kx( , )
U ( , y,1)  1  c
Top plane:
V ( , y,1)  0
W ( , y,1)  0
~
~
T ( , y,0)  T ( , y,1)  0
All fields are x and y periodic
NON-STRATIFIED FLOW ABOVE WAVY
BOUNDARY: COMPARISON WITH DNS BY
SULLIVAN ET AL. (2000)
(<u> + c)+/A
(u'2 + u2w )/u2*
(w'2 + ww2)/u2*
2
-( + w )/u *
8
(à)
Re = 10000, c=0.25
1.5
(b)
6
1
ka = 0.2
kz
4
0.5
2
0
0
1
10
z/z0
100
8
-2
(c)
0
2
4
6
8
1.5
10
(d)
ka = 0.1
6
1
kz
4
0.5
2
0
0
1
10
z/z0
100
-2
0
2
4
6
8
10
NON-STRATIFIED FLOW ABOVE FLAT
BOUNDARY: COMPARISON WITH
PREVIOUS RESULTS
Audin & Leutheusser (1991), Re = 9524
Papavassiliou & Hanratty (1997), DNS, Re = 10640
3
u'/u*
2
v'/u*
w'/u*
1
0
0
40
80
120
z+
160
200
NON-STRATIFIED FLOW ABOVE FLAT
BOUNDARY
Re=80000
Re=40000
25
25
<U+0.5>/u
*
z+
2.5ln z++5.3
20
Mean velocity
profile
20
15
15
10
10
5
5
(a)
0
0
1
10
100
1000
-<U'W'>/u2
*
d<U>/dz/Re/u2
*
total
1.6
Momentum flux
budget
(e)
1
10
100
1.6
1.2
1.2
0.8
0.8
0.4
0.4
(b)
0
(f )
0
1
10
100
0.4
Kinetic energy
budget
1000
1000
1
10
100
1000
0.4
production
dissipation
diffusion
visc. diffusion 0.2
total
0.2
0
0
-0.2
-0.2
(g)
(c)
-0.4
-0.4
1
10
z+
100
1000
1
10
z+
100
1000
Re = 40000:
u* = 0.0256
z* = 0.001
Re = 80000:
u* = 0.0235
z* = 0.00053
MOMENTUM AND HEAT FLUXES BUDGET
IN THE STATIONARY STATE FOR FLAT
BOUNDARY
Heat flux budget:
Momentum flux budget:
T
T
Pvisc
 Pturb
 u T
Pvisc  Pturb  u 2
Pvisc
1 d Ux 

Re
dz
Pturb    U xU z 
T
visc
P
1 d T 

Re Pr dz
T
Pturb
   T 'U z 
KINETIC AND POTENTIAL ENERGY BUDGET
IN THE STATIONARY STATE FOR FLAT
BOUNDARY
Kinetic energy:
d Ux  d
1
  U xU z 

p'U ' z  (U ' 2x U ' 2y U ' 2z )U ' z 
dz
dz
2
1 d  U '  1  U 'i


2
2 Re
Re  x j
dz
2
2
i




2
~
 Ri  T U ' z  0
Potential energy:
d T  1 d
1 d  T'
1
2
  T 'U ' z 

U 'z T ' 

2
dz
2 dz
2 Re Pr dz
Re Pr
2
 T ' 


 x 
 j
2
0
STRATIFIED FLOW ABOVE FLAT
BOUNDARY, Ri = 0.05
Re = 40000
Re = 80000
50
Mean velocity
and temperature
profiles
50
<U+0.5>/u
*
z+
2.5(ln(z+)+2+5z/L)
40
40
30
30
30
20
20
10
10
0
10
100
1000
-<U'W'>/u2
*
d<U>/dz/Re/u2
*
total
1.6
20
10
10
(a)
0
1
10
100
1000
(e)
0
1
10
100
1000
1
1.6
1.2
1.2
1.2
1.2
0.8
0.8
0.8
0.8
0.4
0.4
0.4
0.4
0
(f )
10
100
1000
0
1
10
100
1000
production 0.4
dissipation
buoyancy
diffusion
0.2
visc. diffusion
total
0.4
0.2
0
0
-0.2
-0.2
-0.4
1
10
100
10
z+
100
1000
10
z+
100
1000
1
0
0
-0.2
-0.2
(g)
1
(f )
1000
10
100
1000
production
dissipation
diffusion
0.2
visc. diffusion
total
(g)
(c)
-0.4
1
1000
0
0.2
(c)
100
(b)
0
1
10
-<W'T'>/(u*T*)
1.6
d<T>/dz/Re/Pr/(u*T*)
total
1.6
(b)
Kinetic and
potential energy
budget
30
20
(e)
0
1
Re = 80000
40
<T-1>/T
*
z+
Prt(ln(z+)+5z/L+2.5)
40
(a)
Momentum and
heat flux budget
Re = 40000
1
10
z+
100
1000
1
10
z+
100
1000
RELAMINARIZATION OF STRATIFIED
FLOW ABOVE THE FLAT BOUNDARY
(Re = 40000)
u'
v'
w'
T'
Fluctuations
amplitudes
Ri = 0.05
(a)
0.16
0.16
0.12
0.12
0.08
0.08
0.04
0.04
0
0
0
400
800
t
1200
Ri = 0.1
(b)
1600
Turbulent regime
0
200
400
600
800
t
Laminar regime
RELAMINARIZATION OF STABLY
STRATIFIED FLOW ABOVE THE FLAT
BOUNDARY
Re = 15000
Re = 40000
Re = 80000
U
0.04
T
*
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
0
0.04
0.08
Ri
0.12
0.16
0
*
0.04
0.08
Ri
0.12
0.16
RELAMINARIZATION OF STRATIFIED
FLOW ABOVE THE FLAT BOUNDARY
Re = 80000, Ri = 0.11
Re = 40000, Ri = 0.09
Re = 30000, Ri = 0.075
Re = 15000, Ri = 0.06
Lu*/
400
300
Flores & Riley (2011): L+>100 for
stationary turbulent regime
200
2.5  U ' x U ' z  
L
 Ri  T 'U ' z 
3/ 2
100
0
0.4
N
 
Lu

2
 2.5 Prt
 2.5 Prt  N   2.5 Prt 
2
v
N
 LOZ
0.3
0.2
0.1
0
0.2

/Loz
LOZ   N
0.16
0.12
  
0.08
0.04
0
0.1
1
10
z+
100
1000

1 1 / 2

3 1 / 2

, 
3

1 1 / 4



4 / 3
EFFECT OF THE WAVE
(ka = 0.2, c = 0.05)
u'
v'
w'
T'
Ri = 0.04
0.25
Re = 15000
0.16
0.15
0.12
0.1
0.08
0.05
0.04
0
0
200
0.16
400
Ri = 0.1
0.2
0.2
0
Re = 40000
Fluctuations
amplitudes
600
0
200
0.1
Ri = 0.1
600
Ri = 0.2
0.08
0.12
400
0.06
0.08
0.04
0.04
0.02
0
0
0
200
400
600
800
t
Turbulent regime
0
200
400
t
600
800
Wave-pumping regime
EFFECT OF THE WAVE
(c = 0.05, Re = 15000)
u'
v'
w'
T'
ka = 0.2
Ri = 0.08
0.1
0.025
0.08
0.02
0.06
0.015
0.04
0.01
0.02
0.005
0
0
0
0.1
0.2
0.3
0.4
0.5
0
0.04
Ri
Turbulent
regime
Wave-pumping
regime
0.08
0.12
0.16
0.2
ka
Laminar
regime
Wave-pumping
regime
INSTANTANEOUS VORTICITY MODULUS FIELD:
TURBULENT REGIME
c=0.05, Ri=0.04, t=1000
1
z
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
x, y=0
1
z
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
2
y,x=3
2
1.5
1
0.5
y
0
-0.5
-1
-1.5
0.5
1
1.5
2
2.5
3
3.5
x,z=0.042
4
4.5
5
5.5
6
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
TURBULENT REGIME
(ka =0.2, c = 0.05, Ri = 0.04)
< Ux >
<T>
Ri g
(a)
Mean velocity
and temperature z
profiles
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
Fluctuations
profiles
0
(c)
0.02
0.04
0.06
0.08
(d)
-<U'W'>
d<U>/dz/Re
Pwave
Ptotal
0.0012
Momentum flux
budget
T'
U'x
U'y
U'z
-<W'T'>
d<T>/dz/Re/Pr
PTwave
PTtotal
0.002
0.0015
0.0008
0.001
0.0004
0.0005
0
0
-0.0004
-0.0005
0
0.1
0.2
z
0.3
0.4
0
0.1
0.2
z
0.3
0.4
Heat flux
budget
MOMENTUM AND HEAT FLUX BUDGET IN
THE STATIONARY TURBULENT STATE FOR
WAVY BOUNDARY
Momentum flux:
Heat flux:
Pvisc  Pwave  Pturb  u2
Pvisc
P
1 d  U x 

Re
d
Pturb
Pwave
T
visc
T
visc
P
P
T
turb
 Tu
1 d  T 

Re Pr d

  U xU z 
 z
  z
2

 U x  P   
 U xU z
 
  
P
T
wave

T
Pturb
   T U z 



T
wave
P
 z

 TU x
 
  z
  
 TU z
  



TURBULENT REGIME:COMPARISON
WITH QUASILINEAR THEORY
(ka = 0.2, Ri = 0.04, Re = 15000)
Quasi-linear
theoretical model
(Troitskaya et al. 2013)
0.5
0.5
0.4
0.4
0.3
0.3
Tm
Um
DNS
0.2
0.2
0.1
0.1
0
0
0.001
0.01
0.1
1
Z
-0.0002
u*2-u~w~
0.01
-0.0004
-0.0006
-0.0008
0.1
1
Z
0
<w'T'>-w~T~
0
0.001
0.001
Mean velocity and
temperature profiles
Momentum and
heat fluxes profiles
-0.0002
-0.0004
-0.0006
-0.0008
0.01
0.1
Z
1
0.001
0.01
0.1
Z
1
RELAMINARIZATION OF STRATIFIED
FLOW ABOVE THE WAVY BOUNDARY
0.4
N
Ri = 0.1, Re = 40000, ka = 0.2
Ri = 0.06, Re = 15000, ka = 0.2
Ri = 0.09, Re = 40000, ka = 0
Ri = 0.05, Re = 15000, ka = 0
0.3
0.2
2.5  U ' x U ' z  
L
 Ri  T 'U ' z 
3/ 2
0.1
0
0.1
0.2
1
10
100
1000
/Loz

LOZ   N
0.16

3 1 / 2
0.12
  
0.08
0.04
0
0.1
1
10
100
1000

1 1 / 2

, 
3

1 1 / 4
INSTANTANEOUS VORTICITY MODULUS FIELD:
WAVE-PUMPING REGIME
c=0.05, Ri = 0.08, t=1400
1
z
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
x, y = 0
4.5
5
5.5
6
1
z
0.5
0
-2
-1.5
-1
-0.5
0
y, x = 3
0.5
1
1.5
2
2
1.5
1
0.5
y
0
-0.5
-1
-1.5
0.5
1
1.5
2
2.5
3
3.5
x, z = 0.12
4
4.5
5
5.5
6
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
WAVE-PUMPING REGIME
(ka = 0.2, c = 0.05, Ri = 0.08)
Mean velocity and temperature,
and Rig profiles
<Ux >
<T>
Ri g
(a)
z
Fluctuations profiles
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
T'
U'x
U'y
U'z
2
0
0.01
0.02
0.03
0.04
WAVE-PUMPING REGIME
(ka = 0.2, c = 0.2, Ri = 0.08)
Mean velocity and temperature,
and Rig profiles
<U >
x
<T>
Ri g
(a)
z
Fluctuations profiles
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
T'
U'x
U'y
U'z
3
0
0.01
0.02
0.03
0.04
POWER SPECTRA
(ka = 0.2, c = 0.05, z = 0.08, Re = 40000)
Ri = 0.1
Ri = 0.2
x-sp
y-sp
Ex,y
k -5/3
0.01
x-sp
y-sp
Ex,y
0.01
0.001
0.001
0.0001
0.0001
1e-005
1e-005
1e-006
1e-006
1e-007
1e-007
1e-008
1e-008
1e-009
1e-009
1e-010
1
10
k
100
1
10
k
100
TURBULENT MOMENTUM AND HEAT FLUXES:
EFFECTS OF THE WAVE AND STRATIFICATION
Ri = 0.04, c = 0.05
Ri = 0.06, c = 0.05
Ri = 0.08, c = 0.05
Ri = 0.04, c = 0.2
Ri = 0.08, c= 0.2
Ri = 0.04, no wave
-<U'W'>
-<T'W'>
0.0008
0.0008
0.0006
0.0006
0.0004
0.0004
0.0002
0.0002
0
0
0
0.1
0.2
z
0.3
0.4
0.5
0
0.1
0.2
z
0.3
0.4
0.5
CONCLUSION
The DNS results show that the properties of the boundary layer
flow are significantly affected by stratification. If the Richardson
number Ri is sufficiently small, the flow remains turbulent and
qualitatively similar to the non-stratified case. On the other hand, at
high Ri turbulent fluctuations and momentum and heat fluxes decay to
zero at low wave slope but remain finite at sufficiently large ka
(>0.12). Parameterization of turbulent and heat production, diffusion
and dissipation is also performed by a closure procedure and
compared with the results of DNS. The criteria in terms of the
product of the Kolmogorov time scale and local buoyancy frequency
or/and the ratio of the Kolmogorov vs. Ozmidov lengh scales is
proposed to characterize the different flow regimes observed in
DNS.
Instantaneous vorticity modulus field
(ka=0.2, c=0.05, Ri=0.08):
Wave-pumping regime
Z
Y
X