Measured local thermal dissipation rate in turbulent

Measured local thermal dissipation rate in turbulent - Indico

��
��
��
��
��
��
��
� � � ��
��
� � � ��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
Measured local thermal dissipation rate in
turbulent Rayleigh-Bènard convection
Penger Tong
Department of Physics
Hong Kong University of Science and Technology
OUTLINE:
1. Dissipations in Rayleigh-Bénard convection
2. Measurement of the local thermal dissipation rate
3. Experimental results: spatial distribution and fluctuations
4. Summary
Work done in collaboration with Xiao-Zhou He and Ke-Qing
Xia and was supported by Hong Kong RGC
1. Dissipations in Rayleigh-Bénard Convection
For isotropic turbulence, energy cascades from the large system scale L0
to the small viscous dissipation scale K, at which ReK = KuK/Q 1.
In the inertial range K < r < L0, the kinetic energy cascades at a constant
rate Hu without dissipation and thus u (r) (Hur)1/3.
Important Kolmogorov scales:
uK
K
Van Gogh's painting
Starry Night (1889)
1/ 3
(H uK )
(Q 3 / H u )1/ 4
H u Q (uK / K )
2
3
Q /K
4
For turbulent Rayleigh-Bénard convection, we have
• two local variables, v(r,t) and T(r,t), and two corresponding
dissipation rates, Hu and HT.
• two exact relations:
HN
¢H N (r ,t )²V ,t
H N (r,t)
H0
H0
Nu ( Ra, Pr)
HT (r,t)
N('T / H)2
( N u 1) Ra Pr 2
¢ H u ( r , t ) ² V ,t
Q3/H4
Flow visualization of turbulent
thermal convection in water at
Ra = 2.6u109 (6.5 cm u 4.0 cm)
Understanding heat transport, Nu(Ra,Pr), in turbulent convection
through spatial decomposition of the dissipation rates Hu and HT
Phenomenology I: boundary versus bulk (GL, JFM, 2000; PRL, 2001)
Hu Hu,BL Hu,bulk
Hu,BL Q (u / Ou )2 (Ou / H)
Hu,bulk u3 / H
HT HT ,BL HT ,bulk
HT ,BL N('T / OT )2 (OT / H)
HT ,bulk (uM)'T 2 / H
Phenomenology II: background versus plumes (GL, PoF, 2004)
Hu,BL Q (u / Ou )2 (Ou / H)
Hu,bulk u3 / H
HT , pl N('T / OT )2 (OT / H)Nplsheet
HT ,bg (uM)'T 2 / H
2. Measurement of the local thermal dissipation rate
Instantaneous local viscous dissipation rate:
H u (r , t )
Q
¦
i, j
wui wui
wx j wx j
Instantaneous local thermal dissipation rate:
H T (r , t )
wT 2
wT 2
wT 2
N [(
) (
) (
) ]
wx
wy
wz
Time-averaged local thermal dissipation rate:
¢H T (r, t )² t
wT 2
wT 2
(H T ) m (r ) (H T ) f (r ) N ¦ ¢( ) ² t N ¦ ¢( ) ² t
wxi
wxi
i
i
Total convective heat flux across the cell:
Nu
1
Jz( x, y )dxdy
³³
A
1
V
³³³ ¢H
N
(r, t )² t dr
Temperature gradient probe:
First probe: d = 0.17 mm
Gxi = 0.8 mm, GTmin 5 mK
Second probe: d = 0.11 mm
Gxi = 0.25 mm, GTmin 5 mK
OT 0.8 mm at Ra = 3.6l 109
Rayleigh-Bénard convection cell
A
cooling chamber
Pr
'T
Water
D = 20 cm
H
Ra
D
H
1, 0.5
Q
| 5 .4
N
D gH 3 ' T
QN
heating plate
four ac bridges with lock-in amplifiers
operated at f = 1kHz and ǻf = 100 Hz
Simultaneous dissipation and velocity measurements
top view
side view
Does the invasive probe perturb the local temperature field?
5
10
T4
T3
T2
T1
4
H(GT)/H0
10
3
10
2
10
1
10
0
10
-12
-8
-4
0
4
8
12
GT/VT
Multiple sensor measurements
at the center of the A = 1 cell
with Ra= 4.0 l 109
Single sensor measurements
at the center of the A = 1 cell
3. Experimental results
8
Ra =
8
2.7l109
6
HN(x)
Hn(x)
6
Ra
Õ 9.2l108
2.7l109
Ë 6.6 l109
4
4
2
2
0
0.00
0.25
0.50
x/D
0
0.00
0.25
0.50
x/D
H N (r ) H m (r ) H f (r )
(i) In the bulk region, Hf is
dominant and Hm is negligibly
small.
(ii) Near the sidewall, Hf is
~10 times larger than that
at the cell center.
Ra
Õ 9.2l108
2.7l109
Ë 6.6 l109
HN(z)
50
75
50
25
25
0
0
0.00
0.25
0.50
z/H
0.75
Ra
Õ 9.2l108
2.7l109
Ë 6.6 l109
HN(z)
75
1
10
100
z/G
(iii) Hf increases rapidly in the 1 d z / OT d 10 region and is ~140
times larger than that at the cell center.
4
HN, i(x)
3
Õ

2
Ë
Hz
Hx
Hy
1
Ra = 2.7l109
Õ
30
HN, i(z)
Ra =
40
2.7l109

Ë
20
Hz
Hx
Hy
10
0
0
0.0
0.1
0.2
0.3
x/D
0.4
0.5
1
10
100
z/G
(iv) Hf has three terms, Hf = Hx + Hy + Hz, and the dominant term
is Hz, which is twice larger than Hx and Hy.
At the cell center
Near the sidewall
2
10
HN
HN
1.05l104 Ra-0.33
1.9l103 Ra-0.33
6
1
9
10
10
10
Ra
9
10
10
Ra
(v) Hf ~ Ra-D with the exponent D 0.33 ± 0.03 both at the
cell center and near the sidewall.
10
Measured Hf when the probe is placed ~1 mm above the
lower conducting plate
200
HN
2.05l105 Ra-0.33
100
9
10
10
Ra
10
(vi) Hf ~ Ra-D with the exponent D 0.33 ± 0.03 near the
lower conducting plate.
Inside the thermal boundary layer (almost touching the
lower surface)
3.8l10-4 Ra0.77
4
HN
10
9
10
10
10
Ra
(vii) Hm ~ Ra-E with the exponent E 0.77 ± 0.1 inside
the thermal boundary layer.
5
10
(H/2OT)2 = 5.5l10-2 Ra0.57
HN
Nu2 = 4l10-2 Ra0.57
Hm= 3.8l10-4 Ra0.77
4
10
9
10
10
10
11
10
Ra
(viii) The line of (H/2OT)2 intersects the line of Hm= 3.8l10-4Ra-0.77
at Ra = 6.4l1010.
Fluctuations of the local thermal dissipation rate Hf (r,t)
Near the sidewall at Ra = 3.6u109
GT (K)
1.0
0.5
asymmetric
0.0
-0.5
dT/dx
(K/cm)
5
symmetric
0
dT/dz
(K/cm)
-5
5
slightly
asymmetric
0
-5
-10
HN
200
asymmetric
100
0
0
200
400
t (s)
600
PDF of the local thermal dissipation rate Hf (r,t)
At the cell center
0
10
Ra
f 9.6l108
Õ 2.8l109
Ë 8.2 l109
10
-3
10
Ra
f 8.6l108
Ã 2.5l109
Ë 7.7 l109
-1
10
H(HN)/H0
-1
H(HN)/H0
Near the sidewall
-2
10
-3
10
-4
10
-5
10
-5
10
-7
10
-6
0
20
HN/VH
H( H f ) = H 0 e
40
-D ( H f / V H ) E
with D = 3.9 and E = 0.35
60
10
0
10
20
HN/VH
H( H f ) = H 0 e
-D ( H f / V H ) E
with D = 3.1 and E = 0.47
30
Functional form of the measured H(Hf)
-1
0
10
10
-2
10
Gaussian
poisson
-1
cell center
10
-2
10
10
-4
10
-5
-4
10
-5
10
-6
10
Y
H(HN)/H0
-3
10
-3
Ra
Õ 9.6l108
2.8l109
Ë 8.2 l109
0
1
10
-6
10
-7
10
-8
10
-9
10
-10
2
3
4
10
-10
ln(H1/VH)
H(H f ) = H 0 e
-[ln ( H f / V H ) m ]2 / 2 J 2
-5
-1 0
5
x
log ( H
/V )
f
H
eO O
H (H f ) = H 0
[log ( H f / V H )]!
m = -3.04, J = 1.38 and O = 0.16
Cross-correlation function C(W )= ¢H N (t)G T(t+W )² / V T V H
Ra = 3.6u109
0.8
C(W)
0.6
0.4
0.2
0.0
-10
0
10
W(s)
Correlation amplitude increases from C(0) = 0.2 (cell center) to
C(0) = 0.4 (near the sidewall) and to C(0) = 0.6 (near the lower
conducting plate).
Power spectrum of the local thermal dissipation rate Hf (r,t)
At the cell center
Near the sidewall
5
10
3
10
3
10
Ra =
Ra = 7.7l109
9.3l109
10
PH(f)
PH(f)
1
1
10
-1
10
-1
10
4.0l109
-3
10
-3
0
5
10
15
10
8.6l108
0
5
f
f
PH (f ) = P0 e
-( f / f C )
10
4. Summary
•
Measured thermal dissipation rate can be decomposed into
two terms: HN(r) = Hm(r) + Hf(r), with the mean dissipation
Hm(r) concentrating in the thermal boundary layers and the
fluctuations Hf(r), which are produced by the detached
thermal plumes, occupying mainly in the plume-dominated
bulk region.
•
Measured Hf(r) ~ Ra-D with D 0.33 at the cell center and
near the sidewall and lower conducting plate.
•
Measured Hm(r) ~ Ra-E with E 0.77 inside the thermal
boundary layer.
•
At Ra = 2.7l109, Hf 0.1Nu and Hm = HN - Hf 0.9Nu.
4. Summary - continued
•
Measured histogram of Hf (r,t) has a log-normal or log-Poisson
form.
•
Cross-correlation amplitude C(0) increases from 0.2 (cell
center) to 0.4 (near the sidewall) and to 0.6 (near the lower
conducting plate).
•
Power spectrum of Hf(r,t) has a simple exponential form
PH(f) = P0exp(-f/fc).
•
The experiment reveals an interesting interplay between the
mean dissipation and the fluctuations. A minimum modeling
of the spatial distribution of HN(r) requires a decomposition of
HN(r) into two terms with Hm(r) in the thermal boundary layer
(z/OT 1) and Hf(r) in the bulk region (1< z/OT 10).
Does the invasive probe perturb the local temperature field?
0
0
10
-1
-1
10
-2
H(HN)/H0
H(GT)/H0
10
T4
T3
T2
T1
10
T4
T3
T2
T1
10
-3
10
-2
10
-3
10
-4
10
-4
10
-5
10
-8
-4
0
4
8
GT/VT
Multiple sensor measurements
near the sidewall at Ra= 3.6 l 109
0
4
8
GT/VT
Multiple sensor measurements
near the lower conducting plate
at Ra= 3.6 l 109

Download Report

Measured local thermal dissipation rate in turbulent - Indico

Paperzz.com

Your Paperzz