Folie 1

1
RB Nusselt number, TC torque, pipe
friction Analogies between thermal convection
and shear flows by Bruno Eckhardt, Detlef
Lohse, SGn Philipps-University Marburg and
University of Twente
2
ß
ß
ß
data cryogenic helium approximate power law
only exponent varies with Ra ß(Ra)
3
G 1/2
G 1
G 2
Data acetone, Pr 4.0 (105ltRalt1010)
G. Ahlers, SGn, D. Lohse Physik Journal 1 (2002)
Nr2, 31-37
4
Physical origin of variable exponent Varying
weight of BL and bulk
5
Variable exponent scaling in convective transport
6
Funfschilling, Ahlers, et al. JFM 536, 145
(2005)
7
Very large Ra scaling
8
Nu versus Pr
G. Ahlers et al., PRL 84, 4357 (2000) PRL 86,
3320 (2001) K.-Q. Xia, S. Lam, S.-Q. Zhou, PRL
88, 064501 (2002)
9
M. Couette, 1890 G.I. Taylor, 1923
? r1 / r2 0.83
Conditions for thin BLs d ltlt 2 p ri , G gtgt
1 (2p1)-1 0.14 ltlt ? lt 1 thus 5.4 gtgt s(?) gt
1 s(?) (ra/rg)4 2-1(1?)/v? 4
10
Torque on inner cylinder
Kinematic torque independent of , of ,
if
Experimentally
11
Torque 2pG versus R1 in TC
Lewis and Swinney, PRE 59, 5457 (1999)
r1 16 cm, r2 22 cm, ? 0.724 l 69.4cm, G
11.4 8 vortex state
G R1a
2p
R1 (r1?1d) / ?
log-law
12
Reduced TC torque
data, variable exponent power law, log-law
B.Eckhardt, SGn, D.Lohse, 2006
13
uf-profiles measured by Fritz Wendt 1933 r2
14.70cm, l 40cm, G 8.5 / 18 / 42 r1 10.00cm
/ 12.50cm / 13.75cm ? 0.68 / 0.85 / 0.94



?
?
?


-225

-180
-248
R18.47 104
4.95 104
2.35 104
N?M1/Mlam 52 37
21
Rw 2 670 1 580
820
d/d 1 / 100 1 / 80
1 / 58
14
Exact analogies between RB, TC, Pipe
?
?
,Tas(ra?1d/?)2 s2-1(1?)/v? 4
?
15
Torque 2pG versus R1
data variable exponent log-law
Reduced torque
B.Eckhardt, SGn, D.Lohse, 2006
16
variable exponent
log-law
17
Wind ur-amplitude increases less than control
parameter R1uf
Rw R1 1-? ? 0.10-0.05
friction factor f or cf
18
? dependence of N?
r1/r2 ? 0.500
0.680
0.850
0.935
Data ? 0.935 ? 0.850 ? 0.680
(?LS 0.724)
Fritz Wendt, Ingenieurs-Archiv 4, 577-595 (1933)
19
Skin friction coefficient ? 4 cf
from Hermann Schlichting, Grenzschichttheorie,
Fig.20.1
20
Friction factor pipe flow
data variable exponent log-law
B.Eckhardt,SGn,D.Lohse, 2006
21
Summary
1. Exact analogies between RB, TC, Pipe
quantities Nu Q/Qlam, M1/M1,lam, ?p/?plam or
cf Navier-Stokes based
2. Variable exponent power laws M1 R1a with
a(R1) etc due to varying weights of BLs
relative to bulk
3. Wind BL of Prandtl type, width 1/vRew ,
time dependent but not turbulent, explicit scale
L transport BL width 1/Nu
4. N, N?, Nu as well as ew decomposed into BL and
bulk contributions and modeled in terms of
Rew
5. Exact relations between transport currents N
and dissipation rates ew e elam ew
Pr-2 Ra (Nu1), s -2 Ta (N?-1), 2Re2(Nu-1)
6. Perspectives Verify/improve by higher
experimental precision. Explain very large Ra or
R1 behaviour, very small ?. Measure/calculate
profiles ?(r),ur,uz, etc and Ns, ews. Determine
normalized correlations/fit parameters
ci. Include non-Oberbeck-Boussinesque or
compressibility.