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Discussion of the Klimontovich Theory of Hydrodynamic Turbulence.

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Annalen der Physik. 7. Polge, Band 40, Heft 1, 1983, S. 26-33
J. A. Barth, Leipzig
Discussion of the Klimontovich Theory
of Hydrodynamic Turbulence
By W. EBELING
Sektion Physik der Humboldt-Universitit zu Berlin
A b s t r a c t . The Klimontovich Theory which yields the effective turbulent viscosity as a linear
function of the Reynolds number is compared with experimental data. The new theory seems t o be
able to describe the observed effective viscosity of hydrodynamic vortices and after introducing Home
modifications the flow in tubes up t o very high Reynold numbers.
Uiskussion dcr Klimontovich-Theoric der Hydrodynamischen Turbuleiia
I n h a l t s i i b e r s i c h t . Die Klimontovich-Theorie, welche die effektive turbnlente Viskositiit ids
lineare Funktion der Reynolds-Zahl ausdriickt, wird niit experimentellen Daten verglichen. Die n e w
Theorie scheint zur Beschreibung der effektiven Viskositat von hydrodynamischen Wirbeln und n w h
Einfuhrung einiger Modifikationen aurh zur Dnrstrllung der Striimung in Rohren bis zu sehr Iinhen
Reynolds-Zahlen in der Lage zu sein.
1. Introduction
The probleiii of the transition to turbulence has perplexed many workers since the
original papers of Helmholtz which appeared in 1858 i.e. 12:i years ago and the papers
of Reynolds which appeared 100 years ago in 1883 under the titel “On the experiniental
investigation of the circumstances which determine whether the motion of water shall
be direct or sinuous, and the law of resistance in parallel channels”. The term “sinuous”
has been replaced by “turbulent” nowadays ; however, the problem of turbulence is not
yet solved in spite of the efforts of such excellent physicists like Soninierfeld, Heisenherg,
Weizsiicker, llamb, Landau, Prandtl, Karman, Taylor and others. Here we shall investigate a special new direction of the theory which has been investigated b y K 1 . 1 ~ 0 x TOVICH in 1981 [ I , 2, 31.
2. Entropy Production and Effective Viscosity
Considering an inconipressible liquid KLIMONTOVICH
assiinies that the random coniponent of the velocity produces entropy as well as the mean velocity u. Denoting the
total velocity by ii = u
6u we have for the total entropy production which is a random
f tinct ion
+
where
uii = (aui/ar,
+ aui/ari).
W. EBELINQ
26
Averaging with respect to an ensemble on gets for the average entropy production
KLIMONTOVICH
proposes to consider qt as the turbulent viscosity coefficient. We slightly
generalize the above expressions taking into accoiint velocity gradients in arbitrary
directions which yields
(2.3)
In this way the fluctuation term in the effective viscosity is determined by the dispersion of the velocity gradient tensor. For the regime of developed turbulence KLIMONTOVICH has shown that the nonlinear term in (2.4) can be expressed approximately by the
relation of the Reynolds number to its critical value Re/Rek,. Here Rekr is the critical
value where t,he transition from laminar motions to turbulent motions occurs which is
about 2300 e.g. for the flow in a tube [4] and about 36 for the flow in a vortex [5].
By using the relations of KOLMOGOROV-OBUKHOV,
KLIMONTOMCR
finds the following
approximat,ion
r d r ) = r [1
+Y W l
(2.5)
y ( r ) = Re(r)/Rek,.
Here Re(r) is to be considered as a local Reynolds number and qt(r) as a local effective
viscosity of the turbulent flow. The observation that the effective viscosity increases
Fig. 1. Erperinientd curve for the effective viscosity of hydrodynamic vortices (full line after [ S ] )
in mmpsrision to the Klimontovich formula (dashed line after eq. c2.10))
Discussion of the Klimontovich Theory of Hydrodynamio Turbulenoe
27
linearily with the Reynolds number indeed is not new. The effective kinematic viscosity
of vortices e.g. may be described quite well by the empirical law (see [S] and Fig. 1)
qt/q w, Re/Re,,
if
Re > Rekr
(qt and 7 are the turbulent and laminar viscosities respectively).
(2.6)
The Klimontovich formula reproduces the empirical formula but satisfies also the
necessary condition
qt+ 7
if
Re+ 0.
(2.7)
In the transition region Re = 10 - 100 the empirical behaviour (full line in Fig. 1)
shows a more abrupt change from
qtlr
==
1
(2.8)
to the linear law
qt/q = (Re - RO)/Rekr
than that described by the Klimontovich formula
qt/q =
+
(2.9)
(2.10)
where Re is a mean Reynolds number for the vortex flow. However the overall behaviour
is quite correct described by this law. In a similar way the relation of the turbulent
entropy production to the laminar entropy production is well described by
(2.11)
Ut/Ol = 1 f Re/Rekr.
The good agreement with experimental data from tube flow measurements (Fig. 2)
demonstrates again the soundness of the assumption (2.5).
Fig. 2. Relation of the turbulent to the laminar entropy production (full line
after eq. (2.11),crosses - experimental values after NIKURADSE/~TZT
[GI)
- theoretical
curve
3. The Turbulent Flow in Tubes
The hydrodynamic flow in tubes is possibly the most extensively and intensively
studied example of a turbulent motion [6- 91. Therefore tube flow experiments are the
best test object for the verification of a new theoretical formula. KLJMONTOVICH
assumes
W. EBELING
28
for a Poiseuille flow the following radius-dependent Reynolds number
y ( r ) = y* [l - +/a21
y*
(3.1)
Re* = (v*a/2v)
= (Re*/Rekr);
( a - radius of the tube). The characteristic velocity is open so far. KLIMONTOVICH
proposes
L'*= [Apa/2e11"2
(3.2)
where Ap is the pressure difference on a length 1 of the tube. Now the local critical
Reynolds number Rekr is still open. We take the view here that the critical Reynolds
numher is a local quantity which characterizes the boundary layer flow. Therefore we
assume
y* = xav*/2v,
x = (Rekr)-l
(3.3)
where x is still to be specified.
We obtain in this way a n r-dependent effective viscosity
?]It@)
=
7
+ y * ( l - r2/a2)1
(3.4)
and get the velocity profile
U(T) =
Ap$
1
-* -In [I
417 Y*
+ y*( 1 - r2/u2)].
(3-5)
From this follows the total flow through the t'ubc per unit time [2]
dpne a 4
417 Y*
&=-.-
(.[: - + 1
1
ln(l+y*)-l
By introducing the distance from the boundary y
the boundary layer flow
-u(Y) _ u*
x
[
In 1
+ x-
"I:
1
=
(3.6)
u - r we obtain from eq. (3.4)
(3.7)
Further we get the drag coefficient of the tube
(3.8)
where
(3.9)
is the Reynolds number from the average velocity ii.
Let US compare these results with the Prandtl law for the boundnry layer flow 16, 71
(3.1.0)
and for the drag coefficient of the tube
1
2.5 In (Re 1
6)- 0.8.
F=G'
(3.11)
29
Discussion of the Klimontovich Theory of Hydrodynamic Turbulence
The Klimontovich formula is in agreement with the main (logarithmic) term in the
Prandtl law if we make the choice
Rekr = x-l = 2.5,
x = 0.4.
(3.12)
However the constants which have t o be added to the logarithmic terms differ considerably. Therefore the Kliinontovich formulae agree with the experimental data only
a t very small and at very high Reynolds numbers. In order to get agreement in the
whole region of Reynolds numbers we have to modify the basic assumptions. Let us
assume in the following that the effective, viscosity shows the radial distribution
(3.13)
x(e)-+x
if
e 2 p*,
x(p)-+B
if
e < Q*, p
< 1.
In other words t,he slope of the relative effective viscosity is very small (or probably
even zero) within the laminar sheet of the boundary layer and nearly constant outside
n=3
1
2
3
1
5
lg
D
Fig. 3. Relative viscosity and its slope in dependence on the distance from the wall. (Dashed line:
Klimontovich theory with x = 0.4, Full line: modified theory with distance-dependentslope n = 3,
Dash-dotted line: dit.0 with n = 2)
W. EBELINQ
3()
the laminar sheet (Fig. 3). For convenience we assume here the form
1
1
Aixi
= - 21 r(e) x
1 xie
+
ZAi
+
=
(3.14)
1, ZAixi = X , ZA& = /%
and have now the free constants A i and xi. Since the Navier-Stokes equation for the
flow in tubes reads
(w*r/av)
- - = (3.15)
dr
a v*
1 y(e)
we get by integration
Y')
+
(3.16)
30
2(
1(
I .
/!--
10
1
2
I
0.8
1
0.6
3
L
t
I
0.L
0.2
rla
Big. 4. Radial profile in the tube for Re = lo6 (curve a) and boundary law (curve b). Full line
experimental curve after [7], dash-dotted line: modified theory with n = 3
Discussion of the Klimontovich Theory of Hydrodynamic Turbulence
31
Introducing here eq. (3.13) the radial law follows
1
-2 Ai In (1+ xi@)
x
v*
(3.17)
In the limit e --f 00 we get from eq. (3.17) t,he boundary layer law
n
1
..(e!= [In e + 2 A, lnxi - 11.
(3.18)
v*
x
i=l
By comparision with the empirical law (3.10) we have obtained two sets for the free
constants (surely there are still better variants)
n = 2: A , = 6, A, = -6, x, = 0.1, x, = 0.04
n = 3: A , = 312.4, A, = -277.3, A, = -84.1, x1 = 0.16, x, = 0.16,
x, = 0.12.
In Fig. 4 is shown that the modified radial flow distribution agrees quite well with the
experimental data. Finally we calculate the mean velocity in the tube zi and the drag
coefficient 3, from
a
(3.20)
dr 2nru(r)/4naa
ii =
0
)/z 2 1/2v*l;ii.
=
In this way we get the new drag law
(3.21)
1 -
\'.'
\
0-
c
3
L
5
6
7 lg Re
Fig. 6. Drag coefficient for the flow in tub- with smooth d a m (full line - experimental 0-0,
dashed line - original Klimontovich theory after eq. (3.8),daah-dottedline - modified theory after
eq. (3.21), dotted line - laminar behaviom)
W. EBELTNO
32
where
(3.22)
The new drag law for the flow in tubes is also in sufficient agreement wit.h the experimental dat,a (Fig. 5).
4. Coilelusions
The close inspection of the new theory of turbulent flows presented by KLIMONTO[1--3] shows that this theory reflects the general behaviour of the effective viscosity and the turbulent entropy production observed experimentally [5, 101 in a correct
way. However, in order to describe the finer details of turbulent flows as e.g. the radial
profile and the drag coefficient of the flow in tubes, several modifications have to be
introduced. In the modified theory a t least two phenomenological constants appear
which may be identified with the Karman-Prandtl constants in the empirical laws characterizing turbulent boundary layer flows. The examples considered here are concerned
with smooth surfaces only, an extension to rough surfaces as well as to other more
complicated boundary flows [6, 7, 111 seems to be possible.
Let us summarize the main conclusions again :
(i) The Kliinontovich theory which assumes linear laws for the dependence of the
effective viscosity on the Reynolds number and on the distance from the wall gives a
good overall picture of the properties of turbulent flows which yields especially the correct limits at small and at high Reynolds numbers.
(ii) In order to get good agreement in the transition region too, we assume in this
work a different behaviour in two regions:
a) in the laminar region near the wall the effective viscosity is constant or eventually
very slowly increasing
VICH
qt m q
if
0 5 (a - r ) < y* = OY/V*.
(4.1)
b) In the turbulent region at greater distance the viscosity is assumed to be linearily
increasing with the distance from the wall
W*
qtwvr] [ 1 + 0 . 4 T ( a - r - y * )
]
if
(a - r ) > y*.
(4.2)
(iii) The modified theory seems to be able to describe the available experimental
data for tube flows in the whole region of Reynolds numbers. Especially the observed
velocity profile is quite well represented.
In conclusion we want to underline that the theory presented here is still semiphenomenological since there appear phenomenological constants. The developement
of a statistical theory which is able to calculate the averages in eq. (2.4) and in this way
to derive the effective local viscosity would be highly desirable.
M. ARTZT
A c k n o w l e d g e m e n t . The author has to thank Yu. L. KLIMONTOVICH,
and G. SEIFERT
for fruitful discussions.
References
[l] KLIMONTOVICH,
Yu. L.: Letters to Shurn. t e c h . Fiz. 7 (1981) 118.
[2] KLIMONTOVICH,
Yu. L.: Ann. Physik (Leipz.) 89 (1982) end Preprint 127 Sektion Physik 04,
Humboldt-Universiti-tBerlin 1982.
[3] KLIMONTOVICH,
Yn. L.: Statistical Physics. Moscow: Nauka 1982.
[a] LANDAU,
L. D.; LIFSCFIITZ,
E. M.: Hydrodynamik, Berlin 1981.
Discussion of the Klimontovich Theory of Hydrodynamic Turbulence
33
[5] ALBRINQ,W.: Wiss. Z. TU Dresden 24 (1975) 677;
ALBRING,
W. : Elementarvorgange fluider Wirbelbewegungen, Berlin: Akademie-Verlag 1981;
KRAFT,J. : Beschreibung turbulenter Stromungen, Dissertation, TU Dresden 1969;
HAMA",J.; ORZSCEIQ,W.: Ein Beitrag zur Modellierung der Turbulenz. Dissertation, TH
Leuna Merseburg 1982.
[6] PRANDTL,
L.: Fiihrer durch die Stromungslehre, 6. Aufl., Braunschmeig: Vieweg 1965.
[7] SCHLICFITING,
H. : Grenzschicht-Theorie. Karlsruhe: Verlag Bmun 1951.
[8] STRUMINSKY,
V. V., et al. (Ed.): Turbulent Flows (in Russian), Moscow: Nauka 1974.
[9] HEINRICH, M.; ULBRICFIT,
H.: Mechanik der Kontinua, Berlin: Akademie-Verlag 1981.
[lo] ARTZT,M. : Modellierung turbulenter Stromungen mit Hilfe der Thermodynamik irrevermbler
Prozesse und der Kontinuumsmechanik, Dissertation, Berlin 1982.
[ll] HOFFMEISTER,
M.: Technische Mechanik 1 (1980) 1; Tagungsband Turbulenzmodelle und ihre
Anwendung in der Technik, Berlin 1982.
Bei der Redaktion eingegangen am 17. Dezember 1982.
Anschr. d. Verf.: Prof. Dr. W. EBELING
Sektion Physik der Humboldt-Universitiit zu Berlin
DDR-1040 Berlin, Invalidenstr. 42
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