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IDET KONGELIGE NORSKE VIDENSKABERS SELSKAB
532.51
Some Remarks on Reynolds Number
By
R. GRAN OLSSON
DET KONGELIGE NORSKE VIDENSKABERS SELSKAB FORHANDLINGER BD XXV, 1952 Nfl 3
532.51
Some Remarks on Reynolds Number
B
R. GRAN OLSSON
(Fremlagt i Fellesmotet ilte februar 1952)
1. General considerations.
As well known, Reynolds number is defined as the ratio of
inertia force to shear force for fluid flow. The former is,
ex-pressed as the product of mass X acceleration, i. e. X v2/l, where is the density of the fluid, 1 a characteristic length, and
v a characteristic velocity. If one considers only the dimensions,
then mass and acceleration v2/l.
The shear forces can be expressed as the product of a shear stress i and an area which can be denoted by 12.
Accordingly, the ratio of inertia to shear force becomes
where R is Reynolds number. This expression can be rewritten
by introducing a velocity v, defined by
which leads to another form for Reynolds number,
(la)
R=-as the ratio of the squares of the characteiistic velocity v and the velocity v,, of the shear stress. The latter is the geometric mean of the turbulent velocity fluctuations u'. and v' in two dimen-sional flow, i. e. [1]
(2a)
As is well known, the flow pattern changes with certain values of Reynolds number: for low values of R the flow is laminar, while it is turbulent for higher values. This transition from laminar to turbulent flow does not occur for a fixed value of
R but rather for values of R within a certain range dependent
on the varying of the conditions for the flow.
For laminar flow the shear stress r can be expressed by means
of the equation
SIt, T U
D. K. N. V. S. Forhandlinger Bd XXV, Nr 3
where is the viscosity and y the distance from the wall to a
fluid particle. Dimensionally the shear stress can be written as
I,
(3a)
and Reynolds number now becomes [1] (2b)
I,
where v = is the flUid's kinematic viscosity.
This expression for Reynolds number is
valid only as long as relatio,n (3) for r is
va-lid, i.e.on.ly for laminar flow.
Returning to eq. (2a) one sees that as long as the velocity v of the shear stress is large, Reynolds number is small and the flow is laminar due to the stabilizing action of the shear force. For large Reynolds numbers, where the characteristic velocity v is dominant to ui., the stabilizing action of the shear effect is not sufficient and the flow becomes turbulent. In the turbulent motion, it is not the actual viscosity of the fluid which produces the friction but rather that an apparent friction is produced. by the turbulent velocity fluctuations u' and v'.
2. Application to the turbulent flow in a pipe.
In a pipe of radius r and length 1 with a pressure difference zip over the, length 1 the following relation holds for stationary
motion
p7r9 = r, 2nrl
where is the shear stress along the wall. In technical hydrau-lics one uses the resistance coefficient 1 defined by the equation
el,2 1
'2d
Using the value of zip from eq. (4) in eq. (5), then 2r,l ov*21
(5a) r
and quoting the two expressions for zip gives
.
turb8ijSR
This expression for 2 is also valid for laminar flow, where the shear stress at the wall i's given by
dv 4v 8v
(3b) ' .
R. Grañ Olsson:, Some Remarks on Reqnolds Number
32u1
i"l
(4a)
and so the resistance coefficient
Alam
vd Lid In this case Reynolds number becomes
r, 8
8'
which gives for the resistance coefficient the same for laminar
flow
(6a)
A1=8R1
i.e. the resistance coefficient is the samefor
both laminar and turbulent flows.
Using a logarithmic scale one has logA+logR=1og8
and the resulting curve for the resistance coefficient as a func-tion of Reynolds number is a straight line fOr the active flow pattern (fig. 1).
3. The relatIon between the Reynolds
nnm-bers corresponding to laminar andturbulent
flows.
For laminar flow Reynolds number Riam
=
is used while for turbulent flow RtUrb=_!_, where v is found from the shear stresses at the wall,=
The measurements of the velocity in turbulent flow lead to the
velocity at the center of the pipe given by [1, p. 145]
Vmax = (. in +5,5)
The variation of velocity as a function of the distance y from
the wall can be written in the form [1, p. 142]
Umax_*F()
where F () is a function of the dimensionless quantity only. Eq. (8) extended to apply for an arbitrary distance y becomes
10 D. K. N. V. S. Forhandlinqer Rd XXV. Nr 3 ,0
'(10)
Fig. 1. The resistance coefficient as
Re
function of Reynolds number.
(8a) v=v*(2,5 +5,5)
which together with the equation, resulting from using eq (8) in eq. (9) gives
(9a) 1'max - V = 2,5 V.
-Since the velocity distribution is known one can calculate the average velocity by means 'of the integral
r-0
Using the velocity distribution av given in (9a) results in
(9b) U=Vmax3,75V*
A better value was found from experiments of' J. NIKIJRADSE
according to which [2]
/000 .900 00 700 00 500 400 300 200 /00
/
-v.d 3)
Fig. 2. Relation between Rturb (turbulent flow) and
=
(laminar flow).
/000
/00 doo
12 D. K. N. V. S. Forhandiiner Bd XXV, Nr 3
[fl W. F. DURAND: Aerodynamic Theory, Vol. 111, Berlin 1935, Div. G: The Mechanics of Viscous Fluids by L. PaNDm, p. 128.
[21 J. NIKUBADSE: V. D. J. Forschungscheft 356, Berlin 1932.
Trykt 4de april 192
kommijon ho F. Bruxa Bohhandel
AktSetrykkeHet I Troii.dhjem
By means of eqs. (9a) and (9b) one can express Rturb in terms
of Riam . Eq. (8) with (9b.) gives
v, (2.5 in + 1,75)
Introducing T5/v=V Rturb in eq. (9d) and writing
rv
yr
Y
one gets
V RtUrb 2,5 In vr + 1,75
"VRu.
having divided by v, in eq. (9d). Since vr/v=4R1 one has
V=2,51n-,--m +5,22
or, using the logarithm with base 10
9g) r = 5,76 In __ + 5,22
where the constant term would be 4,90 on the basis of
NIKURAD-SE'S experiments.
This equation shows that Rrb as a function of in (Riath/Rrb) is a straight line which is also a good approximation for small values of Riam, even beyond the lower range where the theory should be used. Fig. 2 shows graphically the relation between Rlurb and Riam. This shows that for increasing Riam, Rurb appro-aches constant value. This agrees with the known fact that the coeffiëient of resistance, A, approaches a constant value as the Reynolds number increases: also that the resistance of the flow
increases as the square of the velocity such as is actually the case
