The presence of two stable regimes offers the basis for assuming that there exists an "intermediate" unstable regime. Computations with given initial perturbation of the ve!oci~y field were carried out in order to determine this unstable stationary motion. Depending on the amplitude of the initial perturbation, either regime I or regime II was established. Changing the amplitude of the initial perturbation, one could obtain the characteristics of regime III. A similar procedure was used in [5] for obtaining an unstable stationary state~ The behavior of curves II and III indicates that the multivortex motion is excited "stiffly" and has an end point near G: = 2275. We note that the break-up of the convective structure in the layer heated from below, accompanied by an increase of the heat flux, has been observed in a numerical experiment [6]. The author thanks G. Z. Gershuni and M. I. Shliomis for attention to the work and for the discussion of the results. LITERATURE CITED i.
2. 3. 4. 5. 6o
G . B . Petrazhitskii and E. V. Bekneva, "Experimental and theoretical investigations of heat transfer in natural convection in a closed annular region," Tr. Mosk. Vyssh. Tekh. Uchilisshche im. N. E. Baumar, No. 170 (1973). R . E . Powe, C. T. Carley, and E. H. Bishop, "Free convective flow patterns in cylindrical annuli," Trans. ASME, Ser. C., J. Heat Transfer, 91, No. 3 (1969). S . H . Yin, R. E. Powe, J. A. Scanlan, and E. H. Bishop," Natural convection flow patterns in spherical annuli," Int. J. Heat Mass Transfer, 16, No. 9 (1973). R . E . Powe, C. T. Carley, and S. L. Carruth, "A numerical solution for natural convection in cylindrical annuli," Paper Amer. Soc. Mech. Eng., No. WA/HT-9 (1970). T . P . Lyubimova, "On convective motions of a Newtonian fluid in a closed cavity heated from below," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1974). V . G . Vasin and M. P. Vlasyuk, On Wavelengths of Two-Dimensional Convective Motions in a Horizontal Layer of Liquid Heated from Below [in Russian], Preprint No. 84, Inst. Problo Mekh. (1974).
EFFECT OF THE VISCOSITY ON THE READINGS OF TOTAL-HEAD TUBES WITH S~IALL VELOCITIES OF THE FLOW N. P. Mikhailova and E. U. Repik
UDC 532.5.032:.532.574.2
It is well known that the use of the Bernoulli equations with measurement of the velocity of a gas flow using a total-head tube is valid only with large Reynolds numbers. With small Reynolds numbers the action of the forces of viscosity, which become commensurate with the inertial forces, leads to an increase in the stagnation pressure Po in comparison with the value calculated using the Bernoulli equation, while the pressure coefficient Cp nust be greater than unity. In this case, the readings of the total-head tube will depend both on the Reynolds number and on the geometric form of the tube. The effect of the Reynolds number and the geometry of the total-head tube on its readings will manifest itself in the following cases: a) with the measurement of a very small flow velocity (U < i m/see) or with the measure ment of the velocity under the flow conditions of a very viscous liquid; b) with measurement of the velocity in the boundary layer in the immediate vicinity of the wall or in channels using very small total-head tubes. The existing theoretical solutions [i, 2] and experimental data [3-5] are in poor agreement, which renders their practical use difficult. The most careful study of the effect of Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 136-139, January-February, 1976. Original article submitted January !0, 1975. This material is protected b y copyright registered in the name o f Plenum Publishing Corporation, 2 2 7 West 1 7th Street, N e w York~ IV. Y. 10011. No part o f this publication m a y be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, el~,ctronic, mechanical, photocopying, microfilming, 'recording or otherwise, w i t h o u t written permission o f the publisher. A c o p y o f this article ~ available f r o m the publisher f o r $ Z 5 0 .
1t7
the viscosity on the readings of a cylindrical total-head tube with small flow velocities was made in the experiments of [4, 5]. However, the experiments of [4] are in poor agreement with the experiments of [5], although in both cases the investigations were made using total-head tubes with exactly the same cylindrical form, having a flat outlet cross section. The difference consists only in the fact that the total-head tubes had a different ratio of the inside diameter of the receiving opening of the tube d to the outside diameter D. In the experiments of [4] the ratio d/D = 8 = 0.74, while in the experiments of [5] it was equal to 0.6. It can be postulated that this difference is the reason for the scatter in the values of Cp obtained in [4, 5]. A dimensionless
analysis shows that for cylindrical tubes
c~=c~ (Be, 6) H o w e v e r , t h e r e i s no i n f o r m a t i o n ings of a total-head tube.
in the literature
(1)
on t h e d e g r e e o f e f f e c t
o f B on t h e r e a d -
It must also be pointed out that the results o f t h e e x p e r i m e n t s o f [ 4 , 5] d i f f e r n o t only quantitatively [ t h e v a l u e s o f (Cp -- 1) d i v e r g e by m o r e t h a n a f a c t o r o f 2 ] , b u t h a v e a different c h a r a c t e r o f t h e d e p e n d e n c e o f Cp on t h e R e y n o l d s n u m b e r . T h u s , t h e d r o p o f Cp b e l o w u n i t y i n t h e r e g i o n o f a c h a n g e i n t h e R e y n o l d s n u m b e r s f r o m 60 t o 1000, o b t a i n e d i n t h e e x p e r i m e n t s o f [ 4 ] , was n o t c o n f i r m e d b y t h e l a t e r e x p e r i m e n t s o f [ 5 ] , i n w h i c h s p e c i a l attention was p a i d t o t h e c o n f i r m a t i o n o f t h i s somewhat u n e x p e c t e d r e s u l t . M o r e o v e r , i n [5] the assumption is advanced that the fall of Cp below unity was due to methodological errors, which were present in the experiments of [4].
The present article gives experimental data aiding in the refinement of the existing data on the question under examination and making it possible to establish the reasons for the above-mentioned divergences in the results of the experiments of [4, 5]. To bring out the effect of 8 on the value of Cp, an investigation was made of a series of cylindrical total-head tubes with different values of 8, including the values of 8 which were used in the experiments of [4, 5]. The dimensions of the tubes are given below: Tube No, 1 2 3 4 5
D. mm 1.0085 L0260 1.0i05 0.5i00 1.5770
d. mm B 0.3065 0.304 0.5040 0.492 0.5975 0.592 0.3070 0.602 0.9570 0.607
Tube No, 6 7 8 9 l0
D, mm 0.8t45 i.4025 0.8i50 i.0340 i.603
d, mm 0.5525 0.678 0.9750 0.699 0.6020 0.738 0 . 8 i 6 0 0.789 L 3 6 8 5 0.853
The experimental determination of the values of Cp was carried out in an aerodynamic tube (Fig. la), the scheme of which was analogous to the scheme of the tube described in [5]. The intake of air to the aerodynamic tube was carried out directly from the atmosphere, using a blower installed at the outlet from the working tube. To increase the range of the change in the velocity of the flow in the working part of the tube, and, consequently, also of the Reynolds number, provision was made for the possibility of installing the blades of the working wheel of the blower at different angles to the oncoming flow. The total-head tube under investigation was mounted at the axis of the aerodynamic tube, near the shaped air intake, outside of the limits of the boundary layer, forming at the walls of the tube. An evaluation showed that the deviation in the value of the velocity at the axis of the aerodynamic tube, with the installation of a total-head tube of the largest diameter in the working part of the tube, and in its absence did not exceed 0.25%. It was assumed that, at the axis of the tube, Ps + I/zPU2 = Pa, i.e., the value of the pressure coefficient is equal to c~
Po-Pa
il2pUZ
Po-Pa
I+ - -
I/zpUZ
(2)
The value of the velocity of the flow U at the axis of the tube was determined from the difference between the atmospheric Pa and static Ps pressures. The pressure Ps was measured using a drainage opening I with a diameter of 0.3 mm in the wall of the tube, arranged in the same measurement cross section as the receiving opening of the total-head tube I (Fig. la). The small pressure drops (Po -- Pa) and (Pa -- Ps) were recorded using a high-sensitivity alcohol manometer with a graduation of 0.003 mm H20, in which the monitoring of the level of
118
1.8
1.8
J~q
I.Z
q
#
10
fO
60
100
Z00
Rr
Fig. I the alcohol and read-out and recording of the readings were done automatically, using photodiodes, optical lenses, and a relay system. The electrical signals of the photodiodes were fed to the inputs of lamp amplifiers in the automation block, which control the displacement of the carriages with the photodiodes, depending on the level of the alcohol. The measured pressure drop was recorded with an electromagnetic printing device. A decrease in the random error of the result was achieved as the result of a large number of readings with one measurement (not less than I0 readings). The results of the measurements are given in Fig. ib, where the experimental values of Cp for total-head tubes with different ratios B are given as a function of the Reynolds number Re D = UD/2~; here U = #2(P a -- Ps)/0. [In Fig. ib, each of the curves 2-10 is shifted downwards along the axis of ordinates with respect to the curve i by (n -- 1)'10 -I , where n is the number of the curve.] The mean-square deviations in a determination of the dependence of Cp on ReD from ~r data with Cp < 1.4 does not exceed ~ = 0.01. In Fig. ib, as in all the other figures, the numbers i-i0 denote the number of the tubes, given above. For a more graphic illustration of the effect of B on the character of the dependence of Cp on ReD, Fig. 2a gives the curves shown in Fig. Ib smoothed with respect to the experimental points. The dependence of Cp on $ with a constant ReD number is given in Fig. 3. As can be seen, with an increase in the Reynolds number, the pressure coefficient becomes greater than unity; under these circumstances, the effect of the viscosity appears more strongly for tubes with a small value of B than for thin-walled tubes. The dependence of the value of Cp on B explains the divergence in the values of Cp obtained in [4, 5]. The experiments of [4, 5] (correspondingly, the dotted curves B and C in Fig. 2) are in agreement with the present experiments with the same values of B; here i~ must be emphasized that the methods of investigation used in the present experiments and in the experiments of [4] were significantly different. The same figure gives the calculated curve A [1]. 119
Gp
1.8
\Re D IO =
.
.
.
.
Cp
l.q
I
q
g
18,
7
10
ZO
#O
lO0 Re~ II q
"
E
15
10
i
20
'
Rgt~
Fig. 2
o.2
o.tt
o.#
,[J
Fig. 3
The experiments indicate that in determination of the dependence of Cp on Re D for a cylindrical total-head tube, the characteristic parameter is B, while, at the same time, the absolute dimension of the inside and outside diameters of the tube, which in the present experiments varied by approximately 3 times, had practically no effect on the value of Cp with a given value of the ReD number (Fig. 4). However, it must be pointed out that, if the value of the Reynolds number is determined not from the outer diameter of the tube, but from the inner diameter, the range of the change in Cp as a function of the ratio B is appreciably constricted (Fig. 2b), i.e., in the investigation of the viscosity effect, the parameter Re d = Ud/2v is more universal than the parameter ReD = UD/2v. In the case of a cylindrical total-head tube with B = 0.6, the most widely used value in measurement practice, the following empirical dependence can be proposed: Cp=ReD [1.07RED--3.7]-i w h i c h i s v a l i d f o r 6 < ReD < 30, w h e r e t h e v a l u e s (see the dotted line in Fig. 4).
o f Cp d e v i a t e
(3) from u n i t y
by more t h a n 5%
Another important result is obtaining values of Cp < i for total-head tubes with B > 0.6. The range of change in the Reynolds number with which Cp < i is determined by the value of 8; here Cp rises with an increase in B. The fall of the values of Cp below unity is greater the greater the value of B. With an increase in B, the minimum in the values of Cp is shifted toward the side of lower values of the Re D number. With B = 0.853, which was the greatest value in the present experiments, a value of Cp < i occurs with 17 < ReD < 300; under these circumstances, the minimal value of the pressure coefficient Cpmin = 0.972 and corresponds to Re D = 40, With B ~ 0.6, values of Cp < i were not observed in the whole investigated range of change in the ReD numbers. If it is taken into consideration that the magnitude of the drop in Cp below unity considerably exceeds the scatter of the experimental points (Fig. 5), as well as the fact that this fall in Cp has a completely determined regularity with respect to B and repeatability from experiment to experiment, then a basis is furnished for regarding this experimental fact as completely authentic. Thus, the fall of Cp below unity, which occurred in the experiments 0.74, was not due to methodological errors, as was indicated in [5], but with the law. However, the range of changes in the ReD numbers at which in the experiments of [4] (curve C on Fig. 5) is shifted more toward the numbers than in the present experiments with the same value of ~.
of [4] with B = is in accordance Cp < i was observed side of greater Re D
A rigorous theoretical justification of this phenomenon is not possible. However, it can be postulated that the flow of a viscous liquid around a total-head tube leads not only to a rise in the values of Cp with a decrease in the ReD number, which is obvious from a physical point of view, but simultaneously also promotes a lowering of the stagnation pressure in the tube, due to the suction of the liquid out of the internal cavity of the tube 120
~
q
-7-
5 1.8
-
l.q
1 .
b"
i0
ZO
.
.
50
.
fog
.
ZD17
Ren
Fig. 4
~0
20
FO
Igg
2 ~ ? g Re~
Fig. 5
with its ejection by the viscous liquid flow around the tube. The latter effect should naturally manifest itself to a greater degree for thin-walled tubes than for tubes with a small value of 8, and, in a determined range of Re D numbers (where the action of the main effect has already been weakened), it can lead to a drop of Cp below unity. LITERATURE CITED I. 2. 3. 4. 5.
F. Hamann, "The effect of high viscosity on the flow around a cylinder and around a sphere," NACA Tech. Mem. No. 1334 (1952). W . F . Durand, Aerodynamic Theory, Vol. 3, Springer, Berlin (1935). M. Barker, "On the use of very small Pitot-tubes for measuring wind veloc:ty, " " Proc~ Roy. Soc., Ser. A, i01, No. 712 (1922). C . W . Hurd, K. P. Chesky, and A. H. Shapiro, "Influence of viscous effects on impact tubes," J. Appl. Mech., 20, No. 2 (1953). F . A . McMillan, "Viscous effects on flat-end Pitot tubes at low speeds," J. Roy. Aeronaut. Soc., 58, No. 528 (1954).
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