App1. Sci. Res. 23
Jan u ar y 1971
AN E X P E R I M E N T A L INVESTIGATION OF A WALL J E T IN A TORUS H. W . P I E K A A R
a n d R . W . POLlVIAN
Association Euratom-FOM FOM-Instituut voor Plasma-Fysica Rijnhuizen, Jutphaas, THE NETHERLANDS
Abstract
Results of an experimental investigation of the velocity profile of a turbulent gas injected in a toroidal configuration are presented. The measurements surprisingly show t h a t it is possible to describe the radial distribution of the azimuthal velocity in terms of a plane wall jet discharging in an external stream. The growth of the inner boundary layer, the width of the jet, and the velocity profile are in accordance with the known experimental data on this subject. A fundamentally different relation has been deduced for the decay of the m ax i m u m velocity. Up to now Sigalla's formula Ura/Uj oc ~/a/x is generally accepted. Our data based on an essentially extended range of x/a, correlate with tile exponential relation
Um/Uj =
exp I - - 154(Rez)-0-777 ~ ;
.
Nomenclature a
~kr Ptot Pstat R
Rea Rex u¢
4 U Uj
width of entrance slit Mack number pressure at the stagnation point of the total pressure tube, measured in the entrance slit static pressure at the torus wall minor radius of the torus main radius of the torus Reynolds number with respect to a, Uja/v Reynolds number with respect to x, Ujx/v fluctuation in the azimuthal velocity component mean square of the fluctuation in the azimuthal velocity component azimuthal velocity jet velocity --
393
--
394 gm
Uoo X XO
Y 7
d~
H. W. PIEKAAR AND R. W. POLMAN m a x i m u m a z i m u t h a l v e l o c i t y at a n y station x v e l o c i t y of e x t e r n a l stream (defined as the a z i m u t h a l v e l o c i t y at distance y = 10~1) distance measured along wall from entrance slit value of x where m a x i m u m v e l o c i t y begins to decay distance perpendicular to wail ratio of specific h e a t at c o n s t a n t pressure to specific h e a t at c o n s t a n t volume v a l u e of y where U = U m v a l u e oI y f a r t h e s t from wall, where U = ½Urn k i n e m a t i c viscosity of fluid degree of turbulence
§ 1. Introduction
Recent developments in the research programme of the laboratory of Plasma Physics, Rijnhuizen, Jutphaas, have amongst others been centered around the possibility oI confining thermonuclear plasmas by a cold gas blanket. A plasma confined in this way should not be subjected to the same type of instabilities as a plasma confined by magnetic fields in vacuum, and the gas blanket should provide a barrier against impurities from the wall. Rotation of the gas blanket should be favourable for equilibrium and stability of the plasma column. A toroidal configuration has been chosen in order to avoid end losses. The gas is thought to flow tangentially along the torus wall and by viscous drag should cause the rotation of the gas in the central region oI the torus. In order to preserve toroidal symmetry use has been made of in- and outlet slits covering the entire torus
,!\
R=450mm
ilii
Fig. 1. E x p e r i m e n t a l arranffement.
J
AN EXPERIMENTAL INVESTIGATION OF A WALL JET IN A TORUS 395 length (Fig. 1). Out of plasma physical considerations, irrelevant for this report, a static pressure of 13.2 × 10 a N / m 9' was chosen. This r e p o r t deals with the investigation of the cold gas flow in the toroidal vessel. The resulting flow p r o v e d to be highly similar to the flow of a wall jet along a flat surface as t r e a t e d theoretically b y Glauert [1] and b y Weinhold [2]. The radial distribution of the a z i m u t h a l velocity profile has been m e a s u r e d b y a hot wire a n e m o m e t e r . A w i n d - t u n n e l has been cons t r u c t e d to calibrate hot wire a n e m o m e t e r s at this relatively low pressure. § 2. Apparatus and experimental procedure The m e a s u r e m e n t s h a v e been c o n d u c t e d in a torus (Fig. 1). The main radius was 450 ram, the minor radius 100 ram. Air was blown in t a n g e n t i a l l y t h r o u g h an adjustable slit S1, and r e m o v e d via a fixed slit $9 of w i d t h 7 ram. S1 was adjustable between 0.5 and 5.25 ram. B o t h slits e x t e n d e d over the entire circumference of the torus. As the static pressure in the torus at which the m e a s u r e m e n t s took place was 13.2 × 10s N / m ~,, the jet velocity in the slit S1 could be a d j u s t e d b y v a r y i n g the pressure in front of slit S1 from 13.2 X 103 to 26.4 X 103 N / m 2. The outlet of the torus was conn e c t e d to a v a c u u m vessel of 10 mS, ensuring a flow, s t a t i o n a r y for at least 4 seconds. A valve between $2 and the v a c u u m vessel was necessary to fix the pressure in the torus at 13.2 × 10 a N / m 2. The jet velocity, Uj, was m e a s u r e d b y means of a t o t a l pressure t u b e and r a n g e d from M = 1 to M = 0.26. To connect t o t a l pressure and M a t h n u m b e r the formula
pt2~ _ ( 1 + Pst~t
y_ 1 ~
)7(7--1) M2
(1)
was used, where 7 ---- 1.4. Pressures were recorded with transducers. The radial dependence of the a z i m u t h a l velocity, U(r), was m e a s u r e d at distances v a r y i n g from 75 to 467 m m from slit S1, using a hot wire a n e m o m e t e r , operating at c o n s t a n t t e m p e r a t u r e . The d i a m e t e r and the length of the t u n g s t e n wire were 5 vm and 2.8 m m respectively. At each station a hot wire traverse was m a d e perpendicular to the wall. E a c h of the traverses consisted of 10 readings at 1 m m intervals near the wall and of 15 intervals of 5 m m in
306
H. W. PIEKAAR AND R. W. POLMAN
the "outer" part of the flow. The accuracy of setting the hot wire was 0.1 ram. In interpreting the anemometer signal we assumed that the static pressure was 13.2 × 103 N/m 2 throughout the torus. In reality, due to the centrifugal force, the static pressure at the axis of a rotating gas is less. However, this amounted only to a 0.15 × 10~ N/m 2 in the experiments conducted. The precise static pressure distribution was obtained by using a flat plate parallel to the flow provided with suitable pressure taps. The circular flat plate had a radius slightly less than the minor radius of the torus, and its plane coincided with the plane of Fig. 1. There were 22 static pressure holes, each with a diameter of 2 mm, located at different radii. By rotating the disc in the plane of Fig. 1 the pressure distribution was measured. The width of the jet, ~2, measured from the wall to the point where the velocity has decreased to half the maximum velocity, varied from 5 to 34 mm (Fig. 2). The local maximum velocity, Urn, appeared to be between M = 0.4 and M ---- 0.084. To calibrate the anemometer at 13.2 x 103 N/m 2 a wind-tunnel, especially designed for this experiment, has been employed (Fig. 3). The cone D controls the mass flow, the valve E the pressure. A typical calibration curve is given in Fig. 4. The velocity in the wind-tunnel was measured with an ordinary Pitot tube according to Prandtl's design. Again use has been made of (1).
Fig. 2. Diagram illustrating notations. outside
p r e s s u r e = ~05NI
m2
~-~025
mm
static p r e s s u r e
a t o p e r a t i n g conditions 13.2 x 10 7 N / m e
Fig. 3. Diagram o~ wind-tunnel.
AN EXPERIMENTAL INVESTIGATION OF A WALL J E T IN A TORUS
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I M
.
.
.
.
i
.
.
.
i-,
397
,
Rw=9~
0.4
Rw= 1.5 Ro
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E o = 2.72 (Qt 13.2 x 103 N / m2)//,/ O,3
/
0,2
/
./ /
./!"
/ 0.1
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/ ./.J"
o
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,
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4.5
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Fig. 4. C a l i b r a t i o n c u r v e for a n e m o m e t e r o p e r a t i n g a t c o n s t a n t t e m p e r a t u r e . E O u t p u t v o l t a g e of a n e m o m e t e r device. E0 O u t p u t v o l t a g e of a n e m o m e t e r device for zero v e l o c i t y a t a s t a t i c p r e s s u r e of 13.2 × 10 a N / m 2. Rw R e s i s t a n c e of h o t wire u n d e r o p e r a t i v e c o n d i t i o n . R0 R e s i s t a n c e of h o t wire a t r o o m t e m p e r a t u r e .
§ 3. Experimental results
An experimentally obtained velocity distribution is depicted in Fig. 5. Neglecting the radial velocity component, this distribution actually represents the azimuthal velocity. The abcissa indicates the distance to the wall, the ordinate gives the measured velocity, U, in terms of the speed of sound. Velocities less than M = 0.02 are omitted. No corrections have been made for the effect of turbulence on the anemometer readings. For given x and a the radial distribution of U proved to be similar, meaning that for given x and a the velocity profiles can be approximately generated out of each other b y multiplication. The multiplication factor is a function of Uj. If x and a are varied, a second kind of similarity in the velocity profiles can be obtained, as found, among others, b y Weinhold [21 whose result was based on Glauert's theory for a plane turbulent wall jet discharging in still air. According to Glauert's theory a characteristic velocity profile for a turbulent wall jet can be obtained by plotting U/Um as a function
398
H. W. PIEKAAR AND R. W. POLMAN
I
1
k
t
[
I
i
l
i
,vl
l
0.3
0.2
lOOmm
,3
i
.... xx× entry
\x
x /
1 slit 3,Smm
/ i i
/x
uj : M = 0,55
~
0.1
7 x x I
200
I
I
F
160
×
I
t
120
I
o
×4
r
!
I
40
80
distance to the
wall
I o
(mm)
Fig. 5. E x p e r i m e n t a l a z i m u t h a l v e l o c i t y profiles. T h e a p p r o x i m a t e p o s i t i o n s of t h e a n e m o m e t e r are i n d i c a t e d b y t h e d r a w n lines, d e s i g n a t e d b y 1 a n d 3 in t h e s t y l i z e d cross section.
of y/b~. Weinhold extended the theory by introducing an external stream superimposed on the wall jet and moving in the same direction. The velocity of the external stream Uoo is a function of x. A dimensionless velocity profile is now obtained by plotting
U--Uoo
y
vs
wherein y[(Um + Uoo)/2~ is that value of y where
U--Uoo U m - - Uoo
1 2
Our experimental results are presented in this way in Fig. 6. As in Weinhold's case the velocity at y = 10bl has been chosen as the velocity of the external stream. For those values of x, for which y = 10hi lies in the central region, the influence of the corresponding Uoo is relatively small. For still larger values of x, the distance y = 10bl, becomes larger than the minor radius, and loses its meaning. We did not exceed this limit in our range of parameters. The agreement between the thus obtained dimensionless velocity profile and Glauert's velocity profile is surprising. Further confirmation of the wall-jet character of the toroidal flow is to be
AN EXPERIMENTAL
INVESTIGATION
I
I
8h~ x/o
r
OF A WALL JET IN A TORUS
I
$
I
399
I
velocity profile
--G]auert's • 14
x 33 • •
89
11t <> 169 A 204 o 339 El 406 v 650 + 777
1
0.6 0:8 0.4 I
~_.a+m
0.2 0
I
I
I
2
I
I
1.5
1
0.5 y/y(UA ~U~)
.
I
0
Fig. 6. Comparison of theoretical velocity profile and experiment.
i
l
~
T~--I~[--l-~
O
!
:/
I
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30
T
o ×
20
oeo
x/ x
/
0.6
+++ 2.3 oo® 3.5 A A A 5.25
x x +~10 '~1 ~
00
~=0-040 5 " 4"6
I
I
200
I
f
I
400
I
600
T
I
800
I
x/o
I
1000
F i g . 7. H a l f v e l o c i t y p l a n e .
found in the rate of spread of the jet. The linear increase of
d2
and
Y
2 a
with x/a is indeed in accordance with the experimental results found by Sigalla E3, 41, Bakke [5], and Weinhold E2] (Figs. 7, 8). Of course yE(Um + U~)/21 should never exceed ~2, thus limiting the scope of the relevant empirical formulas in the figures.
400
H. W. P I E K A A R AND R. W. POLMAN
I
I
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I
jo o
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I
I
L
I
:/
l 20
OOO x ~ +++ 000 ~z~a,
Ox, Ox
0.6 1.15 2.3 3,5
5.25
x 10
+x
ti*
o;
~g
Y(
Um÷U x 2 ~ ) / a =0.046~- *0.7
, 4;0
~;o' ~;o ,ooo xtQ
Fig. 8: Rate of spread of ~ jet discharging in an external stream.
Moreover, the relation bl = 0.182x (Umx/v) -(1/5), a s experimentally deduced by Sigalla [31 and theoretically derived b y Myers, Schauer and Eustis [6], enabled us to check the growth of the inner boundary layer hi. Although the measurements were the least accurate near the wall, the formula proved to be a fair representation o f our measurements of dl. The decay of the maximum velocity along the wall is shown in Fig. 9, disclosing a remarkable difference with data previously obtained on this subject. Each experimental point is characterized by its Reynolds number Rex. Each point in the figure represents an experimental point, accompanied by its Reynolds number multiplied b y 10.5 . Although fitting for Sigalla's data, the relation (Um/Uj)(x/a)~ = const, appears to be inapplicable for the toroidal wall jet. The reason is quite simple. The relation for the plane wall jet is theoretically derived b y assuming conservation of the integral momentum flux for the outer layer, while for a toroidal geometry the integral angular momentum flux has to be conserved. If the displacement of the centre of rotation and the divergence and convergence of the flow, inherent to the toroidal geometry, are considered, the experimental profiles (Fig. 5) substantiate the conservation of angular momentum within a few percent. The analysis of the experimental values showed an exponential decrease with Rex = Ujx/~,
AN EXPERIMENTAL INVESTIGATION OF A WALL JET IN A TORUS
10 o
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t
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1
I
(u~m)~ S i g c=3'45('~)-0"5 l l l a U m 4 R -o 777 x . . . . .
cv,=oxo[_l~ e~,
E
r
1
401
"~l--
5x10-1
"N~ ~ 74.
,11.3
2x10 4
4,4
10-1 0
,\ ~ 2 0 0-- \
\, \ , \ , \,l \, ,.~3.~ 400 \ '600 \ 800 P X/O \ \ \ \ \ 4.5 6 7,5 915 1115 ~" 10-5Rex
\
1.5
l°° '
I
I
I
I
T
2" E
l
1.2s.,1,.6,~\\ o.~ ,f&\\\ °'94'~\'t 4"27
2.69,~ ~,,5.01 0.97jT.1.40 .-t ~,.~6~ ,1.22=187 11"90° '.2~90~ ' 'LI 2.64";306" .71
2\0 \ 3~o \400, 1.5
4,5
6 7.5
9.5 11.5
. .,o = tO'5Rex
Fig. 9. Decay of m a x im u m velocity along wall. Each experimental point is characterized by its Reynolds number
Re
z.
402
H. W. P I E K A A R AND R. W. P O LMA N
as parameter:
Uj
exp --154 Rex -o-777 ~
,
(2a)
if Rea = Uja/v is taken as parameter the exponential decrease is represented by Uj
exp --154 Rea -0-777
(-:FI
.
(2b)
In Table I a comparison is made between experimental data and values computed with this formula. In view of the sensitivity of the results for the accuracy at which Uj and the slit width a are known, a mean deviation of less than 0.03 in Um/Uj is highly satisfying. TABLE I am 0.710 0.816 0.648 0.603 0.584 0.730 0.594 0.381 0.734 0.428 0.570 0.550 0.472 0.170 0.203 0.700 0.618 0.517 0.534 0.278 0.410 0.512 0.365 0.431 0.590
0.640 0.707 0.603 0.494 0.599 0.673 0.589 0.346 0.668 0.404 0.540 0.573 0.440 0.197 0.211 0.645 0.630 0.540 0.518 0.289 0.406 0.529 0.348 0.515 0.603
0.653 0.474 0.145 0.269 0.159 0.476 0.362 0.625 0.316 0.416 0.352 0.524 0.123 0.382 0.139 0.237 0.477 0.387 0.196 0.261 0.244 0.116 0.110 0.104 0.098
am 0.613 0.506 0.152 0.257 0.165 0.483 0.366 0.599 0.314 0.465 0.350 0.571 0.125 0.469 0.136 0.221 0.553 0.456 0.205 0.274 0.259 0.097 0.108 0.090 0.099
=
154\T /
~/
.J
Decay of m a x i m u m velocity. Comparison between e x p e r i m e n t a l l y obtained values and values derived from the exponential relation.
AN E X P E R I M E N T A L I N V E S T I G A T I O N OF A WALL J E T IN A TORUS
403
Another new aspect is the dependency on the Reynolds number. Myers, Schauer, and Eustis [61 seemed to have suspected such a dependency: " . . . with the higher Reynolds numbers exhibiting a slower velocity decay". Their analysis, nevertheless, failed to show any clearly defined variation of the decay with Reynolds number. However, our range of extended far beyond the range of in their reach. Our data were taken out of experiments covering distances up to 900 slit widths. (Sigalla only 65 and Myeers, Schauer, and Eustis 180). In toroidal geometry the entrance slit is a simulation of an infinitely extended slit, thus preserving the twodimensional character of the wall jet over its full length, enabling us to extend our measurement to a 900 slit widths distance. As shown in Fig. 6 the radial dependence of the azimuthal velocity is fully defined by the wall jet up to a distance of
x/a
x/a
"endless"
y=
1.5y( Urn@U°°) 2
As stated before, we are not dealing with a wall jet discharging in still air but with one discharging in an "external" stream which, due to the toroidal geometry, is generated by the wall jet itself. Hence, in the immediate vicinity of the entrance slit the azimuthal velocity profile is characterized by the mixing of the injected jet and that residue of the fully developed wall jet which has not been carried off through the exit slit $2. Due to the increase of the relative intensity of turbulence with the distance to the wall, the rotating centre of the flow, acting as external stream for the wall jet, can only be discussed in general terms. For this reason it was impossible to ascertain whether or not the centre rotated as a solid body, which was the real objective of this investigation. The only definite statement concerning the rotation of the gas body is that the axis of rotation was outwardly displaced with respect to the vertical through the geometric axis of the minor cross section by 5 to 7 mm.
(u~)~/U
Fig. 10 is a plot of as a function of the distance to the wall. The turbulence measurements in the rotating centre are susceptible to error on account of the nonlinear characteristic of the hot wire. Noting the essentially similar results of Bradshaw [7], and Tailland et Mathieu [8], one m a y assume that the measurements are sufficiently truthful. In the centre the degree of turbu-
404
H. W. PIEKAAR AND R. W. POLMAN
I
L
I
I
I
I
I
l
i
l
l
mm
25
xxxl
exit
I
"
20
15
10
5 XxxX x 0
I
I
200
I
I
I
160
I
I
120
I
t
80
distance
x x×
I
40
to the
walt
;
E I
0 (mm)
Fig. 10. Radial distribution of the degree of turbulence. The approximate position of the a n e m o m e t e r is indicated b y the drawn line, designated b y 1 in the stylized cross section.
oa
E
I
I
I
I
I
/
Z m 0
10o.o
x
L
99.5 x
i
m
/
U o
99.0
x
x ~4 x
m
x x~XXXXXX
i 100
80
X
I
I
60 40 distance
I 20 to t h e w a l l ( m m )
F i g . 1 1. Radial distribution of the static pressure.
lence ran up to values of more than 30%. Theoretically (u~)½/U should tend to infinity on approaching the real axis of rotation. The assumption of uniform static pressure throughout the torus has been justified by the measurements of the static pressure distribution in a torus cross section. A typical radial pressure distribution, showing a pressure difference of only 0.2 x 10~ N/m 2 between centre and wall has been plotted in Fig. 11. The experi-
AN E X P E R I M E N T A L I N V E S T I G A T I O N OF A WALL J E T IN A TORUS
405
mentally obtained pressure and velocity distributions turned out to be connected with the well known formula for the centrifugal force. The dominating frequencies of the turbulent signal ranged from 70 to 1500 Hz and depended on the slit width and on the distance to the wall. The frequencies decreased with increasing distance. § 4. Conclusions
If a rotating gas is turbulent, a solid body rotation is not necessarily implied. The degree of turbulence varied from 5% near the wall to 30~o in the centre (Fig. 10). In view of the above statement the "sagged" velocity profile is not surprising (Fig. 5). The axis of rotation is outwardly displaced with respect to the vertical through the geometric axis of the minor cross section over a distance of 6 ram. The radial distribution of the azimuthal velocity appears to be in accordance with that of a turbulent wall jet discharging in an external stream (Fig. 6). The growth of the jet, a linear function of the distance covered, confirms the wall jet character of the toroidal flow as well (Fig. 8). A quite different functional relationship has been found for the decay of the local maximum velocity Urn. Instead of Sigalla's expression (Urn/Uj) (x/a)°.5 = const., the experimental data correlate with the exponential relation (Fig. 9): Um __ exp --154 (Rex) -0.777
for
Uj
20 <
< 900 a
or
Urn _ e x p
I
--154
/ x \0-2237
) j.
However, due to our maximum realizable values of x, the experiments have only been conducted up to x/a = 90 for a ---- 5.25 mm, x/a = 133 for a = 3.5 mm, x/a = 200 for a = 2.3 mm, x/a = 400 for a = 1.15 mm and x/a = 900 for a ~-- 0.6 ram. As the toroidal aspect ratio (that is the ratio between main and minor radius) could not be varied, this ratio may be hidden in the constants. The same is true for the influence of the ratio between slit width and minor radius. However, the fundamentally exponential decay of the maximum velocity would almost certainly
406
AN E X P E R I M E N T A L I N V E S T I G A T I O N OF A W A L L J E T IN A T O R U S
not be affected b y this ratio. Basic is our extensive range of the parameter x/a, enabling us to detect the clearly defined influence b y the Reynolds number. Sigalla E3, 41, Myers, Schauer, and Eustis E6] performed their measurements up to 60 and 180 slit widths respectively, while our experiments extended beyond as much as 900 slit widths. In spite of the deviation regarding the maximum velocity decay it was concluded from the agreement of the velocity profile and the jet growth with the measurements of other workers, that the toroidal jet investigated in this report can be characterized b y a representation as a two-dimensional wall jet. Acknowledgement
We are grateful to Prof. Dr. C. M. Braams for his stimulating and useful suggestions, as well as to Prof. Ir. J. O. Hinze for his careful reading of the manuscript and his valuable comments. Thanks are also due to Prof. Dr. L. J. F. Broer. We are indebted to Mr. G. K. Verboom for programming the computer, and to Mr. J. Harms for his part in the experimental work. This work was performed as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (ZWO) and Euratom. Received 24 September 1969 In final form 27 February 1970
REFERENCES
[1] GLAUERT,M. B., J. Fluid Mech. 1 (1956) 625. [2] WEINHOLD,K., Rev. Roum. Sei. Techn.-Mec. Appl. 12 (1967) 121. [3] [4] [51 [6]
SIGALLA,A. and M. T. GEE, Aircraft Engineering 30 (1958) 131. SIGALLA, A., J.R.Ae.S. 62(1958) 873. BAKKE, P., J. Fluid Mech. 2 (1957) 467. MYERS, G. E., I. J. SCHAUER and R. H. EI3STIS, J. Basic Eng. 85 (1963) 47. [7] BRADSHAW,P., Turbulent wall jets with and without an external stream, Aero. Res. Coun. 22,008 - F.M. (1960) 2971. [8] TAILLAND,A. and J. MATHIEU, J. de Mdcanique. 6 (1967) 103.