576 On the Static Pressure in Fully-Developed Turbulent Flow By A. F age, A.R.C.S c. ( Communicated byG. I. T a y l o r , F.R.S 1—The present investigationf has a twofold purpose: first, to obtain a measure of the effect of fully-developed turbulence on the reading of a static pressure tube; and second, to obtain information on the dis- tribution of static pressure in the fully-developed turbulent flow in a pipe and in the wake behind a long cylindrical body. The work arose from a paper! by Dr. Goldstein on the measurement of total head and static pressure in a turbulent stream. 2—Static P ressure T ubes A static pressure tube of good design is one for which small local disturbances at the nose, arising from the introduction of the tube into the stream, have time to die away before the static holes are reached. The reading of such a tube in a steady stream depends on the shape of the nose, the distance of the holes behind the nose, and the distance of the supporting stem behind the holes. When the static holes are at least 6 tube-diameters behind the head, and the stem at least 15 stem- diameters downstream from the holes, the reading of the tube is inde- pendent, within wide limits, of the shape of the head.§ Sketches of the static pressure tubes used in the present investigation are given in fig. 1. Care was taken in manufacture that the outside surface of each tube was smooth, and free from burrs at the static holes. Each tube has a hemispherical nose. Tubes A and B have an external diameter 0-086 inch, and static holes 0-015 inch diameter; and tubes C and D an external diameter 0-043 inch, and static holes 0-010 inch diameter. The holes of each tube were about 9 tube-diameters behind the nose, and about 36 stem-diameters forward of the stem. Each tube t The investigation was carried out in the Aerodynamics Department of The National Physical Laboratory, and permission to communicate the results was kindly granted by the Aeronautical Research Committee. t The preceding paper. § Ower and Johansen, * Aero. Res. Ctee.,’ Rep. and Mem. No. 891, pp. 985-996 (192-526). on August 24, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from
Transcript
576
On the Static Pressure in Fully-Developed TurbulentFlow
By A. Fage, A.R.C.Sc.
( Communicated by G. I. Taylor, F.R.S.—Received February 28, 1936)
1—The present investigationf has a twofold purpose: first, to obtain a measure of the effect of fully-developed turbulence on the reading of a static pressure tube; and second, to obtain information on the distribution of static pressure in the fully-developed turbulent flow in a pipe and in the wake behind a long cylindrical body. The work arose from a paper! by Dr. Goldstein on the measurement of total head and static pressure in a turbulent stream.
2—Static Pressure Tubes
A static pressure tube of good design is one for which small local disturbances at the nose, arising from the introduction of the tube into the stream, have time to die away before the static holes are reached. The reading of such a tube in a steady stream depends on the shape of the nose, the distance of the holes behind the nose, and the distance of the supporting stem behind the holes. When the static holes are at least 6 tube-diameters behind the head, and the stem at least 15 stem- diameters downstream from the holes, the reading of the tube is independent, within wide limits, of the shape of the head.§
Sketches of the static pressure tubes used in the present investigation are given in fig. 1. Care was taken in manufacture that the outside surface of each tube was smooth, and free from burrs at the static holes. Each tube has a hemispherical nose. Tubes A and B have an external diameter 0-086 inch, and static holes 0-015 inch diameter; and tubes C and D an external diameter 0-043 inch, and static holes 0-010 inch diameter. The holes of each tube were about 9 tube-diameters behind the nose, and about 36 stem-diameters forward of the stem. Each tube
t The investigation was carried out in the Aerodynamics Department of The National Physical Laboratory, and permission to communicate the results was kindly granted by the Aeronautical Research Committee.
t The preceding paper.§ Ower and Johansen, * Aero. Res. Ctee.,’ Rep. and Mem. No. 891, pp. 985-996
(192-526).
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was carried in a holder, so that the azimuth position of the static holes could be varied, and airtightness at the common surface was obtained with a film of grease. Tube A has 12 holes arranged in three closely spaced sections, the four holes in any one section being spaced at 90° to each other, and at 30° to those of the neighbouring section. Tubes B and D have four closely spaced holes on a generator; and tube C has four holes in one section, spaced at 90°.
STATIC PRESSURE TUBES.
4 holes spaced 0'05
Fig. 1.
3—Reading of a Static Pressure Tube
The reading of a static pressure tube in a turbulent stream gives a measure of the average total pressure inside the tube, and differs from the true average static pressure by a pressure arising from the impact of the fluctuating cross velocities on the tube and its holes. The difference in reading due to this “ impact ” pressure depends on the design of the tube, especially on the number, size, and arrangement of the static holes, and on the magnitude and frequency of the cross velocities. If a tube has a large number of small holes equally spai^ed around its periphery (tube A), the reading with the tube aligned in the mean direction of flow is independent of the azimuth position of the holes. It is to be expected that the relation between the reading of the tube S, and the true average static pressure p, can be written in the form
S = p + K P p + w*],( 1)
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where v and w are the cross components (direction specified later) of the fluctuating velocity, the bars denote average values, and K has a characteristic value for the same tube in turbulent streams of the same kind. This relation can be written in the form
for turbulent flow in a pipe, where pU*2 = intensity of friction at the wall, and the suffix c denotes values on the axis of the pipe.
It does not appear possible to obtain a reliable prediction of K without recourse to experiment: and in the present investigation the scheme adopted is to determine K from values of S, t?2, and w2 measured in turbulent streams for which theoretical relations for in terms of pt>2 and pw2 are known.
The problem was discussed with Professor G. I. Taylor, who informed the writer that such theoretical relationships can be obtained for turbulent flows in circular and flat rectangular* pipes, on the assumption that the stresses due to viscosity are small compared with the Reynolds apparent stresses pv2 and pw2. The relation for a circular pipe is
where v is the radial component and w the tangential component of the disturbed velocity at radius r.
The relation for a flat rectangular pipe is
where v is the component of the disturbed velocity at right angles to the wider wall. The second of these arises directly from Reynolds’s method of averaging the equations of motion. The first is found by expressing Reynolds’s equations in cylindrical coordinates.
The form in which these relations will be used in the present analysis are:
(S — Sc) = (p — p c) , K f (v2 w2) _ (v2 + w2)c-pu* 2 Pu* 2 T L u * 2 u* 2 J (2)
r "T \~ + v* = w2 — v \dr Lp
v2 = const, P '
r JL r P_L» v% 1 _ (w2— i;2)Br LpU*2 ^ U*»J U*2 ’
(3)
andP | _£l = const. (4)
pU*2 T U^2
f I.e., a pipe whose section is a very elongated rectangle.
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Traverses of static pressure tubes A and B and a small total head tube were made for air flowing through a long circular pipe of diameter 5 inches. Fig. 2a gives a sketch of the pipe, with the principal dimensions. The pipe was fitted with a faired entry, and a uniform flow of air was maintained by an airscrew fan, fitted at the exit end and driven by a D.C. motor (battery supply). The fan and motor were carried on a stand, separated from the pipe. The overall length of the pipe was 650 inches (130 D). The exploration section was 508 inches (102 D) from the entry and 142 inches
------------ 650"-
56*__ ,
Fig. 2.
(28 D) from the exit. Each tube was traversed across the pipe by a micrometer screw. A tube, before introduction into the pipe, was aligned parallel to a straight edge carried on the micrometer holder, and afterwards, with the tube in the pipe, this straight edge was aligned parallel to a generator of the external surface of the pipe. The pressure in a tube was measured against a datum pressure taken at a hole in the side of the pipe. This hole was taken 7*4 inches (60 stem-diameters) forward of the stem, and about 3 inches forward of the exploration section, so that the stem (the exposed length changed during the traverse of a tube) should not interfere with the datum pressure.
The readings of a static pressure tube were taken on a Chattock gauge modified, in the manner suggested by Falkner,f to obtain high sensitivity. This modification was the introduction of a bubble of a lighter liquid
t ‘ Aero. Res. Ctee.,’ Rep. and Mem. No. 1589, p. 849 (1933-34).
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(50% Nujol, 50% paraffin) in a heavier liquid (water), which allowed the control force on the bubble to be kept small, because the forces of gravity and surface tension were in opposition. Additional sensitivity was obtained by the use of large cups, and by decreasing the effective distance between the cups by inclining the axis of the glass work at a small angle to the axis of rotation. With this gauge a pressure difference of about 0-00001 inch of water could be detected. The measurements of total head were taken on a standard 13-inch Chattock gauge. The range of U 0D/v covered was 36,800 to 136,000, where U 0 is the mean rate of flow.
Table I—V alues of (S — Sc)/pU2
The Reynolds number U 0D/v given at head of columnStatic tube B
The results are specified with reference to a rectangular system of axes XYZ, where OX is taken coincident with the axis of the pipe, and OY is normal to the wall. Table I gives the pressure readings, means of values for ± r, in the form (S — Sc)/pU2, where S is the reading at radius r, Sc is the reading on the axis against the same datum, and U is the mean velocity at radius r.
For tube A, the value of (S — Sc)/pU2 falls with an increase in r, until a minimum is reached at about r — 0-7R, and then rises. For tube B, the values of (S — Sc)/pU2 taken with the holes facing the OY direction are lower than those taken with the holes facing the OZ direction.
It is relevant to mention that traverses with the smaller tubes C and D in a l-inch pipe gave results of the same character as those collected in
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Table I. In these tentative experiments a blower was used to draw air through the pipe.
General considerations of fully-developed turbulent flow in a pipe suggest that the best way of presenting the pressure readings is in the form (S — Sc)/pU*2; for then the values obtained should tend to be independent of U 0D/v, for the region where the stresses due to viscosity are negligible. Values of (S — Sc)/pU*2 for tube A are given in Table II. The agreement between the values of (S — Sc)/pU*2 for the same value of r is considered to be satisfactory, in view of the smallness of the pressure readings (S — Sc). A t U 0D/v = 136,000, the greatest value of (S — Sc) is only about —0-0011 inch of water and at U0D/v =36,800 about —0-00009 inch. The mean values of (S — SJ/pU*2 given in the last column of Table II are plotted in fig. 3 ( ).
Static Pressure in Turbulent Flow 581
Table II— V alues of (S — Sc)/pU*2 (tube A)The Reynolds number U 0D/v given at head of column
Maximum values of the cross velocity components (denoted by and Wj) in a circular pipe, obtained from observations taken with an ultramicroscope for U 0D/v = 8090, 13,440, and 18,340, are given in an earlier paper.t Values of v V and V w 2 can be deduced from these measurements on the assumption that the root_mean square values are one-third the maximum values4 Values of V v2/\J# and Vw2/U* for U 0D/v = 13,440 and 18,340, deduced from values of t^/U and wxj\J taken from the faired curves given in fig. 13 of the earlier paper, are plotted in fig. 3 The values for U 0D/v = 13,440 are in fair agreement with those for U 0D/v = 18,340, and the mean curves are considered to be applicable
t Fage, ‘ Phil. Mag.,’ vol. 21, p. 80 (1936).X Townend, ‘ Proc. Roy. Soc.,’ A, vol. 145, p. 180 (1934).
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to the range of Reynolds number over which the pressure measurements were made.
6—Prediction of KThe curve of^p — ^c)/pU* 2 obtained from relation (3), for values of
Vy2/U* and Vw>2/U* taken from the curves of fig. is given in fig. 3b. The value of K is predicted from the relation
582 A. Fage
(S — Sc) _ Q — pc) j _ ^ T(v2 + w2) _ (v2 + w2)e~]pU*2 Pu * 2 L u * 2 u * 2 J ’
Circular pipe
c u r v f f
Fig. 3.
obtained earlier in the paper, for values of (S — Sc)/pU*2 and (p — pc)/pU*2 taken from the curves of fig. 3b and values of
(v2 + w2) (v2 +L u* 2 u* 2 J’
deduced from values of vV/U* and Vw^/U* taken from the curves of fig. 3a. The prediction is confined to the region = 0 • 35 to 0 • 80, for no accuracy is to be expected from the small values of ( pU*2 and(S — Sc)/pU*2 at the central region of the pipe. The results obtained are given in Table III. The value of K is practically constant over the range selected. The mean value can be taken to be 0-28.
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A sketch of the rectangular pipe is given in fig. 2b. .The principal dimensions were: overall length, 336 inches, cross-section, 12 inches (width, 2b) by 2 inches (depth, 2d). The hydraulic mean depth, obtained by dividing the area of the cross-section by its periphery, was 0-857 inch. The working section was 306 inches (357 m) from the entry. The pipe was constructed of five plywood sheets screwed on (2 x 2) quarterings, spaced 12 inches apart internally. A faired entry was fitted and air was drawn through the pipe by an airscrew fan at the exit end. At the working section, a 12-inch length of plywood was cut out of a 12-inch side, and a smooth flat brass plate, carrying the tube micrometer and two nipples Nlv was inserted. Two brass plates (length 3 inches) were also fitted to carry nipples N 2 and N 3 36 and 72 inches forward of nipple Nx. The inner junctions of the brass plates and plywood were faired with wax.
Traverses with static pressure tube A and with a small total head tube were made in an axial plane normal to a 12-inch side, for constant values of the pressure drop (dP/dx) between the holes N 2 and Nx. The static pressure readings were taken against the pressure at the hole Nx, 5 inches forward of the exploration section, with the sensitive gauge (reading to about 0*00001 inch of water) used in the experiments made in the circular pipe. The total head readings were taken on an ordinary 13-inch Chat- tock gauge. The results are specified with reference to a rectangular system of axes XYZ, for which OX is coincident with the axis of the pipe and OY is normal to a 12-inch wall.
Values of (S — Sc)/ pU2 deduced from traverses with static tube A from the axis towards the brass wall are given in Table IV. These traverses were
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made for the exit arrangement shown in fig. A traverse made with two finely-perforated zinc plates spanning the exit cone about midway between the end of the pipe and the fan gave results of the same character.
T able IV—V alues o f (S — Sc)/pU2. (R ecta ng ular P ipe)Static T ube A
Reynolds number U cra/v, where U c is the mean velocity on axis, is givenat head of column
These values were deduced from observations made for water flowing through a brass pipe of section 4*5 x 0-75 inch. The observations were taken with an ultramicroscope at 88 inches (275 m) from the entry (faired). The Reynolds number, U cm/v, was 4200.
moving in the XOY plane towards the wall, Sxy2 is the maximum deviation of particles in the XOY plane moving away from the wall, and 0^ is the maximum deviation of particles moving in a plane parallel to XOZ. Maximum values of the cross velocities (denoted by vx and w ) were calculated from these observations by relations given in the earlier paper on turbulent flow in a circular pipe. These relations are
7) tan and u = *an
The calculated values of yj and E, ranged from about 0*99 on the axis to about 0-90 at y = 0-9 d.
The values of V v2 and V w 2 were taken to be one-third the maximum values vx and wx. The curves of vV /U * and V w 2/JJ^ obtained from faired observations are given in fig.
For a rectangular pipe(p — Pc) = _ ( t f —
pu* 2 \ u * 2
Tire curve of (p — /?c)/pU *2, obtained from this relation for values of V r2/U* and V w 2lU* taken from the curves of fig. is given in fig. 4b.
A. Fage
9—Prediction of KThe value of K for the rectangular pipe was predicted in the same
manner as that for the circular pipe. The results obtained are given in Table VII.
The mean value of K for the range y = 0-35d to 0-80d is 0*22. The value for the circular pipe was 0-28. The mean of the two values is 0-25. The investigation leads, then, to the conclusion that the reading of a small static pressure tube of type A in fully-developed turbulence of the kind experienced in a pipe exceeds the true static pressure by an amount given by 0*25 p [v2 + w2]. In isotropic turbulent flow v2 = w2 and the measured pressure exceeds the true pressure by 0*5 pv2. This is the value taken in Dr. Goldstein’s paper.
the mean direction of flow. Values of andpU2 sin2 0 . pU2 sin2 0
obtained from these readings, where the suffix 0 denotes a reading takenat 0° and the suffix 0 that at zero inclination, are given in Table VIII.The readings of the static pressure tube A inclined to the stream is givenby the relation
S* = S0 — 0-31 pU2 sin2 0 (0 < 20°).
It is seen, then, that a static pressure tube (type A) reads low when inclined in a steady stream. This effect is opposite in sign to that of fully- developed turbulence at zero inclination. Table VIII also shows that the effect of inclination on the reading of a total head tube is smaller than that on the reading of a static pressure tube, especially at a small inclination.
T able VIIIU = velocity of stream (ft /sec)
Values of (S, — S0)/pU2 sin2 0 for Static Tube A0° oTillP
a definite form at a distance of about 30 cylinder diameters downstream; and that the reading of a static pressure tube in this wake is lower than that in the stream outside. The establishment of a definite form of velocity distribution indicates that the flow in the wake has settled down to a state of fully-developed turbulence; and support for this conclusion has been obtained by the writer from turbulence measurements made in the wake behind a long prism.f
An analysis of Schlichting’s pressure readings for the circular cylinder and the writer’s turbulence measurements for the prism is given in Dr. Goldstein’s paper. The value of ( p + -J-p at the centre of the wake, where q denotes the resultant turbulent velocity, was taken to have the same value as that just outside the wake, and the analysis led to the conclusion that the reading of the static pressure tube used by Schlichting gave (p + iptf2). The work just described gives further evidence that this conclusion is probably not far from the truth. It was also assumed that the turbulent velocities for the circular cylinder could be predicted from those measured behind the prism. The validity of this assumption, and the statement that the value of ( + \<f) at the centre of the wake is equal to the value just outside the wake, will now be examined: and for this purpose, measurements of static pressure and turbulent velocities have been made in the wake of a circular cylinder.
12—Pressure Experiments
Traverses of static pressure tubes A, C, D, and a small total head tube were made behind a 0-375-inch circular cylinder extending between, and at right angles to the walls of a small closed-jet tunnel. The cross- section of this tunnel was slightly divergent in the direction of flow. The diameter at the working section was about 14 inches. The traverses were made at 15 and 30 diameters behind the cylinder, along lines in the central plane of the tunnel at right angles to the cylinder axis.
The readings of a static pressure tube were taken, against the static pressure given by a tube well outside the wake, on a 13-inch Chattock gauge with the axis of the glass work inclined at a small angle to the axis of rotation, so as to obtain a very open scale. J The values of (S — S0) given in the paper are the readings in the wake minus the reading taken just outside with the same tube. The total head was measured with a
t * Aero. Res. Ctee.,’ Rep. and Mem., No. 1510, p. 116 (1932-33).% The readings were much greater than those of the pipe experiments, and the
sensitive gauge was not necessary.
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small tube of external diameter 0-086 inch and the values of (H — H0) given are the readings in the wake minus the reading just outside.
The position of a point in the wake is specified with reference to a rectangular system of axes XYZ, for which the origin O is taken in the cylinder axis, XOY is the plane of exploration at right angles to the cylinder axis, and OX is the centre line of the wake. The velocity just outside the wake is denoted by U 0.
13—Wake at 15 D iameters
Values of (S — S0)/pU 02 obtained from the readings taken at 15 diameters behind the cylinder are plotted against y/D, where D is the cylinder
P o i r v t c T u b e . .X Jo
• A O 49 9
o c o 393
X 399
‘w 'O * * 399
>5 -5 0 -25 -20 -15 -10 -05 O 05 IO 15 2 0 25 50 OS
Fig. 5—14-inch closed tunnel; wake at 15 diameters.
diameter, in fig. 5. At the centre of the wake, the values obtained with tube D (holes on a generator) when the holes faced the z-direction are appreciably smaller (note that (S — S0) is negative) than those obtained when the holes faced the y-direction. This result indicates that a state of fully-developed turbulence has not been reached, and that a “ residue ” of the vortex system formed close behind the cylinder is present. The effect of this “ residue ” on the reading of a static pressure tube is not known. It would appear from the nature of the difference between the two curves for tube D that the y-components of the velocity disturbances in the “ residue ” are greater than the z-components, and this condition is one to be expected from the known character of the vortex system. Fig. 5 shows that the curve for tube C (4 holes spaced 90°) lies about
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midway between the two curves for tube D ; and that the readings for tubes A and C are in fairly close agreement.
14—Wake at 30 D iameters
The curves of (S—S0)/pU02 for x — 30 D are given in fig. 6. The curve tube D obtained when the holes faced the j-direction is in close agreement (except for a small difference at the centre of the wake) with that obtained when the holes faced the ^-direction; and it can be concluded that a state of fully-developed turbulence exists. The values of (S — S0)/pU02 for tube C are greater than those for tube D by a small but definite amount
-oooe
F ig. 6— 14-inch closed tunnel; wake at 30 diameters.
(about 0-002). At the centre of the wake, the values for tube A differ from those for tube C ; but the difference is small and probably arises from the fact that the readings were taken at different wind speeds, for the total-head curves taken at these two speeds also show a small difference at the centre of the wake, fig. la.
Schlichting’s distribution of (S — S0)/pU02 obtained for a 1-cm circular cylinder in a jet of 29 metres per sec is given by the dotted line in fig. 6. The curve for tube A resembles Schlichting’s curve, but the value of (S — S0)/pU02 at the centre is lower, and the wake is narrower. It is thought that these differences arise from the fact that the present experiments were made in a small closed jet whereas Schlichting’s experiments were made in an open jet; for experiments made some years ago in a 1 -ft. open jet by Mr. Falkner and the writer gave results which were in
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which allows the movements of small masses of air, heated by electric sparks and made visible by the Schlieren method, to be recorded with a cinema camera. The films were taken by Dr. Townend. An improved method of spark illumination was used. Instead of a single spark, a series of 3 or 4 sparks was passed with the shutter open and the film at rest, so that each picture recorded a succession of positions of a hot spot during its passage downstream. Fogging of the plate, which would occur with the normal adjustment of the Schlieren system if successive views were superposed, was prevented by adjusting the diaphragm to intercept the whole of the light from the source so that the image of a hot spot appeared as a light spot on a dark background, and successive exposures of the background left the impression of a spot unaffected. The time interval between two consecutive impressions of a hot spot was determined from the film from the angular displacement of an arm of a spider rotating at a known speed.
t * Proc. Roy. Soc.,’ A, vol. 145, p. 180 (1934).
VOL. CLV.— A. 2 S
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The records were taken with the 0 -375-inch cylinder mounted in a small 1-ft open-jet tunnel. Two records of the flow were taken at a point x = 28 • 5 D on the^entre line o fthe wake. From the first (cylinder horizontal) values of Vw2/U and Vt>2/U were obtained; and from the second (cylinder vertical) values of Vw2/U and Vw2/U. The value of (H — H 0)/pU 02 measured at the point where the records were taken was — 0-188.
Most of the film measurements were made by Mr. W. S. Walker, and the analyses of these measurements were made by Miss E. E. Cook.
The results obtained are given in Table IX.
T able IX
Position in wake (x = 28 • 5 D, = 0)
PU 02■188; U ° U = 0 -2 1 0
Cof
Film 1—Cylinder horizontal 297 observations analysed
Film 2—Cylinder vertical 377 observations analysed
U = 34-7 ft/sec U = 35-5 ft/sec
V p= 0-148 U* 0-145
006IIr* >_ >
^ = 0-130
Vu2 V pu — 0*55
<U0 - U) ° 55 (U 0 - U)
V p __ 0-^9<u0 - U) 0 55 (U 0 - U)
0 (ignoring sign) = 0-1301A>
(ignoring sign) = 0-116
v p_ = 1 - 1 3V
V^2= 1 1 2
w
Table IX shows that the values of Vw2/U for the two films are in closea/ ? 1 a/ a/ i v2 __
agreement, that -yj- = —q - , and that —q— = 0-88 - p - . The values
of the ratios V v21v and V w2/w (1 -13 and 1-12) are in close agreement with the value 1-11 obtained by Townend from experiments in a square pipe.
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16.—The values of Vw2, Vi?2, and V w 2 taken in Goldstein’s analysis were predicted from ultramicroscope observations of Wi/U, U /U 0, 0 ^ , and taken in the wake behind an isosceles triangular prism mounted in a water tunnel (1*125 inch X 1*125 inch) with a side (0 *04 inch x 1*125 inch) normal to the stream. It was assumed that the root mean square values were one-third the maximum values. The maximum values
and taken were those obtained by the writer from the relations
t?i, Wl = U tan (0*,,, 0**).
The relations vlf w1 = V (U 2 — u 2) ta n (0 ^ , 0^) are now used.Table X gives values of V v 2/U and Vw2/U predicted by these new
relations from observations taken on the centre line of the wake, together with values of Vw2/U, for values of U /U 0 taken from a faired curve given in the earlier paper.
T able X
Centre of Wake (Prism)
u V M* U U V W2 V w*U, u (U 0 -- U) U 0 u (Uo - U ) U 0 u (U 0 - U )
Values of V u 2 J(U0 — U), V t>2/(U0 — U), and U0 — U) are also given, and it is seen that each of these quantities can be taken tobe constant over the range of U /U 0 covered. The mean values of Vwa/(U0 — U), Vjj*/(U0 — U), and V J?/(U 0 — U) for the prism are 0*54, 0*39, and 0-45 respectively. The values obtained for the circular cylinder by the hot-spot method are 0*55, 0*55, and 0*49. The values of a/ m2/(U0 — U) and Vv?/(U0 — U) for the prism are in close agreement with those for the cylinder, whilst the value of V r*/(U0 — U) for the prism is 0*71 the cylinder value.
17— R elation B etw een p a n d
Let the relation between p and p be of the form p + =where kx and k 2 are constants. In the stream just outside the wake the
2 s 2
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On substitution, we get (==-) = 0-624 from (7), = — 0-0176_ \ u 0/ pU0z
sfiTJ2from (6), and ^ . -"2 = 0-0368. The value of from (5) is 0 -48. The
U Uinvestigation leads, therefore, to the conclusion that ( + ip #2) a t the centre is equal to the pressure just outside the wake. Further, it is known that the value of pis a minimum and the value of ip #2a maximum at the centre of the wake, and it may be concluded from the established fact that ( p + ip#2) at the centre is equal to the value just outside the wake, that ( p + ip#2) is constant across the wake. To establish this more general conclusion it would be necessary to measure the distribution of p#2, in addition to the distribution of p. This additional work would take a long time, and it was not attempted because the results obtained for the prismt indicate a fall in ip#2 from the centre outwards, and it does not appear likely that any appreciable departure from a constant value of p -T ip#2 can occur.
18—In a recent paper % Taylor has shown that measured distributions of temperature and velocity in the wake behind a heated circular cylinder confirmed the accuracy of theoretical distributions given by the vorticity transport theory for two-dimensional motion when the turbulent motion is confined to the plane of the mean motion. This theory also predicts, as pointed out by Goldstein, that p + ip # 2 should be nearly constant across the wake.
19—In conclusion, the writer wishes to acknowledge his great indebtedness to Professor G. I. Taylor for the interest he has taken in the investigation, and to Mr. W. S. Walker, who conducted the pressure experiments described in the paper.
Sum m ary
The reading of a static pressure tube in a turbulent stream does not give a true value of static pressure, because of an effect due to the
t The few observations of «X/U, rx/U, and Wx/U given were taken to obtain a rough idea of the width of the wake. Many more observations would be needed to obtain reliable curves of distribution.
t With Appendix, Fage and Falkner,4 Proc. Roy. Soc.,’ A, vol. 135, p. 685 (1932).
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fluctuating cross velocities. The present research has been undertaken to obtain a measure of this effect for fully-developed turbulent flow.
The relation between the reading of a static pressure tube (S) and the true average static pressure ( p) is expressed in the form
S = p + K P \v2 + w2],
where v and w are the cross components of the turbulent velocity, and K has a characteristic value for the same tube in turbulent streams of the same kind. The experiments have been made for the fully-developed turbulent flows in circular and rectangular pipes. A value of K for each of these flows has been predicted from a theoretical relation for , in terms of the Reynolds apparent stresses and pw2, and distributions of S, measured with a sensitive gauge capable of detecting a pressure difference of 0*00001 inch of water. The distributions of pv2 and pw2 were deduced from measurements taken with an ultramicroscope. The values of K obtained for a static tube of common design (12 holes spaced 30°) were 0*28 (circular pipe) and 0*22 (rectangular pipe). It is concluded that the reading of a static pressure tube of this type, in fully- developed turbulence of the kind experienced in a pipe, exceeds the true average static pressure by an amount which is of the order
0-25 p [F2 + w2].
Information has been obtained on the relation between the values of p and p q2,where
q2 — u2 + v2 + w2,
in the turbulent wake behind a long circular cylinder. Traverses made with static pressure tubes of different design in the wake at 15 and 30 diameters behind the cylinder indicate that a state of fully-developed turbulence is reached at 30 diameters. At the centre of this section of the wake, values of w2, v2, and w2 were obtained by the hot-spot method developed by Townend, and a value of p was obtained from the relation
p = S - 0 * 2 5 p ( F + w 2).
The sum of the values of p and ^pq2 was equal to the pressure just outside the wake, and it is concluded from this result, and the known nature of the distributions of p and ^pq2, that (p + %pq2) is constant across the wake.
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