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A Reynolds-number effect in flow past prismatic bodies
MECH. RES. COMM.
A REYNOLDS-NUMBER
Vol.2, 279-282, 1975.
EFFECT
IN F L O W
PAST
Pergamon Press.
PRISMATIC
Printed in USA.
BODIES
J.M. R o b e r t s o n , J.E. C e r m a k a n d S.K. N a y a k Fluid Dynamics and Diffusion Laboratory Colorado State University, Fort Collins, Colorado (Received 15 July 1975; accepted as ready for print 27 July 1975)
Introduction
It is a commonly accepted tenet of fluid dynamics that, for Reynolds numbers above a few thousand, the flow about (and forces on) bodies of sharp corner is independent of Reynolds number. Thus in many flow studies, e.g., scale model studies of wind effects on buildings and structures, measurements on relatively small models are simply scaled to large prototype expectations via the pressure and force coefficients. The drag coefficients of discs and normal plates were early found to be constant at Reynolds numbers greater than about 5000. Structural and building sections of prismatic shape are taken to behave similarly since they have sharp edges clearly fixing the separation locale. Experimental evidence is offered here showing that prismatic flows are not that simple and are subject to a small but significant Reynolds number effect associated with flow reattachment to a side of the F,ow prism. Any prismatic body will experience reattachment of separated flow on one of its sides for some orientations to the flow. The general nature of this occurrence is sketched in Fig. 1 for a rectangular prism at angle ~. The manner in which a wedge's angle, proportions and orientation to the oncoming flow affect this, as analyzed on an ideal-fluid basis, is summarized by Robertson [i]. Viscous and turbulence effects enter to modify the flow but this is more a matter of refinement and filling in the gaps of the idealfluid analysis. A strong indication of the separation-reattachment phenomenon is given by the pressure variation along the side of the prism. Thus, as sketched in
/
FIG. 1 DEFINITION SKETCH
Scientific Communication
CEP75-76JMR-JEC-SKNI 279
280
J.M. ROBERTSON, J.E. CERMAK and S.K. NAYAK
Fig. i, the end of the trapped separationbubble flow is characterized by a rapid rise in the mean pressure coefficient-Cp = 2(p-po)/pV2. Reattachment (R in figure) is before the peak pressure. The separation-reattachment occurrence is basically a function of body geometry and orientation to the oncoming flow. For a right square prism Fig. 2 indicates the angle-ofattack effect in terms of the side-wall pressure variations. Except for the 0 to 13 and 45 degree orientations, reattachment occurred on the side wall, as clearly evidenced by the rapid pressure rise.
Vol.2, No.5/6
Ic
O.,*
¢p
SlB 0
C
-0.5
13 o
-I .O
Experimental Results
20 ° 30 °
Side-wall pressure measurements made on a
~5 o
-i .5
smooth prism of 152 mm side spanning a closed air tunnel of 1.8 m square section when oriented at an angle of 20 degrees for various air speeds yielded the Reynolds number variation indicated in Fig. 3.
FIG. 2 PRESSURE VARIATION ACROSS SEPARATION-REATTACHMENT SIDE
As the Reynolds number
(V = approach air speed, B = side length of prism and
R = VB/~
v = kinematic viscosity)
is increased the reattachment point moves downstream indicating an increase in bubble length.
Quantification of this occurrence for
offered in Fig. 4 where the locale of the point of Cp
~ = 20 degrees
Cp = -0.9
is
and the value of
at the midpoint of the side are presented in their variation with Reynolds number.
s/B 0
O.S
The series 3 results
i.O
do not fully agree with those
oI
of the other series, although showing the same trend, due to
-0.~
R "0.35 x IOs 0.55x I0 I 0.78 x IOs
divergences in setting of the ~%
I 40 x I0 a
Cp
angle of the prism to the flow.*
In regard to the pres-
sure coefficient, •
-I.G
it is to be
nsm
= : 20 °
noted that those of the minimum and maximum pressure of the reattachment and that of
-1.4
FIG. 3 REATTACHMENT-SIDE PRESSURES AT SEVERAL REYNOLDS NUMBERS
An error in ~ of i degree leads to a Cp error of at least 0.i in this region.
Vol .2 , No. 5/6
FLOW PAST PRISMATIC BODIES
2 81
the wake behind the prism are essentially invariant with Reynolds number.
To
check these unexpected results the midpoint pressures were also measured on an 18.6 mm prism in an open-jet tunnel (320 mm jet). shown in Fig. 4, verify the other data at a decrease in
Cp
R
The
Cp
results, also
above 0.3 x 105
and indicate
at lower Reynolds numbers.
Discussion
It is not the purpose of this paper to analyze the full significance of this Reynolds number effect but it obviously influences the forces on the prism. That it is a reattachment-flow effect has been verified through studies on the prism placed symmetrically number.
(i.e., at 45 ° ) where no change appears with Reynolds
Only one other such indication of Reynolds-number effect on reattach-
merit flows has been found.
This is in the wedge-flow studies of Kraemer [2].
For a wedge of 60 degree included angle at an angle of 8 degrees to the oncoming flow, the pressure-rise of reattachment occurred closer to the leading edge at
R = 260,000
than at 26,000.
Roshko [3] addresses the Reynolds-number
question of prismatic bodies but without reaching any conclusions.
Ota and
Itasaka [4] found no Reynolds-number effect on the reattachment locale for an elongated rectangular prism oriented parallel to the flow for 7 x 104 .
R
from 2 to
It seems likely that the Reynolds-number effect is related to the
boundary-layer development along the face of the wedge up to separation as laminar with transition to turbulent in the free shear layer.
Lindsey [5]
found the drag coefficient of a square cylinder in the 0
(o) Locale of Cp = - 0 . 9
degree orientation to increase slowly with
R
up to an
.
~.....L~ ,--'''C~
s/B
R 03 I
of about 0.3 x 105 and attributed this occurrence to "incomplete separation."
Further
-0.8 Cp
study of the flow is obviously needed to clarify the phenome-
f
-
-0.9 -1.0 -I,I
non and delineate the expectations for higher Reynolds num-
(b) Pressure at s / B = 0 . 5
-0.7
O.Ix
~Series;
~,r~/8T6ur~u/e nce 3 Levlll - 0.4 o/.) 4 (Sfrleom TuIrb4Jlance Li'evel 01"33°/~) I I , L I
IOs
0.2 REYNOLDS
bers.
~
0
0.5 NUMBER
I x IO s VB/u
I
I ~_
It does seem clear that
the reattachment of a separated flow to the side of a prism is Reynolds-number dependent.
FIG. 4 REYNOLDS-NUMBER VARIATION OF CHARACTERISTIC FEATURES OF REATTACHMENT PRESSURE RISE ON SQUARE PRISM AT a = 20 DEGREES
282
J.M. ROBERTSON, J.E. CERMAK and S.K. NAYAK
Vol.2, No.5/6
Acknowledgment
This study was supported at Colorado State University under NSF Grant Eng 72-04260-A01, while the first author was a Visiting Professor there.
References
i,
2.
3.
4,
5.
J. M. Robertson, Hydrodynamics in Theory and Application, pp. 531-534. Prentice Hall, Inc., Englewood Cliffs, N. J. (1965). K. Kraemer, Die Druckverteilung am Keil bei inkompressibler Str~mung, ein Beitrag zum Totwasserproblem, Mitt. Max-Planck-Inst. f~r Str~mungsforchung und der Aerodynamischen Versuchsanstalt, No. 30, Goettingen, p. 42. (1964). A. Roshko, On the Aerodynamic Drag o f C y l i n d e r s at High Reynolds Numbers, Proc. Wind Loads on S t r u c t u r e s : U.S.-Japan Res. Seminar (NSF:PB203-449), pp. 8?-96. (1970). T. Ota and M. Itasaka, A Separated and Reattached Flow on a Blunt Fiat Plate, ASME Paper 75-FE-18. (1975). W. F. Lindsey, Drag of Cylinders of Simple Shapes, NACA Report 619. (1938).
A REYNOLDS-NUMBER
Vol.2, 279-282, 1975.
EFFECT
IN F L O W
PAST
Pergamon Press.
PRISMATIC
Printed in USA.
BODIES
J.M. R o b e r t s o n , J.E. C e r m a k a n d S.K. N a y a k Fluid Dynamics and Diffusion Laboratory Colorado State University, Fort Collins, Colorado (Received 15 July 1975; accepted as ready for print 27 July 1975)
Introduction
It is a commonly accepted tenet of fluid dynamics that, for Reynolds numbers above a few thousand, the flow about (and forces on) bodies of sharp corner is independent of Reynolds number. Thus in many flow studies, e.g., scale model studies of wind effects on buildings and structures, measurements on relatively small models are simply scaled to large prototype expectations via the pressure and force coefficients. The drag coefficients of discs and normal plates were early found to be constant at Reynolds numbers greater than about 5000. Structural and building sections of prismatic shape are taken to behave similarly since they have sharp edges clearly fixing the separation locale. Experimental evidence is offered here showing that prismatic flows are not that simple and are subject to a small but significant Reynolds number effect associated with flow reattachment to a side of the F,ow prism. Any prismatic body will experience reattachment of separated flow on one of its sides for some orientations to the flow. The general nature of this occurrence is sketched in Fig. 1 for a rectangular prism at angle ~. The manner in which a wedge's angle, proportions and orientation to the oncoming flow affect this, as analyzed on an ideal-fluid basis, is summarized by Robertson [i]. Viscous and turbulence effects enter to modify the flow but this is more a matter of refinement and filling in the gaps of the idealfluid analysis. A strong indication of the separation-reattachment phenomenon is given by the pressure variation along the side of the prism. Thus, as sketched in
/
FIG. 1 DEFINITION SKETCH
Scientific Communication
CEP75-76JMR-JEC-SKNI 279
280
J.M. ROBERTSON, J.E. CERMAK and S.K. NAYAK
Fig. i, the end of the trapped separationbubble flow is characterized by a rapid rise in the mean pressure coefficient-Cp = 2(p-po)/pV2. Reattachment (R in figure) is before the peak pressure. The separation-reattachment occurrence is basically a function of body geometry and orientation to the oncoming flow. For a right square prism Fig. 2 indicates the angle-ofattack effect in terms of the side-wall pressure variations. Except for the 0 to 13 and 45 degree orientations, reattachment occurred on the side wall, as clearly evidenced by the rapid pressure rise.
Vol.2, No.5/6
Ic
O.,*
¢p
SlB 0
C
-0.5
13 o
-I .O
Experimental Results
20 ° 30 °
Side-wall pressure measurements made on a
~5 o
-i .5
smooth prism of 152 mm side spanning a closed air tunnel of 1.8 m square section when oriented at an angle of 20 degrees for various air speeds yielded the Reynolds number variation indicated in Fig. 3.
FIG. 2 PRESSURE VARIATION ACROSS SEPARATION-REATTACHMENT SIDE
As the Reynolds number
(V = approach air speed, B = side length of prism and
R = VB/~
v = kinematic viscosity)
is increased the reattachment point moves downstream indicating an increase in bubble length.
Quantification of this occurrence for
offered in Fig. 4 where the locale of the point of Cp
~ = 20 degrees
Cp = -0.9
is
and the value of
at the midpoint of the side are presented in their variation with Reynolds number.
s/B 0
O.S
The series 3 results
i.O
do not fully agree with those
oI
of the other series, although showing the same trend, due to
-0.~
R "0.35 x IOs 0.55x I0 I 0.78 x IOs
divergences in setting of the ~%
I 40 x I0 a
Cp
angle of the prism to the flow.*
In regard to the pres-
sure coefficient, •
-I.G
it is to be
nsm
= : 20 °
noted that those of the minimum and maximum pressure of the reattachment and that of
-1.4
FIG. 3 REATTACHMENT-SIDE PRESSURES AT SEVERAL REYNOLDS NUMBERS
An error in ~ of i degree leads to a Cp error of at least 0.i in this region.
Vol .2 , No. 5/6
FLOW PAST PRISMATIC BODIES
2 81
the wake behind the prism are essentially invariant with Reynolds number.
To
check these unexpected results the midpoint pressures were also measured on an 18.6 mm prism in an open-jet tunnel (320 mm jet). shown in Fig. 4, verify the other data at a decrease in
Cp
R
The
Cp
results, also
above 0.3 x 105
and indicate
at lower Reynolds numbers.
Discussion
It is not the purpose of this paper to analyze the full significance of this Reynolds number effect but it obviously influences the forces on the prism. That it is a reattachment-flow effect has been verified through studies on the prism placed symmetrically number.
(i.e., at 45 ° ) where no change appears with Reynolds
Only one other such indication of Reynolds-number effect on reattach-
merit flows has been found.
This is in the wedge-flow studies of Kraemer [2].
For a wedge of 60 degree included angle at an angle of 8 degrees to the oncoming flow, the pressure-rise of reattachment occurred closer to the leading edge at
R = 260,000
than at 26,000.
Roshko [3] addresses the Reynolds-number
question of prismatic bodies but without reaching any conclusions.
Ota and
Itasaka [4] found no Reynolds-number effect on the reattachment locale for an elongated rectangular prism oriented parallel to the flow for 7 x 104 .
R
from 2 to
It seems likely that the Reynolds-number effect is related to the
boundary-layer development along the face of the wedge up to separation as laminar with transition to turbulent in the free shear layer.
Lindsey [5]
found the drag coefficient of a square cylinder in the 0
(o) Locale of Cp = - 0 . 9
degree orientation to increase slowly with
R
up to an
.
~.....L~ ,--'''C~
s/B
R 03 I
of about 0.3 x 105 and attributed this occurrence to "incomplete separation."
Further
-0.8 Cp
study of the flow is obviously needed to clarify the phenome-
f
-
-0.9 -1.0 -I,I
non and delineate the expectations for higher Reynolds num-
(b) Pressure at s / B = 0 . 5
-0.7
O.Ix
~Series;
~,r~/8T6ur~u/e nce 3 Levlll - 0.4 o/.) 4 (Sfrleom TuIrb4Jlance Li'evel 01"33°/~) I I , L I
IOs
0.2 REYNOLDS
bers.
~
0
0.5 NUMBER
I x IO s VB/u
I
I ~_
It does seem clear that
the reattachment of a separated flow to the side of a prism is Reynolds-number dependent.
FIG. 4 REYNOLDS-NUMBER VARIATION OF CHARACTERISTIC FEATURES OF REATTACHMENT PRESSURE RISE ON SQUARE PRISM AT a = 20 DEGREES
282
J.M. ROBERTSON, J.E. CERMAK and S.K. NAYAK
Vol.2, No.5/6
Acknowledgment
This study was supported at Colorado State University under NSF Grant Eng 72-04260-A01, while the first author was a Visiting Professor there.
References
i,
2.
3.
4,
5.
J. M. Robertson, Hydrodynamics in Theory and Application, pp. 531-534. Prentice Hall, Inc., Englewood Cliffs, N. J. (1965). K. Kraemer, Die Druckverteilung am Keil bei inkompressibler Str~mung, ein Beitrag zum Totwasserproblem, Mitt. Max-Planck-Inst. f~r Str~mungsforchung und der Aerodynamischen Versuchsanstalt, No. 30, Goettingen, p. 42. (1964). A. Roshko, On the Aerodynamic Drag o f C y l i n d e r s at High Reynolds Numbers, Proc. Wind Loads on S t r u c t u r e s : U.S.-Japan Res. Seminar (NSF:PB203-449), pp. 8?-96. (1970). T. Ota and M. Itasaka, A Separated and Reattached Flow on a Blunt Fiat Plate, ASME Paper 75-FE-18. (1975). W. F. Lindsey, Drag of Cylinders of Simple Shapes, NACA Report 619. (1938).