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Title Study on the Turbulent Shear Flow Past a Circular Cylinder
Author(s) Kiya, Masaru
Citation 北海道大學工學部研究報告, 50, 1-101
Issue Date 1968-12-20
Doc URL http://hdl.handle.net/2115/40903
Type bulletin (article)
File Information 50_1-102.pdf
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
Study on the Turbulent Shear Flow
Past a Circular Cylinder；’
Masaru KIYA““
（Received August 31， 1968）
Abstract
A two－dimensional fiow pas［ a circular cylinder placed in a turbulent boundary
’layer was described， with special reference to the lift and drag forces acting on the
cylinder． A theoretical analysis of the flow xxras made for the hypothetical inviscid
shear fiow with a linearly varying velocity profile， and experinn ents were performed
with a circttlar cylinder placed ip an artificially produced shear fiow and in a turbulent
boundary layer， respectively．
An explicit solution for the stream function that describes a uniform shear flow
past a circular cylinder subjected to an interference of a plane wall was obtained． lt
was found from detailed numerical calculations that stagnation pressure decreases as
the distance between the cylinder and the plane wall decreases， while in the case of
uniform fiow the stagnation pressure coefllcient remains in unity． The lift force acting
on the cylinder calculated by numerically integrating the pressure on the surface of
the cylinder was found to be positive when there exists a certain amount of velocity
gadient in the transverse direction of flow， which is similar to the case of a practical
boundary layer that develops along a stationary boundary wall．
From the experiment performed in a uniform shear flow artificially produced by
・arranging a grid of rods in a wind tunnel， it was verified that an acceptable agreement
between the theory and the experiment xvas found concerning the stagnation pressure
・and the position of stagnation point， respectively， provided that the clearance between
the cylinder and the plane wall is not too small． lt was also found that the experi－
mental value of Hft coefRcient increases as the cylinder approaches the boundary wall，
一as qualitatively predicted by the theory．
The pressure distribution on the surface of the cylinder placed in an artificially
thicl〈ened turbulent boundary layer of 60 mm in thickness was measured and was
’found to be integrated in the evaluation of the lift and drag forces． lt was found
that the drag and lift coefllcients plotted against the ratio ‘a／（2a）， where A is the clear－
ance between the cylinder and the boundary wal｝ and a is the radius of the cy｝inder，
were represented respectively by a single curve within an acceptable experimental
error， the Reynolds number being in a range of （O．99－3．78）×10‘． With A／（2a） decreas－
ing， the lift coeMcient was found to increase monotonically from zero to the maximum
＊ Some parts of this paper were presented at the Japan－U．S． Seminar on Similitude in Fluid
Mechanics at the lowa lnstitute of Hydraulics Research， University of lowa， lo“Ta City on
September 25， 1967．
＊±’ Fluld Mechanics 1， Department of Mechanical Engineering．
2 Masaru KlyA 21
value of about O．8 at A／（2a）＝＝＝O， and on the contrary the drag coeflicient decreases from
1．2 to the minimum value of O．95 after passing the maximum value of about 1．3 in
the vicinity of d／（2a）＝＝O．16．
Thin circular cylinders were placed in the same turbulent boundary layer 30 mm
apart from the boundary wall to measure the variations of mean velocity， turbulence
and the static pressure in the wake． A complex interaction between the turbulence
in the boundary layer and that generated in the wake was suggested． A drag coeth一一
cient computed from the momentum principle was found， within the limit of experi一・
mental errors， to be the same as that of a cylinder placed in an unbounded uniform
flow． This fact suggests that the normal velocity gradient of the boundary layer exerts
little effect on the drag coeflicient of a thin circular cylinder， when the cylinder is not
in the immediate vicinity of the boundary wall． lt also becomes evident that the
spread of the wake is greic ter towards the side of the smaller velocity than on the
side of the larger one．
These experimental results were satisfactorily interpreted by the theoretical analysis．
that assumes the approaching flow of a constant vertical velocity gradient．
Centents
r．
2．
3，
4．
Introduction ． ， ． ． ． ． ． ． ． ． ． ． i … L L ・ ・ ・ ・ ・ ・ …
Simple shear flow pac st a circular cylinder subjectecl to an
interference of a p｝ane wall ． ， ． ． ． ． ． ． ． ， ． ． ． ． ． ， ， ， ． ．
2．1 Stream function ． ． ， ． ． ． ． ・ ・ ・ ・ 一 ・ ・ ・ ・ ・ ・ J ・ ・ 一 一
2．2 Velocity clistribution ． ． ． ． ， ． ． ． ． ， ， ． ． ． ． ． ． ， ， ． ．
2．3 Pressure distribution ， ， ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ・ ・ ． ．
2，4 Shear flow across two cylinders ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ．
Experiments on the shear fiow past a circular cylinder ． ． ． ， ， ．
3．1 Theory of grid design ， ． ． ． ． ． ． ． ． ． ．・． ． ． ． ， ． ． ． ．
3．2 Production of uniform shear flow ． ． ， ． ． ． ． ． ． ． ， ， ， ． ．
3．3 Uniform shear fiow past ac circular cylinder； experiment ． ． ． ．
Flow arouncl a circular cylinder in a turbulent boundary layer ， ， ．
tl．1 Experimental equipment ． ． ． ． ． ． ． ， ． ・ ・ ・ i J J ・ ・ ・ ・
a’） Air tunnel ．．．．．，．．．．．．．．．，．．．．．，．．．
b） Velocity measurements ． ． ． ． ． ． ． ． ． ． ， ， ． ． ． ． ， ， ．
c） Turbulence measurements ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ， ．
di） Static pressure measurements ． ， ， ． ． ． ． ． ． ， ． ． ． ． ． ，
4．2 Turbulent boundary layer along the floor of the test section ． ．
a） Coordinate system ． ． ． ． ． ． ． ． ． ， ． ． ． ． ． ． ． ． ． ． ，
b） Static pressure clistribution ， ． ． ． ． ， ． ． ． ． ． ． ． ． ． ． ，
c） Two－climensionality of the velocity fie1d ． ． ． ， ， ． ． ． ． ． ．
cl） Characteristics of the boundary layer ． ． ． ． ． ． ． ， ． ， ． ．
4．3 Artificially thiclscened turbulent bounclary layer ． ． ． ． ． ． ． ．
a） Nlethod of artifi6ial thickning of bounclftc ry layer ． ． ． ． ． ． ．
b＞ Static pressure distribution ． ． ． ・ i J J ・ 一 一 ny ・ ・ ・ ・ ・ ・
c） ’1’wo－61imensionality of the aL rtificially thickenecl boundac ry hc yer
cl） Characteristics of the artificially thicl〈ened bounclary layer ． ．
（！） Mean velocity 一． ． ． ． ． ． ． ・ J ・ ・ ・ ・ J J 一 ・ ・ ・ ・ ・ ・
5
7
7
14
15
18・
！9
20
L3
24・
30
3e
30
31
31．
32
32
32
33
33
34
38
38
39
41
41
41
3 Study on the Turbulent Shear Flow Past a Circular Cylinder 3
（2） ’1”urbu｝ence dlistribution ． ． ． ． ・ t ・ ・ ・ ・ ・ ・ ・ ・ 一 ・ ・ ・
（3） Nondimensional plot of mean velocity and turbulence ． ． ．
4，4 Ex． perirnents on the fiow near a circular cylinder in the
turbulent bounclac ry layer ． ， ， ． ． ． ・ ・ ． 」 ・ ・ ・ ・ ・
a）
b）
c）
cl）
Test cylinders ， ， ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ． ， ，
Static pressure dlstribution around the test cylinders ． ． ． ． ，
Lift and drag forces of t’lie circu｝ar cylinder ， ． ． ． ． ． ． ， ，
Estimation of laminar boundary hc yer on the circular cylincler
（1） Estimation due to the Blasius series for asymmetrical case ．
（2） Numerical calculations ． ． ． ． ． ． ， ， ． ・ ・ ・ ・ ・ ・ ・ J ・
（3） Estimation due to the method of Pohl．hausen ancl NValz ． ，
5． Turbulent wake of a circular cylinder located in a turbulent bounclary layer
5．1 VtJake－producting cylinders ． ． ． ． ． ， ， ． ． ． ． ． ， ． ． ・ ． ． 一 ・ ・
5．2 1（［ean velocity distribution ． ． ． ． ． ． ． ． ， ， ． ． ． ． ． ． ． ． ． ， ．
5． 3 Distribution of turbulence ． ． ． ． ． ． 一 ・ 一 ・ ・ J ・ ・ 一 ・ ・ ・ ・ ・ ・ ・
5．4 Estimation of the clrag force of the cylinder by momentum principle ．
5．5 Theoretical consideration on the two－dimensional wake in
6．
a）
b）
c）
d）
Append
a uniform shear flow
Conclucl
ix
The stability of asymmetrical double
a uniform shear flow ． ． ． ． ．
A1，
A2．
A 3．
A4
Mathematical formuhc tion of the problem
Perturbation solution ． ． ． ． ． ， ， ． ． ．
Drag force acting on the cy】三nder．＿ ，
A comparison xKTith experiments ． ． ． ． ．
ing remarks ． ． ． ． ． ． ． ， ・ ・ ・ 一 ・
Configuration of vortex rows ．
Equations of motion of vortices
Condition of stability
The average drag ． ．
．4Xcknowledgements ． ． ． ． ． ，
References ． ． ． ． ． ． ． ． ． ．
rows of vortices 三n
42
43
45
45
45
51
54
54
57
60
64
64
64
68
71
75
75
79
83
84
86
89
89
91
94
98
100
1el
List of symbols
Cl
g
h
hl
ガ
汐
戸
グ
qi
q2
十 一 一 一 一 一
一 一 一 一 一 十
一 一 一 一 一 一
1’ C O
s ．
．
一 一
radius of cylinder
acceleration of gravity
height of air－tunnel
distance between x－axis and the center of cylinder
imaglnary unltpressure， k ff
mean pressurefluctuating pressure
velocity－defect
polar coordinates
complex plane
一4
t ’ ’ ’ ’ ”
zら
ut ，
zム ． ． t
X，Y，2
tre， 2Yc
2 ． ．
／一1 ． ．
C，， ．
」ひ，’乙v ． ．
・びノ @wノ ． ． ’
C刀
Cl；
i 一 一 十
■ 噂 噸 ・
D． ．．．．．
HK
砿 ．．
Re， R ．
R，一，Ra±・，Re ．
．R12
U．
U．
WZ ．
7’ ．
’t） ’
’i＊
・e
g
獅
・］L
’／’t
レ
，R
・rw
． ● ● ● 響
り ◎
i 一
一 一
， 響
i 一
一 一
． 7 ， g o の
一 一 一 一 一 一
To ’
o．9f ．
cv ．
ゴ ．
・e ．
ザ．
．O^v．．
／？）モ．
．
．
■ ウ
一 一
一 一
．
．
．
一 一 一 一 一
一 一 一 一 一
一 一 十 一 一
Masaru KIYA
time
velocity components in the x， y and 2 direction， respectively
fiuctuating components of velocity
shear velocity
cartesian coordinates
coordinates of the center of cylinder
complex planeCDA／ 4ne’o
pressure coeflicient
dra．cr coefficient
lift coeflicient
drag force
shape factor
shape parameter
lift force
torque
Reynolds number
Reynolds number
correlation coeflicient
approaching velocity at the center of cylinder
velocity at the edge of laminar boundary layer
velocity at the edge of turbulent boundary layer
complex fiow potential
shape factor
specific gravity of air
botmdary layer thickness
displacement thickness of boundary layer
nondimensional velocity gradient， eddy viscosity
vortlclty
醒濁護．∴ξ、〃内．ξ了一
；一 cosh a’， tu d／ Ue
dynamic viscosity
kinematic viscosity
density of air
viscous shear stress at the wall
viscous shear stress on the cylinder
velocity potential
stream function
transverse velocity gradient of approaching flow
clearance between the cylinder and the boundary wall
angle
stream function
lmaglnary partreal part
4
5 Stucly on the ’1’urbulent Shear Flow Pac st a Circular Cylincler 5
1． lntroduction
The uniform flow of fiuid around a body submergecl in it has long been
a problem in fiuicl mechanics． Especially the prediction of the drag and lift forces
acting on the body concerned is of great importance for practical purposes， and
a considerable amount of effort has been devoted on this subject．
However， there are many examples in practice where a parallel fiow of fiuicl
is non一一uniform to the exteiit that the velocity of the stream parallel to the fiow
varies in magnitude according to the position measurecl traRsversely to the stream．
Such a flow exhibits shear characteristics， a typical example being the fiow in the
presence of a plane boundary where the velocity increases with the clis£ance from
the bounclary． Other examples of shear flow appear in the wal〈e of a body， in
jets and in the free mixing layer of two styeams with different velocities．
In developing the theory of shear fiow， an undisturbed velocity profile must
be clesribed at first． Since actua！ shear fiows usually have a somewhat complex
velocity profiles， some simplifications are necessary in order to make the problem
anaiytlcally tractable． One of £he simplest models for the general shear flow may
be an inviscicl uniform shear flow in which the velocity increases linearly in
a transverse direction to the stream of a Bon－viscous fluicl． Most theoretical
studies fall under this category．
This paper clescribes theoretical and experimental investigations on the fiow
about a circular cylinder placed in a turbulent boundary layer． The plane boundary
wall placed near the cylinder parallel to the undisturbecl stream out of necessity
comes into the question． The fiow fields near the surface and in the far wake of
the cylinder are considerecl by analytical and experimental means， hence it becomes
possible to determine tlie lift and clrag forces ancl the moment acting on the
cylincler concerned．
The knowledge of the lift and drag forces experiencecl by a bocly or boclies
located in a shear fiow is believed to be indispensable when the behavior of particles
in it is treated， a practical example being the transportotion of smaii particles
through a pipe． by means of air or water fiow． The experimental study （Welschof
［1］） concemiBg the flow of air containing numerous solicl particles shows clearly
the existence of a region of large transverse velocity gradient．
On the other hancl， the diffusion process in the turbulent shear flow is an
important phenomenon in air polution by smoke and in chemical engineering， and
several inves£igations have been performed in these fields． Skramstad ［2］ measured
the temperature distribution behind a heatecl thin wire placed in a turbulent
boundary layer． Corrin ＆ Uberoi ［3］ and Hinze ＆ van der Hegge Zijnen ［4］
made the same kind of measurement in the turbulent mixing zone of a round
free jet and a plane free jet， respectively． They obtained an interesting result in
which the diffusion of heat energy is greater towarcls the side of a larger velocity
than on the side of the smaller one， while the spreacl of wal〈e of a body placed
in a turbulent boundary iayer may be greater towards the side of smaller velocity．
as will be shown in section 5 of Chapter 5．
6 Masaru KIYA 6
The natural wind near the ground has a comparatively large vertical velocity
gradient up to a height of a few hundred meters． The wings of an aircraft which
takes off into the wind will then be affected by the existing shear in the ap－
proaching stream． The effect of shear must also be considered on the aerodynamic
forces experienced by the high architechtural structures， such as skyscrapers，
towers， stacks， etc． Some theoretical investigations of this category have so far
been made by several authors． V． Sanden ［5］ obtained a numerical solution of
shear flow past a wedge－shaped airfoil． Tsien ［6］ determined analytically the flow
field around a two－dimensional symmetrical Joukowsky airfoil placed in a uniform
shear flow， the velocity increasing linearly in the transverse direction to the stream．
He clarifed the effect of shear on the aerodynamic properties of the airfoil con－
cerned， by obtaining the lift force and the moment acting on it． Jones ［7］ studied
the characteristics of thin airfoils placed in a slightly parabolic shear fiow． More
recently， Kotansky ［8］ made an experimental study on an airfoi｝ placed in an
artificially produced shear flow． The analytical treatment given in Chapter 2 of
this paper will give a clue for the determination of the combined effect of both
the shear and the neighboring ground on the aerodynamic characteristics of the
airfoils．
In conjunction with the measurements of velocity distribution in the turbulent
shear fiows， several efforts have been made to obtain the so－ca！led Pitot tube
displacement effect． Arie ［9］ showed that a consiclerable amount of correction of
the velocity profile might be necessary when measurements are performed in a shear
fiow by means of a circular Pitot tube， describing how the location of the stagna－
tion point and the stagnation pressure woulcl be shifted by the presence of a uni－
form shear in the approaching fiow． A three dimensional version of this stucly
was made by Hall ［10］ and Lighthill ［1！］， concerning a sphere in a uniform shear
fiow． Young ＆ Maas ［12］ and Davies ［13］ performed some experimental studies
on this subject．
The dynamics of small particles suspended in a slow shear flow is of great
．importance， for example， in chemical engineering． The behavior of red corpuscles
ln bloocl may be another example． Bretherton ［14］， Saffman ［15］ ancl Kiya ［16］
obtained a theoretical formulae for the clrag and lift forces exerted on a circular
cylinder and a sphere， respectively， placed in a uniform shear fiow． Segre ＆
Silberberg ［！7］ macle extensive experimental investigations on the concentration
distribution of small particles suspendecl in a pipe Poiseuille flow． Eichhorn ＆
Small ［！8］ determined experimentally the lift force acting on a sphere under the
same fiow condition．
In Chapter 2 of this paper， an inviscicl uniform shear flow about a circular
cy｝inder p｝acecl near an infinite boundary plane is treated analytically． The com－
parisons of the theory with experiment are described in Chapter 3， confirming
that the cylider is subjected to a lift force which tencls to move the cylinder away
from the bounclary plane．
Chapter 4 is the description of the experimental investigations of the fiow
near the surface of the circular cylinder placed in the turbulent boundary layer
7 Study on the Turbulent Shear Flow Past a Circular Cylinder 7
that develops along the tunne1 wall． The lift and dyag forces acting on the cyl－
inder are obtained from the pressure distribution as the functions of the spacing
between the cylinder and the bouRdary wall． The laminar boundary layers on
the cylincler surface are computed in detail by means of the Pohlhausen－Walz
method， which permits an estimation of the viscous correction to the lift and clrag
forces． Some remarks on the comparison between the theory and the experiment
are glven．
The flow in the far wake of the cylinder placed in a turbulent boundary ｝ayer
is described in Chapter 5． Measurements of the mean velocity， the turbulence
intensities and the Reynolds stress are made． The theoretical derivation of the
velocity profile in the wake of the cylinder in a uniform shear flow was performed
with the object of interpreting the experimental results． Thei theoretical results
show that the effect of shear on the drag coefficient of the cylinder is of a higher
order of magnitude than the first power of the shear and this fact is confirmed
by the experimental results．
2． Simple Shear Flow past a Circular Cylinder Subjected
to an lnterference of a Plane WaH
In this chapter， a simple shear flow about a circular cylinder is describecl
analytically， together with the interference of a neighbouring bounday plane． This
may be the simplifiecl model of the flow across a circuiar cylinder placed in a tur－
bulent byundary layer． A steady flow of an inviscicl and incompressible fluid is
assumed throughout．
The analytical method thus cleveloped can． also be applied to the analysis of
shear fiow across two cylinclers， and the results coincide with the solutions given
by Lagaliy ［！9］ and MUIIey ［20］ when the oncoming flow is uniform． A proper
technique of conformal transformation of a circle near a plane bounclary may give
a clue to the analysis of the flow across an airfoil in the vicinity of a ground
surface， ancl the flow pattern around the cylinder would reveal an． aspect of the
lirnit of the workability of a Pitot－cylinder in the vicinity of a stationary boundary
frequently encounterecl in the measurement of a two－climensional shear fiow．
Furthermore， the lift force of a cylinder in shear fiow exertecl by the existence
・of a boundary should not be overlookecl in the practical engineering of an efEcient
hydraulic ancl pneumatic transportation of matters．
2．1 Sもream F皿ctio頚
It is wiclely known that the straight lines g’ ＝＝a constant ancl ’ij ＝a constant
・on a s－plane correspond to the two families of circles that intersect at right
．angles with each other on a z－plane when the relation
s・・＝ il。9，社蕉 （2ユ） 之一zc
・exiStS between z and 5－Planes， where 5＝ξ＋勿and 2＝X＋勿， Ifηis des三gnated
to have a coRstant positive value of cr， the equation of the circle on． the x－plane
8 ）！v｛［asaru KIYA 8
that corresponds to rp＝cr（＞O） becomes
‘：’2＋〈・y－ccoth cM）2＝〈c／sinh cu）2 〈2． 2）
The radius of this circle a ancl the distance between its center and x－axis hi are
respectively
a＝ c／sinh ot， h1 ：：’＝ccoth cr （2． 3）
In the same manner， the straight line rp ＝一P（P＞O） on the s－plane gives
x2＋（，y ＋ccoth P）2 ＝＝ （c／sinh P）2 （2． 4）
ゐエ。／sinhβ， ん2＝ccothβ （2．5）
on the 2－plane． Further， x and y can be written in the following forms in terms．
of g and rp by means of Eq． （2． 1）：
一病孟面諭 （・・6・）
…＝。δ無二ξ・ （…b）
Therefore， it can be seen that the x－axis on the x－plane corresponds to the straight
line rp＝O on the s－plane， and the origin on the s－plane goes to the point at infinity
on the 2－plane． The two circles on the z－plane given by Eqs． （2．2） and （2．4） are
shown in Fig． 2． ！． y When the oncoming fiow has a constant
velocity graclient， the velocity clistribution of
the approaching flow suthciently upstream of
the cylinder can be expressed by
z‘。。＝Ue＋ω〃， v。。＝＝＝0 （2．7＞
Then， the stream function to describe the
undisturbed fiow field must be
of］co ＝＝ juco d2J 一＝ u， ，」 ＋一Q…一2，1…E （2． s）
Any arbitrary constant may of course be
added as a constant of integration to the
right－hand sicle of Eq． （2． 8）， but this constant
may be made zero without missing the gen－
era11ty of the present consideration， because
it simply indicates the relative level of the
stream functions． The vorticity included in
the present fiow field can easily be obtainecl
from Eq． （．？．． 7） ：
cgco ＝ r｛1 ’L！20’＝ ww nvl！3’k’‘a－」’ ． ，．， 一 ．，
∂」じ Oy
κ
o・易
⑦
一 一 一
畠一易
O1
o
輪π2 P0
＼
o紡
ら
ユ＿6
02
の
Fig． 2． 1． Definition sketch of two
circu｝ar cylinders．
（2． 9＞
9 Study on the Turbulent Shear Flow Past a Circular Cylincler 9
So for as a two－dimensional uniform shear flow of an inviscid fluid is assumecl，
the vorticity given by this equation must be constant everywhere in the fiow field
now considered． When the stream funct．｛on ¢ at any point in the flow field is
written， for convenience， as
ψ・＝ψ．。＋4f。＋ζグ、
→・・バ）＋！穿＋り （・・！・）
the velocity components and the vorticity at this point are respectiveiy
弓診一一一艦 （・・！・）
ζ一個器一一際＋知 （・・12）
Since C＝：C．． ime 一w （a constant）， one can obtain the following relation by substituting
Eq． （2． 10） into Eq． （2． 12・ ）：
（響＋明り＋c暫墓⊇＋響）一・・ （…3）
σ畷ノ〇＋ωシ
■一…．．…u旧．雪
．、．．立＿．一
Uo：
u＝：uo
一“一
Fig． Z． 2．
Uo ．1一””’itw’…’” X
ヒ｛ 1
（6）
残1瓢αoシ
”竃ωシ
73”u一拳一一．
．一．“一一．X．．ww．．nv一一1一一．．． x
o
（o） （b） （c）死＝41（。。，＋Vee、 残，一・卿㌘
Configuration of shear fiow across a circular cylinder．
Fig． 2．2 shows a schematic physical view of the fiow field expressed by Eq．
（．？．．10）． Fig． 2．2（b） is the case of the first term in the right－hancl side of Eq．
（2． 10）， which is the case of a uniform flow across a circle． Fig． 2．2（c） indicates，
in the same manner， a pure shear fiow across a circle to be introduced by the
second term of Eq． （2． ！0）． These two cases of flow can be superposed to realize
the flow field of a shear flow across a circle shown in Fig． 2．2（a） which is the
case to be solved． Therefore， the relation of Eq． （2．！3） that was deirved by
substituting Eq． （L． 10） into Eq． （2． ！2） leads to the Lap｝ace equations ：
讐＋響一・ （…4＞
饗＋券一・ （…5＞
that can be solved independently for superposition． A definitio．n sketch of a two－
dimensional shear flow across a cylincler in the vicinity of a plane wall is shown
in Fig． 2． 3． The boundary conditions to
be introducecMn this case are： y／a （i） When x一＞oo， namely when s一＞O，
Uf。→0 （2ユ6 a）
er，一）O （2． 16 b）
because there shoulcl be no effect of the
cylinder there．
〈ii） When v＝＝a， namely on the sur－
face of the cylinder， ft x／a ／一／ UJ’o ＝ 一 t｛o［zi］v．，． ，， ＋ ro
Fig． 2． 3． Definition sl〈etch of shear flow
一一野際ξ＋・・（…7・） 忌1・1・議ご謝1・i・th・vi・1蜘
凱一一・．号［藍・γ1
一一蝋己69濃6諭2＋・1 （…7b）
（iii） When rp ＝： O， namely along x－axis in the x－plane that corresponds to the
pla．ne wall， there should be no effect of the cylinder on the stream function，
because the wall is a solid boundary ：
Yfe ＝＝O （2． ！8 a） 蔓ゲ｝：＝0 （2．18b）
ro ancl ”ri in Eqs． （2． ！7 a） and （2．17 b） are respectively the values of stream
funct三〇ns that form a cylinder in shear飾w and pure shear flow． The terms
［？Lx］T・・． and ［tLi］Z．．，． can． be expandecl in Fourier series：
M一［レ・糞1・…7・・c・…ξ］ （…9・）
［・」］1／・，a ＝＝・・［…h・＋・毒、e…7L・（…h・＋・）・・…ξ1 （・・！9b）
The bounclary conditions given by Eqs． （2． 16） ancl （2． 18） and the Fourier series
expansions of ［y］，，，．，． and ［tJ］；．．． suggest a transformation of the inclependent vari－
ables x， ？！ to eH， rp in Eqs． （2．！4） and （2．15）． From the general theory of the
curvi1inear orthogonal coordinates， we have the relation
霧雛＋書雛㍉誹庭（劇烈傷漸劉 （…9・）
where ¢ is an arbitrary continuous function and
［（劉2＋（劉↑・92一［傷）2＋（制2］告 （…9d）
Substitution of Eqs． （2． 6） i・hto；’ Eqs． （2． 19 c） and （2． 19 d） gives
／ θ 1
真
α
ん，々
@｝
0
ll Study on the Turbulent Shear Flow Past a Circu｝ar Cylinder 11
91＝＝ 92＝蕊hη一、d§ξ『
謬＋二一璽shη評馴寒・＋穿）
Accordingly， the govern量ng equa£ions（2．14）and（2．15）take the form
∂；墓・＋∂艶一・ （2・・4・）
…讐Σ＋響一・ （…5・）
．which are again the Laplace equations．
The general solution of Eq．（2．14 a）can be written by the technique of sepa－
ration of variables：
yto＝／lo＋Σ（An cosh 2zηcos 7zξ＋23，、 sinh nηcos 7zξ
？乙＝1
＋C，、 cosh n rp sin／zξ＋D，， sinhηηsin 7．zξ） （2．20）
The solution forψ’ P， to be obtained from Eq．（2．15 a）， also takes the same form．
．When the boundary condition on the surface of the cylinder given by Eq．（2．17 a）
．and the relation given by Eq．（2。19 a）are inspected， it will easily be seen thatξ
is contained in． Eqs．（2．19）玉n the form of cos nξ． Therefore， the compar三son
between Eqs．（2．17 a）and（2．20）gives
C，、xxO， Z）n＝0 （7z＝！，2，3，… ） （2．2！）
Further， when the boundary condition ofψ’ 潤≠n at r7 ＝・ O given by Eq．（2．生8 a）is
．substituted into Eq．（220）together with Eq．（2．2！）， oDe obtains
ノ1。＋Σんcosフzξ一〇 7る照I
Namely，
Ao＝＝0， ／1，1．＝0 （n罵！，2，3，… ） （2．22）
Then， oD． the surface of the cylinder（η＝α）the equat玉。．n of 4㌔given by Eq．（2．20）
beconユes
yf。一ΣB。． sinhηαcos 7zξ （2．23） ？乙＝二1
．On the other hand， the condition given by Eq．（2．17 a）is
・・’・一一・1・・［・＋・薫1幽・…ξ］＋・・ （・・24）
．when Eq．（2．19 a）三s introduced． Therefore， the comparison of coe缶cients三n． Eqs．．
〈2．23）and（2．24）gives
・。一・・。・，B声一メ讐 （・．25）
12 Masaru KIYA 12
Now， the stream function㌍o given by Eq．（2．20）is reduced to嶺e following
form when all of the values of coe備cients thus determined by Eqs．（2．2！），（2．22＞
and（2．25）are substituted：
ト4u・・薫1塾≒欝四一 （・・26）
In the S蹴e manner asψ’。，ψ璽、 can be written aS
．ψ’、＝A6＋Σ（Al、 coshπηcosπξ＋・B；、 sinh 7zηcos／zξ
＋qc・sh・．・ηsin・zξ＋1）；， sinh・・ηsinπξ） （2・27）
and the boundary con．ditions given by Eqs．（2、16 b），（2ユ7 b）and（2．！81））deter－
mine the values of coe備cients l
ヨ Al一・，且；、一・， q一・， Dl汗・一1・・＝一讐cG・h・，（2・28・）
Bト・・・・艶招 （・28b）
Therefore， Eq．（27＞becomes
。Q cothα＋π
Ψ・一一2・・2潟、…～疋r・i・h・・η・・…ξ （2・29）
As a result， the stream functioD． given by Eq、（2．10）to describe the且ow負eld
around a cyl量nder in the vicinity of a plane wall could be determined：
ψ一（姻＋蝋穿÷勾
一卿（o）＋讐ψ働 （・・3・）
where the symbolsψ（o＞andψ（1）are respectively
ψ・・L／る。，謬翻一・薫、一渉曹・・ ξ （・・3・〉
ψ・・L（ sinh rpE16．lg’hli’tL：’i’Lc“os’G’）2一・怠癖謡物…h・・η・・…ξ （・・32）
Sinceγo課μ06 andγ王＝＝（to c2／2）cothα， the quantity of fiuid that fiows through the
clearance between the cylinder and the plane wall can eas三ly be evaluated when．
the stream function along the wall is tak：en as the reference value of zero：
ψ。・・＋・・＝・：・…＋…讐…h・ （・・33）
When the velocity uo in Eq。（2．8）is replaced byθ1一ω1z1， Uo being the．
velocity of apProach at y＝乃、， the stream function g三ven by Eq。（2．30）can be．
reduced tQ the foHow玉ng dimensionless forrn：
濃一（・一・）ψ…s・・h・＋皆ψ（1）s・・1・1・…h・ （・・3・）
13 Study on the Turbulent Shear Flow Past a Circular Cylinder 13
2S
24
20
ノ．6
13
x
12
ノ．2
0salsrl
O4
y
oa一一V 一6
Fig． 2． 4．
34
3e
？6
？2
／8
14
io
O5
一5 －4 －3 －2 一ノ O r／a h，／a＝／25ノ， ！1＝ e．O
Streamlines around a cylinder in a uniform
fiow： hl／a＝1．251， ？，＝L’O．O．
t4
σ
國 … 一
C－en． Lt．e－r ！tlee
3
i・ミ
046卿
。？
／i
Oe一，Xt 一5 4 －3 2 t O ；じ／σ
ん・乃嵩鷹／，n・・a乃
Fig． z． 5． Streamlines around a cylinder in a uniform shear
flow ： hl／tt ＝一L 1．251， 2 ＝i O．75
’where
2 ur＝ e cosh ex ， e ＝＝ to a／ UQ
・Obviously， U，（1－2） means the velocity of approach at the 」u axis as will be seen
in Fig． 2．3． Figs． 2．4 and ．？．．5 are the examples of streamlines computed after
Eq． （2． 34）．
Attention may momentarily be diverted to the writing of the first term of
Eq． （2． 34） by means of the elliptic functions． Since the two terms appearing in
・ip（O＞ are written in the form
。。謡鴇§r＄（ s…cot百）
・喬揖…h・・η・・ ξ一S；（・三論…一）
一α g ＝e
，asCY（2） meaning “the imaginary part of 」i，” the first term uo cip（O） of Eq． （2． 30） can
be defined as the imaginary part of the following complex flow potential ：
“4 ．． M・・a・uKIYA 14
9（・）一一…［…号＋・薫1了≦レ・・一］
The right－hand side of this equation may be rewritten as
ρ（・）…［…音÷噺ぞ♂・・一＋・斯阜・・・…］（・・34・）
On the other hand， the trigonometric series expansions of the ellipticζfunc一．
tion of Welerstrass with the real period 2ω1瓢2πand the imaginary period 2ω3＝4ガα．
assume the forms
ζ（・）一饗一＋吉・・÷・急鶏・，…一 （・・34b）
ワる ζ（・＋・・）一計＋・・＋・妻1Ti，：b，，，，一・・一 （・・34・）
where
η、 ・＝ q（ω1）， η、一ζ（ω、）
The substitution of Eqs，（2．34 b）and（2．34 c＞into Eq．（2．34 a）yields
ρ（の一一・・…［ζ（・）＋ζ（・＋・・）一到
omitting a constant without loss of generality． This is equivalent to the result
obtained by Lagally who considered the potential flow past two c三rcles of arbitrary
rad圭i and arb童trary relative distance．
2．2 Ve璽ocity distribution
The velocity components tt and v can be computed from Eq．（2．11）：
・・一一1・・（纏＋霧・．霧）
・一．99一一一・・C器髪一＋駕一霧）
where
ん＿一cg勉二・g鍾
The terms∂y／∂ξ，∂y／∂η，∂x／∂ξand∂副∂ηcan easily be computed with Eqs．（2．6a＞
ahd（2．6b）， and the velocity components expressed above become
・一÷［…h・…ξ書望＋（・一…h・…ξ）器］ （・・35・）
・一一÷［（・一…h・…ξ）一髪＋…h・…ξ霧1 （・・35b＞
Still， the terms∂ψ／∂ξand∂ψ／∂ηmust further be computed with Eq．（2。30）：
1夢一昧，・）＋離（ξ，・）
15 Study on the Turbulent Shear Flow Past ac Circular Cylinder 15
霧一…噛）＋努㍉・（6， ’o）
wher⑱プi，五，91 and g2 imply
礁・）一群喫器＋・糞、〆ちゴ…hl・η・・…ξ
蜘）薫鵬欝＋・糞、亟9鰹押・・蜘・・…ξ
・1（ξ・・）一陣生欝一・薫〆誓ゴ・噛・・…ξ
・・（g， rp）一2sin認7畿losξ）一・妻、禦禦…h・・η・・ ξ
Accordingly， by introclucing new forms of function
F， （6， ’o） ＝ （！ 一cosh q cos g“）f， （g’， ’o） ＋ sinh rp sin g“ g，（6， rp）
F， （e， rp） 一 （！ 一cosh rp cos g”’）f，（e， rp） ＋ slnh rp sin e g2（g， rp）
G，（gE， ry） ＝＝ 一 sinh ny sin g’ f， （g， rp）＋（1－cosh rp cos g“） gi（e， rp）
G，（6， rp） ＝ 一 sinh rp sin 6f2 （6， ’o）＋（！ 一cosh rp cos 6） g2（e， rp）
one can write the velocity components as
〃一…G・（ξ，η）＋誓G・（ξ，・） （・・36・）
・一一レ・聡・）＋響蜘）］ （・・36b）
Hence， these velocity components can be expressed in the following dimensionless．
form when the reference velocity is taken as Uo shown． in． Fig． 2．3．
え zt． ’ti’：’7，’ ＝（！ ww ？L） Gi（ム⊆，η） 一F ’L’ tanh cr G2（g“， ’o） （2． 37 a＞
都一ト・）聡・〉＋号…h・蜘〉］ （・・37b＞
2．3 Pressure distribution
When the body force is absent， the equations of motion to describe a two一
・dimensional flow of an inviscicl fluid are
磯・＋傷一一一募・ （・・38・〉
・・潔一＋・需一一譜 （・・38b＞
Since the vorticity is
av au c ＝一 一tt：”” 一 ’5／tJ”’
Eqs． （2． 38 a） ancl （2． 38 b） become
16 Masaru KIIYA I6
t・｛建＋・一門一・ζ一詰一蹴 （・・39・）
・驚一＋・寄＋・ζ一一一劣一 （・・39b）
Further， the de丘nition of stream funct三〇n is
確雰．嬬馳吻一ぬ
Accordingly， the sum of the two equations（2．39 a）and（2．39 b）after being muレ
tiplied respectively by dx and dy gives
l「”一． ・一一．lI
I ． 1
．2． 、励，／ン 1
ぼゴ ドボ ピ ドノ
：瑠二を㌧二〕 ｛ ．2 励 ／／25ノ・’ ・！ ミ
一4 「／．／i25／ ‘噛 ド ノ
慧 ／．梛 智一・
糞噌6』／ 一 逃 ？’＝a25
タ 匙6’ q －8 ． ． ， 1 ） ． こし ， 1 ． ）
旨 一8 一1¢ ： ・ 1 ＿ノ4 ． 、 “ ．I l
一ノ0
じ n16』一 ・ E l ・ ＿12
1 旨 I I
し
一・物・一房一一一・・1、1、1／／ 一・Eb。 一一、。．，グ。…ゴσ、b吻
θ画風θ・） θ吻虐・・〉
ノ・ ， 1 旨 、
0一 ・ ・ 「
擁 塗．一：＼．．、励．＿． ． ’ 4026 ・ ヘロ し ド ノ 5 1 ．ぐノ501一 ．
o
－2
一【べ4
×全｝
S－6
－8
－lago
一4
、憲一・tt
＞PS 一8一
ノOI
－12
鴨歴％
125i’一
Pi25 ’
ノ皿ノO
h，／a 一xxxKx p一 ．A，． ， 4026 ・t“NX ’一lv“ i 200e ／50ノ ＿』 ＿ ノ251 ’ ．
1t2S
A＝075 1“X i ． “X X．1 一，a ． ；
i 一，2 1 ，
一6e 一eo o so 6b ＞b ”L4po 一60 一・3’o o’ ’3e 60 θ （グと？yrees♪ θ ぐゴegrees）
Fig． 2． 6． Pressure distribution around a cylinder in a uniform
shear flow： （a） 2＝O．O， （b） O，25， （c） O．75， （d） 1．0．
／17 Study on the Turbulent Shear Flow Past a Circular Cylinder 17
イ青・・＋暑一・ψ）一・
whence
－tl一一 g2＋一e一 一一一ca¢＝constant ’ （2． 40）
where
q ＝ （zt2 ＋ ・v2）｝
Eq． （2．40） is a form of eRergy equation for the shear flow now considered． Since
the velocity components are already obtained， the pressure distribution in the flow
field can easily be computecl． lf the values at a reference point are denoted by
subscription s， the climensionless forna of pressure at any point is
緯一一1一儲ト） （241）
When the point of reference is taken at （oo，hi），
qs ＝ Uo ， Ps ＝Po ， ¢， cr （Uo－to hi） hi＋’wwto／mml’fi
Therefore， the pressure coefllcient C， expressed by Eq． （2．41） becomes
q一…織一・一（園部［ψ一｛（u・一・恥璽｝］
誌輪。灘譜離・lhご留錦慧 ・一…tttttttt一
Ci）c ＝： 1＋2（R－2） （！－tanh cr）m（’tt’，’ ）2 （2． 42）
where q，， is the velocity along the surface of the cylin－
der． Examples of pressure distribution on the surface 2i／
of the cylinder in the vicinity of a plane wall are shown， b
i・Fig．2．6i・t・・m・・f p・ess・・e c・・缶・i・・t・nd・h・d三s一ミ
器b鷲e号、舐二翻e翻監糊。li鴛e欝㍉tainable in terms of lift coefficient Ci： by integrating the
pressure distribution given by Eq． （2．42）：
Ci； E 一pttgi 一一 S’11（． 11，， C，，． sin ede （2． 43）
where e is the angle shown in Fig． 2．3． The results
of this integration are shown in Fig． 2．7． As will be
seen in this figure， the lift is negative in the vicinity
of plane wal｝ so far as 2＝O namely in the case of
一2卜
．4；i rm．ww．．／t．．．．．．．．．．．ll．．．．．．．：i
1 2 3 4
ん，ル
Fig． 2．7． Lift coefficient of
a cylinder in a uniform
shear flow．
18 Masaru KIYA 181
a uniform flow． But the lift becomes positive when there exists a certain amount．
of velocity gradient in the vertical direction of fiow which is similar to the case・
of a practical bounclary layer that develops along a stationary boundary of a wall．．
2．4 Shear Flew across two cylinders
This problem can be solved in the same way as the previous case by slightly’
changing the boundary conditions in solving the Laplace equations given by Eqs．
（2．14a） ancl （2．15a）． The definition sketch of the two cylinders is given in Fig．
Theand the results are
i・1・e ＝・・…鑑㍑1縞
瓦高・［・一・．COI濫謬，器q
Cno＝O， Dno＝O
2．L T｝1e boundary conditions重。 be replaced for Eqs．（2．16），（2．！7）and（2．18）are，
（i）T。→Oandψ◎1→O at5→0（之→・・）
（ii） Sinceη＝αon毛he surface of circle O1，
ψ層。＝一Uo［IJ］η＿α十γ6
一一・…［・＋2薫、・一…c・…ξ］＋・6
T・＝：一号M匙。＋・｛
一一撃［…h・＋・薫、c・…71tr（…h・吻・・一ξ］＋・｛
（iii） rp ＝＝一βon the surface of circle O2， whence
？P．o tt － z‘o［21］η＿一β＋δ6
一・…［・＋・茎、・ ・…ξ］・・6
u］’・一一号Ml一，＋δ｛
一部［…hβ＋・妻、凶…hβ＋・）…小
The s£ream functions 4「o and凱can be obtained by taking the follow圭ng forms
that correspond to Eqs．（2．20＞and（2．27）：
ψ’o＝＝ノ100＋Σ（A，、o cosh 7zηcos／zξ＋㌧8刀．G sinh 7zηcos／zξ
ntt．．．＝1
÷（隔Gcosh 7zηsin nξ＋Z），、o s量nhηηsin 7zξ）
ψ◎1 ・・ fao、＋Σ（A．1 cosh n rp cos n g“＋B．i sinh／zηcosアzξ
7乙＝1
＋Cnl coshηηsin 7エξ＋エ：），、1 sinh 7zηsinπξ）
coefllcients can now be determined by the boundary condition stated above，
19 Study on the Turbulent Shear Flow Past a Circuiar Cylinder 19
A。。訟一U。C＋7’6 ・＝ Zt。C＋δ6罵一Σ、4，、。
？己㍑1
・1・・一 Z1ん・一一讐2c・・h・＋・1一イ…hβ＋・1
／1。1 ””一。。・ごα（goth・＋塾唖＋dggthβ士麺亘！！・． sinh n（a ＋ P）
｛プ竺（cothα＋n）～三～～sh 7z互＝．望．…nP（99憲h．β．ゴ＝7z）£9shぞ弓．蔓．
（2． 44 a）
（2． 44 b）
B。i ＝一tO c2．．
C，，、＝0，
The relations（2．44
γ6一δ6＝2uoc
rl一δ｛ノ葦⊆（…h・一…hβ）
Obviously， Eqs．
cylinders in uni
st／Iln’H’”il””（a””IJPJ一）
D，，， ＝＝ O
a） ancl （．？．．44 b） yield
（．P．． 45 a）
（2． 45 b）
（2．45a）and（2．45 b＞give the quantity of How between two
form flow an．d in pure shear How， respectively． Eq．（2。45 a）is
equivalent to the result obtained by Lagally［19】．
Finally the stream function． becomes
ψ一・…ψ②＋誓ψ（3）
where
ψ・・L講≧論ξ＋・糞1藩努1鵠（…h・・η・・…ξ一・）
一・蓼惚1獅噛β一・］・・鋤・・…ξ
ψ・・L（謡勢翻2
一・煮8…πα（cothα＋ノz）旦i書痴誤簿othβ鍍三in惚（…h・・η・・…ξ一・）
一・煮、鷺…”α（cgthα＋71）col認三三塾oshア！“…h・・η・・…ξ
3． Experiments on the Uniform Shear
Flow past a Circular Cylinder
The experiments were conducted for the purpose of verifying the trend of
change in the lift coethcient of a circular cylinder in a uniform shear flow clevel－
oped in an air tunnel installecl at the Fluid Mechanics Laboratory of the Faculty
of Engineering， Hokkaiclo University． The air tunnel is of a blow－off type， the
flow being procluced by a centrifugal blower driven by a 100 kW electric motor．
The velocity of air flow can be adjusted in three ways ； by changing the revolu－
tion of the blower ancl by changing the valve and gate opening installed respec－
tively at the suction tower of the blower and at the exit of the tunnel． The cross
2e Masaru KlyA 20
section of the test section is 50 cm x 50 cm， and the side walls over the en£ire
length of 5 m are transparent．
Th，e experiments were made in a uBiform shear fiow artificially generated by
arrariging a grid of rods of 3．2 mm diameter over an entire section of the ap－
proaching flow after Owen and Zienkiewicz’s ［21］ theory．
．3．1 Theory of GTid Design
The test section of the wind tunnel may be treated as a long channel， as
sketched in Fig． 3．！， with walls y＝＝O， y：：：：h， iB which a grid is placed in the
plane x＝O． At great distances upstream the velocity is uniform and has the
valueび；far downstream the velocity is again parallel to the walls and its mag．
．nitude is given by
u一＝ U＋ tu（y一’／．？．1一’） （3． ！）
ダ『
Fig． 3． 1． Arrangement of the grid and the coordinate system．
One assumes that the fiuid is inviscid and that tu hf U is of a smal｝er order
of magnitude than unity， and consequently the departure of any streamline from
a straight line is small． Accordingly， the stream function which describes the
・entire flow field may be written as
ψ＝こノy十ψ’ for ユ〕〈0 （3，2a）
＝一 Uy＋一：tl一（y2－hy）＋¢’ for x＞O （3．2b）
¢’ is the perturbation stream function which represents the effect of the grid， and
satisfies the Laplace equation
履夢＋奪浮’一・ （…）
・owing to the constancy of vorticity throughout the flow field． ip’ is everywhere
small compared with Uh． Appropriate solutions satisfying the boundary conditions
∂ψ’1鯉一型1鯉一・
．for all x are
拐、か遜・・響一，x〈・ （・・4・）
21 Stucly on the ’1］urbulent Shear Flow Past a Circular Cylinder 21
一薫、8・・蒜・・雫・x＞・ （…b）
where the constants A． and B． may be cleterminecl by the condition of flow at
the gricl．
The presence of the gricl requires that （i） a¢／0y is continuous through x＝：O・
（continuity）， （ii） the velocity component a¢／0x obeys a certain refraction condition
across x＝＝O， and （iii） the difference between the total pressures on a given stream－
line for large positive and negative values of x must be equal to the resistance
per unit area imposed by the grid． To these may be adcled the conclition that the
static pressures far upstream and far clownstream of the grid are independent of iy．．
Conclition （i） leads to
・薫1凪C・・丑秀鉱一・蓉1・・Bn…撃一傷一助誓
or， settlng
樋＋h・蓉、G…丑秀鉱
ωん
熟・rB・・＝cバび （3・5）
The refraction condition （ii） requires the y－component of the velocity to
change by a factor of in passing through the gricl． a for a grid of uniform resist一・
ance is known to be 1．1（！＋K）”ii2， where K is the resistance coeflicient
K一（p．． 一p ，．．）／佳脚）］・）
p．． and p． ． being the pressures far upstream and far downstream of the grid． lt
is unlikely that this value will be seriously in error for a gricl with slightly gradecl
resistance provided that K is given its loca1 value． lf the resistance grading is
reprentecl by
K一瓦［・＋・（司 （・・6）
such that E（2y） is small and
∫1・（・」）姻
it follows that
・r、端再［i 一 一li｝一d，EII一： eK－6一）1 一t一 o （，2） （・・7＞
With cr given by Eq． （3．7）， the refraction condition is
妻1・・B…s・・撃＋（a＋bE）薫、凪S・・撃一・
which， noting that A． and B． are O（to h／U＞， reduces to
22 Mas． aru KIYA 22
B．＋aA．一：O一一〇（e）he／CJ） （3． 8）where
・－批戸一一一亥躊蔽
Finally， one must consider the change in total pressure along a streamline．
Accorc！ing to the assumption of an inviscid fiuid， the total pressure remains cons－
tant along streamlines upstream and downstream of the grid but， at the grid itself，
there is a clecrease in total pressure amounting to K（1／2）P［te（O）］2． Hence
場・ぴラ・一吉卜・（t」→の］2一κ去・囮・）1・ （…）
where po and ？i are respectively the static pressuses far upstream and far down－
stream of the gricl． The streamline which has the ordinate ？yi for large x meets
the grid at y＝yo， and the relation between ye ancl v． i follows from Eqs． （3．2）；
・・忌詞彩輪（劉 （・・！・）
Eq． （3．！0） implies that， to the first orcler in to h／U and g／’IUh， the x component
of the velocity at the gricl is given by
u（o） 一＝ u ＋ e） （y， 一 一・lt… h） 一y 一a－9（一IS－t／L＞－－to） （．3． n）
which， together with Eq． （3． 9）， leads to
継一瓦（ wh1＋・）＋2了（1＋K・）（糊＋響野（3・’2）
Since （Po一？i） is independent of y， Eq． （3．12） resolves into
｛’1［L：O：一tL一一1…1一一K， （3． i3） ’tt”PU2
，and
K・・＋・碧一（・＋K・）（努一儲＋・奢畑野一・ （・…）
The distribution e（2」，） follows directly from Eq． （3．14） if one notes from Eqs．
〈3．4）， （3．5） and （3．9） that
∂響L・ω・・Cゴ翻妻！・C・・…携L・h（rf”a一’）（努一～圭）
（3． 15）
hence，
・（・・）一一・磐一（曇詳丁｝激一助 （・・！6）
23 Study on the Turbulent Shear Flow Past a Circular Cylinder 23
At large Reynolds numbers． the resistance of a grid of rods in a uniform
stream can be predicted with good accuracy if the drag coethcient based on the
blocked area and the average interstitial velocity is taken to be unity （Wieghardt
［22］）． For the present purpose it will be assumed permissible to adopt the same
procedure for each portion of the grid， with the possible exception of places where
the spacing changes rapidly with y， since the scale of variation in the y－direction
of the velocity through the plane of the grid， u（O）， is much larger than the
greatest spacing between adjacent rods． Accordingly，
8 K。．（1＋ε）＝ （1一．e’）2
where
g ＝＝ d／s ；
d is the diameter of each rod and the spacing is s（y）． With e given by Eq． （3．！6）．
one obtains
d－e）2 ＝＝ Ko［i”2 r｛ZihEJ （一k＋’rtir．）（’il“一一1；）］ （3・ i7）
3． 2
ing
Ko＝＝
shown in Fig． 3．2．
The grid was placed at a section 60
cm downstream of the entrance to the
working section， 50 cm square， of the wind
tunnel at the Fluid Mechanics Laboratory．
Fig． 3．3 shows the arrangement of the
grid in the air tunnel． Traverses， across
the working section in planes parallel to
the grid at a number of distances from it
were made with N．P．L． type of Pitot tube
of 8 mm diameter， Pitot coethcient being
unity， at the mean wind speced U of
19．2 m／sec and 22．6 m／sec， respectively．
No variation in static pressure could
be detected from the traverses， which were
confined to distances greater than approxi－
mately one tunnel height from the grid．
The grid could not be approached much
more closely with a conventional static
probe owing to an appreciable inclination
and curvature of the streamlines．
Production of Uniform Shear Flow．
A grid composed of parallel rods of 3．2 mm diameter was constructed accord－
to Eq． （3．17） with tu h／U－O．45 and o．s
1．15． The appropriate values of g are
0．4
A〈0．ヲ
0．2
0’f
O 0，2
Fig． 3． Z．
0．4 0．ti 0．6
、擁
Spacing of rods．
Fig． 3．3．
／O
Grid of rods for the production
of shear flow．
24 ］Vlasaru KlyA 24
Lタ
ノ．2
Lノ
シo
0．9
0．8
0．ク
ー0．4
雛％疋88
05
07’
o．6
4グ0．4
o〃．タ
o．2
びノ
一61．ヲ
Fig． 3． 4・
一一Z．2 一〇，／ 0 0．・f 0．2 0．9 0．4 x／rf
Velocity contours in the plane ．r／lt＝＝3．0，
The contours of tt／U as cleduced from the
Pitot tube traverses in a plane paral｝el to the
grid and 150 cm （x／h＝3．0） downstream of it
are shown in Fig． 3．4． The absence of any
systematic departure from lines parallel to the
x－axis， together with the previously mentioned
fact that the static pressure is constant over
the region， suggests that Bo seconc！ary flow
was present on a large scale．
The profile of zt．IU in the plane of sym－
metry ’of the tunnel， 2 ：O， at clistances of
59．5 cm and ！50 cm from the grid （．zVh ＝＝ 1．19
．and 3．0） are presented in Fig． 3．5． The
agreement between the measured velocity dis一
’tributioB． and that predictecl by the theory is
tolerable．
Ostensib｝y there is no systematic change
in the rate of shear between x／h＝1．19 ancl
x／h ＝＝3．0， and even small irregularities in the
velocity profile appear to suffer no appreciable
decay． The small scale irregularities in the
velocity profile， which were founcl to be re－
peatable in the experiments，
of the gricl．
ミ
f．o
o．s
o，ti
ぐλ4
O，2
Oa7 a6 ag ∠0 〃 〃 ノタ
〃／グ
Fig． 3．5． Velocity profiles inea＄ured
in the plane of syinmetry of t／he
tunnel．：．d／h＝：3．0， 2〃～＝0．o；●， し「．．
22．6 m／sec， o， U＝： 1（ ．2 m／sec，
most probable curve， 一一一 theory．
「D／03
・7，
ク
r
’
／
／ん・
can be attributed to inaccuracies in the manufacture
3．3 Uniform Shear Flow past a Circular Cylinder； Experiment
The plane wall that would be located in the neighborhood of a circular cyl－
inder was representecl by placing a polyvynil chloride plate， 5 mm in thicl〈ness，
paral｝el to the flow， the leading edge being given 26．50 bevel on the other side of
25 Stucly on the Turbulent Shear Flow Past a Circular Cylinder 25
465
Dσ！ど’カ《地だ
@グ〆α〃θどθr4〃
@∠θ〃〆〃 仰。
@ l@ l zヲ。一「
@ 1〃86力汐〃1〃〃〆伽1俳 1
，w
ミ 勲ち励～6@紹〃げ湧 4グ8
@∠θ・桝〃”グ
I@ l
X〃げ／司
@
@ 1
Fig． 3． 6．
a／7！t （．〃〃，
Schematic view of the experimental installation．
the test fiow as wili be seen in Fig． 3．6． The test cylinder was made of a brass
tube， the diameter being 49．3 mm． Fig． 3．7 is given to supplement Fig． 3．6 in
order to show the features of the test． The velocity distributions of the shear
flow at the three sections are shown in Fig． 3． 8． The distance between the test
and the plate was adjusted by i 4i”
cylinder and the plate
a screw device and a static pressure hole
of 1 mm diameter in the central section of
the cylinder was connected to a G6ttingen
type manometer． The angular position of
the pressure hole， read on a protractor，
gave the pressure distribution around the
surface of the cylinder， which was inte－
grated to evaluate the lift and drag forces
acting on the cylinder．
The measured pressure distributions
around the test cylinder are shown in Fig．
3．9 in terms of the pressure coeflicient and
the angle measured from the straight line
parallel to the approaching flow， the pa－
rameter being the ratio of the clearance
between the cylinder and the plane wall
to the diameter of the cylinder． The pres－
sure coethcient is defined by
Cp一ぞ一ρo
PU2 2
where Uo is the approaching velocity at
the center of the cylinder and p， is the
static prassure of the approaching flow．
Since the upstream velocity profile and
the configuration of the test cylinedr yield
the values of A and cr which appeared in
重．1・tl・
IfEllg ll．．
1／i’
甦
慌
Fig． 3．7． Experimental installation．
ヲ9．ノ
ヲ0
“ 20
g巡
ノ0
0
ヒ
』
劇甚
．
o
謬9
十
十十
」醐．0 ／0
u（〃／碗〕
20 90
Fig． 3．8． Velocity prefiles of the artificially
produced shear flow：十，25 cm，○，41 cm
and e， 59 cm from the leading． edge．
26
／，o
〃∂
0，6
0．4
0．2
SC． 0
黛一〇．2
黛
’3－0．4
－0．6
－0．8
一／，0
－f．2
u／．4
Masaru KlyA
0 20 ．40 60
／．0
0．8
0．6一
ごλ4
0．2
0
さ…・・
へ撃一〇．4
一．．‘0．6
…0．8
．・．．．ノ，0
一．．／．2
．…．／．4
一／“6
一．／．8
80 ／00 ／20 ／40 ノ「60 ／80
θ（吻声ee5）
… …
㎜ I I
／80
0 20 40 60 80 ／00 ！「20 ／40 ／？60 1ノ∂∂1．．．tt．．．
（1？ （cVeq， t’ecl，，s）
Fig． 3，9． Pressure distribution around the cylinder．
／60
／80
26一
27 Study on the Turbulent Shear F20w Past a Circular Cylinder 27
f．a
O，8
O．6
登
辻㌦4孕
3
4．2
o
一〇．2
一40 一20 a
θ伽gr8eの
（a）
20 4a
’f．o
O．6
O．6
』聖
喪．．a4
ぎ’
O．2
o
一一 Z．2
一4e 一20 0θ傭多rε85♪
〈b＞
20 40
ノ．0
O，6
O．6
、摘
× O．4
k冷
s
O．2
一〇．2
讐
’ じ2ユ胴4只ヲ〃〃r
ﾚ！を～の・01∂
ﾉ匡oo2
、
’
」
、巳
ノ
亀
●～
、
一4汐 二20 0 20 40
θ侮grεe3）
（c）
Fig． 3．10． Pressure distribution arouncl the test cyiincler：
（a） ti／（2a）＝1．28， （b） O．81 and （c） e．02 ； 一 experi－
ment， 一一一 theory．
28 IMasaru KIYA 28
the theoretical treatment， it becomes possible to compared the theoretical results
with the experiments． The theory is believecl to be valid on the surface of the
cylincler near the stagnation point， so a comparison is made concerning the pressure
distribution in the neighbourhood of the stagnation point． Fig． 3． 10 （a）， （b） and
（c） show that a tolerable agreement is obtainecl for both the stagnation pressure
aiid the position of stagnation point， provided that the clearance between the cyl－
inder ancl the plane wall is not too small． The turbulent boundary layer which
develops along the surface of the plate， as will be seen in Fig． 3．8， may be re－
periment， because
in the boundary layer than in the uniform
it should be noted that the pressure
deviates i ncreasingly from the measured one
of the plate，
sponsible for the large discrepancy between the theory and the ex
the velocity gradient is much larger
shear fiovu’ of the outside layer． Morever，
distribution preclicted， by the theory
with the increase in angle from the stagnation point． This fact may be interpretecl
as the development of a laminar or turbulent boundary layer along the surface of
the cylinder， the clisplacement effect of which produces an effect akin to the increase
in the raclius of curvature of the surface．
ノ．4
／．2
／．0
O．8
O，6
0．4
0． 2
o
Fig． 3． 11．
下丁丁■丁ユ｝．．ドニ「「
0．4 0，8 ／．2 Z6 ti／’（2a ）
ζグ2／Uo＝0．0／b， 2rz ae／ノノ＝・76x／04
Lift and clrag coefficients of the cylinder
in a uniform shear fiow．
←．．．．．圃
The lift and drag forces acting on the cylinder are obtained by integrating
numerically the pressure distribution on the surface of the test cylinder． The
results are shown in Flg． 3．11 in terms of the lift and drag coefllcients Cin C．．
and the distance between the cylinder and the plane wall ln dimenslonless form．
CJ；p and CD． are defined by
c勿「毒r菰 lt’i 一2bu
ol pU 2
L一 sin e de
ぎ
29 Study on the Turbulent Shear Flow Past a Circular Cylinder 29
傷意一青∫：艶一・θ・clθ
where L． and D，， and the lift and drag forces， ancl p， a．nd p，，， are the pressures
on the lower （一1800＄e＄OO） and upper （OO S e S 1800） surfaces of the cylinder，
respectlvely． A sufHx of p is given to the lift L and drag D， because computa－
tions were macle only from the pressure distribution without evaluating the shear
force on the cylincler． As will be eviclent in this figure， the value of ii｛t coefficient
increases when the cylincler approaches the plane wall， again． confirming another
qualitative agreement with the theoretical result．
Finally， it is of some interest to point out that the bounclary layers on the
cylinder seem to separate from the surface after the traBsition from laminar to
turbulent飾ws， for three values of」／（2ζの， namely，0．128，0．066 and O．0． The
Reynolds number 2ct Uo／v for these cases is 7．6×10‘， while the orclinary uniform
flow of the sam．e Reynolds number past a circular cylinder is found not to exhibit
the transition of the bounclary layer on its surface． Fig． 3．！2 shows the relation
between the pressure and shear stress clistributions around the circular cylinder i．n
the uniform fiow obtained experimentally by Fage and Falclner ［23］ for both several
・0
一〇．4
一〇．s
登
幾一ノ．2
準
S 一L6
一20
一2．4
Fig． 3．12一
〃 ” 80 90． ノ00 〃0 だ0 β0 〃0
、。鋼 ’f
5
一 一
／
しbb王ρち
a12x1
！ 〃6矧5
N＼！
、、、
、、、
／0
@ 8潤A 6＼｛～鳶ミ 4ミ 2
亀、
、＼
1
覧
関鳩
、
％ 60 ％’ 刀0 〃0 だリ ノヲ0 〃0 θ（卿・．姻
Relation between the pressure distribution and the
shear stress distribution around a circular cylinder
in a uniform fiow （Fage ancl Fall〈ner ［23］）．
30 Masaru KlyA 30
Reynolds numbers and oncoming fiow conditions． lt is obvious from this figure
that the point of inflexion of the pressure curve near e ＝：＝ 900 implies the transi－
tion point from laminar to turbulent fiows， if the Reynolds number is more than．
！．08 x 105． Also， the point where the pressure becomes almost constant corresponds
to the separation point of the boundary layer． The pressure curve for 1｛ ＝！．06 〉〈 105
implies the laminar separation．
Upon considering the striking resemblance between the pressure clistributions
around the cylinder for A／（2a） ＝：！．！28， O．66 and O．O and those obtained by Fage
ancl Falkner， togethether with the fact that the pressure recoveries at the rear side
of the cylinder are almost the same for both cases， one may say that the transi－
tion of the boundary layer occured even for the subcritical Reynolds numbers
when the cyllnder apporached suthciently to the plane wall in a uniform shear
fiow． The increase in the turbulence level near the plane wall may be responsible
for the transition， although the true mechanisms of the transition from laminar
to turbulent flows are as yet unknown．
4． Flow around a Circuiar Cylinder in
a Turbulent Boundary Layer
Since the velocity gradient in a turbulent boundary layer is several times
greater than that in the artificially generated uniform shear fiow， more effective
comparisons of the theory with experiments may become possible． Thus， experi－
mental studies of the fiow around a circular cylinder placed in a turbulen．t boundary
layer which c！evelops aloRg the tunnel wall were performed in some detail．
4．1 Experimental Eq“ipments
a） Air tt｛nnel
All experiments were carried out in the air tunnel of the Fluid Mechanics
Laboratory of the Faculty of Engineering， Hokkaido University．
Pressure taps of 1 mm diameter installed at intervals of 30 cm， in． the upper
and lower boundaries of the test section over its entire length of 5 m permit the
measuremeut of the static pressure distribution along the tunnel walls． Smooth
poly－vynil chloride plates， 10 mm in thickness， are attachec！ to the fioor of the
tunnel with woocl screws in order to eliminate the effect of roughness of the
wooden floor， because use is made of the turbulent bondary layer which develops
along the lower tunnel wall． Accordingly， the effective area of the test section
becomes 50 cm × 49 cm．
The velocity of the undisturbed fiow in the test section was determined by
／calibration in terms of the pressure difference between two piezometer holes， one
of which was located about 5 cm dow，nstream from the entrance of the test section
and the other was located at the encl of the plenum chamber． The longituclinalturbulence level N’1’2’i’E’^U． is about O．s9（o at an air speecl of U．． ＝＝20 m／sec．
Other dimensions apd characteristics of this tunnel are explained in Chapter 3．
31 Study on the Turbulent Shear Flow Past a Circu］ar Cylincler 31．
∂） ▽lllO（rit：ソバ4easz｛．プe／7zen ts．
The air velocities are measured by means of both standarcl Prandtl－type Pitot
£ube of 8 mm diameter and the total head tube of rectangular cross section O．57 mm
x1・5 mm． The t・tal head tube is ma三nly utilizecl t・・btain・the vel・city distributi・ns
in the turbulent bounclary layer and in the wake of the circular cylinder． Each
of these tubes is mountecl on a traversing mechanism which can be adjusted to
the measuring position with an accuracy of O．！ mm．
The total ancl static pressures are connected to a G6ttingen type manometer
〈lf20 mm） or a Betz type manometer （1／10 mm） to obtain the dynamic pressure－
Allowance must be made for the transverse gradient of velocity， for proximity
to the wall， ancl （strictly， although this is often neglected in practice） for the
ve10c三ty fluCtuatiOnS in a turbulent layer．
The transverse velocity gradient gives rise to a highly complex． flow pattem
in the mouth of the Pitot tube， where the viscosity of fiuid produces a marl〈ecl
effect． As a result， the reading of the tube is not c！ear！y related to the total
pressure of any of the approaching stream filaments and the observed total pres－
sure differs from that of the filament occupying the position o｛ the axis of the
tube before its introcluction into the fiow． The effect can alternatively be repre－
sentecl by a displacement of the effective center of the tube． According to Young
and Maas［ユ2］the e脆ctive displacement A？y fOr a square－cut head in turbulent
flow is towarcls the region of a higher velocity and is given． by the equation
劣一・。・3＋α・8言 （・。・）
where cl and D are the internal ancl external diameters of tube． For want of the
corresponcling formula for the total head tube of rectanguiar cross section， the
．effective displacement was tentatively estimated by means Qf Eq．（4ユ）to obtain
the reasonable results， as will be seen in Section 4． 4． Since cl＝＝O，．35 mm ancl D＝＝
O．56 mm， one obtains A？JID＝＝ O．18． ln the case of turbulent boundary layers， the
corrections were restricted up to the clistance 4D＝L．2 mm from the plane boundayy．
When the tube is cleser to the surface than twice its diameter， the effect of
wall proximitY act in the opposite direction to the effects of velocity graclient and
a further correction may be required． For turbulent pipe fiow MacMi11an ［24］
finds that the correction may conveniently be expressed as an increment A V to
be addecl to the measured velocity， and indicates that a correction of the same
order is applicable in turbulent boundary layer flow as well． Davies’s ［13］ results，
however， suggest that his negligible correction， in bounclary layer fiow is maintained
right to the vxTall for all sizes of the tube． The reason for these discrepancies is
somewhat obscure， thus n．o correctio．ll for prox三mlty to the wall is made三n the
present expenments．
c） Tu7－bulence ’i7iectstt．7’enzents．
The values of s／’L’ti’1’i’2“， s！’1’i’2 and ／tc’ii／ V’ are measured with a constant temperature
type of hot－wire anemometer using a single－hot－wire technique． Tungsten wire
32 iMasaru KlyA 32
of 5 pt diameter was copper plated with
CuSO4 before soldering onto a probe．
When the copper is removed by etching
with HNO3 to produce about 5 ohms of
electric resistance， the length． of the acting
part of the wire is about 1．5 mm and the
total length of the wire is approximately
2mm． The calibration of this hot wire
was macle in a uniform flow in the air
tunnel by comparing with a standardPrandtl－type Pitot tube and the change of
voltage necessary to keep the temperature
of the wire constant for van’ous speeds of
air is obtained． Through calibration the
characteristics of this hot wire is found to
be llnear as intended． One example of the
results of calibration is shown in Fig． 4．1．
d） Static P7”essu7Ae measu7Aements
If it is assumed that the static pres－
sure in a turbulent boundary layer is
of the layer， the static pressure becomes
to the main flow，
walls of the test section．
static pressure holes of a standard
assumption is reasonably satisfied．
喬
R
ノ0
6
6
4
2
0
Fig． 4． 1．
／0 20 ヲO
Yeto・1tyノ〃〃協・・幽
An example of the hot－wire
calibration．
constant and equal to that at the outer edge
constant over each cross－section normal
and consequently may be estimated as the static pressure at the
The results of measurement of the static pressure using
Prandtl－type Pitot tube show that the above
4．2 Turbulent boundary layer along the floor of the test seetion
a） Coo？ndinate system
In the following descriptions of the experimental results， the Cartesian coor－
dinate system x， y and z will be used． The origin of the eoordinate system is
Iocated at the mid－polnt of the tunRel fioor at the entrance to the test section， as
shown iR Fig． 4．2． The x axis is in the downstream direction， the y axis is
verticaliy upwards and the z axis is taken normal to both x and y so as to form
the right－hand system of coordinates．
Fig． 4． 2． Illustration of the coordinate system．
’33 Study・ on the Turbulent Shear Flow Past a Circular Cylinder 331
b） Static p7’essu7’e dist7’ibz｛tion
Owing to the development of the turbulent boundary layer along the tunnel
walis， the main flow velocity increases in a downstream direction， and at the same
time the static pressure decreases in view of Bernoulli theorem． The measured
static pressure distribution is shown in Fig． 4．3， together with the best fit curve
obtained by the method of least squares ；
P，， ＝＝一〇．0050 （x－28．3） mmAq （4． 2）
where x must needs be measured in cm． Eq． （4．2） may be used to compute the
static pressure at any clesired position． The corresponding velocity distribution is
given in Fig． 4．4．
Jt’ ’（cm）
o A ／00 200 300 400 500
Rtや
桑．
s
虞ti）g 一〇v
一〇，2
Fig． 4．3．
● ご2〆〃〃81泣～｛／カケ1伽θ4。〃〃ご拗纏84
、、、、
Static pressure distribution along the tunnel．
／6，2
／6．0
／5．∂
ミ
湿／5．6 g
題／5，4
／5，2
／5．0
／48
0 ／00 200 300 400 ・ ． 500 λゴ〔Cm〕
Fig． 4．4． Velocity distribution along the tunnel．
c） Tzvo－dimensionality of the velocity field．
In the experiments to be described in this chapter， the turbulent boundary
layer along the tunnel floor is used as the approaching shear fiow to the circular
cylinders， then one must examine the two－dimensionality of the velocity field．
Since the air tunnel has a rectangular cross section with the right－angled comers，
secondary flows necessarily occur in any section normal to the main flow． The
“ 皿
1
竹
34 IMasaru KIYA 34
モ
D． 一 ｛ ．一
「一｛ ．曲 ｛ ．
二と・・：．20600η
碕／臨
^．oo
ｿ99
@ …
u．1．…．
@ i ．．1一．．
@ l I@ … 0
o@ Ik
㎜…．r…．
@ 「…罹ｿδ5一
09グ@ iO90080075070
iz〆σあ
O60O．55．065一
一20 一ノ0 o
♂ず （c’m〕
（a）
ノ。 20
ゾOo
80
60
40
20
0
cl）
ミ
戦
’E
5一
轍．．
／oo kIII
OO
狽堰@一一
；1 一一一
60F一・一一一
／1
L
l
ii
20P
1
0L．．
¥〇，95
bl ’1’o
．．t．q．．，〈．”．｛．
Qク
Fig． 4．5．
；多霧．．．．．．．．．．．一」
ｻ 0 〃 20 ．；．塁．ノ 1‘∫〃ノ，
（b）
Velocity coi tours in tlie 2ykN－plane：
（a） ‘v ＝’＝’一 206．0 cm， （b） 338．2 cm．
｝ongitudinal velocity distributions measured at the two sections x＝＝ 206．O cm and
．x’ ur 338．2 cm， respectively， are shown in Fig． 4．5（a） and （b） in terms of ufU．．， y
and z． Although the velocity distributions outside the boundary layer are seen to・
be exactly two－dimensional， those within the boundary ／ayer are somewhat dis－
torted， the velocity near the center line ．or．． m－O being highly clecreased． However，
the velocity field in the vicinity of x＝＝一8cm may be said to be nearly two一・
dimensional， then the developments of the turbu／ent bounclary Iayer are measured
at this section．
d） Cha7”acte7－istics of the tu7”bu／ent boztncla7“cv ／czyer
Fig． 4．6 shows the mean velocity distributions in the turbulent boundary layer
35 Study on the ’1”urbulent Shear Flow Past a Circular Cylinder 35
measured at the section z・・一8cm in terms of z‘／ひ。。（x）and tJ． The displacement
and momentum thicknesses， o“＊’ and e， defined by the equations
・・一∫1（ ul『一砿）吻・・一∫1洗（1一洗）吻
are shown in Fig． 4．7 as functions of u． The vlscous shear stress at the wall T，．
and the shear velocity zt，
T・ ：tU（鋤。。一綱…’i2
may be computed from the measured velocity profiles by means of the well－known
momentum integral equation originally derived by Karman：
識嘉＋（H＋・〉一斗一…総… （・…）
／00 f…一
ハ賢
ミ
．醜
80
60
20
0一 0，2 0．4 Oti a6 Za IO ZO IO ZO ZO ／．0・ ZO IO tO ．翼＿ ．．．“翫
Flg． 4． 6． ］Ntlean velocity distribution in the turbulent boundary
layer along the tunnel wall．
！z（伽）
… ㎜ 買 「 r
｛
？ 内内
c6賃8p
〃努@〃ゑ0 s9 V70 唐Sタ 虫?0 ％4． 物．8 碗．タ
i … I ill i ｝
1 …哲．．．…ド …レ「…㎞… @ l i l ト i i l l l
i 1…1…＿
争 一 …「 …
u解「…P i I
@ し
…．．．．．．．．．．．；．．．＿．＿＿一一．．．．．．・．．．一
…一＿＿．．．．．＿ち．．＿＿＿一
i i」 凸
π ．…
乃
ミ
sも
ピ
§
も
／2 r’
ノO
s ’i
t7
2
／………罫… ユ H．．．．、．．］1……
DrF「一一1． 親
ト 1 1田『田田
u……
i i ↑ 「 トﾂ事θ（〃齢θ晦 ∠力繰8η8の
［
戟@i1 言l l
蕊ヨ 1 ｝ 11i i 1 「 長｛ ． 担
c i l
】 ii l 1＿．一．．
p一．一
@旨 l i l
讐…｝ ．9 l i i1 1δ寮θ 1＼
礁 L ． i ξ
E三ユ i E l l
i
i l I iﾓ＿＿遍
@i i@i i1 i
l l E 一± 0 ／00 200 300 400 5fOO ．z’ c’ cin ．）
Fig． 4． 7． Distrjbution of displacement ancl momentum thicl〈nes＄es．
．36 Masaru KIYA 36
or
蹄鰹田＋（H＋・），各計謝 （4． 3 b）
H is called the shape factor of bonudary layers and defined by
5＊H一 ’i’
沿黶f（4． 4）
The values of u，． fU． thus determined are shown in Fig． 4．8．
Fig． 4．9 shows the same velocity profiles on a non－dimensional basis． ln
order to avoid the arbitrariRess of 6 （boundary layer thickness＞， the nondimensional
distance from the wall y／6＊ is used． lt is seen from Fig． 4．9 that the nondimen－
sional veiocity profiles exhibit a similarity downstream from the section x＝ 65．8 cm．
These profiles， together with the values of H and R， （＝ U．0／v） computed at
each section， may be compared with the data obtained by Klebanoff and Diehl ［25］
which are included in Fig． 4．9．
The wall proxlmity law and the velocity－defect law for the velocity distribu一
惹
鳶
0．06
O，04
O．02
o o o O‘
二。 ’／00 200
g一“
soo
…℃レ．
e
o artificiaU．ノ ご！1ノとぶε〃ε4
0 net thickenedtt．．rmrm．1’’”rj
400 500X． ．（cm）．
Fig・ 4． 8． Distribution of shear velocity．
S
ノ．0
O，8
0．6
0．4
O．・2
O
y／ff＊
Fig． 4．9． Nondimensional mean velocity distribution
in the turbuient boundary layer．
r暇 匠 ’
汽
A
2
f諸
Z
篤
⑦κ‘ε伽。が0観，ρθ罵4ヲ”～””
@ 斤θθ加励’功。〆惚伽浸η勿ρθr
A ノα8か∂ηoが一∠）1θ乃ど，石「θ＝／4850
加．疹．甜η吻ψθr
“ ｝
一
疋（ω 8翠伽）θ伽）〃 £θ? 65L 6 ヲ．51 25／ ノL40 2497
B 227．0 5．57 4．／4 λヲ5 42β6
B ヲ44．0 717／ 566 ／Lβ2 6ソノ2
2 4 6 ∂ ／0 〃
37 Study on the II］tirbulent S， heftc r 1？1．o“r Past a Circular Cylinder 37
ヲ0
26
22
ミ／8
ミ
ノ4
ノ0
6，，L
ヨ ……u 一
暴 一562吻〃罰磐・＋ 4．9 ； ‘！d～’5θr
x紙”伽望ζ＋5／・o爾一 一 一
● ● q〆‘
（
o
9
・5で「胴「7
L！i
ゆ’
〆@！I
i ！@！̂
！
！
’
〆I！ o
，〆！ ● ●
cz働） 1。6翅 …
・ノ0氏グ
怩Q270
Fig． 4．10．
ノkフ0’ ／〃00 〃000
evllr／11
Comparison of experimental results with
the logarithmic law．
／6
，4
／2
／0
“［N 8 E8こ5
tsl 6
4
2
0，tat4rm－LLW．， 一pmide， Y／6
Fig． 4． 11． Comparison of experimental results with
the velocity－defect law．
tion are compared with experimental results in Figs． 4．10 and 4．！！． The mean
velocity distributions for the turbulent boundary layer along the tunBel floor，
together with the law of the wall suggested by C｝auser ［26］ ancl Coles ［27］， are
plotted in the form of z｛／zt． against yzc，／v in Fig． 4．10． The law of the wall
takes the form
一／：Z－！ll一 ＝s．62 10g，， 一一Y5｛’：’一 ＋4．9 （Clauser） 〈4．5a）
or
－f，‘一． 一s．7s log，， 一3一…t9一‘一：’… ＋5．1 （Coles） （4．5b）
In contrast with the measurements for pipes ancl channels， it is seen that there
is a systematic increase in the deviation from the logarithmic law as the Reynolds
number decreases．
In Fig． 4． I！， the boundary ｝ayer results are plotted in the form suggested by
一 ㊥
1・協一〃
、、 瑠 、 、
Q 4τ
√ﾓ
吻陶〃÷獅撫；土器・づ62一斗・・65・瀦辮
》、ミ ⑱■ 0●、
◎
召τ
◎ ●、 o
o
②o 評幽
o z〔oη）
O 6516
E／0515
U2270
、●笥
♂、
炉 ●■
’卜 ｛
o 1＼＼、
38 tMasaru KIYA 38
the velocity－defect law， together with the channel results． lt is seen that the
present experiments give the velocity－defect law between the boundary layer with
zero
case
pressure gradient and the channel results．
is given respectively by
㎜午鮭一一・．621・91。考一÷・．25
（Boundary layer， Klebanoff－Diehl ｛25］）
The velocity－defect law for each
巫譜一一・．621・91。号＋・．65
T
（Channel， Laufer ［281 and D6nch ［29］）
（4． 6 a）
（4． 6 b）
4．3 Artificially thiekened Turbulent Boundary Layer
As seen in Fig． 4． 6， the boundary layer thickness， o“， is about 40 mm at the
position x＝＝：219．8 cm where one intends to set the test cylinders． ln order to
carry out accurate measurements， it is clesirable to use test cylinders of as large
diameter as possible． However， for very large diameters， the finite cross－sectional
area of the tunnel test section necessarily lntroduces complicated wall interference
effects on the flow past the test cylinder， then the maximum diameter of the cyl－
inder must be llmited to one－tenth of the helght of the test section at most．
Since， as a compromise， the maximum diameter is selected to be 36．2 mm， it is
better to have a thicker boundary layer than that described in section 4． 2． This
need leads one to artificially thicken the boundary layer along the tunnel floor
by means of some tripping devices．
α） Method（ゾメ1プ孟晒6ガαZ Thicleening（～ブBoundaノニy Z，ay（？1。
It is at once obvious that the objects protruding outward from a surface
create a larger drag than that of the surface， ancl hence offer a means of producing
a layer of de－energized flow in a comparatively short distance in a streamwise
direction． The energy of the main fiow is diverted into eddy motion in wakes，
and， while the flow i．s turbulent， it does not have the same dynamic structure as
a fully－deveioped turbu｝ent boundary layer over a smooth surface． The question
is， after leaving drag－producing objects， what length of smooth surface is neecled
for the layer to become the same as a smooth surface generated layer． lt would
seem that the lengtlt would be less than that required to generate a layer of the
same thickness， but this is by no means a foregone conclusion．
An attempt of artificial thickening of the turbulent boundary layer with zero
pressure gradient was made by Klebanoff and Diehl ［25］， for the purpose of
facilitating the use of multi－wire hot－wire probes and minimizing errors due to
wire length． Their measurements were performed with regard to the boundary
layer on a fiat plate mounted vertically along the center line of the tunnel test
section． ln one attempt to thicken the layer rods of several diameters were placed
singly in the otherwlse laminar layer and in contact with the surface at 4 feet
from the leading edge． Another method consisted of cementing sand－paper to the
39 Study on the Clrurbulent Shear Flo“r Past a Circular Cylinder 39
surface beginning at the leading edge ancl extencling to clistances up to 2 feet
downstream． By measuring the clistributions of the mean velocity zt，， the intensity
of longituclinal velocity fictuations V’ ^／’i’1””i” and one－climensional spectrum of zt－fiuctua－
tions across the layer， they found that， of these three criterions， the mean velocity
distribution appeared to afforcl a simple means of iclentifying a fully－developed
turbulent bouRclary layer． Each of the above methods was found to produce
a turbulent boundary layer of clesired thickness， name｝y， about two times of the
original layer， at a specified section．
The turbulent boundary layer thus formed was found to beeome characteristic
of that developed from the beginning on a smooth fiat plate after passing over
the required length of smooth surface． For a given boundary layer thickness，
artificial thickening made possible a saving in length provided that there was
enough length for the layer to become free from distortions．
However， in £he present experiment， unsuccessful was an attempt to thicken
the original turbulent boundary layer by cementing rough sand－paper onto the
surface beginning at the entrance of the tunnel test section and exteRcling up to
70cm downstream． Another method was to place a circular rod of 3．2mm di－
ameter on the surface at the entrance of the test section， but this also yielcled
no change in the mean velocity profile． This was not unexpected， because the
boundary layer was already turbulent at the entrance of the test section， contrary
to the case treated by Klebanoff ancl Diehl． Although a much larger rocl of
19 mm diameter arranged in the same manner could thicken the boundary layer
appreciably， the non－dimensional velocity profiles not only were distorted in excess，
but did not attaiR a similarity even at the far downstream position of x＝＝ 250 cm．
Finally， an appropriate position to place this rocl was found by trial and error．
Fig． 4． 12 shows the final configuration of the tripping rod．
砺〃etCt．∫ノク〃〃
sf7ao；500mm
Fig． 4．12． Configuration of the tripping rod．
b） Static p7’essure clisti－ibzt．tion
In order to examine the characteristics of the artificial｝y thickened turbulent
bounclary layer， the static pressure distribution at the tunnel wall was measured
at first． The results are showB in Fig． 4．3． lt will be seen from this figure that
a）
ミ ミ リ 戦 40 20
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」［Cm．］ X（
cm）
Two－
dimensionality
of亡
he
ar亡
ificiaHy
thickened亡
urbulent
boun（
】ary｝
ayer：
（
a）
x＝91，
8cm，
（
b）
216．
5 cm，
（
c）
324，
5 cm
and
（d）
353．
5 cm．
「コ
■
鷹的
^．00－
O99
1 ……zづ5ヌ50刀
i i
」丁
i09グ
i
」
肩：
ogo
．．
．．
一．
一一
．一
一一
ﾑ一一一1
ﾛ」l
I
1P．
．．
．．
．．
．…
@ 1
085
O∂O
盾V5
、
O70－
O65
l ii i
、雌
i0601 ・
壬優
L」20
8 K 釜 秦 6 ゴ さ
4i Study on the Turbulent Shear Flo“r Past a Circular Cylincler 41
the static pressure is identical with that along the original boun．dary layer， and
Eq． （4． ？“） may also be applied to the thickened boundary layer． This is because the
increase in the d三sp工acement thickness due to artici丘ca｝th三ckening is not so large，
namely， 3 mm at the most， corresponding to the increase of O．6％ of the main
flow velocity， that the extra decrease of the static pressure will be submergecl in
the pressure fiuctuations and experimental errors．
C）Tωo－Di’mensionαlitbl（）f the A7ρ孟批ガα殉Tんiclee／zecl BOZ〃zclcw”：y LCtyer
IMean velocity distributions in the artificiaiiy thickened boundary layer are
measured at four sections， that is， x＝＝91．8 cm， 216．5 cm， 324．5 cm and 353．5 cm，
in order to examine the two－dimensionality of the fiow． These are shown in Fig．
4．13． Comparing these figures with Figs． 4．5（a） and （b）， one can see that the
two－climensionality of the flow is remarl〈ably improved by the introduction of the
tripping rod． Especially， the flow fields in the sections x＝nt 91．8 cm and ．zr ＝2i6．5
cm may be said to be almost completely two－dimensional． The equi－velocity iines
at further downstream sections x＝324．5 cm and x＝＝ 353．5 cm are somewhat dis－
torted， and are believed to demonstrate the deve1opment of the secondary flow．
However， since the tvgTo－climensionality of the fiow is satisfactory as a whole， the
development of the boundary layer is measured at the section 2＝＝ 一8cm， as was
done for the boundary layer without the tripping rod．
c／） Cha7’acteristics of the Arti117ciallbl Thicleenecl Boundarbl Labler．
（1） Mean Velocity
The mean velocity profiles in the artificially thickened boundary layer are
sbown in Fig． 4．！4， at a number of sections downstream from x＝203．7cm． From
a comparison of this figure with Fig． 4．6， it wil｝ be seen that the boundary layer
ミ
ミ
ロ嵐
／00 1”um”’
40
20
「下…「「……
u…目←一一．“一一叫一一z（oη）
「鐡7 ＼
L＿」＿．．．一．．＿．．．．．
2ヌ24 ＼ 濯ヨ4 2％1グ 系9ヌ9 ヲ08ヲ 惣ヲ ガヨ0 ヲ烈ヲ 4060
1！i
r－
11■ i「一
@ ヒ
l l i i i堰@l i i l
ll l
…1 1
乖
一 ㎜ 閲
o a2 o，4 o，6 o．6 zo zo． ／o zo ！o zo ／．o ／．o ！o lo or／Uoo
Fig． 4． 14． IVIean velocity distribution in the artificially
thickened turbulent． boundary layer，
42 Masaru KlyA 42
is increased by about 20 mm in thickness at the section x＝＝219．8 cm where the
test cylinders are to be arranged． The values of the displacement and momentum
thicknesses are shown iB Fig． 4．7． The shear velocity is also shown in Fig． 4．8
in terms of u，IU．． and 」c． The values of tt．／U．． are almost identical with those
for the original boundary layer．
（2） Turbulence Distribution
The values of V’：’i7’tumila’， VIS7’E2 and EZtTS’ ｝ were measured with a single hot wire
（Arie ［30］） in order to clarify the characteristics of turbulence of the thickened
boundary layer．
Measurement of VE’；7E”）2 ： The single hot wire was suppoyted vertically in xy
plane． ln this case， the hot wire is sensitive to tL’ and zv’， but when u is large，
the error for u’ which will be caused by w’ is negligible． With the hot wlre
supported in this manner， the mean horizontal velocity ze is also measurable． The
measurement of this component of velocity agreed fairly well with the resu｝ts
previously obtained with the total head tube． The quantity VE’2’L’）一2 can be computed
from
VJrl’ ＝＝ A x V．，，， （4・ 7）in which VRMs is the root－mean－square value of the fiuctuating voltage and A is
the slope of the voltage－velocity characteristics of the hot wire．
Measurement of V“’2 and il’Vi： When the single hot wire is supported at
an angle of 600 against the c axis in the xy plane， the root－mean－square of the
fluctuating voltage gives a clue for obtaining Vi57’i2 and z一｛’v’． ln this case，
（A×VRms）i＝＝（一（’t． sm 6’O－6W：i”S7’”gl’ilLln ttt）’2’）ii2
0r
（A×VRMs）1一（0．866V’it’7S／／’”2）・＋（0．5Vシ2）・＋0。8667夢’
In the same way， when the hot wire is supported at
axis in the xy plane， one obtains
（A×玲Ms）詞0．866V7を）・＋（0．5炉ト0．866、ん’
The sum of Eqs． （4．8） and （4．9） gives
（A×▽ゑMs）1＋（A・VRIIs）1－1．5（Vぞ豊門〉・＋0．5（V扉）・
or
布’2
Since ＞露ノi
When Eq．
tt’び
魂［（丑・㌦ls細・Y・・s）1－！・・（〉をア）2］1！2
（4． 8）
an angle of ！200 to the x
is obtainable with Eq． （4．7）， V’ 狽煤fS’ can be obtained
（4．9） is subtracted from Eq． （4．8）， i’ZilS’｝ is obtained as
＝・
@一1’1’i’g’2’”［（錨・s糊・猪・司
（4． 9）
（4． 10）
with Eq． （4．！0）．
〈4． 11）
43
置
養
0’ fO
0．08
a．06
0御
0． 02
Study on the Turbulent Shear Flo“r Past a Circular Cylinder
i i「1
一．u 1
一＿．L＿一一1－ ｛ ．一
1 憂「
』拓｝u．
¶ ．．一
…「屍1’nvT e一
［III
．i，一．1．wuww．．
20．【．．．．．． ．且
””’ P””’g－r”1－L’
：’ua’iri””
i－rr K’uaiTeKM’
1二「1 繰…
7 羽
一
o 20 ” 12b’”’M－tt’b”一LL’＝gbXto
b．．z’f2．a翠．．諺’．一一諦『”ズ。ア〔鋼
y／o
Fig． 4． 15． Variation ef the fluctuating
turbulent velocity components．
蔭
1
f4
／2
／0
8
6
1；
2
0
x／o－4
「一
●o
禰 … ．L．．一 一 一
●
一
20 40 60 即 卿一上 1．mm一
43
The results of measurements at the section x＝
4． 15 and 4． 16．
（3） Non－Dimensional Plot of Mean Velocity and Turbulence
The non－climensional mean ve1ocity u／U． at various sections are shown in
Fig． 4．17， the abscissa being the nondimensional clistance 2」／S＊ from the wall．
Since the ReyRolds number R， does not differ so much for each section， the
measured velocities fall on the same curve． ln this figure， the non－dimensional
mean velocity distributions for free transition and for a O．25 in． cylindrical tripping
rod proposecl by Klebanoff and Diehl are includccl． lt should be notec！ that their
results for the O．．9．5 in． rocl coincides with that of the 19 mm rod used in the
present experiment． The shape factor of the velocity profile is about ！．3 at each
sectlon．
慧
／．o
0．8
0．6
O．4
O．2
Y （mm） O 0．2 0．4 0，6 0，8 ZO 蜘
Fig． 4．16． Variation of the turbulent
Reynolds stress．
221．3 cm are plotted on Figs．
一一噂 ，o一
燭”，’ ．黛 「
盛
γ，
δ
ジ7 1 i
一一一一Vθ加〃。〃Lρ珀力ご，〆～θニノ4650
@ ルθθご！ヨ〃51ごZo〃σ吻形5∂〃吻∂ρ8
耀8勿η∂が一ρ舟力ど，尺θ躊〃ヲ／0
@ 02ク厩げ〃醒加す認ノηゴ
1@ L⊥1 LL O
Fig． 4．17．
？…Z一一一””’b”T一’Jar，？ 卿享
Nondimensional niean velocity distribution，
44
ミ
ミ
や
思
ヲ0
261
221
さ
Masaru KlyA
ノ0 ？’i”0．．…． ．…．．』．』．．． ／000 万000
yaT／u
Fig． 4． 18． Logarithmic velocity distirbution law．
一 一
曹鼈黶`ど 解θ加ηoガー0ノθ尻
、、@＼、「@ 、
シτ
訷黶`∠
冠・2伽誓・・25・β蜘蹴一つ・2嚇覇・罐耀
●O@ o 、 、＼
＼
～／τ
● 、 、
@ ＼ 、 ＼
■〔oη〕
B20ヲ．7
B2ヲ2．4
E24ヲ．4
・、
@＼ 、
コ、
8。争
8
＼ 、
@ ＼
00／ 〃 ／ － 4
44
蜘
Fig． 4．19． Velocity－defect law．
The non－dimensional mean velocities are plotted in Figs． 4．！8 and 4．19 in
the form of zc／u， against y’t｛．／p and （Uco－tt）／u， against y／6， respectively． ln the
regioR where the law of the wall is valid the mean velocities are well represented
by Clauser’s law of the wall Eq・ o．／o，rrt一一．一一一．一一一一一丁……一一一r一一…r LO
（4．5a）． However， Fig． 4．19 shows
that the artificially thickened bound一 o，os wa一．一L・一一1’一一一一li…一一一一一一， ”（Cm’ 一一一ll一一……一一1’・…’o．s
ary layer has the velocity－clefect law
Eq． （4．6b） for channel fiow rather o，06”e243 一’ la．6
器。警，温州懸。藷1式。、 1、メGrunow and Klebanoff and Dlehl．
Fig． 4．20 shows the longitudinal O・02
turbulence level V77’S2／Uco agalnst the oL一一一L一一一tl…一一一i・一一一2一””””3””’’’””’h”’’’””’’’’’”th’’’’’’’’’”i”’72一 O
non－dimensional distance y／6＊ at y／ifi
various sections， together with the Fig． 4． zo． similarity of the iongituclinal
nondimensional mean velocity u／U．．． component o．f the turbulent velocity，
「．「・
1泣口i｛［…一6『
U
L偽o 唱p 1ミ 1 Ii…≡メ㈲ …引1 に「
@ ヤ ，1 o@ I一．一．．一Ω一〇一
@ ：
… 7
1．黶u陣；：…
ohI … ヨ■iI …i…。24ヲ
B444↑T …lii ｝ 1 1
L二 1@ ………
1ｽ猶．ii ．
1旨じ…1…「一
c …
1 一「．．
Ui i 9 I
奄戟c： hL．．．．．…．．．．．．、
t r
@ ．1．．．…噂’“
k．」…群口
一 …＝．弩 L⊥⊥⊥…．L…！．． ．，．」．．．．…．．．．…穿．一一曼
@ …
D． 旧紅．一当 ． 「 ． r ．．． ．．
45 Study on the Turbulent Shear Flow Past a Circular Cylinde，r 45
It will be seen from this figure that a simiiarity in the turbulent fluctuating
velocities exists， which may suggest that the artificially thickened bounc！ary layer
has fully developed aRd thus attained dynamical equilibrium at the sections of
measurements．
4．4 Experiments on the Flow near a Circular Cylinder in the Turbulent
Boundary Layer
a） Test Cylinclers
The test cylinclers were made of carefully machined brass cylinders and 1 ad
the cliameters of 2a＝rm 36．2 mm， 25．0 mm， 18．5 mm and 9．45 mm， respectively． Each
of the cylinders was mounted horizontally with bearings to the side walls of the
tunnel so that the span coincided with the wiclth of the tunne｝． The cylinders
could be rotated abot their axis in order to obtain any angular positioR of the
pressure hole of O．5 mm diameter． The angular position was read on the protractor
attachecl to the bearing． The distance between the cylinders and the tunnel floor
could be changed arbitrarily by adjusting the position of the bearings．
The angular position of the pressure hole was measured from the stagnation
point when the cylinder was placed in a uniform fiow． The stagnation point was
previously determined by measuring the pressure distribution arouncl the test cylin－
der placed at the outside of the boundary layer．
In order to simulate the flow about a cylinder of an infinite span in a wind
tunnel it is necessary to eliminate the end effects． For this purpose， the span of
the test cylinders was chosen to coincide with the whole width of the tunnel， and
the junction of the test cylinder with the side walls of the tunnel was carefully
sealed with oil clay． Another method was to mount two circular baMe plates of
various diameters and l mm thickness on each side of the pressure hole． The
interval between the ba田e plates was 20 cm and the pressure hole was located at
the micl－point of the baeee pates．
b） Static P7’essttre Dist7’ibzetion ai’oz｛ncl the Test Cylinders
Fig． 4．21 is the definition sketch of the test cylinder placed in the turbulent
boundary layer． 2a is the diameter of the test cylincler， A is the clearance be－
tween the test cylinder and the plane boundary wall， and e is the angle measured
from the stagnation point for uniform fiow to the direction of positive 2」． The
surface of the test cylinder that corresponds to OO S e S！800 will be clenoted by
the suffix “zt．” （upper）， and that for
一 ！800 Se＄OO by the suthx “1” （lower）．
Ue is the approaching velocity at the
center of the test cylinder．
Each test cylinder was mounted
at the section x＝：＝219．8 cm where the
artificially thickened boundary layer
was fully clevelopecl and had the fOl一 Fig． 4．21． Def］nition sl｛etch of the test cy］jncler
lowing properties： placed in the turbulent boundary layer．
a）
f．o
o．，
O．2
o
鷲 9） 一〇．2
一一 o，
n
…O．
6i
一，．2！
一・・y4
0
20
40
tiO
80
fOO
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房 2〃
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づ37
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u「…
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it
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一］
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f．．．．．．一．．．一．．．．．．．．．．．．．．．．
i ：
： ：
…
i
c
［
［
Li 旨i
9
v
@ i
… ．．．．．．．．．．ゴ㌻…．．rr．
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．．．．一
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11
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A ゆ QO 汀 弐 鴇 9 回 ひ 7 ① → 葺 ナ 巳 ① 箕 辺 誌 舞 霊 o ≦ 餌 箕 曳 ○ 環’ a 壁 憶 ○ 曳
く 騨 乱 費 膳 ¢
50 Masaru KIYA se
U．． ＝＝ 15．46 m／sec ，
R， ua一 6！250，
The static pressure
describecl in （a）， and the
respective｝y． The pressure distributions are given in the form of
coefflcient ’＝f71’2t’i’li’pl”il’7”’tT’ against the angle e， the parameter being the
It is seen from these figures that there occurs a systematic clecrease
nation pressure as the clearance A decreases． At the same time，
clistributions around the upper and lower surfaces of the cylinder
deviate systematically from the symmetry．
of both the velocity gradient of the approaching flow and the
wall are clearly seen in the pressure distributioB around the front
cylinder． Name｝y， the velocity gradient tends to increase the
upper surface of the cylinder comparecl with the pressure on the
which is seen in Fig． 4．23（a） and （b），
boundary wall is to increase the pressure on the lower surface．
point shifts to the lower surface of the cylinder as A／〈2a） decreases，
the maximum value of about 一100．
The effect of the baffle plates was tested by
test cylinders with and wi出out bafHe plates． The measurements
the central part of the tunnel where the velocity was uni
shown in Fig， 4．26， together with the results obtained by Fage
It is seen from this figure that the static pressure at the rear of
with the baese plates decreases still further than that of
baMe plates． lt is also seen that the pressure distribution around
without baffie plates coincicles quite well with the data obtained
ノ．Ol
；O．8i
．［
o・6
，．i4i
o．2
01
”e2 1
一 O．4
－O，6
6＝＝ 60 mm． o“＊＝ 7．Lt2 mm． e＝＝ 5．oF 5 mm ） V t 一“一L一 一一i一一4 ）
．1／e，． 一： 7382 ， R， ＝ 5675
distributions were measured for the four test cylinders
results are shown in Figs． 4．22， 4．．？．3， 4．24 and 4．．？一．5，
the pressure
ratio d／（2a）．
in the stag－
the pressure
are found to
It should also be noted that the effects
plane boundary
part of the
pressure on the
lower suface ’
and quite conversely the effect of the plane
The stagnation
and attains
measuring the pressure on the
were made in
form and the results are
and Falker ［23］．
the test cylinder
the cylinder without the
the cylinders
by Fage and
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945．
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ク0 ／00
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6．，｛一4－L？10fttgetx／o41
9．ダ6「7．9ヲx／〃i
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f60 ／6〃
θ吻fiees）
{．・
40 60 eO IOO f20 ／40 f60 ／60 ／SO ISO tt）
F．ffect of the baMe plates on the pressure clistribution
around the circular cylinder，
51 S．t／udy・ on the Turbulent ．Shear Flow Past／ a Circular Cylincler 51
Falkner． ’1］herefore，
overcompensate therear of the cylinder．
Fig． 4． 27 shows
the baffle plates
pressure at the
stagnation presSure （PstagwwPo）／（
against the distance ？y
wall． The clistribution of the dynamic
pressure （一1一一pzc2）／（一S…’一pu2．．） in the un－
disturbecl turbulent boundary layer is
aiso includecl in this figure． lt is
clearly seen that the stagnation pres－
sure takes a lower value for larger
cylinders than for smaller ones when
the distance of the center of the cyl－
inders from the plane wall is the same．
This fact reveals that the undisturbecl
location of the stagnation streamiine is
in size of the diameter of the cylinder．
by the theoretical analysis described in
I－lowever， since the velocity
lac yer， it is slightly diflficult to defi
consicleration． A possible
at the center of the cylinder．
variation of the
↓鯛
from the planeRs眺
ノ00
50
60
40
20
22@ 〃〃
黶Z一 ヲ62
|o一@ノ8ター《トー 只5諄
岬盗‘〃かθ〃
J0勲！γ∠妙σ
gradient
ne the quantity tu
choice may be the un
With the values
nation pressure coefllicient and the location of the stagnation streamline far upstream
of the cylincler are computed from
0 ご22 0．4 0．6 ご7．6 ∠0
吻一ん夢確；夕房鷹
Fig． 4． Z7． Variation of the stagnation pres－
sure of the circu｝ar cylinder placed in the
turbulent boun（lary layer．
nearer to the plane wall with the increase
This phenomenon may also be suggested
chapter 2．
is not constant through the boundary
introduced into the theoretical
disturbed velocity gradient azt／02y
of tu thus cletermined， the stag一
鰍 （え一2）（1－tanh・・）
／／；／7’ ＝ec ’9’9”S’1’i”一g一一［2， ww 1 ＋ （（？， ww 1）2 ＋ 7， （2 一2） tanh ．］it2 ］
T・・va1・…fψ，…一P・）／佳・ひ1）・nd徳・b・・…db・E（［i・・
of the same order of magnitude as
agreement was notto is not aclequate．
backward from Eq．
coefficient， in order
are show．n in Figs．
the ’ 邑stagnatlon polnt ls
c） Lift ancl D7’ag Fo7’ces of the Ci7’czela．7’ Cylii7zclei－
The static pressures on the surface of the test cylinders are
（4・． ！2）
（4． 13）
（4． ！2） and （4． 13） were
those determined experimentally， but a precise
obtained． The reason for this is probably that the choice of
Next an attempt was made to use the values of to countecl
（4． 12） with the experimentally determined stagnation pressure
to compute the theoretical pressure distribution． The results
4．28（a）， （b） ancl （c）． lt should be noted that the position of
is well predicted by the theory．
integrated to
52 Masaru KlyA 52
ito
0．SI
adi’讐
賦
ミ o．4． 1
0．21
o
一〇．2
－40
1二1 1
P
｝ ’｛ ／｛ ’ P．……．…
P 紐づ62伽 …
@ 、・伽・緬i卜 享
･
口／／i凌 樽λ覗2g i i
QLコ、サ
一20 0 20 e （degrees）
（a）
40
ノ．o
0．8
O．6
窟
雫 04一｛i一
0．2
o
m’@OL24tll
F
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一20 0 20θ（卿／でθ5）
（b）
40
覧
妄
／．0
as
o．e
／
rm’垂
’
o．1；．
el一一・7L／
・・
g／一．
， L
oけ
t
7｝
i
2a罵ヲ6．2m！η
〃2の一〇〇ク
A臨aヲ。
一a2
バ、
焔a＼、 ＼
il@ ＼ @
xt
x
一！lO
Ll」40
Fig． 4．28． Theoretical and experimental
pressure distribution arounct a circular
cylinder p］aced in the turbulent bound－
ary layer： （a） A（／2a） ＝＝ O．45， （b） 0．24， （c）
O，09． 一一一一一一一…theory， 一一一一一一一一experiment．
obtain the
coeflicients，
一．20 0 20 θ吻rθ83）
（c）
lift and drag forces acting upon them in the form of lift and drag
Cf．． and CDp：
Ct．，ナ霊Lp
1
2 ρ碍2α
＝＝
@一il” S：’ （C？）i 一 C，，，） sin e d e
C””＝＝一
撃奄狽撃戟fEptii）’IIEII’．2a＝”MltrLSII（Cpi－FCp2‘＞cosede
Suflix i） given to C」； and Ct） means that they are compu£ecl from pressure oniy．
The results are shown in Fig． 4．29， together with the values of Reynolds number．
53 St／udy on the Turbulent Shear Flow Past a Circular Cylinder 53
ノ，4
L2
／，0
O．6
O．6
0，4
0．2
0
∠／（2の
Fig． 4．29． Lift and clrag coefficients of the circular cylinder
placed in the turbulent boundary layer．
It is seen from this figure that， with d／（2a） decreasing， the lift coeflicient increases
moR£onica11y from O to the maximum value of about O．8 at A／（2a）＝＝O， and con－
versely the clrag coeflicient decreases from 1．．P． to the minimum va｝ue of about
O．95 after passlng the maximum value of about 1．3 in the vicinity of Af（2a）＝＝O．6．
The positive value of Ci：． imp｝ies that a circular cylinder in the turbulent bounclary
｝ayer ac cquires a transverse force that tends to move it from the boundary wall，
as was previously predicted qualitatively by the theoretical analysis of the uniform
shear flow． Cf．． vanishes at A／（2a）＝＝：1．0 approximately．
The value of Cth， is about 1．2 for a large A／（2a）， which is equal to the drag
coefllcient of a circular cylinder in an unbouncled uniform flow． The circu｝ar
cylinders with baMe pla£es yie｝d somewhat higher values of CD，， i． e． 1．3 for a large
A／2（a）， which corresponds to a larger decreases in static pressure at the rear．
1－lowever， it is possible to estimate the effect of the baMe plates on the drag
coeflicient by making use of the experimentally cletermined pressure distribution
shown in Fig． 4．26． Since the drag coeflicient is almost but not quite cletemined
by the static pressure at the rear surface of the cylinder， the ratio of Cp． to
（Ct）．）b， the drag coedicient of the cylinder with baMe plates， may be written as
Cpp Cpc，・
（C加）b一（C、、の“
where （C．．．）b ancl C，．． imply the pressure coethcients at the rear of the cylinder
with or w圭thout the ba田e plates． This ratio is found fエom Fig，4．26 to be
approximately O．95． The correctecl values of the drag coefficient are also shown
in Fig． 4．29．
It should be noted in Fig． 4．29 that the data of Cf．． and CD． fall on the
same curves， respectively， with acceptable accuracy． This fact may be of great
practica｝ importance when one studies the motion of solid particles located in the
turbulent boundary layer． SiRce the ratio CJ；．ICD． is an important parameter for
o
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p o
＋！事4
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02 04 06 06
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xﾚ0 ∠2 、ζ4 ノ：6 ∠8 20 22 24 26 28 ヌ0x
54 Masaru KIYA 54
＆
漣
f’0
0．6
0．6
二．
iiiiilill”F
LL．．．L．
ll
’
5…｝．…
@ 1
「
。、㌧雰⊥
rTme1． e’b12s
T0．4F一／ler：
．．L L
o）
Flii
IE．Ei
e
u，
e
L0
Fig． 4． 30・
0．2 0．4 a6 a6 ！．0 f2 zi／r2a）
Varjation of the ratio Czp／CDp with respect to ‘I／（2a）．
practical purposes， its distribution with respect to d／（2a） is plotted in Fig． 4．30．
These data a｝so fall on a single curve．
d） Estimation of Laminar Boundaror Layer on the Circz｛lar Cylinde7－
In the last section， the lift and drag coethcients were computed only from the
static pressure on the circular cylinders， and consequently the effect of viscous skin
friction was not inc1uded． ln this section， the laminar boundary layer developing
along the surface of the circular cylinder was considered to some detail for the
purpose of estimating the contribution of viscous shear stress to the iift and drag
coeflicients． ln order to calculate the laminar boundary layer， use is made of the
Blasius series extended to an asymmetrical case and the approximate method due
to Pohlhausen and Walz， respectively．
（1） Estimation due to the Blasius series for an Asymmetrical Case
The laminar boundary layer on a flat plate piaced parallel to a uniform shear
flow has been considered by several authors． Li ［31］， Ting ［32］， Murray 133］ and
Devan ［34］ treated this problem theoretically on the basis of various procedures
and revealed that the effect of shear on the laminar boundary layer is not of great
importance near the leading edge， but it becomes dominant as the downstream
region is approached． However， an exact analysis of lamiBar boundary layer on
a cylindrical body， such as a circular cylinder， placed in a shear fiow has not been
made as yet， and this problem remains to be solved．
In the present consideration， it is assumed that Prandtl’s bouBdary layer
equations for a two－dimensional steady flow
u dし1 ∂2u o Ou
・∂x＋％rσ…ゐ「巾＋・毎削 （4・！4）
Ov Ou
－5／th一＋一一5－il一 一〇 （4． is）
55 Stucly on the Turbulent Shear Flow Past a Circuiar Cylinder 55
can be applicable， with the boundary conditions
y＝＝O ：u＝O， v＝O； y＝ oo ：u＝U（x） （4．！6）
It should be noted that x denotes here the distance from the stagnation point
measured along the surface of the cylinder and y is the distance measured at
right angles to the surface． u and v are the combponentS’of velocity in the x
and y directions， respectively． lf one defines the non－dimensional variables
jti 一＝ x／（2a）， Q一 （y／（2a））V－R，， isfi 一 u／ U．， w一’ 一 （v／ U．．）VRU，
V＝ UI Ueo 7 P ”： ip／V 2av Uab ， Re ＝＝ 2a Uee／v
where a ls the radius of the circular cylinder and U．． is the velocity outside of
the turbu1ent boundary layer in which the cylinder is placed， Eqs． （4．14） and （4．15）
may be rewritten as
t－s OiZ r． clU 02 i－i o
π栃＋％rU…痂＋砺・． 』 ． （4・17） ∂π ∂汚
一5－i一＋bjt－O ． ・・ ・ （4． is）
Now， it is possible to transform these two equations with two unknowns into one
equation with one unknown by introducing the stream function ip． Putting
，一，＝一一g！／i， t＝一一一一ll／i．・ ・ 一・ （，．，，）
one sees that the continuity equation iS satisfied automatically， and Eq． （4． 17）
becomes
霧轟一騨一ひ募‘・穿 （・・2・）
The boundary conditions become from Eq． （4．16）
y一・・診・・9多一・
，， ＝ ．． ， ugl／i． ＝ u・（．）
It should be emphasized here that， although the external shear does not appear
explicitly三n the mathematical formu｝ation， its effect is treated圭n the form of an
asymmetry of fiow on both sides of the stagnation point， in exactly the same
manner as the effect of the plane wall that exists in the’ vicinity of a circular
cylinder．
Let the velocity outside the boundary layer be given by the series
乙1（記）rπ1菊＋を72あ2＋π3記3十… （4．21）
The coefficients iii， it／2，… depend on the flow about the cyiinder and are to be
considered known． The normal distance from the surface is made climensionless
by assuming
56 Masaru KIYA 56
・癖奪 （・・22）
One adapts癒e following form for the stream function：
ψ一》i斎｛編（η）一←3ぞZ2乏・2プZ（η）＋蝋（・）
＋・纐・）＋嬢（・）＋・・謀（・）・…｝ （・・23）
In order to render the functional coe伍cients fi，プ5，… independent of the particular
properties of the outside flow， i．e． of飢， it12，…， it is necessary to split thern up as
follows：
磁＋一当／・・，五一・・＋薯舞ん・＋蓋ん・，
コへ ヨ 鴻一・・＋鑑乃・＋煮ん・＋一舞髪｝ゐ＋毒，…
f・一・・＋論・＋鷺：k・＋濫ブ・＋額・・＋鍛・Z・＋轟・・，
Hence， denoting di£ferentiatlon with respect toηby a prime：
iv＝2Z1死∫｛十3諏：2露2∫≦＋4π3あ31ブ≦＋5π4詑4プ『h’＋6π5藷5プー9＋77Z6死〔レ『6’＋… （4．24）
藷｛・樋・唖・1…㌍…以＋・…鵡42・・魏…｝
（4．25）
Inserting expressions（4．21），（4．24）and（4．25）三nto Eq．（4．！7＞， and compar三ng terms，
one obtains a systern of s量mukaneous ord三nary differential equations for the func－
tions fi，．f2， g3，ん3，…， the first seven of which are：
f12－fi！｛ノ＝1ザ1”
3f｛fl－2f｛ノf2－fi．fl t＝1十fl”
4fl 99－3f｛〆σ、一f、 gl’ ・＝ 1＋σ1”
轍1－3f｛・h，一f， ・S’ 一 一± ehl〃一号（fl・一f，flt）
5fl gE－4f｛’σ、一f、 9（’一1＋94”
！2 5fl・h（一4f｛’h・一f・・hE’ 一＝ ！＋h（ノL 5（購σ1一％σIL鍍’σ・）
！2 5fl・kl－4fl’1・・一f・ 1・1’一kl”一一’5一（5flh5－2f，h5’一3fl／h・）
The boundary conditions for the functions fi， f2，…follow from Eq．（4．16）and are；
η＝＝＝0：プ竃＝ノ『f＝0， ．ノ～＝プ「6＝・：0， 93＝＝σ≦＝0， h，＝＝1z≦＝0，一・
1 ！
η一。・プf－1・ fS一び・ 9≦一．4・ ／・≦o・…
57 Study on the Turbuient Shear Flow Past a Circular Cylinder 57
Table 4． 1． Universal functions fQr the first six terms
of the asymmetrical Blasius series．
fi’ （o）
fを’（0＞
gS’ （o）
hst （O）
g4’ （o｝
h6’ （O）
ka’（o）
ggt （O）
hg’ （o）
leg’ （o）
la232588
0．798742
0．724447
0．164046
0．673188
0．285916
－O．017072
0．6347e2
0．257808
e．119182
ノ6’（0）
gg’ （o）
g6’ （o＞
ノ～6’（0）
L・6， ｛O）
ノ6’（0＞
g6’ （O）
，，t6’ （O）
n6’ （O）
一〇．033230
－O．OOO840
0．604228
0．L37570
0．206131
－O．021769
－O．009554
0．OOO190
0．0072072
These equations are solved numerically by means of the Runge－Kutta－Gi11 method，
and the resu｝ts are tabu1ated in Table 4． ！ （Arie and lida ［351）．
If one defines the non－dimens1onai shearing stress ？e， displacement thickness
6＊ and momeRtum thickness O by
・。一諾ぜ瓦，δ・一器瓦，θ一髪平瓦
they can be evaluated as follows：
・・一C誇）か。一・・｛耐f’（・）＋・醐・）
＋ 4 iran，， te 3f5’ （O） ＋ 5zz，， thlf，” （O） ＋ 6 z一，， te 5fg’ （O） ＋ 7f，．， fo6f6’ （o） ＋ ・ ・ ・／
δ・一マ毒1∫1（ ？一t．1一一 i． i“ U）吻
θ一マ蓄∫1善（ 1一＞1－e’． U）吻
（2） Numerical Calculations．
The numerical calculations of the laminar boundary layer are performed for
the circular cylinder of P－5．O mm diameter for the case of d／（2a）＝6．836， 1．496，
O．692， O．284， O．160， O．080 and O．036． On the assumption of the Bernoulli theorem
望十聖
the measured values of 1］ressure coefficient C． are used to compute the velocity
a£ the edge of the boundary layer if：
V’＝一tUJr．一．’一 ＝” ［CpstagmCp．（T）］g （4． 26）
The distributions of U（te） aRd d lL7／dhi calculated from Eq． （4．26） are shown in
Fig． 4．3！（a） and （b）． The best fit curves of the sixth degree were obtained by
applying the method of least squares and the coefficients of polynomials are listed
58 Masaru ｝〈IYA 58
／．6
／，2
o．s
0．4
暮・
一〇．4
一as
一L2
一／．6
一一〇．8
2か。
σ θダブ
飢嵩250〃摺
l伽泌ω瑚轍♪｢／6民4／”．96，84
U4解ヲ74ノ．”ｲ29．8！7ラ069O／％ 〃028ﾆ ／65 40〃6mノ45 20008X 以4 訊9び04
ゴ
ごわα
一a4 O x／r2a）
（a）
Fig． 4． 31．
4
4
4
ぼ
シ’ミ4
ミ
4
4
0
一4
a4 0．8 ’ 一一aO 一一〇．4 0 a4 x／c～の
（b）
Variation of （a） surface velocity and （b） its derivative
with respect to x on the circular cylinder．
afi
o
o
o
o
o
o
o
Table 4． 2． Coefficients of the best－fit curves
of the sixth degree．
df（2a） 6．836 O．692 O．080
IZI
配2
廊3
廊4
1Z5
廊6
3．4588
0．oooo
一．？．．2805
0．oooo
－1．0450
0，0000
3．0558
0．4058
－1．918．？．
一！．4471
－e．6756
1．3076
1，7120
1．6359
0．4013
－5．5902
－1．5503
4．7747
in Table 4．2． lt was found that the measured data were wel｝ yepresented by
these polynomials， with the exception of the cases of Af（2a） ＝＝： O．080 and O．036 in
which there exlsts a large asymmetry of tr＝ on both sides of the stagnation point．
The computatlon of io， 6’＊ and O was performed for three cases， i．e． Af（2a）
＝6．836， O．692 and O．080， and the results are shown in Figs． 4．3．？． （a）， （b） and （c）．
The asymmetry of fiow is clearly seen in Figs． 4．32（b） and （c）， while the boundary
layer is practically symmetrical for the case of a／（2a） ＝＝6．836． The calculated
velocitY pro丘lesπ／ひare also shown in Figs．4．33（a）and（b）． Since， owing to
the increase（1 complexity of the differential equations for f． ag． n increases， the
59 Study on the Turbuient Shear Flow Past a Circular Cylincler 59
2．4
2．0
／，6
ノ．2
O．8
0．4
0
一〇，4
一〇，8
一L2
一／．6
一一 Q．0
一24 －as 一〇，6 一a4
2，0
ノ．6
L2
0，8
0．4
o
窃
一NI黛 ℃
嚥ア嘱
ﾂ米A桑
1；
@∬∫
@∬多＼＼ ＼
一22
@θ
翌
一偏2α
1
＼
．∬
ﾚ／
＼、 ∬
～
ゐ
、改
2ご2；瓢0〃〃
ｲ＝顔4伽S讐ノク0．9〃〃｝
S伽♪瓢84N眠 ＼ 4
㎜04
ー一 Z．8
一L2
一／，b
－as 一a6 一a4
一a2 o a2 a4 a6 as x／ua）
（a）
∫＼
・・毒・鷹
@ ・・叢冊_∬・濃冊＼ ＼ ／’一 、曽β／
ノ ｝＼／
イ ・． ！！ 旨
＼ 二二／＼一 一一
、
i ：
5
！7
2α＝25：0〃〃
塔ﾐ四／4．5〃〃～
ﾚ1＝2，0〃〃
ﾚ放忽）一〇紹 ！
一a2 0 0．2 0．4 a6 08 x／（2a）
（c）
2，4
20
／．6
f，2
a6
O．4
o
一〇．4
一〇，8
一1，2
一L6
一2，0
－0．8 一〇，6 一a4 一〇．2 O a2 a4 0，6 a8
x／ua）
（b）
Fig． 4． 32． Variation of shearing stress，
displacement and momentum thick－
nesses on the circular cylinder ：
（a） d／（2a） ＝6．84， （b） O．69， （c） O．08．
Blasius series，
一一一一 Pohlhausen’s method．
evaluation of the functionai coefficients was restricted up to f6， the values of fo，
6＊， O and ii，II－ff obtained for the case of A／（2a） ＝O．080 may not be accurate espe－
clally in the vlcinity of the point of separat1on． This fact was shown by the
velocity profiles calculated from Eq． （4．24）， because the velocity in the boundary
layer near the point of separation became much larger than that at the edge of
the boundary layer， which was not thought acceptable for the case of the steady
two－dimensional boundary ｝ayers． Consequently， the values of 70， 6＊ and e for
most cases of A／（2a．〉 must be computed by other means， for example， the approxi－
mation by Pohlhausen ［36］ and Walz ［37］．
60 Masaru KIYA60
る
慧
江
4．0
3．6
S，2
2．8
24
2． 0，
／．6
／，2
0・ es
O，4
0
0
0
0
0．2 0．4 a6
b！／ガ
（a）
Fig． 4． 33・
ヲ．6
S．2
1「言2，8
雲24慧、。I
tt・i f．61・
1．i
o’1
z
s
O
o
o
122び飯伽〃
ﾕ～2の一〇69
㎜「
1謡 ノ ’
@’@ノ
一町1／〃
．樗，／ ノ γ
瞬 ／
・納／／ 〃／ ／
鯉 ．／
／／ S5
／@ 一〇5’@ 4C 05ノ ∠
／
黷U～4 ∠ 04∠
ﾓヲ@！ノ多
4
ンのノ 〃
覗21@ 矛02 ∠
1仙募 4∠
戟@ o
惚）
nt． zo 0 0．2 0．4 0．6 0．8 XO u／び
（b）
Tangential velocity distribution in the laminar
bounclary layer along the surface o’f the circular
cylinder： （a） A／（2a）＝6．84， （b） O．69．
（3） Estimation due to the Method of Pohlhausen and Walz
This procedure is based on the momen£um integral equation
u’2 一il，1．leL一．一 一t一 （2e＋6＊） u’ ii）Y一，・：：……一一一＝i， （4． 27）
obtained by integratlng the equation of motion with respect to y from the surface
at y＝O to a certain distance outside the boundary layer for all values of x． lf
one assumes a polynomial of the fourth degree for the velocity function in terms
of the dimensioRless distance from the wall rp＝＝y／o“（x） ancl adapts the necessary
boundary conditions， one obtains the relations
号一一騰r窪σ，書一孫r9器一論 （・・28）
そ蓼＊一・＋毯 （・・29）
where a shape factor A is
A＝b・2…tl一一1／／一1…L （4．30）
61 Stucly on the Turbulent Shear Flow Past a Circular Cylinder 61
The second shape parameter K is introcluced by the equation
dV
K一θ2瀟憎 （4・31）
which is connected with the momentum thickness in the same way as the first
shape factor， ／1， was connected with the boundary layer thickness， S， in Eq． （4．30）．
It isseen from Eqs． （4．28）， （4．．9．9）， （4．30） and （4．31） that the shape factors ！1 and
K satisfy the universal relation
K＝（rrgit［lg－7s一一ttltlg－s一一一wh1722）2／1 （4．32）
Denoting
号イ岳一徳σ）（37 A A2315 945 9072）一fi（K＞ （・・33）
一画・一（・＋笥（鑑画一．，6t’，）一f2（K） （・・34）
for the sake of brevity， and substituting K alldθ2， together with fi（K）andプ〉（！（）
from Eqs， （4．33） and （4．34）， respectively， one obtains from the momentum equation
Eq． （4． ．P．7） the relation
・一一 cle2
U・一：〉’iliVi一 一＝ F（K） （4． 3s）
where
F（。κ）＝2、f2（K）一4！（一2／（ニプ〔1（1（）
It was pointed out by Walz that Eq． （4．35） can be reducecl to a simple quaclrature
by the introduction of a further approximation without any appreciable loss of
accuracy． Walz found that the function F（」〈） can be approximated quite closely
by the straight iine
F（K） 一＝ O．470 一6K
In this manner Eq． （4．35） is reduced to
浩伽・〉一…7・一弼・吉盤
This differential equftc tion for UO2 can be integrated explicitly to
びθ・一響∫：ひ・cl・1 （・・36）
The functions K（A）， as well as fi．（K）， f2（K） are tabulated in Schlichting ［38］．
The variations of io， 5’＊ and O at the surface of the circular cylinder calculatecl
by this proceclure are shown in Figs． 4．34（a） ancl （b）． The comparisons between
the results of the approximate calculation and those obtained with the aicl of
a Blasius series containing six terins （up to i’G） are also shown in Figs． 4．32 （a）， （b）
62 Masaru KlyA62
2．4
L6
o．s
澹。量
o，s
一・ ^．6
一2，4
α
6
o
げ
θ
〆
9
9 飢』25：01珊
ズ 烈棚） 」〔〃卯） 」蘭の
0／δヲ．4／”，9 6．64
θ ゐ 4〃 ヲ74 ／．50
げ
cげ 2帰^96
〃ヲ
Vび69
ｿ26
θ ／6．グ 40 α／6
σf ノ45 20 006
6α
9 β．4 09 oo4
一 0・ fi
o．s
0．8
o．s
壊。δ
え§、
蚕・・
ミ
eo．δ
as
0，4
0”ts a4 o
－0．4 0 a4 0．8 礁の え：〃の
（a） （b）Fig． 4．34． Variation of （a） shearing stress and （b） displacement
and momentum thicknesses on the circular cylinder．
0，4
0．8
0．4
o
o
0
o
0
0
oa8
and （c）， and the agreement is not good for the
has been shown by s尺玉lar calculations（Schlic
mate method leads to very satisfactory ・s．o
results， one can say that the Blasius serles
for asymmetrical case does require many
more terms （probably up to 」；．ii in order
to obtain more accurate values of e， 6＊ and 2’O
e for the case of a large asymmetry．
Now it becomes possible to estimate
the contribution of sheariBg stress to the
lift and drag forces and the torque acting
on the circular cylinder． The torque wlll
exist since the distribution of shearing
stress is asymmetrical on bbth sides of the
stagnation point． The lift and drag coefli－
cients， C，n， an（1 CD，， an（1 the coefllcient of
moment C，y， due to the viscous shearing 一zo
stress can be computed as
Cか一 ﾇ、r献・・c・・θdθ
case of A／（2a） ＝＝： O．080． Since it
hting ［38］） that Pohlhausen’s approxi一
o
動
∠eだ5ごα～8
E〉軍」）ワ訪
ア薩。
∠」・? ど8ノど3曜e
吻ガぬ狛 〃㎡A～7＆α2 摩
0 04駈 a8 ∠2 ／6
0’f
一〇v
m／e？a）
Fig． zl・．35． Variation of vig． cous lift and
drag forces ac ncl torque actinss on the
circular cyllnder．
63 Study on the Turbulent Shear Flow Past a Circular Cylinder 63
CDs 一 wwp｛i／pt． ＝一 7k一 S：1’ ？， sin ＠de
G ，撫∫1ン・clθ
where eg’ and 0，rm imply the points of separation on the sides of hi＞O and jE｝〈O，
respectively． The variations of CL，V7？g， CD， VTRZ一 and Cy， VwwRin， with respect to
A／（2a） are shown in Fig． 4．35． The vaiues of C．．， and C．， are of a much smaller
order of magnitude than those of Cf．， and CD，， and C」t， is of a much smaller
order of magnitucle than Cx． and C．，． lt is interesting to note that the variations
of CLs and CDs with respect to A／（2a） have the same tendency as those of C．．，，
and CD．． The corrected vaiues of the lift and drag coeffcients are given in
Table 4．3． The value of C．（＝＝1．23） for A／（2a）＝6．836 is seen to agree quite weil
with that of the drag coefficient of a circular cylinder in an unbounded uniform
flow．
The position of the point of separation can be found from the condition that
the shearing stress must vanish there． The results of computation are listed in
Table 4．4． lt is seen from this tabie that the laminar boundary layer separates
a little earlier on the upper side of the circular cylinder than on the lower one．
Table 4． 3． Li’ft and drag ceefficients corrected for shearing stress．
A／（2a） CDp C加 C．Ds Cns Cl） Ci．
6．836
1．496
0．692
0．284
0．160
0．080
0．036
1．21
1．24
1．31
1．09
1．04
1．04
1．00
o．oo
o．oo
O，03
0．14
0．Ls
O．32
0．38
O．02
0．02
0．02
0．02
0．02
0，02
0．02
OiOO
o．oo
o．oo
o．oo
o，oo
O．Ol
O．O1
1．23
1．26
1．35
1．！！
1．06
1．06
1．02
o，oo
o．oo
O．03
0，14
0．Ls
O．33
0．39
Table 4．4． The position of separation of laminar boundary
layer from the surface of the clrcular cylinder．
ti／（2a） 硲Ot／一 estag
6，836
！．496
0．692
0．284
0．160
0．080
0．036
一82．so
－80．20
－87．oo
－94．oo
－91，70
－91．70
－94．oo
82．so
81・so
82．so
87．oQ
87．oe
88，20
90．so
O．oo
O．oe
一 O．50
一 7．so
ve 7．so
一 8，so
－10．oo
64 Masaru KIYA 64
5一 Turbulent Wake of a Cireular Cylinder located
in a Turbulent Boundary Layer
In the preceding chapters of this paper， the properties of fiow about a circular
cylinder placed in a turbulent boundary layer was described， together with the
effect of a plane bounday wall located in the vicinity of the circular cylinder． ln
this chapter the far wake of the cylinder submerged in the same turbulent boundary
layer as before was investigatecl．
5．i Wake－producing Cylinders
Two circular cylinders of ！．Lt mm and 3．2 mm cliarneters， respectively， were
used as the wake－producing objects， and they were fixed in turn to the wind
tunnel at the position x．＝219．8 cm and tJ，，＝＝ 30 mm， where x． ancl tJ，， denote the
coordinates of the center of each cylinder． The span of each cylinder was 50 cm，
so that it coincided with the width of the tunnel． As previously clemonstrated，
the turbulent boundary layer is fully developed at the section t＝＝2！9．8 cm and at
successlve downstream sectlons where the turbulent wake of the cylinders is to
be developed．
The static pressure along the tunnel was found to be the same as that shown
in Fig． 4．3．
5．2 Mean Veloeity Distribution
The results of measurement of the mean velocity in the wake are shown in，
Figs． 5．1 and 5． L． The distributions of mean velocity far downstream in the
cy｝inders are also plotted in Fig． 5．3 on nondimensional basis， i． e． ze／U．．一2y／o“＊．
It is seen from Fig． 5．3 that although the wake of the cylinder of 3．2 mm cliameter
produces a permanent change in the velocity profile of the disturbecl turbulent
boundary layer， that of the cylinder of 1．2 mm diameter does not pyoduce this
kind of distortion ancl the disturbed boundary layer recovers． This fact implies
欝
E
㎡ 0 σ 0 0 0 0 0 0 0 0 0 0 42 ρ4 86 瑠 ’〃
工一工。
20@．12．5 525 泌
148層 2忽 3〃 卸 423 477 55ノ 刀1 871． ノ〃4
御
2σ嵩〃励力
ｱ置2β～8伽
羅
，勿
％螺2’μ4 46 48 〃 〃 〃 耀 〃 五ρ 〃 〃，〃 ∫o 〃 〃 〃
u／u－
Fig． 5． 1． Distribution （）f mean velocity’ in the wake ot the
circular cylinder of i inni diaineter．
65 ．Study on the ’ll／’i．irbulent Shear Fl（）Nsr Past a Circular Cylinder 65
Fs
畝70 0 0 0 0 0 0 0 σ 0 0 0 0 02 α4 86 618 〃
工一工。
2σ
@ 4Z 〃7 3Z2 553 凋8 〃6 畑 勿7 238 283 32P 4κ 卿
2σ鵠3．2〃7加
H。＝2β～8伽
80
ﾐ40
20
％ 02 04 06 α8 ’．〃 λ0 λ〃 〃 砂 孟0 λρ 〃 〃 〃 〃 〃 〃
Fig． 5． Z．
U／Uoe
Distri｝）ution of meac n velocity in the wake of the
circular cylinder of 3．2 mm diameter．
t）
x
te
ae
偏
．僻
e．2
ah
工一工5× 2σ
E争切1B 工罵406汐仰
躍662，2σ四3．2伽｝
@ ，2σ皿1．2〃掃
@ σ〃φ3fσrδθゴ
@ 伽〃吻γノ解r
ー
Fig． 5・ 3・
2 4 6
y／6’
e le ノ2
Nondimensiona／ plot of mean ve｝ocity at a far
downstreac m section of the turbuient bounclary
iayer clisturbed by the cylinders of 1．2 mm ancl
3．2 mm diameter， respectively．
that the disturbance introcluced by the thinner cylinder is not large enough to
permit an accurate determination of its clrag force to be made from the momentum
principle， since this principle gives the drag force as a residual value． Consequently
the computation of drag force was made only for a cylinder of 3．2 mm cliameter，
and some of the required properties of the flow， i． e． the dlsplacement and momen一
tum thicknesses ancl the shearing stress are shown． in Figs． 5．4（a） and （b）． lt is
seen from Fig． 5．4（a） that the boundary layer is considerably thlckenecl by the
introduction of the cylinder， because a part of the energy of the boundary layer
is divertecl into an eddy motion in the wake． This phenomenon is also we11
represented by the decrease in the shearing stress at the wall， z，．， as will be seen
66 Mftt saru KIyA 66
12
1e
s
偽
頃 6’聾
嶋
4
2
1 0 o
o
・
δ富
O D ，
恚r
I ・
．● ●
@●
9θ
■
・・ﾒ舳め㎡伽繭り・雛。。㎞勿伽祓旧劇砂 32伽伽8αα㈱r耐
卿 卸 蜘x （cmj
（a）
卿 細 卿
×
猷四8
ρ餌4
卿β鰯
ﾐ032 ・繍〃謝副の’雛B伽吻伽r勧創毎?２棚伽血函腔厨4〃2
β〃鐸
4鰯 2り 3即 3砂 初 卿． 5即
Fig． 5． 4．
エ〔cm〕
〈b＞
Variation of （a） displacement ancl momentum
thicl〈nesses and （b） shearing stress in the
turbulent boundary ］ayer disturbed by the
circular cylinder of 3．2 mm diameter．
in Fig． 5．4（b）．
The mean ve｝ocity is plotted in the non－dimensional form of u／u． against
yu．／v in Fig． 5．5． The law of the wall holds even in the turbulent boundary
layer disturbed by the wake of the circular cylinder， and Clauser’s law of the
wall given by Eq． （4．5a） represents the measured data quite closely in this case
as well． For large values of yu．／v the logarithrriic ve｝ocity distribution deviates
from the actual velocity distribution； apparently it no longer holds iR the outer
region of the boundary layer where the distortion introduced by the wake of the
cylinder is considerable． For that region it may be more approprlate to consider
the velocity defect （U．一u）fu． rather than u／u，． The experimental results are
plotted in this form in Fig． 5．6， together with the velocity－defect law given by
Eqs． （4．6a） ancl （4．6b）， respectively． lt should be noted that the experimental data
show a good agreement with the velocity－defect law for the channel flow in the
immediate downstream of the circular cylinder， while an increasing deviation from
this law is’seen as the lowest downstream section is al）proached． Finally， they
a）
b）
26 26 26
k 2！
x こ 20
18
16
i2
io
slb
ミ
◎齪o
oo
oo
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【
佑冨
562
～鰯
，丁＋4ρ
oo
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ooo
工一
工‘
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116
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itMV
26 26 26 8 s 8 s
jE：oe
l
l il…
liil
i
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．．ﾄ．
一「
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蜜岬
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126
26「
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％缶一…鰯睾編・
llli
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9O
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％¢
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5σ9
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283
18
o
o
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207
16
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0O
κ
o
oo
8
12o
0
8
ノ0
o8
o
〃
脚o
搦
yu．／v
Fig．
S．
S．
Logarithmic
law
of
mean
velocity
profile，
a） b）
ktsk
M
、、
o、
、
o
、
o、、
@o 、
@ o
o添
♪
一一
k一一
・・
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62z婦
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κκ
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A
＼o
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E。ト
Oo
、、
、＼
、、
、、
＼@
＼＼
O o
、
、、
、、
、、
o
o
蝕 2σ＼＼
κo
〃o
●
、
、、
●●
、〃
6
、、
@ ＼
、●
o、
訪
、、
、、
＼
o
♂o
。 ㌍7
0
●
亀瀞
嚇〃
シ／δ
1o
llls
；：
＼、
、
＼
＼
、
、
o畦
o一
班思
一ｰ協下÷μ65
74
uκ厚
躍
o
Aぐ
o
ミ。
馬
、＼ 》
＼、
9、代
、、、
o＼《
@ 次
O
o
o
》、
o
o
、o、
、
ミ
盈
20、
、
、
9o
o。
｝
O o
oo
、・
♪、
o 、
o
詳Oo
、、
T09、
6》
o
O o
Oo
＼、
0
oo
o
σ
、痂
、
勧〃
3／
δ1
0
Fig．
5．
6． Velocity－defect law
of mean velocity．
鶏 ω ヨ 窪 団 〇 二 ’ ⑦ 日 ’
煽 ユ Ψ
① 侍 ω 7 ① 『 閃 δ 薫 ℃ 馨 餌 o 旨’ む 三 霞 o 図 ぼ 窪 o 『 9
68 M’asaru KIyA 68
eventually came to be representecl quite ciosely by the velocity－defect law for the
turbu｝ent bounclary layer with a zero pressure graclient． Namely， the bounclary
Iayer distorted by the wake of the cylinder may have the same clynamic structure
as a fully－developed turbulent boundary layer over a smooth flat plate． This fact
may be considerecl to give a clue to a new method of artificial thicl〈ening of the
turbulent boundary layer．
5．3 Dis綬ibution of「馳rbu韮eハce
The values of Vil，75’2 ， Vl’2fe”’i2 and 11Z’i’tr’ in the wake of the cylinder were measured
by means of the single－hot－wire technique， and the results are shown in Figs． 5．7，
a）
t2a
lee
RS tie
紹
2e
狽煤@aas aN aas ale ala
σ ρ 0 0 0 0 0 0 0 η 0 0 〃 2〃 4醒 2％ 8躍 θ〃
JC－3c¢
3紹 477 658 溜 鋳
2σ
ﾈ58125 砿5 アz5 145 148 2〃 3／3
2σ需Z21η解
vじご＝2〃8傭
4〃〃〃ク〃4〃41a〃グ〃〃〃β〃e．〃タ〃嗣u．．
b）
ロミミ
〕6θ
0 0 00 0 0 0 0 0 0 0 0 o o 4躍 碓 2の 召” 卿
20 コじ一ゐ
394 ∬5 沼8 召8 仰 〃7 解 脚 3η 卿 2σﾓ247 〃7 29ノ
〃
勿卸32加加
ﾅ雲2128αη
88
6θ
紹
2θ
F
oρ422卿4 4必ク〃 〃〃 〃〃 卿 卿 4〃 4〃 撰〃 8〃 躍〃 ク〃 凛〃卿 β〃 〃〃
Fig． 5． 7．
ffワu．．
Variation of tt．一component of t．urbuient fluctuation
in the wake of the cylindert of （a） 1．2 mm and （b）
3’ ，2 mm c“arneter， respective｝y，
69 Study on the Turbulent Shear Flow Past a Circular Cylinder 69
s
贈
〃2
0ノ〃
号0 658 4瓢〃ん加x 〃61 ユ2α万0 234．4 〃繍肋謝十243・4励ゲ傾r
o偲
@，駅 o ・
@ ㎡ 翼
誕W 0 篤
@ 其@ ■
躍撃 ●十
， X
催 ● 鴫
o
●卜x
’2
Fig． 5．8． Variation of u－component of
turbulent fiuctuation in the turbulent
bounclary layer before and after the
introduction of a 1．2 mm cylinder．
写δ‘
a）
0／2
〃o
潤ﾝ8 伊
号し噸・
仏一
o％
署ﾁ が
＼、 ＼
oo2
仏 ●●
E ●
＼＼ 、 ＼
●
％ 2ヒ7 40「 60 80 ／6ヒ7
y ［mm）
b）
〃。
V8 伊u鱒
ユ2一工ε
Qσ 一22’
o％
＼、 、
…
o醒
●伊＼＼ ＼
嶋002
N●
％” 40 60 80 〃
y Emm）
c）
α〃
伽
が仏 号血138
o％、
＼へ
o僻
●
@ o怐@ o ●●
@ 伊
＼@ 、 、
＼
〃2
‘し』
、 o_ ㌔、
、、
@ 、％ 2ρ 40 60 80 ！αP
y ［thm］
d）
o〃
鰯
が工一工ビ
Qσ誓ノ79
Nヱん
o％
＼
、、
ω∫
O〃2
、 ＼ 、 ●
伊仏、．
・ ○
E ●
● ． ＼、、
● ●●
％ 〃 40 60 80 脚
Fig． 5．9．
y〔’祠
variation of Vl’17il’IE’）’2’ and V’T：」’；’i’ ac cross the wal〈e；
（a） （x－x．）／（2a） ＝＝19．7， （b） 29，1， （c） 138， （d） 179
70 Masaru KlyA 70
5．8， 5．9 and 5． 10． Figs． 5．7（a） ancl （b） show the distribution． of the intensity of
the u－component of turbulence in the wake of the cylinder of either 1．2 mm or
3．2mm diarnenter， respectively， taking the nondimensional distance from the
cylinder （x－x．）／d as a parameter． One can see from these figures the manner in
which the turbulence in the wake of the cylinder spreads outwards and inter－
mingles with the surrounding turbulehce in the otherwise undisturbed tUrbulent
boundary layer． lt will also be seen that the effect of the cylinder seems to vanish
at the downstream section （x－x．）fd＝＝
a） 1 1 1 1 1 1 500N700， whiclr｝ may be confirmed in turn by the mean velocity profile given
in Figs． 5．1 and 5． 2． According to the
measurements of Townsend ［39］， the ef－
fect of the wake－producing object could
be detected even at sections further clown一
マ
ミ
xlisl”g
1
30
工『ヱじ
20
P97o
一ノ0
一20
|30
〒4ら20 4〃 60 80 ノ〃
y ［mm］
b）
r：
1融
i
一io
20
’o
並 2σ
@2舜’0
’0
o 即 la
シ〔mm〕
60 se Iae
c）
す
婁
摩 1
〃
工一工。
o
2σ
R240
5530
o
フ8．8
0
ノ38
％勿 ぞ0 60 80 ！卯
d）
r“
1
1事
20
10
o 盈 2σ
@1790
2830
379％ 20 40 60 80 脚
y〔伽〕 3励〕
Fig． S． 10． Variation of il’？’V’｝’ across the wake： （a） （x－x．）／（2a）＝19． ．7，
（b） L9．1， （c） 39．4， 55．3， 78，8， 138， （d） 179， 283， 379，
71 Stucly on the Turbulent Sheftc r Flo“T Past a Circuiar Cylinder 71
stream than （x－x．）／cl＝ 1000 in the case of the unbounded uniform fiow of low
turbulence． Consequently it may be concluded that the spreacl and mixing of the
wake are promoted by the action of the surrounding turbulence．
The distribution of V：’i’pa’｝’2’ at the two sections far downstream of the 1．2 mm
cylinder is shown in Fig． 5． 8， together with the values of． V u’2 in the otherwise
undisturbed turbulent boundary layer． The turbulence is seen to recover and
return to its original clistribution．
In F三gs．5．9and 5．10， the variations of V∋万and zん’across the wake are
represented at several sections downstream of the 3．2 mm cylinder， and these
quantities clearly interpret the marked interaction between the wake and the sur－
rounding turbulent boundary layer． lt should be emphasized that although thevalues of Vilt27E’i2 increase considerably at the immediate downstream of the cylinder
as would intuitively be expected，出ose at further downstream sections（say（x－
x．）／d！55．3） are much less than the turbulence of the undisturbed tuybulent
boundary layer in the vicinity of the boundary wall． Some complex features of
this inteyaction of the wake with the surrouncling turbulent flow are also illustrated
by the variation of V：St／1’ across the wake， but the detaiied mechanism of the
interaction still remains to be solved．
5．4 Estimation of the drag force of the cylinder by rnomenturn prineiple
The drag force acting on the cylinder can be estimated by considering the
change in momentum fiux between two vertical sections ｝ocated upstream and
downstream from the cylinder．
The Navier－Stokes equatioB in tensor notation is
O2 zt ・i ∂Zt．、 ！砂 aztt
∂計z‘・∂x、． 万∂c、＋り∂Xk∂x／9
When the relations z｛i nt Ui＋z｛E・ and f＞＝P＋p’ are substituted， this equation becomes
畜（α＋z‘の＋（こみ＋z4）、象・（¢岡一一÷駕碧＋・∂21両差）
where U， and P represent the mean velocity and the mean piezometric pressure，
respectively； zt．i・ and p’ are the fluctuating components of velocity and pressuye，
respectively． Since uS・＝O， OUk／axh＝O， ezt2．／exk 一一〇 and p’＝：O， when one applies
the averaging procedure， one obtains
一霧誓σ畷一＋・磯一一去一器＋・器7 （…）
This is called the Reynolds equation．
A volume integration of Eq． （5． 1） gives
盤面＋∫σ瀞・．＋∫瑠4・一一÷∫誤伽騰沌（…）
where cl（n，v｝ denotes the volume element． Genera｝ly， Green’s theorem states that
72 Masaru KlyA 72
∫一霧・cl・一∫・欄
in which ni are the direction cosiBes of a unit normal vector at the surface
element dS and the positive direction of the normal is understood to be outwarcl
from the surface； G is a function of x，，． Accordingly， Eq． （5．2） takes the form
野面＋∫・z・・u・・u， dS＋∫…繭一一∫・・沼呵・…1急45
Since aUi／6t＝O for a steady fiow， this equation may be reduced to
∫・縄ゐ＋∫麟藤一去∫・澗＋・B継 （…）
This is the general equation of momentum
for steady flow．
In order to obtain the drag force
acting on the cylinder， Eq． （5．3） will be
applied to the control volume ABCD shown
in Fig． 5． 11． Of course， BC denotes the
boundary wall， and AD is taken paral｝e1 Fig． S． l l． Control volume．
to BC． The eoordinates of A and D are
assumed to be （xi， H） and （x2， H）， and the va1ues of velocity and pressure at the
two sections AB and DC are repyesented by the sufllxes 1 and 2， respectively．
Thus if one puts
Ml一∫痢邸・面一∫・・恥
ト青∫縄勘∫…霧45
these take the following forms at the surfaces of the control volume：
（i） surface AB；
Mi＝＝：一Sgu？ci，， Mi一一Ski’1，2’d，
臨l／l肋・砥一一牒4〃
（ii＞ surfce CD ；
・・1∫1σ1吻・ 鵬一∫鯵吻
鵡→∫：tp・ ・1・，晦∫1讐4〃
（iii） surface BC ；
M・…一∫：：σ肱一・・1・L41 ＝一∫：ン～ヂ均一・
ハV
∠
シ’
ｿ コじ’
00
θ
o’
汐 工 崎 妨
73 Study on the Turbulent Shear Flow Past a Circular Cylinder 73
嗣・臨∫讐吻一…努一
where
肌一∫：：繍
denotes the integral of shear stress at the wall．
（iv） surface AD ；
ム孤一∫：：（UV）ADcl℃一42一∫：：・・’漁一・
姻・ 砥一・∫：：寄姻
since one takes the surface AD in the fiow region outside of the boundary layer
in which U is independent of y．
（v） circ｝e L；
Atf，＝O， M，＝＝O， M，＝＝一一tTSiz，．PclS＝＝一一Pp一”
混一・∫（ ∂ひ ∂σ7Zl ww6rmt 一i一 7Z2 ”Oltl｝）dS…一3
wher D． and D， are the pressure and viscous drags of the cylinder， respectively，
and are defined by
D． ＝ 一 Si”P cos ada
単一・∫lc唯一レぬ
The tota｝ drag D actiBg on． the cylinder is given by
D＝ Dp＋Ds
Accordingly， the momentum equation Eq． （5． 3） becomes
Σルti＋Σノ脆瓢・Σノs4，＋Σノ㌧41
0r， after rearranging the terms，
轟＋蕊一∫㌍C結…艦）cl（紛＋∫喉一携）d（髪）
一∫：：1：：（UVUk）AD ・1（署）＋∫：’ftt（毒毒）禰
一一麦∫㌍｛∂懸L∂器）レ（÷） （・・3・）
in Which U．． is the mean velocity at the edge of the boundary layer at the sec－
tion x＝＝ 219．8 cm and R，， denotes the Reynolds number based on U．． and the
radius of the cylinder ancl is equal to 1．5×103． The physica｝ meanings of the
74 Masaru KIYA 74
terms in Eq． （5．3a） are self－evident from their forms．
On the other hancl， when the test cylinder is absent， the shear force Wtt at
the wall may be estimated from
，翫イ還一畿）㌦（＃）＋∫烈岳一一一駿）＊イ暑）
＋∫1職一，砦」禰一∫：：：：：C瓢4㈲
一W｛懲ト∂瓢）｝＊d（ya） （…）
Subtracting Eq． （5．4） from Eq． （5．3a）， one obtains the relation
読。＋㌃塾し∫許。瑳一綴）4傷）一∫fて甚一謙）＊d（9）
＋∫罵乙一，含浸）イ÷）一Sf／’”（P， P2ρσ乱 ρu乙）＊cl（老）
一Slilll（一：；ltl；一）．． ci （｛｝）＋ S：2，lll（一U．一一il；一）1． ci（一ill一） （s． s）
In Eq． （5．4）， the terms including the reciprocal of the Reynolds number and the
turbulent fiuctuating velocities have been neglected， since they are of an order of
magnitude less than ！0”3． As was already meBtioned in 5．2， the static pressure
in the turbulent boundary layer is practically the same before and after the intro－
duction of the cylinder， within a ｝imit of experimental errors． So the third and
fourth terms on the right－hand side of Eq． （5．5） cancel out each other． lf one
takes the sections AB and CD immediate upstream and downstream of the cylin－
der， respectively， the fifth and sixth terrns in Eq． （5．5） may also cancel out each
other and Eq． （5．5） may be reduced to
－nttlligi・i＋一㌃欝一∫∴催）㌦畿｝d（2Ya） （…）
where the velocity Ui immediately upstream of the cylinder is assumed to be the
same． The numerical calulations were performed with xi ＝＝ 219．5 cm， x＝： L2－P・1．3 cm
and H＝＝245 mm， by means of the measured data given in the foregoing sections．
Since the experimental data are a little scattered in the vicinity of the selected
sections， the probable four values of the right hand side of Eq． （5．6） were estimated
as O．769， O．785， O．821 and O．857． The mean value becomes O．808 and the stand－
ard deviation amounts to O．034． Accoydingly， Eq． （5．6） yields
，髭。＋警諾＝一・．・・8土・．・34 （・．・）
The value of （W，一 W．l！‘）IP U2co ct can be computed in terms of the shear velocity；
㌃欝一∫：：：1：絵（罐．．σ乙．．）＊レ㈲一・・…
75 Stucly on the Turbulent Shear Flow Past a Circular Cylinder 75
where the values of it， presented in Fig． 5．4 bave been used． Then， from Eq．
（5．7）， drag coeMcient of the thin cylinder p｝aced in the turbulent boundary layer
may be estimated as
C・一町一購識）2一・・95±…4
since DIP U2． a＝ O．808， U．＝15．46 m／sec and Uo＝＝！4．22 m／sec． The value of CD
・just obtained is practically equivalent to the drag coeflicient of a circular cylinder
in an unbounded uniform flow of the same Reynolds number as Uo afv（＝＝ 1．5 × 103）．
Consequently 1t may be saicl that a thin circular cyllRder placed in a turbu｝ent
boundary layer experiences almost the same drag as that in an uniform flow，
when the cylinder is not in the immediate neighborhood of the boundary wall．
5．5 Theoretical Consideration en the Two－Dimensional Wake in a Uniforrn
Shear Flow
In order to clarify theoretically some of the experimental results obtained in
the foregoing sections， the velocity distribution in the wake of a circular cylinder
located in a turbuient boundary layer is derived on the basis of constant eddy
viscosity． As wei｝ known， the mean veloclty profile of a turbulent boundary
layer has a form that is appoximated quite closely by the ！／7－th power law， but，
for simplicity， it is assumecl here that the approaching flow to the cylinder is an
uniform shear fiow with a constant vorticity． Since this fiow model has a non－
zero transverse velocity gradient as in the turbulent boundary layer， it may be
expected that this model inciudes one of the most essentiai properties of the wake
fiow concerned．
a） Mathematz’cal Formulat’ion of the Problem
The fundamental equations governing the turbulent fiow of an incompressible
fluid are the Reynolds equation and the equation of continuity． For the two－
dimensional case， these equations become
畷＋畷一一溺÷・V・磯｛急一響 （…）
畷＋嶋鋸一÷詫＋vV・U2一讐1艦 （・・9）
鴛一＋霧一・ （・…＞
Let the center of the cylinder be located at the origin of the coordinate system
xi x2， and the velocity profile of the approaching fiow be clescribed by
こ」，。。＝こノ』＋tO x2 （5．11）
The configuration of the problem and the definition of symbols are illustrated in
Fig． 5．12．
In Eqs． （5．．8）， （5．9） and （5． 10）， Ui， U2 imply the components of mean． velocity
ln the wal〈e in the xi and iz／’2 directions， respectively， and ul， ttS are the corre一
76 Masaru KlyA 76
xt
ん 島
妬
20
ﾘ。。属妬＋ω簸
鋳 3’
ん θ，
xt
’
Fig． 5． 12． DefinitiQn sketch of the prob］em．
sponding components of the fluctuating velocity． Let the new varlables qi and g2
be defined by
qi ＝（Uo＋tu x2）一 Ui （5． 12 a）
92 ut U2 （5． 12 b）It is obvious from Fig． 5．12 that gi means the amount of velocity defect in the
wake． Substituting Eqs． （5．12） into Eqs． （5．8）， （5．9） and （5．10） and making some
reductions， one obtains
一（uo＋tox2－qi）一gg／｝i＋g2（tu一一g！／iti）一一一b－S．P一一601／t／i，1一一Qi’llSIE‘S“5 （s． i3）
（U・臓一・・）一絵＋・・総一一調…一塩≦一一tbl’lli2 （・・14）
一一〇6－gi＋k／2－o （s． is）
in which the viscous terms v V2 Ui， v V2 U2 are Reglected， since one assumes such
a large Reynolds number that molecular effects are negligible with respect to the
turbulent effects．
One introcluces length scales Li， L2 for the x”i－and x2－directions， respectively．
Since the turbulent wake region is Barrow in the x2－direction in comparison with
the xi－direction， the length scales satisfy the relation
－2一一一2i一一 ＜＜1 （s． ！6）
Let Vi be a suitable velocity scale for the main velocity Ui ； one may conveniently
take， for instance， the maximum velocity difference across a section perpendicular
to the main－flow diretion． The magnitude of the velocity scale V2 for the velocity
component U2 is no longer independent of the velocity scale Vi， because changes
in the velocities in various directions are not inclependent on the ground of the
equation of continuity． Hence， from Eq． （5．15），
芸一一著・・暑…会 （…7）
77 Study on the ’lrurbuient Shear Flow Past a Circular Cylincler 77
The Reynolds equations of motion contain the turbulence Reynolds stresses
Pte；・teS・， for which， too， one has to introcluce a suitable scale． One knows from
experience that， in anisotropic shear flows， the intensities ・tc12 and zt．S2 are of the
same order of magnitucle （Eli．nze ［40］）． So one may introduce the same scale zt’2
for ／／Z’ P2 and tt52． The magnitude of the shear stress p／iliillZ，1’， compared with P，／Lwwi｝’2
and p uS2， depends on the magnitude of the correlation coefficient RiL，． Thus one
may assume
／2 t2 ／2 ul N ze2” N zc
and
tt｛tcS N Ri2 ze’2 （5． 18）At a large dlstance behind the cylinder qi is small comparecl with Ue， so that one
may put
＞＞ ！ （5． 19） ql
Now， it is possible to write down the equation of motion and， below
each term， its relative orcler of magnitude with respect to the other terms，
expressed in terms of the velocity and length scales． For the ．xi－clirection one
obtains
一（磯＋一・総・・絵）＋（…一・・農）
菟 ．ω五・ 1 堕 1
qi qi qi 一一÷器一一砦一響・ （・。2・）
濃語｝R12暮一号
where dPi means a change in pressure in the ．xi－direction． For the equation of
motion in the x2－direction one obtains
（硫絵＋一・舞一・款）＋・・1塞：
！・一z ∫壺 ！ ！
91 ql
一一罷一難一戦 （・．2！）
濃R、、餐鑛葦
On the gyounds of the assumption that Uo／qi＞＞ ！， the terms qi6gi／O－xi， q2aqJOx2，
qi aq2／Oxi ancl q2 eq2／ax2 on the left－hand side of Eqs． （5．20） and （5．2！） are of
a smaller orcler of magnitucle． lf it is assumecl that the correlatio ncoefficient
R，， is not smal｝ comparecl with unity， （7／／／／111’i’ ^axi in Eq． （5． 20） is small compared
78 1）vlasaru KIYA 78
w呈th∂E～砺／∂x2， and∂zc｛zt1／∂x、 in Eq．（5．2！）is small compared with another tur－
bulence term．
since at the most z♂2／びl may be of the order！， it follows from Eq．（5．20）
that U，／g， cannot be of a higher order of magnitude than L，／L， ； in other words，
Uo／g， must be oi the sarne order of magnitude as Li／L， （Hinze ［40］）． But， if this
is so， the fiyst term on the left－hand side of Eq． （5．21） also remains small com－
pared with the turbulence term a ztS2／0x2． Hence the only other term that can
balance these turbulence terms is the pressure term OPfPOx2．
Consequently， if one retains only the terms of the highest order of magnitude，
the equation of motion Eq． （5． 7“1） may be reduced to
l aP ． O ztS2
－O （5． ．P．2 a） τδ砺＋…∂x2
0r， after integration，
／）十ρ tt≦2＝Po （5．22 b）
where Po ls the pressure at the same section but is outside of the turbu｝ent wake
reg1on．
On the other hand Eq． （5．20） yields the two equations according to the order
of magnitude of tuL2／gi ；
（Case 1）
磯曲÷器＋讐 （・・23・）
for
票一・・r…撃《・ （・・23b）
（Case 2）
（u， ＋ wJc，） st／ti 一wg， ＝＝ 一b一 一81？；：一 ＋ 一Qiltilli2－tii；u’i （s． ．？．4 a）
for
一一f・Z－i」？一一3一… ＞＞！ （5． Lt4 b）
The physical meaning of the conditions given by Eqs． （5．23 b） and （5．24 b） is
i｝lustratecl in Fig． 5．13． ln case 1， the flow in the wake can be treated as if there
is no transverse velocity gradient in the approaching stream． So， in what follows，
attention wili be confined to case 2．
Eq． （5．24 a） can be solved if the function ul・tdE is known， or its relation to
the distribution ofこみ From Eq．（5．22 b）one can calculate the pressure distribu－
tion． The turbulence term can be expressed in terms of the distribution of Ui
by introclucing the eddy－diffusion eoefficient s， if it is assumed that the transport
of momentum is due to the graclient £ype of diffusion：
79 Study on the Tttrbulent Shear 1’low Past a Circular Cylinder 79
Xr
ω∠2
∠2
0・ 島
ω乙2
∠2
0 9，xi
凶ん，～1鋤傭，《1 ω∠、ん》1
Fig． 5．13． lllustration of the physical meaning of
Eqs． （5． 23 b） ancl （5． 23 c）．
一p／Tii’z’；一pE 一［1！9illl－i ＝pE（a）一一ill／ti－i） （s． i？tLs）
For the sake of simplicity， E is assumed a constant here， although it may take
other forms according the various hypotheses proposed by Prandtl （the momentum
transport）， Taylor （the vorticity transport） and Karman （tbe dynamical similarity
hypothesis）．
If one introduces the non－dimensional variables
え一t・cl／しら， ・。一ε／（U。 d）
（＆， 62）＝ （xi／cl， x2／d）， （cNii， cy2） nm （gi／Ue， g2／Uo）
多exIヲ（pug）
Eqs． （5．24 a） and （5． 15） can be rewritten as
（・＋・ξ、）籍一・a・霜＋・・零轟 （・・26・）
一llき＋籍一・ （・・26b）
Eqs． （5．26 a） and （5．26 b） constitute the governing equations．
b）Pe7－tz〃加渉ゼ・7Z・∫・lutio／Z
For such a moderate pressure gradient as involved in the present experiment，
its effect on the wake development may be small enough to be neglectecl． Then
omitting the pressure term in Eq． （5． 26 a）， one obtains the following fundamental
equatlons ：
（1＋2｛？2） 一〇a“／’7Wpm，i 一Ri」2 ＝” ee m｛li’iigC：2，’1一 （5・ 27 a）
一…絶一＋驚・ （・・27b）
Since R is （）1 a sinaller （）rcler of inagnitude than unity， one assumes the
se 1ivlasaru KIYA 80
perturbation solution
～ア1＝〈1｛o）＋え（／ii）十λ2（1i2）＋… （5．28 a）
a2 an qEO）＋7，qSi）＋7，2 qE2）＋… （5． 28 b）
Inserting expressions （5． L8 a） and （5． 28 b） into Eqs． （5． 27 a） and （5． 27 b）， and
comparing the terms of the same power of R， one obtains a system of simultaneous
partial differential equations for the functions g50）， qS’）， 一・・， qSO＞， gSi）， ・・＋ ：
一饗1」・・壁画罷 （・29・，b）
一響＋ξ，讐一・！・・一・、∂滋翌，＠塞ト誓 （・・3・・，b）
∂塞ll＋ξ、讐・1・・一。∂藷i2’，饗lL誓 （・・31・， b）
g［e） and qEO） describe the wake of
flow of which the velocity is unity，
・！・しか・p（一蓋う
g！・・一一鎗一η・xp（rp24Eo）
where
the cylinder placed in an unbounded uniform
and is kRown to have the forms
（5． 32 a）
（5． 32 b）
g“2
「p ＝マ蔭∴ （5・33）
A is the constant of integration and， as will be derivecl later in 5．5 （c）， has the
following relation to the drag coeflicient Cp ；
Aマ総 （・．34）
According to the measurement of the wake behind a circular cylinder placed in
an uniform flow， the value of E’ 潤@is about O．016．
Substitution of Eqs． （5．32a） and （5．3．？．b） into Eq． （5．30a） yields
饗に・・誓一一論・・e・p（ 2．．．？．．．．．r．．．．．．4Ee） （・・35）
If one assumes the solution of the form
qS’一Af（rp） （5． 36）and substitutes this into Eq． （5． 35）， one obtains
幕＋嵐瘍一義exp（ 2．V．．1．．4ee）
This equation can be easily integratecl to
81 St／ucly on t／he Cl）urbulent Shear Fl（’）xv 1）ast／ a Circular Cylincler 81
∫（・）一眠レexp（一訟！・勿ザ（・）∫1．．・xp（一蓋1吻
since！一・O asη→一〇〇． The value of！ノ（0）can be obtained from the cond量tion
that f一＞O as rp 一〉 oo ；
ノ（・）一一、ぎ，1一∫1．．・・e・p（一蓋う吻／∫1．．・・p（一惹）吻一一芸
Accordingly， the so1ution of Eq． （5． 35） becomes
・il・一型）一A∫1．．（k‘，，一一2”一）exp（一ve／”lu）吻 （・・37）
cAi） means a first order correction of the velocity distribution in the wake． The
corresponding component of velocity in the norma｝ direction， （ISi）， can be computed
from Eq． （5． 30 b） in terms of gii｝ ；
qEi’ 一j20fq－t（1’ cie2 ：一 zLS・・一一：e；：：・・一r （sEo＋2rp2＋’tt’1，r） exp（一 nilZ？E＝） （s・ 3s）
An arbitrary function of ei may be adclecl to tlie right－hand side of Eq． （5．38），
but， since there is no clue to determine this function， it will tentative1y be put
equal to zero．
In order to obtaiR a second order correction for the velocity distribution， Eqs．
（5．37） and （5．38） are substituted into Eq． （5．31 a）． After some reductions one
obtains
四一・・穣2L、罐7（…＋誓÷護忌）・xp（一蓋）
If one assumes the so1ution of the form
qi2’ 一＝ AV’g’1’Mg（rp） ’ （5． 39）
g（rp） must satisfy the following ordinary differential equation ；
鶏多＋面一傷詣一一（5， 772 ， 774 ， rp6万＋一9高耐＋『丁6Il．．．』＋一§琶乙百）・・p（一蓋1（・・4・）
Let the particular integral of Eq． （5． 40） be
g． ur （e，i＋exrp2＋一toft ＋一r－sZl） exp（一rr／／S）
where o“， cr， P， and r are the constants to be determined． Substitution of this
expressioR into the ｝eft－hand side of Eq． （5． 40） yields
｛（・・一・）＋C響一画／・・＋C響一一誓）・・争・｝・xp（一蓋う
一画＋誌…＋・、憲＋…義）・xp（一蓋う
By comparing the terms on each side of this equation， one obtains
！9 ！ ． 41 L？！ cu 一＝ wi””’， B ＝” 一tt’g’」ll，’， 7’ ＝＝ ”IEtS？g’， o“ ＝ ”i16r
Accordingly， the particular integral becomes
・。一C錆・・＋墨・・÷書賦＋、鯨蓄）・xp（一｛。） （・…＞
which tends to zero as rp一〉＋oo and 一〇〇， respectively． The complementary func－
tion， g．， of Eq． （5． 40） is the solution of the equation
傷縁。傷，1。一・
If one introduces the new variable T defined by
τ＝ηハ／2εo
the above equation reduces to the form
幽・窪…・
This equation is a special case of the generalized Hermite’s differential equation
c12 G clG 2zt ；一￥ 一F T一：’i；一’W ＋ （s 一F 1） G 一 O （5． 42）
with s＝：：一2． The general solution of Eq． （5．42） is given by
G（・）一As・xp（一t）D・（・）＋Bs・・p侵）D一調
where A，， B， are constants and D，（T） means the Weber function
Ds（・）一・音・音・xp侵）レ｛（丁差，）パ（一養・音・の
一誤1，デ賎号・9）］
Consequently， the complementary function becomes
g． ＝＝ A ．． 2 exp （ 一 一｛1 ） ＋ B．． 2 exp （ 一 一一丁4－1） Di （iT）
＝＝ A ．． 2［exp （ 一 一｛li一） 一V－211」 T （！ 一 I」＝tll ：一 ／i．． co exp （ 一 ｛i2一） clt）］ 一t一 B．一 2 iz
Since g，， must be a real function， it foliows that B．．2＝＝ O． When T一〉＋oo， g，，
certainly tends to zero， but it becomes 一〇〇 when T一〉一〇〇， which is never
acceptable on physical grounds． Then A．2 must also vanish， and the comple－
mentary function will not appear in the final solution．
Accordingly， a second order correction of the longitudinal component of
velocity， qi2）， becomes
gS2’ ：一 A V？1’ g．（rp） （5． 43）
83 Stucly on the Turbulent Sheat r I Iow 1）ast a Circular Cylinder 83
The velocity component iB the x2 direction is obtained from Eq． （5． 31 b） ：
・1・・一・1・・（一・・）一号ll．。（・。一η…制吻
一号∫1．．C紫瑠∵義一＋鵡＋煮r）・・p（一驚う吻（…4）
Since the right－hand sicle of this equation tends to nonzero value （say， cu） when
ry一〉＋oo， one obtains the relation
gS2’（cx））一qS2’（一〇〇）＝＝ cr （5． 45）
Eq． （5．45） shows that the direction of the velocity vector at each edge of the
wake is not parallel to the main flow direc£ion， ancl may be understood to repre－
sent the spread of the wal〈e in the transverse direction． However， one has no
clue to obtain on a theoretical basis the 1）osslble values of qS2）（oo） and gS2）（一〇〇），
respectively， ancl these remain undetermined． Since the same kind of indeterminacy
necessarily appears when one calculates the higher order corrections， it will be
meaningless to extend the calculation to higher order terms． Thus the perturba－
tion solution of the form of Eqs． （5．28 a） and （5．28 b） may be said to give the
correct longltuclinal component of velocity up to the first three terms．
6） Dπ老9Fo7’c（ヲ！L．ctiフzg o22 the Cylincleフ鳥
By means of the momentum principle one can calculate the drag force acting
on the cylinder from the velocity profile in the wake． ln order to determine the
momentum－flux balance one assumes a set of imaginary control planes at large
distances from the cylinder （see Fig． 5．12）． Through the set of planes Ai／12 ancl
Bi B2 there is a fiux of momentum in the xi dlrection：
呵1＝｛（U・鳩）2一（U・＋ω∬・一・・）2｝clX・
≒・呵r。．（Ue鴇）卿・
where h． is the clepth of the control volume． Through the same control planes
AiA2 and Bi B2 there is a defect in mass fiow equal to
呵1．．．｛（U・鳩）一（Ue噛一一・1）｝・IX一・ん∫r。．・幽
This is balanced by an efflux of an equal amount through the set of control
planes AiBi and A2B2， causing a loss in momentum flux in the xi－direction． Upon
remembering that the flow outsicle the wake is uniformly sheared， one iT｝ay write
this loss in momentum flux in th form
一 ph Uo SW．．q， dx，
The sum of these momentum fiuxes must be equal to the drag D of the cylincler：
D一呵雲．．（Uo ＋ 2cax2）…砺
84 Masaru KlyA 84
Accordingly， the clrag coefficient becomes
CD ＝＝ 2SW． ．．（1 一tuz 2？L82） tticlgfi2 （5． s5）
Subsituting Eq． （5． 28 a） into Eq． （5． 55）， one obtains
C戸・∫▽．．・lel d6・÷・・∫［．。．（・ξ・・lo＞＋・i’））礁・＋…
Since， as is easily seen in Eqs． （5．32 a） and （5．37）， the functions gSO） and qii） are
even and odd functions of e2， respectively， the second integral of the above expres－
sion is identically zero． Thus one obtains
Co ＝＝ 2Si． ．． gle’ dgE2 ＝： ．A V’4’1’1’G’ （5， 55 a）
or
且抽象，r （・・56b）
Eq． （5．55 a） shows that the effect of the transverse velocity gradient of the on－
coming fiow on the drag coe伍cient of the cylinder is of a higher order of mag一
’nitucle than the first power of wcl／Uo．
cl） 且Co／7ψα万50πwith E彫）erli7ne／zts
The longitudinal velocity component in the wake is calculated for several
sections downstream of the cylinder， the values of constants being taken as eo＝：
O．016 and 2＝：O．05． The results are presented in Fig． 5． ！4 in the form of q一’i／
（（）D〃4πεo）againstξ2／〉鉱ξr． It will be seen from this丘gure that the spread of
the wake is greater towards the side of the smaller velocity than on the side of
the larger one as would physically be expected， and that the degree of asymmetry
of the wake on both sides of the center line （xi－axis） increases as further down－
stream sections are approached． For the sake of reference， the relation between
the diameter of the cylinder and the inclination of the velocity profile is illustrated
in Fig． 5． 15 for two values of ！， i． e． R＝O．05 and 2＝O．！．
O，12
（ e．08
聾
＼0．04 ト（61
fiO6L
一4
Fig． 5．14．
一2 0 2 4 4．／庶
Distribution of longituclinal component of
velocity in the wal〈e （Theory）．
6
：85 Study on the Turbulent Shear Flow Past a Circular Cylinder 85
The mean ve｝ocity profiles in the wake of
1．2 mm cylinder are shown in Fig． 5．16 for several
・downstream sections， together with the one in the
undlsturbed turbulent boundary layer． As was pre－
viously stated， the center of the cylincler was located
・at a position 30 mm apart from the boundary wall．
Since， as will be seen in Figs． 5． 16， the fiow outside
the wake is considerably accelerated especia｝ly in the
sections （x－x．）／d一一5！．7， ！05 and 148， it is difficult
to define the amount of velocity defect in the wake．
Thus the dotted lines in the figures are tentatively
・chgsen ag ．the refgrence yelocity ．to COMPUte the Fig． s． is． Geometricai relation
velocity defect． The results at the two sections， ’ b?ｔｗｅｅｎ the diameter of cyl－
i・e・ （vwwx．）／d＝！05 and ！48， respectively， are shown inder and the inclination of
in Fig・ 5・ 17 in terms of ij， and e，． lt is clearly velocity profile・
・seen that the wake spyeads more rapidly on the side of lower velocity than on the
．ミ
E
十一5・7Qσ礁孟2ρ卯
4
” 1
忽
如4θ 〃9 〃
UIU．e
（a）
Fig． S．16．
Rs
sa
4e
餐S 30
20
号醐Qσ皿Z2〃塀
1
／
^
室
s
’ρ ゆ
既／娠 〃／σ脚
（b） （c）
Veloclty dlstribution ill the wake of 12 mln cylincler：
（a）（x－x、，y（2a）＝51．7，（b）105，（c＞148，（d）2］LO，
u／U．
（d）
1“ 1，
o．e6
o．e4
e．02
一〇IY＝一E．＃一＝t一一一7一一tT一一）一”Ss 一4 o 4， s
Fig． 5． 17． Velocity defect in the wake of ！．2 mm cylinder．
盈2σ
V5
w加＝12勿〃
86 ）vEasaru KlyA 86
side of the lar，cr．er one， which confirms qualitatively the theoretical prediction．
It was shown in the last section that the drag force of the cylinder will not
be affected by the presence of the transverse velocity gradient of the approaching
stream， at least to the first order of magnitude of the coefficient wcl／Uo． This
fact may be verified experimentally by the result of section 5． 4．
6． Concluding Remarks
This study clescribes a two－dimensional fiow past a circular cylinder placecl
in a turbulent bounclary layer， xvith special reference to the drag and lift forces
actin．a on the cylinder． A theoretical analysis of the fiow was made for a hypo－
thetical inviscid shear fiow with a lineariy varying velocity profile， and experiments
we，re performecl with a circular cylinder placed in an artificially produced shear
Row ac nd in a turbulent boundary layer， respectvively．
The primary resu！ts of these investigations wil｝ be summarizecl in what
follows．
In orcler to simulate theoretically the flow across a two－dimensional body in
a turbuient boundary layer， an analytical method of cletermining the simple shear
flow past two circular cylinders was devised by the use of bipolar coorclinates．
The result inclucles as a special case of solutions of a uniform fiow past two
cylinders that were previously obtained by Lagally ［19］ and Mtiller ［20］， and
therefore constitutes an extension of their problem． By making the radius of the・
lower cylinder infinitely large， one can establish the simple shear flow pase a cir－
cular cylincler subjected to an interference of a plane wa！l by means of the stream
function ：
t’ttb’i”a’ ：＝： a 一 2，） sinh cr¢（e） ＋ ’1／’ sinh cu tanh crip（i）
where
ψ・・L講聖6画一・愚蔓霧響媛
ψ・1L（ sinh nycosh ny 一cos e）2一・黒．c鳴穿：1一一…h・・η・・…ξ
The corresponding pressure coeflicient on the circular cylinder is found to be
’1’lf／．“117”’／／’S，’ ＝” i 一A（2 一2） （i 一tanh cr）一（一tt6．）2
A numerical calculation of the pressure coefiicient was performed in some detail
in order to clarify the combined effect of both the approaching velocity gradient
ancl the plane boundary wall on the pressure clistribution of the cylinder． The
stagnation pressure decreases as the distance between the cylinder and the plane・
wall decreases， while in the case of uniform flow the stagnation pressure coeflicient
is always unity． ln both cases， the pressure distribution becomes asymmetrical on
87 Stucly on the Turbulent Shear Flow Past a Circular Cylinder 87
the upper and lower surfaces of the cylinder， respectively， ancl this fact suggests
that a lift force will act on the cylinder． The lift force is calculatecl by numeri－
cally integratiRg the pressure on the surface of the cylinder， and is found to be
negative in the vicinity of the plane wall so far as tu＝＝O that means the case of
a uniform flow． However， the 1ift force becomes positive when there exists
a certain amount of velocity gradiedt in the vertical cliection of flow， which is
similar to the case of a practical bounclary layer that clevelops along a stationary
boundary wall．
In order to check the theoretical predictions， an experiment was macle with
a cylinder in a uniform shear flow artficially produced by arranging a gricl of
rods over an entire section of a wind tunnel． lt was found that an accept’able
agreement between the theory and experiment was obtained conceming the stag一一
nation pressure and the position of stagnation point， respectively， provicled that
the clearance between the cylinder and the plane wail is not too small． The
experimental value of lift coeflLcient increases when the cylinder approaches the
plane wall， which is the qualitative agreement with the theory．
Klebanoff and Diehl ［25］ have reportecl that it is possible to thicl〈en substan一・
tially a fully clevelopecl turbulent bounclary layer on． a flat plate by placing appro一・
priate tripping devices in contact with the surface， However， it was found from
preliminary measurements that their method was not effectlve when the bounclary
layer to be thickenecl was alreardy turbulent at the section of tripping objects．
After trial and error， the turbulent boundary layer along the tunnel wal！ was
successfully thickenecl by cementing a circular cylinder of 19 mm diameter onto
the tunnel wall in the manner as is shown in Fig． 4． ！2． The layer so formed
becomes almost characteristic of that developed from the beginning on a smooth
fiat plate after passing over the required length of a smooth surface．
Four cylinders of different diameter are mounted in turns in the artificially
thickened turbulent boundary layer of about 60 mm in thickness at the section
where the test cylinder is mounted． The pressure distribution on £he cylinder is
measured to be integrated for the evaluation of the lift ancl drag forces． lt is
found that the drag ancl lift coelllcients plotted against the ratio zl／（2a）， where A
is the clearance between the cylincler and the bounclary wall and a is the radius
of the cylinder， are represented respectively by a single curve within an acceptable
experirnental errors， the Reynolds number being in a range of （O．99N3．78）×104’．
The lift ancl drag coefficients and the Reynolds number have been forrned in terms
of the approaching velocity at the center of the cylinder ancl the diameter of the
cylinder． With A／（2a） decreasing， the lift coefflcient is found to increase mono－
tonically from O to a moximum value of about O．8 at A／（2cz） ＝＝O， ancl on the
contrary the drag coecacient decreases fyom 1．2 to a minimum value of about
O．95 after passing the maximum value of about ！．3 in the vicinity of A／（2a）＝＝O．！6．
The positive value of lift coeflicient implies that the cylinder in the turbulent
boundary layer acquires a transverse force that tends to move it from the boundary
wall， as is preclicted qualitatively by the theoretical analysis of a unlform shear
flow．
，88 Masaru KlyA 88
In order to estimate the contribution of viscous shear stress to the lift and
clrag coethcients， the laminar boundary layer that develops along the surface of
the cylinder is calculated in detail by the use of asymmetrical Blasius series and
Pohlhausen’s approximate method． lt is clarified that， as the distance between
the cylinder and the boundary wall decreases， the distribution of shearing stress
becomes more and more asymmetrical on the upper and lower surfaces of the
・cylincler， and accorclingly a torque acts on the cylinder． Moreover， it becomes
obvious that the laminar boundary layer separates from the surface a little earlier
（about 40 at most when measured from the stagnation point） on the upper side
than on the lower one． The contribution of shearing stress to the lift and drag
coefllcients is founcl to be 2％ at most of the total value in the Reynolds number
range （O．99tw3．78）x10‘， in accordance with the case of uniform flow．
When the cliameter of cylinder becomes very small compared with the thick－
ness of the boundary layer， it becomes diflicult in practice to estimate the drag
force by the measurement of pressure distribution on the cylinder， and use must
be made of the momentum principle． Two cylinders of ！．2 mm and 3．2mm
diameter， respectively， are placed in turns in the turbulent boundary layer of
60mm in thickness and 30 mm apart from the boundary wall to measure the
variations of mean velocity， turbulence and the static pressure in the wake． The
．drag coe鐙cient of the 3。2 mm cylinder computed from the momen加m principle is
found， within a limit of experimental errors， to be the same as that when it is
placed in the unbounded uniform fiow， provided that the Reynojds numbers for
both cases are the same． Hence it may be said that the normal velocity gradient
of the boundary layer exerts little effect on the drag coefficient of a thin circular
・cylincler， when it is not in the immediate vicinity of the boundary wall． lt also
becomes evident that the spread of the wake is greater towards the side of the
．smaller velocity than on the side of larger one．
In an attempt to investigate these facts theoretically， the velocity profile in
the wake is derived for the case of uniform shear fiow that simulates the velocity
profile of the turbulent boundary layer in which the cylinder is placed， the eddy
viscosity being assumed to be a constant． lt is found from the analysis that
’the effect of the transverse velocity gradient of the approaching flow on the drag
．coe蚤cient of the cylinder is of a higher order of magnitude than the丘rst power
・of 2atu／Ue， where tu denotes the velocity gradient． lt is also found theoyetically
that the spread of the wake is greater on the side of lower velocity． Thus the
・experimental results are well interpreted by the theory．
89 Study on the Turbulent Shear Flow Past a Circular Cylinder 89
Appendix
The Stability of Asymmetrical Double Rows of
Vortices in a Uniform Shear Flow
A1． Configuration of Vortex Rows
For a two－dimensional fiow behind a bluff symmetrical obstacle at a Reynolds
number above a certain critical value （about 40）， the well known K6rmfin n vortex
street is formed behind the obstacle． The regular pattern of vortices is in most
cases evident at a distance of four or five diameters behind a solicl body． How－
ever， in suitable circumstances the trail of vortices persists for a consiclerable
clistance behincl the body． For theoretical purposes the system far downstream
was considered by Karman ［41］ to be composed of isolated rectilinear vortices in
two parallel rows． Von Kfirman．’s study was followecl by a widespread and lasting
series of investigations of the subject． For the most part these concernecl them－
selves with experimental comparisons of real vortex streets with Ka’rm6n’s idealized
model， calculations， etc． Rosenhead ［42］ studied the effect of the walls of a chan－
nel on the characteristics and stability of a symmetrically situated vortex street．
Levy and Hooker ［43］ showed that a vortex－street is unstable for the two－
dimensional infinitesimal clisturbances considerecl by Ka’rman if the undisturbed
velocity is not uniform but is greatest along the middle of the street．
In the present analytical and experimental studies on the shear flow past
a eircular cylinder， the Reynolds number of the flow concernecl was of the order
of magnitude between 103 and 104’． For the fiow of lower Reynolcls Rumber，．
a vortex system akin to the K6rman vortex street may be originated even in the
wake of a two－dimensional bluff obstacle in a shear flow． ln this appendix， a flow
model of the vortex system which would be formed behlnd the solid body in
a simp｝e shear flow u一一toy＋U is presentecl， together with a discussion on the
stability of the vortex system concenrned．
As a vortex street in a simple shear flow， one takes the configuration iilus一・
trated in Fig． A．1． 1 and h denote the distance between consecutive vortices in
the same row and the lateral spacing between the rows， yespectively． The circula－
tion of vortices iB the upper row is denoted by 4（1”，＞O）， while that in the lower
Fig． A． 1． Configuration of a vortex－street in a uniform shear fiow．
ge Masaru KIYA 90
one by 一丁2（4＞O）． This fiow model can be reduced to the Karman vortex street
when Ti ＝＝ r2＝ r．
It is noted here that for each single infinite row of equidistant vortices， each
of a strength of L and 一1”2， at distances 1 apart， with the origin shown in Fig．
A．1 and £he axis of x along the row， the complex fiow potentials produced by
each row are
W・一三1・9…穿（・一二 （A・・）
臨膿1・9…チ（ ．hx＋Z 一2一） （八・）
where
z＝x十zy
Then the total fiow field induced by the two rows of vortices can be described
by the sum of VVi and W2：
W＝＝ ur，＋鴛1・9…チ（・一・一染1・9…チ（・吻低・）
It follows easily that when there are two paralle1 rows of equidistant vortices，
the strength being Ti for the upper and 一1一’2 for the lower row， as indicated in
Fig． A． 1， each row of vortices will move towards the negative x direction with
the velocity induced by the vortices in the opposite row． The veloc1ties of advance
of the upper and lower rows are given by
Ui ＝＝ （rEIVI／． ？”）．＝，，，，， 一 一 一S171” tanh 一ZliieZ （A． 4）
U・《響L＿、，．、ガー島…h望 低・）
respectively． Since rlキjP2 in genera1， the two velocitiesころandこノl are not equa1，
and consequently the configuration of vortices shown in Fig． A． 1 cannot persist
in itself． However， if one superposes a uniform shear flow zt＝toy＋U on the
system， each row of vortices can be made to move with the same velocity． The
condition for this requirement is written as
ωん tuh
2＋ひ＋σ，・・＝一2÷ひ＋U・
or
・んメ≠・・nh禦一 （A．・）
Without loss in generality one can put to＞O． Then T2＞ri， that is， the strength
of the vortices in the lower row must be larger than that in the upper row for
the vortex system to exist．
91 Study on the Turbulent Shear Flow Past a Circular Cylinder 91
A2． Equations of Motion of Vortices
When there are n vortices of circulation TL． （le ＝ 1， 2， …， n） at the position
zk ＝ xh ＋ iyk， the complex flow potential W induced at any point z ＝＝ x ＋ iy is
given by
W一飾誰1・9圓
Then the induced velocity at the same point becomes
・吻讐一一薫義．
or， after separating into real and imaginary parts，
・一一絵学一一鹸㌦媒
where
η一m（rcm rk）2＋（鳳）2］1／2
If z coincides with the location z．， of the m th vortex， the velocity iRduced
・on this vortex by the remaining （n－1） vortices can be written as
・e，、、一一鹸㌣血，…曽爾需勉 （A・・7，・）
r7n，k ：：：” ［（xv，一xL一）2＋（y．， nvyA．）2］1／2 （A． g）
where Z’ denotes summation over all possible integral values of li， excludingle ＝ ’2n．
Let the vortices initially on the line y＝＝h／2 be subject to a slight displace－
ment so that the coordinates of the m th vortex are （ml＋x．，， （h／2）＋y．，） where
・one considers the vortex initia1｝y at the point （0，ノz／2） to correspond to クn・a・O．
Also let the vortices on the line y＝＝ 一h／2 be subject to a slight disp｝acement so
・h・tthe c…d・・・・…f・h・紬・・…x・・e｛（ 17Z＋一2”）・＋x；t・一告嚇and・h・
α4・ノ。♪伽ノ＋加4吻） 二Q一．
〈〉＿ 「 4
z6多ぜ6，一一＋施♪2 ＠＋≠）ノ＋ゑτ＋卿
心
4
Fig． A． 2． Configuration of slightly dispiaced vortices．
92 Masaru KIYA 92
vortex initially at （1／2， 一h／2） is considered to correspond to n＝＝O （Fig． A．2）． Let
ztd and w， be the components of velocity of the vortex initially at （O， h／2） induced
by all other vortices that are displaced slightly from their equilibrium position．
From Eqs． （A．7）， （A．8＞ and （A．9）， zt， and ‘u， are seen to tal〈e the forms
・t’i一慣’．．£逸霧砲一。裁…1｝蜘『鶏土ん
V、 ＝＝一坐銑二篶一L蘇蝋1＋塾
where
アー3，，、一（x－xゼ〃n／）2＋（y。一y。、）2
ro2． pt （aro 一 zA 一（n ＋ 一1｝＝） 1］2 ＋ （ye 一y，’， ＋ h）2
1n the same manner， the components of velocity uE， v，， induced on the vortex
initially located at （1／2， 一h／2） by all other displaced vortices become
・4一津．．蕩遍『鞍±㍉菰髪一遍霧鉱
．．s．＝一．i，￥i，，，，．．．．一，i，”．；，．t’一．i．ii，，．，，．rrr一［；．iS－i－1一一・e・一…i・ii・・・…一一Jitl一”””）i・一＋，，t，£，，．．．．一一／／］，f－ts，hn：g，；，：＝z・一…（・・一・一・
in which
1圖∬6一㌃（m一 一1…一） 1］2 ＋ （y6 一 y．， ww 1）・
楊一（妬一四一nl）2＋（y6－yh）2
Expanding 1／7’b2．，， 1／r，2．， ・・一 by the binomial theorem and retainiRg only the first
powers of x．， y．．，， x；， and y；，， one obtains
歳一訓・＋・2（編亙｝，
祐1嘉＋縦琳；（一）一（識一
Upon substituting these relations into the expressions for ui， vi and it．；． tv，， and’
making a small reduction， one obtains
螺壌陣門1即智
93 Study on the Turbulent Shear Flow Past a Circular Cylinder
蛾撫雌認笠1賦）
慧嵩蕩鳳）・
嘱一十悪病＋総揚｛（㌻1豊；押
転｛蹴瀧暁蛤竪
蛎一鉱｛1ヨ輪翻
盤｛（；1無粋織輪髪
Now， the equations of motion of the vortices at （Jvo， hf2＋’yo） and （1／2－y．x6，
十y6）， respectively， are
誓｛（D （一一12’1” ＋ ye） ＋ U＋ ui｝一（・告＋σ＋の
吻。
読一＝z’i
傷L｛・（一嘗＋σ＋’・’・1｝一（一告＋σ＋の
吻6 ， 一…∂r＝Wi
Since it is knowB that
鷲蒔メ 学一一u・
真締一霧＋r金㎞望一一u・
the velocities of displacement of the vortices dxo／clt， clyo／clt， … become
93
一h／2
94 Masaru K工YA 94
讐一…＋撫繊｛（鵠一）
綜鵡ト殉 低・・）
＋蕪｛（綜岬 （A・…）
諜翁＠6一暖辮（八・2）
讐一
÷薫｛（煽勢1轡）一蘇亨（A…3）
A3． Condition er’ Stability
There are an infinite number of such equations but they are reduced to two
if one puts
ll二野雪）φ， ：＝1脚湯■ 鋤
a， P， cr’ and P’ are functions of the time and ¢ is a real constant such that
・≦φ≦・・Th・fac…sc・一φ・・d…（・・＋助φ・・…duceap・…d・・1・yd・p・・d・・g
upon position into the displacements of the vortices； the wave length of this
disturbance being 2r，／f¢ （Rosenhead ［42］）．
Substitution of Eq． （A． ！4） into Eqs． （A． 10） and （A． 11） yields
95 Study on the Turbulent Shear Flow Past a Circular CyJinder 95
窪一（to＋一il．ll＞2一〉， A一一sl．ll12一｝， B）β＋β論・C （八・5）
髪1《，畠・A一、畠・Bン＋at、k．｝・C （A・6）
where
A一．，£ua．EL：’：EllllCO，S2一一”rmZ95 ＝一｝fb（2．一gb）
B一
。一l織唖＋去）φ
n2 cosh k ip n¢ cosh le（rr－ip）
cosh2 1e r， ” cosh k T
le＝；t
To deduce the equations relating to the lower row one has merely to replace Ti
and 4 with 一4 and 一Ti， respectively， and 1nterchange accented and unaccented
letters． Hence
審一十、畠・A＋嘉・B）β’一β，畠2℃ （A・・7）
讐一（一跨A＋，鋸｝・B）at一・妻C （A・・8）
Putting
a＝＝ cro e”t， B＝＝ ／90 e”‘， cr’＝a6 e2‘， P’＝P6 e）’t
in Eqs． （A． 15） and （A． 16）， one obtains
一…＋（・＋嘉A論B）β・＋β1釜｝・C一・
儘・A論・B）・・一β・・＋・6嘉一C一・
一P。妻C一・6・＋（・論調＋論z・B）β6一・
一・・憂C＋（一、制＋一2鼻B）・6一β6・一・
These four homogenous equatiops for cro， Po， cr6 and P6 lead to the condition
一・・＋、嘉｝、ル藁B ・ 2鳥、C
嘉君一嘉B 一・． ，謬》C ・ 一＝ o・
・ 一f｝、C 一・ ・一、鋸1、肝尭B
一憂。 ・ 一、舞識斎B 一・
or， in expanded form
／‘ 一1一 a2，2＋b＝O （A． 19＞where
・一期C・＋・編翫＋研等∫愛幽B・）＋等詔
∂一｛・・一・「皇ノ緬B）＋箒（C・一A・一B・）÷傷郷B｝
・｛急舞（C・一A・一B・）÷贋熟B｝
The solutions of Eq． （A．！9） are
・吐か＝・π面∴・一±．古田そ＝砺ご冴
For stability therefore one must have 一a÷mo and 一a－V712”＝ll－4bT either
negative or zero in the range O＝〈¢：；I T． Hence
－a÷V7！E’＝一’4’D’一：｛O and 一a－V’h2一一4b f｛；O
which are equivalent to the inequalities
alO， blO， a2－4blO （A．20＞For the sake of simplicity， one puts
TiT2 一r， p ＝＝ le rr ， x ＝k（7， 一¢）
Hence， by taking into account the relation
・」猛「…nh摯雛（・一・）ta夢ゑ （A…）
a， b and a2－4b on the left－hand side of Eq． （A．20） can be written in terms of
p， r and x；
堪！4々・・一。。譜ρ（P・s吻…hx－px…hp・醐・
＋（・一・）・P…hp（一歳＋誉㌔憲・P）
一（・・＋・）｛1｛害）2＋誌｝÷。論ず一ぢ） （A22／
97 Study on the Turbulent Shear Flow Past a Circular Cylinder 97
’where
this range into Eqs．
Pwh圭cleft－hand sides
£his manner the curves a＝O，
the area
．x“． lt is found from this proceclure that the area correspond
satisfies
the required area of stability for the vortex system to be stable against disturbances
of any wave－length． This stable area is given by
（・一・）・撃［学一去一（！－tanh2カ）］
弔＋（1一・anh・P）・ド吉≧（・一蜘）≧・ （A・25）
（一1－Trr一／一一，，一i一一4一）2 ksb ＝＝ ［（r一 i）2 p2 tanh2p一 （r 一 i）2p tanh p （一 一tte一 ＋ 一2－i＆ ＋ 一tt一．一一1 一tl’2－p ）
＋・｛。。，歳（ヵ・s吻…hx一匹・hp・磁）一C参一誓）2一で毒｝
＋（・・＋・）傷＋勢δ詞［・｛幽h麺戸井cosh鱒s厘）覧
一（x2 p2P一 2）2一蓋アト（・・＋・）（一争誓）論ア］ （A・23）
（4n2 1‘Ti）2た・（・・一4∂）一［（・＋・）・／誉÷茄銅2
me r6’o－s4’tV， ）” （p2 sinh p sinh x－p」u coshp cosh x）2］
・（・一・）・←ψ《一ぎ＋誓＋蕊鎧）｝2 （A・24）
x can vary in the range O SxSp． lf one substitutes any value of x in
（A．2．？．）， （A．23） and （A．．？．．4）， one can easily obtain the value of
h satisfies the equations a＝：O， b＝O ancl a2－4b ＝O， respective｝y， since the
of these equations are polynomials of the seconcl degree of r． ln
b＝O and a2－4b＝＝O are plotted in the rp－plane ancl
which satisfies the condition Eq． （A． LO） is cletermined for each value of
ing to x＝＝＝O （Le． ip＝＝n）
the condition Eq． （A．20） for all other values of x， ancl consequently is
iil・1，，
4
3
2
／
0
沼 β ｝ 1
5オ∂わzθ
κ
〃フ5虚誕8
11ノ
／1＼
I i
@6、 ＼
@ ＼@ ＼
l
@ i
戟_21
Fig． A． 3．
／ 2
π（励勿
Stable area of the vortex system．
98 Masaru KlyA 98
and is shown in Fig． A．3． The point denoted by K corresponcls to the point of
stability of the ordinary Karman vortex street． Since the bounclary curve of the
stabllity area has an asymptote p＝＝ 1．916， i．e． h／l＝＝O．6！0， the vortex system with．
h／1 more than O．6！0 cannot exist at least on theoretical grouncls．
A4． The Average Drag
If one consiclers the double vortex trail behincl a cylincler to extend without
distortion te an infinite clistance downstream， then by considerations of momentum
oRe can obtain a formu｝a for the average clrag force per unit length of the cylinder．
As previously mentioned in section A 1．， the entire vortex street may be
considered to be moving with the velocity
α一事・柵一身…tanh摯一 （A．25・）
一一去・ゐ・び一揚…・・nh勢… （A．25 b）
iB． the positive x direction． Since the vortices are continuously shecl from the
cyli．nder at interval T， the periocl of the motion wi｝1 be T， and one obtains
T＝＝： 1／ Uv （A・ 26）One takes the x－axis miclway between the rows ancl in the direction of advance．
ene also considers a rectangle ABCD of climensions which are larger as comparecl
with those of the cylinder and h， ／， the side AB being coinciclent with the axis・
？」， which is so chosen that the origin is at the center of the parallelogram whose
vertices are the four vortices nearest to the origi．n （see Fig． A．4．）． The control
volume ABCD is assumecl to be moving with the ’／elocity U．． Then the fluid in
ABCD is acted upon by the force 一D’ by the cylincler， ancl by the pressure over
the externa｝ boundary of ABCD． According to Euler’s momentum theorem and
the theorem of Blasius， the drag force D’ experienced by the cylincler is given by’
D’一竪＋Yxレρ∫，。、C饗）2ゐ｝ （A・27）
0 s
ノノ
コ
0刃
Fig． A．4． Control volume for the evaluation of drag force．
99 Stucly on the Tttrbulent Shear Flow Past a Circular Cy・linder 99・
where A，！． clenotes the ．x－component of the momentum of fluid within ABCD， S
denotes the contour of the rectangle ABCD， and W is the complex fiow potential
of the form
w＝： 一’liill一…i－i…一 iog sin 一1‘一 （2－20）一’！／？，2ttil’1’？一 log sin ’1’f’ （2＋xo） （A・ L8）
／ ．ん
2・＝朋イ朋＋り…
since the velocities on． Bc， cD an．d DA are of an order of 1ケ（ノー嵩（灘2＋の1！2＞，
Eq． （A． 27） can also be written in the form
D’一州繍佳輔環、，C護野・｝ （A・29＞
where L implies the distance between A ancl B．
Since the inotion is peroidic with periocl T， one obtains the mean drag force
D by integrating Eq． （A．29） from O to T with respect to time t；
Dメ・！・7（蜘決佳・ρ短∫笠、隈）2・1・｝ （A・3・））
The ’first terni on the right－hand sicle indicates the increase in ar－monientuin due
to the appearance of two new vortices within the contour in the interval T， and
is given by
魑嬰）㌧望．（畢） （八3！）
To evaluate the integral in Eq． （A． 30）， one has， fro．m Eq． （A． 2・’8），
饗一傷…チ（一・。）一齢…チ（・＋・。）
and therefore
∫胃1／、幽2・レ誰．『ll、、c…チ（2－20）・1・
一 一4！・ 一’…一zi…一 i’i．1．iX， cot2 （2十20） ，1．
＋旧劉lll，、C・・子瞬・）・・婿瞬・）・1・
一一が一豊一五）一誰．繹一五）
＋細く・・…h4一一ゐ）＋／…h勢一｝
Substitution of this expression into Eq． （A． 30） together with Eq． （A． 3！） yields
D一妥（君＋T・）知＋喉漕一二参る…h望．一映必譜㌦
ユ00 Masaru KlyA 10e
or， omitting the last term， one obtains
D一号（T，＋1］，）知＋響≠… ph T， r，
狽狽煤f12’一一
zhtanh 一： ／
（A． 3．？，）
If one puts
tu ＝＝ O， T， ＝＝ 4＝＝ 1’i
砥一揚朋…h亨… （A．33）
Eq． （A．32） reduces to the well known Karman’s formula for the drag force， i． e．
Di，一＝＝：”P．’，ot；／i’］”＋’！”Li－1’”””（U－2U，） （A．34）
Also if one puts
T， ＝＝ r（1－E！ ， r，xlny（！＋E）
where c一 is a small positive constant， one obtains from Eq． （A．32）
D・・＝・D，，・＋・・C盤…＋學の （A・35）
This formula shows that an additional clrag force
・・！器÷學．の
will act on the cylinder placed in a shear fiow． However， if the velocity gradient
of the oncoming stream is not too large， the additional term will make but little
contribution to the drag， because the contribution is of a second order of mag－
nitude compared with Di｛一． Accordingly the cylinder placed in a shear flow with
a moderate normal velocity gradient will be actecl upon by the same drag force
as that of the cylincler in a uniform fiow， if the approaching velocity to the center
of the cylinder is the same in both cases． This result corresponds to the conclu－
sion obtained in chapter 5 for a shear flow of a larger Reynolds number．
Acknowledgements
The author is grateful to Professor M． Arie for his helpful discussions and
encouragements during the progress of this work and especially for his critical
comments on the preparation of the manuscript． The author is also indebted to
the staff of the Laboratory of Fluid Mechanics for the construction of the
experimental apparatus and for the reproduction of some parts of drawings． The
author wishes to add that this paper was prepared for print after the kind encour－
agernent of Dr． Arie．
10i
［！］
［2］
［3］
［4］
［5］
［61
［7］
［8］
［9］
［！0］
［11］
［IL］
［13］
［14］
［15］
［16］
［17］
［18］
［！9］
［20］
［21］
［2L］
［23］
［L，4］
［25］
［26］
［27］
［28］
［29］
［30］
［3！］
［32］
［33］
［34］
［35］
［36］
［37］
［38］
［39］
［40］
［4！］
［42］
．［43］
Stucly on the Tttrbulent Shear Flow Past a Circular Cylinder lel
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