JPRS ID: 10200 TRANSLATION SHIP BOUNDARY LAYER CONTROL BY A.M. BASIN, A.L. KOROTKIN AND L.F. KOZLOV

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JPRS L/ 10200
18 December 1981
Translation
SHIP BOUNDARY LAYER CONTROL
By
A.M. Basin, A.I. Korotkin and L.F. Kozlov
FBIS FOREIGN BROADCAST INFORMATION SERVICE
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JPRS L/1Q200
18 December 1981
SHIP BOUNDARY LAYER CONTROL
Leningrad UPRAVLENIYE POG'_tANICHNYM SLOYEM SUDNA in Russian 1968
pp 2-491
[Book by Abram Moiseyevich Basin, Aleksandr Izrailevich Korotkin,
Leonid Fillippovich Kozlov, Sudostroyeniye, submitted 3 June 1968,
2,800 capies, UDC 629.12: 532. 526]
CONTENTS
Abstract
1
Preface
by the
Authors
2
Introdu
ction
Not included
Chapter
I.
Laminar Boundary Layer Stability
11
�
I.1.
Some General Problems of the Stability oL the Lamtnar
i
Form of Fluid Motion
11
~ �
1.2.
Laminar Boundary Laysr Equations. Initial and Boundary
~
Conditions
16
i �
1.3.
Methods of Calculating the Stability of a Laminar
j
Boundary Layer
23
~ �
1.4.
Small Oscillations Method. Orr-Sommerfeld Equation
38
�
I.S.
Boundary Conditions for the Orr-Sommerfeld Equation
42
! �
1.6.
Construction of Solutions of the Okr-Sommnerfeld Equation
43
�
1.7.
Construction of the Neutral Stability Curve for Giveu
Velocity Profile in the Boundary Layer
51
�
1.8.
Formulas for Finding the Critical Reynolds Number
61
�
1.9.
Construction o� Asymptotic Brancties of a Neutral
Stability Curve
64
�
I.10.
Experimental Confirmation of the Basic Conclusions of
Stability Theory
68
�
I.11.
Application of Stability Theory Results in Calculationa of
the Length of a Laminar Segment of a Boundary Layer
70
9
1.12.
Calculations of Stability oi a Laminar Boundasy Layer with
Suction
75
_ �
1.13.
Stability of the Asymptotic Velocity Prcefile
81
Bibliography
88
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Chapter II. Influence on Surface FlexibiliCy of a Body Over Which Flow
Is Taking Place on Laminar Boundary Layer Stability 36
� II.1. Brief Survey of Studies of the Influence of Surface
Flexibility on rhe Drag of Bodies Moving in a Fluid 96
� 11.2. Boundary Conditions for the Orr-Sommerfeld Equation in
the Case of a Flexible Surface 99
� 11.3.
Method of Calculating the Stability Characteristics of a
Laminar Boundary Layer Developed at a Surface thst is
Compliant in the Normal Direction
104
� 11.4.
Calculations of the Critical Reynolds Number af a Laminar
Boundary Layer as a Function uf the Characteristics of a
~
Compliant Surface fn the Normal Direction and the
Pohlhausen Form Parameter
107
� 11.5.
Method of Calculating the Stability Characteristics of a
Laminar Boundary Layer Develope3 on a Surface Compliant on
the Tangential Direction
109
� 11.6.
Calculation of the Critical Reynolds Number o= a Boundary
Layer as a Function, of the Characteristics of a Surfacz
Compliant in the Tangential Plane and the Pohlhausen
Form Parameter
113
� 11.7.
Energy Exchange of Oscillatory Motions of a Fluid and a
Wall
116
� 11.8.
Examples af Calculating the Lengths of Laminar Segments of
Boundary Layers on Bodies with Deforming Coverings
118
� I1.9.
SomP Structural Diagrams of Flexible Coverings Designed to
Extinguish P�ilsations in a Boundary Layer
122
Bibliography
125
Chapter III. Influence of Variation of the Physical Constants of a
Fluid on Stability of an Incompressible Laminar Boundary
Layer
� III.1. Statement of the Prob]em. Equations of Motion of an
Inhomflgeneous Viscous Fluid 127
� 111.2. Laminar Boundary Layer in an Incompressible Liquid with
Variable Kinematic Viscosity 128
� 111.3. Laminar Boundary Layer in a Ftuid with Variable Density 147
� 111.4. Laminar Boundary Layer Stability in an Incompressible
Fluid with Variable Kinematic Viscosity 152
� 111.5. Stability of a Laminar Boundary Layer with Variable Density
Across the Boundary Layer 156
Bibliography 163
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Chapter IV. Effect of Suction of a Fluid Through a Permeable Surface of a
Body on the Laminar Boundary Layer CharacterisLics 166
- �IV.1.
Simiiar SQiutions of the Laminar Boundary I.ayer Equations
in the Presence of S uction
166
9 IV.2.
Use of Pulse and Ener�y* Equations for Approximar_e Calcula-
tion of a Boundary Layer ;aith Suction or Slowing
171
� IV.3.
Use of the K. K. Fedyayevskiy Method to Calculate a Laminar
Boundary Layer with Oblique Blowing (Suction)
175
� IV.4.
Use of a System of Three Equations of Mnments for Approx-
imate Calculation of a Laminar Boundary Layer
185
, � IV. S.
Approximate 24ethod of Calciilating Optital Suctian of a
Fluid from the Boundary I.ayer of Wing Sections with Porous
Surf ace
154
� IV.6.
Calculaticm of an Axisymmetric Laminar Boundary Layer
in the Preseuce of S uction
201
� IU.7.
Appraximate Method of Calcular_ing a Laminar Boundary
Layer in the Presence or Slot Suction
204
� IV. 8.
De tn_rmination of the Elements of a Slot Suction System
Insuring Boundary Layer Laminarization
211
� IV.9.
Numerical Integration of the Equations of Motion of a
-
Fluid in a Laminar Bounda.rv Layer with Suction ,
217
Bibliography
229
Chapter V.
Laminar -to-Turbulent Boundary Layer Transition
234
�
V.I.
Theory of Laminar Boundary Layer Transition to Turbulent
Under the Effect of Initial Flow Turbulence
234
�
V.2.
Influence of Surface Rot>>-hness on the Laffinar Boundary
Layer Transition to Turbulent
241
�
V.3.
Joint Influence of Initial Turbulence and Surface Roughness
on Transition in the Boundary Layer
248
�
V.4.
Flow Structure and ?'rlctional Drag in the Transition Region
of a Boundary Layer
252
Bib liography
257
Chapter VI.
Controlled Turbulent Boundary Layer
261
�
VI.1.
Some Results of Experimental Studies of a Turbulent Boundary
Layer of a Plate on In3ecting or Removing Fluid at the Wall
261
�
VI.2.
Methods of Calculating a Controlled Turbulent Boundary Layer
-
Which Are a Development of the Truckenbrodt Method
270
�
VI.3.
Application of the K. K. Fedyayevskiy Method to Calciilation
of a Turbulent Boundary Layer in the Presence of Injection and
Suction
281
�
VI.4.
Effect of a High-Speed Jet Aimed Tangentially to a Surfrace
'
over w,;ich Flow Takes Place on the TBL Characteristics for
Positive Pressure Gradient
302
~
VI.S.
Use of High-Molecular Additives to Decredse Surface
Friction in a Turbulent Flow
306
Bibliagraphy
315
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Chapte?- VYI.
Ttao-Phase Boundary Layer
320
� VII.1.
Introduction. General Statement of the Problem of a
Gas Fi lm on a Bo dy
320
� VII.2.
Ttao-Phase Laminar Boundary Layer
330
� VII.3.
CaYculation of a Ztao-Phase Laffinar Boundary Layer for
aa Exponential Velocity Distribution Law at the Outer
Boundary
343
� VII.4.
Integral Expressions for a Two-Phase Boundary Layer
356
� VII.S.
Use of Integral Relatians for Approximate Solution of zhe
Problem of a Two-Phase Laminar Boundary Layer
360
� VII.6.
Turbulent Motion in aTaa-Phase Boundary Layer
(Diffusion Gas Saturation Conditions)
373
Bibliog
raphy
383
Chapter VIII.
Frictional Drag Reduction of a Ship by Creation of Artificial
Cavitie
s (Gas Films) on Its Bottom
385
�VIII.1.
Brief Survey of Experimental-Theoretical Research
385
517III.2.
Investigation of the Shape and Sizes of an Artificial
Cavity Formed on the Lower Side of a Aorizontal Plate
389
' �VIII.3.
Effect of Limitation of the Flow on the Shape and Dimensions
o f an Artificial Cavity Formed on the Lower Side of a
Horizontal Plate
402
�VIII.4.
Results of Expe�imental Studies of the Creation of
Artificial Cavities (Gas Films) on Flat-Bottomed Model
Ships
417
�VIII.S.
Estimation of Drag Reducti.on of a Ship Equipped with a
Device for Creating Artificial Cavities (Gas Films) on
its Bcttom by Model Test Data
423
_
6.
�VIII
Results of Investigating Artificial Cavities Created on
.
the Bottom of Planing Vessels
426
437
Bibliography
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UDC 629.12:532.526
BOUNDARY LAYER CONTROL
Leningrad UPRAVLENIYE POGRANICHNYM SLOYEM SUDNA in Russian 1968 pp 2-491
[Book by Abram Moiseyevich Basin, Aleksandr Izrailevich Korotkin, Leonid Fillippovich
Kozlov, Sudostroyeniye, submitted 3 June 1968, 2800 copies]
, [Text] Abstract: The results of the most significant theoretical and
experimental studies of a new area of modern ship hydrodynamics,
which is of practical importance and is related to modern methods
ef the theory and calculation of a controlled boundary layer, are
discussed in this monograph. Along with theoretical problems,
physical models of the investigated phenomena are considered,
experimental data and structural diagrams of various methods of
' boundary layer control are analyzed. Results are presented from
studying fluid flow stability in a laminar boundary layer, vari-
ous methods of laminarization (suction, flexible surfaces and
' coverings that alter the properties of the fluid), and turbulent
~ boundary layer control by injecting materials with various proper-
ties into the region next to the wall. Practical recommendations
~ are made which will permit the possibilities of one control
method or another to be evaluated, and conclusions are drawn re-
garding the expediency of using boundary layer control for vis-
! cous drag reduction on ships.
1 The book is designed for engineering, scientific workers, post-
~ graduates and students in the higher courses of the shipbuilding
schools and departments.
There are 492 pages, 205 figures, 10 tables and 391 references.
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PREFACE BY THE AUTSORS
At this time applied hydrodynamics has developed to the point that it is passible uot
only to determine the forces acting on a body ist motion, but alao to alter the direc-
tion of t.hese forces in the directioa required in practice. However, whexeas there
are a number of monographs on wave drag on a moving body in which methods of deter-
mination aad reduction of wave drag are presented, the problems of determining
viscous drag are discussed in only individual 3ournal articles, which for the most
part are unsvailable and unknown to shipbuilders at large.
This situation ia explained by the novelty of the problem and absence of a clear
idea about the practical possibilities o� one metlwd or another of reducing viscous
drag.
The monograph presented here is intended to fill this gap to some degree. It dis-
cusses boundary layer control methods and calculation procedures permitting estima--
t-ion of the possibilities of one control method or another. Conclusions are drawn
regarding expediency of boundary layer control to .achieve viscous drag redvction.
Boundary layer contral can theoretically he implemented by two methods. The first
c o n s i s t s i n i n j e c t i n g materials the properties of which differ from the
properties of water at the wall region. The second is based on the idea of lamina-
rization of the boundary layer ; with large Reynolds numbers the friction in the
boundary layer under laminar conditions is less than in a turbulent boundary layer.
' Both of the indicated methods are investigated in the book. When discussing the
first of them, problems are investigated which are connected with atudping a bound-
ary layer both with continuous distribution of the fluid properties acroas the
boundary layer and in the presence of an interface between fluids (an sir or gas
intexlayer between the surface over which flow talces place and the main water flow).
The problems of stability of the fluid flow in a laminar boundary layer and various
methods of laminarization of the boundary layer (suction, flexible coverings, altera-
tion of the fluid properties in the wall region) were investigated in the discussion
of the seconci method.
Structural diagrams are presented for various methods of boundary layer control.
[Fortions of the text are missing]
2
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Diagram 1
Key: a. boundary layer (BL) control
b. increase in lif t of the lif ting
surf aces
c. model basin testing of ships
and foils
d. decrease in drag
e. boundary layer suction
f. boundary layer ventilation
g. turbulixation of the boundary
layer ior proper recalculation
of the residual drag to full
scale
h. turbulizatian of the boundary layer
to prevent laminar separation on
the lifting surfaces
i. decrease in frictional drag
J. d.ecrease in form drag
k. in the forward section of the foil
1. in the af t section of the f a i 1
3
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Reynolda number. Therefore the boundary laqer characteristics of a model and
full scale unit can diff er significantly, which, in turn, can lead to noticea.ble
variation of the force interaction of a body with a fluid and complicates the pro-
cess of recalculating the results of model experiments to full scale. Under full
scale conditions the Reynolds numbers are appreciably higher than during model
- experiments. The boundary layer of full-scale units in practice is completely t�ur-
bulent. On the basis of relative smallness of the Reynolds numbers, in the case of
models the boundary layer can remain laminar at signif icant distances from the for-
ward end, and, consequently, the possibility of laminar separation arises. In
- addition, under laminar flow corrditions the frictional drag is appreciably less
than under turbulent conditions. Accordingly, the people condacting experiments
are faced with the problem of turbulization of the boundary layer which is solved
by various types of turbulence stimulators. ,
The turbulization problem also arises the simulating lifting sur�aces if the experi-
ment is performed with small Reynolda numbers.
For illustration of the effect of the nature of the f low in the boundary layer on
the hydrodynamic properties of lifting surfaces let us consider the results of one
of the experiments performed by G. G. Filippchenko and G. V. Anderaon. A foil having
NACA-2309 section, with 0.5 meter chord and 1 meter epan was moved at a speed of
9.5 m/sec in a stationary air envirnnment. The lift waR measured, and the flaw
separation on the suction face of the foil was recorded by means of silk threads
glued to it. A photograph of the upper surface of a foil aQt at an angle of attack
of 18� is presented in Figure 0.1. The behavior of the silk threads in.dicate6 sepa-
ration of the boundary layer over the grea*er part of the surface. The lift (Cy)
was 0.8 in this case. Figure 0.2 shows the behavior of the silk threads in the
presence of a trip wire 1 mm in diameter positioned with respect to the foil in
accordance with the diagram illustrated in Figure 0.3, Judging by the silk threads,
boundary layer separation is in practice absent over the entire foil although the
towing speed and position of the foil wfth respect to the flow did not change. In
the investigated case the lift was 1.15. Boundar,y layer separatiQn ia eliminated by
turbulization exactly as in Prandtl's well-known experiment with flow over a
sphere.
3. The next important area of application of houndary layer control is the class of
problems connected with decreasing drag. Out of the three components of water re-
sistance to the movement of vessels wave drag, form drag and frictional drag
- first form drag began to be decreased by giving streamlined, smooth lines [o bodieF
movinK in a fluid. It is also possible to reducE frictional drag by de-
_ creasing the wetted surface. In thic case the bodies become relatively short and
the form drag increases sharply as a result of flow separations occurring in the
sft end. Suction or ventilatinn of the boundary.l:e.yer:in thevicinity of the aft end
is proposed to eliminate the indicated separations.
4, The prohlems of influencing frictional drag have heen the least studied. In
practice, shipbuilding has no fully developed methods of reducing fri.ctional drag.
They are all in various stages of testing and determination of the expediency of
their application.
It is the goal of the authors to outline to some degree the class of basic methods
of lowering frictienal drag which appear to be theoretically possible. These methods
have been visually represented in diagram 2, The degree to which individual methods
4
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Figure 0.1. Behavior of silk threads on separation of the
boundary layer over the greater part of a foil surface.
Figure 0,2. Behavior of silic threads in the presence of a
longitudinal trip wire.
of lowering frictional drag have been developed differa. The most prospective and
efficient means is ventilation of the wall region with gases. This follows from a
comparison of the frictional stresses on the wall in fluids with different physical
constants. Therefore the method of reducing frictional drag by creating thin air
films, which has been checked out under full-scale conditions, is the closeat to
practical application. Although suction of the boundary layer to achieve laminari-
zation has shown its best side in aviation, it is premature to recommend this
5
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Oft
L w
PbyMamop
1)
- Figure 0.3. Diagram of trip wire positioned ahead
of a wing.
Key: 1. trip wire
method for practical use in shipbuilding, for there are no corresponding data from
- full-scale experiments.
The method of reducin~hod ahasybeen confirmed by a single experimentaaadLtheoreticall
application. This me
developments.
Along with the indicated methods the classification present.ed in diagram 2 encom-
passes methods which have as their "assets" only theoretical developments (laminari-
zation of the boundary layer by creating a positive kinematic viscosity gradient in
the wall region). Finally, methods of lowering the frictional drag are mentioned
there, the idea behind whlch has still be insuff iciently completely confirmed by
theoretical developmenr_s and experiments (for example, the effect of an elastic wall
on a turbulent boundary layer).
When studyiug diagram 2 it is necessary to consider the known degree of arbitrari-
ness in dividing laminarization methods into mettwds t:zat increase the curvature of
the velocity profile and methods directly influencing pulsating motion in the boun-
dary layer. The presented classification was constructed by the principle of con-
sidering the primary mechanism of controlling diaturbing motion. Thus, with suction,
for example, not only is the steepness of the velocity protile increased, but the
_ centers of pulsating moti1inear tthetwallhetheiPrpfileff ills outts3.gnif i of
dec.reasing the kinematic viscositY
cantly, but the pulsating motion dampa more elowly..
Not all methods of reducing drag preeented in diagram 2 are analyzed in the book.
Only the most prospective in the aense of use for shipbuilding are touched on. ,
Primary attention has been given to discussing new results and investigating the
theoretical aspects of applying one method or another.
The apparatus of stability theory is used for theoretical evaluation of the possi-
bility of boundary layer laminarization. This agproach sometimea encounters ob3ec-
tions in connection with the fact that there is a transition region between the
point of loss of stability defined by this theory and the developed turbulent boun-
dary layer zone. This transition region can also change in ita extent. Therefore
the Alollowing remarks 3re appropriate.
6
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I 00AeHHCHast rpan"ua (H) I
I Harpee (27)I I I
I HCMbI0T0110ECK3A A068EK8 (9)
I TIOAaTIIHCBA rP8XNt1:1 (J.~I 0\ I
f
Diagram 2
Key: 1. decreasing frictional drag
2. laminarization of boundary layer
3. controlling a turbulent boundary
layer
4. controlling disturbances occur-
ring in a laminar boundary layer
5. increasing the curvature of the
velocity profile in a laminar
boundary layer
6. air "lubrication" (creation of
thin f ilms on the surface of
a body)
7. alteratic+n of the physical
constants of the fluid in the
wall region
8. moving boundary
9. nonnewtonian additive
10. compliant boundary
11. magnetohydrodynamic effects
12. creation of a density gradient
in the wall region
13. fluid with unaltered properties
7
C2/
5
w
x
V
S
v
nF
05
a~
0
,
v
s
=
C
O
(3)
G
0
~
~
~
o~
~ y
9
a ~
s
rm
n
O~
a~
s
y
�
9
(1)
:
~
P
C
6
A
n
0
~
~
0
.1
A
n

L
n
5
14, fluid with altered properties
15. boundary compliant in the
direction normal to the siirface
16. boundary compliant in the
direction tangent to the surface
17, suction of fluid
18, pressure gradient
19. nonnewtonian additives
20. creation of a f avorable density
gradient
21. creation of a f avorable kine-
matic viscosity gradient
22, diatributed suction
23. discrete auction
24. injection of thp required com-
ponents at the surface over which
flow is taking place
25. chamical reaction
26, volatilization
27. heating
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Hydrodynamic stability theory has been confirmed by many experiments (see Chapter I)
and correctly indicates the basic trend in the occurrence of turbulence. Based on
the conclusions of stability theory, laminarized foil sectiona were built which un-
der actual conditions demonstrated theoretically predicted better drag character-
istics than ordinary sections (Figure 0.4).
This qualitative agreement between the stability theory conclusions and full-scale
experimentation is also observed in boundary layer laminarization by suction of
fluid from the surface over which flow is taking place.
Thus, the application of hydrodynamic stability theory for qualitative evaluation of
the possibilities of one method of boundary layer laminarization or another appears
to be justified, the more so in that calculatinns performed with respect to the loss
of stability point always give a"margin" in determining the length of the laminar
section (of course, if all of the significant factors of the phenomenon are taken
into account).
10
8
6
5
103Gz
' '4 7-1
1
2
3
I
t
s s Rin 9 .3 4 5 6. 8 10
Ne ~ y
Cp
0,6
014
0,2
0 2 ~
- p,2
- 0,4
Pocnpe8enexue
;(a) aaBner+uu ~
0 I NACA C009 1
-y
0 ~
u ~ NACA 66-0009 ~ 2
0,458
8
i
Figure 0.4. Frictional drag of a sectioa, 1-- ordinary section;
2-- laminarized section; 3�--- friction in the laminar boundary layer
of a plate; 4-- friction in the turbuleat boundary layer of a plate.
Key: a. pressure distribution
Laminarization of a boundary layer requires elimination of quite powerful disturbing
factors: surface vibration and roughness, extraordinary external turbulence, local
boundary layer separations. The effect of some of the indicated factors on transi-
tion of laminar flow to turbulent is discussed in Chapter V.
Hydrodynamic stability theory studies the behavior of small disturbances in a lami-
nar boundary layer. It is obviously impoaeible exactly to define the boundary
which separates small disturbances from finite ones at thia time, It is necessary
to def i.nq the admi,ssi.hle magni,tude of the ahove-enwnexated disturbancea in each
specific case on the basi.s of experimental data.
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It has been pointp3 out that the most prospective neans of reducing the frictional
drag of flat^bottomed transport vessels is ventilation of the wall regian of tbe
flow with low-viscosity and low�-densitp materials, that is, gases.
Accordingly, a theoretical study of a two-phase boundary layer is presented in the
book.
Two boundary layer gas saturation modes are investigated: the film mode in which a
clearly expressed, stable gas-liquid interface is observed, and the diffusian or
bubble mode in which a gas-water mixture moves near the wall, the viscosity and
density of which exceed the viscosity and density of the gas.
From the point of view of decreasing frictional drag the film gas saturation mode
~ is of the greatest interest.
As a result of complexity of studying a two-phase boundary layer, the phase motion
diagram is given, that is, the gas saturation mode (film or bubble) and the mode of
motion uf the liquid and gas (laminar or turbulent) are assumed in advance. This
- approach makes it possible to determine all of the hydrodynamic flow parameters.
The problems of stability of the assumed form of the f low are not considered.
In view of significant difference in physical characteristics of a gas and a liquid
two different methods of treoretical inveatigation of film gas saturation are pre-
sented in the.book.
On the one hand, boundary layer theory is used to study f ilm gas aaturation consid-
ering the eff ect of the gas density and viscosity under the condition of neglecting
the influence of the gas ventilation on the pressure distribution which is assumed
to be given. The form of the interface ia determined by the gas flow rate and
, method of in3ecting it into the f low.
~ On the other hand, for investigation of artificially created gas films the theory
of developed cavitating flow is used in which real gas properties are not coasidered.
; Motion with an interface is considered as motion of an ideal fluid defined by the
cavitation and Froude numbers with a free flow line subject to determination, the
; pressure on which is constant and equal to the gas pressure in the f ilm.
I, Here a study is made of the problem of the limits of applicability of the indicated
, theories important in scientif ic and practical respects.
The efficient use of cavitating gas films to reduce drag is poesible only if the
basic physical laws defining the cavitating flow parameters are known.
In the proposed book a quite detailed discussion is presented of the results of
theoretical and experimental studies determining the parameters of artificial gas
films created on the bottom. The most important results of these studies was the
conclusion of the existence of a limiting Froude number with respect to length of
the gas film or, what amounts to the same thing, with given speed of the vessel
limiting (maximum possible) length of the gas film. In addition, it is demonstra-
ted that limiting gas films can be obtained for very small (thearetically zero) air
flow rates, and the cavitation drag of the fittings used to form the gas f ilms is
very small.
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The results presented in Chapter V'III can be used fcr practical solution of problems
connected with the development of drag reduction devices for flat-bottomed
transport vessels and planing vessels. -
In the Soviet Union, papers by a number of researchers consider the problems connec-
ted with incompressible boundary layer control.
Hydrodynamic stability problene, which are closely intertwined with boundarp layer
laminarization, are being resolved by the Moscow Hydromechanics School under the
direction of academician G. I. Petrov. Boundary layer laminarization by slot suc-
- tion is the subject of papers by K. K. Fedyayevskiy, A. S. Ginevskiy, A. G. Prozorov,
V. N. Nikolayeva, Yu. N. Alekseyev, L. F. Kozlov. Distributed suction from the
boundary layer is represented by studies of Ya. S. Lodorkovskiy, S. S. Zolotov,
' Yu. N. Alekseyev, A. I. Korotkin, L. F. Kozlov. The influence of other discoveries
4n boundary layer laminarization was investigated in papers by A. I. Korotkin, V. B.
Amfilokhiyev and N. A. Sergiyevskiy.
Tur'bulent boundary layer control is represented by papers by S. S. Kutateladze, A. I.
Leont'yeva, L. Ye. Kalikhman, V. P. Mugalev, Yu. V. Lapin, Z. P, Shul'man, A. M.
Basin, Yu. N. Karpeyev, I. P. Ginzburg.
In addition to the early works by L. G. Loytsyanskiy and K. K. Fedyayevskiy on two-
phase boundary layers, it is necessary to note the studiea of G. G. Chernyy, A. N.
Ivanov, A. A. Butuzov, I. D. Zheltukhin, A. M. Basin and V. B. Starobinsltiy. The
influence of high-molecular additives on the characteristics of turbulent flows hae
been studied in experiments performed by a group of scientists under the direction
of G. I. Barenblatt.
liydrotionics problems have been investigated in papers by researchers directed by
A. N. Patrashev.
A bibliography to which references are made in the text is presented at the end of
each chapter. In some of these papera there are exhaustive lists and surveys of
foreign research of the problems discussed ia the book.
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CHAPTER I. LAMINAR BOUNDARY LAYI'sR STABILITY
� 9 1.1. Some General Problems of the Stability of the Laminar Form of Fluid Motion
During fluid flow two forms of motion can be observed in the boundary layer: lami-
nar and turbulent. Tize laminar form of motion of the fluid (from the latin word
Zcnrrina plate) is characterized by the fact that its elements move in an orderly
fashion, in layers, not mixing with adjacent layers. The turbulent form of fluid
motion (from the latin word turbuZentus stormy, disordered) is cha.racterized by
disordered, nonsteady displacements of its elementa along complex trajectories.
Here the fluid particle velocities are of a random nature and vary with high fre-
quency.
It must be emphasized that the presence of disorderliness in the motion of fluid
particles only in time is insufficient for the motion to be considered turbulent [1].
Actually, . o n e c a n c o n c e i v e o f def ined quantity of f lui.d moving randomly as a
solid body in space.
In exactly the same way, it is insuff icient to have disorderliness of motiun of
the fluid particles only in space, for it is possible to imagine steadY motion of
a fluid with disorderly trajectories in any vo].ume.
As an example, let us consider the flow bPhiud a circular cylinder [2]. For small
Reynolds numbers (Figure I.la) the flow in the wake is of a laminar nature. With an
increase in the flow velocity around the cylinder (Figure I.1bTd) a Karman vortex
sheet forms behind the cylinder which, in sgite of a quite complex nonsteady state
law of development in time and space, cannot be classified as turbulent motion be-
cause of the absence of disorderliness of tlie displacements of the fluid elements.
With a further growth of the flow velocity around the cylinder, the motion in the
wake acquires a turbulent nature. An analogous pictuze of the laminar to turbulent
flow transition is observed between coaxial cylindere [3].
_ It is possible to obtain a representation of the nature of velocity pulsations in
the laminar, transition and turbulent regions of the boundary layer from investiga-
tion of some oscillograms (Figure I,2a-c) obtained by V. N. Nikolayeva and N. A.
Sergiyevskiy during wind tunnel testing of a model of a wing usiag a thermoanemo-
metric device.
The existence of two fluid flow conditions natuxallr imposes the probleat of the con-
ditions cauaing transition of one type o� flow iato the other. In practical appli-
cations usually we are dealing with transition of laminary f low into turhulent, in
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I ~ -i a)
.
Re=32
1- 1
Re=35
b)
Re =65
Re=102
dJ
Re=225
cc~
Re -71
Re�16i
~
Re -281
Figure I.1, Nature of motion in the wake behind a cylinder for
various Reynolds numbers Re = ud/v (d is the cylinder diameter)�
conn.ection with which the primary attention of hydro and aeromechanics specialists
is directed toward the discovery and study of the stability conditions of laminar
fluid flow, the conditions of sustaining it [4]. liovever, recently problems have
been discovered in which it was necessary for researchers to deal with transition of
a turbulent flow to laminar [5]-[9]. A discussion is presented below of the prob-
lems connected only with the laminar-turbulent transitioa. The phenomenon of lami-
nar to turbulent transition is highly complex. Thus, for example, for the case of
fluid flow in a boundary layer the following regions are distinguished in the tran-
sition zone from the laminar boundary layer to a developed turbulent boundary layer
[10]. The beginning of the first region (Figure 1.3) is determined by the point of
loss of stability of the laminar boundary layer with respect to small random dis-
turbances incident in the boundary layer. As the experiments of Schubauer and
Skramstad [11] demonstrated, relatively regular oscillations of the laminar flow can
and do exist in this zone. These oscillations, developing downstreamyt increase with
respect to amplitude and, acquiring an irregular nature, make the transition to the
next region characterized by the preaence of turbulent "spots" [4]. Increasing in
size, the turbulent spots gradually f ill the entire wall space, forming a turbulent
boundary layer. Oscillograms of the recordinga of velncity pulsationa in the tran-
sition zone obtained by N. A. Sergiyevskiy using a thermoanemometer when testing
plates in a model testing basin are presented in Figure 1.4.
At the present time only the conditione of occurrence of the first region, that is,
the conditions of stability of a laminar boundary layer with respect to small dis-
turbances [12] have been sufficiently completely studied as applied to the boundary
layer.
From what has been stated it follows that the state of the art in deternaning loss of stability
~in a boundary layer does not offer the possibility of indicating the beginaing of
developed turbulent flow in the layer, for the characteristics of the second
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a)
I C')
IIIII~~~I~AI~I~II~I~A~IIIIII~A~I~~~~I~IAIA~~~~~~~~~~~ LIIIIIAIIwll~llllllll~~ll~l~~~lllll~lll~~IIIU~I~III ~
Figure 1.2. Flow conditions in the boundary layer: a- laminar con-
ditions; b-- transition conditiions; c, turbulent conditions.
The distance between strokes corresponds to 0.002 second.
n
n
r ~c,�n ~
~
c~~e n
on
5 n
cc~c ~c " c cc c ~ n
4 nCrCc ~ c c < < cc~
1 2 3 ccc < r ~ c ~Ccc
c ~ c
~r c~ ~ c rcC~< < c
Figure 1.3. Transition zone in a boundary layer. 1- point of losa:
of stability; 2 developed disturbances in a laminar� boundary
layer; 3- beginning of the turbulent spot region; 4-- turbuleat
spot region; S- beginning of the developed turbulent flow regioa.
transition region have still been inau��iciently completely studi,ed.. Howevex, the
calculation methods of stability theory as applied to a boundary la.yer permit sub--
stantiated comparison of laminar boundary layers developed under various conditions
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~~,;umur,;;mu~~~n~u~u~~~~rr'1II11111111pIpIIIU~IUUlIIIIIUUIIIInllllllllllllllllllltlllllllllllliliiliqlplll~q~
3
~ ~~~illlliilllp~~111141IUUIIIiIIIIIIIIIfIIIIIIIIIiqiiUiulll11111IIIIIIIIUiuUlllutlubqtl4qlUlllllll181lqIh~I
5
7
2
4
IIUin�n~tiul~~uuuuiumimtllpN~~pINIMm~id~NNII~i~IqGppl~ll~ldlp~lll~llpa~r '
6
~~NIIItIlluunnu~Iluuiuu~lul~IIIIIIIIIIIM1~41Ni~~m~~u~n~nlunOUtlBnll~~lll!ll~~~~~~IIpU~I~q
Figure 1. 4. Transition flow conditiona in a boundary layer.
Spacing between strokes corresponds to 0.002 secoad. 1-- laminar
conciitions; 2-6 transition conditions; 1--r turbulent conditions.
from the point of view of their tranaition to the turbulent state, On the basis of
the conclusions of stability theory [13]-[15], laminarized foil sections have been
built which demonstrate better drag characteristica than ordinary sections [16].
Stability theory has also predicted [17], [18] signif icant possibilities of laminari-
zation of bodies by suction of fluid out of the boundary layer, As an example, in
Figure 1.5 we see the results of determining the tranaition point by a thermoanemo-
netric device on a perforated foil with and without suction of air across its sur-
face (the experiment was performed by V. N. Nikolayeva), By the tranaition poiat in
the indicated experiment she meant the point of maximum intensity of the velocity
pulsations recorded by the instrument.
In connection with an investigation of the transition zone it must be noted that the
loss of stability of the initial laminar form of flow in a boundary layer need not
immediately cause turbulization. It can be found that the flow converts to another
laminas form more stable tha.n the initi.al one [19], [20]. An analogous phenomenon
occurs in the walce behind a cylinder (see Figure I.1) when a rectilinear laminar
flow makes the transition to an oscillatory flow which, in turn, develops into a
Karman vortex sheet which is a more atable form of laminar motion than the initial
form. The transition of one laminar fornt of fluid motion into another can be
observed also in experiments with rotating coaxial cylindera [3], [21], wheii the
laminarflow in the gap between cylinders becomes proper vorteac motion (so-called
"Taylor vortices").
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0,6
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30
0,4
0,2
?s A ,~/cew ~ (b) '
Figure I.S. Coordinates of the transition point on a perforated wing
as a furction of the degree of suction. Wing dimensions: chor.d k=
1 m; span b= 1 m; perforated area S= 1 m2; xcr a XcrA'
ICey: a. x cr b. Q, Z/sec c. m/sec
Among researchers studying the problem of transition in a boundary layer there are
- two points of view with regard to its nature, Certain researchers (Taylor [22],
IJieghardt [23]) consider that transition is a consequence of local separations of
the laminar boundary layer occurring under the effect of finite disturbances incideat
' in the boundary layer. Others (Tollmien [24], Schlichting [4], Lin [12]) explain
I transition as a consequence of loss of stability of the laminar boundary layer with
' respect to small random disturbances. Experiments performed under the corresponding
I~ conditions [11], [25], [26] and [27] confirm both theoriea. In reality, obviously
i the situation is such that in the presence of quite large disturbances tranaition
! takes place by a scheme close to the Taylor scheme, and for relatively small dis-
turbances, phenomena develop as predicted by Tollmienr-Schlichting theory.
At this time the Tollmien-Schlichting stability theory based on the classical method
of small disturbances is considered generally accepted. This theory is veYy
strictly substantiated and confirmed by many experiments. As will be obvious from
what follows, it gives a picture of the development of amall random disturbances in
a laminar boundary layer which is quite close to reality.
; When studying different methods of boundary layer laminarization, which will be dis-
; cussed later, it is necessary to have a method of estimating the degree of stability
I of the laminar form of flow under various conditions. The method of small oacilla-
tions, a systematic discussion of which is presented in the following aections, was
selected as this apparatus. Before prdceeding with a description of this metlwd,
it is necessary to indicate its place among the existing methods of calculating
_ stability: the Goertler method and the energy method. A brief description of the
mentioned methods and their areas of application are considered below.
A discussion of various aspects of the problem of boundary layer control and flaw
stability problems in the wall region requires multiple ref erences to the basic
equations of motion of a fluid near boundaries. These equations with the correspoad-
ing boundary and initial conditions are presented in � I.2.in order to facilitate
the subsequent discussion.
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9 1.2. Laminar Boundary Layer Equations. Initial and Boundary Conditions ~
Equations of Motion of a Eluid in a Soundary L.ayer
The general equation of motion of a continuous medium [281-[31] has the form:
, _ _ - - ~
j ~ F - W> 1) i,T ~ - J (IS 0. .
t s -
Here 'i is an arbitrary volume isolated inside a fluid by the aurface S;
p is the fluid particle density;

w is the fluid particle acceleration;
n is the direction of the external normal to the surface S;
-r
F is the mass force vector reduced to a unit mass;
i
pn is the surface force intensity vector.
Let us transform the surface integral entering into the equatioa of motion iato a
volumetric integral. Then we have
i
-
Pn ds IPx Cos (r1, x) Py cos (n, y)
~
aPx avy vp1 ) d~r.
pZ cos (n, z)J dS = J~ vx + ay vt
~
The equation of motion assumes the form;
, ~ i
p + aa" + a,~y + 'P= a I dT = 0.
t
Under the condition of continuity of the investigated functions and their derivatives
the preceding equality exists only in the case where the function under the integral
sign is zero at any time of motion, that is,

F-~,-}- P ~aax + aay.+ aa: 0.
Denoting the projections of the mase force F on~rectangular coordinate axes by F,
Fy, FZ, the projections of the vectors p X, fy, pz by pxx' Pxy' pxzp and so on anff
the projections of the acceleration w by du/t, dv/dt, dw/dt, we find the equations
of moticn in the following form; '
iiu X i- I onxx vau x
F aP:x
P ( dz + dy + dZ ) ;
ilu (~l -f ' 1 y +~~pYY 1 vP?y
vX
ay T vZ
(z.1)
vPx. aPys dPrz 1
F: P\- d.r + dy + oZ J
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The continuity equation must be added to these equations
_oP + v ~ A(e) a (PW) = U. (1.2)
dt dx dy dz
For an incompressible liquid the expressiona for the intensities have the form [28]:
_ ~-u -----vr,
~ - - p 2� a , pxY - p,rx - ~ ~ a~, vx
1~1~~ - P -1- 2Et ~~j-y ; PX: = P:z ~ dz ~z (1.3)
dv dm1
P,: P-'- 2!~ az ; Py= - P=Y - az + ay J'
Here u is the internal friction coefficient or the coefficient of viscosity.
For a uniform viscous incompressible liquid the system (I.1)-(I.3) is simplif ied
all
` FX - _L
a,
+ v Ati;
dt
r
do
= F-~
v
dL
a
v Av;
~i~
t~
y
glu,
i
` F -
=
d''
az
+ ye~,;
(I.4)
~
r
p
du du
dx + ~3y j
dw
dz
- 0.
Here v= u/p is the kinematic coefficient of viscosity.
During motion of a fluid near curvilinear surfaces in some cases it ia expedient to
use curvilinear orthogonal coordinates ql, q2, q3. The corresponding profections
of the velocity vector will be denoted by v, v2, v3. In the indicated coordinates
the equations (I.4) are written as follows }28]:
Jui ui. dui i'= Jul t-3 dui c�~c-_ 0I11 +
dl ~ ll, dy, + N= dq, + Hs dya + Il1H= dqy
v,u;, J/l, ~i dH= ta aH, _ F I I~-
l1, ll, dy, !/iH, dy, fl;,N, dqg 1. P Ht d9t
_ v_ 1 J H_ , a (U~1i,) 113 a (v,n,)
- ii_~~~ I 541 I l~~ri, dq, - HHs a9t ~
J r Hy d(u, ll,) ll_ d(wrll,) 1 (I.5)
~ys L 11,11, J11:1 H~l1 i o9~
- du. ui J;~_ u! du. vi r)._ c'~v:~ dll;,
dl.~Ilt dy_ ll~ Jy3 { lli JqL + Jy,
r)!/: Jll ~ I I dp
-r /!_ll dr/l ~ I t /le.l! dc _ U s d/t ; -  ll J : -
i I : 9~
v
id ll , (t'31:i) 111 d(utll
- -
%/3/li ( Jya [ 11:H3 dq. ll~lla d(h
J (I.6)
J9i [ fl~!!= dql l1~1/x Jy. J
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du~ 3 Ju3 v, du3 V. Jc~. ~ul- Jlli
a~ r c~i, vq, + H, ow~ H. J4�: + iit~,li, v9~ +
u,v, v~~a Ei aH, ~ aH, _ i ~ ap _
+ H~s d9$ H~i a93 tl:Hs a9o F11 p f/a dqo
v r Ha a(viHi) Na a(~3Ns) l
HI He (_.L a9, ~ H~HI dq3 Hs HI a9i J
d r ift a(VsHo) Hi a(V:Hs) l
V, L hsy3 d9s HsHo d93 J}' (1.7)
0 (vj sHa) ~ a(v:HaHi) a(V'H~Hs) = O.
(I. 8)
9l a9s a9~
In equations (I.5)-(I.8) H{ (i = 1, 21 3) are Lsmg coefficients, the values of which
can be obtained using the .:xpression for an element of sn arc in the adopted curvi-
linear coordinate system.
dS2 Hi(dqj)~ Hs(d9z)' -t- Fiatd93)~� (I.9)
S
f
Figure 1.6. Coordinate diagram on a curvilinear
surf ace,
The above-described systems of equations of motion of a viscous fluid are basic when
studying flow over bodies in cases where the forces of inertia in the flow greatly
exceed the viscosity forces everywhere except the thin wall region where the indi-.
- cated categories of forces are comparable with respect to magnitude. The Prandtl
equations describing the motion of a fluid in this region called the boundary layer
with good approximation, can be obtained by different methods [29], [32]--[35].
The derivation of the boundary layer equations for simplest plane-parallel flow
based on the Mises scheme [33], [28], [39] is presented below.
Let us consider a cylindrical body of finite dimenslons wi,th generatrices perpen-
dicular to the plane of the flow. A viscous liquid flows around the cylindrical
body. Let s be the surface of this body. At each point of the surface let us draw
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the external. normal and take values of q= a and q2 = y as the coordinates of any
poir.t M, where x is reckoned along the s~riace from the selected initial point 0
and y is reckoned from the surface to the normal on through the point M(Figure I.6).
- We shall consider that in the flow region of interest to us (0 < x< x; y> 0) the
values of vl, v2, p and their derivatives entering into equations (I.~)-(I.8) are
rinite values. In addition, let us assume for 0< x 0(the symbol m determines to which coefficient : the
~ given convergence radus pertains). Consequentlp [82], in this interval it is
possible to f ind the solution of equation (I.59) 3.n the form of a series with re-
spect to powers of an independent variable. If series for four independeat solu-
I tions of equation (I.59) can be constructed such that they will converge uniformly
in the region of definition of y, the existence of the fundamental system of solu-
tions of equation (I.59) will be proved at the same time. In view of the fact that
the coeff icients in (I.59) are alao regular functions of certain parameters (a, c,
1/aRe), using smallness of some of them (c, 1/aRe) it i.s possible to construct solu-
tions in the form of series with respect to powers of these parameters. Heisenberg
[83] and Lin [67], who investigated the given method in detail proceeded in this
fashion in their research.
Considering that the boundary layer stability is lost for large Re numbers, it is
possible to find the solution of equation (I.59) in the fornn of the series
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~
v (J) = (Po(y)+ aRe '~i(y) ( nRe (I.65)
The parameter 1/aRe is assumed to be a ve ry small value. Substituting (I.65)
in (I.59) and equating the coefficients for identical powers of 1/scRe in the er,pres--
sion obtained, we arrive at the following spstem of differential equations:
(u-c)(yo-a"-(pu) -u �(fo=0;
(ti - c) (ip,"- a2yK) u"q)K _ - i (q1Kv1- 2a3(p^-, + 0~(pK_1).
(I.66)
k = 1,2 . . .
Determining the solutions of the first equation of system (I.66), it is then passible
by the method of variation of arbitrary constants succeseively to determine all the
functions k. flowever, usually we are limited in the series (I.65) to only the
first term, permitting an error in determining ~(y) on the order of (c~,Re)-1, and the
problem thus reduces to integration of the equation
(u - C)((P~ - OC"(Q) - urip _ o. (1.67)
Equation (I.67) determines the development of disturbances in an!.ideal f luid. It
_ must be remembered, of course, that the effect of the viscosity is felt in the for-
mation of the function u(y), which, in turn, determines the vari.ati,on acrose the
boundary layer of the velocity pulsation amplitude. Neglecting the viscosity lowered
the order of equation (I.59), in connection with which it is necessary to use another
approximate method of integration (I.59) discussed below to find the missing solu-
tions. The differential equation (I.67) has a singularity at the point y= yk,
where u(y ) = c. This case corresponds to comparisoa of the propagation rate of
a neutralkdisturbance in the bouudary layer with the basic flow velocity. It is
possible to explain the processes occurring in the layer y= y, which is called the
"critical layer," by the following argument [84]. Let u~s intraduce the disturbing
motion vortex into the investigation
r dx (a(f $ - (pw) eIC
dy
and the basic motion vortex R= r8u/8y. Thea equation (I,67) can be represented as
follows: .
(u - c) ax a 'vd 0.
(I.68)
Equality (I.68) expresses the condition of preservation of the vortices in an ideal
fluid. If for y- yk, 8R/8y # 0, then (8R/8y)vy � 0, for in the general case vY(yk)
# 0. Therefore for observation of the equality (I.68), its first term (u - c)8r/8x
must also be nonzero. For u- c-* 0 this can occur only in the case of 8 r/8x
that is, in a critical layer in the absence of viscosity the vortex of the disturb-
ing mo:ion increaseEr without limit. In order to bring the calculation scheme in
accordance with realit3r in tb.e critical laper it is necessary to consider the
effect of viscosity which will also be done when finding the solutions of equation
(I.59) as a function of viscosity.
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The presence of a pole of the first order at the point y= yl, in the function u"/(u -
c) determines 1851 the forna of the solution of equation (I.67)
(P(J) (y -JK)p(y-JK).
(I.69)
where P(y - yk) is a series with respect to powers of y- yk with coefficients that
depend on the parameter a. Since u(y) is represented in the vicinity of y= yk by
_ a series with respect to powers of y- pk, it is possible to rewrite (I.69) as
follows:
- (P (J) = (u -c) P (Y, u).
Then the possibility of expansion of the 5 unction P(y, a),in a converging series
with respect to powers of the parameter a is proposed. Then, suhstituting
~P(y) (u-c�)Ib`u(J) f algi(J) l... 1.a211 b`', (J) !
in equation (I.67) and equating the coeff icients f or identical powersof a2, we obtain
the system of differential equations
g. (u - c) I 2u'go U
gn (tt - c) 1- 211'g;, = 6'�_i �(ii - c)
(I.70)
i 1, 2, .
Solving the f irst equation of system (I.70), we determine with accuracy to a constant
factor
~
gu (u = I ; br0 (2) ( (u - c)-- dg.
' Knowing the function gn(S�), it is possible to def ine gn+l' usin8 the equation
u,
9ni-t + 214_ C brn+i = gn
SI U
gn+t(J) = I(u.-C)-zdJ~(u-C)28n(!)dJ� (I,71)
u o
The integral in (I.71) is calculated by the formula
~ -
(u' C)-a J(u - c)' k','(J) dy dJ�
u lo Thus, two solutions are found for equation (I.67)
�
~1 lJ~ _ (U - C) v, ~fn (y) CC2n , (1.72)
n=0 :
' N
where
go(y)-= 1; 9.+1 (y)=_r(u-'c)-ady~(u---c)2gn(y)dy;
b
n (I.73)
(P�: (J) _ (ll - C) J, gn (J) ' a2 +
n =0
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where
r!, a I
u - c)-2 dy i (u - C)$ gn (y) (!J�
b'o (y) ~ J (u - c)-'dJ; b''n+j(J) = I (
0 The series (I. 72),(I.73) converge f or any f ixed a, for wi th a suff iciently large n
their terms are majorized 1671 by the terms of the series
A ((LM)x"
(ln)! ~
It is possibla to be convinced of this by substituting the expansion of u- c with
respect to powers of y- yk in the coefficients of the iavestigated series.
Equation (I.67) is second order, in contrast to the fourth-order equation (I.59).
Therefore only two independent solutions of equation (I.59) are obtained by the in-
vestigated method. Two more linearly independent solutions of equation (I.59) can
- be found [12] in exponential form
v
y (J) - exp JP (J) dJ �
u
(I.74)
Substituting (I.74) in (I.59), we obtain the equation for determining the function
P (Y)
P" +
(u - c)(P, -f- P" - a") - u" _ uRe [p; 6p=p' 3p.= _I 41)
+ P" 2a' (P' + P") + a'I .
(I,75)
The solution of equation (I.75) is represented by a series with respect to pawers of
(aRe)-1/2
I
p(!I)==vaRe�Pu(y)+n, (y)~-~- yaRe'Pa(y)-~~
(I.76)
The exponent 1/2 is selected from the condition of equality of the higher-order terms
in both sides of equation (I.75), which corresponds to the assumption of identical
role of viscosity and forces of iner.tia. The function p(yA aRe) in the form of (I.76) has a singularity at Re co. This is necessarY for' the required solution
of (I.74) to vanish when v->- 0. Substitut~g (I.76) in (I.75) and equsting the co--
efficients before identical powers of (CiRe)1 2, we find the system of equations:
(u - c)Po1 - + iPo';
(u - c) (po + 2PoPl) - - i (4pgpi + 6piJ)o);
from which it is possible to determine the following successively:
Po (y) (u c);
Pi(J) - 2 ' Po (I.77)
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Z~ao signs before ttie square root sign in �ormula (1.77) correspond to two linearly
i.~dependent solutions of equation (I.59). Limiting ourselves to the first two terms
in the expansion (I.76), by using (I.74) it is possible to determine:
- s v
c~:~(!I) (u - c) + exp - ~ Via lle(u - c)dy ; (I.78)
- u
s y
(Pa (J) (it - c) ' exp I yia lle (u - c) dy . (1.79)
. o
Approximate methods used to construct the solutions of ~1(y) (I.72), ~2(y) (I.73),
~3(y) (I'78)' ~4(y) (I.79) lead to the appearance of a s3ngularity for y= yk
where u(yk) = c. Here the problem of the sign of the argument of the value of u- c
for u< c remains unexplained. If arg(u - c) _7r, then on integration along the
_ real axis the point y= y must be rounded from above; if arg (u - c) it must
be "rounded from below. ~ To study the behavior ot the solution of equation (I.59)
in the vicinity of tne point y= yk, a new in.dependent variable is introduced j671,
[83]
y - y"
- e '
where e is a small parameter, the magnitude of which is determined f ram the follow-
ing arguments.
Denoting �(y) = x(n), on the basis of equation (I.59) we f ind
(u c) a2e'X) - e2Xu a R t ~ (X" - 2ale2x"-- a'e'X). (1.80)
- e�z
The functions u and u" in the vicinity of the point y= yk can be represented by the
series:
i u - c = u� � er~ + 2 + . . .
~
~ u~ = IIK � K � er~ 2 u~~ (er~)~ + . . . (I.81)
"i
The value af E is assumed to be so small that the series (I.81) converge. The solu-
, tion of equation (I.80) is found in the following form;
T (11) X (TI ) = Xo (yl) + eXi (,q) -F- E`Xs (11) -f- . . . (1.82)
Subsituting 4I.81) and (I.82) in (I.80) and equating the coefficients for ideatical
powers of e, we obtain the system of differential equations for determining the
functions Xn(N :
~XoV
+ uR' -qXo 0,
ixnV U. �TIxn = l.n-1(x), (Il 1); (1.83)
where L_1(X) are linear combinations of the functions Xo, Xl,... ~_l and their
derivat~ves. It is possible to determine the value E from tFe assumption that the
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highest terms ia the left and right sides of equation (1.80) have the same order of
smallness with respect to e
IV
(U-CW(1R2B: x ~ (1.84)
Considering the expansion of (I.81), on the basis of (I.84) we find the value of the
parameter
e = (a Re)-~~' � (1.85)
Usually when determining the function X(n) we are limited to the zero approximatioYY3
that is, it is considered that X(rl) = X00). flere an error on the order of (aRe)
is permitted, which is obvious from (I.82) and (I.85).
Let us consider the solution of the first Qquation of system (I.83). Let us use the
known substitutions [86], reducing it to a Bessel equation. Let us denote XO
-iu'i = a, then the investigated equation becomes a second-order equation.
url* - 0. (I.86)
- which gives two linearly independent solutions to the equation
Xu~ - itl,. �l1Xu = 0� (I.87)
As the two remaining solutions it is possible to take the functions Xp1� 1; Xp2 = t1
which obviously satisfy. equation (I.87). Let ua introduce the new des~red funcf~Con
z = Wyrn-. Here
z _i- Z- � YTl,
j f 1-1 ,
4 (11)-311 z + (11)-1i1.z, 111i2Z ,
and equation (I.86) is written as �ollows
s 1 I
d dz
- z = 0.
+ 4 ' ,1 (I.88)
Let us vary the independent variable, denoting
' _ ~-1/:S . / 3 bt )~=/3 ; d~1 _ a 1/3. r, ~ t \-I/3 .
I ` 2 d~ ~ ~
From (I.88) in the new variables we obtain
dz \ , / t I z 0
a ~~b ~ T` 9~ / ' (I.89)
The solution of equation (I.89) is expresaed in terma of the Besael function of 1/3
order 187]
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Z Cl�JI/7 (b) + C2J-1/3(b),
or, considering the relation of the Bessel functi.ons to the Hankel functions
2 Hiis~~)] ,
t
t I ( ni/:s (  ru/3
J-1/1 l LG ' r! 1 /3~' ' f~l/1~b~ ~
z - D~�lliji~~) IDz�Hiisg)�
Thus, considering the substitutions made, we obtain the following two linearly in-
dependent solutions (I.86):
X11 ~~~~�li;I; f (ip111'112
.I'
y01 li 11� 1/;;, (iPlij`'' I ,
~ a
where it is assumed that rr,; ; arg (--i) - 1 n.
Since
~
O for il oo,
f/1~; 0 for 11 oo,
j then it is possible to take the following expressions as linearly independent solu-
i tions of the f irst equation of system (I.83):
~ Xoi = I ~ Xo: = t~; _ .
~ xW - jdTj j yTi. ti1 [ 3 cip~,~3'2] d~,; (1.90)
~
Ti
, Xo~ d ts L 3 p~I)3~~
f
J 1~~1 � f~ i ia , d~1
a ~
. (1.91)
The solutions of (I.90) and (I.91) have no singularities in th.e critical layer for
r1= 0.
Replacing the Hankel functions in (I.90), (I.91) by their asymptotic representations,
it is possibZe to reduce these expressions to the form (I.78) and (I.79). The region
in which the investigated representat ions are valid thus determines the path of by-
passing the singularity y= yk.
Let us write [87] the asymptotic expansions of the Hankel functions of 1/3 order for
small values of the argument
H\ ,j, l1/2ex 5n1~
P ~ 12/ (I.92)
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' n l argt 0), we arrive at the conclusion that in the vicinity of
the point where Fr = 0.56, w= 2.3, the value of c- 0. Then in the expression
~i~ cu;, � u~: u~ � c
z, ~K + uK I n c-- ~ t- c)- ~x
it is possible to neglect the second term, for c ln c-* 0 for c- 0, and it is
possible to consider uQ = u~. Thus,
z = - I 11�c.
' u '
_ Considering the first equality of (I.106) and the approximate expression c= u0.yk'
we f ind _a + cuu
F,==0,56=C,=-= .
cuo
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It is impossihle to neglect the third term in the formula for z, for a can be quite small. For the Blasius profile u~ a 1.68, and the preceding equality gives the
expression
u . 0,738c. (I.116)
From the expressions w yK (a Re uK)11:1, yK = _L considering (I.116) , we f ind
uK
IZe = 13,8 � a' or IZe = 46,2c'4. (I.117)
For Tollmien [24], the corresponding branch of the neutral stability curve is de-_4
scribed by the formula Re = 46�c-4. On transition to Re*, we obtain Re* = 15.75c
for the Blasius profile. Thus, the equation is obtained for the lower branch of
the neutral curve.
The equation of the upper branch of the neutral etability curve is obtained from
investigation of equation (I.99) in the vicinity of the origin of the coordinates
of a plane [Er, Ei]. Using (I.78), let us calculate the expression
c .
J,; 1 (U) c(I uc 12c
For large vaiues of Re, we f ind
Fr ~ -1- il�'. .
!/h ~ '.~.nC Rc
i
j For the Blasius profile uk =-18.4y~ _-6.55c2, Hence, considering smallness of c,
I it is possible to calculate Ei= 2,6�c�a2. For large Reynolds numbers it is possible
to set Fr = 0, which, using the equality Er = Fr, permits establishment of the re-
! lation of the parameters a and c.
- ~i -I uo. c
. = p, a 11~47.
c�uO
_ In the case of the Blasius prof ile a a 1.68. Considering the expreasioa Fi = Ei,
let us establish the relation Re = Re(d):
1.68 _ 2,6,c.(zs;
c r2ac Re (I.118)
f2e = 2,8a Re = 0,0156c-'u
According to Tollmien's calculations, the upper branch of the neutral stahility
curve is described by the equation Re = 0.0181c-1o -t The neutral atability curve
 for the Blasius profile, the asymptotic branches of which are constructed by for-
mulas (I.117) and (I.118), is shown in Figure 1.20. It is more convenient to use
the following relations for the calculationa:
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f or the lower hranch of the curve
16 fle* Ig 13,8 - 5 Ig , - 4 Ig a*;
for the upper branch of the curve
Iglle* = Ig2,8- ll Iga.- lUlga*.
oc '
0,6
0,5
0,4
0,3
0,2
0,1
U
~
\
~
1
s
7 R
q
1t
i9Rp.
Figure 1.20. Asymptotic branches of the neutral
stability curve.
Let us construct the formulas for the asymptotic branches of a neutral curve of
arbitrary velocity prof ile in a laminar boundary layer analogously. For the lawer
branch of the neutral curve let us consider the vicinity of the iatersection point
of the polar diagram with the real axis where i'V; = 0, 2,294, w =.2,3'. On the
basis of formula (I.111) we have: _
uO.c ~
(1-c)9 ' uo c.uo.uK ' (I.119)
1,3 - - In c
uK uK
Using the smallness of c, we shall conaider uK uu, yK = u� Thea
0
uoc
a=
c�u~ '
2,3- Inc
uo .
2,3 y I 28� uu
Re=r � .
\ yK a �uK CA
-
As a result, the following formulas are obtained for calculiting the lower branch
of the neutral stability curve for small cs
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c.u"' 28�uo
a= 2.3 , Re= L.. �
The upper branch of the neutral curve corresponds to the vicinity of the null point
of the polar diagram. At tihis point F(w) - 0; therefore f~v,.(w) = J. The parameter a
will be defined by a formula analogous to (I.119),
u'c ~
~ � o . c.
u
~I --c~- uu +io�c'uk Inc (I.120)

u, u~'~
For establishment of the relation Re = Re(c) in the vicinity of the origin of the
- coordinates we shall use the equality Ei = Fi
, u,
---.9 �tluf� ,-3 �
c Il~ {Io .11' . , u~l cl~''~xc Rc ~I.121)
. ~ z,)' �c2 's
~ uK
Considering that c� 1, yk-u; = c, uk = u~ and using (I,120) , we f ind
15
Re = 2n.~ ' r r, ~ . (1.122)
.u~, It is possible to obtain formula (I.118) directly from (I.122). Considering expres-
' sion (I.121), it is possible to arrive at the conclusion that in the presence of an
, inflection point in the velocity prof ile (u;T = 0) . the lef t side can approach zero
'
, not only for c-* 0, but also for c-* cH, which corresponda to yl, such that u"(yj) ~
, 0. Therefore for observation of the equality (I.121) when Re the upper branch
i of the neutral curve can approach a# 0. This fact indicates the presence of non-
I viscous instability of the velocity profile with an inflection point [4].
Knowing the position of this point for the velocity prof ile with an inflection point
(X < 0), that is, the value of y, it is poseible to find c~ and, using (I.120),
to calculate the value of e~, wh~ch is approached by the upper branch of the neutral
stability curve for Re - an u0'cIl
= .
(I - cn)a
Proceeding to a*, we obtain
~ uo.lcll a.
a~ -(1 -cii) b .
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g 1.10. Experimental Confirmati,on of th.e Basic Conclusi.ons of Stabilitp Theory
The study of the stability of a laminar boundary layer has developed historically
so that the first results were obtained in the theoretical studies by Tollmien and
Schlichting. It was only after 13 pears tlzat Schubauer and Skramsted, creating a
flow with a low degree of turbulence (0.02X), confirmed all the basic conclusions
of the theoretical research of [4], [11], [16] and [12] well.
Schubauer and Skramsted performed experiments on a plate. Oscillations of different
frequency were created in a boundary layer using a metal plate and electromagnets.
Observing the development of these oscillations, it was possible to determine under
what conditions the oscillations do not damp and do not build. The corresponding
dimensionless frequenc}r of neutral oscillations defined the point in the plane [,Re*,
arv/U2], where Sr = a�c�U0 /8. It is also possible to reconstruct the theoretical
curve Re* = Re*, (a*) in the plane [Re*, a v/U2], for each point (a*, Re*) of the
indicated curve corresponds to a defined 4lue of c(Tigure I.19).
A comparison of the theoretical calculationg with the experimental "ta of Schubauer
and Skramsted is shown in Figure 1.21. Schubauer and Skr:msted made their measure-
ments in the boundary layer using a thermoanemometric device.
Recently Burns, Childs, Nicol and Ross [97] performed analogous atudies i,u the boun-
dary layer. However, for measurementa cf tlie velocity pulsations tizey used a ligIlt
plate which underwent oscillatory movements under the effect af the transverse com-
ponent of the pulsation velocity in the boundary layer. The plate oscillations were
recorded using a ref lected beam. Experimental points are shown on Figure 1.21.
Stability theory also found qualitative confirmation in the exper.imental paper by
Berg [98]. Using smoke plumes,.Berg observed oscillations of a wing in the boun-
dary layer (NACA-0012). A display was obtairied on a stroboscope. The oscillation
source was a loud speaker generating aound of aa�ferent frequency. The oscillations
were observed near the wing surface in the region of positive pressure gradieats.
In the three mention-d e-perimental papers studies were performed in air flows.
Analogous experiments were set up in water by Wortmann [26], who used the telluric
method for visual representation of the flow,'and Hama [99]. In both cases the
oscillations in the boundary layer were created using a vibrating strip by the
scheme first used by Schubauer and Skramsta.d.
The results of Wortmann and Hama, which agreed well with the theoretical results aad
the data of other authors are presented in Figure 1.21.
Investigation of the expression for the amplitude of the disturbing motion curreat
function 1.6 and 9 1.7) indicates that as a result of the presence of a siZigula-
rity at the point y= yk, for y> yk the disturbing motion is described by a for-
mula of the tyne
I F, (J) +Fs (J) In (y - JK)Iera 4x-ra,
and for y< yk the corresnonding expression must have the following structure;
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6
4
aC ~~o
Re
I
� -3
x -6
~ -7
4
o -8
r
~
�
~e~~Oa
?
~ 0
.
�
,
x

1
.
_
.
`
u
z Re' 1O"J
Figure 1.21. Comparison of tive calculated neutral stability curves
for the Blasius profile orith experiateatal data, 1- calculation
by the proposed formulas; 2- Schlichting calculation; 3-,- Lin
calculation; 4-~ Zaat calculation; 5-- Schubauer and Slcramstad
experiment; 6- Burns, Childs, Nicol and Ross experiment; 7--
Hama experiment; 8-- Wortmann experiment.
[ F, (y) -1- Fz (J)1 n I yK - JK i n� F: (y)1 era (x-0).
Thus, for transition through a cri,tical laper, a phase discontinuity ari,ses in the
oscillations determining the disturbing ntotion. Thia conclusi.on waa obtained
theoretically by Tollmien. The experinents of Schuhauer aad Skramstad confirmed
it brilliantly. They were also able to obtain surprising comparison of the amp].itude
distribution of the oscillations across the boundary layer with the SchlicFLtiag
calculations 190].
It is also necessary to point out the recently published results of the experimental
studies of Schilz [27] which agreed satisfactorily with thetheoretical calculations
of the neutral stability curve. Using thermoanemometers, Schilz studied the be-
havior of small disturbances introduced into the boundary layer of a plate, a
surface over which flow was taking place vibrating by a known law.
In conclusion of this section, let us present a statement by one of the founders
of hydrodynamic stability theory, H. Schlichting [4]: "Experimental studies have
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so brilliantly confirmed stability theory that it must be considered a fullp checked-
c+ut component part of hydroaeromecfianics."
9 1.11. Application of Stability Theory Results in Calculations of the Length of a
Laminar Segment of a Boundary Layer
By the length of the laminar segment we mean the distance between the initial (fron-
tal) point of the body over which the flow is taking nlace and the loss nf srab{li*y
p^int of the laminar boundary layer. We exclude the transition zone (see � I.1) from
the investigation, for its extent depends on several factors [4], the most important
of which are the degree of turbulence of the external flow, the external preasure
gradient, and surface roughness.
Failure to consider the transition zone in the boundary layer leads to underestima-
tion of the length of the laminar segment, which goes "into rexerve" for the calcu-
lations connected with laminarization of the houndary layer. For practical calcula-
tions of the loss nf stability point of a laminar houndary layer on hodies wi,th a
pressure gradient along r.he surface, it is necessary to have the function Re* _
Re* (a). Instead of the Pohlhausen form parameter it is possible to use anycother
parameter reflecting the relation between the pressure forces and frictional forces
in the boundary layer. Here it is necessary to coasider that the famili~s c,f pro-
files constructed by diff erent methods using the same parameter X = Ul�8 /Y can
diff er from each other on the basis of the limited nature of the single-parametric
method of approximating velocity profiles in the boundary layer. As an example it
is possible to present a comparison of the family of profiles constructeri by the
Pohlhausen method with the application of a fourth-degree polynomial j13J and the
analogous family in the case of using a sixth-degree polynomial j151.
For the possibility of determining how the form of the approximation of the velocity
prof iles in the laminar boundary layer influences the critical Reynolds numtaer, cal-
culations were made by the scheme in g 1.8 of the functions Re* = P.e* (a) based on
the family of Pohlhausen profiles (sixth degree polynomial), t~ie famify of Howart
profiles and the family of Basin profiles.
In the first case the velocity distribution across the boundary layer is expressed
by the formula
Fe (J) + XGo (J)~
0 0,
the velocity decreases with an incrsase in x. The velocity distribution in the
boundary layer depends on the magnitude of the parameter r. The values of the
functions u(y), u`(y), u"(y) taken from reference [13] are presented in Table 1.2.
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Table 1.3. Functions k'6 and G6
Y/66 I
Fg I
Gfi I
F6
G6
~s
L6
0
0
0
2
0,200
0
-1
0,05
0,100()
0,(0876
1.997
0,150
-0,165
-0,966
0,10
0,194~6
0,01509
1,983
0,104
-0,486
-0,900
0,15
0,2980
0,01918
1,996
0,061
-0,975
-0,793
0,20
0,3938
0,02130
1,884
0,025
-1,536
-0,662
.0,25
0,4858
0.02175
1.793
-0,006
-2.109
-0,528
0,30
0,5726
0,02089
1,674
-0,028
-2,696
-0,388
0,35
0,6528
0,01905
1,528
-0,094
-3,105
-0,253
0,40
0,7252
0,01643
1,366
-0,054
-3,456
-0,126
0,45
0.7891
0,01380
1,185
-0,057
-3,675
-0,015
0,50
0,8438
0,01094
0,999
-0,056
-3,750
-0,067
0,60
0,9253
0,005A4
0,634
-0,044
-3,956
0,170
0,70
0,9685
0,00232
0,326
-0,026
-2,646
0,181
0,80
0,9938
0,00056
0,116
-0,010
-1,536
0,126
0,90
0,y995
0,00005
0,017
-0,002
-0,486
0,044
1,00
1,0000
0
0
0
0
0
The stability calculations for the Howart profi,les wexe performed as a function of
the paramerer r. Racalculation for the Pohlhausen parameter (X ) was made using
the functionai relation r= r(a4) 113] and the approximate relation a6 = 1.152X 4
[15]. The values of r and the values of a6 correspanding to them for which the
stability was calculated appear in Table I.4.
The family of Basin profiles is defined by the folloazing velocity distribution
across the boundary layer [93]:
u = [ 1 X,;� , ( I - stn 2 sin y . (I.123)
The derivatives u' and u" are found by differsntiation of (I.123) with respect to
the y-coordinate. Using (I.123), it is possible to obtain [93] the expressions for
the displacement thickness and pulse loss thickness
0,363 - 0,028X,;
~

~ - 0,0010871'; - 0,0029GXr, + 0,1366.
The stability calculations for the family of Basin profiles were performed for
different values of the parameter XB. The relation between the parameters A 6 and
~g will be found from the condition of equality of the pulse loss thickneases for
the corresponding values of a 6 and a B. Schlichting and Ulrich used the analogous
condition when establishing the relation X 6= a 6 A 4) I151
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b(U, l'366 - O,U0296k6 - U,00108ks) =
afi 985 _ 1911a - 1 ~2 1. (I.124)
( 900J 18018 6935 )
Multiplying both sides of the equality (I,124) by (dU/dx)(1/v), we find the relation:
Xs (0,1366 - 0,002967~6 - 0,00108~,6)2 = (1.125)
_ Xs (0,109 - 0,001054 - O,OOO15546)2. .
The range of variation of X B, ~6 in practice was bouaded by the limits 10 > a6, aB ?
-10. Therefore in equality (I.125) let us retain only the firat-degree terms with
respect to X 6, XB, which leads to an approximate function a$ = 0.64X6 or X6 s 1.56aB.
Table I.A. Relation of the f orm
parameters r and X 6
.

-0,100
3.88
-0,075
2,26
-0,050
1,64
-0,025
0,875
0
0
0,(Y25
-1,0.1
0,050
-2,30
0,075
-3,8G
U,IW
-6,U5
In order to simplify the calculations for each family of profiles graphs of the
variable were constructed
flK
M(C) - n�C�uo� ,-3
uK
for difEerent values of X 69 r, X B (Figure 1.22, 1.23, I.24). This type of graph
greatly facilitates the graphical solution of equations of the type
tI! � --JLI'll~~��%;i �
JI A
The critical Reynolds numbers were calculated by the scheme � 1.8 using formulas
(I.113) and (I.115). On transition from Re~r r.o Re~r, the following expressions
were used:
~ for the family of Pohlhausen profiles [15]
2
b 7
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M(c)
0,6
0, 4
0,
0,4
0,2
0
-0,?
-0,4
-06
M(C) 0, 2 //0, 4 0, S c
-4
,~--6
Figure 1.22. Graph of the function M(c)
for the family of Pohlhausen profiles.
M(c)
1U
0,3
-D,5 _ ,
Figure 1.24. Graph of the function M(c) for the
family of Basin profiles.
for the family of Howarth profiles[13]
1~�
8 = 0,341,
f or the f amily of Basin prof iles 193]
b�
0,363 - 0,028X6.
The functions Re~r = Re~r(a) constructed on the basis of the investigated three-
families of velocity profiles in a ltiminar boundary layer art presented i.n Figure
73
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0
0,2
0
-0,2
-0,4
Figure 1,23, Graph of the function
M(c) for the fau~ily of Howarth profiles.
d=4
-
-
_
-
2
-
-
-
-

~0
'
04
O 1
0?
O,J
0,5
0,6
0, 7
O,
C
-
_
;4 ;1-J
~
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I9 Re.'P
,k
Figure 1. 25. Curves for Re~r(A6~ considering recalculation for the
Pohlhausen parameter. I- family o� Pohlhausen profiles; II
family of Howarth prof iles; III - family of Basin prof iles.
1. 25. Here the parameters r and ag were recalculated by the above-presented formulas
for the parameter a6. Agreement of the curves must be recognized as satisfactory.
If this recalculation is not made, the curves can differ significantly from each
other, which can be seen from a comparison of the curve for the Pohlhausen family
with the corresponding curve from the Basin family (Figure I.26). This fact i.ndi-
cates the necessitp for a careful approach to the problem of which method is used
to obtain the distribution ot the parameter X = U`62/v with respect to length of
the body when performing the stability calculations.
. ~
Figure 1. 26. Curves of Re* (a) without recalculation for the Pohthausen
' parameter. I-- family ofcrBasin profile; It - family of Pohlhausen
profile.
Key: a. cr
Figure 1.27 shows the curves for the critical Reynolds numbers as a function of
the parameter a6 calculated by various authors: Schlichting and Ulrich [15],
Pretsch [14], Finston [94], Zaat [95], Soprunenko [114] and Tetervin [113]. For
comparison, the same graph shows the curve obtained by calculationa by formulas
from g 1.8 for the family of Pohlhausen profiles. The basis for the Schlichting
and Ulrich calculation, just as the Finston calculation uaing approxitnate Lin
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formulas, was the f amily of profiles defined by the sixth--degree Pohlhausen poly-
nomials. In the calculations Soprunenkn used tite Howarthprofiles. The basis for
the Pretsch and Tetervin calculations was the family of Hartree profiles with
power dependence of the velocity at the outer boundary of of the boundary layer on
the longitudinal coordinate. Zaat also selected a single-parametric family of
velocity profiles in the boundary layer for his calculations, but the parameter that
he used [95] differs from the Pohlhausen parameter. Comparison of the curves Recr
Re* (a) indicates that the formulas obtained in g 1.8 for calculations of the
criEical Reynolds number give results that agree quite well with the data of other
authors.
Having the function Re* = Re* (a) available, by the known scheme of [4], [15], it
cr cr
is possible to determine the position of the loss of stability point. Example cal-
culations appear in the following sections.
Key: a. lg Recr
7
- - - - - d
Figure 1.27. Comparison of the curves Re* (A) obtained by different
authors. I Schlichting curve; II Zaat curve; III Pretsch curve;
IV Finston curve; 1-- propused approximate formula; 2-- Teterovih
calculation; 3 Soprunenko calculations [114].
g 1.12. Calculations of Stability of a Laminar Boundary Layer with Suctionl
With relatively small removal of fluid from the wall, the stability of the laminar
boundary layer developed on it increases significantly [4], [17], [18]. During a
theoretical study of this phenomenon all of the researchers known to the authors of
this paper only considered the increase in fullness of the velocity prof ile in the
boundary layer with suction. The stability characteristics were calculated under
the assumption that the transverse velocities are absent in the bnundary layer.
This statemenr of the nroblem maicps it possible to use formulas 9 1.7. 1.8 to calcu-
late the critical Reynolds number and neutral curve of anv velocity profile obtained
during suction.
1The study discuss,~d in this section was performed bv Yu. N. Alek,aeyev and A. I. '
Kcrotkin [11.5]
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A study of the stability of the laminar boundary layer in the presence of constant
transverse velocity in the wall region is presented belaw. The equation for the
current function of the disturbing motion of an incompressible fluid is written
[28] as follows:
~
~Ay ~ JA4p aeo ay~ ao aey _ a~ aee ~
dc + ~y ' vX vX * dy - x � jy Tx- � ay v~Aqp, (1.126)
where 8 is the current function of the undisturbed motion;
~ is the current function of the disturbing motion.
We shall consider that
0 = 01(y) - V.X.
(I.127)
The value of v= const is the transv2rse velocitp component of the main flow in the
boundary layer. An example of a. boundary layer with constant transverse velocity
is the asymptotic boundary layer with fluid suction at the lower boundary.
Substituting (I.127) in (I.126) and considering the usual (g 1.4) expression for the
current function of the disturbing motion
(P y(J)er
.
we obtain the following equation with respect to the amplitude of the curreat func-
tion
( ii b ) ~(NM (r'y)) - cp ~u ((p" - a"(p') _
N)'V
~
where u(y) is the mean velocity distribution in the boundary layer..
Let us taice the velocity at the boundary of the boundary layer as the characteriatic
velocity U and the thickness of the boundary layer as the characteristic dimension
d, and let us rewrite the preceding eQuation fn dimensionless form, retainiag the
previous notation for the dimensionless variable U
( � - c) ( cp" - ac=(p) - u"(p =
- Re ((PI v - 2a=(p" a�q))- A`-u ((P" - aY(p'). (I.128)
The bou.ndary conditions for equation (I.128) will be found from the following pre-
- requisites. On the wall for y= 0 the tangeatial componeat of the velocity of the
disturbing motion is absent on the basis of the adhesioa condition. This leads to
expression (I.61). The presence of a constant transverse velocity of the main flow
for y= 0 theoretically permita the existence of a normal velocity component of the
disturbing motion at the wall v(0) # 0. However, the presence of transverse oscil-
lations under the main flow congitions with a longitudinal velocity gradient for y> 0
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wo u 1 d have to unavoidably cause oscillations of the longitudinal component of *he
uelocity, that is, would contradict the condition v(0) 9 0. Tharefore it is
necessary to consider the idcntity v(0) ff 0, valid,X which gives condition (1.60).
At the boundary of the boundary la+ for y= 1 let us take the condition of smooth-
ness cf the transition of the desired solution of equation (I.128) to the sollition
for an ideal fluid. This condition is written as follows with accuracy to the first
3erivative:
q)(1) = 4'0M: y'(1)=(P0M'
where ~0 (y) is the solution to equation (I.128) written for the region y> 1, where
' u" = 0, u= 1, (aRe)-1 = 0, for the viscosity is not cc.nsidered
i u(1 - c) ~~'u u~'~pu) -I- v(~~ - u'cPo) = 0.
Denoting a 2 ~ _ (D(Y), we arrive at the equation
0
(1)' rQ (I -c) (1) _ U, -
which has the solution
! 1C-11 u y
v
(1)(1/) ' le ,
where Al is an arbitrary cor.itant. Thus, ~.t is po4sible to define the function ~0 (y)
- using tFie following differential equation
~ ~~~--u u
q'o - a q, o = A ie Vn
the general solution of which has the �form
r (c-i) ~
(Po (J) = - utU~ + as (c - 1)- e " ~ + A'-e"u + Age-a'
Since the parameters c, v and a, > 0 are real values, and at a sufficiently large
distance from the surface there should be no disturbances ( 0(y)/Y400 0), A1 = A2 =
for the first teim of the general solution has an oscillatory, nondamping nature
with respect to the variable y, and the second term increases without linit with an
increase in y. Thus, thE smooth transition conditio:is have the form
y~(l)==A9e-", cp'(1) =--uA~e-u.
On the basis of arbitr ariness of the constant A3 the two indicated equalities impose
only one condi,tion (I.63) on the values of ~(1) and which is obtained by find-
ing A3 from the f irst equality and substituting the value found in the second.
' As a result, the boundary conditions of equation (I,128), which are a aeneralization
of t he Orr-Somnerfeld equation to the investigated case, are written in the form af the
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expressions (I.60), (I.61), (I.63) and (I.64). In the lat:ter, it is necessary to
consider the value of iMi ^ E as small as one m:Cght like. Usually this restric-
tion is not required when investigating stability problems by the smail oscilla-
tior method.
'A
Let us note that the relative magnitude of the transverse velocity v in the case of
boundary layer control using laminarization is on the order of (Re)-1, where Re =
Ud/v. AccordinRly, as the f irst two solutions of equation (I.28) we take the
analogous salutions of the Orr-Sommerfeld equation (I.72), (I.73), which are found
under the assumption of smallness of the right hand side of the equation as a
function of viscosity.
_ In order to determine the remaining two solutians of equation (I.128) let us make
the substitution (I.74), which converts (I.128) to the follawing expression:
(rr - c) p= - (A2) - tC -I- iurRc (P" + snn' -I-
- - u Re lnl -F 6P2P'+ 3P~$ -I- 9PP" -f- p"
2u= (Q' -I- nl) + u�I
(I.129)
In turn, finding the solution of equation (I.129) in the form of the series (1.76)
and li.miting ourselves to the first two terias, which gives an error on the order of
(aRe)-1/2, we find (see g I.6): �
lio(J) - � Vi(u-c);
5 Pu u !Ze
l~i ~y~ - 2 � Pu 2 ~
5 U
~P~ (J) ( t~ c) 4 eap c) dy -1- " R(r ,
~ l J , (I.13U)
y4 (y)=(u-c) 4 exp ~jliafte(u-c)dy -f- v2 e ~ .
o (I.131)
When removing fluid the value of v< Q. Let us consider the real part of tfie expres-
sion in the braces in the right-hand side of equality (I.131) for y-* -
y U
) (
/ iulle(u-C)dy-I� u2e yu jliuRe(u-c)dy0 aRe(1-c) aRe(I - c)u Re
, -{=l I/ 2 !y-'+ j/ 2 (y-)+ 2 y ~ .
� y- m
for u(y) = 1 for y> 1.
The value of v is on the order of 1/Re; therefore the product vRe/2 is on the order
of 1(for the asymptotic velocity profile vRe/2 = 2,3). The entire solution is
constructed under the assumption aRe � 1. The relative propagation rate of the
disturbances c is always appreciably less than one. Thus, the real part of the
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exponent in the solution of (I.131) for y is positive and, consequently, ~4
does not satisfy the houndary condition (I.64). Therefore in the general solu-
tion of equat in (1.128)
4
w (y) CK(PK (y)
K=I
the constant c4 = 0.
The formula (I.130) shows that
u Re
w3 (y) _T3o (Y) Q 2 (1.132)
where ~30(y) is the corresponding solution for the Orr-Sommerfeld equation.
In order to determine the arbitrary constants c ci, c3 from the uniform boundary
conditions (I.60), (I.61), (I.64), the cha.racteristc euafion arises which for
neutral disturbances relates the parameters of the disturbing motion a, c to the
Reynolds number and the average motion characteristics. In the discussed case it
has the form (s 1.7)
~ + t 11E-' = 1 -f~ Z,
Y~ ~3 (o) J (I.133)
where the real and imaginary parts of the complex variable z are calculated by the
formulas (I.104), (I.105). Considering (I.132), let us transform the lef t hand side
of (I.133), denoting it by G(w),
G (W) Tao 0) _
9K 'Fao (0) -F- ~ao (0)
u 2 e
1 - F (w) I -1 _ 1 - F (w) mW' -
F(w) u Re JK t- F(w) ( I-{- rnuo) -
~ 2 .
(I- F,W3m) (l- F, - mw3F,) mWyFl (1 1uW3)
~ I- F, - rnw3F')1 Fi (I+ rnuu3)2 ~
F~
(I -F, - nttu3F,)2 + F! (l�}- nIw3)2
where F(w) - Ir(w) + iFi(w) is the Tietjens function (see � I.7),
(Re au~ )113 �
~~yK K ' m= , .
2yKauK
Thus, it is possible to propose the following procedure for calculating the neutral
stability curve.
1. The universal (independent of the velocity profile) functions Gr(w, m), Gi(w, m)
are constructed for a series of values.
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2. Being given the value of Gi, let us deterqine the values of wl, w2 for defined
m by the graph Gi(w, m), and also, using equality Gi = zi(c), let us determine the
value of c. The nature of the function gi(w) is such that in the general case one
value of the function corresponds to two values of the argument.
3. By the value of c found, let us determine yk, u~, for the velocity profile
is known.
4. For the found values of wl, w 2 and selected m let us calculate or define Grl'
Gr2 by the graph.
_ 5. Using equality Gr = 1+ zr, we find otl, a2.
6. By the formula Re y,. )3 u u' we calculate Re1, Re2.
K
7. Let us determine the values of the relative ventilation v= 2ykau.~m, which cor-
respond to the obtained points on the neutral curve.
It is possible to use the indicated scheme without alteration to find the critical
Reynolds number directly if all the calculations are performed for a point on the
curves Gi = Gi(w, m), where Gi = GimaX.
The numerical calculations were performed for the family of Wust profiles
rr (y, u:r (J) ! 110(J) - t1,: l!/)I,
where u*(y) is the velocity distribution in the Blasius profile;
u*(y) is the velocity distribution in the asymptotic velocity profile;
k is the parameter of the family which varied from 0 to 1.
0
The critical Reynolds nwnber Re as a function of the parameters m and k is pre-
sented in Figure 1,28. The dependence of Re on the parameter A and the relative
transverse velocity in the boundary layer iscshown in Figures 1,29, 1,30, Frvm an
investigation of this function it is possihle to draw the conclusion of the exist-
ence of a relative velocity vp for each velocity profile such that for a suction
velocity greater than v the laminar boundary layer retains stability for all Rey-
nolds numbers. The graphs of the function vo = vQ(k) and the corresponding function
Re0 = Re0 are presented in Figures 1.31, 1.32. For suction velocities less than
vo, tnere are two crltical Reynolds numbers the upper Re+ snd lower Re . The flow
in the boundary layer is stable if its Reynolds number is less than Re^ or greater
than Re+. If Re+ > Re > Re_, the flow is unstable.
For more complete explanation of the indicated fact, neutral stability curves were
constructed for the velocity prof ile with the parameter R, = 0.2. Each curve in
Figure 1.33 corresponds to a defined value of the parameter m. If we calculate the
value of relative suction velocity v at each point of the neutral curves and join
the points with v= corst, then closed regions are obtained which are shown in
r'igure 1.33 by the dotted line. Inside these regions the flow is unstable, and
80
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DI
m
Figure 1.28. Critical Reynolds numher as afunctioa
of the parameters m and X. Key: a. cr
I
outside them it is stable. For a defined suction velocity (v0 the instahility
region degenerates to a point. Thus, for distributed suction, for purposes of
laminarization it is meaningful to increase the relative suction velocity only to a
defined amount corresponding to disappearance of the instability regipn, The pre-
sented calculation data allow estimation of the indicated velocity for velocity
profiles from the Blasius prof ile to the asymptotic profile.
g 1.13. Stability of the Asymptotic Velocity Profile
The study of the stability of an asymptotic velocity profile based on an infinite
plate with uniform and constant section with respect to the plate length pla.ys a
special role in studies of the stability of a laminar boundary layer with suction.
The asymptotic profile has the greatest fullness of the velocity diagram, and there-
fore it has the greatest stability by camparison with all velocity profiles formed
on the plate with'constant and uniform suction. The critical Reynolds number was
calculated for asymptotic velocity prof ile in references [17], [18]. Fflr the
critical Reynolds number Pretsch obtained a value of Re* - 5.52�104; Bussman and
Munz found Re~r = 7�104. cr
Since in the calculations of the stability of laminar boundary layers with suction
these figures (more frequently a value of Re* s 7�104) are uaed to estimate tbe
required suction ii'Ztensity and the calculatiSfis by the mentioned authors were
81
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i9 Re.
ts
a,I
~�a~
,
JO-~Af
' 03
t�-D,/0
~S
70
S -6 -
J
~
Figure 1.29. Critical Reynolds numbez as afunction of the parameter Y.
and relative transverse velocity in the bouadary layer (-0.1 :
cxI)l J (M] U. - p._i U')(rx].
Considering the expressions
- e xi) ~ J (Pil + p21 U' ~ dx ] - (U-Y� (U)P~,,

exP j~Pu , P21 U~ dr] _(U')-a((1)-v,
we obtain
- ((J)" rU)P:i U~ pa,(UU/ ~1 'Ps A _ PI: U-Pei (C/~1-Pi I+
_ l plt \ pll ~ ~
r
( , P12P21 l ( (Lr-~1-P~~
\ / j�� - /~tl / J ~IPi ~tl C~X -E- C 1
A' where U0 = U(xc) , U~ = Ul (x0). Let us def ine the constant c from the condition
K(XO~ - KO C _ Ko
(Uo)vjjU~21 '
As a result, we arrive at the following solution of eauation (III.17)):
21= I U' U lPat Pii
h x~- ( h� Pi Uo )PA. C Us / Pit +
x
(Pxz (U't ~-v1~ dx. (III.18)
p, U +v21
x. .
Knowing the function K= K(x), let us determine 8= d(x) from expression (III.15);
then using formulas (III.12) and (III.14) , let us find 8* (x) , d** (x) ,'rW(x) . Using
(III.6), it is possible to construct the velocity distribution in any cross section
of the boundary layer.
If both sides of equality (III.18) are multiplied by the ratio
2
~ a~ )2=~3 37 5~ ,
we find the variation of the form parameter fl = K6**2/d2 along the surface of the
body over which flow takes place. Then if we proceed to a homogeneous fluid,
expressian (II.18) is converted to the formula obtained by Loytsyanskiy [21] with
constants a= 0.47, b= 6.25.
Using the Val'ts results, Schlichting [18] recommends that we set a= 0.47, b= 6.0.
The values of the constants a and b obtained by Basi1, Mel`nik.ov and certain other
authors are presented in the monograph 1211.
. In the case of floar over a plate when Jt = Q, vld/vW � const, K= const, equation
(III.13) is converted to the form
133
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~
.u (37 ti ti= d~1-~ .2
2 \:115 ~Jt~i ~ 9(17�~ ~ 11  � (III.19)
Defining the integration constant from the condition 8(0) a 0, on the basis of
(III.19) we find
/
4 iti _ l
~ ~ .c~�~. ~ 3 ~ t'::~
S � / � 37 _ ti _ .
315 945 91172
Knowing the function 8= 8(x), it is possible to use (III.12) and (III.14) to con-
struct the functions 8*(x), S**(x), TWCx),
When investigating a boundary layer in a fluid with variable kinematic viscosity on
the plate, the system of equations (III.4) can be reduced to the following equation
by introducing the current functicn.
1
0~ a=tp v~ a=,p _ a r dl~
a, dz dy - dx* dy `y dY 1
Introduction of the Blasius variables
- i
~ u 1s
V' (Uxvo) 2 , ~ = y ( ,rv, ~
converts (II1.20) to an ordinary differential equatioa
d- v� ( 2 c vo ) t' - o'
which must be integrated under the boundary conditions:
t (0) _ C'(0) 0. C' IE-.. 1
(iia.2o)
(III.21)
The equation (III.21) was integrated by Z. V. Barisova on the "Ural" computer with
tne following kinematic viscosity variation law:
i vuvu g-B-E,
v
(111.22)
The integration results are presented in Table III.1. For putting formulaQ (III.S)
and (III.22) in correspondence to each other, it is necessary to consider the differ-
ence in scales of the variables r1 = y/d and C. For an identical viscosity distribu-
tion law, the following expression muet be observed
B B, b 1/- U
.rv�
(III.23)
The transition from the n coordinate to the E coordinate is made using the equality
. ~
c l, .rvu
U '
134
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Figures III.2-III.S shDw tte houndaxy layer thickness, the ratios d*/a, 6*/$** and
the friction coefficient o� a plate reduced to a wetted surface as a function of
the parameter vW/v0. The exponent in tfie graphs of the indicated figures is taken
ae B1 = 1.
u
d'
s,sr-
6
S
41/ 1 I I I I I I I I 1
0 / Z J 4 S 6 7 8 9 ~U
vW
~u
Figure 111.2. Boundary layer thickness of a plate
as a function of the ratio vW/V0; B1 = 1.
E'
0
9,4
Q? L-_._-1 1_ I I vw
Figure 111.3. Displacement thicknesa as a function
of the ratio vW/v0 ; Bl = 11
135
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G~
a~~ =N
4
3
1~ ~ I %ow
p s f0 ve
Figure 111.4. The parameter S ~ as a function
of the ratio vw/v0 ; B1 = 0.
z uc
r
201
.

1,
USU ? 4. 6 B ~W %0
Vy
Figure 111.5. Frictioa coefficient of a plate as
a function of the ratio VW/Vo.
136
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V
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~
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6
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4
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Figure 111.12. Critical Reynolds number as a
function of the parameter vo/vo (l0; v'~ ;0) .
n
the amplitude o� t.he current function of the disturbiing motian, Schlichting used the
second scheme mentioned in 9 II.I.4 to determine the relation of the density pulsa-
tions to the velocity pulsations, which led to the appearAnce of a singularity in
the total Orr-Souunerfeld equation.
. Surveys of the principal papers on atability of fluid flows with variable density
appear in the generalizing articles 1401, 141J. A bibliography (primarily of Eng-
lish and American authors) on �low o� inhomogeneous fluids can be found in the
Rurvey j42].
158
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In the following study, the problem of the stabili,ty of a latuinar bouadary layer in
- a fluid witfi variable viscosity is s~~olved under tbe assuiaption that tlie velocity
pulsations do not cause density pulsations.
The system of equations of disturbing motion of an incompxessible fluid that is
inhomogeneous with respect to density based on relations (III.3) is written in the
two-dimensional case as follows:
t ~1v
(~Jt + u~x i uu dy dx ~ F "
d11 dv, du� (III.46)
1- ay ( vy a.r ~ ~ . f)vy Juy ii Juy ) = - d ~ � At/y 2 -~y � Jy ~
dx y
dv, dvY
dz ~ dy
In formulas (III.46) u(y) is the velocity distribution of the main f1aw; p� Q(y);
_ u = u (y) , v = const.
Substituting the f ollowing values in equatiorLs (II.46)
t ~o.,-er.
V. ~ vX (y) c �
vy vy (y),�t (ax-br),
= ,
P* (J)et (a.-ae)
excluding the pressure and ir.troducing the current function of the disturbing motion
using the continuity equation, we arrive at the following differential Pxpression:
111-- i-)lY._'+ P L\u a/ip/-u,(p, =
\
lu -1ll~~Q -1 � A4(Q 2 L~ - Q2y')
-I P (4~` -4- ai(P)j .
(III.47)
Let us reduce equation (III.411) to dimensionlesa form by the usual methnd (see 9 14),
continuing to use the previous notation f4r the dimenaionlesa variables u, 0
(u - c) u=q)j - rip" [(ii - c) y' - u'(P1=
uRr I Iv - 2a2~" 1- al(P ~I- l 2E,S
- a2(Pl) (y" -I u2qP) II.48)
T J .
The solutions to equation (III.48) wi,ll he found by the method discussed in g 1.6.
Let us take the solutions of the equation
159
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'~~((P* - rd Ur(F T t'~~ll --'C) (90,
(III.49)
ag the first two solutions, where H s p`d/p will be a constant if the deasity dis-
tribution is defined bp formula (III.29) S= -B.
Expanding the solution of equation (III.49) ia the form of a series
(p (J) ~ (u - c) (gu (J) + alb'i (y) -1- . . . + a~"b', (J) + . . . j.
for the functions gi(y) (see � 1.6) we obtain the folloving system:
~,`~~J) 1' (2 u ' - c + II ) 9u (J) - 0;
bJn (J) + (2 u u' c + N ) bJn (J) gn-t (J)�
= 1, 2, . .
Integrating the indicated equationa successivelp, we fi.nd;
Y
go t>> (J) - ~ ~ 90 (2) (J) (ie - c)-Ze-H" dy;
" K -
gn+, (y) _ -I~ (u - c)-^~~-tiy ~ (u - c)senyb'' (J) dy dy
b
Thus,
i ~(y) _ - c)ngn (y) u; goi
.
T(z) (J) = (it - c) Fj b', (y) as" ;
n=u
9V2 (y) (u - C)-2e-"yay,
the functions gn(y) are defined by relations (III.50) .
(III.50)
For construction of the remaining two solutiona of equation (III.48) let us use the
expansiong of these solutions in the vicinity of the poiat y m p., iatroducing the
new variable (see g 1.6)
J-yK
e '
where e is a small parameter.
Denoting ~(y) = X(rl), let us tranaform (II1.48) _
(11 c) (X' - u=e'X) - esii"X f- H l(u - c) ex' - u'e'X1 =
- ~ , [X1\' - 2u-e"x u4e'~x
uRce-
2I1(e;~" - u't~x') -I. b-p" ~r.~it uYe�x), . (III.51)
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In equation (III,51)
:r - c== rr,;ei) uK (til)" �
Il~ IIK ll~tl~ ' ~ � � . ~
the values of H= dp`/p, 6 2p"/p for the selected law of density variation (III.29)
are constant. The parameter E is determined from the condition that the largest
terms with respect to e in the left and right-hand sides of equation (III.51) of
the same order, that is, the forces of an inertial nature are commensurate with the
forces of viscosity crigin
~ixei~X ^ Reei Xiv
u
Hence ~
L (ulte)~/:s '
By arguments analogous to the case of a homogeneous flui,d, we arrive at the conclu-
sion that the solutions �3(y), ~4(y) are def ined by formulas (1.90), (I.91). Thus,
the density variability in the adopted stateffient of the problem and for an ordinary
degree of approximation of the solution influences only the "invi scid part"of the
general solution of equation (III.48). Using the generally accepted boundary
conditions (see g 1.5) and the approximate equality c= uDyk, we arrive at the
characteristic equation
c1 (uj) z,
(III.52)
~
where - ,lr, eD
2 CI/p I L (ll - C) �~,1J C).: .
J
. ~
1
The integral ~ eI`"(rr - c)--dy will be calculated i.n the same approximation as in
9 1.7, considering the exnansions
2
0(l);
uK (y - JJ" u, (y - Jj
i,rry (,uy" I /3r.irri� (J JJ ~ . . ,
i~�
y^ --B )(111C
(IIIL53)
The complex equation (III.52) is equivalent to two real equations which considering
(III.53) are written as followst
i,u~ u,,
u ll c)''
ll~ IIK (f','
161
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1 1-
(III.54)
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(w) - j�~,,')e�yK - .
(III.55)
uK upi
On the basis of the system of equations (III.54), (III.55), it is poasible to indi-
cate the following scheme for calculating the neutral stability curve coinciding
in its primary features with the calculation schemP for a homogeneous_fluid (see
� I.7). Being given a defined value of wl, by the graphs of ,ti, (u~), v, (w) we deter-
mine ~S. (wi)- ~'i (W1)� Solving equation (III.55) graphically with known left-hand
side !1~ (WO�for the given velocity distribution and density distribution in the
boundary layer we determine the value of the parameter c, which means the values of
yk, uk corresponding to it. Using (III.54), we find
CuoeB ~ uu !iy
u' -(1 -c)= 1-- uK eK-}- n IriCJ . (III.56)
Then we calculate the neutral Reynolds number
fle = 1 ~c,, 3.
a,,K J.,
It is possible to obtain the critical Reynolds number if all the calculations are
made for values of
.
a
Figure III.13. Neutral stability curves of the bound-
ary layer of a plate in a�luid with variable density.
In Figure 111.13 uze have the neutral stability cuxves foz the Blasius prof ile for
different values of the parameter B. Attenti.on is drawn tq the fact of an increaae
in the flow stability in the case of a negative density gradient (B > 0) in the wall
region. Ohviously this fact can be explained using the scheme for the occurrence of
Reynolds stresses discussed in � 1.3. With negative density gradient the magnitude
- of the Reynolds stresses will be less as a result of variabi,lity of p(y), and f or a
positive gradient (i3 < 0), larger by comparison wS,th a homogeneous fluid. A
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decrease in the value of the Reynolds stresses which transfer the enetgy of the main
flow to the disturbances, naturally leads to an increase in the stabtlity. In the
case of an increase in density witfi respect to the y-coordiaate, th: form of the
neutral curve resembles the corresponding curve for a velocity pro':ile with inflec-
tion point, that is, there is an "inviscid instabilitp," and the region of insta-
bility does not disappear for v-* 0. For the upper branch of the neutral curve when
R the parameter c is defined in this case by the condition uk/u'i - B= 0(B < 0),
and the value af the corresponding wave number a can be determini:d from (III.56) for
0.
BIBLIOGRAPIiY
l. S. M. Targ, OSNOVNYYE ZADAr-HI TEORII LAMINARNYKii TECHENIY ~,Basic Problems of
Laminar Flow Theory), Moscow-Leningrad, GITTL, 1951.
- 2. I. I. Ibragimov, V. P. Kashkarov, Laminar Boundary Layer of a Fluid with Vari-
' able Viscosity," TR. SEKTORA MATEM. I MEICHAN. AN KAZ. SSR (Works of the Mathe-
matical and Mechanical Sector of the Kazakh SSR Academy of Sciences), Vol 2,
1963.
3. V. P. l:ashkarov, A. T. Luk`yanov, "Calculation of Flow of a Drop Liquid with
Variable Viscosity over a Plate," PMTF (Applied Mechanics and Technical Physics),
No 5, 1964.
4. A. K. Pavlin, "A Case of Integration of the Equations of Motion of a Viscous
Fluid with Variable Viscosity Factor," PMM (Applied Mathematics and Mechanics),
Vol 19, No S, 1955.
5. S. A. Regirer, Some Thermohydrodynamic Problems of Steady State Uniform Flow
of a Viscous Drop Liquid," PMM, Vol 21, No 3, 1957.
6. S. A. Regirer, "Thermal Effect on Viscous Drag in a Steady Unif orm Flow of a
Drop Liquid," PMM, Vol 22, No 3, 1958.
7. L. M. Simuni, "Numerical Solution of Some Problems of Motion of�a Fluid with Variable
Viscosity," TEPLO I MASSOPE"ctENOS (Heat and Mass Transfer), Minsk, Belorussian
SSR Academy of Sciences, Vol 5, 1963.

8. 0. T. Hanna, I. E. Myers, Laminar Boundary Layer Flow and Heat Transfer Past a
Flat Plate for a Liquid of Variable Viscosity," AMER. I CH. E. JOURNAL, Vol 7,
No 3, 1961.
9. I. E. Cross, C. F. Dewey, "Sinailar Solutions o� the Lamin.ar Boundary Layer Equa-
tions with Variable Fluid Properties," ARCHIWUM MECHANIKE STOSOWANEJ, Vol 16,
No 3, 1964.
10. H. Schuh, "Uber die L"osung der laminaren Grenzschichlgleichung an der ebenen
Platte fur Geschwindigkeite und Temperaturfeld bei veranderlichen Stoffwerten
und fur das Diffusions feld bei Allheren Konzentrationen," ZAMNI, Vol 25/27, No 2,
1947.
163
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04
11. H. Hausenblas, "Die nichtisotherme laminare Stromuug ei.nex zahea l?lussigkeit
durch enge Spalte und KapillarroIiren,* ING.,ARiCHN., Vol 18, No 3, 1950. .
12. E. W. Adams, "A Class of Similar Solutions for the Velocity and the Temperature
Boundary Layer i.n Planar or Axiallr Symmetric Channel Flow," Z. F. FLUGWISS,
No 3, 1963.
13. D. D. Joseph, "Variable Viscositp Effects on the Flaw and Stability of Flaw ia
Channels anci Pipes," THE PHYSICS OF FLUTAS, Vol 7, No 11, 1964.
14. D. Daniel, "Variable Viscosity Effects on the Flow and Stability of Flow in
Fluids," THE PHYSICS OF FLUIDS, Vol 7, No 11, 1964.
15. B. C. Sadiadis, "Boundary Layer Behavior on Continuous Soli,d Surfaces," I.
AMER. I CH. E. JOURNAL, Vol 7, No l, 1961.
16. B. C. Sadiadis, "IIoundary Layer Behavior on Continuous So1id Surfaces," II. AMER.
I. CH. E. JOURNAL, Vol 7, No 1, .1961.
17. K. Pohlhausen, "Zur naherungsweisen Integration der Differential gleichuag der
laminaren Grezschicht, ZAMM, Vol 1, No 4, 1921.
18. H. Schlichting, TEORIYA POGRANICHNOGO SLOYA (Boundary Layer Theory), Moscow, IL,
1956.
19. K. K. Fedyayevskiy, "Frictional Drag of Wings for Various Origins of the Point
of Transition of a Laminar Boundary Layer to Turbulent," TEKIiNIIZA VOZDUSHNOGO
FLOTA (Air Fleet Engineering), No 7-8, 1939.
20. N. M. Matveyev, METODY INTEGRIROVANIYA OBYKNOVENNYKH DIFFERENTIALINYKH URAVIYENIY
(Methods of Integrating Ordinary Differential Equatione), Izd., LGU, 1955.
21. L. G. Loytsyanskiy, MEKHANIKA ZHIDKOSTI I GAZA (Fluid and Gas Mechanics), Moscow,
GIFML, 1959.
22. H. W. Euanons, D. C. Leigh, "Tabulation of Blasius Function with Blowing and Suc-
tion," AERONAUTIC. RES. COUN. CURRENT PAPER, No 157, 1954.
23. Ye. V. Semenov, "A Problem of Hydrodynamic Theory of Stability in the Case of
Variiible Viscosity," IZV. AN SSSR, MEKH. I MASH. (News of the USSR Academy of
Sciences. Mechanics and Machinery), No 4, 1964.
24. Ye. V. Semenov, Development of Waves on the Surface of Films with Variable
Viscosity with Gas Flow over Them," IZV. AN SSSR, MEKIi. I MASH., No 5, 1964.

25. P. C. Drazin, On Stabi~~ity of Parallel Flow of an Incompressible Fluid of
Variable Density and Viscosity," PROCEEDINGS Ok' THE CAMBRIDGE PHILOSOPHICAL
SOCII;TY (MATH. AND PHYS. SC), Vol 58, No 4, 1962,
26. R. C. Di Prima, D. W. Dunn, "The Ef�ect o� Heating and Cooli,ng on the Stability
of the Boundary Layer Flow of a Liqui.d Over a Curved Suxface," JOURNAL OF THE
AERONAUTICAL SCIEi1CES, Vol 23, No ld, 1956, (Russian translation MEKHANIKA
(Mechanics), collection of translations and surVeys, 1Jo 4, 1956.
164
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27. M. Morduchov, "Stahi.litr of Laminar Boundary Layer near a Stagnati.on Point over
an Impermeahle Wall and a Wall Cooled by Hormal Fluid Injection, ADVTS. COMK.
AERONAUT. TECH. NOTES, No 4037, 1957.(RZH MEKHANIKA (Mechanics Reference Journal)
tlo 1543, 1959.
28. L. A. Dikiy, "Stability of Plane-Parallel Flows of a Inhomogeneous Fluid," PMM,
Vol 24, No 2, 1960.
29. G. I. Taylor, "Effect of Variation in Density on the Stability of Superposed
Streams of Fluid," PROC. ROY. SOC. LONDON, SER. A, Vol 132, 1931, p 499.
30. S. Goldstein, On the Stabflity of Superposed Streams of Fluids of Different Dea-

sities, PROC. ROY. SOC. LONDON. SER. A., Vol 132, 1931, p 524.
31. P. G. Drazin, "The Stability of a Shear Layer in an Unbounded Heterogeneous In-
viscid Fluid," JOURNAL OF FLUID MECHANICS, Vol 4, No 2, 1958.
32. J. W. Miles, "On the Stability of Heterogeneous Shear Flows," J. FLUID MECH.,
Vo1 10, No 4, 1961.
33. L. 'q. I{oward, ":Iote on a Paper of J. W. Miles," J. FLUID MECH., Vol 10, No 4,
1961.
34. J. W. :liles, "On the Stability of Heterogeneous Shear Flows, Part 2," J. FLUID
MECH., Vol 16, No 2, 1963.
35. L. N. Howard, "Neutral Curves and Stability Bour.daries in Stratified Flaw," J.
FLUID MECH., Vol 16, No 3, 1963.
36. J. W. Miles, L. N. Howard,"Note on a Heterogeneous Shear Flow," J. FLUID MECli.,
Vol 20, No 2, 1964.
37. L. Prandtl, GIDRUAEROMEKHANIKA (Hydroaeromechanics), Moscow, IL, 1949.
38. G. Lamb, GIDRODINAMZKA (Hydrodynamics), Moscow-Leningrad, GITTL, 1947.
39. H. Schlichting, "Turbulenz bei Warmeschichtung," ZAMM, Vol 15, No 6, 1935.
40. H. Schlichting, VOZNIKNOVENIYE TURBULENTNOSTI (Occurrence of Turbulence), Moscow,
IL, 1962.
41. I. T. Stuart, "Hydrodynamic Stability," LAMINAR BOUNDARY LAYERS, edited by L.
Rosenhead, Oxford, 1963.
42. D. R. F. Harleman, "Stratified Flow," HANDBOOK OF FLUID DYNAMICS," edited by
V. L. Streeter, New York-London, 1961.
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CHAPTER IV. EFFECT OF SUCTION OF A FLUID THROUGH A PERMEABLE SUkFACE OF A BODiY
OP' THE LAN'''NAR BOUNDARY LAYER CHARACTERISTICS
�IV.1. Simtlar Solutions of the Laminar Boundary Layer Equatione in the Presence
of Suction
Solutions of the laminar boimdary layer equations which define the longitudinal
velocity component profiles having similarity properties, that is, for each cross
section of the boundary layer there is a characteristic linear dimension and
characteristic velocity such that the dimensionless profile obtained with their
help is identical for all cross sectioneof the layer, are called aimilar.
The system of equations of a two-dimensional steady-state boundary layer in dimen-
sional form is described on the basis of (I.18) as follows:
v(I au ap v-11
11 ila + U T
y - p).e + v dy'' ' (IV.1)
du du _ 0' vp U.
d.r f ~y ~ dJ
It is possible to find the longitudinal pressure graclient by using the firet
equation of system (IV.1), considering the identity 8p/8y-0. After defining the
boundary of the boundary layer y=d from the conditiona
z'
(hl lu e ~i~" ~>:a where el, e2 are negligibly small, we find
i vpI i op ~~u UU'
1 � J~ u 6-_ p Jr U dx .
(IV.2)
The f unction U=U(x) in the equality (IV.2) is the velocity distribution on the
outer boundary of the boundary layer. .
If we introduce the current function ~(x, y) on the basis of the continuity equa-
tion, the first equation of system (IV.1) is written as follows:
a
a=~ UU, T v ; '

vX ay ax vy (IV. 3)
166
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It is necessary to integrate equation (IV.3) in the case of a permeable surface
under the following boundary conditions (see �I.2):
0) ' J2kp y Iy=0 ~ ti
v (x' 0) - v jV=0 -f: W;
ii J)d- ~ l y~ p U (x).

The procedure for describing the last of the boundary conditione (IV.4) is of a
somewhat formal nature, for equation (IV.3) itself correctly reflecta the motion
of the fluid only in the wall region and, generally speaking, it is impossible to
consider its solution for y-),-. However, the calculations for the special cases of
flow with boundary conditions (IV.4) give satisfactory correspondence to the
experimental result:;, which is the basis for the application of the methods of an
asymptotic boundary layer in practical applications of the methods in which the
condition for y--~- is used.
Flnding similar solutions, accordin g to the definitian let us introduce [1] func-
tions ~(x) and P(ri) such that

y
tl(i~)
ip (s)
as a result we obtain u/U(x)=(D'(n), that is, the dimensionless velocity profile in
the boundary layer does not depend on thP longitudinal coordinate. In the new
variables equation (IV.3) assumes the form
u~ I ( c~~ _ iW },1(I)" -~L (I". (IV.S) ,
dil� ~ ~ - dx '
In order for the function 0 to depend only on the variable n, the following two
conditions must be observed:
U'
- - u = cunst,

I U' COIISt,
v ( ~p'= q-3 ~
(IV.6)
(IV. 7)
which are the equ , (IV.107)
where vo(x) is the local rate of suction of the fluid from the bounds*v layer.
We shall use the second-moment equation in the following form:
di U� U~ ~~I! H,,cf), (IV.108)
dx U U H� ~
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where H is the boundary layer form parameter;
- c=9.54, HO=2.59 and H4=4 are constante.
After substituting formulas(IV.105) and (IV.106) in equation (IV.108) we have
v clRc.. z vU' nc - 2 //4 t'u (IV.109)
U d.r . Re F n Ho ~ Re - a Ho = 0.
Excluding the value of vo/U from equations (IV.107) and (IV.109), we obtain the
differential equation far calculating the local Reynolds number
a r aH4 -H\
~~Et~~~.~ U' n le-2 1
d.~ U o rle a H = 0, (IV.110)
U +B-2 /o )
where
( (nc - 2) + u (b - 2) H, ~I
A--2 Ho.
f/4 .
F B-2 ll� )
Integrating equation (IV.110) for the boundary crnndition Re**=Reo** for ar-xo,
where xo is the coordinate of the loss of stability point without fluid suction,
we find
Re*'"" ex n� in U (X) I X
p I U(x�) ~ tvlH,(a--1)+aH4l
x c(alld -1- (13 2) ff j U(x)exp In u(x) dx Reo .
~ U (x,) ~ } (IV.111)
.
Calculations of optimal fluid suction f rom the boundary layFr begin with the
P.eynolds number Re0** at the loss of stability point of the layer before which
the f low in the layer is stable with respect r_o small disturbances without suction
of the fluid. For known ve-locity distributian at the outer boundary of the
wing section layer U(x) and given values of the foran parameter H, we calculate
Re** by formula (IV.IIl). Obtaining a family of curves fc,r Re** (x, H) and
local critical Reynolds numbers corresponding to each value of the form parameter
H, let us determine the f unction Re**cr(x) graphically. Here it is recommended
_ that the values of Re**cr(H) be calculated by the following formula [21]:
IZeK~, . _ ~~xp (A~ (3~ t11,
Key: 1. cr
where A1 and B1 are constants.
From the zero moment equation (IV.107) we obtain the formula for calculating the
optimal fluid suction rate distribution
Itc� ~
:'u v d I I ~p rf; � (h - 2 � u I
) ,
Q.c (lf -2) /J 2 (IV.112)
Key: 1. cr
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The first term in the righthand side of formula (IV.112) is dPfined from the
differential equation (IV.110). After corresponding transformations we have
v I I d ReK' _ vU' Ko 00
U Q- 2 ke*'l iLr UA B- 2Re"p +
n Iuf/4 -1- H(B - 2)J 1
~ (B 2) ( Ho (8 - 2) + a/l41 Re..
Kp
Substituting this expression in equation (IV.112) and perforniing the required
calculations, we find the formula in final form for determining the optimal dis-
tribution of the rate of fluid suction fram the boundary layer through a porous
surface along the chord of the wing section
c,,, vu ' (b 2- K�) o�� o(I~ - H,) ~ (IV.113)
U' l3 - l R`"N Iflp (B - 2) -F aH41 R!�p Determining the optimal suction velocity distribution by fonRUla (IV.113) using
the approximate method proposed in reference [25], it is possible to calculate all
the boundary layer characteristics and frictional drag of the wing section.
The discussed method of calculating the distribution of tihe optima.l fluid suction
velocity from the boundary layer was developed as applied to plane flows. Using
the known Ye. I. Stepanov transformation [34), this procedure can be extended to
the case of axisymmetric flows.
For the special case of a porous plate expressions (IV.111) end (IV.113) reduce to
an integral exponential function. The zero moment equation (IV.104) will in this
case be written in the form of an integral expression
N
d
dx j it(U u)cly t~�U = ~ � (IV.114)
Let us transform expression ('IV.114) to the follawing
Ilc". d Re" 1 x.
d R,, ' (IV.115)
By optimal boundary layer suction, just as before, we mean the distribution of the
normal velocity component at the surface of a plate where in each boundary layer
cross section the lo cal Reynolds number is equal to its lower critical value, that
is,
R
e(x) - Rcir� (IV.116)
When calculating the lower critical Reynolds numbers by the small oscillations
method, usually the fluid flow stability in a laminar boundary layer is investi-
gated. This method of investigation is very tedious. In addition it is, however,
possible to use the fact that the influence of various suction velocity distribu-
tion laws along the plate onthe flow stability loss in a laminar boundary layer
can be established using the form parameter H. It is known that the value of the
lawer critical Reynolds number can be calculated by the following approximate
formula:
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lZeH = exp (a - 6H).
(IV.117)
The following two pairs of constants are recommended in references [21], [32]:
a=26.3, b=8 and a=29.1, b=9.
With the first pair of constants formula (IV.117) satisfactorily approximates the
results of calculations of the boundary layer stability of wedges and also the
asymptotic boundazy layer on a porous plate (for H from 2 to 2.8). The data from
calculations by formula (IV.117) with the second pair of constants correspond better
to the lower critica]. Reynolds nwnbers for Schlichting sections with boundarq
_ layer numping and for a six-term Pohlhausen polynomial without suction of the
layer (for H from 2.2 to 2.7).
The results of further calculations depend to a significant degree on the assumed
values of the constants a and b. Accordingly, the data from calculations of the
hydrodynamic stability of a fluid flaw in a laminar boundary layer of a porous
plate were analy zed again with entirely differPnt suction velocity distribution
along the plate surface: namely, with uniform suction v0 =const [28] and with
suction by t.he law v-1/T [36]. This analysis made it possible to establish that
in the range of valu2s of H from 2.15 to 2.80 the mentioned data are approxd.mated
with high accuracy (to 2%) by formula (IV.117) for the next pair of constants:
a=31.3 and b=10. The calculations demonstrated that for optimal sur_tion of the
laminar layer on a porous plate this range of values of H app r4ximately corresponds
to the range of Reynolds numbers ReX from 0.�106 to 1000�106.
In �IV.4 it is demonstrated that for arbitrary suction velocity distribution along
a porous plate for C and H the following approxim,:te formulas ;are valid:
~ = Zu - P!**, (IV.118)
i- c[*".
(IV.119)
Here ~o=0.22, p=0.56, H0=2.59 and c=1.18 are constants, and t**;0.
A comparison of the results of the calculations by formulas (IV.118) and (IV.119)
with the data fram numerical integration by computer of the differential equations
of a laminary boundary layer [2] demonstrated that in the range of values of
t** from 0 to -0.5 the maximum error in formula (IV.118) does not exceed 3%, and
formula (IV.119), 1%.
Substituting formulas (IV.116)-(IV.119) in equation (IV.115) and separating
variables, we obtain an ordinary differential equation
( --hc�) r.xp 12 (u - 11,1,h - bct'")1 dt** d Re..
to - (P I I ) l"s ~
Integrating equation (IV.120) by substitution of the variable z
under the boundary condition: for t**=0,
Rc, = Re,,,
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(IV.120)
2G`�"' --26C1**
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Xe ~
f rI-
U~
4, 6
OS
U4
UJ
Ui
u ~U
~
FOR OFF[C[AL USE ONLY
Ur lU `
1
l
Figure IV.12. Ratio Re**/ReX/2 as a
fimction of the suction parameter t**
Figure IV.13. Comparison of different
values with respect to optimal suction
of a laminar boundary layer.
1-- by formula (IV.113); 2-- by
the data of [21]; 3-- by the data of
[32]
we find that
izC., - Rex, ~ bc, ex p [ 2 (a- /,,,n- p`- )J
. t JJ
(l:;(~~~'i'-~ -260#*)-Ei(P~c_,I ll, (IV.121)
where Ei is an integral exponential function.
Inasmuch as [3) ReH*=225, it is easy to find that Re.,=0.115�10E.
Calculating the f unction Re (t**) by
dependence of Re**/vrR_ex on t** (Figu
possible to determine the local flaw
t'u -
U
formula (IV.121) and using the graphical
re IV.12) proposed in reference [37], it is
rate factor
1�"
Hl'*y Comparison of the functions obtained by different authors for optimal suction
velocity on a porous plate vo/Uo for dif'ferent Reynolds numbers ReeUox/v, where
U0 is the velocity of the oncoming flow, is presented in Figure IV.13. The
results of calculations by the proposed formula (IV.113) (curve 1) and also by the
data of references [21], [32] are plotted on this graph (curves 2 and 3,
respectively). In these calculations it was proposed that the constants A1=26.3
and B1=8. Comparison shows satisfactory ~correspondence of the calculation results
by the proposed formula with the corresponding data from rQference [29]. Tlie
obtained results also make it possible to draw the conclusion th at the distrib ution
of the optimal rate of fluid suction from the boundary layer depends to a signifi-
cant degree on the Reynolds number.
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G
S 6 7 B
6y Rej

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1-~, -~'.nar frictionai drap with o~timal boundary layer suction on a porous
pla*_e ;s calculated bv the formula
X u~x
1 ~ 1 L/.i
o ir
wlich, bv expression (Iv.118), is transformed to the following:
i~ ~�A a e,
f e( IZrA
Lj 2P r z'o ) )
~,u
J- (l ~\1~.1�
Considering that for ReX;ReXO : vO/U0 =O and Re**=0.664 Re , in fir.al form we
obtain
i,4�10~ theiinfluence of
axisymmetric elongated bodies with Reynolds
individual forms of the body on the a�OOftslotsuanddsufluid cked fluidof o~woratesthe
ding
slots is sma11� The obtained numbe
required for boundary layer laminarization differ little from the fO~rBppr ximate
values for a p13te with longitudinal flaw over~e~ fo~wingsnana'bodies of revolu-
estimation of the parameters of the suction sys
tion, it is possible to use the relations obtained for a plate.
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The basis for the calculatians was the Blasius profile which is presented in i
Figure IV.18 in dimensionless coordinates U/Uo, y/a**. The studies of [43] ~
demor.strated that the velocity profile in the indicated coordinates on the p].a:.e
behind the slot very quickly assumes the initial form during suction, then develop-
ing as on an impermiable surface. Therefore, according to the solution of [47],
let us propose that the velocity profile on the entire surface remains unchanged
in dimensionless form, except the pulse loss thickness varies in the vicinity of
the slot, which is defined as the diffe~ence where d** ia the pulse loss
thickness of the profile ABC (Figure IV.18) be?ore the slot, Bdg** is the pulse
loss thickness of the profile AlDBC after the slot. It is considered that the
velocity profile filled out as a result of fluid suction from the boundarq layer
ca.n be obtai.ned, dropping tYe lower part of the initial profile characterized by
the y-axis yp. Thus, -
i u (IV.141)
i
= u, ( - Uo ) ~y;
" (IV.142)
aw f
o Vo l~ Ue)~J�
The amunt of fluid pumped through the slot per second is defined here by the
expression u,
Q= b~ rt drj = Q(Jo), (IV.143)
or in dimensionless form
Q U (y�) y� (IV.144)
Uob~ - f( aB 1�
1 1
The function (IV.144) borrowed from reference [47] is presented in Figure IV.19.
0 0,1 0,10,3 0,40,5 0,60,74 8 0,91,0
U
u.
Figure IV.18. Dimensionless velocity profile on a plate
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'jt
,.,~C � B

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If boundary layer suction tak6s place in such a way tnat an increase in S** i.n
the interval between slots is entirely compensated bq suction, then, using
formula
a+# 0,664 Y' , (zv. i4s)
0
which is valid for the Blasius profile, it is possible to find
1 bu 1 ' i'� "uo (IV.146)
S v,a *I -.r� - `o,sca 1) V - ( 0,664 or l~oS (Reu - (Re~~
v (Zes = 0,441 ' (IV.147)
For description of the boundary layer suction, the dimensionless flow rate is used
Q v. Ren
'�Q U'S _ tluS Uu ReS '
where vo is the mean velocity of the fluid in the slot,
h is the slot width,
S is the distance between slots.
n
p 1? 3 4 ~ 6% b yIu
yu

e
Figure IV.19. Amount of sucked fluid as
a function of the degree of suction
Figure IV.20.
layer suction
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(IV.148)
Slot form for boundary
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1
Rh
Kh

h ~e
b, 0
S, U
40
!0 ?,0 J,u v,u � .,u
yu

e
Figure IV.21. Slot width as a runction Figure IV.22. Radius of xounding of the
of degree of suction and angle of inside edge as a function of the degree
inclinatton of the slot of suction and angle of inclination of
the slot
B}� visual observation of the nature of the flow in a sl.ot (Figure IV.20), in
referen ce [48] the relations were obtained which insure attached flow of the fluid
through a slot of maxLmum width h(Figure IV.21) and minimum radius of rounding of
the entrance edge r(Figure IV.22) as a function of the angle or inclination of the
slot and degree of suction of the fluid from the boundarq layer. Attached fluid
flow in the slot is necessary in connec,tion with the fact that separation of the
bouadary layer. at the entrance to the slot b locks the slot uaeful cross section,
decreasing the flow rate, and it is an addition al source of disturbances in the
layer. The experimental relations h= h( ry.�. in the range
\ bv ~ \ �e 1
of 40�3�106 ii is recommended in reference [21] that we take A=0.765�10
The parametec 6 and the value Re** uniquely related to it are calculated by the
method of :=ucceasive approximations. As the first approximation it ie possible to
take eitiler the function 6(x) calculated for the case vQ=O or the function obtained
under the assumption of constancy of the complex (vo/U) Re**=E=const. The choice
of e+ in the first approximati_on is defined by the conditions of the specific
ca3 culatioi, in particular, the suction or injection velocity distribution law.
For determination of the form parameter H, according to Truckenbrodt, we use the
equation obtained on subtraction of the pu1_ae equation (IV.29), the terms of which
are multiplied by H, from *he energy equation (IV.30),
i L~� (VI.19)
I-lU 1. 2t~- ll 9 I- (I tl) ~ .
d.~ U J.r
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After certain algebraic transformations and introduction af a substitution
r!!I
(11-I)H
(VI.19) is converted to the form
U(.~) 2c _ p1t 1n u_ K(Re*�, y) ip (H) U Re*:.
lix dx
whe re
(2co-11 ~ ~1) i-H
Re�*", y (H) - � (f~ = i)
l~ (Re**, ll) _ - ~H _ Il
(VI.20)
(VI.21)
can be approximated by linear relations (the form parameters H, H and Y. are
Lmiquely related on the basis of (VI.4)):
({Ze�� H) = a (l - b), (VI. 22)
V (11) = - (cl d)� (VI. 23)
In (VI.22) and (VI.23) the coefficients have the following values:
3,04 � 10-=, b- 0,07� Iglle** - 0,230, c= 9,52, d= 1.06.
After linearizatian of the ftmction K and the equation (VI.21) becomes a linear
first-^rder differential equation
~It �(u I - c;) d lnu -ub-d~ (VI.24)
0 1 dx f 0 the common integral of which is the expression
I x x ( x
exp - I P(.r) dx 1(Xl) 9~x) eap ~ I p(z) d.r~ dx ,
~ A� . au ~X~ 1
u ~ C~ U' Gb - lIs io ~*;Fr~
where P - ' 9 : : l~ U ' t . ~ R~
are known functions of the variab le x, for the function A(x) , and, consequently,
also Re**(x) (see formula (VI.11) has already been defined as a result of
integration of equation (VI.S). It is possible to determine the initial value of
R(xl) by the graph of R=Z(H) (Figure VI.13) constructed using expressions (VI.4)
and (VI.20), for the magnitude of H(xl) at the begilning of the permeable section
of the surface over which flow takes place is found as a result of calculating
the TBL on an impermeable wall to the point x-xl. The calculation basically ends
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with determination of F':) and Q(x). All the remaining TBL characteristics such
as Re**, H, H, T0 /pU2 are related to A and Q and are calculated by fortnulas
(VI.11), VI.4) and by the graph presented in Figure VI.13.
For the case of distributed suction or injection with constant velocity imder
plate conditions (u=U,,.; r0-1), equation (VI.15) can be integrated in closed form.
Here, considering the "initial" condition Re**=0; x-=0, it assumes the form
1zi~.~ (V:C. 25)
!Ze�~,~_F~ _ q Re, ~ Uw J Re*'~" dRe.r
0
(here ReX U.x/v).
L
U
0,10
/L
-p;", -
4
n
Figure VI.13. The function Q(Fi)
Single differentiation of ttie components of (VI.25) with respect to ReX leads to
tiie equation
llc, d Itc, (n I)~ U_- IZe~
l~ (GI.26)
1
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the integral of which (for n=1/6) obtained by the method of separation of
variables can be written in the form
Re"i7/6 7 z' 6 15 20 ~
lZex = ,y (s - 1)T ( 6 - 5 z� 4 z~ - 3 zs
2 r" - 6a 1 n z-f- 2409
where 2- I-~- U.
fle**'/b; A== 0,765� 10'=; 0,521 � 10-=.-.
(VI.27)
For vo/U-~0 and, consequently, zil (VI.27) reduces to the expression that follows
from the known Faulkner equality [26]
Re " 0,0153 fte;'7. (VI. 28)
The local friction coefficient is determined from the pulse equation, which for
the given case has the form
ZI d Re' � vu
2 - d Re,, Ud '
Hence, considering (VI.27) we have
~f - �a A Z
- U. [2 (n -f- 1) P t - I
(VI.29)
(VI. 30)
For vo->0, this exnression becomes the expression given by Faulkner
= 0,00655 � fle- 1/6= 0,0132 (Ze.r 1/7 , (VI. 31)
The results of calculating the value of &g by formulas (VI.27)-(VI.30) for the
given v0/U.,,and also the corresponding results of the experimental studies of [1]
are presented in Figure VI.14.
Equdcions (VI.S) and (VI.19) can also be used to investigate the other special
case of control which permits creatian of flow with characteristics close to
separated �low in the entire controlled section of the TBL. This result can be
reached using distributed injection by a defined law. Considering-that the TBL
form parameters in this section are constant and have valuas of H~Hsep and H=HS , and the tangential frictional stress on the wali is close to zero, from
(VI.~I~) we obtain the following expression for the i-njection velocity distribution
law ~1) ~
vo n�� iiu 2 (v1.32)
U (I/oTp - I) U l~X Re"" (I - HoTp) .
(1) (1)
Key: 1. sep
For the special case of a plate (dU/dx-0), from (VI.32) we have
It4~ "11 _ 1~_ _ ~ = const. (VI. 33)
U ~ Horp - I
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}Oa OIFFIciAt. t~RF ONtX
If we assume that the distributed injection velocity is defined in the first
approximation by formula (VI.33), it is possible to find the fuaction 6(x) in
the same approximation
x~ }
x
A U2O+3/0n +1 dX
//oTp - 1 s
x~ ~x) = U2n+3~n+1 0
Key: 1. sep
where 9(x), Re** and d** are uniquely related to each ether. Substitution of
their values in (VI.32) makes it possible to calculate the distributed injection
velocity in the first approxi mation.
The Eppler method [28] belongs to the group of inetLoda following the basic princi-
ples of the paper by Truckenbrodt. The initial equations of the method are also
the pulse and energy equations. Additional semi-emvirical relations, as before,
inc.lude the functions ~g(Re**, H); cD(Re**, H) and H(H). However, the form of
these relations is somewhat altered, beginning with the specific requirements which,
in Eppler's opinion, they must satisfy in the case of suction. Eppler considered
the necessity for expanding the region of apnlication of the above-mentioned func-
tions in the direction of values of HO) leads to protraction of separation.
The methods of calculating a controlled turbulent boundary layer can be generalized
ro the case of a rough surface by the V. F. Droblenkov procedure [37].
When U'(x)>0 it is possible to use the Loytsyanskiy formulas [36] to calculate the
uncontrolled TBL.
Methods of Solving &.e Integral Relation of Pulses for a Controlled Turbulent
Boundary Layer. For complete solution of tihe problem of a turbulent boundary
layer with injection and suction of a fluid it is necessary to define the values
of Re**, z and H=d*/d**. Previously formulas (VI.68) and (VI.70) wexe obtained
which characterize tha values of z and H as:funetions of the form parameter A,
the Reynolde number Re** and the known injection and suction parameters Up/U
ar.d vp/U.
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ThP integral pulse relation which in the case of a uniform nonsteady controlled
boundary layer assumes the following form can be used as the additional equation
_ relating three unknowns Re**, z and H:
J~x* + u ~X (26*~ + a�) + uy a (JS*> _
- (vi
2! ~(1- . io2)
Here 2TO ` 20 7 (V�)=
(VI.103)
After transformations, it is possible to rewrite equation (VI.102)
a ~c** - l (H 1) Re** = ReU x
(VI.104)
x ~ 2 U (1 - U� ~ Us -
Jt
Here
U'= `w � U= u .
ax ' V o
For solution of the pulse equation (VI.104), we use the method of successivE
approximations of K. K. FEdyayevskiy and A. S. Ginevskiy.
Assuming the values of ~f and d* to be known in the first approximation and taking
; t as the parameter, after integrating (VI.104), we obtain
~ vo / i~. ) L ~((f� b+)l X
Ilc*`~ IZc j U I 2 1-- Ul d x,
U. 
A, (vz. ios)
;c cw l:ei� e_(Dlr1�
Here
- ~J (VI.106)
CP
U
The subscript "t" denotes values at the initial point of calculating the controlled
turbulent boundaxy layer.
In *_his case, if the fluid is injected in the entire turbulent boundary layer
regian, then the point of transition of tLie laminar boundary layer to turbulant
is taken as the initial point of the calculation. For a completely turbulen-,
boundary layer Ret*=0; xt=0.
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r
When solving the problem of a nonsteady turbulent controlled boundary layer, the
following functions are assumed given
dU dU (x, f) .
dl dt '
uo = uo(x); vO vO (a),
the number Re=UOL/v and also the contralled boundarq layer parameters at the
initial point in time t=0 are assumed given also. Then using expressions (VI.105)
and (VI.70), on the basis of successive approximations we find the boundary layer
parameter distribution along the body at various points in time. The successive
approximations calculation technique for a nonsteady controlled turbulant boundary
layer does not differ theoretically from the analogous method for a nonsteady
tmcontrolled boundary layer [34].
The results obtained for a plane, nonsteady controlled boundary layer can also be
generalized to the case of mution of a body of revolution with zero angle of
attack if we consider that the boundary layer t'nickness d is appreciabYy less than
the radius of curvature of the transverse cross sectian of the body of revolution
rp (x) .
If
U U.(x)U,.. (1). (VI.I07)
then
00 U,,, U' (.c), "j = U (x) vU,,, (VI.108)
dt Jl
For a solid of ravolution, formula (VI.105), considering (VI.107) and (VI.108) for
a completely turbulent controlled boundary layer assumes the form
where
ro=ro(x) is the radius
The r.emaining formulas
body of revolution.
.'ru
- x.
Ilc:r .u IlCV a~ I !'u~l (�C~ r ~
0 (VI.109)
(U,,,J') -
ji c'P d.r r~' �c-'~' ~
6)
A
~N ~X, i) d.r; cvi. iio,
� u
of transverse cross section of the body of revolution.
of a plane boundaYy layer retain their form also for a
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For the special case of a steady turbulent controlled boundary layer, the integral
relation in formula (VI.102) at3sumes the form
X~ ~ U' r 2 + a~ 1 S** ~ + ~ ( 1 - 'U
l 1
� (VI.111)
or
)d Re" ll~ sr . Yj uO ( po
~l.C +-:1~- ( I+ H) IZe Re U~ Z U 1- ~
(VI.112)
- Equation (VI.112) is solved by the method of successive approximations, assuming
Cf=TO/p/2U2=2x2/zZ and H known in the first appromdmation.
As a result of integration of (VI.112) we obtain
Ile*` - Ile ~ U I z., + U I!-- u} I ev")dz +Iler' e-W(x)�

~ (VI.113)
Formula (VI.113) is simplified when the parameter H differs little along thF outline.
- In this case
ul ~~r�hi~ Ul
~ u (H + 1) d ~ u d lr - i
cu~ c� ~ _ U
and formula (VI.113) assumes the f.orm
u
I A = -
Re J Ull-F-' I I,
r
' (VI.113a)
~ dx U"~ lZet �
The controlled turbulent boundary layer craracteristics are calculated by the
method of successive approximations. The values of zH, ReH*, HH pertaining to the
case of an uncontrolled turbulent boundary layer are taken as the zero approxima-
tion. The method of determining all of the uncontrolled layer characteristics was
presented above. The first approxtmation ior a controlled turbulent boundary
layer i:; outained as follows. Let us cslculate Q by formuia (VI.72), setting
3U/8t=0 in it. By the values of zH, Re**, HH, zO kr_cwn from calculating the zero
approx.imation, let us determine the for~ parameter A in the zero approximation by
the formula (VI.71)
t R~..
z� Iteu vo
Ao= :.-Q -I U
�
x H01
By formula (VI.73) we find the value of thz torm parameter f in the zero approxima- ~
tion 0. :
fo - - Aofl� ( 1� ) .
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zz gl aad H**1 in the
By the knov.�n values of fo and ReH*, we obtain( ~i~,l ~
roximation. Then by the knrwn values of z=zl, H-II1, by formula (YI.113) tion.
firgt app
or (4I.113a), we dtt~ a~~tersWA value
o-f areedeterminedf insthe first~PP
~'he values of the fa p
tion by formulas (VI.71) and (VI.73):
� :
i~ Re; v Rct � zo
) .
Al = ~ H.. Q ~ ZI
~
The controlled turoulent boundary layer characteristics can be calculated in the
second appraximation analogously.
Calculation of a Turbulent Boundary Layer with Injection and Suction fac~g~)aforl
As an example let us consider the special case ooff~~ fover luidainlthe turbulent
which the injection velocity (suction ve~1~o~~~Y~d ,.const.sume For the
the boundary
boimdary layer is constant, that is, v0
form
layer of a plate the basic equations (VI.111) and ( �66) as
~U 1 ! (Vi.114)
'Ix u \ 1 ~ U ~ '
:_e (VI.115)
L,-0.5 the possible velocity profiles are presented in
Figure VII.7 as a function of the value of the parameter to.
The velocity profile for relatively small values of the ventilation parameter is
presented in Figure VII.7, b. This velocity profile, just as the initial one
for 00=0, has a range of variation of the C-axis where the value of 0"(&):0.
With a further increase in (D(0) the velocity profile assumes the form presented
in Figure VII.7, c. This velocity profile is attached and exists in some range
of values of 00. With further increase in the parameter 00, a profile is formed
as presented in Figure VII.7, d.
Let us note that the velocity profiles presented in Figure VII.7, a and b with
return current were obtained on the b asis of the boundary layer equations which
are not applicable in the separation region.
At the same time, the profile presented in Figure VII.7, c is attached,and the
boundary layer equations are applicable to it.
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r~ to
zro ~~o~
6
07
oll
?
1.0
Figure VII.6. Effect of gas injection on frictional drag reduction
o) ~ b) ~ c)
~ - _
~
~ - -
_ ~ -
~ -
- -
~r
Figure VII.7. Velocity p'rofiles for 00, but small; c) and d) attached profiles
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�VII.4. Integral Exnressions for a ltao-Phase Bouadary Layer
Let us consider a two-phase boundary layer conaisting of a wall gas laqer with
density pl and an outer layer of liquid with density p2 (see Figure VII.2).
The Prandtl equation for a two-phase boundary layer can be written in the follaw-
ing form:
a a (uu) ' du i at
vx + ay - U. dx + P,:, ay '
(VII.120)
Hereafter, 3ust as earlier, the index "1" will denote values pertaining to a gas,
and the index "2" the same values for a liquid.
From the continuity equation (VII.43) it is possible to write the following
identity
d (Uu) + v (Uv) dU
Jx dy _ ~l dx '
(VII.121)
Subtracting equation (VII.120) from eqnation (VII.121), we obtain
~r 111(U 11)I + vu ~u (U - u)j (VII.122)
dii i az
~ rf.e (U - Let us find the integral relation for the boundary layer as a rahole. For this
purpose let us integrate equation (VFI.122) from 0 to the thickness of a two-
phase boundary layer d=dl+d2, assuming that far y-0, usup and v=vp.
As a result, we obtain
tlll
-W- J ~ ~i (U - u)~ cl - t~u ~U - u~) J(U - N) dx `!y -
~
u
6, e (VII.12 3)
I dJ - J ~j~ (1Y.
u 61 Equation (VII.123) can be transformed if we consider the known notation
b (VII.124)
U j t-~)dJ,
1 0
0
and consider that
� (VII.125)
J vy dy-==~-TO; J-~ct,=o-s,=-T,.
o e, Substituting (VII.124) and (VII.125) in (VII.123), we obtain the integral relation
of the pulses for a two-phase boundary layer as a whole
ii , , dU z~ ' r l ' i 1 (VII.126)
~Ix (V I U dx 6* - vn (U -1l0) = Pt - T, \ P~ - Ps /
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The integral relation of the pulses for the outer bounda.ry layer will be obtained
as f o l lows .
Let us assume that the normal to the interface of the gas layer and liquid layer
makes a small angle of ddl/dx. Therefore when compiling the integral relation of
the p ulses it is possible to neglect the value on the order of (d6l/dx)2, for in
this case the error is within the limits of accuracy of the assumptions made when
compiling the boimdary layer equations.
He re af ter, when calculating the liquid layer parameters, the transverse coordinate
is reckoned from the liquid-gas interface.
Integrating equation (VII.122) from 0 to d2, we obtain
d y ! I "v~ -I-

dU I d[
(U dy.
u
After transformations, equation (VII.127) assumes the form
,
d
(U'6;') U
~.r
He re
b,; J(1 - dy; b:_' J-~ (1 - dJ�
(VII.127)
(VII.128)
(VII.129)
For an inte rnal boundary layer (gas layer) the integral relatian can be obtained
in two forms.
It is convenient to use the first form of the integral relation in the case where
the velocity distribution is given in the form u/U1 (U1 is the velocity at the
boundary of the gas and liquid layere); the second form, wiien the velocity profile
is given in the form u/U.
In order to obtain [he first type of integraZ relation let us write the continuity
equation (VII.121) in the form
v (Uiu) d (Ulv) ~Ui
_ az ~y- Jx '
Subtracting equetion (VII.120) from equation (VII.130), we obtain
ax (u(U,-�)1 + ay [r,(U,-u)l +
~iu, au dul i aT
a .
+-~ix (U,-u)-+-~U ~X -"U, dX ) P-1 9
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(VII.1?b)
(VII.131
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On the basis of the theorem of differentiation of an integral with respect to a
parameter it is possihee to obtain s~
1 ~x lu(U,-�))dy= ax j�(ul-�) dy-
8, b.
- I( (Ul - tl) tlbl 1 ~ ab0
IY= e, .
dx ~ U,iy=6, dx
In our case when y=d1 and u(U1 u)=0, for u=U1, in addition, d60/dx=0(60=0).
CJ
Thus, we have
e, a ~
J
J d.c l~~ (U, dy dx �(V i- it) dJ.
~ u
(VII.132)
Integrating (VIi.131) from 0 to dl, we obtain the integral relation for the gas
layer
!U
UI dxl A' - vu (Ul - No)
~~u Uflg_o the cavity intersects the bottom line (afjs_0. The value of tre parameter fsflo.0 has the meaning of a limit.
The limiting value of fk or, in other words, the limiting value of the Froude
number Frk=1/ k, for givet speed of the ship Uo corresponds to the limiting cavity
length ~K H -~Hsr.ce, it follows that etien in the investigated
f 1. I'rK
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case with given speed of the ship, a thin cavity of length exceeding the limiting
length cannot be obtained.
Cavities of limiting length are of the greatest interest from the pojnt of view
of using them for drag reduction of a sl:iP inasmuch as these cavities are closed
(6=0) and, consequently, can be ohtainad for very sma11, theoretically zero air
flow rates. The drag of thin appendages behind whicfi these cavities are foYmed
must also be very small ( ~ux ~ 0) .
p-u
Cs ~
n ~ oc
~n
l, l
U 1,0 2,0 JO 4,0 3,0
n =10
~
o--o n -10
n
-a=40
Ot
Figure VIII.3. Curves for the variation of CX/aland S/a as functions
of the parameter f(n is the number of selected calculated points)
A cavity of limiting length corresponds to the limiting cavitation number
ok=crjs_o and the limiting relative thickness of the cavity (nmax)ka(nmax)6=0'
These parameters make it possible to find the gas pressure in a cavity of limiting
length and the maximum thickness of the cavity.
The calciitations demonstrated that variation of the parameter fi (fl=fal) from
zero (a1=0) to infinity (a1=��) , that is, varlation of the Froude number wi.th
respect to the appendage length Fra UO/ gal f:om infinity to zero leads to a mono-
tonic increase in the limiting Froude number along the length of the cavity Frk
from a value of 0.425 to 0.657. The limiting Froude number Frk as a function of
the parameter Fl=gal/U20 and also the parameters of limiting cavities (nmaX/aR)k
and (Q/a)k as a function of the satae parameter are presented in Figure VIII.4.
The application of a modified Ryabuahinekiy system permits not only determination
of the values of the limiting Froude number Frk for the entire range of the ratio
of the appendage len gth to cavity length al=a1/R, but also establishment of the
detailed characteristics of thin cavities of limiting length.
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~ 71 nr~a
~ ot l
10
0,5
6
a
Figure VIII.4. Limiting Froude numher Frk and parameters of the
limiting cavities nmax/aR and v/a as a function of the parameter
f1=ga1/U20
Comparison of Theoretical and Experimental Results. The 2.bove-presented theoreti-
cal data on artificial cavitiea formed on a flat horizontal bottom were checked out
experimentally in a flume with plate equipped with washers 161. The plate was
945 mm long, 200 mm wide and 20 mm thick. The rear tip of the appendage was 315 mm
from the foxward edge of the plate. The cavities were formed by injecting air on
the bottom surface of the plate behind wedge-shaped appendages.
During the air in3ection tests, the follawing va'Lues were measured: the flow
velocity in front of the plate (by a Prandtl tube), the pressure in the cavity (a
micromanometer), the air flow rate (Venturi tube). The geometric p arametera of the
cavities were determined visually, and also by photography. The tests were run in
oncoming flow velocity range of 0.5 to 0.7 m/sec.
Appendages with the following dimensions were tested: 1) a1=100 mmn; a=0.1;
2) a1=50 mm, a=0.12; 3) a1=25 mm, a=0.12; 4) a1-12 mm, a=0.167.
Observations of the cavity development with an increase iii air flow rate confirmed
the main theoretical conclusion regarding the exd.stence of a cavity of limiting
length.
The experiments demonstrated that for very sma11 air flow rates the cavities had a
profile resembling the theoretical profile of limiting cavities. Increasing the
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air flow rate by 20 times led to an increase in the air taken from the en3 of the
cavity region by two loops leading to the side plates, but in practice had no
influence on the geometric dimensions of the aavities. Further significant increase
in the air flow rate led to an increase in cavity thickness and loop thicknesd.
Figure VIII.S shows a comparison of the relations for the values of
Vo Li w:ic !Iw:,c
and as a functian of the parameter fl=gal/U~ obtained
theoretically and by the experimental data for liffiting cavities. In Figure VIII.S
the solid lines are used for the theoretical curves, and dotted lines for the
experimental curves. Experimeatally obtained cavities, the maximum thickness of
which almost do not change at all with an increase-in the air flaw beginning with
almost zero values of the air flow, were considered limiting. Data are preseated
in the same figure foY an appendage a1=12 mm (aal=2 mm).
From Figure VIII.S it is obvious that with the exception of thickneas, according
to the theoretical and experimental data, all of the remaining basic geometr.ic
parameters of limiting cavities (length and width) corresponC to each other. The
difference in the theoretically and experimentally obtained thiclaness of the cavity
becomes significant only for cavities which are long by comparison with the appendage
dimensions, and the appendage dimensions themselves are small. For a small
appendage tlie influence of the surface tension forces is significant. Thus, the
theoretical proposition that the cavity surface is tangent to the appendage surface
is not properly realized.
n~ uT
OC R ?
frn a
07 7
I! 6
0, 5 5
0, 4 y
O,J J
1
u, l F I
o c
uiua
slt ~u,
-0,
-D,
-0
-o,
frA
00
�
~
~ .

! t ~
'
Imas

/ rr rll
~

- -
- - ~
Y
3
~
f~ -

~ ~
�
�
- - ~
'

o 0
�
- ~ - --J~~-
-
TC00lIs

o O
ff, ~ 1170MM, Of � O7
j o �
� �
e =SU +~���Ol~
�
. �
n 73��,���0,12
mrrs
o a
of u, o,.O IPMM
`
.Q. .p.
max ~ - o, 167
au.,..~ .
i
2
a
4
Figure VIII.S. Theoretical and experimental relations for Fr ,
nmaX/aal, nmax/o(a1+R) as functions of the parameter f1=ga1/UO
Key: 1-- theory; 2-- experiment
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u)
100 ;iu 0 X (MI4)SU 100
' ?0
y~MM~
b, P,a7,39; a G=-?,S?; 4=0,04. liter~sec
?00 ISO 100 SD 0x MhQ l00
, 70
2,61; a G _ --2,77; q= 0,24 liter/sec
c) 100 150 100
0 x(MM)

_ ?0 t
(J~MM)
Q�J,78 liter/sec
G d)
,z (hra) /DO
i 100 SO D~.--- SO
~
I 20 '
-I y(""')
P,=2,39;
e)
I 750 ?00 /SO J00
?0
= 7,59; K G- -S,S y(MM)
~ Figure VIII.6. Theoretical and experimental picture of the development
of a cavity with an increase in air flow rate Q: a, b, c experimental
profiles; d, e-- theoretical profiles '
Accordingly, it is of interest to compare the outlines of the cavities obtained
experimentally and tlieoretically for large-sized appendages. A comparison of this
type is presented in Figure VIII.6. Here a, b and c are exnerimental ouClines,
d and e are theoretical outlines. Cases a and b correspond to d, and case c
_ corresponds to e. It is obvious that there is a satisfactory corresQondence
between the theoretically and experimentally obtained values. Thus, all of the
basic theoretical conclusions are qualitatively confirmed by the experimental
results.
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�VIII.3. Effect of Limitation of the Flaw on the Shape and Dimensians of an
Artificial Cavity Formed on the Lower Side of a Horizontal Plate
Solution of the Linear Two-Dimensional Problem. In order to consider the effecC
of limitation of the flow on the form and dimensions of an attificial cavity,
~ according to V. B. Starobinskiy [10], [11], let us consider plane flow of an
ideal, incompressible fluid, but one with weight, in a strip of width H extending
infinitely in bath directions (Figure VIII.7).
Figure VIII.7. Diagram of a cavity in a limited flow .
In the investigated case the problem of cavitating flow around an appendage is
solved under the same assumptions and boundary conditions as for the case o� an
unlimited fluid. In addition to the boundary conditions (VIII.7)-(VIII.12), it is
necessary to consider.an additional boundary conditian at the ',tottom of the body
of water
-I j, O fori/ --c-,�:..r.," ; o ~ it is possible to note that the value of aCx in deep
j,(/~~1.li pl/i)Llf
water at a speed of UO=1.40 m/sEC differs insignificantly from the value of ACX
under shoal conditions at a speed of U0 =1.15 m/sec. Here AR=R-RQ=o the difference
between the model drag with air lubrication and drag for Q=O.
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irM ~ U0
L 4(~
i
.
�-Q-0,4niceK
e-Q.fO (1)
~-a�4,o
(
I
I

~
r

V a
�
~o
I

I i
~ I
0,S
UJ 0,4 0,5
frh& Uo
9(IR'?a)
Figure VIII.19. Experimental dependence of
Uo
Fr� = 1~81N - TI on Frh =I,g (!K 20) .
Key:
1. Z/sec
9
8
~
6
S
4
J
?
1
00, S /,0
R,Kt (a)
!0
I
,
T �?,0.
~
Q.p
K- Q.4,On/C
CK
~
~
~q
Q -!,0
o- Q -0,4
I
/
~
f
~
i
I

I
I
~
I I
I
i
l,3 I
Uo, M/crK (b )
Figure VIII.20. Towrope drag curves for H/T-2.0.
1-- without lubrication; 2, 3, 4-- with lubrication
Key:
a. lt, kg : b m/sec
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nc,r�IO"
0,?
0.1 9,7
1
2
1 -

Ud Q9 !,0 1,J J,Z I,J 1,4 /,J
, w/ceK (a)
Figure VIII.21. Effect of shoal water on drag reduction
- 1-- for H/T=2; 2-- in deep water
Key:
a. m/sec
�VIII.S. Estimation of Drag Reduction of a Ship Equipped with a Device for
Creating Artificial Cailties (Gas Filme; on its Bottom by Model Test Data
Let us discuss an approximate method of calculating the drag reduction of a ship
with a system of gas films arranged on its bottom. This method is based on the
assumption that the effectiveness of the device providing for frictional drag
reduction of a ship will be unchanged for a model and a full-scale ship with
' identical Froude numbers [27].
~ By the efficiency of the device creating the gas film system on the bottom of a
j model, we mean the ratio
k AR�, (ViII. 77)
- nRo .
Here ARm is the experimentally obtained gain with respect to drag;
A is the ideal gain with respect to frictional drag which can be obtained by
ca culating if we assume that the appendage drag is equal to zero and that the
air coN rs the entire flat part of the bottom.
The value of ARO is constant for given velocity, and it is found as the difference
between the frictional drag of a plate with a length equal to the distance between
the foYward and aft perpendiculars of the modelrand beam equal to the distance
between the side kePls and total drag of two plates separated by a solid gas film.
'Ihe resutts of calculating the efficiency k=ORm/DRp for the 461 B barge model are
presented in Figure VIII.22. It is obvious that the highest afficiency of the
device corresponds to speeds close to the calculated speed for which the gas films
cover the greatest Part of the bottom. It is obvious that the efficiency k, whi.ch
contains the experimental value of AR, in the numerator takes into account the
inf.luence of the appendage drag on the dfficiency of the device and also the
inf luence of escape of air from the gas films.
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X~� N~.
M
\1
L ~14!1 4 ~3
"f,0 1,10 !?6 1,JQ I,40 i50 uo,a/ceK ~2~
, Modenu
Figure VIII.22. Efficiency of air lubrication for the 461 B barge
model
Key:
1. 1J0, m/sec, full-scale
2. Uo, m/sec, model
An approximate estimate of the gain with respect to drag of a full-scale vessel can
be made by the formula
eR knR, I~b ` Uirsu� (VIII. 78)
Key: 1. rough
Here ARf is the "ideal" gain with respect to frictional drag calculated for a full-
scale vessel;
Sp is the bottom area in the vicinity of the parallel middle body of a ship covered
with air;
~rough is thP roughness factor of the skin of a ship's hull.
The value of ARf is calculated under the same assumptions as for the case of an
"ideal" gain ARO on a model.
~ yo
(a)
S,0 uo, M/4ac (b )
Figure VIII.23. Gain with respect to drag.
1 calculation; 2 full-scale
Key :
a. km/hr; b. m/hr
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Thc results of calculating the expected absolute gain urith respect to drag
aR with respect to the bare hull towrope drag of the 461 B barge are presented in
Figure VIII.23. It is obvious that the expected drag reduction with respect to
the towrope drag of the barge (curve 1) is 22-32% in the barge speed range of
3.5-5.0 m/sec (12.5-18 km/hr).
A full-scale test of the Offectiveness of the drag reduction of a barge equipped
with a device creating a system of air cavities on its bottom was performed in
the Kuybyshev Reservoir on the Volga in May-June 1965 [27].
Figure VIII.24 shows th e results of full-scale testing of the 461 B barge in the
form of the towrope drag curves for the barge. The middle curve corresponds tQ
the bare hull towrope drag in the presence of a pusher boat, the lower curve was
- obtained when pushing the barge with simultaneous injection of air on its bottom.
The upper curve corresponds to the towrope drag with appendages with which the
narge is equpped to create an air lub rication system.
From a comparison of the curves presented in Figure VIII.24 it is obvious that the
application of air lub rication on the bottom of the 461 B barge led to total drag
reduction of the barge on the order of 23-25y of the bare hull drag in the speed
range of 11-18 km/hr.
N, Kz (a)
%UOU -
6UOU
3000 -
annn - - ~
I
~
/
JUUO
~pOU
fUUU ,
U'
J
._._..__..4
S ua, M/ceK (b )
Figure VIII.24. Results of full-scale testing of the 461 B barge
1-- hull with appendages; 2-- bare hull; 3-- hull equipped with
an air l:lbrication system
Key:
a-- R, kg; b-- m/sec
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Hence, it is possible to draw the conclusion that the full-scale tests demon-
strated the efficiency of using air lubrication for drag reduction of flat-bottomed
heavy-cargo river vessels.
Figure VIII.23 shows a graph of the drap, reduction reduced to thP bare hull drag
(curve 2) constructed using curvES presented in Figure VIII.24, obtained as a
result of analyzing the full-scale tests. Camparison of curves 1 and 2 demn-
strates satisfactory correspondence between the calculated value of the gain with
respect to drag and its value obtained from the full-scale testing in the speed
range of 11-18 km/hr.
�VIII.6. Res ults of Investigating Artificial Cavities Created on the Bottom of
Planing Vessels
In the preceding sections a study was made of the prob lems pertaining to the case
of creating artificial air cavities on the flat bottom of slaw superships having
lar.ge parallel middlebody. The application of air lubrication for these vessels,
in which the frictional drag is a significant part of the total drag, turns out to
be most efficient. Another class of ship in which it is nossib le to expect drag
reduction of the water as a result of artificial cavities is planing vessels.
Planing vessels have frictional drag of about 60% of the total drag. The experi-
ments performed by a collective of authors directed by I. T. Yegorov [23], [24]
demonstrated that it is possible to achieve effective frictional drag reduction on
shallaw-draft planing vessels by injecting air behind the step, that is, by creating
an artificial cavity on the bottom.
For planing vessels the condi tions of creating air cavities differ significantly
from the conditions for slow superships.
In the investigated case of a high-speed vessel the cavity parameters are signifi-
cantly influen ced by the conditions of motion, closeness of the free surface of the
water, variab le trim of the vessel, bottom shape as a whole and center of gravity
position.
Results of a Theoretical Study of Artificial Cavities Created on a Planing Surface.
In order to discaver the laws of formation of an artificial air cavity on a planing
surface A. A. Butuzov solved the two-dimensional linearized problem of cavitating
flow in the case of cavity formation on a body planing over the free surf ace of a
fluid with weight [20]. This probleri is solved under assumptions with respect to
the fluid and the nature of its motion made far the solution of analogous prob lems
presented in �VIII.2.
The planing surface (Figure VIII.25) CDEE'HJ' consists of three rectilinear segments
CD, DE, HJ and the section EE'H of arbitrary shape located inside the cavity. The
free surfaces ahead of and behind the planing body can be cansidered as surfaces
of cavities of great extent, the pressure on which is given (v=0). Accordingly,
they can be taken as bounded, semi-itfinite, horizontal plates A'A and KK'. These
surfaces have been introduced in order not to investigate free surfaces of infinite
length, the introduction of which leads to great difficulties for numerical calcula-
tions.
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~
c~ c K . f' y.,~nr(~'). P'PR
/K /
pt
P-Pu H_`~
ba!'1'~ yD~ P'Po
At A a G I ~ r.~,{ Q yl ~I Ju? D3 aKra
po Va Ko I k, i Ra~ ~ Ra. ~ Ko,:l R~ a Kr.7 Ko,~ ~a - rtn n~
+ I /
(x 00) t
9 9
b)
~
c)
L ~ .
~ ~ ~ -
y,~r,~I~.1 y~~n,(s),p�No
N -uo
~ t~.. R R

x
Po. "o kv,~_
9
Figure VIII.25. Diagram of a planing surface with cavity:
a-- general diagram; b-- simplified diagram; c-- system without
a cavity.
- The vertical position of a planing body and the position of the fictitious hori-
zontal surface IQC' must be determined from solution of the problem inasmuch as
they depend on the given pressure on the f ree surfaces.
In Figure VIII.25, BC, GH and IK denote fictitious planes, the in�lination of which
is selected so as to insure smooth closure of the free surfaces AB, JI and the
cavity surface EG. The distar.ces between the points A and B, J and I are assumed
to be selected quite large, and the dimensions of the plates BC and GH so small
that the adopted schematization of the flow insures absence of influence on the
cavity parameters and pressure distrib ution with respect to the planing surface,
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excluding small vicinities of the points C and H. The fictitious plates GH, JI
are equivalent with respect to their purpose to the fictitious bodies used for
- closure of cavities in the Ryabushinskiy type systems. The sum of the horizontal
components of the pressure forces acting on the plates BC and GH corresponds with
respect to magnitude to the spray drag of a planing surface with cavitq.. Let us
present a solution of the stated problem according to the paper by A. A. Butuzov.
Derivation of the equations of the problem does not differ from the derivation
given in �VIII.2. The equations are a system of inteProdifferential equations with
respect to the contours of the free current lines y=n`i)(x) (i=1, 2, 3, see
Figure VIII.25a):
df iIc'~ (.c) dt ~ In I ~`'~.t x ( ~r -f-
ic ~--x Ri,t-x
.
; IRO. I
1
1-11k ~T ~ n l Rs, r- x l a~ -1
Ru, t, 1 - x
j Inl I?,.~-x I(a*, -a) .
R,,. s - z
He re
ko,k0. is observed.
It must be noted that the given flow system is also realized when the vortices are
replaced by discharges.
The flow parameters for a=0.0278 were calculated by formulas (VIII.92)-(VIII.100).
The resiilts of calculating Rk and Y as a function of v for xf=5.0 and yf=0.2 are
presented in Fip,ure,VIII.31. As is obvious f.rom the p,raph, for any cavitation
number, except Q=O, circulation Y is realized for which the vortex at the point
(5, 0.2) closes the cavity smoothly.
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l" r
Figure VIII.31. Results of calculating flaw parameters
However, when replacing the vortex by a noncavitating profile the range of cavita-
tion numbers for which the cavity is smoothly closed is significantly more narraw
~ and is defined by the relation
i ymb
' YC2tf 1-I-o'
' where b is the chord of the foil reduced to the web length of the wedge;
' Cym is the maximum value of Cy of the f.oil for which cavitation is absent.
The cavity length varies sharply only near a=0.
BIBLIOGRAPHY
1. Ryzhov, L. M. "Problem of Drag Reduction of a Shio by Ir.jecting Water under
the Bottom," GIIVT [Works of the Gor'kiy Institute of Water Transportation
Engineers], Vol 8, 1940.
2. Shanchurova, V. K. "Influence of Water Injection under the Bottom of a Ship
on Water Drag and Speed of the Vessel," RECHNOY TRANSPORT [River Transportation],
No 10, 1958.
3. Fedyayevskiy, K. K. "Frictional Drag Reduction by Varying the Physical
Constants of the Fluid at the Wall," IZV. AN, OTN [News of the USSR Academy of
Sciences, Technical Sciences Division], No 9-10, 1943.
4. Fedyayevskiy, K. K. rrictional Drag Reduction by Varying the Fluid Density
at the Wall. Industrial. Aerodynamics," No 24, LOPASTNYYE MASHINY I TECHENIYA
V KANALAKH [Blade Matars and Flows in Channels], Oborongiz, 1962.
5. Loytsyanskiy, L. G. "Variation of the DraE of Bodies by Filling the Boundary
Layer with a I'luid with Different Physical Constants," PMM [Applied Mathematics
and Mechanics], No 1, 1942.
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6. Butuzov, A. A. "Results of Theoretical Analysis of the Parameters of Cavities
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KORABLYA, MAY 1966 (KRYLOVSKIYE CHTEIJIYA) [Reports at the 16th. -Conference on
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7. Butuzov, A. A. "I,imiting Parameters of an Artificial Cavity Formed on the
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[News of the USSR Academy of Sciences, Liquid and Gas Mechanics], No 2, 1966.
8. Pernik, A. D. PROBLEMY KAVITATSII [Cavitation Problems], Leningrad,
Sudpromgiz, 1963.
9. Tulin, M. P. "Supercavitating Flow Past Foils and Struts. Cavitation in
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10. Starobinskiy, V. B. "Detached Flaw of a Fluid of Finite Depth Over a Thin
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izd. Transport, 1964.
11. Starobinskiy, V. B. "Problem of Frictional Drag Reduction of Vessels Under
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of Such Vessels," DOKLADY K XV KONFERENTSII PO TEORII KORABLYA.
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(Krylovskiy Lectures)], NTOSP, No 64, 1965.
12. "Vorrichtung zur Verringerung des Reibungs - widerstandes von Schiffen mit
einem unter dem Boden angeordneten Luftkissen," RNAT FINE GRAM (DANEMARK).
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PROIZVEDENIY [Tables of Integrals, Stmos, Series and Products], Moscow,
Fizmatgiz, 1962.
14. Sedov, L. I. PLOSKIYE ZADACHI GIDROrMKHANIRI I AERODINAMIKI [Two-Dimensional
Problemg of Hydromechanics and Aerodynamics], Moscow, Fizmatgiz, 1952.
15. Lavrent'yev, M. A.; Shabat, B. V. METODY TEORII FUNKTSIY KOMPLEKSNOGO
PEREMENNOGO [Methods of the Theory of Functions of a Complex Variable], Moscow,
Fizmatgiz, 1958.
16. Muskhelishvili, N. I. SINGULYARNYYE INTEGRAL'NYYE URAVNENIYA [Singular
Integral Equations], r-ostekhizdat, 1946.
17. Ivanilov, Yu. P.; Monseyev, N. N.; Ter-Krikorov, A. M. "Asymptotic Nature
of the M. A. Lavrent'yev Formulas," DAN SSSR [Reports of the USSR Academy of
Sciences], Vol 123, No 2, 1958.
18. Monseyev, N. N. "Asymptotic Methods of the Narrow Strip Tyue," NEKOTORYYE
PROBLEMY MATEMATIKI I MEKHANIKI [Some Prob lems o.f Mathematics and Mechanics],
Novosibirsk, USSR Academy of Sciences, Siberian Division, 1961.
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19. Yakimov, M. L. "Approximgte Formula for Tension During Conformal Manping of '
Regions Having a Narrow Segment," SIBIRSRIY MATEMATICHESRIY ZHURNAL
[Siberian Mathematics Journal], Vol 3, No 6, 1962.
20. Butuzov, A. A. "Theoretical Analysis of Detached Cavitation Created on a
Planing Surface," DOKLADY R XVII NAUCHNO-TEKHNICHESKOY RONFERENTSII PO
TEORII KORABLYA [Reparts at the 17th Scientific-Technical Conference on
Ship Theory], Leningrad, TR. NTOSP [Works of-the Scientific and Techriical
Society of the Shipbuilding Industry], No 88, 1967.
21. Birchhoff, G.; Sarantonello, E. STRUI, SLEDY, I RAVERNI' [Jets, Wakes and
Cavities], Moscow, izd. Mir, 1964.
22. Migachev, V. I. "Symmetric Jet Flow Over a Wedge in the Presence of Vortices,"
TR. LIVT, No 113, 1967.
23. Basin, M. A.; Yegorov, I. T.; Isayev, I. I.; Rramarev, Ye. A.;
Sadovnikov, Yu. M. "Peculiarities of Using Gaseous Media to Alter the Aydro-
dynamic Characteristics of Solid Bodies Moving in a Fluid," ANNOTATSII
DOKLADOV II VSESOYUZNOGO S"YEZDA PO TEORETICHESKOY I PRIKLADNOY MEKHANIKE
[Annotations of Reports of the Second All-Union Conference on Theoretical and
Applied Mechanics], USSR Academy of Sciences, 1964.
24. Isayev, I. I.; Sadovnikov, Yu. M. "Study of the Possibility of Creating
Artificial Cavities on Planing Surfaces," Leningrad, TR. NTOSP, No 88, 1967.
25. Yegorov, I. T.. "Methods of Increasing Lift and Increasing the Lift/Drag Ratio
of Lifting Foils Under Detached Flow Conditions and in the Deep Stage of
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26. Butuzov, A. A. "Artificial Cavitating Flaw Behind a Thin Wedge Placed on the
Lawer Surface of a Horizontal Wall," IZV. AN SSSR, MEKHANIKA ZHIDROSTI I GAZA
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