Table Of ContentSEPARATION OF FLOW
BY
PAUL Κ. CHANG
Professor of Mechanical Engineerings
The Catholic University of America, Washington, D,C., U.S.A.
4
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Preface
THE problem of the separation of fluid flow is one of those viscous flows
which is very important not only for science, but also for practical applica
tion. An attempt has been made to compile references up to date mainly in
the fields of basic physical processes, analyses, and experiments covering the
whole area of interest, giving credit to the original source wherever possible.
The content of material has been used for a graduate course at the Catholic
University of America, Washington, D.C., in 1962-3, also for a special course
at Escuela Tecnica Superior de Ingenieros Aeronauticos, Ciudad Universi-
taria, Madrid, Spain, in 1964-5, where the author conducted classes as a
Fulbright lecturer. Although the material covering the practical application
is limited, this monograph is useful for engineers as a reference.
The unquoted bibliographies have been added in the list of references at
the end of each chapter.
The author wishes to thank the US Air Force, Office of Scientific Research,
for grant AF OSR 80-62, 52-63, 52-64, which made it possible to compile
this monograph under the guidance of Mr. Milton Rogers, Chief of Mechan
ics Division, Capt. Hyman H. Album, Capt. Lucius P. Gregg, and Major G.
Stalk. The Rev. Everett F. Briggs of Monongah, West Virginia, kindly edited
my manuscript. Mr. Richard M. Hartley, of the David Taylor Model Basin,
has rendered most valuable assistance in reading my manuscript, in coopera
tion with Mr. Michael J. Malia of the same institution. Dr. Hyok Sang Lew
of the Catholic Universitv of America, and Mrs. Anne A. Ziegler, as well as
Mr. Louella A. Chatfield, assisted in arranging and typing the manuscript.
To these ladies and gentlemen the author wishes to express his gratitude.
PAUL K. CHANG
Washington, D.C.
List of General Symbols
A wetted area or frontal area
a velocity of sound
CL lift coefficient
Cj^ drag coefficient
Cf skin friction coefficient
Cp pressure coefficient
Cp specific heat at constant pressure
D drag
d diameter
Η shape factor of boundary layer, Η = ٧*/θ
h heat transfer coefficient or height
Le Lewis number
Μ Mach number
Nu Nusselt number
η index of power
Pr Prandtl number
ρ pressure
q dynamic pressure
Re Reynolds niunber
Τ temperature
t time
u streamwise velocity component
V velocity component in >;-direction
w velocity component in z-direction
X coordinate, streamwise direction
y coordinate perpendicular to x-direction
ć coordinate perpendicular to x- and >?-direction
α angle of attack
γ ratio of specific heats
٧ thickness of boundary layer or shear layer
0* displacement thickness of boundary layer
ε eddy viscosity
θ momentum thickness of boundary layer
μ dynamic viscosity
V kinematic viscosity
C-SF 2 xvii
xviii LIST OF GENER AL SYMBOLS
ń density of fluid
τ shear stress
rp steam function
Subscripts
e conditions at the outer edge of boundary layer
max maximum
min τηίηίπΐΐιττΐ
s separation
wall
eo undisturbed free stream
CHAPTER I
Introduction to the Problems of Flow
Separation
Symbols
dm unsteady mass exchange per pulse
A' height
κ turbulent mixing coefficient, κ = also κ = (ό—ό* — ο)/(δ— ό*)
L length of body
Λ index of power
ρ pressure
R reattachment point
S separation point
stagnation temperature with no heating
effective recovery temperature
x' distance from neck to transition location
ao ao = lim —
γο tan yo = lim w/q
A extent of fluctuation
Ap Δρ = p-p^
λ non-dimensional function, A(|) = -21—
αξ
r shear stress
φ angle from stagnation point
2 SEPARATON OF FLOW
Subscripts
Β base
Ο just upstream of separation
Superscripts
— average
The subject of fluid flow separation is one of the many aspects of viscous
flow, which is very important but complicated. Because of flow separation,
energy is lost. In cases of external flow at subsonic speeds, such as in airborne
vehicles, the stream line deviates, the drag increases, the lift decreases, and
reverse flow and stalling occur. In the transonic speed range, control and
structure problems are created by flow separations. For cases of internal flow,
separation can cause reduction in efläciencies. The optimum performance of
fluid handling devices such as fans, turbines, pumps, compressors, etc., can
only be predicted with accurate understanding of flow separation, because the
separation occurs just prior to or at maximum loading. The successful opera
tion of the simplest and most conunon devices can depend also upon flow
separation, as, for example, the throttling action in household water faucets.
Flow separation can also be useful in engineering applications. For example,
a thin airfoil which is suitable for high-speed flight may be made suitable for
low speeds by separation of the flow. If the flow is allowed to separate over
a portion of the upper surface and then reattaches and remains attached, a
very thick pseudo-airfoil results. This thick airfoil is better suited for low-
speed flight.
Another example of desirable flow separation is that caused by placing a
spike in front of a blunt body traveling at supersonic speed. Because of the
presence of the spike, the flow may separate on the spike and form a cone-
shaped flow region in front of the blunt nose. Because of this conical separated
flow region, the shock wave will be changed from one nearly normal to an
oblique one, which reduces the head drag considerably. Escape capsules and
other stores, which must be ejected and recovered from high-speed vehicles,
improve their performance by utilizing flow separation.
Components of high Mach number vehicles, engines, nuclear reactors, re
entry vehicles, etc., operate in high-temperature environments. There
fore, problems of heat transfer are incorporated into the problems of hydro-
and aero-dynamics. Then, heat transfer problems and flow separation prob
lems, each sufficiently difficult separately, must be considered in combination.
As a problem area, flow separation has been worked on by many scientists,
but much work still remains to be undertaken in this portion of the field of
fluid mechanics.
INTRODUCTION TO THE PROBLEM OF FLOW 8ΕΡΑΙ^\Ή0Ν 3
1. Mechanics of Flow Separation
The problem of flow separation is, perhaps, one of the most important
hydrodynamic problems to be investigated intensively in order to find its solu
tion satisfactorily. Because of the complexity of the problem, a rigorous
definition of flow separation and stall should be made. The classical concept
of flow separation is due to viscosity; therefore it is often expressed as
"boundary layer flow separation" or "boundary layer separation". In addi
tion, a necessary condition for flow separation is the adverse pressure gradient.
The details involving these two factors are discussed later in this chapter.
FIG. 1. Flow separation over a smooth surface
FIG. 2. Flow separation over discontinuous surfaces
In a general sense, according to Maskell's concept of separation, flow
separation is inevitable for flow over the finite dimension. Flow will separate
from the solid body surface at the trailing edge and also upstream of it if the
required conditions are met there. The flow separation is not only caused by
a gradual process, which is the case of flow over a smooth surface, but also
by a severe discontinuity of the tangent to the surface (Figs. 1 and 2).
First, the classical concept of flow separation is outlined; then the general
ized definition of flow separation is discussed.
1.1 Qassical concept of onset of flow separation over a smooth
curred body surface
The problem of flow separation is as old as that of boundary layer theory.
Ludwig Prandtl, the father of the boundary layer concept, was concerned
about flow separation before he started his work on the boundary layer. As a
young engineer at the Maschinen Fabrik Augsburg-Nürnberg (MAN),
SEPARATION OF FLOW
Prandtl found that the computed pressure recovery could not be achieved in
actual diffusers. He was occupied for 3 years in figuring out why and how
the flow separations and pressure losses were caused. This problem was solved
finally by his new theoretical concept of the boundary layer [1]. His concept
FIG. 3. Flow in a sharply diverging channel [4]
FIG. 4. The boundary layer is sucked away on both walls; the flow is from left
to right [4]
may be referred to as classical in comparison to the modern development of
separation of flow theory.
The classical concept of flow separation is formed for two-dimensional and
axisymmetric steady flow. Prandtl [2] states clearly that the necessary condi
tion for flow separation from the wall is the increasing pressure in the stream-
wise direction, i.e. positive (or adverse) pressure gradient along the flow
INTRODUCTION ΤΟ THE PROBLEM OF FLOW SEPARATION 5
path (Fig. 3). This statement holds for compressible as well as incompressible
flow. Therefore, it may be said that, in general, flow separation occurs under
adverse pressure gradient and with laminar or turbulent viscosity effects.
If one of these two factors is missing, then the flow does not separate.
For example, by removing the boundary layer the viscosity effect is elimi
nated. As shown in Fig. 4, Prandtl removed the boundary layer along the
wall of a subsonic channel by suction.
In front of the throat, the pressure decreases in the direction of flow because
of the decreasing cross-sectional area of the channel. In this region, the pres
sure gradient is negative (or favorable); hence, flow adheres completely to
FIG. 5. Free stagnation flow without separation, as photographed by
Föttinger [4]
the walls. However, behind the throat, because of a large degree of divergency
or a sufficiently large degree of adverse pressure gradient, the boundary layer
separates from the wall and vortices are formed. But, if the boundary layer
downstream of the throat is removed by suction, the flow reattaches to the
surface. Föttinger [3] provides other experimental evidence to confirm the
two necessary factors which cause flow separation; namely adverse pressure
gradient and viscosity.
Figures 5 and 6 show fluid flowing against two separate walls which were
placed perpendicular to the flow direction.
One wall was plain, the other equipped with a thin protruding plate. With
the plain wall, there was no separation, but with the protruding plate, the
flow separated. The physical phenomena may be explained as follows.
In front of the stagnation point, a considerable pressure rise occurs along
the direction of flow, but flow does not separate because of the absence of
wall friction. Near the wall, the flow does not separate either, since the fluid
