Displacement effect of the laminar boundary layer and the pressure drag

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Title:
Displacement effect of the laminar boundary layer and the pressure drag
Series Title:
NACA TM
Physical Description:
43 p. : ill. ; 27 cm.
Language:
English
Creator:
Görtler, Henry, 1909-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Drag (Aerodynamics)   ( lcsh )
Laminar boundary layer   ( lcsh )
Genre:
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The displacement effect of the boundary layer on the outer frictionless flow is discussed for both steady and unsteady flows. The analysis is restricted to cases in which the potential flow pressure distribution remains valid for the boundary-layer claculation. Formulas are given for the dependence of the pressure drag, friction drag, and total drag of circular cylinders on the time from the start of motion for cases in which the velocity varies as a power of the time. Formulas for the location and for the time for the appearance of the separation point are given for two dimensional bodies of arbitrary shape.
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by H. Görtler.
General Note:
"Report date October 1951."
General Note:
"Translation of "Verdrängungswirkung der laminaren grenzschichten und druckwiderstand." Ingenieur-Archiv, vol. 14, 1944."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779843
oclc - 94226492
sobekcm - AA00006208_00001
System ID:
AA00006208:00001

Full Text
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'72 2. ( 2 3-77 '7 W"


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1315


DISPLACEMENT EFFECT OF THE LAMINAR BOUNDARY

LAYER AND THE PRESSURE DRAG*

By H. Gortler


1. INTRODUCTION


The following considerations refer to two-dimensional, not neces-
sarily steady laminar movements of an infinitely extended fluid without
free surfaces, for very small friction, about a body immersed in this
fluid; they deal with the mutual influence between the boundary layer
developing in the proximity of the body and the outer flow which is
practically free from friction.

We denote by X and Y the coordinates of a Cartesian right-hand
system of the plane fixed with respect to the body, by x and y the
customary boundary layer coordinates defined in a zone near the wall
(that is, an orthogonal right-hand system where x signifies the
wall-arc length measured from a fixed contour point, y the wall distance
measured positively from the wall toward the fluid).

Furthermore, R(x) is assumed to be the radius of curvature of
the wall which we define in the manner that has become customary for
boundary layer investigations, in contrast to the usual mathematical
definition, as positive for walls which are convex with respect to
the fluid.2 Finally, u and v are assumed to be the velocity
components of the boundary-layer flow, U and V the corresponding
components of the potential flow about the body in x or y direction,
U and V the velocity components of the potential flow in X or Y
direction.

*"Verdringungswirkung der laminaren Grenzschichten und
Druckwiderstand." Ingenieur-Archiv, vol. 14, 1944, pp. 286-305.
'The contour of the cylinder cross section in the flow plane
should, if no other requirements are made explicitly, be of continuous
curvature and fre .of multiple point singularities; it may go to infinity.
2In order to have in the defined range of x, y a reversible
one-to-one relation between X, Y and x, y, the requirement must
be made that there R + y # 0. This does not impose any further
limitation, however, since according to presupposition in the boundary
layer region |y/R << 1.






NACA TM 1315


If the well-known boundary layer omissions are permissible, the
general hydrodynamic equations of motion are reduced to the following
boundary-layer equations:


ou 6u 6u 1 aP 2u
+ u + v = +v (l.la)
Ot ox oy P ox 6y2


L-+ = 0, (l.lb)
ox 6y


where t signifies the time, p the constant density, and v the
constant kinematic viscosity of the fluid. Furthermore, p = p(x,t)
is the pressure impressed on the boundary layer by the outer flow.
Insofar as the outer flow noticeably deviates from the regular potential
flow about the body, p(x,t) must be determined on the basis of special
considerations or be taken from pressure measurements. In the present
report, we consider only cases where the potential-theoretical pressure
distribution in first approximation is sufficient for the boundary-layer
calculation according to equations (1.1). In this boundary-layer theory
approximation, the potential velocity prevailing at the "outer edge" of
the boundary layer is replaced by the potential flow prevailing about
the body itself, thus by U(x,0,t), and the pressure term in
equation (l.la) is calculated from Euler's equation of motion with
U(x,O,t) a Uo(x,t) as


1 p 6U0Uo Uo
S = + Uo --. (1.2)
7-+
P Ox at 0 x


The boundary condition required in addition to the adherence conditions
u = v = 0 at the wall y = 0 is that U assume asymptotically for
increasing y- Re the value3 Uo(x,t). (Re is the Reynolds number
formed with a characteristic length and a characteristic velocity.)

The described omissions, correct in the limit for indefinitely
increasing Reynolds numbers, are only an approximation for the large
but still finite Re-values for which laminar flow is plainly still
possible. Particularly the connection between the boundary-layer flow
thus calculated and the outer potential flow, assumed to be undisturbed,
The imaginary line, denoted above, in the customary manner of
expression, as the "outer edge" of the boundary layer, runs at the
distance from the wall y = 8(x,t)("boundary-layer thickness") at
which u practically attains the value Uo(x,t) for the accuracy
requirements in each case.






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does not satisfy continuity. Thus the question arises: What actual
course does the outer, practically frictionless flow which is modified
by the boundary layer take? This problem and the related one regarding
the calculation of the pressure drag will be discussed below. The
considerations will be applied to an example in all details.


2. DISPLACEMENT EFFECT OF THE BOUNDARY LAYER ON THE OUTER FLOW


We assume a body which, from a state of rest, is set in motion
relative to the surrounding fluid. In this case, the boundary layer
gradually developing from the start of the movement will be very thin
at first, so that for sufficiently small times of movement the regular
potential flow about the body itself, as outer flow, may be taken as a
basis for the boundary layer calculation in good approximation. The
well-known investigation concerning the origin of the boundary layers
(H. Blasius ), and later extensionsSo are based on this assumption.
However, with increasing times t the boundary layer increases, for a
finite Reynolds number, up to a noticeable thickness; in general, it
will influence the outer flow by mass displacement away from the wall
to a corresponding extent. For a corresponding improvement of the
customary boundary-layer theory approximation sketched in section 1,
one must therefore consider the mutual influence of the increasing
boundary layer and the outer-flow. It suggests itself to attain such
an improvement in first approximation by using at first the displacement
effect on the basis of the original boundary-layer calculation, and
hence determining a correspondingly corrected course of the outer flow.
In a second step one would then have to calculate the boundary-layer
flow anew, taking the above new outer flow as a basis. However, it will
generally not be possible to carry out this improved boundary-layer
calculation, using the new outer pressure distribution, according to the
old scheme (equations (1.1) with the boundary conditions formulated
there in the text), since the curvature effects, so far neglected in
the boundary-layer equations, generally contribute amounts which can no
longer be neglected within the scope of such an improved boundary-layer
calculation. We shall return to this later on (section 4).

4Blasius, H.: Grenzschichten in FlUssigkeiten mit kleiner Reibung
(The Boundary Layers in Fluids with Little Friction). Dissertation
Gottingen, Zeitschrift fur Mathematik und Physik 56, 1908, p. 1.
(Available as NACA TM 1256.)
-Boltze, E.: Grenzschichten an Rotationskbrpern in Flissigkeiten
mit kleiner Reibung. (Boundary Layers on Bodies of Revolution in Fluids
with Little Friction). Diss. G6ttingen, 1908.
Goldstein, S., and Rosenhead, L.: Proc. Cambridge Phil. Soc. 32,
1936, p. 392.






NACA TM 1315


The improved flow calculation aspired to is of decisive importance
for the numerical determination of the pressure drag7. This becomes
particularly obvious, if one visualizes a body of finite dimension
transferred from a state of rest into a state of rectilinear-uniform
movement relative to the surrounding fluid of infinite extension. For
this terminal state the potential-theoretical pressure distribution, so
far, in first approximation, taken as a basis for the boundary-layer
calculation, does not yield any pressure drag (d'Alembert's paradox).
Thus, small as the changes in pressure gradient at the body caused here
by the boundary layer may be relative to the potential-theoretical
pressure gradient, they alone constitute the pressure drag.

We want to emphasize here once more that, as we discussed before
in the introduction, we do not include in our considerations flows
where, for instance, by pronounced separation of the boundary layer,
the outer frictionless flow is considerably transformed compared to the
potential flow about the body. Of course, precisely such processes are
of foremost importance for the originating pressure drag in many flow
problems. Such processes cannot be included within the scope of the
following considerations which are based on the boundary-layer theo-
retical approximation. However, we should like to refer here to a
related report by M. Schwabe8 where the pressure distribution after
completed boundary-layer separation is determined according to an
empirical formulation for the example of the circular cylinder set into
rectilinear-uniform motion from a state of rest. The space taken up by
the pair of vortices developing on the back of the cylinder after the
separation is determined by observation and then simulated by calculation
by superposition of a suitable time-dependent source-sink flow on the
ordinary potential flow about the cylinder. Dhe then obtains outside
of this space a streamline pattern which corresponds well to actual
conditions, and a pressure drag caused by the nonsteady acceleration
fields.

Furthermore, we want to point out that J. Pretsch9 developed an
approximative method for theoretical determination of the pressure drag

pressure drag" = the component in direction of the movement of
the resultant pressure force on the entire body surface, also called
(less correctly) "form drag". Pressure drag + friction drag = total
drag (presupposing non-existence of free surfaces).
8Schwabe, M.: Uber Druckermittlung in der nichtstationaren ebenen
Str6mung (On the Determination of Pressure in a Nonsteady Two-Dimensional
Flow). Diss. Gottingen. Ing.-Archiv 6, 1935, p. 34.
9Pretsch, J.: Zur theoretischen Berechnung des Profilwiderstandes
(Concerning Theoretical Calculation of the Profile Drag). Diss. G6ttingen,
Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Available as
NACA TM 1009.)







NACA TM 1315 5


of profiles in a steady flow by taking the displacement effect of the
boundary layer into account. In a procedure similar to ours he pre-
supposes that no separation of the boundary layer, or only a slight one,
takes place; in the steady case this signifies, however, a limitation
to slender profiles at a small angle of attack so that here the total
profile drag is due in the greatest part to surface friction. Our
consideration, related in the basic idea, refers at the outset to non-
steady flows as well and aims chiefly at the problem of the drag origin
in movements of cylindrical bodies of arbitrary profile from a state of
rest for small times after the start of motion, naturally without being
limited to these movements. Also, it follows a different method.
Whereas Pretsch makes the additional assumption that the friction losses
in the wake behind the body may be neglected, probably satisfied in
good approximation in view of his presupposition quoted above ("Bodies
of Small Drag"), and then is able to determine, within this scope, the
pressure drag from momentum considerations in a simple and general
manner without actually having to calculate the corrected outer potential
flow in every single case, we abstain from this or a similar simplifying
assumption because we set ourselves a problem of a different type.

In the following considerations we shall first deal with the first
step of problem formulation, namely, the correction of the outer poten-
tial flow by consideration of the displacement effect (resulting from
the boundary-layer flow calculated in first approximation), and shall
discuss a few questions arising in case of nonsteady conditions. After
that the second step which leads to the calculation of the pressure drag
will be treated in general and carried out numerically on an example.


3. CORRECTION OF THE OUTER POTENTIAL FLOW


First, one has to find an appropriate measure for the displacement
effect of the boundary layer on the outer flow. For that purpose the
well-known boundary-layer theoretical length presents itself which is
called "displacement thickness" and denoted by 5*.

The formulation of the boundary condition for u(x,y,t) for
indefinitely increasing y e was based on the conception that the
potential theoretical velocity distribution for the large (though still
finite) values of the Reynolds numbers of interest may be regarded
practically as constant in first approximation for the very small thick-
ness of the boundary layer and may, therefore, be put equal to U(x,0,t).
The definition of the displacement thickness 5* is likewise based on
the conception of this streamline approximation [U(x,y,t) = U0(x,t)
and thus for reasons of continuity V(x,y,t) = -ybU0o/x].






NACA TM 1315


The quantity 5* is explained as follows: The fluid volume trans-
ported at a fixed point x at a fixed time t between the wall y = 0
and the outer edge y = &(x,t) of the boundary layer in unit time
.8
equals u dy. The potential flow about the body would transport, if

the streamline approximation described above were taken for a basis, the
volume U0(x,t)6 through the same cross-section in unit time. The
difference between these two quantities, that is, the loss in rate of
flow per unit time caused by the viscosity effect, leads by virtue of



U0 u dy = UO5*



to the definition of the length 5*



b*(x,t) = / (U u)dy. (3.1)




[Compare figure 1; the hatched areas are equal. Of course, any other
length y = yl > 5 may be selected instead of the upper limit 5 of
the integral in equation (3.1), as long as the velocity U(x,y,t)
in 0 y 5 yl is replaced by Uo(x,t).] According to definition,
5* is, therefore, a measure of the displacement of the streamlines of
the outer potential flow away from the body on the basis of the reduction
(caused by the friction layer near the body) in the quantity of fluid
flowing by the point x at the time t. (Therein a streamline in its
identity for all times is prescribed by the fact that it, together with
the wall y = 0, includes a stream tube of temporally constant through
flow.)

In order to obtain, instead of the potential flow about the pre-
scribed body, a corrected outer potential flow which takes the dis-
placement effect of the boundary layer into account, we visualize the
following model flow: Outside of a line y = 5*(x,t) a potential flow
with y = 5* as streamline is assumed to flow. Within 0 y < 6*
one assumes water at rest relative to the body. In this model flow the
same quantity of fluid is to flow past the body per unit time as in the
viscous fl?--. Equation (3.1) then indicates the value of 6*(x,t) in
first approximation for sufficiently high Re values.







NACA TM 1315


As to the line y = 5* which is to be a streamline of our substitute
flow, it must be taken into consideration that for the general nonsteady
case the course of this line varies with time. The line y = 5* then
is a movable dividing line between potential flow and water at rest.
In contrast to the dynamic dividing lines, it therefore consists in
general not permanently of the same fluid particles, but represents,
for reasons of continuity, a permeable line. In mathematical formulation
the boundary condition to be stipulated expresses that the normal
component of the potential-flow velocity along the dividing line
y = 5*(x,t) should vanish every moment. Then y = 5* is a streamline.

A simple example which we shall investigate more thoroughly later
(section 5) will serve to clear up these conditions. We assume that a
circular cylinder of the radius R is set at the time t = 0, from a
state of rest, relative to the surrounding infinitely extended fluid,
into a rectilinear and uniformly accelerated motion perpendicular to
its axis. The frictionless flow relative to the body is then given by
the velocity potential



0 = bt(r + )cos o
0 r


where r and signify polar coordinates in the flow plane, referred
to the center of the circle, r = R + y, 3 = n x/R, and b denotes
a constant acceleration. This potential flow yields the pressure
variation on r = R required for the usual boundary-layer calculation.
For very small times (small compared to the time t = tA of the start
of separation) this calculation results in a displacement thickness
5* increasing proportionally to vt. For these very small times after
the start of the motion the potential of our improved outer flow
therefore reads


1= bt r ++ cCt)2o s (t > 0, r > R + c -Vt

6* = c v-t, c = const.)






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The line y = c 4~v is a streamline. While it travels out into the
fluid starting from t = 0, the streamline pattern of the entire outer
flow correspondingly varies continuouslyl.

In order to clarify the conception we shall add a few remarks
regarding the mass displacement of the boundary layers. The streamline
displacement thickness 6* does by no means always represent at the
same time a measure for the mass displacement. This becomes immediately
clear from the following simple example. An unlimited plane wall which
up to the time t = 0 is at rest relative to the surrounding infinitely
extended fluid is assumed to be moved in itself, according to an arbitrary
acceleration law, starting from t = 0. A certain velocity profile
(which is known to be easily defined) develops, and one obtains a dis-
placement thickness 6* different from zero. However, a mass displace-
ment away from the wall does not take place due to v = 0 u = u(y,t) .
We could have made clear the difference in the two displacement phenomena
in a perfectly analogous manner on the case of the circular cylinder
set rotating about its axis from a state of rest; thus we should have
spared ourselves the fiction of an infinitely extended body. Here, just
as in the above limiting case of the plane wall, a boundary layer
develops; a mass displacement away from the cylinder in radial direction,
however, cannot take place due to reasons of continuity and symmetry.
If one relinquishes this symmetry by selecting for instance instead of
the circular cylinder a cylinder slightly wavy in comparison, the mass
displacement normal to the contour will be, in general, different from
zero; however, the length b* does not present a measure for it.

10 A
We shall give here for comparison the velocity potential 0 of
the corresponding flow about an expanding impermeable circular cylinder.
If its contour is prescribed by r = R(t), one has


^~ R2(t) dR(t) r
0= btr + R cos a + R(t) In
r tdt R

The additively appearing source yields that additional fluid in the
outer space r = R(t) which the impermeable circular cylinder itself
displaces while expanding


dr\ r ) dR(t)
kdr R(t) dt






NACA TM 1315


Looking for a measure for the mass displacement, one may start from
the relation following from equation (3.1) by differentiation with respect
to x and consideration of the continuity equation

Uo
( (Ub*) = v(x,b,t) + 8 -,x (x,t) (3.2)


(Again any y = yl > may be substituted for 6). The right side
indicates the excess of the v-velocity at the edge of the boundary
layer compared to the V-velocity 85UO/dx of the corresponding
frictionless flow prevailing there. Thus the entire volume displaced
by friction effect between two points x0 and x, per unit time is
at any rate always prescribed by




xx= X=XO



We now visualize, as in the definition of 5*, a model flow. We assume
a frictionless potential flow, with the quantity 86 generally different
from 8*, to be prevailing for y 8&,(x,t); the length 6, is assumed
to be fixed so that precisely the fluid which actually is displaced by
the friction effect would flow through 0 = y = 5 with the
velocity UO. Inside 0 y = 5 we assume for our model flow water
at rest relative to the body. Then there follows from equation (3.2)


U (UO0*,. (3.3)
6x ox(


Hence there results by integration for the "mass displacement
thickness" 68 the statement

(5, -6") = f(t), (3.4)

that is, this difference is only dependent on t.

Hence one can recognize: If one were to proceed in the determination
of the first correction of the outer potential flow described above,






NACA TM 1315


and therewith of the pressure field correction for large Re-values,
in such a manner that one would make not y = 8* but y = 6, for all
times the streamline of the substitute flow, the result would remain
unchanged, according to equation (3.4). Both models differ at any
moment only in the numbering of the streamlines.

As to the relation between 5* and 6 one may state: If one
assumes that at the respective time t the stream tube which includes
at the point x the region 0 y yl with yl = followed up
upstream finally blends completely in a region where the flow is
frictionless (case of approach flow), 5* 6b. If, however, this
presupposition is not fulfilled, as for instance in the examples with
vanishing 56 selected above, the conclusion drawn above is not valid,
either. In every flow of this type the streamtube just considered is
either bounded by the wall in its entire course upstream, or it leaves
the wall in stretches as when a separation of the boundary layer material
from the body with subsequent readherence takes placell upstream
(case of longitudinal flow.) The value of 6b at the time t can be
given for all x if one is able to give, in addition to the variation
of 5*(x,t), known according to equation (3.1), the value of 6, at a
single point x = x0;12 for equation (3.4) then yields only the
difference statement





It is always presupposed that the flow in the x-interval of
interest of the body contour and outside of the boundary layer next to
the body is frictionless. Thus for instance the case where the body
gets into its own wake during the motion is left out of consideration.
12It must be noted, though, that cases exist where it is impossible
on principle to give such a value. Let us visualize for instance the
boundary-layer flow originating when an unlimited wall, deviating from
a mean plane by surface waviness, is moved in its mean plane out of a
state of rest. If no special conditions prevail, the expression for the
total volume displaced at the time t up to a point x in unit time

x(xylt) + _I (x,t) dx

V, ax t


will not even be unique.







NACA TM 1315 11


Therewith we shall close this little digression concerning mass
displacement. Following we shall make use only of the streamline
displacement thickness 6*.


4. CALCULATION OF THE PRESSURE DRAG IN FIRST APPROXIMATION


With the improvement in the calculation of the outer potential flow
a more accurate specification of the pressure distribution in the outer
frictionless flow becomes possible. However, we must warn here against
the following fallacy: In general, one cannot use the resulting pressure
distribution at the edge of the boundary layer as impressed
*pressure p(x,t) for an improved boundary-layer calculation according
to the equations (1.1) and (1.2), for the obtained correction of the
outer pressure gradient is, in general, of the same order of magnitude
as the terms which have been neglected in the equations of motion. To
obtain an improved boundary-layer calculation it would thus be necessary
to take corresponding further terms into consideration in the equations
of motion as well.

The general Navier-Stokes equations of motion and the continuity
equation read in our curvilinear coordinates x,y in full strictnessl3


u R bu bu uv 1 R p + u I
6t R+ y 6x 6y R + y pR + y ox y2

2 .2
1 du R o u
-- +
R + y 0y (R + y)2 jx2


u + 2R V (.la)
(R + y)2 (R + y) ox


R dR
(R + y)3 dx

Ry dR du
(R + y)3 dx x


13Compare, for instance: Tollmien, W.: Grenzschichttheorie
(Boundary-Layer Theory). Handbuch der Experimentalphysik (Manual of
Experimental Physics) Vol. 4, Part I, p. 248, Leipzig 1931.






NACA TM 1315


R
R+y


oyv v
+ 2--+ v=0.
oy R +y


Therein R(x) denotes the radius of curvature of
(contrary to the mathematical definition positive
respect to the flow).


the wall contour y = 0
on walls convex with


If one considers in these equations, on the basis of the customary
estimates founded on the physical picture, only the terms of highest
order for very large Reynolds numbers Re = UL/v (U is a characteristic
velocity, L a characteristic length), one arrives, in the known
manner, at the boundary-layer equations (1.1)14. If one considers in
improved approximation also the terms of the order 0(5/L) compared
with 0(1), there results


S u 1 6p
+ v +
oy p ax


8u
2y
6y


y
R


uv
R


+ y_ pP
pR dx


v du
R by


1 dp u2
P dy R


(4.1)




(4.2)


14Compare, for,instance, W. Tollmien, footnote 13.


dv
U -+
bx


v
dy


u2
R + y


= 1 dp+


p yv




v
(R + y)2

Ry

(R + y)3


. 62v 2R
y2 (R + y)2


R2 2,v
(R + y)2 6x2

R dR
+ u +
(R + y)3 dx

dR .vy

dx Odx


au
-+
dx


(4.2a)


(4.3a)


R
R + y


du
6t


,u
u -
dx







NACA TM 1315


6du + v_= Y u v (4.3)
6x 6y R ox R

On the left side are the terms from equation (1.1), on the right the
newly added terms.

The correction of the outer pressure we obtained above is, in
general, of this same order O(b/L). Thus, if one wants to perform with
this improved outer flow an improved boundary-layer calculation (second
approximation for large Re), one can in general no longer neglect the
right sides and can, therefore, no longer calculate with a
pressure p = p(x,t) impressed on the boundary layer. (Compare addi-
-tional remark at the end of this section.)

For calculation of the pressure drag in first approximation for
large Re, the problem that interests us here, the solution is simpler.
We now know, according to the expositions of section 3, the outer pressure
field, far remote from the body, sufficiently accurately; it only remains
for us to continue this pressure field up to the body surface by deter-
mining the pressure gradient in y-direction through the boundary layer.
Equation (4.2) serves this purpose: It yields op/by sufficiently
accurately for this approximation, if we substitute in it on the right
side u from the first boundary-layer approximation.

If y = 6(x,t) denotes the "outer edge" of the boundary layer,
and p[x,&(x,t),t1 is the pressure distribution of the improved outer
potential flow along this line, one has for 0 = y = 6 in this
approximation



p(x,y,t) = p(x,8,t) R- J 2u dy (4.4)


The pressure drag of the unit length of the cylinder in the flow becomes



WD = p(x,O,t) cos ( dx


-IKE J
(4.5)


= 5.9 px&t) P JOE) u2dy cos p dx,
K nw--0






NACA TM 1315


Wherein ) signifies the angle between surface normal and main flow
direction, and the integral is to be formed over the entire contour K
of the cylinder cross section.15


15Additional remark at the time of proof correction: Regarding
the problem of an improved calculation of the boundary layer flow
(second approximation for moderately large Reynolds numbers), not
followed up further in the present report, the following calculation
procedure seems to me to be promising:

1. Calculation of the boundary layer in the customary first
approximation for very large Re.

2. Hence correction of the outer frictionless flow according to
Ziff's method.

3. Improved calculation of the pressure field p(x,y,t) in the
boundary-layer zone according to equation (4.4).

4. Again calculation of the velocity components u and v of the
boundary layer in second approximation from equations (4.1) and (4.3),
using the pressure field calculated above and replacing the right sides
of equations (4.1) and (4.3) by the known expressions of first approxi-
mation so that the newly added terms of the order 0(6/R) compared with
1 appear in the calculation as prescribed functions.







NACA TM 1315


5. EXAMPLE1I


Following we shall consider as an example a nonsteady flow which
originates if a body is, from a state of rest, set into a rectilinear
motion relative to the surrounding infinitely extended fluid, onward
from t = 0. The relative velocity of the undisturbed approach flow
with respect to the body visualized as.being at rest is assumed to


16The simplest, almost trivial example on which the method developed
may be tested is the case of the plane steady stagnation point flow.
Here the strict solution of the Navier-Stokes equations is known, and
the flow near the wall calculated on the basis of the boundary-layer
theory is known to agree with the exact solution. If one replaces the
potential-theoretical stagnation point flow at the wall by the stag-
nation point flow at the wall shifted by 6*(= const.), one obtains
as the corrected outer flow for y > 6 also full agreement with the
exact solution. Since the outer pressure gradient parallel to the wall
has remained unchanged in this correction and the wall is plane, an
improvement of the boundary-layer calculation proves to be impossible
as it has to be. Thus the exact solution has already been attained with
this one step.

A further example with plane body boundaries (for which the
boundary-layer equations (1.1) therefore are valid except for terms of
the order 0(52/R2) compared with 1) is the case of longitudinal flow
over a plate. According to Blasius, elsewhere, 6* = 1.73 ,vx7,U.
(U. free stream velocity, x distance from the leading edge of the
plate). With the improved outer pressure distribution calculated as
suggested above, there results as the superimposed pressure for an
improved boundary calculation


p(x) = P El + 1.35 Re(x1

(Re(x) = Uox/v). One does obtain here a considerable pressure correc-
tion since the new outer flow has a stagnation point, but one recognizes
that the pressure correction has already dropped at Re(x) = 75 to
1 percent of the stagnation pressure. However, in the proximity of the
leading edge of the plate (small Re(x) values) Blasius' boundary
layer equation cannot be used. An improved calculation of the flow
which in this range does not use the boundary layer approximations is
still lacking. Thus for the time being the method of improvement for
larger Re(x) suggested in this report cannot be utilized for this
example.






NACA TM 1315


be = Cot C = const., n = 0, t ). Then the potential
velocity U0(x,t) along the body boundaries can be represented in the
form


0( for t o, ,
u0(x,t) = t f > (5.1)
q(x)tn for t > 01


We assume *(x,y,t) to be the stream function of the boundary-layer
flow defined by u = d'/oy, v = 6//dx which is obtained when the
pressure gradient to be calculated from equation (5.1) according to
equation (1.2) is taken as a basis, on the strength of the boundary-
layer equations (1.1), thus in the customary first approximation for
very large Re-values. Generalizing the series developments set up
by Blasius1' for the special cases n = 0 (sudden transition from state
of rest to motion at constant velocity) and n = 1 (uniform acceleration)
one obtains the result



x(x,y,t) = 2 rt qtnno(n) + qq't2n+ .,l101 + ..] (5.2)


thus a series development in powers of t n+ (that is, in powers of
the distance covered by the body). The appearing coefficient functions
S (,O l (n), ... with = y/2 fv4t are universal functions of i.
More details may be found in the appendix to this report.

From equation (5.2) follows


qtn i it2n+l +
u = qt n + qq't n '
n.0 nll
(5.3)
v = 2 't n,0 + (q2 + qq)t 2n+ n ,l +
n,OJ


17Compare footnote 4 on page 3.






NACA TM 1315


The boundary conditions are satisfied by virtue of


n,o(O) = n,l(0) = *** =,


(0) = (0) = .. = 0,
n,O n,l


n,O n,9 -


where there is always, here and later on


'1 d .n
n., K d n.K


For the streamline displacement thickness 5*



5*(x,t) = 2 tvtai q'(x)tn+l 1(m)
L .n-n.,l


there results



- ..


with


an =lim I n,0



(A few numerical values an are given in the appendix.)

We consider as an example the flow about a circular cylinder.
the mode of notation of section 3, the velocity potential of the
frictionlezs flow about the circular cylinder ij given by

r2
0 = (t; (r + R-) cos 0
0 r


(5.4)


0, '
J


(5.5)


(5.6)


(5.7)






NACA TM 1315


Hence the velocity at the periphery of the cylinder is calculated as



UO(x,t) = q(x)tn = 2U(t) sin (5.8)


and the resulting pressure distribution at the body surface is



pO(x,t) = pd p[2R(cos 3 + 1) + 2d+ t) sin20 (5.9)


where Pd(t) is the pressure at the forward stagnation point. The
potential-theoretical pressure drag of the length L of the cylinder
at rest is therefore



W = L P cos .3 Rd. = 2pnR2L d. (5.10)
DO J 0 dt


Thus one obtains, as is known, an increase in inertia by double the
amount of the inert mass of the fluid displaced by the cylinder itself.
For the improved calculation of the outer potential flow there results


5*(x,t) = 2 vt a + 2 C (n)tn+l cos + .... (5.11)
L R nl

Following we limit ourselves to small times t for which the two first
terms of the series development represent a sufficient approximation.
In this approximation the line y = 5*(x,t) practically represents a
circle; that is to say, a circle with increasing radius r, the center
of which, for t>d, does not coincide with that of the cylinder cross-
section, but travels slowly downstream. The equation of such a circle
reads formally

r = b cos +
r = b cos 3 + a(l -b sin2 B)2
a/






NACA TM 1315


(a = radius,


b = displacement of the center), and if


r = a + b cos a


[+ ,


We identify


a(t) = R + 2an A TF


4( n,l
b(t) R n,l ~) 0 t


(5.12)


If, temporarily, T7, denote polar coordinates about the center
of this circle in motion relative to the cylinder, the potential flow
appertaining to the latter, the streamline of which is the line
r = R + 5*, that is, r = a(t), is given by the potential



= (= )( + cos T


thus the potential flow here required in the polar coordinate system r,1
fixed relative to the body by


+l = Ul+ 2 (r cos a b)
2 c
r 2br cos 1 + b2 cos b)


(5.13)


At first we introduce dimensionless quantities as
choose R as the characteristic length, and the time
to cover the distance R, starting from the beginning
as the characteristic time, thus


follows: We
T the body requires
of motion t = 0,


(5.14)


T = (n +


1
n+l
1) ---


b << a






NACA TM 1315


Hence results the characteristic velocity
1

R CORn) p T
T \ n+1 l


and the Reynolds number Re formed with this velocity and the length
R becomes8


Re = (R= R~ (5.15)


By making dimensionless the variable lengths x, y, r, a, b by
dividing by R, the time t by dividing by T, the velocity U(t)
by dividing by R/T, the potential by dividing by R2/T, and the
pressure by dividing by pR2/T2 (we denote the dimensionless quantities
by adding a wavy line), we obtain

1 1
= 1 + 2an Re 22
1 (5.12a)
b = 4(n + 1) (o) Re 2t 2;
n,l



S= (n + 1)n 1 + a (r cos x + b).
(1 + 2 br cos x + bl (5.13a)



181f one forms the Reynolds number Rel with the length VvT
one obtains


Re= (RT) 1=Re
Rel = (R/T)4 v = Re2






NACA TM 1315

Furthermore,


1 1
I = Y Re2
2 Jt 2


and


- 4(n + 1)t nn,1(m)n+l cos ].
co xJ.


R= e n
Re-


We consider first the case n = 1 of uniform acceleration from a
state of rest. Following, Pi without more precise data will represent
those additive portions of the pressure pl = p/(pR2/T2) which do not
make any contributions to the pressure and are therefore not of interest
to us. Furthermore, terms of the order 0(52/R2) compared with 1 are
neglected. One obtains for the pressure calculation according to the
Bernoulli equation at the distance from the wall Y = 6/


4 1 + 3alRe t


cos + P


+(R.


S01r 2


- 1)2 10l 2 sin2x + p2+
F2L = R 2R2
)2 ij


Thus one obtains


Pj(, -, ) = 4(l + -3aI le~ ) cos I + 16 2s ~ P + o

(5.16)


(5.11a)


3a~1


g2\
= 0 --'2






NACA TM 1315


According to equation (4.4)


(4. 4a)


In the small times of interest to us there is



n = 4 sin ( 16 sin (16 cos X'I (n),
1,0 1,1


therefore


8/R
- I


-U2 = 16 3sn 8/R
udy = 16 fsin2x
^0


io02d +


128 4sin2x cos x / R d1 -
JO 1,0 1,1


-.3/R
256 6 sin2x cos dy.
o 1,1
(5.17)
Because of lim (I) = 1, lim (' (r ) = 0 the two last integral
T- ) o1,0 1-.> i1,
on the right are, if 6/R is chosen so large that these asymptotic
values are attained with the desirable accuracy numbers independent of
the variation of the boundary-layer edge 6(x,t) with x. The first
integral at the right, together with the corresponding second term at
the right in equation (5.16), may also be combined into an expression
independent of (xE), namely the expression


16 -2sin2 (1- 02d.
0


P(Xr, 0, t = ) = 2dy.
~ -\ /






NACA TM 1315


Thus the variation of &5(,T) over x- does not play any role in
judging what terms make contributions to the pressure drag which is as
it should be. If one finally inserts

1 1
dy = 2 Re 2t2d ,

all together one obtains, therefore,


p( o, Y) =


4(1 +4


3 1e
3acRe t os 0


1 9
+ 256 sina cos 1 Re2 22 J
0


io 1, P4 + P+0(Re-1)
(5.13)


Thus the following pressure drag of a circular cylinder of the length L
for small times after start of the motion and in first order for large
Re-values results


Wp
2pnR 2LCO


1 1 2e+,

-'*~'.9'11 ,`'0 IFn,


(5.19a)


2pnR2LC0 = WD is the potential-theoretical pressure drag. (Compare
equation (5.10).) According to Blasius'calculations 3a1 = 2/rW(=1.128),
furthermore, according to the author's numerical evaluation


0


q, q0 ,ldT = o.09o80






NACA TM 1315


Thus the result in dimensional form is


tL 2
S 0.392 t(5.19)
"D "o R R 2


As friction drag


;-2n
W / a(x, 0, t) sin dx
R J y R

there results from Blasius' calculation results19


WR / D 0.029 2 t (5.20)
DO R (h R2


For sufficiently small times tt << 1) thus WR = WD WDo that
is, the friction drag increases immediately after start of the motion
.ccordinq to the same law as the contribution of the skin friction to
the pressure drag.

The total drag becomes


S= JD+ ijR DO -I + 0,363 R- t4 (5.21)


In figure 2 we represented these results using the dimensionless
t = t/T (with T = f2R /Co for the present case n = 1), thus the
re la 'tion
T 1
= 1 + -t (1.128 + 1.569 4), (5.191)
IDO

Tt, order to obtain the term with t 4 one must here include in
.. (V.21 .lIso the third term of the series. The necessary data
I, "' .. ir. Blasius' report, footnote 4.






NACA TM 1315


1
W = (1.128 0.116), (5.201)
WDo


W = 1 + 17(2.257 + 1.4534). (5.211)
WDO R

Our formulas can be applied with good approximation only for very small
times after start of the motion (solidly drawn parts of the curves),
because we had, for the sake of simplicity, taken into consideration
only a few terms of the development (5.2); but of course, with a little
calculation expenditure they can be easily improved, at least so far
that they are valid up to times shortly after setting-in of the separation.
The value t"= tA(= 0586) = 0.766 plotted in figure 2 indicates
the time of the start of separation, in first approximation, at the
rearward stagnation point according to Blasius. (Compare also appendix.)

For an arbitrary integral n > 0 one obtains correspondingly as
pressure drag with 7i(t) = COtn and WDO = 2pDn2L dU/dt



WD- = 1 + + n + t2n+2 n n,+ n (5.22)
WDO


For the calculation of the coefficient of t2n+2 sufficient numerical
data concerning (_ ) are lacking so far. For this reason we limit
n.,
ourselves in our numerical statements to the first term of the develop-
1
ment with time (term with t2) and put the question whether the law
WR = WD WDo obtained above for n = 1, for times immediately after
start of the motion, is valid also for arbitrary n > 0. The friction
drag has for these small times the value


WR= WD rt (5.23)
Do R 2n






NACA TM 1315


In the appendix the exact expressions for (r) are derived.
n,O
However, the relation


2(2n + l)an = (0) = 22n (n!2 2
nO (2n) qr


(5.24)


is valid; we produce a very simple proof for it in the appendix (one
may confirm it also with the aid of the tables of the appendix we
calculated for n = 1, 2, 3, and 4). Thus, for very small times after
start of the motion, there applies indeed


WE = WD WD


viz:


= WD
O R


nJ'(n 1)
(2n):! F


(n > 0).


(5.25)


(One has 22nn'(n 1)!/(2n)! VfW = 1.1284 for n
0.6018 for n = 3; 0.5158 for n = 4, etc.)


= 1; 0.7522 for n = 2;


One may now consider more general laws of motion of the form


(t)


f


for t 0


(t) for t 0(f(O) = 0)


and carry out corresponding calculations. If one presupposes that the
function f(t) defined in t 0 can be developed into a Taylor series
around t = 0 which converges for the small times 0 = t << tA after
start of the motion which are of interest to us, one may attain the
result quite analogously with a series expression correspondingly
generalized compared to equation (5.2). One can interpret the
law U = Cotn which has been valid so far as the first term of the
development with time of such a general law of motion. Hence it follows







NACA TM 1315


that for all these laws of motion the portion of the pressure drag
caused by friction WD WD increases immediately after the start of
motion according to the same law (5.25) as the friction drag. This is
a noteworthy quality of the circular cylinder.

One may also include the case of a sudden start of motion in these
considerations; it is true that one must then accept, corresponding to
this degeneration of the form of motion at the time t = 0, infinite
pressures and drags at the time t = 0. With U CO = const. for t 0
one obtains


WD = 2pnRE C


4C2
2
+ C


WR = 2pdLCO 4 O i 0


t2 o


Oo 0,1


2 R2 I


The function (0t2a(q) is explained at the end of the appendix. We
did not numerically determine the coefficients at t2 in equations (5.26)
and (5.27). Because of aO = o'(0)(= 1/Ar- = 0.5642) the law stated
above W = W W is valid also in this limiting case n = 0 of
R D DO
the sudden start of motion, for times immediately after the start of
motion. Here in particular WDO = 0 (d'Alembert's paradox).


and


(5.26)


(5.27)






NACA TM 1315


APPENDIX


A FEW CALCULATIONS REGARDING THE DEVELOPMENT OF THE BOUNDARY LAYERS


A few calculations will be given in this appendix which yield,
among other data, those required for the preceding investigation
(section 5) concerning the basic functions tn i(n) of the unsteady
boundary layers; for the rest, they represent merely an extension of
the related calculations by Blasius. We give these calculations apart
from the previous considerations, first, because they would have
disrupted the connection there, and second, because the datu and tables
attained are of interest in their own right.

As assumed in section 5, let a velocity proportional to tn(n > 0)
be imparted to a body from a state of rest relative to the surrounding
indefinitely extended fluid, beginning at t = 0. The potential-
theoretical circumferential velocity U(x,0,t) = U0(x,t) then has the
form (5.1). For calculation of the boundary layer development from
t = 0, if a generalization is made of the series set up by Blasius
for n = 0 and n = 1, the expression



*(x,q,t) = 2 Yt L- tn+X(n+l)X (x,rl)
--O rn,
with (1)


T = y/2 4t


for the stream function of the boundary-layer flow is obtained and
one obtains for the functions Xn, by substituting equation (1)
into the boundary layer differential equation (1.1) a system of
differential equations solved by recursion; we limit ourselves here to
the two first equations of this system which read

a3 n,0 K0 n,O
n- + 2T 4n = 4nq(x),
N3

n,J o ,1 l (2n n,O nO
+ 2 4(2n + 1) = 4(2)
q3 2~ \ an aox

-10 a2 ",o-
_6- c qq 2






NACA TM 1315


We assume first n (later 2n) to be an integer. Then the solutions
may be represented with the aid of Hermite polynomials. With the
statements



SO= q(x)t (T),
n,O n,O


,l = n


one obtains from equation (2) the ordinary differential equations


' + 2I'' 4n' = 4n,
n,O n,0 n,0


'" + 2" 4(2n+ l)C'
nl nl nil


The boundary conditions to be satisfied by n,0
formulated in equation (5.4)


S O2 n -
= 4 nO n,O~n,O- i .


and tn,l


are






NACA TM 1315


In the case of a plane wall moving in its own plane,

S= 2 V%7 qtnfn, () represents the complete solution (because
of q = const.), not only in the boundary-layer theory approximation
set up here, but in strict fulfillment of the complete Navier-Stokes
20
equations. The calculation of tn may take place as follows. The
prary plttn,ng-off
temporary splitting-off


2,
n,O() = 1 e" n()


transforms equation (4) into


-n 2)pn 2(2n + )(n = 0


20Because of U0 = UO(t), u = u(y,t) and v = 0, the Navier-
Stokes equations are for these motions simplified to


bu a2u oU
)- = V +-- +a
t oy ~t
Cy l.


The analogy between


U0 u and the corresponding solutions of the


problem of heat conduction has been known for a long time; it offered
one of the few possibilities of attaining exact solutions of the
Navier-Stokes equations. For the rest, one can see for the present
problem that the first term qt',0 of the series development
following from equation (1) for u as a solution of the above equation
approximately satisfies the boundary-layer equation in the sense that
only the terms of highest order are taken into consideration for small
times after start of the motion, whereas the quadratic inertia terms
are neglected. The iterative improvement of this first approximation
for small times then yields step by step the ascending terms of the
series we set up formally at the outset. This consideration led
Blasius, (elsewhere), at the time to his special series formulations
for n = 0 and n = 1.






NACA TM 1315


For every integral
equation21 is


2n t 0 the general solution of this differential


Ph= n (2Cl + C2 e-d 21


(d )2n
dw)


(C1, C2 are integration constants). As is well known, the .Hermite
polynomials Hm(x) are given by


Hm(x) = e(-


= m(K) (2x)m-2"
0< K<_


(in the form originally
Lm(ix) = iftm(x), thus


given by Hermite). For further use we also put


m() = e- e) 2


.) m 2 ) (2x)m-2 K
0< K.


furthermore


S() ( =o x exdx,
0 -A .


(11)


. d(x) =
dxK


21Compare, for instance, E. Kamke "Differentialgleichungen:
Losungsmethoden und Losungen, I Gewohnliche Differentialgleichungen"
(Differential equations: Methods of solutions and solutions, I.
ordinary differential equations) Leipzig 1942, Part C, No. 2,41, and
put there x = i VI.


(10)


dade-X1






NACA TM 1315


(so that in particular 1 + 00 = 0 represents the error integral, e0
the error function). By using these expressions, a simple recalculation
from equation (8) gives the general solution of equation (4) as



'() = 1 H() + C2 (*)1 +

2n (12)
Ic (7) 7 2n ) ,- (TI ) Hr
K=1 2n-_(TK-1


with C C.
2 2 2

C1 = 0 because of 'n0 (w) = 1 and C2* = -n'/(2n)! because
of S' (0) = 0 and B2n(0) = (2n)./ni. The polynomial sum in
equation (12) disappears for n = 0 and integral n term by term,
since either 2n n or K 1 is an odd number]. Thus the desired
particular integral of (4)22 reads

2n
Sn,0() = 1 + (2n()0 i() 2n (- ()2n-()H-I( (
n,(O + ( 2n l' ( n) -* 0 =

(13)


Because of 0 (7) = (- I)1 ( )H K (n) the solution may also be
written in the following form which is more elegant than equation (13)
but less serviceable for practical calculation
2n
(n= + n! 2 D-)(T). (13a)


22The direct and elementary derivation of the solution for
integral n > 0, and therewith for integral 2n 0, compare below,
fails to work if n does not have this property. But in that case,
too, the solutions are easily found if one makes use of the analogy to
the corresponding solutions of linear heat conduction (compare
footnote 20 to this report) and represents the solution according to
the singularity method.






NACA TM 1315


According to equation (13) one has in particular


' O}C) = 1 + C(TI) = t(n),

' (1) = 1 + (2n2 + 1)$0(n) + nl()
1,0

g,0o() = 1 + 1(4n4 + 12n2 + 3)00 () + (2C2 + 5)i1(TI[,

S(n) = 1 + -1 [8q6 + 6094 + 9012 + 15)00(C) +
3,015
(Ci4+ 28T2 + 33)1(C~],

S,0(n) = 1 + -1- 6i8 + 224n6+ 84oT4 + 840T2 + 105)o(Ti) +

(8,6 + 1084 + 370n2 + 279)7nl (i)1 (
(13b)


Thence one obtains by elementary quadratures n,0(n), likewise
expressed by t0 and T0 with polynomial coefficients. The numerical
evaluation is reproduced in table 123. Because of the special impor-
tance of the solutions ', (n) as boundary-layer profiles u/qtn
n,0
in the case of the plane wall (compare above) we represented them in
figure 3. Owing to 6* = 2\ l (T)) = 2an ;t the stream-
lie ds-laoo e (I 0 ) 2
line displacement thickness 5 can easily be taken from the numerical
cal table 1; compare-also table 2.3
Since it follows directly from equation (7), by single differ-
entiation, that the general integral 1 is the first derivative of
n--
2
the general integral %, with respect to n, it is, with the
boundary conditions satisfied, easy to find as expressions for the basic
functions + 1 (q) with the integral n



o =1- 1 Lno + 2m(1 ,0) (14)

numerical calculations were performed by Miss Ursula Ludewig
23All numerical calculations were performed by Miss Ursula Ludewig.






34 NACA TM 1315

Therein



v" (0) = 22 (n1)2 2 (15)
n,0 (2n)' in

In order to calculate the fundamental functions of the first order
(n,1(), the total course of n,0(r) must be known (according to
equation (5)). Only the additional knowledge of (" (0) is required
n,l
for the problem which is of foremost interest, the question regarding
location and time of a possible separation of the boundary layer in
first approximation.

For because of


(U t :n t n+ltt
S = n- 2 .+ ( + qq'tn" .(0) +
(Yy=/ 2 vt L 0 nPl

one obtains in first approximation the connection
1



(X),T(0)
n+l
--



for location x and time t of the separation. On the other hand,
we are interested in Cn (no) (compare section 5), with a view to the
n,l
calculation of the displacement thickness. These two data can be
determined without solution of the differential equation (5) by a
well-known method as follows: (qT) is assumed to be a function of
in 0 m provided with the continuity properties required for the
following calculation. By partial integration one obtains the following
relations. If Ln and Mn are the differential operators

Ln = 2 + 2n 4(2n + 1),


n d= 2 d 2)'






NACA TM 1315


one has



(B)LI din [ 'n, n li L 1n f0 n Mn0,] dI,
(17)


f(1)L [' ]d-i = ["
OJ nll n nsl


n,1


.O/ n, J h .di


We choose, therefore, I = UO (r) so that


Mn[3n0,O] = 0
with
with


n (0) 1,
nO


a (-o) =
n,0


If then equation (5) and the boundary conditions valid for t (T)
are taken n,to consideration the result is
are taken into consideration the result is


n,1 () =- no O n.02 n,0 n,0 9 ".


On the other hand we choose 8 = 6~m ( s) so that


MOn]- = 1


with


n, (0) = 0,
n'sD


n, (.) finite
n,"D


+ 21-3t' + Mn ]; -
n,l l 0


(18)


(17a)


(17b)


(18a)






NACA TM 1315


Then one obtains, in analogy


(18b)


n = 00 0% O)4(n,02 n0n,0 1)d.


It is easily confirmed that



9n,)(() = 2( 4n + 3) n


furthermore by comparison with equation (7)



n,0() = e2 2 1( 1 ]


(19)


Therewith the desired functions and 3
n,0 n, ao
the known basic functions of zero order +
2q+1,0.


are traced back to


Numerical evaluation yielded for n = 0, 1, 2, 3, and 4 all
together the data here of interest given in table 2. It also shows the
numerical values of an. For the n-values 1, 2, 3, and 4 one finds
the law (5.24) confirmed. A general proof of this law may be produced
with a few calculations on the basis of the known expressions
for no(O(). A much simpler proof of the relation (5.24) will be
presented below. As mentioned above, the expression


u = qtn 10

is the strict solution for the boundary-layer profile on a plane wall
with U = qtn (q = const.) outside of the boundary layer. The wall
shearing stress according to the momentum theorem of the boundary
layers is generally


d /5 _
pudy J pu2dy -
dx .0


SP dy b -P
-J0 it ox


(20)


7 = u x O
0 x J0O


(21)






NACA TM 1315


thus for the above flow, due to the velocity distribution u(n,t) being

independent of x as well as due to p and with
ox dt

n= l- 0 ,0()]

-6 1
T = p (U -u)dy = p (Ub) = pq(2n + l)a tn 2. (22)
6 Ito 60 t n-

On the other hand
1
Tu(o,t) 2n--
T-0 == pqIn (0) 4 t n (23)
6y n,O 2

and thus in combination with equation (15), as asserted,

22n+z(n,)2
2(2n + l)% = I, o(O) = (5.24)


In the case of the flow about a circular cylinder moved rectilinearly
out of a state of rest, investigated in section 5 [compare equation (5.8)]
one has q(x) = 2C0 sin x/R and therefore -q'(x)max = 2C/R
for x/R = i. Thus the separation starts according to equation (16)
at the rearward stagnation point at the time
1
tA = [n,o(O)/2CO 1(O n+l


(compare table 3). The distance covered by the cylinder during that
time is


SA = COtAn+/(n + 1) = Rt" (O)/2(n + 1)t" (0)
Sn,O n,l






NACA TM 1315


Finally we want to give a few indications where to find further
data regarding the basic factions of the plane nonsteady boundary-layer
flow. The series development (5.2) written down up to the third term
reads



4 = 2 t qtn [,0 + tn+lqtn,l + t2(n+1)(qi2 n,2a + q"n,2b) + a..].


Blasius gives, in addition to the functions 0,0(1) and 51,0()
calculated above, the rigorous solutions 0,1(C) and t, (17).
Beyond that, he calculates the numerical values ," 2a(0) and 2b(0)
which are of interest for the determination of the separation.
S. Goldstein and L. Rosenhead24 give the exact expressions for S0,2a(1)
and 0,2b(T). These integrals were, by the way, numerically determined
before by Boltze25 on the occasion of treatment of the corresponding
problem n = 0 of rotationally symmetrical flows which seems to have
escaped the attention of the authors.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics


2%Compare footnote 6 of this report.

2Compare footnote 5 of this report.


__







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NACA TM 1315


TABLE 2


TABLE 3


START OF SEPARATION


tA AND DISTANCE TRAVELLED


SA IN THE


CASE OF A CIRCULAR CYLINDER MOVED RECTILINEARLY

OUT OF A STATE OF REST


n Can On,0(0) 5n,9() Hn,1(0)

0 0.5642 1.1284 0.418 1.607
1 0.3761 2.2568 0.138 0.963
2 0.3009 3.0090 0.072 0.756
3 0.2579 3.6108 0.046 0.632
4 0.2293 4.1266 0.033 0.552


SSA
tAt_} ___ R

0 0.351 0.351
1 1.082 0.586
2 1.258 0.663
3 1.300 0.714
4 1.302 0.748






NACA TM 1315


Figure 1.- Regarding definition of 5*.






NACA TM 1315


0 0.2 0.4 0.6 I0.d6
tA

Figure 2.- Pressure drag WD, friction drag WR, and total drag
W = WD + WR of the circular cylinder for very small times after start
of the motion for uniform acceleration out of a state of rest.







NACA TM 1315


--------- I
o 0.25 05 0.75 .O r n,


Figure 3.- Course of the functions Cn,0(n) for n = 0, 1, 2, 3, and 4.


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