A simple numerical method for the calculation of the laminar boundary layer

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Title:
A simple numerical method for the calculation of the laminar boundary layer
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NACA TM
Physical Description:
47 p. : ill. ; 27 cm.
Language:
English
Creator:
Schröder, K
United States -- National Advisory Committee for Aeronautics
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National Advisory Committee for Aeronautics
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Washington, D.C
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Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
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federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
An iteration difference method for the calculation of the incompressible laminar boundary layer is described. The method uses Prandtl's boundary layer equation and the boundary conditions directly and permits the attainment of an arbitrary accuracy. The method has been tested successfully in the continuation of the Blasius profile on the flat plate, on the circular cylinder investigated by Heimenz and on an elliptical cylinder of fineness ratio 1:4. The method makes possible the testing of previously developed methods, all of which contain important assumptions.
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by K. Schröder.
General Note:
"Report date April 1952."
General Note:
"Translation of "Ein einfaches numerisches verfahren zur berechnung der laminaren grenzschicht." Zentrale f̈r wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Forschungsbericht Nr. 1741, Berlin-Adlershof, February 25, 1943."

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University of Florida
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aleph - 003779544
oclc - 24079797
sobekcm - AA00006198_00001
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AA00006198:00001

Full Text
CA-TlIM-/31-1







,,-- ^ 7 ? 3 ,i~ i



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1317


A SIMPLE NUMERICAL METHOD FOR THE CALCULATION

OF THE LAMINAR BOUNDARY LAYER*

By K. Schroder


ABSTRACT


A method is described which permits an arbitrarily accurate calcu-
lation of the laminar boundary layer with the aid of a difference cal-
culation. The advantage of this method is twofold. Starting from
Prandtl's boundary-layer equation and the natural boundary conditions,
nothing needs to be neglected or assumed, and not too much time is
required for the calculation of a boundary-layer profile development.
So far, the method has been tested successfully in the continuation of
the Blasius profile on the flat plate, on the circular cylinder inves-
tigated by Hiemenz, and on an elliptical cylinder of fineness ratio 1:4.
Above all, this method offers for the first time a possibility of con-
trol by comparison of methods known so far, all of which are burdened
with more or less decisive presuppositions.


OUTLINE


I. INTRODUCTION

II. GENERAL REPRESENTATION OF THE METHOD

III. PRACTICAL EXECUTION

IV. NUMERICAL EXAMPLES AND RESULTS

V. REMARKS REGARDING THE CONVERGENCE OF THE ITERATION PROCESS


*"Ein einfaches numerisches Verfahren zur Berechnung der laminaren
Grenzschicht." Zentrale fur wissenschaftliches Berichtswesen der Luft-
fahrtforschung des Generalluftzeugmeisters (ZWB) Forschungsbericht
Nr. 1741, Berlin-Adlershof, February 25, 1943.






2 NACA TM 1317


I. INTRODUCTION


The flow processes in the laminar boundary layer may be described
by Prandtl's boundary-layer equation. If one limits oneself to the two-
dimensional steady case and introduces, in a suitable region around a
profile contour C situated in a flow, a curvilinear coordinate system
s,n, the coordinate lines of which consist of parallel curves and nor-
mals of C, that equation reads


vs + vn s + p'(s) = 1 2Vs (1)
6s on R an2

when vs, Vn signify the velocity components in the s,n system, R
the Reynolds number, and p = p(s) the pressure distribution along C
taken from a measurement or calculation1. Equation (1) is complemented
by the continuity equation

3V+ n = 0 (2)
s 6n
The transformation

n = n R v = vn l(

yields, instead of equations (1) and (2), the equation system


Vs as+ v ?- + p's) (3)
)OT)


v-.- + 1 0 (4)
ds ar

in which R no longer appears explicitly.

1A mathematically complete derivation of equations (1) and (2)
based on physically plausible assumptions may be found in H. Schmidt's
and K. Schroder's report entitled "Die Prandtlsche Grenzschichtgleichung
als asymptotische Ngherung der Navier-Stokesschen Differentialgleichungen
bei unbegrenzt wachsender Reynoldsscher Kennzahl" (Prandtl's boundary-
layer equation as an asymptotic approximation of Navier-Stokes' differ-
ential equations for indefinitely increasing Reynolds number) Deutsche
Mathematik, 6, Heft 4/5, pp. 307-322. A survey of related literature
is given in H. Schmidt's and K. Schro'der's report "Laminare Grenzschichten,
I. Teil" (Laminar boundary layers, part I) Luftfahrtforschung 19,
Lieferung 3, 1942.







NACA TM 1317 3


The following boundary conditions for the integration of equa-
tions (3) and (4) are usually selected in boundary-layer theory as the
natural ones from the physical point of view. For an initial value s = so
an entrance profile

vs = vs(so,)

is prescribed as a function of I (entrance condition). Furthermore,
in consequence of the adherence of the fluid to the contour, the rela-
tions

fvs] =0 0 = 0
which are to be interpreted as limiting processes, are to be valid along
C (adherence condition). Finally, for s-values larger than or equal
to so the velocity component vs is to converge for Tr ---) toward
the velocity U(s) which is connected with the prescribed pressure
distribution p(s) by

U(s) U'(s) = p'(s) (5)

(transitional condition).

The general significance of these boundary conditions will be dis-
cussed more thoroughly in the second part of the Luftfahrtforschung
report quoted in footnote 1. Here we shall only point out that the tran-
sitional condition formulated for 9 --- m must not be confused with a
condition for n --> since for the latter limiting process the veloc-
ity components converge toward those of the basic flow The limiting
process rj -- o denotes, on the contrary, the asymptotic transition
to the boundary values, resulting along C of the outer potential flow
obtained for R--* 0. This can best be made clear by the example of
the stagnation-point flow at the flat plate, treated in the second report
indicated in footnote 1 (by the author and H. Schmidt). Whereas the
quantity 5 there specified as boundary-layer thickness tends
like 1/VR toward zero, a quantity d tending toward zero, for
instance, like l/ R, can be prescribed in such a manner that,the flow
outside of a layer of the thickness d adhering to the contour for
R -- converges toward the outer potential flow. However, to the
asymptotic transition toward the boundary values of this assumed potential

2It is assumed, of course, that this limiting process is meaning-
ful.







NACA TM 1317


flow alonL: C then there corresponds the limiting process


lim n = lim d/iR =
R---) R--;*

So far, an appropriate existence and uniqueness theorem for this
boundary-value problem does not exist. However, the results obtained
with the new method described below show that the statement of the
problem is perfectly sensible.

In the literature it has been pointed out more than once3 that for-
mal power series developments of the function representing the solution
with respect to n make the fact plausible that the entrance profile
cannot be selected completely arbitrarily, but that it is dependent on
the pressure distribution p = p(s).

Our method for the determination of the velocity profiles yields a
numerical solution of the mentioned boundary-value problem with the aid
of the difference calculation; it is superior to other methods because
it requires no assumptions beyond equations (3) and (4) and the boundary
conditions. In our method, the boundary-layer bonds of the entrance
profile do not appear directly and thus do not cause any difficulties in
the numerical calculation. A severe violation of these bonds causes,
in our method, the variation of the successive boundary-layer profiles
to become completely disordered. Small violations of these bonds, in
contrast, do not exert any considerable effect on the further develop-
ment of the profile4.



3Compare S. Goldstein "Concerning Some Solutions of the Boundary-
Layer Equations in Hydrodynamics," Proc. Cambridge Phil. Soc. 26, 1930,
pp. 1-30, L. Prandtl, "Zur Berechnung der Grenzschichten" (Concerning
Calculation of the Boundary Layers) ZAMM. 18, 1938, pp. 77-82, (NACA
TM 959) and H. Gdrtler, "Weiterentwicklung eines Grenzschichtprofils
bei gegebenem Druckverlauf" (Further development of a boundary-layer
profile for prescribed pressure variation) ZAM4. 19, 1939, pp. 129-140.

4L. Prandtl and H. Gortler (reports quoted in footnote 3) arrive
at the same conclusion, although on another basis.






NACA TM 1317


II. GC~ERAL REPRESENTATION OF THE METHOD


If one introduces into equation (3), instead of s, the new inde-
pendent variable


S= J dt (6)
so U(t)

under the assumption that U(s) / 0 for s = so whereby

dS 1
--u-U
ds U(s)

is valid, and if one uses the new designations

U( ,iT) = vs(s,q), U(M) = U(s(0)), u*(C,TI) = u(,i) U(()

there follows from equations (3) and (4) by way of


with equation () taken into consideration dour initial equation
w eu o (o ao

with equation (5) taken into consideration, our initial equation


bu* u 2u* 1 5u
oj a2 U() ai


" d- -
J' 6k


u* u
U(0) M


According to the statement of the problem in the introduction, we
have to find a solution


u = u (,1l)


of equation (7) for all points ( ,|)
the plane of the rectangular Cartesian
approaching the straight lines


in the right upper quadrant of
,T] coordinates which in


S= 0 or T = 0 respectively


5If separation phenomena appear, the solution will, in general, be
of interest only up to the separation point or possibly a little way
beyond it.






NACA TM 1317


tends toward prescribed functions:



lim u*(,t) = vs(so,y) U(so) ( 0) (8)

or



lim u*(,t) = -() ( > o0) (9)
TI--O 0
and which vanishes for '-->m



lim u*(,n) = 0 (t 0) (10)


The fundamental formulation of our method consists in using the
functional relation (7) in the sense of the known method of successive
approximations for the calculation from a prescribed approximate solu-
tion which already satisfies the indicated boundary conditions of a
sequence of corrected functions which converges toward the actual solu-
tion of the problem; one substitutes the last obtained approximate solu-
tion every time on the left side of equation (7) and integrates the
resulting partial differential equation of the type of the inhomogeneous
heat conduction equation.

The examples so far calculated numerically showed that the iteration
process is obviously convergent. Nevertheless, a general proof of this
fact would be very desirable and we reserve returning, in a given case,
to a mathematical examination of these problems. (Compare also Section V.)

One may characterize the method by stating as the desired result a
continual improvement of a given approximate solution in the sense of
Oseen's method of linearization. Then this linearization of the hydro-
dynamic equations of motion (which, of course, for the boundary-layer
flow taken by itself is not permissible) consists in introducing the
velocity loss u* and in neglecting all nonlinear terms in u*,v and
their derivatives. From equation (3) one would thereby obtain


U(s) aLu 62u + u*a = 0
6B 2 dB

thus on the left side (aside from the term u*" which, however, does
ds
not alter the character of the equation) precisely the expression which






NACA TM 1317


also appears on the left side of our initial equation (7).
In integrating (under the boundary conditions (8), (9), and (10))
the differential equation

u* 2 = f(S,) (11)
at an2

into which had been introduced for abbreviation the function

f ".UI-16u _n U 6u (12)
f(l f) --(- -( --U (12)
U( 6y)J0o 6 u() OE
to be regarded as known in the sense of our approximations, one may now
use successfully the difference calculation. For the homogeneous equa-
tion this has been done, simultaneously with a proof of convergence, by
R. Courant, K. Friedrichs, and H. Lew For the inhomogeneous equation
here dealt with, the proof of convergence together with a formula for
error estimation may be found in a paper by L. Collatz7.
If one covers the right upper quadrant of the E.n plane by a net
of lattice points with the coordinates

Ep = pk Ta = aZ (p,a = 0, integers)
(compare fig. 1) and introduces at the same time, with a view to later
applications, the new designations

up,o = u(tp,1r), u*p, = u*(Pp,7i)

[Laul ru] Iul = EPIJ
[ J p,4m o LO[jpJLl [ojitp,= H] p,o


6R. Courant, K. Friedrichs, and H. Lewy: "Uber die partiellen
Differenzengleichungen der mathematischen Physik" (On the partial differ-
ence equations of mathematical physics), Math. Ann. 100, 1928, pp. 32-74,
particularly pp. 47-52.

7L. Collatz: "Das Differenzenverfahren mit hoherer Approximation
fur lineare Differenzengleichungen" (The difference method with higher
approximation for linear difference equations), Schriften des Math. Sem.
u.d. Inst. f. agewandte Math.d.Univ. Berlin, Bd. 3, Heft 1, 1935.







8 NACA TM 1317


there corresponds to the differential equation (11) the difference equa-
tion of first approximation


U p+l,C U*p,o
k


u*p,a+l 2u*,a


If one selects the step magnitudes k and I
not independent of each other but so that


2
2


in | and n direction


(14)


equation (13) is transformed into the simpler difference equation


* U*p,a+1 + u*p,a-l
u p+l,o = + kfp,


(15)


It can be shown that the solution of equation (15) for the corresponding
boundary-value problem for 1--*O and therewith also for k--wO con-
verges toward the known solution of the boundary value problem of equa-
tion (11).

Since the values


u p,o (P ? o)


u*0,a (a o)


fp,o (P oa ro0)


are known, one may, according to equation (15), successively calculate
all values


u p,


(p 0,0 2 o)


progressing stepwise from lattice point column to lattice point column.

Actually, however, we apply another correction at every step in order
to compensate the systematic error originating by the fact that the
derivative appearing on the left side of equation (11)

1[u*

[^JP, a


+ u*p,a-l


= fp,a


(13)






NACA TM 1317


was replaced by the difference quotient of first approximation

u*p+l,a u*p,a
k

(Compare the following section.)

One notes that due to the transitional condition (10) for the entrance
profile u 0 necessarily must vanish for a-a-o dnd that fp
likewise vanishes for c---, since even the approximate solution used
for the formation of fp was supposed to satisfy the condition (10);
hence one recognizes that the corrected solution (obtained with the aid
of the difference calculation in the manner described above) also satis-
fies the transitional condition (10).


III. PRACTICAL EXECUTION


In practice one may vary the method in such a manner that one does
not at all require an approximate solution prescribed at the outset in
the first quadrant of the E,rn-plane; one rather determines this approx-
imate solution for every step and then improves it to the desired accuracy
before passing on to the next step. Thus one applies a combined system
of continuation and correction.

If one deals with the flow about a profile contour, the initial pro-
file at the point s = so is best taken from the well-known power-series
developments by Blasius-Hiemenz, the coefficients of which for the first
three terms were given in table form by Howarth8. For reasons of con-
vergence, these broken-off series will represent a good approximation of
the solution of the boundary-layer equation only at a small distance from
the forward stagnation point (s = 0) of the outer potential flow. In
the permissible range they represent, as it were, an improved stagnation
point flow.

Our calculations so far have shown that the series are serviceable
up to s-values for which the "first boundary layer bond"


=O p(s) =- d (16)



8Compare L. Howarth: On the calculation of steady flow in the
boundary layer near the surface of a cylinder in a stream. R & M no.
1632, 1934.






10 NACA TM 1317


which is a direct result of equations (3) and (6) is satisfied with
sufficient accuracy.

In practice, one has therefore to start the calculation by approxi-
mating the function U(s) for small s-values as well as possible by a
polynomial of the form


U(s) = uls + u3s3 + u S5

for the case of a profile symmetrical in free-stream direction, or
respectively, of the form


U(s) = uls + us2 + u3 3

for the case of a profile unsymmetrical in free-stream direction; one
may sometimes get by with only two terms.

After having determined, in the manner described above, the value
so > 0 at which the continuation method may start, one first sets up
the connection (given by equation (6))


=- dt (s > so)
SU(t)

by evaluating the integral on the right side, for instance according to
the trapezoidal rule. One graphically represents the functions t = E(s)
and U = U(s) in a common diagram so that U = U(0) can be immediately
taken from it.

The step magnitudes k and 1, connected by equation (14), must be
selected so that, first, a sufficient number of subdivision points are
distributed over the profiles to be calculated, and second, a sufficiently
rapid continuation in t direction is possible. When profiles of not
too pronounced S-shape (near the separation point) are to be calculated,
eight to ten equidistant subdivision points generally will be sufficient
to define the profile. In upward direction (that is, for large i values)
one will have to take so many subdivision points that the profile dies
out sufficiently gradually toward the asymptotic value U. This provides
a first indication for the selection of Z and therewith also of k.
It should finally be remarked regarding the step magnitude k that it
must be at least large enough to make, for fixed E and variable -,
the derivatives (obtained in first approximation by formation of

difference quotients) take a reasonably regular course (compare the
following discussion). Hence the lower limit is set for k and there-
with also for Z.







NACA TM 1317 11


On the other hand, one will be forced to choose the smallest possible
step magnitude k at points t where the curves u = u(a,const) exhibit
great curvatures (which occurs particularly directly ahead of the separa-
tion point), in order to make a sufficiently exact calculation of the
profiles possible. There 1, too, will necessarily be small. Since,
however, the boundary-layer thickness has greatly increased at the sepa-
ration point, one will have there a great many subdivision points dis-
tributed over the profile. This is in one respect convenient the posi-
tion of the separation point is better defined. On the other hand, the
expenditure of work increases at such points. However, at the end of
this section we shall point out a possibility of reducing the steps
in 5 direction without necessarily having to accept a step reduction
in j direction. At the same time we shall then be able to indicate a
criterion by which the necessity of a step reduction in t direction
may be recognized.

Once a certain selection of step magnitudes has been decided upon,
it is a question of obtaining a first approximation for the values fo 0
appearing in equation (15), in order to be able to execute the first step
in t direction. It should be noted that together with the initial pro-
file at s = so also the values of u for values s < so may be taken
from the series developments. Particularly the values u _C (that is,
-1, a
the profile one step ahead of the initial profile) are thus known.

We then'put for a first approximation of the -ul occurring

in fo,a:i

ul u0, UO-1,
[o0,a k


Therewith jdl too can be evaluated numerically. Our calculation

experience has shown that this integration may be very conveniently carried
out with sufficient accuracy by use of the trapezoidal rule with the aid
of the present subdivision; this can be done purely schematically by
calculation according to tables. For at the t points where the deriva-

ous u d? come to
tives u become very large whereby the values = dqO come to

be of great importance in the calculation of the profiles and must be
determined relatively exactly as for instance in the neighborhood of the
separation point it will be necessary to select small k (and there-
with also Z) values so that a sufficient number of subdivision points
are distributed over the profile to allow application of tie trapezoidal
rule with sufficient accuracy.






12 NACA TM 1317


If one puts, furthermore, with good approximation


u= ua+l uO',-1 (17)


u lo and ul, may be calculated in first approximation.

The values thus obtained will be denoted by k* 11, and [ul 1,J
It will be best to arrange the entire calculation procedure in the form
of a table (compare table I on page 31). With the values obtained
i]1a one will form corrected values of the derivatives -in
according to the scheme

u
ul [Ul] 1,c 1,0
W[ O0,o 2k

whereupon one obtains (with the aid of table II on page 31) a second
approximation [],o for the values Ul., with the values 70a,a,
and Ca taken from the first table. Whereas the derivatives [I0jl

formed in first approximation might show at a few points a an irregular
course, this will generally no longer be the case for the corrected deriva-

tives I The columns for the quantities DCJ, 12j] ,a and
030

2 1,Co occurring further on in table II will be explained only later.
This procedure is continued until the values obtained in the third-
from-last column of the table no longer vary in the desired decimal. In
the examples we calculated the iteration was carried so far that for every
step E0 the values up no longer varied except for an error of
about 1/4 to 1/2 percent of the maximum velocity U(tp) in each case.
For the selected step magnitude k this was the case after two to three
iterations.

Due to the favorable position of the errors, the profiles calculated
in the manner described generally show a very smooth course. If the
u = u(F,const) are concave in respect to the E axis, as is the case
for instance in the flow about the circular cylinder or the ellipse near
the separation point (compare fig. 7 and fig. 12), the convergence occurs
only on one side in the direction from larger to smaller values for u.
The opposite behavior exists when the course of this curve is convex with
respect to the F axis as is the case for instance in the boundary-layer
flow at the flat plate.






NACA TM 1317


If one wants to obtain with the described procedure a calculation
of the u variation as accurate as possible without selecting too small
a step magnitude k, thereby increasing too much the expenditure in cal-
culation, one will find it necessary (as mentioned before) to make at
every step a correction which takes the fact into account that in setting
up the basic equation (15) the difference quotient of first approximation

only was substituted for the derivative 16L


If one were to select instead the representation of higher approxi-
mation
*
uu P+l,a u p-l,a

p, a 2k
one would obtain by maintaining equation (14)

u u* 2ki()
pu+lia = u p-l.o + u p,+l + u pa-1 2up, + p (18)

instead of equation (15).

Since this relation, however, (as can be seen immediately) behaves
considerably less favorably regarding propagation of errors than equa-
tion (15), the profiles calculated with its aid will no longer show the
smooth course mentioned before. Calculation practice has shown that one
obtains very smooth curves if one writes instead of equation (18)

[2 *1
S+ 2k P. u + 2kfP (19)


and forms the second derivative appearing in it according to the scheme


2 pp,+l21


from the first derivative T] already calculated in good approxi-

mation according to equation (17) by Jumping over. However, the case a = 1






14 NACA TM 1317


requires special consideration since I-] is not known at first.
But if one takes into consideration that according to equation (16)


1H,2 p,0 d-
one may put

H ,u p1 p1 jcu
Ip, LIpl $ pI O
iu + = p + ) -) (21)
6Ip, dP 1 p,l 21

and hence calculate the value on the left from 1U already known
LJp,i
according to equation (17).
We now use the relation (19) not as a substitute for (15) in the
sense that the entire calculation is to be made with (19), for it became
clear particularly near the separation point where the derivatives
6u become very large that the convergence relations here can be easily
blurred (unless an especially small k value was selected); the values
UPYU obtained by iteration do not remain quite fixed, but creep on
continuously, although only by small amounts (compare also the remarks
in section V).
Rather we use equation (19) for making a correction in the values
Ulge obtained after the last iteration in the manner described above.
With the aid of the value (l (a 1), (already contained in the
fourth column of table I) to which we add the value 1[ Just cal-
L1_0,0
culated according to equation (21) we determine (taking equations (20)
and (14) into consideration) the values

D u+ [2u -* + ( -
Do = u -1, + 2k = u .-_1 + l -
IN 0O a+1 0[a-I






NACA TM 1317


We now assume, for instance, that the values [u2 ,a prescribed by
table II were the final values even in the first procedure; we then
insert the values Da in table II and calculate with the quantities
A0 + B. appearing in them the values (corrected with respect to 1


21, = Da + 2(Au + Ba)
these values, too, we note in the table. In the last column of this
table we write the values


[2i1,0 = +( *) 1,+


If the corrected values a deviate too much (that is, by more

than 1/4 to 1/2 percent of U) from [u2]l, we calculate with the deri-
Svatives

t 2lJ, l,
6t o,o 2k

once more corrected values according to table III, p.31. The values

[3U1 then represent the final values for the profile at the point
5 = 1.*

For calculation of every step in 5 direction one must, therefore,
calculate three to four of the calculation tables mentioned. The time
expenditure may be estimated at approximately three to four hours per
step. It should be stressed that all calculation operations are of
purely schematic character and can therefore readily be performed by
assistants,

The values obtained are plotted on millimeter graph paper and the
curve drawn through them. If slight scatter has resulted, after all, at
one point or the other, one eliminates it with the aid of the drawing
before starting on the next step.

If the graph of the profile calculated just now shows that the curve,
due to the increase in boundary-layer thickness, at the upper end no
longer dies out gradually enough toward the asymptotic value U, one adds,






NACA TM 1317


in calculating the following step, and T subdivision point in upward
direction.

The following condition should be mentioned which became evident in
the practical calculation. If a step magnitude not sufficiently small
is selected, two successive profiles may, due to accumulation of errors,
show points where they are somewhat too close, or else somewhat too dis-
tant from each other, compared to their actual course. In the calcula-
tion this can be recognized by the fact that the third profile following
these two profiles shows a behavior, at these points, compared to the
second profile opposite to the behavior of the first compared to the
second profile. For T values at which the first two profiles were
too close one notices a gap somewhat too wide between the last two and
vice versa. If one does not want to repeat the calculation with smaller
steps, one may, as was found practically to be useful, once omit the
corrective calculation mentioned before for the profile to be calculated
next, thereby eliminating the fluctuating of the profiles, and may then
continue calculating in the normal manner.

According to our calculation experiences one can recognize that the
step magnitude k must be reduced in t direction by the fact that the
two i values obtained in the corrective calculation which pertain to
the same i( (thus in the example considered above the values o2]

and [3],i) deviate from each other by considerably more than 1/4 to
1/2 percent of the pertaining U value.

If a new step magnitude in the E direction, kl, is selected kl< k
(for instance, kl =-) there appears as a result, because of equation (14),

also a new step magnitude.


11 = 2k


in I direction. If one wants to continue the calculation with the
smaller steps kI for instance starting from S = r one needs as the
initial values for further calculation the numbers


u(Eroll) u(r-klall) (a = 1,2, .)

The first named numbers may be read off directly on the profile curve
for t = tr already obtained. In order to obtain the latter, a double
graphic interpolation must be made. One plots versus F the values






NACA TM 1317 17


u tp zl) (a = 1,2 .) read off for the values of = 5p r)
from the curves of the previously calculated profiles. Generally it
will be sufficient to do this for the values u( I aoll) of three suc-
cessive profiles, thus for 0 = r 2, r 1, r. From the curves drawn
through them u = u(t,all) (a = 1,2 .) one may then read off the
values u. lr-klal) (a = 1,2 .).

If a boundary layer is to be calculated up to the separation point,
it will in general be necessary to select, in the proximity of the sepa-
ration point, rather small steps k. Since, however, due to the large
increase in boundary-layer thickness, the profiles are here very elongated,
one would obtain, because of the small step magnitude I in q direc-
tion, a very great number of subdivision points over the profile; this
would of course increase the time expenditure for the calculation of a
step. However, one may save a great deal of calculation expenditure by
selecting, instead of equation (7), for instance

1 au* 62u* 1 du d + + 1 (22)
2 0 atN2 u(t)'70To d (I 2) 2 d

as the initial equation, and then performing the integration as before.
If one again denotes the step magnitudes in t direction by k, those
in I direction by i, one obtains instead of equation (14) the relation

z2
k -


To the same I as in the first considered case, therefore, there corre-
sponds half the step magnitude in t-direction.

In this manner the step reductions were carried out for the following
examples of boundary-layer flow on the circular and elliptic cylinder.
The convergence of the iterations now occurred no longer only on one side
toward the limit but alternately (except for the values assumed for small
.a.

The numerical calculation showed further that a further step reduc-
tion in t direction, still for the same 2, for instance with the aid
of the initial relation

i u* 8-lu* 3 -u a 3 dU()
4as nt2 (sadvi abec e 4

was not advisable because the values assumed in the upper profile parts







NACA


'T .. 7 1

TM 1317


on the right side are given as differences of two (approximately equal)
large numbers and therefore scatter widely; by this the convergence rela-
tions may be concealed. Thus, if one is forced to reduce the step mag-
nitude k still further, one will do so in the manner described above
with the aid of the relation (22). It was found that one arrived in this
manner, even for the extreme example of the circular cylinder, at a
tolerable work expenditure even for the steps immediately ahead of the
separation point.

The separation point k = ta (and therewith a = Ba) is found by

graphic interpolation, or extrapolation, of the values L 0

contained in the tables.

The example of the Blasius flow at the flat plate shows very clearly
the high degree of accuracy attained with this method. Here the profile
obtained by continuation could be compared with the exact profile. After
calculation of six steps, the calculated values deviated so little from
the exact ones that they could hardly be distinguished within the scope
of drawing accuracy. The differences amount to less than 1/2 percent
referred to U.

In order to enable following the mode of calculation in detail, we
add the complete calculation of the first step in the continuation of a
Blasius profile at the flat plate.


IV. NUMERICAL EXAMPLES AND RESULTS

1. Continuation of a Blasius Profile at the Flat Plate.


The value s = 0 is to correspond to the leading edge of the plate.
For the boundary-layer equation (3) which because of p'(s) = 0 is
simplified to

v a- + v -, = V2-3


together with the continuity equation


hav av
Vs + = 0


1







NACA TM 1317


then exists according to Prandtl-Blasius,
of the form


Vs= :1 U'(0)
2


for which applies


v --40


for


for


as is well known, a solution



^^ k


with


for


S--10



T--4cO


and all T



and all s 0


The function 9 = c(p) satisfies the ordinary differential equa-
tion of the third order

(p = Vp,,


and the boundary conditions


9(0) = 0


p'(o) = 0


lim 9'() = 2.


The values of ('(5) are


C j q'(P )


0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0


0
0.0664
0.1328
0.1989
0.2647
0.3298
0.3938
0.4563
0.5168
0.5748
0.6298


We choose U = 1


to be taken from the following table:


1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0


1 j'(C)


0.6813
0.7290
0.7725
0.8115
0.8460
0.8761
0.9018
0.9233
0.9411
0.9555


2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0


0.9670
0.9759
0.9827
0.9878
0.9915
0.9942
0.9962
0.9975
0.9984
0.9990


so that we may put


S= S


v,--4U







NACA TM 1317


and start our continuation procedure at s = 1. As step magnitude in
q direction we take Z = 0.6 so that k becomes equal to 0.18. The
initial profile then may be taken directly from the above table, whereas
the profile one step farther back, thus the profile at s = 0.82, is to
be obtained from this table by graphic interpolation. The values are
contained in the following table:


TI

0
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4


u(0.82,ri)


0
0.225
0.430
0.626
0.782
0.894
0.955
0.982
0.995
1


T I u(1,l)


0
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.0
6.6


0
0.1989
0.3938
0.5748
0.7290
0.8460
0.9233
0.9670
0.9878
0.9962
0.9990
1


Six steps (that is, up to s = 2.08) were calculated by the method
described. The calculation of the first step is contained completely in
the table added at the end of the report. The results are represented
in figure 2.


2. Circular Cylinder According to Hiemenz.


Hiemenz9 measured
of diameter 2r = 9.75
the flow at a velocity


the pressure distribution on a circular cylinder
centimeters immersed in water and approached by
of 19.2 centimeters per second.


In order to make the quantities appearing in the basic equations
dimensionless, one introduces the reference length 2 = 1 centimeter
and the reference velocity Vo = 7.151 centimeters per second which corre-
sponds for v = 0.01 centimeter2 per second to a Reynolds number


R = -- = 715.1



9K. Hiemenz: Die Grenzschicht an einem in den gleichfdrmigen Flius-
igkeitsstrom eingetauchten geraden Kreiszylinder (The boundary layer on
a rectilinear circular cylinder immersed in the uniform fluid flow).
Dissertation G'3ttingen, 1911, published in Dingler's polytechn. J. Vol. 326,
1911, pp. 321-342.







NACA TM 1317


then the velocity distribution measured for 0 = s = 7, that is up to
the separation point, observed shortly before s = 7 (corresponding to
an angle a of 800 to 82 from the forward stagnation point) may be
represented satisfactorily by the polynomial


U(s) = s 0.006289 s3 0.000046 s5

On the basis of the previous indication, the solution of Blasius-
Hiemenz could be used up to the value s = 4.5 (a ~ 550) so that our
calculation starts at s = 4.5 (as does GBrtler'slO). The value I = 3.4,
and thus k = 0.08 were selected as step magnitudes for the first steps.
The representation of


S= 1 dt and U = U(s)

againstt s may be seen from figure 3.
against s may be seen from figure 3.


The initial profiles at E = 0.08 and =

tables, are compiled, together with the values of

from the same tables, in the following table:


0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
00


u(-0,08,T)


0
1.282
2.229
2.871
3.269
3.488
3.596
3.649
3.667
3.672
3.674


0
1.289
2.2268
2.948
3.384
3.628
3.749
3.813
3.834
3.839
3.842


0, taken from Howarth's
--4. resulting
6814.5


[ s ]
ss s4.5j


0
0.008
0.104
0.249
0.367
0.448
0.497
0.529
0.538
0.540


When the latter values are used, the calculation of the first step
requires only one worksheet of the type described before. With the step
magnitudes indicated, first four steps (up to t = 0.32) were calculated.


10See footnote 3.






22 NACA TM 1317


The profiles obtained are represented together with the initial profiles
in figure 4 (partly displaced with respect to each other).

With the initial relations (22) as a basis, five further steps
(up to E = 0.52) were calculated for the same 2 = 0.4 and the required
k = 0.04. Likewise with the use of equation (22), one step (E = 0.54)
with I = 0.08 = 0.283 and k = 0.02 and finally two more steps
with I = 0.2, k = 0.01 (up to 5 = 0.56) were calculated. The pro-
files are also represented in figure 5.

By plotting of the values the separation point was found
to be ksep = 0.5697, that is Ssep = 6.87 (compare figure 6). Thus
all together twelve steps were to be calculated. Figure 7 shows the
curves u = u(t ,const). Their steep decline in the neighborhood of the
separation point is remarkable.

Figure 8 shows a comparison of a few of the profiles obtained by us
(-S) with those of Blasius-Hiemenz (-- --B-H), Pohlhausen (- -P), and
Gortler (- -G) which were obtained for the same pressure distribution.
The comparison shows, first of all, that the Blasius-Hiemenz solution
becomes insufficient in the neighborhood of the separation point; the
reason obviously lies in the fact that the series developments used con-
verge for large s only slowly, if at all, and that, therefore, with -
merely the first three terms the actual course is not satisfactorily
represented there.

Our values agree best and most systematically with those obtained
by GCrtler. The differences are increasingly noticeable toward the sepa-
ration point. The deviations from the values obtained by Pohlhausen,
considered as a whole, remain for this example within tolerable limits
although a systematic variation of the differences cannot be determined.
It is remarkable that the differences assume higher values precisely in
the proximity of the velocity maximum (t ~ 0.36, s ~ 6, a ~ 710) (com-
pare the curve for E = 0.32 represented in figure 8) while again sub-
siding to some extent toward the separation point.

The separation point was found according to Girtler in good agree-
ment with our value Ssep = 6.8, according to Hiemenz at Ssep = 6.98,
and according to Pohlhausen at Ssep = 6.94. An approximately correct
position of the separation point is, therefore, by itself not yet deci-
sive for the usefulness of a method.

llCompare K. Hiemenz, paper quoted in footnote 9, H. Gortler, paper
quoted in footnote 3, and K. Pohlhausen, "Zur naherungsweisen Integration
der Differentialgleichung der laminaren Grenzschicht" (On the approximate
integration of the differential equation of the laminar boundary layer),
Z.A.M.M. Bd. 1, 1921, pp. 252-268.


'-






NACA TM 1317 23


3. Elliptic Cylinder of the Aspect Ratio 1:4.

As a further example, we calculated the boundary layer for an ellip-
tic cylinder of the aspect ratio 1:4, taking as a basis the pressure
distribution resulting from the potential theory.


S= zn and Vo = 4.3 Uo
10

were chosen as reference quantities for the introduction of.dimensionless
quantities, with lo being half the circumference of the ellipse and
Uo the free stream velocity. The dimensionlesss) velocity at the edge
of the boundary layer could be taken directly from a table by Schlichting
and Ulrich.12 It is represented in figure 9 together with the function

dt
= .(s) =4.2 RY
0.2 UV t)

In the interval 0 < a 0.2 it was possible to represent U = U(s)
satisfactorily by the polynomial


U(s) = s 5.116 s3

The initial profile, however, was chosen at s = 0.163 (E = 0.25) for
the reasons mentioned before. For the first six steps 1 = 0.5 and
k = 0.125 were taken as step magnitudes. The two initial profiles are
represented in the following table, together with the values

as = 0.163

T u(-0.375,i) u(-0.25,i) -s =0.163
a s j= 0.163

0 0 0 0
0.5 0.0573 0.0589 0.0949
1.0 0.0951 0.0990 0.2672
1.5 0.1166 0.1226 0.4235
2.0 0.1268 0.1340 0.5197
2.5 0.1310 0.1389 0.5703
3.0 0.1323 0.1403 0.5858
0.1327 0.1408

12H. Schlichting und A. Ulrich, "Zur Berechnung des Umschlages laminar-
turbulent" (On the calculation of the transition from laminar to turbu-
lent) Bericht S 10 der Lilienthal-Gesellschaft (1940), pp. 75-135-






24 NACA TM 1317


From 9 = 0.5 onward the steps could, first, be increased. The
following further steps were calculated: Seven steps with
Z = 0.5 = 0.7071 and k = 0.25 (up to t = 2.25), three steps with
Z = 1, k = 0.5 (up to t = 3.75), four steps with I = 2 = 1.414
and k = 1 (up to t = 7.75), and nine steps with Z = 2 and k = 2
(up to 5 = 25.75).

With the aid of the starting equation (22) another step reduction
was made. The further steps were: Three steps with I = 2 and k = 1
(up to t = 28.75) and finally one step with Z = 42 = 1.414 and k = 0.5
(up to E = 29.25). A complete calculation was thus made of 33 steps
altogether.

It became clear that selection of larger steps is not advisable,
particularly at the point where the curve U = U(s) turns from its
steep ascent to the flatter course (compare figure 12).

On the "high plateau" of velocity distribution itself one could
have chosen steps somewhat larger but they would have had to be reduced
again when approaching the separation point. A large number of the pro-
files we calculated can be seen in figure 10. The separation point was

determined from the variation of [- as ssep = 8.475 (compare

figure 11).

Since Schlichting and Ulrich completely calculated13 the same
example once according to the ordinary Pohlhausen method (P4-method),
and then according to a Pohlhausen method modified by taking a polynomial
of the sixth degree as a basis (P6-method), the comparison could be made
for a number of profiles. The results are compiled in figures 13 and 14.

As far as the pressure minimum the deviations between our curves
and the P4- and P6-curves are not too large. However, larger deviations
appear in the proximity of the separation point. There the profiles of
the original Pohlhausen method agree with ours better than the profiles
of the P6-method, especially for small q values. The resulting separa-
tion point was, according to the P4-method, at sep = 8.38, according
to the P6-method, at Ssep = 8.26; thus these values (especially that
of the P4-method) do not deviate too widely from our value.


13Quoted in footnote 12.






NACA TM 1317


V. REMARKS REGARDING THE CONVERGENCE OF THE ITERATION PROCESS

Regarding the conditions of convergence of the iteration process
described in section III, we can prove the following theorem which will
probably be sufficient for the requirements of practical calculation.
If there applies for the profile at the point t = Ep for all I 1oo


0 [u] < U(yp)


and


f( p)


[<] t p,T
< I +


the sequence of the velocity values obtained by the iteration process

["nup+l,a (a$ ao)

converges with increasing n.
Thus one is always able to predict, when calculating a new step,
whether the iteration process will converge. The presuppositions of
the theorem are satisfied with certainty when for all 9 s1 oo

0 o [u: < U(Cp) and [u 0
-pp,

apply as is the case for instance for the profiles before the pressure
minimum; for then


is 5a s cothr
I tp, q TJ p,1

is valid if n* is'selected so that


uETPI, =


= [U]pn < u U(p)


(TI T oo)


T, 1 l6 p,






26 NACA TM 1317

The presuppositions are satisfied also for the S-shaped profiles
beyond the pressure minimum when -u does not become too large. For
the examples we calculated, boundary-layer flow for circular and elliptic
cylinder, the presuppositions are satisfied, up to the separation point.
Proof of the theorem: If one takes into consideration that according
to the procedure of section III one has to put


[u n+1lp+l,


p,o+1 4 u*p,a-l
= +


k [oLu 1, p+ T up-1,T




22k


k iUu]p+l,g Up-l,a
U(--- uI
U(5p) 2k


there follows with


= Max
T=1,2,... ,


u IlP1,T un-.lp+1T I I


(a o)


with the presuppositions taken into consideration, obviously


uul+1 P+1. u ]p+1,l


1
2U( 1p)


2- (IU(p) upl)
2U(p)


dN +1,0a


a- [du] ,


dn]+1,0


Slu1 dp]+1a






NACA TM 1317


with 0 < a < 1 and a behind independent of n. Furthermore, one can
see for oneself that a (0 < a < 1, independent of n) can be chosen
so that simultaneously the estimations


uLu p+l, a p+I12 L c1




uuljp+la Iu p+la a [dup+i,a

exist so that

[du+1]pl+ a [dn pl,

must be true as well.

Hence there exists the limit


up+ ,T = im [in]p+l,






I\-^ L + 1
= [uIp+la + ([u2+1,0 [u p+ ,a +



up+1 -un-i p ) + ..' (a co)
since the series at right may be majorized by a convergent geometrical
series.
One recognizes further that if equations (18) or (19) are used
instead of equation (15) if convergence of the iteration process
can be proved under the assumptions that for t = Fp and all q 4 io


0 Is (U]tp,, < U()






NACA TM 1317


and


since one then minds the estimations[u
since one then finds the estimations


(u 1 p,~


U(p) [ult


[u ,+1,


[uu+ p+,a- u]P1,o <


U(p- ]p) Id~
+ ^e )1^1"


thus


[u+1 p+ 4-C [ iduop+,a

again with 0 < a < 1 and a independent of n. These presuppositions
are satisfied for instance for the profiles before the pressure minimum.
However, in the proximity of the wall, if the profiles there show an
approximately rectilinear course, the convergence will take place only
very slowly.
Beyond the pressure minimum one can, therefore, not arrive at a
general statement on the convergence. As mentioned before, our calcu-
lations in the proximity of the separation point showed that the case
of divergence may actually occur. Therewith the procedure we selected,
using the relation (19) merely for the correction calculation, proves
to be perfectly reasonable also from the general point of view now
considered.
Correspondingly, one recognizes that the iteration process performed
with the aid of relation (22) certainly is convergent at the point
S= Sp for o S ao if there for all T) T oo

o u]pn < U(p)


ulp.+1 1 up 1,I<






NACA TM 1317

and


(1) <
U(p)


+ u for -l


1 + for >1
2 U( P) f (Ep ) 2


If one takes generally the initial relation


1 u* 62u* 1 3u 11u u_
I aU

(1i- i) d


convergence at the point t = tp for a = ao
the following sufficient conditions: For all
valid


would be assured under
Y= nuo there shall be


)+ for 1
m' ( ) U()


* UC1 P) U(t-) M


6(


(m> 1)


and


CU(p)


0 < []p,< ( Ep)






30 NACA TM 1317

Thus, if one wants on the right side a positive limit also for


[u],1 (Y---O) U(o

m must be not greater than 3.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics












NACA TM 1317


.-4

+
II
0
-. +


- r4
5


o0 *


4
+


i4






















92
-*1




4
















B?5


0
o





5I
I l
0


4+


0







MI
o
































SO


|I


0 *


-4




0
.-4
r 0


I rI
,^ r

11


I,


-0
fI









0a
o +










S0











III
C























r1 .
0
















2ii


i
o

























.5o
03











^ 5


- 0
* 5 .s
a9 m"


In


0
4 .2


0


0






C
o





c[
*^ 0





0
OK




D

''


0 .


o *






0 -
01


a .


0 .


0 .


S .


a a .





o -
01


01

0 +
S*
r-d, 0
u
4f, ft
F1
Is
-'
0
0
C
II


01
4
-- ^ ------4


* *


1-1















II

o o o, a o o



8



0 0 a 9 9 0, 9 7 o o
9 9 9 0






0a a 1' % 0' ID m W
L- N 0 -- -* I 0% C 0 00 0C











9_o* 9 9 9 9 9 9 0
S 8






o







-, 9 C 9 o!
0 0C8 8








o N 0r-0 0 0
0 0
C3 C 0 0 C
t C 0? 0 0 0 0 0 0
0 C Y\ r C





0 0 0 0 0 0 0 0


H 0 0 0 0 0 0 0







S0' 0" 0 0 00 o






--- S -i< -. -I .0 .^fl co
O ? l i?' r- r 9 a 0' ?r H














S05 0 0* 0 0 0 8
S6 0o o o 0 o




o, 9u CU! .4 1' (41






d 9 0

'.V0 -7 C .0 Cs 0 a C (
.




0 I 0 a a'4 01 -m rr4 c- 0 0 0 0 r-0 *
n t| d0' d d d (b-^d si1 ~ 00co ^0 Co C 0
a < :0 '0 *' < 0 ^ o? f '

0lROh~

11 r 0
a1 Q" ~- F* 1*' -T ^- P-0 r m O'lN i- 3
0 pp o u^^o 3^1 'toid 'iiQFI.^o i o
rf ?^,8 ^fi ,^ ^? g8



c^ V C f O < O ^ O ^ A


NACA TM 1317


_I







1
1
r


I
i


.I
I










i
1


I







1


;I








'I











1;1


b
0








Sg
U
0








S.I










NACA TM 1317


AL a 0 H 0% 0 -
N-r- 'D inn 0 NQ in 0C
-4 in if\ 0 in % 0


II






9 N- H 0% C! 9 C o
r 1. 0 0. 0 0







--- 0 0 0 0 0 0 0 0
c o no o 0N
Sio a 0 0 0 0 0
9I 9 9 9 9 9o 9 o9 o o o


H0




0 0 0 0 0 0 0 0 0 D 0 0 H













II
04 0 o i N C- cu in H 7
HO C N 4 Nm0

0 0 9 0 9 0 0 9 0 9 0









in
+ 0 9 9 0 0 0 h 0 0
I 0
ts







09 m 9P 9 9 0 0 0 0
U i m 9 0 0 0






0---M r 01 n N N in r 0 0 0 r i O % 0% 0%
+ '-s i NH C C' C Ci 0' 1i in 0





















l ----0 Oj 0 H0 0w 0 0 0 0 00 0 -
0 N 0 0 d, N n 0 0a






cu0 0 H 0 H N0 0 0 0 0 -
CQ 0 0 0 0 0 0 0 0









,-, o oy oo '" 0 of 9 o o 0











iifS ~ ~~

^o (\ co r o do fi w -^ o ^ c










34 NACA TM 1317


11 1



SII





S-
r-I b 0- ( o o1- -i M m c co
0 k Fl N h u rn 0
S0 0 C 0 0 0 0 H





II
0 +






H 0 r 0 t l n .- 0 0 0 0
0 CU 0 H 0 0 C H 0






S0 0 0 0 0 0 0 0 0 0 0
a n 00 N00

S- p0 ,0 -r 0 0 0
4 4 4 4 4


0 a O n 0 0 0


C C.) 0I ? I a I I

a, -" 0 I- 0 0U0 \ 0 0\ L0
o -to 0 0o o 9o o- o- H
0 c H -4 C 0D
0 C CU C O O 0 0 0
Sa o o a o o o o o




0 O O 4 l CO 9 O O O
b 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0






0 -1 Lr, 4 N HO m m m fl UN E O\ ONO\

II 9 I I I C
0

r o 4^ 0 n 4 w (
U 44 O0 f HCUa H NO





__ M j a U\ 44 0 0A 400 0 00
0
0





0D HH; 0. 44% 44 q C?0 rlO ? C CUCO o o

n N4 Cr CP CUC4 HOJ HH 00 00 00




V.0 CU 90 4 0 '1 CU _O 4 0 '\.\0 CUL


it





MACA TM 13.7 35





71






____________ 2_.3)
A


Figure 1.- Tre net of the lattice points.






36 NACA TM 1317



8




7




6




5
Calculated according to
1/ the difference method
--- Exact solution
4




3
3 2.08
1.90
1.72
1.54
2 ---- w 1.36 -
1.18








0 0.2 0.4 0.6 0.8 1.0
u


Figure 2.- Continuation of a Blasius profile at the flat plate.







NACA TM 1317 37










"-C-


-r4




r4t
0


ODO

I0 1

S\ II






,UI




--- __ _- __- q-


0\
S0D o





4. 4. I. I
D <> o 0 0.


i d j i w' o







38 NACA TM 1317










ro
-- U %







CD
00
0 ,









_o / I i\- i








0
7-I



co0 o ~ O
4 -- j O a
Yt;

to cu







NACA TM 1317 39


Figure 5.- Further profiles for the circular cylinder.







.NACA TM 1317


.52 .53 .54 .55


.56 .5697


Figure 6.- Determination of the separation point for the circular cylinder




&,/3.6 3. ,4.0

4
S- --------- 2

0 '2.0


3



U((,const)











0 0.2 0.4 0.6


Figure 7.- The curves u = u(S const.)


for the circular cylinder.







NACA TM 1317 41



0


lo


LO











0 .0










.4
--


Sdt
0





0' Li
CL NL





0 0.

I II \ C
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s---^!~ l







NACA TM 1317




2i









2.
*
-J


Figure 9(a).- The functions t = E(s) and
cylinder.


32

U(s)
0.29r- 28


0.28

U (s)

0.27


U = U(s) for the elliptic


Figure 9(b).- The functions 5 = E(s) and U = U(s) for the elliptic
cylinder.


0.27


0.23
U(s)
0.19


0.15


0.26 -



0.25-







NACA TM 1317 43


0.1 0.2
U

Figure 10.- Velocity profiles for the elliptic cylinder.






NACA TM 1317


0.015


0012


0.009
[dul
[d1J 71--0



0.003


0


Figure 11.- Determination of the separation point for the elliptic cylinder.






NACA TM 1317 45






8



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///S
C-,




















O o
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0 0


II
8 0 0 0
1-I







46 NACA TM 1317








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L











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o, m

a



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\b
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MID







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i s ^\\\
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----- N v cn







NACA TM 1317


0 0.1 0.2

Figure 14.- Comparison of the profiles for the elliptic cylinder.
Figure 14.- Comparison of the profiles for the elliptic cylinder.


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UNIVERSITY OF FLORIDA


262 0801 09 0