Behavior of the laminar boundary layer for periodically oscillating pressure variation


Material Information

Behavior of the laminar boundary layer for periodically oscillating pressure variation
Series Title:
Physical Description:
9 p. : ill ; 27 cm.
Quick, August Wilhelm
Schröder, K
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


An extract is presented of theoretical results from a more detailed report on the growth of the laminar boundary layer on an undulated surface. For a surface which was undulated about the mean of a flat plate, the drag was shown to be appreciably less than that for a flat plate at very low Reynolds numbers; however, the onset of transition on the undulated plate occurred at lower Reynolds numbers than on the flat plate.
Includes bibliographic references (p. 42).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by August Wilhelm Quick and K. Schröder.
General Note:
"Report No. NACA TM 1228."
General Note:
"Report date September 1949."
General Note:
"Translation of "Verhalten der laminaren Grenzschicht bei periodisch schwankendem Druckverlauf." Ludwig Prandtl zum 70. Geburstage, Schriften der Deutschen Akademie der Luftfahrtforschung, pp. 247-255. (To Ludwig Prandtl upon his 70th birthday, Publications of the German Academy for Aviation Research), Berlin 1945."

Record Information

Source Institution:
University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003759951
oclc - 85847564
sobekcm - AA00006234_00001
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Full Text





By August Wilhelm Quick and K. Schr6derl

The calculation of the phenomena within the boundary layer of bodies
ianmersed in a flow underwent a decisive development on the basis of
L. Prandtl's trains of thought, stated more than forty years ago, and
by numerous later treatises again and again touching upon them. The
requirements of the steadily improving aerodynamics of airplanes have
greatly increased with the passing of time and recently research became
particularly interested in such phenomena in the boundary layer as are
caused by small external disturbances. Experimental results suggest that,
for instance, slight fluctuations in the free-stream velocities as they
... occur in wind tunnels or slight wavelike deviations of outer wing contours
From the prescribed smooth course as they originate due to construction
inaccuracies may exert strong effects on the extent of the laminar
boundary layer on the body and thus on the drag. The development of
'turbulence in the last part of the laminar portion of the boundary layer
is, therefore, the main problem, the solution of which explains the
behavior .of the transition point of the boundary layer. A number of
reports in literature deal with this problem, for instance, those of
"Tollmien, Schlichting, Dryden, and Pretsch. The following discussion
of the behavior of the laminar boundary layer for periodically
oscillating pressure variation also purports to make a contribution to
that subject.

The attempts to calculate such phenomena as undertaken in literature,
for instance, by Dryden and Pretsch, were based on the calculation method
by Pohlhausen. Very early separation phenomena resulted for very slight
variations of the existing pressure distribution; thus there came up for
discussion doubts uttered by L. Prandtl among others as to the admissi-
bility of the Pohlhausen method which, due to its using only a single
parameter, was possibly not capable of fully embracing all phenomena.

"nVerhalten der laminaren Grenzschicht bel periodisch schwankendem
Druckverlauf." Ludwig Prandtl zum 70. Geburtstage, Schriften der
Deutachen Akademie der Luftfahrtforschung, pp. 247-255. (To Ludwig
Prandtl upon his 70th birthday, Publications of the German Academy for
Aviation Research), Berlin 1945.

i The more detailed original report was published under the same
:i:itle as UT 1257.

I Y "I'll 7 11 5- 7c'T '


NACA TM 1228

Since a method of Schr6der recently described elsewhere permits a reliable
calculation of the laminar boundary layer, It could be shown with the
aid of examples that the boundary layer actually Is very sensitive to
slight oscillations of pressure and tends easily toward separation.2

It was assumed for the calculated examples that the flow concerned
is the flow about a plate where the undulation starts only after an
initial plane section. After a portion with constant flow velocity U(s)
(as customary, let a and n be understood as a system of generally
curvilinear tangential and normal coordinates) a sinusoidal U(s)-distribution
then sets in. The boundary layer calculation was based on dimensionless
quantities, with the velocities referred to the constant free stream
velocity Uo and the lengths referred to the length of the starting
distance L, beginning at the leading edge of the plate.

Figure 1 shows the results of the calculation on an example with the
wave length X = 0.072 and tl.h maximumm fluctuation in the U(s)-distrlbution
equalling 1/2 percent. Although the velocity profiles were measured
at the position s indicated numerically at each profile a clearer
picture was obtained by grouping neighboring profiles according to rising
or falling variation of U(s) as indicated by the arrows above each

One recognizes from this example that the boundary layer overcomes
three waves merely by periodic deformations of the profiles; at the
fourth wave, however, a very weak reverse flow begins to appear but
subsides again upon increase of U(s), that is, upon pressure drop.
However, at the next wave the picture changes completely. A strong
reverse flow now appears which subsides only in the profile parts next
to the wall, whereas in the central profile parts the reverse flow
becomes still stronger.

From this and further fully calculated examples we may conclude
that every undulation, even the weakest, in the U(s)-dlstribution finally
leads to reverse flow, if only the calculation is carried sufficiently
far. Figure 2 shows the corresponding streamline pattern on which one
can see particularly clearly the setting-in of the reverse flow and also
the formation of a small vortex.

A few remarks concerning the physical interpretation and thus the
quantitative evaluation of the calculation results are to follow. The
boundary layer actually proves to be extremely sensitive to small
periodical oscillations in pressure. Strong variations of the displacement
thickness are connected with it. Figure 3 shows the displacement thickness
for the example described compared to that for undisturbed pressure
distribution; the periodic variation and the considerable amplitudes

2We had the great privilege of several interesting and stimulating
discussions with Prandtl on the subject of this report.

NACA TM 1228

of the displacement thickness can be recognized. These periodic variations
of the displacement thickness cannot continue without retroaction on the
flow since they obviously must be superimposed on the wall undulation and
lead to a modification of the external-pressuro distribution. TllU. c.hango
in the external-pressure distribution can, however, be avoided by
amplifying the wall undulation by the amounts of the boundary layer
undulation so that for each Rcynolds number a new wall results. Thus a
boundary layer calculation yields, according to the selected Reynolds
number, a series of boundary layer flows along walls of different undulation.
In this sense figure 4 illustrates for the present example walls corro-

sponding to Reynolds numbers R = 10 105, and 10 Taken as a basis of
comparison, the effective wall, which corresponds to the U(s) distribution
for a potential flow, shows an undulation so weak that it is hardly
recognizable in the figure. The figure shows further that for small
Reynolds numbers a relatively strongly unduilated wall is levelled by the
boundary layer, whereas for large Reynolds number a relatively weakly
undulated wall is equalized by the boundary layer. Hence it follows
that for a constant undulation and altered Reynolds number for instance
velocity increase the boundary layer becomes less effective toward
equalizing the wall undulation and thus the effective undulation increases
with growing Reynolds number. The starting point of the reverse flow
on such a wall should, therefore, travel forward with increasing velocity.
The corresponding facts apply to the opposite case of a plane wall with flow. Here also the starting point of the reverse flow must
travel toward the front wit', increasing velocity when the oscillation of
the free stream is held constant. Both behaviors agree with the test
experience. In the described manner one succeeds in obtaining data on
the behavior of the boundary layer flow on undulated walls, in particular
on the setting-in of the reverse flow and thus of a vortex formation in
dependence on the Reynolds number.

One obtains a particularly instructive picture of the boundary layer
flow by plotting the streamline pattern not in a rectilinear s, n system,
but as in figure 5 against the wall corresponding to a certain R.
The undulation of the streamlines then resulting at the edge of the
boundary layer must correspond to the pressure distribution taken as a
basis, thus must be practically rectilinear in the present example.

This is being rather well confirmed. Furthermore one can recognize
with particular clarity that the flattening of the undulation takes place
in the immediate proximity of the wall which Is shown by the early
smoothed-out course of the streamlines. One can also see that the
customary concept of the superimposabllity of a wall con.our on the
displacement thickness is admissible, since this flattening of the
streamlines has been completed at a distance from the wall in the order
of magnitude of the displacement thickness.

NACA TM 1228

In figure 3 a comparison of the drag conditions of the undulated with
those of the plane plate has been performed which shows that in this
example a local drag reduction of about 25 percent occurs; this reaction
is caused by the fact that the regions where the shearing stress is reduced
by the undulation compared to the plane plate outweigh the regions where it
is increased. However, it is not advisable to utilize this effect in
other than regions with pressure drop where there is no danger of a premature
boundary layer transition (caused by the undulation) to turbulent state.

The calculations and considerations performed in a more voluminous
report and given here in the form of an extract may be summarized in the
following conclusions:

1. The laminar boundary layer proves to be actually extremely
sensitive to slight variations in the pressure distribution, and tends
easily toward separation.

2. The start of the reverse flow may be calculated as a function
of the Reynolds number and the pressure oscillation; the retroaction of
the undulation of the displacement thickness on the pressure distribution
must be taken into consideration.

3. The results of the calculation, in agreement with test results,
lead to the interpretation that the transition of the boundary layer from
laminar to turbulent is caused by the onset of reversal flow followed
by a vortex formation which, in turn, may be produced by fluctuation of
the free stream and by wall roughness. If a monotonously increasing
pressure'rise exists, the point of transition, caused by additional pressure
oscillation, will lie generally ahead of the separation point of the
laminar boundary layer which results by calculation with the undisturbed
pressure distribution..

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

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