Wind-tunnel corrections at high subsonic speeds particularly for an enclosed circular tunnel

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Title:
Wind-tunnel corrections at high subsonic speeds particularly for an enclosed circular tunnel
Series Title:
NACA TM
Physical Description:
43 p. : ill ; 27 cm.
Language:
English
Creator:
Göthert, B
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Airplanes -- Fuselage   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
A review of existing publications on wind-tunnel corrections is followed by an approximate method for determining the corrections due to model and wake displacement and to lift. Relations are investigated for fuselages and wings of various spans in closed circular tunnels. A comparison is made between the computations and the tests in the DVL high-speed wind tunnel.
Bibliography:
Includes bibliographic references (p. 33).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by B. Göthert.
General Note:
"Report date February 1952."
General Note:
"Translation of "Windkanalkorrekturen bei hohen Unterschallgeschwindigkeiten unter besonderer Berücksichtigung des geschlossenen Kreiskanals." Forschungsbericht Nr. 1216, Deutsche Versuchsanstalt für Luftfahrt, E. V., Institut für Aerodynamik, Berlin-Adlershof, May, 1940."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
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oclc - 156940315
sobekcm - AA00006188_00001
System ID:
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Full Text
NicpkA -\300







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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1300


WIND-TUNNEL CORRECTIONS AT HIGH SUBSONIC SPEEDS

PARTICULARLY FOR AN ENCLOSED CIRCULAR TUNNEL
rI
By B. Gothert


SUMMARY

After a review of existing publications on wind-tunnel correct-
ions at high Mach numbers, an approximate method is given for deter-
'mining the corrections due to model displacement and wake displacement
behind resistance bodies and due to the lift. The correction compu-
tations are first carried out for the incompressible flow. According
to the Prandtl principle, the models and the wind tunnels in the
incompressible flow are made to correspond to the models and the wind
tunnels in the compressible flow; between the corrections of these two
correlated tunnels definite relations then exist. Because of the Prandtl
principle is applied only to the flow at a large distance from the
models, the wind-tunnel corrections can also be computed if the
assumptions in the Prandtl principle are not satisfied in the
neighborhood of the model. The relations are investigated with
particular detail for fuselages and wings of various spans in closed
circular tunnels. At the end of the report a comparison is made
between the computations and the tests in the DVL high-speed wind
tunnel.

I. STATEMENT OF PROBLEM AND REVIEW OF LITERATURE

If a model of finite thickness is mounted in a closed tunnel,
the tunnel at the model cross section is narrowed by a definite amount
because of the displacement of the model. The air is now forced by
this contraction to flow around the model with greater velocity than
would be the case in an unlimited air stream. In the application of
the results of measurement to the free air stream, the measured values
are therefore to be correlated with a higher velocity than corresponds
to the velocity in the tunnel without the model.

'Windkanalkorrekturen bei hohen Unterschallgeschwindigkeiten unter
besonderer Berucksichtigung des geschlossenen Kreiskanqls."
Forschungsbericht Nr. 1216, Deutsche Versuchsanstalt fur Luftfahrt,
E. V., Institut fdr Aerodynamik, Berlin-Adlershof, May, 1940.





I:


"l


- qa /os3C








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For wind tunnels with small velocities, as have so far been
predominantly used, the increase of the velocity due to the model is
so small in view of the generally small model dimensions that it may
generally be neglected. For higher tunnel velocities the velocity
correction increases very rapidly, however, and at a more rapid rate
the more nearly the velocity approaches that of sound. This fundamental
behavior is shown, for example, by Ferri (reference 1, p. 112) by
computing the velocity increase due to a definite amount of tunnel
narrowing on the assumption of a uniform velocity distribution over
the narrowed cross section (see fig. 1). From these computations it
is found, for example, that a narrowing by the model of the tunnel by
1 percent gives, at small Mach numbers, an increase in velocity of
1 percent; for the Mach number M = 0.8, 2.7 percent; and for M = 0.90,
as much as 11.0 percent.

This simple rough computation indicates the absolute necessity,
on the one hand, of computations for the correction of the tunnel
velocity due to the model displacement and, on the other hand, the need
for choosing the dimensions of the model in relation to those of the
tunnel so as to be smaller the nearer the velocity of sound is
approached.

Accurate computations for flows in a compressible fluid, on
account of the complicated considerations, currently offer little
promise of success when it is considered that even the simple case of
the two-dimensional flow about a wing in an infinite compressible air
stream already lies at the limit of present-day computation possibili-
ties. Approximations must therefore be sought that come as close
as possible to the actual facts.


1. Approximate Computation of Lamla

A simple estimate of the correction for two-dimensional flow was
given by Lamia (reference 2), who in his computations took into
account the compressibility of the air up to terms of higher order and
made use of the following concept as a basis:

If a wing with infinite span is placed in an air stream (without
lift), two streamlines lying symmetrical to the wing at a large distance
ahead of and behind the wing are separated by a distance hbI, while in
the plane of the wing they are displaced by the greater distance
hom + Ln. Between two streamlines at distance hm therefore, the
flow in the model cross section no longer transports the same quantity
of air G = as between the streamlines the same distance apart at a
large distance ahead of or behind the wing, but a smaller quantity of
air GCD- LG. Lamla takes the approach velocity v in the free air









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stream to be increased by the amount Av0 until at the model cross
section between two streamlines at distance ho the quantity G,
is transported. The required increase in the velocity is considered
as an approximate value for the velocity correction to be applied to
the closed tunnel; the distance between the two streamlines ho,
are set equal to the distance between the tunnel walls.

An important result of this estimate of Lamla was that the
velocity corrections given by Ferri were recognized as upper limiting
values, which can be attained only for very slender models with very
large chord in comparison with the tunnel diameter. In by far the
majority of cases, however, the chord of the model is so small in
comparison with the tunnel diameter and its thickness ratio so large
that the disturbance velocities produced by the model from the
immediate neighborhood of the model up to the tunnel wall are greatly
decreased and therefore also the effect on the flow produced by the
wall becomes considerably smaller. For example, with 4 percent cross-
sectional narrowing by the model, the velocity correction for the Mach
number M = 0.75 is given by (from fig. 8 of reference 2):

Model Thickness ratio, Velocity correction
d/t Lv/v0
(percent)
Plate in direction of -- 0 11.4 according to
flow with finite Ferri
thickness 11.4 according to
Lamla
Elliptical cylinder 0.1 11.4 according to
Ferri
4.2 according to
Lamla

The numerical values found by Lamla, on account of the great
simplification used, can only serve as guides for the approximate
values of the corrections. The tunnel boundary conditions enter his
computation only partly because equal flow was assumed through the
cross section at the model position and the cross sections far ahead
of and behind the model. This inexact taking into account of the
effect of the tunnel wall led to the result that, in accordance with
his computations for all Mach numbers, as for example for M = 0,
the closed and the open tunnel have equal corrections of the velocity
but with opposite sign. For compressible flow it is known, however,
that the open tunnel requires only about one-half or one-fourth as
large corrections as for the closed tunnel (reference 3, pp. 54-58).








4 NACA TM 1300


2. Approximate Computation of Franke-Weinig

A further approximation for the two-dimensional problem is given
by Franke and Weinig (reference 4). In their investigation they
strictly consider the boundary conditions along the tunnel wall and
take into account the compressibility in a manner similar to Prandtl.
The velocity correction due to the displacement of the model is given
in the following form:


Avx = F
vo 6 h2 (1-M2)3/2


1 1
3 abcve + below V0

where

F cross-sectional area of profile

h distance of tunnel wall

upper and Vlower velocity at upper and lower tunnel wall,
respectively, in plane of model

vO velocity far ahead of model
Mach number, tunnel velocity
v/a velocity of sound

The second form of the velocity correction given previously,
which refers to the disturbance velocity measured at the wall, is in
many cases found to be very useful. This form is referred to also in
the following investigation (see section IV) in cases in which the
assumptions of the Prandtl principle in the neighborhood of the model
are no longer satisfied although, with the aid of the wall velocity,
a correction according to the Prandtl principle is still possible.


3. Purpose of the Present Investigation

The purpose of the present investigation is to give a useful
approximation of the velocity and angle-of-attack correction, in
particular for the circular-shaped closed wind tunnel. Because at
high Mach numbers the wake produced by the model resistance greatly
increases, an approximation to take account of this wake is developed.
Further, the results obtained are checked with the aid of wind-tunnel









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measurements so far as the available measurements permit such comparison.

In contrast to the work so far done, the computation is first
carried out completely for the incompressible flow. The results
thereby obtained are applied to the flow in compressible media by
correlating each flow picture of the compressible flow with a definite
flow picture in the incompressible flow with suitably modified tunnel
and model dimensions so that for both flow pictures the corrections
are either the same or stand in definite relation to each other. This
computation method has the advantage of great clarity of computation,
for the investigation is split up into two independent partial investi-
gations. Furthermore, a large number of already existing correction
computations for incompressible flows can, according to the same
principles, be applied in a simple manner to compressible flows so that
a good portion of the existing data is not lost but can be further
utilized.

The investigation was carried out in general form both for bodies
of rotation and for wings of various spans. The numerical data given
refer, however, predominantly to the case of the flow with rotational
symmetry. In a further report that is soon to follow the numerical
data will be supplemented.


II. CORRECTION OF THE FLOW VELOCITY DUE TO MODEL

IN CASE OF INCOMPRESSIBLE FLOW

1. Equivalent Dipole or Source Strength of Model

(a) Equivalent dipole strength of model without wake.

The flow about profiles or about bodies of arbitrary shape can,
as is known, be simulated by definite arrangements of sources and
sinks or of dipoles (doublets), the strength and distribution of which
are so chosen that the resulting streamline coincides with the outline
of the body investigated. For large distances from the dipole or
source-sink system, the disturbance velocities produced by them can be
shown to be equal to that of a single equivalent dipole that is located
at the center of gravity of the s stem of singularities and whose
strength is determined as follows:

Intensity of equivalent dipole

M+ = 22(a Q) = M


'The equation given holds not only for doubly symmetrical but also for
arbitrary body shapes.









6 NACA TM 1300


where

Q strength of source or sink

2a distance between source and corresponding sink

M moment of elementary dipole

On the assumption that the dimensions of the model are small as
compared with the diameter of the wind tunnel, the disturbance velocity
of a model at the tunnel wall can be represented by the effect of the
previously defined equivalent dipole NMT. For large models, whose
chord is comparable with the tunnel diameter, the simple estimate
previously given of the equivalent dipole strength is insufficient; for
this case a computation will later be given that permits estimating the
deviation from the simple equivalent dipole.

The equivalent dipole for various shapes of bodies was computed
by Glauert as a function of the maximum thickness and the ratio of
thickness to length of the body (reference 3). In the present report,
Clauert's results are so modified that the equivalent dipole strength
becomes a function primarily of the volume of the displacing body; the
effect of the thickness ratio and the difference between two-dimensional
and three-dimensional flow then remain very small for slender bodies.
In this new representation the dipole intensity is given by2:


2For cylinders, Glauert gives the equivalent dipole strength as

M+ = b r/2 dmax2 v0


(reference 5, p. 53). The given value referred to the volume hV
is therefore connected with the X value of Glauert by the relation
3
1 dmax2 b Y
V 2 V

For the circular cylinder, X = 1 and therefore v = 2. For bodies
with rotational symmetry,

M+ = dm3 v0


(reference 5, p. 59) and therefore
V 3
4 max
kv 4 v --


For the sphere, Glauert gives X = 1 so that XV = 1.5.





]--


HACA TM 1300 7


M= 2 (aQ) = v V v, (1)

where

V volume of body

vO approach velocity

For bodies with rotational symmetry the equivalent dipole is to be
applied at the center of gravity of the body. For cylinders (t/b--C)
the previously computed equivalent dipole strength is to be distributed
along the axis of gravity of the cylinder so that along this axis there
is a uniform dipole distribution dMt = db M+/b

The factor XV depends on the shape of the body and the thickness
ratio. For very slender bodies (d/t--0) in two-dimensional and
rotational symmetry cases, XV has the same value kV = 1.

.For bodies with elliptical outline, the factor XV is given in
figure 2 as a function of the thickness ratio for the two limiting
cases of rotational symmetrical bodies and elliptical cylinder (t/b-0).
It is seen from the figure that, for slender bodies such as occur in
airplane structures, XV differs only by a slight amount from unity,
for example,

wing with d/t = 0.15 Vy = 1.15

fuselage with d/t = 0.25 XV = 1.065

For the very thin plate transverse to the air stream, the volume
of the body is zero. Because, however, such a plate will nevertheless
give a displacement of the streamlines, the factor XV must tend to
infinity. If the computations in the literature3 are used for the flow
about a plate, the product is obtained for the plate with constant
height h and very large span b (h/b--0):

xk V= h2 b

and for the circular disk with the diameter d:

v= d
3or very wide plate: reference 5. or circular plate: reference

3For very wide plate: reference 5. For circular plate: reference 6.









8 NACA TM 1300



(b) Equivalent source strength to take into account the

wake displacement behind a drag body

Behind bodies which a flow drag connects with an energy loss
(but not an induced drag), as is known, a wake or dead-water region
is formed that evidently increases the displacement of the body.
(See fig. 3.) This dead-water region has its source in the neighbor-
hood of the body and extends downstream to infinity, with increasing
mixing with the normal flow. It is useful to represent the effect of
this dead water for points at a great lateral distance away by a
system of sources that arise at the place of the drag body and displace
the normal stream behind the body by the same amount as the wake.
Between the strength Q of this source system and the drag of the body,
under certain simplifying assumptions, a relation can be given which
may serve as an estimate for the displacement due to the wake4.

4According tb the momentum theorem, if the undisturbed pressure is
assumed to prevail at the cross section considered behind the body, the
drag W of a body is given by


W= J dm Lv = mass flow per second x velocity loss

= p (VO v) v r = p 0 v1) v0 (0 vl)

= VoJ (Vo vl) df


(See fig. 3.) The integral contained in the last equation gives,
however, precisely the additional strength per unit volume Q of the
equivalent source that, on the assumption of potential flow, displaces
the streamlines at a great lateral distance from the wake by the same
amount as the wake, that is,

Q =J(vO v) df or W = p v Q


By transformation
W vO
Q = = fws vo/2


is obtained. The same relation between the source strength and the
drag has already been given by reference 7 (p. 32).









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On the assumption that the static pressure in the wake differs
only slightly from the pressure of the undisturbed flow, there is
obtained, by neglecting small quadratic terms, equivalent source
strength:
1
Q = 0 fws (2)


where fyw = c, F represents the harmful drag of the body (without
account taken of the induced drag). This relation between the source
strength and the drag area is independent of the shape of the body and
holds both for two-dimensional and for three-dimensional flow.

The exact location of this equivalent source is not uniquely
determined. It appears admissible, however, to assume the source as
located at the center of gravit: of the voilune. Although the position
of the equivalent source rnay not be correctly given by this assumption,
the error thus introduced is small provided that the equivalent is used
only to represent the flow relations at a great distance from the source.
This assumption is, however, satisfied for obtaining the tunnel correct-
ions because the model dimensions are always small in comparison with
the tunnel diameter.

In the derivation of the preceding equivalent source strength, the
assumption was made that the static pressure in the wake differed only
slightly from the pressure of the undisturbed stream. At a large
distance behind the drag body this assumption holds quite well, so that
to a very good approximation the previously given equivalent source
strength remains the same although the dead-water region constantly
expands. Immediately, behind the wing, however, the static pressure
may differ greatly from the undisturbed pressure of the flow so that
deviations from the simple relation given above between the drag and
the source strength are obtained. It was shown by M-uttray that the
source strength decreases rapidly from a maximum value at the wing
trailing edge to a final constant value behind the wing (reference 7,
fig. 22). This decrease means, however, that, to the previously
computed equivalent source strengt-h, a system of additional sources and
sinks behind the wing is to be added. At a large distance from the
wing, as for example at the tunnel wall, this additional source system
acts approximately like a single dipole. The action of this additional
wake dipole is taken into account in the following correction compu-
tations if the magnitude of the tunnel corrections from the disturbance
velocities measured at the wall is determined according to equation (4).
The error made in assuming that the location of this wake dipole does
not, coincide with the profile center of gravity is only of small
significance on account of the effect of the wake corrections.








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The same consideration applies also to the additional displacement
due to the compression shocks, so that their effect can enter approxi-
mately into the corrections provided the shock length is small as
compared with the tunnel diameter.


2. Disturbance Velocity at the Tunnel Center and at the Wall

Due to the Model and the Dead Water

Because, according to the assumption made, the model dimensions
are small in comparison with the tunnel diameter, the disturbance
velocities at the tunnel wall can be represented by the previously
determined equivalent dipole and the equivalent source. The boundary
condition which is to be satisfied by the presence of the tunnel wall
for the closed tunnel is that all velocity components normal to the
tunnel wall must vanish.

This requirement is satisfied, as is known, by superimposing an
auxiliary flow with a velocity field on the flow about the model in
the free airstream so that the composite flow satisfies the previously
given condition. The computaton for the circular tunnel is quite
complicated. In the present case the solution was determined by a
method given in reference 8 (p. 250) of developing the field of the
radial velocities along the tunnel wall and the velocity field of the
auxiliary flow into a Fourier series. By comparing the coefficients
of the two Fourier series, the constants for the velocity field of
the auxiliary flow were then obtained so that the velocity at
arbitrary points of the flow field could be determined. As velocity
correction, that velocity is taken which induces the auxiliary flow at
the location of the model.

The results of this computation are given in the present report
for the additional velocities at the tunnel center and for the
symmetrical case (with the equivalent dipole and equivalent source
at the tunnel center) and for the additional velocity at the tunnel
wall, because for these arrangements the computation is still rela-
tively simple, only the first term of the Fourier series being taken
into account. For several nonsymmetrical arrangements with respect
to the excess velocity at the tunnel wall, numerical values are also
given for which the first four Fourier terms entering the computation
were taken into account. It is still necessary to check whether the
further Fourier terms have an effect on the results5.

5A further report will be issued on the required corrections of the
given numerical data when the higher Fourier terms are taken into
account in which the computation procedure will also be described in
detail.









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(a) Disturbance velocity due to the model.

In the notation of Glauert, the velocity correction to be applied
at the tunnel center is written as follows:

AVx V (3)
vo V D-' ( )

where

V volume of body

D tunnel diameter

As has been previously explained, XV represents a factor that
takes into account the model shape. (See fig. 2.) The factor TV
depends on the tunnel shape and on the ratio of the model dimensions
to the tunnel diameter. For a closed circular tunnel the factor TV
is given in table 1 for several typical cases.

In the method given by Weinig, the additional velocity at the
center of the tunnel may also be represented as a function of the
additional velocity at the wall of the tunnel. As in the two-
dimensional case computed by Weinig, it may also be shown for the
closed circular tunnel that the disturbance velocities occurring at
the plane of the model at the tunnel wall Avxw due to the model
bear a definite ratio to the disturbance velocity Avx at the tunnel
center. (See fig. 4.) If, for the comparison velocity Avx, the
arithmetical mean of the velocities above and below the model is
chosen, this mean value is independent of the lift because a potential
vortex at the model at these two points induces equal but opposite
circulation velocities. The wall velocity, according to figure 4,
is composed of two parts: namely, a part that directly represents
the disturbance velocity of the equivalent dipole and a part that is
due to the wall effect. The required velocity Avx at the tunnel
center may therefore be written as

"Vx Lvxw
m (4)
Vo V0

The factor m in this equation depends on the tunnel shape and on the
ratio of the model dimensions to the tunnel diameter. For the closed
circular tunnel the factor m for several typical cases is given in
table 2.









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For comparison, it may be mentioned here that for a wing with infinite
span between two walls, Weinig gives the value m = 1/3. The factor m
is likewise found to depend to a relatively large degree on the tunnel
shape.


(b) Correction for models of very large chord

The numerical values given in the preceding section (a) for the
velocity corrections were computed on the assumption that all the model
dimensions except the wing span were small compared to the tunnel
diameter. It is often necessary, however, to obtain an idea of how
the data are modified if the preceding assumption no longer holds true
with respect to the model chord.

In order to investigate this effect on bodies of rotation in the
center of the tunnel, the additional velocities at the tunnel center
and at the wall were computed for various distances of the source-sink
system replacing the body by the method given in section (a). The
distribution of the sources and sinks for these bodies was chosen as
shown in figure 5. With this arrangement, the ratio Fprofile/d t
was found to be equal to 0.75 as corresponds to the usual shapes in
airplane structures and to wings and fuselages. Because for these
bodies the magnitude of the equivalent dipole is known from equation (1)
M4 = E (2aQ), the disturbance velocities at the center of the tunnel
and at the wall can likewise be immediately given by equations (3) and
(4) if the effect of the large model chord is neglected. Through a
comparison of the results, the numerical values for TV and for the
ratio m (equations (5) and (4)) were found to require a correction.
The correction is represented in figure 6. For very large model chord,
the additional velocities at the tunnel center are seen to differ only
by a small amount from the value for vanishingly small models. The
wail velcoity sAvx and therefore also the ratio m of wall velocity
to center velocity depend, however, to a great extent on the model
chord. For example, for a wing whose chord is equal to the tunnel
diameter, the center velocity is 89 percent of the velocity obtained
on the assumption of vanishingly small model chord, whereas the wall
velocity is 66 percent of the corresponding value.

The previous investigations for models of large chord refer only
to bodies with rotational symmetry and to wings with vanishingly small
span in comparison with the tunnel diameter. Strictly speaking, it
would be necessary to carry out also corresponding computations for
wings with various ratios of the span to the tunnel diameter. Before
the results are extended in this respect, however, it may be assumed
as a useful approximation that the correction values given in figure 6
hold also for wings with finite ratio of span to tunnel diameter.









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(c) Disturbance velocity due to the wake

' The effect on the tunnel wall of the dead-water region formed
behind a drag body and therefore the effect on the tunnel correction
can be given approximately by an equivalent source at the location of
the model. (See section II, 1, (b).) If, for this flow, the boundary
conditions at the tunnel wall are satisfied by superposition of the
source flow in the free air stream on an additional velocity field
according to section II, 2, (a), it is readily seen that in the closed
Tunnel at the model location no velocity component arises in the
approaching flow direction because of the additional flow field; only
for unsymmetrical arrangement of the source in the tunnel will a
velocity component arise at right angles to the approaching flow
S direction, which leads to a change of angle of attack. This change of
angle of attack is also, however, in general without significance, for
models are practically never mounted unsymmetrically in the wind tunnel.

Although no velocity component in the the flow direction is induced
by the additional velocity field at the model location, a correction for
the approach velocity in the closed tunnel is necessary. The relations
are seen most simply with the aid of figure 7 in the case of the
infinitely wide tunnel. The velocity field arising from the source
in the tunnel and from the external sources reflected in the wall is
characterized by the fact that the velocity components in the model
plane in the flow direction vanish. At an infinite upstream and
downstream distance, however, a parallel flow is formed with the
velocity v = 1 Q For the flow in the tunnel at a large distance from
FK
the source this means that the approach velocity is no longer v'O but
v = 1 Q- The velocity v0 therefore increases up to the plane of
v Vo 2FK "
the model by the amount ,-vx = F and up to a section very far behind
the body by Av = Q/FK. The required approach-velocity correction in the
closed tunnel is, therefore, with the air of equation (2), obtained as

VAx 1 Q 1 ws
v0 2 FK v0 4 FIK

where

'f, = Cw F harmful drag area


tunnel cross section









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The distribution of this correction velocity over the model cross
section is uniform; that is, at the tunnel center and at the wall there
is the same velocity increment fvx. In the equation

xAv/v0 = fm VxV,/VO

the factor m therefore has the value m = 1.

This equation for the velocity correction due to the wake and
magnitude of the factor m is independent of the shape of the closed
tunnel and therefore holds for square as well as for circular tunnels,
etc.

In a corresponding manner it may be shown simply that in wind
tunnels with open test section no velocity correction is required to
take into account the displacement due to the wake. The boundary
condition of the free jet would be satisfied by reflecting sources
and sinks alternately at the tunnel wall, in a similar manner to that
indicated in figure 7. At the center of the tunnel no direct velocity
component is obtained in the flow direction. At an infinite distance
upstream and downstream of the plane of the source the velocity
components due to the sources and the sinks reduce to zero as the
velocities induced by them mutually cancel.


III. VELOCITY CORRECTION DUE TO THE MODEL AND THE WAKE AT

HIGH SUBSONIC VELOCITIES WITHIN THE REGION OF VALIDITY

OF THE PRANDTL PRINCIPLE

The equations given above for the tunnel correction hold only for
the incompressible flow. With the aid of the Prandtl principle,
however, a given tunnel with the model in a compressible flow can be
correlated with another tunnel with models of modified dimensions in
an incompressible flow. Between the velocities in the two correspona-
ing tunnels definite relations hold so that, with the aid of these
relations, the flow in the compressible medium can be reduced to the
flow in the incompressible medium. The form of the Prandti principle
which is here used is as follows (reference 9):

At each point of the compressible flow the same potential holds
as compared with the corresponding point of the incompressible flow
and the same velocity in the flow direction (x-direction) with the
velocities at right angles to this direction reduced by the factor
I/1-Mz. The corresponding points in the two flow fields are connected
by the following relations:









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Xincompr = Xcompr
Al--
Yincompr = Ycompr


Zincompr = VM zcompr

where
tunnel velocity
M Mach number,
velocity of sound


1. Model Displacement

For a wing with large aspect ratio B/t in a closed wind tunnel,
the potential field for incompressible flow is shown in figure 8. The
model is assumed to be slender enough and to possess a sufficiently
sharp leading edge that the Prandtl principle is satisfied along the
entire profile contour. The profile contour is given by the boundary
streamline of a definite source-sink arrangement so that the potential
lines are continued to the interior of the surface enclosed by the
boundary streamline. On the potential lines 0 = const so that they
must intersect the tunnel wall and the profile surface at right angles,
for at these bounding surfaces no normal velocity components exist.

If this potential field is now transformed according to the
Prandtl principle, that is, if the lines 0 = const are separate
farther apart in the y and z directions by the factor 1/ ~-M ,
and the x component is kept constant, a new potential field is
obtained for the compressible flow. (See fig. 8.) In this distorted
potential field, condition for the tunnel wall, namely, that all
potential lines must be at right angles to the flow direction, can be
seen to be satisifed along a new c tour whose y and z coordinates
are increased by the factor 1//V1-M6. For circular tunnels this
satisfaction means, for example, that the diameter of the tunnel in
the compressible flow is greater than the comparison tunnel in the
incompressible flow in the ratio 1/1/1-M2.

In the transformation according to the Prandtl principle the wing
in the tunnel receives definite modifications. If, for the slender body,
the assumption is made that, within the bounding potential line which
replaces the profile, the potential lines may be replaced by their
tangents at their points of intersection with the x-axis, it is seen
that the inclination of this tangent becomes steeper in the ratio
1//1 -M This assumption means, however, that the stream lines in
the neighborhood of the x-axis, as also in particular the bounding








16 NACA TM 1300


stream, have flatter slopes dy/dx by the factor VT-M2 than before
the distortion. The profile shape determined by the bounding stream-
line is therefore thinner after the distortion in the ratio Vl--M.
The span of the wing with large aspect ratio B/t is determined by
the length of the segment with sources and sinks or dipoles distributed
on it. This d stance is likewise increased by the distortion in the
ratio 1/4/-M2 so that the comparison wing in the compressible flow
has a greater span in the ratio 1/4/1-M2.

Because the profile section has, through the transformation, become
more slender in the ratio -M2 but the span has become larger in the
ratio 1/4/1-MT, the initial wing and the corresponding wing of the
compressible flow have the same volume. It is also known for slender
bodies of rotation that in the transformation according to the Prandtl
principle their contour is, to a first approximation, unchanged so
that in this case also the volume undergoes no change (reference 8).

Because in the Prandtl transformation the velocity components in
the flow direction remain the same, a condition that holds also without
restriction for the velocity components due to the model and the tunnel
wall, the same corrections on the approach velocities are to be used
for the bodies in the compressible flow as for a body transformed in
the manner explained above in an incompressible flow in a tunnel
reduced by the factor '1/1W.

If, therefore, a model that at small velocities has the approach-
velocity correction Avx/vo for small velocities is placed in a closed
tunnel and the Mach number continuously increased, the velocity
correction according to equation (3) increases in the ratio
1/(1-M2)3/2, that is,


(Avx 1 lx (6)
V0 compr (1-M2)5/2 \v) incompr

In the preceding relation, the factors Ty and AV contained
in equation (3) were assumed not to vary in the transformation accord-
ing to the Prandtl principle. For XV a small displacement actually
occurs that is due to the change in the effective thickness ratio.
According to figure 2, however, XV depends only to a slight extent
on the thickness ratio, so that the change in XV may in general be
neglected without great error.

The factor TV does not vary as long as the model chord is small
in comparison with the tunnel diameter. With increasing Mach number,
however, the effective ratio of model chord to tunnel diameter rises as
1/ 47-7, because the model chord remains unchanged while the tunnel
diameter becomes smaller by the factor l T-M2. When the Mach number is
increased, the question is thus raised of the correction of models with
large chord in a closed tunnel. According to figure 6, the correction









NACA TM 1300 17


of the factor rV can be estimated by letting the effective ratio of
the model chord to the tunnel diameter increase by the factor 1/ 1-M2.

As long as the model chord can still be considered small in compari-
son with the tunnel diameter, the factor m in equation (3) undergoes
no change. In the transformation according to the Prandtl principle,
the same ratio of model span to tunnel diameter occurs for the compari-
son tunnel as for the initial tunnel, so that for both systems the same
factor according to table 2 is to be used. If in the transformation
too high ratios of chord to tunnel diameter result, however, then the
value of m is to be corrected with the aid of figure 6.


2. Additional Velocity Due to the Wake

The same consideration for the computation of the velocity correct-
ion due to the model displacement can also be applied for the additional
displacement due to the wake. In a purely formal manner, on the
assumption that the drag surface does not vary with increasing tunnel
velocity, there is obtained from equation (3):


(LvX 1 'Lx
O compr VO incompr (7)

It is thus seen that, in contrast to the cases so far given, the
tunnel correction does not increase by l/(l-M2)3/2 but by 1/(l-M2).
Because, in this computation, the effect of the wake was replaced by the
flow about a source which, as is known, gives in a parallel stream a
bounding stream line in the form of a "half-bod-y," the preceding result
means that for a body of small chord, for example, a fuselage in a
tunnel, the velocity corrections increase by the factor 1/(l-M2)3/2;
for a-half-body in a closed tunnel the velocity corrections increase
only by the factor l/(l-M2).

S/////////////////// ///////////////
-- Fuselage in -- Half-body
= ---tunnel in tunnel




The reason for this different behavior of the two body shapes is
to be sought in the fact that for flow with rotational symmetry, the
radial velocities acting on the wall at points a small distance upstream
or downstream of the model for short bodies (equivalent dipole) decrease
approximately in inverse ratio to the fourth power of the distance from
the body, whereas for the half-body (equivalent source) they decrease
only in the inverse ratio of the second power of the distance from the
body. If, therefore, corresponding to the Prandtl principle, the tunnel









NACA TM 1300


diameter in an incompressible flow is reduced for both bodies to the
same extent, it is seen immediately that in the case of the half-body
the wall effect must increase by a less amount than for the short body.

If the velocity correction is again referred to the wall velocity,
the factor m in equation (4) is still to be set equal to 1 for the
region of validity of the Prandtl principle, because when the wake is
taken into consideration, this factor is independent of the magnitude
of the tunnel diameter. (See section II, 2, (c).)


IV. VELOCITY CORRECTION DUE TO THE MODEL AND THE WAKE FOR

THE CASE WHERE THE PRANDTL PRINCIPLE ALONG THE MODEL

SURFACE DOES NOT HOLD TRUE

The assumption for the validity of the Prandtl principle is that
the additional velocities due to the body are small compared with the
tunnel velocity. In many cases, particularly at high Mach numbers,
this assumption is not satisfied along the surface of the body. At

6The radial velocities of source and dipole decrease according to these
considerations, for small values of x, by the second or fourth power
of R; the additional velocities increase, however, with the second or
third power of 1//l-M2. This difference in the exponential relation
is to be ascribed to the fact that at a great distance ahead of or
behind the source and the dipole, the radial velocities in each case
vary by the same amount so that an equalization of the power exponent
occurs. A simple check computation for the amount of the velocity
increase is possible for the half-body. If a half-body is in a closed
tunnel, the stream is parallel far ahead of and behind its nose. The
velocity increase is computed simply by the decrease in the cross
section df. From the known equations

dp k M2 di
P 1-M2 f

and dp k 2 dv
p v

in agreement with the relation previously found from the Prandtl
principle:
dv 1 df
v = f ff7









.. NACA TM 1300 19


some distance ahead of or behind the body the additional velocities
have become small enough, however, that for the flow outside of a
definite boundary line the Prandtl principle may again be applied.
(See references 10 and 4.) In figure 9, for example, the region
enclosed by the boundary line no longer follows the Prandtl principles
so that the transformation law is unknown for the streamlines within
the boundary and for the body contour.

SThis fact is now applied to the flow in a tunnel, first for the
simplest case of the flow in an infinitely wide tunnel with parallel
walls. In an incompressible flow the boundary conditions at the
tunnel wall are satisfied for the dipole replacing the body by an
infinite reflection of the equivalent dipole at the tunnel wall. In
S the neighborhood of the dipole the disturbance velocities are so large,
however, that within a certain limiting region the Prandtl principle
is no longer applicable (fig. 10). The transformation of this
incompressible flow field according to the Prandtl principle is possible,
however, for the regions lying outside with boundary lines; within
the boundary lines, as stated above, a transformation must be made
according to an as yet unknown law so that the body contour for the
compressible flow is likewise unknown. According to the Prandtl
principle, the strength of the equivalent dipole can no longer be
determined by computation for a given body. This gap can however
be closed for the wind-tunnel test by a simple measurement. Because
the additional velocity at the tunnel wall above and below the model
is a measure of the effective dipole strength and the tunnel walls,
moreover, lie within the region of validity of the Prandti principle,
S the effective dipole strength can be determined by measuring the
S velocity at the wall. The required tunnel-correction velocity at the
S tunnel center, is as is known, the sum of the induced velocities of
all reflected dipoles whose intensity is known from the measurement
of the wall velocity and for which the tunnel center belongs to the
S region of validity of the Prandti principle. This knowledge means,
S however, that between the wall velocity and the approach-velocity
S correction at the tunnel center the same relation holds as in an
unrestricted region of validity of the Prandtl principle, that is,


Lvx/V0 = m Lvxw/vO


This relation is not restricted to only the flow between two
walls. As can be readily seen, it can be applied also for the closed
S tunnel of arbitrary cross-sectional shape, for example, for the
circular tunnel. The only difference consists in the fact that in the
flow between two walls the additional potential to satisfy the tunnel
i. wall conditions is produced by the reflected singularities, whereas
in the most general case the additional potential arises also from the









NACA TM 1300


singularities outside of the tunnel; the strength and the location of
these singularities cannot be found, however, by simple reflection.

As long as the model dimensions are small as compared with the
tunnel diameter the values to be assigned to m are those given in
table 2. For models with large chords and for high Mach numbers,
however, an estimate for the correction of the factors m due to very
large ratio of the chord of the model to the tunnel diameter must be
made with the aid of figure 6.

The relations given between the velocity at the wall and the
tunnel-correction velocity for flows about models, for which in the
neighborhood of the model the Prandtl principle no longer holds, can
also be derived for the velocities induced by the wake.

The previously described extension of the tunnel-correction
computations to Mach numbers for which the Prandtl principle in the
neighborhood of the model no longer holds is naturally inapplicable
without restriction up to a Mach number M = 1. An upper limit of the
applicability is given, for example, if the sound velocity is attained
or exceeded at the tunnel wall. In this limiting case the assumptions
of the Prandtl principle are no longer satisfied in the neighborhood
of the tunnel wall so that the entire transformation process is
inapplicable. How closely this upper limit may be approached in the
wind tunnel measurement without fundamentally altering the pressure
field about the profile can be determined by experiment. This limiting
Mach number will depend, among other factors, on the degree of
obstruction of the tunnel by the model and on the angle of attack of
the model. Corresponding tests to determine admissible model dimensions
for definite limiting Mach numbers and angles of attack are at present
being conducted at the DVL.


V. TUNNEL CORRECTIONS DUE TO LIFT AT HIGH SUBSONIC VELOCITIES

1. Wing in Free Air Stream

In order to consider briefly the flow relations about a wing with
lift in a compressible medium, the wing will first be considered replaced
by a vortex filament in a free air stream. The flow field about a
potential vortex in an incompressible flow is schematically represented
in figure 11. If the velocities along the closed bounding line ABCD
are added, the following integral represents the circulation of the
vortex;

v de = Z = circulation









NACA TM 1300 21


The magnitude of the circulation is, as is known, independent of
the path of integration as long as the vortex lies within the boundary
line. In a medium of constant density there is then obtained for the
lift per unit length

A = p Z v0

where p is the air density.

If the flow field of this vortex is now transformed according to
the Prandti principle where there is again excluded a region in the
immediate neighborhood of the vortex center, the control surface ABCD
is extended in the direction of the y axis. (See fig. 11.) At the
same time, however, the velocity component v,, is reduced at corres-
ponding points in the ratio /- so that the product of the path
element by the velocity at the normal boundary lines undergoes no
change as a result of the transformation. Because, however, in the
x direction neither the lengths nor the velocities have changed, the
circulation integral along the lines ABCD or A'B'C'D' remains
unchanged. If the line A'B'C'D' is taken at a very large distance
from the vortex, where the disturbance velocities and therefore the
pressure and density changes in the flow have been reduced to vanishingly
small values, the compressible flow in the neighborhood of the control
line completely resembles an incompressible flow, which means that for
the lift per unit length the familiar relation again holds:


A=p vO Z

where p is the density of the medium at a great distance from the
vortex.

From the preceding consideration, the conclusion can therefore
be drawn that in the Prandtl transformation the circulation and the
lift referred to the span element are unchanged.

For compressible flow the same lift per unit span is, as known,
attained at an angle of attack reduced in the ratio AI-MW. This is
seen from the fact that in the transformation the velocity components
normal to the flow direction become smaller while the components in
the flow direction remain unchanged. The slope of the stream lines
and therefore also the angle of attack thus become smaller in the
same ratio.

The same considerations may be applied also for a wing of finite
span with elliptical lift distribution. A wing in a compressible flow
is then to be compared with a wing in an incompressible flow for which
the span of the second wing is reduced in the ratio n/1-M2. At each








22 NACA TM 1300


section at corresponding distances from the center of the wing the
circulation and the lift per unit span element then agree for both
wings with elliptical lift distribution, whereas in the incompressible
flow the angle of attack, both the geometrical and the induced, are
increased in the ratio 1/ /VM2. The comparison wing in the
incompressible flow then has an aspect ratio (span b/chord t) imparied
by the factor r/1-M2. For the induced drag in the compressible flow
the equation
2
ca F
cwi = ca ai= b

nevertheless still holds where

F wing area in compressible flow

b span in compressible flow

because for the comparison wing ip the incompressible flow the induced
angle of attack Aai is increased by 1/s1-M while through the
transformation to a compressible flow this angle of attack is again
reduced by l-M 2. The two effects thus mutually cancel.

The equations for the induced angle of attack

Ca F
AcL= b2

and the induced drag
Ca2 F
cwi -- r
3T b
retain their validity also for compressible flow.


2. Wing with Lift in the Wind Tunnel

As has already been stated in corresponding cases, a wing in the
wind tunnel with compressible flow can be made to correspond to a
definite wing with equal lift in a similar reduced wind tunnel with
incompressible flow. Between the correction velocities of the compari-
son tunnel and the initial one the relation then exists that the vy
corrections in the compressible flow must be made smaller in the ratio
4/1-M2. In this manner all correction computations can be carried over
to the compressible flow.









S NACA TM 1300 23


The results of the existing wind-tunnel-correction computations
are, in general, represented in the following form:

Ca F
Angle of attack correction: a = --

where

5 correction factor, function of ratio wing span to tunnel diameter,
tunnel shape, and lift distribution

F wing area

FK tunnel cross section

In the Prandtl transformation the value of the factor 6 is not
S changed because the ratio of span to tunnel diameter, which determines
its value, remains the same and so do the tunnel shape and the lift
distribution. The wing area, however, becomes smaller in the ratio
lfM2, the tunnel cross-sectional area in the ratio (1-M2) so that
for the comparison tunnel in the incompressible flow the angle of
attack is greater in the ratio 1/ /1-M2. This increase is, however,
again canceled in converting to the co'mpressible flow because all vy
velocities and angles of attack become smaller in the Prandtl trans-
formation by -M.

The important result is thus found that for elliptical lift
distribution the angle-of-attack corrections and therefore also the
S corrections of the induced drag can be dealt with in the same way as
for incompressible flow.

The assumption underlying the preceding general result is,
however, that the dimensions of the wing with the exception of the
span are small compared with the tunnel diameter and only the corrections
at the location of the wing are considered. If such is not the case,
as, for example, in the correction of a wing with large chord due to
the stream curvature in the wind tunnel or in the computation of the
downwash behind a wing, it is again advisable to make use of the idea
of a comparison tunnel with its corrections determined. It is thus
found, for example, that the correction due to the stream curvature
for large chord wings increases 2s 1/A/1-, the profile chord remain-
ing the same in the Prandtl transformation, while the tunnel diameter
becomes smaller by the factor q/1-M2. With increasing Mach number the
ratios of the wing chord to the tunnel diameter also increase in the
comparison. The corrections due to the flow curvature ray therefore
at high Mach numbers be of significance, although for an incompressible
flow they may be entirely negligible.









NACA TM 1300


VI. APPLICATION OF THE WIND TUNNEL CORRECTIONS DERIVED PREVIOUSLY

1. Superposability of the Individual Corrections

In the present work the different factors for the wind tunnel
corrections, like model displacement, wake displacement, and lift,
were treated as though only one of these magnitudes was alone effective
(for example, a displacement body without lift and drag, or a lifting
vortex without displacement, and so forth). In the wind tunnel test,
however, the various factors enter in general together so that the
question arises of the superposability of the individual corrections.

In the incompressible flow the question can immediately be
answered in that the individual factors are simply to be superposed
linearly. Each individual correction can, as is known, be computed
from the corresponding potential field. Because for incompressible
flow the potentials can be superposed linearly, the corrections derived
from them can similarly be superposed linearly.

In the case of the compressible flow, a flow picture in the
incompressible flow was made to correspond to each flow picture, with
the aid of the Prandtl principle. Between the velocities and angles
of attack of the two flow fields the familiar relations of the Prandtl
principle hold. In the incompressible comparison flow field, the
individual factors and the corrections may again be linearly superposed.
Because the required corrections in the compressible flow differ from
these comparison corrections only by a factor, the law of linear
superposition of the corrections holds also for the compressible flow.


2. Application of the Corrections in the Wind-Tunnel Test

(a) The corrections due to the lift are obtained according to
the known equations from the measured lift. The corrections do not
increase, for equal lift coefficient, with increasing Mach number as
long as the model chord in the comparison tunnel remains small as
compared with the tunnel diameter. The corrections due to flow
curvature for large-chord models increase, however, with increasing
Mach number in the ratio 1/'1/-M so that these corrections can
become of significance although for the incompressible flow they are
entirely negligible.

(b) The corrections due to the measured drag are computed
according to equations (5) and (7). They increase for equal drag area
fws = cw F as 1/(1-M2).









NACA TM 1300


(c) The corrections due to the model displacement are, for small
Mach numbers, estimated according to equations (3) and (6), as long as
it is assumed that along the body contour the assumptions of the Prandtl
principle are satisfied with sufficient accuracy. The corrections
increase as l/(l-M2)3/2. If at high Mach numbers in the neighborhood
of the model the Prandtl principle no longer holds, the correction
velocity at the tunnel center is obtained from the additional velocity
measured at the wall above and below the model with the aid of equa-
tion (4) and the factor m given in table 2. From the additional
velocity measured at the wall, the part due to the drag is to be
subtracted as this correction is already taken into account. In
addition, the effect of the mounting is naturally. to be taken into
account as is done most simply by a calibration measurement.

At high Mach numbers the previously given corrections depend on
measurements that are obtained at a large distance from the model and
that there have the same values as in the neighborhood of the model.
Because the assumptions for the Prandtl principle used in these correct-
ions are well satisfied at a large distance from the model, the previous
corrections are also admissible when the Prandtl principle in the
neighborhood of the model no longer holds.


3. Sample Computations and Comparison with the Approximate

Computations of Ferri and Lamia

For the velocity correction due to the model displacement the
previously derived equations give higher approximations than those of
Ferri or of Lamla. In order to obtain an idea of the admiissibility
of the assumptions of Ferri or Lamla it is therefore convenient to
compare, with the aid of a few examples, the various approximate
computations. (See table 3.)

From the comparison of the values in the table it is seen that,
in accordance with expectation, the wind-tunnel corrections of Ferri
are greatly overestimated. The values of Lamia, particularly for the
tunnel with open section, also still lie considerably higher than the
more accurate values of Franke-Weinig or Glauert-Gothert.

The comparison of the closed circular tunnel with the closed
tunnel with plane walls shows that in spite of the same ratios of
model thickness to tunnel height the wind-tunnel corrections consider-
ably deviate from each other.


~_ I








NACA TM 1300


4. Relations from the Adiabatic Equation as an Aid to

Wind-Tunnel Corrections

For the correction of dynamic pressure and Mach number, several
of the relations obtained from the adiabatic equations are of import-
ance. These are:

Correction of the dynamic pressure:


dq/q = (2-M2) dv/v = 1/2


(2-M2) dp/q


Correction of the Mach number:


dM/M = (1+ k M2)


dv/v 1/2 1 M2)
dv/v = 1/2 (1+ -9-- W)


where

p static pressure


q pv2/2, dynamic pressure

v velocity

k 1.405 for air


The value of the expression in parenthesis (1 +
sented in figure 12. The factors 1~-M2, 1-M2
occur frequently, are also plotted in figure 13.


k-1 2
-- M2) is repre-
and (1-M2)3/2, which


VII. COMPARISON OF THE COMPUTED CORRECTIONS WITH

WIND TUTNEL CORRECTIONS

To check the tunnel corrections determined above for high subsonic
velocities, the high-speed DVL wind tunnel was available. This wind
tunnel has a closed measuring section of 2.7-meter diameter and attains
at about 50 percent of the available driving power the velocity of
sound in the measuring section.


1. Changes in the Wall Pressure through the Mounting of

Rectangular Wings of Various Chords

In order-to learn the effect of the tunnel wall on the measure-
ment values at high subsonic velocities, four rectangular wings of


dp/q









NACA TM 1300


equal profile, NACA 0015-64, with various chords were investigated in
this tunnel; the chords were t = 350, 500, 700 and 1000 millimeters.
The span (B = 1.35 m) and the mounting were the same for all four wings
investigated. The additional velocities produced by the tunnel walls
cannot be directly measured. At most the increase in the surface
pressure at various Mach numbers could be compared insofar as the.change
in the surface pressures in the free air stream at high subsonic
velocities could be considered as known from some computations. In
this case too, however, the observed increase in the surface pressure
is due partly to the increase of the flow about the profile as a result
of the compressibility effect ard partly to the effect of the tunnel
wall, so that the part due to the wall effect, obtained through split-
ting the measured values, is at least uncertain.

A useful method of checking the corrections is offered, however,
by the measurement of the wall pressure in the model plane. The
computation of the wall pressure depends on the same assumptions as the
computation of the correction velocities at the tunnel center, as has
already been explained in the previous sections. This close connection
expresses itself also in the fact that the pressure changes due to the
model measured at tie wall stand in quite definite relation to the
correction velocities at the center of the tunnel. (See equation 4.)
In addition, the pressure changes at the tunnel wall are always greater
than the pressure changes entering the correction computation so that
the wall pressures are more easily susceptible to an accurate measure-
ment. The tests were conducted b- measuring the wall pressures pI,
PIII, and pw for various dynamic pressures. (See fig. 14.) The
measuring stations for the pressures pI and PIII were uniformly
distributed over the entire circular cross-section. For the measure-
ment of Pw in the test section, three close-lying holes were bored -
in the wall in the model plane above and below the model and these were
combined to give the arithmetic mean. It may be shown that the dynamic
pressure and the Mach number in the plane of the model and the wall
pressure p! without the model were functions orLy of the ratio
(pI PIII)/PI" These relations were determined by tests and plotted
as calibration curves. The values determined from this ratio for the
dynamic pressure and the Mach number are, in what follows, denoted as
the uncorrected test-section values. If a model is mounted in the test
section, the indicated wall pressure pw changes for equal pressure
ratio (pI PIII)/PI. From this change of the wall pressure, the
corrections to be applied can be determined.

The wall-pressure changes measured for wings of various chords
are plotted in figure 15 as functions of the corrected Mach number in
the test-section center. In order to eliminate the effect of the
mounting, the wall-pressure change was not referred to the wall pressure
of the free test section but to the wall pressure for a mounted wing









28 NACA TM 1300


of 350-millimeter chord. In order to obtain a computational comparison
the Prandtl principle was assumed to be valid also in the neighborhood
of the model. Although this assumption does not hold with certainty
if compression shocks occur at the model, that is, at a Mach number
above approximately 0.76, it may nevertheless be assumed that the
admissibility of the assumptions can be checked from the trend of the
computed and measured curves. This assumption holds at least for the
greater part of the curve, which is below the critical Mach number of
0.76. The computation of the wall-pressure changes was then carried
out with the aid of equations (3), (5), (6), and (7) and tables 1 and 2,
first without taking account of the measured wing drag and again with
the drag taken into account. It was found that for the wings investi-
gated, the wing drag becomes of significance only if it has increased
greatly' because of the compression shock. Nevertheless it even then
has a small effect so that the error in taking account of the wing drag
has not too great an effect on the curves.

The effective Mach number with which the rise in the wall pressure
is to be computed is not unique, particularly at high Mach numbers.
The Prandtl principle requires that the mean Mach number of the flow
field under consideration be substituted. In order to show the effect
of this uncertainty, two curves were computed, one for the corrected
Mach numbers at the tunnel center, which in any case had to be considered
as too small in the neighborhood of the model. For the other curve the
Mach numbers computed at the wall were used, which in the neighborhood
of the body came close to the mean Mach number. From the curves in
figure 15 it is seen that the computed curves on the whole agree well
with the measurement points. The existing deviations all lie within
a scatter range that can be explained by an inexact estimate of the
effective Mach number.

It may therefore be said in conclusion that the computations
throughout agree with the measurements to the degree that may be
expected from the assumptions made.

With regard to the previously mentioned uncertainty in the deter-
mination of the effective Mach number, it is further to be added that
this uncertainty does not exist in the correction of the dynamic
pressure and the Mach number. The principal factor of importance,
namely, the ratio of measured wall-pressure change to the tunnel cor-
rection, is independent of the corresponding Mach number. Only in the
determination of the correction factors mcorr/m and rvcorr/v at
large model chord and in the taking into account of the wake displace-
ment does the Mach number enter. The deviation of the computation
through uncertainties in these corrections should not be of great
significance.









NACA TM 1300 29


2. Effect of the Lift on the Wall Pressure

The measurements described above on the rectangular wing were
carried out for a symmetrical flow about the wing, that is, for zero
lift. The fact that a change in the angle of attack to a first approxi-
mation has no effect on the mean wall pressure is to be expected from
the considerations of section II, 2. The circulation due to the lift
produces on the upper and lower sides equal and opposite velocity and
pressure changes so that the disturbance velocity through the circu-
lation in forming the mean of the wall pressures on the top and under
sides drops out. There is only an effect due to changes in the wing
drag as a result of different angle of attack, which, however, because
of the small effect of the drag on the wind-tunnel corrections, cannot
be of great significance.

These facts could be confirmed by tests. Figure 16 shows, for a
rectangular wing of 500-millimeter chord, the wall pressure p, as a
S function of the pressure difference PI PIt, already discussed
previously for 0 and 5 angles of attack. It is seen that there are
no systematic deviations between the measuring points of the various
angles of attack that could not be ascribed to errors in measurement. The
S 5 angle of attack means, however, that at high Mach numbers, in general,
the limit of the angle of attack is of greatest interest in the wind
tunnel experiment. This independence of the angle of attack means a
considerable simplification in evaluating the results because the cali-
bration curves for dynamic pressure and Mach number need not be corrected
for each investigated angle of attack.


3. Pressure Drop in the Test Section Due to Wake Displacement

In order to check equations (5) and (7) for taking into account
the wake behind the drag bodies, use was made of the fact, discussed
in section II, 2, c, that at a large distance behind the model the
disturbance velocity due to the wake is just twice as great as the
corrections at the wing location according to the equation (5). It
was further assumed that at the end of the test section, that is, about
two wing chords behind the investigated wing of 500-millimeter chord,
this final value is already practically attained. Under these assump-
tions the additional pressure drop due to the wake as compared with the
test section without obstacles could be computationally, estimated. For
the effective Mach number there were again substituted two values which
correspond respectively to the Mach number in the test section and the
Mach number at the end of the test section.

For this comparison too (fig. 17), the measuring values lie within
the region that is described by the computed curves. The residual
deviations can be ascribed to measuring errors or deviations in the
effective Mach number.


V ,








NACA TM 1300


VIII. SUMMARY

(1) For wings with finite ratio of span to tunnel diameter and
for bodies of rotation in closed circular tunnels, the wind-tunnel
corrections due to the model displacement in an incompressible flow
are computed.

The corrections, in contrast to Glauert's method, are given as a
function of the volume of the displacing body. In this method the
effect of the contour shape for slender body shapes becomes vanishingly
small; in particular in the limiting case of very slender bodies the
same form factor is obtained for three-dimensional and two-dimensional
flows.

(2) For incompressible flow the additional velocities due to the
dead-water region behind the resistance bodies is represented by a
simple equation.

(3) With the aid of the Prandtl principle it is shown that for
compressible flow the tunnel corrections due to the model displacement
increase as l/(1-M2)3/2 and due to the wake for equal drag area as
1/(1-M2). The corrections due to the lift remain, for equal lift
coefficient, unchanged provided the wing chord is small compared to
the tunnel height.

(4) On increasing the Mach number the corrections to take account
of the stream curvature for models with large chord rise as 1//1-M2
so that these corrections have significance at high Mach numbers even
though they are negligible at small Mach numbers.

(5) The derived corrections for high Mach numbers remain valid
if along the model surface the assumptions of the Prandtl principle
are no longer satisfied. The limiting Mach number up to which the
method is applicable is to be determined by wind-tunnel tests.

(6) A comparison between computation and measurement shows good
agreement as far as may be expected from the assumptions made.

Translated by S. Reiss
National Advisory Committee
for Aeronautics









NACA TM


1300


TABLE I FACTOR TV FOR THE CLOSED CIRCULAR TUNINEL1 (EQUATION 3)

[B, span of rectangular wing; D, tunnel diameter]


Model shape Body of Rectangular wing
rotation
B/D = 0 B/D = 0.25 B/D = 0.50 B/D = 0.75
Factor TV 1.02 1.02 1.04 1.06 1.10


iGlauert (reference 3, p. 58) gives for the factor Tv after suitable
conversion for bodies with rotational synmmetry:

Ty = 0.797 X 4/r = 1.016

for closed circular tunnel,

7V = -0.206 x 4/i = -0.263

for open circular tunnel. The deviations of these values from those
in table 1 lie within the accuracy of computation.


TABLE II FACTOR m FOR THE CLOSED CIRCULAR TUINUEL (EQUATIIO 4)

[B, span of rectangular wing; D, tunnel diameter]

Model shape Body of Rectangular wing
rotation
B/D = 0 B/D = 0.25 B/D = 0.50

Factor m 0.45 0.45 0.46 0.49


L


*f









NACA TM 1300


TABLE III VELOCITY CORRECTION IN VARIOUS WIND TUNNELS FOR

A WING WITH ELLIPTICAL CROSS-SECTION WITH d/t = 0.10

FOR A MACH NUMBER OF 0.75


Closed tunnel
with parallel
walls;
h = constant;
b ---o


11.4
4.2
2.5


-- Ferri
-- Lamla
-- Franke-
Weinig
and Glauert-
GCthert


Open free jet 4 1 -4.2 -- Lamla
with parallel
walls; -1.3 -- Glauert-
h = constant; G8thert
b --!

Closed circu- 4 0.25 1.2 -- Gothert
lar tunnel









NACA TM 1300


REFERENCES

1. Ferri, A.: The Guidonia High-Speed Tunnel. Aircraft Engineering,
vol. XII, no. 140, Oct. 1940, pp. 302-305.

2. Lamla, E.: Der Einfluss der Strahlagrenze in Hochgeschwindigkeits-
Windkanalen. F.B. 1007, Luftfahrtforschung, Dez. 15, 1938.

3. Glauert, H.: Wind Tunnel Interference on Wings, Bodies, and
Airscrews. R. & M. No. 1566, British A.R.C., 1933, pp. 54-58.

4. Franke, A., and Weinig, F.: The Correction of the Speed of Flow
and the Angle of Incidence Due to Blockage by Aerofoil Models
in a High Speed Wind Tunnel with Closed Working Section.
F.B. 1171, Rep.& Trans. 259, British M.A.P., April 1946.

5. Fuchs, Richard, und Hopf, Ludwig: Handbuch der Flugzeugkunde,
Bd. II. Aerodynamik. Richard Carl Schmidt & Co. (Berlin),
1922.

6. Lamb, H.: The Hydrodynamic Forces on a Cylinder Moving in Two
Dimensions. R. & M. No. 1218, British A.R.S., 1929.

7. Mattray, H.: Ueber die Anwendung des impulsmessverfahrens zur
unmittelboren Ermittlung des Profilwiderstandes bei
Windkanaluntersuchungen. GDC 10/107T, ZWB Rep. 824/2,
Nov. 15, 1937.

8. Lotz, J.: Korrektur des Abwindes in Windkanalen mit kreisrunden
oder elliptischen Querschnitten. Luftfahrtforschung, Ed. 12,
Nr. 8, Dez. 25, 1935, S. 250-264.

9. Gothert, B.: Einige Bemerkungen zur Prandtl'schen Regel in Bezug
auf ebene und raumliche Stromung. (ohne Auftrieb), F.B. 1165,
Institute fur Aerodynamrik, Berlin-Adlershof, Dez. 30, 1939.

10. Prandtl, L.: General Considerations on the Flow of Compressible
Fluids. NACA TM 805, 1936.








ACA TM 1300

NACA TM 1300


Vo Vo + Av




'0


Figure 1. Wind tunnel with model.


One-dimensional velocity distribution according
to Ferri.


1.4


Figure 2. Equivalent dipole


.OC .16 .24 .32 .40

Thicnmess ratio, d/t

strength for elliptical cylinder and ellipsoid
(recomputed from reference 3).


of revolution


Trailin vortices




- .Mod e 1


FiZure 3. Model with vortex wake.









NACA TM 1300

SA

Through model-, Tihrough wall






Figure 4. Velocity distribution for a model in the ind tunnel
Figure 4. Velocity distribution for a model in the wind tunnel.


Figure 5. Distribution of the sources and sinks for the bounding
streamlines under consideration.


1.4

1.2

1.0

.8

.6

.4

.2


0 .1 .4 .5 .6


. .9 1.0 1.1 1.2


Figure 6. Effect of the model chord on the factors TV (equation (3))
and m (equation (4)) for source-sink body with ratio F/d x t = ~ 0.75
(rotational symmetry or wing with ratio of span/tunnel diameter --0).


m /m
corr /m
W..W WO


2


corr/ 7
I I l

TV bzw m = Factors for minutely small model chord
corr bz corr = Factors for finite model chord


Model chord/tunnel diameter t/D
I I I I I


-7


i;.










NACA TM 1300















Source




~- Reflected
.- S h t t r mirrored
--- ource

Figure 7. Source in the tunnel with reflected sources.


(a) Incompressible medium.


(b) Compressible medium, potentials of
(a) distorted according to Prandtl.
Figure 8. Wing in closed tunnel for incompressible and compressible flow according to the
Prandtl principle.


1ine


Figure 9. Dipole in parallel flow.
Figure 9. Dipole in parallel flow.









NACA TM 1300 37


---- --r -'- --- -- -


PReflected
mirrored
dipole






Boundary line -
Dilpole
------n the
t urnel









Reflected
rlic,re.
--- -dipo ie




Figure 10. Dipole inr tunnel with boundary lines for the case of the
validity of the Prandtl principle.




Compressible
Sincompressible A' B'

I. B




-II -I



D C -


Figure 11. Potential vorte:-: in the parallel flow.









NACA TM 1300


1.4

1.2


1.0
k-1 2
8C = 1 + 2
.8


.6


.4


.2
"- Mach number, M

0 .2 .4 .6 .8 1.0

Figure 12. Factor C as a function of the Mach number.
dl/4 = C dv/v = -1/2 C dp/ V2:




IF







NACA TM 1300 39


o














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ft
// ,











0 *I






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4 r--
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0 J 0









NACA TM 1300


Figure 14. Wall pressures for determining the dynamic pressure.









NACA TM 1300


.lach number at tLfurfi center, v-a corr


Figure 15. Comparison of the computed and measured wTli-pressure .chsarge. ttirouh
mountrng of rectangular wirgs (NACA c00i5-64) with various chords.








NACA TM 1300


Figure 16. -


400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 26
mm H20
Wall pressures at the measuring section at 00 and 50
angles of attack of a rectangular wing (NACA 0015-64)
with 500-millimeter chord.










NACA TM 1300 43














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