Correction factors for wind tunnels of elliptic section with partly open and partly closed test section

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Title:
Correction factors for wind tunnels of elliptic section with partly open and partly closed test section
Series Title:
NACA TM
Physical Description:
18 p. : ill. ; 27 cm.
Language:
English
Creator:
Riegels, Friedrich Wilhelm
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Wind tunnels   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Jet-boundary corrections for partly open and partly closed elliptical wind tunnels for the cases of one and two solid wall segments are presented. Also presented are the combinations of model span and extent of the solid portion of the tunnel wall for which the average correction factor is zero.
Bibliography:
Includes bibliographic references (p. 11).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by F. Riegels
General Note:
"Report date March 1951."
General Note:
"Translation of "Korrekturfaktoren für windkanäle elliptischen querschnitts mit teilweise offener und teilweise geschlossener mess-strecke." Luftfahrtforschung Bd. 16, Lfg. 1, 1939."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779928
oclc - 95136045
System ID:
AA00009234:00001


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Full Text
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1310


CORRECTION FACTORS FOR WIND TUNNELS OF ELLIPTIC SECTION

WITH PARTLY OPEN AND PARTLY CLOSED TEST SECTION*

By F. Riegels


SUMMARY


A wind tunnel of elliptic section with partly open and partly
closed test section contains a wing with rectangular lift distribu-
tion. The additional flow caused by the wall interference is determined
by conformal representation.

The correction factor for an elliptic jet of 1:/2 axial ratio is
plotted for several span-channel width ratios and several included
angles, that is, the angles which the lines connecting the end points
of the solid part of the tunnel boundary form with the center of the
ellipse. (Compare figs. 1 and 5.) As on the circular jet (refer-
ences 5 and 6), it is found that the correction for angle of attack.
and drag becomes zero at a certain included angle. This angle varies
with the wing span.

The theory is so applied that it can be utilized -also for elliptic
tunnels with different axial ratio. An extension to include wings
suspended over the median plane of the tunnel is likewise easily
possible.


I. INTRODUCTION


The finite boundary of jets causes additional velocities in the
flow, especially at the wing. As a consequence, the angle of attack
and the drag coefficient measured in the jet at a given lift must be
corrected. Calculations dealing with the effect of the jet boundaries
are numerous. Open and closed jets have been explored and also jets

*"Korrekturfaktoren fMr Windkanlle elliptischen Querschnitts mit
teilveise offener und teilweise geschlossener Mess-strecke."
Luftfahrtforschung Bd. 16, Lfg. 1, 1939, pp. 26-30.







NACA TM 1310


whose boundaries consist partly of solid walls and partly of free jet
boundaries (references 1 to 6).

The last arrangements have the advantage that they make it easy
to conduct a greater number of tests and also permit choosing a
boundary in such a way that the mean additional velocity at the wing,
and hence the correction factor, becomes exactly zero for the measure-
ments. If the wing model is suspended with the suction side downward,
jets with fixed boundaries produce additional downward velocities,
those with free jet boundaries, upward velocities. Therefore, it can
be expected that, with suitable distribution of fixed and free boundaries
over the section, no additional velocity is produced at all.

The present article deals with the effect of a partly open and
partly closed jet of elliptic section.


2. ELLIPTIC JET WITH ONE SOLID WALL (FIG. 1)


With the assumption of small additional velocities whose squares
are negligible, and a nondeformed jet boundary, the determination of
the additional flow can be reduced to the consideration of the flow
condition in a section infinitely far downstream, where an additional
velocity exists which is twice as great as that at the wing (reference 1).

Suppose that the jet has the elliptic section with the axes 2a'
and 2b' shown in figure 1; the wing of span b = 2s and rectangular
lift distribution is mounted in the center of the jet section. The
chosen system of coordinates (x,y) has its origin in the center of
the ellipse, so that the shed vortices of the wing push through the
section at the points x = ts.

Now, the problem is to define an additional flow in such a way
that the boundary conditions are satisfied. They are: for the solid
part, disappearance of the normal velocity component; for the free
jet boundary, constant pressure, which for the assumed smallness of
the velocity is identical with the requirement that the tangential
velocity component shall disappear (reference 1). But, as such an
additional flow is difficult to define in the z-plane, it is attempted
to find a plane by conformal representation in which the potential
of the required additional flow is easily obtainable. Now, reference 5
cites a report by K. Kondo which treats the corresponding problem for
circular jet. The mapping function

z" = c tan t

is used which maps the inside of a circle in the z"-plane on the inside







NACA TM 1310


of a strip in the s-plane in such a way that a part of the circumference
is changed in the one, the remaining part in the other of the straight
lines bounding the strip (fig. 3). But in the plane of the strip, an
additional potential that satisfies the boundary conditions is easily
indicated and, if it succeeds in mapping the inside of the ellipse on
the inside of a circle, the aforementioned representation is fundamentally
accomplished.

The procedure is developed step by step. With the aid of the
function


z = k sn(2 arc sin ) (1)


the inside of the ellipse in the z-plane is mapped on the unit circle
in the z'-plane (fig. 2), where 2K is the half real period, K the
modulus of the Jakobian elliptic function sn Z and


e = a'2 b'2

half the distance of the foci of the ellipse.

As to the theory of this mapping function, the reader is referred
to the report by de Haller (reference 7) and the "Schwarzschen
Abhandlungen" (reference 8).

Next, the plane z' is rotated about n/2 and followed by a
translation


z" = I c2 iz' (2)

as a result of which the axis of the ordinates of the new z"-plane
exactly separates the fixed part of the circumference from the free
part. (Compare figs. 2 and 3.) Both parts meet in the points z" = tic.
The subsequent transformation


z" = c tan ( (3)


projects these two points to infinity, where the arcs of the circle
change into straight lines of distance n/2 (fig. 3). Combining these






4 NACA TM 1310


transformations produces finally the mapping function of the z-plane
on the (-plane


S= arc tan [1 c2 irk en ? arc sin (4)
c Zn e

Since the derivative of this function is used later on, it is
given here


-i.? 1k ccnZ dn Z
d 1 rt (5)
dz z2(1 ksn2Z + 2i k(l c2)snz)

where, for abbreviation, ?K arc sin M = Z is introduced.
it e

The boundary conditions, disappearance of the normal component
on the fixed part and disappearance of the tangential velocity on the
free part of the circumference of the ellipse, can be satisfied now
in the s-plane by repeated reflection of the original vortex doublet
at the boundaries. The result is the vortex system represented in
figure 4. The potential of this flow is readily defined, since the
vortex rows can be added up in horizontal direction (reference 5):

tan ill' tan + 2 + V + in'
tan tan
Pi 2 2
FO = log (6)
t' + iq' tan + 2o + El i'
tan 2tan2
2 2

with I' + iq' and i' ir' representing the points corresponding
to the vortex points z = Ts in the s-plane. The potential of the
additional flow Is obtained by subtracting the potential of the
original flow in the z-plane from the potential Fo


tan ( iT)' tan 4 + +21o1
log + + 2t + V- in)' 2
2 2
F = T-3 log ----------------- l log z + s ( )
2rt ( 6 + IT!' ( + 2io + *- T* 2n z s
tan tan
2 2









NACA TM 1310


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    NACA TM 1310


    For the calculation, the following relations are added.

    For points of the major axis, the mapping function (equation (1))
    changes to


    x' = /k sn( arc sin A; k)
    V IT e


    (12)


    for 0 < x < e, and


    x' =


    Ln arc cosh -; k'
    n e


    (13)


    for e < x
    with k' denoting
    The points z = is
    equations (12) and
    then at the points


    t' = arc tan


    1' = arc tanh



    For the extreme case of
    becomes


    = a'b's
    4(a,2 b'2)


    the modulus k' = f1 k2
    in which the vortices in
    (13) to z' = s,; in the


    complementary to k.
    the z-plane lie change
    t-plane, the vortices


    1 2c2 + s 12 + 1V + s12(2 4c2 + 512)

    2+11 c2



    1 + B12 + /l + sl2(2 +12 12)


    2cs


    a wing with zero span, formula (11)


    fK k 4 c + 2c2 + C


    vbwhere


    k (K22
    k 4K2


    6 1 2- + k2)
    6 b1.2 A2
    a,2L


    1 -


    is a constant solely dependent on the axial ratio of the elliptic
    tunnel section.


    (14)


    (15)


    (16)






    NACA TM 1310


    3. ELLIPTIC JET WITH TWO SOLID WALLS (FIG. 5)


    In this case, it is appropriate to map the inside of the elliptic
    section on the inside of a rectangle (fig. 5). The fixed walls
    correspond then to two opposite sides of the rectangle and the free
    jet boundaries to the other two sides. The first step is the same as
    before, namely, the inside of the ellipse is mapped on the inside of
    the unit circle with the aid of the mapping function



    z' = k sn arc sin z; k) (17)


    given by equation (1). The inside of the unit circle becomes the
    inside of a rectangle by means of the function


    ( i=' 2z'
    sn; cos -) = 2 (18)
    2 1 + z'2


    if C represents the coordinate of the plane of the rectangle and
    cos is the modulus of the elliptic function sn (compare
    reference 6, p. 170), where c, is an angle specifying the tunnel
    section opening ratio in the plane of the circle. (Corpare fig. 5.)
    The connection between the s-plane and the original z-plane is
    therefore given by the mapping function


    cos = 2/k sn (Z; k)
    sn(; cos (19)
    2) 1 + ksn2(Z.; k)


    2K z
    where, for abbreviation, arc sin = Z, as before. The modulus k
    Ir e
    is again dependent on the axial ratio of the elliptic section. The
    square of the derivative of this function Is given by

    16K2 2
    6K 2 kcn 2Z dn2 Z
    (dt 2 A2 (20)
    S= (e2 z2)(1 2k cos sn2Z + k2snZ)







    8 NACA TM 1310


    The conditions at the jet boundaries are satisfied by repeated
    reflection of the vortex doublet on the sides of the rectangle, while
    the sense of rotation of the vortices is inverse for reflection at the
    sides that correspond to the fixed walls and remains the same for
    reflection on the sides of the rectangle that correspond to the free
    jet boundaries. The potential of such a system of vortices is given
    by (compare reference 6)



    sn -c n
    F0 i 2 2
    dshS-Lk^-
    Fo 2# g g

    2

    s 4-p enF
    + log (21)
    dnrF-
    2


    where T6' is the location of the original vortex in the t-plane (in
    the z-plane, the wing is in the center of the jet, hence z = Ts).
    Subtracting from this expression the potential of the original vortex
    in the z-plane


    ri z + s
    F1 = r log z- (22)
    S2i z s


    leaves the potential of the looked-for additional flow



    F = FO F1 (23)


    Since F = 0 + it, the stream function can be deduced, and so yields
    the mean downwash over the span of a wing suspended in the center of
    the section







    NACA TM 1310


    1 *(-s) *(s)
    2 2s


    Iir dt\ cnfl
    w log s
    4ns Fdz z = s snt'dnt'

    With
    CnVt
    r =a-
    2


    and from the relation for the additional angle of attack


    - c
    v a F
    V 8 F0


    (24)


    (25)


    the correction factor 5 follows as


    2KcnZ dn Z(l k2snZ)
    -, a'b' x
    F = 2s2- log
    e2 s2snZ(l 2k cos esn2Z + k2s n Z)


    . (26)


    where k is the modulus of the present elliptic function. For the
    extreme case of a wing with disappearing span, this equation simplifies
    to


    5 = alb' 1 K2 (1 6k cos 3- + k2)
    6(a'2 b' 2


    (27)







    NACA TM 1310


    4. RESULT


    For an elliptic tunnel of l: 2 axial ratio, the correction
    factors 3 were computed by equations (11) or (16) and (26) or (27)
    for wings with rectangular lift distribution for the span-major ellipse
    axis ratios b- = 0, 0.2, 0.4, 0.6, and 0.8 and plotted in
    2a'
    figures 6 and 7 against an angle p or I which is decisive for the
    ratio of fixed wall and free jet boundary. It is noted that the
    correction factors for rp = 0 or -S = it and p = 2v or b = 0, that
    is, for completely open and completely closed jets, assume the values
    given by Sanuki and Tani (reference 9).

    The diagrams indicate the distance which the solid part of the
    jet boundary must reach to produce zero correction. The amount of
    this favorable coverage varies with the span of the wing, as seen
    from figure 8 where the angles for which 5 = 0 are plotted against
    the span-tunnel width ratio.

    Clearer than the plotting of the angles is the representation of
    the ratio d/b', where d is the distance between the major axis and
    the point at which solid and free boundary meet (fig. 9).

    In general, the span of a model wing for a tunnel of l:V2 axial
    ratio amounts to about b = 0.7 x 2a'. So in this case with the use
    of one solid wall the covering would have to extend to d = 0.385 to
    b'
    produce zero correction. By choosing a partial covering of the jet
    boundaries with two solid walls, the correction would disappear for
    d4 = 0.64, so that the last arrangement seems more promising in many
    b*
    respects as far as experimental technique is concerned, but naturally
    calls for more elaborate test preparations.

    Translated by J. Vanier
    National Advisory Committee
    for Aeronautics







    NACA TM 1310


    REFERENCES


    1. Prandtl, L., and Betz, A.: Vier Abhandlungenr zur Hydrodynamik und
    Aerodynamik. Kaiser Wilhelm Instituts fUr Stromungsforschung,
    GCttingen, 1927.

    2. Theodorsen, Theodore: The Theory of Wind-Tunnel Wall Interference.
    NACA Rep. 410, 1931.

    3. Theodorsen, Theodore, and Silverstein, Abe: Experimental Verifi-
    cation of the Theory of Wind-Tunnel Boundary Interference.
    NACA Rep. 478, 1934.

    4. Tani I., and Taima, M.: Two Notes on the Boundary Influence of Wind
    Tunnels of Circular Cross Section. Rep. Aeronaut. Res. Inst.
    Tokyo, Nr. 121, 1935.

    5. Kondo, K.: The Wall Interference of Wind Tunnels with Boundaries
    of Circular Arcs. Rep. No. 126 (vol. X, 8), Aero. Res. Inst.
    Tokyo Imperial Univ., Aug. 1935.

    6. Kondo, K.: Boundary Interference of Partially Closed Wind Tunnels.
    Rep. No. 137 (vol. XI, 5), Aero. Res. Inst., Tokyo Imperial Univ.,
    Mar. 1936.

    7. de Haller, P.: L'Influence des Limites de la Veine fluide sur les
    characteristiques aerodynamiques d' une surface portante. Comm.
    de 1' institute d' Aerodynamique de 1' Ecole Polytechnique
    Federal, Zirich 1934.

    8. Schwarz, H. A.: Gesammelte Math Abhandlungen, Bd.II, p. 102.

    9. Sanuki, M. and Tani, J.: The Wall Interference of Wind Tunnel of
    Elliptic Cross-Section. Proc. Phys. Math. Soc. Jap. III.s., Bd 14,
    1932, p. 592.







    NACA TM 1310


    \ /



    2a' -


    Figure 1.- The elliptic jet with one solid wall.


    /--


    1


    I- I -


    Figure 2.- Mapping of z-plane on the z'-plane.


    Ile


    N





    NACA TM 1310


    - -
    N.

    0


    0


    Figure 3.-
    Figure 3.-


    0


    i;,1



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    -b- ___
    / ^0c

    /Mapping of the z "-plae on the -plane.

    Mapping of the z "-plarne on the s- plane.


    I


    q


    q


    0I


    Figure 4.-


    The reflection system for compliance with the boundary
    conditions.


    C)


    0









    NACA TM 1310


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    NACA TM 1310


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    / r 2s

    -Jos./


    Figure 6.- The correction factor 6 plotted against angle p one
    solid wall.


    -08


    -10


    -1 .4






    NACA TM 1310


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    I-.-------- 1 4 4 4 4 -~.---------~. 4t.&..-1 4


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    Figure 7.- The correction factor plotted against angle a two
    solid walls.


    N


    \


    200


    600


    08

    06

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    NACA TM 1310


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    120





    90





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    300




    ~ S
    a' -

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    Figure 8.- The values of and d for zero correction factor plotted
    against the span-tunnel width ratio.







    NACA TM 1310


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    UNIVERSITY OF FLORIDA


    31262 08105 805 8




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