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# On the contribution of turbulent boundary layers to the noise inside a fuselage

## Material Information

Title:
On the contribution of turbulent boundary layers to the noise inside a fuselage
Series Title:
NACA TM
Physical Description:
43 p. : ill ; 27 cm.
Language:
English
Creator:
Corcos, G. M
Liepmann, H. W ( Hans Wolfgang ), 1914-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

## Subjects

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

## Notes

Abstract:
The concept of attenuation is abandoned and instead the problem is formulated as a sequence of two linear couplings: the turbulent boundary-layer fluctuations excite the fuselage skin in lateral vibrations and the skin vibrations induce sound inside the fuselage. The techniques used are those required to determine the response of linear systems to random forcing functions of several variables. A certain degree of idealization has been resorted to. Thus the boundary layer is assumed locally homogeneous; the fuselage skin is assumed flat, unlined, and free from axial loads; and the "cabin" air is bounded only by the vibrating plate so that only outgoing waves are considered. The results, strictly applicable only to the limiting cases of thin boundary layers, show that the sound pressure intensity is proportional to the square of the free-stream density, the square of cabin air density, and inversely proportional to the first power of the damping constant and to the second power of the plate density. The dependence on free-stream velocity and boundary-layer thickness cannot be given in general without a detailed knowledge of the characteristics of the pressure fluctuations in the boundary layer. For a flat spectrum the noise intensity depends on the fifth power of the velocity and the first power of the boundary-layer thickness. Thus it appears that boundary-layer removal is not an economical means of decreasing cabin noise.
Bibliography:
Includes bibliographic references (p. 27).
Funding:
Statement of Responsibility:
by G.M. Corcos and H.W. Liepmann.
General Note:
"Report date September 1956."

## Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874415
oclc - 156938071
System ID:
AA00009199:00001

Full Text
kck74-~m -I/ ac

TECfJICAL MEMORANDUM 1620

ON THE CONTRIBUTION OF TURBULENT BO~NDAR(Y

LAYERS TO THIE NOISE INSIDE A FUSELACE*~

by

G. M. Corcos"H
and
HI. W. Liepmann**

September, 1956

"Un~edited by the NACA (the Committee
takes no responsibility for the
correctness of the author 's
statements. )

Aerodynamics Research Group
Douglas Aircraft Company, Inc .
Santa Monica, California
California Institute of Technology
(Consultant, Aerodynamics Research
Group, Douglas Aircraft Company, Inc .
Santa Maonica, California)

NACA TM 1420

TABLE OF COjNTENTS

ABSTRACT.............................. i

INTRODUCTIONI .. .. .... .. .. .. .. .. .. .. .. 1

I. THIE ACOUSTIC COUPLING OF A RANDOMLY
VIBRATING PLATE WITH AIR~ AT REST .. ... ... .. .. 4

II. THE DYNAMIC BEHAVIOR OF THIE SKIN ... .. .. .. .. .. 8

a) The Mean Acceleration .. .. .. .. ... 9)

b) The Length Scale A .. .. .. .. .. 12

III. THE FORCING FUNCTION .. .. ... .. ... .. .. 14

IV. SPECIAL CASES .. .... .. .. .. ... 2C

1. Convectred Turbulence ... .. ... .. .. .. 2C

a) The coupling of the plate with air at
rest in the case of convected turbulence .. 20

b) The response of the plate .. .. .. 21
2. The case of zero scale .. ... .. .. .. .. 23

V. SUMMARY OF RESULTS AND DISCUSSION .. .. .. .. .. .. 24

A Note on Testing ... .. .. .. .. .. 25

VI. REFERENICES .. .. .. .. ... .. ... ... .. 27

APPENDIX I: THE RANDOM RADIATION OF A PLANE SURFACE:

A. Four Limiting Cases .. .. .. .. .. ... 28

B. The Noise Generated by Skin Ripples
of Fixed Velocity .. .. ... .. .. 35

C. The Generation of Noise by Plate
Deflection~ of Zero Scale ... .. .. .. 38

APPENDIX II: THE SIMPLIFICATION OF THE PLATE RESPONSE INTEGRAL 40

APPENDIX III: THE EVALUATION OF THIE INTEGRAL SCALE .. 42

Digilized by ille Internel Archive
I i~n201wll MII1undIn~ from
UniverslIv of Florida. George A. Smather Llbraries wyith support from L'rRASIS and the Sloan Foundalion

hilp: wwwv\~.archlve.or g delailS oncontrlb ullonol00ulnlt

NACA TM 11+20

ABSTRACT

The following report deals in preliminary fashion with the transmission
through a fuselage of random noise generated on the fuselage skin by a tur-
bulent boundary layer. The concept of attenuation is abandoned an instead
the problem is formulated as a sequence of two linear couplings: theturbu-
lent boundary layer fluctuations excite the fuselage skin in lateral vira-
tions and the skin vibrations induce sound inside the fuselage. The techniques
used are those required to determine the response of linear systems to random
forcing functions of several variables. A certain degree of idealization has
been resorted to. Thus the boundary layer is assumed locally homogeneous,
the fuselage skin is assumed flat, unlined and free from axial loads and the?
"cabin" air is bounded only by the vibrating plate so that only outgoing
waves are considered. Some of the details of the statistical description
have been simplified in order to reveal the basic features of the problem.

The results, strictly applicable only to the limiting case of thin
boundary layers, show that the sound pressure intensity is proportional to
the square of the free stream density, the square of cabin air density
and inversely proportional to the first power of the damping constant and
to the second power of the plate density. The dependence on free streams
velocity and boundary layer thickness cannot be given in general without a
detailed knowledge of the characteristics of the pressure fluctuations in
the boundary layer (in particular the frequency spectrum). For a flat
spectrum the noise intensity depends on the fifth power of the velocity
and the first power of the boundary layer thickness. This suggests that
boundary layer removal is probably not an economical means of decreasing
cabin noise.

In general, the analysis presented here only reduces the determionat
of cabin noise intensity to the measurement of the effect of any one of
four variables (free stream velocity, boundary layer thickness, plate
thickness or the characteristic velocity of propagation in the plate).

The plate generates noise by vibrating in resonance over a wide!
range of frequencies and increasing the damping constant is consequently
an effective method of decreasing noise generation.

One of the main features of the results is that the relevent quatitis
upon which noise intensity depends are non-dimensional numbers in which boundary
layer and plate proper-ties enter as ratios. This is taken as an indication
that in testing models of structures for boundary layer noise it is not
sufficient to duplicate in the model the structural characteristics of the
fuselage. One must match properly the characteristics of the exciting
pressure fluctuations to that of the structure.

Ii

TECHNICAL MFJEORAN~DUM 1420

INTRODUCT~IIO

In his efforts to minimizer the noise levels for which, he is responsible,
the airplan~e designer has had to pay increasing attention to a source of
noise which until recently had been ignored. This is the boundary laye~r.
The boundary layer will generate noise whenever it is the seat of any
fluctuating phenomenon. In particular it will nurture random pressure
fluctuations whenever it is turbulent.

The designer's interest will, naturally center on the characteristics
of that part of the boundary layer noise which has been transmitted through
the fuselage skin and into the cabin rather than on that part which is
is, as we shall see, a rapidly increasing function of the velocity of Uthe
boundary with respect to the air and to a likely observer outside a fuelage
either the relative velocity of the plane is low (as near a take-off or
landing) or the plane is considerably distant.

As a consequence the practical question which has to be~ raised concerns
the effects on a fuselage skcin, and on the air which it encloses, of the
boundary layer pressure fluctuations acting directly on the skin.*

Two features of this problem are worth noting: To begin with, the
fuselage will transmit noise only by deflecting laterally. The thickne~ss
of the skin is very small when compared to the wave length in metal of
audible sound waves so that, effectively there are no fluctuating pressure
Gradients within the skin and hence the latter will not oscillate in lateral
compression.

In the second place, the turbulent pressure fluctuations in the bounary
layer are random both in space an~d in time. The fluctuaions are generated
locally. If they are measured simultaneously at two different points of
the boundary layer, say on the skin itself, they are found to hav nro
relationship with each other unless the two points are separated by a very
short distance. Alike the value of the pressure fluctuation, at a given
point soon loses correlation with itself.+-

"Noise intensity is defined in this report as pi2 foai. It is assumed th
this is the physical quantity of interest. It ha's the dimensions of an
energy flux; but it is not necessarily equal to the energy flux at some
point in the field, nor is it necessarily equal to the density of
energy radiated away (lost) by the fuselage.

"Recently two authors (Refs. 5 & 6) have suggested that the ranomes in
time is not independent of the randomness in space; i~e., that the pressure
fluctuations at the wall are created by the convection at a single speed
of a "frozen" pattern of pressure disturbances. Some attention is paid
later to this eventuality which is treated as a special case.

2 NACA TM 1420

Wle are thus led to visualize the process of transmission of boundary
layer noise through the metal stein of' the fuselage as follows: A multitude
of external pressure pulses push the elastic skein in and out and the skin,
in turn, not unlikte a set of distributed pistons, creates inside the fuse-
la~se pressure waves which propagate and superimpose. This constitutes cabin
noise.

It is of course desirable to determine the characteristics of this
nroi se. One should point out that the data for` the problem are not complete
and are not likely to be so in the near future. Specifically the structure
of the turbulent mechanism within the boundary layer and, in particular
of the coupling between pressure fluctuations, velocity fluctuations and
temperature fluctuations is not enough explained or measured to define
wholly our forcing function. As a conseqqg ,ce it is not now possible to
define say av:erage cabin noise intensity pi' as a function of say, free
stream Mlach number, Rey~nolds number and plate characteristics. It is
however possible and it is the purpose of this report to indicate the
approximate functional dependence- of p; on these quantities anrd thus
to give similarity rules which will reduce to a minimum the amount of
testing required.

Hie assume at the outset that the boundryis: layer unsteady pressure
field is known and that it induces small deflections in the s~in. As a
consequence

(a) The skinr dynamics are described by a linear equation
(b) The generation of' a random pressure field inside the cabin

Thus the mathematical techniques used are those required to obtain
the response of linear sy~stems to stochastic forcing functions of several.
variables.

We also assume the fuselage to be a large flat plate. This assumption
is not necessary but it simplifies the discussion and allows us to present
mrore clearlyr th~e new features of the problem.

The material in this report is presented as follows: First we study
the radiation of sound from a randomly vibrating plate. It is found that
the sound levels in the cabin are de-fined byr the intensity and the scales
of the plate normal accelerations.

Second, generalized Foulrier analysis is put to use in order to relate
the normal acceleration of the plate to the forces exerted on it by the
boundary layer flow.

Third, the bolundary layer forces are defined in terms of flow character-
istics, and dimensional similarity is used to determine the significant
parameters.

Finally, the functional form of the noise intensity in the cabin is
given save for an unknown function oftone non-dimensional parameter. This
function depends on the frequency spectrum of the pressure fluctuations
in the boundary layer. Hlo measurements yielding this spectrum have been

NACA TM 1620

reported to date and speculations concerninC: it would introduce in the
analysis both complication and uncertainty.

A summary of results is given at the end of the report. Som
derivations and some of the longer arguments have been presented as
appendices to the text.

NACA TM 1420

I. THE ACOUSTIC COUIPLING OiF A RANDO3MLY

VIBRATINGC PLATE WITHI AIR AT REST

when the sound is generated in an otherwise unbounded stationary gas by a
large~ flat plate or dise oscillating normally to its plane (Ref. I page 107).

where :

the static pressure p has been broken into a steady part p and
a fluctuating part, pi:

p = p ~j(x,y,z) + pi(x,y,z,t)

V, = the normal velocityI of the plate

do* = an element of plate surface area

ai = the speed of sound in the air

S = the vector position over which the integration is carried out

=(x',o,Z')

if)~ = the distance between source and field points

= (x-~y~ -x') 2 z-'2

A = the total area of the plate

The subscript i refers to properties inside the cabin or on the side of the
plate on which air is not flowing.

The normal acceleration bVn/8t is a random fumetion of time and space
and so is pi. W~e wish to evaluate the mean square of the pressure (a quantity
to which sound intensity is directly proportional).

W~e notice that for a sum

alike, for the integral in (1))*
+This discussion follows closely the arguments set forth in ref. 3 and the
background of information on statiat~ical methods can be found in ref. 2.

NACA TM 1420

and since aX = arif a does no depend on timet, we~ can write

No teexpression

is the correlation (-i.e. the time average of the product) of the normal
acceleration of the plate surface at two points 81 n 2adttw
different times(t- Ryan) and (t- 9:)&}. If the two points vibrate
completely independently from each other, this expression is zero and
if the two points are brought topothLEr (Sy-wc therefore cl- r2) the
-orrelation function is simply(ids/Db)l Now we assume that the average
properties of the plate motion are the same anywhere on its surface and
at any time (we assume statistical homogeneity and stationarity). Then the
expression above for the correlation becomes merely a function of the
distance ( -5\) between the two points and of('L-Ra We call this

then

IAA
Now we can evaluate pi2 (Y) under a variety of assumptions for for the
plate area A and for the distance Y between the plate and the observer. We
will consistently hold the view that the normal acceleration at most points
of the plate surface are not correlated ( ~= 0) and that two points of
the plate, in order to show appreciable correlation, must be a small
fraction of the total plate size away from each other.

We define as h thema distance over which~bd~ is strongly
correlated (i.e. g (10#dj)~ and call it the integral (length) scale*.
w The integral scale is given, say in the x-direction by

We should note that
The integral in (4) is not finite if the plate area A is infinite
The length scale h plays no role in the geometry of the problem.
The integral is a function of Y, the distance to the plate, and the plate
dimensions only. For instance if the plate is circular and of radius R

+ This case is treated more rigorously in Appendix IA

NACA TM 1420

Our hypothesis can then be expressed as

where R is the average linear dimension of the plate.

A representative case+
Suppose that the observer distance Y to the plate is such that

and that no appreciable phase difference can arise at Y between two strongly
correlated signals (i.e., between sound pulses originating within jL of
each other). Then, if we examined equation (3)

ve see that the inner integral contributes very little except when the
point 3t is approximately within a distance ), of \$ Then, approximately
'Ls and 1 P(bVag Thus the inner integral is approximately

and equation (3) can be rewritten

I;' "~
47\:'

dsl

h' (dv,)nl
dr

NACA TM 1420

It is apparent from this result that the distance Y from the plate to
the observer is measured in terms of plate diameters, and not in terms
of average correlation lengths or wave lengths )L This result holds for
all the cases considered (see Appendix I) and depends only on the
assumption that at a given time the plate vibrations are largely un-
correlated or incoherent."

to define a strength.

Equation (4) indicates that in order to evaluate the pressure
intensity for the case considered above, one must first determine (bS /3)0
the mean square normal acceleration of the plate and \ the length
scale for the plate deflections. Other cases (i.e. cases for which
either the plate deflects differently or the observer moves closer to
it) are treated in Appendix IA. For some of them the time scale or
mean period NON is required as well as L -

We rewrite Equation (4)

where la (pV is a function of the plate geometry and of the distance
between the observer and the plate. It is defined from equation (4) as

Notice that if the integral scale Is not the same in the x' and in
the z' direction we may simply substitute in equation (6) hrr' and h S/
for h\

+ This result holds, as Appendix IB shows, even when the pressure is
generated by the travel through air at rest of a "randomly bumpy plate"
i.e., when the time-wise and space-wise variations of upwash are not
independent. The latter example is therefore quite distinct from the
flow of an infinite (periodically) wavy wall for which the only
characteristic length is the wave length.

NACA TM 1420

II. THIE DYNAMIC BEHAVIOR OF THE SKIN

The response of the bare fuselage skin to the random pressure field
of' the boundary layer is given, in the absence of axial loads, by a plate
equation. For a flat plate this equation is

where x and : are the coordinates along the plate surface (x being the
free stream direction), y the deflection of the plate at a point, E the
modulus of elasticity of the plate material, e" its density, hL Poisson's
ratio, 2h the thickness of the plate, # a damping constant which has
dimensions (1/T). Damping may be present because energy is absorbed either
within the skin or by the air." For air limping P0 /' gy) ~

Hlotice that to allow for air damping is to provide for a feedback in
the coupling between the plate and the air at rest. On the other hand
we exclude feedback between the plate and the boundary layer. In other
terms we are not considering the possibility that the plate vibrations
are large enough to induce time-dependent pressure gradients of the same
order of magnitude as our forcing function. Such a feedback would amount
to panel flutter. It cannot be handled by the present method.

r(x,z,t) is the random forcefLunit mass exerted by the pressure
fluctuations on the plate surface. It is characterized by a power spectral
density. F(kl,k2,0;) which is a continuous function of the wave numbers kl
(in the x direction), k2 (in the z, direction) and of the frequency co.
The coefficienrt E/30-(1-f2) has the dimensions of a velocity squared
and it is defined as c2.

*The skin construction may be such that the damping it causes is primarily
viscous or primarily flexural. In the latter case it seems more appropriate
to write with Ribner (ref. 5)

where fig is the flexural damping constant due to the plate and/Q the
damping due to the energy radiated to the air. As is shown in Appendix IB
the noise intensity within the fuselage may or may not be related to the
acoustic energy radiated by the plate and thu~~tcee~ aitdb h lt n hus P may or may not be

NACA IM 1420

We still have to specify the space-wise boudr conditions on the
plate and we are led, for the sake of simplicity, to either one of two
limiting cases. In the first case, the~ forcing; funtion (the random
pressure field in the boundary layer) is characterized by an integral
scale so large that at a given time, a skin are~a between two stiffeners
(assumed rigid) is very likely to be subjected to a pressure load of the
same sig~n (see Fig. Ia). This allows us to exrpress y and f as functions
of t only.

Z OR X & OR X

F-clG C F us. Ib

In the second case, the integral scale of the forcing function, is very
small in comparison with the distance between two stiffeners an the
behavior of the skin is in the ave~rage very much as though the supports
were removed to infinity (see Fig. Ib). The real case will in general
be intermediate between these two limiting examples. However, the first
case seems to apply to boundary layers of excessive~ thickness: A
reasonable guess for the average correlation length might be one dis-
placement thickness s +; for d+ to be~ larger than the spacing between
stiffeners (of the order of a foot) the boundary layer thickness Al would
have to be of the order of five! feet or more. This unlikely case is treted
in reference (7).

On the other hand, the second limiting case (Fig. Ib) would seem
to provide a reasonable model for boundary layer thickness no0t exceeding
one foot. This is the model discussed now.

a) The Mean Acceleration

According to our assumption, the average motion

NACA TM 1420
10I

is not sensibly affected by the presence of stiffeners. A large number
of: pulses act on the sktin at a given time between two consecutive
stiffeners. The random pulses may be positive orr negative and thus
there will be a large number iof load rev~ersals between supports. Then
the effect of the boundary conditions can be expected to become small,
in the average. Consequently one can define a generalized admittance and
use it in much the same way as is often done in one-dimensional problems."
For instance, the mean square plate displacement is given by

Here the mean square of' the forcing function f' is related to the
spectrumn by:

l;ffis the generalized admnittance, and kl, k2 and Wt are respectively
thel wave number In thle x( direc-tiojn, the wave number in the z directions
and the frequency. The determination of If)(is easy once it is realized
that this expression is the square of' the Fourier transform of the
fundamental solution and so can be written by inspection. Thus an
average solution of (7) is:

and

--- fr.uw~ h Bl (9)

Equation (9) gives the mean square response of an unbounded plate to
a random forcing function. One should notice that the plate will always
exhibit resonance no matter what value the dam~ping constant /8 may have.
This resonance occurs, not at a given frequencyi or at a set of discrete

+ See in particular reference (2).

NACA TM 1420

WITH 1~= 0.1
ano +Z = A

THE ADMITTANCE MAP FOR AN UNBOUNDED PATE

FIGURE 2

//X C~~4
e n2 5!
C'h',3;IC~2 -Gi

NACA TM 1420

frequencies but over the whole frequency spectrum, whenever the follow-
ing relationship obtains between frequencies and wave numbers:+

We can visualize the resonance condition as a crest or ridge in the
wave space (see Fig. 2) which originates at the line k2 = (3/chf2 and
which becomes hiigher and steeper as the wave number and the frequency
increase. Thus the effective damping is a function of the exciting
frequency.

b) The Length Scale A

Equation (6) shows that h2 is needed as well as ( VI/}t)2
Now A is a length scale. It was defined, say in the x direction as

Ewa

and could be termed the equivalent length of perfect correlation.

There are various ways of evaluating the integral scale. Perhaps
the most convenient one for our purpose is that (found for instance in
Ref. 2 Eq. IIS) which is derived from the relationship between correla-
tion functions and spectral functions. Thus if a stationary random fune-
tion J(t) possesses a correlation function which is sufficiently well
behaved,

+ H~ere resonance is defined as the maximum of the response curve 1/j(k)
holdinggg) constant. The locus of maximae holding k constant is given
by:

These maximae correspond only for zero damping.

NACA TM 14c20

one can define a spectral density function

Nov for the particular casek 4 = this gives

Since

n, =- *~~
d] (o )

we have the result that

z Pe)

(10 )

We~ hv already obtained a forma representation for the spectral density
function of the plate. It is the integrand of Eq. (9) so we! can write,
in view of (10)

(11)

and a similar expression for hy .

**j O /t,,c

31 [n-lh e- z r SE'A4,s.

/ bVn 2.

NACA TM 11620

III. THE FORCING FUNJCTION

We suppose that a turbulent boundary layer develops on the skin of
the airplane~ (on one side of our plate). The forces which excite the
plate are the pressure fluctuationls experienced by the plate itself.
We assume that all characteristics of the boundary layer are fixed once
we have specified the boundary lawyer thickness b, the free stream
ve~locity Ueo and density In terms of pressure fluctuations this
implies that at a fixed point of the "vetted" surface of the skin, we
have for the mean square pressure fluctuations

~- fo Ut
and the: integral scales, i.e.

are proportional to b. Also the relative contribution to pressure
intensity of the various frequency bands must be a function of Ugg,5,
amndtJ only so that f aC is a random function of three independent
variables, x,z,t, PIis related to a three-dimensional spectrum by:

anc that :

NACA TM 1420

Loosely speaking, this means that a characteristic frequency for pressure
fluctuations is proportional to Use /, and a characteristic wave length is
proportional to d Now the forcing function of Eq. (7) is a forcefunit
mass so that according to our similarity hypothesis

B'

One can thus define a spectral function associated with the forcing
function f(x,z,t):

and thus

S~i"~b
6lk'L

(12)

Here F2 is a function of K1, K2, and .11 only and these are non-dimensional
variable s:

Jr.= o/ U,

In terms of these non-dimensional units, Eq. (8) becomes:

~y 91 U~

Ua,

NACA TM 11s20

and Eq. (9) becomes

which can be written

Again, F2 is a function of the3 integration variable only, so that one
can write

b(*l" po', U + ch (5 &4

Equation (14) yields the two-non-dimensional parameters upon which the
plate dynamics depend. The first one, Ch /SJo 04 i the product of a
speed of free stream
Mach umer, ( ) and a
Speed of propagation of waves in the plate
thickness ratio (to~dr )ai crrs The second one, pbu
plate thickness
is a non-dimensional damping parameter which is, alike, a function of
plate and free stream properties.

If we treat the equation for the integral scale (Eq. 11) in the
same way, we notice that no new non-dimensional parameter occurs, so
that, at most

SL c (15)
7- 5GS T

Cao4jLr(~ d~acboK
ck ~=rp~'rtpr

I SUao nU JL
F(Kln)r(L ak 'e

NACA TM 1420

We now vish to investigate the form of the functions H and L in
Eas. (14) an (15) respectively. :First, we make an assumption which
is not strictly necessary but which simplifies the manipulation of
Eq. (12). We take the function F(KI, KtlR) to be symrmetrie in KI
and Kg, which leads us to define a new wave number.

and to write

Thus, Eq. (13) becomes

Now, the dapng
circumtances it

S ~
'L~~IC~o Uao
bth"

paramneter 13141D is assumed small and under these
can be shown (see Appendix II) that

(16)

The small difference between these two integrals can easily be evalaed
for arbitrarily small values of pllUa even though both integral are
unbounded as P-*** This leads us to believe tha for low damping the
main contribution to the inner integral comes from the resonansce condition

Thus, if the spectral function F( L,K) is reasonably wide, i~e. ifbF bK((1
over a large range of K Eq. (16) suggests tha we write

(r7)

1:JL"drr IraDIc CCIH du(hc Fl kJT)
r ILC I
c Ke ~r
lb'0~; Uclr,

Kdl

NACA TM 1420

The requirement that F be flat in KC when compared to 1/X(K) is equivalent
to the requirement that the average correlation distance or integral scale
for the boudr layer pressure fluctuations be small compared to integral
scale of the plate defl-ection. Translated in physical terms the simplifica-
tion suggested here is prompted by the following remark: If the plate
has some stiffness, it makes little difference whether the forcing
function is assumed to be distributed over smal distances or made of con-

FIGURE 3

Thus a satisfactory model for the problem at hand would
rain drops on a metal roof. Equation (16) allows us to
to get

be the impact of
integrate over K,

blbo
FlJb~"~;E~J' '

~" 9."uf
t 4 a2h~+

~:~~ ~oC
---
a' h'e ~

dJL

(18)

(~~3~ns \ loI"n

")dn

bV~ndt ~

bu n
ch
~-(~,

NACA TM 1-420

The expression

can be evaluated only when F(K n) is known. It may be an increasing
or adeceasig fncton olinh .In the absence of data on the
spectral function F, we will not tep odfn t

The function H defined by Eq. (14) can be written:

c~~~h & ']I .

where f,, is an unspecified function related by (18) to the bounar
layer pressur-e spectrum.

In order to determine p 2 we need to find out, in addition, wha
quantities the integral scale h depends on. Here we make use of con-
siderations which are similar to those yielding Eq. (16;) (see Appendi III).
The result is that

whref is another function related to the boundary layer pressure
spectrum by 1II(4). Hlow we are able to write Eq. (6) as

(21)

here

Expression (21)gves the functional dependence of pressure intensity "inside"
on boundary layer parameters for a typical case. The only quantity, not
immediately available is h(b */ck). It is probable that we shall have to
await experimental data to define its numerical value reliably.

NACA TM 1420

IV. SPECIAL CASES

1. Convected Turbulence

Two authors (ref. 5 & 6) have recently suggested that the boundary
layer pressure fluctuations at any point of the fuselage skin are caused
essentially by the passage over the point of a fixed (i.e. time inde-
pendent) pattern of pressure disturbances carried downstream at a fixed
convective velocity. So far, experimental evidence in proof or disproof
is lacking. However, it is interesting to incorporate this special case
in the general formulation which has been presented. Both the response
of the plate and the coupling of the plate with the air at rest must
then be reconsidered.

a) The coupling of the plate with air at rest in the case of
convected turbulence

If a fixed spatial pressure distribution is carried downstream
on the surface of the plate, it is easy to show that the (infinite) plateY
response will be of the same kind, i.e., that it will consist of ripples
which are randomly distributed in space but which travel through the plate
at the same convective '.elocityr as the boundary layer disturbance. The
determination of the pressure field inside the fuselage is not in principle
different for this case and has been carried out in Appendix I~B*t. The
result is that for both subsonic moving ripples (with convection velocity
U, C ELi ) and moderately supersonic ones:

where I

For higher supersonic speeds, the function of geometry and Mach number
appearing as an integral is more complicated. The equation (I.10) above
has the same form as equation (6). On the other hand there is a sharp
difference in terms of energy radiated by the plate between the subsonic
and the supersonic case, since no energy at all is radiated by subsonic
ripples while the supersonic ones do generate some. One must, then, make

"Here the presence of transversal bulkheads will change the picture
because of multiple reflections of the ripple.

**cThis problem can also be viewed as a steady (randomly bumpy) wing
problem from the standpoint of a stationary observer.

MIACA TM 1420

a distinction between the results in terms of pressure intensity (the
quantity of practical interest) and in terms of energy radiation. This
distinction stems from the fact that (as is pointed out on page 7) the
acoustical field investigated is truly a near field.
b) The response of the plate
According to the convective hypothesis, time is not an independent
variable once the convective velocity [f, is fixed. Translated in tenns
of the spectral density T'1~(c,kl,k2) of the pressure fluctuations, this
means that I (A) ,hlak2)is zero, except whenP -= U, g, or in non-
dimensional form, when n.s rum We rewrite equation (12) for this
special case.

4
3,' u,
QS kL

'SS-~ k,, a,, ,>

Here 6 [- KsL-~ is the Dirac delta function of the variableR..
Then the plate response becomes

bVl~n \~t

at 1

99~ u~
QL~'

(23)

he re

Now if we assume as before that F is symmetric in K1 and K2 and
substitute

~= .kh;l

KI

a26 7(~11 K,~~n~61"-
Uao ~Uo~

R+ F(~,,HIJL)b~-*~
II a, d~cl ace, ~ao[' klL(22)
urf JL-

F~ ~ K,,~~ dn<~ dKL
q 6 YI
4p~ ~r LII\ r ~~hZ r~trK~ r.
~r h~

L ~~, i~~ r.~ crJ~I;

Fl K,,~z~ UL~bp)

NACA TM 1420

we~ finally get

Here

and

~bzUf d' BZ
E
a~ k+ c)p

ItrCosCa Tr(Beosesuslde
c

(24)

3 = *,/ U,

T.(aeo581~)-- F(K,,~, U,

with

KIL K+ = CoS

is indicated by the following dimensional argument.

and

Th length scale X

The mean correlation length or integral scale is a weighted average
of all wave lengths, so that dimensionally
I
Since resonance dominates the plate response, = is given from the
plate response equation (equation 23) byr the resonance condition

h

A similar reasoning would have yielded, in the non-convective case,
C~(h/,,5) VE instead of eq. (20).
Combining (24) and (25) according to (I.10) we notice that we can
still write as in equation (21).

(25)

h'L~ p

4:, 0^

(24

Here 61Is a weak function of the kach number as seen from (I.10).

NAC TM 16420

2. The case of zero scale

Under some circumstarnces it is possible that the space average of
th plate motion vanishes, i.e.:

This does not ean tha th norma accelerations at two neighboring points
show no correlation, but that the
correlation function becomes negative J
as indcated in Fig. (4) and in suc
a way that its space integral vanishes.
We can then consider the! normal
accelerations as dipoles rather tha
sources and we are led to a slightly
IC shows, however, that if one defines
a length 2 ba such that

FIGURE 4

The results are again ide:ntical in forn with those of equation (6). B~ere? hL
can be viewed as the mean moment arm of deflection moments. A~lternately
one can redefine the integral scale as

,== CCl o (27)

where c is a constant. Equation (27) can thus be used to define the!
integral scale in any event.

NACA IN 1420

V. SUMMARY OF RESULTS AND DISCUSSION

Appendix I discusses in addition to the cases mentioned in the text
a few examples which provide different limiting conditions. Thus the
observer is brought close to the plate (Y(( 7\ ). A short time scale is
considered etc..... The common feature of all these analyses is that the
resulting mean noise intensity can always be represented, say by equation
(26). We shall therefore retain this equation:

as ~ r~st
Here

Pi"

Q)i
a;

96

6

Il~op

6

zh

r"
v

MI

UI

c

general statement that we can make at the present time.

=mean square noise intensity inside

=air density inside

=speed of sound inside

= air density in the free stream

=plate density

= free stream velocity

=boundary layer thickness

=plate thickness

=viscous damping constant (of units 1/time)

=perpendicular distance between observer and fuselage

=geometry of the plate

= Mach number Ula

convectivee velocity of turbulence pattern

=characteristic velocity in the plate

NCACA TM 1420 25

For all but high supersonic- velocities, the dependence of on Na~ is
quite small and can be disretgarded. The function@~ (Y,g), a quantity
which does not depend on the dynamics of the problem but only on its
geometry should be modified to take into account the fact that the
fuselage is a cylinder and not a large flat plate.

Thne form of the function 5 cannot be given here both because no
information is yet available on boundary layer pressure spectra an
because S depends too critically on the tyipe of model assumed. How-
ever, if is measured while any one of the four vaiable defining
S (6,Uap or b) is varied, then the functional forn of thte noise
intensity inside a fuselage can be determined. Thus the main contribution
of the analysis is to diminish the extent of the testing required.

One of the conclusions which can be drawn frau the foregoing equation
is that unless the boundary layer pressure spectrum is a ve sharp
function of frequency (which would makre S very sensitive tolUIICht ) it
is not practical to decrease cabin noise by boundary layer suction: Since
the noise intensity is a weak function of boundary layer thickness,
decreasing appreciably cabin noise would involve the remloval of a
prohibitive amount of air.

Another conclusion is that increasing the damping is a very effec-
tive way of limiting the production of noise of all frequencies, since
the structure transmits sounds essentially by resonance.

The analysis which has been presented deliberately omitted some of
the features of the problem which would influence the results an intro-
duce new parameters. For instance, the fuselage of commrc-ial airplanes
is usually subjected to an axrial. tension as well as other loadis. In
addition the skin is curved. To account for these featurs of the prob-
lem one would introduce further terms in the differential equation
describing the plate and one could treat it in much the same way as
has been done here.

to the study of a germane problem, the fatigue of panels which are
buffeted by a turbulent boundary layer.

A NOTE ON TESTING

The discussion of the various limiting solutions makes it clear
that for the transmission of boundary layer noise through a structure~,
the ratio of outside (boundary layer) noise to inside (cabin) noise is
in general a function of boundary layer as well as structural character-
istics.

This is to say, first, that an attenuation coefficient cannt be
defined by testing the structure alone with a standard noise source.
Thus accurate testing requires at the outset that th model be tested

NACA TM 1420

for transmission of a noise similar to boundary layer noise. The main
property of such a noise, as we have seen is that it must be random in
space as well as in, tim, whch precludes the use of one or a few
concentrated sources as noise generators. The only proper substitutes
for boundary layer pressure fluctuations are forcing functions whose
effects on a fuselage are local.* The impact of water drops for instance
might be found adequate simulation. Further, similarity in testing
requires the matching of' parameters which are ratios of plate and forcing
funtion properties. For instance if the forcing function used in the
test is a turbulent boundary layer, similarityr parameters are:

*This is not true of jet noise which is Generated away from the fuselage.

NaACA TM 1420

VI. REFERIEICES

1. Lord Rayleigh: The Theory of Sound. Dover Publications,
New York. Volume II (1945).

2. Liepmann, H. W.: Aspects of the Turbuence Problem (Part I)
ZAMP, Volumet III (1952) pp 321-342.

3. Liepman, H.W.: On the Application of Statistical. Concepts
to the Buffeting Problem. Journal of the Aeronautical Sciences
19 (1952) pp 793-800.

4. Timoshenko 8. and Young, D. A.: Advanced Dynamics. Mc~ratw-Kll,
New Yorkt (1948).

5. Ritne~r, H. S.: Boundar~y-Layer Induced Noise in the Interior of
Aircraft. UTIA Report No. 37 (April 1956).

6. Kraichnaun, R. H.: Iloise Transmission from Boundary Laye~r Pressure
Fluctuations. To be published.

7. Co~rcos G. H. and Liepmuann, H. W.: On the Transmission Through
a Fuselalle Wlall of Boundary Layer Noise. Douglas Report Noe.
SM-19570. (1955)

MACA TM 1420

AiPPENDIX I

PHE RANDOM RADIATION OF A PLANE SUIRFACE:

A. FOUR LIMITING CASES

In order to determine the couling between. fuselage vibrations and
cabin air one has to choose a model for the correlation 7 1 between the
normal accelerations at two different points of the plate. The model
which was discussed and for which equation (4) was made plausible is
predicated upon two conditions:

A. That the observer is distant enough so that a large number
of plate elements vibrating independently contribute sound
in comparable amounts, i.e.

Here as before, h is the integral (length) scale for the plate
normal accelerations and Y is the perpendicular distance between
the observer and the plate.

B. That the time scale of the phenomenon is large enough so that
the differences in phase (introduced by the. unequal distance
from the point at Y to the? various points of a plate element
of length PL) are unimportant, i.e.

ai is the speed of sound in the fuselage air, and 8 is the
integral (time) scale for the phenomenon:

Then one can choose a s;imrple model for the :orrelation function 1 '

where a is the delta function. The norma accelerations are assumed
perfectly correlated within a length Lh and not at al for distances

NACA TS( 1c20

greater than PL Thlen

rr
ZI Rt

FICUTRE 5

Under these conditions the~ noise at the microphone is contributed primarily
from a single plate element which in the average vibrates in phase. The
evaluation of this contribution is particularly simple. We can vaite, very
nearly

andupon integrating

which is equation (4). This case,h 4))\ 44 4)X 1.<\$ ,;Ls)4
coragendsi to the following conditions. The passenger (or the mderophone)
is far from the plate (in terms of PL ), the boundary layer is thick and
the(3 airpane velocities low. One may well wondr about cases for which
these conditions do not aply. Whle it apears difficult to answer such
a qury with Genrerality it is possible to consider other limiting cases.

For instance let us assume that condition 2 still applies but that
our observer is extremely close to the plate. This would correspond to
the following physical case: A thin fuselage skin, a thick boundary
layer, a low airplane velocity and we are3 measuring neise by placing a
imicroprhone very close to the skin and insulating it on all sides except
the side which faces the skin. Then X7\pY }\44 CL'shSM

O IRCA TM 1420

(I.2)

and

If, for the sake of definiteness, we assume the element circular, then

and since '} ) Y it is permissible to write

The pressure intensity is therefore given as

Thus Eq. (6) applies for the very close as well as for the very far
field when phase effects are not important ( A Regi *

Now assume that we carry on the same experiment but that the boundary
layer is thin and that the velocity of the airplane is high so that the
exciting frequencies are high. Let us assume in addition that the skin
is thick, so that y (4~ h : < C /A i

FUSELAGE~ BOUym9RY LAYER

FIGURE 6

NACA TM 1420

N~ow the time scale of the plate motion is short and phase effects are
prevalent. We define a simple time history in analogy to the space
description of Eq. (I.1)

The microphone still receives signals effectively only from one plate
element an al points within that element vibrate in phase but the
pressure pulses originating from that element do not arrive at the
microphone in the sam time. Then:

......... 6( I 3)
4Is6 q*** St* at, rs0.
4~r 4t ~ICt

A' is simply Equation (I.3) is evaluated by noticing that:

Here f; are the real roots of C( ) = 03 which are included in the interval
between and and b. W~e only have one root, namely rl = r2. If we choose
to integrate, say, with respect to s2 first we get (assumning again that
the element is circular

bIr A'Wed. 4, I v.l r 9.& #./t~'+Y;
4- (I.5)

and according to (I.4) the inner internal yields:

NACA TM 1420

so that

The time scale appears explicitly in the answer. For the unbounded plate
however it is simply proportional to f /Uao just as the time scale f or the
boundary layer pressure fluctuations.

Finally we may consider a physical case for which phase effects are
important and for which the microphone has been placed a large distance
away from the plate. i.e.: )19 <( A /4*, j r 4 Y

FL./SELAGE SA/JV

MICROPHONESE~

FIGURE 7

NAA M1420

Now the contribution from each sub-element of vibrating plate is still
in the average independent from that of the next one. However, there are
in addition cancellations from. withiin one element just as in the previous
case. This will happen if the boundary lawyer is thin, the airplane
velocity is high LI small) an the observer is far from the fuselage wall.

In ordr to evaluae thiis limiting case we first specify the tim
behavior of the correlation function: wert write

3 (P,-h e,-ejJd(qp~(Lt-h, e..e) (1[.0

so that

4A

and we integrate first with resptect to .SL Using the same techniques as
in the previous exape, we ge~t:

Nov ve assume that

Integrating with respect to 02, 91 1 successively:

ata Je(o 'ho a

NACA TM 1420

For a circular plate of radius R, this would give

9;'
1TT

i~g (~,w O) h( o)

In general, and defining

a function. of the plate geome~try and of the distance Y only, we have

10;"
~icx~;

(Is)

-"( b\rtbtt

c" le d

ld-~

~(%lx~~

NA~CA TM 1420

APPED~IX I
B. TH NOISE GEEAE BY SKIN RIPPLES OF FIXED VELOCIT:

If the turbulence pattern is frozen, as discussed in section IV-1
ripples will travel through the (infinite) skin at a fixed convective
velocity. Then the~ correlation f~unction.'t mut be written differenty:

where UI is the speed of propagation of the rippe ( turbulence
convective speed) and therefore

where now

~lr J ~,'Z ~ Etl''L~t ~~

The inner integral is of' the form

which can be written

as in part A. The expression

I
l~s)

hz ~; 'IbJ1I dLllldrt:
r \~e
CI

1 dre ~' skz~l~,-~,~

NACA TM 1420

has either one or two real roots depending as 41,41 or Cfl

respectively.

For M ( 4

Sthe only real root is

and thus the inner integral yields

Il J~'

so that:

equation (4) for low

to the root X, w 5s

X"p;.t

e-rr2~

(I.lOa)

notice that equation (I.10a) above tends to
convective speeds.

For NI ( (I.9) has, in addition
root Given by

Another

.~ 5 (M:CI)PI C -L Llrl
plZ~- I

ICt is easy to show that this root exists for al values of xl. In order to
simplify the integration let us assume slightly supersonic conditions; i.e.
let us write

where

f. ~e ~

Then

'12 ~ C~tlL) lfl*4~

~)~

(Ca)

NACA TM 1420

and the inner integral =

I

so that for the supersonic case:

I
a (*Ml, j

Ms

(I.10b)

A result which is save for a constant coefficient the same as I.10a.

i
'LIL ~i

M,[r,+ U;xl')

*~L I 4 v
at]

NACA TM 11420

APPER3DIX I
C. THE GENERATIONI OF NOISE BY PLATE DEFLECTION'S OF ZERO SCALE

If the space average of the correlation function is zero and if
the plate vibrations are isotropic in x' and z' one can define a new
length scale as a moment arm:

( s is a fixed point)

alternatively one can define a modified integral scale

h"; = I 1 ( 7, ) 1

Here

h''=: C ~CI

where c is a constant.

Then, one can idealize

provided

It follows that

and integrating with respect to

h'29c~
e-~~t

51\

~, bp, bet
Ilt

9~ dp,
hi

Now, unless a or b = 0

t
"I 16 PILP'.)"P

the correlation function as

= (~6V~~~2

P('.~qb'(L-P~

Jb (1) 5()b~~J.t6t

NACA M 1)+20

and thus

Is,4ip. keplcy b 9((II-~d9~
rPI 1;
ILs.

4sro) I Idj S~'bP ,,,~~ 3
rt~ zi~ rc'Sq;

-h')S(a6F~)~
Leae

h'~4L~!YTtz. \t 51

IOi~'

If t-here is a (time) microscale
0) (OrC

P~ d~
ILu

~7;5_

Z~C

h'"Pr"rb~rcr'c4;"
4" \0"/

Here the plate has been assumed circular and R is its diame~ter.

;2

j* "'9[6~
)LZ

t~F-l

APPETH)IX I~I

THIE SIMPLIFICATION OF THlE PLATE RESPONSE INTEGRAL
(Equation 16)

We consider the approximation equation

NACA TM 1420

(16)

The righit-hand side is clearly un~bounded as the damping constant /3 --L O
since its value is explicitly proportional to 1/;a (see for instance
EQ.(18)). On the other hand the difference between the left and the
right-hiand integrals is finite for (3 = lj. To show this we write

Then the left-iland side becomes fojr (9 = 3

suchitd a I)

(II1)

and the right-ha~nd side becomes; for 9 = 0

as

C on

CL

(II2)

( ~t1)2

f
(~_()2 (~+l)e

I

so tha the difference D between expressions (Ill) and (II2) is

anCi+~scr

dc ho

& 0..

(II4)

on L~

KAM

NACA TM 1420 41

This e~xprssion is finite and of course independent of fS so that we can
concude that the left-hand integral of (16) is unbounded fo /i = O.
Further, it is clear that D is a regular function of/C3 so tha th ratio
of th lef't-han~d side to the right-hand side of Eq. (16) can be made ar--
bitraily close to unity, by choosing arbitrarily small /3. If a corree-
tion is desired a numerical check indicates that Eq. (IIA) gives a good
appr-oximation to the error made even with moderately large damping.

NIACA TM 1420

AP~PEIJDIX III

THE EVALUATION OF THE INTEORAL SCALE

Our starting point is Eq. (ll). In terms of non-dimensional variables
it becomes

We now simplify the denominator by writing successively

Then we define

(III1)

(III2)

RJLK~Z

Equation (III2) can now be written

and if we replace K.2 b;, its value at resonance," namely

+ The justification for that step is identical to that advanced in
Appendix II.

Ch ..

(fl As h4.) 12

J2. F ,

NACA TM 14210

We can write

> V. *

rr p~l~s
,,
8 a'h'

~To
P6

(III3)

If we compare (III]) to (183) we Get immediately

F( )~da

cks
U,

9.

(II14)

NACA LangLey Field Va.

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i 9,'R., T(~l go v. LI d,,l (

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{Ch!\
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B'CI- C 6010
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