Boeing document no. D6-81571,. 1994. [42] Frendi, A., and Maestrello,. L., "On the combined effect of mean and acoustic exci- tation on structural response.
NASA
Contractor
Report
201631
On the Coupling Between a Supersonic Turbulent Boundary Layer and a Flexible Structure
Abdelkader Analytical
Frendi Services
Contract
NASl-19700
October
1996
National
Aeronautics
& Materials,
and
Space Administration Langley Research Center Hampton, Virginia 23681-0001
Inc., Hampton,
Virginia
ON THE COUPLING BETWEEN A SUPERSONIC TURBULENT BOUNDARY LAYER AND A FLEXIBLE STRUCTURE Abdelkader Analytical
Services
Frendi & Materials,
Hampton,
Inc.
VA 23666
Abstract
A mathematical
model
and a computer
code
have
been
vibration of an aircraft fuselage panel to the surrounding layer and acoustic fluid. The turbulent boundary layer decomposition equations.
agreement
supersonic
panel.
1.
earlier
and
are
and
flexible
used.
leads
number
Results data.
the
to the
from
this
response
in acoustic
increases
couple
the
that
are
in the
and radiation
damping
acoustic
resulting model
It is shown
to lower
to be due to an increase Mach
averaging
numerical
panel
to fully
flow field, turbulent boundary model is derived using a triple
a conditional
equations
experimental of the
the
applying
acoustic
is believed
Increasing
with
and
existing
full coupling
This
regime.
agreement
panel with
regime,
the
in this
flow variables
Linearized
in good from
of the
developed
on the
damping,
panel
which
is in
work.
Introduction Future
efficient,
civilian
ments,
a substantial
in various Speed
aircraft,
less expensive
subsonic
and
faster
amount fields.
Transport
aircraft
With
noise
and sonic
fatigue.
The
source
of aircraft
interior
fuselage.
These
vibrations
turbulent
boundary
level,
it is imperative
from
the
turbulent
that
mechanics
Early
on the
tural
vibration
either boundary
and
experimental layer.
on
One
development
renewed research
noise
caused
aircraft.
layer
the
phenomenon, effects
of turbulent noise
or empirical
of the
turbulence,
widely
boundary
generation models used
been
pressure
to reduce
pressure
focussed
mainly
on
the
pressure
in the
literature
is the
noise
is transmitted problem
structure field
mostly
interior
understood
the
area
skin of the
in the
fluctuations
to describe
models
outer
noise
least
in the
the
is a challenging
layer
of a High
increasing of the
is the
fuel
require-
to be accomplished
by which
that
more
stringent
fluctuations
in order This
quieter,
development
vibrations
mechanisms
interior.
be
needs
in the have
the
by the
to
of these
work
interests
Therefore,
to the
have
In light
interest
is generally
we understand
interior data
and
the
in turn
the
boundary
involves a fluid mechanics. work
are
layer
aircraft.
(HSCT),
of interior
will
today's
of research
engineering
Civil
or supersonic,
than
in the
Corcos
as it in fluid
on strucand
used
turbulent model
[1].
This tral
model
density
was derived
of the
based
pressure
on experimental
_(w)
describes
distribution
and
the above
equation,
distances
and
Wooldridge
the
the
w¢ = ¢(_)ACNIB(N
frequency
exponential
Ue is the
the
cross
spec-
convection
functions
W_)ezp(-i_)
content
term
ta is the frequency,
[2] the
and gives
as;
r(¢,_,_) where
observations
(or
auto-spectrum),
represents
the
¢ and r/are
the streamwise
velocity.
A and B
(1)
Based
were
A
convection
on
the
represented
and
of the
B
the
pressure
and spanwise
experiments
spatial field.
In
separation
of WiUmarth
by decaying
and
exponentials
of the
form;
N) = where data.
the
constants
In recent [3-10]. that
years,
Graham
13 are arbitrary
many
improvements
[11] performed
for aircraft
to account
a and
interior
for the
noise
problems, of the
that
The
performance
superior
model
was
to these 14].
the Eflmtsov's
derived
models,
This
multidimentional
Gaussian
It is important determine
field
constants
in the
turbulent
is therefore coupling
This
is not important
Monte
Carlo
with
and become
layer
above
(low speeds),
to the
to solve field
models
Since
He
fact
that
this
In addition this
to be
need
surface
high
the
useful
however,
thickness.
experiments.
problem
[12-
a homogeneous,
experimental
pressure
speed
constants
and
can be
of its inability
layer
zero mean.
the
is fixed
proposed concluded
be used for such problems.
method
for compressible
approach
model
pressure
useful.
once
been and
because
attributed
laboratory
layer
all of the
In addition,
boundary
decoupled.
the
have
models
on boundary
was
than
set of experimental
model
various
is inadequate
length
model
process
especially
limited.
Corcos'
of the Corcos
boundary that
to fit a given
of the
model
rather
used the
random
experimentally, is further
data
to emphasize
the various
to obtain models
studies
idealized
Corcos'
Efimtsov
aircraft
several
approach
study
correlation
extension
of the
from
of the original
a comparative
dependence
recommended
and are chosen
(21
flows,
data the
are determined
structural-fluid in some
speeds
use
of these
the
pressure problem
problems
the
to
is difficult
interaction
engineering
at supersonic
data
coupling
where between
fluid and structure is very important. As was shown by Lyle and Dowell [15], acoustic damping on the structure becomes dominant as the speed is increased from subsonic to supersonic.
This
showed
in a supersonic
of the
that
structural
In order the
complete
appropriate Stokes The
is further response
to account set
With
at higher
the recent
calculations
fluid-structure equations
The fluid
structural
of Wu and
the coupling
Macstrello
term leads
[16] who
to an over-estimate
modes.
differential
requirements the
by the neglecting
conditions.
and
computational
equations.
regime,
for the various
of partial
boundary
equations,
confirmed
for solving
for the
motion
response
advancements
interaction
by
and
the
a problem
in computers, 2
fluid
is described
is given such
effects,
the
the
one needs structure
by the nonlinear nonlinear
will
plate
be dictated
full Navier-Stokes
to solve with
the
Navierequations.
by the
fluid
equations
have
been
integrated
is referred severely the
for a restricted
to in the limited
by the
computational
sible
simple
Since dynamics equations. Eddy
spent
This In DNS,
the
Kolmogorov
limited
to those
subgrid
scales)
in the early
are
the
testing
before
view
point,
DNS
flows,
near-wall
calculation
However,
idea but
for solving expensive
problems
and
a large
Over
infancy,
into
engineering
number over
DNS
of problems.
restriction
is not
but
scales
(or
models
and
need
From
[17]
developed
these
performs models
how
an equivalent
as it is for
DNS.
engineering
numerous;
geometries,
to
extensive
a computational
than
as severe
of LES are
to are
does a poor
been
LES a preferred
it has yet to be used in complex
down scales
small
Though
less expensive
The weaknesses
scales
have
one
these
Large
of resolved
resolved
layers
problems.
do not make
as the
tool
it is still
too
models
are
spatial
etc.
a century
quantities
of averaging
in fluid
by Smagorinsky
[18,20].
In addition,
to various
all the
new models
insight
kind
unresolved.
Reynolds
computationally,
still in their
as the
known
well in free shear
physical
bases
Navier-Stokes
of the unresolved
Recently
data
is in the number
was introduced
size of
developed.
approach
and the
is
is impos-
researchers of the
to resolve
is one order of magnitude
improvements variety
more
such
remain
the
reasonably
by the
being
forms
are coarser
The contribution
layers.
with
can be applied
all these
enough
of the first LES models
and
tool,
simplified
grids
approach
problems
models
DNS
a method
This
to generating
engineering
of a new
be small
boundary
an LES calculation
restricted
LES and
in LES the
performs
i.e.
region,
they
should
One
This model
sophisticated,
the
between
Such
(DNS).
engineering
turbulence
a useful
developing
than the grid size.
Smagorinsky
more
treat
used
however,
is modeled.
1960s.
various
of time
geometries.
can be resolved
been
led to the development
grids
larger
have
being
The difference
scale;
job in wall bounded using
has
LES.
deal
that
use in practical
method
from
and
Simulation
numbers its
to validate
away
a great
effort
Simulation,
scales. the
in order
is decades
have
of Reynolds of this
of flows
Numerical
Therefore,
The uses
flows
DNS
number
as Direct
range
domain.
at this point.
for some
literature
ago,
a time
O. Reynolds
averaged
[21] proposed
mean,
denoted
a decomposition
by an overbar,
of the
and
turbulent
flow
a fluctuation;
1=i+1'. Using the
this
decomposition,
contribution
became
known
used
to solve
only
for the
flow.
from as the a wide
mean
In order
decomposition
he derived the turbulent Reynolds variety
Averaged they
set of equations
fluctuations
(RANS)
problems.
cannot
provide
this deficiency,
for the
mean
had to be modeled.
Navier-Stokes
of engineering
quantities,
to overcome
a new
(a)
W.C.
Since
information
Reynolds
and
new
equations
these
any
quantities
These
and
equations on the
Hussain
have are
dynamics
[22-24]
where
equations
used
been
derived of the a triple
of the form .f = f+.f+f"
to study
small
represents
the
conditional motion. scale
amplitude
averaging Liu [25] used
wavelike
wave
wave motion
eddies.
and
disturbances
(4)
in turbulent
jr" is the turbulent
were used to arrive
at a set of dynamic
a similar
to study
Merkine
approach and
Liu
the
flows.
In equation
A combination equations
the near-field
[26] studied 3
shear
fluctuation.
describing
jet noise
development
due
(4),
of time
f and
the wave
to the large-
of noise-producing
large-scale
wavelike
[28] and
Gatski
fine-grained
which RANS
last
few
they
researchers
the models etc.
have
The
is described,
results
The
more
noise
from
Merkine
between
Gatski
[27], Alper
large-scale
[30] calculated shear
coupled
unsteady
coherent
turbulence
remarks
are:
of this paper
are
to a Very
and the
is organized
describes
are given given
Large
as follows;
the
method
in section
in section
developed
in section
or VLES
used
(4) and a summary
the
or DNS, geometry,
mathematical
to solve
of the
the
results
model,
and
Boundary
Layer
triple decomposition as follows;
g represents
Equations
proposed
by W.C.
Reynolds
[24], the
flow
quantities
a flow quantity
and (_) its Favre
(4) averaged
mean
defined
by
- (pg) p When
decomposing
of Morkovin's therefore;
some
(5).
g = .0+ .0+ g" where
by
[32]. The
LES
for any
(2)
of the models
identified
either
be used
of solution
been
has been than
can
acoustic
The extension with the new
Simulation
less costly method
Bastin
Lighthil]'s
have
method
Eddy
production
recently,
structures.
models
RANS
it is computationally
extensively
(3)
unsteady
Liu and
Model
Turbulent
Using the are decomposed
equivalent
tested
section
The
sound
More
with
and
structures
the
layer.
approach
advanced
physics.
VLES
and discussions
Mathematical
2.1
mixing numerous
been
remainder
concluding 2.
flow.
modeling
years,
as being
of using
the
Liu and
in a free turbulent
the jet
contain
advantages
model
shear
structures
jet.
interactions
led to the solution of a wide variety of engineering problems. method to unsteady problems has been made less difficult
because many
in a free
the
a semi-deterministic
to calculate
In the
turbulent
[29] studied
coherent
[31] used
analogy
in a plane
Liu
turbulence
due to large-scale et al.
eddies
and
the
hypothesis
density,
the
[33] which
(5)
turbulent
fluctuations,
has
recently
been
p", verified
are by
neglected Sommer
by et al.
virtue [34],
(6) In equations (4) and (6), (_, p) is the low frequency is the turbulent fluctuation. By
defining
the
total
mean
(4)
part
of the
mean
and
(g")
to be
G =0+0, equation
variation
(7)
becomes g=G+g". 4
(8)
Equation time
(8)
has
a similar
independent In the
mean
derivation
averaging
was
form
while
in (8),
of the
introduced.
to equation G is time
dynamic
Some
(3);
used
of this
>=0
only
difference
is that
in (3),
f is a
dependent.
equations
properties
] =0
(10)-(13),
(u_',
defined
momentum,
CO
P-(y-1)[E-p(U_-I-U_-t-U_+-t-)/2].
(p, Ud, E, P)
e u, p")
axe the
P/(7
1) + p(u_
averaged
-
(12)
represent
turbulent
stress
the
total
fluctuations.
+ u_ + u_)/2.
means The
of the
variables
variable
In equations
(13)
ill)
(e) and
(p, ud, e, p),
is the (12),
total
energy
< 7-dj > is the
tensor,
(14)
Moo [ _,kco.j+-Sg.,,/ _ COU_ COUj'_ --_,-G;_ 20Uk _f,q,] < r,j >=--_ where
Moo,
viscosity, stress
ReL
respectively.
tensor
..
and
and
# axe the freestream The
term
is modeled
Mach
number,
< _i_j - "- " > of equation
in the
same
way.
Reynolds (11)
In equation
number
is similar
(12),
and to the
< :-uuj'" >,
molecular Reynolds
< P-"'uy" > and
H
< _-Ouj > have to be modeled. Invoking the Boussinesq approximation to the
mean
strain-rate
tensor,
leads
that
the Reynolds
stress
tensor
is proportional
to 2
1 (15)
where and
k =< u_-"-u_" > /2 is the S O is the
mean
strain-rate
turbulent tensor
kinetic given
energy,/zt
is the
turbulent
eddy
viscosity
by,
(16)
5
The
turbulent
eddy
viscosity
is obtained
from
the
relation
(17)
_,= c;-_ where
C_ is a constant,
w the
The conservation
specific
equations
dissipation
for k and
rate
(elk)
with
e being
the
dissipation.
w are;
a(pk) + _= (pujk)
au,
a (_,, a_)
(18) (19)
a (_.aw) In equations (18) C_, t = 0.09 (same model model. 2.2
described
and (19), pa = C_npCk/_a) and C,., 1 = C,,,tc2/CCC_t)°'%%), as C_, in the log layer), crk = c% = 2, C_ 2 = 0.83 and _ = 0.41. above
The
Plate
The
out-of-plane
was
derived
by Wilcox
[35] and
plate
displacement,
DA2w
plate
as the
(k-w)
turbulence
Equation w, is given _w
where Young
is known
with The
D is the Modulus, material
and
+ p_,h--_- i- + F_
1" is the structural
A=
The right hand side, 6p, of equation fluids and can be written as
biharmonic
equation,
Ow
plate stiffness obtained from D h the plate thickness and vp the density
by the
= EphS/12(1 Poisson ratio.
damping.
c_
(20)
= 5p
04
The biharmonic
= _-_+ 2o--;/_, +_ (20) represents
- v_), with In equation
Ep being the (20), pp is the
term
is defined
c_
(21)
the pressure
loading
due to the adjacent
6p = f - pb_ with
(p=)
pressure 2.3
being calculated
The
Acoustic
In order arrive
at (see
the
radiated from
the
acoustic model
Radiation
to calculate
the
pressure
in section
and
as
(22) (pbZ)
the
turbulent
(22),
Kirchhoff's
[WU(TR"z')]
dz'dz'27r
boundary
layer
(2.1).
Equation pressure
pa of equation
formula
is used
to
[36] for details)
Pa(z'Y'z't)
= P°° /
/D
(23)
where
the
square
integration
brackets,
domain,
[.], are
D, is the
used
whole
to denote
plate
a retarded
°_ = "_-r.
and wit time,
In equation
(23),
the
i.e.
=', z')] = wil(t - Rtcoo, ',z') R = [(z -
where
to a point (23)
is used
When the
using
3.
the
plate
"I- (z -
(z',
to calculate
boundary
layer
layer
Navier-Stokes
code
order
accurate
volume
finite
that
is based
centrally
split
with
a good
separation. profiles
Two
a smaller
out
state
(25),
normal
a large
mean
that
that
using
and
Pahl
of Simpson's
profile
conditions,
are ones
the
Using turbulent plate
the
This
the
the
presence
In the
following where
those
considered
with
state
mean
of the
minimal velocity
plate,
calculation
boundary
then
is carried
as follows
(25) using
of the
method
pe,
equations
(10)-(13),
is specified
surface
edge
shock,
flexible
is simply
plate
supported.
equation
as
(20)
developed
through layer
is by
a combination
pressure
fields
acceleration
(18),
(19)
fields,
at every
of a leading
steady
in time.
pressure
is repeated
plate
dynamics
and
the
the
edge
and boundary velocity
called
leading
is obtained
step plate
routine In the
The
rule
IMSL 0.25.
for structural
a trapezoidal
new
an
0.05 and
plate's
with
plate
only
flows edge
terms
spatially
retains
an unsteady
domain
results, the
viscous
an implicit,
steady
edge is turbulent.
procedure
computational
a second an upwind
is generally
the leading
generated
acoustic
layer
to obtain
the
to be between
pressure,
and time
boundary
equation.
case
and
previous
shows
in one
acoustic
in space
plate
the
dimensional
domain
supersonic.
rule
This
first
edge
upstream
difference
radiated
with
thin-
uses with
all the
aerodynamic
includes
chosen
region
finite
method
approximation
at the inflow
of the leading
are integrated
the
except
The
between the
a two top
flow in the
downstream
[39].
(23)
to update The
of
coincides
three-dimensional
while
in time
case;
number
amplitude
an implicit
as follows.
equation
expansion
point
+ eR,,(y, z, t).
is a random
integration
Coupling tained
shows
the
while
integrated ttoff
series
are dJscretized
surface.
of the leading
velocity
the
method,
body
for each
domain
downstrean
e is a small
calculation,
laminar,
Equation
and in the far-field.
observer
numerical
terms
splitting
to the
are made
tLa(y,z,t)
[38] and
The
The thin-layer
= In equation
[37].
using
for high-Reynolds-number
using the
solved
are integrated
scheme.
calculations domain
are
flux difference
derivatives
by perturbing
RNNOF
plate
the
(x,y,z)
point
of sound.
a Taylor
0 (when
The convective
The equations
approximation
are obtained
using
speed
of the
pressure,
at R :
as CFL3D
scheme.
on Roe's
differenced.
terms
to be
surface
surface
equations
known
approximate-factorization
viscous
the
an observer
coo is the
of Solution
The turbulent
are
on the
the singularity
from
distance (24),
source).
Method
scheme
both
to calculate
to avoid
is the
In equation
pressure
(23)
is used
z')l] ill
0, z').
the
equation
integrand
with
z') 1 -t- yi
on the
(24)
different
and
these time
is ob-
as boundary the are
acoustic then
step.
used
Figure
mathematical
models.
all the
studied
because is clamped
cases
between
two
rigid
1
4.
Results
and
This a typical given
section is divided into several fully coupled run are presented.
in subsection
and radiation are
on the
response
given
Results The
The
(4.2). and
and
flow
mean
flow
finally
in subsection
structural
parameters
are;
results
on the
response
from two runs at two different
the
effects
of plate
boundary
=
in this
1.98
case
Much
conditions
are taken
from
MaestreUo
Much
number),
Tt
(free-stream
(Total temperature), T,, = 550 R (Wall temperature) and Re/ft number
plate
Case
used
Moo
of coupling
analyzed.
Coupled
properties
In the subsection (4.1) results from to existing experimental data are
effects
(4.5)
are
Fully
the
(4.4),
radiation
a Typical
and
(4.3),
In subsection
acoustic
From
subsections. Comparisons
In subsection
axe discussed.
numbers
4.1
Discussions
=
[40]. 563
R
= 3.73xi06 (Reynolds
per foot). A titanium plate is used in this study with the following parameters;
length, a -- 12 inches, width b = 6 inches, thickness h = 0.062 inches, density per unit area pph - 2.315x10 -s Ibf-scc2/in3, stiffnessD - 345. Ibf-inand damping r - 7.4x10 -4 lbf.sec/in s. The parameter e of equation (25) is set to 0.25. The plate is oriented such that
the
length,
The vertical
computational directions,
83 x 53 x 83. while ft
a, is along
used
respectively,
Equally
upstream
streamwise
domain spaced
grid stretching
vertical
the
is used
of the
direction
inflow
direction.
is 2 x 1 x 2 ft
the
number
points
are
of points
used
in the vertical
boundary.
is located
used
in the
6000 Hz) to both
the
level
numerical
advanced
turbulence
experimental
fluctuation
on the
measurement
that
based
the
structure
with
Effect The
plate
flow
step;
plate,
layer,
and
the
layer plate
little
the
center
the
with
effect
on the
response
of the
an
indication
that
shown
in Fig.
the
lower
of added before 4.4
uncoupled
plate
13. This plate
out. the
is directly 11 shows
the
center
for both
the
two
PSDs. layer.
on Fig.
one
critical
9 show
for
a good (25)
levels.
plate
fluid. back PSD
is higher
response
result
is important significantly
where
noise
Therefore
one
on noise
of Mach
in the
control
reduction
same
the
of the
fully
coupled.
The the
plate
whole
case
for interior
noise
is typically to consider
vibration response
to notice range.
and
at that
This
therefore
in higher
considerations.
both
plate
is
less
[42], this difference would peak responses occurs. As
results
interior the
layer There
frequency
structure
latter
cases.
It is important
on the
uncoupled
the
time
This
boundary
uncoupled displacement
the
at each
fluid flow.
The PSD
through
turbulent
calculation,
since
of the
response
the
turbulent
and
as those
on the
This is expected
damping
techniques
the
other
to the
coupled
overpredicts
needs
are
In the into
12 for the two cases.
in more
are
calculation,
transmitted
Figure
plate
In flexible
communicated
results
analysis
frequencies weight.
deciding
Effect
point
e of equation
of full coupling
the surrounding
boundary
is shown
calculation
effects
carried
fleld
the
turbulent
in this the
is not
between
coupling
higher
an uncoupled
depended
are
determined
response. Based on the results obtained by Frendi and MaestreUo be even greater at higher excitation levels where shifting in the the
surface
at this
that
parameter
experimentally
field and
with
at the plate
of the plate
used
are
pressure
to as uncoupled.
the
expected
level
of Fig.
an
Recent
of the
peak
frequencies
The
and
frequencies.
to mention results
to be due
resolution
behavior
of the
numerical
range.
to access
acoustic sides
response
or no difference
has
at high
frequency
4.1,
The
is believed
grid
similar
It is important
frequency
calculations
radiated
fluctuations
is little
Hz.
properties
In order
on both
the
case is referred pressure
fuselage.
agreement
plate
two
boundary
however
better
(4.1).
communicates
turbulent
agreement
[41] show
in subsection
for this
Higher
Coupling
mean
of a flexible
in better
up to 2000
experiments
in subsection
boundary
presented
in level
used.
[40]. The
on the
range
decrease
model
Concorde
of Maestrello
's location
to obtain
of
result
on the
results
This rapid
turbulence
would
to that
are in the
can be adjusted
given
model
on the
agreement
rapidly. and
measurements
pressure
4.3
decreases
resolution
noise
difficult
levels,
In other
levels, and
response
noise
as
words,
particularly
expensive
of a fully
at
in terms
coupled
plate
to be used.
Number
Two Mach number cases have been studied, 1.98 and 3.03, using the mean flow properties given in the above subsections. Figure 14 shows the PSD of the surface pressure at the
center
of the two
plate
displacement
cases
is higher, Mach
of the
number.
Lyle and
response
at low Fig. Dowell
for the
frequencies.
15. This
This result
two
cases.
at the
plate
However
is attributed is consistent
[43] who showed
that
The two center
at high
show
little
10
the
the
The
response
for Mao with
number
and
PSDs
between
damping
of Frendi
Mach
identical.
or no difference
in acoustic
the results
increasing
are nearly
frequencies,
to an increase with
PSDs
-- 1.98
increasing
Maestrello
increases
the
[42] and the
acoustic
damping cases.
on the Since
pressure 4.5
PSDs
Effect
17 shows
lent boundary response plate The
shows
for
due mainly to show
side are
a higher show
simply
supported
effect
A model
the
three
the
two
radiated
involving
be made
plate
changing
using
this
other
lower The
plate
radia(4.1).
plate
from
plate
objective
radiated
properties
frequency.
(such
to lower
indicated
of this
on the
supported
content
is also
turbu-
displacement
modal
frequency level
in order
the
The simply
at the lowest
parameter
approach
18.
in the
19.
and acoustic as in subsection
of the
Fig.
response
a shift
Fig.
any
of the
the PSD
A slightly
resolution,
of changing
that
couples
and
is
calculation pressure
as mass,
to analyze
their
a supersonic
test
of the
experimental
code. results
boundary
have
been
The
given
by
layer.
layer
a flexible
number
damping
radiated be used made
the
the
boundary
pressure to assess
to support
results
is important
increases
lowers
carried this
for the structural
Additional
• As the
Mach
results
both
of the plate
• Changing
boundary
calculations
• Full coupling calculations.
acoustic
turbulent
with
field has been developed. A three dimensional code The code is a modified version of CFL3D. Extensive
dimensional
turbulent
for the the
is field.
stiffness,
effect
on
the
Remarks
capability
existing
case.
vibration
center
as expected,
and
can
pressure
flow properties
However,
content also
calculations
Concluding
identical.
pressure
the radiated acoustic to solve this model. and
nearly frequency
frequencies
direct
radiated
at low frequencies,
on plate
at the
radiated
the
damping...etc) radiation field.
conditions
the same mean
of the pressure
to lack of frequency
the
Additional
5.
of edge using
PSDs
natural
of the
frequencies
effect
a different
lower
PSDs
the the
layer
shows
of the
Conditions
are made
that
the PSDs
in the response
identical.
Edge
to access
16 shows
or no difference
are nearly
two calculations
Figure
Figure
is little
of Plate
In order tion,
structure.
there
the
model
response
showed
acoustic
conditions
show
20
good
summarizes
agreement
pressure
response
damping
especially of the
Figure
and
and
with
fluctuations
in
that;
for accurate
response
out.
plate
has been written two dimensional
on the plate
high
plate
and acoustic
radiation
increases.
The
modes.
changes
frequency content. This shows that the best noise reduction mechanisms.
the
response
and
the
a computational tool can Further test runs can be
this idea.
Acknowledgements The author of NASA Thomas me
during
would
Langley Gatski the
like to acknowledge
Research
Center
for his availability course
under
the support contract
to discuss
the
of this work.
11
of the
NASl-19700. problem
Structural
Acoustics
I would
and the
helpful
Branch
like to thank advice
Dr.
he gave
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yf i Leading edge
i_
_
_111
i shock 5
I
Navier Stok_--_'_
i D_2w + pphWtt + l'w t = p-. p+
I
+
I+x
:+.,.,o +1--.,..,_,° --I-.,o,o +: x0
xo+ L i
.
P®rr
P :'_'.UD
Figure
1: Two
dimensional
X
[w_ (,;,x', z')] _
dx'dz'
computational
domain
15
and the
mathematical
model.
0.30
I
I
I
!
0.15
_ I_ '
II '
l'
-
1
i
0.00
k -0.15
i
-0.30 0.000
l
0.025
0.050 _r"qe,
Figure the
center
2a:
Time of the
history
of the
T
turbulent
plate.
16
0.075
O. 1O0
sec
boundary
layer
surface
pressure
at
0.0010
/
'
,
,
r 0"0005 "_.
1
i!
0.0000
j
-0"0005
I
-0.0010
/
i
0.000
0.025
0.050 Time,
Figure
2b:
Time
history
of the
,
center
17
0.075
O. O0
sec
plate
displacement
response.
0.0008
O.O0O4
-O.OOO4
Figure
2c:
Time
history
of the
radiated
pressure
18
one
foot
away
from
th_ plate
center.
....
125
I
........
I
'
'
'
121
1t7
C!] -o
t12 fl_
108
104
100 100
1000 Frequency,
Figure pressure
3a:
Power
at the
Spectral
center
Density
(PSD)
of the plate.
19
10000 Hz
of the turbulent
boundary
layer
surface
10
10
10
-8
-9
-10
N
-1-
c •-
10
10
10
10
-11
#V
-12
-13 -
-14 0
1000
2000 Frequency,
Figure
3b:
_J
Power Spectral
Density
(PSD)
3000
4000
Hz
of the center plate displacement
20
response.
9° h ....
I
........
I
........
8O
7O
D
m -o
_-_ 6O fl_
5O
4O
50
....
I
........
I
1O0
1000 Frequency,
Figure 3c: Power the plate center.
Spectral
......
Density
(PSD)
21
10000 Hz
of the radiated
pressure
one foot away from
1.2
C9 O_ c1. CU
Figure
4a:
Probability • at the
center
distribution of the
of the turbulent
plate.
22
boundary
layer
surface
pressure
1.2
0.9
O 0_
0.6
0.3
0.0 -0.002
0.000 Wj
Figure
4b:
Probability
distribution
of the
23
0.002
[rt
center
plate
displacement
response.
1.2
0.9-
O
0.6-
0.3
0.0 -0.005
Figure
4c:
Probability plate
0.005
distribution
of the radiated
center.
24
pressure
one foot
away
from
the
1.2
(0)
'
'
'
1
I
1.0
0.8
O O
0.6 e_ e_
0.4
0.2 0.0
_
0.00
,
0.05
0.10
0.15
0.20
_', ft
1.2
I
I
I
I
1
1.0
0.8 0
0.6
0 ci. a.
n,-
0.4
0.2 0.0
0.00
0.05
0.10
0.15
0.20
1?, ft
Figure direction,
5: Two-point (b)
in the
correlations spanwise
of the
wall pressure;
direction.
25
(a)
in the
streamwise
0.0 0.000
0.025
0.050 _,
Figure
6:
0.075
0.100
msec
Auto-correlation
of the
26
wall pressure.
i::1
(1) I (.n
i
(1) I
8 {'_
I
t'_
I I _P
_o
o _,=ll
2'7
35
......
|
......
i
........
I
5O
25 fA
2O
15
10
5
........
I
I
........
I
10
........
100
I
i
•
1000
i
t
iI,
1OOOO
y÷
Figure 8: Mean velocity -Numerical results.
profile, A Experimental
28
results
(Maestrello
[40]),
10
-9 l
I
___
10
-10
Experimental
(Ref.
[40])
Numerical m
10
-11
--
I
!2
N
T
-... ._c
" I
10
-12
_/I
vl
f \1
/
II Ii II I
10
i{
I
-14 \1
iII_
i!
I 1_ l
10
-15
, ,,,,I,,,,
,,,,,
,1 I
1000
r,_',
9:
Comparison displacement
of the
Power
Spectral
response.
29
I I I I II
I
, ,,,_,li,,
2000 Frequency,
I
rill
I
, , ,,
0
Figure
I
,,
3000
, ,
I!
4000
Hz
Density
(PSD)
of the
center
plate
125
II
!
a I J
a
a
___
Experimental
__
Numerical
!
!
!
!
!
(Ref.
! i
[40])
120
115 m "t3
v
n
110
105
1 O0 1 O0
1000 Frequency,
Figure
10: Comparison ary
layer
of the Power Spectral surface
pressure
at
the
10000 Hz
(PSD)
Density center
30
of
the
of the turbulent
flexible
plate.
bound-
....
I
--
'
'
......
I
I
,
i
i
i
|
i I
Uncoupled
121 .......... Fully Couolec
117
rm ,--7
112
108
104.
I00 100
1000 Frequency,
Figure 11: Comparison of the PSDs of the surface for the uncoupled and coupled cases.
31
10000 Hz
pressure
a.t,the center
of the plate
-7
10
10
-8 __Uncoupled .......
I0
N
I-
Plcte
Fully
Couplet
P!ote
-9
-I0
10
c-
V
-11
lO
10
-12
|
',:,,:.,' '_,'1
!
0
I
I
I
i
I
|
f
I
I
I
I
I
I
i
I
1000
I
I
I
I
!
!
I
I
i
i
2000 Frequency,
Figure 12: Comparison of the PSDs uncoupled and coupled cases.
I
I
I
I
I
3000
I
!
t
!
i
•
" I
[
|
i
4000
Hz
of the center plate displacement
32
for the
l
i
I
!
i
i
I
,
i
I
i
i
i
i
I
i
7O C_
Uncoupled Fully
Couplet
2O 1O0
1000 Frequency,
10000 Hz
Figure 13: Comparison of the PSDs of the radiated plate for the uncoupled and coupled cases.
33
pressure one foot away from the
109
-
103
r',n
98 n
92
Mcch
Numbe-
__3.03
86
....... 1.98 i_
'lll
8O
I
I I I 'Ill
100
JI
1000 Frequency,
0000 Hz
Figure 14: Comparison of the PSDs of the surface pressure at the center of the plate for two different Mach numbers.
34
10
-8
I [
I
J
!
|
Math
I
j
Number
__3.03
10
.........
1.98
°,:°
10
-16 0
2000
4000 Frequency,
Figure different
15: Comparison Mach numbers.
of the
PSDs
of the
35
6000
8000
Hz
center
plate
displacement
for two
100
I
I
I
I
]
I I
Mach
I
I
I
Number"
_303 8O
.......
198
rr_
6O fi_
4O
2O 100
Figure the
plate
16: Comparison for two different
i000
of the Mach
10000
Frequency,
Hz
PSDs of the numbers.
radiated
36
pressure
one
foot
away
from
125
I
....
Plate
........
I
Boundcries
___
Clamped
.......
Simply
120 Suppcr:ea
11,5 r_ 'o
s
: ii 110
.::,:.:,
ii: i.,"....."
105
IO0
'.°°
I
'
i
°.
I
I J
I
I
I
I'
'
100
1000 Freq_e,_cy,
Figure for two
17: Comparison edge conditions
_ I J
of the PSDs of the plate.
0000 Hz
of the surface
37
pressure
at the
center
of the
plate
-7
10
10
-8
I
tI
I
|
I
I
I
I
I
_
|
I
I
|
|
|
i
..:
I
t
_
I
I
#lcte
I
|
|
I
I
$
|
I
L
|
I
I
I
I
_
I
I
Roundcries
!i
o-9 i.J:l_ ,_'.._._
_
Clcrnped
.......
Simoly
o Suppor:_d
N
"t" c
10-10
I
10
'
1
_IIIflff_llrJt
0
r
1000
r
f
f
_
I
f
_
!
t
t
itlrllt_l_tll
2000 Frequency,
Figure 18: Comparison conditions of the plate.
i!..
of the PSDs
.3000
ii
'
4000
Hz
of the center plate displacement
38
for two edge
100
1
!
I
I
I
I
l
1
_
1
I
I
I
I
I
I
I
I
I
I
I
go
80
70 r_
60 V
0._
50
4O Clampe_
30
...........
Simply
Supported
20 10000
Figure 19: Comparison of the PSDs of the plate for two edge conditions of the plate.
39
radiated
pressure
one foot
away
from
the
U
O
0
..,,...,
O i=,
N
i=,
REPORT
DOCUMENTATION
PAGE
Form
Approved
OMB
No.
0704-0188
Public repotting burden for this coIleot_n of in_ormat=0n is estimated to average 1 hour pet ret4_0_se, _',cluding the time 'lot revmwmg instruct_rta, searching existing 0ala sources. gathering and mainla_ing the data needed, and coml[_eting and revmw_g the collection of information Send comments regarding this burden estimate or any other aspect of this colteotion of information, including suggestions fc_ re0ucing this burden, to Washington Hla_:lUarle_ Services, Directorate for Inlon'nat=onO_eratmrrs and Repoffs. 1215 Jefferson Davis Highway. Suite 1204. Arlington. VA 22202-4302, and to the Offce of Management arcl Budget. Paperwork Re0uctK_ Pro mct (0704-0188). Wast_ngton. DC 2C_03. 1. AGENCY
USE
ONLY
(Leave
blank)
2. REPORT
DATE
3. REPORTWPE
October 1996 4. TITLE
AND
AND
DATES
COVERED
Contractor Report
SU_TiTIJE
5.
On the Coupling Between a Supersonic Turbulent Boundary Layer and a Flexible Structure
FUNDING
NUMBERS
C NAS1-19700 WU 537-06-37-20
6. AUTHOR(S)
Abdelkader
Frendi
7. PERFORMINGORGANIZATIONNAME(S)AND ADDRESS(ES) Analytical
Services
Hampton,
VA
& Materials,
8. PERFORMING ORGANIZATION REPORT NUMBER
Inc.
23666
9. SPONSORING I MONITORING AGENCYNAME(S) ANDADDRESS(ES)
10.
SPONSORING/MONITORING AGENCY
National Aeronautics and Space Administration Langley Research Center Hampton, VA 23681-0001
REPORT
NUMBER
NASA CR-201631
11. SUPPLEMENTARY NOTES Langley Technical Monitor: Final Report 12a.
D_i
HII_UTION
I AVAILABILITY
Kevin P. Shepherd
STATEMENT
12b.
DISTRIBUTION
CODE
Unclassified - Unlimited Subject Category 71
13.
ABSTRACT
(Maximum
200
words)
A mathematical model and a computer code have been developed to fully couple the vibration of an aircraft fuselage panel to the surrounding flow field, turbulent boundary layer and acoustic fluid. The turbulent boundary layer model is derived using a triple decomposition of the flow variables and applying a conditional averaging to the resulting equations. Linearized panel and acoustic equations are used. Results from this model are in good agreement with existing experimental and numerical data. It is shown that in the supersonic regime, full coupling of the flexible panel leads to lower response and radiation from the panel. This is believed to be due to an increase in acoustic damping on the panel in this regime. Increasing the Mach number increases the acoustic damping, which is in agreement with earlier work.
14. SUBJECT YP.HMS
15.
High-Speed Research, High-Speed Civil Transport, Acoustic Damping, Plate Response, Coupling
Supersonic,
Turbulent,
NUMBER
OF PAGES
41 16.
PRICE
CODE
A03 17.
SECURITY
CLASSIRCATION
OF REPORT
Unclassified NSN
7540-01-280-5500
18.
SECURITY CLASSIRCATION OF THIS PAGE
Unclassified
19.
SECURITY
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OF ABSTRACT
OF ABSTRACT
Unclassified Standard Form 298 (Rev. 2-89) Prescr=bed byANSI St_l Z39-18 298-102