Indian Journal of Engineering & Materi als Sc iences
Vo l. 9. December 2002, pp. 409-41 3
Simulation of small disturbance waves over alternate rigid and compliant panel s t
P K Sen", S Hegde" & P W Carpenterh
" Department of App li ed M echanics, Indian Institute of Technology,
h
ew Delhi 11 0016. India
Department of Engineeri ng, Uni versity of Warwi ck, Coven try. CV4 7 AL. UK
Receil'ed 13 Decell/ber 200 1; accepted 16 October 2002
In thi s paper. th e simul ati on o f two-dimensionnl smnll disturbnn ee waves propngming over aiternnte ri gid nnd
cO lllplian t panels is di scussed. Due to the abrupt change in the material properti es o f the wal l at the juncti on of ri gidcO lllpli ant pa nels. a sudden change is ex pec ted in th e amp litude of the oncom ing wave. An ana lyti ca l-cuill -nuilleri ca l
method is de veloped for th e calc ulati on of aillplitude rati os at th e two sides of ri gid-compliant wall junctions.
The use of co mpliant wa ll s fo r drag reduction , by
keeping th e flow laminar for large Rey nolds numbers,
is we ll resea rched 1.2. Davies and Carpenter) studi ed
th e evolution of two dimensional T-S waves over
alternate ri gid-compliant panels through numerical
simulations. It was noticed in their simulations that
th e propagating T-S wave exh ibits a sudden change in
the am plitude at th e ri gid-compliant wa ll junction. It
is necessa ry to calcul ate a relation between th e
amp litudes on two sides of th e junction. A ' vibrating
ribbon' type transfer interface is assumed at the wall
j uncti on in th e present work. Once the amplitude
rati os are determined at the lead ing and trailing edges
of th e compliant panel, it is poss ible to simulate th e
T-S wave and other waves over th e entire stretch of
ri gid and compl iant panels. Simulation s are carried
out fo r a few typ ical cases of propagatin g disturbance
waves.
Formulation of the Problem
A chann el flow is considered here. The ri gidcompli ant-rigid region is formed ha ving a compliant
panel insert as show n in Fi g. I . The compliant wa ll
properties are app li cable in th e reg ion from X=XI to
.1'=.\2. The dornai n from v=O to Y=YI corresponds to th e
half wid th of the channel. The two junctions, at X=XI,
th e rigid-comp liant wa ll interface and at .\=X2, th e
co mpliant-ri gid interface, produce two different kind s
or 'j ump' phenome non . Also, we need to look at what
portion of th e onco min g wave is transmitted and what
amount of it gets reflected at these interfaces. The
governin g eq uati on is th e linearised Navier-Stokes
th
t prese nted at th e 28 Nat ional Conference on Fluid M echani cs &
Fluid Power. held nt Punjab Engi neering Coll ege. Chandi garh ,
durin g 13- 15 December 200 I
equation , written below in vo rti city form, in terms of
the disturbance stream function lJI= lJI(Y,x,I):
-a (\7 2 VI ) +
at
-
11 -
a (\7 2Ij/ ) -
-H
11
ax
alj/- - \I 7 4 Ij/=O
-
ax
... ( I )
R
Owing to periodic forcing in time, at constant
temporal (ci rcu lar) frequencyw. lJI may be written in
th e following form:
V/( y, x,t)= f(y,X)e -iWI
... (2)
For the sake of distingui shing the ri gid and
co mpli ant sides, th e function j(y, x) is definedin two
halves:
f(y, x)= H (- x}l- (y, x) + H (x)f+(y, x)
. .. (2a)
where, in Eq. (2a), H(x) is the Heaviside step
funct ion . Substitutin g Eq. (2) in Eq. ( I) yields the
operative differential eq uat ion for the problem:
.-j'\72r) +11- -a('\v72f) -
- /U~v . '
-
1./
ax
#
-af - - I crlf
v ' =0
... (3)
ax R
Veloci ty profile
leading edge
tra iling edge
• -- - . - - , . - . - - - . - . - - - - - -- . - .' - . - . -- ;-_-I
:
I
smail
:.. .-
~.- . .. -""
"
Y-Y1
:
:
... ...:
disturbacel .. . . . ..
____ --!
--. ~
'"
:
··· ·~
~
i~
>=0
x=o
Rigid wall
x=x1
Compliant wall
x=x2
Rigid wall
Fig. I - Ri gid-co mpliant-rigid wa ll arrangement
x=x3
410
INDIAN J. ENG. MATER . SCI., DECEMBER 2002
A Fourier transform is now defined for J(y,x), in two
halves, with a slit at the location of the joint, viz., at
x=o
u
F_(y, a) =
f I _(y ,x)2 -iU.'dx;
-00
< x:<:; 0;
0 :<:; x < 00 ;
... (4a)
... (4b)
We now take the Fourier tran sform of Eq. (3)
separately for the domains -oo<x :::;0 and OSx<oo . For
both sets of transforms, we need to know the value of
the functionfiy,x) at x=O. For both halves this is given
as V 2f _(y,x =0)=Q, and also, V 2f+(y,x= 0)=Q.
Both halves of the Fourier transformed equation (3),
therefore, reduce to an Orr-Sommerfe ld like equation
with a forcing term given in term s of cp. The equations
are as follows :
. .. (7a)
Unlike as in a Weiner-Hopf type of problem,f andf+
are considered complete in themselves, so that the
inverse tran sform is given separately for -00<.,c;0 and
OSx<oo. The invers:': tran sforms are respectively given
as:
. .. (Sa)
L(a )F+ =
where, in Eqs (7a,b), the operator L(a) is the OrrSommerfeld operator. The expanded fo rm of thi s
operator is given below:
L(a)F = i(aii -w)(F" _a 2 F)
- iaii"F -
\ - ia x uX;
A
0<
i .+ = - IfF' + (Y , x ;e
_
2n
X
< 00 ;
.. . (5b)
c"
where, in C, ,Cb are suitable contours enclosing the
po les . The above method of representation is most
conveni ent if the j unction is to be modelled as a
"vibrat ing ribbon". ff, for instance, we are dealing
with j ust a pair of waves, viz., the rigid TS mode for
- oo<.x:::;O and the compliant TS mode for O::;'x<oo,
then:(i ) th e ribbon is "created" by the incident wave
fro m the ri gid side; and, (ii) the compliant TS mode
x~O tunes in fro m the " vibrating ribbon ".
We now attempt to g ive an essential mode l of the
" vibrat in g ribbon " . The two frin ges on either side of
the ribbo n merge into the upstrea m and downstream
side waves respectively. ft now remains to look at a so
called 'essential for m' or 'elemental fo rm ' fo r the
ribbo n fun ction. Due to rapid change in x at around
x=O, o r the ribbon location, V 2lj1
. .. (7b)
-Qii;
~(F"" R
2(1.2 F" + a 4 F)
. . . (8)
Now, let us consider the rigid side, i.e. , - 00<.,C;;0,
with boundary conditions applicable to the rigid side
applied to F_.
Let the eigenvalue corresponding to the rigid side
be <X r • fn which case, in the neighbourhood of values
close to a" Eq (7a) may be expanded as follows:
... (9a)
Sim ilarl y, corresponding to the compli ant side, i.e.,
for OSx<oo, and wi th appropriate boundary conditions
for the compliant side applied to F+, and remembering
that the eigen-value for the compliant s ide is given by
a c , one obtains:
... (9b)
is essentially
T he operator L2 (a) in (9a,b) is defined as follows:
[ ~.:~ l~o
2
L2(a)F = i [uF'" - u"'F - 3a uF + 2waF]
T his sugges ts th e form for the ribbon function cp as
1 [--4aF '" + 4a 'J F ]
--
... (1 0)
R
... (6)
where Q "'" (
cr{1
l ax
\ ~O
We note that substituting Eq. (6) for cp in Eqs (9a,b)
results in:
... ( II a)
SEN el at.: SIMULATION OF SMALL DISTURBANCE WA YES
4.a r )F+ +(a-ar)~(ar)F+ =-UC
... (lIb)
where, the constant C is defined as C=Q(y). In other
words, essential content of Q is a constant.
Calculation of jump in the amplitude
According to the method of solution based on
adjo ints, the 'jump' in the amplitude of the
disturbance wave may be generically defined by a
factor A, as follows. Consider, for example, the rigid
side disturbance wave defined as
... (12)
where, over-line (-) over </> will generically denote a
suitably normalised </>, e.g., </> = 1+iO at y= I .
Thereafter the magnitude of A will generically define
the magnitude of the jump. Using thi s convention, and
using methods based on adjoint theory, we note that
1
C
f e , udy
o
A,
1
I [L
2
(a,)iP,
... (13)
p,dy
where, in Eq. (13) and subsequently, S, will
(generically) imply the adjoint eigen-function to <l>,.
Actually Eq. (13) depicts the 'creation' of the
' ribbon'. We may define the magnitude of the rigid
side wave as A = I, whereupon, we obtain the value of
Cas
I
C =
f [L
2
(a r)f,.
Prdy
..:!o_ _ _ _ _ __
I
foeJ:idy
411
where, in Eqs [(12)-(15)], the subscripts rand c
respectively denote quantities on the rigid side and the
compliant side. But on the compliant side the T-S
wave may show up in combination with other modes.
To determine the 'jump' corresponding to each
individual mode we need to run the program with
appropriate £Xc.
The other Eigen modes
Once we know the amplitude of the T-S wave Ac on
the compliant side for a given case, it is possible to
determine the amplitudes for other eigen modes on
the compliant side as well. For this we utilise the
boundary conditions available to us at the two ends of
the compliant panel. We note that at the leading edge
the wall displacement, r] = 0 and by assuming hinged
joint at x=O, we obtain the second condition as
r]" = O. Similarly, these same conditions are satisfied
at X=Xb the trailing edge also. On the compliant side,
along with the T-S mode (az), we need to consider
other eigen modes like the vorticular mode (az), long
wave (a3) and evanescent modes (lX4 and a5) also.
The vorticular wave is like the T-S wave but
originates at the trailing edge and is an upstream
travelling wave. The other mode is a long wave with a
smaller wave number. Apart from these three modes,
two more local evanescent modes in the close vicinity
of the edges are also noticed. The combined
amplitude of the wall motion at the two ends of the
compliant panel is calculated by adding up the
individual contributions of all the relevant modes.
This gives rise to four relations with four unknown
amplitudes as follows:
... (16)
... (14)
... (17)
... (18)
Similarly using Eq . (13), we may now determine the
magnitudes of the jumps Ac on the compliant side as
follows:
I
C
fe eudy
o
I
fL
°
2
(a , )fce c dy
. . . (15)
... (19)
where, in (16)-(19) al. az. a3. a4 and a5 are the
amplitudes of the component waves, viz, T-S wave,
vorticular wave, long wave, leading edge evanescent
mode and trailing edge evanescent mode respectively .
The amplitudes are computed by solving the above
system of Eqs [( 16)-( 19)] using Gaussian elimination .
INDIAN J. ENG. M AT ER. SC I., DECEMB ER 2002
4 12
Results and Discus;;ion
The fo rmul ati on is verifi ed by runnIng the
comp uter program ror di fferen t cases. Th ere is very
good agreement between th e present results and th e
numeri ca l simul ati ons carri ed out by Dav ies and
Ca rpenter' . Firstl y, th e simul ation below cut-off
frequen cy is in ves ti oated. T he simulati ons above cutoIl freq uency are ca rri ed out in more detail as it ca n
point to i mportant inform ation about th e stabili zing
aspects of th e co mpliant surfaces . Th e Reyno lds
Ilumber is taken as R=1 2000 for all th e cases . A gain,
th e \va ll param eter values are taken as in Davi es and
Carpenter" [n this case and th e subseq uent cases th e
(non dimensional ) wa ll parameters used are: III , d, B,
T and K w hich respec ti vely denote th e density,
dampin g coe ffi cient, fi ex ural ri gidity, strea m wise
tension per unit length and spring stiffness per unit
area.
T-S wa ve below cut-off frequ ency
For thi s case the wa ll parameters are taken <!s:
111=0.33333, T=0, d=O, B= 1.92* 10 7 , K=4B.The nondimensiona l frequency is f3=0 .24, and, from th e
present ca lcul ati ons , th e eigen values for th e ri gidside and for the co mpliant-side are respec ti ve ly gi ve n
as fo ll ows:
Ct , :::
1.03 17 - 0.00931;
Ct , :::
0.9395 + 0.0 109 i.
For thi s case th e wa ll param eters are taken as :
VO RT; . ... _LW .., ... CV1 .;
',: \Mik
. '. '"
j,jll/ V,' , ~, ~~
' \1 "
0.5
- 1.0 .
· 1.5'--_
20
,: \
:
I
1
I
~. .
I
J
~ !
!
. i
'-
I
....!
150
i
+ 0 .00018
r--~-~----~----~---~
_
L1
-. -- L2
100
I
0
-50
~\~~~,
-100
L--_~_~
o
.
i
- 0 .01 380
As can be seen from the simul ati on results shown
in Fi gs 2 and 3, there is a very close agreement in th e
answers, w ith th ose publi shed by Dav ies and
Carpenter' . An important thin g to bear i n mind here,
is to chose an appropriate initi al phase o f th e wave
w hi ch matches bes t with th e simil ar simul atio ns in
D av ies and Carpenter' .
The indi vi dual modes on th e co mpl iant side are
show n in Fi g. 2. Th e dom inant among th ese seem to
be T -S mode and vorticular modes. The res idual part
of the T -S wave is transmitted across th e trailing
edge.
Th e case of a longer panel is shown in Fi g. 3
w herein th e wa ll vorti ci ty pl ots for a ri gi d-compliant-
-150
. ~2:,
+ 0. 02125 i
= 0 .920795
1.1
ex "0' = - 0 .833 156
ex lev = - 0 . 145 876
w
111=0.33333 , T=O , (kO, K=I.92 * 10 7 , B=4K. The three
15 ~
, ; ~TS"
Ct
50
T-S wave above cut-off frequ ency
10.
eigen va lues for T-S mode, vo rticul ar mode and lono
c
wave mode are ca lcul ated respecti ve ly as:
50
__
100
~_~
150
____
200
X
250
~
300
_ _ __
350
400
a
15 0
'------r---o---~-~--~---r--,
:>-.
,
--'-_ _-'-_ _ _-'-_ _L-_--'-_ _-'-_ - "
40
60
80
100
120
140
160
x
a
-150
__
100
'--_-'-_~
a
50
_ __'____~_~_ _~_...J
350
400
300
150
200 X 250
~
b
b
Fi g. 2-Colllplialll I,va ll e ige n Ill odes(a) prese nt res ults (b)
co rrespo nd in g resu lts fro m Dav ies & Ca rpe nte r]
Fig. 3- Wall vorti c it y plo ts Ino ll -dime ns io na l len gr h of th e pane l
= 340, jU ll cti o ns are loca ted a t x=40 and x==360 1 (a) f resent rcsult s
(b ) co rres po ndin g res u lts from Dav ie s & Ca rpe ntc r'
SEN el 01. : SIMU L ATION OF SMALL DISTU RBANCE WAVES
4 13
rigid region are depi cted. W e noti ce a clear jump in
the leading edge j unct ion and another simil ar but
opposite kind of jump at th e trailing edge.
present approach as a tool for transItIon prediction
and control, holds great potenti al for cngi neeri ng
appli cati ons in vo lving th e use of co mpli ant surfac es .
Conclusions
References
The resul ts from the present work match very we ll
with the direct nu meri ca l sim ulati on results of Dav ies
and Carpente r'. T he agreemen t i n resul ts vali dates th e
'v ibratin g ri bbon' model for the j uncti on and also th e
so lut ion meth odo logy based on the adjoint. Th e
Ca rpe nter P W & Ga rrad P J. J Fluid Mech. 155 ( 1985) 4fl5SIO.
2 Ca rpe nte r P W & Ga rrad P J. J Fluid Mech, 170 ( 1986) 199232 .
3 Da vies C & Carpent e r P W. J Fluid Mech, 335 ( 1997) 36 1392.