NASA Contractor Report 187579

ICASE Report No. 91-50

ICASE

THE UNSTEADY LAMINAR BOUNDARY LAYER ON AN

AXISYMMETRIC BODY SUBJECT TO SMALL AMPLITUDE

FLUCTUATIONS IN THE FREE-STREAM VELOCITY

Peter W. Duck

Contract No. NAS1-18605

June 1991

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

National Aeronautics and

Space Administration

Lsngley Reseoreh Center

Hampton, Virginia 23665-5225

t_A-:,,.-_: '-]i ," ,-z _ l!_i_ U' _r _..J¥ ! ,,_i ,',

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Ir_f_ 7 _'!, - r" J,"; i: LI£Fi-Y ':in _I -',"l)tJrt

tims., _ ) : :+ < LSCt 3 -,I _-;l
j_
THE UNSTEADY LAMINAR BOUNDARY LAYER ON AN

AXISYMMETRIC BODY SUBJECT TO SMALL AMPLITUDE

FLUCTUATIONS IN THE FREE-STREAM VELOCITY

Peter W. Duck"

Department of Mathematics

University of Manchester

ENGLAND

ABSTRACT

The effect of small amplitude, time-periodic, freestream disturbances on an

otherwise steady axisymmetric boundary layer on a circular cylinder is considered.

Numerical solutions of the problem are presented, and an asymptotic solution,

valid far downstream along the axis of the cylinder is detailed. Particular

emphasis is placed on the unsteady eigensolutions that occur far downstream, which

turn out to be very different from the analogous planar eigensolutions. These

axisymmetric eigensolutions are computed numerically and also are described by

asymptotic analyses valid for low and high frequencies of oscillation.

*Research was supported by the National Aeronautics and Space Administration under NASA

Contract No. NAS1-18605 while the author was in residence at the Institute for Computer

Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton,

VA 23665.

1. Introduction

The cffcct of time-periodic disturbances in the frecstrcam of an

otherwise steady boundary layer has rcccived considcrabic

atlcnlion over Ihc years. This work was initiated by Lighlhill (1954),

who considered Ihc Flow past a semi -inlini tc I1;11 plalc, wi 111 a sinai I

amplitude, time-periodic frccslrc_lm disturbance, and oblaincd solutions

close to and far from the leading cdgc. This work was later extended

by Roll and roscnwcig (1960), I.am ,Ind Roll (1960) and Ackerbcrg and

Phillips (1972). Of parlicular interest arc Ihc unsteady eigcnsolutions

that Form part of the far-downstream flow. One set of these was

studicd by l.am and Roll (1960), Ackcrbcrg and Phillips (1972) and

Goldstcin (1983) and has cxponcnlially decaying solulion downstream (sec

(8.1) below), with [hc fcaturc of decreasing dccay ratc wilh int;rcasing

ordcr; these cigcnsolulions arc dctcrmincd primarily by conditions close

to the wall. A second set of cigcnsolulions was constructed by Brown

and Stewartson (1973a,b) and has the feature of increasing decay rate with

increasing order; these eigcnsolul ions arc determined from condi lions far

away from the wall, in the outer reaches of lhc boundary layer.

Indeed, these seemingly diverse characteristics of the cigcnsolutions

have been the subjcct of some controversy over the years. Ilowcver,

Goldslcin et al (1983) include a quite detailed discussion of this

dichotomy; briefly, these aulhors expound the argument thai the two sets

of cigensolutions are in fact, cquivalenl, hul with the Brown and Stewarlson

(1973 a,b) expansions being valid at much longer distances

(O(In 1⁄2 x)1⁄2 >> 1) downstream, than Ihe Lain and Rott (1960) eigensolutions

(which are valid for O(x) >> I). Further, Goldstein el al (1983) poinl

out Ih;.ll as the order of the l.am and Roll (1960) eigensolutions increases,

the asymptotic behaviour o1 the (inner) solulion is likely to be achieved

at progressively largcr values of x, sincc, for x >> I, thc scale of the
region associated with the eigensolutions moves away from the wall with

increasing order. This, in some ways is not inconsistent with the fact

that the Brown and Stewartson (1973a,b) eigensolutions are centered at the

outer edge of the steady boundary layer. Goldstein et al (1983) also

conclude, using these arguments, that the limit as x ---, _ and the limit

as n ---, _ (where n is the order of the eigensolution) cannot be

interchanged, tiowever, and significantly, Goldstein (1983) went on to

illustrate the physical importance of the Lam and Rott (1960)

eigensolutions, by showing how these develop, far downstream, into unstable

Tollmien-Schlichting waves.

The problem of ''order-one'' unsteady, freestream disturbances (but

such that the freestream does not reverse direction) has been Considered

by a number of authors. Pedley (1972) considered this problem,

asymptotically close to and far from the leading edge, whilst Phillips

and Ackerberg (1973) presented numerical solutions to the problem for

locations from the leading edge to far downstream, their method being

based on a time-marching scheme. More recently, Duck (1989) presented

a new numerical method to tackle this problem, based on a spectral

treatment in time, and a spatial finite-difference scheme, which properly

takes into account regions of reversed flow that inevitably occur.

The problem of steady flow along a circular cylinder (in particular

far downstream along the axis of the cylinder) is itself interesting, partly

because it is so very different in nature from that of planar (i.e. Blasius

type) flow. Early investigations of this problem include the work of

Glauert and Lighthill (1955) and Stewartson (1955), whilst Bush (1976) has

presented a more modern approach to the problem. Notably, in the far

downstream limit, the problem becomes double structured, with an inner

layer (comparable in thickness with the radius of the body) which is

predominantly viscous in nature, and an outer layer (much larger than the
radius of the body) which is a region of predominantly uniform flow (sce

Section 4 for fuller details).

In this paper wc investigate the cffcct of small amplitude, timeperiodic, frccstrcam disturbances on the axisymmetric boundary layer on

a circular cylinder. Particular emphasis is placed on the cigensolutions

relevant to the far-downstream Ilow, which turn out I(.}be markedly

different from the analogousplanar cigensolulions of Lainand Roll (1960),

and possess someinlcrcsting properties. Further, since ;_nadditional

Icnglhscale is present in the prnblcm ti.c. the body radius), a second

non-dimensional parameter (in addition to the Reynolds number) is prcscnt,

and we arc able to exploit this parameter from an asymptotic point of view.

The layout of the paper is as follows. In Section 2 the problem is

formulatcd, and in Section 3 a fully numerical finite-difference scheme

for the steady and unsteady problem is described, and rcsults for the wall

shears are presented, for axial Ioc_llions from the leading edge to far

downstream. In Section 4 the development of Ihc (inhomogeneous) component

of the flow is described. In Section 5 the presence of eigcnsolutions far

downstream is elucidated, the cigcnproblcm is formulated and expanded in the

form of an asymplolic series. In Sect ion 6 numerical solutions of the

(leading-order ) c igcnproblem arc described, whilst in Section 7 the

eigcnproblcm is considered in the asymptotic limits of high and low

frccstream oscillation. In Section 8 the conclusions of the p:_pcr are

presented.
2. Formulation

We introduce a cylindrical polar coordinate system (ar, O, az), where

a is the radius of the body (assumed constant), and the z axis lies

along the axis of Ihe body, with z = 0 corresponding to tile tip of

I he body.

Suppose Ihal Ihc fluid is incompressible and of kincmat c viscosity

v, the freeslream velocity is laken Io he purely in the z direction,

and of tile form W_,(l+_ cos (,)I'j, where Woo, 8 and 0) are constants,

with _ << I. Note that although in all the ensuing analys s wc shall

confine our attention exclusively to frcestream velocities of tile above

form, it is relatively straightforward to extend our ideas to other

spatial and temporal (periodic) variations.

The velocily field is written as Woo (u,O,w), and non+dimensional time

as t = tot* Further it is assumed that u w and indeed the entire

• 0 t 1

solution is independent of 0, implying axial symmetry.

In this problem there are two fundamental non-dimensional parameters,

namely a Reynolds number based on cylinder radius

R = Woo;-.I._, (2.1)

V

which will be assumed to be large Ihroughout this paper, together with

a frequency parameter

l_- v

(0a _-

The usage of Ihe boundary-layer approximation requires lha[

Z = R-lz

is the key axial lengthscale, and

U = Ru

is the imporlanl order-one radial velocity scale.

equal ions thcn become ( to Icading order )

(2 2)

(2 3)

(2 4)

Thc bounda ry- I ayc r
l /)w 3w 3w O2w

+w--+U -

13 3t 3Z 3r 3r 2

together with

• Z (rw) + _ (rtl) = O.

! _w 1 r_w

+ - -- + - -- (r --.4 _.), (2.5)

r Or 13_t

(2.6)

Since it will be assumed that 8 << 1, the unsteady component of the flow

may be taken to be a small perturbation about the steady solution

(a similar treatment has been used in many of the related planar studies

cited in the previous sect ion, for cxample I.am and Rott 1961"), Lighlhill

1954, Ackerbcrg and Phillips 19721. Specifically

Ll(r,Z,t) = U0(r,Z ) + 8 Re {[_(r,Z)c it} + 0(821, (2.7)

w(r,Z,t) = w(l(r,Z ) + 8 Re {w(r,Z)c it} + 0(821, (2.8)

The steady component of the solution is described by

wo 3w 0 + UO _ = 32w 0 + / 3wO, (2.9)

r')Z 3r 3r 2 r r)r

3 3

r'J"Z (r wo) + _ (rUo) = O, (2.10)

with wo(r=l) = Uo(r=l) = O,

w0 ---_ 1 as r ---, _, (2.111

whilst the unsteady perturbation to this flow is given by

¢-)w _ 1 3w i

i..._ + wo _ + _, 3w 0 + UO _3w + U 3wO - --32w + - _ + -,

3Z 3Z 3r 3r 3r 2 r 3r 13

(2.12)

(r_) + (rU) = O, (2.13)

subject to

_(r=l) = U(r=l) = O,

---* i as r _ _. (2.14)

To close lhe problem we further suppose thai ag Z _ (1, planar

conditions prevail, with the boundary-layer thickness becoming

negligible in this limit. A similar procedure was followed by Scban and
Bond (1951) and was further utilised in a related problem by Duckand

Bodonyi (1986). The (steady) system (2.9) (2.11) then reduces to the

Blasius (planar) problem as Z----_O, with corrections due to curvature

effects given by Scban and Bond (1951). As Z---¢ _, the far downstream,

double-structured solution of Glauert and Lighthill (1955),

Stewartson (1955) and Bush (1976) cmcrges from this system.

Regarding the unsteady system (2.12) (2.14), this becomes quasi-steady

in form as Z---* O, with the time derivative term vanishing in this limit.

In the following section fully numerical solutions to both the steady

and unsteady system and considered, and in the later sections of this paper

the far-downstream behaviour of the unsteady component of the flow is

investigated in some detail.
3. Numerical solution of the problem.

In this scction we considcr fully numerical solutions to systems

(2.9)-(2.11) and (2.12)-(2.14).

Two slrcamlunclions are inlroduced, onc for tile steady component of

the flow, the olher for tile unsleady componenl, viz

given by

-I a'l' I O'l'

UO = 7 _'TZ' wo = 7 _-17'

U -I a_ w- l a_

= 7 rTZ' r 717'

The problem determining u? and _ is then

Vrrr v__ + _r

r r 2

r

+ _ = / [_r _rZ + _r _rZ]

_rrr r r 2 r

+ i -]3..-,ir

with 'e = _r = _ = _r = 0 on r = 1,

_r, _r '-+ r as r ---+ _,.

Anticipating a Blasius-type solution as

0 < Z <_ 1 was cast in terms of

and _, respectively

(3.1)

(3 2)

(3 3)

(3 4)

(3 5)

(3.6)

Z ---+ O, the problem for

= (r-I)/Z1⁄2, _ = Z1⁄2, (3.7)

as the independcnl variables, with the dependent variables taken as

F0 and F, where

v = _ F0 (_,_), _ = _P(_,_). (3.8)

For Z > I, _(r,Z) and _(r,Z) were treated as the unknown variables.

In both Z g 1 and Z > 1 the systems were written as a system of

first order equations in r (or _). Ilaving solved the problem for
Z = O, a Crank-Nicolson procedure in Z (or _) was adopted. Overall,

the numerical differencing scheme was based on that of Keller and Cebeci

(1971). At each Z (or _) station, first the steady system was

computed, with Newlon ileralion being used to treat the non-linearily

in the problem. Once convergence was achieved, the (linear) unsteady

sysLcm was then computed in a st raighl forward manner.

Results for _rr Ir= 1 (essentially the steady component of wall shear)

along the cylinder arc shown in Fig. I. This illustrates the (Blasius-type)

singularity as Z---, 0, together with a monotonic decline as Z increases.

Figure 2 shows the results for the real and imaginary components of

_rr Ir= 1 (esscntially the unsteady componcnt of wall shear) for 13 = 0.25.

This shows how the real component exhibits an inverse square singularity

as Z 0 (in line with that of Wrr Ir= 1) whilst the imaginary

component drops to zero at the leading edge. This occurs because as

stated previously, as Z _ O, the system determining F(q,_) becomes

quasi-steady, with the unsteady velocity perturbation tnoving entirely in

phase across Ihe boundary layer. For Z _> I, both the real and imaginary

components rapidly approach constant values. This aspect is dealt with

in the following section.

Figures 3,4 and 5 show the corresponding distributions for

13 = 1,2 and 5 respectively, all of which exhibit similar qualitative

features to the 13 = 0.25 results, although the asymptotic amplitude

of _rr]r= I is seen to diminish as [_ incrcascs, in the following

section the asymptotic form of the flow structure, far downstream of

the leading edge is considered.
4, The far downstream develovment of the flow.

In this section we investigate tile Z >> I solution for the

(unsteady) system (2.12)-(2.14). It was shown by Glauerl and Lighthill

(1955), Stewarlson (1955) and Rush (1976) that tile steady solution

oblaincd from (2.9)-(2.11) divitlcs into Iwo layers far (Iownstrcam.

Specifically, for r = (l(I) it was shown that

whc re

oo n+l

'1' = 7., e 't'

n=l) On

(r)

n+2 oo

+n-L-',l_ 't'ln(r)

+ 0 (._2), (4.1)

2 (4 2)

1_ = _*

and whe re

_on(r) = KOn {1⁄2r 2 log r 21" I r2+_ }

KO0 = I, KO1 = (_- - log 2),

y = 0.5772 .... (Eulers constant). (4.3)

_PIo = KI(I {1⁄2 r2 log r

7

KIO = _l '

1 r2 + _}

(4.4)

(Note that in Stewarlson 1955, the lasl Icrm in his equation (3.20) should

be a logarithmically squared term, and not as shown). It is also found

r r

2 Y()l) r

! l

r ^

+ KO0 r log r I '.'--_ K r dr + KII r log r ,

I _Po0

(4.5)

implying that for r >> 1,

9
_ll - _ Z ajn rJ (log r) 2-n

j=0 n=0

Consider now the outer layer, wherein

tl = r/Z1⁄2 = t)(I)

(consistent with (3.3)), whcrcin

^ A

't' = Z { 'YOO('q) + _ 't'01(q) + O(e2) }

+ _2 _yl()(rl ) + 0(83).

It is found _()0(_) = } .q2,

11 __q2

e

,e01(n ) = _ q2 I _ dn

+ c- _r12 I,

(4.6)

(4.7)

(4.8)

WlO(tl) = _Ol(q) 1⁄2. (4.9)

For comparison wilh our fully numerical results, asymptotic

approximations Io Ihe basic flow determined from

Wrr Ir= 1 - _ _l)Orr (r=l) + c 2 _POIrr (r=l)

=• _: + KO I _2 (4.10)

are shown on Fig.1 as a broken line.

Now consider Ihe Z _ [ solul ion to the system given by

(2.12)-(2.141. This turns out to bc quite straightforward.

first (and most importantly) the radial scale

smallness of r, w is expeclcd 1o develop as

_(r,Z) = wo(r ) + O(e), (4.11)

where w0 is to bc determined from

+ / a.__w0 _ i.__wO=_ i (4.12)

ar 2 r 3r 13 13

the appropriate solution of which is simply

Con s i de r

r = 0(11" Ihen due to the

10
I10(2)[_ "-- ] (4.13)

_,o(r) = I

ito(2) [,./-_ 1

i.e. the axisymmelric Stokes shear-wave solution, where ito(2)(z) denotes

the second |lankcl function, of order zero and argument z. Note also that

as [_ _ O, the planar Stokes shear solution is retrieved (in accord with

tile work of Ackcrberg and Phillips 1972). In this limit, a thin Stokes

layer forms on tile surface of tile body and consequently curvature

effccts become less important.

It is a routine mailer to continue this solution to higher orders

of e; however little additional insight is gleaned from this, and instead

we go on to consider (briefly) the outer laycr, where q = 0(1) (see

(4.3)).

Writing

= wo(q) + e Wl(q) + ()(e2), (4.14)

then

wo(n) = _,

Wl(_) = w2(rl) = ..

In fact the correction to

Z-1

= O.

^

wo(rl)

(4.15)

(4.16)

can only hc algebraically small in

Results obtained using this asymptotic structure (in particular

(4.8)) are shown for comparison with the fully numerical results as

broken lines on Figs. 2-5; the agreement is seen to he satisfactory.

ltowever, since the Z >> I structure delailcd above is obtained

without any recourse to upstream conditions, there must be a further

element to the downstream flow, not reflected in the above analysis

(see also the comments of Ackerbcrg and Phillips 1972). This

arises from cigcnfunctions of lhc syslcm (2.12)-(2.14). This aspect

is investigated next, in some detail.

11
5. The form of the eigensolutions as 7.--_ _.

tlere the form of (exponenlially small) eigensolulions

as Z-.-, _o is sought. Specifically, we investigate eigensolutions of

(3.4), with the basic flow described by Seclion 4.

As a firsl approximation to the form of Ihese cigensolulions,

consider the scale r = (1(i), and suppose _ in (3.4) is replaced by

_Yoo(r), and terms 0(_) anti smaller are ncglcctcd. This yields

r - _ _YOOr rZ

c _Z I 'eOOrr - _r.-_ I = 0 r

Assuming a solulion for

= f(z) w(r),

then

(5.1)

by separation of variables, namely

(5.2)

Yrrr _rr_2r + _r-_- i

c { 7 _[)Or _r + _ I _OOrr _ I}

r

= O. (5.3)

A solution of the assumed form is possible only if

rz +_ f : o, (5 4)

E

where A is a constant. Recalling the definition of t_ in (4.2),

(5.4) integrales to give

f(Z) = exp{-,_ [Z log Z-Z]}

A

,2-z _z

: Z e (5.5)

Here it is required that Re(A) > 0 to ensure decay as Z _ _,, and

the arbitrary multiplicative constant in _(r) has been included.

Iiowever (5.2) and (5.5) arc correct only to leading ordcr in c

and Z. It turns out the form of _ required for r = 0(1)

12
is

= h(Z) f(Z) Z p (log z)q {Voo(r)

+ c _()i(r) + ()(E 2)

i

+Z Ic _lO(r) + O(c2)l

+ 0(1/Z2)},

f(Z) is given by (5.5) and h(Z)

Further it is found necessary to expand

whe re

log Z.

ascending powers of _:, viz

A = A0 + _:AI + c 2 A2 + 0(_3). (5.7)

p and q are constants to be determined at some later stage.

(5.6)

is smaller than any power of

A itself in terms of

In view

of our comments regarding Re(A), then Re(Ao) > 0. The form of (5.6)

and (5.7) is necessitated because of Ihe series development of tile

basic flow in powers of r and I/Z, and is found to be essential for

solubility al higher orders of the solution. (Indeed Goldstcin 1983

pointed out the omission of algebraic lerms in tile strcamwisc development

of Ihe planar cigcnsolutions in Ihe work of Ackerbcrg and Phillips 1972,

which contained only the exponential development of the flow).

Substitution of (5.6) and the results of Section 4 into (3.4), and

®a¢

taking terms ()Ill(Z) Z e ZI' (log Z)ql yields the following i, J

equation for VO0

l.{tt'o0} = O,

1.{_00} = _00 _-_JQ' r + _()0' 72 + -r _t'00

where

Recalling the form of _1'()O,

t i w

L { _00 } = VO0

(5.8)

Aovoo [ _-_r ''- -_r ']'

(5.9)

given in (4.3), Ihen

___O_O.'r + _()0 [ 12- 1], + AO log r ]

A0 _ = O. (5 10) r

13
The boundary conditions to be applied to this system are those of

no-slip and impermeability on r = 1, i.e.

_00 (r=l) = _00' (r=l) = O, (5.11)

whilst as r ----4 _ _Ol} should n_n_9_l_be exponentially large. To be

more precise on this last point, the three linearly indepen{lenl solutions to

(5.10) in this limit take the form

i

¥00 A -AO{I A {log r _ }, (5.12)

r

iJ IA{} log r ]J dr

AO0 B I

_ooB ~ (log r)3/4 e , (5.13)

r

-i I IA{} log r l1⁄2 dr

Ao{}C 1

_{}oC - (log r)3/4 e (5.14)

Clearly either one of (5.13) or (5.14) is inadmissible (if A{} is complex)

due to the r >> I condition, and so

I" i 2

= AO{} /log r - -- +

I

13A{} r2AoIog r

! )]

+ O( r2(Iog r) 2

(5.15)

in this limit, _here AO(} is an arbitary constant (amplitude). The

system (5.10) , (5.11} and (5.15) represents an cigenvalue problem for

A(}. ltowever we defer discussion of this problem until the following

sect ion (where a dctai led invest igal ion is carried out {}f this aspect ).

Instead, let us turn to consider higher order terms in the expansions

(5.6) and (5.7). Taking terms

O{h(7,} Z- -_ e _ Zp (log Z) q-1 ]

in (3.4) yields

14
L{_OI} = A1 {- ¥00'_00' + _00 [ _00''

r r

+Ao( ,oo',o,' ,,,oo (

r r

r 2

r 2

(5.16)

llowever, on account of (4.3) this equation may be written as

L{WOI} = (A 1 + KoIA O) {

_gOo"Po0'

r

r r 2

The boundary condilions for this system are essentially the same as

those for _()(1; these can only be satisfied if

A 1 = KO1 AO, (5.18)

implying that

_Ol(r) -- AO1 _o0(r),

where AO1 is a constant (amplitude).

determine higher order terms in 1he A

For example

A2 = AO KOI2 _ AO KO2 _ 1⁄2 K()I AO (5.20)

and hence

_02 = A02 ¥o0(r) • (5.21)

Indeed, the following general result is applicable

_On = AOn _o0(r) • (5.22)

To progrcss further, in parlicular to determine terms thai arc

O(Z -I ) smaller than those considered already, let us investigate terms

- p-I l

O[h(Z) Z "_ e _ Z (log Z) q- ]

in the governing equation. This yields Ihc following equation for _1(1

L{_I( )} = p R 1 A0 R2, (5.23)

whe re

(5.19)

It is straightforward to

expansion, in a similar fashion.

15
R1 = _00'_'00' + _00 _00'

r r 2

R2 = _O0'_YlO ' + _O0_YlO '

r r 2

in view of (4.4)

and so

I 0 = K I 0 _00,

R2 = KIO RI.

Repealing the arguments used In determine

t hen

p = Ao K 10

7

= _ AO.

_/O0_Y()()' '

r

_O_lO'' (5.24)

r

AI

applicable to the

in particular its

then for 11 = ()(I)

form

(5.25)

(5.26)

and A2 previously.

(5.27)

Finally for this section, let us consider briefly the outer solution,

11 = 0(1) scale. In view of the r = 0(1) solution,

O(Iog r) behaviour as r _ _,, together with (5.6),

tile solution is expected to develop in the following

_01,Z) = g(Z) h(Z) Zp (log z)q _[)(rl,e) + Z£ _I(11'e)

+ O(Z2)}, (5.28)

where

g(Z) = f(Z)/e.

It is then possible Io obtain an exact solution for

matches on to Ihe r = 0(1) solution.

vo =Tq) I Ao -n- I

)I

where AO is a constant, and

oo

tY0 = Y_ cn lYOn(q).

n=(I

If we now expand

_0

This is given by

(5.29)

which

(5.30)

(5.31)

16
oo

n=O

then it is straighlforward Io show lha!

_00 = AO0,

(which maLches on to (5.15)), and

n

where AOI is an :lrbitary (.'onslan[.

(5.32)

(5.33)

(5.34)

Olhcr Icrms may be obtained similarly.

In the following section we go on Io consider numerical solutions

to (5.10). The value of q is delermined in the Appendix.

17
6. Numerical solutions of the ei2envalue Droblem (5.10)-f5.11)

The problem was tackled using three separate numerical techniques.

The first comprised a fourth order Runge-Kutta technique, shooting

inwards from r = too (chosen to be suitably large). In particular,

the technique i.volved (i) imposing a solutio, of the form (5.12)

at ro,,, generating values of _()()A (r = 1) and _00 A' (r=l) and then

(ii) imposing a solution of the form (5.13) at ro. (or 5.14) depending

on the sign of Rc{iA01⁄2}), generating values of _0() B (r=l) and

_()()B, (r=l) (or _()()C (r=l) and _00 C' (r=l)). Tile value of

A0 was then chosen by Newton iteration by imposing (5.11), by forcing

the determinant

or

_/00 A ( r = I )

_to()A' ( r=l )

_/00 B ( r= I )

VO0 B' ( r=l )

(6.1)

I _0() _/00 AA' (r=l) VO0C (r=l) (r=l) V00 C' (r=l) (6.2)

tO zero.

The second numerical scheme employed involved using a second-order

fini te-di fference approximal ion to (5.10), construct ing a quadra-diagonal

system (corresponding to the appro×imation to (5.10), together with the

boundary conditions (5.11). (5.15)) and the determinant of this system was

forced to zero by adjusting A0 by Newton iteration.

The third numerical scheme used was a direct (global) finite-difference

approach; using the same finite-difference scheme as our second scheme,

the system was instead written in Ihe form

A - A0 B = O, (6.3)

where A and B are both squarc matrices. The A0 were determined

18
by using NAGroutine FO2GJF,suitable for solving generalised eigenvalue

problems of this kind. This schemehas two distinct advantages

(i) of not requiring iteration and (ii) generating multiple values (if

present) of A0 simultaneously, howeverit can require substantial

computer storage.

Results from all three schemeswcrc found to agree; in practice the

procedure wasusually to obtain eslimales to the values of A0 using the

third scheme. If these were then deemedof to be insufficient accuracy,

enhancedsolutions (oblainecl on a finer and/or moreextensive grid) were

obtained using the secondscheme(i.e. the local finite-difference

scheme).

Results were obtained for a range of I_. It was found that at all

the values of _ investigated, there are many(probably an infinite

number) values of AO. Further all three methodsdid yield a large number

of spurious modes, tlowever these were usually readily identifiable, being

strongly dependentupon grid size and range, whilst genuine modeswere

comparatively grid insensitive.

Results for Rc{AO} are shownin Fig.6 and for Im{AO} in Fig.7.

Just the first four modesare shownin each case higher modesbecome

extremely difficult to compute(and, indeed distinguish from each other

and also the previously described spurious modes), particularly

in the limits of l____ooand _ .--4(). Ilowcvcr the trends are clear, namely

that IAoI _ _, as I_ _ 0 and IAoI _ 0 as _ _ _, for all modes.

In the following section we investigale these two limits asymptotically.

19
7. As,nnptotic solutions of the ei_envalue problem (5.10)

In this section tile limits _ ---, _ and _ ---, 0 in

equation (5.10) are considered, for which some analytic progress is

possible.

7.1. The limit I_ _ -

Physically, this corresponds to a low frequency limit to the

problem. The numerical rcsulls prcsenlcd in the previous section

indicate that (all the) A(I _ 0 as [_ _ ,,_. Consequcnlly if r = 0 (l),

then to leading order (assuming A0 = o(l))

_00'" - Voo" + __L_' = 0,

r r 2

with ¥00 (I) = _#00' (I) = O. (7.1)

The solution to this system is then

r 2 I

_/00 = Bo { r2 log r - _. + _ }, (7.2)

where B0 is some arbitary constant. This solution must ultimately

cease to be a valid approximation to (5.10) as r _ oo specifically when

r = 0(A0-1⁄2 ). Considering the particular development of A0 as [_ _ _,,

this is found Io take on the following form, in order to obtain a consisLent

and meaningful asymptotic solut ion

^ ^

A0 = y([3) [_-0 + _ _-I + 0(_2)], (7.3)

2

where _: = - -, (7.4)

log y

and y(13) must be determined from

log (y-1⁄2) = _-I (7.5)

This is a transcendental equation for Ihe small parameter y (see

Duck 1984, Duck and ltall 1980 for similar examples). In order Io obtain

a meaningful balance of terms when r = 0(A(I-1⁄2), it is necessary that

10 = i (7.6)

(the leading lerm in Ihe expansion for AO).

2O
r

define the outer lengthscalc

p = _ r = o(I).

where the following problem must be considered

I I

XPPP _pp

i_ _ O,

P

with

In view of these comments, and the above comments regarding the scale of

for which (7.2) ccases to bc a valid approximation to (5.10), wc

X - Cl Ilog p i ill as p ___, oo,

(7.7)

(7._)

Z = 0 (p2) as p --_ O, (7.9)

whcre c 1 is an arbitary constant, and Z is related to _00 by

X - 1⁄2_¢()0" (7.10) log 7-

the

in the previous section (indeed (7.8) is very similar to (5.10), and is

of about the same computational complexity, savc for the absence of any

physical paramclers).

Values for the first fcw _.l'S arc tabulatcd in Table 1 (accuracy

to at least the number of digits shown). It appears that all the

kl'S possesscd the _ real value (and hcnce decay ratc) to within the

accuracy of Ihc computation. Thc evidence was that a large (probably

infinite) number of thcsc modes cxisl; these higher modes wcrc difficult

to compute accurately, requiring small grid sizes and extensive grid

domains. Further. with increasing ordcr, Ihc imaginary part of the kl'S

became progrcssivcly more negative, although the difference betwecn modes

did diminish. Indeed. thcsc trcnds can bc confirm(d, asymptotically,

by carrying out a I_11 >> I analysis on (7.8)-(7.9). In this

limit, a WKB solution to (7.8) cxisls of Ihe form

21

The system (7.7) - (7.8) represents a well-posed eigenvalue problem for

kl'S which was solved using the three nt.mcrical tcchniqucs describcd
BlP{

X=

I i logp+_! 13/4

+ ell logp+i_.l ]

for Re {Iogp-i_.l] > O

region) where

i_. 1

P() = e

(and we expect c I = o (BI)), whilsl

X = ell Iogp-ikl I

p1⁄2

f eP() A I

i i Iogp+k 113/4 1,

P

I i l i logp+_, llj dp

e PO

(7.11)

(and tile p;llh of inlegralion lies within Ihis

P

I ililogp+kll1⁄2 dp

P

"I ilil°gp+kll1⁄2 dp

P()

+ g2 c

(7.12)

for Re {Iogp-iZi} < O

region).

A routine Ircatment of Ihe transition layer about

reveals

A2 = iA 1 .

To proceed furlher, consider an inner layer wherein

wi th

(7.131

(and the path of integration lies within this

P = PO

(7.14)

Pl = _11⁄2 p = O(I), (7.151

satisfying the following e(luation to leading order

I + ( 1⁄2 + I) = O, (7.16)

XPlPlPl P XPlPl XPl Pl

the solution of which is

i

Xpl = _" BOPl JO(Pl),

(the second solution of this equation involving

22

YO(Pl )

(7.17)

is neglected
on account of (7.9)). Taking the limit of (7.17) as Pl _ oo gives

P/-fii"-P"/_'2_-- n (7 18)

Xp 1 - B0 cos (PI-_) •

The limit of (7.13) as p --40 is

X _ Cl Ilogp-ikll

Alp1⁄2 ell { ikl1⁄2p+ -21+ikl1⁄2p } (7 19)

+ lilogp + kl] 3i4 e- ie

whcrc

0

I! =I

PO

ililogp+kll1⁄2 tip,

with the integration path lying within

is to match with (7.18) then

21

e = -1

which leads to

7[

ki = 21 " i log 12 ,./-_nl,

where n is a (large) positive integer.

B0 = 2i _ AO.

kl1⁄2

Re {Iogp-ikl} < O.

(7.2(1)

If (7.19)

(7.21)

(7.22)

For consistency, we also require

(7.23)

The formula represented by (7.22) was used to obtain asymptotic

estimates to the results shown in Table I. Mode 11 corresponds to

n = 1, mode III corresponds to n = 2 and so on; it is seen the

agreement between the computed asymptotic results is most satisfactory,

(_- = 0.785...). It is quite clear lhal Ihis asymplolic form will fail when

n = o( -1 ).

The leading order terms, namely Re(A()) = _1"Y _: and Im(Ao) -

are shown on Figs.6 and 7 respectively, for comparison with the

numerical solutions obtained from the full equation, (5.10). The

results are not contradictary, given the "largeness" of the small

parameter ¢. Indeed, computations for A0 from (5.10) at larger

23
values of 13 did become exceedingly difficult, due to the large

lengthscalc (O(y-I)), together with mode "jumping" caused by the

close proximaty of modes, which made the use of grid refinement with

the local method impractical.

7,2. The limit g---+ O

This corresponds to the high frequency limit of the problem.

According to the numerical results presented in Section 6 IAoI increases

as 13 .--+ O. This limit is now invcsligatcd.

It is possible to write a WKB-typc approximate solution to (5.10)

(assuming IA(}I and 13-1 arc both large) as

_()o(r) = B1 [log r - _-_ ]

r

-5,4 il [Ao I°g r l_ ]1⁄2 dr

O

r

A'' r- ]

(7.24)

(for Re {AoIogr

this region

B3 =

and

B I =

i

rr } < O, and the palh of inlegralion lies wilhin

P

whe re

3/2

3/2

'1 ;!

i

(7.25)

+ B3 e -il ], (7.26)

(7.27)

is the turn ng poinl, and

r 0

1 = - I [Ao log r

I

]1 cir. (7.28)

24
(7.25) and (7.26) are obtained by imposing boundary conditions on r = 1,

and the integration path lies within Re {AoIogr - _} < O.

Re {Aologr- _} > O the WKB-typeapproximate solution can be Fo r

written

_oo(r) = I_'1 [l°g r 7@ ]

r 1⁄2

4- B4rt O

5/4 c (7.29)

[_-Aolog r I

where it has been assumed Re{(-Ao)t } < 0 (otherwise we require tile

negative root inside the integral), B i is given by (7.26), and the

integation path lies within Rc [AoIogr _} > O. In order that

(7.29) matches to (7.24) across the transition layer of thickness

O(Ao - 1 /3) , (routine) treatment (see also tile analysis for

below) of the latter yields

B I =- iB 2, (7.30)

and the following dispersion relationship for AO results

I ;I

It turns out that there arc two distinct families of solulion as

The first fami ly of solut ions as [_ _ O

More speci fically

^o= I_-_ [ ^o+ _/3 ^, +... ].

^

whe re AO, AI

r O- 1 = O( I ).

2il

e = -i.

tlowever it appears I = 0([_ -1⁄2),

which can only be avoided i I

A0 = 0([3 .3/2 )

corresponds to AO = O(13-1).

(7.32)

are generally O(1) quantities. This implies that

Consequently to leading order (7.31) reduces to

(7.33)

and so there is a contradiction,

25
or

() r

ro

J [ A0 1 r

og

1

i

-m--.-

AO

I e -p p1⁄2 dp = O,

0

i] 1⁄2 dr = O, (7.34)

(7.35)

y [3_, -,,-- i ] = O, (7.36)

AO

where y(Zl,Z 2) represents tile incomplete gamma function. This represents

an eigenvalue problem for A0, which was solved numerically using a

combination of trapezoidal quadrature and Newton iteration; results

^

for the first few A0 are shown in Table 2. Note that there appear

to be many values (probably an infinite number), although these

seem to be concentrated within a finite annular region in the complex

^

A0 plane. As the ordcr increased, the values become very close to

neighbouring values, and the computation became exceedingly difficult;

however, with increasing order tile values of Ao do sccm to be

approaching a finite value (indecd the author was unable to find

any solution for IAoI < 0.098). Note too that it is easy to show

using integration by parts that there are no solutions to (7.35) as

^

IA()I _ **, whilst using the asymptotic expansion for the incomplete

gamma function (Abramowilz and Stcgun 1964) it is also possible to show

that no solutions exist as IAoI _ 0 either; this then confirms our

statement about the values of A0 being confined to an annular region in

complex A0 space.

Note also that both A0 and -complex conjugate {Ao} are roots of

(7.35)" however the latter family of solutions may bc disregarded since

in all cases we require A0 to possess a positive real part.

26
The second family of solut ions for A0 occurs when

A0 = 0([3-3/2). In this case, from (7.27), It0-11 << 1,

and indeed the wall (r=l) lies in._L__nLL_the transition layer.

Consequently, wc are unable Io use (7.31), hut must consider the

transition layer in detail (allhough this is quite a routine task).

Suppose

A(1 = [_-3/2 A0, (7.37)

where AO = 1)(1)" Then defining

= (r-l) 1_-t (7.38)

to leading order (5.10) reduces to

^^^ ^ ^

voo;;_ + voo; (X0;-i )

- A0 _00 = 0. (7.39)

Writing _ = (_S0)-1/3 _ + i , (7.40)

_0

and differentiating (7.3()) will] respect to _, yields

_()()c_o(_o o ¥0000 = 0. (7.41 )

The required solution (that is not exponentially large as c_ _ oo)

is

(7.42) ¥0oo = D Ai (o),

(where D is independent of o).

The implementation of the boundary conditions on

Ai' [ -i (-A 0)113 ] = 0. (7.43)

So

Now since the zeroes of the Airy function and ils derivative are

confined exclusively to the negative real axis, then

Ai' (_n) = O, (n=1,2,3 .... ) (7.44)

where the _n arc real and positive and tabulated by Abramowitz and

= 0 requircs

27
Stegun (1964). Consequently

AO- 21⁄21+in3/2+_ 0(_1⁄2), (7.45)

where the appropriate roots have been chosen to ensure boundcdncss

of the Airy I'unct ion.

it is inleresling (allhotJgh, in some ways not too surprising) that

(7.45) is identical to the corresponding expression found in the

analogous planar study (l.am and Roll 1900, Ackcrbcrg and l'hill ips

1972 and Goldslein 1993), allhoul:h of course the corresponding ffZ)

is quite different in the prcsenl case.

As a check on the numerical results as ]3 I_ O, on Fig

wlriation of [_3/2 At) with [_ is shown (first three modes).

clcar that thcsc rcsults approach those givcn by (7.4.5) as _ _ O.

Thc 0(]3 -I ) family of results rcfcr to higher modcs, and thus it is

nol realistically possible to compare our numcrical rcsults with this

family.

In the following scclion wc draw some conclusions from this work.

I hc

It is very

28
8. Conclusion

In this papcr thc cffcct of small amplitudc frcestrcam oscillations

on an othcrwise stcady boundary laycr on an axisymmctric body has bccn

invcstigatcd. Particular attcntion has bccn focuscd on thc fardownstream cigcnvalucs and cigcnsolulions. As notcd in Section 1, in

[hc case of the planar problem, Iwo distinct families of cigcnsolutions have

bccn presented, namcly those originally considcrcd by l.am and Rott (1960)

and those considcred by Brown and Slcwartson (1973a,b), wilh thc former

family having dccay rates Iha! decrease with increasing order, whilst the

laltcr family have dccay rates thai intircas¢ with increasing ordcr.

In the present study, cigenvalues appear to occur with decreasing decay

rate with increasing order. IIowcvcr, some of the asymptotic work in

Section 7 (in particular that relevanl Io [3 _ oo with A0 = 0([3-1))

does strongly suggcst that a finite value of A0 is bcing approachcd

with increasing ordcr. Indccd, the author was unable to obtain a

consistent asymptotic solution to (5.10) for [3 = 0(i), A0 _ O, again

suggesting the finite limit of A(I with increasing ordcr. This, in

some ways may be regarded as a rather more satisfactory statc of

affairs than Ihal found with the Lam and Rot! (1960) eigensolutions,

which have decay rates that become diminisingly small with increasing order

(although see our comments, allribuled Io Goldstein et al 1983, in

Section 1). Further the [3 ----, 0 work of Section 7 does suggest

tha! all modes possess the same decay tale in Ihis limit up to at least

second order.

Itowever, it may well be that the planar work of Brown and Stewartson

(1973a,b) could pcrhaps be extended Io include the effects of curvature, to

yield a further (perhaps relaled) Iamily of eigensolutions. A further

interesting study would bc an investigation of the far-downstream evolution

of the cigcnsolutions. Just as in the planar case, these all bccome

29
increasingly oscillatory far downslrcam, and will, as a consequence,

ultimately cease to be valid approximations to the Navier Stokes equations.

This will lead, presumably, to the formation of unstable TollmicnSchlichting waves, ill a manner analogous to that described by Goldstein

(1983) in tile planar case.

llowcvcr, Ihcrc are a number of (other) important differences between the

planar and the axisymmetric cigcnsolutions and eigenvalues. Most

important ly the downst ream (i .e. axial ) behaviour of these cigcnsolutions

(dcscribcd by f{Z)) which is quile different in tile two cases, in the

axisymmctric case being given by (5.6) whilst in the planar Blasius case

f(x) = c-AX3/2x p, (8.1)

as shown by Goldstcin (1983), (where x is the strcamwisc coordinate).

Note that if the basic flow wcrc of the form _' = xm F(q), with q = y/x m, y

being Ihe transverse boundary layer variable, then using arguments similar

to this paper,

f(Z) = x p cxpI-A x2n-re+l}, for 2n-m+l > 0. (8.2)

30
Appcndix

Consider terms

O{h(Z) Z _e _ p-I q-I Z (IogZ) }

in (5.1) to enable us to delermine the value of q.

It is found, afler some algebra

L{_¢II} = P{ U/OO'_Ol'r + _POI r'_O0'

I vOl" Zo_t.... _tO0 L

+ q { _()0

'

q'O0 , '2, ¥00

2r r r 2

r r r r 2

r r 2

+ AO {_ _lO'r_POl' 'Plo'¥Or 1' _ell'_OO'r

r r 2 r r 2

r r 2

_ l__2r _l)O' q'O0' + _i W()() [ _O0''r

(A.I)

Recalling tile expressions obtained already for P, A1, _/01, _10,

(A.I) leads to the following (slightly simplified) equation

L{_gll} : O(r) { A0 KIO AOI + _ + A0 Koi alO - (Alo + KIO ) A0 }

r 1 r r 2

i

+- 1 vo' *o(,' + 1⁄2 _'(,(, [ v_O_O' ' _00'], (A.2)

2r r r 2

where

31
Q<r)=*oo'Voo'voo[ "

r r r 2

It is quite clear that _ll = 0(r2(l°g r) 2)

simplify arguments later, we write

VII = VIII(r) + q _1111(r),

(A.3)

as r _ _. To

(A.4)

whe re villi(r) is any regular function which has the following behaviour

as r-_

Vll ll(r) = " 2 + o(I).

The governing equation for VII I

I.l_ll I} = R3(r) + q R4 (r),

(A.5)

whc re

and

and that

may then be written in the form

(A.6)

R3(r) = Q(r) { AO KIt ) AO1 + AO k() I A10

(A10 + K10)Ao } _ _YlI'VOO'r

+,oo *-EL']

r r 2

i * •

2r r r 2

1

r4(r) = - 7 Q(r) L[VllII). (A.7)

The boundary conditions that must be applied to this system are

VIii(I) = Villi(l)

VIii'(1) = villi'(1)

VII I --_ O as r _ _. (A.8)

The value of q is then determined by the condition that a solution

to this system exists.

Consider now the (complex conjugate of the) adjoint to the

system (5.1(I), denoted by _+(r), :lnd dctcrmincd from

32
t

V+''' + _ff+' + xg+' [ A0 log r i 1 ]

-r -;-; 2

+ 2 AO V+ = 0 (A.9)

r

subject to the boundary conditions

_'(r=l) = O, V+'(r=l) = 1 (say), (A. lO)

¥+ = o ((logr) -2) as r _ _. (A.11)

If we now further suppose, as we are quite at l ibery to so do (although

this simplifies, but is not crucial for our arguments) that

then

VII I1(I) = VII II'(1) = O, (A.12)

i R3(r) v+(r) dr

q = _ 1 (A. 13)

_ R4(r) v+(r) dr

1

(llartman 1964, for example).

This (at leas! in principle) delermines, or provides a means of

determinining the index of the logari!hmic term multiplying f(Z).

At this stage it would also appear to be legitimate to set the function

h(Z) in (5.6) equal to a constant, although categorical determination

of this poinl seems difficult because of tile algebraic complexity

in extemling the analysis to higher order.

33
Acknow I edgcmen t

The computations presented in this paper were carried out with

computer time provided by the Universily of Manchester and SERC Grant

No. GR/E/25702. This research was also supported by the National

Aeronautics and Space Administration under NASA contract No.NASI-18605

while the author was in residence at tlle Inslilute for Compute Applications

in Science and Engineering (ICASE), NASA Langley Research Center, Hamplon,

VA23665. The author wishes Io thank the referees for some very

helpful comments on an earlier version of this paper, including

pointing out equation (7.22).

34
References

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Functions, with Formulas, Graphs and Mathematical Tables,

Washington D.C.: U.S. Government Printing Office.

ACKERBERG, R.C. and PttlLI.IPS, J.II. 1972. The Unsteady laminar boundary

layer on a semi-infinite flat plate due to small fluctuations in the

magnitude of the freestrcam velocity. J. Fluid Mech. 51, 137.

BROWN, S.N. and STEWARTSON, K. 1973a. On the propagation of disturbances

in a laminar boundary layer I. Proc. Camb. Phil. Soc. 73, 493.

BROWN S.N. and STEWARTSON, K. 1973b. On the propagation of disturbances

in a laminar boundary layer II. Proc. Camb. Phil. Soc. 73, 503.

BUSII. W.B. 1976. Axial incompressible viscous flow past a slender body

of revolution, Rocky Mount. J. of Math. fi, 527.

DUCK, P.W. 1984. The effect of a surface discontinuity on an axisymmetric

boundary layer. Quart. J. Mech. Appl. Maths. 37, 57.

DUCK, P.W. 1989. A numerical method for treating time-periodic boundary

layers. J. Fluid Mech. 204, 544.

DUCK P.W. and BODONYI, R.J. 1986. The wall jet on an axisymmctric

body. Quart. J. Mech. Appl. Maths. 34, 407.

DUCK P.W. and IIAI.L, P. 1989. On Ihe interaction of acxisymmetric

Tollmien-Schlichting waves in supersonic flow. Quart. J. Mech.

Appl. Maths. 42, 115 (also ICASE report no. 88-10)

GLAUERT, M.B. and LIGHTIIILL, M.J. 1955. The axisymmetric boundary layer

on a long thin cylinder. Proc. R. Soc. Lond. A230, 188.

GOLDSTEIN, M.E. 1983. The evolution of Tollmien-Schlichting waves near a

leading edge. J. Fluid Mech. 127, 59.

GOLDSTEIN, M.E., SOCKOL, P.M. and SANZ, J. 1983. The evolution of

Tollmien-Schlichting waves near a leading edge. Part 2.

Numerical determination of amplitudes. J. Fluid Mech. 129, 443.

KELLER, H.B. and CEBECI, T. 1971. Accurate numerical methods for

boundary-layer flows. Part I. Two-dimensional laminar flows.

Proc. Inf. Conf. on Numerical Methods in Fluid Dynamics

(Published as Lecture Notes in Physics. Springer).

LAM, S.H. and RO'IT, N. 1960. Theory of linearised time-dependent boundary

layers. Corncll University GSAE Rcp. AFOSR. TN-60-1100

I.IGIrFlllLI., M.J. 1954. The response of laminar skin friction and heat

transfer to fluctuations in the stream velocity. Proc. R. Soc. A224, 1.

PEDI.EY, T.J. 1972. Two-dimensional boundary layers in a frcc stream which

oscillate without reversing. J. Fluid Mech. 55, 359.

35
PHILLIPS, J.H. andACKERBERG, R.C. 1971. A numerical method for

integrating the unsteady boundary-layer equations whenthere are

regions of backflow. J. Fluid Mech. 58, 561.

RO'I_, N. and ROSENWEIG, M.L. 1980. On the response of the laminar boundary

layer to small fluctuations of the free-stream velocity. J. Aero.

Sci. 27, 741.

SEBAN, R.A. anti BOND, R. 1952. Skin friction and heat transfer

characteristics of a laminar boundary layer on a cylinder in axial

incompressible flow. J. Aero. Sci. 10, 171.

STEWARTSON, K. 1955. The asymptotic boundary layer on a circular cylinder

in axial incompressible flow. Quart. Appl. Maths. 13, !13.

36
Mod c

!

II

Ill

IV

V

Vi

VII

VIII

_-! Asymptotic _!

.785 + .160i

/t

.785 I .282i _1" I .266i

K

.785 - 1.934i _l 1"95()i

/t

.785 - 2.340i _ 2.364i

.785 - 2.631i _1" 2.652i

/t

• 785 - 2.858i _f 2. 875i

K

.785 - 3.044i 2[ 3.057i

/t

.785 - 3.200i _ 3.211i

4

Tablc ! Valucs of Li

AO

.1408 + .2262XI0-Ii

.7455×I0 -I + .7991XI0-2i

.5076XI0-I + .4181XI0-2 i

.3848XI0 -I + .2603XI0-Ii

3099XI0 -I + .1790XI0-2i

2594XI0 -I + .1313XI0-2i

2231XI0 -I + .I007XI0-2i

1952XI0 -I + .7996XI0-3i

1742XI0 -I + .6514XI0-3i

1571XI0 -I + .5418X10-3i

Tablc 2 Valucs of A0

37
q_Jrrr = 1

1.5

1.0

I I I

1 2 3

Fig. 1.

_- Asymptotic

\

\

I I I I I I i

4 5 6 7 8 9 10 Z

Variation of _Orrlr= 1

_rrlr : 1

2.0

1.5

1.0

0.5

Lk_,...,_,, /'- Re{_Srrlr = 1} Asymptotic

"_ _-- ImC_rrlr = 1}

Asymptotic

I I I I I I I I I I I

1 2 3 4 5 6 7 8 9 10 11Z

Fig. 2. Variation or _rrlr= ! , 13= t

38
1.5

1.0

0.5

0

//- Re{_rrlr = 1} Asymptotic

_- Im{_rrjr = 1}

Asymptotic

I I I I I I I t I I I

1 2 3 4 5 6 7 8 9 10 11 Z

Fig. 3. Variation or _rrlr= i, 13 = 1

_rrlr = 1

2.0_

0.5

0

Re{_l/rrlr = 1} Asymptotic

Asymptotic

_-\_-- Iml_ffrrlr = 1}

/ , i

I I I I I I I I I __

1 2 3 4 5 6 7 8 9 10 11 z

Fig. 4. Variation of _rrlr= ! , _ = 2

39
_rrlr = 1

t

2.0 _-

t

1.5 -t

\

1.0-

0.5

0

Re{_l/rrlr = 1}

Asymptotic

//-- lm{_/rrlr = 1} Asymptotic

1 2 3 4 5 6 7 8 9 10 11 12 Z

Fig. 5. Variation of _rrlr= 1, _ = 5

Re{Ao}

!

-I

I

1.2"-"

1.0

0.8

0.6

0.4

0.2

0

Mode 1

I

_: ....... Mode 2

,, Mode 3

it

"' .............. Mode 4

__ Asymptotic

,

S', \ "--..

. _._-..__ ".--......

-('";'_ _,_,_ _---_"r_- .L__..I I -

1 2 3 4 5 6 7

Fig. 6. Variation of Re{A()} with _.

40
Im{Ao}

1.2 !I

1.0_i Model

:i Mode 2

0.8 3

° Iii

0.4

0.2

0

.............. Mode 4

"i_,_ Asymptotic

",%

", ,%

•..,.,-.::_.:,,.., .

1 2 3 4 5 6 7 8

Fig. 7. Variation of Im {AO} with I_.

r

_'\O = AO

.8 --

.7 _ _ -- Re{A°} mode 1

.6

_- Im{Ao} mode 1

15

.4 --

.3 _-Im{Ao} mode3

\

,2 --

.1 -- \

I I I

0

! Im{Ao} mode 2 Re{Ao} mode 2

-- Re{Ao} mode 3

1 i I I >

•001 .002 .003 .004 .005.006,007

Fig. 8. Variation of 133/2 A0 with I_, first thrcc modes.

41
Report Documentation Page %,]lOq,]l _or_3,Jl_-s aric,

1, Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA CR- 187579

ICASE Report No. 91-50

4. Title and Subtitle

THE UNSTEADY LAMINAR BOUNDARY LAYER ON AN AXISYMMETRIC

BODY SUBJECT TO SMALL AMPLITUDE FLUCTUATIONS IN THE

FREE-STREAM VELOCITY

7. Author(s)

Peter W. Duck

_ Performing Or_nization Name and Addre_

nstitute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, VA 23665-5225

12, Sponsoring A_ncy Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

5. Report Date

June 1991

6. Performing Organization Code

8. Performing Organization Report No.

91-50

10, Work Unit No.

505-90-52-01

11. Contract or Grant No.

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring/_gency Code

15. Sup_ementa_ Notes

Langley Technical Monitor:

Michael F. Card

To appear in Journal of Fluid

Mechanics

Final Report

16. Abstract

The effect of small amplitude, time-periodic, freestream disturbances on an

otherwise steady axisymmetric boundary layer on a circular cylinder is considered.

Numerical solutions of the problem are presented, and an asymptotic solution, valid

far downstream along the axis of the cylinder is detailed. Particular emphasis is

placed on the unsteady eigensolutions that occur far downstream, which turn out to

be very different from the analogous planar eigensolutions. These axisymmetric

eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation.

17, Key Words(Sug_st_ by Author(s})

receptivity, free-stream disturbances

18. Distribution Statement

Ol - Aeronautics (General)

Unclassified - Unlimited

19. SecuriW Cla_if. (of this report)

Unclassified

_. Securi w Cla_if. Iof this _}

Unclassified

21. No, of pa_s _. Price

43 A03

NASA FORM 1(i_ OCT

NASA-Langley, 1991