ON THE IMPULSIVE MOTION OF A FLAT
PLATE IN A VISCOUS FLUID. II
By K. STEWARTSON
(Department of Mathematics, University College, London)
[Received 29 March 1972]
SUMMARY
The manner in which the variable x, measuring distance from the leading edge, first
enters into the solution of the title problem is elucidated. The structure of the
corresponding eigenfunctions is different in each of four regions of the boundary
layer and the corresponding contribution to the velocity is weaker than any finite
power of — x + Ut, where t measures time and the variable x first enters at x = Ut.
A comparison between the predicted variations of displacement thickness and of the
skin friction with the computed values obtained by Dennis (2) is favourable.
1. Introduction
I N part I of this paper Stewartson (6) considered the development of the
unsteady boundary layer on a semi-infinite flat plate immersed in an
infinite viscous fluid. The plate is given, impulsively, a velocity U at time
t = 0 in a direction parallel to its length and thereafter moves in the same
direction with the same velocity. It was found that the structure of the
boundary layer at a fixed distance x from the leading edge is different,
according as T ( = Utjx) Jjl. If T < 1 it is the same as for an infinite plate
and independent of x. If T > 1, x enters the structure, which approaches
the steady Blasius form asr->-oo. Physically, the significance of T = 1 is
due to the wave-like character of the governing equation when regarded as
a function of x and t. This means that the disturbance caused by the
presence of the leading edge travels through the boundary layer with the
maximum local velocity at any station of x, i.e. the mainstream velocity.
Hence, the effect of the leading edge is felt when Ut > x, i.e. T > 1. Two
interesting questions which naturally arise relate to the mathematical
mechanism by which the variable x enters the solution at T = 1+ and the
variable t disappears as T->-OO. An answer to the second question was
indicated in the paper but very little progress was made with the first
question.
Since then the problem has aroused considerable interest and of the
subsequent contributions we mention especially the first full numerical
solution by Hall (3), the numerical solution of the similarity form of the
governing equation by Dennis (2) and the completion of the mathematical
description of the solution as T -> oo by Watson. This study has just been
[Quart. Journ. Mecb. and Applied Math., Vol. XXVI, Pt. 2, 1973]
144
K. STEWARTSON
published as an Appendix to Dennis's paper, which also includes a full
bibliography to which the reader is referred for further details of the work
that has been carried out.
The purpose of this paper is to answer the first question posed above
by elucidating the manner in which the variable x enters the solution at
T = 1 + . Referring to (6, section 5) the basic problem is to find a function
^r(£, T) satisfying
9£2 8r'
together with the boundary conditions tft = difi/d£ = 0 when £ = 0,
1 as £ -> oo and as r -*• oo,
The dependence of 0 on T is weak when T— 1 is small and positive; we may
therefore linearize the equation by writing
^ = ^ 0 (£)+Y(£,T)
(3)
2
and neglecting T . Then Y satisfies
with boundary conditions T = 0 when T = 1, T = S¥/d£ = 0 when t, = 0,
and ar/9£->0as £->oo.
The appropriate structure for Y is found using a method devised by
Brown and Stewartson (1) in a related problem, to which the reader is
referred for further details. There is a critical value of £, around which
the whole solution pivots, that at which the coefficient of 32XF/3££r in
(4) vanishes. Since
A i £ » )
when£>l
(5)
and T—1 <^ 1, we can, to a close enough approximation, define this point
to be£ = £0, where
We now divide the range of £ into four parts, each of which requires a
different choice of scaling and a different treatment.
2. Region I: £ > £0
This is the outermost part of theflowfield,where the velocity is almost
the same as that in the main stream; we re-write (4) as
dr
ON T H E IMPULSIVE MOTION OF A FLAT PLATE
145
and, as a first approximation, we neglect the right-hand side. A solution
of the reduced equation is needed in which 9T/9£ is exponentially small as
£ -> oo and also oscillates near £ = £0, in order to provide the basis for an
eigenfunction. If £0 ^> 1, as is the case, the appropriate solution is
»A1Ai(Yp)e^p{^+^r)},
(8)
where Ai is Airy's function,
I = £0+r(8/£0)*,
( T - 1 ) dp/dr = &Q+l+(Ko)*P,
(9)
p is a slowly varying function of £0, which tends to afinitelimit as £0 -»- oo,
to be determined later, and An, for any integer n, denotes a multiplicative
constant, whose value may or may not be determined by the methods of
this paper. The relative error in (8) is difficult to assess without explicitly
computing the next term in the expansion, which is a formidable task, but
it is expected to be 0(£^"*) when Y = 0(1). Further the solution needs
modifying in the neighbourhood of £ = £0, because the terms on the righthand side can no longer be neglected. It is convenient to discuss the
appropriate changes which then arise in a separate region of the boundary
layer.
3. Region II: 1—tyo\dt, « T—1
Here we introduce a new variable
so that
9
£ = Co-rP°g(Ho^)+log-8f]+o (£„-»+«)
(11)
to
for any e > 0. In this region, which must overlap with Region I, we require
S r^j \ and so the operator
| = - K o ^ + o(£ 0 —).
(12)
We take the appropriate form for T in this region to be
T = h(r)H(8),
(13)
to make a match with (8) feasible, and in addition, using (8), we anticipate
Now, on substituting (13) into (4), wefindthat
S
5W7.2
146
K. STEWABTSON
the relative error being o (£o~t+*)> « being any positive number, when S <->-' 1
and 0{t^2S) when 8 > 1. We need a solution of (15) to match with (8) as
S -> 0 and to die away as 8 -»• oo, on the physical ground that the eigenfunction Y must be centred on £ = £0. Mathematically this condition as
8 -> oo turns out to lead to a consistent description of the eigenfunction and
Region I I can, from some points of view, be thought of as a boundary layer
below the inviscid solution in Region I. The appropriate solution of (15) is
^
= - ^ U Sh(r) j 8ft USD dSv
(16)
where Ko is the Bessel function of zero order, of the second kind and with
imaginary argument, which duly dies away exponentially as 8 -> co. In the
limit S -> 0
(17)
where y = 0-577 is Euler's constant, and in terms of £ this reduces to
....
(18)
}.
(19)
Moreover, when Y is small, (8) implies that
°±
Comparing (18) and (19) and assuming that Regions I and II overlap we
see that a match is possible if
Ai(-p) = -i[2y+2-log(2C0V7r)](8/£0)*Ai'(-:p),
or
^ = /c n +2-»|"2 y +2-log4V 7 r-iloglog-^ T ]/log-l T H
(20)
neglecting higher powers of {log(r—I)"1}"1, where xn (n = 1,2,...) are the
zeros, in ascending order, of Ai(—x) = 0 and K1 = 2-238, K2 = 4-09, etc.
Since the number of roots of Ai(—x) is infinite and * n ->oo with n this
argument leads to a physically and mathematically reasonable set of
eigenvalues and eigenfunctions but we cannot claim as yet that the set is
complete.
An immediate consequence of (20) is that we can make an estimate of the
contribution to the displacement thickness from \F when T— 1 is small. Since
the eigenfunction is assumed to be centred on £ = £0 we must have Y -> 0 as
8 -> oo, for otherwise we should not be able to satisfy the boundary condition at £ = 0. Hence Y is known when 8 ~ 1, from (16), and as S -*• 0
ON THE IMPULSIVE MOTION OF A FLAT PLATE
147
and, by comparing (18) with (19),
MT) = 4 e ( T - l K 0 - " « e X p / i ( T ) ;
(21)
incidently this result confirms (14), in view of (9). With this value of *F as
boundary condition, we can integrate (8) across Region I, to obtain
YtT.ooJW^T-l^o-'/'exp^T),
'
(22)
because the contribution from Region I is of relative order £,f * compared
with that from Region II.
On the assumption that the error in (I(T), as defined in (9) and (20), is
0(£$), and arises from the error in (8) when T ~ 1 and from the matching
procedure between Regions I and II, we have extended the formula for /X(T)
by one further term. If we define
^ ! ^ - ^ ^ ,
(23)
the extra term is the final term of (23) apart from the change of sign and
logT(T,oo)-log8* = o{log 1/(T— l)}*+«
(24)
for any e > 0. The determination of the next term in the expansion ia
likely to be formidable since there are many sources of the error.
Professor Dennis, in a private communication, has kindly supplied
further data from his numerical calculations, which has been used to infer
the computed value of the displacement thickness and hence the deficit
under the Rayleigh value when T > 1.. A comparison between the computed
value 8^ of this deficit and our asymptotic formula is shown in Table 1
below.
TABLE 1
10
105
I'lO
I-I
o
5
2-996
••94
35
•'5
1-20
261
2-303
1-897
I-6IO
486
1387
2-22
2-25
2-26
228
1-40
145
2272
1 50
00
1-30
796
i-35
1196
1688
1-204
1-050
0917
2-29
2-30
232
5
1 25
2948
0-696
2-34
0-799
2-33
The variation in the ratio 8^/8* between T = 1-1 and T = 1-5 is about 10%
which can be regarded as satisfactory since the value of 8^ varies by a factor
of over 80 in the same interval.
When S > 1, (16) implies that
?
-
,
S
*
)
:
(25)
148
- •' '
- K. S T E W A R T S O N . .
but on examining the relative size of the terms neglected in writing down
(15) wefindthat those 0(l,^z $) now become significant. We are led therefore
to introduce a new region of the range 0 < £ < oo in which S = 0{?%).
4. Region III: S ~ $
The new term which becomes significant in this region comes from the
term 92XF/3£&T in (4) and in addition, when S^> 1, the only other term of
primary significance is 331F/3£3. Further, in the coefficient of d^/d^dr, the
variation with respect to r is of secondary importance so that the governing
equation reduces to
(K«)
(26)
which may be simplified further to
3*0
8 0 0 ^ 3O
8
{
'
2
on writing s = TT~*exp(—££ ), replacing £ by £0 except when it occurs as an
exponent, setting
a = (r-l)Aog - A - ) * , .
O = SVjd^
(28)
and neglecting terms of relative order £<f2.
We note that whereas in all other regions the characteristic independent
variable is 0(1), for example T in Region I and £ in Region IV below, here
the characteristic variable s is small. The reason is that it enables the
governing equation (27) of this region to be written in a simple form and is
purely a matter of convenience. The range of values of s for this region is
1, i.e.
The lower limit on a is equivalent to requiring S !> 1 and we see, from
(25), that then sd/ds ^> 1, which is assumed in deriving (26) and enables us
to neglect the terms proportional to dW/dt,2 and 9Y/3T in (4). It may now
be verified that (27) is satisfied by
<D 0 =
where Bn (n = 1, 2, 3...) is a relatively slowly varying function of £ and r,
provided a£(f2 <^s <^ a, which agrees with (25) and satisfies |s9<J>0/ds| ^> |O 0 |.
An appropriate solution of (27), valid for all s in the rangeCT£<J"B<^ s <^ 1, is
=
JJ
c—ioo
C1(8,w)exp{wo-2(8w)i-b(logw)t-3K12-t(±logco)t}da>,
(30)
ON T H E I M P U L S I V E MOTION OF A FLAT P L A T E
149
where Cx is a slowly varying function of a and to, and c is a real positive
constant. We estimate (30) using the method of steepest descents.
When ot^a < 8 < a the saddle is at
o, = J_ log ! + £ - (I log !)* + ...
4cr
CT
2»a \ 4
(31)
a/
and (30) then reduces to (29). When a :>~cr£o> however, the saddle is at
(32)
and
« = B,«ip|_£_2|log^'_3« 1 2f(jlogi|)*-...},
(33)
Further details, including a more precise determination of B2, may be
obtained by continuing the expansion further and including higher powers
of £<f r but it rapidly becomes extremely complicated and, since £0 is itself
only a slowly varying function of T, there seems no advantage at this stage
in writing them down.
In terms of T, £ (33) implies that, on leaving Region III,
,,
1 ,log 8exp(^—
I
—=• = BJX, r)exp - J log —
8exp(-K 2 )
(34)
The exponent in (34)' consists of two series of descending powers of
log 1 /(T — 1); in the first the coefficients of the series are independent of £ and
the terms neglected are o{log 1/(T—1)}1+* (e > 0); in the second the terms
neglected are o{log 1/(T—1)}*. In addition the opportunity has been taken
to replace part or all of the powers of log 1/(T— 1) in the definition of a by
corresponding powers of J£2 instead of ££*, as the definition (6) would imply.
The change makes a negligible difference to 3T/3£ in Region i n and anticipates the need for matching the solution when s/a ^> 1 with the solution in
Region IV as £ -> oo.
5. Region IV: £ ~ 1
The lowest region of interest extends from the lower edge of Region III
right down, to the plate and, taking the hint supplied by the necessity of
150
K. STEWARTSON
matching our solution with (34) as £ -> oo, we assume that
(35)
where P, Qlt Q2, R are functions of £ only and Xv Aj are constants. The
exponent in (35) contains two series in descending powers of log 1/(T—1);
in the one whose coefficients are independent of £, terms o{log 1/(T—l)} 1+e
are neglected and in the one whose coefficients are functions of £, terms
o{log 1/(T— 1)}' are neglected, where e is any positive number. Theoperator
S/d£ is assumed to be 0(1) in this region and to a sufficient accuracy we may
take
, ,™
Y« —.
(36)
y
2PP' 31,
'
On substituting (35) into the basic equation (4) for T , equating ascending
powers of T—1 and descending powers of log 1/(T—1) in appropriate succession, we obtain
,n~\
4 p , 2 _ , _ ,,
and
2 P P ' & = A 1 (l-&),
(38)
2PP'Q'2 = |A a (l-0i),
(39)
-4:PP'B'-(2P'*+2PP")R-t,PP'R
= -(l-ti)Q1R+(l-2tit)RP»-p0RPl2P',
(40)
where primes denote differentiation with respect to £. From (37) we have
if(l-&)*d£;
(41)
the choice made of the upper limit of integration permits (41) to match with
the corresponding part of (34) as £ -> oo, for
From (38) and (37)
Qx = 2A 1 logP+Q 10 ,
(43)
where Q10 is a constant; also
02 = fA 2 logP+e M ,
(44)
where Qi0 is a constant.
Finally, from (39)
iA- «-^-i J J 4
The dominant factors of (34), i.e. those in the exponent which depend on
ON THE IMPULSIVE MOTION OF A FLAT PLATE
151
T— 1, can now be matched to the dominant factors of (32) when £ ^> 1 but
5
>^if
A1=-i
and
As = - § * ! .
(46)
We are now in a position to consider the boundary conditions
T = 9Y/S£ = 0
at £ = 0. Although in (35) T is exponentially small when £ = 0, it is
not zero and it is necessary to add two other solutions of (4) to it in order
that these conditions may be satisfied. One new solution is provided by
taking the opposite sign to P from that chosen in (41) but now we shall need
to add to P a constant in order that the coefficients of (T— I)" 1 in (35) and
this second solution be equal when £ = 0. Thus the new solution is domina(47)
and is negligible in comparison with (35) as soon as £ > 0. I t is noted that
the corresponding form for R becomes highly singular as £ -> oo but for
similar reasons to those stated in (1) it is not thought that the resulting
modifications to T are significant. A third solution is given by
T = L(T)+&(Z)M(T)
(48)
by analogy with the solution given by Lam and Rott (4) of a related
equation, where M is arbitrary and
- T 2 i ' = \M+TM'.
(49)
Taking L and M to be 0{exp[—P^O)/^— 1)]} we can now satisfy all the
boundary conditions at £ = 0 and can be confident that the dominant form
of *F for £ > 0 is given by the structure found in this paper. I t follows that
the dependence of the skin friction on T near T = 1 is described by
i,
( — 0-850
1 \2
1/,
3KJ.
1 U . /,
1 \1+t\ , _ - .
J ^ e x p l — ^ t o g — ) - ^ ( l o g _ ) f +o(log—) j . (50)
since ^ ( 0 ) = 0-850, where e is any positive number. The leading term of
(50) has also been obtained in a partial investigation of the problem by
Smith (5).
Dennis (2) has given a table of values of the skin friction computed from
the numerical solution of the impulsive flat plate boundary layer and in
Table 2 below a comparison is made between his results and the predictions of (50).
TABLE 2
T
FDx 10*
- l o g FD
-log*1
log F/FD
i-o
I- i
1-3
i- 3
i
I
I
2
n-5
8-9
IO •8
11 5
CO
11
•5
16 •5
5 •8
5 •o
14
8
9-4
4' 1
53
I5
24
8-3
30
53
I'D
52
7-5
23
52
152
K. STEWARTSON
In this table FD denotes the excess in the value of dhpldt,2 at £ = 0 over
that predicted by Rayleigh's solution, as computed by Dennis. The values
of FD when r— 1 is very small are clearly too small to be significant. In the
range 1-3 ^ T < 1-6, logF/FD is fairly constant which is encouraging. Its
value (-^ 5) is rather higher than that (/-^ 2-3) for log 8*I8D given in Table 1,
but the discrepancy might possibly be removed if as many terms in the
series for the exponent were retained in (50) as were retained in (23).
REFERENCES
1.
2.
3.
4.
5.
6.
S. N. BSOWN and K. STBWABTSON, Proc. Camb. phil. Soc. (1973), to appear.
S. C. R. DENNIS, J. Inst. Math. Applies. 10 (1972) 105-17.
M. G. HAT.T,, Proc. B. Soc. A 310 (1969) 401-14.
S. H. LAM and N. ROTT, Cornell Univ. G.S.A.E. Rep. AFOSR TN-60-1100 (1960).
S. H. SMITH, SI AM J. appl. Math. 22 (1972) 148-54.
K. STBWABTSON, Q. Jl. Mech. appl. Math. 4 (1951) 182-98.