Stability of laminar shear layer between parallel uniform streams

Faculty WbMT

Dept. of Marine TechnobgY Mekelweg 2, 2628 CD Deift

The NetherafldS

Reports of Research Institute for Applied Mechanics Vol. VIII, No. 30, 1960

STABILITY OF LAMINAR SHEAR LAYER

_BETWEEN

PARALLEL UNIFORM STREAMS (I)

By Hikoji YAMADA*

Resume _{First paper on the stability of a shear layer between two}

parallel uniform streams, whose velocity distribution is erf-function. In the former half inviscid fluid is discussed by_{means of polynomial approximation,}

dividing the half domain of existence O y < into three sub-domains.

Instability is exhibited only by antisymmetrical_{oscillations, the wave number} of neutral stability being a lidIe larger than I.

In the latter half we concern with _{cases of finite Reynolds number and} employ the method of expansion by means of Hermite functions. _{To see} the pertinence of our method inviscid flow is_{treated again by this procedure,} results according well with the former ones. _{Stability curve given by our}

method seems satisfactory except when the Reynolds number is small and shall be the subject of next paper.

Stability of a laminar shear layer between two parallel uniform streams is a well-known problem and several efforts have been devoted for solution of this

difficult problem. Among these efforts those of Helmholtz, Rayleigh, and_Rosenfeld

are well-known and they dealt with a separation surface between two inviscid liquid flows. _{Recent investigations of viscous flow on the basis of the Orr-Sommerfelds}

stability equation are those of Lessen0>, _{Carrier>2, Esch>3, and of Tatsumi-Gotoh>4>.}

By these works the region of large Reynolds _{number and that of extremely small}

one have become clear, but the region between them which has an intermediate

Reynolds number remains obscure. _{In Esch's paper whole region has really been}

dealt with, but haply by reason of the _{singular velocity distribution he adopted}

his

stability curve presents a few aspects which are alluded by Tatsumi-Gotoh and, we

think, require further discussions. _{We then have intended to study this}

problem

and alike in some details, this _{report being the first one. in this and a following}

i

Now at Kyoto University. _{This work was carried out when}

he was a member of the Research Institute for Applied Mechanics.

(I) _{M. Lessen : On stability of free laminar}

boundary layer between parallel _streams, NACA Report no. 979 (1950).

G. Carrier _{Interface stability of the Helmholtz}

type, Los Alamos Internal Report

(1954).

R. E. Esch : _{The instability of a shear layer}

between two parallel strcams, .1. Fluid Mech., vol. 3, pp. 289-303 (1957).

T. Tatsumi and K. Gotoh : _{The stability of free boundary layers between two uniform}

we discuss stability of a more natural shear layer,assuming the erf-function

distribu-tion of mean flow velocity

w (y) = d (1)

which is the one used by Carrier in his inviscid case (a fact we read in the Esch's

paper). Not only we may assume sufficient approximation to a real shear layer,

but also (1) is free of any singular point from the mathematical point of view.

§ 1. Stability of inviscid flow (1)As our mean flow w(y) isparallel we

adopt the coordinate-system, in which x-axis is along and y-axis is across the flow.

The stream function 'fr of disturbance, which is superimposed ori mean flow, is

assumed of the form:

1fr _=

ip (y) et''

(2)

and the determination of amplitude .' (y) is to be done by means of the

Orr-Som-merfeld's equation:

L ()

(Qf - 2a" + a4)

(w - c) (v" -

)w"ç2=O. (3)

All the quantities are reduced in non-dimensional forms by thecharacteristic length

L and characteristic velocity U; R is the Reynolds number UL/z, y the kinematic

viscosity of the liquid; dash denotes differentiation by y. At infinity (y -= ± c)

ço and çY have to vanish.

As is well known, this problem has to be unterstood as a characteristic value

problem for c and c, being a and R two given real positive constants. Usually

c has a complex value C=Cr± ICj and according as c is positive, zero, or negative

the disturbance increases, remains constant, or decreases. The middle case i.e. the

case of neutral stability divides stability from instability and the condition c1 =O

imposes a functional relation between a and R, which is called stability curve.

Now turning to our problem we determine, at first, the wave number a of

neutral stability (c = O) of inviscid flow (R = cx). This number has been

cal-culated by G. Carrier and value a little larger than i reported (so we read in

Esch's paper). His paperbeing, however, beyond our reach we calculate this case

again as follows.

In this case stability equation (3) reduces to

(w e) (w" - a2ç) - w"g' = 0, (4)

and by means of the well-known theorem of Tollmien characteristic value e has to

be equal to the velocity of flow at a point ye, where w" vanishes. In our case as

w" vanishes at the origin (y O) we have c = O, and then (4) rewrites itself into

çY'a2+F(y)ço0,

(5)

and

N

STABILITY OF LAMINAR SHEAR LA YER BETWEEN PARALLEL

UNIFORM STREAMS (1) ₃

F(y)

--- =-_yeU

₍₆₎

by (1).

In solving the characteristic problem (4) we note that the equation admits

even and odd functions. _{We are then to manipulate only with the half infinite}

region y O with the boundary condition aty=O:

for even ç (y) ça,' (0) =0, (7)

for odd ç (y) _{ç (0) =0. (7')}

We divide then the region Oy <cx _{into three subregions I(Oy < 1.6), II (1.6}

.:ly <3.2), and Ill (3.2<y <

c) ;

in I and II we approximate F(y) by poly-nomials of 6 degrees, and in Ill by _{identical zero, polynomials being determined}

by the values of F at five points of equal _{intervals and the values of F' (y) at}

both end-points (y= 0, 1.6 and y = 1.6, 3.2). _{In Table 1 compared are the}

approxi-mate values of F(y) with the exact ones, in which _{we see sufficient accordance.}

Table 1.

y w(y)

_(')eXi

f

ox.

Using these expressions of F(y) into (5), _{and changing y-variable into ':}

y .

y-1.6.

'7

-6

-

1 6 in

we have

i :

_{-fi ()}

ÇO = 0, _{fi('7) =À1+ :_} ¡9(I) n ₍₈₎

f W"\' H I''.''' 0.0_0.2 0.0000_0.2227 -2.0000 -2.0000 0.0000 -1.9473 -1.9475 0.4 04284 -1.7956 -1.7956 0.6 0.6939 -1.5644 -1.5642 0.8 0.7421 -1.2829 -1.2829 1.0 0.8427 _{-0.9852 -0.9855} 1.2 0.9103 -0.7049 --0.7049 1.4 0.9523 -0.4673 -0.4670 1.6 0.9764 _{-0.2859 -0.2859 0.7617} 1.8 0.9891 _{-0.1608 -0.1610} 2.0 0.9953 -0.0831 -0.0831 2.2 0.998 I -0.0393 -0.0393 2.4 0.9993 -0.0171 -0.0171 2.6 0.9998 -0.0068 -0.0068 28 0.9999 -0.0025 -0.0023 3.0 1.0000 -0.0008 -0.0002 3.2 « -0.0003 0.0000 0.0000 3.4 _-00001 3.6 _{-0.0000 'i} I II III

lI:

_-f2(?7) 'Pz 0, f2 () = 22 +

flnri,

n=1

where , q are amplitude functions in the regions I and 11, and the constants 2's

and fl's are as follows:

= 2.56(r2 - 2.0000); 9(t) = 0.0000, ß2' = 8.6088,

fl) = 1.1346, ßW == 9.3330, ì951 =4.0378, ß6t1 = 0.0601.

22=2.56(a2-0.2859) ; = 3.1193, fl2 = 4.7618,

1.8528, /94 -2.7749, j9(2) =3.2666, /96(2) =1.0132.

As we know from (8) and (8') that ço and ç are intergral functions of ,

we assume power series expansions:

= a,1iì' (1=1, or 2),

(9)

and insert them in each equation, obtaining

= (n+1)(n±2) ,o

a,,,

9)

(j = 1, or 2), (10)

which determine the expansion coefficients a's. The first two of them, i.e.

a1(-), which remain arbitrary, are fixed once to (1, 0) and the other Linie to (0,

1), the resulting functions being denoted by ç' and

respectively. a's are

determined up to n 20, and each of them, being polynomials of ,(,, up to

(see below).

In the region 111 y is replaced by = (y - 3.2)/1.6 and then (5) reduces

to

2

2.56 c2ç' = 0;

solution of this, which fits to our condition at infinity, is evidently

Now we have to join together the solutions of three regions. At first we

take up the symmetrical solution çc(y), which is equal to Aço1 in 1. In lI and Hl

it can be expressed as B2+CcP?> and Dço3 respectively, and analytical

continua-Lion of them requires, as is well known, that the functions and their first derivatives

(8')

where dash denotes differentiation with regard to

The condition of solvability of this system of linear homogeneous equations

should have the same values at the joining points, i.e. y =

we have - Bq72('(0) + Cç2t(0) -A1t'(l) Bço2(8)'(0)±C472)'(0) Bç2t'(1) ± Cq'2(a) (1) + D'p3 (0) -= ± Dço3'(0) = 1.6 0, 0, 0, 0,

and 3.2, and thus

06

0

02

STABILITY OF LAMINAR SHEAR LA VER BETWEEN PARALLEL

UNIFORM STREAMS (1) 5

regarding to A, B, C, D is

çoi')(l) I O O

D'> (a) =

_=0,

₍₁₄₎

00 0 20 30 40

Fig. I. Amplitude functions of neutral stability; 'p and ço

are inviscid, c' H- içot is for aR 11.50, a 0.800,

50

which develops easily into

D(')(a) = ç'(')(I) 1.6aco2(0)(l) + co2"'(1)

± coit-''(1) _1.6aco2')(1) + c2ta)'(1) = 0, (14') and is nothing but the characteristic equation for the determination of a, through

2 and 2. The antisymmetric solutìon _{(y) is dealt with in the same way and}

the characteristic equation reads:

D('(a) = 9(a)(l) 1.6aço2(0)(1) ± co2t''(1)}

±çolt'(1)l.6aco2t)(1) ± co"'(l) = 0.

(15)

To solve (14') and (15) we notice at first that the roots a of these equations

are confined within (0, /2), for if a

is not less than 2 a2±F is everywhere non-negative and original equation (5) indicates that co does not vanish at infinity, whose vanishing being one of our boundary conditions. When a2<2 the magnitudes

of and ìì _{do not exceed 5.12 by definition and power series defining a,s can be}

cut off at a proper exponent. _{If we permit inaccuracies in the 4th decimal places}

of each numerical value of co(i), and accordingly in the 3rd decimal places of

ço'(i), a,'s have to be taken into _{up to n = 20, and every a, up to 2, thus each}

ç in D(a) being polynomials of 2i or ' of fifth degree.

Using these expressions D (a) can easily be calculated _{for each value of}

a, arbitrarily given, and we know that D('(a) = O has only one root a0 in the

neighborhood of a = 1.00, and

D((a) =

O has no root except a = 0. By trial and

q1t'(1)

o

i

o

O

co'(I) co(l)

I

error a0 is fixed to 1.035. We know then that antisymmetrical disturbance (i.e. ç)

is instable when 0a<a0 and stable when cr<a, and symmetricaldisturbance (i.

e. ç) is always stable.

The amplitude of neutral stability oscillation ç(y, a0) is shown in Fig. 1, in

an arbitrary scale, denoting it by i. To obtain this function we determine B, C

and D (assuming A = I say) by (13),

Ç'xÍ _{the coefficients ço s being fixed by use of}

the values a = 1.035

2 = - 2.378 and

-2! = 2.0 10. With B, Cand D thus

deter-0.3788 mined ç'(y a0) is expressed by -0.5932 0.3810 -0.0019 -0.2407 0.2356 -0.0784 -0.0582 0.0911 -0.0525 0.0060 0.0159 -0.0146 0.005 1 0.0014 ç2i(s)(,) in I,

B.('() ± Cçs2(°)() :--

() jfl JI

2.018e1'6 in III.

Of course çi's in these expressions are functions defined in (9) and now poly-nomials of 7J of degree 20. Their

coef-ficients are fixed with values of and

22 above given, and shown in Table 2.

-0.0029

0.0017

-0.0003 § 2. Method of solution in general

case- Solution ço of (3) i.e. L(ç)=0 is in

-0.0001 general a linear combination of four fun-darnental solutions, the two of them being

so-called inviscid integrals, the other two being viscous integrals. Now the

ex-isting region of oscillation being an entire space (- - <y < oo), the latter two,

however, drop by reason of the boundary condition at infinity (y = ± cxD) ; the

remaining two are slowly varying functions and vanish exponentially at infinity (ç9 -.exp (± ay)). Then ç is quadratic integrable and can be approximated arbitrari-ly accuratearbitrari-ly in the mean by a linear combination of functions which belong to a complete set of functions.

We take as such a set of functions the Hermite functions5

I' (y) = KuHn (y)e'f°12, K, =- (2"n! V)-t17, n = 0, , (16)

which are not only complete but orthogonal:

JV'm (y)

(y) dy = mn, (17)

and approximate ç (y) by a linear combination of them:

ç (y) anVt, (y), (18)

n=0

0) Cf. Courant-Hilbert Methoden der mathematischen Physik, vol. 1 (2nd ed.), chap. 2 §9, and Jahnke-Emde-Lösch: Tafeln höherer Funktionen (Teubner 1960).

Table 2. ç91(s ao 1.0000 a1 0.0000 -1.1883 as 0.0000 a4 0.9527 a5 0.0567 -0.7275 a7 0,0608 0.3743 ag - 0.0469 a10 -0.1768 a11 0.0284 a12 00809 a13 -0.0228 a14 -0.0273 a15 0.0097 a16 0.0093 a7 -0.0041 a2o 0.0007

STABILITY OF LAMINAR SHEAR LA YER BETWEEN PARALLEL

UNIFORM STREAMS i 7

the number of terms being appropriately chosen. _{As ç vanishes more slowly at}

large y according as a becomes smaller, more terms of the series should have_been

taken in for an approximation required. _{Here we use (18) with the number of}

terms fixed, and cases of small _{will be discussed separately in another way in a}

subsequent paper.

Several formulas concerning to 'fr, (y) are, by the difinition (16), derived

from those of so-called Hermite polynomials H,, (y):

H,, (y) =

_{(_)'l eY(f) e}

(n>0), ₍₁₉₎

FJ,,(y) = 2n H; (y)

_(n0),

_(19')

H1 (y) - 2yH (y) + 2n H,1 (y)

=O (n>0), (19")

where H_1 is identically zero. From these we have following two relations:

i/2(n+ 1) *+i - 2yifr +V_i O

(n>0), (20)

d

_/

n+1

dy

-

r

which are fundamental, !r_1 being identically zero. _{By the former we calculate}

numerical values of Hermite functions successively and by the latter we express the

derivatives of a Hermite function by the functions themselves. _{Especially by}

re-peated use of it we have

,,V(n-1)n,

2n _V'(n+1)(n+2)

-

2 n-1 2 " 2

'I'',"

i/(n-3)(n-2)(n-1)n

/

V'(n-1)n(4n-2)

+

/n(n+l) (4n+6)

v'(n+1) (n+2) (n+3)(n+4) 4, + 4 +

(n2),

_(20') (21) (21')

which we will require immediately, functions with negative _{suffixes being identically}

zero. Evidently ,, is even or odd function of y according as n is even or odd

integer.

Each term of (3) i.e. L(ç') vanishes exponentially when y tends to infinity

and then L_{(q) is expressible by means of 'fr,,, i.e.}

L()=

ßm'fr,n(Y), (22)

and (3) is equivalent to

where ¡9,,'s are:

/

i 19m=

I

¡nR - 2a2çe" ± (r4ç9) J

\(wc)

(ç"a2ç9)

w'çci dy.

When we integrate every terni of this expression by parts several times the

deriva-tives of ço in the integrand are reduced to itself, and afterwards we have to use

the expansion (18); in this way we know that the degree of approximation of our calculation depends on the accuracy of the expansion (18), and not on those of its derivatives. The result is the same as direct insertion of (18) and termwise in-tegration, such that

19m

a(-_A1.

- Bm,, +

rCn)

,=0 if)

where

Am,, = Am =J 2a2'r,," -F

a4')dy,

Cmn= Cnm =

J

ifr(i/r.,,//a2,,)dy, and

Bm =J

- Wi"'n1 dy; (25) the notations:

i =

= ₂

c;

(26) w[=1' _{w =}

Jed

being used.

Using (23) into (22') we have a set of linear homogeneous equations for

the expansion coefficients a,,s, and the condition for non-zero solution is the

de-terminant equation:

Am,, B,,,,, + r = O. (27)

¡p

This equation determines numerical values of r i.e. C = Cr -F-icL, for a given pair

of values (a, p) i.e. (a, R). Especially for the case of neutral stability (c)=O) r is real (unknown), and the equation separates into two real equations for p and r, a being a given wave number. Stability curve a(R) and neutral wave velocity e (a) follow from them.

The constants A,,,, and C,,,,, are easily calculated by the formulas (21) and

1

then

STABILITY OF LATvfJNAR SHEAR LA YER BETWEEN PARALLEL

UNIFORM STREAMS (1) 9

(21'), and all vanish except the followings:

V(n-3)(n-2) (n l)n

An_4 n - fl.fl4

-A2nAn,n_2=I/(fl_

A,,,,, = --- (2n2+2n-F1) ± (2n+1)a2+4; C-2.n = C,fl_2

_-

₂ ' 2n±1- -

a,

_2.

the ones which have negative suffixes being zero.

Determination of the constants B,,,, _{is a little lengthy. We replace 'Ir,," jn}

(25) by (21) and use the notations:

I,,,,,

=

f

W1 /1,dy, (29)

j,,,,,

=

f

iJr,,, wi" if,, dy (29e)

V(n-1)n

2n+1

,'

V'(n+1)(n±2)

mn - mn-2 - + _{) mn} + ,n,n-I-2Jm,,,,

(30) and we have to calculate 'mn and J11,. For Jm,Z we rewrite the relations (19),

(19') for Hermite polynomials into those of Hermite functions:

((m_ie12)

_{= - V2m 'l'me}

_(m1),

₍₃₁₎

,

e'i)

/2n 'fm-i e"1 (n>O), (31')

and use in (29). Then we see easily that

¡mn = -

('t,, e) W1 'fi, e" dy

= _{[m-1,nJ +j/_-Im_i,n_}

_(in,n1),

₍₃₂₎

where another notation

(28)

J

j.

where [m + n] is an integral of the type:

[ni, n] =J e , dy (33)

being used. With this recurrence formula and relations: [m-1,01,

i/2m ¡0,0=0, (32')

which can easily be proved, all 'm,, are reduced to the integrals of the type [ni, n].

The same is true for the integrals We use the explicit expression of w" i.e.

-2ye' in (29') and rewrite 2yç by means of (20), resulting into

Jm,i

=v'2(m±l) [m+l,nI-V2m [m-1,n] ,

(34)

and especially

Jo,0 = 0. (34')

We have thus been led to integrals of the type m, n], and these, in turn, are reducible to simpler ones, for by means of (19)

[ni, n]

(_)m+fl KKJ (_) e-°2 (-i) e-° dy

d m+n

( )"'KrnKnJ

e° (

_{d )} e-°'dy= (- [m±nI,

and the latter, having a simple recurrence formula:

[ml =- (ni-1) [m-2]

(35)

[ni] e2°2 H (y) dy, (36)

(36')

which can easily be proved by use of (19") and then (19'), we can evaluate start-ing from the special cases:

ill = 0, [0] ₌ . (36")

We have then completed the integrations. First the table of [ni], and then

table of [ni, n] are constructed. With the latter I,,,, and J are easily calculated,

and by use of these into (30) the required numerical values of B,,, are found, as

we see in Table 3.

§ 3. Stability of inviscid flow (2)-- To see the pertinence of our method

of integration and to obtain some indications for management of genaral cases

of finite Reynolds number, we have engaged ourselves again to the inviscid flow,

which had been discussed in section 1. In this case equation has been taken up in the form (4), i.e.

0 2 3 4 5 6 7 8 9 lo -0.28814--0.03423a2 1.99599 + 0.14120 a2 -2.24695 -0.55028a2 - 1.52795 -0.55256a2 2.71675 + 0.15999 a2 -0.90072 -0.07589a2 O -0.57120--00489 la2 2.56856 + 0.14977 a2 -2.63305--0.55256a2 -1 .90772-O.55420a2 3.24121 +0.16329 a2 3.76612 + 0.16589 a2 -0.37500-0.50000a2 0.62500 - 0.50000 a2

-0.04420 - 0.53033 a2 0.12758 + 0.10206 a2 0.63152 + 0.12758 a2 0.05419 - 0.03423 a-° -0.08208--004891 a2 -0.04423-H0,0I 321a2 -0.00198+0.02101 a2 0.02703 - 0.00545 a2 0.0 1598 - 0.00947 a2 Table 3. Bm 0.14068 + 0.01321 a2 -0.85093--0.05883a2 3.12684 ± 0.15567 -3,01615-0.55420a2 -2.28660--0.55544a2 2 - I .01646-0.53033a2 -0.36988 -0.54127a2 1.14875 ± 0.14120 a2 -0.26264---0.05883a2 0.07455 + 0.02724 a2 0.73995 + 0.10206 a2 1.39699 ± 0.12756 a2 -1.45241-0.541 27a2 - 1.85548-0.54688a2 -0.761 74-0.54688a2 -1. 14642-0.55028a-1.66988 + 0.14977 a2

-2.19280 + 0.15567 a2 -0.46457--0.06605a2

--0.67876-0.071 55a2 0.17259 + 0.03232 a2 0.29212 + 0.02101 a2 -l.12504-0.06605a2 3.67664 + 0.15999 a2 -3.39747--0.55544a2 I -0.08330-0.00545a2 0.45589 + 0.02724 a2 l.39426-_0.Ø7155a2 4.22092 + 0.16329 a2

-_3.77754__O.55640a2 -2.66476-0.55640a2 -3.04248- 0.557l7a -0. 15840-0.00947a2 062510 0.03232 a2 -1 .65962-0.07589a2 4.76148 -- 0.16589 a2 -4.15681 -0.55717a2 3 4 5 6 7 8 9 IO

L(ç:) = w(ç" - a2) - w"ç = 0

(37) which continues directly to the case of finite Reynolds number (3), and not to

(5). As the waves of neutral stability have zero wave velocity (c=O) in general,

which we will show in next section, we have taken it in (37) beforehand.

As in section 1 q' is symmetrical or antisymmetrical with regard to x-axis,

and then we have

= aoiJro ±a2'/r2

+ ±

a5ijr + . (38)

= aj'!'j ± a3iJr3

-F

+ ag/r9

±-

(38')

in place of (18). Inserting these in (37) in turn, L(q') is antisymrnetrical or

sym-metrical, and (37) is replaced by the orthogonality of L(q') to the set of functions

(un,V'e,ujt5, ..) or to ('/'o,V'2, 1i'4, ..), i.e.

B2m+i,,ja2 = O (m=0, 1, 2, ...), (39)

fl=t)

or

a2,+I = O (rn=0, 1, 2, ), (39')

each being nothing but the condition (22') and (23) ; evidently A,,,, and C,,, do

not appear in present case.

Eliminating ao,,'s from (39) and a2,,1's from (39') we have two characteristic equations:

D(')(a) B2m+i,2 = 0, (40)

D(")(a) -T: B2m,2,,+1 = 0, (40')

which by virtue of Table 4 can be developed into power series of a2. We cut

short the expansions (38) and (38') to first five terms, and consequently D's in

and (40') are determinants of fifth rank, which when developed give

D(')(a) - 1.1733 - 0.6779a2 ± 0.5903'-o4

+ 0.6434a° ± 0.12991a6 + 0.007788a'°, (41)

= 1.3 173 + 9.694la2 + l0.0900a4

± 3.2640a° + 0.3861a8 -f- 0.01446a'°. (41') gives one and only one root a,, = 1.067 and (41') no root; this character is

the same as in section 1 and the wave number a0 of neutral disturbance is

identi-cal within about 3 % error.

Making use of the value a,, in (39) we obtain the coefficients a2,/a,, (n = 1,

2, 3, 4), and then by (38) the neutral oscillation amplitude çry)

= 1.3719'!',, + 0.13821!r2 + 0.1033'fr4 ± O.03Ol',,, O.0205'/r,,,

a,, being fixed so as q',,(0) = 1.0000. This function, calculated by means of a table

of Hermite functions, is drawn in Fig. I as ç',, scale being changed arbitrarily.

perti-STABILITY OF LAMINAR SHEAR LA YER BETWEEN PARALLEL

UNIFORM STREAMS (1) 13

nence of our method of approximation.

Thus we have recognized that the expansion of ço into a series of Hermite

functions is a convenient method of approximate calculation of our problem, and

series with a few terms would be sufficient in the general case of not very small

Reynolds number. We proceed then to this case.

§ 4. Calculation of general case As we sec in (27), (28), and in Table

3, elements of the determinant (27) have such a special character that (ip) A,,,0

± rC,,,, and B,,,, _{have non-vanishing elements alternately in rows, and also in}

columns, so that (27) is written in

/Aoo + rCoo, B,,1, -Ao2 H- rC,,2, B03,

- B10, -J--Aji H- rC1i, - B32, -4ia H- iCia,

1 423 + rC20, - B21, _{A + rC22,} B23, LP LP B30, -A31 + rC3j, - B32, A33 + TC33,

= o,

and then, if we multiply rows of even number by i, the imaginary unit, and also

columns of odd number by i, _{the latter multiplication being equivalent to the}

in-troduction of new coefficients _{= ia,,1 in places of the coefficients of}

expan-sion of odd number a,,+i, i.e. equivalent to the adoption of expansion:

= a,,'!',, ± ia1'"1 -1- a2fr2 + 1a3'fr3 + ... (43)

and elimination of a0, a1, a2, a3, from simultaneous linear equations, we have H- irC,,,,, B,,1, -A03 H- irC,,2, B,,3,

- B10, --Aii + irC1i, - B12, A13 + irCIl,

- A20 ± irC2o, _{B21, --A22 + 11C22,} B23,

- B30, ---A3j + irC3j, - B32, 1

A33 + irCjj,

= o. (44)

For a real value of a A,,,,,, B,,,,,, C,,,,, are all real, as will be seen from

(44) has to reduce into two independent real equations for two real unknowns r

and p'.

As one of these equations manifests the vanishing of the imaginary part of (44), and as imaginary unit i appears in (44) always combined with r, we see the imaginary part giving a root r = O certainly, irrespective of value a.

Non-exisence of another real root is very probable in view of the single root r = O

when R cc, but not being proved here; for proof it seems necessary to discuss

and solution p i. e. R for an arbitrarily given a determines the required stability curve.

p1

- 0, (44') becomes

and in this case the system of linear equations for the coefficients aIm, a,,,41 divides

into two independent sets, the one the set of a3,,,1, the other that of a21, the

former giving an antisymmetrical function ç(y), the latter a symmetrical one This character had been used in section 3 at the outset and two determinants D1'

(a), D()(a) there appeared are now the constituents of (45), such that (45) is

equivalent to

D(a)(a)

Dm(a) = 0.

(45')

Turning to the case of finite R we have to abridge the infinite determinant

(44') to one of such a few rank, that we are able to advance numerical

calcula-(6) This was also the case in the Esch's paper, cited in footnote (3).

When R cc i.e. B10, O, B2, O, B30, 0, 832, 0, to O, B01, O, B03, O, B21, 0, B23, _{= O,} (45) sole root r = O.

(44) numerically, which would be a difficult task. We here assume6 simply the

Substituting r = O in (44), characteristic equation for f) reduces to

A00,

-

B10, B01,

A,

B12, 413, A20, B21, 1A22, B23, B30, _i_A31, B32, 1 A02, B03, = O, (44')

STABILITY OF LAMINAR SHEAR LA YER BETWEEN PARALLEL

UNIFORM STREAMS (1) 15

tions. _{As we have seen in section 3, the cut off there adopted is sufficiently good,}

and it is then natural in present case to accept it also, i.e. to adopt determinant

composed of the first ten or eleven rows and columns. _{When we choose eleven,}

attention has to be paid on the selection of rows, if we want to have the proper

decomposition (45') in the limiting case p -' m. At least when p is not large,

however, such caution is meaningless, and larger rank of determinant will promise

more accurate result.

We have adopted first ten, and then first eleven, and developed them into

algebraic equations of 5th degree in 2, both being the same degree. Their

coef-ficients, which are polynomials of a2, are too complex to be obtained in their

general form, so that we assigned several numerical values for _{a at first, reducing}

thus A,y1,,, Bmn to numerical values, and then developed the determinants. _This

de-velopment has been the most tedious task in our calculations. Result is expressed

in the form:

G,(p) - I -F C1p2 + C2p4 -F- Cj96 -F C4p -F C5p10 = 0, (n = 9, 10) (50)

and the coefficients c0 are shown in Table 4. The roots_{p of these equations, then}

Table 4.

aR, and then R, are shown also in that table. For the case n=9, to each a which is less than a certain number a0 one and only one p was found, and none above

n 9 ( 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 1.00 1.05 1.065 1.067 C1 -0.2753 -0.03935 +0.1574 +0.2773 -F03245 +0.3255 -F02930 -F0.2861 -F02759 +0.2713 +0.2424 +0.2461 lo -1.047 -0.6209 -0.2574 -0.02144 +0.09886 +0.1422 +0.1364 +0.1375 +0.1320 +0.1299 +0.1174 +0.1141 c3X103 -4.162 -2.730 -1.481 -0.6371 -0.1683 +0.04896 +0.1168 +0.1358 -F0.1364 +0.1377 +0.1284 +0.1256 c4Xl05 -2.750 -1.865 -1.078 -0.5350 -0.2220 -0.06399 +0.002300 +0.01887 +0.02420 +0.02710 -j-0.02951 +0.02975 C5X108 -3.291 -2.225 -1.303 -0.6792 -0.3169 -0.1320 -0.04640 -0.02409 -0.01497 -0.01040 -0.00181 -0.00016 0.00000 p 1.413 1.862 2.765 4.428 7.370 12.975 24.03 35.57 45.56 55.24 I.30x102 4 X102 00 1.252 1.651 2.450 3.924 6.532 11.50 21.30 31.52 40.38 48.96 1.15x102 4 X102 oc, R 4.173 4.126 4.901 6.541 9.331 1437 23.67 33.18 41.20 48.96 IA X102 4 xl02 00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.964 0.98 1.00 -0.859 -0.802 -0.629 -0.331 -0.00527 +0.2302 +0.3605 +0.3972 +0.3851 +0.3508 +0.3279 +0.3148 +0.3089 -2.80 -2.66 -2.28 -1.629 -0.8894 -0.3237 -1-0.01265 +0.1671 +0.2126 +0.2046 +0.1914 +0.1822 +0.1788 -12.75 -12.20 -10.69 -8.03 -4.92 -2.474 -0.961 -0.1711 +0.1586 +0.2546 +0.2586 +0.2534 +0.2530 -11.8 -11.3 -10.0 -7.69 -4.899 -2.653 -1.229 -0.4489 -0.08981 +0.04888 +0.07485 +0.08259 +0.08718 -22.9 -22.0 - 19.6 -15.17 -9.85 -5.53 -2.75 -1.177 -0.4180 -0.09093 -0.01476 0.00000 +0.01445 +0.02863 0.945 0.970 1.056 1.274 1.75 2.68 4.36 7.365 13.32 29.73 73.49 oc,

-

-0.838 0.860 0.936 1.13 1.55 2.37 3.87 6.53 11.80 26.35 65.13

-a0. a0 is here nothing but the wave number corresponding to the inviscid liquid,

and is of course equal to a,, = 1.067 found in section 3.

Some remarks are necessary for the case n = 10. As is said above, values

of p which are not too large are almost equal to the corresponding values of n

9, and are expected to be more accurate ones than the latters. There will then

be not any solid reason to reject G,0(p) = O as irrational in cases of large p, and especially of f) -= In fact the determinant of first eleven rows and columms

does vanish and not determine the corresponding wave number when p=cu. This

means zero absolute term in the expansion of determinant into a polynomial of p', and results in an algebraic equation of 10th degree in p, i.e. G,o(p) = 0, one

degree smaller than the rank of determinant. In this case, as in the other case,

a0 is to be determined as a vanishing point of the coefficient c5 of G,0 (p), and

interpolating values given in Table 4, we find a0' = 0.964,which can be accepted

as a rough approximation to a0 = 1.035 of section 1.

If we write a'2,+1/p instead of a2fl+, in (43) we have the characteristic

equa-which replaces (44'), and the development of this equation, cut off to first eleven

rows and columns is, as is easily understood, nothing but the equation G,o(p)=0.

Above all, c, is proportional to the left hand side of (51), abridged of course to eleven rank, and made p' vanish ; this determinant really vanishes at a0'=0.964.

The characteristic function corresponding to this a0' is, as seen from the

substitu-tion above introduced, in a form:

43 a0','r0 + a2iIr2 -I- aIr4 _f-... , (51')

a fact which accords well with the result of the other case in section 3.

The value a,' depends, however, on the elements A2,,,2,, (m, fl=0, 1, 2, 3, 4, 5),

this character being irrational from theoretical point of view. Presumably this value

approaches gradually to cr,, = 1.035 or so, we think, when the rank of determinant

is increased, and becomes free of the elements. If we cut off (51) to first ten

rows and columns, and make p' vanish, it must be identical with the

correspond-ing one of (45). Apparent difference between them is the appearance of A2rn,2 in the former, which has no effect on the symmetrical characteristic function, and excludes unnecessary discussion of antisymmetrical one.

The stability curve a as a function of aR, above obtained, is shown in Fig.

2 and Fig. 3, in the former the general feature and in the latter the feature when

a is small being indicated. In these figures we see no hump in the intermediate

region of aR which has been a remarkable point of Esch's results, and see rather

tion A,0, B4O, A20, B01,

pAj,,

B21, p2A,,, A,2,

B,2,

A22, B32, B03, f)2A13, B23, p2Au, 0, (51)

11 08 05 02 08 06 04 02

STABILITY OF LAMINAR SHEAR LA YER BETWEEN PARALLEL UNIFORM STREAMS (I)

11

10

09

5 JO 50 100 500 1000 _{5000 10000(o}

Fig. 2. _{Stability curves; solid curves arc those of § 4, broken line is Esch's} approximation, _{that of Esch by computer, inviscid point of § 1.}

17 cx n=q n=Io

,/'

//

/

,''i

0

;,..

1/ J I JI I I

JIll

I I_____l_ 0< / ,,/ / /'0 00 2 _{4 6} 8 10

an accordance with that of Lessen in its general feature.

When a is small our approximation is expected to be rough, as is stated at

the outset. Compared with Esch's first approximation for very small a, which has

acquired more general meanings through the work of Tatsumi and Gotoh, and a

value ( at one point aR= 1, which has been calculated by Esch by means of an

electronic digital computer, both being shown in the figures, our stability curve

seems to be rough when R is 3 or smaller (a is 0.55 or smaller). These cases of

small a will be the principal theme of the second paper.

The velocity distribution of neutral waves, which is a stationary flow pattern

by reason of e = 0, is given by (2), and c(y) there by (43). Now the characteristic

value p being inserted, linear equations for a0, a1, a2, a:1, have all the coefficients

real (set of these coefficients forms the determinant (44')), and therefore aa,,,, are also all real (a0= i assumed). Then we have

(y) =ço,(y) + iç(y);

ÇOr(y) = a0"!'0 + a2"!'2 +... , (52) ço(y) = a1"!'1 + al"!'9

-and the disturbance stream function (2) rewritten into

= ç9,.(y)cos(ax) -çe(y) sin (ax). (53)

As an example we have taken the case (a = 0.800, aR 11.50), in which

a's are as follows:

a0 = 1.0000, a1 = 0.0916, a7 0.2821, a3 -= 0.0172,

a4 0.1119, a = 0.0199, a5 0.0624, a7 = 0.0085, a5 = 0.0228, a9 0.0121.

Using a table of 1-lermite functions we calculate numerical values of ço,. and ç,

which we see indicated in Fig. 1, in comparison with ç and Ç1ii. Stream lines

of these disturbance flows will be given in the second paper. Remarkable is the

small imaginary part, and antisymnietricity of flow about x-axis (the centre line of

our shear layer) seems nearly conserved down to a pretty smallReynolds number,

and therefore flow pattern remaining almost the same (To be coniinued)7. (Received Sept. 20, 1960)

(7) Almost all numerical computationsof the paper are undertaken by Miss S. Hoshino, to whom our thanks are due.