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,
0}
the set of u with mass flux discrepancy u>0. Then, by (31.2),
9[u; p] -P(p)=0
(31.3)
when u is a statistically stationary solution. Of course u is also an element of the larger set of admissible fluctuations fin X. Hence P-',<maxHr,,Y-'[u; p].
(31.4)
maxnr,. r?3 - t [u; p] ,<maxRY -' [u; p] = P- t
(31.5)
Moreover,
because all of the functions in the intersection of H and ,At are in H. The variational problem could be posed as
P = minji. [u; p]
(31.6)
if only those u for which p>0 are allowed in the competition. Clearly, 2 All of the required cylinder averages exist for almost periodic functions and the variational problem seems well posed in this set of functions. We do not presume to take a position on the extent to which "true turbulence" could be almost periodic. Other settings for the variational problem which lead to the same analysis could be defined; an alternate procedure which is commonly adopted is to leave the function space incompletely specified, specifying only that the fields u are uniformly bounded. Assumptions are then required at a later stage.
102
IV. Friction Factor Response Curves for Flow through Annular Ducts
Statistically stationary turbulent Poiseuille flow with mass flux discrepancy µ cannot exist when
P P(0) = P,
(31.8)
where P.. is the first critical pressure gradient of energy theory,
&2_a (u)
dP dp
(iii) dP
dp
(Lwu + 2rlnri2
-4 \-
(h(r)wu)Z
(31.9) *
is a decreasing function of p.
(31.10)
Proof: The first critical pressure gradient of energy theory is given by
P 1 =maxN (h(r) wu> 2(f 0uj2) Noting now that H cH (since zero mean values are not required in H) we have
P, 1=maxN' `[u,0]-< maxHc3 `[u,0]=P-1(0),
(31.12)
proving (31.8). Actually, equality holds in (31.1.2); to prove it, we must show that the maximizing element in H has a zero cylinder average (Exercise 31.1). To prove (31.9), imagine ut and u2 are minimizing vectors for (31.6) belonging
to µ, and µ2. Then 3[u1; 92] -?[u1; ut ] % P(uz) -P(µ1)
.[u2; p2]
p1] %.i[u2; p2] _ l [u2; pt]
4([wu+ ru) 2rlnt? ={pz-pt} \ 2 /
2/
(31.13)
32. Turbulent Plane Poiseuille Flow- An Lower Bound for the Response Curve
103
Dividing (31.13) by u2=ut and passing to the limit uz-'ut we prove (31.9). Returning to (31.13) with (31.9) we find that
4P(u)
dP(u2)
2-fit) du >1 (1 z-I ) du
(31.14)
proving (31.10).
The variational problem for the values P(p) have been considered by Busse (1970A, 1972A) and Howard (1972) for flow in a round pipe and by Busse (1969A, 1970A) for flow in a channel. It is well to note that the structure of the variational
problem for turbulent flow has much in common with observed properties of real turbulence. We have not here emphasized this most interesting aspect of the variational theory as a model theory for turbulence. This aspect is stressed by Busse and Howard, in the notes to this chapter, and in our discussion of the turbulence problem for porous convection which is considered in Chapter XII. Exercise 31.1: Show that the maximizing field for (31.11) has a zero horizontal mean. Hint: prove that the maximizing element in the almost periodic class has a nonzero wave number (Exercise 44.1). Exercise 31.2 (Busse, 1969A, 1970A): Consider a homogeneous incompressible fluid between two parallel rigid plates of height d. The plates move with velocity Va relative to one another. Show that every statistically stationary solution of the Navief-Stokes equations has a positive fluctuation mo-
mentum transport (wu)>0. Show that every statistically stationary solution with µ=(wu)>0 has
R=
Vod
> R(p)
V
R(I.L)=mines
vuz
(wi!)' +
u12((wu-(wu))Z+(w5-(wa))2>
(31.15)
Prove a monotonicity-convexity theorem for R(u).
Exercise 31.3: Find Euler's equations for the minimum value P(y) given by (31.6).
Exercise 31.4 (Hayakawa, 1970): Formulate a variational problem for the response curve relating the mass flux discrepancy to the pull of gravity in open channel flow when the axis of channel makes an angle a with the horizontal.
§ 32. Turbulent Plane Poiseuille Flowan Lower Bound for the Response Curve In the last section we formulated a variational problem which leads to a lower bound on the pressure gradient when the mass flux discrepancy is given. We wish to give an explicit estimate of this bound. We will restrict the analysis to the case p-->1; that is, to plane Poiseuille flow. The main result to be obtained is first stated, then proved, below.
104
W. Friction Factor Response Curves for Flow through Annular Ducts
The response function for statistically stationary flow P(R),
(32.1 a, b)
,u = < U. - P.>
satisfies the inequality x
(32.2)
P(p) >' P(lC) >' P(µ)
where
P(0) + 576u/(P(0) + 48) for
P(u)-l2µ={48v1t-48
for
jl < y* µ>,u*,
(32.3)
y* =(P(0)+48)2/576 , P(0) = Pe = min H
-
793-6.
(32.4)
Before constructing the estimate (32.2) we note that P=8 Umv/l2 (see (17.7)); hence P- 8(U I/v). Therefore P,=8R. where R, (rl =1) = 99.21 is given in Table (22.1). For a fixed fluid the pressure gradient P is an increasing function of the mass flux discrepancy < Ux - VX> on the bounding curve P(p). The bounding curve is
most easily interpreted in the coordinates of the response curve of Fig. 27.2. This interpretation will be given in § 35. The proof of (32.2, 3) starts with the relation (30.13)
P=
+y)2>+)2> 2
(32.5)
where
-i
+y-
2
follows from (32.5) in the following way: first,
P-12y
y
]2>
-12y
second, using =12, we note that < [wu - (wu> -12z ] 2> = - <wu> 2 -12 2
= ([wi -<wu>]2>-122.
(32.6)
32. Turbulent Plane Poiseuille Flow -An Lower Bound for the Response Curve
105
This proves (32.6). From the inequality (C.11) proved in Appendix C, ([rvu - <wu> -12z] 2>
576
2
D+48
where D=/
.
It follows that
P-12p,>D+(576p/(D+48))>,minD[D+(5761t/(D+48)]
.
(32.7)
This minimum is attained when D+48 = ^576µ = 24µ'l2 and is equal to
2 576p-48=481-48. But, by (32.4) D>P(O); therefore, D+48 can never equal 24p'/2 if 24p'12 (b2-a2)
(33.7 c)
where
1(!. a Y
ar ax
a
a2
aW'a aX or
aW' r
ax
and
d
_
02
art
1
r Or + i?X2
The total mass flux =2(P[V.j-P,[V,])b()l).
(33.11)
The proof (33.11) is left as Exercise 33.2. The bifurcation problem for the double-primed variables may be obtained from (33.7) using (33.9 a, b) and the relation
P"=s W=2sRP
(33.12)
where R is defined by (28.3) and E22
=2(P[V,,]
By choosing a normalizing condition in the form = b(rl)/2,
(33.13 a)
we find that (33.13 b)
E2 =,f -.f,
is the friction factor discrepancy where 2
.f=P[1222D5/[b22a2SurVxdr
_ _ = 4P[Vx]/(V>2
(33.14)
and Dh = 2(b - a) = 2l is the hydraulic diameter (the ratio of four times the area of the cross section of the annulus to the wetted perimeter) and P=PI'/v2. We rescale the time:
34. Construction of the Bifurcating Solution
109
With these definitions established we may now state the bifurcation problem in terms of the friction factor discrepancy. Thus, a
[at
+Uo
o
ax
1
1 dUn O% +EI('Y,dY')- d2'Y=0 2R dr [r dr ] Ox d
in
r?Yf
Yr= ar =0 at r=a,b, and
Yr is 27r-periodic in ax and cot.
(33.13 e)
All solutions of (33.13) have the same mass flux _ . Eqs. (33.13) are a complete statement of the mathematical problem for the bifurcating solution. With a given, we seek solutions [{x,r;E,a,n), R{E2,a,rl), co{EZ,a,>j)] of (33.13). The bifurcating solution is necessarily time-periodic when the solution of the
spectral problem (Orr-Sommerfeld problem, see § 34(a)) is both unique and time-periodic. There is a unique dimensionless mass flux O of linearized stability for laminar flow. Motions existing in this (R, a) region are called subcritical. On the other hand, the points on the surface of periodic solutions neighboring OF are in the unstable region for laminar flow and are supercritical. The properties of the sketch may be verified by inspecting the values in Table 34.1 noting that aR (0, a)/0e =0 and aR(O'a)
a(e2)
-y
a2R(001)
ae2
_ - [It(O,a)]2:t2 . z
At the point C, Z'=0, and it is preferable to represent the surface of periodic solutions as f(a,R,e2)=0. This may be solved for a(R,e2); as/dR=O and ca/00>0 at C. It follows that nonlinear periodic solutions exist with larger wave numbers (a(R,e2)>athan the largest value a= a. found on the neutral curve. The bifurcation theory assumes that the eigenvalues a are simple at criticality.
The simplicity assumption has not yet been justified theoretically3. Computations for axisymmetric bifurcations are consistent with the assumption of simplicity and it has always been possible to find one and only one normalized solution of the Orr-Sommerfeld problem for each pair (R,a). Given simplicity, theory guarantees that, apart from phase, only one axisymmetric periodic solution (with a fixed) bifurcates. The Floquet analysis of the stability of bifurcating solutions which was developed in § 14 applies here. According to the factorization theorem, subcritical ' The restriction of the stability and bifurcation analysis to axisymmetric disturbances is artificial and undesirable. Three-dimensional disturbances, even when spatially periodic, will introduce another wave-number parameter and a, at criticality, may lose simplicity for certain values of the parameters. The restriction to two-dimensional solutions for the plane Poiseuille flow problem is less restrictive than might at first be supposed. This follows from Squire's theorem (Exercise 34.2) which guarantees that the values R(a) lying on the lower branch of the neutral curve (arcCF) are the smallest of the critical values R for three-dimensional disturbances. It follows that the twodimensional disturbances will bifurcate first at the smallest values of R(a); in particular, the solution which bifurcates at the minimum critical value R,, (at B) is two dimensional.
35. Comparison of Theory and Experiment
119
solutions are unstable so long as the friction factor discrepancy (e2) is a decreasing function of the Reynolds number R. For larger values of e2 the friction factor bifurcation curve is expected to turn back around (see Figs. 27.1 and 27.3) and to regain stability. Stability is in the sense of linear theory; moreover, here, stability is only with respect to small two-dimensional disturbances, 2n-periodic in ax. Therefore, the recovery of stability of supercritical solutions with larger amplitudes (see Fig. 15.1) may hold only in a restricted class of disturbances. The numerical results of Table 34.1 show that for most values of x on the neutral curve of Fig. 34.1, the bifurcating solution is subcritical, unstable, and will not be seen in experiments. It is possible that the unstable time-periodic bifurcating solution can be observed as a transient when the parameters lie on or near the surface of periodic solutions. Exercise 34.1: Formulate the linearized stability problem for Poiseuillc flow through annular ducts for three-dimensional disturbances. Show that the assumption that the velocity components of the disturbance are independent of 0 implies that the azimuthal (0) component of the disturbance must vanish.
Exercise 34.2 (Squire, 1933): Consider the linearized stability problem for plane Poiseuille flow. Show that for disturbances which are proportional to exp[i(xx+ fly)],
L(aR,xz+f2,v/x)w=O,
w=Aw=O at Z=±2'-
Prove that R(O,a) on the lower branch of the neutral curve for two-dimensional (/1=0) disturbances (CF in Fig. 34,1) is also the locus of the smallest critical values for three-dimensional disturbances ($ O). Proofs of this theorem are in Lin (1955). Stuart (1963) and 13etchov and Crimiinalc (1967). Exercise 34.3 (Yih, 1973): Prove that min [Us]
where D[U=] is the stretching tensor of the basic parabolic Poiseuille flow and not the mean motion. To obtain (**) from (*) one first notes that if the disturbance is infinitesimal, the fluctuation may be dropped relative to the quadratic terms; then D;J[V=] =D;J[U ]. For infinitesimal fluctuations the mean motion coincides with the basic motion. Reynolds reasons that the fluctuation dissipation integrals (for example, the last term of (30.7)) which have been neglected in going from (*) to (**) are of one sign and always contribute to a decrease in 0 should be stable to axisymmetric disturbances. On the other hand, the argument also shows how flows with an adverse distribution of angular momentum (QC 0
for all eigenvalues tr. At criticality, (v1,A,B)=0, and the eigenvalue problem (39.1) may be reduced to ordinary differential equations by Fourier decomposition into normal modes proportional to e`tx`+nef `at) Elimination of the pressure and the axial disturbance u, using
-iaru=DJ'+inv where J'=wr, annulus as an instability. For the Poiseuille flow, the discriminant 12C can have the correct (negative) sign
for instability only if the spiral disturbance angle is of opposite sign on the two walls. These two different spirals, one on each wall, are observed in Nagib's (1972) experiment and can he seen in Figs. (46.4, 5).
Another striking example of Rayleigh's mechanism is exhibited in the photographs (Figs. 53.12 and 53.13) of Taylor instability near the equator of rotating spheres. The Taylor vortices appear near the equator of the sphere where QC0 and a vertical secondary motion does not develop (sec Exercise 53.3).
V. Stability of Couette Flow between Rotating Cylinders
142
leads to the following:
v L2 f - 4L
Df +inv r
2inBD f rs
= ikLf -
2iic
--(D f + inv) r
f +2(n 2 r (Ar+B 2 2)
2
-2inAD
r
r
r
]-
r L} (n2 +azr2)
f)
1
r
v, (39.2)
+L2!
f -2a 2
D(rc)l
=x(n2+a2r2)Df +inv -a2r_D r
n
n
f +2a2Af
z
where K=n A+-2 -cJ, L
r D(rD)- r2 -a2 and f=Df =v=O at r=a,b.
Eqs. (39.2) determine eigenvalues v(a, n) = v(- a, n)
.
(39.3)
Assuming that v(oc, n) is a simple eigenvalue of (39.2) we have real eigenfunctions of (39. 1.) in the form, say,
rw(r, 0, x)=f(r) expi(ax+nO+iwt)+c.c. (39.4)
+ cf (r) exp i(- ax + nO + 1d t) + c.c.
where c is an arbitrary complex constant and "c.c." stands for complex conjugate. In general, v(a, n) is a multiple (double) eigenvalue corresponding to the superposition of two waves. Results for (39.2) may, without loss of generality, be stated under the restriction S01 >0; the stability boundary for S21 0 (this is the region between the lines defining solid rotation and potential flow). Taylor found axisymmetric, steady solutions (n=6=0) for A 0 and for A Gv<w2> 0 and Q t > Sl2 ,
(40.15)
and
8'v(0) < (vl Gv)2
.
(40.16)
40. Generalized Energy Analysis of Couette Flow; Nonlinear Extension of Synge's Theorem
151
When these conditions hold,
ev(t)
[v-GvP
(
gv(0}
(t)][v-Gve
2
t / 2 (0)] 2exP{
vAy/2t
(b-a)
I
(40.17)
The stability criterion A>0, Q1 > Q2 was established for axisymmetric solutions of the linearised stability equations by Synge (1938 A, B). The present result (Joseph and Hung, 1971) is a nonlinear extension of Synge's result.
Proof: To prove the conditional stability theorem (II) for Couette flow, we shall need to establish the following evolution equation:
d9v dt
WV2\_v -2B(\ r3 / V
(40.18)
This equation is the sum of A times (40.2) and Eq. (40.19) below. Eq. (40.19) follows
from the integration of {
V
r
v (40.1 b) }
over a period cell; in carrying out the
integration, we calculate, using V2 ru =0,
( r V(u' v2)v/ - B
Y32)
(v(V_)v) = -\K IV2v
/+2B\Y/-\l YZ/
and
2B`
rorU)
Combining the last two equations we find that
Kv(v_)v)= -\ V (a v)2)-IY t rc.v\2\ and
(40.19)
We may simplify the computation by introducing variables
-r 0, V
rAu=^r.
V. Stability of Couette Flow between Rotating Cylinders
152
In these variables (40.18) becomes 2
id /,1,2+"l2 +(rV)02>+A
= -v(IV24/12+IV27I2+
+(rV)IV20I2)
-
dr(rV)= d (Ar2+B)=2Ar>0, we have (since A>0) that b2Q2>rV>a2S2t
.
Hence, we may continue the inequality (40.11) as
/04\ oe(b i 2)+1 and the remainder of the proof leading to (40.17) follows along the path leading from (40.11) to (40.14). Exercise 40.1: In the conditional stability theorems (I) and (II), we used two of the three energy identities (40.2), (40.3) and (40.18). Consider the functional formed as a linear combination of these three identities. Prove a conditional stability theorem for this functional. Form a variational problem for the "optimum" stability limit for the sum of the three energy identities. Formulate the problem of finding the values of the coupling constants which give the largest region of conditional stability with monotonic decay. Exercise 40.2 (A maximum principle for the total angular momentum; Rabinowita. 1973): The operator e2
L = Y3 _ Q; {X, t) 1
cx;cx;
1
R
(?
2
ox;
Of
is said to be uniformly parabolic in a four-dimensional (x,t) domain (xeY/-, 0ovg(A),
(41.1)
the "optimum stability boundary". It turns out that the value of 1 can be selected so that v, and the linear limit VL for Taylor vortices are virtually indistinguishable when b2Q2 0. The following result will help to characterize the optimizing value Lemma:
/wo\
n
(41.2a)
;+1= A<wo) When
h2522 < a2Qt
WO
r /} = <wo)lrr
.
and .522/Qt > 0,
r, a,/]
(44.5 a)
where (44.5 b)
u" = u cos fi + v+ sin /J
and the maximum is over solenoidal fields u = (u, v, w) which vanish on r=a, b. In general, (44.5 a) will have its largest value among functions which allow
one to raise the value of 1wu"I on each cylinder r=const. without increasing the value of the dissipation denominator IVul2 too sharply. The value of the dissipation cannot be easily controlled among functions w since w and aw/cr both vanish at the walls. To maximize (44.5) we need to give u' the greatest freedom to take on the values which increase lwu'"1 at the smallest dissipation cost. Of course, u" is not free when w is given since the vectors u which compete for the maximum have div u = 0. If e;,= sin/J,ee+cos fige,,
e,..=cos/ige0-sin fl
e,,
(44.6)
are orthogonal energy-spiral directions on the cylinder of radius r the only derivative of u' which appears in divu=0 is 8u"/ax". If, on some cylinder, r=-F,
'd) =P,,)
(44.7)
then du"/0x"=0 and 1wu'"1 can hold to its largest value on each energy-spiral line y" = y' = const. on the cylinder T The argument implies that the critical disturbance of energy theory should vary only slightly along energy-spiral lines. (44.7) can be proved when 7=a/b-d (see § 51) and it is supported by numerical analysis for general. values of a/h (Joseph and Munson, 1970). Exercise 44.f (Almost periodic solutions of (44.2)): All vectors ae Hx may be represented as
u(r,g,x)...Yr--. E'---
ui."(r)e'('e+a,=),
(44.8)
where, since u is real-valued,
a;=a_, and Here, the overbar means complex conjugate, and uJ
1
2;[
Iimt'. 1 2L
are Fourier coefficients. Show that 1
= max,., A(a,,n) =
n)= A(a5, n,) ,
(44.9)
165
44. Eigenvalue Problems of Energy and Linear Theory
where, below, u=u,,,(r) and A(a,n) = -/V,[u,a] where, since we averaging over functions of r alone,
2(1-rf) i
r dr
'=I D(rD)-(n2+1)/r2-x2,
D=
d
L=.9+1/r2,
,
(44.10)
and
9.[u; a] = -<w2'i >
- -tin( -)
= \
rnv-iw (44.11)
r
r
Hint: Follow the'proof of 20.7. Exercise 44.2 (Euler equations for the Fourier coefficients): Show that the Enter equations for (44.9) may be reduced to two equations for the unknown coefficients u(r) and f = rw(r) where u(r)=(u(r), 14r), w(r)):
n2 f (D U)/r = - 2A {iarDLJ'-(n2+22 r") Lu - 2a2rDu} +anJrD(V;r) ,
-r 'DU{(n2+a2r2)u+iarDf}-i2AarfD
1
r
(44.12 a)
DU
{
+{n (n2+a2r2)urD(V/r)- P f D(rD(Vir))-' (2n2+a2r2)DfrD(V/r)} = 2A(L2f +4ixLu)
111
(44.12 b)
At the boundaries, r=a,b,
u=f =Df=O.
(44.12 c)
(ii) The Spectral Problem
The spectral problem may be obtained from (44.1) by substituting solutions + uu, into the equation which is obtained from (44.1) proportional to e a by linearization,
-cu+2ii A
-Vp+IVZU
(44.13)
where uEH, The critical values of I =1r are those for which (1)=0. Spiral Couette flow is stable by the criteria of the spectral problem when (I)>0 for all eigenvalues a(a.). Spiral flow loses stability when R exceeds R, =7.,i;' where IL' is the first critical Reynolds number of the spectral problem.
166
VI. Global Stability of Spiral Couette-Poiseuille Flows
The equations for the Fourier coefficients of u and p in the spectral problem (44.13) are
-w+i9w-2Vv+Dp-I{ Pw- Z v1=0, inp
W
Yv+
2in r2
w
(44.14 a)
0,
(44.14b)
i.fu+wDU+ ixp -1Lu =0 , 1
r
(44.14c)
D(rw)+inv+iau=0,
(44.14d)
r
w=v=u=0 at r=a,b.
(44.14 e)
where £ and L are defined by (44.10) and
£=nV+aU-w, r
V(r)-V(r)+Or.
(44.15)
The first critical value of linear stability theory may be obtained from the principal eigenvalue of (44.14) I
RL
= max,
n
(44.16)
1,(2, n) =I(xL+ nL)
Some interesting features of (44.14) are most easily detected by examination of energy integrals for (44.14). These are formed as follows: 0 = + +
=i-2 'wv )+( vcD(rV+
/ +iLl ;u]-5. r
}
(44.17)
The real part of (44.17) is
u]-=0
(44.18)
and the imaginary part of (44.17) gives
+((wv-u`v)D(r4V ))
=0.
(44.19)
167
44. Eigcnvalue Problems of Energy and Linear Theory
It is clear from the form (44.15) of .q' that the frequency uw can be regarded as a
wave on the spiral 9=const. Eq. (44.18) can be made the basis for estimates of the decay constant a, n and . are given; when S =0
when
=J[u]/9,,[a; n] tan,
PP _ -1.
(48.2)
We next change coordinates so that x' lies in the direction of the stream at an angle z from x U'=UCOSX+vsiny,
v''= -usin j+vcos y, and w' = w to find that I
R-=maxH
(48.3)
,)-maxnu+a2
(48,12)
and HB is as before except that the continuity equation now becomes Dw + iau = 0 .
(48.13)
Suppose now that (u, v, w) are maximizing functions for (48.12). We may decompose these functions into even and odd parts; e. g., w = we +wo where
VI. Global Stability of Spiral Couette-Poiseuille Flows
184
2we=w(z)+w(-z), etc. Elimination of u in (48.12) using (48.13) followed by this decomposition leads one to the following:
_1_ R(a, i1,)
1- eostfi(.o1 [We, Dw o] + .41 [wo, ww]) + sin D,
[we, ve] +,f 2 [ wo, vo]) 2[we]+`' 1[ve]+sin2{p-2[wo]+
1[vo]' (48.14)
where 0 < u < it/2, . 1 [w, Dw] = a ,
f2[w, v] =a2
,
`' t[v]=a2+a4 and
92[w] = 0, 0 ' =
I
where
1
R(a, i fi)
=max0-0-,"12
(48.17)
R(a, i/i)
Then by (48.17) and (48.15), respectively,
0. Furthermore, they show that -1F is the product of the overall angular velocity and the total vorticity; that is, IF is Rayleigh's discriminant. In the general case >X # + 2fly Cy,
IF (VIA, X, (l)
,
IF is still Rayleigh's discriminant; that is, (50.1)
,
where fly. = ey, ft
and y- = ey. - (2 S1 + curl U)
are components of the overall angular velocity (ft) and of the total vorticity (211 + curl U) in the direction y' normal to the direction x' in which disturbances do not vary. To verify (50.1) note that
Srt= -0ex,
U=(e,,cosy+eysinZ)(1-z)
and use the geometry of Fig. 49.1. Numerical analysis of (49.5,6) gives solutions whenever IF >0 and not otherwise (see Table 51.1). Problem (49.5,6) contains plane Couette flow (IF=0) as a special case.
The determination of the wave speed im(a) of the most persistent small disturbance has a particularly simple solution. The answer is that a
im(o) _ - 2 sin(x -'') .
(50.2)
The argument leading to (50.2) starts with the observation that every solution of (49.5, 6) or the equivalent problem (49.2 a, b, c) has
-<J(50.4b)) to produce =1. 01 =acos(g-4).
(50.10)
alp
Show that when I - 4' =rr/2 the disturbance angle which minimizes R also maximizes Rayleigh's discriminant. Does a formula like (50.10) follow from (50.8)?
§ 51. The Energy Problem for Rotating Plane Couette Flow when Spiral Disturbances are Assumed from the Start A distinguished spiral direction (es) is found as a part of the solution of the linear stability problem. This direction is determined by the wave number which gives the largest critical value 1. for a neutral solution (re(o)=o) of the spectral problem. The spiral disturbance is also observed in experiments. It is, therefore,
reasonable to examine the consequence of assuming the preferred direction from the start. The aim here is a nonlinear analysis which takes advantage of the presumed spiral form for the disturbance.
It will be convenient to decompose the motion along the normal to the direction x' in which u, v, w and p do not vary; for example, u - u(y', z, t) U(z) = (1- z) [(cosy cos * + sin )C sin 0) ex. +(sin ;t cos 4 - sin $ cost)
=(1-z)[cos( -4')ex +sin(y-4t)ey.],
(51.1 a)
and
- ft =fUcos4'ex -C)ey. sin* =flex .
(51.1 b)
The governing equations for the x'-independent disturbances are
°t
-cos( - +})w=i OZ u,
+(u+U). Vv+(2(lcos4-sin(; -$))w= - 00-p +ISv ay
+usin$)= -
at where
2-a2
a2
O2 - ay 2 +3z2
0Z
+7.oZw,
(51.2 a)
(51.2 b)
(51.2 c)
VI. Global Stability of Spiral Couctte-Poiseuille Flows
192
and av
ey'
aw +-0. az
(51.2 d)
The boundary conditions are
u=0 at
z=0,1
(51.2 e)
and u is almost periodic its y'. The disturbance velocity component u cannot be driven by a disturbance pressure gradient because the assumption that du/ox'=0 implies that 02 p/c'x Z=0. Then O p/dx' = K(y', z) and since p is almost periodic in x', it is bounded as
x'2-, co and it follows that K-0. There are several consequences of the independence of p upon x'. One consequence is the existence of two energy identies: one governing the energy of the longitudinal component of velocity
i at+(21)sin* -cos(z-4+))(wu>=-7 ,
(51.3 a)
and one governing the evolution of the energy of the transverse components,
t<w2+v2>-sin(x-4)(wv>-2f)sin$<wu>=-'. 2
(51.3 b)
Eqs. (51.3) are the subject of analysis of this section. We form the sum (51.3 b) +A(51.3 a)
with t>0 and let
to obtain
dt <w2 + v2 + 02> - sin(Z - i) (wv> - Wcos(x - 0) <wq5>
-2f2sin*1_-V )<wc >= -7r . \V"
(51.4)
J
This is the equation governing the evolution of the disturbance energy when spiral disturbances are assumed from the start; it can be written as (51.5)
51. The Energy Problem for Rotating Plane Couette Flow
193
where
e=<w2+v2+02>/2,
-,;=-2flsin
(
-q = ,
-
and
Az=maxH2
(51.6)
where H2 is the set of x'-independent kinematically admissible vectors; that is, vectors a satisfying divu=0 and boundary and periodicity conditions. The energy inequality (51.5) integrates to
9(t)2A& for all ueH2 and provided that R 0 we cannot select a positive value of A which will make the coefficient of <w4,> in (51.6) vanish. Then the optimizing value for (51.8) 2=2
is sought as the root of 0
'OAAA
OA
195
51. The Energy Problem for Rotating Plane Couette Flow
Since u is a maximizing vector, c
(-
.
'1
z)
o(-fix)
1
1
f
2
_ 2fl s i n r{ 1 + ? J+cos(x- fir)
0. In contrast, if axial symmetry is not assumed from the start, the coincidence of the energy and linear limits occurs only
when 0=4,
52. Numerical and Experimental Results
197
Exercise 51.1: Show that the maximum value of F(x,$,Q) considered as a function of 0 is COS2(X-V), and
R_<max R(Z,O,I2)=K(sin( -O)) where R(r), r=sin(z-P) is given by the function R(t) whose values are found in Table 48.1. Exercise 51.2: Find the parabolic equation governed by
q(.v',z,t)=u(y,z,t)+{20sini,-cos(z-4i)iz, where u is any solution of (51.2a) satisfying the boundary conditions (see Exercise 40.2). Show that q(y',z,t) lies between the maximum and minimum values of [q(y , 0, t), q(Y ,1, t), q(Y , z, 0)]
§ 52. Numerical and Experimental Results In previous sections of this chapter we analyzed RPCF. This flow has been interpreted as a limiting case, r) - * 1, of spiral Couette flow between rotating and sliding cylinders. The main qualitative properties of spiral Couette flow depend only weakly on the relative gap size. A brief summary of some of the results which hold when q is unrestricted is given in this section. We first consider energy stability analysis for three-dimensional disturbances.
Some of the results are given in Fig. 52.1. The energy stability limit ft,x) depends only weakly on z and il. Numerical results reported by Joseph and Munson (1970) but not shown here show that there is a mean radius at which
s',=$ as was assumed by (44.7). An energy stability theory for x'-independent disturbances has not yet been given for general values of n. Linear stability limits for 'i =0.5 and n =0.8 are given in Hung, Joseph and Munson (1972). The instability limit RL, the wave speed im(t) and the azimuthal periodicity >7 of the spiral disturbance are important experimental observables. The number of vortices seen in any given experiment should correspond to the number of zeros (2n) of cosnO of the eigenfunction belonging to RL. We shall now consider experimental results for spiral flow between closely spaced cylinders. We compare these experiments with theory for rotating plane Couette flow. Here the linear theory gives instability when F >0 and stability when F -,;O. The equality may be thought to represent the situation in which either the angular momentum of the basic spiral flow is constant in planes parallel to the plane of the disturbance, or it can represent plane Couette flow in a non-rotating system fl = F =0. The condition F Rj are sufficiently representative of physically realizable initial conditions. It would appear from experiments that even if such energetic disturbances are realizable, they are globally stable and eventually decay. where the surfaces 6=const. Exercise 52.1: Define a system of orthogonal spiral coordinates are swept out by radial vector e,(t) which rotates and translates at constant speed. Write the IBVP for disturbances of spiral now in these coordinates. Assume that along one spiral direction the disturbance does not vary. Form two energy identities for these spiral disturbances.
Chapter VII
Global Stability of the Flow between Concentric Rotating Spheres
In this chapter we shall consider the steady laminar motion of an incompressible viscous fluid contained between two concentric spheres which rotate about a common axis with fixed, but different, angular velocities (see Fig. 53.1). Some results are given for the flow between eccentric spheres (inner sphere displaced along the axis of rotation).
Fig. 53.1 Parameters for the flow
between concentric spheres
§ 53. Flow and Stability of Flow between Spheres (a) Basic Flow
The basic laminar flow in a spherical annulus is a function of two spatial variables, r and 0, and in non-dimensional form is strongly dependent on the Reynolds number. This Reynolds number dependence makes the study of flow in a spherical annulus different from, and perhaps more typical than, problems like spiral flow between cylinders which are independent of Reynolds number. Since there is no simple solution for flow between spheres, various approximate solutions, valid over various ranges of the parameters, have been obtained.
53. Flow and stability of Flow between Spheres
203
These include (1) analytic perturbation solutions for small Reynolds numbers (Haberman, 1962; Ovseenko, 1963; and Munson and Joseph, 1971A), (2) Galerkin-
type solutions for moderate Reynolds numbers (Munson and Joseph, 1971A), (3) finite-difference numerical solutions of the non-linear partial differential equations (C. Pearson, 1967; Greenspan, 1975) and (4) singular perturbation solutions (boundary layer, inviscid core) for large Reynolds numbers (Proudman, 1956). We stress the first two methods since they show most of the phenomena associated with flow in a sperical annulus and are in a form that is convenient for the stability analysis. The basic flow is obtained from the governing Navier-Stokes equations which can be written in terms of a stream function in the meridian plane, 4i, and an angular velocity function, Q, as follows (Rosenhead, 1963, p. 131)
_ 10e-000r r2sin6 r3sin20
+
1 DzK2
'
R
[S2,rcos0-0e sin 0] -
rzsin0
['(I5)
2Dz 1i
1
r3 sin 'O
R
D4
(53.1)
where a2
1
D2ar2+_r2
a2
a
1
ae2r2eoto00
The non-dimensional velocities are related to 0 and 0 as follows:
Ur2sin0'
UB-
ao/ar, rsin0
_=
UO
s rsinO'
(53.2a)
The fluid volume is 'V_ {r,0,0:rl
(53.4 a)
where H is the collection of smooth solenoidal vectors u which vanish on the boundary sphere, VU is the velocity gradient of the basic flow and < > is a volume-averaged integral over the spherical annulus; 0.7, or so). According to Munson and Menguturk (1975) the bifurcating flow which replaces the basic flow appears to be steady when R is slightly above the first critical value .observed in the experiments. For Reynolds numbers above the first critical value, various other modes of flow are observed for the narrow gap case. Zierep and Sawatzki (1970) catalog these flow configurations for two different narrow gap situations (n=0.95 and n = 0.848) and present photographs of the flow. A somewhat similar situation for flow between rotating cylinders is well documented by Coles (1965). Munson and Menguturk (1975) note that the various transitions in the flow are easily observed not only by flow visualization techniques, but also by the
fact that the torque characteristics change quite noticeably when the gap is narrow. If the gap is wide, the transitions are much different and more difficult to observe. It has been noted that the flow in a wide gap is considerably different from that
in a narrow gap. In fact, several persons have studied the flow in a wide gap experimentally and were unable to detect any transitions by either torque measurements or flow visualization techniques. The apparent lack of transitions is due to the fact that the changes in the flow that occur due to a transition in the flow become much smaller as the gap size increases. Careful experiments (Munson and Menguturk, 1975) suggest that transitions in the flow do occur as the Reynolds number is increased, but their effects are not nearly as large for flow in a wide gap. A series of experiments using the apparatus shown in Fig. 53.9 for n=0.44 showed that four break points (or transitions in the flow) occur between Reynolds numbers corresponding to the basic laminar flow and those for which complete
turbulence was obtained. It was found that the flow in the wide gap is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy theory, R.. The experimental torque results (m = m.(R)) for n = 0.44 as shown in Fig. 53.10 seem to indicate a "crisis free" (no transition) situation since the curve appears
214
VII. Global Stability of the Flow between Concentric Rotating Spheres
to be smooth. However, upon a closer look at the torque data, several transitions are observed. Fig. 53.11 is a greatly enlarged portion of the torque curve in the
neighborhood of R=540, the second transition point for this flow. Although the change in the torque curve is not nearly as large as those for flow in a narrow gap careful experiments show that the data is repeatable and free of hysteresis. The first and third break points, occurring at R = 290 and R = 900 for this radius ratio (n =0.44), produce similar small changes in the torque characteristics. The fourth and more pronounced, break occurs at R=2100, at which point the flow suddenly becomes turbulent. 0.50
q = 0.44
P=O
0.45
I
_
I
I
I
i
I
I
600
500
R
Fig. 53.11: Magnification of the response curve of Fig. 53.10 in the neighborhood of the second point of transition
According to energy theory (Munson and Joseph, 1971B), flow in a spherical annulus is stable to any disturbance (large or small) if. R < Rp 310 when the tj=0.44 and µ=0. (This value is obtained by an extrapolation from the known
results for n=0.5.) The close agreement between the critical value R,r 310 of energy theory and the first break point, R=290, is noteworthy. Since linear theory indicates that RL : 1300 when a principle of exchange of stability is assumed, we may tentatively conclude that transition from the basic laminar flow is subcritical and, therefore, is caused by finite disturbances under conditions for which infinitesimal disturbances decay. Careful observation of the flow field by using aluminum flakes suspended in the fluid was carried out in order to observe the nature of the various transitions in the flow. No instabilities were observed for Reynolds numbers near the first break point (R = 290) of the torque curve. It is possible that the flow following this instability may occur in a form similar to the secondary motion of the basic flow and thus be indistinguishable from the basic flow, although it would produce a break in the torque curve. At Reynolds numbers corresponding to the second break point (R = 540), an instability appears in the form of small spots or puffs of turbulence. These
53, Flow and Stability of Flow between Spheres
215
Fig. 53.12: Taylor vortices between rotating spheres. The inner, aluminum sphere (a=4.118 inches) is rotating and the outer, plexiglass sphere (b=4.634 inches) is stationary, a2ro1/v=1078 (MoralesGome?, 1974)
Fig. 53.13: Wavy Taylor vortices between rotating spheres. The axisymmetric Taylor vortices lose stability to wavy vortices when a2w1/vx1400. In this photo a2w1/v=2696 (Morales-Gomez, 1974)
VII. Global Stability of the Flow between Concentric Rotating Spheres
216
turbulent spots occur at the center of the secondary basic flow swirl and rotate about the axis of rotation at a rate corresponding to the angular velocity of the primary flow at that location. At Reynolds numbers corresponding to the third break point (R = 900) a slight waviness or unsteadiness is observed near the equator. This unsteadiness increases and spreads toward the poles as the Reynolds number is increased. Finally, for Reynolds numbers corresponding to the fourth break point (R=2100) the flow becomes completely turbulent. For the first three break points the change in the dimensionless torque occurs as a decrease in m, whereas the fourth break (transition to turbulence) occurs as an increase in A. Munson and Menguturk note that the magnitude of these breaks in the torque curve for n=0.44 is considerably smaller than those which occur for larger values of n.
Thus the transitions involved in the wide gap appear to have a different character from those in the narrow gap (in particular there are no Taylor type vortices near the equator) and the instability is initiated as a subcritical one near the critical value of the energy theory. A summary of the various theoretical and experimental results is shown in Fig. 53.8. Munson and Menguturk (1976) have obtained interesting stability results for the flow between eccentric rotating spheres. This flow was described briefly in
subsection (a). The experiments of Munson and Menguturk indicate that the stability of the flow in the eccentric spheres is dependent upon the magnitude of the eccentricity and the radius ratio. When the gap is narrow, the Reynolds number for the initial transition of the basic flow (from the original laminar flow to Taylor type vortices) is insensitive to the amount of eccentricity. This result is not unexpected since the region near the equator (in which the flow first becomes unstable is similar to rotating cylinders whether the spheres are concentric or eccentric (see Figs. 53.12, 0). When the gap is wide the various transitions depend strongly upon the eccentricity. This dependence on eccentricity is dramatically illustrated in Fig. 53.5. This figure shows that the flow in the region where the gap is small is laminar whereas the flow in the region where the gap is large is turbulent. The two flows are separated by a non-propagating laminar-turbulent interface. Exercise 53.1: Consider the possibly unsteady axisymmetric motion of an incompressible viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities. Let (r, 0, 0) be polar spherical coordinates. The velocity components and the pressure are independent of 0 but are otherwise arbitrary. Show that the total angular momentum rU,sinO is bounded above and below by the maximum and minimum values which this quantity takes on in the annulus at t=0 and at the boundary of the spheres for t>O.
Exercise 53.2 (Munson and Joseph, 1971A): Consider the series solution (53.2b). Show that to leading order U, - Q0(r,O)/r,
sin fl = sin 0 f (r)/r
where
f(r)=Ar+ Br, b3a3(W1a -WZb) -h3
-a'
--
(53.6)
53. Flow and Stability of Flow between Spheres
217
and f02b4-o 1a4
B=
-b3-a3.
Formulate and solve the perturbation problem for 0,(r,0). Exercise 53.3 (Rayleigh's discriminant): Define cylindrical coordinates (P, z,
"r =rsinO,
z =rcosO,
in the sphere. Show that instability according to the criterion of Rayleigh's discriminant is given by Ub) s=0,
and since ¢' is now sufficiently arbitrary (B 3.4)
where we have put u, n is = 0 and on S,. Eq. (B 3.4) expresses natural boundary conditions; natural in the sense that the kinematically admissible competitors uE H for the minimum (B 1.3) need not satisfy (B 3.4), but the winner u=u, does need to satisfy (B 3.4). In sum, the eigenvalue problem which is equivalent to (B 1.3) is
V2u+Au+Vp=O,
divu=0,
(B 3.5)
u1s,=0
(B 3.6 a)
and (B 3.6 b)
In the same way but more easily, one finds that
v20+A0=0,
(B 3.7)
pls,=0
(B 3.8 a)
and
n+ fh8ls-s, =0,
(B 3.8 b)
is equivalent to the problem (B 1.4). The eigenvalue problem (B 3.5, 6) can also be derived using fundamental lem-
ma 1 instead of lemma 2. But then the components of u=u, +$' must be in H, and it may be true that div u = div O :A 0 (or equivalently, _ - «. Vp> # 0
for arbitrary peC'(-r)). It is then natural to seek the stationary value of A[u, p;
*.., A] = 2 + s - 2 (B 3.9)
226
Appendix B. Variational Problems for the Decay Constants and the Stability Limit
for arbitrary preassigned pEC'(Y1) and u;=u;+E4,eH. The vector a(p) which makes
[u+E0,p;Y',A]=0
(B 3.10)
when e =O will not, in general, be solenoidal, but we can select p=p such that and A[a, P; Y', A] =AI [u,Y'', it]. In seeking a solution u for (B 3.10) and a p such that V a = 0, we come back to the original problem. In the problem (B 1.5), it is necessary that the competitors 0 for the minimum satisfy an orthogonality condition =0. Attempting to proceed as before, we set 0 = 02 + eo a H, where 02 EH solves (B 1.5) and 0 e H is arbitrary. But 0
cannot be arbitrary because ==O. Then, the differentiation of A2(02+EC 'Y',A) with respect to sat a=0 leads to the expression
-«(V202+A262)> + (0
(a02
+ thO2))
s =0
but since 0 is not arbitrary, we cannot use lemma 1. To resolve the problem, we let 0=62+E4'eH and do not require =0
but only that =0. Then ¢ is arbitrary but, of course, the condition =0 must be dropped and (B1.5) must be replaced with a different but equivalent problem. In the same spirit that the arbitrary function p was introduced above,. one here introduces the real (Lagrange) multiplier C, and seeks the minimum of the functional { + it j
B4. Domain and Transfer Constant (A) Dependence of the Decay Constants
227
since the addition of a constraint could only raise (not lower) the minimum. In bounded domains and for periodic functions in cylinders and strips, it is known that this process of raising the eigenvalue does not get "stuck", so that the eigenvalues tend to infinity with the addition of more and more orthogonality conditions (there is no accumulation point at any finite point of the spectrum). This implies that the eigenvalues are discrete and denumerable, and that there are at most a finite number of eigenvectors for each eigenvalue. The eigenvectors can all be made orthonormal, and the resulting set of orthonormal vectors is complete. Exercise B 3.1 (Galdi, 1975): Show that the minimum value of the ratio it [A, 1]=JI 012dz/!/ ()2dz
among functions for which 0(1)=0(0)=0 is tr2/12. Show that infA[0,:oo]=0 among functions for which 0(0)=A(ro)=0.
Hint: Consider the function e-2p:-e`s= with J3>0.
B 4. Domain and Transfer Constant (A) Dependence of the Decay Constants First we shall show that the membrane eigenvalues increase with A at a decreasing rate (see Fig. B. 2). Consider the principal eigenvalue i1, =A expressed variationally A(A) = mines 1 +
Aj+
111 + (Bl'' acon)
(B 4.6)
S
+A
((2)
-
+2At"0 for which A may increase. Exercise B4.3: Let u=0 and 0=0 on S. Note that 2
(B 5.4)
over functions 0=0o+srl, where s and the Lagrange multiplier "a" are real numbers, 0 vanishes at the boundary and 0o solves the minimum problem (B 5.3). Of course =0, but this property is not necessarily shared by 7, which is now sufficiently arbitrary for the fundamental lemma I to be applied to the relation 1). Alternately, one can prove this same result directly in the strip of unit height.
Here, an acceptable periodic solution of V2 P +A P =O, 'P=0 at z= 0,1 is W= sinirzcosaxcos/3y, A=1r2+a2+/32. Moreover, this solution is orthogonal to every function of z alone on the period cell P(a, J1). In the case when the boundaries of the strip are free, one cannot even guarantee the existence of a nonzero decay constant. Here,
lim,.t (1-)2A1(j) =0. In the unit strip one finds that P= cos ax cos /3y solves V 2'P+ (x2 + f 2) T = 0 and
OW/Oz=0 at z=0,1. Hence, both n2(I'=O at z =0,1) and 0(8'P/8z=0 at z=0,1) are limiting eigenvalues of acceptable eigensolutions when a2+#2 -4 0. The eigensolutions which we have found are such that ( =0 (ut = 0), so that the radial velocity component is identically zero. However, one can find velocity fields having small nonzero values of ul, i. e., JP,., 0) u2 = k; -40, where P(a,11) is
the period cell of the last paragraph, and such that the ratio 2 JP(5,p) D [u]: DLu]/fp(, 1) Jul'
differs by terms of order r from the minimum value of this ratio which is attained when a=0. This means that we can get stronger values for At only among fields with no vertical vorticity (u2=0). Such fields sometimes occur in stability problems (see Chapters IX and X).
B 8. The Energy Eigenvalue Problem We shall now leave the problem of the decay constant and turn to the problem of the energy stability limit. This limit is defined by (4.6) as
v.=max,
(here uls=0).
(B 8.1)
When written in variables made dimensionless by introducing a length scale I and a velocity scale Um, (B 8.1) becomes
R,
=maxa(-/)
(B 8.2)
where the same notation has been used for dimensionless and dimensional variables.
B8. The Energy Eigenvalue Problem
241
We want to convert (B 8.2) into an eigenvalue problem, to characterize the set of eigenvalues with respect to completeness and to show that R. defined by (B 8.2) can also be found as the principal eigenvalue of a partial differential equation. Again keeping in mind problems of convection where the boundary conditions may be more complicated, we shall consider a slightly more general problem than (B 8.2), i. e.
= max, (f/s)
P
(B 8.3)
where
f = -,
.9 =+.4(hIuI2>S_S1
and H is the linear vector space defined by (B 1.6). Suppose that the maximum of (B 8.3) is attained when u =u. Consider the
is an arbitrary admissible value of f/9 when u; = ut + an, where n, = u` as t_o vector (it satisfies Eq. B 8.1). For each fixed m we have 1
.f (a)
P(a)
-9(a)
(B 8.4)
Clearly 1/p(a) is a maximum when s=O and d E
de
0(p(a)-I)=P(0)
dl
d2
as - as
=0.
(B 8.5)
Using (B 8.3) we may write (B 8.5) as
)+A(huu} =0.
p(u D[U]
(B8.6)
Here all quantities are evaluated at P=O and we have used the symmetry of D to write
KDUi.u)=(u.D[U1.). Eq. (B 8.6) may be regarded as Euler's functional equation. It holds for every kinematically admissible vector aau/as. To convert this equation into an eigenvalue problem for a system of differential equations, we note that
2\D[u]:GDau])
2
0 `D [ii] de}) -(zu'OZa)>
242
Appendix B. Variational Problems for the Decay Constants and the Stability Limit
and apply the divergence theorem and the fundamental lemma 2 of the calculus of variations (Section B 2). This leads one to the equation
u-D[U] -
V u=-Op
(B 8.7 a)
and the natural boundary conditions (B8.7b)
The Euler eigenvalue problem is defined by (B 8.7 a, b) and the admissibility conditions
u1s,=0,
divu=0.
(B8.7c,d,e)
B9. The Eigenvalue Problem and the Maximum Problem Let there be two different eigensolutions u, and u,, belonging to different eigenvalues p, 36 pr of the same Euler eigenvalue problem (B 8.7). By comparing the problem satisfied by the two different solutions it is easy to show that (P, - pr) 1 eigenfunctions (p is an eigenvalue of multiplicity n), then these eigenfunctions can be formed into an orthogonal set. Hence, eigenfunctions of finite multiplicity may be assumed to satisfy orthogonality conditions.
It is easily shown that if u, is an eigensolution with eigenvalue pl, then (no summation over 1)
-
i
(B 9.3)
2+A2
Choose A so as to minimize this expression, namely set
A =(J (t) _f, 3(t)/3) 0 cannot have a positive maximum; it follows that y(x)>0 for all x>a. Typically, the stability problems are of the form .N20+%t f(x)w=0,
(D.9a)
/1,22w-14S(x)0=0
(D.9b)
O = w=Dw =OIx=a,6
(D.9 c)
This problem is equivalent to the pair of integral equations 0 = - Ab G2(x, xo) f (xo) w(xo) dxo
and w=.? .V G4.(x,xo)5z(xo)8(xo)dxo
.
These equations are equivalent to either one of the pair of equations
6= -A2 5b Kl(x,xl)p(xl)0(xl)dxl or
w= -12 J.' K2(x, xl) /(xi) w(xl)dxl where
K1(x,xl) _ .V/(xo)G2(x,xo)G4(xo,xl)dxo
(D.10a)
and
K2(x,xl) _
5s(xo)G4.(x,x0)G2(x0,xl)dx0
(D.10b)
We would like next to apply the theorem concerning (D.1) to the problems (D.9a,b). To do this we need to show that Kl and K2 are oscillation kernels and, of course, we require that /(x) and(x) are nonnegative on (a,b). That K 1 and K.
are oscillation kernels follows (essentially from the fact that the product of oscillation matrices is again an oscillation matrix (Gantmacher and Krein, 1960, pg. 86). The composition Kl or K2 of oscillatory Green functions is again
Appendix D. Oscillation Kernels
255
oscillatory (see Ivanilov and Iakovlev, 1966, or Karlin, 1968). Hence, the nice spectral properties apply for 2, w and 0 of Eq. (D. 9) if /and g are nonnegative.
Among the problems which may be reduced to integral equations with oscillation kernels are: Poiseuille flow problems: u, f satisfying (21.5 a, b) and
u= f =Df =0 at r=n(1-q)-t and
(D.11)
u,f satisfying (22.4a,b) and (D.11) when DU is negative. (b) Couette flow problems:
f, v satisfying (39.2) when K=n=0 and B>0, A0.
rZV'
(c) Rotating Poiseuille flow problemsf, u satisfying (46.3) when Co = 0. (d) The generalized Benard problem for convection in a porous container: X, satisfying (73.8) when .f and G' are positive. satisfying (73.9) when f and G' are positive X*, (e) The Benard problem in a spherical annulus: ll, satisfying (74.11) and (74.13) when g(r) and e(r) are positive, Y
satisfying (74.12) and (74.13) when g .'{r + / e(r) is positive.
(f) The generalized Benard problem: y,
V
satisfying (75.13, 15) when 1/+j+j/ (z)>0,
z, B satisfaing (75.14,15) when 1 +y'g (z) > 0;
(g) The Benard problem in a spherical cap of porous material: 9* satisfying (74.31) when 8(r) and g(r) are positive. X, 9,
Appendix E
Some .Aspects of the Theory of Stability of Nearly Parallel Flow
There are strictly parallel flows (for example, Poiseuille-Couette flow down a sliding annulus) and "nearly" parallel flows. "Nearly" parallel flows are those for which the derivatives along the "streaming' axis are small relative to the transverse (r) derivatives. Flow in a boundary layer or jet is an example of nearly parallel flow.
It is customary to treat nearly parallel flows as if they were exactly parallel. There is no strict justification for this procedure and it is almost certainly not justified in all cases. However, the instability results which follow upon treating nearly parallel flow as strictly parallel are sometimes in good agreement with experiment. Since so many nearly parallel motions might occur, it is natural to study the stability problem for arbitrarily given "parallel" flows U(r). This leads to the study of the Orr-Sommerfeld (OS) problem in the cylindrical annulus. The theory of the OS equation relies on asymptotic analysis and is very complicated
(see Lin, 1955 for the older references; for more recent work see Reid, 1965); asymptotic methods were once used in computations but now computations are more easily, quickly and accurately done by numerical methods. It is possible however, to obtain delicate properties of solutions to the OS problem from extremely simple arguments. In § E1 we give these arguments and show how they are supported by numerical analysis of the exact problem. The most important question to ask of the theory of arbitrarily given parallel flow is not how solutions behave but whether and in what sense these solutions are valid approximations to the problems they are claimed to represent. This question is discussed in § E 2.
§ E 1. Orr-Sommerfeld Theory in a Cylindrical Annulus For some parallel motions the critical Reynolds number of linear theory is very high, even infinite. When R is large we expect the flow field to divide into inviscid
and viscous regions. In the viscous regions, like those in the neighborhood of solid boundaries, the velocity must change rapidly and derivatives of velocity are so large they cannot be neglected even when they are divided by the large Reynolds number.
257
E1. Orr-Sommerfeld Theory in a Cylindrical Annulus
Linear stability theory leads to such a division into inviscid and viscous regions. The structure of the eigensolutions is so delicately balanced in this division that even slight changes in the problem can induce big changes in the linear stability results. In contrast, slight changes in the problem do not strongly effect the results of energy analysis (Carmi, 1969). The hypersensitivity of the linear stability limit to slight changes in the velocity profile is related to the fact that the high critical Reynolds numbers are usually not observed. Naturally, if the limit for stability to small disturbances is large, it is more likely that the nonlinear terms in the disturbance equations will lead to instability under subcritical conditions. To understand the delicate balance between viscous layers and the inviscid interior we shall study the Orr-Sommerfeld problem (34.6 a). Though a complete analysis of this problem is beyond the scope of this work some of the more important properties of solutions can be determined by elementary analysis. This 1
analysis starts from equation (34.6a) with R = 2R and w = - - a'Y'/ax, r
i(U-c)w'w-i!I' wI2=(aR)-1w92w
(E1.1)
-M 'c=Cr+ic;=Ufia a (r- r ri)- 1 and b=(1
where IF(r) is defined by (44.29a), <e = r
and w and Dw vanish at a =r(1 -
-
-Jri) -1. The basic variable of
this analysis is the Reynolds stress distribution. a =_ - i [wDw -wDw] .
(E 1.2)
This variable arises naturally when the complex conjugate of (E 1.1) is added to (E 1.1). This addition leads to (44.26a, b). At the border of stability and instability c;=0, and (44.26a) may be written as 1
_
,7R
'-z /
(E 1.3)
.
The local energy transfer between the basic and disturbed motion depends on the sign of rDU; when positive, energy is transferred from the basic motion to the disturbance motion. z vanishes strongly at a rigid wall (see E1.1). We wish to determine the distribution of -r(r) across the annulus. Since neutral disturbances are generally found only when aR is large it is reasonable to assume
that a small disturbance of the basic shear flow U(r) is strongly influenced by shearing forces only in boundary layers where steep gradients are required to bring the velocity to zero. One of the main mathematical difficulties in the theory
of the Orr-Sommerfeld equation is that this assumption is not quite correct; there is a second type of layer in which the viscosity is important. The second type of viscous layer, which is called a critical layer, occurs at each root r, a < r [rr,]=0.
(E1.8)
E1. Orr-Sommerfeld Theory in a Cylindrical Annulus
259
Suppose that U(r) is a monotone profile. Then there can be only one value of for which U(r)-cr=0. The jump formula (E1.7) then shows that a neutral eigensolution of Rayleigh s problem can exist only if Y',=0 (the possibility that [wr]? =0 can be excluded by the argument given in Exercise (E 1.1). For plane parallel flow the same conclusion about monotone velocity holds; in this case,
however, the condition 'I',=0 reduces to Rayleigh's inflection point criterion U' =0. Rayleigh's problem for symmetric profiles in a channel can be shown to be equivalent to the monotone case and the condition U" =0 is necessary and also sufficient (Tollmien, 1935) for the existence of neutral eigensolutions of Rayleigh's problem.
Flows which are unstable by the criteria of Rayleigh's inviscid theory are relatively unstable by the criteria of the full Orr-Sommerfeld theory. This relative instability is particularly evident in the stability properties of the flow as R-sce. When Rayleigh's instability criterion is satisfied there is a band of wave numbers a for which the basic flow is judged unstable by the Orr-Sommerfeld theory. On
the other hand, if Rayleigh's criterion is not satisfied the basic flow is always judged stable by the Orr-Sommerfeld theory in the limit aR->oo. The relative instability of flows satisfying Rayleigh's instability criterion is reflected in an appreciable lowering of the critical Reynolds numbers as is indicated in Fig. E 1.1.
Fig. E1.1: Schematic sketch comparing the neutral curves of two flows; one flow satisfies the inviscid instabil-
ity criterion of Y'(r,)=0 of Rayleigh, the other low !P(r,.)00 is judged stable by Rayleigh's inviscid theory
The lowering of the critical Reynolds number when a vorticity maximum is introduced in the basic flow can be seen in the study of retarded boundary layer flows by Obremski, Morkovin and Landahl (1969). The critical Reynolds number of the Orr-Sommerfeld theory is very sensitive to changes in velocity profile. Small changes of U(r) in round pipes (Gill, 1965) and in channels (Fu and Joseph, 1970) can lead to large changes in the critical Reynolds number. It must be remembered, however, that arbitrary profiles U(r) are not steady solutions of the full Navier-Stokes equations. Though such profiles could arise locally as a perturbation of some basic shear flow due to, say, a long bump on the annulus wall, it would not be possible to maintain this perturbation further downstream; local instability produced in one part of the annulus would disappear further downstream where the basic flow is again of a stable type. In the absence of a maximum of the weighted vorticity (E 1.8) must hold. The vanishing of the sum of the jumps of the Reynolds stress at each critical
layer is a more stringent condition than one would expect to hold generally. For example, the two jumps which would occur when U(r,) = cr in plane Poiseuelle flow could not sum to zero. Thus, at first glance, the inviscid analysis leading to (E 1.7) would appear to have only a limited applicability to the Orr-Sommerfeld problem.
260
Appendix E. Some Aspects of the Theory of Stability of Nearly Parallel Flow
The conclusion just reached goes too far; the reason is that the viscous boundary
layers have been ignored; the wall layers imply that (E1.7) could not hold very near the wall and (E1.8) cannot then be deduced from (E1.7). Considerable insight into the Orr-Sommerfeld problem at finite Reynolds numbers follows from (E t.7) and an elementare analysis of the wall boundary layer given below. Consider a velocity profile U(r)>0, U(a)=U(b)=0 (see Fig. E1.3a). Imagine that there is an inviscid neutral solution in the interior of the annulus and that the instability wave moves forward with velocity c, where 0 < c, < max U(r),
a 0 near the inner and outer wall-
It follows that the energy supply near the boundaries is positive. This important result was first found by C. C. Lin (1954) by a somewhat different analysis from
E1. Orr-Sommerfeld Theory in a Cylindrical Annulus
261
10
Fig. E1,2: The disturbance boundary-layer analysis and the jump condition for the Reynolds stress imply cona ditions (E1.13). These conditions do not determine the sign or magnitude of the Reynolds stress in the inviscid center. If the sign of the Reynolds stress is assumed to be positive, we have the situation sketched in (a) whereas a negative Reynolds stress implies the sketch in (b). Both distributions occur (see Figs. E1.4 and E1,5). A good qualitative picture of these distributions (which were calculated numerically by finite differences) also follows upon completion of the sketches of Fig. E1.2 as smooth curves in the most obvious way b
the one just given. Lin's result was motivated by Prandtl's conjecture: by promoting a positive energy supply at the wall viscosity can have a destabilizing effect.
The delicately balanced structure of solutions to the Orr-Sommerfeld equation at finite Reynolds numbers, to which we alluded at the beginning of this section,
is implied in the inviscid jump formula (E1.7) and the wall Reynolds stress results (E 1.12), We have found that for profiles U(r)>,O which increase from zero at the inner wall and decrease to zero at the outer wall (i) a > 0 near the inner wall and r 0.
Exercise E1.3: Evaluate (E1.5) for Hagen-Poiseuille flow and plane Couette flow. Construct an argument to support the conclusion that these two flows are absolutely stable to infinitesimal disturbances.
264
Appendix E. Some Aspects of the Theory of Stability of Nearly Parallel Flow
§ E 2. Stability and Bifurcation of Nearly Parallel Flows It is very hard to decide about the validity of the assumption that the stability nearly parallel flow may be studied through the OS equation. Given the mathematical difficulty involved in studying stability problems governed by nonseparable partial differential equations, it is perhaps natural to seek justification through comparison with experiments. In fact the comparisons are often indecisive and the question of validity is left unanswered. The difficulty in assessing validity through experiment is compounded by nonlinear effects; even in strictly parallel
shear flows there is no apparent agreement with experiment because the bifurcating solution is unstable and the stable solutions are not "close" to eigenfunctions of the OS equation. Thus the problem of validity involves discussion of the parallel flow approximation in the linear and nonlinear theory. Some persons believe that it is possible to take advantage of the basic directionality of nearly parallel flow by a perturbation method based on the notion of two space scales. Formal theories of this type have been given by Bouthier (1972, 1973) and Ling and Reynolds (1973) for the linear part of the problem and by Joseph (1974 B) for the linear and bifurcation part of the problem. The existence of two space scales in nearly parallel flow is motivated by the boundary layer equations. In two-dimensions these equations are in the form: UU.'U+VeyU= voyyU ,
(%U+ayV=o . Solutions of these equations are often in the form U = U(vx, Y) ,
V=vV(vx, y) .
In the study of the stability of these solutions the dependence of the nearly parallel basic flow on the axial variable x is always in the form vx. In the two scale method X = vx and x are treated as independent variables; x is a "slow" variable if v is small. The choice of a proper zeroth order in a bifurcation theory for nearly parallel
flow is very important. The earliest mathematical study of the effects of nonparallelism (Lanchon and Eckhaus, 1964) already indicated that though the use of the OS theory at zeroth order is valid for the Blasuis boundary layer, this same approximation could not be expected to give correctly the linear stability limit for flows like those in jets. Even earlier, Tatsumi and Kakutani (1958) noted that the OS theory might not apply to jets. If a flow is not well represented at the zeroth order, it cannot be approximated by perturbations. The problem of the correct zeroth order is basic in developing a perturbation theory which will apply equally to flows in boundary layers and jets. Haaland (1972) has noted that the difference between flows of the boundary layer type and flows of the jet type can be characterized by the behaviour of the velocity component V at large distances from the axis y = 0 of the basic flow. The boundary layer grows by the diffusion of vorticity and does not require inflow from infinity.
265
E2. Stability and Bifurcation of Nearly Parallel Flows
On the other hand, the conservation of the axial momentum of the jet together with the slowing of the jet with distance x downstream requires the entrainment of new fluid. The spreading of the jet implies a non-zero inflow (V00) at infinity. In his study of the linear theory of stability of nearly parallel flows, Haaland modifies the Orr-Sommerfeld theory to include some of the effects of inflow. The effect of retaining these terms is to confine the vorticity of disturbances to the regions of the main flow where the flow is essentially irrotational. These inflow terms make a big difference in the critical Reynolds numbers especially when the wave numbers are small (see Fig. E2.1).
1.4
1.2
(a) Conventional Orr-Sommerfeld equation (b) Modified Orr-Somn?erfeld equation 0.2
10
20
30
40
R
Fig. E2.1: Neutral curves for the Shear layer and the Bickley jet (after Haaland, 1972).
A bifurcation theory for nearly parallel flow should therefore allow for a certain flexibility in the choice of the zeroth order. The bifurcation theory of Joseph (1974B) allows this flexibility of choice; the theory then corrects this zeroth
order for effects of nonlinear terms and of linear terms which are neglected at the zeroth order (the extra terms). We shall not give this formal bifurcation theory in detail here. The basic mathematical procedures are as follows: first, one introduces two scales x and x and in the equation o
o
0
ox
ox
0x'
at the zeroth order some terms containing v are retained and some are neglected. The second step then is to introduce a false parameter v replacing v wherever it is going to be set to zero at zeroth order. Having done this we may then introduce co and s as in the perturbation theory of Chapters Il and IV. At this point we have
a well-defined perturbation problem depending on parameters s and v. The
266
Appendix B. Some Aspects of the Theory of Stability of Nearly Parallel Flow
bifurcation problem, so defined, is not a true problem because the coefficients of the extra terms has been introduced artificially; some of the terms which like are neglected in the OS theory are kept at the zeroth order whilst the others appear in the theory at higher orders, as corrections. In the perturbation the parameters v(&, f) and w(e, v) are developed into a double power series along the lines set out in Chapter II. The solution of the false problem is supposed to solve the true problem when v(c,v)=v". This method is called the method of false problems. The technical details are in the paper by Joseph (1974B).
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Subject Index
Absolute stability 10 of Couette flow in the linearized approximation 197 - of Hagen-Poiseuille flow in the linearized approximation 120
- of Poiseuille flow to x,-independent distur-
supercritical
-, natural
bances 83 Almost periodic
1, 4, 5, 42
- symmetry breaking, into spiral vortices, see Taylor vortices Boundary conditions - at the origin, in cylindrical coordinates 73 - for toroidal and poloidal potentials 237 225
Busse's theorem on preferred disturbances
- disturbances 67, 70-71, 164-165
183-186
- functions and turbulence 101 - properties of functions 218--230 Amplitude equation 56, 147 Attracting radius 9, 31 -32, 91, 128 Attractors
-, non-periodic 60 - phase mixing 145 -, quasi-periodic 145,
-, strange
- cylinder
-, first, of energy theory 1, 11 -, first, of spectral problem 1, 27 -, first, of spectral problem for time-periodic
59
Auto-correlation function Average
Center-manifolds 59 Critical amplitude 148 Criticality 38 Critical viscosity 1, 10, 27
220
68, 94
horizontal 94 time-cylinder 97
- volume
7
Basic flows
1, 11, 21, 66, 159-161, 202-206,
solutions 27
-, second, of energy theory 19 -, second, for Poiseuille flow 42 - theorem, comparing values of 30 Decay constants 13, 15, 19 and Hadamard's formula 229
- in a box 261
monotonicity and convexity of, in the transfer constant h 228, 243
Bifurcation - curve 5, 35
- , direction of
- into tori
34
59
of circular Couette flow 143 of nearly parallel flow 264 - of Poiseuille flow 91, 106-119 -, one-sided 49 -, quasi-periodic 53, 59 solutions 32
- theory 4-6
-, time-periodic 34-43, 106-118
-, steady 48-51 -, subcritical
231
- in a spherical annulus 237 -, minimum problem for 221
4, 5, 41
- in the domain 229 Decay of stable disturbances 85 - of transverse components of a fully-developed disturbance 82 Dissipation 12, 164 Differentiation of stationary functionals with respect to parameters 230 Disturbances almost periodic 67
-, arbitrary initial 8, 10 - compared with fluctuations -, decay of stable 85
106
Subject Index
279
Disturbances
Experiments
- in entrance region of a pipe 85-89 -,initially increasing 3,15,20,24,69,156-
- on disturbances in the entrance of a pipe 85 -88 - on friction factors in turbulent pipe flow
157,180
- neutral
91-93, 125-126
26
-, nonaxisymmetric
118, 120, 142-143
-, small 2, 4, 25 -, spiral 140, 193-194
122
see Taylor vortices Domain dependence of eigenvalues
225, 229
Eigenvalue bounds for the Orr-Sommerfeld equation 168-170 Energy
-, evolution equation for 11, 16, 68
- functionals 3, 193 - method of Stuart 55 - of an irrotational disturbance - production integral 12, 13 - stability limit 10
- on the instability of Couette flow between cylinders 131, 136-138, 143-146 on the instability of spiral Couette flow 161, 197-201 on the instability of Poiseuille flow down a rotating pipe 174-177 - on the instability of flow between rotating spheres 210-216 Extremum problems - for bounding the pressure gradient in turbulent Poiseuille flow 101
- for circular Couette flow 132, 147, 150 - for decay constants 11, 14, 72, 221-223,
13
239
- stability theorems for basic flows
13, 14,
17
- supply at wall, Lin's formula - supply for parallel flow 260 - theory, history of 127-128
120-
- on the instability of Poiseuille flow
261
Equivalence of maximum problem and Euler eigenvalue problem in the computation of the energy stability limit 243 Euler eigenvalue problem for decay constants 224 - for energy theory of Poiseuille flow 73 - for energy theory of circular Couette flow
- for eigenvalues and critical values of the spectral problem for spiral flow 168-170 - for flow between rotating spheres 207 - for Poiseuille flow 69, 70, 72, 173 - for rotating plane Couette flow 179, 182, 194
- for spiral flow
163
- for the first critical viscosity of energy theory 15,241 - for the second critical viscosity of energy theory
17
- for the optimum energy 153,-193
134, 153
- for energy theory of spiral flow 163, 181, 195
- for energy theory of flow between rotating spheres 208
- for energy theory in functional form 241 - for energy theory in differential form 242 - in the calculus of variations 223-224,
242-243 Evolution equation for the energy of an arbitrary disturbance
- multipliers 28, 29, 44
- theory 27-30
Fluctuations 23, 92 Fredholm alternative, see solvability Frequencies of quasi-periodic functions
27, 54,
58
11
for the energy of a disturbance of Poiscuille flow
Factorization theorems for the stability of steady and time-periodic bifurcating solutions 45, 47, 50, 126 Floquet exponents 27, 28, 30, 44
69
- for the energy of an x,-independent disturbance of Poiseuille flow 81 for the energy of the fluctuations and the mean motion in turbulent Poiseuille flow 96
- for the energy of axisymmetric disturbances of circular Couette flow 147 for the energy of spiral disturbances of rotating plane Couette flow 192 Exchange of stability 27, 54
- of time-periodic bifurcating flow Friction factor discrepancy 108
125
- vs. Reynolds number 91-93 upper bounds 92, 123 Frictional resistance to flow in pipes 89 Fundamental lemmas of the calculus of variation 223
Generalized energy functionals
3
in circular Couette flow 40-57 - in rotating plane Couette flow 191-197
Subject Index
280
Generic properties of dynamical systems 59 Global
- and local analysis 2 - extensions of Hopfs local theorem see factorization stability and linearized stability
- stability theorems
two times and multiple scales 56, 264266
31
- Liapounov and Schmidt 56
10, 11, 17
Power series 56
- for Hagen-Poiseuille flow
- false problems
81
- of oscillation type 252-254 Hopfs bifurcation theorem 55 13, l4, 20, 30, 119, 132-133,
167-170, 244-258 Instability due to inflection points or viscosity
-
48 167,
- of bifurcating time-periodic flow 47, 50 - of bifurcating Poiscuille flow 125-126 - of circular Couette flow 136-138, 141-
of the spectral problem for circular Couette flow
143
Necessary and sufficient conditions for global stability 4, 15
- of parabolic Poiseuille flow 173 - of rotating plane Couette flow 188, 196
- of spiral flow
146
- of laminar Poiseuille flow 91-93, 111, 120 - 122
- of laminar flow between rotating spheres
208-217 - of parallel and nearly-parallel shear flow 257-263 - of rotating axisymmetric flow 138-141 - of rotating plane Couette flow 189 191 - of rotating Poiseuille flow 173-177 of spiral Couette flow 197-201 Kinematically admissible disturbances
266
Monotonicity and convexity of variational functionals 17, 102-103, 227-228 Monodromy matrix 28 Multi-a solution 60, 128 Multiple eigenvalucs
- in the stability of bifurcating solutions 44,
257-263 - of basic steady flows 4, 10, 25 27 - of basic time-periodic flows 4, 27-30 - of bifurcating steady flow 50-51
171
Neutral stability 26, 31, 261 - curves comparing flows with and without points of inflection 259 curves for boundary layers 263 - curves for plane Poiseuille flow 111 - curves for the shear layer and the Bickley jet 265 Nonuniqueness of flow 11, 33, 128, 131, 144145
Normal modes 70-71
3, 14,
15, 31, 222
Krein's criterion for oscillation Green functions 251
Laminar-turbulent comparison theorems 99t00 Limiting flows
155
- energy 2-4
5
Green functions
Inequalities
Mean motion 23 Mean-radius approximation Method of
Optimum stability boundaries
153-157,
191-196 Orr-Sommerfeld equation
111
-, eigenvalucs of 168 170 - in a cylindrical annulus 256 modified for nearly parallel-flows with inflow 265
11, 21
sensitivity to changes in the velocity proMaximum principle
152
- applied to oscillation kernels 253 - for disturbances of rotating plane Couette flow
197
- for the total angular momentum in circular Couette flow
152
- for the total angular momentum between rotating spheres 216 Maximum and minimum problem, see Extremum problems Marginal stability 26 see neutral stability
file
259
-, simplified analysis of 260 Orthogonality - and solvability 39, 49, 114, 228-229 - of eigenfunctions of Euler's energy stability problem 242
- of eigenfunctions of the membrane equation
226
- of the toroidal and poloidal potentials 237-
238
Oscillation kernels and Green functions 154, 251-255
80,
Subject Index
281
Reynolds number 23, 73, 94, 130, 161 Reynolds stresses in the linear theory of paral-
Phase mixing 60, 145 Poiscuille flow
- bifurcating, time-periodic
106-120
lel flow
- stability of laminar 64- 89, 120 -122 stability of rotating 173-177 stability of rotating parabolic
257
-,jump in 258
-, Hagen 64, 66, 71, 76- 78, 79, 120 - in the entrance region 85-88 parabolic 76-79 173
-, turbulent 90-109, 120-126 Power series
- for bifurcating Poiseuille flow 113 - for bifurcating, steady flow 49 - for bifurcating, time-periodic flow 35 - for laminar flow between rotating spheres 204
Lin's formula for 258 simplified criteria for obtaining the distribution of 261 Sign of the motion 50 Simple eigenvalue 33, 80, 252 Slow motions t 1 Snap-through instability 5, 53, 126 Solvability lemma for time-periodic bifurcating solutions 39, 114
- for steady bifurcating solutions 49 Spectral problem 26
- used by different authors to study bifurca-
-, adjoint
tion 56 Principal eigenvatues and eigenfunctions 27, 49, 80, 226 Principle of linearized stability, see conditional stability
adjoint for Poiseuille flow 27 - for circular Couette flow 142, 153
37
- for flow between rotating spheres 206 - for Poiseuille flow 109 - for rotating plane Couette flow 187
- for spiral flow 173 - for time-periodic flow 27
Quasi-periodic
Rayleigh's discrimininant 138-139
Spectrum of integral equations with oscillation kernels 252 Spiral Poiscuille-Couette flow 160 flow angles 159, 162
-, Coles form of
- flow directions
- manifold of solutions 2 - bifurcating solutions 6, 53-54, 58-60
139
effect of viscosity on
139
- for the stability Couette flow 141 - for the stability of flow between rotating spheres 217 - for the stability of rotating plane Couette flow
140
- for the stability of rotating Poiseuille flow 140
limits of applicability for 141, 143 Rayleighs inflection point criterion for the stability of shear flow 167, 259, 263 Recovery of stability 49-50 Repeated branching - in Couette flow between rotating cylinders 145
- in flow between rotating spheres 214 - in the theory of Ruelle and Takens 59, 145 - in the transition to turbulence 1 -4, 52 Response curves 5, 42
- for flow between rotating cylinders
100,
for flow between rotating spheres 212 - for flow between shearing planes 100
- for open channel flow 103 - for pipe flow 3, 90- 93, 100 - for thermal convection 3, 90 Reynold's discriminating equation
- Poiscuille-Couette flow
127
160
vortices 162 see Taylor vortices Stability
-, absolute, see absolute stability -, asymptotic 9 -, conditional 4, 9, 27, 30-32, 55, 147, 150
-, criteria 9, 10
- global -, limit
9, 10, 15 10
-, local and global theories of 2 -, monotonic 9, 15 -, orbital asymptotic 29 -, unconditional 9 Stability of
- basic flows 8, 15, 17 - bifurcating solutions
- steady
6
51
time-periodic
136, 138
161
flow directions which minimize subcritical instability 171
47
- boundary layers 61, 263 - circular Couette flow 130, 137, 142 - flow around bodies 61 - flow between rotating spheres 206-211 - nearly parallel flow 167, 264 - nearly rigid Couette flow 134-135
Siubject Inilex
282
- laminar comparison theorems 99-100
Stability of
- null solution - parallel flow
8, 15 256
- Poiseuille flow, see Poiseuille flow igid moti ons 16
-r
- rotating plane Couette flow 180-182,
-
186-189, 191-197
- spiral fl ow
107
-, selection principles for realized 128 -, sensitivity to initial values in 22 -, stable 86, 94-92,120, 125 , sta tistica lly station ary 71 , 90-1OS -, transition to, in pipes 23 -, transition to, through repeated branching
-
1-3, 59-63
201
- turbulence 92, 125 Statistical stationarity 90, 92, 97, 117
-, transition to, through snap-through instabilities 52-53, 59
Subcritical instability 5, 42
-, variational analysis of 3, 100-105, 127-
-,extent of
53
128
- in boundary layers 263 - in circular Couette flow
138, 141, 143
- in flow between rotating spheres 216 - in Poiscuille flow 42, 53, 58, 61, 91-93, - in rotating- Poiseuille flow
- in spiral flow
173
171
Synge's theorem about conditions for stability of circular Couette flow
Taylor number Taylor vortices
Uniqueness
- of bifurcating solutions
111, 118, 121
Squires theorem
Unbounded domains 20,24
150
84, 119
140
- between rotating cylinders 131 - between rotating spheres 215 - in double sets in rotating Poiseuille flow 176-177
33
- of flow of compressible fluids 24 - of solutions of the initial-value problem 21,24
-for inviscid fluids 24 - of steady, time-periodic and almost-periodic solutions at low Reynolds numbers 22, 24 Upper bounds for functionals of statistically stationary turbulence -, additional constraints 4 - for friction factors in Poiseuille flow in channels
92, 123
- for heat transport by convection
3, 90 - for mass-flux in plane Couette flow 103-
-, non-uniqueness of 145 -, spiral vortices 140, 162, 197-199
- for momentum transport in plane Couette
Toroidal and poloidal potentials 209 -, boundary conditions.for 237
- for torque in Couette flow between
- in a bounded domain 236 - in a period cell 236 - in a spherical annulus 234 Turbulence -, almost periodic
101
- between rotating cylinders 145 - between rotating spheres 214, 216
-, flashes of 88
105
flow
100, 103
cylinders 100
Variational theory of turbulence 128, 138 Velte's method of weakened constraints 231 Wave number selection three* staibiity Weinbergas theorem 59
144
