J Eng Math (2007) 57:205217

DOI 10.1007/s10665-006-9096-4
ORI GI NAL PAPER
Nonlinear critical layers in the boundary layer
on a rotating disk
J. S. B. Gajjar
Received: 5 May 2006 / Accepted: 11 September 2006 / Published online: 10 November 2006
Springer Science+Business Media B.V. 2006
Abstract The work of Gregory, Stuart and Walker (1955, Proc R Soc Ser A 406:93106) and Hall (1986,
Phil Trans R Soc London Ser A 248:155199) is extended to include nonlinear effects for the stationary
cross-owvortex. It is shown that amplitude-dependent neutral modes are described by a forced Haberman
equation. The corrections to the neutral wavenumbers and waveangles are derived and it is suggested that
the nonlinear neutral modes can have wavenumbers decreased by an O(1) amount as compared to linear
theory.
Keywords Hydrodynamic stability Rotating disk ow Cross-ow Nonlinear
1 Introduction
In one of the rst detailed experimental and theoretical investigations of three-dimensional boundary-layer
stability, Gregory, Stuart and Walker [1] (hereafter referred to as GSW), highlighted the importance of the
cross-ow mechanism and its relevance to the stationary mode observed in rotating-disk ow. Cross-ow
instabilities are characteristic of a fully three-dimensional boundary-layer ow such as that occurring on
a swept wing. The basic mechanism involved is that, in appropriate ow directions, the effective velocity
prole appears to be inexional, and hence prone to inexional instability. One particular prole is such
that the point of inexion coincides with the point of zero velocity, thus giving rise to neutral disturbances
with zero phase speed, and commonly termed the stationary mode.
The paper by GSW [1] outlined the basic inviscid theory for the stationary mode. Neutral curves cal-
culated in Malik [2] for the stationary mode showed that, for large Reynolds number, the neutral curve
had two distinct branches, one the upper-branch corresponding to the inviscid mode of GSW, and another
which was termed the lower-branch mode. Hall [3], using asymptotic methods, obtained the correction
terms to the inviscid mode of GSW and showed also how the lower-branch mode of [2] could be identied
with wall modes.
J. S. B. Gajjar (B)
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
e-mail: j.gajjar@manchester.ac.uk
206 J Eng Math (2007) 57:205217
An asymptotic description of the linear and nonlinear properties of non-stationary modes is given in
[4]. In [5], the properties of unsteady nonlinear critical layers are used to study the evolution of growing
modes.
Our aim in this paper is to extend the work by GSW and that of [3] to include nonlinear effects for the
upper-branch stationary mode. Nonlinear calculations for the lower-branch mode are given in [6].
It is clear fromthe work of [3] (hereafter referred to as I), that the importance of the nonlinear terms will
be enhanced in the critical layer. The structure of the solution given in [3] is very similar to that occurring in
two-dimensional boundary-layer ows. The extension to include nonlinear effects is therefore very similar
to the studies of nonlinear critical layers in [712]. In the canonical problem, however, there are differences.
The governing equation here turns out to be a forced Haberman equation, see [7], the forcing arising from
the Coriolis terms. The other difference from two-dimensional ows is the important role played by the
mean-ow interaction terms. As in I, the corrections to the wavenumber and wave-angle stem from a
balancing of the wall-layer and critical-layer phase shifts. In the Haberman stage these corrections are
amplitude-dependent. With increased nonlinearity, the critical-layer phase shift approaches zero and the
structure of the modes is similar to that discussed in [8] for two-dimensional ows. In this limit it is found
that there are O(1) corrections to the wavenumber and wave-angle. A signicant result here is that there
is a cross-over from the inexional prole to one in which U

B
is non-zero at the critical layer. Here U
B
(y)
is the effective cross-ow prole.
In Sect. 2 we discuss briey the critical-layer details for the structure given in [3]. The nonlinear theory
is described in Sect. 3. Throughout this work the Reynolds number R is taken to be large.
2 Basic equations and linear theory
Consider a disk, located at z = 0, which rotates about the z-axis with angular velocity . With axes rotating
with the disk, the NavierStokes and continuity equations, suitably non-dimensionalised, are:
u.u +2(

k u) 2r r = p +
1
R

2
u, (1)
.u = 0. (2)
Here u = (u, v, w) are the velocity components in cylindrical polar coordinates (r, , z) and ( r,

,

k) are
the corresponding unit vectors in the respective coordinate directions. Also R =
2
l/ is the Reynolds
number, l is some reference lengthscale, and is the kinematic viscosity of the uid.
The basic steady ow is given by the von Krmn [13] solution as,
u = U
B
= (r u( ), r v( ), R

1
2
w( )),
where z = R

1
2
=
3
, = R

1
6
and u, v, w satisfy,
u
2
( v +1)
2
+ u

w u

= 0,
2 u( v +1) + v

w v

= 0,
2 u + w

= 0.
Primes above denote differentiation with respect to . The boundary conditions on U
B
are
u = 0, v = 0, w = 0, at = 0,
u 0, v 1 as .
J Eng Math (2007) 57:205217 207
2.1 Linear theory
Following GSW and I, we perturb the basic ow by writing
(u, p) = (U
B
, p
B
) +(U, V, W, P) . . . , (3)
where 1 is the innitesimal disturbance size and p
B
denotes the basic pressure. For linear theory to
hold, the requirement that O(
2
) is needed to ensure that nonlinear balances in the critical-layer
region remain negligible; see later. The disturbance quantities (U, V, W, P) are now expressed as
U = U
0
cos x +U
1
+ ,
V = V
0
cos x +V
1
+ ,
W = W
0
sin x +W
1
+ ,
P = P
0
cos x +P
1
+ , (4)
where, as in I, we have introduced x-variations such that

r
=
3
(
0
+
1
+ )

x
+

r
,

=
3
(m
0
+m
1
+ )

x
.
Here =
0
+
1
+ , m/r = (m
0
+ m
1
+ )/r are the scaled wavenumbers in the r,

directions,
respectively. The x-variable here accounts for fast variations in the direction of propagation of the wave
which makes an angle tan
1
(/r) with the radial direction. The disturbance eigenfunctions U
0
, V
0
, W
0
, P
0
all depend on (as well as r), but not on x. In later sections the explicit dependence on r is suppressed.
The main objective of our work is to discuss how the neutral wavenumbers and waveangles change with
amplitude of the disturbance.
Substitution of (4) in the NavierStokes and continuity equations yields at leading order,
U
0B
U
0
+rW
0
u

=
0
P
0
,
U
0B
V
0
+rW
0
v

=
m
0
r
P
0
,
U
0B
W
0
= P

0
,
U
00
+W

0
= 0,
(5)
where primes denote differentiation with respect to , and we have dened U
nB
= r
n
u + m
n
v and
U
nk
=
n
U
k
+(m
n
/r)V
k
. Thus U
0B
can be regarded as the effective or cross-ow velocity. Manipulation
of (5) gives the Rayleigh equation for W
0
, namely:
U
0B
[W

0

2
0
W
0
] U

0B
W
0
= 0, (6a)
where
2
0
=
2
0
+(m
2
0
/r) is the effective wavenumber.
We are interested in stationary modes, so following GSW and I, we choose
0
, m
0
such that
U
0B
= U

0B
= 0 at =

. (6b)
The equation for W
0
, (6a) together with (6b) and the boundary conditions
W
0
= 0 at = 0, , (6c)
208 J Eng Math (2007) 57:205217
species an eigenvalue problem for determining
0
. The solution is given in I and it is noted that

0
= 1.16,
0
/m
0
= 4.26/r,

= 1.46.
The behaviour of the eigenfunctions locally near =

are required in the subsequent analysis and it is
easily deduced that
U
0

[
0
P
0
rW
0
u

]
B
01
(

)
+ , (7a)
V
0

[(m
0
/r)P
0
rW
0
v

]
B
01
(

)
+ , (7b)
as

. The constants B
jk
are dened by
B
jk
= r
j
a
ku
+m
j
a
kv
,
where
a
ku
=
_
d
k
u
d
k
_
=

, a
kv
=
_
d
k
v
d
k
_
=

.
Note that, whereas U
0
, V
0
are singular at the critical layer, W
0
and U
00
are regular at =

. From (6b),
B
00
= B
02
= 0. The eigenfunction W
0
is normalised such that W

0
( = 0) = 1.
The viscous terms in the full equations are needed to smooth out the singularities at the critical level
=

and a balance of inertial and viscous terms shows that the thickness of the critical layer is O().
At the next order in the inviscid region, substituting (4) in the NavierStokes equations and writing
W
1
= W
1s
( ) sin x +W
1c
( ) cos x,
we nd that W
1c
satises the same equation as W
0
. However, W
1s
satises
U
0B
[W

1s

2
0
W
1s
] U

0B
W
1s
= 2U
0B
(
0

1
+m
0
m
1
)W
0
+
_

m
1

0
m
0
_
r
_
u

0B
u
U
0B
_
W
0
. (8)
The second term on the right-hand side of (8) gives a logarithmic singularity for W
1s
with the behaviour
W

1s
r
(m
0

1
m
1

0
)(B
01
a
2u
B
03
a
0u
)
m
0
B
2
01
(

)
(W
0
)
=

, (9)
as

+. The continuation below the critical layer ( <

), see Sect. 3.1 below, gives the well-known
phase jump
W
1
k
0
(

) log(

) sin x >

,
W
1
k
0
(

)[log(

) sin x + cos x] <

,
where k
0
is the coefcient of (

)
1
in (9). Hence
[W

1c
]
+

= k
0
, [W

1s
]
+

= 0, (10)
with []
+

denoting the jump across =

.
J Eng Math (2007) 57:205217 209
Before considering the critical layer, the wall-layer problem, see I, where =
4
and
U =

U
0
+ , V =

V
0
+ , W =

W
0
+ , (11)
shows that W
0
satises a forced Airy equation and, in particular, the solution properties imply that
W
0


r
sin x
3Ai

(0)

1
3
cos(x +/3) as 1. (12)
Here we have dened (> 0) by =
0
u

(0) +(m
0
/r) v

(0). The phase-shift given by (10) matches with the

wall-layer phase shift in (12) and gives rise to the eigenrelations. In fact, a solvability condition on W
1
in
the inviscid region shows that
[W

1c
W
0
W
1c
W

0
]

+
+[W

1c
W
0
W
1c
W

0
]

0
= 0, (13a)
[W

1s
W
0
W
1s
W

0
]

+
+[W

1s
W
0
W
1s
W

0
]

0
= 2
_

1
+
m
0
m
1
r
2
_
_

0
W
2
0
d
+rm
0
_

1
m
0

m
1

0
m
2
0
_
_

0
W
2
0
_
u

U
0B
U

0B
u
U
2
0B
_
d . (13b)
Using (10), (12) and evaluating (13a), we obtain
(W
2
0
)(

)r
_

m
1

0
m
0
__
a
2u

B
03
a
0u
B
01
_

B
01
=
3Ai

(0)
2
1
3
. (14)
Similarly, (13b) gives
3

3Ai

(0)
2
1
3
= 2
_

1
+
m
0
m
1
r
2
_
_

0
W
2
0
d
+
_

1
m
0

0
m
1
m
2
0
_
rm
0
_

0
W
2
0
_
u

U
0B
U

0B
u
U
2
0B
_
d . (15)
The relations (14, 15) are equivalent to the relations derived by Hall [3]. We note that the jump across the
critical layer xes the correction to the wave-angle, and the conditions on W
1s
determine the correction to
the wave-number.
Our calculations show that
I
1
=
_

0
W
2
0
d = 0.091, I
2
=
_

0
W
2
0
_
u

U
0B
U

0B
u
U
2
0B
_
d = 0.0596,
which leads to

1
+
m
0
m
1
r
2
= 9.2r

1
3

0
, (16a)
_

1
m
0

m
1

0
m
2
0
_
r = 17.5r

1
3
. (16b)
210 J Eng Math (2007) 57:205217
The numbers in (16a, 16b) are different from I, because in I the jump across the critical layer has the
wrong sign in view of the basic ow properties near the critical level.
We turn next to the details of the critical layer.
3 Critical-layer analysis
3.1 Linear critical layer
Let z =
3

+
4
. For the linear critical layer analysis only, it is more convenient to work in terms of
complex quantities and from (7, 9) the expansions in the critical layer region take the form
U = (
1

U
0
+

U
1
+

U
2
+ )e
ix
+c.c.,
V = (
1

V
0
+

V
1
+

V
2
+ )e
ix
+c.c.,
W = (

W
0
+

W
1
+
2

W
2
+ )e
ix
+c.c.,
W = (

P
0
+

P
1
+
2

P
2
+ )e
ix
+c.c.,
where c.c. denotes the complex conjugate.
Substitution of the above in (1, 2) yields

P
0
=

P
1
= const,

W
0
=
i
2
0

P
0
B
01
, r
0

U
0
+m
0

V
0
= 0,
and

U
0
satises the equation
i(B
01
+B
10
)

U
0

d
2

U
0
d
2
= i(
0

P
0
+

W
0
ra
1u
).
The solutions for

U
0
then give the required behaviours (7) as . At the next order

W
1
has a trivial
solution.

W
2
satises the equation
i(B
01
+B
10
)
d
2

W
2
d
2
+
d
4

W
2
d
4
= i
2
0
(B
01
+B
10
)

W
0
+2i
2
0
(1 +a
0v
)

0
d

V
0
d
i

W
0
(B
03
+B
12
). (17)
The second term on the right-hand side of (17) arises from the inuence of the Coriolis force and can be
removed by writing
d
2

W
2
d
2
=
2
+
_

2
0
+
B
03
B
01
_

W
0
+
2
0
(1 +a
0v
)
B
01

0
d
2

V
0
d
2
.
Here
2
satises
i(B
01
+B
10
)
2

d
2

2
d
2
= i

W
0
_
B
12

B
03
B
10
B
01
_
. (18)
The solution for
2
,

2
= i|B
01
|
2/3
_
B
12

B
03
B
10
B
01
__

0
e
t
3
/3
e
i(+
B
10
B
01
)|B
01
|
2/3
t
dt
J Eng Math (2007) 57:205217 211
gives the continuation below the critical layer and the required i phase jump. This completes the
details of the linear critical layer.
We consider next what happens when the disturbance size is increased. As in previous critical-layers
studies, nonlinear balances will rst signicantly alter the properties of the critical layer, and this happens
when = O(
2
).
4 Nonlinear critical layer
4.1 Inviscid region
Here, with z =
3
, the above discussion suggests that in this region the expansions for the ow quantities
will take the form
u = r u + u
M
( ) +
2
( u
0
+u
M1
( )) +
3
u
1
+
4
u
2
+ , (19a)
v = r v + v
M
( ) +
2
( v
0
+v
M1
( )) +
3
v
1
+
4
v
2
+ , (19b)
w =
2
w
0
+
3
w
1
+ , (19c)
p =
2
p
0
+
3
p
1
+ . (19d)
The additional terms u
M
, v
M
at O() are necessary because of the properties of the nonlinear critical layer,
see [7, 9, 10]. It is noted also that the disturbance for the w-component of velocity is larger than the basic
ow in this direction. Next writing,
( u
0
, v
0
, w
0
, p
0
) = A(u
0
( ) cos x, v
0
( ) cos x, w
0
( ) sin x, p
0
( ) cos x), (20)
and substituting in the NavierStokes equations then shows that w
0
( ) satises the Rayleigh equation (6a)
with boundary conditions (6b, 6c). In (20) the amplitude constant A is O(1), given that the disturbances
represented by the terms of ( u
0
, v
0
, w
0
, p
0
) are of O(
2
).
At next order the cos x component of w
1
, say w
1c
, satises the same equation as w
0
. The sin x component
of w
0
, w
1s
now satises
U
0B
(w

1s

2
0
w
1s
) U

0B
w
1s
= 2A
_

1
+
m
0
m
1
r
2
_
W
0
A
(U
0M
+U
1B
)
U
0B
_
U

0B
U
0B

(U

0M
+U

1B
)
(U
0m
+U
1B
)
_
W
0
, (21)
where U
kM
=
k
u
M
+
m
k
r
v
M
. The appearance of the mean-ow terms in (21) will thus alter the eigenre-
lations determining the corrections to the wavenumber and wave angle. The eigenrelations will also be
affected by the properties of the nonlinear critical layer, and it to this that we turn our attention next.
4.2 Critical-layer expansions
The expansions in the critical layer region are now, with z =
3

+
4
,
u = ra
0u
+ u
0
+
2
u
1
+
3
u
2
+ , (22a)
v = ra
0v
+ v
0
+
2
v
1
+
3
v
2
+ , (22b)
w =
2
w
0
+(
3
w
1
+a
0w
) +
4
( w
2
+a
1w
) + , (22c)
p =
2
p
0
+
3
p
1
+
4
p
2
+ . (22d)
212 J Eng Math (2007) 57:205217
The a
0w
, a
1w
terms in (22c) come from expanding the basic ow for the w-component of velocity. The
leading-order problem with u
jk
=
j
u
k
+
m
j
r
v
k
is
u
00
x
+
w
0

= 0, (23a)
(B
10
+ u
00
)
u
0
x
+ w
0
u
0

=
0
p
0
x
+

2
u
0

2
, (23b)
(B
10
+ u
00
)
v
0
x
+ w
0
v
0

=
m
0
r
p
0
x
+

2
v
0

2
, (23c)
p
0

= 0. (23d)
These equations may be solved for the cross-ow component, by taking
0
times (23b) added to m
0
/r times
(23c). After matching with the inviscid regions, this gives
w
0
=

2
0
B
01
p
0
x
=

2
0
Ap
00
B
01
sin x, p
0
= Ap
00
cos x, p
00
= p
0
(

), u
00
= B
01
+C
00
, (24)
and C
00
=
0
u

M
(

) +(m
0
/r)v

M
(

) is independent of and therefore does not jump across =

. Writing

B
10
= B
10
+C
00
, v
0
satises,
(

B
10
+B
01
)
v
0
x
+ w
0
v
0

=
m
0
r
p
0
x
+

2
v
0

2
. (25)
Equation 25 can be written in a more standard form with the normalisations:
v
0
= ra
1v
+K
0
(Y V

0
), K
0
=
_
m
0
r
p
00
A

2
0
Ap
00
ra
1v
B
01
_
1
|B
01
|C

, C

Y =

B
10
B
01
+,
C
2

=

2
0
Ap
00
B
2
01
, B
01

c
= C
3

. (26)
and then V

0
satises the Haberman [7] nonlinear critical layer equation,
Y
V

0
x
+sin x
V

0
Y

c

2
V

2
= 0, (27a)
with boundary conditions,
V

0
Y +
v
M

)
K
0
+
cos x
Y
+ as Y , (27b)
to match with the inviscid zone. The parameter
c
arising from the normalisations gives a measure of
nonlinearity in the sense that linear theory corresponds to
c
and strong nonlinearity when
c
0.
The properties of Eq. 27 are well known; see [7, 9, 11]. In particular, V

0
plays the role of the vorticity here,
so that with V

0
=
2

0
/Y
2
, and

0
Y

Y
2
2
+
v

M
(

)
K
0
Y +cos x ln |Y| +B

0
(x) as Y .
J Eng Math (2007) 57:205217 213
It can be shown that
2
c
v
+
M
(

) v

M
(

)
K
0
=
1

_
2
0
(B
+
0
B

0
) sin x dx = . (28)
In the limit that A 0,
c
and corresponding to linear theory. For A 1,
c
0 and

c
C
(1)
with C
(1)
= 5.516. In the limit
c
0 the solution for V

0
is given by
V

0
= L
0
(), =
Y
2
2
+cos x, L() = L
0
+
_

1
L

0
() d, (29)
with
L() =
2
_
2
0

2( cos x)
1
2
dx
for
_
Y >

2( cos x)
1
2
Y <

2( cos x)
1
2
,
L() = L
0
inside the cats eyes |Y| <

2( cos x)
1
2
.
In particular, from (28)
(V

0
)

=
1
K
0
_
v
+
M
(

) v

M
(

)
_
=
1
2
C
(1)
for A 1. (30)
Thus, the problem for V

0
shows that, even though the jump in the effective mean ow is zero, the separate
components u
M
, v
M
do suffer jumps across

.
For O(1) values of
c
, a numerical solution of (27) is necessary, which can be found in, for example, [7].
The second-order problem gives rise to the set of equations,

x
( u
01
+ u
10
) +
w
1

= 0,
N
1
( u
1
, u
0
) +r
_
a
2
0u

_
1 +a
2
0v
__
=
0
p
1
x

1
p
0
x
,
N
1
( v
1
, v
0
) +2ra
0u
(1 +a
0v
) =
m
0
r
p
1
x

m
1
r
p
0
x
,
with the operator N
1
dened by
N
1
( u
1
, u
0
) = (

B
10
+B
01
)
u
1
x
+(B
20
+u
10
+u
01
)
u
0
x
+ w
0
u
1

+( w
1
+a
0w
)
u
0

2
u
1

2
.
These equations can be solved by setting
u
01
+ u
10
=

1

, w
1
=

1
x
to get
u
01
+ u
10
= B
01
d(x, r), (31)
where d(x, r) can be determined by matching, although we do not need it explicitly.
214 J Eng Math (2007) 57:205217
The continuation below the critical layer requires the solution of the third-order problem, where
( u
2
, v
2
, w
2
, p
2
) satisfy the equations

x
( u
02
+ u
11
+ u
20
) +
w
2

= 0, (32a)
N
2
( u
2
, u
1
, u
0
) + u
0
a
0u
2a
ov
v
0
2 v
0
+ra
0u
u
0
r
=
_

0
p
2
x
+
1
p
1
x
+
2
p
0
x
_
, (32b)
N
2
( v
2
, v
1
, v
0
) +a
0u
v
0
+2 u
0
+a
0u
v
0
+a
0v
u
0
+ra
0u
v
0
r
=
_
m
0
r
p
2
x
+
m
1
r
p
1
x
+
m
2
r
p
0
x
_
, (32c)
(B
01
+

B
10
)
w
0
x
=
p
2

. (32d)
The operator N
2
above is dened by
N
2
( u
2
, u
1
, u
0
) = (B
01
+

B
10
)
u
2
x
+(B
20
+ u
10
+ u
01
)
u
1
x
+(B
30
+ u
20
+ u
11
+ u
02
)
u
0
x
+ w
0
u
0

+(w
1
+a
0w
)
u
1

+( w
2
+a
1w
)
u
0

2
u
2

2

2
0

2
u
0
x
2
.
If we let

= u
02
+ u
11
+ u
20
,

2
x
= w
2
,
then, using (24, 31) and after some manipulation, it can be shown that
2
satises
(B
01
+

B
10
)

x
_

2
_
+ w
0

3


4

4
= B
03

2
0
(B
01
+

B
10
) w
0
+
2(1 +a
0v
)
2
0

0
_
v
0

ra
1v
_
. (33a)
The boundary conditions on
2
to enable a match with the solutions outside the critical layer are

B
03

3
6
+
_
B
12
+C

02
_

2
2
+
_
D

01
+C

11
+B
21
_

+
B
01
_
B
12
+C

02
_
B
03

B
10
B
2
01
_

2
0
P
00
A
B
01
_
cos x log || +U

(x), (33b)
as . Here
C

jk
=
_

j
u
(k)
M
(

) +
m
j
r
v
(k)
M
(

)
_

(34a)
D

01
=
_

0
u
(k)
m1
(

) +
m
0
r
v
(k)
m1
(

)
_

. (34b)
Equation 33a is the nonlinear counterpart of (17). We note that the /r terms have cancelled out when
working in terms of the cross-ow components, but they do not disappear in the equation for u
2
, v
2
. Also
the terms in (33a) stemming fromthe Coriolis forces, cannot be removed, unlike in the linear case, primarily
because the equation here is nonlinear. This and the B
03
term (arising from the basic ow) provide the
forcing in the resultant Haberman equation below. The C

02
, D

01
contributions stem from the mean-ow
terms in the inviscid region, as can be seen from (19).
J Eng Math (2007) 57:205217 215
The normalisations (26) together with

2
=

2

2
+

2
0
B
01
p
0
,

2
= C
3

_
B
12

B
03

B
10
B
01
_

2
, (35)
reduces (33) to the canonical form,
Y

x
_

2
Y
2
_
+sin x

2
Y
3

c

2
Y
4
= D
3

c
+

_
1
V

0
Y
_
, (36a)

2
Y
D
3
Y
3
6
+(1 +b

2
)
Y
2
2
+
_
d
1
+b

1
_
Y
+
_
1 +b

2
_
cos x log |Y| +YF
2
(x) +F

1
(x) as Y . (36b)
Here

c
=
B
01
B
12
B
03

B
10
B
01
,

=
2(1 +a
0v
)
2
0
K
0

0
|B
01
|
c
C
3

, D
3
=
B
03
C

c
, b

2
=
C

02

c
,
C

D
1

c
= B
21
+
B
03

B
2
10
2B
2
01

B
10
B
12
B
01
, C

c
b

1
=

B
10
C

02
B
01
+D

01
+C

11
. (37)
Equation 36 is a forced Haberman [7] equation and is different from previous studies on critical layers
because of the forcing terms in (36a), and the extra Y
3
, b

2
contributions in the boundary conditions.
Nevertheless (36) does have properties similar to the Haberman [7] equation.
Identities relating the jump in the velocity to that in the mean vorticity can be derived as for the
Haberman equation. These are obtained by integrating (36) with respect to Y, then over a period; see ([7]).
It is found that

c
_
b
+
2
b

2
_
=

K
0
_
v
+
M
v

M
_
, (38)
and
2
c
(b
+
1
b

1
) =

_
2
0
(B
+
0
B

0
) dx. (39)
The quantity =
1

_
2
0
sin x(F
+
1
F

1
) dx is the jump in the sin x component of
2
/Y
We consider next how the jump in the velocity modies the eigenrelations.
4.3 Nonlinear eigenrelations
The leading-order wall displacement is basically the same as in the linear case. The expansions in the wall
layer (z =
4
) are
u = r u(0) +
2
u
0
+ ,
v = r v(0) +
2
v
0
+ ,
w =
3
w
0
+ ,
p =
3
p
0
+ .
216 J Eng Math (2007) 57:205217
Solutions for the fundamental are as in the linear case, see (11), and after matching with the w
1
component
in the inviscid regions gives
( w
1
)( = 0) =
3Ai

(0)

1
3
Acos
_
x +

3
_
. (40)
The critical-layer problem (36) determines the jump conditions on w
1
/ and with the above normalisa-
tions we nd that
_
w

1c
_
+

= C
2

()
B
12
B
01
B
03

B
10
B
01
, (41a)
(w

1s
)
+

= 0. (41b)
Hence the resulting eigenrelations are

3Ai

(0)
2
1
3
= ()w
2
0
(

)
_
B
12
B
01
B
03

B
10
B
2
01
_
, (42a)
and
3

3
2
1
3
Ai

(0) = 2
_

1
+
m
0
m
1
r
2
_
_

0
w
2
0
d

_

0
_
U

0B
(U
0M
+U
1B
) U
0B
_
U

0M
+U

1B
__ w
2
0
( )
U
2
0B
d . (42b)
Now

c
B
01
=
B
12
B
01
B
03

B
10
B
2
01
=
r
m
0
_

m
1

0
m
0
_
_
a
2u
B
01
B
03
a
0u
B
2
01
_
m
0

B
03
C
00
B
2
01
. (43)
The last term in (43) involving C
00
is not present in the linear case and represents the effects of the
mean-ow term.
From (16), (40), (42) we nally obtain
_

1
m
0

0
m
1
m
2
0
_
= 17.5r

1
3
_

_
+
B
03
C
00
m
0
(a
2u
B
01
B
03
a
0u
)
(44a)
_

1
+
m
0
m
1
r
2
_
= 9.2r

1
3
_

1
2J
1
_
J
2
m
0
_
B
03
C
00
a
2u
B
01
B
03
a
0u
_
J
3
_
, (44b)
where
J
1
=
_

0
w
2
0
d , J
2
= m
0
_

0
U
0B
u

0B
u
U
2
0B
d , J
3
=
_

0
U

0B
u
0M
U
0B
u

0M
U
2
0B
d .
The corrections to the wave number and wave angle given by (44) are the main results in this paper and
show the dependence of the wavenumber and angle on the neutral amplitude A via the phase jump , and
the mean ow terms. For a given A and hence
c
, a numerical solution of the forced Haberman equation
(36a) is required to determine . If
c
, corresponding to A 0, it can be shown that and
a match with linear theory is achieved. For large amplitudes A 1 then
c
0 and the solution of the
J Eng Math (2007) 57:205217 217
Haberman equation gives rise to the cats eye structure with approaching zero. The limiting structure is
akin to that of the strongly nonlinear critical layer [8]. The full details of this limit are omitted here, (these
are available by request from the author), but the analysis suggests that there is a cross-over from the
inexional instability to BenneyBergeron type modes. These are amplitude-dependent stationary neutral
modes.
5 Discussion
In this paper we have discussed critical layers in rotating-disk ow and have shown how the calculation
of the wavenumber and waveangle depend crucially on the properties of the nonlinear critical layer. For
very small amplitudes A(R

1
3
) the stationary modes correspond to inexional modes and are the same
as those rst discussed in [1]. For increased amplitudes A R

1
3
these modes are again inexional but are
now amplitude-dependent and require the solution of the forced Haberman equation with novel boundary
conditions. For larger amplitudes A R

1
3
, specically A R

2
9
, there is a cross-over fromthe inexional
modes to BenneyBergeron-type modes. If we assume C
00
= 0 in (44a) and b
+
2
= b

2
to leading order
in (38), then in this second stage the wavenumber is decreasing as the amplitude increases, and an O(1)
change in the wavenumber is predicted when A R

2
9
. The decreased wavenumber, corresponding to
fewer vortices, is in qualitative agreement with experiments.
In GSW it was suggested that viscosity would account for the discrepancy between the observed and
calculated values of the wavenumber from linear theory. The inuence of nonlinearity, as shown here, can
signicantly alter the wavenumber and it is possible that the experimentally observed modes are highly
nonlinear stationary modes of the type discussed here. Of course a more quantitative comparison with
experiments requires a solution of the nonlinear forced Haberman problem which is left open.
Acknowledgements The author would like to thank Professor P. Hall for suggesting the problem and discussions on his
work. Professor F.T. Smith FRS is thanked for suggesting many improvements to an earlier draft of the paper.
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