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OCR for page 553
Three-Dimensional Instability Modes
of the Wake Far Behind a Ship
G. Triantafyllou (Massachusetts Institute of Technology, USA)
1 Abstract
The three-dimensional linear stability of the viscous wake far
behind a ship is investigated. The Euler equations are linearized
around the time-average flow in the wake, and the stability problem
is formulated as an eigenvalue problem for waves travelling
parallel to the course of the ship. It is shown that the complex
phase velocity of the unstable waves satisfies Howard's semi-circle
theorem. For a self-similar velocity profile, a numerical solution is
obtained by expanding the perturbation pressure in a Fourier series
and solving a set of simultaneous ordinary differential equations. It
is found that the wake becomes unstable in its antisymmetric
pressure mode for a range of wavenumbers less than a "cut-off"
value. In the zero Froude number limit, the frequency and phase-
velocity of the unstable gravity waves are determined entirely by
the characteristics of the shear flow in the wake, a fact allowing the
derivation of a simple approximation for the eigenvalues and
eigenvectors. As the Froude number is increased, the growth-rates
of the unstable waves, and the "cut-off" value are reduced,
indicating a stabilizing effect on this mode. In the infinite Froude
number limit the wake becomes unstable in a different mode. The
wake does not exhibit self-excited behaviour, because the
instability of the shear flow is of the convective type at all Froude
numbers. External noise, like ambient waves in the ocean, can
drive the wake instability producing spatially growing waves. The
free-surface manifestation of the spatially unstable waves exhibits a
characteristic staggered pattern of alternating "hills" and "valleys".
2 Introduction
The best known feature of the flow behind a ship is the
Kelvin wave pattern. The Kelvin wave pattern has been
extensively studied, because it is a very significant source of ship
resistance at high speeds. The viscous wake of the ship has
received much less attention, since it is assumed that it is for most
ships thin, and its influence on the wavemaking of the ship has
routinely been neglected, with some notable exceptions
(Tatinclaux, 1970, Peregrine, 1971). In recent years, however, the
viscous wake of the ship has attracted a considerable amount of
attention, both as a basic fluid mechanical problem, and in
connection with the problem of wake imaging. As aerial pictures
of the ocean have revealed, viscous wakes of ships are visible at
very large distances behind the ship (see, among others, Peltzer et
al., 1978, Milgram, 1988, and Skop et al., 1990). Thus, the viscous
wake, even though relatively thin, leaves a very persistent "trace"
on the ocean surface, and offers an effective means of ship
detection. The problem has several different aspects, including the
backscatter of electromagnetic waves from the ocean surface
(Valenzuela, 1978), and is currently extensively studied.
From the fluid mechanics point of view, which is mainly of
interest in the present paper, investigations of the interaction of
vertical flows with a free surface have revealed several interesting
new properties: Benney and Chow, 1986, Sarpkaya, 1986, Lugt,
1987, Oikawa et al., 1987, Tryggvason, 1988, Willmarth et al.
1989, Bernal and Kwon, 1989, Triantafyllou and Dimas, 1989
Liepmann, 1990. The basic hydrodynamics of wake/free surface
interactions are not understood well enough yet to provide a full
description of the complex phenomena involved, nor an
explanation of the aforementioned persistence of ship wakes. A
solution of the problem from first principles, through direct
simulation of the Navier-Stokes equations, is still impossible,
owing to the combined complexity provided by the very high value
of the Reynolds numbers of ships, typically 109, and the presence
of a moving boundary, the free surface, which renders the
computational domain time-dependent. It appears therefore that
the problem has to be approached in successive stages.
In this paper a specific aspect of the wake/free surface
interaction is addressed, namely the linear hydrodynamic stability
of wake, seen as a three-dimensional shear flow. It has been well
known that in supercritical transitions, linear theory can determine
whether a certain flow state is unstable or not. In recent years, it
has become increasingly clear that linear theory can also provide a
good description of the "shape" of the unsteady flow patterns that
result from the instability, whereas the amplitude of the patterns is
determined by non-linear effects. (See for instance Koch, 1985,
Triantafyllou et al., 1986, Triantafyllou et al., 1987, Chomaz et al.,
1988, Unal and Rockwell, 1988, Karniadakis and Triantafyllou,
1989, Hanneman and Oertel, 1989, Triantafyllou and Karniadakis,
1990). Linear theory has thus become a very useful conceptual tool
in interpreting the physics of unsteady viscous flows. A
fundamental concept in the linear instability theory in media that
are unbounded in the direction of propagation of the instability
waves is the distinction between absolute and convective
instabilities (see the review article by Bers, 1983). Absolutely
unstable flows are self-excited, and a localized perturbation leads to
growing motions at any fixed location in space. Convectively
unstable flows on the other hand remain steady in a noise-free
environment, because all localized perturbations are convected
away. It is interesting to investigate to what extent these concepts
can elucidate the problem of shear flow/free surface interaction.
For the two-dimensional wake/free surface interaction
problem, it has been recently shown (Triantafyllou & Dimas, 1989)
that the vicinity of a free surface drastically alters the instability
properties of two-dimensional shear flows, and renders an
absolutely unstable flow convectively unstable. As a result, high
Reynolds number wakes of floating objects remain steady at low
Froude numbers, and have the fonn of steady recirculating flows.
Few things have been known for the considerably more complex
three-dimensional problem, studied here. We consider the space-
time evolution of perturbations around the mean flow in the wake.
The perturbations have the form of waves that propagate parallel to
the course of the ship, and have an eigenfunction type of
dependence in the other two directions. The presence of the free
553
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surface is acknowledged through the kinematic and the dynamic
boundary conditions. An eigenvalue problem is thus obtained for
the frequency which depends parametrically on the Froude number
of the flow. The eigenvalues and eigenvectors are solved for
numerically. From the computed eigenvectors, the shape of the
free-surface manifestations of the instability waves is determined.
The issues that are of interest here are: (i) The shape of the
unsteady patterns that result from the wake/free surface interaction,
and (ii) Whether these patterns can be self-excited (which, as
mentioned before, is related to the absolute versus convective
instability distinctions).
3 Three-dimensional shear flow/free surface
interactions
In this section we discuss the formulation of the shear
flow/free surface interaction problem in general. Let x,y,z be a
system of coordinates, with the x-axis parallel to the direction of
the flow, the z-axis parallel and opposite to the direction of gravity,
and the y-axis perpendicular to the other two. The unit vectors
along the x, y, z axes are i, j, k respectively (figure 1). For the
problem of interest here, the frame of reference is fixed with
respect to the ship.
We assume that all velocities have been non-dimensionalized
with respect to some reference velocity UOO, (in this case the speed
of the ship), the pressure with respect to pu`~,2 (p is the density),
and all lengths with respect to the width b of the wake. Consistent
with this non-dimensionalization, the acceleration of gravity g is
replaced by 1/F2, where F is the Froude number of the flow,
defined as F = UOO/~.
The non-dimensional Euler equations, linearized around a
parallel flow U(y,z), can be written as follows (Drazin and
Howard, 1966):
aa +uaa )U+vvu+aap = 0
a'+Uaax )V+VP = 0
(1)
where u is the component of the perturbation velocity parallel to
the x-axis, v = ( v, w) is the projection of the perturbation velocity
vector in the y,z plane, and p is the perturbation pressure field;
also, V=(a/3y, 3/3z). In equation (1) p is the dynamic pressure,
i.e. the totalpressurep minus the hydrostatic: p=p+F-'z. The
incompressibility condition requires that the perturbation velocity
has to be divergence-free:
aU+vv = 0
ax
(2)
At the free surface we have the kinematic and dynamic
conditions for the free-surface elevation q(x,y,t). They can be
written as follows:
al1 +u3~ ) = vk = w (3)
a' ax
p F-2 ~
(4)
U<-
k ~ j x
1
~ I 1 _ _
~ ~U(y,z)
Figure 1: Definition sketch.
From the momentum equation in the z-direction (the second of (1)),
we have that:
aW+ uaW ) = _a
at ax az
We use (5) to eliminate w from (3), in order to combine the two
boundary conditions into a single condition for the pressure:
F2(a3+Uaa )2p+3aP=0 (6)
The dispersion relation of the flow can now be obtained by
considering wavy perturbations, i.e. by setting u,v,w,p into the
momentum and continuity equations proportional to exp(i(kx-c3t)),
where ~ is the frequency and k the wavenumber. Then the
momentum equations (1) become:
i(kU-ce)u+vVU+iLp = 0
i(k U-~) v + Vp = 0
(7)
The incompressibility condition for the perturbation velocity
v = (u, v,w) is written as:
iku+ V v=0
The boundary condition (6) at the free surface is written as:
(8)
F2(kU_ ce)2p__= 0 (9)
az
We also impose the condition that the perturbation decays far
outside the wake, i.e. u,v,w,p - 0 when ~y2+z2 - 0O.
We have used for notational simplicity in (7), (8), (9) the
same symbols for the perturbation quantities in the frequency and
in the time-domain; this does not cause confusion since we will
work mainly in the frequency domain.
We multiply the first of (7) by ik, operate on the second of
(7) with V, add the two, and use the incompressibility condition (8)
to obtain a single second-order partial differential equation for the
dynamic pressure p:
554
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(kU-~)( V2p-k2p ) -2kVpVU = 0 (10)
Equation (10) subject to the boundary condition (9) plus the
condition p ~ O for by 1,1 z I ~ on defines an eigenvalue problem for
co, which depends parametrically on the Froude number. If for
some range of real wavenumbers complex frequencies with
positive imaginary parts exist, the flow is unstable. In fact, since
the flow is inviscid, complex eigenvalues appear in conjugate pairs,
corresponding to one growing and one decaying mode. Finally,
when the flow is uniform, i.e. dU/dr=O, the basic equation (10)
reduces to Laplace's equation, and from the free surface boundary
condition the classical dispersion relation for deep water waves is
recovered.
We will now show that Howard's semi-circle theorem is
valid for the complex eigenvalues. To this purpose, we first recast
the basic equation (10) into the following form:
V( P 2) - k2 P 2 = 0 (11)
where c=~/k is the phase velocity of the wave. We multiply (11)
by the complex conjugate of p, px and integrate along the y,z plane.
This gives:
\\
V l l 2 where Q stands for:
IAdYdz(Pxv(< P 2) - k2 P ) = 0 (12)
Since unstable waves have complex phase velocity, U- c does not
vanish anywhere in the flow field, and the integral in (12)
converges. For the first term of the integrand we write:
px V ( Vp 2 ) = V ( ~ P ~ 2) - ~ U- c ~ 2
(13)
We substitute into (12), and apply the divergence theorem for
the integral of the first term.
dydzV( P Vp )= dl n P Vp (14)
IA (U C)2 J-s (U C)2
where S represents the boundary of the domain A, consisting of the
free-surface and a line at infinity (figure 2), and n is the outward
pointing unit vector. We use the boundary condition at the free
surface, and that p tends to zero at infinity to obtain:
I dYdZ(U-C)2( IVPI2+k2IPI2)I U C l-4 =
A
I CO
= k2F2 dYIP(Y 0)12 (15)
With c=cr+ici, I ci l > 0, we can separate the real and
imaginary parts of (15) as follows:
|AdYdz((u-cr)2-ci2)Q= (16)
= k2F2J dylp(y 0)12
_co
\
y
~-
/ S
Figure 2: Integration contour in the y, z plane.
| dydz(U-cr)Q = 0 (17)
A
Q=( IVPI2+k2lPI2)l U C 1-4 2 0 (18)
Also, following Howard, 1961, we note that, if Umin,U,,~
denote, respectively, the minimum and maximum velocity in the
flow field, we have:
| dy dz ( U-Umin ) ( U- Um~C) Q < 0 (19)
We use (16), (17) to eliminate the integrals |U2Q |UQ from the
left side of (19). This yields:
0 2 | dydz(U-Umin)(U-Um~v`)Q = | dydz x
A A
((Cr + Ci ) - (Umin + Um`Dc) Cr + UminUm~c ) Q +
I oo
+ k2F2 dy IP(Y 0) 12 2
_oo
2 | dydz((Cr_ min m~X )2+c 2
A
U i -U (20)
which implies, since Q is positive, that:
Umin+ Umax )2+C 2 ( min m )2 ~ O (21)
Equation (21) is Howard's semi-circle theorem (Howard, 1961,
Drazin and Howard, 1966), for the three-dimensional shear-flow
free surface interaction; it states that the vector cr,ci lies within a
555
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circle with center at the average of the maximum and minimum of
the velocity, and radius half their difference. Equation (21)
generalizes an earlier result by Yih, 1972 for the instability of two-
dimensional shear flows with a free-surface.
Before proceeding with the instability of the wake, it is
useful to discuss briefly the zero and infinite Froude number limits
of the formulation. In the former the free surface condition is
reduced to ap/az = 0, whereas in the latter to p = 0. It is reasonable
to assume that on the free surface Du/az=0. In this case, we can
define the "double-flow" in the whole space through the extension
U(y,z)=U(y,-z). Then the zero Froude number limit
corresponds to an instability mode with pressure that is symmetric
around the z=0 plane in the unbounded fluid, and the infinite
Froude number limit to a mode with pressure that is antisymmetric
around z=0. In two-dimensional flows, for instance, the zero
Froude number limit corresponds to the "varicose" mode in
unbounded fluid, and the infinite Froude number limit to the
"sinuous" mode. For wake flows, the first is connectively unstable,
and the second absolutely unstable. The instability properties of
the wake as a function of the Froude number give a smooth
transition from the one limit to the other (Triantafyllou and Dimas,
1989).
4 Instability of the far wake
Far behind the ship, where the effects of the details of the
ship hull form have diffused, we assume that the average flow has
become self-similar, independent of the angle ~ = atan(z/y) in the
y,z plane. We consider, in other words, a time-average velocity of
the form U = U(r). The reasoning behind this assumption is that, at
low Froude numbers, the time-average flow can be approximated
by half of that behind the "double-body", ( a fictitious object that is
twice the submerged part of the ship), which far behind the object
asymptotically tends to acquire an axisymmetric form. This
assumption is supported by the numerical computations of Swean,
1987, who computed the steady flow past a ship using the
parabolized Navier-Stokes equations, and a K-£ model of
turbulence. Swean's results suggest that indeed the steady flow
tends to become self-similar; his computational results show good
agreement with the experimental results of Mitra et al., 1985, and
Mitra et al., 1986.
Because of the self-similarity assumption, it seems natural to
work in polar coordinates. Equation (10) for the perturbation p(r,0)
is written in polar coordinates as follows:
(kU-~)(aa P2+laaPr+ 120eP2-k2p)-2kddrU3aPr =0
(22)
defined for r 2 0, -A < ~ ~ 0. For the boundary conditions on the
free surface, we note that on ~ = 0, -it, we have:
COs(~) a a
.,
Equations (22), (24), (25) define for any given k an eigenvalue
problem for the frequency of; that is to say, they constitute the
dispersion relation for gravity waves propagating above the wake
of the ship.
5 Fourier Series Solution
Because of the linearity of the problem, and the symmetry of
the average flow U(r) around the plane y =0, an arbitrary
perturbation can be decomposed into two parts: One in which the
pressure is anti-symmetric around y = 0, referred to as Mode I, and
one in which the pressure is symmetric around y = 0, referred to as
Mode II. Thus, given that the free surface elevation is proportional
to the value of the dynamic perturbation pressure there, Mode I
disturbs the free surface in an antisymmetric manner around y = 0,
whereas Mode II disturbs the free surface in a symmetric manner
(figure 3). The fact that the two modes are separable facilitates the
numerical solution of the problem.
L
/
/
/ \ Y
'
'/
/
MODE I
-
y
\J ' ~ \ /
MODE 11
(23) Figure 3: Free surface elevation for Mode I (antisymmetric), and
Mode II (symmetric).
Consequently, the boundary condition on the free surface becomes:
cos(~)(kU-cl))2F2p--aaP~ = 0 on ~ = 0, -'c (24) 5.1 Mode I
Finally, we impose the condition that far from the wake the
perturbation vanishes, i.e.
p(r,f3) ~ O
r ~ so (25)
556
We start with mode I, which satisfies the following
symmetry relations:
p(r,-~) = - p(r,O)
aaPe(r.-~) = aP~(r,O)
(26)
OCR for page 557
Since the initial flow is independent of the angle 8, it is convenient
to expand p(r,8) in a cosine-series in the interval -a < ~ c 0. In
accordance with the antisymmetric character of Mode I, the Fourier
series will contain odd-order coefficients only. Direct substitution
of a Fourier series in the basic partial differential equation (22) is
not applicable, because, for non-zero values of the Froude number,
p satisfies a mixed boundary condition on the free surface, and its
Fourier series can not be differentiated twice.
We thus make the following substitution:
p(r,8) = q(r,8) + aP~(r,°) (~+2 ) (27)
or, after using the free surface boundary condition (24):
where in (33) Lo stands for the following operator:
Lo = d + ~ d _ k2 (34)
In the derivation of (33) the following result has been used:
J: do ( ~ + 2 ) cos (n 8) = ~ 2 ~ (35)
We also need to express G in terms of the Fourier
coefficients qn. This can be done by substituting (31) into (28) and
the result into (29); we obtain:
pf r, 8 ) = q fir, id ) + G ( r) ( f3 +72t ~(28) G = F 2 rick U- 03~ 2 ( I, q,'+ J2C G ~(36)
where G(r) is defined by:
G(r) = F2r(k U- ce ) 2 p(r,O) (29)
e=0,-~:
The new variable q(r,8) satisfies the following conditions at
a~(r,O) = aa~q(r,-~) = 0 (30)
The function 0+~/2 is antisymmetric around the y=0 plane.
Consequently, the new variable q can be expanded in a cosine
Fourier series containing odd terms only. The Fourier series is
twice differentiable with respect to ~ because of the boundary
conditions (30). We write for q:
00
q(r,{~) = ~' qn( r ) cos (n 0) (31)
no=
Where in (31) it is implied that the summation is carried over all
odd n only. The same convention applies for the rest of the section
too. The coefficients qn(r) of the Fourier series in (31) are given
by:
qn( r) = tic ~ q(r,0) cos (nod) do (32)
In order to obtain the differential equations satisfied by the
coefficients qn, we substitute (28) into (10) and use the finite
Fourier transform method, i.e. we multiply both sides of (10) by
cos(n8), and integrate with respect to ~ from -a to 0. In an
unbounded fluid this procedure yields an infinite set of uncoupled
ordinary differential equations for qn(r), compeletely equivalent to
those obtained for the perturbation velocity by Batchelor and Gill,
1962. In the problem considered here, however, owing to the
presence of the free surface, the equations for the coefficients of the
Fourier series are coupled, as follows:
( kU-o) ( Loqn - 2 qn + 2LoG ) - (33)
- 2k d ( d + 2 dr ) = 0
or, after solving with respect to G:
G =
2 2 qn (37)
1 - (~t/2)F rake-I) n
Equation (37) expresses G(r) in terms of qn(r), but is not
very appropriate for the numerical solution of the problem, since it
depends non-linearly on the frequency m. In order to obtain
relations between the unknown variables that are linear in in, we
introduce an auxiliary unknown function H(r) defined as follows:
H = F(kU-~)p(r,O) (38)
Now equation (36) can be replaced by the following pair of
coupled equations:
H = F(kU-m)( I, qn+2G )
n
G = F(kU-~)rH (40)
Also, because of the antisymmetry of the free surface elevation
around the plane y=O, we have the following condition for the
coefficients of the Fourier series:
qn(r=O) = G(r=O) = 0, n= 1,3,... (41)
Equations (39), (40), and (33), subject to the conditions (41), and
that the unknown functions H(r), G(r), qn(r) n = 1, 3, 5,... vanish as
r ~ on, define for a given k an eigenvalue problem that depends
linearly on the eigenvalue m. Consequently, if we truncate the
domain to O < r < R. and use finite-differences to approximate the
derivatives in (33) at specified points ri, i = 1,2,..-N, the discretized
versions of equations (39), (40) and (33) define a generalized
algebraic eigenvalue problem for m. The latter can be solved using
a standard Q-Z algorithm.
557
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5~2 Mode II
Mode II satisfies the following symmetry relations:
p(r,O) = p(r,-rc) (4V
aaPe(r °)=-aP~(r,-~)
The eigenvalue problem for Mode II can thus be formulated in a
similar manner as for Mode I, with the important difference that the
Fourier series has to be chosen in a way that is compatible with the
symmetry conditions (42).
We thus use the following substitution (instead of (28)):
p(r,f3)=q(r,8)+~; aP~(r~o)(o+2)2 (43)
As with Mode I, the derivative of q(r,8) satisfies the boundary
conditions (30), and can be expanded in a cosine Fourier series:
q(r,[~) = 2qo(r)+~ qn(r)C°S(n0) (44)
n
where the coefficients qn are given by (32), and the summation in
(43) is carried now over all even n only. We substitute (43) into
(10), and apply the finite Fourier transform technique, noting that
for n even:
1 ( ~ +72` ) 2cos (n 8) do = 2 (45)
Using the same procedure as for Mode I we obtain the
following equations:
H=F(kU-(~)(2q°+2-qn+4G) (46)
G = F(kU-~)rH (47)
for n = 2,4,... we have:
(kU-(~) ( LoQn- 2qn+ 2LoG )- (48)
-2k d ( d" + 2 d ) = 0
whereas for n=0 we have:
(kU-(~) ( LoQo+6LoG + 7` 2)~2k dr ( dr
+ ~6 dG ) = 0
(49)
in equations (47), (48), and (49) G(r)=F2r(kU-~)2p(r,0), and Lo
is defined as before by equation (34). Finally, instead of (41), we
have that the slope of the unknowns vanishes at r=O:
qn(r=0) = ddG(r=0) = 0 n=0,2,4,... (50)
and that the unknown functions H(r), G(r), and qn(r), n= 0, 2, ...
vanish as r ~ oo.
6 Numerical results
In order to solve for ~ as a function of k, we truncate the
Fourier series for q(r,6~) after M terms, and use a finite-difference
scheme to discretize the ordinary differential equations on a grid
that has N points. We then form a compound eigenvector X of
order (M+2)xN as follows: The first N positions of the
eigenvector are occupied by the values of H(r) at the N
discretization points ri, i = 1,2,.. N. the next N positions by the
values of G(r), the next N positions by the values of q~(r), and so
on; finally, the last N positions are occupied by the values of qM(r)
at the discretization points. The discretized equations can then be
combined into a single matrix equation of the form:
A.X = mB.X
(51)
where A. B are compound matrices of order
((M+2) N) x ((M+2) ~).
In general, depending on the value of M which is required
for the convergence of the Fourier series, the order of the
eigenvalue problem can become quite high, and require enormous
amounts of computation (the number of operations is of order
(M+2)3N3). We note however that the coefficients of the Fourier
series are not directly coupled with each other, but, indirectly,
through the variable G. which is of order F2. As a result, for low
values of the Froude number F. which are mainly of interest here,
the coupling between the coefficients qn is very weak. This,
combined with the fact that for F=0 only the n=1 mode is unstable,
allows an accurate representation of the Fourier series using only a
low number of terms. The decomposition of the pressure field into
modes I and II proves quite helpful in that respect too, since, say
for mode I, truncating the series at the (2M+l)-th term requires the
use of M coefficients in the Fourier series. For the finite difference
grid, the domain was truncated at R = 6, and N= 80 grid points were
used in a second-order finite difference scheme. At the two ends of
the interval all perturbation quantities were set equal to zero.
The free surface elevation A(r) can easily by determined
from the computed values of H(r), which occupy the first N
positions of the compound eigenvector x, as follows:
P(, ) (kU(r3-~) (5V
For a self-similar average flow, the following non-
dimensional velocity distribution can be used:
U(r) = 1 rum exp(-ar2) (53)
where um, a are constants. In this paper the values
um= 0.368, a= 0.89 were used. Those are the values for the
self-similar profile measured by Ogata and Sato, 1966, far behind
an axisymmetric body in unbounded fluid. The numerical results
obtained using (53) are discussed in the next two subsections. We
only discuss Mode I, since Mode II was found for this velocity
profile to be stable. For velocity profiles with larger velocity
deD~cits, both modes can become unstable.
558
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6.1 Form of the instability waves
When the Froude number is equal to zero, the Fourier
coefficients become uncoupled, i.e. each one represents a different
wave, and only the n=1 coefficient represents an unstable wa-ve.
The eigenvalue problem is therefore identical with the one in
infinite fluid, which is unstable, with the maximum amplification
occuring for a wavenumber equal to 0.55, and giving a complex
frequency (0.4524, 0.0172) The variation of the amplitude of the
fast coefficient as a function of r is shown in figure 4a (semi-log
scale). Subsequently, a finite value of the Froude number was
used, using three terms in the Fourier expansion: n= 1, n=3, n=5.
The series was thus truncated at the seventh term. The variation of
the amplitude of the three coefficients qn as a function of r is
shown in figure 4b. The first coefficient is indistinguishable from
the one found in the F=0 case. The other two coefficients are
extremely small; the maximum value of the n=3 coefficient is for
example about 104 times smaller than the n=1. The Fourier series
can accurately be represented using the n=1 term only, as further
attested by the fact that the complex frequency has changed by less
than one per cent from its F=0 value. As the Froude number is
further increased, the importance of the other terms is gradually
increased. At F=1.5 the computation was repeated using five terms
(n= 1, 3, 5, 7, 9. The maximum value of the n=3 term is now about
103 times smaler than that of the first (figure tic). The complex
frequency has changed somewhat, to a value (0.4526, 0.0166); the
real part has thus changed by less than one per cent, whereas the
imaginary part has been reduced by four per cent. This implies that
increasing the Froude number has a stabilizing effect on Mode I. At
F=2.5 the n=3 term is still considerably more than 10+2 times
smaller than the first. The contribution of the Fourier coefficients
with n ~ 1 is more visible in the far field. The complex frequency
has changed to the value (0.4527, 0.0155), showing a decrease in
the growth-rate of about eight per cent from the F=0 value.
F=0
-
o.
D1
1 E-2 _
-
1 E 3 _
1 E-4 _
l E S _
-
, 1 , ~. 1 , 1 ,
3 4 5 6
1 E-6
0 1 2
\
Figure 4a: Variation of the amplitude of the most unstable
pressure eigenmode with r for zero Froude number (semi-log
F=O.S
O.1
~E-:
lE-3t
lE 4'
1 E-!
Figure 4b: Variation of the amplitude of the Fourier coefficients
n= 1,3,5 with r for the most unstable wave at F=0.5 (semi-log
scale).
F=1.5
E-2 _
OFT
559
n=
Figure 4c: Variation of the amplitude of the Fourier coefficients
n=1,3,5,7,9 with r for the most unstable wave F=1.5 (semi-log
scale).
OCR for page 560
1~
o.'
1 E-; _
i
I.
x
-
-
lF_1
1 E-'
1F'
,'_'
r
Figure 4d: Variation of the amplitude of the Fourier coefficients
n=1,3,5,7,9,11,13 with r for the most unstable wave at F=2.5
(semi-log scale).
The variation of the free surface displacement generated by
the most unstable wave along an x = constant plane as a function of
the coordinate y is shown in figure Sa for Froude number equal to
0.5. (Only the part 0 < y < so is shown). The free surface elevation
basically follows the F=0 pressure eigenmode and decays
exponentially with the distance y far from the region where shear is
present. The phase of the free surface elevation of the same
eigenmode is shown in figure Sb (again only the part 0 c y < oo is
shown). The free surface elevation in the middle of the wake,
where the fluid velocity is reduced, lags behind the elevation
outside the wake, where the fluid velocity has its free-stream value.
This is basically the variation of the phase of the pressure
eigenmode for zero Froude number. As the Froude number is
increased, the free surface elevation is also increased,
proportionally to the square of the Froude number. The shape of
the elevation changes however by very little, owing to the
aforementioned separation in magnitude between the n=1 and the
subsequent Fourier coefficients. It is only after F=2.5 that the latter
become significant enough to start altering the shape of the
elevation. This can be seen in figure 6a, where the free surface
elevation divided by the square of the Froude number Is plotted as a
function of r for F-0.5, 1.5,2.5. The effect of the Froude number
is more visible in the phase of the free surface elevation (figure fib).
n, ~
. .
O.OB _
A common feature of figures 4a-4d, where the Froude
number is not too high, is that there is a wide separation in 0.06
magnitude between the n=1 and the subsequent coefficients. For
large n, the Fourier series converges faster than n~3; this can be
seen in figure 4e, where the Fourier coefficients for F=2.5 have
been multiplied by n3 and replotted as a function of r in a linear
scale. In fact equations (33), (34) suggest that, with the exception
of the region around r=0, the convergence is like n - . The
convergence rate is therefore algebraic, and we can safely say that
the main advantage in using the Fourier series expansion, as
opposed to a direct numerical solution of (10), lies in the wide
separation in magintude between the first and the subsequent
Fourier coefficients, which allows an accurate representation of the
series using very few coefficients.
n rid
0.03 _
0.0
~E
-
n=?'
~:=~=3~,
, . 1 , 1 , 1 , 51
2 3 4 5 6
Figure 4e: Plot of qnxn3, n= 3,5,7,9,11,13 as a function of r,
for the same wave as in figure 4d, exhibiting a convergence rate
faster than n~3.
560
0.04
_
O.- ~
~\\
Figure Sa: Amplitude of the free surface elevation as a function
of y, for the most unstable wave at F= 0.5.
o.lS ~
O.1 _
0.05
~.1
-0.15 _
-0.2 , 1 ,
s
Figure Sb: Phase of the free surface elevation as a function of y
for the most unstable wave at F= 0.5.
OCR for page 561
nit.
- ~ -
0.25
no
11 t'
0.1
n,
0.2
0.15
n1
_
_._ _
_ .
.. .1 ~'.-', 11
'- l /'(~-~
~,~
, 1
l
0 1 2 3
an'
-
1 , 1 , ~1
4 5 6
Figure 6a: Amplitude of the free surface elevation as a function of
y for F= 0.5, 1.5, 2.5.
.
_
....
.. _
... _
, 1 , 1_
, 1 2
(I 41h
.
and;
11 t
1
-0.2
Finally, in the infinite Froude number limit there is a
symmetric mode that becomes unstable. More specifically, when
F = on, the boundary condition on the free surface becomes p=O.
Then an exact solution of the problem is given by:
p(r,0) = q 1 (r) sin(~) (54)
where q ~ ( r) is the pressure eigenmode in the zero Froude number
case. The frequency of (54) is the same as the F=O. The fact that
the F= oo mode is just the F=0 mode rotated by ~/2 in the y,z
plane is due to the axisymmetry of the basic flow. In a non-
axisymmetric flow the two limits will be different.
0.02aCF
0.02:
3- 0.015
OF ~
, 1 ,
3
y
play
4 5 6
Figure fib: Phase of the free surface elevation as a function of
y for F= 0.5, 1.5, 2.5.
As discussed above, the growth-rate of the unstable waves is
consistently decreased as the Froude number is increased, showing
that the latter has a stabilizing effect. This can be seen in figure 7,
where the whole unstable frequency range has been plotted for
three different values of the Froude number, F= 0, 0.5, 1.5. The
cut-off frequency, marking transition from stable to unstable waves
is reduced as the Froude number is increased, and the growth-rates
of the unstable waves are also decreased. Increasing the Froude
number causes therefore a, slow, but clear, "shrinking" of the
unstable wave range. At the cutoff frequency, the frequencies of
the growing and the decaying mode coalesce. Consequently, the
condition OD/3co = 0 is satisfied there, where D ( in, k ;F ) = 0
denotes the dispersion relation of the flow at Froude number F.
This condition is typical of the onset of instability in non-
dissipative media.
a_ F=0.5
i, l~,\
"\
__-3 _ / ' .
o {,,,_ , __1, , 1 ,1~*
n t`oC ^= 075
0.25 0.5
lo,
arias = 0
g ~, I
Figure 7: Growth-rate as a function of frequency in the unstable
wave range, for F=0,0.5,1.5. (Note, that at the "cut-off' point
aD/a~=o).
6.2 Spatial instability
The physical character of the wake instability was
determined next, i.e. whether it is of the absolute or of the
convective type. As mentioned before, if mapping of the k-real
axis through the dispersion relation into the cl)-plane yields
frequencies with positive imaginary parts, the flow is unstable.
This was done in the previous subsection, where an unstable
wavenumber range was found, by solving the dispersion relation
with respect to the frequency. In order to distinguish whether this
instability is absolute, or convective, the "pinching" double roots
(Bers, 1983) of the dispersion relation have to be determined.
More specifically, one has to determine a complex pair (mo,ko)
that satisfies:
D(mo,ko;F) = ak (C)o~ko;F) = 0 (55)
plus the "pinching" requirement, stating that the double root should
be formed from the coalescing of a right-travelling with a left-
travelling wave. Then it can be shown (Bers, 1983) that, if the
imaginary part of me is positive, any localized perturbation leads to
561
OCR for page 562
a growing motion at any location in space, and the instability is 0-03
termed absolute; if the imaginary part of NO is negative, any
localized perturbation will be convected away in finite time,
leading to decaying motions at any fixed location in space; the
instability is then termed convective.
It can be further shown that, if the instability is absolute, the
long-time response of the flow develops a normal-mode with a
characteristic frequency and wavenumber. After non-linearities
saturate the growth of the linear instability, the flow settles into a
self-sustained oscillation at approximately this frequency and
wavenumber. Absolutely unstable media are for this reason called
"oscillators". Convective instabilities are on the other hand very
receptive to persistent external excitations: There exists a range of
excitation frequencies which lead to asymptotic states that are
oscillatory in time and grow in space along the direction of
propagation of the instability waves. Convectively unstable flows
are thus characterized as "amplifiers". This distinction is
fundamental in discussing the spontaneous appearance of unsteady
patterns in viscous flows.
The double roots were determined using the procedure
suggested in Triantafyllou et al., 1986, and Triantafyllou et al.,
1987, in which the complex wavenumber plane is mapped through
the dispersion relation into the complex frequency plane; the pinch
point type of double roots are located from the local angle-doubling
property of the map. This procedure is particularity appropriate for
the present problem, where it is possible to solve for co as a
function of k, in the manner described before, but in general it is
extremely difficult to do it the other way around. That the so
determined double roots are of the "pinch-point" type can be
verified from the fact that there is only one unstable branch of the
m(k) function (Triantafyllou et al., 1987).
For the velocity profile (53) the instability was found to be
convective at all Froude numbers. This is due to the fact that the
instability is convective when the Froude number is equal to zero,
which is the most unstable case. The presence of flow reversal is
required to produce an absolute instability in the self-similar wake,
but such a velocity profile can not exist far behind the ship. We
note that, if such a profile were present, it would remain absolutely
unstable for low Froude numbers.
Since the flow is connectively unstable, the question of
spatial instability to an excitation periodic in time is very
important. The spatially unstable waves for this problem can be
determined using an iterative procedure: A complex wavenumber
is assumed, the frequency is determined by solving the eigenvalue
problem as explained in the previous section, and then a new guess
is made untill a wave with complex wavenumber and real
frequency is obtained. That the complex wavenumber corresponds
to a growing, as opposed to an evanescent, mode can be verified
again from the fact that only one unstable mode exists for real k.
For Froude number equal to 0.5, the absolute value of the growth
rate of the spatial mode as a function of frequency is shown in
figure 8. (The growth-rates are actually negative, showing that the
wave are amplified as x ~ on J. The most amplified spatial mode
has complex wavenumber (0.65, -0.02217) and real frequency
0.5306. The real parts of its frequency and wavenumber differ thus
somewhat from those of the most amplified temporal mode. The
variation of the amplitudes of the Fourier coefficients of the
spatially unstable mode as a function of r is shown in figure 9. The
amplitude and phase of the free surface elevation are shown in
figures lea, and 10b respectively. The more clear difference
between the spatially unstable mode and the temporal modes
discussed before can be seen in the variation of the phase with the
distance from the wake axis. Figure 10b shows that a phase-lag is
present in the spatial mode over a larger distance than in the
temporal mode, which implies that the wavecrests of the spatial
mode will be more curved.
562
1 E-4
1F
/~ \
0.02
0 0.2
1 , 1
0.4 0.6
11 , 1
0.8 ~
(1),
Figure 8: Absolute value of the spatial growth-rate as a function
of frequency for F = 0.5.
0.1 _
1
1 E-2 _
1 E-3 _
-
-
~1
4 56
Figure 9: Variation of the amplitude of the Fourier coefficients
n = 1, 3, 5 with r for the most unstable spatial mode at F=O.S (semi-
log scale).
Given the convective character of the wake instability, it is
the spatial modes that are the physically significant ones. It is
therefore interesting to obtain a visual picture of how their free
surface manifestation can be expected to look like. A perspective
view of the free surface elevation caused by the most amplified
spatial mode at Froude number F=0.5 is shown in figure 11. The
plot has been constructed in a frame of reference that moves with
the phase velocity of the wave. In this frame of reference the wave
appears stationary in time, and its spatial growth can be seen more
clearly. The free surface elevation thus consists of two parallel
series of alternating hills and valleys the height of which increases
exponentially with x. The wave grows indefinitely according to
linear theory, but in reality the growth will be eventually saturated
by non-linear effects, unless the free surface wave breaks before
that.
OCR for page 563
0.1
odors
o.os
0.02s
o
t, 1 7 ~
4 ~
~ = _
Figure 10a: Amplitude of the free surface elevation as a function
of y, for the most unstable spatial mode at F=0.5.
0.2
O.'
0.05
.05
-0.15
4~.2 J_. _ 1 I I __l__l
01 2 3
y
_, 1_ 1 1 1 , _
4 5
Figure 10b: Phase of the free surface elevation as a function of y,
for the most unstable spatial mode at F=0.5.
Finally, it should be mentioned that we have investigated the
presence and form of spatially unstable modes in the wake of the
ship, without discussing the mechanics of excitation of these modes
by ambient waves. This is a subject that little is known about, and
will probably attract attention in the near future. A related problem
is the response of the wake to excitations in the stable wavenumber
range: An incident harmonic wave will be then partly reflected and
partly transmitted when it meets the region of shear flow, whereas a
stable wave will be excited above the wake. It will be interesting to
determine the reflected and transmitted waves in relation to the
energy carried by the wave propagating above the shear flow. A
study of the stable response of the wake to water waves will
complement the results presented here.
7 Conclusions
The instability of the wake behind a ship has been
investigated numerically. The main outcome of this investigation
is that the unstable waves are antisymmetric about the centerplane
of the wake. The free surface manifestation of the instability wave
develops a pattern consisting of two parallel series of alternating
"hills" and "valleys". An interesting observation is that, at low
Froude numbers, the frequency and phase velocity of the instability
waves is controlled by the characteristics of the shear flow in the
wake, and is practically the same as in the F=0 case. This allows
the derivation of a very simple approximation for the unstable
modes, as follows: We first solve for the eigenvalue co and
eigenvector q'(r) for F=O; this requires the numerical solution of
only an NxN eigenvalue problem. Then, for low F. the unstable
eigenmode of the same flow with a free surface can be
approximated by the first term of the Fourier series:
p(r,~) = at (r)cos(~) +
F2r(kU-~)2q~.(r)
1-(~/2)F2r(kU-~)2 2
(56)
Equation (56) satisfies exactly the linearized boundary conditions
at the free surface, (as in fact will any truncated series for q in
(28)), and approximately the basic partial differential equation (10).
Equation (56) becomes asymptotically exact as the Froude number
tends to zero.
The fact that the instability of the wake is convective, implies
that unstable wavepackets are convected with the mean flow, and
the flow can not be self-excited. Even a connectively unstable flow
can, however, be driven by background noise, to generate a
spatially growing response. For the wake/free surface interaction
problem, a source of persistent "noise" is provided by the ambient
waves, which are almost always present in the ocean. Thus, if the
frequencies of the ambient waves cover the range of spatial
instability, they can excite waves that grow in the streamwise
direction. The vortices that are formed by this noise-driven
instability can cause a local mixing in that part of the wake where
shear exists, creating a region of fluid with somewhat different
properties than the surrounding fluid (e.g. lower temperature, since
cooler fluid from below has been brought upwards). This region,
which has a width roughly equal to the width of the wake, can
therefore become visible to scientific instruments and the human
eye. The formation of such a region might account for the
observation that ship wakes remain visible long after the passage of
the ship, when all hydrodynamic disturbances have presumably
been dissipated.
ACKNOWLEDGEMENTS
Most computations were performed using the Cray-2 of the
MITSF. This work is supported by the Office of Naval Research,
under Contracts N00014-87-K-0356 and N00014-90-J1312, and
the National Oceanic and Atmospheric Administration, under Sea-
Grant Contract NA86AA-D-SG089.
8 References
1. Batchelor, G.K. and Gill, A.E., 1962, J. Fluid Mech.,
14, p. 529.
2. Benney, D.J., and Chow, K., 1986, Stud. Appl. Math.,
3. Bernal, L.P., and Kwon, J.T., 1989, Phys. Fluids A, 1,
p.449.
563
OCR for page 564
Figure 11: Perspect~ve view of the spat~ally amplified wave.
4.Bers, A., 1983, in Handbook of Plasma Physics,
M.N. Rosenbluth and R.Z. Sagdeev, gen. eds., vol.l,
Ch 3.2, North Holland.
5. Chomaz, J.M., Huerre, P., and Redelropp, L.G., 1988,
Phys. Rev. Lett., 60, p. 25.
6. Drazin, P.G., and Howard, L.N., 1966, in Advances in
Appl~ed Mechanics, 7, p. 1.
7. Hanneman, K., and Oertel, H., 1989, J. Fluid Mech.,
199, p. 55.
8. Howard, L.N., 1961, J. Fluid Mech., 10, p. 509.
9. Karniadakis, G.E., and Triantafyllou, G.S., 1989,
J.Fluid Mech., 199, p. 441.
10. Koch, W., 1985, J. Sound Vibr., 99, p. 53.
11. Liepmann, D., 1990, Phd Thesis, University of
California San Diego, La JolLa, California.
12. Lugt, H.J., 1987, Phys. Fluids, 30, p. 3647.
13. Milgram, J.H., 1988, J. Ship Res., 32(1), p.54.
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aero-146, Virginia Polytechnic Institute and State
University.
15. Mitra, P.S., Neu, W.L. and Schetz, J.A., 1986, VPI
aero-153, Virginia Polytechnic Institute and State
University.
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Appl. Math., 76, p. 69.
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Int. J. Remote Sensing, 8, p. 689.
564
18. Peregrine D.H., 1971, J.Fluid Mech., 49, p. 253.
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on Naval Hydrodynamics, Berkeley, California, p. 38.
20. Sato, H., and Okada, O., 1966, J. Fluid Mech., 26, p.
237.
21. Skop, A.R., Griffin, O.M., and Leipold, Y., 1990, J.
Ship Res., 34(1), p. 69.
22. Swean, T.F. 1987, NRL Memorandum Report 6075.
23. Tatinclaux, J.C. 1970, J. Ship Res., 14, p. 84.
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Fluids A,1,p. 1813.
25. Triantafyllou, G.S., and Karniadakis, G.E., 1990,
Phys. Fluids A, 2, p. 653.
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OCR for page 565
DISCUSSION
John P. McHugh
The University of New Hampshire, USA
I have recently considered very similar problems as in this paper, and
I have a comment and a question. The comment is that the reduction
of the quadratic eigenvalue problem to a linear eigenvalue problem
may be accomplished another way. Instead of your analytical version
(Eq. 38, p. 55), you can create a matrix version of the quadratic
problem: [A) + B.) + C](y) = 0. Then use Z = Ay in the first two
terms. The question concerns neutral modes. Did you find any
neutral modes where the wavespeed is outside the range of velocity
of the primary hull? I have found such waves in similar cases.
AUTHORS' REPLY
First, regarding the comment, the proposed alternative linearization
of the eigenvalue problem seems equivalent with what I have done.
I believe that the treatment I have presented is closer to the physics
of the problem, but the final choice is probably a matter of
preference. I am not familiar with the work of the discusser, and as
he gives no specific reference for his work, I cannot make a more
detailed comparison. Regarding the question, I have determined the
neutrally stable modes of this problem for various Froude numbers by
considering the limit of the unstable modes as their growth-rate goes
to zero (see Figure 7). The phase velocities of these modes are inside
the range of velocity of the basic flow. In fact, from Howard's
theorem, it is straightforward to see that in the limit as the growth-
rate of the unstable wave tends to zero the phase velocity has to
remain within the range of the basic flow. Consequently, I do not
see how in the similar problem that the discusser has considered that
the phase velocity of neutrally stable modes can possibly lie outside
this range. I would like to thank Prof. McHugh for his interest and
comments.
DISCUSSION
Ali H. Nayfeh
Virginia Polytechnic Institute and State University, USA
The results presented in the paper show a weak influence of the
Froude number on the growth-rates and bandwidth of unstable
disturbances, contrary to the known results about the influence of the
Mach number on the stability of compressible boundary layer. Are
these results due to neglecting the influence of the Froude number on
the mean-flow profile U(y,z)?
AUTHORS' REPLY
At.
Increase of the Froude number clearly results in a reduction of the
growth-rates and of the unstable wavenumber range, and, in the
infinite Froude number limit, Mode I gets stabilized. Overall,
therefore, the influence of the Froude number cannot be considered
weak. The influence is weak only for low Froude numbers, a result
that simply reflects the physics of the problem: at low Froude
numbers, the presence of the free surface reduces the motion of the
fluid in the vertical direction. As a result, the motion of the fluid is
confined mostly in the horizontal direction, and the free surface is
merely deformed to accommodate the non-uniform pressure caused by
the fluid motions. This behavior does not change until the Froude
number becomes high enough for the free surface to behave like an
almost perfectly compliant boundary, and accounts for the weak
influence of the Froude number on the instability mode. I believe,
therefore, for low Froude numbers, the weak influence of the Froude
number on the instability is a realistic result. I do agree, however,
that, above some value of the Froude number, the average flow itself
might start to change significantly in a manner that would accelerate
the stabilization. There are no data available, to my knowledge, as
to what that value may be; existing steady computations of the wake
behind a ship (Ref. 22) show a velocity profile similar to the one
used in this paper. I would like to thank Prof. Nayfeh for his interest
and comments.
565
OCR for page 566
Representative terms from entire chapter:
surface elevation
